MODELING PACKAGED HEAT PUMPS IN
A QUASI-STEADY STATE ENERGY
SIMULATION PROGRAM
By
TANG, CHIH CHIEN
Bachelor of Science
Oklahoma State University,
Stillwater, Oklahoma
2003
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May, 2005
ii
MODELING PACKAGED HEAT PUMPS IN
A QUASI-STEADY STATE ENERGY
SIMULATION PROGRAM
Thesis Approved:
Dr. Daniel Fisher
Thesis Adviser
Dr. Jeffrey Spitler
Dr. Ronald Delahoussaye
Dr. A. Gordon Emslie
Dean of the Graduate College
iii
ACKNOWLEDGEMENTS
This page is dedicated to everyone who has somehow laid a print in my life both
directly involved in this research and those who stand by the sidelines cheering me on
towards the end. First of all, thank you God for bringing me to form and giving me
talents that are limited only by my own imagination. Dr. D.E. Fisher, you are the best
advisor a graduate student could have, thank you for your generous support, optimistic
attitude, guidance, and not to forget the Thanksgiving dinners. My committee members,
Dr. J.D. Spitler and Dr. R.D. Delahoussaye for their constructive guidance and expertise.
Thank you Mum and Dad for your sacrifices to put me through college. My
brother and his wife, Clement and Sue, and Yee Shyen for being the cheerleading squad.
Calvin for guidance and valuable inputs; Shawn and Ben for helping with the
experimental instrumentation; Chanvit for programming tips. Also a note of appreciation
to all my colleagues; Xiaobing Liu, Dongyi Xiao, Haider, Muhammad, Wei Xiu, Xiao
Wei, Brian Kastl, Arun Shenoy and Sankar for their friendships and the wonderful
memories.
I would also like to express my gratitude to York International Unitary Product
Group especially to Nathan Webber, Messrs C. Obosu and M. Chitti for providing the
measured data and their expertise on air-to-air heat pumps. Special thanks also to
ClimaterMaster especially to L.N. Nerurkar for providing the experimental data for
water-to-air heat pumps. Last but not least, financial support from the U.S Department of
Energy and advice from the EnergyPlus development team are gratefully acknowledged.
iv
TABLE OF CONTENTS
Chapter Page
1.0 Introduction................................................................................................................ 1
1.1. Background............................................................................................................. 1
1.2. Objective ................................................................................................................. 2
1.3. Scope....................................................................................................................... 3
2.0 Review of Heat Pump Models in the Literature ........................................................ 6
2.1. Steady State Air-to-Air Heat Pump Models ........................................................... 6
2.1.1 EnergyPlus Model............................................................................................. 6
2.1.2 Detailed Deterministic Model by Iu et al........................................................ 13
2.2. Steady State Water-to-Air Heat Pump Models..................................................... 15
2.2.1 Jin & Spitler Model......................................................................................... 15
2.2.2 Lash Model ..................................................................................................... 16
2.3. Heat Pump Cycling Models.................................................................................. 20
2.3.1 Time-Constant Models.................................................................................... 20
2.3.2 Part-Load Fraction Model............................................................................... 22
2.3.3 Part-Load Fraction Model by Katipamula and O’Neal (1992)....................... 25
2.3.4 Henderson and Rengarajan Model.................................................................. 26
3.0 Simulation of Cycling Equipment in a Quasi-Steady State Simulation
Environment............................................................................................................. 28
3.1. Overview of the EnergyPlus Quasi-Steady State Simulation Methodology ........ 28
3.1.1 Successive Substitution with Lagging ............................................................ 29
3.1.2 Ideal Controls.................................................................................................. 29
3.1.3 Variable Time Step ......................................................................................... 31
3.2. Simulation of Unitary Equipment in EnergyPlus ................................................. 31
3.2.1 Zone/Air Loop Interactions............................................................................. 32
3.2.2 Unitary Equipment Simulation Manager........................................................ 35
4.0 Implementation of Heat Pump Models in EnergyPlus ............................................ 38
4.1. Curve-Fit Water to Air Heat Pump Model ........................................................... 39
4.1.1 Modification of Lash (1992) and Shenoy (2004) ........................................... 39
4.1.2 Catalog Data Points......................................................................................... 46
4.1.3 Model Implementation in EnergyPlus ............................................................ 47
4.2. Parameter Estimation Based Water-to-Air Heat Pump Model............................. 50
4.2.1 Model Development........................................................................................ 51
4.2.2 Parameter Estimation Procedure..................................................................... 58
4.2.3 Model Implementation.................................................................................... 64
4.2.4 Accounting for Fan Heat................................................................................. 66
v
4.3. Part-Load Latent Degradation Model ................................................................... 68
4.3.1 Model Development........................................................................................ 69
4.3.2 Modification of Part-Load Latent Degradation Model for Cycling Fan......... 78
4.3.3 Model Implementation.................................................................................... 79
4.3.4 Model Sensitivity Analysis............................................................................. 83
4.4. Curve-Fit Water-Water Heat Pump Model........................................................... 88
4.4.1 Model Development........................................................................................ 88
4.4.2 Model Implementation into EnergyPlus ......................................................... 92
5.0 Validation of the Heat Pump Models....................................................................... 93
5.1. Steady-State Air-to-Air Heat Pump Model Validation......................................... 93
5.1.1 The Experimental Facility............................................................................... 93
5.1.2 Experimental Procedure.................................................................................. 95
5.1.3 Experimental Validation Results .................................................................... 96
5.1.4 Investigation of Compressor Shell Heat Loss............................................... 101
5.1.5 Summary of Air-to-Air Heat Pump Validation ............................................ 104
5.2. Steady-State Water-to-Air Heat Pump Model Validation .................................. 106
5.2.1 Experimental Validation Results for Cooling Mode .................................... 107
5.2.2 Experimental Validation Results for Heating Mode..................................... 115
5.2.3 Model Performance Beyond Catalog Range................................................. 120
5.2.4 Summary of Water-to-Air Heat Pump Validation ........................................ 129
5.3. Preliminary Verification of Curve-Fit Water-to-Water Heat Pump Model........ 131
5.3.1 Curve-Fit Model Verification with Catalog Data ......................................... 131
5.3.2 Comparisons of Curve-Fit Model and Parameter Estimation Based Model. 135
5.3.3 Summary of Water-to-Air Heat Pump Validation ........................................ 140
6.0 Conclusion and Recommendations........................................................................ 142
6.1. Summary of Results............................................................................................ 142
6.2. Future Work........................................................................................................ 143
REFERENCES ............................................................................................................... 146
APPENDIX A: Generating Coefficients for EnergyPlus Curve-Fit Air-to-Air Heat
Pump Model........................................................................................................... 149
APPENDIX B: Generating Coefficients for EnergyPlus Curve-Fit Water-to-Air
Heat Pump Model .................................................................................................. 158
APPENDIX C: Generating Parameters for EnergyPlus Parameter Estimation
Based Water-to-Air Heat Pump Model.................................................................. 163
APPENDIX D: Coefficients and Parameters for Water-to-Water Heat Pump
Models.................................................................................................................... 166
APPENDIX E: Proposal for New Curve-Fit Air-to-Air Heat Pump Model Based
on Lash (1992) Approach ...................................................................................... 167
vi
APPENDIX F: Failure in Generalized Least Square Method (GLSM) for Fixed
Inlet Conditions...................................................................................................... 169
vii
LIST OF TABLES
Table Page
Table 2.1: Recommended Part-Load Fraction Parameters by DOE-2, Henderson
et al. (1999) ..................................................................................................... 23
Table 4.1: Summary of Heat Pump Models in EnergyPlus .............................................. 38
Table 4.2: Comparison of Cooling Catalog Data and Simulation Results for
Curve-Fit Water-to-Air Heat Pump Model by Shenoy(2004): Eq 2.25-
2.27.................................................................................................................. 40
Table 4.3: Comparison of Heating Catalog Data and Simulation Results for
Curve-Fit Water-to-Air Heat Pump Model by Shenoy(2004): Eq 2.28-
2.29.................................................................................................................. 40
Table 4.4: Comparison of Cooling Catalog Data and Simulation Results for
Curve-Fit Water-to-Air Heat Pump Model Version 1: Eq 4.1-4.3 ................. 42
Table 4.5: Comparison of Heating Catalog Data and Simulation Results for
Curve-Fit Water-to-Air Heat Pump Model Version 1: Eq 4.4-4.5 ................. 42
Table 4.6: Comparison of Cooling Catalog Data and Simulation Results for
Curve-Fit Water-to-Air Heat Pump Model Version 2: Eq 4.7-4.10 ............... 45
Table 4.7: Comparison of Heating Catalog Data and Simulation Results for
Curve-Fit Water-to-Air Heat Pump Model Version 2: Eq 4.11-4.13 ............. 45
Table 4.8: Comparison of Fan Mode Operating Mode..................................................... 68
Table 4.9: Base Parameter Values for Model Sensitivity Analysis .................................. 83
Table 5.1: Percentage RMS error for Curve-Fit Model and Detailed Model ................. 100
Table 5.2: Parameter/Coefficient Generator Outputs Compared with Catalog Data
(Cooling)....................................................................................................... 107
Table 5.3: Comparison of Water-to-Air Heat Pump Models using Catalog Data
with Experimental Measurements (Cooling)................................................ 108
Table 5.4: Parameter/Coefficient Generator Outputs Compared with Experimental
Data (Cooling) .............................................................................................. 111
viii
Table 5.5: Comparison of Water-to-Air Heat Pump Models using Experimental
Data with Experimental Measurements (Cooling) ....................................... 112
Table 5.6: Parameter/Coefficient Generator Outputs Compared with Catalog Data
(Heating) ....................................................................................................... 115
Table 5.7: Comparison of Water-to-Air Heat Pump Models using Catalog Data
with Experimental Measurements (Heating) ................................................ 115
Table 5.8: Parameter/Coefficient Generator Outputs Compared with Experimental
Data (Heating)............................................................................................... 117
Table 5.9: Comparison Water-to-Air Heat Pump Models using Experimental Data
with Experimental Measurements (Heating) ................................................ 118
Table 5.10: Catalog Data and Input Data Range for Cooling Mode .............................. 121
Table 5.11: Catalog Data and Input Data Range for Heating Mode............................... 121
Table 5.12: Heat Pump Performance Range in Catalog and Input Data ........................ 122
Table 5.13: Parameter/Coefficient Generator Outputs Compared with Input Data........ 122
Table 5.14: Result Summary of Heat Pump Models Operating Beyond Catalog
Range for Cooling Mode .............................................................................. 125
Table 5.15: Result Summary of Heat Pump Models Operating Beyond Catalog
Range for Heating Mode............................................................................... 127
Table 5.16: Parameter/Coefficient Generator Outputs Compared with Input Data
for 2-ton and 6-ton Heat Pumps.................................................................... 128
Table 5.17: Result Summary of Heat Pump Models Operating Beyond Catalog
Range for 2-ton and 6-ton Heat Pumps......................................................... 128
Table 5.18: Comparison of Cooling Catalog Data and Simulation Results for
Curve-Fit Water-to-Water Heat Pump Model .............................................. 132
Table 5.19: Comparison of Heating Catalog Data and Simulation Results for
Curve-Fit Water-to-Water Heat Pump Model .............................................. 132
Table 5.20: Result Summary of Water-to-Water Heat Pump Models Compared
with Catalog Data ......................................................................................... 139
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LIST OF FIGURES
Figure Page
Figure 3.1: Zone Equipment/Air Primary Loop Interaction ............................................. 33
Figure 4.1: Information Flow Chart for Curve-Fit Water-to-Air Heat Pump Model ....... 48
Figure 4.2: Flow Diagram for Curve-Fit Water-to-Air Heat Pump Model ...................... 49
Figure 4.3: Water-Air Heat Pump Configuration ............................................................. 51
Figure 4.4: Flow Diagram for Estimating the Load Side Exterior Heat Transfer
Coefficient....................................................................................................... 61
Figure 4.5: Flow Diagram for Parameter Estimation Program......................................... 63
Figure 4.6: Information Flow Chart for Parameter Estimation Based Water-to-Air
Heat Pump Model ........................................................................................... 64
Figure 4.7: Flow Diagram for Parameter Estimation Based Water-to-Air Heat
Pump Model, Jin(2002) .................................................................................. 65
Figure 4.8: Concept of Moisture Buildup and Evaporation on Coil................................. 70
Figure 4.9: Linear Decay Evaporation Model .................................................................. 73
Figure 4.10: Information Flow Chart for Latent Degradation Model............................... 81
Figure 4.11: Interaction of the Latent Degradation Model with Water to Air Heat
Pump Cooling Coil Subroutine....................................................................... 82
Figure 4.12: Sensitivity of Part-Load Latent Degradation Model to wet t for
Continuous Fan ............................................................................................... 84
Figure 4.13: Sensitivity of Part-Load Latent Degradation Model to wet t for
Cycling Fan..................................................................................................... 85
Figure 4.14: Sensitivity of Part-Load Latent Degradation Model to fandelay t for
Cycling Fan..................................................................................................... 86
x
Figure 4.15: Sensitivity of Part-Load Latent Degradation Model to γ for
Continuous Fan ............................................................................................... 87
Figure 4.16: Sensitivity of Part-Load Latent Degradation Model to γ for Cycling
Fan................................................................................................................... 87
Figure 4.17: Information Flow Chart for Water-Water Heat Pump Simple ..................... 91
Figure 5.1: Schematic of the Test Loop, Iu et.al (2003) (Used with permission) ............ 94
Figure 5.2: Uncertainty for Measuring Device ................................................................. 95
Figure 5.3: Experimental Test Matrix for Validation of Air-Air Heat Pump
Models............................................................................................................. 96
Figure 5.4: Validation of Curve-Fit Model and Detailed Model for Total Cooling
Capacity (Cooling Mode) ............................................................................... 97
Figure 5.5: Validation of Curve-Fit Model and Detailed Model for Sensible
Cooling Capacity (Cooling Mode).................................................................. 98
Figure 5.6: Validation of Curve-Fit Model and Detailed Model for Compressor
Power (Cooling Mode) ................................................................................... 98
Figure 5.7: Validation of Curve-Fit Model and Detailed Model for Heating
Capacity (Heating Mode)................................................................................ 99
Figure 5.8: Validation of Curve-Fit Model and Detailed Model for Compressor
Power (Heating Mode).................................................................................... 99
Figure 5.9: Percentage of Compressor Shell Heat Loss in Cooling Mode ..................... 103
Figure 5.10: Analysis of Compressor Shell Heat Loss................................................... 104
Figure 5.11: Validation of Curve-Fit Model and Jin(2002) for Total Cooling
Capacity using Catalog Data for Generating Parameters & Coefficients..... 109
Figure 5.12: Validation of Curve-Fit Model and Jin(2002) for Sensible Cooling
Capacity using Catalog Data for Generating Parameters & Coefficients..... 109
Figure 5.13: Validation of Curve-Fit Model and Jin(2002) for Power
Consumption using Catalog Data for Generating Parameters &
Coefficients ................................................................................................... 110
Figure 5.14: Validation of Curve-Fit Model and Jin(2002) for Heat Rejection
using Catalog Data for Generating Parameters & Coefficients .................... 110
xi
Figure 5.15: Validation of Curve-Fit Model and Jin(2002) for Total Cooling
Capacity using Experimental Data for Generating Parameters &
Coefficients ................................................................................................... 113
Figure 5.16: Validation of Curve-Fit Model and Jin(2002) for Sensible Capacity
using Experimental Data for Generating Parameters & Coefficients........... 113
Figure 5.17: Validation of Curve-Fit Model and Jin(2002) for Power
Consumption using Experimental Data for Generating Parameters &
Coefficients ................................................................................................... 114
Figure 5.18: Validation of Curve-Fit Model and Jin(2002) for Heat Rejection
using Experimental Data for Generating Parameters & Coefficients........... 114
Figure 5.19: Validation of Curve-Fit Model and Jin(2002) for Heating Capacity
using Catalog Data for Generating Parameters & Coefficients .................... 116
Figure 5.20: Validation of Curve-Fit Model and Jin(2002) for Power
Consumption using Catalog Data for Generating Parameters &
Coefficients ................................................................................................... 116
Figure 5.21: Validation of Curve-Fit Model and Jin(2002) for Heat Absorption
using Catalog Data for Generating Parameters & Coefficients .................... 117
Figure 5.22: Validation of Curve-Fit Model and Jin(2002) for Heating Capacity
using Experimental Data for Generating Parameters & Coefficients........... 119
Figure 5.23: Validation of Curve-Fit Model and Jin(2002) for Power
Consumption using Experimental Data for Generating Parameters &
Coefficients ................................................................................................... 119
Figure 5.24: Validation of Curve-Fit Model and Jin(2002) for Heat Absorption
using Experimental Data for Generating Parameters & Coefficients........... 120
Figure 5.25: Performance of Water-to-Air Heat Pump Models Beyond Catalog
Range for Total Cooling Capacity ................................................................ 123
Figure 5.26: Performance of Water-to-Air Heat Pump Models Beyond Catalog
Range for Sensible Cooling Capacity ........................................................... 123
Figure 5.27: Performance of Water-to-Air Heat Pump Models Beyond Catalog
Range for Heat Rejection.............................................................................. 124
Figure 5.28: Performance of Water-to-Air Heat Pump Models Beyond Catalog
Range for Cooling Power Consumption ....................................................... 124
Figure 5.29: Performance of Water-to-Air Heat Pump Models Beyond Catalog
Range for Heating Capacity.......................................................................... 126
xii
Figure 5.30: Performance of Water-to-Air Heat Pump Models Beyond Catalog
Range for Heat Absorption ........................................................................... 126
Figure 5.31: Performance of Water-to-Air Heat Pump Models Beyond Catalog
Range for Heating Power Consumption ....................................................... 127
Figure 5.32: Comparison of Cooling Load Side Heat Transfer Rate for Simulation
Results with Catalog Data............................................................................. 133
Figure 5.33: Comparison of Cooling Source Side Heat Transfer Rate for
Simulation Results with Catalog Data .......................................................... 133
Figure 5.34: Comparison of Cooling Power Input for Simulation Results with
Catalog Data.................................................................................................. 134
Figure 5.35: Comparison of Heating Load Side Heat Transfer Rate for Simulation
Results with Catalog Data............................................................................. 134
Figure 5.36: Comparison of Heating Source Side Heat Transfer Rate for
Simulation Results with Catalog Data .......................................................... 135
Figure 5.37: Comparison of Heating Power Input for Simulation Results with
Catalog Data.................................................................................................. 135
Figure 5.38: Performance of Water-to-Water Heat Pump Models in Simulating
Load Side Heat Transfer Rate (Cooling) ...................................................... 136
Figure 5.39: Performance of Water-to-Water Heat Pump Models in Simulating
Source Side Heat Transfer Rate (Cooling) ................................................... 137
Figure 5.40: Performance of Water-to-Water Heat Pump Models in Simulating
Power Consumption (Cooling) ..................................................................... 137
Figure 5.41: Performance of Water-to-Water Heat Pump Models in Simulating
Load Side Heat Transfer Rate (Heating) ...................................................... 138
Figure 5.42: Performance of Water-to-Water Heat Pump Models in Simulating
Source Side Heat Transfer Rate (Heating).................................................... 138
Figure 5.43: Performance of Water-to-Water Heat Pump Models in Simulating
Power Consumption (Heating) ..................................................................... 139
xiii
NOMENCLATURE
Symbols
BF = bypass factor
p C = specific heat, J/(kg-K)
COP = coefficient of performance
dp = dew point temperature, °C
EIR = energy input ratio
FMF = flow modifying factor curve
h = enthalpy, J/kg
co o h A = load side external surface heat transfer coefficient, W/K
LHR = latent heat ratio
air m = air mass flow rate, kg/s
w m = water mass flow rate, kg/s
r m = refrigerant mass flow rate, kg/s
Mo = moisture holding capacity of the coil, kg
max N = heat pump cycling rate, cycles/hr
NTU = number of transfer units
c P = condensing pressure, Pa
e P = evaporating pressure, Pa
dis P = discharge pressure, Pa
suc P = suction pressure, Pa
Power = power consumption, W
PLR = part-load ratio
PLF = part-load fraction
xiv
SHR = sensible heat ratio
fandelay t = fan delay time, s
on t = duration of time the compressor is on, s
off t = duration of time the compressor is off, s
wet t = the ratio of the moisture holding capacity of the coil to the steady
state latent capacity of the heat pump
0 t = time for condensate removal to begin, s
db T = dry-bulb temperature, °C, °F or K
wb T = wet-bulb temperature, °C, °F or K
TMF = temperature modifying factor curve
UA = heat transfer coefficient, W/K
comp W = compressor work, W
e Q = initial evaporation rate when the compressor off, W
h Q = total heating capacity, W
lat Q = latent capacity, W
sens Q = sensible cooling capacity, W
source Q = source side heat transfer rate, W
total Q = total cooling capacity, W
X = runtime fraction
loss W = compressor power losses due to mechanical and electrical
losses, W
τ = heat pump time constant, s
γ = the ratio of the initial evaporation rate and steady-state
latent capacity
ε = heat transfer effectiveness
w = humidity ratio, kg/kg
xv
η = efficiency
1
1.0 Introduction
The rise in oil and energy prices has prompted the Department of Energy to
increase research funding for renewable energy and increase the efficiency of unitary
equipment systems. EnergyPlus, an hourly building simulation program funded by DOE,
is one such endeavor that allows building and system designers to design better building
envelops and unitary systems that are energy efficient and low in first cost.
1.1. Background
In hourly energy simulations, it is essential to accurately predict the performance
of heat pumps over the range of full and part-load operating conditions. A number of heat
pump models have been proposed by researchers over the years ranging from detailed
deterministic models to simple curve-fit models. Detailed deterministic models are based
on thermodynamic laws and heat transfer relations applied to individual components. The
models generally require a lot of parameters or input data and require longer simulation
times. On the other hand, simple curve-fit model treats the heat pump as a black box and
the system performance is predicted using equations generated from the heat pump
performance curve provided by the manufacturer’s catalog.
However, the suitability of these models for incorporation into EnergyPlus has to
be evaluated based on simulation run time, availability of data or required parameters,
accuracy and stability of the models and ease of use. In short, the heat pump model
2
should be relatively easy to use, reasonably accurate and have a short simulation run
time.
1.2. Objective
This research is focused on building upon previous heat pump models that have
been developed by previous researchers in the form of validation, improvement, and
implementation into EnergyPlus. From this research, the selection and implementation of
heat pump models in EnergyPlus will be justified on the basis of models’ accuracy, ease
of use, and simulation run time.
Unitary heat pump models are discussed in Chapter 2 together with related
models developed by researchers. The heat pump models implemented in EnergyPlus are
steady state models. Since a properly size heat pump operates mostly at part-load
conditions, the outputs for full load conditions need to be adjusted for part-load
operation. Several methods developed by researchers, ranging from time-constant models
to part-load fraction models are evaluated based on their adaptability to the EnergyPlus
simulation environment. In addition, a part-load latent heat model transfer by Henderson
and Rengarajan(1996) was incorporated in the water-air heat pump model to allow better
prediction of the latent capacity at part-load condition.
The simulation environment for cycling unitary equipments in EnergyPlus is
discussed in Chapter 3. The heat pump models modifications and implementation in
EnergyPlus are described in Chapter 4. A curve-fit water-to-water heat pump model is
developed based on the same approach used by Lash(1992) for the curve-fit water-to-air
heat pump model.
3
The performance of the curve-fit air-to-air heat pump model in EnergyPlus is
compared to a detailed deterministic model proposed by Iu et. al (2003) using
experimental data obtained from the OSU test rig and the manufacturer. On the other
hand, two water-to-air heat pump models were implemented in Energyplus: a parameter
estimation based model by Jin (2002) and a curve-fit model based on Lash (1992). The
two models are compared to experimental results obtained from the manufacturer in
Chapter 5.2. In addition, the newly proposed curve-fit water-to-water heat pump is
verified by comparison with the parameter estimation based water-to-water heat pump
developed by Jin (2002)
At the end of this research project, EnergyPlus users will not only have a selection
of heat pump models best suited to their needs but also have full confidence in the
simulation results. Lastly, recommendations for future work and further validation of the
models is proposed.
1.3. Scope
The scope of the research work is summarized and categorized based on the type
of heat pump model. For air-to-air heat pump model, the main objective is to investigate
the performance of EnergyPlus curve-fit model and the following tasks have been
completed:
• Ran the OSU test facility and collected validation data for cooling mode.
• Conducted a preliminary study on the compressor shell heat loss.
4
• Modified the EnergyPlus curve-fit model to account for the effects of indoor dry-bulb
temperature in heating mode.
• Validated the EnergyPlus curve-fit model with experimental data together with a
detailed deterministic model by Iu et. al.(2003).
• Proposed a new curve-fit air-to-air heat pump model based on Lash (1992)
approach.
For water-to-air heat pump model, the goal is to continue the work by previous
researchers and implement the models into EnergyPlus simulation environment. The
completed tasks are as follows:
• Implemented the parameter estimation based model by Jin (2002) into
EnergyPlus.
• Modified Shenoy (2004) curve-fit model and finalized the implementation into
EnergyPlus.
• Developed an Excel spreadsheet for generating parameters/coefficients for the
curve-fit model and the parameter estimation based model.
• Modified the latent degradation model by Henderson and Rengarajan (1996) to
include cycling fan operation mode. Conducted a parametric study of the model.
• Implemented the part-load fraction model and the latent degradation model for
water-to-air heat pump model.
• Validated the curve-fit model and the parameter estimation based model using
experimental measurements obtained from the manufacturer.
• Investigated the performance of both models beyond the catalog range.
5
For water-to-water heat pump model, the main objective is to develop and
implement a new curve-fit model into EnergyPlus to accompany the parameter estimation
based model which was implemented by Murugappan (2002). The completed works are
as follows:
• Proposed a new curve-fit water-to-water heat pump model based on Lash (1992)
approach and implemented the model into EnergyPlus.
• Developed an Excel spreadsheet for generating parameters/coefficients for the
curve-fit model and the parameter estimation based model.
• Conducted a preliminary verification of the curve-fit model and compared its
performance with the parameter estimation based model by Jin (2002).
6
2.0 Review of Heat Pump Models in the Literature
A number of heat pump models have been proposed by researchers over the years.
These generally fall into two extremes: detailed deterministic models and simple curve-fit
models. Detailed deterministic models are generally complicated models requiring
numerous generally unavailable inputs. This makes them unfavorable to building
simulation programs like EnergyPlus and DOE-2. In addition, it is generally accepted that
simple curve-fit models tend to fail when operating beyond the catalog data. In recent
years, parameter estimation based models have been developed by Oklahoma State
University. These compares fairly well with detailed deterministic models while retaining
the strength of the curve-fit model with easily accessible inputs.
2.1. Steady State Air-to-Air Heat Pump Models
2.1.1 EnergyPlus Model
The air-air heat pump model in EnergyPlus uses empirical functions for capacity
and efficiency from DOE-2 (DOE 1982) in conjunction with the apparatus dew point
(ADP)/bypass factor (BF) relations to determine the off-design performance (Henderson
et. al 1992). The approach is analogous to the NTU-effectiveness calculations based on
the sensible-only heat exchanger calculations extended to a cooling and dehumidifying
coil.
7
The heat pump performance at off-design conditions is computed by adjusting the
capacity and energy input ratio (inverse of COP) at rated conditions to the temperature
modifying factor, TMF and flow fraction modifying factor, FMF. The TMF and FMF are
non-dimensional factors or performance curves obtained from the heat pump catalog
data. The TMF curves adjust the heat pump performance due to variation in air
temperatures from the rated conditions. On the other hand, the FMF curves adjust for the
performance effects of variation in air flow rate from the rated conditions.
For cooling mode, the rated condition (80°F [26.7°C] indoor dry bulb and 67°F
[19.4°C] wet bulb; 95°F [35.0°C] outdoor dry bulb; 350~450 cfm/ton [0.047~0.06 m3/s
kW]) is essentially the ARI “A” Cooling Steady State Condition which is the standard
rating conditions for air-source heat pumps. The TMF curves for total cooling
capacity, ( , ) C f iwb odb and energy input ratio, ( , ) EIR f iwb odb are functions of the indoor
wet-bulb temperature and outdoor dry-bulb temperature. Both of them are formulated in
similar fashion as shown below.
( ) ( ) ( )2 ( ) ( )2 ( )( )
1 2 3 4 5 6
,
, C
C
C rated
f iwb odb C a a iwb a iwb a odb a odb a iwb odb
C
= = + + + + + (2.1)
( ) ( ) ( )2 ( ) ( )2 ( )( )
1 2 3 4 5 6
,
, C
EIR
C rated
f iwb odb EIR b b iwb b iwb b odb b odb b iwb odb
EIR
= = + + + + + (2.2)
where
iwb = indoor wet-bulb temperature, °C
odb = outdoor dry-bulb temperature,°C
C EIR = cooling energy input ratio
C,rated EIR = rated cooling energy input ratio
8
C C = total cooling capacity, W
C,rated C = rated total cooling capacity, W
To generate the TMF curves, data points at the rated air flow rate but at indoor air
wet-bulb and outdoor air dry-bulb temperatures that vary from the rated conditions are
selected from the manufacturer’s catalog. The ratio of the respective total capacity to
rated total capacity is calculated. Then, the coefficients for ( , ) C f iwb odb are calculated
using the generalized least square method. The same approach is also used for calculating
the coefficients for ( , ) EIR f iwb odb .
The FMF curves are functions of the ratio of air flow rate to the rated air flow
rate. The equations below show the FMF curves for the cooling capacity, ( / ) C rated f Q Q
and energy input ratio, ( / ) EIR rated f Q Q :
( ) ( ) ( )2
1 2 3
,
/ C / /
C rated rated rated
C rated
f Q Q C c c Q Q c Q Q
C
= = + + (2.3)
( ) ( ) ( )2
1 2 3
,
/ C / /
EIR rated rated rated
C rated
f Q Q EIR d d Q Q d Q Q
EIR
= = + + (2.4)
where:
Q = indoor air volumetric flow rate, m3/s
rated Q = rated indoor air volumetric flow rate, m3/s
The FMF curves for the cooling capacity, ( / ) C rated f Q Q and energy input ratio,
( / ) EIR rated f Q Q can be obtained by plotting capacity ratios with their respective flow
fractions in Excel. The data points selected from the catalog must be at rated indoor wet-bulb
and outdoor dry-bulb temperatures but cover a range of indoor air flow rates. Using
9
the TMFs and FMFs, the cooling capacity and energy input ratio at rated conditions are
adjusted for the off-rated conditions as follows:
( ) ( ) , , / C Crated C = C ⋅ fC iwb odb ⋅ fC Q Qrated (2.5)
( ) ( ) , , / C Crated EIR EIR rated EIR = EIR ⋅ f iwb odb ⋅ f Q Q (2.6)
In order to accurately predict the humidity level, the heat pump model must
properly predict the split between sensible and latent capacity over a range of operating
conditions. The sensible and latent fractions of the total capacity are determined by the
apparatus dew point/bypass factor (ADP/BF) approach (Carrier et al. 1959). The
approach is analogous to the NTU-effectiveness calculations used for sensible-only heat
exchanger calculations extended to a cooling and dehumidifying coil. The rated total
capacity and rated sensible heat transfer rate is used to determine the ratio of the change
in the air humidity ratio to the change in the air-dry bulb temperature, known as
SlopeRated.
, ,
in out
db in db out rated
SlopeRated w w
T T
−
= −
(2.7)
where:
in w = humidity ratio of air entering the cooling coil at rated conditions, kg/kg
out w = humidity ratio of air exiting the cooling coil at rated conditions, kg/kg
db,in T = dry-bulb temperature of air entering the cooling coil at rated conditions, kg/kg
db,out T = dry-bulb temperature of air exiting the cooling coil at rated conditions, kg/kg
10
The apparatus dew point is the point on the saturation curve where the slope of the line
between the point on the saturation curve and the inlet air conditions matches the
SlopeRated. Once the apparatus dew point is found, the coil bypass factor at rated
conditions rated BF is calculated using the equation below:
,
,
out rated ADP
rated
in rated ADP
h h
BF
h h
−
=
−
(2.8)
where:
out,rated h = enthalpy of air leaving the cooling coil at rated conditions, J/kg
in,rated h = enthalpy of air entering the cooling coil at rated conditions, J/kg
ADP h = enthalpy of saturated air at the coil apparatus dew point, J/kg
For an air-to-refrigerant heat exchanger, the BF can be defined in terms of the number of
transfer unit (NTU) as follows:
BF =1− e−NTU (2.9)
where:
BF = bypass factor
NTU = number of transfer units
Equation (2.9) can be further extended and formulated in terms of the constant, o a , and
the indoor air flow rate, Q, as follows:
1 1
o
air p
UA a
BF e m C e Q
− −
= − = − (2.10)
where:
UA = heat transfer coefficient, W/K
air m = air mass flow rate, kg/s
11
p C = air specific heat, J/(kg-K)
Q = indoor air volumetric flow rate, m3/s
For a given coil geometry, the bypass factor is only a function of mass flow rate
and the constant, o a , which is determined from the rated conditions by rearranging
Equation (2.10) to:
log ( ) o e rated rated a = − BF ⋅Q (2.11)
where:
rated BF = rated bypass factor
rated Q = rated indoor air volumetric flow rate, m3/s
Then the temperature and humidity of the air leaving the cooling coil are calculated with
the ADP and BF approach shown below:
(1 ) exit inlet ADP T = BF ⋅T + − BF ⋅T (2.12)
(1 ) exit inlet ADP w = BF ⋅w + − BF ⋅w (2.13)
where:
w = absolute humidity, kg/kg
T = dry-bulb temperature, °C
inlet = evaporator inlet
exit = evaporator outlet
ADP = average saturated conditions at evaporator surface
For heating mode, the TMF curves for heating capacity and heating energy input
ratio are functions of the indoor dry-bulb temperature and outdoor dry-bulb temperature.
12
The FMF curves are only functions of the ratio of the air flow rate to the rated flow rate.
The rated conditions (70°F [21.1°C] indoor dry bulb and 60°F [15.5°C] indoor wet bulb;
47°F [8.33°C] outdoor dry bulb and 43°F [6.11°C] outdoor dry bulb; 350~450 cfm/ton
[0.047~0.06 m3/s kW]) are the Standard Rating Conditions specified by ARI (2003).
Both modifying factors and calculations for the heating capacity and heating energy input
ratio are shown below.
( ) ( ) , , / H H rated C = C ⋅ fH idb odb ⋅ fH Q Qrated (2.14)
( ) ( ) , , / H H rated EIR EIR rated EIR = EIR ⋅ f idb odb ⋅ f Q Q (2.15)
idb = indoor dry-bulb temperature,°C
odb = outdoor dry-bulb temperature,°C
Q = indoor air volumetric flow rate, m3/s
H C = heat pump total heating capacity, W
H EIR = heating energy input ratio
( ) ( ) ( )2 ( ) ( )2 ( )( )
1 2 3 4 5 6
,
, H
H
H rated
f iwb odb C e e idb e idb e odb e odb e idb odb
C
= = + + + + + (2.16)
( ) ( ) ( )2 ( ) ( )2 ( )( )
1 2 3 4 5 6
,
, H f f f f f f
EIR
H rated
f iwb odb EIR idb idb odb odb idb odb
EIR
= = + + + + + (2.17)
( ) ( ) ( )2
1 2 3
,
/ H / /
H rated rated rated
H rated
f Q Q C g g Q Q g Q Q
C
= = + + (2.18)
13
( ) ( ) ( )2
1 2 3
,
/ H / /
EIR rated rated rated
H rated
f Q Q EIR h h Q Q h Q Q
EIR
= = + + (2.19)
EnergyPlus curve-fit air-to-air heat pump model requires 8 distinct curves
to simulate the heat pump performance in both cooling and heating mode. The data points
selected from the catalog data must meet the requirement of the respective curve which
can be tedious. The main advantage however, is that this model requires very few data
points. This could be a disadvantage as well since few data points meet the curve’s
requirement especially for the FMF curves which could lead to insensitivity of the model.
2.1.2 Detailed Deterministic Model by Iu et al.
The heat pump model proposed by Iu et al (2003) is a steady state multi-component
based model that simulates the heat pump system as four main components,
namely compressor, expansion device, and two heat exchangers (condenser and
evaporator), as well as the distributor, and interconnecting lines. The model is capable of
predicting the capacity and pressure drop effects of different circuit designs. The heat
pump model uses different types of models ranging from semi-empirical to curve-fit
equations to predict the heat transfer processes in each component
The compressor model uses two 10-coefficient polynomial equations from
ARI (1999) to predict the refrigerant mass flow rate and the power consumption. The
polynomial equations are generated from the compressor manufacturer’s catalog. A semi-empirical
equation from Arron and Domanski (1990) is used to model the short tube
orifice, and the distributor pressure drop is obtained from the manufacturer’s catalog data.
The heat exchangers (condenser and evaporator) and interconnecting lines (i.e. discharge,
14
suction and liquid lines) are modeled using 1st principles approach, which means
thermodynamic laws and heat transfer relations are used to predict the performance of the
coil.
In short, the heat pump model is a detailed deterministic model that
requires numerous physical inputs that are not available in the catalog data provided by
the heat pump manufacturer. The model was primarily developed for advanced heat
pump design and simulation. However, the model is not suitable for hourly energy
simulation program due to long computational time and generally unavailable inputs. The
model however serves as a benchmark for the verification of the EnergyPlus curve-fit air-to-
air heat pump model.
15
2.2. Steady State Water-to-Air Heat Pump Models
2.2.1 Jin & Spitler Model
The water-air heat pump model developed by Jin(2002) is a parameter estimation
based model which includes several unspecified parameters that are estimated from the
heat pump catalog using a multi-variable optimization procedure. No additional data is
required besides the heat pump catalog which makes this model attractive for building
system designer and simulation users. In addition, the model retains the physically based
representation of the heat pump compared to equation-fit models. Jin(2002) claims that
this allows extrapolation beyond the catalog data without catastrophic failure compared
to equation-fit models.
Besides that, Jin & Spitler (2002) claims that the parameter based model for
water-water heat pump has relatively the same RMS error as detailed deterministic
models and performs better than equation-fit models. However, they didn’t make similar
comparisons for their water-air heat pump model. The assumptions made in developing
the model are as follows:
• The expansion process is isenthalpic.
• Expansion and compression in the compressor are isentropic processes with
equal and constant isentropic exponents.
• The isentropic is exponent dependent on the refrigerant type, and the value is
obtained from Bourdouxehe et al. (1994).
• No heat loss from the system (e.g. no heat loss from the compressor).
16
• Oil has negligible effects on the refrigerant properties and compressor
operation
• The pressure drop at the discharge and suction valves are equal, constant and
isenthalpic.
The heat pump model consists of four major components: compressor, evaporator,
condenser and expansion device. Other components are neglected due to their
comparatively small influence on the thermodynamic cycle or performance. The type and
number of parameters depends on the operating mode, compressor type and source side
fluid type. The parameters are estimated using a Nelder Mead Simplex routine converted
to VBA(Visual Basic for Applications) from (Kuester and Mize 1973). A detailed
description of the model is presented in Chapter 4.
2.2.2 Lash Model
Lash (1992) proposed an equation-fit model that uses five non-dimensional
equations to predict the heat pump performance in cooling and heating mode. The heat
pump performance is based on entering air temperatures, entering water temperatures and
the inlet mass flow rate of water. The coefficients for the non-dimensional equations are
obtained from the manufacturer’s data using the generalized least squares method. The
equations used to predict the heat pump performance in cooling and heating mode are
shown below:
17
Cooling Mode:
,
, ,
total 1 2 w in 3 ref w
total ref ref wb w ref
Q T T m A A A
Q T T m
= + +
(2.20)
,
, , ,
sens 1 2 w in 3 ref w 4 ref w
sens ref ref wb w ref db w ref
Q T T m T m B B B B
Q T T m T m
= + + +
(2.21)
,
, ,
c 1 2 w in 3 ref w
c ref ref wb w ref
COP T T m C C C
COP T T m
= + +
(2.22)
Heating Mode:
,
, ,
h 1 2 w in 3 ref w
h ref ref db w ref
Q T T m D D D
Q T T m
= + +
(2.23)
,
, ,
h 1 2 w in 3 ref w
h ref ref db w ref
COP T T m E E E
COP T T m
= + +
(2.24)
Where:
A1- E3 = Equation fit coefficients for the cooling and heating mode
ref T = 283K
w,in T = Entering water temperature, K
w m = Mass flow rate of water through the heat pump
w,ref m = Base mass flow rate of water through the heat pump
db T = Entering air dry bulb temperature, K
wb T = Entering air wet bulb temperature, K
total Q = Total cooling capacity, W
total,ref Q = Reference total cooling capacity, W
18
sens Q = Sensible cooling capacity, W
sens,ref Q = Reference sensible cooling capacity, W
h Q = Total heating capacity, W
h,ref Q = Reference total heating capacity, W
h COP = Heating coefficient of performance
h,ref COP = Reference heating coefficient of performance
c COP = Cooling coefficient of performance
c,ref COP = Reference cooling coefficient of performance
The Lash (1992) equation fit model did not account for the effects of variable air
flow rate on the capacities. Thus the model is insensitive to the variation of air flow rates
from the base condition. However, Shenoy(2004) proposed the following equations to
incorporate the effect of the air mass flow rate in the heat pump performance.
Cooling Mode:
, ,
, ,
total 1 2 w in air ref 3 ref w
total ref ref air wb w ref
Q T m T m A A A
Q T m T m
= + +
(2.25)
, ,
, , ,
sens 1 2 w in air ref 3 ref w 4 ref w
sens ref ref air wb w ref db w ref
Q T m T m T m B B B B
Q T m T m T m
= + + +
(2.26)
, ,
, ,
c 1 2 w in air ref 3 ref w
c ref ref air wb w ref
COP T m T m C C C
COP T m T m
= + +
(2.27)
Heating Mode:
19
, ,
, ,
h 1 2 w in air ref 3 ref w
h ref ref air db w ref
Q T m T m D D D
Q T m T m
= + +
(2.28)
, ,
, ,
h 1 2 w in air ref 3 ref w
h ref ref air db w ref
COP T m T m E E E
COP T m T m
= + +
(2.29)
20
2.3. Heat Pump Cycling Models
On-off cycling is a major contributor to the degradation of performance in heat
pumps. The U.S. Department of Energy (1979) proposed a test procedure for estimating
the seasonal energy efficiency ratio (SEER) for air conditioners and heat pumps operating
under cyclic conditions. Manufacturers are required by law to label the heat pump with
SEER for heating and cooling mode. Researchers have acknowledged that on-off cycling
of the heat pump to meet the cooling load has a detrimental effect on the heat pump
performance. Work has been done to analyze the effect of percent on-time, cycling rate
and thermostat control on transient sensible and latent load. Cycling models were
evaluated for incorporation into the EnergyPlus steady-state heat pump models in order to
predict the heat pump performance at part-load.
2.3.1 Time-Constant Models
Basically cycling models can be categorized as time-constant models and
detailed models. Time-constant models are extensions of steady state models created by
modeling the system performance using empirical functions. During startup, the system
capacity or temperature across the indoor coil could be modeled as a first-order system
(single time constant) which was done by Groff and Bullok (1979). The heat pump
response at start up can be modeled as first-order whereby the instantaneous cycling
capacity at time, t is as follows:
1
t
cyc ss Q Q eτ
−
= −
(2.30)
where:
21
ss Q
= steady-state or full-load capacity, W
cyc Q
= instantaneous cycling capacity at respective time, W
t = time after compressor is turned on, s
τ = heat pump time constant, s
O’Neal and Katipamula (1991) also claimed that the single-time constant model is
adequate for simulating general cyclic performance of the heat pump even though it
ignores the actual physical phenomena that occurs. Mulroy and Didion (1985) proposed
a two-time constant model that allows estimation of the impact of refrigerant migration
and thermal mass on cyclic performance.
1 1 1 2
t t
cyc ss Q Q e τ Ae τ
− −
= − +
(2.31)
where
τ 1 = first time constant, s
τ 2 = second time constant, s
A = constant parameter
The first constant is to capture the capacity delay due to the mass of the heat
exchanger. And the second constant is for the time delay to pump the refrigerant from the
evaporator to the system. In addition, the two-time-constant model requires regressive
curve fitting of the constants and is not explicitly derived from heat pump capacity like
the single-time constant model. The two-time-constant is better at characterizing
individual unit’s dynamic performance.
22
2.3.2 Part-Load Fraction Model
At part-load conditions, steady state models require some sort of correlation to
estimate the performance of the model. The part-load fraction (PLF) correlation takes
into account the efficiency losses due to compressor cycling. Part-load ratio (PLR) is the
ratio of the average heat pump’s capacity at part-load to the full-load capacity as follows:
PLR Part Load Capacity
Steady State Capacity
−
=
−
(2.32)
In EnergyPlus, the average heat pump part-load capacity is essentially the demand load of
the heat pump for the respective time step. The steady-state capacity is the output of the
heat pump model at full load. Part-load fraction (PLF) is defined as the ratio of the part-load
energy efficiency ratio to the steady-state energy efficiency ratio as follows:
PLF Part Load EER
Steady State EER
−
=
−
(2.33)
The energy efficiency ratio is the ratio of the heat pump capacity to the heat pump
power consumption. Henderson and Rengarajan (1996) give a detailed theoretical
derivation of the PLF equation which is shown below:
max
1
4 (1 / )
max 1 4 (1 / ) 1 N PLR PLFold
new old PLF τ N PLR PLF e τ
−
−
= − − −
(2.34)
The part-load fraction model requires two parameters: heat pump cycling rate
(Nmax) and heat pump time constant (τ ) to calculate for heat pump PLF from the PLR.
Given PLR from the heat pump simulation and the two parameters, new PLF is calculated
using an initial guess of 1 old PLF = . For the second iteration the calculated new PLF is used
23
as old PLF and the iteration continues until convergence is achieved. Boone et. al. (1980)
noted that with this model the PLF is overestimated at low loads for the following
reasons:
• Power input due to crankcase heater, controls, fans in the off-period is large
compared to the heating/cooling capacity.
• Heating/cooling capacity increases slowly when the compressor cycles on
especially at low PLR, whereas the power input reaches steady state almost
immediately.
Boone et. al. (1980) adjust the heat pump efficiency for the off-cycle power consumption
using the fraction of on-cycle power use (pr) shown below:
Adjusted PLF '
1
PLF PLR
PLR PLR pr
PLF PLF
= =
+ −
(2.35)
The parameters required to calculate the adjusted PLF are based on field tests and
experiments. Henderson et al. (1999) categorize the parameters with respect to the heat
pump condition and recommends some values for use in DOE-2.
Condition Cycling Rate,
Nmax Time Constant, Fraction of on-cycle
power use,pr
Poor Heat Pump 3 60 0.03
Typical Heat Pump 2.5 60 0.01
Good Heat Pump 3 30 0.01
τ
Table 2.1: Recommended Part-Load Fraction Parameters by DOE-2, Henderson et al.
(1999)
Instead of iterating for PLF from PLR as described in Equation (2.34), the part-load
fraction model is incorporated in DOE-2 and EnergyPlus as a curve-fit equation. Using
the parameters recommended in Table 2.1, the adjusted PLF is calculated beforehand as a
24
function of the PLR using Equation (2.34) and Equation (2.35). The relationships
between the adjusted PLF and PLR for the three heat pump conditions are then curve-fitted
and represented by the following equations:
( ) ( )2 ( )3 PLF ' = a + b PLR + c PLR + d PLR (2.36)
Only the coefficients: a,b,c and d are provided to the user based on the heat pump
conditions. Although this method is less computationally expensive, the user is required
to generate the curve if the conditions/parameters are different from what is listed in
Table 2.1.
After calculating the adjusted PLF from the PLR, the run-time fraction, X can be
calculated from the equation below:
'
on
cycle
X PLR t
PLF t
= = (2.37)
where:
X = compressor runtime fraction
on t = compressor on-time (duration of on-time), s
cycle t = compressor cycle-time (duration of one cycle: on-time and off-time), s
Mathematical derivation of the run-time fraction is shown by Henderson and Rengarajan
(1996). The run-time fraction is essentially the percent on-time which is the ratio of the
on-time to the cycle time.
The part-load ratio and runtime fraction are used to adjust the steady state heat
pump model’s capacity and power consumption for part-load conditions. Unlike the heat
pump capacity, the power consumption of the heat pump reaches steady-state almost
instantaneously after the heat pump is turned on. Thus the heat pump power input is
25
adjusted based on the run-time fraction and the heat pump capacity is adjusted based on
the part-load ratio as follows:
part load ss Q Q PLR − = × (2.38)
part load ss Power Power X − = × (2.39)
2.3.3 Part-Load Fraction Model by Katipamula and O’Neal (1992)
Katipamula and O’Neal (1992) conducted a series of tests by varying variables
that influence the PLF of the heat pump in cooling mode. The tests were conducted
according to the standard heat pump test procedure, and the functional relationship of
each independent variable (fraction on-time, indoor dew-point, cycling rate, indoor dry-bulb
temperatures, and outdoor dry-bulb temperature) with the dependent variable (PLF)
was evaluated. From the study, they found that the indoor dry-bulb temperature and
outdoor dry-bulb temperature had no influence on the PLF thus they are omitted. The
final expression of their model is shown below:
PLF =α0 +α1 (1− exp(−k /τ )) +α2dp +α3N max (2.40)
where
k = heat pump on-time, s
τ = heat pump time constant, s
dp = dew point temperature, °C
max N = heat pump cycling rate, cycles/hr
26
Katipamula and O’Neal (1992) concluded that the fraction on-time affected the
PLF the most while the dew point temperature and cycling rates are not very significant.
The model requires several selected tests at various off-design conditions in order to
determine the functional relationship of the PLF with the fraction on-time, dew point
temperature and cycling rate.
2.3.4 Henderson and Rengarajan Model
The part-load fraction models discussed in Section 2.3.2 and Section 2.3.3 do not
account for re-evaporation of moisture from the cooling coil back into the air stream
when the compressor is shut off. Henderson and Rengarajan (1996) proposed a model
based on the single-time constant model to predict latent capacity at part-load condition
with constant fan operation. The model can be applied both to air-air heat pumps and
water-air heat pumps but the model requires measured field data which are not available
from the catalog or the manufacturer.
The model predicts the latent heat ratio (LHR) at part-load conditions as a
function of the run-time fraction ( X ). The model requires parameters at rated conditions
such as the maximum cycling rate of the thermostat ( max N ), the heat pump time constant
(τ ), the ratio of the initial evaporation rate to the steady-state latent capacity (γ ), and the
ratio of the moisture holding capacity of the coil to the steady state latent capacity of the
heat pump ( wet t ). The assumptions made in developing the model are as follows;
27
• The model assumes that the cooling coil can only hold an amount of water,
(Mo). Additional condensate will drains from the coil once the maximum
amount, Mo, has been reached.
• Condensate removal begins once (Mo) is reached. The hysteresis effect due to
previous wetting, surface tension, and a dirty coil is negligible.
• The time constant for the total, sensible and latent capacity at start-up is the
same.
The assumptions enable the model to be used for calculating the net latent
capacity at quasi-steady cyclic conditions. Since the process of moisture evaporation
from the deactivated coil is complex, they proposed three simplified evaporation models
which are referred to as exponential decay, linear decay and constant evaporation. The
advantage of this model is that it can be applied to any kind of system simulation model
as long as the parameters, run-time fraction ( X ) and the latent heat ratio at steady-state
conditions are available. The latent degradation model is described in more detail in
Chapter 4.3.
28
3.0 Simulation of Cycling Equipment in a Quasi-Steady State
Simulation Environment
This chapter describes the EnergyPlus simulation environment and the
methodology employed in solving the simulation state variables at the inlet and outlet of
each component. The zone/air loop interaction is investigated mainly focusing on the
unitary equipment simulation manager.
3.1. Overview of the EnergyPlus Quasi-Steady State Simulation
Methodology
EnergyPlus is an integrated simulation environment whereby the major parts,
zone, system and plant are solved simultaneously based on fundamental heat balance
principles. In EnergyPlus (2004), interactions between zones, system and plant are
achieved by fluid loop models which calculate the simulation state variables at the inlet
and outlet of each component.
The simultaneous solution of the zones, systems and plant is controlled by an
integrated solution manager which relies on successive substitution iteration using the
Gauss-Seidel philosophy of continuous updating. The integrated solution manager will
drive the zone cooling demand, system supply capacity and plant capacity to convergence
given the zone thermostat set point. EnergyPlus yields more realistic and accurate
simulation results compared to building simulation programs such as BLAST(Building
Loads Analysis and System Thermodynamics) or DOE-2, which use the sequential
29
simulation method with no feedback from one part of the simulation to another. For the
sequential simulation method, the zone cooling demand is fed to the air handling systems,
but the response from the system is not used to update the zone conditions. This can lead
to nonphysical results.
3.1.1 Successive Substitution with Lagging
The zone and system integration uses a shortened simulation time step, typically
between 0.1 and 0.25 hours and the IBLAST’s time-marching method by Taylor et. al.
(1990, 1991) with the zone conditions lagged by one time step. The error associated with
this approach depends on the time step, with shorter time steps resulting in higher
accuracy but longer computation time. Zone air capacity was introduced into the heat
balance to allow the maximum increase in the time step without jeopardizing stability.
The resulting method called “lagging with zone capacitance” allows the dynamic
processes in the zone to be captured more precisely compared to the sequential programs
that use time steps of one hour.
3.1.2 Ideal Controls
In real buildings, the thermostat serves as the basic control for most systems by
taking samples of the air temperature and sending signals to the control unit. For unitary
equipment in EnergyPlus, the temperature predictor-corrector serves as the thermostat
and control unit and is responsible for controlling the air system to meet the desired zone
temperature. Real controllers sample the zone conditions at shorter intervals than the
30
characteristic response time of the zone system. This results in a well controlled slowly
oscillating zone temperature. However, the simulation model can only sample the zone
conditions based on the system’s variable adaptive time step described further in Section
3.1.3.
The zone load is used as a starting point to place a demand on the air system. The
system simulation determines the actual supply capability, and the zone temperature is
adjusted according to the system response.
Heat pump models in EnergyPlus have two types of fan operating modes which
are cycling fan (AUTO) and continuous fan. For continuous fan mode, the supply air
temperature is a continuous function of the zone temperature. The fan is kept running at
constant spend and the zone temperature is kept within the desired range by switching the
compressor on and off. The fraction of the time step that the compressor is turned on is
known as the run-time fraction. For cycling fan(AUTO) mode, both supply air
temperature and supply air volume are continuous functions of the zone temperature.
Although the fan operates at constant speed, the intermittent fan acts as a variable volume
system by adjusting the air flow rate based on the heat pump part-load ratio, PLR.
Zone humidity is also an important factor that should be simulated to achieve
desirable thermal comfort. A methodology similar to the temperature predictor-corrector
is used to simulate the humidity of the zone. The idea is to predict the moisture load from
the scheduled latent loads, zone infiltration and outside air. Then, system components
with moisture control such as cooling coils, dehumidifiers, and humidifiers will try to
meet the predicted moisture load and provide feedback from the system to update the
zone conditions. This process is repeated until convergence is achieved.
31
3.1.3 Variable Time Step
Initially, developers of IBLAST used a fixed time step of 0.25 hours to update the
zone temperatures, but instabilities occurred after integration of the central plant
simulation. Very short time steps were required to keep the simulation stable and
eventually the simulation became too computationally expensive. An adaptive variable
time step was proposed to maintain the stability of the simulation when the zone
conditions are changing rapidly and speed up the computation when the conditions are
fairly consistent.
EnergyPlus adopted the two time step approach from IBLAST. For stability
reasons, a variable adaptive time step determined by the program is used for updating the
zone temperature and system response. However, the adaptive variable time step
approach could not be applied easily to the surface heat transfer calculations. Thus the
contributions to the zone loads from the surface heat transfer, internal gains, and
infiltration are updated at a default or user specified time step that is constant. This
approach yields simulation stability and accuracy while keeping computation time at a
minimum.
3.2. Simulation of Unitary Equipment in EnergyPlus
Unitary system models including the air-air heat pump and water-air heat pump
implemented in EnergyPlus are steady state models. However, the heat pump will operate
mostly at part-load by cycling on and off to keep the zone temperature at the thermostat
set point. The output of the model will represent the actual performance of the unitary
32
equipment at full load. Thus cycling models described in Chapter 2 must be used to
predict the performance of the heat pump at part-load. The cycling models used in
EnergyPlus is the part-load fraction model and the latent degradation model by
Henderson and Rengarajan (1996).
3.2.1 Zone/Air Loop Interactions
The air loop simulation can be divided into the air primary loop and zone
equipment. The supply side or air primary loop is defined by the section starting from the
node after the return streams from the zone are combined until just before the air streams
are branched off to individual zone as shown in Figure 3.1. The demand side or zone
equipment is the rest of the zone/air loop which includes everything from the point where
the duct is split to serve various zones to the point where the return ducts are mixed to a
single return duct.
33
Zone 2 Terminal Air
Unit
Terminal Air
Unit
Zone 1
(Control
Zone)
Heating
Coil
Cooling
Coil
Supplementary
Heating Coil
Fan
Air Primary Loop /
Supply Side
Zone Equipment /
Demand Side
Outside Air
Relief Air
Mixing
Box
Figure 3.1: Zone Equipment/Air Primary Loop Interaction
According to EnergyPlus(2004), the air loop simulation uses an iterative method
to solve the algebraic energy and mass balance equations combined with the steady state
component models. The zone simulation and air handling system simulation are
integrated by calling the system simulation from the zone air heat balance to determine
the system respond to the zone load. During each simulation time step of the air primary
loop, the zone temperatures and humidity ratios are held constant. The simulation of the
air primary loop iterates until the zone load is met or system capacity is exceeded. Then
the zone temperatures and humidity ratios are corrected based on the simulation results of
34
the primary air system. The full air loop simulation is managed by 2 managers:
ManageAirLoops (simulates the primary air systems or supply side) and
ManageZoneEquipment (simulates the zone equipments or demand side).
ManageAirLoops manages the primary air system side or supply side which
includes the mixing box, fan and coils. The subroutine SimAirLoops does the actual
simulation of the primary air systems and tries to converge on the zone load served by the
systems. More details on the iterative procedure is discussed in Section 3.2.2
ManageZoneEquipment simulates all the equipments directly connected to the
zone. Initially, the supply air plenum and zone splitters are simulated based on the
primary air system outlet conditions. Then each air terminal unit modulates its air flow
rate in order to satisfy its zone load. Air outlet conditions and flow rates from the zone
are passed to the mixer and eventually back to the return air node which is the inlet of the
primary air system.
The direct air unit model may be used to describe the simple system configuration
typical of residential unitary equipment based systems. The direct air unit allows the
primary air system to supply air directly to the zone without any zone level control or
tempering. The direct air unit requires specification of the zone inlet node which acts as
both the zone inlet node and the outlet node of the zone splitter. For instance, the water-to-
air heat pump unit acquires the zone load demanded by the “control zone” and
modulates the air flow rate and capacity to meet the load required. The heating or cooling
capacity delivered by the heat pump is distributed to all the zones by the direct air units
serving each of the zones.
35
3.2.2 Unitary Equipment Simulation Manager
HVACFurnace is a subroutine under SimAirLoops which manages the simulation
of air-to-air and water-to-air heat pumps. The EnergyPlus air-to-air and water-to-air heat
pump models are “virtual components” that consists of an ON-OFF fan, a heating coil, a
cooling coil, and a gas or electric supplemental heating coil.
A single heat pump may be configured to serve multiple zones as shown in Figure
3.1, with one thermostat located in the “control zone”. One of the parameters required is
the fraction of the total system volumetric airflow that is supplied to the control zone. The
heat pump cooling or heating load is determined by the control zone cooling load and the
fraction of the total system volumetric airflow that is supplied to the control zone:
Heat Pump Cooling Load Control Zone Cooling Load
Control Zone Air Flow Fraction
= (3.1)
The heat pump model is capable of simulating two fan operating modes: cycling
fan (AUTO) and continuous fan. In cycling fan mode, the fan cycles on and off together
with the compressor. The fan heat contributes to the sensible heat balance of the air
primary systems. The algorithm for the heat pump simulation in cooling mode is listed as
follows:
1. In cooling mode, the heating coil and the supplementary heating coil are turned
OFF, and the coil inlet conditions are passed to the outlet nodes. The sensible
capacity of the cooling coil is determined using two steps. First the full load
output of the heat pump is calculated as shown in Equation (3.2). Then a
simulation is performed with the compressor OFF and the fan on as shown in
Equation (3.3).
36
( )( ) , min - air out full load control zone HR FullCoolOutput = m h h (3.2)
( )( ) , min - air out coil off control zone HR NoCoolOutput = m h h (3.3)
where:
air m = air mass flow rate kg/s
out,coil off h = enthalpy of the air exiting the heat pump at full-load conditions, J/kg
out,coil off h = enthalpy of the air exiting the heat pump with cooling coil OFF, J/kg
min HR = enthalpy evaluated at the minimum humidity ratio of the heat pump
exiting air which is constant
The cooling coil sensible capacity is calculated as:
HPCoilSensCapacity = FullCoolOutput − NoCoolOutput (3.4)
2. Determine the part-load ratio, PLR or the heat pump,
(HeatPumpCoolingLoad NoCoolOutput )
PLR
HPCoilSensCapacity
−
= (3.5)
3. Based on the PLR, calculate the part-load fraction and the runtime fraction using
the part-load fraction model discussed in Section 2.3.2.
4. Simulate the cooling coil again using the calculated PLR and runtime fraction.
The heat pump model is a steady state model thus the cooling coil output is at full
load conditions. The PLR and runtime fraction is used to adjust the heat pump
outputs to part-load conditions based on the operating mode of the fan. For AUTO
fan, the heat pump design air flow rate is multiplied by the PLR to determine the
average air flow rate for the entire time step. The cooling coil outlet conditions
37
such as the enthalpy and humidity ratio are not adjusted and represent the full-load
values. For continuous fan, the outlet air flow rate is kept constant at the
design air flow rate, while the other cooling coil outlet conditions are calculated
as the “average” conditions over the simulation time step. The outlet conditions
are averaged using the PLR as follows,
( ) ( ) , 1 houtlet = PLR houtlet full load + − PLR hinlet (3.6)
( ) ( ) , 1 outlet outlet full load inlet w = PLR w + − PLR w (3.7)
5. For AUTO fan mode, the part-load performance of the cooling coil and fan is
non-linear thus iterations are required until the heat pump output matches the
required cooling load. The PLR is adjusted accordingly and if the heat pump
capacity is unable to meet the required load, PLR is set to 1 with the heat pump
running at full load. For continuous fan mode, the fan heat remains constant since
the air flow rate is constant at the design air flow rate. Thus the calculated
FullCoolOutput and NoCoolOutput in Step 1 are constant. No iteration is required
for continuous fan mode since the PLR is also constant.
For heating mode, the algorithm for the heat pump simulation is similar to the cooling
mode. The only difference is the remaining heating load will be passed to the
supplementary heating coil when the heat pump heating coil can not meet the zone
demand.
38
4.0 Implementation of Heat Pump Models in EnergyPlus
The heat pump models that have been implemented in EnergyPlus consist of
curve-fit and parameter estimation models. The model developer and documentation for
each heat pump models is shown in Table 4.1.
Heat Pump Model Developer Implemented into E+ by Current Reference for Model
Implemetation
Air-to-Air
Curve-Fit Model Adopted from
DOE 2 Buhl & Shirey EnergyPlus(2004)
Water-to-Air
Curve-Fit Model Lash (1992) Shenoy(2004) & Tang Shenoy(2004)
Parameter Based
Model Jin (2002) Fisher and Tang EnergyPlus(2004)
Water-to-Water
Curve-Fit Model Tang Tang N/A
Parameter Based
Model Jin (2002) Murugappan (2002) EnergyPlus(2004)
Table 4.1: Summary of Heat Pump Models in EnergyPlus
In this research project, the heat pump models that have been developed by
previous researchers have been modified and implemented in EnergyPlus. The models
are chosen for their robustness and generally available parameters. In addition to that,
these models are further validated using measured data from the OSU laboratories and
from heat pump manufacturers. This chapter gives a brief summary of the models with
modifications. A curve-fit water-to-water heat pump model is also proposed based on the
same approach used in the curve-fit water-to-air heat pump by Lash (1992).
39
4.1. Curve-Fit Water to Air Heat Pump Model
The curve-fit water-air heat pump model is based on Lash (1992) and Shenoy
(2004). However, further analysis of the model for several different heat pumps shows
that the model has some problem capturing the heat pump performance for heating and
cooling mode. This section describes the modifications of the model and changes to the
implementation procedure into EnergyPlus as previously explained in Shenoy (2004).
4.1.1 Modification of Lash (1992) and Shenoy (2004)
The water-air heat pump model was proposed by Lash (1992) and later improved
by Shenoy (2004) to include variable air flow rate. Using Equation 2.25-2.29 and the
Generalized Least Squares method, the proposed model is tested for 1-ton, 2-ton, 3-ton
and 5-ton heat pump for both cooling and heating mode. The model is evaluated for a
range of heat pump capacities to evaluate the robustness of the model.
For cooling mode, 54 data points are obtained from the manufacturer catalog data
with entering water temperatures of 30°F to 110°F, two sets of air flow rates, entering air
dry-bulb temperature of 80°F and an entering air wet-bulb temperature of 67°F. The data
points are then extended to 810 points for a range of entering air dry-bulb and wet-bulb
temperature using the correction factors provided in the catalog. As mentioned in
Appendix B of Jin (2002), some points in the dataset are invalid because the relative
humidity of the exiting air exceeds 100%. The relative humidity of the exiting air is
calculated from the inlet air conditions, latent capacity and sensible capacity indicated in
the catalog. These data points are not included in the coefficients computation; data sets
with no latent capacity are also excluded. As a result, the number of data points used to
40
generate the coefficients for cooling mode varies for different heat pumps. The
percentage error of the calculated performance compared to the catalog data is shown in
the table below:
1-ton 2-ton 3-ton 6-ton
Number of Data Points 466 510 348 468
Total Capacity RMS error (%) 10.40 10.27 9.31 9.26
Sensible Capacity RMS error (%) 8.99 10.80 8.10 8.99
Heat Rejection RMS error (%) 7.03 6.49 6.17 6.75
Total Power Input RMS error (%) 28.52 31.65 26.48 26.09
Cooling
Table 4.2: Comparison of Cooling Catalog Data and Simulation Results for Curve-Fit
Water-to-Air Heat Pump Model by Shenoy(2004): Eq 2.25-2.27
Table 4.2 shows that there is no significant trend of deterioration in performance as the
model is used to simulate a range of capacities. However, the model does a poor job of
simulating the power consumption with percentage RMS error of more than 25%.
For heating mode, 44 data points are obtained from the catalog data for entering
water temperatures of 30°F to 110°F, two sets of air flow rates and an entering air dry-bulb
temperature of 80°F. The data points are then extended to 252 data points to account
for variation of entering air dry-bulb temperature using the correction factors provided by
the manufacturer. The simulation results using the generated coefficients are compared
with the catalog data and the results are shown below in Table 4.3:
1-ton 2-ton 3-ton 6-ton
Number of Data Points 252 252 252 252
Heating Capacity RMS error (%) 24.61 21.48 20.56 21.31
Heat Absorption RMS error (%) 36.14 29.30 28.36 30.95
Total Power Input RMS error (%) 9.42 11.40 11.53 11.15
Heating
Table 4.3: Comparison of Heating Catalog Data and Simulation Results for Curve-Fit
Water-to-Air Heat Pump Model by Shenoy(2004): Eq 2.28-2.29
41
As shown, the model did a poor job simulating the heating capacity and the heat
absorption or source side heat transfer rate of the heat pump. The heating capacities have
RMS errors of more than 20%, and the heat absorption results have RMS errors of more
than 28%. However, the accuracy of the model seems to be consistent and insensitive to
the variation of the heat pump capacity.
Table 4.2 and Table 4.3 show that the model requires improvement to
achieve simulation results with percentage RMS errors of at less than 10%. It is noted
that heating capacity and heat absorption are a strong function of the water inlet
temperature and a weak function of the air flow rate. Combining the water inlet
temperature and the air flow rate together under a single coefficient, D2 and E2 will
result in coefficients that are unable to capture the change in heat pump performance with
respect to the change in the inlet water temperature.
Thus Equation 2.25 - 2.29 is modified by including an additional term in each
equation which results in the equations below:
Cooling Mode:
, ,
, ,
total 1 2 w in 3 ref w 4 air ref
total ref ref wb w ref air
Q T T m m A A A A
Q T T m m
= + + +
(4.1)
, ,
, , ,
sens 1 2 w in 3 ref w 4 ref w 5 air ref
sens ref ref wb w ref db w ref air
Q T T m T m m B B B B B
Q T T m T m m
= + + + +
(4.2)
, ,
, ,
c 1 2 w in 3 ref w 4 air ref
c ref ref wb w ref air
COP T T m m C C C C
COP T T m m
= + + +
(4.3)
42
Heating Mode:
, ,
, ,
h 1 2 w in 3 ref w 4 air ref
h ref ref db w ref air
Q T T m m D D D D
Q T T m m
= + + +
(4.4)
, ,
, ,
h 1 2 w in 3 ref w 4 air ref
h ref ref db w ref air
COP T T m m E E E E
COP T T m m
= + + +
(4.5)
The equations above are implemented into the model and it is simulated for the
same data points. The simulations results for the heating and cooling mode are shown in
Table 4.4 and Table 4.5:
1-ton 2-ton 3-ton 6-ton
Number of Data Points 466 510 348 468
Total Cooling RMS error (%) 5.70 5.50 5.03 5.77
Sensible Cooling RMS error (%) 8.81 9.76 8.09 8.77
Heat Rejection RMS error (%) 9.16 6.82 5.86 6.60
Total Power Input RMS error (%) 28.34 20.74 17.06 14.70
Cooling
Table 4.4: Comparison of Cooling Catalog Data and Simulation Results for Curve-Fit
Water-to-Air Heat Pump Model Version 1: Eq 4.1-4.3
1-ton 2-ton 3-ton 6-ton
Number of Data Points 252 252 252 252
Heating Capacity RMS error (%) 1.71 1.25 1.12 1.24
Heat Absorption RMS error (%) 7.17 7.67 6.23 7.97
Total Power Input RMS error (%) 8.90 8.86 8.51 8.98
Heating
Table 4.5: Comparison of Heating Catalog Data and Simulation Results for Curve-Fit
Water-to-Air Heat Pump Model Version 1: Eq 4.4-4.5
Comparing Table 4.2 and Table 4.4, the percentage RMS error for the sensible
capacity improved to about 5% from 10% while there is no significant improvement to
the total cooling capacity, heat rejection and the power consumption. However, there is a
huge improvement for the heating capacity and heat absorption for the heating mode by
comparing Table 4.3 and Table 4.5. Error for heating capacity dropped from over 20% to
43
about 1% while the heat absorption dropped from about 30% to around 7%. No
significant improvement is observed for the power consumption.
Table 4.4 and Table 4.5 show that the model is still unable to simulate the
compressor power well especially for cooling mode. In order to calculate the power
consumption, the product of the calculated COP and calculated capacity has to be taken
as shown in the equation below:
Wcalculated = COPcalculated ×Capacitycalculated (4.6)
This procedure allows propagation of error from the capacity to the power consumption.
In order to prevent the propagation of error, a new equation is proposed as a substitute for
Equation 4.3 and 4.5 by fitting the coefficients directly to the heat pump power
consumption.
Since most heat pump catalog data gives the air and water volumetric flow rate
instead of the mass flow rate, the air mass flow rate and water mass flow rate are
converted to volumetric flow rate for convenience. In addition, it also prevents the
discrepancies in the fluid property routines employed in the Coefficient Calculator
Program and EnergyPlus.
In addition, the source side heat transfer rate or heat rejection for cooling mode is
calculated in Table 4.2, Table 4.3, Table 4.4 and Table 4.5 by adding the total power
consumption and the load side heat transfer rate. The heat pump manufacturer assumed
that there are no losses as reflected in the catalog data. As a result, addition of the power
input and the load side heat transfer rate always equal to the source side heat transfer rate
44
for cooling mode. A new set of equations for simulating the source side heat transfer
curve is proposed to account for the compressor shell loss. Most researchers assume that
the compressor shell loss is about 10% of the compressor power input.
By observing the heat pump catalog and correction factors, the source side heat
transfer rate is a function of the water inlet temperature, inlet wet bulb temperature, and
load side and source side mass flow rates. Thus the formulation of the source side heat
transfer rate equation is similar to the total cooling and heating capacity which are
influenced by the same variables. With more experience in the governing equations and
the numerical solver, it is possible to formulate all the equations in a simpler and standard
form. For maximum capability in capturing the heat pump performance curve, one
coefficient is assigned to each variable which reduces the error drastically based on
observation. The equations below shows the reformulation of the entire set of governing
equations in its final form;
Cooling Mode:
,
, , ,
total 1 2 wb 3 w in 4 air 5 w
total ref ref ref air ref w ref
Q T T V V A A A A A
Q T T V V
= + + + +
(4.7)
,
, , ,
sens 1 2 db 3 wb 4 w in 5 air 5 w
sens ref ref ref ref air ref w ref
Q T T T V V B B B B B B
Q T T T V V
= + + + + +
(4.8)
,
, , ,
c 1 2 wb 3 w in 4 air 5 w
c ref ref ref air ref w ref
Power T T V V C C C C C
Power T T V V
= + + + +
(4.9)
, ,
, , , ,
source c 1 2 wb 3 w in 4 air 5 w
source c ref ref ref air ref w ref
Q T T V V D D D D D
Q T T V V
= + + + +
(4.10)
45
Heating Mode:
,
, , ,
h 1 2 db 3 w in 4 air 5 w
h ref ref ref air ref w ref
Q T T V V E E E E E
Q T T V V
= + + + +
(4.11)
,
, , ,
h 1 2 db 3 w in 4 air 5 w
h ref ref ref air ref w ref
Power T T V V F F F F F
Power T T V V
= + + + +
(4.12)
, ,
, , , ,
source c 1 2 db 3 w in 4 air 5 w
source c ref ref ref air ref w ref
Q T T V V G G G G G
Q T T V V
= + + + +
(4.13)
Using the same data points, Equation 4.7-4.10 are used to simulate the heat pump for
cooling mode and Equation 4.11- 4.13 for heating mode. The results are shown in Table
4.6 and Table 4.7 below:
1-ton 2-ton 3-ton 6-ton
Number of Data Points 466 510 348 468
Total Cooling RMS error (%) 3.12 2.76 2.68 2.72
Sensible Cooling RMS error (%) 4.46 5.13 4.49 4.25
Heat Rejection RMS error (%) 2.22 2.37 2.19 2.19
Total Power Input RMS error (%) 5.68 4.16 1.86 2.97
Cooling
Table 4.6: Comparison of Cooling Catalog Data and Simulation Results for Curve-Fit
Water-to-Air Heat Pump Model Version 2: Eq 4.7-4.10
1-ton 2-ton 3-ton 6-ton
Number of Data Points 252 252 252 252
Heating Capacity RMS error (%) 1.60 1.59 0.84 1.02
Heat Absorption RMS error (%) 2.50 2.24 1.61 1.59
Total Power Input RMS error (%) 0.83 1.28 1.47 0.91
Heating
Table 4.7: Comparison of Heating Catalog Data and Simulation Results for Curve-Fit
Water-to-Air Heat Pump Model Version 2: Eq 4.11-4.13
46
Table 4.6 and Table 4.7 show that increasing the number of coefficients improved
the model accuracy for both heating and cooling mode with RMS error of less than 6%.
The governing equations have a general form whereby the inlet conditions are divided by
the rated inlet conditions. This form results in each term having a uniform value of about
1.0. Thus the coefficient for the inlet variable indirectly shows the sensitivity of
calculated output to the respective inlet variable. The coefficient for the inlet variable will
have a negative sign if the inlet variable is inversely proportional to the calculated output.
4.1.2 Catalog Data Points
Unlike the parameter estimation based model, the curve-fit model uses matrix
functions to calculate the coefficients. The “knowns” and “unknowns” are formulated in
matrix form and solved using the generalized least square method as described by Shenoy
(2004). Thus the number of data points required is essentially based on the number of
coefficients. For example, the minimum number of data points for cooling mode is 6
because there are 6 coefficients required in calculating the sensible cooling capacity
(Equation 4.8). The general mathematical rule of requiring n equations to solve for n
unknowns applies to this model. The equations are essentially the data points obtained
from the catalog data.
In order for the generalized least square method to work properly, the data points
obtained from the catalog data should vary all model variables. For instance, the inlet air
flow rate should not be fixed at a certain flow rate. Using fixed inlet air flow rate will
cause the model to be insensitive to the variation of air flow and might even cause
problem in calculating the coefficients. The generalized least square method uses matrix
transpose, inverse and multiplication to calculate the coefficients thus one might
47
encounter a scenario of “division by zero” or huge errors if the inlet conditions are fixed.
This problem is a major drawback to the model where the catalog data does not have
varying inlet conditions. The failure of the generalized least square method is illustrated
in Appendix F.
However, a quick check on the following heat pump manufacturer’s catalog data;
Addison, ClimateMaster, Trane, and Florida Heat Pump(FHP), only FHP does not
publish heat pump performance data at varying air and water flow rates on their website.
Software is available on the FHP website which could be used to generate the heat pump
performance at different inlet conditions. The other heat pump manufacturers provide
corrections factors to adjust for either the air temperatures or flow rates which will give
the heat pump performance variables at varying inlet conditions. In short, the catalog data
points used for the coefficient generator should have varying inlet conditions and one
should expect the model not to perform as expected if the inlet conditions are fixed.
4.1.3 Model Implementation in EnergyPlus
Implementation of the curve-fit water-to-air heat pump model in EnergyPlus is
generally similar to Shenoy (2004). The only changes to the model are the governing
equations and the required performance coefficients. In addition, the proposed source
side curve was not implemented in EnergyPlus because simulating the source side heat
transfer rate individually would cause the heat balance equations to be out of balance
which is discomforting to some users. Figure 4.1 below shows the performance
coefficients, inputs and outputs of the model:
48
Curve-Fit Water to Air Heat Pump
Model
A1-A5
Total Cooling
Capacity
Coefficients
C1-C5
Power
Coefficients
W,ref V
Cooling Mode
Reference
Conditions
air,ref V
total,ref Q
c,ref Power
E1-E5
Heating
Capacity
Coefficients
F1-F5
Power
Coefficients
W,ref V
air,ref V
h,ref Q
h,ref Power
Heating Mode
Reference
Conditions
Inputs
Outputs
Inlet Air
Dry-Bulb
Temp
(K)
Water
Inlet
Temp
(K)
Air
Volumetric
Flow Rate
(m/s)
Water
Volumetric
Flow Rate
(m/s)
Total Cooling
/ Heating
Capacity (W)
Power Input
(W)
Source Side
Heat Transfer
Rate (W)
Sensible
Cooling
Capacity (W)
Sensible
Capacity
B1-B6
sens,ref Q
ref T ref T
Inlet Air
Wet-Bulb
Temp
(K)
Figure 4.1: Information Flow Chart for Curve-Fit Water-to-Air Heat Pump Model
As described earlier in Chapter 3, the Furnace Module will call the heat pump
model to simulate the performance of the heat pump at the zone sensible demand and the
49
corresponding compressor runtime fraction. The figure below shows the flow diagram of
the curve-fit water-to-air heat pump model.
CalcFurnaceOutput
SimWatertoAirHPSimple
GetInputFlag GetWatertoAirInput
CalcHPCoolingSimple CalcHPHeatingSimple
yes
no
Runtimefrac <=0
ZoneSensDemand=0
ZoneSensDemand <0
ZoneSensDemand >0
yes
no
UpdateSimpleWatertoAirHP
yes yes
Inputs:
Runtimefrac
ZoneSensDemand
InitSimpleWatertoAirHP
Figure 4.2: Flow Diagram for Curve-Fit Water-to-Air Heat Pump Model
50
In addition, the latent degradation model by Henderson and Rengarajan (1996) is
incorporated to simulate the latent and sensible capacity of the heat pump at part-load
conditions. Refer to Chapter 4.3 for details on interaction between the heat pump cooling
coil subroutine or CalcHPCoolingSimple and the latent degradation model. For the sake
of brevity, more details on the input data file structure (IDF), input data dictionary (IDD),
and output reports can be obtained from the EnergyPlus website.
4.2. Parameter Estimation Based Water-to-Air Heat Pump Model
The Parameter Estimation Based Water-Air Heat Pump Model was developed by
Jin (2002). The model is capable of simulating performance of heat pump under heating
and cooling mode and the usage antifreeze as the source side fluid. The table below
shows the comparison for the requirements to implement curve-fit and parameter
estimation heat pump models in EnergyPlus:
Curve-Fit Model Parameter Estimation Model
Requires coefficients generated using
Generalized Least Square Method.
Does not require refrigerant property
routines.
No successive substitution method is
required.
Requires 8-10 parameters depending on the
compressor type and source side fluid.
Requires refrigerant property routines
Successive substitution method is required
to drive the model to convergence.
51
4.2.1 Model Development
This section gives a brief outline of the model which is described in detail
by Jin (2002). Generally, the heat pump is modeled as 4 major components: the
compressor, expansion device, evaporator and condenser. The thermodynamics process
that might occur in the refrigerant lines, accumulator and etc. are ignored due to their
small contribution. The diagram below shows the configuration of the water-air heat
pump in cooling mode.
Expansion Compressor
Device
water m
air m
L Q
S Q
comp W
Figure 4.3: Water-Air Heat Pump Configuration
The heat pump model is capable of handling reciprocating, scroll and rotary
compressors. The refrigerant mass flow rate for each compressor is computed as shown
in Equation 4.14-4.16. The work done by each compressor is modeled as shown in
Equation 4.17-4.18.
52
Reciprocating:
1 1
1
1 dis
r
suc suc
m PD C C P
v P
γ
= + −
(4.14)
Rotary:
d
r
suc
m V
v
=
(4.15)
Scroll:
2
r c
r
suc e
m V C P
v P
= −
(4.16)
where:
r m = refrigerant mass flow rate, kg/s
PD = piston displacement, m3/s
suc ν = specific volume at suction state, m3/kg
1 C = clearance factor
dis P = discharge pressure, Pa
suc P = suction pressure, Pa
γ = isentropic exponent
r V = the refrigerant volume flow rate at the beginning of the compression, m3/s
2 C = coefficient to define the relationship between pressure ratio and leakage rate
c P = condensing pressure, Pa
e P = evaporating pressure, Pa
53
d V = displacement of rolling piston compressor, m3/s
Reciprocating and Rotary:
−
−
−
= 1
1
1
γ
γ
γ
γ
suc
dis
t r suc suc P
P
W m P v (4.17)
Scroll:
1 1 1 1
1
c
e
t e r i
i
P
P
W PV v
v
γ γ γ
γ γ γ
−
− = + −
−
(4.18)
where:
t W
= theoretical power, W
γ = isentropic exponent
r m = refrigerant mass flow rate, kg/s
suc P = suction pressure, Pa
suc ν = specific volume at suction state, m3/kg
dis P = discharge pressure, Pa
r V = the refrigerant volume flow rate at the beginning of the compression, m3/s
c P = condensing pressure, Pa
e P = evaporating pressure, Pa
i ν
= ‘built-in’ volume ratio
54
A simple linear representation is used to estimate the actual required power input to the
compressor by taking account of the efficiency of the compressor and electro-mechanical
power loss shown in the following equation:
t
loss
W W W
η
= +
(4.19)
where W
is the compressor power input, η is the efficiency of the compressor and
loss W
is the constant part of the electro-mechanical power losses.
The source side heat exchangers in both heating and cooling mode, as well as the
load side heat exchanger in heating mode are identified as sensible heat exchangers.
Sensible heat exchangers only have phase change on the refrigerant side. Sensible heat
exchanger is modeled as a counter-flow heat exchanger with negligible pressure drop and
the thermal effectiveness is as calculated follows:
ε =1− e−NTU (4.20)
F pF
NTU UA
m C
=
(4.21)
where
ε = heat transfer effectiveness
NTU = number of transfer units
UA = heat transfer coefficient, W/K
F m = water mass flow rate or air mass flow rate in case of heating mode, kg/s
pF C = water or air specific heat, J/(kg-K)
55
Under extreme operating conditions, anti-freeze is added to the water loop to
prevent it from freezing. Addition of the antifreeze changes the heat transfer coefficients
and hence the performance of the heat pump. The overall heat transfer coefficient for the
mixture can be computed as follows:
( ) _ 0.8
3
2
1
total antifreeze UA
C V C
DF
− =
+
(4.22)
where
V = fluid volumetric flow rate, m3/s
DF = degradation factor
0.8
3 C V − = estimated coolant side resistance, K/W
2 C = estimated resistance due to refrigerant to tube wall convection, tube wall
conduction and fouling, K/W
The coefficients 2 C and 3 C is estimated from the catalog data that uses pure water as the
working fluid. Thus the performance of the heat pump with various percentage of
antifreeze can be evaluated once the coefficients 2 C and 3 C are known and the
degradation factor, DF can calculated as follows:
0.47 0.8 0.33 0.67
,
,
antifreeze antifreeze antifreeze p antifreeze antifreeze
water water water p water water
h C k
DF
h C k
μ ρ
μ ρ
−
= =
(4.23)
In cooling mode, the load side heat exchanger is modeled as a direct expansion
cooling coil. The coil is assumed to be completely wet or completely dry. The total
56
cooling capacity is calculated by the ‘enthalpy method’ developed by McElgin and Wiley
(1940). The total heat transfer for the completely wet coil is,
wet wet air ( a,i s,e ) Q =ε m i − i (4.24)
The heat transfer effectiveness, wet ε based on the enthalpy potential method is as follows:
a i s e
a i a o
wet i i
i i
, ,
, ,
−
−
ε = (4.25)
where:
1 ( NTUwet )
wet ε = − e − (4.26)
ia,i = enthalpy of moist air at inlet state, J/kg
ia,o = enthalpy of moist air at outlet state, J/kg
is,e = enthalpy of moist air at evaporating temperature, J/kg
The overall number of transfer units, wet NTU , is based on the outside and inside surface
heat transfer coefficient as following:
( )
( )
,
1 ps
c o o pa i
wet
air pa
C
h A C UA
NTU
m C
+
=
(4.27)
where:
Cps is the specific heat of saturated air defined by:
T Te
s
ps dT
C dh
=
=
hc,oAo = external surface heat transfer coefficient, W/K
(UA)i = inside surface heat transfer coefficient, W/K
air m = air mass flow rate, kg/s
57
Cpa = air specific heat, J/(kg-K)
The number of transfer units can be simplified by grouping the inside and outside heat
transfer coefficients as an overall heat transfer coefficient, ( )tot UA .
( )
( ) tot
wet
air pa
UA
NTU
m C
=
(4.28)
Equation 4.20-4.26 is used to calculate the total heat transfer, and a method is
required to split the total heat transfer into the sensible and latent heat transfers. The
effective surface temperature, s,e T , based on the analysis of dehumidifying coils in
ASHRAE Handbook of HVAC Systems and Equipment (ASHRAE 2000) is used to
determine the sensible heat transfer rate of the cooling coil. The enthalpy of the saturated
air is as follows:
−
−
−
= −
maC pa
hc o Ao
e
i i
i i a i a o
s s e a i ,
1
, ,
, , , (4.29)
The effective surface temperature, s,e T , is calculated iteratively from the corresponding
enthalpy of saturated air, s,s,e i . After computing the effective surface temperature, the
sensible heat transfer rate can be computed using the following equation:
( )
,
, , 1
hc oAo
mairCpa
sen air pa a i s e Q e mC T T
−
= − −
(4.30)
58
4.2.2 Parameter Estimation Procedure
The heat pump model requires distinct parameters based on the operating mode,
compressor type and the type of fluid. The general parameters required in cooling mode
are shown below:
tot UA = load side total heat transfer coefficient, W/K
co o h A = load side external surface heat transfer coefficient, W/K
ΔTsh = superheat temperature at the evaporator outlet, °C
loss W = compressor power losses due to mechanical and electrical losses, W
η = compressor’s efficiency, dimensionless
The parameters required by the respective compressor models are as follows:
Reciprocating Compressor:
PD = compressor piston displacement, m3/s
ΔP = compressor suction/discharge pressure drop, Pascal
1 C = compressor clearance factor, dimensionless
Rotary Compressor:
PD = compressor piston displacement, m3/s
ΔP = compressor suction/discharge pressure drop, Pascal
Scroll Compressor:
r V = refrigerant volume flow rate at the beginning of the compression, m3/s
i v = built-in-volume ratio, dimensionless
59
2 C = leak rate coefficient the relationship between pressure ratio and leakage rate,
dimensionless
As shown in Equation 4.22, additional parameters are required to calculate the source
side heat transfer coefficient for use of an antifreeze mixture as the source side fluid. The
parameters needed for water and antifreeze are as follows:
Pure water:
s UA = source side heat transfer coefficient, W/K
Mixture of antifreeze and water:
1 C = source side heat transfer resistance1
2 C = source side heat transfer resistance2, K/W
All the parameters are required in cooling and heating mode except for the load
side exterior heat transfer coefficient, co o h A . The load side external heat transfer
coefficient, co o h A , is only required in cooling mode to determine the sensible heat and
latent heat for the dehumidifying cooling coil. The load side exterior heat transfer
coefficient, co o h A , can be estimated separately using the golden search minimum method
to find the optimal values that gives the lowest sum of squares of relative errors for both
sensible and latent heat.
( ) ( )
( )
( ) ( )
( )
2 2
, ,
1 , ,
N sens cat i sens i lat cat i lat i
i sens cat lat cat i i
Q Q Q Q
SSE err
= Q Q
− − = + ≤
Σ
(4.31)
where
err = tolerance error
60
sens,cat Q
= catalog sensible capacity, W
sens Q
= calculated sensible capacity, W
lat,cat Q
= catalog latent capacity, W
lat Q
= calculated latent capacity, W
The procedure for estimating the load side exterior heat transfer coefficient is outline in
the flow diagram below:
61
Data from catalog data: load side
entering dry bulb/wet bulb temperatures,
air flow rate, total cooling capacity,
sensible capacity and latent capacity.
Initial Guess: c,o o h A
Calculate the inlet and outlet air
enthalpy difference from total
cooling capacity and air flow rate.
Calculate air side
effectiveness,
,
1
hc oAo
maCpa e
−
−
Calculate enthalpy of saturated air, Eq 4.29
Calculate the corresponding effective surface
temperature, s,e T iteratively.
Calculate sensible heat transfer rate, Eq 4.30
The latent load is lat total sens Q = Q −Q
New Guess:
c,o o h A
Output:
Optimal value of c,o o h A
no
yes
Converges on the
tolerance error, Eq 4.31
Figure 4.4: Flow Diagram for Estimating the Load Side Exterior Heat Transfer
Coefficient
62
For the case of reciprocating compressor with pure water as the source side fluid,
the rest of the parametersUAs , UAtot , ΔTsh , loss W , η , PD, ΔP and 1 C are searched for the
optimal values to converge on the heat transfers and compressor power. Nelder Mead
Simplex is used to estimate the parameters that will give the minimum value of the
following objective function.
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
, ,
2
, ,
2 2 2
1
cat L cat L S cat S
cat L cat S cat
N W i W i Q i Q i Q i Q i SSE
i W i Q i Q i
− − − = Σ + + =
(4.32)
where:
cat W
= catalog compressor power consumption, W
W
= calculated compressor power consumption, W
L,cat Q
= catalog load side heat transfer rate, W
L Q
= calculated load side heat transfer rate W
S ,cat Q
= catalog source side heat transfer rate W
S Q
= calculated source side heat transfer rate W
The parameter estimation procedure is outlined in the following flow diagram,
63
Data from catalog data:
, , , ,
,
, , ,
, , , ,
air in DB air in WB air
water in water L S comp
T T m
T m QQW
Initial Guess: UAs , UAtot , ΔTsh , loss W , η , PD, ΔP and 1 C
Effectiveness of evaporator, Eq
4.26 and condenser, Eq 4.20.
Calculate evaporating and condensing
temperature from the effectiveness
Calculate refrigerant state at condenser and
evaporator outlets
Calculate the refrigerant mass flow rate, r m using
respective compressor model, Eq 4.14-4.16
Calculate compressor power
consumption, Eq 4.17-4.18
New estimation of
the parameters
Output:
Optimal values of s UA , tot UA ,
sh ΔT , loss W , η , PD, ΔP and 1 C
no
yes
Converges on the
tolerance error, Eq 4.32
Calculate total cooling
capacity
Figure 4.5: Flow Diagram for Parameter Estimation Program
64
4.2.3 Model Implementation
The two objective functions described earlier are combined into a single
program that uses the parameters generated to solve for the heat transfer rates and
compressor power given the inlet conditions. The program requires two nested iterative
loops to solve for the load side heat transfer rate and the source side heat transfer rate
using the successive substitution method. Figure 4.6 shows the inputs, outputs and the
parameters required by the heat pump model. The algorithm for the model is shown in
Figure 4.7.
L Q
S Q
(UA)L
Cooling
Model
Implementation
DBiL T WBiL T wiS T wis m iL V
( )s UA
SH ΔT
ΔP
C
PD
( )tot UA
c o o h A ,
W
loss W
η
( )S UA
SH ΔT
ΔP
C
PD
loss W
η
Heating
Figure 4.6: Information Flow Chart for Parameter Estimation Based Water-to-Air Heat
Pump Model
65
Given:
Parameters for cooling and heating mode.
, , , ,
,
, , ,
, , , ,
air in DB air in WB air
water in water L S comp
T T m
T m QQW
Initial Guess: Qsource,guess
Effectiveness of evaporator and condenser
Effective surface temperature
Sensible heat transfer, sen Q
no
yes
Pevap < Low pressure cutoff
Pcond >High pressure cutoff
Calculate Qtotal
Initial Guess: Qtotal,guess
Evaporating and condensing temperature
Evaporator and condenser outlet
Suction and discharge pressure
Calculate the refrigerant mass flow rate, r m
Calculate Wcomp and Qsource
ABS(Qtotal,guess-Qtotal)<err
ABS(Qsource,guess-Qsource)<err
yes
Outputs: Qsource, Qsen,Qtotal,Wcomp
no
END yes
no
Figure 4.7: Flow Diagram for Parameter Estimation Based Water-to-Air Heat Pump
Model, Jin(2002)
66
High pressure cutoff and low pressure cutoff is the maximum allowable
condenser pressure and minimum allowable evaporator pressure. These two parameters
are required to increase the robustness of the program for extreme operating conditions.
EnergyPlus uses successive substitution with lagging to converge on the system, zone
and plant. The inlet flow rates and inlet temperatures to the heat pump model vary every
iteration until convergence is achieved. Thus the heat pump model might attempt to use
physically unrealistic values which will results in unrealistic results or errors in the
refrigerant properties. Physical heat pumps in the industry also possess this safety
measure to protect the heat pump from overly high or low operating pressure. If the
maximum allowable condenser pressure or minimum allowable evaporator pressure is
exceeded, the heat pump model will be shut off and the outlet conditions will be set equal
to the inlet conditions.
4.2.4 Accounting for Fan Heat
The cooling capacity and heating capacity reported in the catalog data includes
the contribution of heat from the indoor fan. Note that the total power input in the catalog
data includes the fan power, fan W and compressor power, comp W . The manufacturers
conduct the experiment in an enclosed chamber and assume no heat loss from the
packaged heat pump. The heat balance equation reflected in the catalog data are as
follows:
Cooling Capacity:
( TotalCool,coil fan,heat ) source ( comp fan ) Q −Q = Q − W +W
67
Heating Capacity:
( Heat,coil fan,heat ) source ( comp fan ) Q +Q = Q + W +W
All the fan power input, fan W will eventually be converted to heat, fan,heat Q and
reflected in the load side heat transfer rate. In reality, some fan and compressor shell
energy will be lost to the environment. The compressor shell heat loss is about 10% of
the compressor power input based on experiments conducted by other researchers and
here at OSU. However, the amount of fan heat lost to the environment is usually
negligible since the fan is mounted in the air stream. The manufacturers’ experimental
data balance of within 5% for the rating conditions.
Unfortunately, the fan power consumption is not reported in the manufacturer
catalog data. This causes a problem for the parameter estimation based model because the
model can only take account of the coil heat transfer. Besides that, the model can only
model the compressor power input but the manufacturers provide the total power input
which includes both the compressor and the fan power. Given the lack of information,
contribution from the fan is included in the parameter calculation. Thus the model outputs
reflect contributions from the fan in both the coil capacity and power consumption. The
model works reasonably well but the model tends to show insensitivity in the power
calculation beyond the catalog data range as discussed in Section 5.2.3.
68
4.3. Part-Load Latent Degradation Model
Khatar et al. (1985) investigated the effect of fan cycling on air conditioner latent
load. They found that 19% of the moisture accumulated during the compressor “ON”
cycle is re-evaporated back to the air stream during the compressor “OFF” cycle. In
addition, they found that at low run time fractions, the moisture removal rate for fan
“AUTO” mode is 2.5 times higher than for fan “CONTINUOUS” mode. However, at
high run time fraction, the moisture removal for both fan modes is about the same. The
table below shows the advantages and disadvantages of both fan control modes.
Fan "ON" mode Fan "AUTO" mode
Comfort Air flow rate remains the same,
provides some degree of comfort.
False thermostat reading due to
pockets of warm air.
Fan Power More fan power consumption. Less fan power consumption.
Moisture Removal
During compressor "off" cycle,
moisture from cooling coil and drain
pan re-evaporate back to zone.
Oversized system with high
compressor cycling rate would cause
humidity problem.
Moisture drains out. Oversized
system with high compressor cycling
rate would cause humidity problem.
Humidity Control
Harder to mantain. Condensed water
evaporates back to air stream.
Thermostat set to lower temperature
to elminate extra humidity leads to
more energy consumption.
Easier to mantain. Amount of
condensed water re-evaporating
back to air stream is minimal.
Sensible Cooling
Provides cooling when compressor
cycles off. But more compressor work
to bring the coil temperature back
down when it cycles on.
No cooling or air flow when
compressor cycles off.
Air Infiltration Indoor air fan induced air infiltration. Indoor air fan induced air infiltration
is reduced.
Sound Fan noise on all the time. Fan noise switching on and off. May
be disturbing.
Table 4.8: Comparison of Fan Mode Operating Mode
69
4.3.1 Model Development
In order to account for the moisture that is re-evaporated back into the air stream,
Henderson and Rengarajan(1996) proposed a part-load latent degradation model for
continuous fan mode. The model assumes that the cooling coil can only hold a certain
amount of water and additional condensate will drain out once the maximum amount has
been exceeded without any hysteresis effects from previous wetting, surface tension and
surface dirt. Besides that, the latent capacity, total capacity and sensible capacity take the
same amount of time to reach steady state, and thus havethe same time constant based on
the single time constant model described in Section 2.3.1.
Figure 4.8 shows the phenomena of the moisture building up in the coil when the
compressor turned on. The latent capacity response of the coil can be modeled by the
single time constant method discussed in Section 2.3.1. The latent capacity at time, t is
as follows:
( ) 1
t
L L Q t Q eτ
= −
(4.35)
where:
( ) L Q t = latent capacity at t time, W
L Q = steady-state latent capacity, W
τ = heat pump time constant, s
After the moisture had exceeded the maximum moisture holding capacity of the
coil, o M , condensates starts to drain from the coil. All the latent capacity of the coil from
time 0 t onwards is considered to be useful. When the compressor cycle off, the moisture
that is held in the coil, o M is evaporated back into the air stream. If the off-time of the
70
compressor is long, the amount of moisture evaporated back into the air stream is equal
to o M .
Latent Capacity (W)
Time (seconds)
Mo
Evaporation
toff
ton
QL
Time (t Qe o) when
condensate first falls
from drain pan
twet = Mo / QL
y = Qe / QL
Net Moisture
Removal
Figure 4.8: Concept of Moisture Buildup and Evaporation on Coil
Symbols used in Figure 4.8:
on t = duration of time the compressor is on, s
off t = duration of time the compressor is off, s
L Q = steady-state latent capacity, W
e Q = initial evaporation rate after compressor shut off, W
o M = maximum moisture holding capacity of the coil, J
0 t = time when condensate first falls from the drain pan, s
71
wet t = the ratio of the moisture holding capacity of the coil, o M to the steady-state latent
capacity, L Q , s
γ = the ratio of the initial evaporation rate, e Q to the steady state latent capacity, L Q
The model calculates the time 0 t to estimate the amount of useful moisture
removal or effective latent capacity. The model uses two non-dimensionalized parameters
wet t andγ . Henderson and Rengarajan (1996) believe that the values for both parameters
are similar for a large class of cooling coils with the same coil geometry and features.
Henderson et.al (2003) conducted several test for different coil geometry at the nominal
conditions of ASHRAE Test A conditions. From their study, they found that the mass of
moisture retained in the coil is mostly a function of the coil surface geometry with some
secondary dependence on the entering dew point and face velocity. On the other hand, the
moisture evaporation rate during the off-cycle is function of the wet-bulb depression or
the difference between the wet-bulb and dry-bulb temperatures of the entering air as
follows:
( )
e e,rated ( )
rated rated
DB WB
Q Q
DB WB
−
=
−
(4.36)
where:
e Q = initial evaporation rate after compressor shut off, W
e,rated Q = initial evaporation rate after compressor shut off at nominal conditions, W
DB = inlet air dry-bulb temperature, °C
WB = inlet air wet-bulb temperature, °C
rated DB = rated inlet air dry-bulb temperature, 26.7°C
72
rated WB = rated inlet air wet-bulb temperature, 19.4°C
Since the parameters are similar for the same coil geometry, the parameters can be
calculated by adjusting the parameters γ rated and wet,rated t at the nominal conditions to the
respective inlet air conditions as following:
( )
,
, ,
L rated
wet wet rated
L
Q
t t
Q DBWB
= (4.37)
( )
( ) ( )
,
,
L rated
rated
rated rated L
DB WB Q
DB WB Q DB WB
γ γ
−
=
−
(4.38)
where:
rated γ = parameter γ at nominal conditions
wet,rated t = parameter wet t at nominal conditions, s
L,rated Q = steady-state latent capacity at nominal conditions, W
( , ) L Q DBWB = steady-state latent capacity at actual operating conditions, W
Three possible evaporation models were proposed which are exponential decay,
linear decay, and constant evaporation. Henderson and Rengarajan (1996) suggested that
the linear decay model appears to be the most physically realistic during the off cycle and
also results in “middle of the road” performance. Based on recommendations by the
researchers, the linear decay evaporation model shown in Figure 4.9 was selected for
EnergyPlus.
73
Figure 4.9: Linear Decay Evaporation Model
The linear decay evaporation model assumes that the wetted surface area
decreases with the amount of water left on the coil. The evaporation rate, q(t) at time, t
is shown below:
2
( )
2
e
e
o
q t Q Q t
M
= −
(4.39)
where:
q(t) = evaporation rate at time, t , W
The amount of moisture evaporated from the coil,M(t) can be calculated by taking the
integral of the evaporation rate, q(t) as following;
( )
2
2
0
( ) ,
4
t
e o
e
o e
M t q t dt Q t Q t t M
M Q
= = − ≤
∫ (4.40)
The maximum moisture holding capacity of the coil, o M before condensate
removal begins at time, 0 t = t , is equal to the amount of moisture remaining in the coil
74
when the compressor is first activated, i M and the addition of moisture to the coil from
time, t = 0 to 0 t = t . The amount of moisture added to the coil from time, t = 0 to 0 t = t
can be calculated by taking the integral of the heat pump latent capacity response given in
Equation (4.35). Equation (4.41a) and Equation (4.41b) below shows the derivation of
the maximum moisture holding capacity of the coil, o M .
0
1 o
t t
o i L M M Q eτ dt
= + −
∫ (4.41a)
1
to
o i L o M M Q t τ e τ −
= + + −
(4.41b)
where:
i M = amount of moisture remaining in the coil when the compressor is first activated, J
0 t = time when condensate first falls from the drain pan, s
The amount of moisture remaining in the coil when the compressor is first
activated, i M is calculated by deducting the amount of moisture evaporated from the coil
during the off-cycle, ( ) off M t from the maximum moisture holding capacity of the coil,
o M . The amount of moisture evaporated from the coil back into the air stream can be
calculated from Equation (4.40). Equation (4.42a) and Equation (4.42b) below shows the
derivation for the amount of moisture remaining in the coil when the compressor is first
activated, i M
( ) i o off M = M −M t (4.42a)
2
2,
4
e o
i o e off off off
o e
M M Q t Q t t M
M Q
= − + ≤
(4.42b)
75
( ) off M t = amount of moisture evaporated from the coil during the off-cycle, J
off t = duration of time the compressor is off, s
The duration of the compressor on-time, on t and off-time, off t can be calculated from the
heat pump cycling rate, max N and the run-time fraction, X . The part-load fraction model
discussed in Section 2.3.2 is employed to calculate the run-time fraction, X . Parameters
max N and τ can be obtained from the recommended values in Table 2.1. The compressor
on-time, on t and off-time, off t are calculated as follows:
( ) max
1
on 4 1 t
N X
=
−
(4.43)
( ) max
1
off 4 t
N X
= (4.44)
where:
on t = duration of time the compressor is on, s
off t = duration of time the compressor is off, s
max N = heat pump cycling rates, cycles/s
X = compressor run-time fraction
By equating Equation (4.41b) and Equation (4.42b), i M and o M are eliminated and the
value o t can be computed as follows:
2
1 1 2 1 , 2
4
j
to
j e o
o e off off off
L o e
t Q t Q t e t M
Q M Q
τ τ + −
≤
= − − −
(4.45)
76
The time when condensate removal starts is at 0 t and it is determined by successive
substitution, where 0
t j is used to calculate 1
0
t j+ . Substituting the two non-dimensionalized
parameters wet t andγ into Equation (4.45) resulting in Equation (4.46)
2 0
1 2
0 2
1 , 2
4
t j
j wet
off off off
wet
t t t e t t
t
τ γ
γ τ
γ
+ −
= − − − ≤
(4.46)
By knowing 0 t , the net amount of moisture removal for each cycle indicated by the
shaded area in Figure 4.8 is given below:
( ) L L on 0 q = Q t − t (4.47)
where:
L q = net amount of moisture removal for each cycle, J
L Q = steady-state latent capacity, W
on t = duration of time the compressor is on, s
0 t = time for condensate removal to begin, s
The equation above only applies for on 0 t > t or the net latent capacity is zero. With the
assumption that the time constant (τ ) is similar for total, latent and sensible capacity, the
integrated total capacity for each on-cycle is given by:
0 0
1 1 on on
t t t t
T S L q Q eτ dt Q eτ dt
= − + −
∫ ∫ (4.48a)
( ) 1
ton
T S L on q Q Q t τ e τ −
= + + −
(4.48b)
where:
77
L q = integrated total capacity for each on-cycle, J
L Q = steady-state latent capacity, W
S Q = steady-state sensible capacity, W
Thus rearranging Equation(4.45) and Equation(4.48b), the latent heat ratio for each cycle
can be determined as following:
0
1
on
L L on
eff t
T L S
on
q Q t t LHR
q Q Q
t τ e τ
+
−
−
= = + + −
(4.49a)
0
1
on
eff on
t
ss
on
LHR t t
LHR
t τ e τ
+
−
−
=
+ −
(4.49b)
where:
eff LHR = effective latent heat ratio due to cycling
ss LHR = steady-state latent heat ratio
on 0 t t + − indicates that the equation is only valid if on 0 t > t . For cases where
on 0 t < t , the effective latent heat ratio at part-load is equal to zero because the amount of
moisture in the coil did not reach the maximum moisture holding capacity of the coil,
o M thus no moisture is drained from the coil. From their sensitivity analysis, the LHR
function is affected the most by wet t and max N . The effect of γ is reduced at lower
runtime fraction because the evaporation is completed before the end of the off cycle.
The heat pump time constant, τ has little effect on the LHR function.
78
4.3.2 Modification of Part-Load Latent Degradation Model for Cycling Fan
For cycling fan operation or fan “AUTO” mode, the heat pump control has a built
in delay time for the evaporator fan to shut off after the compressor cycles off. Fan time
delay is preprogrammed into the heat pump control to save energy by extracting sensible
heat from the cool coil after the compressor has shut off. Although fan delay allows more
sensible heat transfer, it is at the expense of the fan power and latent heat transfer. The
built in time delay for the fan can usually be obtained from the heat pump manual. For
example, the fan time delay for the 3-ton York heat pump in the OSU laboratory is 60
secs.
The model proposed by Henderson and Rengarajan (1996) is based on continuous
fan operation with evaporation of moisture from the coil taking place for the entire
compressor off cycle period, off t . The amount of moisture that evaporates from the coil
back to the air stream is calculated by taking the integral of the evaporation rate over the
entire off-time, off t shown in Equation (4.40).
For cycling fan operation or fan AUTO mode, EnergyPlus assumes that there is
no evaporation of the moisture back to air stream, thus LHReff = LHRss . This can be a
source of error since moisture is evaporated back to the air stream both by natural
convection during the entire heat pump off cycle period and forced convection during the
fan time delay period. In cycling fan operation, forced evaporation from the coil can be
accounted for by applying the fan delay time, fandelay t to the model proposed by
Henderson and Rengarajan (1996).
79
By assuming that there is no evaporation of moisture from the coil by natural
convection, the amount of moisture evaporated back to the air stream is calculated by
taking the integral of the evaporation rate over the fan delay time, fandelay t .
( ) 0
t fandelay ( )
fandelay ∫ q t dt = M t (4.50)
The steps required to calculate the LHR ratio is similar to Henderson and Rengarajan
(1996) with the exception that off-time, off t in all the equations is replaced the fan delay
time, fandelay t . Equation (4.46), which calculates the time when condensate removal starts
is altered to the following form:
2 0
1 2
0 2
1 , 2
4
t j
j wet
fandelay fandelay fandelay
wet