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IMPROVEMENTS TO THE RADIANT TIME SERIES METHOD COOLING LOAD CALCULATION PROCEDURE By BEREKET ASGEDOM NIGUSSE Bachelor of Science in Chemical Engineering Addis Ababa University Addis Ababa, Ethiopia 1989 Master of Engineering Science in Mechanical Engineering University of New South Wales Sydney, Australia 1998 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2007 ii IMPROVEMENTS TO THE RADIANT TIME SERIES METHOD COOLING LOAD CALCULATION PROCEDURE Dissertation Approved: Dr. Jeffrey D. Spitler Dissertation Adviser Dr. Daniel E. Fisher Dr. Lorenzo Cremaschi Dr. Alan Noell Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENTS First I would like to express my deepest gratitude to my advisor Dr Jeffrey D. Spitler, for his continuous guidance and support over the course of my Ph.D. degree study. I am very grateful for his constructive advice and criticism, without which my success would have been impossible. I would like to this opportunity to thank Dr Daniel Fisher, Dr Alan Noell, and Dr Lorenzo Cramaschi for their time in serving as my advisory committee members. Next I would like to thank the US State Department for the twoyear financial support as Fulbright Scholar. Much of the work was funded by ASHRAE 1326RP. I am also grateful to ASHRAE for the further financial support provided to me as student GrantinAid. Finally I would like to express my deepest appreciation for my mother, sisters, brothers, and family members for their encouragement and unconditional love. And my special appreciation goes to my wife Aida Mebrahtu for her support and encouragement. iv TABLE OF CONTENTS Chapter Page I. INTRODUCTION......................................................................................................1 1.1 Background........................................................................................................1 1.2 Objectives ........................................................................................................13 II. REVIEW OF LITERATURE..................................................................................17 2.1 The Radiant Time Series Method ....................................................................18 2.1.1 The RTSM Procedure .............................................................................20 2.1.2 Heat Transfer Phenomena.......................................................................22 2.1.3 RTF Generation ......................................................................................27 2.1.4 Limitations of the Radiant Time Series Method.....................................28 2.2 Dynamic Modeling of Thermal Bridges ..........................................................30 2.2.1 OneDimensional Conduction Transfer Function...................................32 2.2.2 Steady State Conduction Models ............................................................36 2.2.3 Multidimensional Conduction Dynamic Models ..................................39 III. RTS METHOD IMPROVEMENTS......................................................................51 3.1 New RTF Calculation Engine..........................................................................53 3.1.1 The Mathematical Model –Reduced Heat Balance Method ...................55 3.1.2 Validation of the New RTF Engine ........................................................70 3.1.3 1D Finite Volume Method PRF Generation ...........................................74 3.1.4 1D RTF Generation in Different Programming Environment................76 3.2 Improved Fenestration Model..........................................................................78 3.2.1 Development of Improved Fenestration Model......................................79 3.2.2 Radiative  Convective Split in the RTSM .............................................84 3.2.3 Application of Fenestration Model without Internal Shade....................98 3.2.4 Application of Fenestration Model with Internal Shade.........................99 3.3 Heat Losses in the RTSM Procedure.............................................................100 3.3.1 Derivation of the Mathematical Algorithm...........................................101 3.3.2 Dimensionless Loss Conductance.........................................................115 3.3.3 Performance of Improved RTSM Procedure ........................................118 3.3.4 Heat Losses in the RTSM and TFM Procedures ..................................122 3.3.5 Conclusion and Recommendation ........................................................125 3.4 Summary and Conclusions ............................................................................127 v Chapter Page IV. PARAMETRIC STUDY OF THE RTSM PROCEDURE..................................131 4.1 Parametric Run Generation............................................................................132 4.2 Test Zone Parameters.....................................................................................134 4.2.1 Zone Geometry and Construction Fabric..............................................135 4.2.2 Thermal Mass Types.............................................................................138 4.2.3 Internal Heat Gains and Schedules .......................................................139 4.2.4 Glazing Types .......................................................................................140 4.2.5 Interior Shade Model ............................................................................142 4.2.6 RadiativeConvective Split ...................................................................143 4.2.7 Solar and Radiant Heat Gain Distribution ............................................144 4.2.8 Design Weather Days ...........................................................................145 4.3 Methodology: HBM and RTSM Implementation..........................................147 4.3.1 The HB Method Code...........................................................................148 4.3.2 The RTS Method Code .........................................................................150 4.3.3 The RTSM and HBM Models Comparison ..........................................151 4.4 Results and Discussion Original RTSM.......................................................153 4.4.1 RTSM Peak Design Cooling Load Prediction ......................................154 4.4.2 Conclusion and Recommendation ........................................................163 4.5 Results and Discussion – Current and Improved RTSM...............................166 V. DYNAMIC MODELING OF THERMAL BRIDGESMETHODOLOGY ........175 5.1 Introduction....................................................................................................175 5.2 The Equivalent Homogeneous Layer Wall Model ........................................175 5.2.1 Steady State Rvalue .............................................................................178 5.2.2 StepByStep Procedure........................................................................178 VI. DYNAMIC MODELING OF THERMAL BRIDGES  VALIDATION ...........183 6.1 Experimental Validation ................................................................................184 6.1.1 Guarded Hot Box Dynamic Response Test Facility .............................184 6.1.2 The Test Procedure and Specimens ......................................................186 6.1.3 Experimental Determination of the CTFs.............................................190 6.1.4 The Experimental Validation Procedure...............................................194 6.1.5 The Equivalent Walls............................................................................197 6.1.6 Comparison of Conduction Heat Gains ................................................217 6.2 InterModel Validation ..................................................................................223 6.2.1 Numerical Validation............................................................................224 6.2.2 The Rvalues and the Equivalent Walls ................................................227 6.2.3 Performance of the Equivalent Walls ...................................................228 vi Chapter Page 6.2.4 Summary and Conclusion.....................................................................231 6.3 Conclusions and Recommendations ..............................................................233 6.4 Recommendations for Future Work...............................................................236 VII. CONCLUSIONS AND RECOMMENDATIONS.............................................237 7.1 Conclusions – RTSM Improvements............................................................ 237 7.1.1 Accounting Space Heat Losses.............................................................238 7.1.2 Improvements to the RTSM Fenestration Model .................................238 7.1.3 Improvements to the RTF Generation ..................................................239 7.1.4 Developments to RTSM Implementation .............................................239 7.1.5 Parametric Study of the Performance of RTSM...................................240 7.2 Conclusion: Dynamic Modeling of Thermal Bridges....................................243 7.3 Recommendations for Future Work...............................................................245 REFERENCES ..........................................................................................................249 APPENDIX................................................................................................................259 APPENDIX A: THE NEW RTF ENGINE VALIDATION......................................259 APPENDIX B: 1D FINITE VOLUME METHHOD PRF GENERATION .............264 B.1 Derivation of 1D Finite Volume Numerical Model ......................................264 B.2 Finite Volume Method PRF Generation Validation .....................................277 APPENDIX C: RTSM IMPLEMENTATION IN OTHER COMPUTING ENVIRONEMNTS..............................................................................................279 APPENDIX D: FENESTRATION MODELS FOR HEAT BALANCE AND RTS METHODS ................................................................................................285 vii LIST OF TABLES Table Page Table 3.1 Description of zone constructions for RTF generation engine validation.................................................................................................70 Table 3.2 RMSE of the RTF of the New RTF Engine............................................71 Table 3.3 Recommended radiative / convective spits for the RTSM procedures ...............................................................................................98 Table 4.1 Test Parameter range and levels ...........................................................134 Table 4.2 Construction Type Materials.................................................................137 Table 4.3 Thermal Mass Type materials...............................................................138 Table 4.4 Thermal and Optical Properties of glass window .................................141 Table 4.5 Optical properties of shade layers.........................................................143 Table 4.6 Mean Lighting Heat Gain Parameters from ASHRAE 1282RP..........144 Table 4.7 Design weather conditions for the fourteen USA locations..................146 Table 4.8 RTSM and HBM component models ...................................................152 Table 4.9 Month of Annual Peak Cooling Load for zones with single pane clear glass..............................................................................................159 Table 4.10 RTSM peak cooling load extreme over predictions for glazing without interior shades..........................................................................168 Table 4.11 RTSM peak cooling load extreme over predictions for glazing without interior shades..........................................................................171 Table 6.1 Layerbylayer descriptions of the ASHRAE RP515 test walls..........189 viii Table Page Table 6.2 SurfacetoSurface Rvalues of ASHRAE RP515 Test Walls.............204 Table 6.3a Equivalent walls computed with experimentally determined Rvalues ....................................................................................................209 Table 6.3b Equivalent walls computed with handbook Rvalues ...........................210 Table 6.4a AirtoAir CTSF of the test walls generated from experimentally determined conduction transfer functions.............................................212 Table 6.4b AirtoAir CTSF of the EHL walls determined with experimentally determined Rvalues.....................................................213 Table 6.4c AirtoAir CTSF of the EHL walls determined with handbook Rvalues ....................................................................................................214 Table 6.5 Results summary of peak heat gains and time shift..............................221 Table 6.6 Test walls construction description intermodel validation ..................225 Table 6.7 Surfacetosurface Rvalue of intermodel validation test walls...........227 Table 6.8 Equivalent walls of intermodel validation test walls...........................228 Table 6.9 Peak heat gains and time shift for intermodel validation ....................229 ix LIST OF FIGURES Figure Page Figure 2.1 Radiant Time Series Method represented as a nodal network. A single wall is shown with the outside surface on the left (Rees at al. 2000) ...............................................................................................19 Figure 2.2 The original RTSM cooling load calculation method represented as flow diagram (Rees at al. 2000).......................................................21 Figure 2.3 Lineartriangular ramp temperature pulse representation....................41 Figure 3.1 Adiabatic boundary condition for RTF generation: Tso=Tsi .................58 Figure 3.2 Flow chart of solution scheme I...........................................................69 Figure 3.3 Solar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior facade.........................................72 Figure 3.4 Nonsolar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior facade ...................................72 Figure 3.5 Solar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior facade.........................................73 Figure 3.6 Nonsolar RTF for lightweight construction zone with carpet for 50% glazing fraction of the exterior facade.........................................73 Figure 3.7 Peak heat gains calculated using finite volume method versus the State Space method PRFs ....................................................................76 Figure 3.8 Absorbed component as a fraction of total solar heat gain for different fenestration classes................................................................83 Figure 3.9 TsIn and the corresponding MRT of heavyweight construction opaque exterior surfaces and 24°C room air temperature....................86 x Figure Page Figure 3.10 TsIn and the corresponding MRT for an opaque surface at peak load for three aspect ratios and 24°C room air temperature ................87 Figure 3.11 TsIn and the corresponding MRT for single pane clear glass fenestration and 24°C room air temperature........................................89 Figure 3.12 TsIn and the corresponding MRT for south facing fenestration at peak load for three aspect ratios and 24°C room air temperature........90 Figure 3.13 Radiative fractions for fenestration in a heavyweight construction zone and single pane clear glass with 90% glazing fraction................91 Figure 3.14 Radiative fractions for fenestration in a heavyweight construction zone and single pane clear glass with 50% glazing fraction................92 Figure 3.15 Radiative Fraction against glazing fraction of exterior facade for heavyweight zone for five glazing types .............................................93 Figure 3.16 Radiative fractions against percent glazing of exterior facade for lightweight zone for five different glazing types at peak cooling load condition.......................................................................................96 Figure 3.17 Radiative fractions against percent glazing of exterior facade for heavyweight zone for five different glazing types at peak cooling load condition.......................................................................................96 Figure 3.18 RTSM versus the HBM peak cooling loads for single pane clear glass without interior shaded fenestration............................................99 Figure 3.19 Current RTSM versus the HBM peak cooling loads for single pane clear glass with interior shaded fenestration .............................100 Figure 3.20 Representation of fenestration inside surface heat balance ...............103 Figure 3.21 The improved RTSM cooling load calculation method represented as flow diagram ..............................................................115 Figure 3.22 Dimensionless loss conductance against glazing fraction for zones with unshaded fenestration ......................................................117 Figure 3.23 Dimensionless loss conductance against glazing fraction for zone with two exterior facades and interior shaded fenestration ...............117 xi Figure Page Figure 3.24 RTSM peak cooling load vs. HBM for light and heavyweight zone for single pane clear glass without internal shade.....................119 Figure 3.25 Hourly cooling load profile for lightweight zone at 50% glazing fraction for single pane clear glass in Chicago, Illinois.....................120 Figure 3.26 Hourly cooling load profile for lightweight zone at 90% glazing fraction for single pane clear glass in Chicago, Illinois.....................120 Figure 3.27 RTSM peak cooling load vs. HBM for light and heavyweight zone for single pane clear glass with internal shade ..........................121 Figure 4.1 Schematic of Parametric Run Generator............................................133 Figure 4.2 Zone orientation and number designations ........................................136 Figure 4.3 Structure of Heat Balance Method for a Zone ...................................149 Figure 4.4 The current RTSM cooling load calculation method represented as flow diagram..................................................................................150 Figure 4.5 RTSM annual peak cooling load versus the HBM for the USA weather locations for single pane clear glass.....................................160 Figure 4.6 RTSM annual peak cooling load versus the HBM for the USA weather locations for double pane clear glass....................................161 Figure 4.7 RTSM annual peak cooling load versus the HBM for the USA weather locations for double pane lowe glass ..................................161 Figure 4.8 Maximum RTSM peak cooling load over prediction against glazing fraction for the three glazing types........................................162 Figure 4.9 Current RTSM annual peakcooling load maximum and average over prediction for zone without interior shade.................................169 Figure 4.10 Improved RTSM annual peakcooling load maximum and average over prediction for zone without interior shade ...................169 Figure 4.11 Current and Improved RTSM annual peakcooling load versus HBM for lightweight zones single pane clear glass without interior shade......................................................................................170 xii Figure Page Figure 4.12 Current and Improved RTSM annual peakcooling load versus HBM for heavyweight zones with single pane clear glass without interior shade......................................................................................170 Figure 4.13 Current RTSM annual peakcooling load maximum and average over prediction for zones with dark roller interior shade...................172 Figure 4.14 Improved RTSM annual peakcooling load maximum and average over prediction for zones with dark roller interior shade .....173 Figure 4.15 Current and Improved RTSM annual peakcooling load versus HBM for lightweight zones with single pane clear glass and dark roller interior shade ............................................................................173 Figure 4.16 Current and Improved RTSM annual peakcooling load versus HBM for lightweight zones with single pane clear glass and dark roller interior shade ............................................................................174 Figure 6.1 Sectional view of guarded hotbox facility (Brown and Stephenson 1993b) ................................................................................................186 Figure 6.2 Flow chart of the experimental validation procedure ........................196 Figure 6.3 Thermal bridge types: (a) sandwiched type; (b) exposed type ..........198 Figure 6.4 ASHRAE RP515 Test Walls ............................................................203 Figure 6.5 CTSF plot for the steel stud wall (Wall#1)........................................215 Figure 6.6 CTSF plot for the steel stud wall (Wall#4)........................................216 Figure 6.7 CTSF plot for hollow block with insulation and brick exterior finish (wall#5)....................................................................................216 Figure 6.8 Heat gain for insulated steel stud wall with stucco exterior finish (Wall#1) .............................................................................................217 Figure 6.9 Heat gain for precast reinforced concrete slab with steel furring and insulation covered with gypsum board on the exterior (Wall#2) .............................................................................................218 Figure 6.10 Heat gain for precast reinforced concrete slab with steel furring and insulation with gypsum board (Wall#3)......................................218 xiii Figure Page Figure 6.11 Heat gain for insulated steel stud wall mounted on reinforced concrete slab (Wall#4) .......................................................................219 Figure 6.12 Heat gain for hollow concrete block with insulation and brick on the exterior and gypsum board (Wall#5) ...........................................219 Figure 6.13 Heat gain for insulated steel stud wall with brick exterior finish (Wall#6) .............................................................................................220 Figure 6.14 Heat gain for solid concrete block wall with insulation and granite veneer exterior finish (Wall#7)..............................................220 Figure 6.15 Summary of peak heat gains for the seven ASHRAE test walls .......222 Figure 6.16 Handbook Rvalue errors of the seven ASHRAE test walls..............222 Figure 6.17 Wood and steel stud walls construction details .................................226 Figure 6.18 Heat gain of a wood stud wall for periodic solair temperature boundary condition ............................................................................229 Figure 6.19 Heat gain of a warm steel stud wall for steady periodic solair temperature boundary condition ........................................................230 Figure 6.20 Heat gain of a cold steel stud wall for steady periodic solair temperature boundary condition ........................................................230 Figure 6.21 Heat gain of a hybrid steel stud wall for steady periodic solair temperature boundary condition ........................................................231 Figure 6.22 Handbook Rvalue errors compared to that of the 2D finite volume method...................................................................................232 Figure 6.23 Peak cooling load prediction error of the equivalent walls compared to the 2D finite volume method ........................................232 xiv 1 CHAPTER I 1 INTRODUCTION 1.1 Background Design cooling load calculation methods have evolved since their inception during the 1930’s. The historical development of cooling load calculation procedures has been strongly influenced by the development and availability of digital computing facilities, and by the desire to provide methods that are of utility to average practicing engineers that can be used with tabulated data (Rees et al. 2000a; Romine 1992). It is useful to define the terms “heat gains” and “cooling load” and the relationship between them in the context of load calculations. Heat gain is defined as the instantaneous heat flow into a space by conduction, convection and radiation. Cooling load is defined as the amount of heat removed from a space to keep the space air at a fixed desired temperature. Therefore, all heat gains do not necessarily become cooling loads: convective heat gains become cooling load instantaneously, while radiant heat gains are first absorbed by the structure and then released by convection to become a cooling load at a later time. Absorption and reradiation of radiant heat gains among the surfaces in the zone continues as long as temperature difference exits. Under some circumstances, some of the heat gains may be conducted back out of the space. 2 The challenge in the early days of the cooling load calculation was primarily to develop procedures to quantify the heat gains. In the 1930s peakcooling loads were over predicted due to failure to account for thermal mass effects of construction in the load calculation (Houghten et al. 1932; James 1937; Kratz and Konzo 1933). Analytical equations for computing transient conduction heat gains through homogeneous layer constructions exposed to solar radiation were developed. Houghten, et al., (1932) used Fourier analysis and assumed sinusoidally varying outside surface temperatures. Alford, et al., (1939) improved this by assuming sinusoidally varying outdoor air temperature and accounting for solar radiation separately. Despite an effort to develop a rigorous analytical procedure for computing transient heat conduction, there was little success in establishing a general quantitative relation suitable for practicing engineers. The electric analogy method of predicting heat flow through walls based on the identity of the transient heat flow and flow of electricity can be implemented experimentally and can closely match direct thermal measurements (Paschkis 1942). An electric analog thermal circuit of an embedded tube cooling slab model was developed using electrically equivalent resistance, capacitance, and source terms (Kayan 1950). This allowed determination of the slab surface temperatures, temperature isotherms in the slab and heat transfer rates. By the mid 1940s, the American Society of Heating and Ventilation Engineers (ASHVE), a predecessor of the American Society of Heating, Refrigeration and Air Conditioning Engineers (ASHRAE), developed a manual method for calculating the heat gain through 3 various external surfaces with equivalent temperature differentials (ETD) values. The ETD values were often 20 to 40 degrees Fahrenheit above the difference between outside and inside air temperatures (Rees et al. 2000a; Romine 1992). In the ETD method two procedures were involved: the ETD were generated from experimentally measured surface temperatures and conductance (Rees et al. 2000a) for transient conduction heat gain, and the instantaneous solar heat gains through glazing were calculated using heat fluxes and shading coefficients. The ETD method excessively overestimated cooling load due to the assumption that the heat gains instantaneously caused cooling loads on the system. The delays of solar heat gains before becoming cooling load were well understood but simple quantitative relations for these effects were not available until the 1940s and 1950s. Designers made various approximations to compensate for the over prediction of cooling loads (Romine 1992). Transient conduction heat gain calculation procedures through external surfaces developed using Fourier analysis assumed periodic variation of solair temperature as the external driving temperature, constant indoor air temperature, and fixed outside and inside conductance1 (Mackey and Wright 1944; Mackey and Wright 1946). The solair temperature is a concept derived from the equivalent temperature (Billington 1987) used then in UK. It is defined as the temperature that would give the same amount of heat transfer as that of the actual outdoor air temperature and solar radiation incident on the surface. Mackey and Wright (1946) formulated semiempirical relations to estimate inside surface temperatures for multilayered walls based on an analytic solution for 1 A fixed value of combined outside conductance of 4.0 (Btu/hr⋅ft2⋅°F) is still commonly used after 60 years. 4 multilayered walls. The damping and delay effects of the surface thermal mass on the inside surface temperature were accounted using a decrement factor and time lag. Developing an equation for the inside surface temperatures using the solair temperature to account for the incident solar flux provided the first convenient manual procedure for computing instantaneous heat gains. The heat gains were computed from the inside surface temperature and room air temperature, assuming a fixed combined inside conductance. Later, Stewart (1948) used this procedure to tabulate the ETD for various construction assemblies, surface exterior colors, surface orientations, latitude angles and hours of the day. The tabulated ETD values were adjusted for use with walls and roofs overall heat transfer coefficient, instead of combined inside conductance. This concept was then adopted by ASHRAE as the total equivalent temperature difference and time averaging (TETD/TA) method in the 1960s. The TETD/TA load calculation method first introduced in the 1967 Handbook of Fundamentals (ASHRAE 1967; Rees et al. 2000a; Romine 1992). The TETD/TA method mainly involves two steps: calculation of heat gains components from all sources and conversion of these heat gains into cooling loads. The TETD replaced the ETD with improved tables and equations for the equivalent temperature differences. Walls and roofs were characterized by two parameters decrement factor (ratio of peak heat gain to the peak heat gain that would occur with no thermal mass in the wall) and time lag (delay in peak heat gain compared to peak solair temperature). The TETD could then be calculated knowing sol 5 air temperature, room air temperature, and decrement factors and time lags. Conversion of the instantaneous heat gains into cooling loads using the time averaging technique is a two step procedure: first, split the instantaneous heat gain into convective and radiant components using recommended radiative /convective splits; second, the radiative component of the heat gain is time averaged depending on the thermal mass of the construction to get the cooling loads. For lightweight construction, the hourly radiant cooling load is the radiant component of heat gain time averaged over a 2 to 3 hour period prior to and including the time of maximum load conditions. For heavyweight construction, the hourly radiant cooling load is the radiant component of heat gain time averaged over a 5 to 8 hour period prior to and including the time of maximum load conditions (ASHRAE 1967). The total hourly cooling load is the sum of the convective component and the hourly radiant cooling load. The work described above did not explicitly consider interactions between heat gain components. The earliest attempt to model zone dynamics involving conduction through the envelope, solar heat gains and the radiant exchange among surfaces and convection between surfaces and room air utilized physical (electric and hydraulic) analogies in the 1940s and 1950s. However, the analogies remained research tools as it was not feasible for practicing engineers to build electric circuits, nor were the insights gained reduced to manual calculation procedures. Leopold (1948) used a hydraulic analogy to investigate zone dynamics. The model included thermal storage, radiation, convection, and conductions. Thermal capacitance 6 was represented by vertical tubes in series connection attached to a distribution header connected to a storage tank, and resistances were represented by restricted tube. Radiation absorbed by surfaces was represented by liquid flow from a pump through a calibrated restriction, and temperatures were represented by fluid pressure. The hydraulic model demonstrated dynamics of zones and gave some insights to the limitations the load calculation procedures. Despite all efforts to improve the accuracy of load calculation procedures, peak cooling load computed using the ASHVE Guide 1952 over predicted by 16 to 32 % compared to values measured in a small single story residential house with large glass exposure due to failure to account for the storage effect (Gilkey et al. 1953). Similarly, a field survey made on single family houses over a wide range of climates and construction fabrics revealed over sizing of cooling equipment capacity due to failure to account for the thermal mass effects of building structures (Willcox et al. 1954). Dynamic modeling of thermal mass effects of structures and furnishing in a building was attempted using analog computers by solving the electrical equivalent thermal circuit of actual buildings (Willcox et al. 1954). The model used pure resistances to represent doors, windows, blinds and infiltration. Distributed resistances and capacitances were used to represent walls, roofs and partitions. The outdoor and indoor temperatures were represented by potential differences. With this approach, the authors found it difficult to construct a circuit that both had a onetoone physical correspondence with the building, and which gave a good match to transient thermal measurements. They did find that they 7 could “tune” a simpler circuit to give the correct dynamic response, but this has limited usefulness for design load calculations. However, they had better success with an analog computer, which utilizes amplifiers and allows better measurement of intermediate values. The analog computer’s calculated response was only 7% higher than the actual thermal measurements. The work of Brisken and Reque (1956), in developing what they called the ‘Thermal Response Method’, was the first attempt to use digital computers by representing a wall using twolump (oneresistance and twocapacitance) thermal circuit that was connected to outdoor solair temperature and indoor air temperature nodes using outside and inside combined conductance. The two differential equations for the twolump thermal circuit were solved using the Laplace transform method to determine the room response to a unit square pulse applied at the solair temperature while the room air temperature was constant. The method was not adopted in the ASHVE Guide, but the approach later became the basis for development of the conduction transfer function method with a unit triangular pulse adopted by ASHRAE for transient conduction heat gain calculations. A procedure for computing room response factors using a detailed thermal circuit model involving radiation exchange among inside surfaces and room furnishings, convection between surfaces and room air, and various room heat sources was developed by Mitalas and Stephenson (1967). An effort to provide a more rigorous load calculation procedure led to the development of conduction transfer functions for transient conduction through homogeneous multilayered constructions (Stephenson and Mitalas 1971). The transfer 8 function method (TFM) for computing zone thermal response and cooling load was first published in the 1972Handbooks of Fundamentals (ASHRAE 1972). The method relied on a set of tabulated room transfer function coefficients. Given the enormous (in the 1970s) computational efforts required by the TFM and the lack of computer resources and skills of practicing engineers there was a need for a method that could be used manually. As a result, a simplified procedure called the Cooling Load Temperature Difference / Cooling Load Factor (CLTD/CLF) method was developed under ASHRAE RP138 by Rudoy and Duran (1975). The CLTD/CLF method is a single step load calculation procedure. CLTD values were calculated by dividing the cooling load due to a particular wall or roof using the TFM by the Uvalue of the constructions. Due to its simplicity, the CLTD/CLF method replaced the TETD/TA methods as the ASHRAErecommended manual load calculation procedure. However, the CLTD/CLF method had limitations due to a lack of tabulated CLTD/CLF design data that matched the wide range of design conditions faced by practitioners. Thus, designers showed continued interest into TETD/TA method due to its flexibility for manual load calculations and adaptations for various building envelope assemblies and design locations (Romine, 1992). ASHRAE’s continued commitment to refine load calculation procedures, to investigate effects of different building design parameters, and to provide accurate design data led to new research directions in the 1980s. ASHRAEfunded research project 472RP characterized room response based on fourteen building design parameters. Generating, 9 tabulating, and printing the whole range of the CLTD/CLF data on the basis of the fourteen design parameters became an impractical task (Sowell 1988c). However, ASHRAE maintained the CLTD/CLF method, which later became the Cooling Load Temperature Difference /Solar Cooling Load / Cooling Load Factor (CLTD/SCL/CLF) method, as a manual load calculation procedure by tabulating CLTDs for representative families of walls and roof assemblies and developing a mapping procedure for the actual constructions. Software for generating CLTD and CLF data based on the weighting factors and conduction transfer function coefficients developed in ASHRAE RP−472 was developed as part of ASHRAE RP626 (Spitler et al. 1993b). Spitler, et al. (1993a) introduced a new factor, the solar cooling load (SCL), for converting solar heat gain into cooling load. Though the TFM required high computational resources, it remained the only computational design cooling load calculation procedure recommended by ASHRAE until the late 1990s. The Transfer Function Method was not well received (Romine 1992) by practicing engineers for the following reasons: • Intimidating look of the equations • Required iterations and convergence may take three to five successive design day calculations • Computer resources and a lack of computing skills also limited its implementation for load calculations A simple and yet reasonably accurate load calculation procedure that did not involve iterative processes was highly desired by ASHRAE to replace the manual procedures. An 10 ASHRAE funded project (RP875) for continued improvements of load calculation procedures led to the development of the Heat Balance Method (HBM) (Pedersen et al. 1997) and the Radiant Time Series Method (RTSM) (Spitler et al. 1997) for calculating peak cooling loads. The HBM was first implemented in the 1960s by Kusuda in NBSLD, later by Walton in 1980s in Building Loads Analysis and System Thermodynamics (BLAST) and in Thermal Analysis Research Program (TARP) as cited by Pedersen, et al. (1997). However, a complete description of the procedure for load calculation purposes had not been available. The first complete description of the heat balance method formulation starting from the fundamental principles, and covering implementation and solution techniques as applied for peak cooling load calculation was presented by Pedersen, et al., (1997). Since the heat balance method is based on the fundamental principles of the physics involved, it is commonly used as a reference model for simplified load calculation programs. The RTSM closely followed the HBM hourly cooling load profile and in most cases slightly overpredicted the peak cooling load; however, the over predicted peak cooling load was significant for zones with large amount of single pane glazing and cool design weather conditions (Rees et al. 1998). The radiant time series method (RTSM) was developed as a spreadsheet method intended to replace the TETD/TA and the CLTD/SCL/CLF methods. It also effectively replaced the TFM. The radiant time series method (RTSM) as a simplified load calculation procedure was adopted as a 11 nonresidential building load calculation procedure by ASHRAE and published in Pedersen, et al. (1998) and the 2001 −Handbook of Fundamentals (ASHRAE 2001). Experimental validation of the heat balance and the radiant time series methods has been done in test cells at Oklahoma State University (Chantrasrisalai et al. 2003; Iu et al. 2003). ASHRAE research project RP942 compared the peak cooling load predictions made with the RTSM to those made with the heat balance method (HBM) using a parametric run investigation tool (Rees et al. 1998; Spitler and Rees 1998). Although ASHRAE 942RP identified building design parameters that lead to over predictions of peak cooling load, the project did not result in design guidance for practicing engineers. The radiant times series method (RTSM) has effectively replaced the manual load calculation procedures and has attracted interest due to: Its amenability to spreadsheet implementations as opposed to the Transfer Function Method, which requires iteration. Captures and depicts the physics involved in the Conduction Time Series Factor (CTSF) and Radiant Time Factor (RTF) coefficients, unlike the Transfer Function Method. Has essentially the same accuracy as the TFM. 12 However, the RTSM also has the same approximations as the TFM that, in some cases, lead to over prediction of peakdesign cooling load: The RTSM replaces the outside heat balance by an exterior boundary condition known as the solair temperature, which allows the use of fixed combined conductance of convection and radiation. The RTSM computes the radiant heat gain from the interior surfaces as if they all radiate to the room air temperature instead of performing inside surface and room air heat balances. This allows treatment with a linearized radiation coefficient, which is combined with the convection coefficient. This assumption can over predict the instantaneous heat gain, which again contributes to the RTSM peak cooling load overprediction. The RTSM uses an adiabatic boundary condition when computing Radiant Time Factors (RTF), causing the RTF to always sum to one. When these RTF are used, this approach conserves the entire solar and internal heat gains during conversion to cooling load, and there is no way that the RTSM can account for any heat gains conducted back out. As a result, the RTSM tends to over predict the peakcooling load when there is a large amount of single pane glazing or other highly conductive surfaces. The resulting over predictions was shown in 942RP to be as high as 37%. It would be very helpful for designers to have guidance as to when the RTSM is likely to gives significant overprediction. 13 1.2 Objectives The previously published research in the RTSM cooling load calculation procedure has only identified the likely over of peak cooling load and the conditions favorable for over prediction but non them provided a procedure for accounting the heat gain loss and did not provide guidance on the limitation of the RTSM. Therefore, one of the objectives of this thesis is to develop an algorithm that reduces the RTSM peak cooling load likely over prediction significantly and establish the limitations of the RTSM in a form of design guidance. Furthermore, ten years of experience with the RTSM has indicated several improvements that would be helpful for design engineers. These include an improved RTF generation procedure, developing a numerical procedure for periodic response factor generation, updated fenestration modeling and investigated a procedure for treating thermal bridges. These improvements are discussed briefly below. The RTSM needs radiant time factors (RTF) for the zone to be analyzed. The ASHRAE Handbook of Fundamentals (ASHRAE 2001; ASHRAE 2005) has given tabulated RTF for specific cases, but the accuracy resulting from users choosing the “nearest” zones has not been investigated. The original presentation of the procedure utilized a full blown HBM program to generate the RTF. While this approach works, the HBM program has many features and data that are not needed for generating RTF. Therefore, one of the objectives of this thesis is to develop a simplified procedure and implement the algorithm for computing RTF, as described in Section 3.1. Also investigated is a direct method of calculating periodic response factors as an alternative to converting conduction transfer function coefficients back to response factors for use in the RTF generation procedure. A 14 onedimensional finite volume numerical fullyimplicit formulation for generating periodic response factor will be developed. The finite volume periodic response generator and the simplified RTF generator algorithm will also be implemented in other computing environments. Furthermore, the RTF generation has been investigated with constant radiation coefficient with the intent of reducing the computational time. The radiant time series method load calculation procedure was developed based on the shading coefficients and optical properties of doublestrength glass for computing solar heat gains. Developments in fenestration models and availability of a new set of fenestration data – solar heat gain coefficients  replaced the use of shading coefficients in fenestration modeling. Moreover, the shading coefficient data are no longer available. Therefore, one of the objectives of this thesis is investigation of a new fenestration model using window manufacturer’s data and the new set of tabulated glazing and fenestration data available in the ASHRAE’s Handbook of Fundamentals as presented in Section 3.2. Improved fenestration model for the RTSM will be investigated and integrated that make use of these new developments. Moreover, a new set of radiative / convective splits for fenestration solar and conduction heat gains will be established. It has been identified that the likely over prediction of the RTSM procedure is due to failure to account for the space radiant heat gains conducted back to the outside. In this thesis an algorithm for accounting the radiant heat gain loss by conduction will be derived and investigated in Section 3.3.1. The procedure accounts for solar and internal radiant heat gains conducted back out through fenestrations and highly conductive mass 15 less surfaces. Furthermore, the likely overprediction of peak cooling load by the RTSM will be investigated parametrically over a wider range of building design parameters to characterize its limitations and develop design guidance for practicing engineers. This is covered in Chapter Four. Another challenge faced by designers is the treatment of thermal bridges in wall and roof constructions. Steady state treatment of thermal bridges is covered in the ASHRAE Handbook of Fundamentals (ASHRAE 2005). Dynamic modeling of thermal bridges has been a research interest in building energy and load calculation applications for about two decades. Despite repeated efforts to develop multidimensional conduction models capable of dynamic modeling of thermal bridges, these models have never been integrated into design load calculation procedures for several reasons. Therefore, another objective of this thesis is to investigate an approximate onedimensional dynamic model of thermal bridges that can be directly implemented into design cooling load calculation procedures, and develop design recommendations usable by practicing engineers. This is covered in Chapters Five and Six. Before addressing these three chapters, the thesis gives an indepth literature review of the Radiant Time Series Method and the treatment of thermal bridges in Chapter Two. Improvements to the Radiant Time Series Method procedure are discussed in Chapter Three. Parametric investigation of the Radiant Time Series Method to establish the limitations based on adapted fenestration model is covered in Chapter Four. Chapters Five and Six deal with approximate onedimensional dynamic modeling of thermal 16 bridges methodology and the validation, respectively. Conclusions and recommendations for future work are given in Chapter Seven. 17 CHAPTER II 2 REVIEW OF LITERATURE The literature review covers two separate topics related to building energy analysis and load calculation methods. The first section deals with developments in Radiant Time Series method, and the second section deals with dynamic and steady state modeling of thermal bridges. The first part of the literature survey (Section 2.1) describes the development of the Radiant Time Series Method (RTSM) as a simplified design cooling load calculation procedure, discusses the key assumptions introduced to derive the RTSM procedure, and explains in detail the limitation of the RTS method in predicting the peak design cooling, and discusses the necessary conditions for the RTSM peak cooling load overprediction. The second part of the literature survey (Section 2.2) deals with dynamic and steady state modeling of thermal bridges in relation to building energy analysis and load calculation program. It discusses the importance of steady state and dynamic modeling of thermal bridges in building energy analysis and load calculation application, reviews previously published dynamic modeling techniques for thermal bridges and explains why these models have not been adopted. 18 It also presents a brief summary of response factors and conduction transfer functions and the associated onedimensional conduction modeling procedures. In addition it summarizes recommended approximate procedures for steady state analysis of thermal bridges. 2.1 The Radiant Time Series Method The Radiant Time Series Method (RTSM) was introduced as a simplified design load calculation procedure (Spitler et al. 1997). The RTSM was intended to replace ASHRAE’s simplified load calculation procedures: the cooling load temperature difference/solar cooling load/cooling load factor (CLTD/SCL/CLF) method, the total equivalent temperature difference/time averaging (TETD/TA) method, and the Transfer Function Method (TFM). The radiant time series method can be thought as a twostage process (Spitler et al. 1997). The first stage of this process is to calculate all the radiant and convective heat gains of the zone. The second stage is the conversion of these gains into contributions to the cooling load on the zone air. Several key approximations have been employed in simplifying the RTSM in order to avoid the iteration steps in the procedure so as to make the method suitable for spreadsheet implementation. The first simplifying assumption in the radiant time series method is the treatment of exterior and interior convection and radiation coefficients using combined constant conductance. This simplification in effect entails the assumption that surfaces exchange long wavelength radiation with air node. This assumption allows 19 the replacement of individual surface heat balances and represents each wall by a simplified nodal network as shown in Figure 2.2 (Rees et al. 2000a). Qia Qinf riQir rsQSi QPa QSol TSA 1/hcoA 1/hciA Qcond,out Qcond,in Figure 2.1 Radiant Time Series Method represented as a nodal network. A single wall is shown with the outside surface on the left (Rees et al. 2000a) The second approximation is periodicity of the design weather conditions. The radiant time series method takes advantage of the periodicity of the design day solair temperature and constant room air temperature to develop the periodic response factor. Conduction heat gains are calculated by periodic response factors (PRF) or Conduction Time Series Factors (CTSF) driven by the difference between the design day periodic Solair temperature TSA and room air temperature Ta, which is assumed constant. The periodic response factors replace the CTF in the heat balance method and eliminate the iterative conduction heat gain calculation, which is inherent in load calculation methods involving transfer functions. This assumption is key in that it avoids the iteration step and hence makes the RTSM suitable for spreadsheet implementation (Rees et al. 2000a; Spitler et al. 1997). The CTSF are determined from periodic response factors divided by the overall Uvalue of the construction. 20 The third simplifying approximation in the RTSM is the conversion of the radiant components of the heat gains into cooling load using the radiant time factors (RTF), which replaces the air heat balance. The radiant gains at each hour are converted by a series of twentyfour room response factors known as the radiant time factors (RTF). The contribution of the internal heat gains Qir and the transmitted solar heat gains QS to the load appear at the room air node as shown in the nodal network diagram (Figure 2.1) but multiplied by the radiant time factors ri and rs, respectively. These contributions are summed up to get the total hourly load. Cooling load is defined as the rates at which heat must be removed from the space to maintain a constant room air temperature and is represented as QPa in Figure 2.1. The fourth approximation in the RTSM is that solar and internal heat gains are divided into radiative and convective components using fixed radiative / convective splits (Rees et al. 1998; Spitler et al. 1997). 2.1.1 The RTSM Procedure The RTSM procedure, in terms of processing the input data and steps to arrive at the 24 hourly cooling loads, is described as follows and the calculation flow diagram is shown in Figure 2.2. The first step of the RTS method is calculation of all internal heat gains. This is done in exactly the same way as for the Heat Balance Method. All gains that are independent of the zone surface temperatures hence can be computed at the beginning of the simulation and stored as hourly values for later use. These include solar gains through 21 glazing, infiltration (assuming fixed internal air temperature), and internal gains, which are determined from the 24 hours schedule and peak internal heat gain. Determine Lighting, equipment & occupant heat gains at each hour Calculate conduction heat gain for each hour using windows response factor. Calculate conduction heat gain for each exterior opaque surface using wall/roof response factor, Calculate transmitted solar heat gain for each window for each hour Split all heat gain into radiant and convection components Determine the infiltration heat gains for each hour Sum all convective components for each hour Process the radiant heat gains using the Radiant Time Factors: either nonsolar or solar. The result is the hourly loads due to the radiant heat gain Σ Calculate absorbed solar heat gain for each window for each hour Figure 2.2 The original RTSM cooling load calculation method represented as flow diagram (Rees et al. 2000a) The periodic response factors operate on the solair and internal dry bulb temperatures. The hourly values of the solair temperature and the room air temperatures are known at the beginning of the calculation. Once the conduction heat gains through the individual surfaces have been calculated, the next important step is to divide all the gains into 22 radiant and convective components. This is done using fixed radiative / convective splits for each type of heat gains. The second stage of the RTS calculation procedure is to convert all the heat gains into contributions to the load at the air node. Convective components of the gains make instantaneous contributions to the cooling load while the radiant components of the heat gains are converted to cooling loads by means of the radiant time factors (RTF). The hourly contributions of the radiant gain to the cooling load are calculated from the 24 hourly radiant gains and the RTF. The radiant time factors are zone dynamic response characteristic, which are dependent on the overall dynamic thermal storage characteristics of the zone and defines how the radiant gain at a given hour is redistributed in time to become contributions to the cooling load at future hours. The contributions of the past and current radiant gains are simply added to the hourly convective gains to give the hourly cooling load. 2.1.2 Heat Transfer Phenomena This section describes the specific practices and assumptions used by the RTSM to model some of the principal zone heat transfer mechanisms. Exterior Convection and Radiation The RTS Method uses a fixed exterior surface conductance combined with a solair temperature to model exterior convection and radiation. This is one of the first simplifying assumptions of the radiant time series method. 23 Transient Conduction Heat Transfer The RTS Method treats external and internal excitation of conduction heat flow separately. In the RTS procedure, transient conduction heat transfer due to external excitation is modeled using a set of 24 periodic response factors. Given the constant zone air temperature Ta and the current and 23 past values of solair temperature TSAθ, the current hour’s conduction heat gain per unit surface area is given by: ( ) Σ= − = − 23 0 , , '' , , j cond i t Pj SA i j a q Y T T θ δ & (2.1) where, '' cond ,i, t q& = the current hour conduction through the ith surface, Btu/h⋅ft2 (W/m2) Pj Y = the periodic response factors at j hours from the present, Btu/h(W) SA i θ jδ T , , − = the solair temperature of the ith surface j hour from the present, °F(°C) a T = the constant room air temperature, °F(°C) The periodic response factors YPj include both the interior and exterior surface conductance. Periodic response factors can be computed from response factors (Spitler et al. 1997), from the generalized form of the CTFs (Spitler and Fisher 1999a), and frequency domain regression method (Chen and Wang 2005). The solair and inside temperatures are known at the beginning of the calculation, therefore the heat gains due to conduction can be calculated straightforwardly without the need for any iteration, which makes the RTSM amenable for spreadsheet implementation. These gains subsequently have to be divided into radiant and convective components. 24 Interior Convection and Radiation The RTS Method uses fixed combined interior radiation and convection conductances. The convection and radiation coefficients are added (as a resistance) into the wall. This approach, though it simplifies the procedure, has the effect of having the wall radiating to the zone air temperature. In most cases, this causes the RTSM to slightly overpredict the peak cooling load (Rees et al. 2000a). The RTS Method uses radiant time factors (RTF) to convert and redistribute the radiant part of the conducted gain. Analogous to periodic response factors, radiant time factors are used to convert the cooling load for the current hour based on current and past radiant gains. The radiant time factors are defined such that r0 represents the portion of the radiant gains convected to the zone air in the current hour. r1 represents the portion of the previous hour’s radiant gains that are convected to the zone air in the current hour, and so on (Spitler and Fisher 1999b). The cooling load due to radiant heat gain is given by: Σ= = − 23 j 0 Qt rjqt jδ (2.2) Where Qt = the current hour cooling load, Btu/h(W) qt− jδ = the radiant gain at j hours ago, Btu/h(W) j r = the jth radiant time factor, Btu/h(W) 25 Transmitted and Absorbed Solar Radiation Calculation of transmitted and absorbed solar radiation associated with fenestration is a very important part of the design cooling load calculation procedure. The response of the zone is dependent not only on the value of the transmitted and absorbed solar energy but also on its distribution in the zone. Two simple procedures applicable for load calculation purposes have evolved: (1) the use of normal solar heat gain coefficient and transmittance and absorptance correction for angle dependence using a reference standard DSA glass angle correction coefficients (Spitler et al. 1993a), (2) the use of angle dependent beam solar heat gain coefficient and constant diffuse solar heat gain coefficient tabulated values (ASHRAE 2005). The first approach allows separate treatment of transmitted and absorbed solar radiation. Though transmitted and absorbed components are calculated separately, the procedure is based on approximate analysis analogous to the concept of shading coefficient. This was adopted as a standard procedure but with demise of the shading coefficients a new procedure is needed. The second approach is used in a combined treatment of transmitted and absorbed components. In the second approach the solar heat gain coefficient includes both the transmitted portion of the solar heat gain and the inward flow fraction of the absorbed component. This therefore precludes the separate treatment of the absorbed solar heat gain, which has both radiative and convective components. Likewise, the RTSM uses the solar radiant time factor to convert the beam and diffuse solar heat gains into cooling loads. The diffuse solar gains are treated in a similar way to internal short and long wavelength radiant gains. As noted previously in the discussion on internal convection 26 and radiation heat transfer, some of the solar radiation that is reradiated can be conducted to the outside. The RTSM cannot account for this, and so for some zones and design weather condition tends to overpredict the cooling loads. Internal Heat Gains In the radiant time series method the hourly schedules and peak gain rate for the three type of internal heat gains (e.g. people, lights, and equipment) are specified by the user along with the respective radiative/convective splits. Though the split between radiative and convection actually depends on the zone airflow rates and surface temperatures, constant values are used even in detailed building energy analysis programs. In the RTSM the radiative component heat gain contribution on the cooling load is estimated with the radiant time factors. The RTSM does not account for the portion of the radiant gain that is conducted to the outside and so for some zone constructions tends to overpredict the cooling loads. The degree of overprediction depends on the zone construction conductance, and design weather conditions. This has been one of the limitations of the RTSM procedure and is discussed in Section 2.1.4. 2.1.3 RTF Generation Radiant time factors (RTF) are dynamic response characteristics of a zone when a zone is excited by unit heat gain pulse. (Spitler et al. 1997) described two procedures for generating RTF coefficients. The first method uses a load calculation program based on the heat balance method (Pedersen et al. 1997). 27 The radiant time factors are generated by driving a heat balance model of the zone with a periodic unit pulse of radiant energy under adiabatic wall conditions. The radiant time factors are therefore different for every combination of zone construction and geometry. In principle, they are also different for every chosen distribution of radiant pulse. Thus far two types of distributions have been commonly used for a given zone (Spitler et al. 1997). One is found assuming an equal distribution (by area) of radiant pulse on all zone surfaces and is used for all diffuse radiant gains. A second set is found with the unit pulse of radiant energy added at the floor surface and in some cases to the furniture as well to treat beam solar gains. The conversion of radiant gains by the use of radiant time factors, where there is no requirement for knowledge of past temperatures or cooling loads, again avoids the iteration processes. The second method demonstrated by (Spitler and Fisher 1999b) is to generate radiant time factors directly from a set of zone weighting factors using the existing ASHRAE database (Sowell 1988a; Sowell 1988b; Sowell 1988c). This approach would use a computer program to map a given zone to the fourteen zone characteristic parameters in the database and transform the weighting factors to radiant time factors using matrix manipulation. However, the custom weighting factors do not represent all possible zone constructions. Use of a weighting factor database requires some approximations to fit the fourteen selection parameters. Therefore, development of an RTF generating tool that fits practical design condition is essential for RTSM implementations. 28 One important assumption in calculating the radiant time factors is imposing adiabatic boundary condition for all surfaces in the zone. As the consequence of this assumption the radiant pulse used to generate the radiant time factors is then only redistributed in time, otherwise its energy is entirely conserved in the zone. In the RTSM, since no solar and internal radiant heat gains are conducted out of the zone, this often leads to slight overprediction of the peakcooling load. However, for zones with large amount single pane glazing, and cooler summer design weather conditions, a significant portion of the radiant heat gains can be conducted out, and those never become part of the cooling load. In these cases a much larger overprediction relative to the heat balance method is expected (Rees et al. 2000a; Rees et al. 1998; Spitler et al. 1997). 2.1.4 Limitations of the Radiant Time Series Method Quantitative comparison with the heat balance method shows that the RTSM tends to over predict the peak cooling loads (Rees et al. 1998; Spitler et al. 1997). Parametric investigations conducted for 945 zones cases showed that the peak load is slightly over predicted (Spitler et al. 1997). The heat balance method uses a detailed fundamental and rigorous mathematical model for the outside and inside surface heat balance. For medium and light weight construction, in particular zones with large amount of single pane glazing, the peak loads were over predicted significantly. In another similar study (Rees et al. 1998) made quantitative comparison of 7,000 different combinations of zone type, internal heat gains, and weather day. The result shows that the RTSM cooling load profile closely follows that of the heat balance cooling load; however, it over predicted the peak load for majority of the test cases when a radiative heat gain is large and zones 29 are made with large amount of single pane glazing. For a heavy weight construction midfloor, northeast corner zone, with 90% of the exterior wall area consisting of singlepane glass (Rees et al. 1998) the RTSM over predicted the peakcooling load by 37%. Three main reasons have been pointed out for peak cooling load over prediction: (1) the use of adiabatic boundary condition for the RTF generation, (2) combined treatment and constant assumption of convection and radiation coefficients, which makes the zone internal surfaces to radiate to the room air, and (3) simplification of the solair temperature calculations. Rees et al. (1998) concluded that the RTS method enforces conservation of radiant heat gains by ignoring the heat gain conducted to the outside environment as the principal reason for over prediction of peak cooling load. For internal surfaces with conditioned adjacent zones, the adiabatic boundary condition is a reasonable approximation; however, for external surfaces the adiabatic boundary condition in some cases very conservative approximation. Zones for which the peak design cooling load occurs in winter or zones located at lower design weather temperatures can be shown (with the HBM) to conduct a large amount of heat gains through the exterior surfaces with very low conductance (e.g. single pane glazing windows). On the other hand, the RTSM conserves the entire radiant heat gains and has no procedure to account for the heat gain conducted to the outside. Therefore, the RTSM over predicts the peak design cooling load slightly for hot and warm cooling design weather locations, while it tends to over predicts more and more for cold design weather conditions. 30 Experimental validation of radiant time series cooling load calculation method revealed that reflection loss of solar heat gain from the zone with high glazing fraction is significant (Iu et al. 2003). Though the rereflection and direct transmission losses can be computed they require detailed input data of glazing optical properties, zone geometry and orientations. In fact this phenomenon is likely to cause significant loss only in highly glazed buildings. 2.2 Dynamic Modeling of Thermal Bridges Dynamic modeling of thermal bridges has been an area of interest in building energy analysis and design load calculation programs. Building energy analysis and load calculation programs developed in the USA use onedimensional conduction transfer functions to predict heat conduction through the building envelope. However, many wall and roof constructions contain composite layers (e.g. steel studs, and batt insulation) that lead to local multidimensional heat conduction. The element with very high thermal conductivity is often referred to as a thermal bridge. Thermal bridges are important for both steady state and dynamic heat conduction. Several publications (Brown et al. 1998; Carpenter et al. 2003b; Kosny and Christian 1995b; Kosny et al. 1997b; Kosny and Kossecka 2002; Kosny et al. 1997c) indicate that onedimensional approaches cannot predict heat transmission through building envelopes without errors, especially for walls with thermally massive elements and a high disparity in the thermal conductivity of layer materials. Numerical studies indicate that thermal bridge effects of steel stud walls can reduce the thermal resistance of the clear wall by up 31 to 50% (Kosny et al. 1997a). Similar studies on metal frame roofs showed that the thermal bridge effect reduces the effective thermal resistance of the clear cavity values by as high as 75% (Kosny et al. 1997c). However, there is a limitation in the use of onedimensional response factor or conduction transfer functions methods when it comes to analysis of composite walls such as stud walls. This is a common problem in modeling heat conduction in steel stud walls and the ground where onedimensional analysis cannot predict the heat conduction without significant error. Multidimensional heat conduction effects are either ignored or not accounted properly. The onedimensional analysis may be valid for homogeneous layer wall; however, at the edges and corners, heat transfer significantly deviates from that of the onedimensional analysis. In practice, the edge and corner effects are simply ignored. Numerical and experimental investigations showed that ignoring the edge effects could under predict the heat transmission by over 10% (Davies et al. 1995). However, for portions of walls not near the edges, onedimensional analysis can be a reasonable approximation for lightweight walls without significant thermal conductivity disparity, such as those made from wood studs (Davies et al. 1995). Therefore, the need for multidimensional transient heat conduction models in building energy analysis and load calculation programs is crucial for accurate prediction of building energy consumption and peak load estimation; hence, it is also necessary for reliable HVAC equipment sizing and thermal comfort prediction. The following section discusses the onedimensional dynamic conduction modeling commonly used in load calculation and energy analysis programs in the USA. 32 2.2.1 OneDimensional Conduction Transfer Functions Transient conduction heat transfer through building envelopes can be calculated using lumped parameter methods, numerical methods, frequency response methods and conduction transfer function methods. Conduction transfer functions have been used most commonly in lead calculation and building energy analysis programs due to their computational efficiency and accuracy. The response factors are time series solutions of transient heat conduction that relate the current heat flux terms to current and past temperatures. Conduction transfer function coefficients are derived from response factors, which are determined using Laplace transform method (Kusuda 1969; Mitalas 1968; Stephenson and Mitalas 1971), or numerically (Peavy 1978). Conduction transfer function coefficients can be also determined directly using frequencydomain regression (Wang and Chen 2003), stable series expansion based on the Ruth stability theory (Zhang and Ding 2003), and State Space method (Seem 1987; Strand 1995). The next sections presents the use of response factor and transfer function coefficients in onedimensional conduction. Heat conduction through building structures is represented by onedimensional partial differential heat equation and the Fourier’s law of heat conduction as follows: t c T x t x q x t p ∂ ∂ = ∂ ∂ '' ( , ) ( , ) ρ (2.3) x q x t k T x t ∂ ∂ '' ( , ) = − ( , ) (2.4) 33 Where q” = is the heat flux, (W/m2 K) T = is the temperature, (oC) k = is the thermal conductivity, (W/m K) ρ = is the density, (kg/m3) cp = is the specific heat of the solid, (kJ/kg K) The solution of equations 2.3 and 2.4 can be represented as time series solutions called response factors. The time series solution of the heat conduction equation is determined for a unit triangular ramp excitation of the temperatures on both the internal and external surfaces of a wall. The response factors can be determined using Laplace Transform method (Clarke 2001; Hittle 1992; Kusuda 1969; Stephenson and Mitalas 1971), numerical methods (Peavy 1978), and time domain methods (Davies 1996). The current heat flux at interior surface of the wall '' q&i,t in terms of current and past boundary temperatures as inputs and the response factors is given by: Σ Σ∞ = − + ∞ = = − + − 1 , 1 1 , 1 '' , n n o t n n n q&i t Ti t n Z T Y (2.5) The heat flux at the external surfaces '' q&o,t is given by: Σ Σ∞ = − + ∞ = = − + − 1 , 1 1 , 1 '' , n o t n n n q&o t Ti t n Yn T X (2.6) 34 Where '' q&i,t : heat flux at the interior surfaces at time step t, W/m2 '' q&o,t : heat flux at the external surfaces at time step t, W/m2 Ti,tn+1: interior boundary temperature at time step tn+1, oC To,tn+1: exterior boundary temperature at time step tn+1, oC X: is the selfcoupling response factor; the heat flux at the exterior surface for triangular ramp input of the exterior boundary temperature and zero interior boundary temperature (W/m2 K). Y: is the cross coupling response factor; the heat flux at either surface for triangular ramp input of the boundary temperature at the other surface, (W/m2 K). Z: is selfcoupling response factor of the interior surface for triangular ramp input of the boundary temperature at the interior surface and zero exterior boundary temperature, (W/m2 K). The primary advantage of the response factor method is that for a given building structure, assuming constant thermophysical properties, the response factors need to be determined only once. Numerical methods such as finite difference or finite element methods generally require high computational time; however, they allow variable time step and variable thermophysical property simulations. Conduction transfer function methods are further refinements of response factor methods. The replace many of the higher order terms of the response factors and the past 35 temperatures with the past heat fluxes; hence, use fewer coefficients (Hittle, 1992) (McQuiston et al. 2005; McQuiston 2000). The heat fluxes at the interior and exterior surfaces of a wall in terms of conduction transfer functions are given by: Σ Σ Σ = = − + − = = − + − + M m M m k m o t m m i t m M m qi t Zk mTi t m Y T F q 1 1 , , 1 , 1 , , 1 '' , (2.7) Σ Σ Σ = = − + − = = − + − + M m M m k m o t m m o t m M m qo t Yk mTi t m X T F q 1 1 , , 1 , 1 , , 1 '' , (2.8) Where Xk,m: the mth conduction transfer coefficient for the exterior selfcoupling term of order k, (W/m2 K) Yk,m: the mth conduction transfer coefficient for the cross coupling term of order k, (W/m2 K) Zk,m: the mth conduction transfer coefficient for the interior selfcoupling term of order k, (W/m2 K) Fm: is defined as the flux history term coefficients (). 2.2.2 Steady State Conduction Models Multidimensional and in particular twodimensional steady state conduction models have been used to study and investigate the accuracy of the approximate onedimensional thermal resistance calculation procedures for thermal bridges. Studies made on multidimensional steady state heat conduction analysis of composite walls have indicated that the heat fluxes deviate significantly from that of an approximate onedimensional heat 36 conduction models that ignore the thermal bridges. This has been demonstrated using numerical and experimental analysis of steady state heat conduction in building wall specimens (Barbour and Goodrow 1995; Brown et al. 1998; Carpenter and Schumacher 2003; Kosny and Christian 1995a; Kosny and Christian 1995c; Kosny et al. 1997a; Kosny et al. 1997b; Thomas and Antar 1998). Compared to dynamic analysis, steady state models for heat transfer of thermal bridges are well developed. The next section briefly discusses ASHRAE’s recommended approximate onedimensional steady heat conduction models. Approximate Steady State Models Steady state heat transfer through composite material walls is commonly treated with onedimensional models that utilize some approximations in representing the thermal resistance of composite walls. ASHRAE recommends the following methods: isothermal plane method, parallel path method, the zone method and modified zone method and insulation/framing adjustment factor method to compute the overall thermal resistance based on qualitative criteria (ASHRAE 2005). Isothermal Plane Method This method assumes that for layer materials with high thermal conductivity the temperature at each plane remains isothermal. Composite layers sandwiched in between these two isothermal plane layers are combined using area weighted parallel heat flow path method. Then the overall resistance is determined from layer resistances using a series sum of resistances. This method is recommended for concrete blocks where web 37 and the core section are combined using areaweighted parallel heat flow path method and then combined in series with the face and air film resistances. For widely distributed metal members with high crosssectional area constructions such as roof decks the isothermal plane method underpredicts the overall resistance; hence, the zonal method is recommended (ASHRAE 2001). Parallel Path Method The parallel heat flow path method assumes no heat flow in the lateral direction; hence, the heat flow path in the construction is principally longitudinal. The resistance is calculated from the areaweighted average of the individual thermal transmittances of the different parallel heat flow paths in the construction. This method predicts the overallresistance of a construction with reasonable accuracy for wood frame or wood stud walls, where the disparity in thermal conductivity between the wood and the cavity insulation is small. The actual overall thermal resistance lies in between the isothermal plane and the parallel path methods (Barbour and Goodrow 1995; Brown et al. 1998; Gorgolewski 2005; Thomas and Antar 1998). Another method developed as extension of the parallel path method is parallel path correction factor method (ASHRAE 2005). The thermal resistances along the stud and center of wall are area weighted to get the overall average thermal resistance. The parallel path correction factor Fc, method is recommended for the metal stud walls. 38 Modified Zone Method For building envelopes with widely spaced metal members such as steel stud walls, the actual overall thermal resistance lies in between the isothermal plane and parallel path methods (Barbour and Goodrow 1995; Gorgolewski 2005). The ASHRAE zone method was introduced for calculating overall resistance for such constructions (ASHRAE 2005). The zone method extends the parallel heat flow path method to account the local highly conductive region as a separate path for the heat flow and divides the construction into two zones. The zone method determines the width of zone containing the metal element as function of the distance from the stud face to the surface of the construction. Studies have shown that zone method does not consider the thermal bridge region of influence or the metal zone width dependency on stud spacing, stud depth and sheathing thermal conductivity (Barbour and Goodrow 1995; Kosny and Christian 1995a). The modified zone method was introduced to improve the zone method by including the metal zone area dependency on: ratio of resistivity of cavity insulation to sheathing materials, thickness of sheathing insulation, and stud flange area (Kosny and Christian 1995a). Insulation / Framing Adjustment Method Thermal resistance of wall assemblies containing metal framing can be calculated using insulation /framing adjustment factors. Such framing factors are provided by ASHRAE / IESNA standard 90.12004 (ASHRAE 2005). The adjustment factor corrects the resistance of the core insulation for the metal frame effect. It is also called correction factor method. 39 Gorgolewski Method The Gorgolewski (2007) proposed a semiempirical correlation for computing the steady state Rvalue of light frame steel stud walls. This procedure uses weighted average of isothermal plane and parallel path method Rvalues. The weighting parameter is calculated from semiempirical correlation that depends on the geometry of the steel frame and the isothermal and parallel path methods Rvalues and hence is suitable for programming application as it does not involve subjectivity. The following section discusses development in multidimensional conduction dynamic modeling method and the barriers for their implementation. 2.2.3 Multidimensional Conduction Dynamic Models There have been repeated efforts to develop multidimensional dynamic heat transfer model that produce CTFs in onedimensional form for energy analysis and load calculation programs. Previously published methods for dynamic modeling of multidimensional conduction proposed for use in building energy analysis and load calculation programs include: numerical methods, numerical CTF method, equivalent wall methods and statespace method. The later methods were developed with the intention to use in developing onedimensional CTF coefficients. However, their adoption has been delayed for several reasons. Nevertheless, there remains a clear need for a simple onedimensional approximate dynamic model for modeling of construction with thermal bridges. The next section discusses previously published multidimensional conduction dynamic models and their limitations. 40 Numerical Methods Numerical models of multidimensional conduction heat transfer have been developed, but are still limited to research use (Davies, et al., 1995). Numerical methods include finite difference, finite volume and finite element techniques. The third generation building simulation program, ESPr, uses the finite control volume energy conservation method (Clarke 2001; Nakhi 1995). ESPr, a whole building energy simulation program developed at the University of Strathclyde offers multidimensional heat conduction analysis of walls, edges, corners and the ground; however, the multidimensional heat conduction model has limitations on the composite layer specification and, at best, is difficult to use. Numerical methods require high computational time since it involves solving the nodal variables at each time step. Therefore, implementation of multidimensional heat conduction finite difference or finite element methods for real composite walls requires higher computational time and computer memory. The lack of graphical user interface for automatic building geometry and construction material acquisition has been a hurdle for the development of spatial discretization for use in the multidimensional conduction model. Burch et al. numerical CTF method Burch et al. (1992) presented a numerical procedure for calculating CTF coefficients that accounts for thermal bridge effects of metal studs, aluminum frame windows and metal frames on office building envelopes. The method solves the conduction equation numerically using finite difference techniques by applying linearly varying boundary conditions that replicate the triangular ramp temperature boundary conditions as shown in 41 Figure 2.3. Then the principle of superimposition is used to determine the response factors from the three linear temperature excitations at base time steps of 2δ. The heat flux at a particular surface yields the required response factors. The numerical procedure of determining the response factors for multidimensional conduction models can be summarized as follows (Burch et al. 1992): i. Develop the triangular ramp unit excitation (Figure 2.3) representation of the boundary temperature at one of the surfaces of interest while the other face of the surface is kept at zero temperature (Hittle, 1992). ii. Determine the numerical solution of the heat flux at the surface interest by summing the individual heat fluxes of the cells or nodes for each excitation. This yields one of the response factors. In a similarly way the other response factors can be determined. iii. The CTFs coefficients are determined from the response factors using recursive algorithms (Hittle, 1992). 1Δ 1Δ 2Δ Elapsed Time Temperature 1 Figure 2.3 Lineartriangular ramp temperature pulse representation 42 The Burch et al. (1992) finite difference procedure based CTF coefficients determination method requires a separate standalone transient heat conduction analysis program with a spatial discretization scheme. Thus, the Burch et al. method is difficult to be integrate into existing building energy analysis and load calculation programs without significant modifications of their codes (Burch et al. 1992). Equivalent Wall Method Kossecka (1998) and Carpenter et al. (2003a) developed the concept of an equivalent wall, which replicates multidimensional thermal dynamics of the complex composite wall with a simple homogeneous layer wall. The generated equivalent wall, which has the same dynamic behavior to that of the real composite wall, is represented by onedimensional response factors or conduction transfer function that can be implemented in the commonly used building energy analysis and load calculation programs (Kosny and Kossecka 2002). Generating equivalent walls requires proper identification of the thermal mass and negligible mass resistance components from the construction layer configuration. The thermal characterization of constructions can be defined by a parameter called thermal structure factor (Kossecka 1998). The concept of thermal structure factor is presented next. Thermal Structure Factors Thermal structure factors, which are dimensionless parameters, define the thermal energy storage characteristic of building structures when it goes through two successive steady state ambient temperature transitions (Kossecka 1998; Kossecka and Kosny 2002). 43 Thermal structure factors of a wall depend on the resistance and thermal capacities of the layers and their sequence of arrangements in the wall. Thermal structure factors (φ) of building structures (Carpenter et al. 2003a; Kossecka 1998; Kossecka and Kosny 2002) are given by: c ( ) dx C L ii p 2 0 ϕ = 1 ∫ρ 1−θ (2.9) c ( )dx C L ie p ϕ = 1 ∫ρ θ 1−θ 0 (2.10) c dx C L ee p 2 0 ϕ = 1 ∫ρ θ (2.11) Besides, the following identity needs to be met by the thermal structure factors + 2 + = 1 ii ie ee ϕ ϕ ϕ (2.12) Where C Overall thermal capacity of the wall, (J/m2 K) cp specific heat of a layer in the wall, (J/kg K) ρ density of a layer in the wall, (kg/m3) L thickness of the wall, (m) θ dimensionless temperature, () φii Interior structure factor () φie core structure factor () 44 φee exterior structure factor () A high interior thermal structure factor (φii) implies that the higher thermal mass layer is located near the interior surface and most of the resistance is located near the exterior surface of the wall. Vice versa, a high exterior thermal structure factor (φee) indicates that the higher thermal mass layer is located near the exterior surface and most of the resistance is located near the interior surface of the wall. And a high core thermal structure factor (φie) implies that the higher thermal mass layers are located at the center of the wall and the resistances are placed symmetrically on both sides of the wall. The relationships between response factors and thermal structure factors (Carpenter et al. 2003a; Kossecka 1998; Kossecka and Kosny 2002) are given: ii n n C nX ϕ δ − = Σ∞ =1 (2.13) ie n n C nY ϕ δ = Σ∞ =1 (2.14) ee n n C nZ ϕ δ − = Σ∞ =1 (2.15) Where Xn: the nth term exterior selfcoupling response factor, (W/m2 K) Yn: the nth term cross coupling response factor, (W/m2 K) Zn: the nth term interior selfcoupling response factor, (W/m2 K) C Overall thermal capacity of the wall, (J/m2 K) 45 φii Interior structural factor () φie core structural factor () φee exterior structural factor () δ time step, (s) Equivalent wall Generation Procedure The equivalent wall generation requires five steps (Carpenter et al. 2003a). These steps are summarized as follows: (1) Develop a threedimensional model of the building envelope to exact dimensions using multidimensional dimensional heat conduction solver computer programs. (2) Generate threedimensional response factors using numerical methods and use them to determine the three dimensional conduction transfer function; (3) Calculate the thermal structure factors of the composite wall using the three dimensional response factors determined in step 2 and the thermal structure factor identity; (4) Generate the fictitious equivalent wall. The equivalent wall generation requires selecting random set of resistances for each layers of the wall and calculating the capacitance or randomly specifying the capacitance and calculating the resistances. A three layer fictitious wall is recommended for simplicity (Carpenter et al. 2003a). The material layer configuration, i.e., the relative position of the high thermal mass and the high resistance layers in the fictitious wall layers configuration must resemble that of the actual wall. 46 (5) The transfer function coefficients for the threedimensional numerical model and the equivalent wall model matching is done by trialanderror until reasonable accuracy is achieved by adjusting the resistance and/or capacitance of the equivalent wall layers. The steady state resistance, thermal response factors and structure factors of the real wall and the fictitious equivalent wall must be the same. State Space Method The state space method can be used to determine conduction transfer function coefficients that can represent multidimensional transient heat conduction in walls. The state space method is based on first order differential equation representation of transient heat conduction by spatially discretizing the conduction domain and representing the inputs by continuous, piecewise linear functions (Seem 1987). The advantage of the state space method compared to the Laplace transform method (Hittle, 1992) is that it can be extended to solve multidimensional transient heat conduction problems. The state space method is represented in the form of equations as follows: [ ][ i ] [ ][ b ] i A T B T dt dT = + (2.16) [ ] [ ][ ] [ ][ ] q = C T + D Tb (2.17) Where Ti vector of n interior node temperatures as state variables, °C (°F) 47 A a constant coefficient transition matrix with size of nxn t the time, (s) B the constant coefficient matrix of input vector of size (nxp) Tb vector of p boundary temperatures as inputs, °C(°F) q vector of p heat fluxes as outputs, W/m2°K(Btu/h⋅ft2⋅°F) C a constant coefficient matrix of the output vectors of size (mxn) D constant coefficients matrix of the input vector of size (mxp) Solution of equations 2.16 and 2.17 for constant elements matrix of A and B in a compact representation is given by: Σ( ) Σ = = = = = − − − j n j j n j qt S jTt j Fjqt j 0 1 δ δ (2.18) Where Sj are the conduction transfer function coefficients, (W/m2 K) Fj are the coefficients of the past heat flux history terms, () Twodimensional state space method Seem (1987) demonstrated that the state space method can be applied to model multidimensional transient heat conduction in building envelopes. In the state space method, the heat conduction domain is discretized in multidimensions; hence, the multidimensional heat conduction effects in composite walls such as those with steel studs can 48 be modeled accurately. Therefore, the state space method can model dynamics of walls; however, as the conduction domain becomes complex, the number of nodes required for accurate modeling also increases. Consequently, evaluation of the exponential matrix becomes cumbersome or sometimes almost impossible (Amjad et al. 2003). Barrier to Use of Dynamic Multidimensional Models Three multidimensional dynamic thermal bridge models, which could be integrated into existing onedimensional conduction transfer function procedures, have been proposed. The multidimensional dynamic CTF models are: the equivalent wall method, the state space method, and the numerical CTF method. Beyond other reasons, implementation and integration of these models into the existing programs has been delayed due to inherent limitations in the development of the multidimensional dynamic procedures, and high computational resource requirement. These barriers to implementation in whole building load calculation / energy simulation programs include: The equivalent wall and the Burch et al., numerical CTF models require either separate standalone multidimensional transient heat conduction analysis numerical programs or integration of a significant multidimensional conduction subprogram. Models with spatial discretization requirements need both an implementation of the discretization procedure and user interface to support this level of details. Given the complexity of providing an interface to specify a whole building, this may be too much of a refinement. 49 The equivalent wall method thermophysical properties determined by trialanderror could be out of range for typical building materials properties. Although the state space method does not require a separate standalone numerical program, a spatial discretization tool is necessary. On the other hand, the state space method can be integrated with existing building energy analysis and load calculation programs most conveniently if an automatic building envelope discretization and/or interactive user interface for material specification and construction model reduction program is made available. However, the multidimensional state space method will suffer from a numerical accuracy problem that grows as the number of nodes increases. 50 51 CHAPTER III 3. Radiant Time Series Method Improvements This chapter covers several improvements to the Radiant Time Series Method (RTSM). These improvements include: new algorithm for generating Radiant Time Factors (RTF) and developing a onedimensional finite volume numerical method periodic response factor generating procedure, adapting an improved fenestration model and establishing radiative / convective splits, developing a procedure for accounting heat losses through fenestration to the outside, and facilitating implementation of the RTSM procedure in different computation environments. There are several approaches for generating the RTF for a given building zone. Since these procedures have been adopted directly or indirectly from a fullblown heat balance method (HBM) procedure, they tend to have some unnecessary overhead and a simpler method developed specifically for RTF generation is highly desirable. Derivation of the reduced HBM RTF engine is described in Section 3.1. The new radiant time factors (RTF) generating algorithm utilizes periodic response factors (PRF) to model transient conduction as described in Section 3.1.1. Currently available PRF generation procedures convert conduction transfer function coefficients generated using Laplace and State Space method. 52 This procedure is undesirable for use in VBA and SCILAB type computational environments. Therefore, a onedimensional finite volume numerical procedure for computing periodic response factors has been implemented and investigated. The finite volume numerical procedure implementation is discussed in Section 3.1.2. Derivation of the algorithm and validations are given in APPENDIXB. As part of this research, an improved fenestration model compatible with currently available fenestration data will be adapted to the RTSM. Furthermore, a new set of radiative / convective splits compatible with the improved fenestration model has been established. The improvement in the RTSM fenestration model is described in Section 3.2. The previously published RTSM cooling load procedure2 (Rees et al. 2000a; Spitler et al. 1997) does not account for the solar and internal radiant heat gains conducted back out of the zone. Ignoring these back losses is the principal reason that the RTSM over predicts the peakcooling load. An approximate algorithm that accounts for zone radiant loss in the RTSM procedure has been derived and is described in Section 3.3. For all improvements described in this section, results and discussed are provided for each Sections. However, for improvement to the fenestration model and the heat losses accounting procedure, it is highly desirable to demonstrate satisfactory performance over a wider range of cases. Such a study is the subject of Chapter 4. 2 Referred to as the “original RTSM procedure” in this thesis. 53 3.1 New RTF Calculation Engine The Radiant Time Series Method (RTSM) converts the radiant component of the heat gains into cooling loads using the Radiant Time Factors (RTF), a 24term series. The 24 hourly radiant time factors describe the dynamic response characteristics of a zone. Two procedures were developed for RTF generation (Spitler et al. 1997). Currently there are five approaches available for generating the radiant time factors: (1) software that comes with the book Cooling and Heating Load Calculation Principles (Pedersen et al. 1998), (2) software that comes with the book by McQuiston et al. (2005), (3) software developed as based on the ASHRAE toolkit (Iu 2001), (4) tabulated RTF values in the ASHRAE Handbook of Fundamentals, and (5) RTF generated from zone heat gain weighting factors. The first four approaches use the full heat balance method as a calculation engine to compute the RTFs. The HBM programs that come with the books by Pedersen et al. and McQuiston et al. are limited to twelve surfaces only for any zone and the RTF generating software developed by Iu is limited to six surfaces. The McQuiston et al. and Pedersen, et al. programs are full load calculation programs that treat the RTF as an auxiliary output. The McQuiston et al. and Iu programs have interfaces that allow users to select material layer thermophysical properties from a database. The interface used by McQuiston et al. accepts much more information than is actually necessary to generate the RTF. The tabulated values in the Handbook of Fundamentals were generated for limited building design conditions using Pedersen et al. program. But the accuracy of the RTSM peak cooling load resulting from the users choosing the “nearest” zone has not been investigated. 54 An alternative approach (Spitler and Fisher 1999b) is to generate radiant time factors directly from a set of zone heat gain weighting factors using the existing ASHRAE 472 RP database (Sowell 1988a; Sowell 1988b; Sowell 1988c). These weighting factors were developed for use with the Transfer Function Method (TFM). This approach would use a computer program to map a given zone to the fourteen zone parameters in the database and transform the weighting factors to radiant time factors using matrix manipulation. However, the 472RP weighting factors do not represent all possible zone designs and construction types. ASHRAE’s 472RP weighting factors were generated for combination of discrete building design parameters; hence, the RTF generated from these weighting factors represent only specific buildings. User judgment is required to match an existing building to one of the combinations of discrete building design parameters. In conclusion, the existing approaches all suffer from being too cumbersome, requiring too much user judgment, or too limited with regard to the number of zones surfaces. Therefore, development of an RTF coefficientgenerating tool that handles a wide range of practical design conditions is desirable. Such a tool should meet the following requirements: i. capable of generating RTF for a wide variety of practical building constructions, including a practical number of building surfaces, i.e. more than twelve. ii. simple user interface that does not require unnecessary information. iii. can be integrated with other applications such as spreadsheets by eliminating unnecessary features and source code. It would be preferable to 55 minimize the required calculations and eliminate the use of DLLs if possible. iv. The methodology should take advantage of the steady periodic nature of the boundary condition. When this is done, the methodology can benefit from matrix algebra easily accessed in modern computing languages and environments such as SCILAB or MATLAB. The new RTF calculation engine is derived from an inside surface heat balance and room air heat balance for adiabatic zone. The simplified procedure eliminates several steps of the HBM, particularly the exterior surface heat balance and eliminates input data that are not necessary for the RTF generation. The new RTF generating program is first developed as a FORTRAN DLL which may be called from a spreadsheet. Then, it will be implemented in VBA and SCILAB. The following section describes the derivation of the mathematical algorithm for the RTF generation program. 3.1.1 The Mathematical Model Reduced Heat Balance Method The formulation of the mathematical model for the radiant time factor generating procedure makes use of the major assumptions used in the Heat Balance Method (HBM) (Pedersen et al. 1997); that surfaces (walls, roofs, windows, etc) can be treated as having uniform surface temperatures, uniform long wave and short wave length radiation; surfaces are gray; onedimensional conduction is valid and surfaces are exposed to steady periodic boundary conditions. The simplified heat balance procedure for RTF generation uses an inside surface heat balance, steady periodic boundary condition, constant 56 convection coefficients and constant room air temperature. The simplified HB procedure for RTF generation eliminates the following procedures that are part of the full heat balance method: outside surface heat balance, weather data, solar radiation calculations, shading calculations, infiltration and ventilation. Because this simplified version of the HBM uses a smaller number of heat balance steps, it will be referred to here as the “Reduced Heat Balance Method” (RHBM). In the next section, the RHBM is described stepbystep. The inside surface heat balance for the ith surface is given by: '' 0 , '' , '' , '' , + + + = conv i cond i rad i gain i q& q& q& q& (3.1) Where, '' conv,i q& = convection heat flux from the room air to the ith surface, Btu/h⋅ft2 (W/m2) '' q&cond ,i = the ith surface conduction heat flux from the outside surface to the inside surface, Btu/h⋅ft2 (W/m2) '' rad ,i q& = radiant heat flux from other internal surfaces to the ith surface, Btu/h⋅ft2 (W/m2) '' gain,i q& = radiant heat gain flux of the ith surface from lights, people and equipments, Btu/h⋅ft2 (W/m2) 57 Conduction Model The RTSM uses airtoair periodic response factors to compute conduction heat gain driven by steady periodic exterior solair temperature and a constant indoor air temperature. For RTF generation, surfacetosurface periodic response factors are used with steady periodic surface temperatures computed with the reduced heat balance method. The transient conduction heat flux at the inside surface using surface periodic response factors (Spitler et al. 1997) for steady periodic boundary conditions is given by: Σ Σ = − = = − − 23 0 , , 23 0 , , '' , , j Pj In i t j j q&cond i t YPjTOut i t jδ Z T δ (3.2) Where, In i t jδ T , , − = inside surface temperature jth hours before the current hour, °F (°C) Out i t jδ T , , − = outside surface temperature jth hours before the current hour, °F (°C) Pj Y = surfacetosurface cross periodic response factor, Btu/h⋅ft2⋅°F (W/m2⋅°C) Pj Z = surfacetosurface inside periodic response factor, Btu/h⋅ft2⋅°F (W/m2⋅°C) The transient heat conduction equation can be simplified further using the appropriate boundary conditions. 58 Boundary Conditions For all surfaces in a zone, the outside surface temperature is assigned the inside surface temperature to maintain the equivalent temperature as the boundary condition. This approach emulates an adiabatic boundary condition by forcing an equal amount of heat flow from the other side of the surface, hence balancing the heat flow into the construction. This condition is represented as follows: ( ) Σ= = − − 23 0 , , '' , , j q&cond i t YPj ZPj TIn i t jδ (3.3) Figure 3.1 Adiabatic boundary condition for RTF generation: Tso=Tsi. Convection Model The RTSM procedure is formulated to use fixed convection coefficients. This assumption is not required in the Heat Balance Method. Heat transferred from the room air to the zone surfaces by convection using a constant convection coefficient is given by: TSO TMRT T 1/hcIniAi Si 1/hradiAi '' cond ,i q& '' gain,i q& '' conv,i q& '' rad ,i '' q& cond ,o q& 59 qconv i t hcIn,i (Tr TIn,i,t ) '' & , , = − (3.4) Where, In i t T , , = inside temperature of the ith surface at time t, °F (°C) Tr = constant room air temperature, °F (°C) hcIn,i = convection coefficient of the ith inside surface, Btu/h⋅ft2⋅oF (W/m2 °C) '' conv,i,t q& = convection heat flux at the ith surface at time t, Btu/h⋅ft2 (W/m2) Internal Longwave Radiation Model The internal long wave radiation model assumes zone air is transparent to long wave radiation and considers the zone surfaces as gray and the long wavelength radiation as diffuse. With these assumptions, long wavelength radiation in building can be modeled using the uniform radiosity method, the total gray exchange factors method (Hottel and Sarofim 1967), the mean radiant temperature and balance (MRT/balance) method (Walton 1980) or the Mean Radiant Temperature Network (MRTNet) method (Carroll 1980). The uniform radiosity and the total gray exchange factor method require exact view factors. The uniform radiosity method involves solving the radiosity at every time step hence the method is computationally intensive, which makes it unsuitable for building applications. The total gray exchange factor method combines the surface properties and geometries into the gray exchange factors, which can be computed at the beginning of the 60 simulation and eliminates the simultaneous solution at every time step. However, it requires exact view factors. In real buildings, calculation of exact view factors is computationally intensive and the furnishings and other internal heat sources cannot be easily represented and are mobile during the lifetime of the buildings. Hence, any gains in accuracy facilitated using the exact view factors are unlikely to be realized in practice. Therefore, the extra effort introduced in specifying locations and dimensions of furnishings is unlikely to be rewarded with any tangible benefit. The advantage of the MRT/ balance and MRTNet methods is that both use approximate view factors based on area and emissivity and allow an approximate representation of furnishings and partitions surfaces. In the MRT/balance method each surface exchanges radiation with a fictitious mean radiant temperature calculated from areaemissivitysurface temperature product weighted of the remaining surfaces viewed by the surface. The radiation flux imbalance resulting from approximate view factors in the MRT/balance method is balanced by redistributing it to the surfaces. In the MRTNet method each surface exchanges radiation with a single fictitious mean radiant temperature of the zone that is computed from all surfaces. The radiation coefficient linking the each surface is corrected for each surface depending on the surface and MRT temperature of the zone and hence balances the zone radiation exchange. Both MRT methods essentially have the same accuracy (Liesen and Pedersen, 1997). Therefore, due to simplicity, the mean radiant temperature network (MRTNet) method (Carroll 1980) has been selected for use in the RTF generation algorithm. With the 61 MRTNet method, each surface in the zone is linked to a fictitious single radiant node. The radiation heat transfer from the fictitious node to the surface is given by: q&rad ,i,t = hrad ,i,t (TMRT ,t − TIn,i,t ) (3.5) Where TMRT ,t = mean radiant temperature of the zone at time t, °F (°C) rad i t h , , = radiation coefficient of the ith surface at time t, Btu/h⋅ft2⋅°F (W/m2⋅°C) The MRT radiation model (Carroll 1980) requires a two step update of the radiation coefficients for each time step. First, initialize the radiation coefficients for each surface to the reference temperature as follows: ( ) i i i ref rad ref F T h ε ε σ 1/ 1 / 4 3 , + − = (3.6) Where Tref = mean reference temperature in absolute scale, °R (°K) hrad ,ref = radiation coefficient at reference temperature of 300K, Btu/h⋅ft2⋅°F (W/m2⋅°C) Fi = the MRTNet view factor for ith surface, (). i ε = the longwave emissivity of ith inside surface, (). 62 The MRTNet view factor is an approximate view factor that compensates for the selfweighting in the mean radiant temperature, TMRT, is given by: 1/(1 / ) 1 Σ= = − N i Fi AiFi AiFi (3.7) Where N = the number of surfaces in the zone Since the Fi appears on both sides of the equation, iteration is required after setting the initial values of Fi to unity (Carroll 1980). However, the MRT network view factors can be calculated at the beginning of the simulation during the initialization phase. The hourly radiation coefficient that links each surface to the single fictitious mean radiant temperature in Celsius and Fahrenheit scales, respectively, is updated at each time step as follows: rad i t In i t rad ref h T h , , , ' , , = (0.865 + / 200) (3.8a) rad i t In i t rad ref h T h , , , ' , , = (0.775 + / 360) (3.8b) Then the mean radiant temperature, TMRT, is calculated from weighted average of the product of surface temperatures, surface area and the updated radiation coefficient as follows: 63 Σ Σ = = = N i In i t i rad i t N i MRT t i rad i t T A h T A h 1 ' , , , , 1 , , ' , / (3.9) Again the radiation coefficients are adjusted using the updated mean radiant temperature in Celsius and Fahrenheit scales, respectively, as follows: ' , , ' , , , (0.865 / 200) rad i t MRT t rad i t h = + T h (3.10a) ' , , ' , , , (0.775 / 360) rad i t MRT t rad i t h = + T h (3.10b) Then the corrected radiation coefficient of the individual surface is used to compute the room mean radiant temperature (TMRT,t) as follows: Σ Σ = = = N i In i t i rad i t N i MRT t i rad i t T A h T A h 1 , , , , 1 , , , / (3.11) So, for each iteration of the heat balance, a fixed twostep iteration to determine the radiation coefficient and the MRT is done as given in Equations 3.7 to 3.11. Radiant Heat Gain Distribution Model Computing the precise distribution of longwave radiation from internal sources requires knowledge of the exact location / position, surface area and temperature of the sources. This approach creates additional complexity to internal radiation exchange models. Therefore, the conventional approach is to distribute the internal radiant heat gains 64 uniformly to all surfaces in the zone, based on an area weighted or an areaabsorptance product weighted distribution model. Similarly, internal short wave radiation heat gain can be reasonably represented by uniform distribution as most of the cases have diffuse sources. For transmitted solar radiation an accurate distribution model could track the sun’s position and the resulting sun patch hourbyhour. However, partial surface irradiation is incompatible with the onedimensional and uniform surface temperature assumptions used in the conduction and radiation exchange submodels. Therefore, the most commonly used distribution model is to distribute the transmitted beam solar on the surfaces that are most likely to intercept the solar radiation  the floor and the furnishings. Transmitted diffuse solar heat gain is treated like long wavelength radiant heat gains and distributed uniformly. The two different distributions are the genesis of the two different RTF series. Long wave length RTF is generated by distributing the heat gain pulse uniformly to all surfaces in the zone. The solar RTF is generated by distributing the heat gain 50% to the floor and 50% to the furniture. Derivation of the Reduced HBM Algorithm Substituting the individual heat balance components into the heat balance equation (3.1) for the ith surface yields: ( ) ( ) , , ( , , , ) ) , , 0 23 0 , , , , , + − + = − + − Σ= − i rad i t MRT t In i t i gain i t j i cIn i r In i t i Pj Pj In i t j A h T T A q A h T T A Y Z T & δ (3.12) 65 For each surface the heat balance equation 3.12 can be reduced to the form shown below: ( ) ( ) '' , , , , , , 23 1 , 0 0 , , , , , , i cIn i r i rad i t MRT t i gain i t j i cIn i P P rad i t In i t i Pj Pj In i t j A h T A h T A q A h Z Y h T A Z Y T = + + & − + + − + Σ= − δ (3.13) Solution Schemes At least two solution schemes for solving Eqn. 3.13 can be developed: solution scheme I involves setting up the reduced heat balance equation for all 24 hours for each surface and then marching through each surface until the 24hourly surface temperatures for all surfaces converge. Solution scheme II involves setting up the heat balance equation so as to solve for the inside temperatures of all the surfaces in the zone at every hour, and then march through each hour. These two solution schemes are described in the next two sections. Solution Scheme I For a particular surface, the reduced heat balance (equation 3.13) can be written for 24 hours in compact matrix notation: [Ω][T ] [Γ] In = (3.14) The Matrix Ω will have dimensions of 24 by 24 and is given by the following expression: 66 − − − − + + − − − − − − − + + − − − + + − − − + + − − − Ω= P P P P P P P P radi cIni P P P P P P P P P P P P P P radi cIni P P P P P P radi cIni P P P P P P radi cIni P P P P P P Z Y Z Y Z Y Z Y h h Z Y Z Y Z Y Z Y Z Y Z Y Z Y h h Z Y Z Y Z Y h h Z Y Z Y Z Y h h Z Y Z Y Z Y 23 23 22 22 21 21 0 0 , ,24 , 22 22 21 21 20 20 23 23 2 2 1 1 0 0 , ,3 , 3 3 1 1 0 0 , ,2 , 23 23 2 2 0 0 , ,1 , 23 23 22 22 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) L M M M M O M L L L (3.15) The Matrix Γ is a 24element column vector and is given by the following expression: + + + + + + + + + + Γ = '' , , ,24 ,24 , ,24 '' , , ,23 ,23 , ,23 '' , , ,3 ,3 , ,3 '' , , ,2 ,2 , ,2 '' , , ,1 ,1 , ,1 cIn i r rad i MRT gain i cIn i r rad i MRT gain i cIn i r rad i MRT gain i cIn i r rad i MRT gain i cIn i r rad i MRT gain i h T h T q h T h T q h T h T q h T h T q h T h T q & & M & & & (3.16) The inside surface temperatures for a particular surface are determined from Eq. 3.17 as follows: [T ] [Ω] [Γ] In −1 = (3.17) In this scheme, the 24 hourly values of surface temperature are solved for each surface sequentially; this is done iteratively until all surface temperatures for all hours are converged. The radiation coefficients and the Ω matrix elements need to be updated at each iteration step. Thus, the repeated matrix inversion is computationally intensive. Further 67 simplification of the RTF generation algorithm has been investigated using fixed radiation coefficients. This is presented in Section 3.1.4. Solution Scheme II In solution scheme II the surface temperatures are solved for all surfaces at every hour. This solution scheme also can be formulated in matrix notation; however, it is formulated as an iterative procedure. For an hourbyhour march through all the surfaces at each step, the heat balance equation can be solved iteratively from the following equation: ( ) ( cIn i P P rad i t ) rad i t MRT t gain i t j cIn i r Pj Pj In i t j In i t h Z Y h h T Y Z T h T q T , 0 0 , , '' , , , , , 23 1 , , , , , + − + + − + + = Σ= − δ & (3.18) As with solution scheme I the radiation coefficients and the mean radiant temperatures must be updated at each iteration step until the surface temperature converges. The convergence criterion can apply on the surface temperature or the RTFs. The zone radiant time factor is determined from the inside surface temperatures, zone air temperature, convection coefficients and zone surfaces inside areas. The sum of the convection heat transfer from each surface per unit heat gain pulse of the zone is the hourly radiant time factor coefficient and is computed as follows: [ ] [ ] [ ] In cIn gain t T Ah q r = Δ ⋅ & 1 (3.19) 68 Where gain q& = heat gain pulse with which the zone is excited, Btu/hr (W) [ ] t r = a column vector of 24element radiant time factor coefficients, () [ ] cIn Ah = a row vector of Nelements of the product of inside surface area and convection coefficients, () [ΔTIn ] = a 24 by N matrix of the difference between inside surface temperature and the room air temperature, (K) The convergence criteria employed is that when the change in the sum of the RTFs changes less than 0.00001 between the successive iteration steps, the solution is considered converged. Solution scheme I is adapted in all subsequent sections and implementations in all other computing environments. 69 Figure 3.2 Flow chart of solution scheme I Get Surface Parameters Compute Periodic Response Factors Initialize the calculation Beginning Iteration Loop Start Surface Loop I = 1, Nmax Set up system matrix Eqns. 3.15 & 3.16 Compute Surface Temperatures Eq. 3.17 i = Nmax Compute RTFs Eq. 3.19 No ΣRTF =1 Update TMRT and hrad No convergence 70 3.1.2 Validation of the New RTF Engine The new RTF generating engine has been validated against the fullblown heat balance method FORTRAN program originally developed by Pedersen et al. (1997). The test zone geometry and construction fabrics are given in Table 3.1. Three construction types: light, medium and heavy weight constructions were used for the validation. Each zone has a single exterior surface with single pane clear glass and 50% glazing fraction of the exterior facade. Zones were modeled with and without carpeting. Table 3.1 Description of test zone constructions for RTF generation algorithm validation Fabric Element Lightweight Mediumweight Heavyweight External wall steel siding, 2 in insulation, air space, ¾ in gypsum 4in face brick, 2 in insulation, air space, ¾ in gypsum 4in face brick, air space 2 in. insulation, 8 in HW concrete, ¾ in gypsum Roof/Ceiling 4 in. LW concrete, ceiling air space, acoustic tile 4 in. HW concrete, ceiling air space, acoustic tile 8 in. HW concrete, ceiling air space, acoustic tile Partition ¾ in. gyp, air space, ¾.in. gypsum ¾ in. gyp, air space, 3/4.in. gypsum ¾ in. gyp, 8 in. HW concrete block, 3/4.in. gypsum Floor Acoustic tile, ceiling air space, 4 in. LW concrete Acoustic tile, ceiling air space, 4 in. HW concrete Acoustic tile, ceiling air space, 8 in. HW concrete Furnishing 1 in. wood @ 50% of floor area 1 in. wood @ 50% of floor area 1 in. wood @ 50% of floor area Carpeting Resistance layer of 2.73 ft2 h ºF/Btu Resistance layer of 2.73 ft2 h ºF/Btu Resistance layer of 2.73 ft2 h ºF/Btu Notes: 1. Surface layers are listed in order from the outside of the room to the inside of the room. 2. The test zone is 15ft x 30ft x 9ft high. The test zone has one exterior wall, 30 ft long. 3. The % glazing is fraction of the exterior facade. 4. Long wavelength absorptance of 0.9 were used for all inside surfaces The following two heat gain pulse distribution models were used for the RTF generation: (1) area weighted uniform distribution model for nonsolar RTF, and (2) for the solar RTF generation 50% to the floor and 50% to the thermal mass surfaces. The RTF plots are shown in Figures 3.3 to 3.6. The RTF plots for medium and heavyweight construction zones are shown in Appendix A. Root mean square errors (RMSE) of the 71 RTF were computed for the 24hourly values of the New RTF Engine and the HVAC Load Explorer, a fullblown heat balance program, that come with a book by McQuiston et al (2005). The RMSE for three test zone construction types is given in Table 3.2. Table 3.2 RMSE of the RTF of the New RTF Engine Without Carpet With Carpet Zone Construction Nonsolar Solar Nonsolar Solar Heavyweight 0.00013 0.00022 0.00014 0.00027 Mediumweight 0.00008 0.00012 0.00008 0.00015 Lightweight 0.00027 0.00037 0.00007 0.00017 The RMSE is calculated as follows: ( ) Σ= = − 23 0 2 , , 24 1 j NewRTF Engine j HVAC Load Explorer j RMSE RTF RTF (3.20) The RMSE of the RTF computed using the New RTF engine are within the convergence limits of the program used to generate the reference RTF. The maximum errors are observed for most of the cases in the first three terms of the RTF as is evident from the plots in Figures 3.3 to 3.6 and Figures 1A to 8A shown in the Appendix A. 72 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1 3 5 7 9 11 13 15 17 19 21 23 Hour RTF New RTF Engine HVAC Load Explorer Figure 3.3 Nonsolar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior facade 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1 3 5 7 9 11 13 15 17 19 21 23 Hour RTF New RTF Engine HVAC Load Explorer Figure 3.4 Solar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior façade 73 0.00 0.10 0.20 0.30 0.40 0.50 0.60 1 3 5 7 9 11 13 15 17 19 21 23 Hour RTF New RTF Engine HVAC Load Explorer Figure 3.5 Nonsolar RTF for lightweight construction zone with carpet for 50% glazing fraction of the exterior facade 0.00 0.10 0.20 0.30 0.40 0.50 0.60 1 3 5 7 9 11 13 15 17 19 21 23 Hour RTF New RTF Engine HVAC Load Explorer Figure 3.6 Solar RTF for lightweight construction zone with carpet for 50% glazing fraction of the exterior facade 74 3.1.3 1D Finite Volume Method PRF Generation Although, Spitler et al. (1997) demonstrated the use of nonperiodic response factors to generate PRF, most others implementations have used an existing Laplace or State Space method based CTF generation procedure, then converted the CTF to PRF using the Spitler and Fisher (1999b) procedure. This procedure may be less than ideal for two reasons: 1. Computing CTF then converting to PRF for use in RTSM involves an extra effort. 2. Since the RTSM is intended to be a spreadsheet method, it would be best if the entire procedure could be encapsulated within the spreadsheet. With Microsoft Excel, the VBA programming language allows procedural programming “within” the spreadsheet. Therefore, computation of PRF is possible within the spreadsheet. Laplace and State Space methods for CTF generation might be implemented but significant complexity is a formidable barrier to such an implementation. Consider that the FORTRAN 90 implementations (Iu et al. 2004) of the Laplace and State Space Methods are 1000 and 2000 lines long, respectively. Therefore, a simpler approach is investigated here, using a 1D finite volume method fully implicit scheme. For comparison purposes, the implementation investigated here was written in SCILAB and only takes 150 lines. In FORTRAN 90 it takes about 450 lines. A uniform gridding scheme, in each layer, and zero thickness boundary nodes (Patankar 1991) are used, which allows the imposition of realistic boundary conditions. 75 Moreover, a higher order treatment is used for flux calculation. This method is described fully in Appendix B. Validation of Periodic Response Factor Generation The 1D finite volume method (FVM) periodic response factor generation algorithm has been validated against the Spitler and Fisher (1999a) procedure, which is converting CTFs generated using the State Space method
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Title  Improvements to the Radiant Time Series Method Cooling Load Calculation Procedure 
Date  20071201 
Author  Nigusse, Bereket Asgedom 
Keywords  Cooling Load, Fenestration Model, Radiant Time Series Method, Thermal Bridges 
Department  Mechanical Engineering 
Document Type  
Full Text Type  Open Access 
Abstract  The objective of the study was improvement of the Radiant Time Series Method (RTSM) cooling load calculation procedure. A series of investigations were carried out, including: improvements to peak cooling load prediction performance, development of a simplified procedure for Radiant Time Factor (RTF) generation, adaptation of the fenestration submodel to be compatible with currently available manufacturer's data, establishing appropriate radiative fractions for all heat gain types. The simplified RTF generation procedure facilitates implementation in a range of computing environments. Furthermore, an approximate onedimensional dynamic model for thermal bridges has been investigated and validated against both experimental results and detailed computational results.\nThe Radiant Time Series Method was modified by introducing a dimensionless loss conductance that is readily calculable from the zone geometry and surface characteristic. Along with the revised fenestration submodel, newlyrecommended radiative fractions, and simplified RTF generation procedure, the RTSM performance was validated against the reference method, the heat balance method (HBM) with a large parametric study. The parametric study compared peak cooling loads determined with the improved RTSM procedure to those determined with the HBM for a wide range of building design parameters, for which all combinations formed 2,867,200 different zones. Use of the dimensionless loss conductance caused the maximum overprediction to go from 37% to 18%.\nSeveral additional simplifications were made to facilitate implementation in a range of computing environments. These simplifications included development of a onedimensional finite volume numerical procedure for generating periodic response factors. This reduced the number of lines of sourcecode required with the previous statespace method from 2000 to 500 in Fortran 90. In a computing environment with embedded matrix algebra, such as SCILAB, the number of lines of sourcecode can be further reduced to 150. The new formulation of the RTF generation procedure allows very compact implementation (150 lines of source code, compared to the original procedure, implemented in approximately 4000 lines of Fortran 90 source code) in computing.\nThe approximate onedimensional dynamic model of thermal bridges has been validated against published experimental results and also against numerical results generated with a 2D finite volume program. The approximate model can predict peak heat gains within ±2.6% of the experimental measurements, provided accurate steady state thermal resistances are used. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  IMPROVEMENTS TO THE RADIANT TIME SERIES METHOD COOLING LOAD CALCULATION PROCEDURE By BEREKET ASGEDOM NIGUSSE Bachelor of Science in Chemical Engineering Addis Ababa University Addis Ababa, Ethiopia 1989 Master of Engineering Science in Mechanical Engineering University of New South Wales Sydney, Australia 1998 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2007 ii IMPROVEMENTS TO THE RADIANT TIME SERIES METHOD COOLING LOAD CALCULATION PROCEDURE Dissertation Approved: Dr. Jeffrey D. Spitler Dissertation Adviser Dr. Daniel E. Fisher Dr. Lorenzo Cremaschi Dr. Alan Noell Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENTS First I would like to express my deepest gratitude to my advisor Dr Jeffrey D. Spitler, for his continuous guidance and support over the course of my Ph.D. degree study. I am very grateful for his constructive advice and criticism, without which my success would have been impossible. I would like to this opportunity to thank Dr Daniel Fisher, Dr Alan Noell, and Dr Lorenzo Cramaschi for their time in serving as my advisory committee members. Next I would like to thank the US State Department for the twoyear financial support as Fulbright Scholar. Much of the work was funded by ASHRAE 1326RP. I am also grateful to ASHRAE for the further financial support provided to me as student GrantinAid. Finally I would like to express my deepest appreciation for my mother, sisters, brothers, and family members for their encouragement and unconditional love. And my special appreciation goes to my wife Aida Mebrahtu for her support and encouragement. iv TABLE OF CONTENTS Chapter Page I. INTRODUCTION......................................................................................................1 1.1 Background........................................................................................................1 1.2 Objectives ........................................................................................................13 II. REVIEW OF LITERATURE..................................................................................17 2.1 The Radiant Time Series Method ....................................................................18 2.1.1 The RTSM Procedure .............................................................................20 2.1.2 Heat Transfer Phenomena.......................................................................22 2.1.3 RTF Generation ......................................................................................27 2.1.4 Limitations of the Radiant Time Series Method.....................................28 2.2 Dynamic Modeling of Thermal Bridges ..........................................................30 2.2.1 OneDimensional Conduction Transfer Function...................................32 2.2.2 Steady State Conduction Models ............................................................36 2.2.3 Multidimensional Conduction Dynamic Models ..................................39 III. RTS METHOD IMPROVEMENTS......................................................................51 3.1 New RTF Calculation Engine..........................................................................53 3.1.1 The Mathematical Model –Reduced Heat Balance Method ...................55 3.1.2 Validation of the New RTF Engine ........................................................70 3.1.3 1D Finite Volume Method PRF Generation ...........................................74 3.1.4 1D RTF Generation in Different Programming Environment................76 3.2 Improved Fenestration Model..........................................................................78 3.2.1 Development of Improved Fenestration Model......................................79 3.2.2 Radiative  Convective Split in the RTSM .............................................84 3.2.3 Application of Fenestration Model without Internal Shade....................98 3.2.4 Application of Fenestration Model with Internal Shade.........................99 3.3 Heat Losses in the RTSM Procedure.............................................................100 3.3.1 Derivation of the Mathematical Algorithm...........................................101 3.3.2 Dimensionless Loss Conductance.........................................................115 3.3.3 Performance of Improved RTSM Procedure ........................................118 3.3.4 Heat Losses in the RTSM and TFM Procedures ..................................122 3.3.5 Conclusion and Recommendation ........................................................125 3.4 Summary and Conclusions ............................................................................127 v Chapter Page IV. PARAMETRIC STUDY OF THE RTSM PROCEDURE..................................131 4.1 Parametric Run Generation............................................................................132 4.2 Test Zone Parameters.....................................................................................134 4.2.1 Zone Geometry and Construction Fabric..............................................135 4.2.2 Thermal Mass Types.............................................................................138 4.2.3 Internal Heat Gains and Schedules .......................................................139 4.2.4 Glazing Types .......................................................................................140 4.2.5 Interior Shade Model ............................................................................142 4.2.6 RadiativeConvective Split ...................................................................143 4.2.7 Solar and Radiant Heat Gain Distribution ............................................144 4.2.8 Design Weather Days ...........................................................................145 4.3 Methodology: HBM and RTSM Implementation..........................................147 4.3.1 The HB Method Code...........................................................................148 4.3.2 The RTS Method Code .........................................................................150 4.3.3 The RTSM and HBM Models Comparison ..........................................151 4.4 Results and Discussion Original RTSM.......................................................153 4.4.1 RTSM Peak Design Cooling Load Prediction ......................................154 4.4.2 Conclusion and Recommendation ........................................................163 4.5 Results and Discussion – Current and Improved RTSM...............................166 V. DYNAMIC MODELING OF THERMAL BRIDGESMETHODOLOGY ........175 5.1 Introduction....................................................................................................175 5.2 The Equivalent Homogeneous Layer Wall Model ........................................175 5.2.1 Steady State Rvalue .............................................................................178 5.2.2 StepByStep Procedure........................................................................178 VI. DYNAMIC MODELING OF THERMAL BRIDGES  VALIDATION ...........183 6.1 Experimental Validation ................................................................................184 6.1.1 Guarded Hot Box Dynamic Response Test Facility .............................184 6.1.2 The Test Procedure and Specimens ......................................................186 6.1.3 Experimental Determination of the CTFs.............................................190 6.1.4 The Experimental Validation Procedure...............................................194 6.1.5 The Equivalent Walls............................................................................197 6.1.6 Comparison of Conduction Heat Gains ................................................217 6.2 InterModel Validation ..................................................................................223 6.2.1 Numerical Validation............................................................................224 6.2.2 The Rvalues and the Equivalent Walls ................................................227 6.2.3 Performance of the Equivalent Walls ...................................................228 vi Chapter Page 6.2.4 Summary and Conclusion.....................................................................231 6.3 Conclusions and Recommendations ..............................................................233 6.4 Recommendations for Future Work...............................................................236 VII. CONCLUSIONS AND RECOMMENDATIONS.............................................237 7.1 Conclusions – RTSM Improvements............................................................ 237 7.1.1 Accounting Space Heat Losses.............................................................238 7.1.2 Improvements to the RTSM Fenestration Model .................................238 7.1.3 Improvements to the RTF Generation ..................................................239 7.1.4 Developments to RTSM Implementation .............................................239 7.1.5 Parametric Study of the Performance of RTSM...................................240 7.2 Conclusion: Dynamic Modeling of Thermal Bridges....................................243 7.3 Recommendations for Future Work...............................................................245 REFERENCES ..........................................................................................................249 APPENDIX................................................................................................................259 APPENDIX A: THE NEW RTF ENGINE VALIDATION......................................259 APPENDIX B: 1D FINITE VOLUME METHHOD PRF GENERATION .............264 B.1 Derivation of 1D Finite Volume Numerical Model ......................................264 B.2 Finite Volume Method PRF Generation Validation .....................................277 APPENDIX C: RTSM IMPLEMENTATION IN OTHER COMPUTING ENVIRONEMNTS..............................................................................................279 APPENDIX D: FENESTRATION MODELS FOR HEAT BALANCE AND RTS METHODS ................................................................................................285 vii LIST OF TABLES Table Page Table 3.1 Description of zone constructions for RTF generation engine validation.................................................................................................70 Table 3.2 RMSE of the RTF of the New RTF Engine............................................71 Table 3.3 Recommended radiative / convective spits for the RTSM procedures ...............................................................................................98 Table 4.1 Test Parameter range and levels ...........................................................134 Table 4.2 Construction Type Materials.................................................................137 Table 4.3 Thermal Mass Type materials...............................................................138 Table 4.4 Thermal and Optical Properties of glass window .................................141 Table 4.5 Optical properties of shade layers.........................................................143 Table 4.6 Mean Lighting Heat Gain Parameters from ASHRAE 1282RP..........144 Table 4.7 Design weather conditions for the fourteen USA locations..................146 Table 4.8 RTSM and HBM component models ...................................................152 Table 4.9 Month of Annual Peak Cooling Load for zones with single pane clear glass..............................................................................................159 Table 4.10 RTSM peak cooling load extreme over predictions for glazing without interior shades..........................................................................168 Table 4.11 RTSM peak cooling load extreme over predictions for glazing without interior shades..........................................................................171 Table 6.1 Layerbylayer descriptions of the ASHRAE RP515 test walls..........189 viii Table Page Table 6.2 SurfacetoSurface Rvalues of ASHRAE RP515 Test Walls.............204 Table 6.3a Equivalent walls computed with experimentally determined Rvalues ....................................................................................................209 Table 6.3b Equivalent walls computed with handbook Rvalues ...........................210 Table 6.4a AirtoAir CTSF of the test walls generated from experimentally determined conduction transfer functions.............................................212 Table 6.4b AirtoAir CTSF of the EHL walls determined with experimentally determined Rvalues.....................................................213 Table 6.4c AirtoAir CTSF of the EHL walls determined with handbook Rvalues ....................................................................................................214 Table 6.5 Results summary of peak heat gains and time shift..............................221 Table 6.6 Test walls construction description intermodel validation ..................225 Table 6.7 Surfacetosurface Rvalue of intermodel validation test walls...........227 Table 6.8 Equivalent walls of intermodel validation test walls...........................228 Table 6.9 Peak heat gains and time shift for intermodel validation ....................229 ix LIST OF FIGURES Figure Page Figure 2.1 Radiant Time Series Method represented as a nodal network. A single wall is shown with the outside surface on the left (Rees at al. 2000) ...............................................................................................19 Figure 2.2 The original RTSM cooling load calculation method represented as flow diagram (Rees at al. 2000).......................................................21 Figure 2.3 Lineartriangular ramp temperature pulse representation....................41 Figure 3.1 Adiabatic boundary condition for RTF generation: Tso=Tsi .................58 Figure 3.2 Flow chart of solution scheme I...........................................................69 Figure 3.3 Solar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior facade.........................................72 Figure 3.4 Nonsolar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior facade ...................................72 Figure 3.5 Solar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior facade.........................................73 Figure 3.6 Nonsolar RTF for lightweight construction zone with carpet for 50% glazing fraction of the exterior facade.........................................73 Figure 3.7 Peak heat gains calculated using finite volume method versus the State Space method PRFs ....................................................................76 Figure 3.8 Absorbed component as a fraction of total solar heat gain for different fenestration classes................................................................83 Figure 3.9 TsIn and the corresponding MRT of heavyweight construction opaque exterior surfaces and 24°C room air temperature....................86 x Figure Page Figure 3.10 TsIn and the corresponding MRT for an opaque surface at peak load for three aspect ratios and 24°C room air temperature ................87 Figure 3.11 TsIn and the corresponding MRT for single pane clear glass fenestration and 24°C room air temperature........................................89 Figure 3.12 TsIn and the corresponding MRT for south facing fenestration at peak load for three aspect ratios and 24°C room air temperature........90 Figure 3.13 Radiative fractions for fenestration in a heavyweight construction zone and single pane clear glass with 90% glazing fraction................91 Figure 3.14 Radiative fractions for fenestration in a heavyweight construction zone and single pane clear glass with 50% glazing fraction................92 Figure 3.15 Radiative Fraction against glazing fraction of exterior facade for heavyweight zone for five glazing types .............................................93 Figure 3.16 Radiative fractions against percent glazing of exterior facade for lightweight zone for five different glazing types at peak cooling load condition.......................................................................................96 Figure 3.17 Radiative fractions against percent glazing of exterior facade for heavyweight zone for five different glazing types at peak cooling load condition.......................................................................................96 Figure 3.18 RTSM versus the HBM peak cooling loads for single pane clear glass without interior shaded fenestration............................................99 Figure 3.19 Current RTSM versus the HBM peak cooling loads for single pane clear glass with interior shaded fenestration .............................100 Figure 3.20 Representation of fenestration inside surface heat balance ...............103 Figure 3.21 The improved RTSM cooling load calculation method represented as flow diagram ..............................................................115 Figure 3.22 Dimensionless loss conductance against glazing fraction for zones with unshaded fenestration ......................................................117 Figure 3.23 Dimensionless loss conductance against glazing fraction for zone with two exterior facades and interior shaded fenestration ...............117 xi Figure Page Figure 3.24 RTSM peak cooling load vs. HBM for light and heavyweight zone for single pane clear glass without internal shade.....................119 Figure 3.25 Hourly cooling load profile for lightweight zone at 50% glazing fraction for single pane clear glass in Chicago, Illinois.....................120 Figure 3.26 Hourly cooling load profile for lightweight zone at 90% glazing fraction for single pane clear glass in Chicago, Illinois.....................120 Figure 3.27 RTSM peak cooling load vs. HBM for light and heavyweight zone for single pane clear glass with internal shade ..........................121 Figure 4.1 Schematic of Parametric Run Generator............................................133 Figure 4.2 Zone orientation and number designations ........................................136 Figure 4.3 Structure of Heat Balance Method for a Zone ...................................149 Figure 4.4 The current RTSM cooling load calculation method represented as flow diagram..................................................................................150 Figure 4.5 RTSM annual peak cooling load versus the HBM for the USA weather locations for single pane clear glass.....................................160 Figure 4.6 RTSM annual peak cooling load versus the HBM for the USA weather locations for double pane clear glass....................................161 Figure 4.7 RTSM annual peak cooling load versus the HBM for the USA weather locations for double pane lowe glass ..................................161 Figure 4.8 Maximum RTSM peak cooling load over prediction against glazing fraction for the three glazing types........................................162 Figure 4.9 Current RTSM annual peakcooling load maximum and average over prediction for zone without interior shade.................................169 Figure 4.10 Improved RTSM annual peakcooling load maximum and average over prediction for zone without interior shade ...................169 Figure 4.11 Current and Improved RTSM annual peakcooling load versus HBM for lightweight zones single pane clear glass without interior shade......................................................................................170 xii Figure Page Figure 4.12 Current and Improved RTSM annual peakcooling load versus HBM for heavyweight zones with single pane clear glass without interior shade......................................................................................170 Figure 4.13 Current RTSM annual peakcooling load maximum and average over prediction for zones with dark roller interior shade...................172 Figure 4.14 Improved RTSM annual peakcooling load maximum and average over prediction for zones with dark roller interior shade .....173 Figure 4.15 Current and Improved RTSM annual peakcooling load versus HBM for lightweight zones with single pane clear glass and dark roller interior shade ............................................................................173 Figure 4.16 Current and Improved RTSM annual peakcooling load versus HBM for lightweight zones with single pane clear glass and dark roller interior shade ............................................................................174 Figure 6.1 Sectional view of guarded hotbox facility (Brown and Stephenson 1993b) ................................................................................................186 Figure 6.2 Flow chart of the experimental validation procedure ........................196 Figure 6.3 Thermal bridge types: (a) sandwiched type; (b) exposed type ..........198 Figure 6.4 ASHRAE RP515 Test Walls ............................................................203 Figure 6.5 CTSF plot for the steel stud wall (Wall#1)........................................215 Figure 6.6 CTSF plot for the steel stud wall (Wall#4)........................................216 Figure 6.7 CTSF plot for hollow block with insulation and brick exterior finish (wall#5)....................................................................................216 Figure 6.8 Heat gain for insulated steel stud wall with stucco exterior finish (Wall#1) .............................................................................................217 Figure 6.9 Heat gain for precast reinforced concrete slab with steel furring and insulation covered with gypsum board on the exterior (Wall#2) .............................................................................................218 Figure 6.10 Heat gain for precast reinforced concrete slab with steel furring and insulation with gypsum board (Wall#3)......................................218 xiii Figure Page Figure 6.11 Heat gain for insulated steel stud wall mounted on reinforced concrete slab (Wall#4) .......................................................................219 Figure 6.12 Heat gain for hollow concrete block with insulation and brick on the exterior and gypsum board (Wall#5) ...........................................219 Figure 6.13 Heat gain for insulated steel stud wall with brick exterior finish (Wall#6) .............................................................................................220 Figure 6.14 Heat gain for solid concrete block wall with insulation and granite veneer exterior finish (Wall#7)..............................................220 Figure 6.15 Summary of peak heat gains for the seven ASHRAE test walls .......222 Figure 6.16 Handbook Rvalue errors of the seven ASHRAE test walls..............222 Figure 6.17 Wood and steel stud walls construction details .................................226 Figure 6.18 Heat gain of a wood stud wall for periodic solair temperature boundary condition ............................................................................229 Figure 6.19 Heat gain of a warm steel stud wall for steady periodic solair temperature boundary condition ........................................................230 Figure 6.20 Heat gain of a cold steel stud wall for steady periodic solair temperature boundary condition ........................................................230 Figure 6.21 Heat gain of a hybrid steel stud wall for steady periodic solair temperature boundary condition ........................................................231 Figure 6.22 Handbook Rvalue errors compared to that of the 2D finite volume method...................................................................................232 Figure 6.23 Peak cooling load prediction error of the equivalent walls compared to the 2D finite volume method ........................................232 xiv 1 CHAPTER I 1 INTRODUCTION 1.1 Background Design cooling load calculation methods have evolved since their inception during the 1930’s. The historical development of cooling load calculation procedures has been strongly influenced by the development and availability of digital computing facilities, and by the desire to provide methods that are of utility to average practicing engineers that can be used with tabulated data (Rees et al. 2000a; Romine 1992). It is useful to define the terms “heat gains” and “cooling load” and the relationship between them in the context of load calculations. Heat gain is defined as the instantaneous heat flow into a space by conduction, convection and radiation. Cooling load is defined as the amount of heat removed from a space to keep the space air at a fixed desired temperature. Therefore, all heat gains do not necessarily become cooling loads: convective heat gains become cooling load instantaneously, while radiant heat gains are first absorbed by the structure and then released by convection to become a cooling load at a later time. Absorption and reradiation of radiant heat gains among the surfaces in the zone continues as long as temperature difference exits. Under some circumstances, some of the heat gains may be conducted back out of the space. 2 The challenge in the early days of the cooling load calculation was primarily to develop procedures to quantify the heat gains. In the 1930s peakcooling loads were over predicted due to failure to account for thermal mass effects of construction in the load calculation (Houghten et al. 1932; James 1937; Kratz and Konzo 1933). Analytical equations for computing transient conduction heat gains through homogeneous layer constructions exposed to solar radiation were developed. Houghten, et al., (1932) used Fourier analysis and assumed sinusoidally varying outside surface temperatures. Alford, et al., (1939) improved this by assuming sinusoidally varying outdoor air temperature and accounting for solar radiation separately. Despite an effort to develop a rigorous analytical procedure for computing transient heat conduction, there was little success in establishing a general quantitative relation suitable for practicing engineers. The electric analogy method of predicting heat flow through walls based on the identity of the transient heat flow and flow of electricity can be implemented experimentally and can closely match direct thermal measurements (Paschkis 1942). An electric analog thermal circuit of an embedded tube cooling slab model was developed using electrically equivalent resistance, capacitance, and source terms (Kayan 1950). This allowed determination of the slab surface temperatures, temperature isotherms in the slab and heat transfer rates. By the mid 1940s, the American Society of Heating and Ventilation Engineers (ASHVE), a predecessor of the American Society of Heating, Refrigeration and Air Conditioning Engineers (ASHRAE), developed a manual method for calculating the heat gain through 3 various external surfaces with equivalent temperature differentials (ETD) values. The ETD values were often 20 to 40 degrees Fahrenheit above the difference between outside and inside air temperatures (Rees et al. 2000a; Romine 1992). In the ETD method two procedures were involved: the ETD were generated from experimentally measured surface temperatures and conductance (Rees et al. 2000a) for transient conduction heat gain, and the instantaneous solar heat gains through glazing were calculated using heat fluxes and shading coefficients. The ETD method excessively overestimated cooling load due to the assumption that the heat gains instantaneously caused cooling loads on the system. The delays of solar heat gains before becoming cooling load were well understood but simple quantitative relations for these effects were not available until the 1940s and 1950s. Designers made various approximations to compensate for the over prediction of cooling loads (Romine 1992). Transient conduction heat gain calculation procedures through external surfaces developed using Fourier analysis assumed periodic variation of solair temperature as the external driving temperature, constant indoor air temperature, and fixed outside and inside conductance1 (Mackey and Wright 1944; Mackey and Wright 1946). The solair temperature is a concept derived from the equivalent temperature (Billington 1987) used then in UK. It is defined as the temperature that would give the same amount of heat transfer as that of the actual outdoor air temperature and solar radiation incident on the surface. Mackey and Wright (1946) formulated semiempirical relations to estimate inside surface temperatures for multilayered walls based on an analytic solution for 1 A fixed value of combined outside conductance of 4.0 (Btu/hr⋅ft2⋅°F) is still commonly used after 60 years. 4 multilayered walls. The damping and delay effects of the surface thermal mass on the inside surface temperature were accounted using a decrement factor and time lag. Developing an equation for the inside surface temperatures using the solair temperature to account for the incident solar flux provided the first convenient manual procedure for computing instantaneous heat gains. The heat gains were computed from the inside surface temperature and room air temperature, assuming a fixed combined inside conductance. Later, Stewart (1948) used this procedure to tabulate the ETD for various construction assemblies, surface exterior colors, surface orientations, latitude angles and hours of the day. The tabulated ETD values were adjusted for use with walls and roofs overall heat transfer coefficient, instead of combined inside conductance. This concept was then adopted by ASHRAE as the total equivalent temperature difference and time averaging (TETD/TA) method in the 1960s. The TETD/TA load calculation method first introduced in the 1967 Handbook of Fundamentals (ASHRAE 1967; Rees et al. 2000a; Romine 1992). The TETD/TA method mainly involves two steps: calculation of heat gains components from all sources and conversion of these heat gains into cooling loads. The TETD replaced the ETD with improved tables and equations for the equivalent temperature differences. Walls and roofs were characterized by two parameters decrement factor (ratio of peak heat gain to the peak heat gain that would occur with no thermal mass in the wall) and time lag (delay in peak heat gain compared to peak solair temperature). The TETD could then be calculated knowing sol 5 air temperature, room air temperature, and decrement factors and time lags. Conversion of the instantaneous heat gains into cooling loads using the time averaging technique is a two step procedure: first, split the instantaneous heat gain into convective and radiant components using recommended radiative /convective splits; second, the radiative component of the heat gain is time averaged depending on the thermal mass of the construction to get the cooling loads. For lightweight construction, the hourly radiant cooling load is the radiant component of heat gain time averaged over a 2 to 3 hour period prior to and including the time of maximum load conditions. For heavyweight construction, the hourly radiant cooling load is the radiant component of heat gain time averaged over a 5 to 8 hour period prior to and including the time of maximum load conditions (ASHRAE 1967). The total hourly cooling load is the sum of the convective component and the hourly radiant cooling load. The work described above did not explicitly consider interactions between heat gain components. The earliest attempt to model zone dynamics involving conduction through the envelope, solar heat gains and the radiant exchange among surfaces and convection between surfaces and room air utilized physical (electric and hydraulic) analogies in the 1940s and 1950s. However, the analogies remained research tools as it was not feasible for practicing engineers to build electric circuits, nor were the insights gained reduced to manual calculation procedures. Leopold (1948) used a hydraulic analogy to investigate zone dynamics. The model included thermal storage, radiation, convection, and conductions. Thermal capacitance 6 was represented by vertical tubes in series connection attached to a distribution header connected to a storage tank, and resistances were represented by restricted tube. Radiation absorbed by surfaces was represented by liquid flow from a pump through a calibrated restriction, and temperatures were represented by fluid pressure. The hydraulic model demonstrated dynamics of zones and gave some insights to the limitations the load calculation procedures. Despite all efforts to improve the accuracy of load calculation procedures, peak cooling load computed using the ASHVE Guide 1952 over predicted by 16 to 32 % compared to values measured in a small single story residential house with large glass exposure due to failure to account for the storage effect (Gilkey et al. 1953). Similarly, a field survey made on single family houses over a wide range of climates and construction fabrics revealed over sizing of cooling equipment capacity due to failure to account for the thermal mass effects of building structures (Willcox et al. 1954). Dynamic modeling of thermal mass effects of structures and furnishing in a building was attempted using analog computers by solving the electrical equivalent thermal circuit of actual buildings (Willcox et al. 1954). The model used pure resistances to represent doors, windows, blinds and infiltration. Distributed resistances and capacitances were used to represent walls, roofs and partitions. The outdoor and indoor temperatures were represented by potential differences. With this approach, the authors found it difficult to construct a circuit that both had a onetoone physical correspondence with the building, and which gave a good match to transient thermal measurements. They did find that they 7 could “tune” a simpler circuit to give the correct dynamic response, but this has limited usefulness for design load calculations. However, they had better success with an analog computer, which utilizes amplifiers and allows better measurement of intermediate values. The analog computer’s calculated response was only 7% higher than the actual thermal measurements. The work of Brisken and Reque (1956), in developing what they called the ‘Thermal Response Method’, was the first attempt to use digital computers by representing a wall using twolump (oneresistance and twocapacitance) thermal circuit that was connected to outdoor solair temperature and indoor air temperature nodes using outside and inside combined conductance. The two differential equations for the twolump thermal circuit were solved using the Laplace transform method to determine the room response to a unit square pulse applied at the solair temperature while the room air temperature was constant. The method was not adopted in the ASHVE Guide, but the approach later became the basis for development of the conduction transfer function method with a unit triangular pulse adopted by ASHRAE for transient conduction heat gain calculations. A procedure for computing room response factors using a detailed thermal circuit model involving radiation exchange among inside surfaces and room furnishings, convection between surfaces and room air, and various room heat sources was developed by Mitalas and Stephenson (1967). An effort to provide a more rigorous load calculation procedure led to the development of conduction transfer functions for transient conduction through homogeneous multilayered constructions (Stephenson and Mitalas 1971). The transfer 8 function method (TFM) for computing zone thermal response and cooling load was first published in the 1972Handbooks of Fundamentals (ASHRAE 1972). The method relied on a set of tabulated room transfer function coefficients. Given the enormous (in the 1970s) computational efforts required by the TFM and the lack of computer resources and skills of practicing engineers there was a need for a method that could be used manually. As a result, a simplified procedure called the Cooling Load Temperature Difference / Cooling Load Factor (CLTD/CLF) method was developed under ASHRAE RP138 by Rudoy and Duran (1975). The CLTD/CLF method is a single step load calculation procedure. CLTD values were calculated by dividing the cooling load due to a particular wall or roof using the TFM by the Uvalue of the constructions. Due to its simplicity, the CLTD/CLF method replaced the TETD/TA methods as the ASHRAErecommended manual load calculation procedure. However, the CLTD/CLF method had limitations due to a lack of tabulated CLTD/CLF design data that matched the wide range of design conditions faced by practitioners. Thus, designers showed continued interest into TETD/TA method due to its flexibility for manual load calculations and adaptations for various building envelope assemblies and design locations (Romine, 1992). ASHRAE’s continued commitment to refine load calculation procedures, to investigate effects of different building design parameters, and to provide accurate design data led to new research directions in the 1980s. ASHRAEfunded research project 472RP characterized room response based on fourteen building design parameters. Generating, 9 tabulating, and printing the whole range of the CLTD/CLF data on the basis of the fourteen design parameters became an impractical task (Sowell 1988c). However, ASHRAE maintained the CLTD/CLF method, which later became the Cooling Load Temperature Difference /Solar Cooling Load / Cooling Load Factor (CLTD/SCL/CLF) method, as a manual load calculation procedure by tabulating CLTDs for representative families of walls and roof assemblies and developing a mapping procedure for the actual constructions. Software for generating CLTD and CLF data based on the weighting factors and conduction transfer function coefficients developed in ASHRAE RP−472 was developed as part of ASHRAE RP626 (Spitler et al. 1993b). Spitler, et al. (1993a) introduced a new factor, the solar cooling load (SCL), for converting solar heat gain into cooling load. Though the TFM required high computational resources, it remained the only computational design cooling load calculation procedure recommended by ASHRAE until the late 1990s. The Transfer Function Method was not well received (Romine 1992) by practicing engineers for the following reasons: • Intimidating look of the equations • Required iterations and convergence may take three to five successive design day calculations • Computer resources and a lack of computing skills also limited its implementation for load calculations A simple and yet reasonably accurate load calculation procedure that did not involve iterative processes was highly desired by ASHRAE to replace the manual procedures. An 10 ASHRAE funded project (RP875) for continued improvements of load calculation procedures led to the development of the Heat Balance Method (HBM) (Pedersen et al. 1997) and the Radiant Time Series Method (RTSM) (Spitler et al. 1997) for calculating peak cooling loads. The HBM was first implemented in the 1960s by Kusuda in NBSLD, later by Walton in 1980s in Building Loads Analysis and System Thermodynamics (BLAST) and in Thermal Analysis Research Program (TARP) as cited by Pedersen, et al. (1997). However, a complete description of the procedure for load calculation purposes had not been available. The first complete description of the heat balance method formulation starting from the fundamental principles, and covering implementation and solution techniques as applied for peak cooling load calculation was presented by Pedersen, et al., (1997). Since the heat balance method is based on the fundamental principles of the physics involved, it is commonly used as a reference model for simplified load calculation programs. The RTSM closely followed the HBM hourly cooling load profile and in most cases slightly overpredicted the peak cooling load; however, the over predicted peak cooling load was significant for zones with large amount of single pane glazing and cool design weather conditions (Rees et al. 1998). The radiant time series method (RTSM) was developed as a spreadsheet method intended to replace the TETD/TA and the CLTD/SCL/CLF methods. It also effectively replaced the TFM. The radiant time series method (RTSM) as a simplified load calculation procedure was adopted as a 11 nonresidential building load calculation procedure by ASHRAE and published in Pedersen, et al. (1998) and the 2001 −Handbook of Fundamentals (ASHRAE 2001). Experimental validation of the heat balance and the radiant time series methods has been done in test cells at Oklahoma State University (Chantrasrisalai et al. 2003; Iu et al. 2003). ASHRAE research project RP942 compared the peak cooling load predictions made with the RTSM to those made with the heat balance method (HBM) using a parametric run investigation tool (Rees et al. 1998; Spitler and Rees 1998). Although ASHRAE 942RP identified building design parameters that lead to over predictions of peak cooling load, the project did not result in design guidance for practicing engineers. The radiant times series method (RTSM) has effectively replaced the manual load calculation procedures and has attracted interest due to: Its amenability to spreadsheet implementations as opposed to the Transfer Function Method, which requires iteration. Captures and depicts the physics involved in the Conduction Time Series Factor (CTSF) and Radiant Time Factor (RTF) coefficients, unlike the Transfer Function Method. Has essentially the same accuracy as the TFM. 12 However, the RTSM also has the same approximations as the TFM that, in some cases, lead to over prediction of peakdesign cooling load: The RTSM replaces the outside heat balance by an exterior boundary condition known as the solair temperature, which allows the use of fixed combined conductance of convection and radiation. The RTSM computes the radiant heat gain from the interior surfaces as if they all radiate to the room air temperature instead of performing inside surface and room air heat balances. This allows treatment with a linearized radiation coefficient, which is combined with the convection coefficient. This assumption can over predict the instantaneous heat gain, which again contributes to the RTSM peak cooling load overprediction. The RTSM uses an adiabatic boundary condition when computing Radiant Time Factors (RTF), causing the RTF to always sum to one. When these RTF are used, this approach conserves the entire solar and internal heat gains during conversion to cooling load, and there is no way that the RTSM can account for any heat gains conducted back out. As a result, the RTSM tends to over predict the peakcooling load when there is a large amount of single pane glazing or other highly conductive surfaces. The resulting over predictions was shown in 942RP to be as high as 37%. It would be very helpful for designers to have guidance as to when the RTSM is likely to gives significant overprediction. 13 1.2 Objectives The previously published research in the RTSM cooling load calculation procedure has only identified the likely over of peak cooling load and the conditions favorable for over prediction but non them provided a procedure for accounting the heat gain loss and did not provide guidance on the limitation of the RTSM. Therefore, one of the objectives of this thesis is to develop an algorithm that reduces the RTSM peak cooling load likely over prediction significantly and establish the limitations of the RTSM in a form of design guidance. Furthermore, ten years of experience with the RTSM has indicated several improvements that would be helpful for design engineers. These include an improved RTF generation procedure, developing a numerical procedure for periodic response factor generation, updated fenestration modeling and investigated a procedure for treating thermal bridges. These improvements are discussed briefly below. The RTSM needs radiant time factors (RTF) for the zone to be analyzed. The ASHRAE Handbook of Fundamentals (ASHRAE 2001; ASHRAE 2005) has given tabulated RTF for specific cases, but the accuracy resulting from users choosing the “nearest” zones has not been investigated. The original presentation of the procedure utilized a full blown HBM program to generate the RTF. While this approach works, the HBM program has many features and data that are not needed for generating RTF. Therefore, one of the objectives of this thesis is to develop a simplified procedure and implement the algorithm for computing RTF, as described in Section 3.1. Also investigated is a direct method of calculating periodic response factors as an alternative to converting conduction transfer function coefficients back to response factors for use in the RTF generation procedure. A 14 onedimensional finite volume numerical fullyimplicit formulation for generating periodic response factor will be developed. The finite volume periodic response generator and the simplified RTF generator algorithm will also be implemented in other computing environments. Furthermore, the RTF generation has been investigated with constant radiation coefficient with the intent of reducing the computational time. The radiant time series method load calculation procedure was developed based on the shading coefficients and optical properties of doublestrength glass for computing solar heat gains. Developments in fenestration models and availability of a new set of fenestration data – solar heat gain coefficients  replaced the use of shading coefficients in fenestration modeling. Moreover, the shading coefficient data are no longer available. Therefore, one of the objectives of this thesis is investigation of a new fenestration model using window manufacturer’s data and the new set of tabulated glazing and fenestration data available in the ASHRAE’s Handbook of Fundamentals as presented in Section 3.2. Improved fenestration model for the RTSM will be investigated and integrated that make use of these new developments. Moreover, a new set of radiative / convective splits for fenestration solar and conduction heat gains will be established. It has been identified that the likely over prediction of the RTSM procedure is due to failure to account for the space radiant heat gains conducted back to the outside. In this thesis an algorithm for accounting the radiant heat gain loss by conduction will be derived and investigated in Section 3.3.1. The procedure accounts for solar and internal radiant heat gains conducted back out through fenestrations and highly conductive mass 15 less surfaces. Furthermore, the likely overprediction of peak cooling load by the RTSM will be investigated parametrically over a wider range of building design parameters to characterize its limitations and develop design guidance for practicing engineers. This is covered in Chapter Four. Another challenge faced by designers is the treatment of thermal bridges in wall and roof constructions. Steady state treatment of thermal bridges is covered in the ASHRAE Handbook of Fundamentals (ASHRAE 2005). Dynamic modeling of thermal bridges has been a research interest in building energy and load calculation applications for about two decades. Despite repeated efforts to develop multidimensional conduction models capable of dynamic modeling of thermal bridges, these models have never been integrated into design load calculation procedures for several reasons. Therefore, another objective of this thesis is to investigate an approximate onedimensional dynamic model of thermal bridges that can be directly implemented into design cooling load calculation procedures, and develop design recommendations usable by practicing engineers. This is covered in Chapters Five and Six. Before addressing these three chapters, the thesis gives an indepth literature review of the Radiant Time Series Method and the treatment of thermal bridges in Chapter Two. Improvements to the Radiant Time Series Method procedure are discussed in Chapter Three. Parametric investigation of the Radiant Time Series Method to establish the limitations based on adapted fenestration model is covered in Chapter Four. Chapters Five and Six deal with approximate onedimensional dynamic modeling of thermal 16 bridges methodology and the validation, respectively. Conclusions and recommendations for future work are given in Chapter Seven. 17 CHAPTER II 2 REVIEW OF LITERATURE The literature review covers two separate topics related to building energy analysis and load calculation methods. The first section deals with developments in Radiant Time Series method, and the second section deals with dynamic and steady state modeling of thermal bridges. The first part of the literature survey (Section 2.1) describes the development of the Radiant Time Series Method (RTSM) as a simplified design cooling load calculation procedure, discusses the key assumptions introduced to derive the RTSM procedure, and explains in detail the limitation of the RTS method in predicting the peak design cooling, and discusses the necessary conditions for the RTSM peak cooling load overprediction. The second part of the literature survey (Section 2.2) deals with dynamic and steady state modeling of thermal bridges in relation to building energy analysis and load calculation program. It discusses the importance of steady state and dynamic modeling of thermal bridges in building energy analysis and load calculation application, reviews previously published dynamic modeling techniques for thermal bridges and explains why these models have not been adopted. 18 It also presents a brief summary of response factors and conduction transfer functions and the associated onedimensional conduction modeling procedures. In addition it summarizes recommended approximate procedures for steady state analysis of thermal bridges. 2.1 The Radiant Time Series Method The Radiant Time Series Method (RTSM) was introduced as a simplified design load calculation procedure (Spitler et al. 1997). The RTSM was intended to replace ASHRAE’s simplified load calculation procedures: the cooling load temperature difference/solar cooling load/cooling load factor (CLTD/SCL/CLF) method, the total equivalent temperature difference/time averaging (TETD/TA) method, and the Transfer Function Method (TFM). The radiant time series method can be thought as a twostage process (Spitler et al. 1997). The first stage of this process is to calculate all the radiant and convective heat gains of the zone. The second stage is the conversion of these gains into contributions to the cooling load on the zone air. Several key approximations have been employed in simplifying the RTSM in order to avoid the iteration steps in the procedure so as to make the method suitable for spreadsheet implementation. The first simplifying assumption in the radiant time series method is the treatment of exterior and interior convection and radiation coefficients using combined constant conductance. This simplification in effect entails the assumption that surfaces exchange long wavelength radiation with air node. This assumption allows 19 the replacement of individual surface heat balances and represents each wall by a simplified nodal network as shown in Figure 2.2 (Rees et al. 2000a). Qia Qinf riQir rsQSi QPa QSol TSA 1/hcoA 1/hciA Qcond,out Qcond,in Figure 2.1 Radiant Time Series Method represented as a nodal network. A single wall is shown with the outside surface on the left (Rees et al. 2000a) The second approximation is periodicity of the design weather conditions. The radiant time series method takes advantage of the periodicity of the design day solair temperature and constant room air temperature to develop the periodic response factor. Conduction heat gains are calculated by periodic response factors (PRF) or Conduction Time Series Factors (CTSF) driven by the difference between the design day periodic Solair temperature TSA and room air temperature Ta, which is assumed constant. The periodic response factors replace the CTF in the heat balance method and eliminate the iterative conduction heat gain calculation, which is inherent in load calculation methods involving transfer functions. This assumption is key in that it avoids the iteration step and hence makes the RTSM suitable for spreadsheet implementation (Rees et al. 2000a; Spitler et al. 1997). The CTSF are determined from periodic response factors divided by the overall Uvalue of the construction. 20 The third simplifying approximation in the RTSM is the conversion of the radiant components of the heat gains into cooling load using the radiant time factors (RTF), which replaces the air heat balance. The radiant gains at each hour are converted by a series of twentyfour room response factors known as the radiant time factors (RTF). The contribution of the internal heat gains Qir and the transmitted solar heat gains QS to the load appear at the room air node as shown in the nodal network diagram (Figure 2.1) but multiplied by the radiant time factors ri and rs, respectively. These contributions are summed up to get the total hourly load. Cooling load is defined as the rates at which heat must be removed from the space to maintain a constant room air temperature and is represented as QPa in Figure 2.1. The fourth approximation in the RTSM is that solar and internal heat gains are divided into radiative and convective components using fixed radiative / convective splits (Rees et al. 1998; Spitler et al. 1997). 2.1.1 The RTSM Procedure The RTSM procedure, in terms of processing the input data and steps to arrive at the 24 hourly cooling loads, is described as follows and the calculation flow diagram is shown in Figure 2.2. The first step of the RTS method is calculation of all internal heat gains. This is done in exactly the same way as for the Heat Balance Method. All gains that are independent of the zone surface temperatures hence can be computed at the beginning of the simulation and stored as hourly values for later use. These include solar gains through 21 glazing, infiltration (assuming fixed internal air temperature), and internal gains, which are determined from the 24 hours schedule and peak internal heat gain. Determine Lighting, equipment & occupant heat gains at each hour Calculate conduction heat gain for each hour using windows response factor. Calculate conduction heat gain for each exterior opaque surface using wall/roof response factor, Calculate transmitted solar heat gain for each window for each hour Split all heat gain into radiant and convection components Determine the infiltration heat gains for each hour Sum all convective components for each hour Process the radiant heat gains using the Radiant Time Factors: either nonsolar or solar. The result is the hourly loads due to the radiant heat gain Σ Calculate absorbed solar heat gain for each window for each hour Figure 2.2 The original RTSM cooling load calculation method represented as flow diagram (Rees et al. 2000a) The periodic response factors operate on the solair and internal dry bulb temperatures. The hourly values of the solair temperature and the room air temperatures are known at the beginning of the calculation. Once the conduction heat gains through the individual surfaces have been calculated, the next important step is to divide all the gains into 22 radiant and convective components. This is done using fixed radiative / convective splits for each type of heat gains. The second stage of the RTS calculation procedure is to convert all the heat gains into contributions to the load at the air node. Convective components of the gains make instantaneous contributions to the cooling load while the radiant components of the heat gains are converted to cooling loads by means of the radiant time factors (RTF). The hourly contributions of the radiant gain to the cooling load are calculated from the 24 hourly radiant gains and the RTF. The radiant time factors are zone dynamic response characteristic, which are dependent on the overall dynamic thermal storage characteristics of the zone and defines how the radiant gain at a given hour is redistributed in time to become contributions to the cooling load at future hours. The contributions of the past and current radiant gains are simply added to the hourly convective gains to give the hourly cooling load. 2.1.2 Heat Transfer Phenomena This section describes the specific practices and assumptions used by the RTSM to model some of the principal zone heat transfer mechanisms. Exterior Convection and Radiation The RTS Method uses a fixed exterior surface conductance combined with a solair temperature to model exterior convection and radiation. This is one of the first simplifying assumptions of the radiant time series method. 23 Transient Conduction Heat Transfer The RTS Method treats external and internal excitation of conduction heat flow separately. In the RTS procedure, transient conduction heat transfer due to external excitation is modeled using a set of 24 periodic response factors. Given the constant zone air temperature Ta and the current and 23 past values of solair temperature TSAθ, the current hour’s conduction heat gain per unit surface area is given by: ( ) Σ= − = − 23 0 , , '' , , j cond i t Pj SA i j a q Y T T θ δ & (2.1) where, '' cond ,i, t q& = the current hour conduction through the ith surface, Btu/h⋅ft2 (W/m2) Pj Y = the periodic response factors at j hours from the present, Btu/h(W) SA i θ jδ T , , − = the solair temperature of the ith surface j hour from the present, °F(°C) a T = the constant room air temperature, °F(°C) The periodic response factors YPj include both the interior and exterior surface conductance. Periodic response factors can be computed from response factors (Spitler et al. 1997), from the generalized form of the CTFs (Spitler and Fisher 1999a), and frequency domain regression method (Chen and Wang 2005). The solair and inside temperatures are known at the beginning of the calculation, therefore the heat gains due to conduction can be calculated straightforwardly without the need for any iteration, which makes the RTSM amenable for spreadsheet implementation. These gains subsequently have to be divided into radiant and convective components. 24 Interior Convection and Radiation The RTS Method uses fixed combined interior radiation and convection conductances. The convection and radiation coefficients are added (as a resistance) into the wall. This approach, though it simplifies the procedure, has the effect of having the wall radiating to the zone air temperature. In most cases, this causes the RTSM to slightly overpredict the peak cooling load (Rees et al. 2000a). The RTS Method uses radiant time factors (RTF) to convert and redistribute the radiant part of the conducted gain. Analogous to periodic response factors, radiant time factors are used to convert the cooling load for the current hour based on current and past radiant gains. The radiant time factors are defined such that r0 represents the portion of the radiant gains convected to the zone air in the current hour. r1 represents the portion of the previous hour’s radiant gains that are convected to the zone air in the current hour, and so on (Spitler and Fisher 1999b). The cooling load due to radiant heat gain is given by: Σ= = − 23 j 0 Qt rjqt jδ (2.2) Where Qt = the current hour cooling load, Btu/h(W) qt− jδ = the radiant gain at j hours ago, Btu/h(W) j r = the jth radiant time factor, Btu/h(W) 25 Transmitted and Absorbed Solar Radiation Calculation of transmitted and absorbed solar radiation associated with fenestration is a very important part of the design cooling load calculation procedure. The response of the zone is dependent not only on the value of the transmitted and absorbed solar energy but also on its distribution in the zone. Two simple procedures applicable for load calculation purposes have evolved: (1) the use of normal solar heat gain coefficient and transmittance and absorptance correction for angle dependence using a reference standard DSA glass angle correction coefficients (Spitler et al. 1993a), (2) the use of angle dependent beam solar heat gain coefficient and constant diffuse solar heat gain coefficient tabulated values (ASHRAE 2005). The first approach allows separate treatment of transmitted and absorbed solar radiation. Though transmitted and absorbed components are calculated separately, the procedure is based on approximate analysis analogous to the concept of shading coefficient. This was adopted as a standard procedure but with demise of the shading coefficients a new procedure is needed. The second approach is used in a combined treatment of transmitted and absorbed components. In the second approach the solar heat gain coefficient includes both the transmitted portion of the solar heat gain and the inward flow fraction of the absorbed component. This therefore precludes the separate treatment of the absorbed solar heat gain, which has both radiative and convective components. Likewise, the RTSM uses the solar radiant time factor to convert the beam and diffuse solar heat gains into cooling loads. The diffuse solar gains are treated in a similar way to internal short and long wavelength radiant gains. As noted previously in the discussion on internal convection 26 and radiation heat transfer, some of the solar radiation that is reradiated can be conducted to the outside. The RTSM cannot account for this, and so for some zones and design weather condition tends to overpredict the cooling loads. Internal Heat Gains In the radiant time series method the hourly schedules and peak gain rate for the three type of internal heat gains (e.g. people, lights, and equipment) are specified by the user along with the respective radiative/convective splits. Though the split between radiative and convection actually depends on the zone airflow rates and surface temperatures, constant values are used even in detailed building energy analysis programs. In the RTSM the radiative component heat gain contribution on the cooling load is estimated with the radiant time factors. The RTSM does not account for the portion of the radiant gain that is conducted to the outside and so for some zone constructions tends to overpredict the cooling loads. The degree of overprediction depends on the zone construction conductance, and design weather conditions. This has been one of the limitations of the RTSM procedure and is discussed in Section 2.1.4. 2.1.3 RTF Generation Radiant time factors (RTF) are dynamic response characteristics of a zone when a zone is excited by unit heat gain pulse. (Spitler et al. 1997) described two procedures for generating RTF coefficients. The first method uses a load calculation program based on the heat balance method (Pedersen et al. 1997). 27 The radiant time factors are generated by driving a heat balance model of the zone with a periodic unit pulse of radiant energy under adiabatic wall conditions. The radiant time factors are therefore different for every combination of zone construction and geometry. In principle, they are also different for every chosen distribution of radiant pulse. Thus far two types of distributions have been commonly used for a given zone (Spitler et al. 1997). One is found assuming an equal distribution (by area) of radiant pulse on all zone surfaces and is used for all diffuse radiant gains. A second set is found with the unit pulse of radiant energy added at the floor surface and in some cases to the furniture as well to treat beam solar gains. The conversion of radiant gains by the use of radiant time factors, where there is no requirement for knowledge of past temperatures or cooling loads, again avoids the iteration processes. The second method demonstrated by (Spitler and Fisher 1999b) is to generate radiant time factors directly from a set of zone weighting factors using the existing ASHRAE database (Sowell 1988a; Sowell 1988b; Sowell 1988c). This approach would use a computer program to map a given zone to the fourteen zone characteristic parameters in the database and transform the weighting factors to radiant time factors using matrix manipulation. However, the custom weighting factors do not represent all possible zone constructions. Use of a weighting factor database requires some approximations to fit the fourteen selection parameters. Therefore, development of an RTF generating tool that fits practical design condition is essential for RTSM implementations. 28 One important assumption in calculating the radiant time factors is imposing adiabatic boundary condition for all surfaces in the zone. As the consequence of this assumption the radiant pulse used to generate the radiant time factors is then only redistributed in time, otherwise its energy is entirely conserved in the zone. In the RTSM, since no solar and internal radiant heat gains are conducted out of the zone, this often leads to slight overprediction of the peakcooling load. However, for zones with large amount single pane glazing, and cooler summer design weather conditions, a significant portion of the radiant heat gains can be conducted out, and those never become part of the cooling load. In these cases a much larger overprediction relative to the heat balance method is expected (Rees et al. 2000a; Rees et al. 1998; Spitler et al. 1997). 2.1.4 Limitations of the Radiant Time Series Method Quantitative comparison with the heat balance method shows that the RTSM tends to over predict the peak cooling loads (Rees et al. 1998; Spitler et al. 1997). Parametric investigations conducted for 945 zones cases showed that the peak load is slightly over predicted (Spitler et al. 1997). The heat balance method uses a detailed fundamental and rigorous mathematical model for the outside and inside surface heat balance. For medium and light weight construction, in particular zones with large amount of single pane glazing, the peak loads were over predicted significantly. In another similar study (Rees et al. 1998) made quantitative comparison of 7,000 different combinations of zone type, internal heat gains, and weather day. The result shows that the RTSM cooling load profile closely follows that of the heat balance cooling load; however, it over predicted the peak load for majority of the test cases when a radiative heat gain is large and zones 29 are made with large amount of single pane glazing. For a heavy weight construction midfloor, northeast corner zone, with 90% of the exterior wall area consisting of singlepane glass (Rees et al. 1998) the RTSM over predicted the peakcooling load by 37%. Three main reasons have been pointed out for peak cooling load over prediction: (1) the use of adiabatic boundary condition for the RTF generation, (2) combined treatment and constant assumption of convection and radiation coefficients, which makes the zone internal surfaces to radiate to the room air, and (3) simplification of the solair temperature calculations. Rees et al. (1998) concluded that the RTS method enforces conservation of radiant heat gains by ignoring the heat gain conducted to the outside environment as the principal reason for over prediction of peak cooling load. For internal surfaces with conditioned adjacent zones, the adiabatic boundary condition is a reasonable approximation; however, for external surfaces the adiabatic boundary condition in some cases very conservative approximation. Zones for which the peak design cooling load occurs in winter or zones located at lower design weather temperatures can be shown (with the HBM) to conduct a large amount of heat gains through the exterior surfaces with very low conductance (e.g. single pane glazing windows). On the other hand, the RTSM conserves the entire radiant heat gains and has no procedure to account for the heat gain conducted to the outside. Therefore, the RTSM over predicts the peak design cooling load slightly for hot and warm cooling design weather locations, while it tends to over predicts more and more for cold design weather conditions. 30 Experimental validation of radiant time series cooling load calculation method revealed that reflection loss of solar heat gain from the zone with high glazing fraction is significant (Iu et al. 2003). Though the rereflection and direct transmission losses can be computed they require detailed input data of glazing optical properties, zone geometry and orientations. In fact this phenomenon is likely to cause significant loss only in highly glazed buildings. 2.2 Dynamic Modeling of Thermal Bridges Dynamic modeling of thermal bridges has been an area of interest in building energy analysis and design load calculation programs. Building energy analysis and load calculation programs developed in the USA use onedimensional conduction transfer functions to predict heat conduction through the building envelope. However, many wall and roof constructions contain composite layers (e.g. steel studs, and batt insulation) that lead to local multidimensional heat conduction. The element with very high thermal conductivity is often referred to as a thermal bridge. Thermal bridges are important for both steady state and dynamic heat conduction. Several publications (Brown et al. 1998; Carpenter et al. 2003b; Kosny and Christian 1995b; Kosny et al. 1997b; Kosny and Kossecka 2002; Kosny et al. 1997c) indicate that onedimensional approaches cannot predict heat transmission through building envelopes without errors, especially for walls with thermally massive elements and a high disparity in the thermal conductivity of layer materials. Numerical studies indicate that thermal bridge effects of steel stud walls can reduce the thermal resistance of the clear wall by up 31 to 50% (Kosny et al. 1997a). Similar studies on metal frame roofs showed that the thermal bridge effect reduces the effective thermal resistance of the clear cavity values by as high as 75% (Kosny et al. 1997c). However, there is a limitation in the use of onedimensional response factor or conduction transfer functions methods when it comes to analysis of composite walls such as stud walls. This is a common problem in modeling heat conduction in steel stud walls and the ground where onedimensional analysis cannot predict the heat conduction without significant error. Multidimensional heat conduction effects are either ignored or not accounted properly. The onedimensional analysis may be valid for homogeneous layer wall; however, at the edges and corners, heat transfer significantly deviates from that of the onedimensional analysis. In practice, the edge and corner effects are simply ignored. Numerical and experimental investigations showed that ignoring the edge effects could under predict the heat transmission by over 10% (Davies et al. 1995). However, for portions of walls not near the edges, onedimensional analysis can be a reasonable approximation for lightweight walls without significant thermal conductivity disparity, such as those made from wood studs (Davies et al. 1995). Therefore, the need for multidimensional transient heat conduction models in building energy analysis and load calculation programs is crucial for accurate prediction of building energy consumption and peak load estimation; hence, it is also necessary for reliable HVAC equipment sizing and thermal comfort prediction. The following section discusses the onedimensional dynamic conduction modeling commonly used in load calculation and energy analysis programs in the USA. 32 2.2.1 OneDimensional Conduction Transfer Functions Transient conduction heat transfer through building envelopes can be calculated using lumped parameter methods, numerical methods, frequency response methods and conduction transfer function methods. Conduction transfer functions have been used most commonly in lead calculation and building energy analysis programs due to their computational efficiency and accuracy. The response factors are time series solutions of transient heat conduction that relate the current heat flux terms to current and past temperatures. Conduction transfer function coefficients are derived from response factors, which are determined using Laplace transform method (Kusuda 1969; Mitalas 1968; Stephenson and Mitalas 1971), or numerically (Peavy 1978). Conduction transfer function coefficients can be also determined directly using frequencydomain regression (Wang and Chen 2003), stable series expansion based on the Ruth stability theory (Zhang and Ding 2003), and State Space method (Seem 1987; Strand 1995). The next sections presents the use of response factor and transfer function coefficients in onedimensional conduction. Heat conduction through building structures is represented by onedimensional partial differential heat equation and the Fourier’s law of heat conduction as follows: t c T x t x q x t p ∂ ∂ = ∂ ∂ '' ( , ) ( , ) ρ (2.3) x q x t k T x t ∂ ∂ '' ( , ) = − ( , ) (2.4) 33 Where q” = is the heat flux, (W/m2 K) T = is the temperature, (oC) k = is the thermal conductivity, (W/m K) ρ = is the density, (kg/m3) cp = is the specific heat of the solid, (kJ/kg K) The solution of equations 2.3 and 2.4 can be represented as time series solutions called response factors. The time series solution of the heat conduction equation is determined for a unit triangular ramp excitation of the temperatures on both the internal and external surfaces of a wall. The response factors can be determined using Laplace Transform method (Clarke 2001; Hittle 1992; Kusuda 1969; Stephenson and Mitalas 1971), numerical methods (Peavy 1978), and time domain methods (Davies 1996). The current heat flux at interior surface of the wall '' q&i,t in terms of current and past boundary temperatures as inputs and the response factors is given by: Σ Σ∞ = − + ∞ = = − + − 1 , 1 1 , 1 '' , n n o t n n n q&i t Ti t n Z T Y (2.5) The heat flux at the external surfaces '' q&o,t is given by: Σ Σ∞ = − + ∞ = = − + − 1 , 1 1 , 1 '' , n o t n n n q&o t Ti t n Yn T X (2.6) 34 Where '' q&i,t : heat flux at the interior surfaces at time step t, W/m2 '' q&o,t : heat flux at the external surfaces at time step t, W/m2 Ti,tn+1: interior boundary temperature at time step tn+1, oC To,tn+1: exterior boundary temperature at time step tn+1, oC X: is the selfcoupling response factor; the heat flux at the exterior surface for triangular ramp input of the exterior boundary temperature and zero interior boundary temperature (W/m2 K). Y: is the cross coupling response factor; the heat flux at either surface for triangular ramp input of the boundary temperature at the other surface, (W/m2 K). Z: is selfcoupling response factor of the interior surface for triangular ramp input of the boundary temperature at the interior surface and zero exterior boundary temperature, (W/m2 K). The primary advantage of the response factor method is that for a given building structure, assuming constant thermophysical properties, the response factors need to be determined only once. Numerical methods such as finite difference or finite element methods generally require high computational time; however, they allow variable time step and variable thermophysical property simulations. Conduction transfer function methods are further refinements of response factor methods. The replace many of the higher order terms of the response factors and the past 35 temperatures with the past heat fluxes; hence, use fewer coefficients (Hittle, 1992) (McQuiston et al. 2005; McQuiston 2000). The heat fluxes at the interior and exterior surfaces of a wall in terms of conduction transfer functions are given by: Σ Σ Σ = = − + − = = − + − + M m M m k m o t m m i t m M m qi t Zk mTi t m Y T F q 1 1 , , 1 , 1 , , 1 '' , (2.7) Σ Σ Σ = = − + − = = − + − + M m M m k m o t m m o t m M m qo t Yk mTi t m X T F q 1 1 , , 1 , 1 , , 1 '' , (2.8) Where Xk,m: the mth conduction transfer coefficient for the exterior selfcoupling term of order k, (W/m2 K) Yk,m: the mth conduction transfer coefficient for the cross coupling term of order k, (W/m2 K) Zk,m: the mth conduction transfer coefficient for the interior selfcoupling term of order k, (W/m2 K) Fm: is defined as the flux history term coefficients (). 2.2.2 Steady State Conduction Models Multidimensional and in particular twodimensional steady state conduction models have been used to study and investigate the accuracy of the approximate onedimensional thermal resistance calculation procedures for thermal bridges. Studies made on multidimensional steady state heat conduction analysis of composite walls have indicated that the heat fluxes deviate significantly from that of an approximate onedimensional heat 36 conduction models that ignore the thermal bridges. This has been demonstrated using numerical and experimental analysis of steady state heat conduction in building wall specimens (Barbour and Goodrow 1995; Brown et al. 1998; Carpenter and Schumacher 2003; Kosny and Christian 1995a; Kosny and Christian 1995c; Kosny et al. 1997a; Kosny et al. 1997b; Thomas and Antar 1998). Compared to dynamic analysis, steady state models for heat transfer of thermal bridges are well developed. The next section briefly discusses ASHRAE’s recommended approximate onedimensional steady heat conduction models. Approximate Steady State Models Steady state heat transfer through composite material walls is commonly treated with onedimensional models that utilize some approximations in representing the thermal resistance of composite walls. ASHRAE recommends the following methods: isothermal plane method, parallel path method, the zone method and modified zone method and insulation/framing adjustment factor method to compute the overall thermal resistance based on qualitative criteria (ASHRAE 2005). Isothermal Plane Method This method assumes that for layer materials with high thermal conductivity the temperature at each plane remains isothermal. Composite layers sandwiched in between these two isothermal plane layers are combined using area weighted parallel heat flow path method. Then the overall resistance is determined from layer resistances using a series sum of resistances. This method is recommended for concrete blocks where web 37 and the core section are combined using areaweighted parallel heat flow path method and then combined in series with the face and air film resistances. For widely distributed metal members with high crosssectional area constructions such as roof decks the isothermal plane method underpredicts the overall resistance; hence, the zonal method is recommended (ASHRAE 2001). Parallel Path Method The parallel heat flow path method assumes no heat flow in the lateral direction; hence, the heat flow path in the construction is principally longitudinal. The resistance is calculated from the areaweighted average of the individual thermal transmittances of the different parallel heat flow paths in the construction. This method predicts the overallresistance of a construction with reasonable accuracy for wood frame or wood stud walls, where the disparity in thermal conductivity between the wood and the cavity insulation is small. The actual overall thermal resistance lies in between the isothermal plane and the parallel path methods (Barbour and Goodrow 1995; Brown et al. 1998; Gorgolewski 2005; Thomas and Antar 1998). Another method developed as extension of the parallel path method is parallel path correction factor method (ASHRAE 2005). The thermal resistances along the stud and center of wall are area weighted to get the overall average thermal resistance. The parallel path correction factor Fc, method is recommended for the metal stud walls. 38 Modified Zone Method For building envelopes with widely spaced metal members such as steel stud walls, the actual overall thermal resistance lies in between the isothermal plane and parallel path methods (Barbour and Goodrow 1995; Gorgolewski 2005). The ASHRAE zone method was introduced for calculating overall resistance for such constructions (ASHRAE 2005). The zone method extends the parallel heat flow path method to account the local highly conductive region as a separate path for the heat flow and divides the construction into two zones. The zone method determines the width of zone containing the metal element as function of the distance from the stud face to the surface of the construction. Studies have shown that zone method does not consider the thermal bridge region of influence or the metal zone width dependency on stud spacing, stud depth and sheathing thermal conductivity (Barbour and Goodrow 1995; Kosny and Christian 1995a). The modified zone method was introduced to improve the zone method by including the metal zone area dependency on: ratio of resistivity of cavity insulation to sheathing materials, thickness of sheathing insulation, and stud flange area (Kosny and Christian 1995a). Insulation / Framing Adjustment Method Thermal resistance of wall assemblies containing metal framing can be calculated using insulation /framing adjustment factors. Such framing factors are provided by ASHRAE / IESNA standard 90.12004 (ASHRAE 2005). The adjustment factor corrects the resistance of the core insulation for the metal frame effect. It is also called correction factor method. 39 Gorgolewski Method The Gorgolewski (2007) proposed a semiempirical correlation for computing the steady state Rvalue of light frame steel stud walls. This procedure uses weighted average of isothermal plane and parallel path method Rvalues. The weighting parameter is calculated from semiempirical correlation that depends on the geometry of the steel frame and the isothermal and parallel path methods Rvalues and hence is suitable for programming application as it does not involve subjectivity. The following section discusses development in multidimensional conduction dynamic modeling method and the barriers for their implementation. 2.2.3 Multidimensional Conduction Dynamic Models There have been repeated efforts to develop multidimensional dynamic heat transfer model that produce CTFs in onedimensional form for energy analysis and load calculation programs. Previously published methods for dynamic modeling of multidimensional conduction proposed for use in building energy analysis and load calculation programs include: numerical methods, numerical CTF method, equivalent wall methods and statespace method. The later methods were developed with the intention to use in developing onedimensional CTF coefficients. However, their adoption has been delayed for several reasons. Nevertheless, there remains a clear need for a simple onedimensional approximate dynamic model for modeling of construction with thermal bridges. The next section discusses previously published multidimensional conduction dynamic models and their limitations. 40 Numerical Methods Numerical models of multidimensional conduction heat transfer have been developed, but are still limited to research use (Davies, et al., 1995). Numerical methods include finite difference, finite volume and finite element techniques. The third generation building simulation program, ESPr, uses the finite control volume energy conservation method (Clarke 2001; Nakhi 1995). ESPr, a whole building energy simulation program developed at the University of Strathclyde offers multidimensional heat conduction analysis of walls, edges, corners and the ground; however, the multidimensional heat conduction model has limitations on the composite layer specification and, at best, is difficult to use. Numerical methods require high computational time since it involves solving the nodal variables at each time step. Therefore, implementation of multidimensional heat conduction finite difference or finite element methods for real composite walls requires higher computational time and computer memory. The lack of graphical user interface for automatic building geometry and construction material acquisition has been a hurdle for the development of spatial discretization for use in the multidimensional conduction model. Burch et al. numerical CTF method Burch et al. (1992) presented a numerical procedure for calculating CTF coefficients that accounts for thermal bridge effects of metal studs, aluminum frame windows and metal frames on office building envelopes. The method solves the conduction equation numerically using finite difference techniques by applying linearly varying boundary conditions that replicate the triangular ramp temperature boundary conditions as shown in 41 Figure 2.3. Then the principle of superimposition is used to determine the response factors from the three linear temperature excitations at base time steps of 2δ. The heat flux at a particular surface yields the required response factors. The numerical procedure of determining the response factors for multidimensional conduction models can be summarized as follows (Burch et al. 1992): i. Develop the triangular ramp unit excitation (Figure 2.3) representation of the boundary temperature at one of the surfaces of interest while the other face of the surface is kept at zero temperature (Hittle, 1992). ii. Determine the numerical solution of the heat flux at the surface interest by summing the individual heat fluxes of the cells or nodes for each excitation. This yields one of the response factors. In a similarly way the other response factors can be determined. iii. The CTFs coefficients are determined from the response factors using recursive algorithms (Hittle, 1992). 1Δ 1Δ 2Δ Elapsed Time Temperature 1 Figure 2.3 Lineartriangular ramp temperature pulse representation 42 The Burch et al. (1992) finite difference procedure based CTF coefficients determination method requires a separate standalone transient heat conduction analysis program with a spatial discretization scheme. Thus, the Burch et al. method is difficult to be integrate into existing building energy analysis and load calculation programs without significant modifications of their codes (Burch et al. 1992). Equivalent Wall Method Kossecka (1998) and Carpenter et al. (2003a) developed the concept of an equivalent wall, which replicates multidimensional thermal dynamics of the complex composite wall with a simple homogeneous layer wall. The generated equivalent wall, which has the same dynamic behavior to that of the real composite wall, is represented by onedimensional response factors or conduction transfer function that can be implemented in the commonly used building energy analysis and load calculation programs (Kosny and Kossecka 2002). Generating equivalent walls requires proper identification of the thermal mass and negligible mass resistance components from the construction layer configuration. The thermal characterization of constructions can be defined by a parameter called thermal structure factor (Kossecka 1998). The concept of thermal structure factor is presented next. Thermal Structure Factors Thermal structure factors, which are dimensionless parameters, define the thermal energy storage characteristic of building structures when it goes through two successive steady state ambient temperature transitions (Kossecka 1998; Kossecka and Kosny 2002). 43 Thermal structure factors of a wall depend on the resistance and thermal capacities of the layers and their sequence of arrangements in the wall. Thermal structure factors (φ) of building structures (Carpenter et al. 2003a; Kossecka 1998; Kossecka and Kosny 2002) are given by: c ( ) dx C L ii p 2 0 ϕ = 1 ∫ρ 1−θ (2.9) c ( )dx C L ie p ϕ = 1 ∫ρ θ 1−θ 0 (2.10) c dx C L ee p 2 0 ϕ = 1 ∫ρ θ (2.11) Besides, the following identity needs to be met by the thermal structure factors + 2 + = 1 ii ie ee ϕ ϕ ϕ (2.12) Where C Overall thermal capacity of the wall, (J/m2 K) cp specific heat of a layer in the wall, (J/kg K) ρ density of a layer in the wall, (kg/m3) L thickness of the wall, (m) θ dimensionless temperature, () φii Interior structure factor () φie core structure factor () 44 φee exterior structure factor () A high interior thermal structure factor (φii) implies that the higher thermal mass layer is located near the interior surface and most of the resistance is located near the exterior surface of the wall. Vice versa, a high exterior thermal structure factor (φee) indicates that the higher thermal mass layer is located near the exterior surface and most of the resistance is located near the interior surface of the wall. And a high core thermal structure factor (φie) implies that the higher thermal mass layers are located at the center of the wall and the resistances are placed symmetrically on both sides of the wall. The relationships between response factors and thermal structure factors (Carpenter et al. 2003a; Kossecka 1998; Kossecka and Kosny 2002) are given: ii n n C nX ϕ δ − = Σ∞ =1 (2.13) ie n n C nY ϕ δ = Σ∞ =1 (2.14) ee n n C nZ ϕ δ − = Σ∞ =1 (2.15) Where Xn: the nth term exterior selfcoupling response factor, (W/m2 K) Yn: the nth term cross coupling response factor, (W/m2 K) Zn: the nth term interior selfcoupling response factor, (W/m2 K) C Overall thermal capacity of the wall, (J/m2 K) 45 φii Interior structural factor () φie core structural factor () φee exterior structural factor () δ time step, (s) Equivalent wall Generation Procedure The equivalent wall generation requires five steps (Carpenter et al. 2003a). These steps are summarized as follows: (1) Develop a threedimensional model of the building envelope to exact dimensions using multidimensional dimensional heat conduction solver computer programs. (2) Generate threedimensional response factors using numerical methods and use them to determine the three dimensional conduction transfer function; (3) Calculate the thermal structure factors of the composite wall using the three dimensional response factors determined in step 2 and the thermal structure factor identity; (4) Generate the fictitious equivalent wall. The equivalent wall generation requires selecting random set of resistances for each layers of the wall and calculating the capacitance or randomly specifying the capacitance and calculating the resistances. A three layer fictitious wall is recommended for simplicity (Carpenter et al. 2003a). The material layer configuration, i.e., the relative position of the high thermal mass and the high resistance layers in the fictitious wall layers configuration must resemble that of the actual wall. 46 (5) The transfer function coefficients for the threedimensional numerical model and the equivalent wall model matching is done by trialanderror until reasonable accuracy is achieved by adjusting the resistance and/or capacitance of the equivalent wall layers. The steady state resistance, thermal response factors and structure factors of the real wall and the fictitious equivalent wall must be the same. State Space Method The state space method can be used to determine conduction transfer function coefficients that can represent multidimensional transient heat conduction in walls. The state space method is based on first order differential equation representation of transient heat conduction by spatially discretizing the conduction domain and representing the inputs by continuous, piecewise linear functions (Seem 1987). The advantage of the state space method compared to the Laplace transform method (Hittle, 1992) is that it can be extended to solve multidimensional transient heat conduction problems. The state space method is represented in the form of equations as follows: [ ][ i ] [ ][ b ] i A T B T dt dT = + (2.16) [ ] [ ][ ] [ ][ ] q = C T + D Tb (2.17) Where Ti vector of n interior node temperatures as state variables, °C (°F) 47 A a constant coefficient transition matrix with size of nxn t the time, (s) B the constant coefficient matrix of input vector of size (nxp) Tb vector of p boundary temperatures as inputs, °C(°F) q vector of p heat fluxes as outputs, W/m2°K(Btu/h⋅ft2⋅°F) C a constant coefficient matrix of the output vectors of size (mxn) D constant coefficients matrix of the input vector of size (mxp) Solution of equations 2.16 and 2.17 for constant elements matrix of A and B in a compact representation is given by: Σ( ) Σ = = = = = − − − j n j j n j qt S jTt j Fjqt j 0 1 δ δ (2.18) Where Sj are the conduction transfer function coefficients, (W/m2 K) Fj are the coefficients of the past heat flux history terms, () Twodimensional state space method Seem (1987) demonstrated that the state space method can be applied to model multidimensional transient heat conduction in building envelopes. In the state space method, the heat conduction domain is discretized in multidimensions; hence, the multidimensional heat conduction effects in composite walls such as those with steel studs can 48 be modeled accurately. Therefore, the state space method can model dynamics of walls; however, as the conduction domain becomes complex, the number of nodes required for accurate modeling also increases. Consequently, evaluation of the exponential matrix becomes cumbersome or sometimes almost impossible (Amjad et al. 2003). Barrier to Use of Dynamic Multidimensional Models Three multidimensional dynamic thermal bridge models, which could be integrated into existing onedimensional conduction transfer function procedures, have been proposed. The multidimensional dynamic CTF models are: the equivalent wall method, the state space method, and the numerical CTF method. Beyond other reasons, implementation and integration of these models into the existing programs has been delayed due to inherent limitations in the development of the multidimensional dynamic procedures, and high computational resource requirement. These barriers to implementation in whole building load calculation / energy simulation programs include: The equivalent wall and the Burch et al., numerical CTF models require either separate standalone multidimensional transient heat conduction analysis numerical programs or integration of a significant multidimensional conduction subprogram. Models with spatial discretization requirements need both an implementation of the discretization procedure and user interface to support this level of details. Given the complexity of providing an interface to specify a whole building, this may be too much of a refinement. 49 The equivalent wall method thermophysical properties determined by trialanderror could be out of range for typical building materials properties. Although the state space method does not require a separate standalone numerical program, a spatial discretization tool is necessary. On the other hand, the state space method can be integrated with existing building energy analysis and load calculation programs most conveniently if an automatic building envelope discretization and/or interactive user interface for material specification and construction model reduction program is made available. However, the multidimensional state space method will suffer from a numerical accuracy problem that grows as the number of nodes increases. 50 51 CHAPTER III 3. Radiant Time Series Method Improvements This chapter covers several improvements to the Radiant Time Series Method (RTSM). These improvements include: new algorithm for generating Radiant Time Factors (RTF) and developing a onedimensional finite volume numerical method periodic response factor generating procedure, adapting an improved fenestration model and establishing radiative / convective splits, developing a procedure for accounting heat losses through fenestration to the outside, and facilitating implementation of the RTSM procedure in different computation environments. There are several approaches for generating the RTF for a given building zone. Since these procedures have been adopted directly or indirectly from a fullblown heat balance method (HBM) procedure, they tend to have some unnecessary overhead and a simpler method developed specifically for RTF generation is highly desirable. Derivation of the reduced HBM RTF engine is described in Section 3.1. The new radiant time factors (RTF) generating algorithm utilizes periodic response factors (PRF) to model transient conduction as described in Section 3.1.1. Currently available PRF generation procedures convert conduction transfer function coefficients generated using Laplace and State Space method. 52 This procedure is undesirable for use in VBA and SCILAB type computational environments. Therefore, a onedimensional finite volume numerical procedure for computing periodic response factors has been implemented and investigated. The finite volume numerical procedure implementation is discussed in Section 3.1.2. Derivation of the algorithm and validations are given in APPENDIXB. As part of this research, an improved fenestration model compatible with currently available fenestration data will be adapted to the RTSM. Furthermore, a new set of radiative / convective splits compatible with the improved fenestration model has been established. The improvement in the RTSM fenestration model is described in Section 3.2. The previously published RTSM cooling load procedure2 (Rees et al. 2000a; Spitler et al. 1997) does not account for the solar and internal radiant heat gains conducted back out of the zone. Ignoring these back losses is the principal reason that the RTSM over predicts the peakcooling load. An approximate algorithm that accounts for zone radiant loss in the RTSM procedure has been derived and is described in Section 3.3. For all improvements described in this section, results and discussed are provided for each Sections. However, for improvement to the fenestration model and the heat losses accounting procedure, it is highly desirable to demonstrate satisfactory performance over a wider range of cases. Such a study is the subject of Chapter 4. 2 Referred to as the “original RTSM procedure” in this thesis. 53 3.1 New RTF Calculation Engine The Radiant Time Series Method (RTSM) converts the radiant component of the heat gains into cooling loads using the Radiant Time Factors (RTF), a 24term series. The 24 hourly radiant time factors describe the dynamic response characteristics of a zone. Two procedures were developed for RTF generation (Spitler et al. 1997). Currently there are five approaches available for generating the radiant time factors: (1) software that comes with the book Cooling and Heating Load Calculation Principles (Pedersen et al. 1998), (2) software that comes with the book by McQuiston et al. (2005), (3) software developed as based on the ASHRAE toolkit (Iu 2001), (4) tabulated RTF values in the ASHRAE Handbook of Fundamentals, and (5) RTF generated from zone heat gain weighting factors. The first four approaches use the full heat balance method as a calculation engine to compute the RTFs. The HBM programs that come with the books by Pedersen et al. and McQuiston et al. are limited to twelve surfaces only for any zone and the RTF generating software developed by Iu is limited to six surfaces. The McQuiston et al. and Pedersen, et al. programs are full load calculation programs that treat the RTF as an auxiliary output. The McQuiston et al. and Iu programs have interfaces that allow users to select material layer thermophysical properties from a database. The interface used by McQuiston et al. accepts much more information than is actually necessary to generate the RTF. The tabulated values in the Handbook of Fundamentals were generated for limited building design conditions using Pedersen et al. program. But the accuracy of the RTSM peak cooling load resulting from the users choosing the “nearest” zone has not been investigated. 54 An alternative approach (Spitler and Fisher 1999b) is to generate radiant time factors directly from a set of zone heat gain weighting factors using the existing ASHRAE 472 RP database (Sowell 1988a; Sowell 1988b; Sowell 1988c). These weighting factors were developed for use with the Transfer Function Method (TFM). This approach would use a computer program to map a given zone to the fourteen zone parameters in the database and transform the weighting factors to radiant time factors using matrix manipulation. However, the 472RP weighting factors do not represent all possible zone designs and construction types. ASHRAE’s 472RP weighting factors were generated for combination of discrete building design parameters; hence, the RTF generated from these weighting factors represent only specific buildings. User judgment is required to match an existing building to one of the combinations of discrete building design parameters. In conclusion, the existing approaches all suffer from being too cumbersome, requiring too much user judgment, or too limited with regard to the number of zones surfaces. Therefore, development of an RTF coefficientgenerating tool that handles a wide range of practical design conditions is desirable. Such a tool should meet the following requirements: i. capable of generating RTF for a wide variety of practical building constructions, including a practical number of building surfaces, i.e. more than twelve. ii. simple user interface that does not require unnecessary information. iii. can be integrated with other applications such as spreadsheets by eliminating unnecessary features and source code. It would be preferable to 55 minimize the required calculations and eliminate the use of DLLs if possible. iv. The methodology should take advantage of the steady periodic nature of the boundary condition. When this is done, the methodology can benefit from matrix algebra easily accessed in modern computing languages and environments such as SCILAB or MATLAB. The new RTF calculation engine is derived from an inside surface heat balance and room air heat balance for adiabatic zone. The simplified procedure eliminates several steps of the HBM, particularly the exterior surface heat balance and eliminates input data that are not necessary for the RTF generation. The new RTF generating program is first developed as a FORTRAN DLL which may be called from a spreadsheet. Then, it will be implemented in VBA and SCILAB. The following section describes the derivation of the mathematical algorithm for the RTF generation program. 3.1.1 The Mathematical Model Reduced Heat Balance Method The formulation of the mathematical model for the radiant time factor generating procedure makes use of the major assumptions used in the Heat Balance Method (HBM) (Pedersen et al. 1997); that surfaces (walls, roofs, windows, etc) can be treated as having uniform surface temperatures, uniform long wave and short wave length radiation; surfaces are gray; onedimensional conduction is valid and surfaces are exposed to steady periodic boundary conditions. The simplified heat balance procedure for RTF generation uses an inside surface heat balance, steady periodic boundary condition, constant 56 convection coefficients and constant room air temperature. The simplified HB procedure for RTF generation eliminates the following procedures that are part of the full heat balance method: outside surface heat balance, weather data, solar radiation calculations, shading calculations, infiltration and ventilation. Because this simplified version of the HBM uses a smaller number of heat balance steps, it will be referred to here as the “Reduced Heat Balance Method” (RHBM). In the next section, the RHBM is described stepbystep. The inside surface heat balance for the ith surface is given by: '' 0 , '' , '' , '' , + + + = conv i cond i rad i gain i q& q& q& q& (3.1) Where, '' conv,i q& = convection heat flux from the room air to the ith surface, Btu/h⋅ft2 (W/m2) '' q&cond ,i = the ith surface conduction heat flux from the outside surface to the inside surface, Btu/h⋅ft2 (W/m2) '' rad ,i q& = radiant heat flux from other internal surfaces to the ith surface, Btu/h⋅ft2 (W/m2) '' gain,i q& = radiant heat gain flux of the ith surface from lights, people and equipments, Btu/h⋅ft2 (W/m2) 57 Conduction Model The RTSM uses airtoair periodic response factors to compute conduction heat gain driven by steady periodic exterior solair temperature and a constant indoor air temperature. For RTF generation, surfacetosurface periodic response factors are used with steady periodic surface temperatures computed with the reduced heat balance method. The transient conduction heat flux at the inside surface using surface periodic response factors (Spitler et al. 1997) for steady periodic boundary conditions is given by: Σ Σ = − = = − − 23 0 , , 23 0 , , '' , , j Pj In i t j j q&cond i t YPjTOut i t jδ Z T δ (3.2) Where, In i t jδ T , , − = inside surface temperature jth hours before the current hour, °F (°C) Out i t jδ T , , − = outside surface temperature jth hours before the current hour, °F (°C) Pj Y = surfacetosurface cross periodic response factor, Btu/h⋅ft2⋅°F (W/m2⋅°C) Pj Z = surfacetosurface inside periodic response factor, Btu/h⋅ft2⋅°F (W/m2⋅°C) The transient heat conduction equation can be simplified further using the appropriate boundary conditions. 58 Boundary Conditions For all surfaces in a zone, the outside surface temperature is assigned the inside surface temperature to maintain the equivalent temperature as the boundary condition. This approach emulates an adiabatic boundary condition by forcing an equal amount of heat flow from the other side of the surface, hence balancing the heat flow into the construction. This condition is represented as follows: ( ) Σ= = − − 23 0 , , '' , , j q&cond i t YPj ZPj TIn i t jδ (3.3) Figure 3.1 Adiabatic boundary condition for RTF generation: Tso=Tsi. Convection Model The RTSM procedure is formulated to use fixed convection coefficients. This assumption is not required in the Heat Balance Method. Heat transferred from the room air to the zone surfaces by convection using a constant convection coefficient is given by: TSO TMRT T 1/hcIniAi Si 1/hradiAi '' cond ,i q& '' gain,i q& '' conv,i q& '' rad ,i '' q& cond ,o q& 59 qconv i t hcIn,i (Tr TIn,i,t ) '' & , , = − (3.4) Where, In i t T , , = inside temperature of the ith surface at time t, °F (°C) Tr = constant room air temperature, °F (°C) hcIn,i = convection coefficient of the ith inside surface, Btu/h⋅ft2⋅oF (W/m2 °C) '' conv,i,t q& = convection heat flux at the ith surface at time t, Btu/h⋅ft2 (W/m2) Internal Longwave Radiation Model The internal long wave radiation model assumes zone air is transparent to long wave radiation and considers the zone surfaces as gray and the long wavelength radiation as diffuse. With these assumptions, long wavelength radiation in building can be modeled using the uniform radiosity method, the total gray exchange factors method (Hottel and Sarofim 1967), the mean radiant temperature and balance (MRT/balance) method (Walton 1980) or the Mean Radiant Temperature Network (MRTNet) method (Carroll 1980). The uniform radiosity and the total gray exchange factor method require exact view factors. The uniform radiosity method involves solving the radiosity at every time step hence the method is computationally intensive, which makes it unsuitable for building applications. The total gray exchange factor method combines the surface properties and geometries into the gray exchange factors, which can be computed at the beginning of the 60 simulation and eliminates the simultaneous solution at every time step. However, it requires exact view factors. In real buildings, calculation of exact view factors is computationally intensive and the furnishings and other internal heat sources cannot be easily represented and are mobile during the lifetime of the buildings. Hence, any gains in accuracy facilitated using the exact view factors are unlikely to be realized in practice. Therefore, the extra effort introduced in specifying locations and dimensions of furnishings is unlikely to be rewarded with any tangible benefit. The advantage of the MRT/ balance and MRTNet methods is that both use approximate view factors based on area and emissivity and allow an approximate representation of furnishings and partitions surfaces. In the MRT/balance method each surface exchanges radiation with a fictitious mean radiant temperature calculated from areaemissivitysurface temperature product weighted of the remaining surfaces viewed by the surface. The radiation flux imbalance resulting from approximate view factors in the MRT/balance method is balanced by redistributing it to the surfaces. In the MRTNet method each surface exchanges radiation with a single fictitious mean radiant temperature of the zone that is computed from all surfaces. The radiation coefficient linking the each surface is corrected for each surface depending on the surface and MRT temperature of the zone and hence balances the zone radiation exchange. Both MRT methods essentially have the same accuracy (Liesen and Pedersen, 1997). Therefore, due to simplicity, the mean radiant temperature network (MRTNet) method (Carroll 1980) has been selected for use in the RTF generation algorithm. With the 61 MRTNet method, each surface in the zone is linked to a fictitious single radiant node. The radiation heat transfer from the fictitious node to the surface is given by: q&rad ,i,t = hrad ,i,t (TMRT ,t − TIn,i,t ) (3.5) Where TMRT ,t = mean radiant temperature of the zone at time t, °F (°C) rad i t h , , = radiation coefficient of the ith surface at time t, Btu/h⋅ft2⋅°F (W/m2⋅°C) The MRT radiation model (Carroll 1980) requires a two step update of the radiation coefficients for each time step. First, initialize the radiation coefficients for each surface to the reference temperature as follows: ( ) i i i ref rad ref F T h ε ε σ 1/ 1 / 4 3 , + − = (3.6) Where Tref = mean reference temperature in absolute scale, °R (°K) hrad ,ref = radiation coefficient at reference temperature of 300K, Btu/h⋅ft2⋅°F (W/m2⋅°C) Fi = the MRTNet view factor for ith surface, (). i ε = the longwave emissivity of ith inside surface, (). 62 The MRTNet view factor is an approximate view factor that compensates for the selfweighting in the mean radiant temperature, TMRT, is given by: 1/(1 / ) 1 Σ= = − N i Fi AiFi AiFi (3.7) Where N = the number of surfaces in the zone Since the Fi appears on both sides of the equation, iteration is required after setting the initial values of Fi to unity (Carroll 1980). However, the MRT network view factors can be calculated at the beginning of the simulation during the initialization phase. The hourly radiation coefficient that links each surface to the single fictitious mean radiant temperature in Celsius and Fahrenheit scales, respectively, is updated at each time step as follows: rad i t In i t rad ref h T h , , , ' , , = (0.865 + / 200) (3.8a) rad i t In i t rad ref h T h , , , ' , , = (0.775 + / 360) (3.8b) Then the mean radiant temperature, TMRT, is calculated from weighted average of the product of surface temperatures, surface area and the updated radiation coefficient as follows: 63 Σ Σ = = = N i In i t i rad i t N i MRT t i rad i t T A h T A h 1 ' , , , , 1 , , ' , / (3.9) Again the radiation coefficients are adjusted using the updated mean radiant temperature in Celsius and Fahrenheit scales, respectively, as follows: ' , , ' , , , (0.865 / 200) rad i t MRT t rad i t h = + T h (3.10a) ' , , ' , , , (0.775 / 360) rad i t MRT t rad i t h = + T h (3.10b) Then the corrected radiation coefficient of the individual surface is used to compute the room mean radiant temperature (TMRT,t) as follows: Σ Σ = = = N i In i t i rad i t N i MRT t i rad i t T A h T A h 1 , , , , 1 , , , / (3.11) So, for each iteration of the heat balance, a fixed twostep iteration to determine the radiation coefficient and the MRT is done as given in Equations 3.7 to 3.11. Radiant Heat Gain Distribution Model Computing the precise distribution of longwave radiation from internal sources requires knowledge of the exact location / position, surface area and temperature of the sources. This approach creates additional complexity to internal radiation exchange models. Therefore, the conventional approach is to distribute the internal radiant heat gains 64 uniformly to all surfaces in the zone, based on an area weighted or an areaabsorptance product weighted distribution model. Similarly, internal short wave radiation heat gain can be reasonably represented by uniform distribution as most of the cases have diffuse sources. For transmitted solar radiation an accurate distribution model could track the sun’s position and the resulting sun patch hourbyhour. However, partial surface irradiation is incompatible with the onedimensional and uniform surface temperature assumptions used in the conduction and radiation exchange submodels. Therefore, the most commonly used distribution model is to distribute the transmitted beam solar on the surfaces that are most likely to intercept the solar radiation  the floor and the furnishings. Transmitted diffuse solar heat gain is treated like long wavelength radiant heat gains and distributed uniformly. The two different distributions are the genesis of the two different RTF series. Long wave length RTF is generated by distributing the heat gain pulse uniformly to all surfaces in the zone. The solar RTF is generated by distributing the heat gain 50% to the floor and 50% to the furniture. Derivation of the Reduced HBM Algorithm Substituting the individual heat balance components into the heat balance equation (3.1) for the ith surface yields: ( ) ( ) , , ( , , , ) ) , , 0 23 0 , , , , , + − + = − + − Σ= − i rad i t MRT t In i t i gain i t j i cIn i r In i t i Pj Pj In i t j A h T T A q A h T T A Y Z T & δ (3.12) 65 For each surface the heat balance equation 3.12 can be reduced to the form shown below: ( ) ( ) '' , , , , , , 23 1 , 0 0 , , , , , , i cIn i r i rad i t MRT t i gain i t j i cIn i P P rad i t In i t i Pj Pj In i t j A h T A h T A q A h Z Y h T A Z Y T = + + & − + + − + Σ= − δ (3.13) Solution Schemes At least two solution schemes for solving Eqn. 3.13 can be developed: solution scheme I involves setting up the reduced heat balance equation for all 24 hours for each surface and then marching through each surface until the 24hourly surface temperatures for all surfaces converge. Solution scheme II involves setting up the heat balance equation so as to solve for the inside temperatures of all the surfaces in the zone at every hour, and then march through each hour. These two solution schemes are described in the next two sections. Solution Scheme I For a particular surface, the reduced heat balance (equation 3.13) can be written for 24 hours in compact matrix notation: [Ω][T ] [Γ] In = (3.14) The Matrix Ω will have dimensions of 24 by 24 and is given by the following expression: 66 − − − − + + − − − − − − − + + − − − + + − − − + + − − − Ω= P P P P P P P P radi cIni P P P P P P P P P P P P P P radi cIni P P P P P P radi cIni P P P P P P radi cIni P P P P P P Z Y Z Y Z Y Z Y h h Z Y Z Y Z Y Z Y Z Y Z Y Z Y h h Z Y Z Y Z Y h h Z Y Z Y Z Y h h Z Y Z Y Z Y 23 23 22 22 21 21 0 0 , ,24 , 22 22 21 21 20 20 23 23 2 2 1 1 0 0 , ,3 , 3 3 1 1 0 0 , ,2 , 23 23 2 2 0 0 , ,1 , 23 23 22 22 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) L M M M M O M L L L (3.15) The Matrix Γ is a 24element column vector and is given by the following expression: + + + + + + + + + + Γ = '' , , ,24 ,24 , ,24 '' , , ,23 ,23 , ,23 '' , , ,3 ,3 , ,3 '' , , ,2 ,2 , ,2 '' , , ,1 ,1 , ,1 cIn i r rad i MRT gain i cIn i r rad i MRT gain i cIn i r rad i MRT gain i cIn i r rad i MRT gain i cIn i r rad i MRT gain i h T h T q h T h T q h T h T q h T h T q h T h T q & & M & & & (3.16) The inside surface temperatures for a particular surface are determined from Eq. 3.17 as follows: [T ] [Ω] [Γ] In −1 = (3.17) In this scheme, the 24 hourly values of surface temperature are solved for each surface sequentially; this is done iteratively until all surface temperatures for all hours are converged. The radiation coefficients and the Ω matrix elements need to be updated at each iteration step. Thus, the repeated matrix inversion is computationally intensive. Further 67 simplification of the RTF generation algorithm has been investigated using fixed radiation coefficients. This is presented in Section 3.1.4. Solution Scheme II In solution scheme II the surface temperatures are solved for all surfaces at every hour. This solution scheme also can be formulated in matrix notation; however, it is formulated as an iterative procedure. For an hourbyhour march through all the surfaces at each step, the heat balance equation can be solved iteratively from the following equation: ( ) ( cIn i P P rad i t ) rad i t MRT t gain i t j cIn i r Pj Pj In i t j In i t h Z Y h h T Y Z T h T q T , 0 0 , , '' , , , , , 23 1 , , , , , + − + + − + + = Σ= − δ & (3.18) As with solution scheme I the radiation coefficients and the mean radiant temperatures must be updated at each iteration step until the surface temperature converges. The convergence criterion can apply on the surface temperature or the RTFs. The zone radiant time factor is determined from the inside surface temperatures, zone air temperature, convection coefficients and zone surfaces inside areas. The sum of the convection heat transfer from each surface per unit heat gain pulse of the zone is the hourly radiant time factor coefficient and is computed as follows: [ ] [ ] [ ] In cIn gain t T Ah q r = Δ ⋅ & 1 (3.19) 68 Where gain q& = heat gain pulse with which the zone is excited, Btu/hr (W) [ ] t r = a column vector of 24element radiant time factor coefficients, () [ ] cIn Ah = a row vector of Nelements of the product of inside surface area and convection coefficients, () [ΔTIn ] = a 24 by N matrix of the difference between inside surface temperature and the room air temperature, (K) The convergence criteria employed is that when the change in the sum of the RTFs changes less than 0.00001 between the successive iteration steps, the solution is considered converged. Solution scheme I is adapted in all subsequent sections and implementations in all other computing environments. 69 Figure 3.2 Flow chart of solution scheme I Get Surface Parameters Compute Periodic Response Factors Initialize the calculation Beginning Iteration Loop Start Surface Loop I = 1, Nmax Set up system matrix Eqns. 3.15 & 3.16 Compute Surface Temperatures Eq. 3.17 i = Nmax Compute RTFs Eq. 3.19 No ΣRTF =1 Update TMRT and hrad No convergence 70 3.1.2 Validation of the New RTF Engine The new RTF generating engine has been validated against the fullblown heat balance method FORTRAN program originally developed by Pedersen et al. (1997). The test zone geometry and construction fabrics are given in Table 3.1. Three construction types: light, medium and heavy weight constructions were used for the validation. Each zone has a single exterior surface with single pane clear glass and 50% glazing fraction of the exterior facade. Zones were modeled with and without carpeting. Table 3.1 Description of test zone constructions for RTF generation algorithm validation Fabric Element Lightweight Mediumweight Heavyweight External wall steel siding, 2 in insulation, air space, ¾ in gypsum 4in face brick, 2 in insulation, air space, ¾ in gypsum 4in face brick, air space 2 in. insulation, 8 in HW concrete, ¾ in gypsum Roof/Ceiling 4 in. LW concrete, ceiling air space, acoustic tile 4 in. HW concrete, ceiling air space, acoustic tile 8 in. HW concrete, ceiling air space, acoustic tile Partition ¾ in. gyp, air space, ¾.in. gypsum ¾ in. gyp, air space, 3/4.in. gypsum ¾ in. gyp, 8 in. HW concrete block, 3/4.in. gypsum Floor Acoustic tile, ceiling air space, 4 in. LW concrete Acoustic tile, ceiling air space, 4 in. HW concrete Acoustic tile, ceiling air space, 8 in. HW concrete Furnishing 1 in. wood @ 50% of floor area 1 in. wood @ 50% of floor area 1 in. wood @ 50% of floor area Carpeting Resistance layer of 2.73 ft2 h ºF/Btu Resistance layer of 2.73 ft2 h ºF/Btu Resistance layer of 2.73 ft2 h ºF/Btu Notes: 1. Surface layers are listed in order from the outside of the room to the inside of the room. 2. The test zone is 15ft x 30ft x 9ft high. The test zone has one exterior wall, 30 ft long. 3. The % glazing is fraction of the exterior facade. 4. Long wavelength absorptance of 0.9 were used for all inside surfaces The following two heat gain pulse distribution models were used for the RTF generation: (1) area weighted uniform distribution model for nonsolar RTF, and (2) for the solar RTF generation 50% to the floor and 50% to the thermal mass surfaces. The RTF plots are shown in Figures 3.3 to 3.6. The RTF plots for medium and heavyweight construction zones are shown in Appendix A. Root mean square errors (RMSE) of the 71 RTF were computed for the 24hourly values of the New RTF Engine and the HVAC Load Explorer, a fullblown heat balance program, that come with a book by McQuiston et al (2005). The RMSE for three test zone construction types is given in Table 3.2. Table 3.2 RMSE of the RTF of the New RTF Engine Without Carpet With Carpet Zone Construction Nonsolar Solar Nonsolar Solar Heavyweight 0.00013 0.00022 0.00014 0.00027 Mediumweight 0.00008 0.00012 0.00008 0.00015 Lightweight 0.00027 0.00037 0.00007 0.00017 The RMSE is calculated as follows: ( ) Σ= = − 23 0 2 , , 24 1 j NewRTF Engine j HVAC Load Explorer j RMSE RTF RTF (3.20) The RMSE of the RTF computed using the New RTF engine are within the convergence limits of the program used to generate the reference RTF. The maximum errors are observed for most of the cases in the first three terms of the RTF as is evident from the plots in Figures 3.3 to 3.6 and Figures 1A to 8A shown in the Appendix A. 72 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1 3 5 7 9 11 13 15 17 19 21 23 Hour RTF New RTF Engine HVAC Load Explorer Figure 3.3 Nonsolar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior facade 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1 3 5 7 9 11 13 15 17 19 21 23 Hour RTF New RTF Engine HVAC Load Explorer Figure 3.4 Solar RTF for lightweight construction zone with no carpet for 50% glazing fraction of the exterior façade 73 0.00 0.10 0.20 0.30 0.40 0.50 0.60 1 3 5 7 9 11 13 15 17 19 21 23 Hour RTF New RTF Engine HVAC Load Explorer Figure 3.5 Nonsolar RTF for lightweight construction zone with carpet for 50% glazing fraction of the exterior facade 0.00 0.10 0.20 0.30 0.40 0.50 0.60 1 3 5 7 9 11 13 15 17 19 21 23 Hour RTF New RTF Engine HVAC Load Explorer Figure 3.6 Solar RTF for lightweight construction zone with carpet for 50% glazing fraction of the exterior facade 74 3.1.3 1D Finite Volume Method PRF Generation Although, Spitler et al. (1997) demonstrated the use of nonperiodic response factors to generate PRF, most others implementations have used an existing Laplace or State Space method based CTF generation procedure, then converted the CTF to PRF using the Spitler and Fisher (1999b) procedure. This procedure may be less than ideal for two reasons: 1. Computing CTF then converting to PRF for use in RTSM involves an extra effort. 2. Since the RTSM is intended to be a spreadsheet method, it would be best if the entire procedure could be encapsulated within the spreadsheet. With Microsoft Excel, the VBA programming language allows procedural programming “within” the spreadsheet. Therefore, computation of PRF is possible within the spreadsheet. Laplace and State Space methods for CTF generation might be implemented but significant complexity is a formidable barrier to such an implementation. Consider that the FORTRAN 90 implementations (Iu et al. 2004) of the Laplace and State Space Methods are 1000 and 2000 lines long, respectively. Therefore, a simpler approach is investigated here, using a 1D finite volume method fully implicit scheme. For comparison purposes, the implementation investigated here was written in SCILAB and only takes 150 lines. In FORTRAN 90 it takes about 450 lines. A uniform gridding scheme, in each layer, and zero thickness boundary nodes (Patankar 1991) are used, which allows the imposition of realistic boundary conditions. 75 Moreover, a higher order treatment is used for flux calculation. This method is described fully in Appendix B. Validation of Periodic Response Factor Generation The 1D finite volume method (FVM) periodic response factor generation algorithm has been validated against the Spitler and Fisher (1999a) procedure, which is converting CTFs generated using the State Space method 



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