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CFD MODEL OF FLOW THROUGH AIR FILTER PLEATS By CHARLES B. TEBBUIT Bachelor of Arts Columbia College, Columbia University New York 1988 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 July 1995 OKLAIIOMA STATE tJNIVERSITY CFD MODEL OF FLOW THROUGH AIR FIL TER PLEA TS Thesis approved: ~ Thesis Advisor ~~~ Dean of the Graduate College n ACKNOWLEDGMENTS I would like to express my gratitude to Dr. Frank W. Chambers for inviting me to participate in this filtration project and for his guidance in the production of this work. Dr. Chambers possesses an extensive knowledge of fluid dynamic subject matter. He approaches his courses and students with earnest zeal. I thank Dr. Ronald L. Dougherty for his support during work on the filtration project and for his detailed review of this work which detected many inconsistencies I had overlooked. I also thank Dr. A.J. Ghajar for participating on my committee and for his suggestions on the improvement of the paper. I express my appreciation for the assistance of my coworkers in this study: Guojiang Liu, Rob Duran, Faqiu Liang, and Rob Newman. Finally, I would like to express my deep gratitude to my wife, Sharon. She has selflessly supported our family during my studies at Oklahoma State. She provides love, horne, food, and joy. iii TABLE OF CONTENTS 1. Introduction ............................................................................................................. 1 1.1 Function of the Air Filter ........................................................................... 1 1.2 Filter Design and Pleating .......................................................................... 1 1.3 Motivation for Research ............................................................................ 3 1.3.1 Applications of Simulated Pleat Flowfields .................................. 3 1.3.2 Objective of this CFD ModeL ..................................................... 3 1.4 Filter Geometry ......................................................................................... 4 1.5 Models of Flow Across Pleated Filters ....................................................... 6 1.5.1 Review of Porous Media Flow  Darcy's Law ............................. 6 1.5.2 The Work of Gurumoothy ........................................................... 7 1.5.3 The Work of Cai ......................................................................... 8 1.5.4 The Work of Chen, Pui, and Liu .................................................. l0 1.6 Description of this CFD Model .................................................................. 11 1.6.1 General Method .......................................................................... 11 1.6.2 Key Assumptions ........................................................................ 12 II. Derivation of Numerical Method ............................................................................ 13 2.1 Overview of Method .................................................................................. 13 2.2 Flow Outside of the Filter .......................................................................... 13 2.2.1 Viscous Flow Equations .............................................................. 13 2.2.2 Turbulence Considerations .......................................................... 16 2.3 Flow Within the Filter ................................................................................ 22 2.3.1 Development of the IntraFilter Momentum Equation .................. 22 2.3.2 Calculation of Darcy Parameters .................................................. 26 2.4 Grid Development ..................................................................................... 27 2.4.1 Range of Coverage ...................................................................... 27 2.4.2 Calculation of Grid Geometry ...................................................... 30 2.5 Finite Difference Equations ....................................................................... 34 2.5.1 Overview ................................................................................... 34 IV 2.5.2 Finite Difference Formulations for Convective Terms ................. 36 2.5.3 Finite Difference Approximations for Other Terms ..................... 37 2.5.4 Complete Form of the Finite Difference Equations ...................... 39 2.6 Boundary Conditions ................................................................................. 40 2.7 Solution Method ........................................................................................ 41 2.8 Output of Program .................................................................................... 43 III. Investigation of Parameters ................................................................................... 44 3.1 Introduction ............................................................................................... 44 3.2 A Sample Run ........................................................................................... 45 3.3 The Upstream Differencing Parameter (n) ................................................. 54 3.4 The Mixing Length Constant (y) ................................................................ 59 3.5 Upstream and Downstream Coverage (htup and htdown) ........................... 62 3.6 The Grid Expansion Coefficients (eta and zeta) .......................................... 65 3.7 The Pleat Height (ht) ................................................................................. 66 3.8 The Grid Density (n/if x) ........................................................................... 69 3.9 Selecting a Convergence Criterion (epsi) ................................................... 71 3.10 The Square Pleat Geometry ..................................................................... 72 IV. Flow Simulations and Discussion .......................................................................... 73 4.1 Introduction ............................................................................................... 73 4.2 Flow Through the AF3192 Filter ............................................................... 74 4.3 Inlet Velocity ............................................................................................. 77 4.4 Pleat Angle ................................................................................................ 81 4.5 Pleat Height ............................................................................................... 84 4.6 Simulated Dust Loading ............................................................................. 85 4.7 Flow Through a Square Pleat. .................................................................... 88 4.8 Discussion of Results ................................................................................. 94 4.8.1 Suggested Experimental Confrrmation ......................................... 94 4.8.2 Reexamination of the Results from Cai.. ..................................... 95 4.8.3 Problems with the Boundary Condition at the Filter Interface ...... 95 4.8.4 Problems with the Turbulence Model .......................................... 96 V. Conclusion and Recommendations .......................................................................... 97 5.1 Conclusions From Study ............................................................................ 97 v 5.2 Further Refinements to Model ................................................................... 98 References ................................................................................................................... 100 Appendix  Experiment to Determine K and b for Specific Media ................................ 102 VI LIST OF FIGURES Fig. 1.1 Pressure drop vs. pleat density for given media type and pleat height, adapted from Brown [1993, p. 65] ................................................................ 2 1.2 Filtration efficiency vs. flow velocity, adapted from Stenhouse [1975]. The total efficiency E is the product of the efficiency via separate mechanisms: adhesion Eadh and collection EcoJl ................................... 3 1.3 Pleat shape .................................................................................................... 4 1.4 Schematic of filter .......................................................................................... 5 1.5 Square pleat geometry used by Chen et al. [1993] .......................................... 10 1.6 Assumed flow through pleats ......................................................................... 12 2.1 Coordinate orientation with reference to filter ................................................ 14 2.2 Expected flow streamlines through pleat, adapted from Brown [1993, p. 65] ................................................................. 14 2.3 Wake flow ..................................................................................................... 19 2.4 Jet flow ......................................................................................................... 19 2.5 Mixing layer flow ........................................................................................... 19 2.6 Flow profiles downstream of fllter ................................................................. 20 2.7 Choosing a characteristic width for the jet model ........................................... 21 2.8 Critical Rayleigh number vs. K/ J3, adapted from Katto & Masuoka [1966] ......................................................... 25 2.9(a) Grid simulation (triangular pleat) ................................................................... 27 2.9(b) Grid simulation (square pleat) ........................................................................ 27 2.10 Variable location in cell. ................................................................................. 28 2.11(a) Grid showing key parameters (triangular pleat) .............................................. 28 2.11(b) Grid showing key parameters (square pleat) ................................................... 29 2.12(a) Actual air / filter interface (triangular pleat) .................................................... 29 2. 12(b) Actual air / filter interface (square pleat at corner) .......................................... 30 vii 2. 13(a) Schematic showing geometric parameters (triangular pleat) ........................... 31 2.13(b) Schematic showing geometric parameters (square pleat) ................................ 31 2.14(a) Schematic showing expanded grid (triangular pleat) ....................................... 32 2.14(b) Schematic showing expanded grid (square pleat) ............................................ 32 2.15 2.16 3.1(a) 3.1(b) 3.1(c) 3.1(d) 3.2 3.2(a) 3.2(b) 3.3 3.3(a) 3.3(b) 3.3(c) 3.4 3.4(a) 3.4(b) 3.4(c) 3.4(d) 3.5 3.6(a) Local cell orientation ..................................................................................... 35 Boundary conditions ...................................................................................... 41 Sample of a vector plot, in proportion to the actual size of the flowfield. The actual flow data is 45° above horizontal. Although the magnitude of the flowfield varies, the vectors are set equallength; thus they represent direction only ................................................................................................. 49 Sample of a vector plot for same data, with the ydimension expanded 200%. Note the magnitude of the vectors is the same as above, but the direction is weighted equal in amount to the expansion ................................. .49 Same as (a), but the vectors are scaled linearly to represent magnitude ........... 50 Same as (a), but the vectors are logscale. This is better to represent magnitude if the magnitude within the flowfield varies greatly ........................ 50 Vector flowfields for 45° pleat, 2 mm high ..................................................... 51 Full and proportional flowfield, linearscaled vectors ...................................... 51 Pruned and proportional flowfield, linearscaled vectors ................................. 51 Vector flowfields for 10° pleat, 8 mm high ..................................................... 52 Full and proportional flowfield, linearscaled vectors ...................................... 52 Pruned and proportional flowfield, linearscaled vectors ................................. 52 Full and expanded flowfield [yweighted 500%], linearscaled vectors ........... 52 Vector flowfields for 3° pleat, 3 em high ........................................................ 53 Full and proportional flowfield, linearscaled vectors ...................................... 53 Pruned and proportional flowfield, linearscaled vectors ................................. 53 Full and expanded flowfield [yweighted 1600%], linearscaled vectors .......... 53 Full and expanded flowfield [yweighted 1600%], logscaled vectors ............. 53 Solution of Id convectiondiffusion problem with uniform grid and solutions east and west of the variable equal to 1 and 0 respectively, adapted from Patankar [1980, p. 96] .............................................................. 55 Flow through 3 ° pleat as in Fig. 4.4, a = 0 (full and expanded [yweighted 1600%], linearscale vectors) ....................... 56 viii 3.6(b) Flow through 45° pleat as in Fig. 4.2, a. = 0 (full and proportional, linearscale vectors) ................................................... 56 3.7 Flow downstream of filter .............................................................................. 59 3.8 The effect of "I on the downstream flow ......................................................... 61 3.8(a) Flowfield for "I = 0 (full and proportional, linearscale vectors) ....................... 61 3.8(b) Flowfield for "I = 0.098 (full and proportional, linearscale vectors) ................ 61 3.8(c) Flowfield for "I = 0.196 (full and proportional, linearscale vectors) ................ 61 3.9 Freelength as related to filter height ............................................................... 67 3.10 Effect of pleat height on the flowfield 45° pleat (pruned and proportional, linearscale vectors) ................................ 68 3.1O(a) ht = 1.79 mm, 0 freelength cells, upstream pressure = 414 Pa ......................... 68 3.1O(b) ht = 2.50 mm, 2 freelength cells, upstream pressure = 377 Pa ......................... 68 3.1O(c) ht = 3.60 mm, 7 freelength cells, upstream pressure = 349 Pa ......................... 68 3.11 Single geometry with grid fineness varied, 3° pleat (pruned and expanded [yweighted 700%]) ..................................... 70 3.11 (a) nfil_x = 4 ....................................................................................................... 70 3.11 (b) nfu_x = 6 ....................................................................................................... 70 3.11(c) nfil_x = 8 ....................................................................................................... 70 4.1 4.1(a) 4.1(b) 4.1(c) 4.2 4.2(a) 4.2(b) 4.2(c) 4.2(d) 4.3 4.4 4.5. Simulated flow through an AF3192 filter pleat (4° pleat, 3 em high) ............... 75 [YWeighted 327%, Linearscale Vectors] ..................................................... 75 [YWeighted 327%, Logscale Vectors] ........................................................ 75 [Proportional, Equallength Vectors] ............................................................. 75 Flowfields for two different inlet velocities ..................................................... 79 Inlet velocity of 10 m/s [Y Weighted 273%, Logscale Vectors] .................... 79 Inlet velocity of 0.5 m/s [Y Weighted 273%, Logscale Vectors] ................... 79 Inlet velocity of 10 m/s [Proportional, Equallength Vectors] ......................... 79 Inlet velocity of 0.5 m/s [Proportional, Equallength Vectors] ........................ 79 Pressure drop vs. inlet velocity for varying pleat angles. The curves are each separately scaled versus their first datum point at V =0.5 m/s ............ 80 Control volume for flow entering fI1.ter pleat. ................................................. 81 Angle vs. pressure drop. The value at the far right represents the analytical value of flow through a flat pleat; it is connected linearly to the tails of the curves for demonstration only ....................................................... 82 ix 4.6 4.6(a) 4.6(b) 4.6(c) 4.7 4.8 4.9 4.9(a) 4.9(b) 4.9(c) 4.9(d) 4.10 4.11 Comparison of flowfields for different pleat angles ......................................... 83 3.0° pleat, 3 em high [YWeighted 330%, Linearscale Vectors] .................... 83 4.2° pleat, 3 em high lYWeighted 330%, Linearscale Vectors] .................... 83 13.6° pleat, 1 em high [YWeighted 330%, Linearscale Vectors] .................. 83 Height vs. pressure drop for two pleat angles ................................................ 84 Permeability vs. pressure drop for two pleat angles (log scale) ....................... 86 Comparison of flowfield for clean and simulated dirty filter (4° pleat, 3 em high) ...................................................................................... 87 Clean fIlter, K = 7.8ell m2 [YWeighted 312%, Linearscale Vectors] ........ 87 Dirty fIlter, K = 7.8e12 m2 [YWeighted 312%, Linearscale Vectors] ......... 87 Clean fIlter, K = 7.8el1 m2 [Proportional, Equallength Vectors] ................. 87 Dirty filter, K = 7.8e12 m2 [Proportional, Equallength Vectors] ................. 87 Comparison of angles (8) for triangular and square pleat.. .............................. 88 Pleat angle vs. pressure drop for two geometries. (Square pleat angle converted to triangular equivalent.) ................................................................ 91 4.12 Flow through square pleat, 4° angle, 3 em high ............................................ 92 4. 12(a) [YWeighted 330%, Linearscale Vectors] ..................................................... 92 4.12(b) [YWeighted 330%, Logscale Vectors] ........................................................ 92 4.12(c) [Proportional, Equallength Vectors] ............................................................. 92 4.13 Flow through square pleat, 13° angle, 1 em high .......................................... 93 4.13(a) [Proportional, Linearscale Vectors] .............................................................. 93 4.13(b) [Proportional, Logscale Vectors] .................................................................. 93 4. 13(c) [Proportional, Equallength Vectors] ............................................................. 93 A.l Setup for experiment to determine Darcy parameters ..................................... 103 A.2 Curve fit for one layer of filter media ............................................................. 104 x Table 2.1 3.1(a) 3.1(b) 3.2 3.3 3.4 3.5(a) 3.5(b) 3.6 3.7 3.8 3.9 4.1 4.2 4.3(a) 4.3(b) 4.4 4.5 LIST OF TABLES Program parameters ....................................................................................... 33 Input to a sample run ..................................................................................... 46 Raw data from file OUTPUT.DAT ..................................................................... 47 Input summary for a 45° pleat, 2 mm high ...................................................... 51 Input summary for a 10° pleat, 8 mm high ...................................................... 52 Input summary for a 3° pleat, 3 cm high ......................................................... 53 Effect of a on program stability, 3 ° pleat ....................................................... 57 Effect of a on program stability, 45° pleat. .................................................... 58 The effect of htup on the upstream flow ......................................................... 63 The effect of htdown on the downstream flow ................................................ 64 Effect of eta on the flow upstream of pleat.. ................................................... 65 Effect of zeta on the flow downstream of pleat .............................................. 66 Input parameters for AF3192 filter flow simulation ........................................ 74 Flow direction angle from AF3192 simulation ................................................ 76 Flow direction angle for inlet flow = 10 rn/s (4° pleat, 3 cm high) ................... 80 Flow direction angle for inlet flow = 0.5 rn/s (4° pleat, 3 cm high) .................. 80 Input data for square pleat simulation (3 cm high, 3° angle) ........................... 89 Input data for square pleat simulation (1 cm high, 10° angle) ......................... 89 xi b e edown cup D d, d(x) dele delx dely dnlgth e eta f flgth freelength g h NOMENCLATURE inertial factor in fIlter momentum equation packing factor, equal to 1/0; dummy constant in code, number of cells downstream relative to ipleat in code, number of cells upstream relative to ipleat average diameter of fibers in media width of jet used in turbulence model, equal to, the halfpleat width; called wd in code in code, cell dimension along pleat length (idirection) in code, cell dimension (idirection) in code, cell dimension (idirection) in code, total length downstream of filter variable used in calculation of Darcy parameters, K and b in code, upstream expansion coefficient for grid in code, xmomentum equation; also variable used in calculation of Darcy parameters, K and b in code, total length of flowfield number of flowwise cells within pleat not directly influenced by pleat folds in code, ymomentum equation; gravity in code, continuity equation xii ht htdown htup ibar icell idown iml imax ipleat j J jbar jcell jml jmax K lmfp lmix L, I nfily in code, pleat height in code, length of downstream flowfield relative to pleat height in code, length of upstream flowfield relative to pleat height in code, cell counter in xdirection in code, number of cells within the flowfield in the xdirection in code, first i cell of pleat (xdirection) in code, number of cells downstream of filter relative to ipleat in code, imax  1 in code, number of cells flowwise including boundaries in code, number of i cells within pleat in code, cell counter in ydirection unit flow direction vector in code, number of cells within the flowfield in the jdirection in code, first j cell of pleat (ydirection) in code, jmax  1 in code, number of cells widthwise including boundaries filter media permeability molecular mean free path mixing length descriptive length in code, for square geometry, number of cells spanning length of pleat median in code, for square geometry, number of cells spanning pleat section oriented normal to flow for triangular geometry, number of cells spanning pleat thickness in code, for square geometry, number of cells spanning width of pleat median xiii p p,Pij Q R t tp tx ty U up/gth u, Uij v VD Vmix Vth v, Vii wd x, Xi z zeta mean pressure in code, mean pressure normalized for density volume flow rate fiber radius thickness of the filter media; time in code, pleat thickness in code, for triangular geometry, pleat thickness in idirection in code, for triangular geometry, pleat thickness in jdirection mean velocity in flow direction (xdirection) in code, length of flowfield upstream of filter Reynold's averaged turbulent velocity fluctuations in code, mean velocity in flow direction (idirection) mean velocity in transverse direction (ydirection) Darcian velocity (areaaveraged velocity) mixing velocity pore velocity molecular velocity in code, mean velocity in transverse direction (idirection) in code, pleat width flow direction transverse direction direction along pleat in code, downstream expansion coefficient for grid xiv ex o 'Y Ilf Ilt Vf Vt e eo e~ p qs v ~ o in code, upstream differencing coefficient porous media porosity, equals lIc in code, constant to determine mixing length in turbulence model dynamic viscosity of fluid turbulent dynamic viscosity kinematic viscosity of fluid turbulent kinematic viscosity pleat angle pleat angle for square geometry pleat angle for triangular geometry density dummy variable gradient represents triangular geometry represents square geometry xv Chapter 1 Introduction 1.1 Function of the Air Filter The combustion engine derives its energy from the exothermic reaction of fuel and air. As this reaction takes place within a collection of carefully gauged moving parts, it is important that the reactants are free from particulate matter that could damage the engine over time. Therefore filters are necessary to glean particles from the air and fuel as well as the lubricating oils used within the engine. Air filter systems are designed to rid the air of all particles larger than roughly 1 Jlm in diameter, as these exceed the oil film thickness between moving parts. Smaller particles can also cause problems, so their removal is sought as well [Jaroszczyk et aI., 1993]. 1.2 Filter Design and Pleating Air filter design is guided by the sometimes opposing concerns of high efficiency and low cost. Thereby, the following conditions and requirements guide filter design: • small space available within engine compartment; • high required flow rate to fuel engine; • low pressure drop across filter to reduce energy required to supply flow of air; • high filtration efficiency for particles to submicron level; • long filter lifetime, i.e. have large dust load capacity; • low cost. 1 One of the key design features of air filters is pleating of the filter media. Pleating helps fulfill the above design specifications in several ways. It increases the effective area of filtration which immediately increases filter capacity and filter efficiency. Pleating also serves to reduce the flow velocity through the filter media for a given flow rate. This decreases the pressure drop across the filter, thus requiring less energy to supply a given flow rate to the engine. A schematic of the effect of pleating on the pressure drop across a filter is shown in Fig. 1.1. This shows that the more compact the pleating, the lower the pressure drop, up to the point where the pleats begin creating a restriction in the flow, thereafter the pressure drop rises [Brown, 1993, p. 641. pleats / unit length Fig.1.1 Pressure drop vs. pleat density for given media type and pleat height, adapted from Brown [1993.p. 65J. The relationship between velocity and filtration efficiency is shown in Fig. 1.2. The velocity across the filter is slowed by pleating. So pleating can be manipulated to increase filter efficiency. 2 veloctty  Fig.1.2 Filtration efficiency vs.flow velocity, adapted from Stenhouse [1975]. The total efficiency E is the product of the efficiency via separate mechanisms: adhesion Eadh and collection Ecoll .. 1.3 Motivation for Research 1.3.1 Applications of Simulated Pleat Flov.fields Some of the advantages of fIlter pleating mentioned above are based on certain assumptions regarding flow through the pleats. Flow through pleats is assumed to be oriented normal to the pleats, thus the effective area of the filter is presumed to be the unpleated, flattened area of the filter. Also, the efficiency of a filter is a function of the velocity; when calculating filter efficiency using a theoretical model, it is necessary to know the velocity through the pleats. In addition, as the velocity will vary at different points along the pleat, efficiency can be gauged on a local basis within the filter. A flowfield model can also give a theoretical estimate of the pressure drop across a filter for different pleat configurations. 1.3.2 Objective of this CFD Model It is difficult to access the flow near and through a pleated filter with flow measuring equipment. In this project, a flow simulation program called PLEA TFLO is developed to 3 determine the actual path and magnitude of the flow through the pleats. This simulation program will allow the analysis of flow through pleats under varying conditions and varying geometries. It will also give a measure of the pressure drop across a filter. 1.4 Filter Geometry The automobile air filter generally consists of the pleated filter media supported by a wire mesh encased in a rubberized frame. The media is machine pleated and attached to the base by glue. The actual shape of the pleats is between a square and triangular wave. triangular wave square wave approximate pleat shape Fig.1.3 Pleat shape. The pleat angle of automobile engine air filters is generally near vertical. This serves to maximize filter area, reduce pressure drop, and lower intrafilter velocity. A schematic of 4 the Purolator AF3192 filter is shown below. The pleats are 3 cm high with a 3 mm pitch. The media is about 635 !lm thick. Bottom View Top View Side View Fig.l.4 Schematic offilter 5 rubberized frame 1.5.2 The Work of Guru moo thy Gurumoothy [1990] modeled the flow through an entire air induction system (AIS), including the filter. The flow was calculated with the PHOENICS code which solves fluid flow using the transient viscous flow equations with a kE model for turbulence. The 3 dimensional simulation included a boundary condition that accounted for the induction system walls. The filter was treated as a separate region in the flow, solving the flow according to the macroscopic qualities of the ftlter region. The equation solved within the filter region was this extended form of the Darcy equation. (12) This includes the direct relation between pressure drop and velocity related by filter resistance K which is the basic Darcy law. It also includes an account for pressure drop due to flow inertia related by a factor b. The velocities are all macroscopic values. An experimental method to calculate the parameters K and b for a specific media was offered by Gurumoothy [1990]. The two parameters are related to pressure drop in 1 dimensional form by (13) Integration of the equation over the filter width L and the substitution of QIA for U yields (14) This is of the form 7 (15) Using this relation K and b can be calculated experimentally. For the measurement of the width, L, the height of the entire pleated filter was used, thus the media region included a good deal of nonmedia space. The values Gurumoothy obtained were K = 8.3561e09m2 b = 1. 7875ge+03m1 (16) Gurumoothy validated his program using comparison with experimentally determined wall and internal pressure values found within the AIS. The CFD prediction was generally within 10% of the experimental value. It is also notable that the CFD values were found to be grid dependent, although to a small extent. 1.5.3 The Work ojCai Cai [1993] created a transient flow simulation program based on the SOLA program to analyze the detailed flow phenomena within the pleats. This program again applies the basic continuity and momentum equations of viscous flow over the region outside the filter; and turbulence was calculated with a kE model. Within the filter a momentum equation that preserved the basic viscous flow formulation was introduced. The equation was adapted from Vafai & Tien [1981]. Vafai & Tien start with the Darcy law for pore velocity, which is the actual velocity within the media pore, as opposed to the "Darcian velocity" which is a macroscopic velocity for a given cell (Vdarcy = o· Vpore ,0 == media porosity). The coefficient function of the second term on the right is the equivalent of b used above. 8 This basic momentum equation is then refonnulated for use within the porous media by volume averaging the variables to take into account pore space and media space. With V a small volume and Vr the part of the volume containing fluid, the volume average of a variable 'I' is ('I') = ~ J ~ 'I'd /I. This essentially reestablishes macroscopic variable values. Applying these to the NavierStokes fonnulation within the media yields p((V. V)V) = V(P}f + IlV2(V) + filter resistance (18) The filter resistance is incorporated using the Darcy tenns (Eq. 17) with the same volumeaveraged velocity as introduced above. pD(V=) V(P)f +g+IlV2 (V) IBl · (V) Fp B2 . ((V) . (V) ) (V) Dt K JK f I(V)I (19) Cai used this equation to represent momentum within the filter. The main difficulties in doing a transient analysis of detailed flow through a pleat are the large gradients encountered and the draconian stability criteria required as a consequence of these gradients. A transient analysis requires a sufficient number of time steps to reach a steady state flow condition. Cai's results seem to be limited by an exceedingly small time forwarding criterion required for stability. 9 1.5.4 The Work a/Chen, Pui, and Liu Chen, Pui, & Liu [1993] created a finite element method to solve for flow through pleated media. For flow within the media, an expression that combines the basic laminar momentum equation with Darcy's law is used. P (V .)v V =VP~ V +~V 2 V 82 D D K D 8 D (110) This is similar to the previous formulation except that it is steadystate and does not include an inertial resistance term, as their research involved only low flow velocities. The authors examined a square pleat geometry. inlet flow + + + pleat 'head" downstream channel channel pleat 'median' pleat 'bottom' Fig. 1.5 Square pleat geometry used by Chen et al. [1993). A description of their findings for the velocity flowfield follows. At the entrance of the pleat a portion of the flow passes through the pleat head, but the bulk of the flow enters the pleat channel. In the channel they found the tangential velocity at the pleat median to be near zero, there is very little horizontal flow. In the downstream channel the flow 10 exhibits similar characteristics. The flow emits from the downstream channel in a jetlike manner. The authors found the pressure drop to be linearly related to velocity. This suggests the dominance of the Darcy term. Pressure drop over a pleat geometry is dominated by media resistance at low pleat count (pleats per unit length) and viscous drag at high pleat count. An optimal pleat count (i.e. that which yields minimal pressure drop) was found where the combined effects of media resistance and viscous drag were minimized. Data were nondimensionalized to create a correlation curve for normalized pressure drop as a function of various filter parameters for various media types. This results in curves similar to that of Fig 1.2. 1.6 Description of this CFD Model 1.6.1 General Method In this CFD code, the steady state equations are solved directly. Three partial differential equations (PDEs) are used. These are the twodimensional continuity equation and the xand ycomponents of the momentum equation. The latter equation has two forms, one for extrafilter flow, one for intrafilter flow. The extrafilter form of the momentum equation is for viscous flow and includes a simple algebraic model for turbulent flow. The intrafilter equation includes the Darcy law modified to include the effects of flow inertia. The PDEs are translated into finite difference equations (FDEs) to be solved for a grid covering the flow region. The nonlinear FDE matrix is solved iteratively. The grid is sized based on given filter geometry and dimensions. The grid cells can expand up and downstream of the filter. The modeled filter can have either triangular or square pleats 11 (see Fig. 1.4). The results are written to files that can be viewed or further analyzed with vector plotting software. J .6.2 Key Assumptions As filter systems are generally given rather cramped spaces, the flow entering the filter chamber is not generally normal to the filter. In addition, the flow entering the filter comes down piping, so pipe geometry shapes the flow profile entering the filter. However, in this simulation we assume the flow enters the filter chamber normal to the filter. The inlet flow is assumed to be uniform flow with free slip at the boundaries (i.e. no normal flow component). The pleating is assumed to run infinitely. The permeability is assumed constant even though a reduction might occur at the comers due to glue and folding. Finally the fluid is taken to be clean single phase air. Fig. 1.6 Assumedflow through pleats. 12 Chapter 2 Derivation of Numerical Method 2.1 Overview of Method For the present study, a numerical solution is to provide velocity and pressure throughout the flowfield upstream, within, and downstream of a pleated filter. To arrive at a solution, the differential equation forms of the fundamental viscous flow equations are translated into finite difference equations (FDEs). The FDEs are then fit to a grid representing the flowfield, and they are solved for each gridpoint in the field. This method is contained in a FORTRAN code program called PLEA TFLO. 2.2 Flow Outside of the Filter 2.2.1 Viscous Flow Equations Fluid flow is governed by the laws of conservation of mass, momentum, and energy. Air flow at low Mach numbers can be considered an incompressible Newtonian flow. The viscosity can be assumed constant because the temperature varies insignificantly. Thus velocity and pressure can be found without the energy equation. Conservation of mass is expressed in the continuity equation. VV=O (21) 13 Conservation of momentum takes the fonn of the NavierStokes equations. DV 1 _ n2 =V'P+g+vjv V Dt p (22) Certain assumptions about the flowfield affect the final appearance of the FDEs. • Flow is sought in its steadystate condition. Time is not a factor. • Gravity plays an insignificant role. • Flow is considered to be two dimensionaL The velocity component along the pleat (z) is considered constant or zero, so the NavierStokes equation will have only two plane component directions (x and y). Fig.2.1 Coordinate orientation with reference to filter. • As seen in the direction of the expected flow streamlines of Fig. 2.2, the velocity gradients around the pleat will be large, so turbulent effects can be expected to be significant. This will affect the mean flow parameters, so it must be considered. Thus the fundamental equations are taken in their turbulent fonn. This is accomplished by splitting the Upstream AIr Flow ) ) ) Ftlter Streamlines Fig. 2.2 Expected flow streamlines through pleat, adapted from Brown [1993,p. 65}. 14 equation variables into mean (capital letters) and fluctuating (small letters) components, then timeaveraging the equations. This adds turbulent stresses (uiuj ) to the momentum formulation: (23) Taking into account these four assumptions, the fluid dynamic equations are represented with these three component equations: continuity: xmomentum: ymomentum: au + av =0 ax ay These three equations contain six unknowns: • the mean velocity terms, U, V; • the mean pressure, P; • the turbulent fluctuations of velocity, UU, UV, vv . 15 (24) (25) (26) So their solution requires further information. 2.2.2 Turbulence Considerations Turbulent fluctuations cannot be calculated directly, short of direct numerical simulation. However, they can be modeled fairly successfully. Models up to recent times have been based on the Boussinesq eddyviscosity approximation. This assumes a turbulence viscosity (Ilt or v t) that is analogous to molecular viscosity, except that it is based on the scale of the local turbulence rather than molecular scales. Similar to molecular viscosity's role in relating shear stress to the velocity gradient ( 'txy = 11' dU / dy), turbulent viscosity is used to relate the turbulent shear stress (UiUj) to the velocity gradient. dU UV=VI (27) dy However, unlike molecular viscosity, turbulent viscosity is not a constant property of the fluid. It varies with the flow; its value is a function of the flow. Various models of turbulence have been devised to solve for Ilt. Some of the more accurate models involve the addition of partial differential equations (PDEs) to the solution set, such as the k£ model which adds the turbulent kinetic energy (TKE) equation and the turbulent dissipation (TD) equation to the three conservation equations already given above, and solves for Ilt as a function ofTKE and ID locally. These additional PDEs complicate the solution matrix and increase computation time for solution. Moreover, Cai's [1993] computations made with the kE model showed turbulence to have little effect on the flow distribution through the filter. For these reasons, the use of extra PDEs was avoided. The "algebraic" models of turbulence are so called because they are based on geometric approximations of turbulence. They add no PDEs to the solution. Turbulent viscosity is modeled by analogy to molecular viscosity for a dilute gas following Wilcox [1993, p. 27 16 30]. Just as molecular viscosity is a product of the mean free path and an average molecular velocity, 1 ~ = pvlhlmfp Vlh == molecular velocity lmfp == mean free path (28) 2 so the turbulent viscosity is calculated as an analogous function of a length and a velocity. 1 ~I = pvmixlmix 2 (29) The mixing length (lmix) is an estimate of the distance over which an eddy maintains its directional momentum. The mixing velocity (vmix) is the product of the mixing length and the velocity gradient. so dU Vmix = C·lmixdy 1 2 dU ~I=c·p·lmix  2 dy (210) 1 2 dU or VI = C·lmix  2 dy (211) The constant and the ~ are absorbed in the mixing length. The turbulent viscosity is then substituted in Eq. (27) to give the formulation of the shear stress. (212) This yields a formulation for turbulent shear stress as a function of the mean flow variables. As velocity along the pleat length is considered to be zero or constant (dU / dz = 0), the shear stresses in the zdirection (uw) are insignificant [Townsend, 17 1976, p.196]. Turbulent normal stresses (UiUi ) are also less significant [ibid, p. 1901 so that in the fluid momentum equations, UV» UU, VV,UW, vw (213) It remains for the mixing length (lmu) to be determined. Wilcox [1993] has reviewed mixing length models calibrated for specific empirical models of selfsimilar turbulent flows. For selfpreserving free shear flows the mixing length is calculated as a product of a constant (y) and the width of the flow phenomenon (d(x)). lmix = y. d(x) (214) Free shear flows are qualified as turbulent flows not bounded by walls but bounded by a nonturbulent ambient fluid. The general classifications of these consist of wake, jet, and mixing layer flows. 18 ) ) ) ) ) ) ) ) Fig.2.3 Wake flow. Fig.2.4 letflow. Fig.2.5 Mixing layer flow. 19 Execution of the numerical method using the laminar equations showed that the flow emanating downstream from the filter has a profile similar to that shown in Fig. 2.6. The square pleat produces a similar, even stronger gradient. Fig.2.6 Flow profiles downstream o/filter. A series of strong and weak flow sections can be seen. The flow does not qualify as a mixing layer because the initial "ambient" velocities (Uj and U2 ) would change moving downstream. The wake model cannot be applied because the magnitude of the "defect" in the downstream flow is too large to be considered a wake [White, 1991, p. 259]. The flow is, however, similar to the general shape of a jet profile. It is strong in the middle and tapers off toward the sides, and the relative strength of the middle compared to the sides weakens as the flow moves downstream. There are several aspects in which the downstream filter flow is not analogous to the jet model. Each section is not bounded by a nonturbulent ambient fluid; it is bounded by an identical flow. Also, the flows spread into one another, so the profiles are unable to spread 20 out moving downstream. Moreover, a selfpreserving shape does not develop until about 20 diameters downstream [White, 1991, p. 471], farther than we wish to consider; however this "jet" does have a somewhat developed profile to begin with. Even with these shortcomings, the magnitude of the turbulent stresses should be reasonably gauged by the jet model. The main divergence with the archetype is that the flow width is constant rather than continually spreading. The practical result of this is that the scale of the turbulent stress will shrink moving downstream. This would be expected to occur as the velocity gradient decreases downstream, and the flow profile flattens out. So, to apply the jet model, a flow halfwidth (d) must be specified. The choice is either (a) to consider the flow as a positive jet stretching between low velocity points or (b) to view the flow as a set of positive and negative jets relative to the inlet flow. The former is chosen because that profile is truer to a jet profile, i.e. the velocity gradient inverts at the edges. Moreover, with this model the filter is considered to stretch infinitely, so the outflow need not be related to the magnitude of the inlet flow. Uo (0) (b) Fig.2.7 Choosing a characteristic width for the jet model. 21 Wilcox's value of'Y for a plane jet (0.098) is applied to the flow simulation. Since the flow width does not expand downstream, d(x) is considered a constant equal to half the width of a jet which equals the width of a halfpleat. A halfpleat is one half of the "V" that makes up the pleat. d(x) = d = halfpleat width (215) So the mixing length approximation is lmix = 0.098* d (216) and the turbulent shear stress is  (uv) = ( 0.098*d )2 dU dU dy dy (217) 2.3 Flow Within the Filter 2.3.i Development of the intraFilter Momentum Equation Within the filter, the flow obeys the same physical rules as outside the filter. Continuity still holds in the same form, but the momentum equation must be reformulated to take into account the fibrous media as well. Instead of entering the fibers into the flowfield, a macroscopic model of flow through a porous media can be applied. For a flow where viscous effects far outweigh inertial effects, known as Stokes flow, dimensional analysis of such a flow through porous media shows that the pressure drop across the media is directly proportional to the macroscopic velocity. The filter variables involved are media thickness (t), average fiber radius (R), and packing factor (c) (the proportion of media I space) [Brown, 1993, p. 33]. VP= ~tV fCc) R2 (218) 22 The filter variables are grouped inversely under one variable K that represents the permeability of the filter media. The result is Darcy's law for Stokes flow which shows the pressure drop to be a direct function of velocity (Darcian velocity). Vp=Il v K D (219) When the velocities are higher, inertia becomes appreciable and must be considered in a model of fluid momentum. A Reynolds number for flow through a filter can be calculated using fiber diameter (D) or the square root of penneability (.JK). VD ReD = vI (220) If Re is less than one, a condition of Stokes flow exists, and inertia is insignificant. As our work on this project has shown, actual inlet velocity over an air filter can vary significantly, from the Stokes flow range to the inertial range (see Sabnis [1993] and Newman [1994] for experimental flow regimes). From this, it can be assumed that the maximum velocity would be on the order of 10 rn/s. The fibers in the AF3192 filter media average approximately 40 Ilm in diameter [Sabnis, 1993]. The penneability for the media is 7. 8e 11 m2 • Thus the expected maximum Re D and Re.fK are 27 and 6 respectively. This is beyond the realm of Stokes flow, thus inertia can be expected to be significant and will be considered. However, with pleating effects, intrafilter velocities are often less than 1 rn/s, so in practice a condition near to Stokes flow will often exist. Vafai & Tien [1981] show inertia in porous media flow to be affected by penneability and a function based on penneability, a penneabilitybased Reynolds number (ReK), and the media geometry relating the layout of the fibers. (221) 23 with] = VDflV D I, a unit direction vector. These filterbased coefficients in front of _ inertia can be grouped together under one inertial coefficient, b. VP=b p( V ) D .VD J 2 (222) Note that the flow velocities found through the filter are area averaged, that is they represent the velocity in the cell assuming the media resistance exists but the fibers do not. Vafai & Tien term this the "Darcian fluid velocity". The actual velocities through the filter would be a function of the porosity, O. Thus, following continuity, the actual "pore" velocity would be  VD Vpore =8 However, Darcian velocities are used throughout this analysis. (223) Clearly, flow through the filter is laminar (Re  10). So by including the Darcy terms, the momentum equation for flow within the filter becomes P() IJ. 2 IJ. b( ) 0 V .V V =VP+V V V P V .V J 2 D DOD K D 2 D D (224) An analysis of the magnitude of the various terms allows for some pruning of this equation. Katto and Masuoka [1966] devised a criterion for the onset of convective flow within a porous medium. The criterion is a function of the Rayleigh number Ra, media permeability K, and media thickness L. 24 10' '" " '" 1 "~ I ~VE ~C I ~fs ~ ; I "\ i I'\. I 103 ! 106 10~ 104 10'" 102 10" 10° 10' KIt. Fig.2.8 Critical Rayleigh number vs. K/ I3 , adaptedfrom Katto & Masuoka [1 966/. Obviously, as the temperature gradient is assumed to be slight across the flowfield, Ra = O. However, to get an idea of the particular magnitude of the Rayleigh number for this problem, a ~ T of 1°C is assumed and the thermal diffusivity of the saturated media kmedia is assumed equal to that of air. From this we get K/ I3 "" 2.0e04 and Ra ::;: 2. 5e  02, well under the limit even if a significant discrepancy exists in the assumptions. Although flow within the media pores is certainly in the low Reynolds number regime, the magnitude of the viscous tenu in relation to the Darcy terms is minimal. Assuming extreme gradients (U  1 dU  1 d 2 U  1 dx  1. Oe  04 ), the magnitude of the viscous term is still well below those of the Darcy tenus'" . 11 d 2U ::;: 2400 ~U::;: 300 000 P b U 2 ::;: 42 000 B dx2 K ' 2 ' Viscous diffusion Darcy resistance Inertial Resistance ** Values of Darcy parameters to be derived below. 25 Thus the final momentum equation includes the pressure as a function of the Darcy resistance and flow inertia. This is referred to as the extended Darcy equation. (225) This is a macroscopic momentum equation for flow through porous media. The terms are negative because the pressure drops moving downstream in a flow. 2.3.2 Calculation of Darcy Parameters It remains for the Darcy parameters for particular media to be determined. Gurumoothy [1990] has demonstrated how to organize the Darcy equation as a function of pressure drop and flow rate. The Darcy equation is integrated over the media thickness (t), then velocity is replaced with the flow rate divided by the flow normal area (QI A). This yields This equation is of the form M=eQ+ fQ2 (226) (227) This is a 2nd degree polynomial. An experiment to measure pressure drop across a filter media versus flow rate was completed to derive e andf and thus K and b. The experiment was done using the media from the AF3192 filter. The values for K and b found from the experiment were K=7.8ell m2 b = 6.8e+04 ml (228) The experimental method is detailed in the appendix. These values differ considerably from those of Gurumoothy (Eq. 16) as that integration was carried over the whole height of the fIlter, while here the integration was done over the thickness of the media. 26 2.4 Grid Development 2.4.1 Range of Coverage Grid generation algorithms are created for both triangular and square pleats. The scope of the flowfield needs to be minimized as much as possible in order to permit as detailed an analysis as possible. The inlet flow is uniform, so, because of the symmetry of the geometry, coverage can be limited to a half pleat width. Filter Cells Grid Filter Fig.2.9(a) Grid simulation (triangular pleat). Filter Cells Grid Fig.2.9(b) Grid simulation (square pleat). 27 To set positioning points for the finite difference equations, a grid is set up. A rectangular grid cell is used with pressure in the center, uvelocity on the front wall and vvelocity on the top wall. v(t.j} o u(il J} p(1.j} u(1.j) IL.....~JI v(1J1) Fig.2.10 Variable location in cell. The key control parameters and grid are shown for each geometry below. The parameters are defined in Table 2.1. 1 jmax 1++ jml 1++ jbar I+++ 1 \. nfl[x Fig.2.l1(a) Grid showing key parameters (triangular pleat). 28 1 jmax t+tff.t.o jml jbor jcell 1 ... Fig.2.11(b) Grid showing key parameters (square pleat). Using this grid the mooeled filter / air interface would seem to be a jagged line for the triangular pleat and a smooth interface for the square pleat. However the actual interface is more closely related to a line connecting the pressure points at the cell centers, as the velocities on the left and lower sides of the cell have the characteristic (air or filter region) of the previous celL So the grid of the square pleat lacks the consistent cell symmetry of the grid of the triangular pleat. j i Fig.2.12(a) Actual air / filter interface (triangular pleat). 29 j i Fig.2.12(b} Actual air / filter interface (square pleat at corner). 2.4.2 Calculation of Grid Geometry The key criteria for setting up grids for each geometry are the angle of the pleating (8), the height of the pleating (ht), and the thickness of the pleat media (tp). The fineness of the grid is detennined by the number of cells across the pleat. There are three fineness parameters for the square pleat (nfil_x, nfil y, nfil_ c), and only one for the triangular pleat (rifil_x). The number of cells widthwise (jbar) and the number of cells along the pleat Upleat) as well as the cell dimensions are determined using this data (see Table 2.1 below). A key difference between the two geometries is what the angle 8 represents. For the triangular pleat, the angle 8A represents the slope of the media face versus the freestream. For the square pleat, the angle 80 is the tangent of the halfpleat width over the height. These are not the same; the difference is intensified for shorter pleats. 30 ) ) ) , ) , ( ~ ) 0 , .0 \ ",""sJ , , , , , ht , , , , , t~I4 f " , p X " ~.~ ............. nfil:.A, ....... ... , , , , , , , , , , , , , , , , , , , , , wd , , dely delx  icell ipleat idown Fig.2.13(a) Schematic showing geometric parameters (triangular pleat). ) r ]I hrtp , ) I I ;~ L ____________ ) ) ) )nllLY{ +, wd ~ ~nflCC dely}:a delx ~~~ ipleat Fig.2.13(b) Schematic showing geometric parameters (square pleat). 31 The number of cells up and downstream of the pleat are calculated relative to the number cells along the pleat Upleat) using the multipliers cup and cdown respectively. The grid is expanded up and downstream using the expansion coefficients eta and zeta respectively. I I I [.I. 111 Ud;:;I;:rn11ll I I I Fig.2.14(a) Schematic showing expanded grid (triangular pleat). 11111111111 tllllllill I ~~ ipleat Fig.2.14(b) Schematic showing expanded grid (square pleat). The table on the following page shows the derivation of all the geometric parameters. The userdefmed media thickness (tp) and pleat angle (8) are always maintained. However, as the precision of the settings is limited by the fineness of the grid, the pleat height (ht) can only be as precise as the grid dimension delx. If delx equals 1.0, then ht cannot equal 8.5, it can only be 8.0 or 9.0. Also note that the width of the flowfield is jbar* dely, this will be used as the jet width in the formulation for the mixing length [mix. 32 Parameter Summary PARAMETER SYMBOL SHAPE DERIVATION pleat angle e 8,0 pleat thickness tp 8,0 pleat height ht 8,0 cells across media (xdir.) filii x 8,0 cells across media (ydir.) nfil y 0 cells along media (xdir.) nfil c 0 relative # cells upstream cup 8,0 relative # cells downstream cdown 8,0 cell width across filt. (xdir.) delx 8 tp / sin(e ) / nfil_x 0 tp / nfil x cell width along flit. length dele 0 (ht  2tp2delx) I nfil_c (xdir.) cell width (ydir.) dely 8 delx * tan(e ) ° tp/ nfil y upstream expansion coelf. eta 8,0 downstream expansion coeff. zeta 8,0 length upstream of filt. uplgth 8,0 icell2 delx· Letak k=O length downstream offilt. dnlgth 8,0 idownl de/x· L zetak k=O pleat heights upstream htup 8,0 uplgthlht pleat heights downstream htdown 8,0 dnlgthlht total length of fiowfield figth 8,0 hHuplgth+dnlgth pleat width wd 8 ht * tan(e )  tp I cos(e ) 0 ht * tan(e ) Table 2.1 Program parameters (continued on next page). 33 first cell oiJzleat (ydir.) jcell 0 ((jbar+2)12)(nfil vI2)+ I # cells widthwise jbar ~,D wd I del}' # cells in pleat flowwise ipleat ~ nfil_x + jbar 0 2 * nlil x + 2 + nfil c first cell of pleat (xdir.) icell ~,D cup * ipleat + 1 + 0.5 # cells downstream offilter idown ~,D cdown * ipleat + 0.5 # cellsflowwise ibar ~,D icell  2 + ipleat + idown # cells widthwise inc. jmax ~,D jbar + 2 boundaries # cells flowwise inc. imax ~, 0 ibar + 2 boundaries Table 2.1 Program parameters (continued from previous page). 2.5 Finite Difference Equations 2.5.1 Overview The PDEs to be solved are as follows. Continuity Momentum outside the filter including turbulence model, conservative fonn (l(U') + (l(UV) +~ (lp _ v (l2U + (l2U ) _ ~(l .... )2 (lUi (lU = 0 ax ay p ax J ax2 dy2 dy ay dy a(uv) + a(v2) +~ ap _VJ (a2v + a2v )~(lmlX)2 au au =0 ax dy p dy ax2 ai ax ay ay 34 (229) (230,231) Momentum inside the filter (232,233) Each grid point has the continuity equation and the two components of the particular momentum equation to solve for three unknowns. These equations must be translated into finite difference equations based on grid location. A location within the cell was chosen as the central point for finite difference approximation for each equation. The xmomentum equations if) were centered about the uvelocity at the front wall; the ymomentum equations (g) were centered about the vvelocity at the top wall; and the continuity equation (h) was centered about the pressure variable location at the center of the cell. Finite difference approximations were made with central differencing and simple forward differencing. However, for the convective terms in the fluid momentum equations (d(UPj }Jdxj ), some amount of upstream differencing is necessary in order to maintain stability. The following diagram is a reference for orientation. v(I. 1.1+1) v(IJ+l) v(i+l.j+l) 0 0 0 u (Iol )+1) p(l·1J+l) u(l1 )+1) pO.j+1) uo. 1) p(l+1.J+1) uO+ ~ I) v(l1J) v(I.j) yO+1J) 0 0 0 v(Ir» PO·l.» u(I J) p(l.j) u j) p(1+1.j) u(Hl j J) v(I.1.j.l) yOJI) v(I+1.j·l) 0 0 0 0· .j·l) pO·l.J1) u(I ]1) PO.)1) u(. 1) p(i+1.J1) u{i+ .j·1 yO·l.j2) v(i+l.j2) i Fig.2.15 Local cell orientation. 35 2.5.2 Finite Difference Formulations/or Convective Terms For higher Reynolds number flows, a central difference representation of convection yields an unstable result. A way to ensure stability is to use upstream differencing  derivatives made with upstream and center stream grid points only. Physically, a central difference representation of convection is inappropriate, because, in fact, convection is really "received" from upstream and "transmitted" downstream [White, 1991, p. 199]. However, although sole use of upstream differencing ensures stability, it can lead to "an unnecessary amount of numerical smoothing" [flirt et aI., 1975]. So a combination of upstream and central differencing is used. The proportion of upstream differencing is specified by the parameter a. The FDEs for the convection terms are as follows [ibid]. d(UU) 1 2 '' = {(Ui, j + Ui + l,j) + a*IUi, j + Ui + l,jl(Ui,j  Ui + 1,j) dx 4.1x (234) (Ui l,j + Ui,j)2  a*lw  l,j + W,jl(Ui l,j  Ui,j)} d(UV) 1 = {( Vi,j + Vi + l,j)(Ui,j + Ui,j + 1) + alVi,j +Vi + l,jl(Ui,j  Ui,j + 1) dy 4~y (235) ( Vi,j 1 +Vi + l,j l)(W,j l+Ui,j)  alVi,j  1 + Vi + l,j  Jj(W,j  1  W,j)} d(UV) 1 '':'= {(w,j + w, j + 1)( Vi,j + Vi + l,j) + alw,j + W,j + Ij( Vi, j Vi + l,j) dx 4.1x (236) (w, j 1 + Ui,j + 1)( Vi  l,j +Vi,j) alw,j 1 + Ui,j + Jj( Vi  l,j  Vi,j)} d(W) 1 2 = {( Vi,j +Vi,j + 1) +alVi,j +Vi,j + Jj(Vi,j Vi,j+1) dy 4~y (237) ( Vi,j 1 + Vi,j)2  alVi,j 1 + Vi,jl( Vi,j 1 Vi, j)} Analysis of these equations shows that the a terms serve to cancel out the downstream (i+ 1) terms in the central difference approximations. 36 2.5.3 Finite Difference Approximationsfor Other Terms The remaining terms in the NavierStokes equations are centered on the particular variable within the cell as stated earlier. The pressure terms are represented with forward differencing. Pressure in the program is normalized with density, sop = t p . 1 ap (pi,j + 1 Pi,j) =~.:.... p ay ~y (238) The diffusion terms use central differencing. The turbulent shear stress formulations were also represented with central differencing. (241) !uv= !(k..': :)= ( 'b A)2 IUi,j+1Ui,A(Ui,j+1Ui,j)IUi1,j+1Ui1,jI(Ui1,j+1Ui1,j) 'Y' ] ar·uy . 2 ~.(~y) (242) 37 In the Darcy equation, the inertia tenn requires a calculation of total velocity liD; of course liD = ~U; + V; . The xDarcy equation is centered at the uvelocity variable position in the cell (refer to Fig. 2.10). So it is necessary to create a fonnulation for the vvelocity component at the/ront of the cell. So V is taken as an average of the v's to the northwest and southeast of the upoint, as these points run parallel to the filter face. ( ) Vi,j+Vi+l,jl v at Uij =  2 Likewise, ( ) Ui  J, j + 1 + Ui, j U at Vij =  2 This fonnulation is better suited for the triangular pleat, as there is no mixing of filter and nonfilter cells. The inertia tenns in the Darcy equation are thus: b( VD UD) = bU i,j (Vi,j+V2i+l,jl)2 + (Ui,j")2 2 2 (243,244) b( V b (Uil,j+l+Ui,j)2 ( ..) 2 D • VD ) = Vi,j + VI,) 2 2 2 38 2.5.4 Complete Form of the Finite Difference Equations Taking all components together, the full FDEs are as follows. Equation "f' outside of filter (fluid momentum xcomponent) (245) (W,j +W+ 1,j)2 + alw,j + Ui+ 1,jl(w,j w+ 1,j) (w  J,j+W,j)2 alw l,j+ud(w  J,jUi,i) 4tu { ( Vi,j + Vi + l,j)(W,i + Ui,i + 1) + alVi,j + Vi + l,A(w,j  Ui,j + 1) } ( Vi,j 1 + Vi + l,j l)(W,j 1 + Ui, j)  alVi,j  i+Vi + i,j ll(Ui,j 1 Ui, j) +~~~~ 4~y + v + ~..:,,~ (pi+i,j Pi,j) (W+I,j2W,j+WI,j Ui,j+I2W,j+Ui,jIJ tu f (tu)2 (~y)2 ( 'b A)2 IW,j+1w,A(w,j+1Ui,j)IUi,jW,jJj(W,jW,j1)_o  a·] ar·uy . (~y)3  Equation "g" outside of filter (fluid momentumycomponent) { (W,j + Ui, j + 1)( Vi,j + Vi + i,j) + alw, j + W,j + Jj( Vi, j  Vi + 1, j) } (W,j  i+ W,j + i)( Vi  i,j + Vi,i)  alW,i  i + Ui,j + il( Vi  i,j Vi,i) 4tu (246) (Vi,j + Vi,j + i)2 +alVi,j + Vi,j + il( Vi,j  Vi,j + i)  (Vi'i  i + Vi,j)2  alVi,j  i + vi.iI( Vi,j 1 Vi,j) +~~~~~~~~~~~ 4~y + V +.."... (pi,j + 1 pi,i) (Vi + i,j  2Vi,j + Vi1,j Vi,j + i  2Vi,j + Vi,j lJ ~y f (tu)2 (~y)2 ( 'b A)2 IW,j+lUi,A(w,J+lUi,j)lwl,J+lWi,iI(Wl,i+ 1 Ui  1,J) 0  a·] ar· uy . 2 = tu·(~y) 39 Equation ''/' in filter (Extended Darcy xcomponent) ( ) A~ vI A_b (Vi+l,jJ+Vi,j)2 .. 2 0 (') 47) Pi+l,j Pi,j +LUUi,j+LUUi,j +U',j   K 2 2 Equation "g" in filter (Extended Darcy ycomponent) ( ) A VI A b (Ui_l,j+l+Ui,j)2 2 0 Pi,j+lPi,j +LlYVi,j+LlYVi,j +Vi,j = K 2 2 (248) Equation "h" (continuity inside and outside of filter) Ui,jUil,j Vi,jVi,jl 0 + = & ~Y (249) 2.6 Boundary Conditions At the upstream position of the flow, the unifonn inlet velocity is imposed. At the final downstream column, a boundary condition of continuitive flow is assumed. This assumes the velocity gradient over the exit boundary is zero. The length of the grid should be sufficient so that these conditions do not influence the flow immediately near the boundary, i.e. the velocity gradients should be zero at the upstream and downstream boundaries. However, as it worthwhile to reduce the grid size as much as possible to thereby reduce run time, the effects of imposing these upstream and downstream boundary conditions over too tight a grid are analyzed in the next chapter to see the effect over the area of importance, viz. the pleat. If the effect is minimal, the smaller number of grid cells could be used. 40 At the downstream boundary, a pressure reading of zero is prescribed to provide a reference to flow pressure upstream. Along the sides of the flow, free slip symmetry is assumed. So at the edges of the halfpleat, crosswise velocity is assumed to be zero, and velocities on either side of the pleat are assumed to be reflective. This symmetry condition requires that the inlet velocity have no crossflow component. inlet velocity Fig.2.16 Boundary conditions. continuitive outflow pressure = 0 Lastly, it is noted that at the air / filter interface there is no boundary condition to prevent variables on either side from being included in both the viscous flow and filter momentum equations. This betrays itself in certain aspects of the solution. 2.7 Solution Method Solving for the FDEs is somewhat complicated by the nonlinearity of the equations. The following summary of the solution method is from Gerald & Wheatly [1994] pp. 1657. 41 There are three equations to solve for three unknowns per cell. feu, v,p) = xmomentum = 0 g(u,V,p) = ymomentum = 0 h(u,v) = continuity = 0 The solution to the equations is (ur, vr' Pr)' With an initial guess of the solution (ui' Vi' Pi)' the functions can be expanded to a truncated Taylor series. f(Ur, Vr, pr) = 0 = f(Ui, Vi, pi) + fu (ui, Vi,pi)(Ur  ui) + Iv (ui, Vi, pi)(Vr  Vi) + fp(Ui, Vi, pi)(pr  pi) g(Ur, Vr,pr) = 0 = g(Ui, Vi,pi)+ gu(Ui, Vi,pi)(UrUi)+ g)Ui, Vi,pi)(VrVi)+ gp(Ui, Vi,pi)(pr pi) h(Ur, Vr, pr) = 0 = h(Ui, Vi, pi) + hu (ui, Vi,pi)(Ur  ui) + hv (ui, Vi, pi)( Vr  Vi) + hp (Ui, Vi,pi)(pr  pi) (250) Here fx == df , etc. These equations can thus be solved for the difference between the dx solution and the estimate. where l/u (Ui, Vi, pi) gu(Ui, Vi,pi) hu (Ui, Vi, pi) Iv (Ui, Vi, pi) gv(Ui,Vi,pi) hv(Ui,Vi,pi) (251) The .1. tenns are solved by Gaussian elimination and added to the original estimate of the solution. (252) 42 Then equation (251) is solved again using the new estimates. This iteration is continued until convergence. Convergence is obtained when the difference between two iterations reaches an arbitrarily small value. This criterion is discussed in Chapter 3. 2.8 Output of Program The program outputs data to two files. The file OUTPUT.DAT has the final values displayed for each gridpoint, including: • u and v velocities, • angle of flow, • pressure, • and continuity. The file VELOCITY.DAT and other variants contain columnated position and velocity data to be output to a vector plotting utility. 43 Chapter 3 Investigation of Parameters 3.1 Introduction The following section examines some of the parameters used in the PLEA TFLO program with the triangleshaped pleat geometry. Parameters are examined in two categories: parameters directly affecting program output and parameters affecting program run time. Lastly, the squarewave pleat geometry is examined separately. As one of the difficulties in implementing a CFD method for viscous fluid flow is program stability, the key parameter in maintaining stability, the upstream differencing proportion a., is studied to find the best value. Also, as the applied turbulence model is not tailored for the particular flow in this study, the effect of the turbulence parameter that determines the mixing length ('y) is discussed. The area of the flow that is of particular interest is the entrance and exit to the pleat. In order to have a more detailed and faithful simulation of the flow, a fine grid is desired. However, the greater the number of grid points, the more computer resources required. So this part of the parameter analysis examines ways to minimize coverage of the less crucial areas of the flowfield, while increasing detail around the filter pleat. Testing is done to optimize code settings to: • minimize the distance covered upstream (htup) and downstream (htdown) of the filter; 44 • maximize the grid expansion up (eta) and downstream (zeta) of the filter; • minimize the pleat height (htup) used to examine a specific pleat angle; • increase the density of the grid (nfil_ x); • maximize the convergence criterion (epsi) for faster solution. 3.2 A Sample Run Before examining the parameters, sample runs are done to demonstrate the format of the output. The output format takes two forms: a printout of the data and a file formatted for creating a vector plot. The typewritten printout of the input and output data is contained in the file OUTPUT.DAT. The output data include u, v, p, continuity, and angle for every cell in the flowfield. The velocities are in meters per second; the normalized pressure (pressure divided by density) is in meters squared per second squared; the continuity is taken for each cell; the angle is in degrees counterclockwise from a due downstream flow. A sample is shown for an abbreviated flowfield. The input parameters are contained in the box below. This tells the geometric configuration. This also tells whether convergence was attained and the total number of iterations to obtain convergence, or, if convergence was not attained, then it gives the final value of the convergence criterion. It also gives a value for upstream pressure. 45 Pleat Shape: :~iangular Paramer.eys ::"n meters, aeqrees , seco::cs r3ase Gcome'try Ja:::.2_,,__ ~~_=, De~::"vec Geo~,e:.ry ;~~1:.a l)pleat heiq~t. h':. ,.242EO: xr # cells lrl x c:~· ib2.~ 2) pleat media wic~h cp ; .635E03 x) # cells in y c'r 31 clea~ angle ,checad 3.008 1 x) # xcells to 4) reI * ce~: upstrm cup i .600 x) # vcells to fLt 5) rel # ce:i dnstrm: cdowni .700 i x) x cell wdth 6) # of x fii:: cells' nf11 x' 4 x) y cell wdth 7) ~ of y filt cells r.~il::::y :J x) c cell wdtr. 8) ~ o~ c fj~t cellsinfiJ CI 0 i x) pleat width 9) up expans coeff eta I 1.500 i x) total flow Igth 10) down expans 8oef' zeta 1.500 i x) filt hts upstrm 11) max # of i tee :cntmx' 20 x) file, rots d:lstrrr. 12) convergence cril, epsi 1.100E02 ! x) est iter runtime 13) x inlet veloc ! uin I 3.000 x) est max runtime s c de ..... x .383E;J2 Qe~y .::'59£03 dele • OOCE:>t co wd , .636£03 I f1gth i .485£0: htup .986 i htdown: ~, . 600 lestimel.323E+Ol ' I estot 1.645£+02 ! ii 14) kin. visc'ty nu ,.151E04 I :'7) x permeability K x .780E10 15) upstrm flux coefl alphal 1.000 1 18) Y permeability i Ky i .780EcO 1 16) turb coeff i gamma! .098 I 19) x inertia factor! bx 1.680E+05 i I I 20) Y inertia faceo, b:::y, .680E+05 1 I Reached Convergence Criterion? Yes. No. of iter = 5 I Upstream Pressure = .34877E+04 Pa 1 Table 3.l(a) Input to a sample run. The values on the right numbered "x)" are derived values, all others can be input directly. Once cell dimensions are established (defx and defy), the height and width are recalculated based on a set thickness (tp) and angle (8). The exactness with which the height can be specified is limited by the fineness of the grid (nfil_ x) (also see Section 2.4.2) Some of these parameters are further examined in this chapter. It is only noted that the thickness of the media specified here (6.35e03 m) is the approximate thickness of the media used in the Purolator AF3192 filter, a passenger car engine air filter. Next OUTPUT.DAT gives the variable values per cell. The filter region lies between the slash '\" marks. U velocity 1 2 3 5 6 7 8 9 6 3.000 3.000 2.999 2.994 .027 .082 .555 3.004 10.229 3.000 3.000 2.999 2.994\ .027 .082 .555 3.004\10.229 4 3.000 3.000 3.000 2.997 2.582\ .081 .547 2.996 .579 3 3.000 3.000 3.000 3.002 4.214 4.577\ .533 2.996 .592 2 3.000 3.000 3.001 3.006 5.177 7.260 10.365\ 3.004 .599 1 3.000 3.000 3.001 3.006 5.177 7.260 10.365 3.004 .599 10 11 12 13 14 15 16 17 18 6 7.694 5.831 4.494 3.775 3.375 3.161 3.057 3.009 3.009 5 7.694 5.831 4.494 3.775 3.375 3.161 3.057 3.009 3.009 4\ 4.233 4.079 3.489 3.261 3.134 3.061 3.023 3.004 3.004 46 .0 5\ 2.09 .0 7 .00 .0 7 .00 V velocity 6 5 4 3 2 .000 .000 .000 ~ooo .000 .000 10 6 .133 .000 4\ .133 3 .059\ .029 .000 .~oo .000 .000 .ooe .ooc .098 .000 .098 .106 .002\ .000 angle (positive 6 4 3 6 5 4\ 3 2 1 2 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 10 l.0 .0 1.8 58.9\ 38.6 .0 11 l.0 .0 1.'; 2.9 75.6\ .0 continuity 10 11 2.407 2.690 2.849 2.935 2.97 1.610 2.273 2.642 2.844 2.94 1.610 2.273 2.642 2.844 2.94 5 6 .occ .ooe .ooe .oce .occ .oce .156 .003 .008\ .000 .000 .oce .156\ .003 .ooe .177 .128\ .oeo .114 .109 .000 .000 .000 :2 .070 .000 .070 .101 .084 .000 =3 .025 .000 .025 .033 .023 .000 14 .009 .000 .009 .012 .009 .000 horizontal equals 0) 345 2 .0 .0 80.3 .0 .0\ .0 .0 .0 3.4\ .0 .0 2.4 .0 .0 l.3 .0 .0 .0 12 .9 l.1 2.4 3.0 .0 12 13 .4 .0 .4 .7 .6 .0 3 14 .2 .0 .2 .2 .2 .0 15 .003 .000 .003 .004 .G03 .000 6 2.0 .0 2.1 1.6\ .9 .0 15 .1 .0 .1 .1 .v 7 .025 .000 .025 .049 .163\ .000 16 .001 .000 .001 .001 .001 .000 7 2.6 .0 2.6 5.3 .9\ .0 16 .0 .0 .0 .0 .0 .0 2.996 2.996 2.99: 2.99: 2.99: 2.99: 8 .128 .COG\ .:28 .257 .386 .000 17 .000 .000 .000 .000 .000 .000 8 2.4 .0\ 2.5 4.9 7.3 .0 17 .0 .0 .0 .0 .0 .0 .379 .000 .379 .252 .126 .000 18 .000 .000 .000 .000 .000 .000 9 2.1 .0 33.2 23.0 : .. 9 .0 18 .0 .0 .0 .0 .0 .0 6 9 6 .OOOOE+OO .OOOOE+OO .OOOOE+OO 3 .OOOOE+OO 2 .OOOOE+OO .OOOOE+OO .1962E13 .7422E13 .1962E13 .7422E13 .7910E14 .3839E13 .1648E13 .3567E14 .2106E13 .1857213 .OOOOE+OO .OOOOE+OO .1066£12 .1137E12 .3553E14 .2842E13 .0000£+00 .4547E12 .1066E12\.1137E12 .3553E14 .2842E13 .0000E+00\.4547E12 .1094E12 .8527E13\ .OOOOE+OO .OOOOE+OO .OOOOE+OO .1137E12 .1030E12 .5684E13 .1421213\ .2274E12 .2274E12 .1137212 .1201E12 .2274E12 .OOOOE+OO .1137E12\.4547E12 .0000£+00 .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE,OO 10 6 .1137E12 5 .1137E12 4\.2274E12 3 .OOOOE+OO\ 2 .2842E13 4 3 2 .OOOOE+OO pressure 1 .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO 10 11 12 13 .4547E12 .2274E12 .8527E13 .4547E12 .2274E12 .8527E13 14 15 .7105E14 .7105E14 .7105E14 .7105E14 16 .1688E13 .1688E13 .1847E12 .5684F13 .5684E13 .3553E14 .1865E13 .l421E13 .OOOOE+OO .7105[13 .7105E14 .3553E14 .1421£:3 .2354E13 .3553£14\.1137E12 .5684E13 .7105E14 .3191£13 .7105E14 .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO (normalized) 2 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 11 3 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 12 .3488E+04 .3488E+04 .3488E+04\ .3488E+04 .3488E+04 .3488E+04\ .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 13 14 6 .3472E+04 .3472E+04 .3472E+04 .3472E+04\ .3472E+04 .3472E+04 15 7 .3423E+04 .3423E+04 .3424E+04 .3426E+04 .3426E+04\ .3426E+04 16 17 18 .3553E14 .2109£+01 .3553E14 .2109£+01 .2265£13 .8272E+00 .8660E14 .8641£+00 .3553E14 .2072,,+01 .OOOOE+OO .OOOOE+OO .3067E+04 .3747E+03 .3067E+04\ .3747E+03 .3072E+04 .3900E+03 .3082E+04 .4002E+03 .3098E+04 .4053E+03 .3098E+04 .4053E+03 17 18 6 .1366E+02 .5003E+01 .4553E+01 .1221E+01 .3800E+00 .1467£+00 .8942E01 .7840E01 .0000£+00 5 .1366E+02 .5003E+01 .4553E+01 .1221E+01 .3800E+00 .1467E+00 .8942EOl .7840E01 .OOOOE+OO 4\ .1413E+02 .4934E+Ol .4535E+01 .1212E+01 .3787E+00 .1465E+00 .8939E01 .7840EOl .0000£+00 3 .1603E+02\.4854E+Ol .4519E+Ol .1203E+01 .3770E+00 .1462E+00 .8936EOl .7840E01 .0000£+00 2 .1698E+02 .4794£+01\.4499E+Ol .1197E+01 .3761£+00 .1460E+00 .8934E01 .7841EOl .OOOOE+OO 1 .1698E+02 .4794E+01 .4499E+Ol .1197E+01 .3761E+00 .1460E+00 .8934EOl .7841E01 .OOOOE+OO Table 3.1 (b) Raw datafromfile OUTPUT.DAT. 47 Results are also output as a set of x,y position and u,v velocity data to be used for vector plotting. A more intuitive understanding of the flow can be gained through a vector plot of the flowfield. For the triangular pleat configuration, the points are taken at the uposition in the cell (see Fig. 2.10). The vvelocity at the uposition is averaged from the two vvelocities northwest and southeast of the uposition, because these are aligned in the filter direction and assure that extra and intrafilter data are not mixed (for the triangular pleat). Ideally, vector flowfields would be displayed exactly proportional to the modeled flowfield. However, it is difficult to produce a clear picture when the pleats are nearly vertical and the flowfield is 100 times longer than it is wide. So the flowfield is modified in various ways: either disregarding much of the flow up and downstream of the filter (xpruning), or exaggerating the ydimension (yweighting). The latter method also weights the vvelocity component. An example is shown (Fig. 3.1) to exhibit its effect. The expansion only weights the directional component of v not the magnitude. The vectors' magnitude can be equalweighted (linearscale), logweighted (logscale), or ignored (equallength). Examples are shown in Fig. 3.1 (c), (d), and (a), respectively. Following Fig. 3.1 are sample outputs from the PLEA TFLO program representing the range of angles to be tested. Examples using the different vector graphing methods are shown. 48 5 I   4 ~   3 ~   2 I   1 I   /  _:.1' __ ~ _ ~ _ y: _ / __ :.!' _ ;/ _ ~ _ / __ /  ~ ~  ~  ~  ':/ j' ~  jt  /  I I , I I I I I I I I I , I I I I I I t z / ~ / :/ Y I! / '/ z  '/  _:£ ~ ~  ~ / ~ ~  ~ '/   O~~~I~I~~I~~I~wul~~I~~I~~~I~~I~~I~ o 2 3 4 5 6 7 8 9 10 11 Fig.3.l(a) Sample ofa vector plot, in proportion to the actual size of the flow field. The actual flow data is 45° above horizontal. Although the magnitude of the flowfield varies, the vectors are set equallength; thus they represent direction only. 5 4 3 I I I I I I I I I I 2 1_ _1 _1 __1 __ 1_ _/ __ / __ / __ ! _! _ I I I I • , I I I , o 2 3 4 5 6 7 8 9 10 11 Fig. 3.1 (b) Sample of a vector plot for same data, with the ydimension expanded 200%. Note the magnitude of the vectors is the same as above, but the direction is weighted equal in amount to the expansion. 49 5  o  ~ ..0 0 , ~  If_ _ !.t ~ _v:_~ _ ~  ~ ,  I I , 0 0 I I _ L :" ~ ~ ~   ...    ~  _ ....   0 0 I 0 4    ~ ,   I I I I ~ /: "'f   "1  ,,   ~   f   ~  0 , I I 3    ~ I 0 . 0 0 • ..,   1   f  f   t   / / , 2    1   .. •   ~   ~   r'   r'   ,t.   , / / 0 ; 0 0 0 I 0 ; ; I I I I I I o ; ~~wuwu~~~~~~~~~wuwu~~~~~~~wu~~~ o 2 3 4 5 6 7 8 9 10 11 Fig. 3.1 (c) Same as (a). but the vectors are scaled linearly to represent magnitude. 5 1  ~  ,  Y  ~  ~  ~  _0/ _ _ 0/_ ~  ~  0 0 I 0 ~ ~ 4 1  ~  ~  Y  ~  ~  ~ _ 0/_ _ o£   0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 I 3 f  ~  J_  Y _'/.. ~  _'/  '/  / _0/ ~ 0 I 0 I 0 0 0 0 0 , 0 0 I 0 I 0 / 0 2 1   1  ? r r '1  :/ '1 / / 0 I I I 0 0 0 0 ~ ~ '/  It   /  / / / 1 1  1'  /  : : : : 0 I I J 0 I 0 2 3 4 5 6 7 8 9 10 11 Fig.3.1(d) Same as (a), but the vectors are logscale. This is better to represent magnitude if the magnitude within the flowfield varies greatly. 50 Parameters =:" mete:!':"s, aeg~ees, sec:c:',(1s Base Geometry Data Oerivec Geomet:::y ::lat.a 1 ) ple2t heigh:: hr. I .202£02 Xl # cel.Ls i~ x a~r ib2:C 2~ 2) pleat. media ",idth, t.p I . 635t:03 x) # ceils i ~: y di:c JDa~ 5 3) pleat angie ilhetadi 45.00C ! x) if xcells to filt., icel~ 8 4 ) rel # cell upstrml C'.1p 1 .800 x) # ycells to fi:t jcel: Q 5) reI # cell anstrml edownl 1.000 x) x eel: wdth I delx 1 .225;;',03 6) # of x fLt. cells I nUl x '" x) y cell wdth dely 1 .225£03  7) # of y fi 1"C cells I nfil  y' C x) c cell wdth dele i _ OCOETOC 8) # of e filt cellslnfil el C I x) pleat width I wd ! • ::"12E:02 9) up expans eoeff I eta I 1.000 x) total flow 19tD flgth i .540E02 10 ) down expans coef i zeta I 1.000 x) filt ht.s upstrm htup i .786 11) max # of iter I jcntmx . 25 x) E~t hts dnstrm htdowni :.01C 12) convergence criti epsi .:'00£02 x) est:. iter r:.:::.time ,estimej .:}8E"t02 13 ) x inlet veloc I uin ! 3.000 x) est max runtime 1 estoc 1 .294E+03 !i 1 Flow Parameters 1 14) kin. visc'ty nu i.151E04! 17) x permeab~.Lty K x 1.780E10 15) upstrm flux coefl alphal :'.000 1 18) Y permeability : Ky i. 780E10 i 16) turb coeff I gamma 1 .098 I 19) x inertia factorl bx 1.680E+05 : 1 I 20) Y inertia factor, b=y i. 680E+05 1 I Reacned Convergence Criterion? Yes. No. of iter ~ 18 I 1 Upstream Pressure = .49748E+03 Pa i !1 0.001 0.0008 0.0006 0.0004 0.0002 Table 3.2 Input summary for a 450 pleat, 2 mm high. ' ., ..... ....: t ...' O~~~~~~r_~~~~~~_r~~~~~~r_~~~~~_r~~ o 0.001 0.002 0.003 0.004 0.005 0.006 Fig. 3.2( a) Full and proportional flowfield, linearscaled vectors. 0.001 ' 0.0008  0.0006    /'  0.0004 ,,/ '    ?' 0.0002  ..!.   , , . , Orrrrr~I~~,,~I~~~rrIr~·~~I,,~~~I~rr~I~~,,~ 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 Fig. 3 .2(b) Pruned and proportional flowfield, linearscaled vectors. Figs. 3.2 Vector flowfieldsfor45° pleat, 2 mm high. 51 Plea~ Shape: Triang~:ar Parameters in meters, deqrees, seconas I 3ase Geometry Da:.a De,iveci Geome~ry Ja~a 1 ) pleat neig!".: l.,i .... , .822E 02 x) # ce:~s L' x d i::" ioar 27 2) pleat.. rr,edia wlci:.r: tp I .635E03 x) # ce ~ ~ s lr. y di :: jbar 5 3) plea':: thetad 10.000 x) # xcells ~o c • ~ce:: 9 4 ) reI ~ . ~pstrIT', cup I .900 x) # ycells to .,:::' + I jce:l, C '~I 5) ~el * cell dnsl::7'L cdown, 1. 200 x) x cell wdtr. delx i ~914~G3 6) # of x f i 1 ~ ce.; .... s i nfil XI 4 x) y cell wdt.~. dely i .161£03  7) # of y fLl~ cel~slnfil yi 0 x) c cel" WQt:h delc I .000EcOO  8) # of c f i _ :: ce:lslnfil Ci a x) pleat: w~dti; I wd I .805£03 9) up expans coeff eta i 1.000 x) tot.al flow 19c~. f1gth I .247E01 10) down expans coefl zeta I 1.000 x) filt hts upstrm I htup i .892 11) max # of iter 1 jcntrnxi 25 x) filt hts dnstrm Ihtdownl 1.226 12) convergence critl epsi i .100E02 I x) est iter runtime lestimei .149;;:+02 I 13) x inlet veloc 1 uin 3.000 I x) est max runtime ! estotl .372E+03 ! i, 1 Flow Parameters 1! 14) kin. visco ty 1 nu 1.151E04 1 17) x permeability K xi. 780E10 15) upstrm flux coefl alphal 1.000 I 18) Y permeability Ky I. 780E10 I 16) turb coeff 1 gammai .098 1 19) x inertia factor, bx 1.680E+05 1 I 1 20) Y inertia factor! b y I. 680E+05 ' 1=1 1 Reached Convergence Criterion? Yes. No. of iter 11 1 Upstream Pressure ~ .62404E+03 Pa i Table 3.3 Input summary for a 100 pleat, 8 mm high. o. 00080;;;;;;L@ 1Eii........E~E. ..§... .....iii§T"""""§y§......,.......~ ,....i.,...b...,.f_froir: ....i.rt=;;...;;;. , ,. iT., I Iii I i r f i ""'[:i9.".", "f~if~if fi_ r'" ¥...,.i. ..WjrW ii..jriL¥ +If; ....Ii,i..;. .......,,;r ""I,§ 1 o 0.005 0.01 0.015 0.02 0.025 Fig.3.3(a) Full and proportionaljlowjield, linearscaled vectors. 0.0008 ~ >  I ~ " ." , r 0.0004 ::0. • ::,... .. ~  T= , . , 0. .. 0  .:...." . ."  i I i i I i , i i I i I i i I i 0.005 0.007 0.009 0.011 0.013 0.015 0.017 Fig. 3 .3(b) Pruned and proportional jlmifield, linearscaled vectors. 0.0008..............  ,, .. , ...     0.0006 .. ," ~ .. _ _ _ _  0.0004 .... " ..... ' . ".... .....   ... 0.0002 .............. .....:: ' O+rrrrr_rr_~~~~~'r_~~~·~.. · ·~~~~~~~~~~~~ I I I I o 0.005 0.01 0.015 0.02 0.025 Fig.3.3(c) Full and expandedjlowfield [yweighted 500%], linearscaled vectors. Figs. 3.3 Vectorjlowfieldsfor 100 pleat, 8 mm high. 52 Plea~ Shape: :rianq~lar Paramet.ers :;'. met.ers, degrees, seco:'.QS 1 3ase GeomeLry Data Derived Geometry ~ata J)plea::: heigh~ 1 ';~ 1.273£01 )() * cells in x dir ~.,..bar=2~7 2) pleat media '''ldth i :CD 1.635203 x) # cecls in y a~r ="ar 5 3) plea\. angle 1 :cne:.ad, 3.000 x) # xcell s to :':i 1 t: icel ~ . 9 4) reI # cell upstrrn! cup I .900 x) # ycells to filt jce':... ;, 5) reI # cell dnstrmi cdown: 1.200 x) x cell wdth delx ,.303£82 6) # of x fil::: cellslnfil xi 41 x) y cell wdtr. I dely :.:59:::03 7) # of y filt: cells 1 nfi ly i 0 x) c cell wdth I dele i. 0002:+00 8) # of c :llt cellsinfil::ci 0 x) pleat width wd 1.795t:C3 9) up expans coeff eta 1.000 x) total flow Igth flgth: .818E01 10) down expans coef: zeta I 1.000 x) filt hts upstrm i htup I .809 11) max # of iter I jcntmx: 25 I x) filt hts dnstr,n Ihtdownl 1.112 12) convergence crit: epsi 1.100£02 i x) est ieer runtime iescimei.149E+02 13) x inlet veloc uin 1 3.000 x) est max runtime estotl.372E+03 11 1 Flow Parameters 1, 14) kin. viscty 1 n~ 1.151E04 1 17) x permeability K x i .780£10 15) upstrm flux coefi alpha! 1.000118) Y permeability Ky i.780EI0 16) turb coeff I gamma 1 .098 1 19) x inertia factor' bx i .680E+05 I 1 ! 20) Y inertia factor i b=y I. 680S+05 1..  1 Reached Convergence Criterion? Yes. No. of iter = 6 1 Upstream Pressure = .14310£+04 Pa 11 Table 3.4 Input summary for a 3° pleat, 3 em high. o OOOR I"" • pi , II II • II . tj. i II I I i• I¥ IS • d iii i T F'i"i P I , i • I , I • I, ii• T i • i , '1W jI i o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Fig. 3.4( a) Full and proportional flowfield, linearscaled vectors. 0.0008o I ; = :s.   ~  L I I r z ,   ~ ~ SF I Iii T,r i if i  i~ . iii I I I I liT i 'j er i I ay 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0008 o Fig. 3 A(b) Pruned and proportional flowfield, linearscaled vectors. ....  ....  ff  i I r  o I 0.01   ....' ,. ..... '\," ,. .... " " , ...... "" .' ,. ............... '"'"  , .......,. . " .... . , ,,  ~ ...... I 0.02 I 0.03 I 0.04 I 0.05 I 0.06 I I 0.07 0.08 I 0.09 I 0.05 0.09 Fig.3A(c) Full and expandedflowfield [yweighted 1600%], linearscaled vectors. 0.0008 o f~; , 11, _______________ _ ~ ,,,.' ),  f '" "r ',,:.., ...... _ I "'If' I        " ....... '"', I. , :. "            t  __ . __ ~I _', __________ _ I o 0.01 I 0.02 I 0.03 I 0.04 I 0.05 I 0.06 I I 0.07 0.08 0.09 Fig.3.4(d) Full and expandedflowfield [yweighted 1600%], logscaled vectors. Figs. 3.4 Vector flowfields for 3° pleat, 3 em high. 53 The significance of the results is discussed in chapter 4. The examples of a 45° and 3° pleat represent the extremes to be observed. Note that the height of the 45° example is significantly less than that of the 3° pleat. This is because the x cell dimension (defx) is set by the pleat thickness (tp) and the prescribed fineness (nfil_ x), and then since the cells must align diagonally to fit the prescribed pleat angle, the y cell dimension defy is set by the angle (e~). If, from this, defy turns out to be O.lmm, and a 45°, 3 em high pleat is sought, 300 cells would be needed lengthwise and widthwise to cover just the pleat region of the flowfield. If defy is 0.1 rum, and a 3°, 3 em high pleat is sought, less than two cells would be needed lengthwise and widthwise to cover the pleat region; by virtue that for the 3° angle, deLx is 19 times dely, and the width of the flowfield would be only 0.2 mm. One other remark is made about the presentation of the flowfield. With the smaller pleat angles, the flow through the pleats appears rather minimal, and one might wonder how continuity is maintained. With the smaller pleats the cells are much longer (flowwise) than they are wide. Thus only a small degree of cross velocity is necessary to counter a large influx of flowwise velocity. The magnitude of the continuity error calculated cellbycell shown in Table 3.1(b) is typical of all program runs. 3.3 The Upstream Differencing Parameter (a) The convective terms of the NavierStokes equations are translated into FDEs using some proportion of upstream differencing represented by a. The form of upstream differencing has been adapted from the SOLA program [Bin et aI., 1975]. In that CFD code for viscous flow, transient effects were also included, thus making the stability criteria different than those for the present steadystate analysis. However, instability still exists in centraldifference representations of convection [Patankar, 1980, p. 83f), and upstream differencing is necessary to maintain stability. Although a thorough stability analysis is not 54 done here, empirical testing is done over the range of 8x and 8y used in practice. Any departure away from full upstream differencing brings on the likelihood of instability. However, although full upstream differencing is always stable, it is not always accurate. (J) Exact .0 0 1 .:  ~  (J) 0 c .(.J..) :m 0 .! c a iI " .... ' ~ , ... '\ ' : \ ''''...UPwind \ "\ \ 2 a 2 Peclet Number  ...... _ Fig.3.5 Solution of ld convectiondiffusion problem with uniform grid and solutions east and west of the variable equal to 1 and 0 respectively, adapted from Patankar [l980, p. 96J. Patankar shows results using different differencing schemes [Patankar, 1980, p. 961 (Fig. 3.5). From this it can be seen what the upstream flux coefficient is doing  essentially providing an approximation of the exact solution using the central and upstream differenced results. To get a clearer view of the effect of the upstream differencing parameter for this CFD code, some runs are made with varying values of a.. If a. is set at zero, correspondent to full central differencing for the convective terms, the flowfield is as follows. 55 0.0008 0.0006 0.0004  , ""..." ,. ... ........ \ ". .   , "',. ... ," ...    . .... ""......... ......... ....... ...... ... " " .... 0.0002 " ...... ~: . .. ... .:........!. ....... o ~r~~I~~r~Ir~~I~~~~Ir~~Ir~~~Ir~~~I~~~ o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Fig.3.6(a) Flow through 3° pleat as in Fig. 4.4, a ::: 0 (full and expanded [yweighted J600%J.linearscale vectors). 0.08 The instability can be seen in the periodic waving of the vectors upstream of the pleat. The instability is even clearer with the solution of the 450 pleat found with full central differencing. 0.001   ...... '" ...  ... 0.0008 0.0006 , , 0.0004 0.0002 . ..... ... ......... ' ,,' , , ),~               .. , ..I .. _ _ ...  • ...... ~ ': ' , iI'" ,. •     ~   . .'. O~~~r~~~~~~~~~r~~~~~~~~~r~~~ o 0.d01 0.d02 0.d03 0.d04 o.dos· 0.d06 Fig.3.6(b) Flow through 45° pleat as in Fig. 4.2, a ::: 0 (full and proportional, linearscale vectors). As a is increased, the magnitude of the instability is less discernible. It can best be 0.007 observed by viewing the numbers themselves. Below are the uvelocity values for the cells upstream of the pleat for varying values of a for the 30 pleat. 56 keacned Convergence cri:erion? Yes. Upstrea~ ?ressu~e = .:3506E+O' Pa U veloc':ty 3.000 3.000 3. DOC 3.00C 3.000 3.000 3.000 a = 0.5 2.709 2.709 2.830 3. 003 3.172 3.286 3.286 .182 .182 3.110 3.007 2.891 .8: ~ 2.8; t, 2.596 /.596 2.770 .008 3.23:) 3.390 3.390 3.458 3.458 3.276 3.009 2.731 2.525 2.S2S 2.221 2.221 2.579 3.033 3.447 3.721 3.721 ~o. of it.e~ 8 4.131 .020 .C~2 4.131\ .020 .01,7 3.687 2.075\ .042 3.024 3.549 3.:19\ 2.336 4.46~ j.302 1.82: 4.89: 6.49~ 1.821 4.89: 6.49, .:~> .87\J ~:.~C;C 9.3!,C ',09'J .~S:; .878\" .488 9.34::; 1.09~ .153 .8.',;) .866\ 5.343 5.169 .149 .8?~ .888 .102\ 2.7'.. 7 ~.:)96\ ,798 .912 .106 .GI0\ 8.9,.,7 :1.66"7\ .936 .109 .010 8.94 7 :1.66) .936 .109 .010 : 5 ~.69 .69"1 .~8C, 3. :.',: . ~76 .00' \ . DC: l8/ .9, oJ : 9, .. 67 ':, .. 62' 11 Reached Convergence Criterion: Yes. No. of iter = i upstream Pressure  .13914E+04 Pa i1 U velocity 1 2 4 2.992 2.992 2.995 3.000 3.005 3.008 3.008 789 3.427 .020 .041 3.427\ .020 .041 3.282 2.082\ .041 3.030 3.525 3.095\ 2.744 4.447 5.26~ 2.517 4.926 6.558 2.517 4.926 6.558 10 .156 .156 . 1 ~4 .150 11 12 13 14 1':' 16 4.507 4.507 3.000 2.999 3.000 2.999 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.001 3.000 3.001 3.002 3.002 3.001 3.000 2.999 2.998 2.998 3.030 3.030 3.019 3.000 2.982 2.969 2.969 2.883 2.883 2.930 3.001 3.070 3.116 3.116 .896 11.302 9.330 7.126 5.718 .896\11.302 9.330 7.126 5.718 .87: .891\ 5.342 5.149 4.573 .817 2.963 2.:50 _. :J63 5.577\ .847 .825 .912 .935 .106\ 2.706 3.126 .110 .009\ 1.58, 8.963 11.561\ .960 8.963 11.56' .960 .112 .112 a = 0.8 1j i Reached Convergence Cri~erion: Yes. No. of iter 6 1 Upstream Pressure = .14153E+04 Pa 1 U velocity 2 7 8 9 10 .156 .156 .154 11 12 13 .911 11.245 9.325 .911\11.245 9.325 .886 .906\ 5.341 .009 .ODJ\ .009 .001 . ~63 14 15 16 7.143 5.733 4.551 7.143 5.733 4.551 5.139 4.566 3.833 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.001 3.001 3.001 3.000 2.999 2.999 2.986 2.986 2.991 3.000 3.008 3.015 3.146 .020 .041 3.146\ .020 .041 3.099 2.072\ .041 3.013 3.499 3.081\ 2.912 4.439 5.245 2.830 4.970 6.592 .151 .863 5.568\ .841 8.971 11.499\ .927 .949 .974 .974 .108\ 2.700 3.116 2.956 .112 .009\ 1.586 2.128 .115 .009 .001\ 1.532 3.000 3.000 3.000 3.000 2.999 3.015 2.830 4.970 6.592 8.971 11.499 .115 a = 0.9 11 I Reached Convergence Criterion: Yes. No. of iter = I Upstream Pressure = .14232E+04 Pa 11 U velocity 10 11 12 13 7 6 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 2.997 2.997 2.998 3.000 3.002 3.003 3.003 3.067 .020 .041 3.067\ .020 .041 3.045 2.068\ .041 3.006 3.490 3.077\ 2.960 4.437 5.239 2.922 4.986 6.603 2.922 4.986 6.603 .157 .916 11.226 9.323 .157 .916\11.226 9.323 .155 .891 .911\ 5.340 .151 .868 .931 .108\ 5.565\ .846 .954 .113 8.973 11.479\ .978 .115 8.973 11.479 .978 a = 1.0 11 I Reached convergence Criterion: Yes. No. of iter = 6 ! I Upstream Pressure = .14310E+04 Pa I 11 U velocity 1 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 7 2.993 .020 2.993\ .020 2.994 2.063\ 2.999 3.480 3.005 4.436 3.009 5.002 3.009 5.002 .041 .041 .041 3.072\ 5.233 6.614 6.614 11 12 .157 .920 11.208 .157 .920\11.208 .155 .896 .916\ .151 .873 .936 5.562\ .851 .958 8.976 11.459\ .983 8.976 11.459 .983 .115 13 9.321 9.321 5.340 .109\ .114 .116 .116 .009 .001 1.532 14 15 16 7.148 5.738 4.564 7.148 5.738 4.564 5.136 4.564 3.838 2.699 3.113 2.954 .009\ 1.586 2.121 .009 .001\ 1.523 .009 .001 1.523 14 15 7.152 5.743 7.152 5.743 5.133 4.562 16 4.577 4.577 3.842 2.697 3.110 .952 .009\ 1.586 2.115 .009 .001\ 1.Sl~ .009 .001 1.514 Table 3.5(a) Effect ofa on program stability, 3° pleat. 57 Note that there is instability even when a is set as high as 0.90 (see columns 6 & 7). For a 45° pleat, the instability is greater at Iowa's, but seemingly entirely damped out at an a of only 0.50. !1 I Reached convergence Criterion? Nc. Final epsi =.61£011 i Upstream Pressure = .48964E+03 Pa ! .. velocity 1 3.000 8.722 3.000 8.722 3.000 2.020 3.000 1.181 3.000 1. 506 3.000 3.934 3.000 3.934 a = 0.5 4.438 4.438 2.351 2.353 2.651 3.206 3.206 4 7.624 7.624 1.339 .338 1.962 3.737 3.73'7 3.615 3.615 4.033 1.587 2.767 2.998 2.998 6.020 6.020 1.213 2.603 1.712 3.453 3.453 7 2.429 2.429 5.000 1. 622 3.204 2.744 2.744 4.903 4.903\ .921 4.032 1.695 3.449 3.449 9 10 11 .830 1.505 2.228 .838 1.505 2.228 5.416\ 1.375 !.947 1.593 5.839\ 1.765 4.368 2.038 5.919\ 2.793 4.243 3.141 2.793 4.243 3.141 12 13 3.184 5.042 3.184\ 5.042 14 15 16 5.457 6.070 .OJ2 5.457 6.070 6.012 2.372 2.131\ 3.403 3.990 4.189 2.105 2.136 1.847\ 1.978 2.489 2.084 2.407 2.003 1.425\ 1.510 5.256\ 3.283 2.290 1.537 .800\ 5.256 3.283 2.290 1.537 .800 ,1 I Reached Convergence Criterion? No. Final epsi =.14E021 i ups~ream Pressure = .49242E+03 Pa ,1 U velocity 1 3.000 2.996 3.000 2.996 3.000 2.998 3.000 3.000 3.000 3.002 3.000 3.004 3.000 3.004 a = 0.8 2.987 2.987 2.992 3.000 3.008 3.012 3.012 4 2.970 2.970 2.981 3.001 3.019 3.029 3.029 2.928 2.928 2.965 3.005 3.041 3.060 3.060 2.878 2.878 2.921 3.009 3.077 3. 11:) 3.11S 7 2.665 2.665 2.919 3.050 3.157 3.210 3.210 2.614 .833 2.614\ .833 2.686 3.582\ 3.068 3.410 3.269 3.570 3.363 3.604 3.363 3.604 10 11 12 13 14 1.512 2.231 3.177 5.005 5.683 1.512 2.231 3.177\ 5.005 5.683 1.387 1.958 2.384 2.164\ 3.186 4.362\ 1.783 2.119 2.148 1.841\ 3.806 4.864\ 2.100 2.411 2.003 3.934 4.163 5.220\ 3.271 .286 3.934 4.163 5.220 3.271 .286 15 16 5.972 6.078 5.972 6.078 3.799 4.134 2.263 2.518 1.429\ 1.468 1.53" .801\ 1.537 .801 11 I Reached Convergence criterion? Yes, No. of iter = 22 ! i Upstream Pressure = .49494E+03 Pa 11 U velocity 1 2 7 3.000 2.995 2.984 2.961 2.918 .833 2.657 6 3.000 2.995 2.984 2.961 2.918 2.833 2.657 3.000 2.997 2.991 2.978 2.955 2.915 2.854 3.000 3.000 3.001 3.002 3.005 3.016 3.050 3.000 3.003 3.010 3.023 3.049 3.097 3.186 10 11 12 13 14 15 16 2.305 .832 1.513 2.232 3.174 4.992 5.673 5.963 .069 2.305\ .832 1.513 2.232 3.174\ 4.992 5.673 5.963 6.069 2.772 3.267\ 1.389 1.961 2.388 2.174\ 3.191 3.79: 4.118 3.140 3.529 4.020\ 1.789 2.125 2.153 1.845\ 2.278 2.529 3.347 3.655 4.017 4.558\ 2.109 2.414 2.005 1.430\ 1.481 3.000 3.005 3.015 3.036 3.073 3.139 3.253 3.437 3.717 4.062 4.460 5.204\ 3.267 2.286 1.538 3.000 3.005 3.015 3.036 3.073 3.139 3.253 3.437 3.717 4.062 4.460 5.204 3.267 2.286 1.538 .803\ .803 a= 1.0 ,1 I Reached Convergence Criterion? Yes. No. of iter = 20 I Upstream Pressure = .49656E+03 Pa I 11 U velocity 1 3.000 3.000 3.000 3.000 3.000 3.000 3.000 2.995 2.995 2.997 3.000 3.003 3.005 3.005 2.982 2.982 2.989 3.001 3.011 3.017 3.017 2.957 2.957 2.975 3.002 3.026 3.040 3.040 2.908 2.908 2.949 3.006 3.055 3.082 3.082 2.813 2.813 2.904 3.018 3.109 3.157 3.157 2.617 2.617 2.833 3.056 3.209 3.284 3.284 2.169 2.169\ 2.770 3.174 3.395 3.492 3.492 10 11 .833 1.514 2.232 .833 1.514 2.232 3.121\ 1.392 1.963 3.531 3.843\ 1.792 3.716 4.074 4.398\ 3.799 4.177 4.614 3.799 4.177 4.614 12 13 14 15 16 3.172 4.982 5.665 5.957 6.063 3.172\ 4.982 5.665 5.957 6.063 2.390 2.180\ 3.194 3.787 4.110 2.128 2.157 1.848\ 2.286 2.535 2.115 2.416 2.007 1.431\ 1.489 5.194\ 3.265 2.286 1.539 5.194 3.265 2.286 1.539 .803\ .803 Table 3.5(b) Effect oia on program stability, 45° pleat. 58 In order to ensure stability under all configurations, a is set at unity consistently. It is borne in mind that this could produce excessive damping under certain conditions. 3.4 The Mixing Length Constant (y) The model used to determine the turbulent stresses calculates a mixing length that is a direct function of the flow halfwidth (d) (see Section 2.2.2).  2 dU dU uv = lmi.J:   where lmi.J: = y. d and d = flow half  width dy dy d Fig.3.7 Flow downstream offilter. This turbulence model is based on an idealized jet flow (Fig. 2.4). There are several conditions in the ideal model that are not met in the flow conditions downstream of the filter. • The jet is not surrounded by nonturbulent flow. • Its width does not spread moving downstream. • Velocities within the jet stream range above and below the inlet flow velocity, rather than being solely greater or less than an ambient velocity. 59 • The flow is not a fullydeveloped selfpreserving flow. The main divergence with the idealized model is that instead of a single jet in an ambient fluid, there are a series of jet flows lined up beside one another; hence the jets are unable to expand widthwise. In the selfpreserving flow of an expanding jet, the effect of width expansion (see Eq. 2.14) is to increase the magnitude of the turbulent stress moving downstream. The effect of containing the width of the jet downstream is therefore to decrease the turbulent stresses downstream. This makes sense as the stresses would be expected to lessen as the velocity gradient weakens. So, although the model does not simulate the exact conditions, it is still reasonable to apply this algebraic model because: • It is still a jetlike flow. • The mixing length is still assumed to be based on a direct relation to the jet halfwidth. • The magnitude of the turbulence constant ("I) will still be essentially the same. Even for widely varying types of freeshear flows, the values of "I are all of the same magnitude, from 0.071 for mixing layers (which can produce the largest velocity gradients) to 0.180 for wakes (which produce small velocity gradients). The value Wilcox [1993] suggests for a plane jet is y=0.098 (31) The main consequence of turbulent stress on the mean flow is an increased viscous effect that acts to flatten out the velocity profile. The effect can be seen in the comparison of the flowfield for the 45° pleat derived assuming the following turbulent stress conditions: laminar flow, our modeled turbulence coefficient, and double that value. 60 0.001 0.0008 0.0006 0.0004 0.0002  ..."..        ~,' ." .".. . ,  ~>" ~ ..~",,,., " ' ~",."."" ... ____ " " ~411* __ ., .. 0~~~~~~~~~~~~~~~_r~T_~_r~~~~~~~~ o 0.001 0.002 0.003 0.004 0.005 0.006 0.001 0.0008 0.0006 0.0004 Fig.3.8(a) Flowfieldfor y = 0 (full and proportional, linearscale vectors). " "  ...... ......:' " ..,..,..  " 'A ______________ _ , , , ', .,      ...1" , ." "" _ .. ___ _ 0.0002 '" 0.007          ~ ....   ., ..... 0~~~~~1~r~~1r~~~1~r~·~~1~r~~1~r~~1r~~~ o 0.001 0.002 0.003 0.004 0.005 0.006 0.001 0.0008 0.0006 0.0004 0.0002 Fig.3.8(b) Flowfieldfor y = 0.098 (full and proportional, linearscale vectors).       ."... ,. ~>"  ~,' ",. ;' "' "       ~~", ~/~~,  .....  ~/ ;' , ,..,.               " " ~ .....   .'. 0.007 0~~~~~1~~r~1r~~~1~r~~~1~r~~1~rr~1r~~~ o 0.001 0.002 0.003 0.004 0.005 0.006 0.007 Fig.3.8(c) Flowfieldfory = 0.196 (full and proportional, linearscale vectors). Figs.3.8 The effect oiY on the downstream flow. One side effect of applying this turbulence model in all nonmedia regions of the flowfield is that it is operative in the crevasse of the pleat as well. This increases the viscous and turbulent drag within the crevasse even though the region may be prohibitively small for the development of turbulence. One solution would be to tum the turbulence model off within the pleat crevasse. However, this was not done here. The effects of this phenomenon are noted in Section 4.4. 61 3.5 Upstream and Downstream Coverage (htup and htdown) The remaining parameters examined affect the run time of the program. As the ponion of the flowfield that is of panicular interest is the flow immediately entering, within, and exiting the filter media, the flow far upstream and downstream of the filter is only of interest insofar as it affects the filter region. The flow upstream of the filter is affected by the lefthand boundary condition, a uniform inlet flow. If this boundary condition is set too close to the filter, an anificially large velocity gradient is created. Ideally, there should be enough distance between the lefthand boundary and the filter so that the lefthand boundary condition has no effect on the gradient. As this effect is difficult to distinguish in a vector plot, the values of the uvelocity before the pleat are shown for two values of htup (0.5 and 1.63) for a 45° pleat (Table 3.6). The numbers are aligned so that the filter regions coincide. As can be observed, a very short runup (htup = 0.50) to the filter forces the velocity gradient; this has some effect on the intrafilter flow as well. With a long runup (htup = 1.63) to the filter, the velocity gradient develops naturally. On the downstream side, upstream differencing used in the finite difference approximation for the convective terms limits the influence of downstream phenomena on upstream locations. The righthand boundary condition assumes the velocity gradient has disappeared before reaching the exit. However, this is not true unless a very lengthy exit length is used. If a gradient does actually exist at the exit, the continuitive outflow condition will anificially force the gradient at the exit. 62 The effect of this artificial condition can be observed by comparing a flowfield solved for different values of htdown (Table 3.7). It can be seen that the exit boundary condition does affect the flowfield near the exit, but barely affects the flow near and within the filter. htup = 0.50: htup = 1.63: 1 1 Reached Convergence Criterion? Yes. 1 No. of iter ~ 20 1 Upstream Pressure ~ .49653E+03 Pa 1 11 U velocity 2 3 4 3.000 3.000 2.999 2.998 6 3.000 3.000 2.999 2.998 5 3.000 3.000 2.999 2.999 4 3.000 3.000 3.000 3.000 3 3.000 3.000 3.001 3.001 2 3.000 3.000 3.001 3.002 1 3.000 3.000 3.001 3.002 11 5 6 7 8 9 10 ! Reached Convergence Criterion? Yes. 7 2.996 2.992 2.985 2.971 2.946 2.897 I I No. of iter ~ 20 6 2.996 2.992 2.985 2.971 2.946 2.897 ! Upstream Pressure ~ .49755£+03 Pa 1 2.997 2.995 2.991 2.983 2.968 2.942 11 3.000 3.000 3.000 3.8C1 3.002 3.006 3 3.003 3.005 3.009 3.018 3.033 3.06? U velocity 2 3.004 3.008 3.015 3.028 3.051 3.093 1 3.004 3.008 3.015 3.028 3.051 3.093 2 3 5 6 11 12 13 14 15 16 7 3.000 2.821 2.325 .843 1. 518 2.234 7 2.802 2.609 2.163 .832 1. 514 2.232 6 3.000 2.821 2.325\ .843 1.518 2.234 6 2.802 2.609 2.163 \ .832 1. 514 2.232 5 3.000 2.948 2.887 3.244\ 1.393 1.965 2.898 2.827 2.764 3.116\ 1.391 1. 963 3.000 3.047 3.191 3.599 3.934\ 1.793 3.018 3.055 3.173 3.528 3.840\ 1.792 3 3.000 3.088 3.292 3.666 4.075 4.435 3 3.115 3.215 3.399 3.718 4.074 4.397 2 3.000 3.096 3.305 3.648 4.079 4.573 2 3.167 3.294 3.500 3.806 4.182 4.616 1 3.000 3.096 3.305 3.648 4.079 4.573 1 3.167 3.294 3.500 3.806 4.182 4.616 7 9 10 11 12 17 18 19 20 21 22 7 3.174 4.984 5.667 5.958 6.064 6.062 7 3.172 4.982 5.665 5.957 6.063 6.061 6 3.174\ 4.984 5.667 5.958 6.064 6.062 6 3.172\ 4.982 5.665 5.957 6.063 6.061 2.391 2.180\ 3.194 3.787 4.110 4.282 5 2.390 2.180\ 3.194 3.787 4.110 4.282 4 2.129 2.157 1.847\ 2.286 2.535 2.667 4 2.128 2.157 1. 848\ 2.286 2.535 2.667 3\ 2.llS 2.416 2.006 1. 431 \ 1.488 1.452 3\ 2.ll5 2.416 2.007 1.431\ 1.489 1. 4 02 2 5.192\ 3.263 2.285 1.538 .803\ .537 5.195\ 3.265 2.286 1.539 .803\ .538 5.192 3.263 2.285 1.538 .803 .537 5.195 3.265 2.286 1. 539 .803 .538 Table 3.6 The effect ofhtup on the upstreamjlow. 63 htdown = 0.88 htdown = 3.38 U velocity U velocity 1 3 6 2 6 7 3.000 2.99" 2.982 2.957 2.908 2.8:3 7 3.00C 2.995 2.982 2.957 7.9G8 2.8: 6 3.COO 2.995 2.982 2.957 2.908 2.813 6 3.00C 2.995 2.982 2.957 2.908 2.8:3 3.000 2.997 2.989 2.975 2.949 2.904 5 3.000 2.997 2.989 2.97" 2.949 2.904 3.000 3.000 3.00: 3.002 3.006 3.018 4 3.000 3.000 3.00" 3.002 3.0C6 3.018 3 3.000 3.003 3.8:1 3.026 3.055 3.109 3 3.000 3.003 3.012 3.026 3.055 3.109 2 3.000 3.005 3.017 3.040 3.082 3.157 2 3.000 3.005 3.017 3.040 3.082 3.157 3.000 3.00:: 3.017 3.040 3.082 3.157 3.000 3.005 3.017 3.040 3.082 3.157 7 8 10 11 12 8 9 10 .c ~ .? 7 2.617 2 .169 .833 1. 514 2.232 3 .172 7 2.617 2 .169 .833 1. 5" 4 2.232 3.172 6 2.617 2.169\ .833 1. 514 2.232 3.172 6 2.617 2.169\ .833 1.514 2.232 3.172 5 2.833 2.770 3.121\ 1.392 1.963 2.390 5 2.833 2.770 3.121\ 1.392 1. 963 2.398 4 3.056 3.174 3.53: 3.843\ 1. 792 2.128 4 3.056 3.174 3.531 3.843\ 1.792 2.128 3 3.209 3.395 3.716 4.074 4.398\ 2.115 3 3.209 3.395 3.716 4.074 4.398\ 2.115 2 3.284 3.492 3.799 4.177 4.614 5.194 2 3.284 3.492 3.799 4.177 4.614 5. J 94 1 3.284 3.492 3.799 4.177 4.614 5.194 3.284 3.492 3.799 4.177 ~ ~ 61 ~ 5.194 13 14 15 16 17 18 13 14 15 16 17 18 7 4.982 5.665 5.957 6.062 6.056 5.983 7 4.982 5.665 5.957 6.063 6.061 5.995 6\ 4.982 5.665 5.957 6.062 6.056 5.983 6\ 4.982 5.665 0.957 6.063 6.061 0.995 5 2.180\ 3.194 3.787 4.110 4.282 4.363 5 2.180\ 3.194 3.787 4.110 4.282 4.366 4 2.157 1.848\ 2.286 2.536 2.668 2.755 4 2.157 1. 848\ 2.286 2.535 2.667 2.749 3 2.416 2.00·1 1. 431 \ 1.490 1.454 1.436 3 2.416 2.007 1.431\ 1.489 1.452 ... 431 2\ 3.265 2.286 1. 53 9 .803\ .539 .463 2\ 3.265 2.286 1. 539 .803\ .538 .459 3.265 2.286 1. 539 .803 .539 .463 3.265 2.286 1. 539 .803 .538 .459 19 20 21 22 23 :'9 20 21 22 23 24 7 5.868 5.721 5.534 5.279 5.279 7 5.896 5.781 5.660 5.538 5.420 5.308 6 5.868 5.721 5.534 5.279 5.279 6 5.896 5.781 5.660 5.538 5.420 5.308 5 4.378 4.349 4.288 4.199 4.199 5 4.388 4.373 4.338 4.293 4.2,3 4.192 4 2.819 2.866 2.908 2.960 2.960 4 2.809 2.849 2.875 2.892 2.905 2.914 3 1.464 1. 530 1. 626 1. 760 1.760 3 1. 448 1.493 1.550 1. 612 1.673 I. "132 2 .470 .534 .645 .802 .802 2 .459 .504 .577 .665 .759 .853 .470 .534 .645 .802 .802 .459 .S04 .577 .665 .759 .853 25 26 27 28 29 30 7 5.202 5.103 5.010 4.925 4.845 4.771 6 5.202 5.103 5.010 4.925 4.845 4.771 5 4.143 4.096 4.051 4.009 3.970 3.934 4 2.922 2.928 2.934 2.939 2.943 2.948 3 1.789 1.842 1.891 1. 938 1.981 2.02l 2 .945 l.031 1.113 1.189 1.261 1.327 .945 1.031 1.113 1.189 1. 261 1.327 31 32 33 34 35 36 7 4.702 4.637 4.577 4.521 4.467 4.417 6 4.702 4.637 4.577 4.521 4.467 4.417 5 3.900 3.868 3.838 3.810 3.783 3.758 4 2.951 2.954 2.958 2.960 2.963 2.965 3 2.058 2.093 2.125 2.156 2.185 2.212 2 1.389 1.448 1. 502 1.553 1. 602 1. 648 1.389 1.448 1. 502 L553 1. 602 :.648 37 38 39 40 41 42 7 4.369 4.323 4.277 4.229 4.175 4.107 6 ,.369 4.323 4.277 4.229 4.175 4.107 5 3.734 3.710 3.687 3.663 3.636 3.606 4 2.968 2.970 2.972 2.975 2.979 2.987 3 2.238 2.263 2.289 2.315 2.345 2.382 2 ".691 1.734 1.775 1.818 1.864 1.917 1 1. 691 1.734 1.775 1.818 1.864 1.917 (col umn 43 ~ 42) Table 3.7 The effect ofhtdown on the downstreamJlow. 64 3.6 The Grid Expansion Coefficients (eta and zeta) One theoretically simple method to deal with the entrance and exit boundary conditions is to introduce an expandable grid. As detail becomes less important moving up and downstream of the filter, the cells can be expanded in those directions. In this way, the effect of the boundary conditions at the up and downstream positions is diminished, while keeping the cell count low. The cell expansions begin at the xcells two positions upstream and downstream of the pleat. The initial sample of the 10° pleat (Fig. 3.3) had a long enough runup to the filter so that the flow could develop naturally from the upstream boundary condition. For comparison, eta is increased to 1.2 and icell is adjusted (via cup) so that the upstream distance (htup) is kept nearly the same; so only five upstream cells are used on the right but eight on the left. eta = 1.00, cup = 0.90, htup = 0.892 eta = 1.24, cup = 0.50, htup = 0.897   U velocity U velocity 3 4 3.000 3.000 3.000 3.000 3.000 2.999 6 3.000 3.000 2.999 3.000 3.000 3.000 3.000 3.000 3.000 5 3.000 3.000 2.999 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.001 2 3.000 3.000 3.000 3.000 3.000 3.001 2 3.000 3.000 3.001 10 11 12 7 6 2.992 2.880\ .047 .095 .360 1. 490 2.986 2.876\ .047 .095 .360 1. 490 2.995 2.927 2.392\ .088 .329 1.283 2.992 2.925 2.391 \ .088 .329 1. 283 3.000 3.005 3.621 3.390\ .275 1.096 4 3.000 3.004 3.620 3.390\ .275 1.096 3 3.005 3.074 4.299 5.207 5.697 \ .916 3 3.008 3.076 4.299 5.207 5.697\ .916 2 3.008 3.114 4.64 : 6.220 8.339 10.216 2 3 014 3.119 4.642 6.221 8.339 10.216 13 14 15 16 17 18 10 11 12 13 14 15 6\ 9.638 9.135 ".566 6.401 5.582 5.029 6\ 9.638 9.135 7.566 6.401 5.582 5.029 5 1. 084 \ 5.025 5.019 4.647 4.289 4.036 5 1. 084 \ 5.025 5.019 4.647 4.289 4.036 4 1. 238 .225 \ 2.400 2.785 2.859 2.905 4 1. 238 .225\ 2.400 2.785 2.859 2.905 3 1. 415 .290 .003\ 1.182 1. 589 1.882 3 1. 415 .290 .003\ 1.182 1.589 1.882 2\ 1. 624 .326 .012 .015\ .681 1.147 2\ 1.624 .326 .012 .015\ .681 1.147 19 20 21 22 23 24 16 17 18 19 20 21 6 4.642 4.359 4.145 3.979 3.847 3.740 4 642 4.359 4.145 3.979 3.847 3.740 5 3.854 3.719 3.614 3.531 3.464 3.408 3.854 3.719 3.614 3. ~31 3.464 3.408 4 2.935 2.954 2.967 2.975 2.981 2.985 4 2.935 2.954 2.967 2.975 2.981 2.985 2.090 2.242 2.358 2.448 2.521 2.579 2.090 2.242 2.358 2.448 2.521 2.579 2 1. 4 79 1.726 1.917 2.067 2.187 2.287 2 1. 4 79 1.726 1.917 2.067 2.187 2.287  Table 3.8 Effect of eta on the flow upstream of pleat. 65 It can be seen that the velocity values entering the pleat and within the pleat are the same. Likewise, an increased zeta grid expansion coefficient downstream can allow the flow profile distance enough to reestablish a flat profile under viscous forces. Note that the expanded grid on the right reaches the standard uniform flow condition. zeta = 1.0, cdown = 1.2, htdown = 1.226 u velocity 2 5 6 6 3.000 3.000 3.000 3.000 3.000 2.999 5 3.000 3.000 3.000 3.000 3.000 3.000 4 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.001 9 10 11 12 6 2.992 2.880\ .047 .095 .360 1.490 5 2.995 2.927 2.392\ .088 .329 1.283 3.000 3.005 3.621 3.390\ .275 1.096 3.005 3.074 4.299 5.207 5.697\ .916 2 3.008 3.114 4.641 6.220 8.339 10.216 13 14 15 16 6\ 9.638 9.135 7.566 6.40~ 1.084\ 5.025 5.019 4.647 4 1. 238 3 1.415 2 \ .624 .225\ 2.400 2.785 .290 .003\ 1.182 .326 .012 .015\ 17 5.582 4.289 2.859 1. 589 .681 18 5.029 4.036 2.905 1.882 1.147 19 20 21 22 23 24 6 4.642 4.359 4.145 3.979 3.847 3.740 5 3.854 3.719 3.614 3.531 3.464 3.408 2.935 2.954 2.967 2.975 2.981 2.985 2.090 2.242 2.358 2.448 2.521 2.579 1.479 1.726 1.917 2.067 2.187 2.287 25 26 27 28 29 6 3.652 3.578 3.511 3.415 3.415 3.362 3.323 3.288 3.237 3.237 2.989 2.991 2.994 2.997 2.997 2.628 2.669 2.706 2.761 2.761 2 2.369 2.439 2.500 2.590 2.590 zeta = 1.8, cdown = 1.2, htdown = 89.42 U velocity 4 5 3.000 3.000 3.000 3.000 3.000 2.999 3.000 3.000 3.000 3.000 3.000 3.000 4 3.000 3.000 3.000 3.000 3.000 3.000 3 3.000 3.000 3.000 3.000 3.000 3.000 2 3.000 3.000 3.000 3.000 3.000 3.001 7 8 10 11 12 6 2.992 2.880\ .047 .095 .360 1.490 5 2.995 2.927 2.392\ .088 .329 1.283 3.000 3.005 3.621 3.390\ .275 1.096 3.005 3.074 4.299 5.207 5.697\ .916 2 3.008 3.114 4.641 6.220 8.339 10 216 13 14 15 16 6\ 9.638 9.135 7.566 6.400 1.084\ 5.025 5.019 4.647 4 1.238 .225\ 2.400 2.786 3 1.415 .290 .003\ 1.183 2\ 1.624 .326 .012 .016\ 17 5.561 4.283 2.868 1.602 .685 18 4.823 3.945 2.930 1.995 l.307 19 20 21 22 23 24 6 4.177 3.684 3.346 3.142 3.043 3.009 5 3.627 3.375 3.196 3.083 3.026 3.005 2.964 2.984 2.994 2.998 3.000 3.000 3 2.342 2.611 2.799 2.915 2.974 2.995 1.890 2.346 2.665 2.861 2.957 2.991 25 26 27 28 29 6 3.001 3.000 3.000 3.000 3.000 5 3.001 3.000 3.000 3.000 3.000 4 3.000 3.000 3.000 3.000 3.000 3 2.999 3.000 3.000 3.000 3.000 2 2.999 3.000 3.000 3.000 3.000 Table 3.9 Effect o/zeta on the flow downstream o/pleat. 3.7 The Pleat Height (ht) As the number of cells widthwise in the grid (jbar) for a given pleat angle (8) and given pleat thickness (tp) is a function ofthe pleat height (ht) (see Table 2.1), reducing the pleat 66 height is one way of reducing the number of grid cells. Obviously, reducing the pleat height will create a different (higher) pressure drop across the pleat, but it may be an effective way to observe flow trends just the same. With regard to the latter, the main concern with this grid reduction is its affect on distinguishing the flow trend through the section of the filter away from the direct effect of the pleat folds, defined as the free length of the pleat. freelength = ht 2tx freelength freelength 2ht Fig.3.9 Freelength as related to filter height. If the height of the pleat is reduced so that ht is in the same range as tx, then the flowfield is dominated by the pleat folds. If we are trying to simulate a flow for a pleat of a greater height, the simulation will not be realistic. In the case of a small pleat angle (i.e. 3°) with the triangle geometry, the pleat folds do dominate, and this is a realistic portrayal of the flow. For larger pleat angles, the pleat fold area should not dominate. So if ht = 2tx, there is little freelength. For comparison, flowfields are obtained for a 45° pleat with varying pleat heights. Note that the increase in freelength leads to a decrease in the influence of the pleat folds and an increase in the filter area which in turn decreases the overall pressure drop. 67 0.001 0.0008 0.0006 0.0004 0.0002 0 0.001 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 , ' '"     ~~....  ....... ' ,  , , .,../. //'' //', /'   ,   ,, /' /' .... "  , ... !.. ~  .. , , p 1 , 0.0025 0.d03 0.0035 , 0.0015 0.002 0.004 0.0045 Fig.3.1O(a) ht = 1.79 mm. 0 freelength cells. upstream pressure = 414 Pa. , , ..  , """', '" ,  '   , / , ' / ,  , " " "  ), .... / /, " " ,  ,     ",   , "  . , O~~~~~I~~~~~'~~~~'~~~~~'~~~~'~~~~~'~~~~ 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Fig.3.l0(b) ht = 2.50 mm. 2 freelength cells. upstream pressure = 377 Pa. , 0.0025   ..... , ..  ... ,     , ...... ",,' , , r __ _ , 0.002  , , .,' , , "",,'" 0.0015 : 0.001 f , 0.0005 f l" , , '" , o ,  ,  1 I "1 I   ,  i'" "I " 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.0055 0.006 0.0065 0.007 0.0075 Fig. 3.l0( c) ht = 3.60 mm. 7 freelength cells. upstream pressure = 349 Pa. Figs. 3.10 Effect of pleat height on theflowfield 45° pleat (pruned and proportional, linearscale vectors). 68 It can be seen that with no area of the filter outside of the influence of the folds (ht = 1.79 mm), there is no region of the flow that orients normal to the pleat. Whereas with a larger freelength (ht = 2.50 or 3.60 mm), there is a region of flow oriented normal to the pleat. Judging from the similarities seen in the freelength region of the last two plots, it is suspected that this flow trend continues for greater pleat heights as well. So the pleat height can be set for a minimal number of freelength cells (2  4), where the number of freelength cells is given by: # of freelength cells = jbar  (nnode + 1) 3.8 The Grid Density (njiCx) Obviously, the denser the grid, the clearer the trends in the flow and the better the approximations to the PDE are, and vice versa. On the other hand, the denser the grid is, the longer a solution will take. Thus the effects of varying the fineness of the grid (determined by nfil_x only, for the triangular pleat) are examined to see the consequences. It is kept in mind that there comes a point where the grid is too fine. Recall that the velocity within the filter represents an areaaverage macroscopic velocity. The real velocity is related to the macroscopic velocity by media porosity: v = VdaTCY actual 8 (32) So if the grid were to shrink below the actual size of the filter pore, the velocities would no longer be describing a true macroscopic velocity for the cell. Thus a minimum cell dimension is specified by the pore size. The average distance between fibers given an 69 average fiber diameter of 39 !lm and a porosity of 0.77 [Sabnis, 1993] is about 32 !lm, assuming a facecentered cubic geometry. The average pore size based on the square root of the permeability of the media, following Bejan r 1984], is 9 !lIn. As the media thickness considered in this report is only about 600 !lm, this would set a limit to the grid detail specified. For the 3° pleat, runs are made with the grid fmeness varied (Fig. 3.11). 0.0008 0.0006 0.0004 0.0002 0 0.02 0.0008  0.0006  0.0004  0.0002_ 0 0.02 ...  , ,   , I I 0.025 0.03 .. '" ... . .... ... ..... : .....  .... :.. . ' , .  '   , , . :........ I I 0.035 0.04 Fig. 3.11 (a) nfil_x = 4. .....  , .... ,   , , ........ :a ., , ,  I 0.045 ..... _ ......... . .... ....... I 0.025 .: __ __ ___ ..a. ... f ...    I 0.03 I 0.035 I 0.04 Fig. 3.11 (b) nfily = 6. . ,. I 0.045 . ... .." I 0.05 I 0.05 0.055    0.055 o. 00081==~::::,:,:. , ::::::::::=:::,;:::,::_::;:===_=_=:=_=======~:::::j 0.0006= =.. .~. ..:.. ..:. ........ :. ..... .. . !." ;...... .:..:.. :::=:: =    0.0004 =:::::::.::.::.:' ':. ,: : " '::' : :: :: :: ::  ...... ..... .. ..... ..... ........ ...... ........ ... 0.0002        " .' ,. • O~~~~;~~T~~rT=~=r~;T~=r~~~~~~~~~r·'~~~,~~ I I I I I I 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Fig.3.11(c) nfily = 8. Figs.3.11 Single geometry with gridfineness varied, 3° pleat (pruned and expanded [yweighted ~700%]). 0.055 Although the increased resolution reveals greater detail, it does not reveal phenomena otherwise hidden. The main quality discovered is that the flow is oriented more normal to 70 the filter at the pleat entrance and exit, however this can be ascertained without the finer grid. As the greater detail comes at a high cost (computing time), it is not always sought on its own merit. 3.9 Selecting a Convergence Criterion (epsi) As the solution of the set of nonlinear equations is approached iteratively, some arbitrary condition of solution convergence must be specified to halt the iteration. Any or all of the three variables being solved (P,u,v) can be used in the convergence criteria. The key standard for the criterion is that it ensures iteration has been carried far enough so that a certain level of precision has been reached and will not be further influenced by continued iteration. Pressure could be used as a standard for convergence, but detennining a standard value is not possible as the magnitude of the pressure would vary greatly with d
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Title  CFD Model of Flow Through Air Filter Pleats 
Date  19950701 
Author  Tebbutt, Charles B. 
Document Type  
Full Text Type  Open Access 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  CFD MODEL OF FLOW THROUGH AIR FILTER PLEATS By CHARLES B. TEBBUIT Bachelor of Arts Columbia College, Columbia University New York 1988 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 July 1995 OKLAIIOMA STATE tJNIVERSITY CFD MODEL OF FLOW THROUGH AIR FIL TER PLEA TS Thesis approved: ~ Thesis Advisor ~~~ Dean of the Graduate College n ACKNOWLEDGMENTS I would like to express my gratitude to Dr. Frank W. Chambers for inviting me to participate in this filtration project and for his guidance in the production of this work. Dr. Chambers possesses an extensive knowledge of fluid dynamic subject matter. He approaches his courses and students with earnest zeal. I thank Dr. Ronald L. Dougherty for his support during work on the filtration project and for his detailed review of this work which detected many inconsistencies I had overlooked. I also thank Dr. A.J. Ghajar for participating on my committee and for his suggestions on the improvement of the paper. I express my appreciation for the assistance of my coworkers in this study: Guojiang Liu, Rob Duran, Faqiu Liang, and Rob Newman. Finally, I would like to express my deep gratitude to my wife, Sharon. She has selflessly supported our family during my studies at Oklahoma State. She provides love, horne, food, and joy. iii TABLE OF CONTENTS 1. Introduction ............................................................................................................. 1 1.1 Function of the Air Filter ........................................................................... 1 1.2 Filter Design and Pleating .......................................................................... 1 1.3 Motivation for Research ............................................................................ 3 1.3.1 Applications of Simulated Pleat Flowfields .................................. 3 1.3.2 Objective of this CFD ModeL ..................................................... 3 1.4 Filter Geometry ......................................................................................... 4 1.5 Models of Flow Across Pleated Filters ....................................................... 6 1.5.1 Review of Porous Media Flow  Darcy's Law ............................. 6 1.5.2 The Work of Gurumoothy ........................................................... 7 1.5.3 The Work of Cai ......................................................................... 8 1.5.4 The Work of Chen, Pui, and Liu .................................................. l0 1.6 Description of this CFD Model .................................................................. 11 1.6.1 General Method .......................................................................... 11 1.6.2 Key Assumptions ........................................................................ 12 II. Derivation of Numerical Method ............................................................................ 13 2.1 Overview of Method .................................................................................. 13 2.2 Flow Outside of the Filter .......................................................................... 13 2.2.1 Viscous Flow Equations .............................................................. 13 2.2.2 Turbulence Considerations .......................................................... 16 2.3 Flow Within the Filter ................................................................................ 22 2.3.1 Development of the IntraFilter Momentum Equation .................. 22 2.3.2 Calculation of Darcy Parameters .................................................. 26 2.4 Grid Development ..................................................................................... 27 2.4.1 Range of Coverage ...................................................................... 27 2.4.2 Calculation of Grid Geometry ...................................................... 30 2.5 Finite Difference Equations ....................................................................... 34 2.5.1 Overview ................................................................................... 34 IV 2.5.2 Finite Difference Formulations for Convective Terms ................. 36 2.5.3 Finite Difference Approximations for Other Terms ..................... 37 2.5.4 Complete Form of the Finite Difference Equations ...................... 39 2.6 Boundary Conditions ................................................................................. 40 2.7 Solution Method ........................................................................................ 41 2.8 Output of Program .................................................................................... 43 III. Investigation of Parameters ................................................................................... 44 3.1 Introduction ............................................................................................... 44 3.2 A Sample Run ........................................................................................... 45 3.3 The Upstream Differencing Parameter (n) ................................................. 54 3.4 The Mixing Length Constant (y) ................................................................ 59 3.5 Upstream and Downstream Coverage (htup and htdown) ........................... 62 3.6 The Grid Expansion Coefficients (eta and zeta) .......................................... 65 3.7 The Pleat Height (ht) ................................................................................. 66 3.8 The Grid Density (n/if x) ........................................................................... 69 3.9 Selecting a Convergence Criterion (epsi) ................................................... 71 3.10 The Square Pleat Geometry ..................................................................... 72 IV. Flow Simulations and Discussion .......................................................................... 73 4.1 Introduction ............................................................................................... 73 4.2 Flow Through the AF3192 Filter ............................................................... 74 4.3 Inlet Velocity ............................................................................................. 77 4.4 Pleat Angle ................................................................................................ 81 4.5 Pleat Height ............................................................................................... 84 4.6 Simulated Dust Loading ............................................................................. 85 4.7 Flow Through a Square Pleat. .................................................................... 88 4.8 Discussion of Results ................................................................................. 94 4.8.1 Suggested Experimental Confrrmation ......................................... 94 4.8.2 Reexamination of the Results from Cai.. ..................................... 95 4.8.3 Problems with the Boundary Condition at the Filter Interface ...... 95 4.8.4 Problems with the Turbulence Model .......................................... 96 V. Conclusion and Recommendations .......................................................................... 97 5.1 Conclusions From Study ............................................................................ 97 v 5.2 Further Refinements to Model ................................................................... 98 References ................................................................................................................... 100 Appendix  Experiment to Determine K and b for Specific Media ................................ 102 VI LIST OF FIGURES Fig. 1.1 Pressure drop vs. pleat density for given media type and pleat height, adapted from Brown [1993, p. 65] ................................................................ 2 1.2 Filtration efficiency vs. flow velocity, adapted from Stenhouse [1975]. The total efficiency E is the product of the efficiency via separate mechanisms: adhesion Eadh and collection EcoJl ................................... 3 1.3 Pleat shape .................................................................................................... 4 1.4 Schematic of filter .......................................................................................... 5 1.5 Square pleat geometry used by Chen et al. [1993] .......................................... 10 1.6 Assumed flow through pleats ......................................................................... 12 2.1 Coordinate orientation with reference to filter ................................................ 14 2.2 Expected flow streamlines through pleat, adapted from Brown [1993, p. 65] ................................................................. 14 2.3 Wake flow ..................................................................................................... 19 2.4 Jet flow ......................................................................................................... 19 2.5 Mixing layer flow ........................................................................................... 19 2.6 Flow profiles downstream of fllter ................................................................. 20 2.7 Choosing a characteristic width for the jet model ........................................... 21 2.8 Critical Rayleigh number vs. K/ J3, adapted from Katto & Masuoka [1966] ......................................................... 25 2.9(a) Grid simulation (triangular pleat) ................................................................... 27 2.9(b) Grid simulation (square pleat) ........................................................................ 27 2.10 Variable location in cell. ................................................................................. 28 2.11(a) Grid showing key parameters (triangular pleat) .............................................. 28 2.11(b) Grid showing key parameters (square pleat) ................................................... 29 2.12(a) Actual air / filter interface (triangular pleat) .................................................... 29 2. 12(b) Actual air / filter interface (square pleat at corner) .......................................... 30 vii 2. 13(a) Schematic showing geometric parameters (triangular pleat) ........................... 31 2.13(b) Schematic showing geometric parameters (square pleat) ................................ 31 2.14(a) Schematic showing expanded grid (triangular pleat) ....................................... 32 2.14(b) Schematic showing expanded grid (square pleat) ............................................ 32 2.15 2.16 3.1(a) 3.1(b) 3.1(c) 3.1(d) 3.2 3.2(a) 3.2(b) 3.3 3.3(a) 3.3(b) 3.3(c) 3.4 3.4(a) 3.4(b) 3.4(c) 3.4(d) 3.5 3.6(a) Local cell orientation ..................................................................................... 35 Boundary conditions ...................................................................................... 41 Sample of a vector plot, in proportion to the actual size of the flowfield. The actual flow data is 45° above horizontal. Although the magnitude of the flowfield varies, the vectors are set equallength; thus they represent direction only ................................................................................................. 49 Sample of a vector plot for same data, with the ydimension expanded 200%. Note the magnitude of the vectors is the same as above, but the direction is weighted equal in amount to the expansion ................................. .49 Same as (a), but the vectors are scaled linearly to represent magnitude ........... 50 Same as (a), but the vectors are logscale. This is better to represent magnitude if the magnitude within the flowfield varies greatly ........................ 50 Vector flowfields for 45° pleat, 2 mm high ..................................................... 51 Full and proportional flowfield, linearscaled vectors ...................................... 51 Pruned and proportional flowfield, linearscaled vectors ................................. 51 Vector flowfields for 10° pleat, 8 mm high ..................................................... 52 Full and proportional flowfield, linearscaled vectors ...................................... 52 Pruned and proportional flowfield, linearscaled vectors ................................. 52 Full and expanded flowfield [yweighted 500%], linearscaled vectors ........... 52 Vector flowfields for 3° pleat, 3 em high ........................................................ 53 Full and proportional flowfield, linearscaled vectors ...................................... 53 Pruned and proportional flowfield, linearscaled vectors ................................. 53 Full and expanded flowfield [yweighted 1600%], linearscaled vectors .......... 53 Full and expanded flowfield [yweighted 1600%], logscaled vectors ............. 53 Solution of Id convectiondiffusion problem with uniform grid and solutions east and west of the variable equal to 1 and 0 respectively, adapted from Patankar [1980, p. 96] .............................................................. 55 Flow through 3 ° pleat as in Fig. 4.4, a = 0 (full and expanded [yweighted 1600%], linearscale vectors) ....................... 56 viii 3.6(b) Flow through 45° pleat as in Fig. 4.2, a. = 0 (full and proportional, linearscale vectors) ................................................... 56 3.7 Flow downstream of filter .............................................................................. 59 3.8 The effect of "I on the downstream flow ......................................................... 61 3.8(a) Flowfield for "I = 0 (full and proportional, linearscale vectors) ....................... 61 3.8(b) Flowfield for "I = 0.098 (full and proportional, linearscale vectors) ................ 61 3.8(c) Flowfield for "I = 0.196 (full and proportional, linearscale vectors) ................ 61 3.9 Freelength as related to filter height ............................................................... 67 3.10 Effect of pleat height on the flowfield 45° pleat (pruned and proportional, linearscale vectors) ................................ 68 3.1O(a) ht = 1.79 mm, 0 freelength cells, upstream pressure = 414 Pa ......................... 68 3.1O(b) ht = 2.50 mm, 2 freelength cells, upstream pressure = 377 Pa ......................... 68 3.1O(c) ht = 3.60 mm, 7 freelength cells, upstream pressure = 349 Pa ......................... 68 3.11 Single geometry with grid fineness varied, 3° pleat (pruned and expanded [yweighted 700%]) ..................................... 70 3.11 (a) nfil_x = 4 ....................................................................................................... 70 3.11 (b) nfu_x = 6 ....................................................................................................... 70 3.11(c) nfil_x = 8 ....................................................................................................... 70 4.1 4.1(a) 4.1(b) 4.1(c) 4.2 4.2(a) 4.2(b) 4.2(c) 4.2(d) 4.3 4.4 4.5. Simulated flow through an AF3192 filter pleat (4° pleat, 3 em high) ............... 75 [YWeighted 327%, Linearscale Vectors] ..................................................... 75 [YWeighted 327%, Logscale Vectors] ........................................................ 75 [Proportional, Equallength Vectors] ............................................................. 75 Flowfields for two different inlet velocities ..................................................... 79 Inlet velocity of 10 m/s [Y Weighted 273%, Logscale Vectors] .................... 79 Inlet velocity of 0.5 m/s [Y Weighted 273%, Logscale Vectors] ................... 79 Inlet velocity of 10 m/s [Proportional, Equallength Vectors] ......................... 79 Inlet velocity of 0.5 m/s [Proportional, Equallength Vectors] ........................ 79 Pressure drop vs. inlet velocity for varying pleat angles. The curves are each separately scaled versus their first datum point at V =0.5 m/s ............ 80 Control volume for flow entering fI1.ter pleat. ................................................. 81 Angle vs. pressure drop. The value at the far right represents the analytical value of flow through a flat pleat; it is connected linearly to the tails of the curves for demonstration only ....................................................... 82 ix 4.6 4.6(a) 4.6(b) 4.6(c) 4.7 4.8 4.9 4.9(a) 4.9(b) 4.9(c) 4.9(d) 4.10 4.11 Comparison of flowfields for different pleat angles ......................................... 83 3.0° pleat, 3 em high [YWeighted 330%, Linearscale Vectors] .................... 83 4.2° pleat, 3 em high lYWeighted 330%, Linearscale Vectors] .................... 83 13.6° pleat, 1 em high [YWeighted 330%, Linearscale Vectors] .................. 83 Height vs. pressure drop for two pleat angles ................................................ 84 Permeability vs. pressure drop for two pleat angles (log scale) ....................... 86 Comparison of flowfield for clean and simulated dirty filter (4° pleat, 3 em high) ...................................................................................... 87 Clean fIlter, K = 7.8ell m2 [YWeighted 312%, Linearscale Vectors] ........ 87 Dirty fIlter, K = 7.8e12 m2 [YWeighted 312%, Linearscale Vectors] ......... 87 Clean fIlter, K = 7.8el1 m2 [Proportional, Equallength Vectors] ................. 87 Dirty filter, K = 7.8e12 m2 [Proportional, Equallength Vectors] ................. 87 Comparison of angles (8) for triangular and square pleat.. .............................. 88 Pleat angle vs. pressure drop for two geometries. (Square pleat angle converted to triangular equivalent.) ................................................................ 91 4.12 Flow through square pleat, 4° angle, 3 em high ............................................ 92 4. 12(a) [YWeighted 330%, Linearscale Vectors] ..................................................... 92 4.12(b) [YWeighted 330%, Logscale Vectors] ........................................................ 92 4.12(c) [Proportional, Equallength Vectors] ............................................................. 92 4.13 Flow through square pleat, 13° angle, 1 em high .......................................... 93 4.13(a) [Proportional, Linearscale Vectors] .............................................................. 93 4.13(b) [Proportional, Logscale Vectors] .................................................................. 93 4. 13(c) [Proportional, Equallength Vectors] ............................................................. 93 A.l Setup for experiment to determine Darcy parameters ..................................... 103 A.2 Curve fit for one layer of filter media ............................................................. 104 x Table 2.1 3.1(a) 3.1(b) 3.2 3.3 3.4 3.5(a) 3.5(b) 3.6 3.7 3.8 3.9 4.1 4.2 4.3(a) 4.3(b) 4.4 4.5 LIST OF TABLES Program parameters ....................................................................................... 33 Input to a sample run ..................................................................................... 46 Raw data from file OUTPUT.DAT ..................................................................... 47 Input summary for a 45° pleat, 2 mm high ...................................................... 51 Input summary for a 10° pleat, 8 mm high ...................................................... 52 Input summary for a 3° pleat, 3 cm high ......................................................... 53 Effect of a on program stability, 3 ° pleat ....................................................... 57 Effect of a on program stability, 45° pleat. .................................................... 58 The effect of htup on the upstream flow ......................................................... 63 The effect of htdown on the downstream flow ................................................ 64 Effect of eta on the flow upstream of pleat.. ................................................... 65 Effect of zeta on the flow downstream of pleat .............................................. 66 Input parameters for AF3192 filter flow simulation ........................................ 74 Flow direction angle from AF3192 simulation ................................................ 76 Flow direction angle for inlet flow = 10 rn/s (4° pleat, 3 cm high) ................... 80 Flow direction angle for inlet flow = 0.5 rn/s (4° pleat, 3 cm high) .................. 80 Input data for square pleat simulation (3 cm high, 3° angle) ........................... 89 Input data for square pleat simulation (1 cm high, 10° angle) ......................... 89 xi b e edown cup D d, d(x) dele delx dely dnlgth e eta f flgth freelength g h NOMENCLATURE inertial factor in fIlter momentum equation packing factor, equal to 1/0; dummy constant in code, number of cells downstream relative to ipleat in code, number of cells upstream relative to ipleat average diameter of fibers in media width of jet used in turbulence model, equal to, the halfpleat width; called wd in code in code, cell dimension along pleat length (idirection) in code, cell dimension (idirection) in code, cell dimension (idirection) in code, total length downstream of filter variable used in calculation of Darcy parameters, K and b in code, upstream expansion coefficient for grid in code, xmomentum equation; also variable used in calculation of Darcy parameters, K and b in code, total length of flowfield number of flowwise cells within pleat not directly influenced by pleat folds in code, ymomentum equation; gravity in code, continuity equation xii ht htdown htup ibar icell idown iml imax ipleat j J jbar jcell jml jmax K lmfp lmix L, I nfily in code, pleat height in code, length of downstream flowfield relative to pleat height in code, length of upstream flowfield relative to pleat height in code, cell counter in xdirection in code, number of cells within the flowfield in the xdirection in code, first i cell of pleat (xdirection) in code, number of cells downstream of filter relative to ipleat in code, imax  1 in code, number of cells flowwise including boundaries in code, number of i cells within pleat in code, cell counter in ydirection unit flow direction vector in code, number of cells within the flowfield in the jdirection in code, first j cell of pleat (ydirection) in code, jmax  1 in code, number of cells widthwise including boundaries filter media permeability molecular mean free path mixing length descriptive length in code, for square geometry, number of cells spanning length of pleat median in code, for square geometry, number of cells spanning pleat section oriented normal to flow for triangular geometry, number of cells spanning pleat thickness in code, for square geometry, number of cells spanning width of pleat median xiii p p,Pij Q R t tp tx ty U up/gth u, Uij v VD Vmix Vth v, Vii wd x, Xi z zeta mean pressure in code, mean pressure normalized for density volume flow rate fiber radius thickness of the filter media; time in code, pleat thickness in code, for triangular geometry, pleat thickness in idirection in code, for triangular geometry, pleat thickness in jdirection mean velocity in flow direction (xdirection) in code, length of flowfield upstream of filter Reynold's averaged turbulent velocity fluctuations in code, mean velocity in flow direction (idirection) mean velocity in transverse direction (ydirection) Darcian velocity (areaaveraged velocity) mixing velocity pore velocity molecular velocity in code, mean velocity in transverse direction (idirection) in code, pleat width flow direction transverse direction direction along pleat in code, downstream expansion coefficient for grid xiv ex o 'Y Ilf Ilt Vf Vt e eo e~ p qs v ~ o in code, upstream differencing coefficient porous media porosity, equals lIc in code, constant to determine mixing length in turbulence model dynamic viscosity of fluid turbulent dynamic viscosity kinematic viscosity of fluid turbulent kinematic viscosity pleat angle pleat angle for square geometry pleat angle for triangular geometry density dummy variable gradient represents triangular geometry represents square geometry xv Chapter 1 Introduction 1.1 Function of the Air Filter The combustion engine derives its energy from the exothermic reaction of fuel and air. As this reaction takes place within a collection of carefully gauged moving parts, it is important that the reactants are free from particulate matter that could damage the engine over time. Therefore filters are necessary to glean particles from the air and fuel as well as the lubricating oils used within the engine. Air filter systems are designed to rid the air of all particles larger than roughly 1 Jlm in diameter, as these exceed the oil film thickness between moving parts. Smaller particles can also cause problems, so their removal is sought as well [Jaroszczyk et aI., 1993]. 1.2 Filter Design and Pleating Air filter design is guided by the sometimes opposing concerns of high efficiency and low cost. Thereby, the following conditions and requirements guide filter design: • small space available within engine compartment; • high required flow rate to fuel engine; • low pressure drop across filter to reduce energy required to supply flow of air; • high filtration efficiency for particles to submicron level; • long filter lifetime, i.e. have large dust load capacity; • low cost. 1 One of the key design features of air filters is pleating of the filter media. Pleating helps fulfill the above design specifications in several ways. It increases the effective area of filtration which immediately increases filter capacity and filter efficiency. Pleating also serves to reduce the flow velocity through the filter media for a given flow rate. This decreases the pressure drop across the filter, thus requiring less energy to supply a given flow rate to the engine. A schematic of the effect of pleating on the pressure drop across a filter is shown in Fig. 1.1. This shows that the more compact the pleating, the lower the pressure drop, up to the point where the pleats begin creating a restriction in the flow, thereafter the pressure drop rises [Brown, 1993, p. 641. pleats / unit length Fig.1.1 Pressure drop vs. pleat density for given media type and pleat height, adapted from Brown [1993.p. 65J. The relationship between velocity and filtration efficiency is shown in Fig. 1.2. The velocity across the filter is slowed by pleating. So pleating can be manipulated to increase filter efficiency. 2 veloctty  Fig.1.2 Filtration efficiency vs.flow velocity, adapted from Stenhouse [1975]. The total efficiency E is the product of the efficiency via separate mechanisms: adhesion Eadh and collection Ecoll .. 1.3 Motivation for Research 1.3.1 Applications of Simulated Pleat Flov.fields Some of the advantages of fIlter pleating mentioned above are based on certain assumptions regarding flow through the pleats. Flow through pleats is assumed to be oriented normal to the pleats, thus the effective area of the filter is presumed to be the unpleated, flattened area of the filter. Also, the efficiency of a filter is a function of the velocity; when calculating filter efficiency using a theoretical model, it is necessary to know the velocity through the pleats. In addition, as the velocity will vary at different points along the pleat, efficiency can be gauged on a local basis within the filter. A flowfield model can also give a theoretical estimate of the pressure drop across a filter for different pleat configurations. 1.3.2 Objective of this CFD Model It is difficult to access the flow near and through a pleated filter with flow measuring equipment. In this project, a flow simulation program called PLEA TFLO is developed to 3 determine the actual path and magnitude of the flow through the pleats. This simulation program will allow the analysis of flow through pleats under varying conditions and varying geometries. It will also give a measure of the pressure drop across a filter. 1.4 Filter Geometry The automobile air filter generally consists of the pleated filter media supported by a wire mesh encased in a rubberized frame. The media is machine pleated and attached to the base by glue. The actual shape of the pleats is between a square and triangular wave. triangular wave square wave approximate pleat shape Fig.1.3 Pleat shape. The pleat angle of automobile engine air filters is generally near vertical. This serves to maximize filter area, reduce pressure drop, and lower intrafilter velocity. A schematic of 4 the Purolator AF3192 filter is shown below. The pleats are 3 cm high with a 3 mm pitch. The media is about 635 !lm thick. Bottom View Top View Side View Fig.l.4 Schematic offilter 5 rubberized frame 1.5.2 The Work of Guru moo thy Gurumoothy [1990] modeled the flow through an entire air induction system (AIS), including the filter. The flow was calculated with the PHOENICS code which solves fluid flow using the transient viscous flow equations with a kE model for turbulence. The 3 dimensional simulation included a boundary condition that accounted for the induction system walls. The filter was treated as a separate region in the flow, solving the flow according to the macroscopic qualities of the ftlter region. The equation solved within the filter region was this extended form of the Darcy equation. (12) This includes the direct relation between pressure drop and velocity related by filter resistance K which is the basic Darcy law. It also includes an account for pressure drop due to flow inertia related by a factor b. The velocities are all macroscopic values. An experimental method to calculate the parameters K and b for a specific media was offered by Gurumoothy [1990]. The two parameters are related to pressure drop in 1 dimensional form by (13) Integration of the equation over the filter width L and the substitution of QIA for U yields (14) This is of the form 7 (15) Using this relation K and b can be calculated experimentally. For the measurement of the width, L, the height of the entire pleated filter was used, thus the media region included a good deal of nonmedia space. The values Gurumoothy obtained were K = 8.3561e09m2 b = 1. 7875ge+03m1 (16) Gurumoothy validated his program using comparison with experimentally determined wall and internal pressure values found within the AIS. The CFD prediction was generally within 10% of the experimental value. It is also notable that the CFD values were found to be grid dependent, although to a small extent. 1.5.3 The Work ojCai Cai [1993] created a transient flow simulation program based on the SOLA program to analyze the detailed flow phenomena within the pleats. This program again applies the basic continuity and momentum equations of viscous flow over the region outside the filter; and turbulence was calculated with a kE model. Within the filter a momentum equation that preserved the basic viscous flow formulation was introduced. The equation was adapted from Vafai & Tien [1981]. Vafai & Tien start with the Darcy law for pore velocity, which is the actual velocity within the media pore, as opposed to the "Darcian velocity" which is a macroscopic velocity for a given cell (Vdarcy = o· Vpore ,0 == media porosity). The coefficient function of the second term on the right is the equivalent of b used above. 8 This basic momentum equation is then refonnulated for use within the porous media by volume averaging the variables to take into account pore space and media space. With V a small volume and Vr the part of the volume containing fluid, the volume average of a variable 'I' is ('I') = ~ J ~ 'I'd /I. This essentially reestablishes macroscopic variable values. Applying these to the NavierStokes fonnulation within the media yields p((V. V)V) = V(P}f + IlV2(V) + filter resistance (18) The filter resistance is incorporated using the Darcy tenns (Eq. 17) with the same volumeaveraged velocity as introduced above. pD(V=) V(P)f +g+IlV2 (V) IBl · (V) Fp B2 . ((V) . (V) ) (V) Dt K JK f I(V)I (19) Cai used this equation to represent momentum within the filter. The main difficulties in doing a transient analysis of detailed flow through a pleat are the large gradients encountered and the draconian stability criteria required as a consequence of these gradients. A transient analysis requires a sufficient number of time steps to reach a steady state flow condition. Cai's results seem to be limited by an exceedingly small time forwarding criterion required for stability. 9 1.5.4 The Work a/Chen, Pui, and Liu Chen, Pui, & Liu [1993] created a finite element method to solve for flow through pleated media. For flow within the media, an expression that combines the basic laminar momentum equation with Darcy's law is used. P (V .)v V =VP~ V +~V 2 V 82 D D K D 8 D (110) This is similar to the previous formulation except that it is steadystate and does not include an inertial resistance term, as their research involved only low flow velocities. The authors examined a square pleat geometry. inlet flow + + + pleat 'head" downstream channel channel pleat 'median' pleat 'bottom' Fig. 1.5 Square pleat geometry used by Chen et al. [1993). A description of their findings for the velocity flowfield follows. At the entrance of the pleat a portion of the flow passes through the pleat head, but the bulk of the flow enters the pleat channel. In the channel they found the tangential velocity at the pleat median to be near zero, there is very little horizontal flow. In the downstream channel the flow 10 exhibits similar characteristics. The flow emits from the downstream channel in a jetlike manner. The authors found the pressure drop to be linearly related to velocity. This suggests the dominance of the Darcy term. Pressure drop over a pleat geometry is dominated by media resistance at low pleat count (pleats per unit length) and viscous drag at high pleat count. An optimal pleat count (i.e. that which yields minimal pressure drop) was found where the combined effects of media resistance and viscous drag were minimized. Data were nondimensionalized to create a correlation curve for normalized pressure drop as a function of various filter parameters for various media types. This results in curves similar to that of Fig 1.2. 1.6 Description of this CFD Model 1.6.1 General Method In this CFD code, the steady state equations are solved directly. Three partial differential equations (PDEs) are used. These are the twodimensional continuity equation and the xand ycomponents of the momentum equation. The latter equation has two forms, one for extrafilter flow, one for intrafilter flow. The extrafilter form of the momentum equation is for viscous flow and includes a simple algebraic model for turbulent flow. The intrafilter equation includes the Darcy law modified to include the effects of flow inertia. The PDEs are translated into finite difference equations (FDEs) to be solved for a grid covering the flow region. The nonlinear FDE matrix is solved iteratively. The grid is sized based on given filter geometry and dimensions. The grid cells can expand up and downstream of the filter. The modeled filter can have either triangular or square pleats 11 (see Fig. 1.4). The results are written to files that can be viewed or further analyzed with vector plotting software. J .6.2 Key Assumptions As filter systems are generally given rather cramped spaces, the flow entering the filter chamber is not generally normal to the filter. In addition, the flow entering the filter comes down piping, so pipe geometry shapes the flow profile entering the filter. However, in this simulation we assume the flow enters the filter chamber normal to the filter. The inlet flow is assumed to be uniform flow with free slip at the boundaries (i.e. no normal flow component). The pleating is assumed to run infinitely. The permeability is assumed constant even though a reduction might occur at the comers due to glue and folding. Finally the fluid is taken to be clean single phase air. Fig. 1.6 Assumedflow through pleats. 12 Chapter 2 Derivation of Numerical Method 2.1 Overview of Method For the present study, a numerical solution is to provide velocity and pressure throughout the flowfield upstream, within, and downstream of a pleated filter. To arrive at a solution, the differential equation forms of the fundamental viscous flow equations are translated into finite difference equations (FDEs). The FDEs are then fit to a grid representing the flowfield, and they are solved for each gridpoint in the field. This method is contained in a FORTRAN code program called PLEA TFLO. 2.2 Flow Outside of the Filter 2.2.1 Viscous Flow Equations Fluid flow is governed by the laws of conservation of mass, momentum, and energy. Air flow at low Mach numbers can be considered an incompressible Newtonian flow. The viscosity can be assumed constant because the temperature varies insignificantly. Thus velocity and pressure can be found without the energy equation. Conservation of mass is expressed in the continuity equation. VV=O (21) 13 Conservation of momentum takes the fonn of the NavierStokes equations. DV 1 _ n2 =V'P+g+vjv V Dt p (22) Certain assumptions about the flowfield affect the final appearance of the FDEs. • Flow is sought in its steadystate condition. Time is not a factor. • Gravity plays an insignificant role. • Flow is considered to be two dimensionaL The velocity component along the pleat (z) is considered constant or zero, so the NavierStokes equation will have only two plane component directions (x and y). Fig.2.1 Coordinate orientation with reference to filter. • As seen in the direction of the expected flow streamlines of Fig. 2.2, the velocity gradients around the pleat will be large, so turbulent effects can be expected to be significant. This will affect the mean flow parameters, so it must be considered. Thus the fundamental equations are taken in their turbulent fonn. This is accomplished by splitting the Upstream AIr Flow ) ) ) Ftlter Streamlines Fig. 2.2 Expected flow streamlines through pleat, adapted from Brown [1993,p. 65}. 14 equation variables into mean (capital letters) and fluctuating (small letters) components, then timeaveraging the equations. This adds turbulent stresses (uiuj ) to the momentum formulation: (23) Taking into account these four assumptions, the fluid dynamic equations are represented with these three component equations: continuity: xmomentum: ymomentum: au + av =0 ax ay These three equations contain six unknowns: • the mean velocity terms, U, V; • the mean pressure, P; • the turbulent fluctuations of velocity, UU, UV, vv . 15 (24) (25) (26) So their solution requires further information. 2.2.2 Turbulence Considerations Turbulent fluctuations cannot be calculated directly, short of direct numerical simulation. However, they can be modeled fairly successfully. Models up to recent times have been based on the Boussinesq eddyviscosity approximation. This assumes a turbulence viscosity (Ilt or v t) that is analogous to molecular viscosity, except that it is based on the scale of the local turbulence rather than molecular scales. Similar to molecular viscosity's role in relating shear stress to the velocity gradient ( 'txy = 11' dU / dy), turbulent viscosity is used to relate the turbulent shear stress (UiUj) to the velocity gradient. dU UV=VI (27) dy However, unlike molecular viscosity, turbulent viscosity is not a constant property of the fluid. It varies with the flow; its value is a function of the flow. Various models of turbulence have been devised to solve for Ilt. Some of the more accurate models involve the addition of partial differential equations (PDEs) to the solution set, such as the k£ model which adds the turbulent kinetic energy (TKE) equation and the turbulent dissipation (TD) equation to the three conservation equations already given above, and solves for Ilt as a function ofTKE and ID locally. These additional PDEs complicate the solution matrix and increase computation time for solution. Moreover, Cai's [1993] computations made with the kE model showed turbulence to have little effect on the flow distribution through the filter. For these reasons, the use of extra PDEs was avoided. The "algebraic" models of turbulence are so called because they are based on geometric approximations of turbulence. They add no PDEs to the solution. Turbulent viscosity is modeled by analogy to molecular viscosity for a dilute gas following Wilcox [1993, p. 27 16 30]. Just as molecular viscosity is a product of the mean free path and an average molecular velocity, 1 ~ = pvlhlmfp Vlh == molecular velocity lmfp == mean free path (28) 2 so the turbulent viscosity is calculated as an analogous function of a length and a velocity. 1 ~I = pvmixlmix 2 (29) The mixing length (lmix) is an estimate of the distance over which an eddy maintains its directional momentum. The mixing velocity (vmix) is the product of the mixing length and the velocity gradient. so dU Vmix = C·lmixdy 1 2 dU ~I=c·p·lmix  2 dy (210) 1 2 dU or VI = C·lmix  2 dy (211) The constant and the ~ are absorbed in the mixing length. The turbulent viscosity is then substituted in Eq. (27) to give the formulation of the shear stress. (212) This yields a formulation for turbulent shear stress as a function of the mean flow variables. As velocity along the pleat length is considered to be zero or constant (dU / dz = 0), the shear stresses in the zdirection (uw) are insignificant [Townsend, 17 1976, p.196]. Turbulent normal stresses (UiUi ) are also less significant [ibid, p. 1901 so that in the fluid momentum equations, UV» UU, VV,UW, vw (213) It remains for the mixing length (lmu) to be determined. Wilcox [1993] has reviewed mixing length models calibrated for specific empirical models of selfsimilar turbulent flows. For selfpreserving free shear flows the mixing length is calculated as a product of a constant (y) and the width of the flow phenomenon (d(x)). lmix = y. d(x) (214) Free shear flows are qualified as turbulent flows not bounded by walls but bounded by a nonturbulent ambient fluid. The general classifications of these consist of wake, jet, and mixing layer flows. 18 ) ) ) ) ) ) ) ) Fig.2.3 Wake flow. Fig.2.4 letflow. Fig.2.5 Mixing layer flow. 19 Execution of the numerical method using the laminar equations showed that the flow emanating downstream from the filter has a profile similar to that shown in Fig. 2.6. The square pleat produces a similar, even stronger gradient. Fig.2.6 Flow profiles downstream o/filter. A series of strong and weak flow sections can be seen. The flow does not qualify as a mixing layer because the initial "ambient" velocities (Uj and U2 ) would change moving downstream. The wake model cannot be applied because the magnitude of the "defect" in the downstream flow is too large to be considered a wake [White, 1991, p. 259]. The flow is, however, similar to the general shape of a jet profile. It is strong in the middle and tapers off toward the sides, and the relative strength of the middle compared to the sides weakens as the flow moves downstream. There are several aspects in which the downstream filter flow is not analogous to the jet model. Each section is not bounded by a nonturbulent ambient fluid; it is bounded by an identical flow. Also, the flows spread into one another, so the profiles are unable to spread 20 out moving downstream. Moreover, a selfpreserving shape does not develop until about 20 diameters downstream [White, 1991, p. 471], farther than we wish to consider; however this "jet" does have a somewhat developed profile to begin with. Even with these shortcomings, the magnitude of the turbulent stresses should be reasonably gauged by the jet model. The main divergence with the archetype is that the flow width is constant rather than continually spreading. The practical result of this is that the scale of the turbulent stress will shrink moving downstream. This would be expected to occur as the velocity gradient decreases downstream, and the flow profile flattens out. So, to apply the jet model, a flow halfwidth (d) must be specified. The choice is either (a) to consider the flow as a positive jet stretching between low velocity points or (b) to view the flow as a set of positive and negative jets relative to the inlet flow. The former is chosen because that profile is truer to a jet profile, i.e. the velocity gradient inverts at the edges. Moreover, with this model the filter is considered to stretch infinitely, so the outflow need not be related to the magnitude of the inlet flow. Uo (0) (b) Fig.2.7 Choosing a characteristic width for the jet model. 21 Wilcox's value of'Y for a plane jet (0.098) is applied to the flow simulation. Since the flow width does not expand downstream, d(x) is considered a constant equal to half the width of a jet which equals the width of a halfpleat. A halfpleat is one half of the "V" that makes up the pleat. d(x) = d = halfpleat width (215) So the mixing length approximation is lmix = 0.098* d (216) and the turbulent shear stress is  (uv) = ( 0.098*d )2 dU dU dy dy (217) 2.3 Flow Within the Filter 2.3.i Development of the intraFilter Momentum Equation Within the filter, the flow obeys the same physical rules as outside the filter. Continuity still holds in the same form, but the momentum equation must be reformulated to take into account the fibrous media as well. Instead of entering the fibers into the flowfield, a macroscopic model of flow through a porous media can be applied. For a flow where viscous effects far outweigh inertial effects, known as Stokes flow, dimensional analysis of such a flow through porous media shows that the pressure drop across the media is directly proportional to the macroscopic velocity. The filter variables involved are media thickness (t), average fiber radius (R), and packing factor (c) (the proportion of media I space) [Brown, 1993, p. 33]. VP= ~tV fCc) R2 (218) 22 The filter variables are grouped inversely under one variable K that represents the permeability of the filter media. The result is Darcy's law for Stokes flow which shows the pressure drop to be a direct function of velocity (Darcian velocity). Vp=Il v K D (219) When the velocities are higher, inertia becomes appreciable and must be considered in a model of fluid momentum. A Reynolds number for flow through a filter can be calculated using fiber diameter (D) or the square root of penneability (.JK). VD ReD = vI (220) If Re is less than one, a condition of Stokes flow exists, and inertia is insignificant. As our work on this project has shown, actual inlet velocity over an air filter can vary significantly, from the Stokes flow range to the inertial range (see Sabnis [1993] and Newman [1994] for experimental flow regimes). From this, it can be assumed that the maximum velocity would be on the order of 10 rn/s. The fibers in the AF3192 filter media average approximately 40 Ilm in diameter [Sabnis, 1993]. The penneability for the media is 7. 8e 11 m2 • Thus the expected maximum Re D and Re.fK are 27 and 6 respectively. This is beyond the realm of Stokes flow, thus inertia can be expected to be significant and will be considered. However, with pleating effects, intrafilter velocities are often less than 1 rn/s, so in practice a condition near to Stokes flow will often exist. Vafai & Tien [1981] show inertia in porous media flow to be affected by penneability and a function based on penneability, a penneabilitybased Reynolds number (ReK), and the media geometry relating the layout of the fibers. (221) 23 with] = VDflV D I, a unit direction vector. These filterbased coefficients in front of _ inertia can be grouped together under one inertial coefficient, b. VP=b p( V ) D .VD J 2 (222) Note that the flow velocities found through the filter are area averaged, that is they represent the velocity in the cell assuming the media resistance exists but the fibers do not. Vafai & Tien term this the "Darcian fluid velocity". The actual velocities through the filter would be a function of the porosity, O. Thus, following continuity, the actual "pore" velocity would be  VD Vpore =8 However, Darcian velocities are used throughout this analysis. (223) Clearly, flow through the filter is laminar (Re  10). So by including the Darcy terms, the momentum equation for flow within the filter becomes P() IJ. 2 IJ. b( ) 0 V .V V =VP+V V V P V .V J 2 D DOD K D 2 D D (224) An analysis of the magnitude of the various terms allows for some pruning of this equation. Katto and Masuoka [1966] devised a criterion for the onset of convective flow within a porous medium. The criterion is a function of the Rayleigh number Ra, media permeability K, and media thickness L. 24 10' '" " '" 1 "~ I ~VE ~C I ~fs ~ ; I "\ i I'\. I 103 ! 106 10~ 104 10'" 102 10" 10° 10' KIt. Fig.2.8 Critical Rayleigh number vs. K/ I3 , adaptedfrom Katto & Masuoka [1 966/. Obviously, as the temperature gradient is assumed to be slight across the flowfield, Ra = O. However, to get an idea of the particular magnitude of the Rayleigh number for this problem, a ~ T of 1°C is assumed and the thermal diffusivity of the saturated media kmedia is assumed equal to that of air. From this we get K/ I3 "" 2.0e04 and Ra ::;: 2. 5e  02, well under the limit even if a significant discrepancy exists in the assumptions. Although flow within the media pores is certainly in the low Reynolds number regime, the magnitude of the viscous tenu in relation to the Darcy terms is minimal. Assuming extreme gradients (U  1 dU  1 d 2 U  1 dx  1. Oe  04 ), the magnitude of the viscous term is still well below those of the Darcy tenus'" . 11 d 2U ::;: 2400 ~U::;: 300 000 P b U 2 ::;: 42 000 B dx2 K ' 2 ' Viscous diffusion Darcy resistance Inertial Resistance ** Values of Darcy parameters to be derived below. 25 Thus the final momentum equation includes the pressure as a function of the Darcy resistance and flow inertia. This is referred to as the extended Darcy equation. (225) This is a macroscopic momentum equation for flow through porous media. The terms are negative because the pressure drops moving downstream in a flow. 2.3.2 Calculation of Darcy Parameters It remains for the Darcy parameters for particular media to be determined. Gurumoothy [1990] has demonstrated how to organize the Darcy equation as a function of pressure drop and flow rate. The Darcy equation is integrated over the media thickness (t), then velocity is replaced with the flow rate divided by the flow normal area (QI A). This yields This equation is of the form M=eQ+ fQ2 (226) (227) This is a 2nd degree polynomial. An experiment to measure pressure drop across a filter media versus flow rate was completed to derive e andf and thus K and b. The experiment was done using the media from the AF3192 filter. The values for K and b found from the experiment were K=7.8ell m2 b = 6.8e+04 ml (228) The experimental method is detailed in the appendix. These values differ considerably from those of Gurumoothy (Eq. 16) as that integration was carried over the whole height of the fIlter, while here the integration was done over the thickness of the media. 26 2.4 Grid Development 2.4.1 Range of Coverage Grid generation algorithms are created for both triangular and square pleats. The scope of the flowfield needs to be minimized as much as possible in order to permit as detailed an analysis as possible. The inlet flow is uniform, so, because of the symmetry of the geometry, coverage can be limited to a half pleat width. Filter Cells Grid Filter Fig.2.9(a) Grid simulation (triangular pleat). Filter Cells Grid Fig.2.9(b) Grid simulation (square pleat). 27 To set positioning points for the finite difference equations, a grid is set up. A rectangular grid cell is used with pressure in the center, uvelocity on the front wall and vvelocity on the top wall. v(t.j} o u(il J} p(1.j} u(1.j) IL.....~JI v(1J1) Fig.2.10 Variable location in cell. The key control parameters and grid are shown for each geometry below. The parameters are defined in Table 2.1. 1 jmax 1++ jml 1++ jbar I+++ 1 \. nfl[x Fig.2.l1(a) Grid showing key parameters (triangular pleat). 28 1 jmax t+tff.t.o jml jbor jcell 1 ... Fig.2.11(b) Grid showing key parameters (square pleat). Using this grid the mooeled filter / air interface would seem to be a jagged line for the triangular pleat and a smooth interface for the square pleat. However the actual interface is more closely related to a line connecting the pressure points at the cell centers, as the velocities on the left and lower sides of the cell have the characteristic (air or filter region) of the previous celL So the grid of the square pleat lacks the consistent cell symmetry of the grid of the triangular pleat. j i Fig.2.12(a) Actual air / filter interface (triangular pleat). 29 j i Fig.2.12(b} Actual air / filter interface (square pleat at corner). 2.4.2 Calculation of Grid Geometry The key criteria for setting up grids for each geometry are the angle of the pleating (8), the height of the pleating (ht), and the thickness of the pleat media (tp). The fineness of the grid is detennined by the number of cells across the pleat. There are three fineness parameters for the square pleat (nfil_x, nfil y, nfil_ c), and only one for the triangular pleat (rifil_x). The number of cells widthwise (jbar) and the number of cells along the pleat Upleat) as well as the cell dimensions are determined using this data (see Table 2.1 below). A key difference between the two geometries is what the angle 8 represents. For the triangular pleat, the angle 8A represents the slope of the media face versus the freestream. For the square pleat, the angle 80 is the tangent of the halfpleat width over the height. These are not the same; the difference is intensified for shorter pleats. 30 ) ) ) , ) , ( ~ ) 0 , .0 \ ",""sJ , , , , , ht , , , , , t~I4 f " , p X " ~.~ ............. nfil:.A, ....... ... , , , , , , , , , , , , , , , , , , , , , wd , , dely delx  icell ipleat idown Fig.2.13(a) Schematic showing geometric parameters (triangular pleat). ) r ]I hrtp , ) I I ;~ L ____________ ) ) ) )nllLY{ +, wd ~ ~nflCC dely}:a delx ~~~ ipleat Fig.2.13(b) Schematic showing geometric parameters (square pleat). 31 The number of cells up and downstream of the pleat are calculated relative to the number cells along the pleat Upleat) using the multipliers cup and cdown respectively. The grid is expanded up and downstream using the expansion coefficients eta and zeta respectively. I I I [.I. 111 Ud;:;I;:rn11ll I I I Fig.2.14(a) Schematic showing expanded grid (triangular pleat). 11111111111 tllllllill I ~~ ipleat Fig.2.14(b) Schematic showing expanded grid (square pleat). The table on the following page shows the derivation of all the geometric parameters. The userdefmed media thickness (tp) and pleat angle (8) are always maintained. However, as the precision of the settings is limited by the fineness of the grid, the pleat height (ht) can only be as precise as the grid dimension delx. If delx equals 1.0, then ht cannot equal 8.5, it can only be 8.0 or 9.0. Also note that the width of the flowfield is jbar* dely, this will be used as the jet width in the formulation for the mixing length [mix. 32 Parameter Summary PARAMETER SYMBOL SHAPE DERIVATION pleat angle e 8,0 pleat thickness tp 8,0 pleat height ht 8,0 cells across media (xdir.) filii x 8,0 cells across media (ydir.) nfil y 0 cells along media (xdir.) nfil c 0 relative # cells upstream cup 8,0 relative # cells downstream cdown 8,0 cell width across filt. (xdir.) delx 8 tp / sin(e ) / nfil_x 0 tp / nfil x cell width along flit. length dele 0 (ht  2tp2delx) I nfil_c (xdir.) cell width (ydir.) dely 8 delx * tan(e ) ° tp/ nfil y upstream expansion coelf. eta 8,0 downstream expansion coeff. zeta 8,0 length upstream of filt. uplgth 8,0 icell2 delx· Letak k=O length downstream offilt. dnlgth 8,0 idownl de/x· L zetak k=O pleat heights upstream htup 8,0 uplgthlht pleat heights downstream htdown 8,0 dnlgthlht total length of fiowfield figth 8,0 hHuplgth+dnlgth pleat width wd 8 ht * tan(e )  tp I cos(e ) 0 ht * tan(e ) Table 2.1 Program parameters (continued on next page). 33 first cell oiJzleat (ydir.) jcell 0 ((jbar+2)12)(nfil vI2)+ I # cells widthwise jbar ~,D wd I del}' # cells in pleat flowwise ipleat ~ nfil_x + jbar 0 2 * nlil x + 2 + nfil c first cell of pleat (xdir.) icell ~,D cup * ipleat + 1 + 0.5 # cells downstream offilter idown ~,D cdown * ipleat + 0.5 # cellsflowwise ibar ~,D icell  2 + ipleat + idown # cells widthwise inc. jmax ~,D jbar + 2 boundaries # cells flowwise inc. imax ~, 0 ibar + 2 boundaries Table 2.1 Program parameters (continued from previous page). 2.5 Finite Difference Equations 2.5.1 Overview The PDEs to be solved are as follows. Continuity Momentum outside the filter including turbulence model, conservative fonn (l(U') + (l(UV) +~ (lp _ v (l2U + (l2U ) _ ~(l .... )2 (lUi (lU = 0 ax ay p ax J ax2 dy2 dy ay dy a(uv) + a(v2) +~ ap _VJ (a2v + a2v )~(lmlX)2 au au =0 ax dy p dy ax2 ai ax ay ay 34 (229) (230,231) Momentum inside the filter (232,233) Each grid point has the continuity equation and the two components of the particular momentum equation to solve for three unknowns. These equations must be translated into finite difference equations based on grid location. A location within the cell was chosen as the central point for finite difference approximation for each equation. The xmomentum equations if) were centered about the uvelocity at the front wall; the ymomentum equations (g) were centered about the vvelocity at the top wall; and the continuity equation (h) was centered about the pressure variable location at the center of the cell. Finite difference approximations were made with central differencing and simple forward differencing. However, for the convective terms in the fluid momentum equations (d(UPj }Jdxj ), some amount of upstream differencing is necessary in order to maintain stability. The following diagram is a reference for orientation. v(I. 1.1+1) v(IJ+l) v(i+l.j+l) 0 0 0 u (Iol )+1) p(l·1J+l) u(l1 )+1) pO.j+1) uo. 1) p(l+1.J+1) uO+ ~ I) v(l1J) v(I.j) yO+1J) 0 0 0 v(Ir» PO·l.» u(I J) p(l.j) u j) p(1+1.j) u(Hl j J) v(I.1.j.l) yOJI) v(I+1.j·l) 0 0 0 0· .j·l) pO·l.J1) u(I ]1) PO.)1) u(. 1) p(i+1.J1) u{i+ .j·1 yO·l.j2) v(i+l.j2) i Fig.2.15 Local cell orientation. 35 2.5.2 Finite Difference Formulations/or Convective Terms For higher Reynolds number flows, a central difference representation of convection yields an unstable result. A way to ensure stability is to use upstream differencing  derivatives made with upstream and center stream grid points only. Physically, a central difference representation of convection is inappropriate, because, in fact, convection is really "received" from upstream and "transmitted" downstream [White, 1991, p. 199]. However, although sole use of upstream differencing ensures stability, it can lead to "an unnecessary amount of numerical smoothing" [flirt et aI., 1975]. So a combination of upstream and central differencing is used. The proportion of upstream differencing is specified by the parameter a. The FDEs for the convection terms are as follows [ibid]. d(UU) 1 2 '' = {(Ui, j + Ui + l,j) + a*IUi, j + Ui + l,jl(Ui,j  Ui + 1,j) dx 4.1x (234) (Ui l,j + Ui,j)2  a*lw  l,j + W,jl(Ui l,j  Ui,j)} d(UV) 1 = {( Vi,j + Vi + l,j)(Ui,j + Ui,j + 1) + alVi,j +Vi + l,jl(Ui,j  Ui,j + 1) dy 4~y (235) ( Vi,j 1 +Vi + l,j l)(W,j l+Ui,j)  alVi,j  1 + Vi + l,j  Jj(W,j  1  W,j)} d(UV) 1 '':'= {(w,j + w, j + 1)( Vi,j + Vi + l,j) + alw,j + W,j + Ij( Vi, j Vi + l,j) dx 4.1x (236) (w, j 1 + Ui,j + 1)( Vi  l,j +Vi,j) alw,j 1 + Ui,j + Jj( Vi  l,j  Vi,j)} d(W) 1 2 = {( Vi,j +Vi,j + 1) +alVi,j +Vi,j + Jj(Vi,j Vi,j+1) dy 4~y (237) ( Vi,j 1 + Vi,j)2  alVi,j 1 + Vi,jl( Vi,j 1 Vi, j)} Analysis of these equations shows that the a terms serve to cancel out the downstream (i+ 1) terms in the central difference approximations. 36 2.5.3 Finite Difference Approximationsfor Other Terms The remaining terms in the NavierStokes equations are centered on the particular variable within the cell as stated earlier. The pressure terms are represented with forward differencing. Pressure in the program is normalized with density, sop = t p . 1 ap (pi,j + 1 Pi,j) =~.:.... p ay ~y (238) The diffusion terms use central differencing. The turbulent shear stress formulations were also represented with central differencing. (241) !uv= !(k..': :)= ( 'b A)2 IUi,j+1Ui,A(Ui,j+1Ui,j)IUi1,j+1Ui1,jI(Ui1,j+1Ui1,j) 'Y' ] ar·uy . 2 ~.(~y) (242) 37 In the Darcy equation, the inertia tenn requires a calculation of total velocity liD; of course liD = ~U; + V; . The xDarcy equation is centered at the uvelocity variable position in the cell (refer to Fig. 2.10). So it is necessary to create a fonnulation for the vvelocity component at the/ront of the cell. So V is taken as an average of the v's to the northwest and southeast of the upoint, as these points run parallel to the filter face. ( ) Vi,j+Vi+l,jl v at Uij =  2 Likewise, ( ) Ui  J, j + 1 + Ui, j U at Vij =  2 This fonnulation is better suited for the triangular pleat, as there is no mixing of filter and nonfilter cells. The inertia tenns in the Darcy equation are thus: b( VD UD) = bU i,j (Vi,j+V2i+l,jl)2 + (Ui,j")2 2 2 (243,244) b( V b (Uil,j+l+Ui,j)2 ( ..) 2 D • VD ) = Vi,j + VI,) 2 2 2 38 2.5.4 Complete Form of the Finite Difference Equations Taking all components together, the full FDEs are as follows. Equation "f' outside of filter (fluid momentum xcomponent) (245) (W,j +W+ 1,j)2 + alw,j + Ui+ 1,jl(w,j w+ 1,j) (w  J,j+W,j)2 alw l,j+ud(w  J,jUi,i) 4tu { ( Vi,j + Vi + l,j)(W,i + Ui,i + 1) + alVi,j + Vi + l,A(w,j  Ui,j + 1) } ( Vi,j 1 + Vi + l,j l)(W,j 1 + Ui, j)  alVi,j  i+Vi + i,j ll(Ui,j 1 Ui, j) +~~~~ 4~y + v + ~..:,,~ (pi+i,j Pi,j) (W+I,j2W,j+WI,j Ui,j+I2W,j+Ui,jIJ tu f (tu)2 (~y)2 ( 'b A)2 IW,j+1w,A(w,j+1Ui,j)IUi,jW,jJj(W,jW,j1)_o  a·] ar·uy . (~y)3  Equation "g" outside of filter (fluid momentumycomponent) { (W,j + Ui, j + 1)( Vi,j + Vi + i,j) + alw, j + W,j + Jj( Vi, j  Vi + 1, j) } (W,j  i+ W,j + i)( Vi  i,j + Vi,i)  alW,i  i + Ui,j + il( Vi  i,j Vi,i) 4tu (246) (Vi,j + Vi,j + i)2 +alVi,j + Vi,j + il( Vi,j  Vi,j + i)  (Vi'i  i + Vi,j)2  alVi,j  i + vi.iI( Vi,j 1 Vi,j) +~~~~~~~~~~~ 4~y + V +.."... (pi,j + 1 pi,i) (Vi + i,j  2Vi,j + Vi1,j Vi,j + i  2Vi,j + Vi,j lJ ~y f (tu)2 (~y)2 ( 'b A)2 IW,j+lUi,A(w,J+lUi,j)lwl,J+lWi,iI(Wl,i+ 1 Ui  1,J) 0  a·] ar· uy . 2 = tu·(~y) 39 Equation ''/' in filter (Extended Darcy xcomponent) ( ) A~ vI A_b (Vi+l,jJ+Vi,j)2 .. 2 0 (') 47) Pi+l,j Pi,j +LUUi,j+LUUi,j +U',j   K 2 2 Equation "g" in filter (Extended Darcy ycomponent) ( ) A VI A b (Ui_l,j+l+Ui,j)2 2 0 Pi,j+lPi,j +LlYVi,j+LlYVi,j +Vi,j = K 2 2 (248) Equation "h" (continuity inside and outside of filter) Ui,jUil,j Vi,jVi,jl 0 + = & ~Y (249) 2.6 Boundary Conditions At the upstream position of the flow, the unifonn inlet velocity is imposed. At the final downstream column, a boundary condition of continuitive flow is assumed. This assumes the velocity gradient over the exit boundary is zero. The length of the grid should be sufficient so that these conditions do not influence the flow immediately near the boundary, i.e. the velocity gradients should be zero at the upstream and downstream boundaries. However, as it worthwhile to reduce the grid size as much as possible to thereby reduce run time, the effects of imposing these upstream and downstream boundary conditions over too tight a grid are analyzed in the next chapter to see the effect over the area of importance, viz. the pleat. If the effect is minimal, the smaller number of grid cells could be used. 40 At the downstream boundary, a pressure reading of zero is prescribed to provide a reference to flow pressure upstream. Along the sides of the flow, free slip symmetry is assumed. So at the edges of the halfpleat, crosswise velocity is assumed to be zero, and velocities on either side of the pleat are assumed to be reflective. This symmetry condition requires that the inlet velocity have no crossflow component. inlet velocity Fig.2.16 Boundary conditions. continuitive outflow pressure = 0 Lastly, it is noted that at the air / filter interface there is no boundary condition to prevent variables on either side from being included in both the viscous flow and filter momentum equations. This betrays itself in certain aspects of the solution. 2.7 Solution Method Solving for the FDEs is somewhat complicated by the nonlinearity of the equations. The following summary of the solution method is from Gerald & Wheatly [1994] pp. 1657. 41 There are three equations to solve for three unknowns per cell. feu, v,p) = xmomentum = 0 g(u,V,p) = ymomentum = 0 h(u,v) = continuity = 0 The solution to the equations is (ur, vr' Pr)' With an initial guess of the solution (ui' Vi' Pi)' the functions can be expanded to a truncated Taylor series. f(Ur, Vr, pr) = 0 = f(Ui, Vi, pi) + fu (ui, Vi,pi)(Ur  ui) + Iv (ui, Vi, pi)(Vr  Vi) + fp(Ui, Vi, pi)(pr  pi) g(Ur, Vr,pr) = 0 = g(Ui, Vi,pi)+ gu(Ui, Vi,pi)(UrUi)+ g)Ui, Vi,pi)(VrVi)+ gp(Ui, Vi,pi)(pr pi) h(Ur, Vr, pr) = 0 = h(Ui, Vi, pi) + hu (ui, Vi,pi)(Ur  ui) + hv (ui, Vi, pi)( Vr  Vi) + hp (Ui, Vi,pi)(pr  pi) (250) Here fx == df , etc. These equations can thus be solved for the difference between the dx solution and the estimate. where l/u (Ui, Vi, pi) gu(Ui, Vi,pi) hu (Ui, Vi, pi) Iv (Ui, Vi, pi) gv(Ui,Vi,pi) hv(Ui,Vi,pi) (251) The .1. tenns are solved by Gaussian elimination and added to the original estimate of the solution. (252) 42 Then equation (251) is solved again using the new estimates. This iteration is continued until convergence. Convergence is obtained when the difference between two iterations reaches an arbitrarily small value. This criterion is discussed in Chapter 3. 2.8 Output of Program The program outputs data to two files. The file OUTPUT.DAT has the final values displayed for each gridpoint, including: • u and v velocities, • angle of flow, • pressure, • and continuity. The file VELOCITY.DAT and other variants contain columnated position and velocity data to be output to a vector plotting utility. 43 Chapter 3 Investigation of Parameters 3.1 Introduction The following section examines some of the parameters used in the PLEA TFLO program with the triangleshaped pleat geometry. Parameters are examined in two categories: parameters directly affecting program output and parameters affecting program run time. Lastly, the squarewave pleat geometry is examined separately. As one of the difficulties in implementing a CFD method for viscous fluid flow is program stability, the key parameter in maintaining stability, the upstream differencing proportion a., is studied to find the best value. Also, as the applied turbulence model is not tailored for the particular flow in this study, the effect of the turbulence parameter that determines the mixing length ('y) is discussed. The area of the flow that is of particular interest is the entrance and exit to the pleat. In order to have a more detailed and faithful simulation of the flow, a fine grid is desired. However, the greater the number of grid points, the more computer resources required. So this part of the parameter analysis examines ways to minimize coverage of the less crucial areas of the flowfield, while increasing detail around the filter pleat. Testing is done to optimize code settings to: • minimize the distance covered upstream (htup) and downstream (htdown) of the filter; 44 • maximize the grid expansion up (eta) and downstream (zeta) of the filter; • minimize the pleat height (htup) used to examine a specific pleat angle; • increase the density of the grid (nfil_ x); • maximize the convergence criterion (epsi) for faster solution. 3.2 A Sample Run Before examining the parameters, sample runs are done to demonstrate the format of the output. The output format takes two forms: a printout of the data and a file formatted for creating a vector plot. The typewritten printout of the input and output data is contained in the file OUTPUT.DAT. The output data include u, v, p, continuity, and angle for every cell in the flowfield. The velocities are in meters per second; the normalized pressure (pressure divided by density) is in meters squared per second squared; the continuity is taken for each cell; the angle is in degrees counterclockwise from a due downstream flow. A sample is shown for an abbreviated flowfield. The input parameters are contained in the box below. This tells the geometric configuration. This also tells whether convergence was attained and the total number of iterations to obtain convergence, or, if convergence was not attained, then it gives the final value of the convergence criterion. It also gives a value for upstream pressure. 45 Pleat Shape: :~iangular Paramer.eys ::"n meters, aeqrees , seco::cs r3ase Gcome'try Ja:::.2_,,__ ~~_=, De~::"vec Geo~,e:.ry ;~~1:.a l)pleat heiq~t. h':. ,.242EO: xr # cells lrl x c:~· ib2.~ 2) pleat media wic~h cp ; .635E03 x) # cells in y c'r 31 clea~ angle ,checad 3.008 1 x) # xcells to 4) reI * ce~: upstrm cup i .600 x) # vcells to fLt 5) rel # ce:i dnstrm: cdowni .700 i x) x cell wdth 6) # of x fii:: cells' nf11 x' 4 x) y cell wdth 7) ~ of y filt cells r.~il::::y :J x) c cell wdtr. 8) ~ o~ c fj~t cellsinfiJ CI 0 i x) pleat width 9) up expans coeff eta I 1.500 i x) total flow Igth 10) down expans 8oef' zeta 1.500 i x) filt hts upstrm 11) max # of i tee :cntmx' 20 x) file, rots d:lstrrr. 12) convergence cril, epsi 1.100E02 ! x) est iter runtime 13) x inlet veloc ! uin I 3.000 x) est max runtime s c de ..... x .383E;J2 Qe~y .::'59£03 dele • OOCE:>t co wd , .636£03 I f1gth i .485£0: htup .986 i htdown: ~, . 600 lestimel.323E+Ol ' I estot 1.645£+02 ! ii 14) kin. visc'ty nu ,.151E04 I :'7) x permeability K x .780E10 15) upstrm flux coefl alphal 1.000 1 18) Y permeability i Ky i .780EcO 1 16) turb coeff i gamma! .098 I 19) x inertia factor! bx 1.680E+05 i I I 20) Y inertia faceo, b:::y, .680E+05 1 I Reached Convergence Criterion? Yes. No. of iter = 5 I Upstream Pressure = .34877E+04 Pa 1 Table 3.l(a) Input to a sample run. The values on the right numbered "x)" are derived values, all others can be input directly. Once cell dimensions are established (defx and defy), the height and width are recalculated based on a set thickness (tp) and angle (8). The exactness with which the height can be specified is limited by the fineness of the grid (nfil_ x) (also see Section 2.4.2) Some of these parameters are further examined in this chapter. It is only noted that the thickness of the media specified here (6.35e03 m) is the approximate thickness of the media used in the Purolator AF3192 filter, a passenger car engine air filter. Next OUTPUT.DAT gives the variable values per cell. The filter region lies between the slash '\" marks. U velocity 1 2 3 5 6 7 8 9 6 3.000 3.000 2.999 2.994 .027 .082 .555 3.004 10.229 3.000 3.000 2.999 2.994\ .027 .082 .555 3.004\10.229 4 3.000 3.000 3.000 2.997 2.582\ .081 .547 2.996 .579 3 3.000 3.000 3.000 3.002 4.214 4.577\ .533 2.996 .592 2 3.000 3.000 3.001 3.006 5.177 7.260 10.365\ 3.004 .599 1 3.000 3.000 3.001 3.006 5.177 7.260 10.365 3.004 .599 10 11 12 13 14 15 16 17 18 6 7.694 5.831 4.494 3.775 3.375 3.161 3.057 3.009 3.009 5 7.694 5.831 4.494 3.775 3.375 3.161 3.057 3.009 3.009 4\ 4.233 4.079 3.489 3.261 3.134 3.061 3.023 3.004 3.004 46 .0 5\ 2.09 .0 7 .00 .0 7 .00 V velocity 6 5 4 3 2 .000 .000 .000 ~ooo .000 .000 10 6 .133 .000 4\ .133 3 .059\ .029 .000 .~oo .000 .000 .ooe .ooc .098 .000 .098 .106 .002\ .000 angle (positive 6 4 3 6 5 4\ 3 2 1 2 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 10 l.0 .0 1.8 58.9\ 38.6 .0 11 l.0 .0 1.'; 2.9 75.6\ .0 continuity 10 11 2.407 2.690 2.849 2.935 2.97 1.610 2.273 2.642 2.844 2.94 1.610 2.273 2.642 2.844 2.94 5 6 .occ .ooe .ooe .oce .occ .oce .156 .003 .008\ .000 .000 .oce .156\ .003 .ooe .177 .128\ .oeo .114 .109 .000 .000 .000 :2 .070 .000 .070 .101 .084 .000 =3 .025 .000 .025 .033 .023 .000 14 .009 .000 .009 .012 .009 .000 horizontal equals 0) 345 2 .0 .0 80.3 .0 .0\ .0 .0 .0 3.4\ .0 .0 2.4 .0 .0 l.3 .0 .0 .0 12 .9 l.1 2.4 3.0 .0 12 13 .4 .0 .4 .7 .6 .0 3 14 .2 .0 .2 .2 .2 .0 15 .003 .000 .003 .004 .G03 .000 6 2.0 .0 2.1 1.6\ .9 .0 15 .1 .0 .1 .1 .v 7 .025 .000 .025 .049 .163\ .000 16 .001 .000 .001 .001 .001 .000 7 2.6 .0 2.6 5.3 .9\ .0 16 .0 .0 .0 .0 .0 .0 2.996 2.996 2.99: 2.99: 2.99: 2.99: 8 .128 .COG\ .:28 .257 .386 .000 17 .000 .000 .000 .000 .000 .000 8 2.4 .0\ 2.5 4.9 7.3 .0 17 .0 .0 .0 .0 .0 .0 .379 .000 .379 .252 .126 .000 18 .000 .000 .000 .000 .000 .000 9 2.1 .0 33.2 23.0 : .. 9 .0 18 .0 .0 .0 .0 .0 .0 6 9 6 .OOOOE+OO .OOOOE+OO .OOOOE+OO 3 .OOOOE+OO 2 .OOOOE+OO .OOOOE+OO .1962E13 .7422E13 .1962E13 .7422E13 .7910E14 .3839E13 .1648E13 .3567E14 .2106E13 .1857213 .OOOOE+OO .OOOOE+OO .1066£12 .1137E12 .3553E14 .2842E13 .0000£+00 .4547E12 .1066E12\.1137E12 .3553E14 .2842E13 .0000E+00\.4547E12 .1094E12 .8527E13\ .OOOOE+OO .OOOOE+OO .OOOOE+OO .1137E12 .1030E12 .5684E13 .1421213\ .2274E12 .2274E12 .1137212 .1201E12 .2274E12 .OOOOE+OO .1137E12\.4547E12 .0000£+00 .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE,OO 10 6 .1137E12 5 .1137E12 4\.2274E12 3 .OOOOE+OO\ 2 .2842E13 4 3 2 .OOOOE+OO pressure 1 .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO 10 11 12 13 .4547E12 .2274E12 .8527E13 .4547E12 .2274E12 .8527E13 14 15 .7105E14 .7105E14 .7105E14 .7105E14 16 .1688E13 .1688E13 .1847E12 .5684F13 .5684E13 .3553E14 .1865E13 .l421E13 .OOOOE+OO .7105[13 .7105E14 .3553E14 .1421£:3 .2354E13 .3553£14\.1137E12 .5684E13 .7105E14 .3191£13 .7105E14 .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO .OOOOE+OO (normalized) 2 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 11 3 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 12 .3488E+04 .3488E+04 .3488E+04\ .3488E+04 .3488E+04 .3488E+04\ .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 .3488E+04 13 14 6 .3472E+04 .3472E+04 .3472E+04 .3472E+04\ .3472E+04 .3472E+04 15 7 .3423E+04 .3423E+04 .3424E+04 .3426E+04 .3426E+04\ .3426E+04 16 17 18 .3553E14 .2109£+01 .3553E14 .2109£+01 .2265£13 .8272E+00 .8660E14 .8641£+00 .3553E14 .2072,,+01 .OOOOE+OO .OOOOE+OO .3067E+04 .3747E+03 .3067E+04\ .3747E+03 .3072E+04 .3900E+03 .3082E+04 .4002E+03 .3098E+04 .4053E+03 .3098E+04 .4053E+03 17 18 6 .1366E+02 .5003E+01 .4553E+01 .1221E+01 .3800E+00 .1467£+00 .8942E01 .7840E01 .0000£+00 5 .1366E+02 .5003E+01 .4553E+01 .1221E+01 .3800E+00 .1467E+00 .8942EOl .7840E01 .OOOOE+OO 4\ .1413E+02 .4934E+Ol .4535E+01 .1212E+01 .3787E+00 .1465E+00 .8939E01 .7840EOl .0000£+00 3 .1603E+02\.4854E+Ol .4519E+Ol .1203E+01 .3770E+00 .1462E+00 .8936EOl .7840E01 .0000£+00 2 .1698E+02 .4794£+01\.4499E+Ol .1197E+01 .3761£+00 .1460E+00 .8934E01 .7841EOl .OOOOE+OO 1 .1698E+02 .4794E+01 .4499E+Ol .1197E+01 .3761E+00 .1460E+00 .8934EOl .7841E01 .OOOOE+OO Table 3.1 (b) Raw datafromfile OUTPUT.DAT. 47 Results are also output as a set of x,y position and u,v velocity data to be used for vector plotting. A more intuitive understanding of the flow can be gained through a vector plot of the flowfield. For the triangular pleat configuration, the points are taken at the uposition in the cell (see Fig. 2.10). The vvelocity at the uposition is averaged from the two vvelocities northwest and southeast of the uposition, because these are aligned in the filter direction and assure that extra and intrafilter data are not mixed (for the triangular pleat). Ideally, vector flowfields would be displayed exactly proportional to the modeled flowfield. However, it is difficult to produce a clear picture when the pleats are nearly vertical and the flowfield is 100 times longer than it is wide. So the flowfield is modified in various ways: either disregarding much of the flow up and downstream of the filter (xpruning), or exaggerating the ydimension (yweighting). The latter method also weights the vvelocity component. An example is shown (Fig. 3.1) to exhibit its effect. The expansion only weights the directional component of v not the magnitude. The vectors' magnitude can be equalweighted (linearscale), logweighted (logscale), or ignored (equallength). Examples are shown in Fig. 3.1 (c), (d), and (a), respectively. Following Fig. 3.1 are sample outputs from the PLEA TFLO program representing the range of angles to be tested. Examples using the different vector graphing methods are shown. 48 5 I   4 ~   3 ~   2 I   1 I   /  _:.1' __ ~ _ ~ _ y: _ / __ :.!' _ ;/ _ ~ _ / __ /  ~ ~  ~  ~  ':/ j' ~  jt  /  I I , I I I I I I I I I , I I I I I I t z / ~ / :/ Y I! / '/ z  '/  _:£ ~ ~  ~ / ~ ~  ~ '/   O~~~I~I~~I~~I~wul~~I~~I~~~I~~I~~I~ o 2 3 4 5 6 7 8 9 10 11 Fig.3.l(a) Sample ofa vector plot, in proportion to the actual size of the flow field. The actual flow data is 45° above horizontal. Although the magnitude of the flowfield varies, the vectors are set equallength; thus they represent direction only. 5 4 3 I I I I I I I I I I 2 1_ _1 _1 __1 __ 1_ _/ __ / __ / __ ! _! _ I I I I • , I I I , o 2 3 4 5 6 7 8 9 10 11 Fig. 3.1 (b) Sample of a vector plot for same data, with the ydimension expanded 200%. Note the magnitude of the vectors is the same as above, but the direction is weighted equal in amount to the expansion. 49 5  o  ~ ..0 0 , ~  If_ _ !.t ~ _v:_~ _ ~  ~ ,  I I , 0 0 I I _ L :" ~ ~ ~   ...    ~  _ ....   0 0 I 0 4    ~ ,   I I I I ~ /: "'f   "1  ,,   ~   f   ~  0 , I I 3    ~ I 0 . 0 0 • ..,   1   f  f   t   / / , 2    1   .. •   ~   ~   r'   r'   ,t.   , / / 0 ; 0 0 0 I 0 ; ; I I I I I I o ; ~~wuwu~~~~~~~~~wuwu~~~~~~~wu~~~ o 2 3 4 5 6 7 8 9 10 11 Fig. 3.1 (c) Same as (a). but the vectors are scaled linearly to represent magnitude. 5 1  ~  ,  Y  ~  ~  ~  _0/ _ _ 0/_ ~  ~  0 0 I 0 ~ ~ 4 1  ~  ~  Y  ~  ~  ~ _ 0/_ _ o£   0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 I 3 f  ~  J_  Y _'/.. ~  _'/  '/  / _0/ ~ 0 I 0 I 0 0 0 0 0 , 0 0 I 0 I 0 / 0 2 1   1  ? r r '1  :/ '1 / / 0 I I I 0 0 0 0 ~ ~ '/  It   /  / / / 1 1  1'  /  : : : : 0 I I J 0 I 0 2 3 4 5 6 7 8 9 10 11 Fig.3.1(d) Same as (a), but the vectors are logscale. This is better to represent magnitude if the magnitude within the flowfield varies greatly. 50 Parameters =:" mete:!':"s, aeg~ees, sec:c:',(1s Base Geometry Data Oerivec Geomet:::y ::lat.a 1 ) ple2t heigh:: hr. I .202£02 Xl # cel.Ls i~ x a~r ib2:C 2~ 2) pleat. media ",idth, t.p I . 635t:03 x) # ceils i ~: y di:c JDa~ 5 3) pleat angie ilhetadi 45.00C ! x) if xcells to filt., icel~ 8 4 ) rel # cell upstrml C'.1p 1 .800 x) # ycells to fi:t jcel: Q 5) reI # cell anstrml edownl 1.000 x) x eel: wdth I delx 1 .225;;',03 6) # of x fLt. cells I nUl x '" x) y cell wdth dely 1 .225£03  7) # of y fi 1"C cells I nfil  y' C x) c cell wdth dele i _ OCOETOC 8) # of e filt cellslnfil el C I x) pleat width I wd ! • ::"12E:02 9) up expans eoeff I eta I 1.000 x) total flow 19tD flgth i .540E02 10 ) down expans coef i zeta I 1.000 x) filt ht.s upstrm htup i .786 11) max # of iter I jcntmx . 25 x) E~t hts dnstrm htdowni :.01C 12) convergence criti epsi .:'00£02 x) est:. iter r:.:::.time ,estimej .:}8E"t02 13 ) x inlet veloc I uin ! 3.000 x) est max runtime 1 estoc 1 .294E+03 !i 1 Flow Parameters 1 14) kin. visc'ty nu i.151E04! 17) x permeab~.Lty K x 1.780E10 15) upstrm flux coefl alphal :'.000 1 18) Y permeability : Ky i. 780E10 i 16) turb coeff I gamma 1 .098 I 19) x inertia factorl bx 1.680E+05 : 1 I 20) Y inertia factor, b=y i. 680E+05 1 I Reacned Convergence Criterion? Yes. No. of iter ~ 18 I 1 Upstream Pressure = .49748E+03 Pa i !1 0.001 0.0008 0.0006 0.0004 0.0002 Table 3.2 Input summary for a 450 pleat, 2 mm high. ' ., ..... ....: t ...' O~~~~~~r_~~~~~~_r~~~~~~r_~~~~~_r~~ o 0.001 0.002 0.003 0.004 0.005 0.006 Fig. 3.2( a) Full and proportional flowfield, linearscaled vectors. 0.001 ' 0.0008  0.0006    /'  0.0004 ,,/ '    ?' 0.0002  ..!.   , , . , Orrrrr~I~~,,~I~~~rrIr~·~~I,,~~~I~rr~I~~,,~ 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 Fig. 3 .2(b) Pruned and proportional flowfield, linearscaled vectors. Figs. 3.2 Vector flowfieldsfor45° pleat, 2 mm high. 51 Plea~ Shape: Triang~:ar Parameters in meters, deqrees, seconas I 3ase Geometry Da:.a De,iveci Geome~ry Ja~a 1 ) pleat neig!".: l.,i .... , .822E 02 x) # ce:~s L' x d i::" ioar 27 2) pleat.. rr,edia wlci:.r: tp I .635E03 x) # ce ~ ~ s lr. y di :: jbar 5 3) plea':: thetad 10.000 x) # xcells ~o c • ~ce:: 9 4 ) reI ~ . ~pstrIT', cup I .900 x) # ycells to .,:::' + I jce:l, C '~I 5) ~el * cell dnsl::7'L cdown, 1. 200 x) x cell wdtr. delx i ~914~G3 6) # of x f i 1 ~ ce.; .... s i nfil XI 4 x) y cell wdt.~. dely i .161£03  7) # of y fLl~ cel~slnfil yi 0 x) c cel" WQt:h delc I .000EcOO  8) # of c f i _ :: ce:lslnfil Ci a x) pleat: w~dti; I wd I .805£03 9) up expans coeff eta i 1.000 x) tot.al flow 19c~. f1gth I .247E01 10) down expans coefl zeta I 1.000 x) filt hts upstrm I htup i .892 11) max # of iter 1 jcntrnxi 25 x) filt hts dnstrm Ihtdownl 1.226 12) convergence critl epsi i .100E02 I x) est iter runtime lestimei .149;;:+02 I 13) x inlet veloc 1 uin 3.000 I x) est max runtime ! estotl .372E+03 ! i, 1 Flow Parameters 1! 14) kin. visco ty 1 nu 1.151E04 1 17) x permeability K xi. 780E10 15) upstrm flux coefl alphal 1.000 I 18) Y permeability Ky I. 780E10 I 16) turb coeff 1 gammai .098 1 19) x inertia factor, bx 1.680E+05 1 I 1 20) Y inertia factor! b y I. 680E+05 ' 1=1 1 Reached Convergence Criterion? Yes. No. of iter 11 1 Upstream Pressure ~ .62404E+03 Pa i Table 3.3 Input summary for a 100 pleat, 8 mm high. o. 00080;;;;;;L@ 1Eii........E~E. ..§... .....iii§T"""""§y§......,.......~ ,....i.,...b...,.f_froir: ....i.rt=;;...;;;. , ,. iT., I Iii I i r f i ""'[:i9.".", "f~if~if fi_ r'" ¥...,.i. ..WjrW ii..jriL¥ +If; ....Ii,i..;. .......,,;r ""I,§ 1 o 0.005 0.01 0.015 0.02 0.025 Fig.3.3(a) Full and proportionaljlowjield, linearscaled vectors. 0.0008 ~ >  I ~ " ." , r 0.0004 ::0. • ::,... .. ~  T= , . , 0. .. 0  .:...." . ."  i I i i I i , i i I i I i i I i 0.005 0.007 0.009 0.011 0.013 0.015 0.017 Fig. 3 .3(b) Pruned and proportional jlmifield, linearscaled vectors. 0.0008..............  ,, .. , ...     0.0006 .. ," ~ .. _ _ _ _  0.0004 .... " ..... ' . ".... .....   ... 0.0002 .............. .....:: ' O+rrrrr_rr_~~~~~'r_~~~·~.. · ·~~~~~~~~~~~~ I I I I o 0.005 0.01 0.015 0.02 0.025 Fig.3.3(c) Full and expandedjlowfield [yweighted 500%], linearscaled vectors. Figs. 3.3 Vectorjlowfieldsfor 100 pleat, 8 mm high. 52 Plea~ Shape: :rianq~lar Paramet.ers :;'. met.ers, degrees, seco:'.QS 1 3ase GeomeLry Data Derived Geometry ~ata J)plea::: heigh~ 1 ';~ 1.273£01 )() * cells in x dir ~.,..bar=2~7 2) pleat media '''ldth i :CD 1.635203 x) # cecls in y a~r ="ar 5 3) plea\. angle 1 :cne:.ad, 3.000 x) # xcell s to :':i 1 t: icel ~ . 9 4) reI # cell upstrrn! cup I .900 x) # ycells to filt jce':... ;, 5) reI # cell dnstrmi cdown: 1.200 x) x cell wdth delx ,.303£82 6) # of x fil::: cellslnfil xi 41 x) y cell wdtr. I dely :.:59:::03 7) # of y filt: cells 1 nfi ly i 0 x) c cell wdth I dele i. 0002:+00 8) # of c :llt cellsinfil::ci 0 x) pleat width wd 1.795t:C3 9) up expans coeff eta 1.000 x) total flow Igth flgth: .818E01 10) down expans coef: zeta I 1.000 x) filt hts upstrm i htup I .809 11) max # of iter I jcntmx: 25 I x) filt hts dnstr,n Ihtdownl 1.112 12) convergence crit: epsi 1.100£02 i x) est ieer runtime iescimei.149E+02 13) x inlet veloc uin 1 3.000 x) est max runtime estotl.372E+03 11 1 Flow Parameters 1, 14) kin. viscty 1 n~ 1.151E04 1 17) x permeability K x i .780£10 15) upstrm flux coefi alpha! 1.000118) Y permeability Ky i.780EI0 16) turb coeff I gamma 1 .098 1 19) x inertia factor' bx i .680E+05 I 1 ! 20) Y inertia factor i b=y I. 680S+05 1..  1 Reached Convergence Criterion? Yes. No. of iter = 6 1 Upstream Pressure = .14310£+04 Pa 11 Table 3.4 Input summary for a 3° pleat, 3 em high. o OOOR I"" • pi , II II • II . tj. i II I I i• I¥ IS • d iii i T F'i"i P I , i • I , I • I, ii• T i • i , '1W jI i o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Fig. 3.4( a) Full and proportional flowfield, linearscaled vectors. 0.0008o I ; = :s.   ~  L I I r z ,   ~ ~ SF I Iii T,r i if i  i~ . iii I I I I liT i 'j er i I ay 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0008 o Fig. 3 A(b) Pruned and proportional flowfield, linearscaled vectors. ....  ....  ff  i I r  o I 0.01   ....' ,. ..... '\," ,. .... " " , ...... "" .' ,. ............... '"'"  , .......,. . " .... . , ,,  ~ ...... I 0.02 I 0.03 I 0.04 I 0.05 I 0.06 I I 0.07 0.08 I 0.09 I 0.05 0.09 Fig.3A(c) Full and expandedflowfield [yweighted 1600%], linearscaled vectors. 0.0008 o f~; , 11, _______________ _ ~ ,,,.' ),  f '" "r ',,:.., ...... _ I "'If' I        " ....... '"', I. , :. "            t  __ . __ ~I _', __________ _ I o 0.01 I 0.02 I 0.03 I 0.04 I 0.05 I 0.06 I I 0.07 0.08 0.09 Fig.3.4(d) Full and expandedflowfield [yweighted 1600%], logscaled vectors. Figs. 3.4 Vector flowfields for 3° pleat, 3 em high. 53 The significance of the results is discussed in chapter 4. The examples of a 45° and 3° pleat represent the extremes to be observed. Note that the height of the 45° example is significantly less than that of the 3° pleat. This is because the x cell dimension (defx) is set by the pleat thickness (tp) and the prescribed fineness (nfil_ x), and then since the cells must align diagonally to fit the prescribed pleat angle, the y cell dimension defy is set by the angle (e~). If, from this, defy turns out to be O.lmm, and a 45°, 3 em high pleat is sought, 300 cells would be needed lengthwise and widthwise to cover just the pleat region of the flowfield. If defy is 0.1 rum, and a 3°, 3 em high pleat is sought, less than two cells would be needed lengthwise and widthwise to cover the pleat region; by virtue that for the 3° angle, deLx is 19 times dely, and the width of the flowfield would be only 0.2 mm. One other remark is made about the presentation of the flowfield. With the smaller pleat angles, the flow through the pleats appears rather minimal, and one might wonder how continuity is maintained. With the smaller pleats the cells are much longer (flowwise) than they are wide. Thus only a small degree of cross velocity is necessary to counter a large influx of flowwise velocity. The magnitude of the continuity error calculated cellbycell shown in Table 3.1(b) is typical of all program runs. 3.3 The Upstream Differencing Parameter (a) The convective terms of the NavierStokes equations are translated into FDEs using some proportion of upstream differencing represented by a. The form of upstream differencing has been adapted from the SOLA program [Bin et aI., 1975]. In that CFD code for viscous flow, transient effects were also included, thus making the stability criteria different than those for the present steadystate analysis. However, instability still exists in centraldifference representations of convection [Patankar, 1980, p. 83f), and upstream differencing is necessary to maintain stability. Although a thorough stability analysis is not 54 done here, empirical testing is done over the range of 8x and 8y used in practice. Any departure away from full upstream differencing brings on the likelihood of instability. However, although full upstream differencing is always stable, it is not always accurate. (J) Exact .0 0 1 .:  ~  (J) 0 c .(.J..) :m 0 .! c a iI " .... ' ~ , ... '\ ' : \ ''''...UPwind \ "\ \ 2 a 2 Peclet Number  ...... _ Fig.3.5 Solution of ld convectiondiffusion problem with uniform grid and solutions east and west of the variable equal to 1 and 0 respectively, adapted from Patankar [l980, p. 96J. Patankar shows results using different differencing schemes [Patankar, 1980, p. 961 (Fig. 3.5). From this it can be seen what the upstream flux coefficient is doing  essentially providing an approximation of the exact solution using the central and upstream differenced results. To get a clearer view of the effect of the upstream differencing parameter for this CFD code, some runs are made with varying values of a.. If a. is set at zero, correspondent to full central differencing for the convective terms, the flowfield is as follows. 55 0.0008 0.0006 0.0004  , ""..." ,. ... ........ \ ". .   , "',. ... ," ...    . .... ""......... ......... ....... ...... ... " " .... 0.0002 " ...... ~: . .. ... .:........!. ....... o ~r~~I~~r~Ir~~I~~~~Ir~~Ir~~~Ir~~~I~~~ o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Fig.3.6(a) Flow through 3° pleat as in Fig. 4.4, a ::: 0 (full and expanded [yweighted J600%J.linearscale vectors). 0.08 The instability can be seen in the periodic waving of the vectors upstream of the pleat. The instability is even clearer with the solution of the 450 pleat found with full central differencing. 0.001   ...... '" ...  ... 0.0008 0.0006 , , 0.0004 0.0002 . ..... ... ......... ' ,,' , , ),~               .. , ..I .. _ _ ...  • ...... ~ ': ' , iI'" ,. •     ~   . .'. O~~~r~~~~~~~~~r~~~~~~~~~r~~~ o 0.d01 0.d02 0.d03 0.d04 o.dos· 0.d06 Fig.3.6(b) Flow through 45° pleat as in Fig. 4.2, a ::: 0 (full and proportional, linearscale vectors). As a is increased, the magnitude of the instability is less discernible. It can best be 0.007 observed by viewing the numbers themselves. Below are the uvelocity values for the cells upstream of the pleat for varying values of a for the 30 pleat. 56 keacned Convergence cri:erion? Yes. Upstrea~ ?ressu~e = .:3506E+O' Pa U veloc':ty 3.000 3.000 3. DOC 3.00C 3.000 3.000 3.000 a = 0.5 2.709 2.709 2.830 3. 003 3.172 3.286 3.286 .182 .182 3.110 3.007 2.891 .8: ~ 2.8; t, 2.596 /.596 2.770 .008 3.23:) 3.390 3.390 3.458 3.458 3.276 3.009 2.731 2.525 2.S2S 2.221 2.221 2.579 3.033 3.447 3.721 3.721 ~o. of it.e~ 8 4.131 .020 .C~2 4.131\ .020 .01,7 3.687 2.075\ .042 3.024 3.549 3.:19\ 2.336 4.46~ j.302 1.82: 4.89: 6.49~ 1.821 4.89: 6.49, .:~> .87\J ~:.~C;C 9.3!,C ',09'J .~S:; .878\" .488 9.34::; 1.09~ .153 .8.',;) .866\ 5.343 5.169 .149 .8?~ .888 .102\ 2.7'.. 7 ~.:)96\ ,798 .912 .106 .GI0\ 8.9,.,7 :1.66"7\ .936 .109 .010 8.94 7 :1.66) .936 .109 .010 : 5 ~.69 .69"1 .~8C, 3. :.',: . ~76 .00' \ . DC: l8/ .9, oJ : 9, .. 67 ':, .. 62' 11 Reached Convergence Criterion: Yes. No. of iter = i upstream Pressure  .13914E+04 Pa i1 U velocity 1 2 4 2.992 2.992 2.995 3.000 3.005 3.008 3.008 789 3.427 .020 .041 3.427\ .020 .041 3.282 2.082\ .041 3.030 3.525 3.095\ 2.744 4.447 5.26~ 2.517 4.926 6.558 2.517 4.926 6.558 10 .156 .156 . 1 ~4 .150 11 12 13 14 1':' 16 4.507 4.507 3.000 2.999 3.000 2.999 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.001 3.000 3.001 3.002 3.002 3.001 3.000 2.999 2.998 2.998 3.030 3.030 3.019 3.000 2.982 2.969 2.969 2.883 2.883 2.930 3.001 3.070 3.116 3.116 .896 11.302 9.330 7.126 5.718 .896\11.302 9.330 7.126 5.718 .87: .891\ 5.342 5.149 4.573 .817 2.963 2.:50 _. :J63 5.577\ .847 .825 .912 .935 .106\ 2.706 3.126 .110 .009\ 1.58, 8.963 11.561\ .960 8.963 11.56' .960 .112 .112 a = 0.8 1j i Reached Convergence Cri~erion: Yes. No. of iter 6 1 Upstream Pressure = .14153E+04 Pa 1 U velocity 2 7 8 9 10 .156 .156 .154 11 12 13 .911 11.245 9.325 .911\11.245 9.325 .886 .906\ 5.341 .009 .ODJ\ .009 .001 . ~63 14 15 16 7.143 5.733 4.551 7.143 5.733 4.551 5.139 4.566 3.833 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.001 3.001 3.001 3.000 2.999 2.999 2.986 2.986 2.991 3.000 3.008 3.015 3.146 .020 .041 3.146\ .020 .041 3.099 2.072\ .041 3.013 3.499 3.081\ 2.912 4.439 5.245 2.830 4.970 6.592 .151 .863 5.568\ .841 8.971 11.499\ .927 .949 .974 .974 .108\ 2.700 3.116 2.956 .112 .009\ 1.586 2.128 .115 .009 .001\ 1.532 3.000 3.000 3.000 3.000 2.999 3.015 2.830 4.970 6.592 8.971 11.499 .115 a = 0.9 11 I Reached Convergence Criterion: Yes. No. of iter = I Upstream Pressure = .14232E+04 Pa 11 U velocity 10 11 12 13 7 6 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 2.997 2.997 2.998 3.000 3.002 3.003 3.003 3.067 .020 .041 3.067\ .020 .041 3.045 2.068\ .041 3.006 3.490 3.077\ 2.960 4.437 5.239 2.922 4.986 6.603 2.922 4.986 6.603 .157 .916 11.226 9.323 .157 .916\11.226 9.323 .155 .891 .911\ 5.340 .151 .868 .931 .108\ 5.565\ .846 .954 .113 8.973 11.479\ .978 .115 8.973 11.479 .978 a = 1.0 11 I Reached convergence Criterion: Yes. No. of iter = 6 ! I Upstream Pressure = .14310E+04 Pa I 11 U velocity 1 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 7 2.993 .020 2.993\ .020 2.994 2.063\ 2.999 3.480 3.005 4.436 3.009 5.002 3.009 5.002 .041 .041 .041 3.072\ 5.233 6.614 6.614 11 12 .157 .920 11.208 .157 .920\11.208 .155 .896 .916\ .151 .873 .936 5.562\ .851 .958 8.976 11.459\ .983 8.976 11.459 .983 .115 13 9.321 9.321 5.340 .109\ .114 .116 .116 .009 .001 1.532 14 15 16 7.148 5.738 4.564 7.148 5.738 4.564 5.136 4.564 3.838 2.699 3.113 2.954 .009\ 1.586 2.121 .009 .001\ 1.523 .009 .001 1.523 14 15 7.152 5.743 7.152 5.743 5.133 4.562 16 4.577 4.577 3.842 2.697 3.110 .952 .009\ 1.586 2.115 .009 .001\ 1.Sl~ .009 .001 1.514 Table 3.5(a) Effect ofa on program stability, 3° pleat. 57 Note that there is instability even when a is set as high as 0.90 (see columns 6 & 7). For a 45° pleat, the instability is greater at Iowa's, but seemingly entirely damped out at an a of only 0.50. !1 I Reached convergence Criterion? Nc. Final epsi =.61£011 i Upstream Pressure = .48964E+03 Pa ! .. velocity 1 3.000 8.722 3.000 8.722 3.000 2.020 3.000 1.181 3.000 1. 506 3.000 3.934 3.000 3.934 a = 0.5 4.438 4.438 2.351 2.353 2.651 3.206 3.206 4 7.624 7.624 1.339 .338 1.962 3.737 3.73'7 3.615 3.615 4.033 1.587 2.767 2.998 2.998 6.020 6.020 1.213 2.603 1.712 3.453 3.453 7 2.429 2.429 5.000 1. 622 3.204 2.744 2.744 4.903 4.903\ .921 4.032 1.695 3.449 3.449 9 10 11 .830 1.505 2.228 .838 1.505 2.228 5.416\ 1.375 !.947 1.593 5.839\ 1.765 4.368 2.038 5.919\ 2.793 4.243 3.141 2.793 4.243 3.141 12 13 3.184 5.042 3.184\ 5.042 14 15 16 5.457 6.070 .OJ2 5.457 6.070 6.012 2.372 2.131\ 3.403 3.990 4.189 2.105 2.136 1.847\ 1.978 2.489 2.084 2.407 2.003 1.425\ 1.510 5.256\ 3.283 2.290 1.537 .800\ 5.256 3.283 2.290 1.537 .800 ,1 I Reached Convergence Criterion? No. Final epsi =.14E021 i ups~ream Pressure = .49242E+03 Pa ,1 U velocity 1 3.000 2.996 3.000 2.996 3.000 2.998 3.000 3.000 3.000 3.002 3.000 3.004 3.000 3.004 a = 0.8 2.987 2.987 2.992 3.000 3.008 3.012 3.012 4 2.970 2.970 2.981 3.001 3.019 3.029 3.029 2.928 2.928 2.965 3.005 3.041 3.060 3.060 2.878 2.878 2.921 3.009 3.077 3. 11:) 3.11S 7 2.665 2.665 2.919 3.050 3.157 3.210 3.210 2.614 .833 2.614\ .833 2.686 3.582\ 3.068 3.410 3.269 3.570 3.363 3.604 3.363 3.604 10 11 12 13 14 1.512 2.231 3.177 5.005 5.683 1.512 2.231 3.177\ 5.005 5.683 1.387 1.958 2.384 2.164\ 3.186 4.362\ 1.783 2.119 2.148 1.841\ 3.806 4.864\ 2.100 2.411 2.003 3.934 4.163 5.220\ 3.271 .286 3.934 4.163 5.220 3.271 .286 15 16 5.972 6.078 5.972 6.078 3.799 4.134 2.263 2.518 1.429\ 1.468 1.53" .801\ 1.537 .801 11 I Reached Convergence criterion? Yes, No. of iter = 22 ! i Upstream Pressure = .49494E+03 Pa 11 U velocity 1 2 7 3.000 2.995 2.984 2.961 2.918 .833 2.657 6 3.000 2.995 2.984 2.961 2.918 2.833 2.657 3.000 2.997 2.991 2.978 2.955 2.915 2.854 3.000 3.000 3.001 3.002 3.005 3.016 3.050 3.000 3.003 3.010 3.023 3.049 3.097 3.186 10 11 12 13 14 15 16 2.305 .832 1.513 2.232 3.174 4.992 5.673 5.963 .069 2.305\ .832 1.513 2.232 3.174\ 4.992 5.673 5.963 6.069 2.772 3.267\ 1.389 1.961 2.388 2.174\ 3.191 3.79: 4.118 3.140 3.529 4.020\ 1.789 2.125 2.153 1.845\ 2.278 2.529 3.347 3.655 4.017 4.558\ 2.109 2.414 2.005 1.430\ 1.481 3.000 3.005 3.015 3.036 3.073 3.139 3.253 3.437 3.717 4.062 4.460 5.204\ 3.267 2.286 1.538 3.000 3.005 3.015 3.036 3.073 3.139 3.253 3.437 3.717 4.062 4.460 5.204 3.267 2.286 1.538 .803\ .803 a= 1.0 ,1 I Reached Convergence Criterion? Yes. No. of iter = 20 I Upstream Pressure = .49656E+03 Pa I 11 U velocity 1 3.000 3.000 3.000 3.000 3.000 3.000 3.000 2.995 2.995 2.997 3.000 3.003 3.005 3.005 2.982 2.982 2.989 3.001 3.011 3.017 3.017 2.957 2.957 2.975 3.002 3.026 3.040 3.040 2.908 2.908 2.949 3.006 3.055 3.082 3.082 2.813 2.813 2.904 3.018 3.109 3.157 3.157 2.617 2.617 2.833 3.056 3.209 3.284 3.284 2.169 2.169\ 2.770 3.174 3.395 3.492 3.492 10 11 .833 1.514 2.232 .833 1.514 2.232 3.121\ 1.392 1.963 3.531 3.843\ 1.792 3.716 4.074 4.398\ 3.799 4.177 4.614 3.799 4.177 4.614 12 13 14 15 16 3.172 4.982 5.665 5.957 6.063 3.172\ 4.982 5.665 5.957 6.063 2.390 2.180\ 3.194 3.787 4.110 2.128 2.157 1.848\ 2.286 2.535 2.115 2.416 2.007 1.431\ 1.489 5.194\ 3.265 2.286 1.539 5.194 3.265 2.286 1.539 .803\ .803 Table 3.5(b) Effect oia on program stability, 45° pleat. 58 In order to ensure stability under all configurations, a is set at unity consistently. It is borne in mind that this could produce excessive damping under certain conditions. 3.4 The Mixing Length Constant (y) The model used to determine the turbulent stresses calculates a mixing length that is a direct function of the flow halfwidth (d) (see Section 2.2.2).  2 dU dU uv = lmi.J:   where lmi.J: = y. d and d = flow half  width dy dy d Fig.3.7 Flow downstream offilter. This turbulence model is based on an idealized jet flow (Fig. 2.4). There are several conditions in the ideal model that are not met in the flow conditions downstream of the filter. • The jet is not surrounded by nonturbulent flow. • Its width does not spread moving downstream. • Velocities within the jet stream range above and below the inlet flow velocity, rather than being solely greater or less than an ambient velocity. 59 • The flow is not a fullydeveloped selfpreserving flow. The main divergence with the idealized model is that instead of a single jet in an ambient fluid, there are a series of jet flows lined up beside one another; hence the jets are unable to expand widthwise. In the selfpreserving flow of an expanding jet, the effect of width expansion (see Eq. 2.14) is to increase the magnitude of the turbulent stress moving downstream. The effect of containing the width of the jet downstream is therefore to decrease the turbulent stresses downstream. This makes sense as the stresses would be expected to lessen as the velocity gradient weakens. So, although the model does not simulate the exact conditions, it is still reasonable to apply this algebraic model because: • It is still a jetlike flow. • The mixing length is still assumed to be based on a direct relation to the jet halfwidth. • The magnitude of the turbulence constant ("I) will still be essentially the same. Even for widely varying types of freeshear flows, the values of "I are all of the same magnitude, from 0.071 for mixing layers (which can produce the largest velocity gradients) to 0.180 for wakes (which produce small velocity gradients). The value Wilcox [1993] suggests for a plane jet is y=0.098 (31) The main consequence of turbulent stress on the mean flow is an increased viscous effect that acts to flatten out the velocity profile. The effect can be seen in the comparison of the flowfield for the 45° pleat derived assuming the following turbulent stress conditions: laminar flow, our modeled turbulence coefficient, and double that value. 60 0.001 0.0008 0.0006 0.0004 0.0002  ..."..        ~,' ." .".. . ,  ~>" ~ ..~",,,., " ' ~",."."" ... ____ " " ~411* __ ., .. 0~~~~~~~~~~~~~~~_r~T_~_r~~~~~~~~ o 0.001 0.002 0.003 0.004 0.005 0.006 0.001 0.0008 0.0006 0.0004 Fig.3.8(a) Flowfieldfor y = 0 (full and proportional, linearscale vectors). " "  ...... ......:' " ..,..,..  " 'A ______________ _ , , , ', .,      ...1" , ." "" _ .. ___ _ 0.0002 '" 0.007          ~ ....   ., ..... 0~~~~~1~r~~1r~~~1~r~·~~1~r~~1~r~~1r~~~ o 0.001 0.002 0.003 0.004 0.005 0.006 0.001 0.0008 0.0006 0.0004 0.0002 Fig.3.8(b) Flowfieldfor y = 0.098 (full and proportional, linearscale vectors).       ."... ,. ~>"  ~,' ",. ;' "' "       ~~", ~/~~,  .....  ~/ ;' , ,..,.               " " ~ .....   .'. 0.007 0~~~~~1~~r~1r~~~1~r~~~1~r~~1~rr~1r~~~ o 0.001 0.002 0.003 0.004 0.005 0.006 0.007 Fig.3.8(c) Flowfieldfory = 0.196 (full and proportional, linearscale vectors). Figs.3.8 The effect oiY on the downstream flow. One side effect of applying this turbulence model in all nonmedia regions of the flowfield is that it is operative in the crevasse of the pleat as well. This increases the viscous and turbulent drag within the crevasse even though the region may be prohibitively small for the development of turbulence. One solution would be to tum the turbulence model off within the pleat crevasse. However, this was not done here. The effects of this phenomenon are noted in Section 4.4. 61 3.5 Upstream and Downstream Coverage (htup and htdown) The remaining parameters examined affect the run time of the program. As the ponion of the flowfield that is of panicular interest is the flow immediately entering, within, and exiting the filter media, the flow far upstream and downstream of the filter is only of interest insofar as it affects the filter region. The flow upstream of the filter is affected by the lefthand boundary condition, a uniform inlet flow. If this boundary condition is set too close to the filter, an anificially large velocity gradient is created. Ideally, there should be enough distance between the lefthand boundary and the filter so that the lefthand boundary condition has no effect on the gradient. As this effect is difficult to distinguish in a vector plot, the values of the uvelocity before the pleat are shown for two values of htup (0.5 and 1.63) for a 45° pleat (Table 3.6). The numbers are aligned so that the filter regions coincide. As can be observed, a very short runup (htup = 0.50) to the filter forces the velocity gradient; this has some effect on the intrafilter flow as well. With a long runup (htup = 1.63) to the filter, the velocity gradient develops naturally. On the downstream side, upstream differencing used in the finite difference approximation for the convective terms limits the influence of downstream phenomena on upstream locations. The righthand boundary condition assumes the velocity gradient has disappeared before reaching the exit. However, this is not true unless a very lengthy exit length is used. If a gradient does actually exist at the exit, the continuitive outflow condition will anificially force the gradient at the exit. 62 The effect of this artificial condition can be observed by comparing a flowfield solved for different values of htdown (Table 3.7). It can be seen that the exit boundary condition does affect the flowfield near the exit, but barely affects the flow near and within the filter. htup = 0.50: htup = 1.63: 1 1 Reached Convergence Criterion? Yes. 1 No. of iter ~ 20 1 Upstream Pressure ~ .49653E+03 Pa 1 11 U velocity 2 3 4 3.000 3.000 2.999 2.998 6 3.000 3.000 2.999 2.998 5 3.000 3.000 2.999 2.999 4 3.000 3.000 3.000 3.000 3 3.000 3.000 3.001 3.001 2 3.000 3.000 3.001 3.002 1 3.000 3.000 3.001 3.002 11 5 6 7 8 9 10 ! Reached Convergence Criterion? Yes. 7 2.996 2.992 2.985 2.971 2.946 2.897 I I No. of iter ~ 20 6 2.996 2.992 2.985 2.971 2.946 2.897 ! Upstream Pressure ~ .49755£+03 Pa 1 2.997 2.995 2.991 2.983 2.968 2.942 11 3.000 3.000 3.000 3.8C1 3.002 3.006 3 3.003 3.005 3.009 3.018 3.033 3.06? U velocity 2 3.004 3.008 3.015 3.028 3.051 3.093 1 3.004 3.008 3.015 3.028 3.051 3.093 2 3 5 6 11 12 13 14 15 16 7 3.000 2.821 2.325 .843 1. 518 2.234 7 2.802 2.609 2.163 .832 1. 514 2.232 6 3.000 2.821 2.325\ .843 1.518 2.234 6 2.802 2.609 2.163 \ .832 1. 514 2.232 5 3.000 2.948 2.887 3.244\ 1.393 1.965 2.898 2.827 2.764 3.116\ 1.391 1. 963 3.000 3.047 3.191 3.599 3.934\ 1.793 3.018 3.055 3.173 3.528 3.840\ 1.792 3 3.000 3.088 3.292 3.666 4.075 4.435 3 3.115 3.215 3.399 3.718 4.074 4.397 2 3.000 3.096 3.305 3.648 4.079 4.573 2 3.167 3.294 3.500 3.806 4.182 4.616 1 3.000 3.096 3.305 3.648 4.079 4.573 1 3.167 3.294 3.500 3.806 4.182 4.616 7 9 10 11 12 17 18 19 20 21 22 7 3.174 4.984 5.667 5.958 6.064 6.062 7 3.172 4.982 5.665 5.957 6.063 6.061 6 3.174\ 4.984 5.667 5.958 6.064 6.062 6 3.172\ 4.982 5.665 5.957 6.063 6.061 2.391 2.180\ 3.194 3.787 4.110 4.282 5 2.390 2.180\ 3.194 3.787 4.110 4.282 4 2.129 2.157 1.847\ 2.286 2.535 2.667 4 2.128 2.157 1. 848\ 2.286 2.535 2.667 3\ 2.llS 2.416 2.006 1. 431 \ 1.488 1.452 3\ 2.ll5 2.416 2.007 1.431\ 1.489 1. 4 02 2 5.192\ 3.263 2.285 1.538 .803\ .537 5.195\ 3.265 2.286 1.539 .803\ .538 5.192 3.263 2.285 1.538 .803 .537 5.195 3.265 2.286 1. 539 .803 .538 Table 3.6 The effect ofhtup on the upstreamjlow. 63 htdown = 0.88 htdown = 3.38 U velocity U velocity 1 3 6 2 6 7 3.000 2.99" 2.982 2.957 2.908 2.8:3 7 3.00C 2.995 2.982 2.957 7.9G8 2.8: 6 3.COO 2.995 2.982 2.957 2.908 2.813 6 3.00C 2.995 2.982 2.957 2.908 2.8:3 3.000 2.997 2.989 2.975 2.949 2.904 5 3.000 2.997 2.989 2.97" 2.949 2.904 3.000 3.000 3.00: 3.002 3.006 3.018 4 3.000 3.000 3.00" 3.002 3.0C6 3.018 3 3.000 3.003 3.8:1 3.026 3.055 3.109 3 3.000 3.003 3.012 3.026 3.055 3.109 2 3.000 3.005 3.017 3.040 3.082 3.157 2 3.000 3.005 3.017 3.040 3.082 3.157 3.000 3.00:: 3.017 3.040 3.082 3.157 3.000 3.005 3.017 3.040 3.082 3.157 7 8 10 11 12 8 9 10 .c ~ .? 7 2.617 2 .169 .833 1. 514 2.232 3 .172 7 2.617 2 .169 .833 1. 5" 4 2.232 3.172 6 2.617 2.169\ .833 1. 514 2.232 3.172 6 2.617 2.169\ .833 1.514 2.232 3.172 5 2.833 2.770 3.121\ 1.392 1.963 2.390 5 2.833 2.770 3.121\ 1.392 1. 963 2.398 4 3.056 3.174 3.53: 3.843\ 1. 792 2.128 4 3.056 3.174 3.531 3.843\ 1.792 2.128 3 3.209 3.395 3.716 4.074 4.398\ 2.115 3 3.209 3.395 3.716 4.074 4.398\ 2.115 2 3.284 3.492 3.799 4.177 4.614 5.194 2 3.284 3.492 3.799 4.177 4.614 5. J 94 1 3.284 3.492 3.799 4.177 4.614 5.194 3.284 3.492 3.799 4.177 ~ ~ 61 ~ 5.194 13 14 15 16 17 18 13 14 15 16 17 18 7 4.982 5.665 5.957 6.062 6.056 5.983 7 4.982 5.665 5.957 6.063 6.061 5.995 6\ 4.982 5.665 5.957 6.062 6.056 5.983 6\ 4.982 5.665 0.957 6.063 6.061 0.995 5 2.180\ 3.194 3.787 4.110 4.282 4.363 5 2.180\ 3.194 3.787 4.110 4.282 4.366 4 2.157 1.848\ 2.286 2.536 2.668 2.755 4 2.157 1. 848\ 2.286 2.535 2.667 2.749 3 2.416 2.00·1 1. 431 \ 1.490 1.454 1.436 3 2.416 2.007 1.431\ 1.489 1.452 ... 431 2\ 3.265 2.286 1. 53 9 .803\ .539 .463 2\ 3.265 2.286 1. 539 .803\ .538 .459 3.265 2.286 1. 539 .803 .539 .463 3.265 2.286 1. 539 .803 .538 .459 19 20 21 22 23 :'9 20 21 22 23 24 7 5.868 5.721 5.534 5.279 5.279 7 5.896 5.781 5.660 5.538 5.420 5.308 6 5.868 5.721 5.534 5.279 5.279 6 5.896 5.781 5.660 5.538 5.420 5.308 5 4.378 4.349 4.288 4.199 4.199 5 4.388 4.373 4.338 4.293 4.2,3 4.192 4 2.819 2.866 2.908 2.960 2.960 4 2.809 2.849 2.875 2.892 2.905 2.914 3 1.464 1. 530 1. 626 1. 760 1.760 3 1. 448 1.493 1.550 1. 612 1.673 I. "132 2 .470 .534 .645 .802 .802 2 .459 .504 .577 .665 .759 .853 .470 .534 .645 .802 .802 .459 .S04 .577 .665 .759 .853 25 26 27 28 29 30 7 5.202 5.103 5.010 4.925 4.845 4.771 6 5.202 5.103 5.010 4.925 4.845 4.771 5 4.143 4.096 4.051 4.009 3.970 3.934 4 2.922 2.928 2.934 2.939 2.943 2.948 3 1.789 1.842 1.891 1. 938 1.981 2.02l 2 .945 l.031 1.113 1.189 1.261 1.327 .945 1.031 1.113 1.189 1. 261 1.327 31 32 33 34 35 36 7 4.702 4.637 4.577 4.521 4.467 4.417 6 4.702 4.637 4.577 4.521 4.467 4.417 5 3.900 3.868 3.838 3.810 3.783 3.758 4 2.951 2.954 2.958 2.960 2.963 2.965 3 2.058 2.093 2.125 2.156 2.185 2.212 2 1.389 1.448 1. 502 1.553 1. 602 1. 648 1.389 1.448 1. 502 L553 1. 602 :.648 37 38 39 40 41 42 7 4.369 4.323 4.277 4.229 4.175 4.107 6 ,.369 4.323 4.277 4.229 4.175 4.107 5 3.734 3.710 3.687 3.663 3.636 3.606 4 2.968 2.970 2.972 2.975 2.979 2.987 3 2.238 2.263 2.289 2.315 2.345 2.382 2 ".691 1.734 1.775 1.818 1.864 1.917 1 1. 691 1.734 1.775 1.818 1.864 1.917 (col umn 43 ~ 42) Table 3.7 The effect ofhtdown on the downstreamJlow. 64 3.6 The Grid Expansion Coefficients (eta and zeta) One theoretically simple method to deal with the entrance and exit boundary conditions is to introduce an expandable grid. As detail becomes less important moving up and downstream of the filter, the cells can be expanded in those directions. In this way, the effect of the boundary conditions at the up and downstream positions is diminished, while keeping the cell count low. The cell expansions begin at the xcells two positions upstream and downstream of the pleat. The initial sample of the 10° pleat (Fig. 3.3) had a long enough runup to the filter so that the flow could develop naturally from the upstream boundary condition. For comparison, eta is increased to 1.2 and icell is adjusted (via cup) so that the upstream distance (htup) is kept nearly the same; so only five upstream cells are used on the right but eight on the left. eta = 1.00, cup = 0.90, htup = 0.892 eta = 1.24, cup = 0.50, htup = 0.897   U velocity U velocity 3 4 3.000 3.000 3.000 3.000 3.000 2.999 6 3.000 3.000 2.999 3.000 3.000 3.000 3.000 3.000 3.000 5 3.000 3.000 2.999 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.001 2 3.000 3.000 3.000 3.000 3.000 3.001 2 3.000 3.000 3.001 10 11 12 7 6 2.992 2.880\ .047 .095 .360 1. 490 2.986 2.876\ .047 .095 .360 1. 490 2.995 2.927 2.392\ .088 .329 1.283 2.992 2.925 2.391 \ .088 .329 1. 283 3.000 3.005 3.621 3.390\ .275 1.096 4 3.000 3.004 3.620 3.390\ .275 1.096 3 3.005 3.074 4.299 5.207 5.697 \ .916 3 3.008 3.076 4.299 5.207 5.697\ .916 2 3.008 3.114 4.64 : 6.220 8.339 10.216 2 3 014 3.119 4.642 6.221 8.339 10.216 13 14 15 16 17 18 10 11 12 13 14 15 6\ 9.638 9.135 ".566 6.401 5.582 5.029 6\ 9.638 9.135 7.566 6.401 5.582 5.029 5 1. 084 \ 5.025 5.019 4.647 4.289 4.036 5 1. 084 \ 5.025 5.019 4.647 4.289 4.036 4 1. 238 .225 \ 2.400 2.785 2.859 2.905 4 1. 238 .225\ 2.400 2.785 2.859 2.905 3 1. 415 .290 .003\ 1.182 1. 589 1.882 3 1. 415 .290 .003\ 1.182 1.589 1.882 2\ 1. 624 .326 .012 .015\ .681 1.147 2\ 1.624 .326 .012 .015\ .681 1.147 19 20 21 22 23 24 16 17 18 19 20 21 6 4.642 4.359 4.145 3.979 3.847 3.740 4 642 4.359 4.145 3.979 3.847 3.740 5 3.854 3.719 3.614 3.531 3.464 3.408 3.854 3.719 3.614 3. ~31 3.464 3.408 4 2.935 2.954 2.967 2.975 2.981 2.985 4 2.935 2.954 2.967 2.975 2.981 2.985 2.090 2.242 2.358 2.448 2.521 2.579 2.090 2.242 2.358 2.448 2.521 2.579 2 1. 4 79 1.726 1.917 2.067 2.187 2.287 2 1. 4 79 1.726 1.917 2.067 2.187 2.287  Table 3.8 Effect of eta on the flow upstream of pleat. 65 It can be seen that the velocity values entering the pleat and within the pleat are the same. Likewise, an increased zeta grid expansion coefficient downstream can allow the flow profile distance enough to reestablish a flat profile under viscous forces. Note that the expanded grid on the right reaches the standard uniform flow condition. zeta = 1.0, cdown = 1.2, htdown = 1.226 u velocity 2 5 6 6 3.000 3.000 3.000 3.000 3.000 2.999 5 3.000 3.000 3.000 3.000 3.000 3.000 4 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000 3.001 9 10 11 12 6 2.992 2.880\ .047 .095 .360 1.490 5 2.995 2.927 2.392\ .088 .329 1.283 3.000 3.005 3.621 3.390\ .275 1.096 3.005 3.074 4.299 5.207 5.697\ .916 2 3.008 3.114 4.641 6.220 8.339 10.216 13 14 15 16 6\ 9.638 9.135 7.566 6.40~ 1.084\ 5.025 5.019 4.647 4 1. 238 3 1.415 2 \ .624 .225\ 2.400 2.785 .290 .003\ 1.182 .326 .012 .015\ 17 5.582 4.289 2.859 1. 589 .681 18 5.029 4.036 2.905 1.882 1.147 19 20 21 22 23 24 6 4.642 4.359 4.145 3.979 3.847 3.740 5 3.854 3.719 3.614 3.531 3.464 3.408 2.935 2.954 2.967 2.975 2.981 2.985 2.090 2.242 2.358 2.448 2.521 2.579 1.479 1.726 1.917 2.067 2.187 2.287 25 26 27 28 29 6 3.652 3.578 3.511 3.415 3.415 3.362 3.323 3.288 3.237 3.237 2.989 2.991 2.994 2.997 2.997 2.628 2.669 2.706 2.761 2.761 2 2.369 2.439 2.500 2.590 2.590 zeta = 1.8, cdown = 1.2, htdown = 89.42 U velocity 4 5 3.000 3.000 3.000 3.000 3.000 2.999 3.000 3.000 3.000 3.000 3.000 3.000 4 3.000 3.000 3.000 3.000 3.000 3.000 3 3.000 3.000 3.000 3.000 3.000 3.000 2 3.000 3.000 3.000 3.000 3.000 3.001 7 8 10 11 12 6 2.992 2.880\ .047 .095 .360 1.490 5 2.995 2.927 2.392\ .088 .329 1.283 3.000 3.005 3.621 3.390\ .275 1.096 3.005 3.074 4.299 5.207 5.697\ .916 2 3.008 3.114 4.641 6.220 8.339 10 216 13 14 15 16 6\ 9.638 9.135 7.566 6.400 1.084\ 5.025 5.019 4.647 4 1.238 .225\ 2.400 2.786 3 1.415 .290 .003\ 1.183 2\ 1.624 .326 .012 .016\ 17 5.561 4.283 2.868 1.602 .685 18 4.823 3.945 2.930 1.995 l.307 19 20 21 22 23 24 6 4.177 3.684 3.346 3.142 3.043 3.009 5 3.627 3.375 3.196 3.083 3.026 3.005 2.964 2.984 2.994 2.998 3.000 3.000 3 2.342 2.611 2.799 2.915 2.974 2.995 1.890 2.346 2.665 2.861 2.957 2.991 25 26 27 28 29 6 3.001 3.000 3.000 3.000 3.000 5 3.001 3.000 3.000 3.000 3.000 4 3.000 3.000 3.000 3.000 3.000 3 2.999 3.000 3.000 3.000 3.000 2 2.999 3.000 3.000 3.000 3.000 Table 3.9 Effect o/zeta on the flow downstream o/pleat. 3.7 The Pleat Height (ht) As the number of cells widthwise in the grid (jbar) for a given pleat angle (8) and given pleat thickness (tp) is a function ofthe pleat height (ht) (see Table 2.1), reducing the pleat 66 height is one way of reducing the number of grid cells. Obviously, reducing the pleat height will create a different (higher) pressure drop across the pleat, but it may be an effective way to observe flow trends just the same. With regard to the latter, the main concern with this grid reduction is its affect on distinguishing the flow trend through the section of the filter away from the direct effect of the pleat folds, defined as the free length of the pleat. freelength = ht 2tx freelength freelength 2ht Fig.3.9 Freelength as related to filter height. If the height of the pleat is reduced so that ht is in the same range as tx, then the flowfield is dominated by the pleat folds. If we are trying to simulate a flow for a pleat of a greater height, the simulation will not be realistic. In the case of a small pleat angle (i.e. 3°) with the triangle geometry, the pleat folds do dominate, and this is a realistic portrayal of the flow. For larger pleat angles, the pleat fold area should not dominate. So if ht = 2tx, there is little freelength. For comparison, flowfields are obtained for a 45° pleat with varying pleat heights. Note that the increase in freelength leads to a decrease in the influence of the pleat folds and an increase in the filter area which in turn decreases the overall pressure drop. 67 0.001 0.0008 0.0006 0.0004 0.0002 0 0.001 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 , ' '"     ~~....  ....... ' ,  , , .,../. //'' //', /'   ,   ,, /' /' .... "  , ... !.. ~  .. , , p 1 , 0.0025 0.d03 0.0035 , 0.0015 0.002 0.004 0.0045 Fig.3.1O(a) ht = 1.79 mm. 0 freelength cells. upstream pressure = 414 Pa. , , ..  , """', '" ,  '   , / , ' / ,  , " " "  ), .... / /, " " ,  ,     ",   , "  . , O~~~~~I~~~~~'~~~~'~~~~~'~~~~'~~~~~'~~~~ 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Fig.3.l0(b) ht = 2.50 mm. 2 freelength cells. upstream pressure = 377 Pa. , 0.0025   ..... , ..  ... ,     , ...... ",,' , , r __ _ , 0.002  , , .,' , , "",,'" 0.0015 : 0.001 f , 0.0005 f l" , , '" , o ,  ,  1 I "1 I   ,  i'" "I " 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.0055 0.006 0.0065 0.007 0.0075 Fig. 3.l0( c) ht = 3.60 mm. 7 freelength cells. upstream pressure = 349 Pa. Figs. 3.10 Effect of pleat height on theflowfield 45° pleat (pruned and proportional, linearscale vectors). 68 It can be seen that with no area of the filter outside of the influence of the folds (ht = 1.79 mm), there is no region of the flow that orients normal to the pleat. Whereas with a larger freelength (ht = 2.50 or 3.60 mm), there is a region of flow oriented normal to the pleat. Judging from the similarities seen in the freelength region of the last two plots, it is suspected that this flow trend continues for greater pleat heights as well. So the pleat height can be set for a minimal number of freelength cells (2  4), where the number of freelength cells is given by: # of freelength cells = jbar  (nnode + 1) 3.8 The Grid Density (njiCx) Obviously, the denser the grid, the clearer the trends in the flow and the better the approximations to the PDE are, and vice versa. On the other hand, the denser the grid is, the longer a solution will take. Thus the effects of varying the fineness of the grid (determined by nfil_x only, for the triangular pleat) are examined to see the consequences. It is kept in mind that there comes a point where the grid is too fine. Recall that the velocity within the filter represents an areaaverage macroscopic velocity. The real velocity is related to the macroscopic velocity by media porosity: v = VdaTCY actual 8 (32) So if the grid were to shrink below the actual size of the filter pore, the velocities would no longer be describing a true macroscopic velocity for the cell. Thus a minimum cell dimension is specified by the pore size. The average distance between fibers given an 69 average fiber diameter of 39 !lm and a porosity of 0.77 [Sabnis, 1993] is about 32 !lm, assuming a facecentered cubic geometry. The average pore size based on the square root of the permeability of the media, following Bejan r 1984], is 9 !lIn. As the media thickness considered in this report is only about 600 !lm, this would set a limit to the grid detail specified. For the 3° pleat, runs are made with the grid fmeness varied (Fig. 3.11). 0.0008 0.0006 0.0004 0.0002 0 0.02 0.0008  0.0006  0.0004  0.0002_ 0 0.02 ...  , ,   , I I 0.025 0.03 .. '" ... . .... ... ..... : .....  .... :.. . ' , .  '   , , . :........ I I 0.035 0.04 Fig. 3.11 (a) nfil_x = 4. .....  , .... ,   , , ........ :a ., , ,  I 0.045 ..... _ ......... . .... ....... I 0.025 .: __ __ ___ ..a. ... f ...    I 0.03 I 0.035 I 0.04 Fig. 3.11 (b) nfily = 6. . ,. I 0.045 . ... .." I 0.05 I 0.05 0.055    0.055 o. 00081==~::::,:,:. , ::::::::::=:::,;:::,::_::;:===_=_=:=_=======~:::::j 0.0006= =.. .~. ..:.. ..:. ........ :. ..... .. . !." ;...... .:..:.. :::=:: =    0.0004 =:::::::.::.::.:' ':. ,: : " '::' : :: :: :: ::  ...... ..... .. ..... ..... ........ ...... ........ ... 0.0002        " .' ,. • O~~~~;~~T~~rT=~=r~;T~=r~~~~~~~~~r·'~~~,~~ I I I I I I 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Fig.3.11(c) nfily = 8. Figs.3.11 Single geometry with gridfineness varied, 3° pleat (pruned and expanded [yweighted ~700%]). 0.055 Although the increased resolution reveals greater detail, it does not reveal phenomena otherwise hidden. The main quality discovered is that the flow is oriented more normal to 70 the filter at the pleat entrance and exit, however this can be ascertained without the finer grid. As the greater detail comes at a high cost (computing time), it is not always sought on its own merit. 3.9 Selecting a Convergence Criterion (epsi) As the solution of the set of nonlinear equations is approached iteratively, some arbitrary condition of solution convergence must be specified to halt the iteration. Any or all of the three variables being solved (P,u,v) can be used in the convergence criteria. The key standard for the criterion is that it ensures iteration has been carried far enough so that a certain level of precision has been reached and will not be further influenced by continued iteration. Pressure could be used as a standard for convergence, but detennining a standard value is not possible as the magnitude of the pressure would vary greatly with d 



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