A semi-empirical pressure drop model: Part 1--pleated filters.
Overview of the Research
The demand for improved indoor air quality (IAQ) has created a need for gas phase filtration units capable of removing contaminants such as volatile organic compounds (VOCs), tobacco smoke, carbon monoxide, and formaldehyde. Strategies to remove these harmful contaminants include employing a packed-bed or an adsorbent-entrapped filtration media such as microfibrous sorbent-supported media (MSSM). Through a wet-laid process, MSSM's sinter-locked matrix of micron-sized fibers can entrap sorbent particles with diameters as low as 30 microns, leading to better chemical removal efficiency and higher sorbent utilization than a traditional packed bed. The disadvantages of adsorbent-entrapped media are a high pressure drop created by small, entrapped sorbent particles and a low saturation capacity due to the relatively thin thickness of the media (Harris et al. 2001).
New tactics for building more efficient gas phase filters need to be researched in order to maximize the usefulness of adsorbent-entrapped media. Pleated and V-bank filters are two designs that can improve both pressure drop performance and overall capacity for filtration units made from these materials. By understanding the pressure drop limitations within these filtration systems, additional media and adsorbent material can be packaged into a unit to increase the contaminant removal capacity while maintaining an acceptable resistance.
The following article, Part 1, discusses the creation and utilization of a model that can predict initial flow resistance in a pleated filter with a depth of 89 mm (3.5 in.) or less. The second article, Part 2, will extend the model presented below to include V-banks composed of multiple pleated filters.
The flow resistance of a filter is a critical design and operational parameter. A large pressure drop across the filter can overload the air handler unit and reduce airflow. More importantly, the pressure drop is directly related to the energy consumption of the filtration system. Energy consumption can account for 80% of the total expenses, while labor and procurement costs account for the remaining 20% (Arnold et al. 2005).
Numerous filter designs are commercially available, yet pleated filters are one of the more popular styles due to their unique performance benefits. A pleated filter uses a highly folded media to increase the available filtration area and extend the filter's useful life. The extra area also bestows the additional advantage of reducing the pressure drop and energy consumption of the filter. The resistance across a pleated filter fits a second order polynomial composed of a geometric ([K.sub.G][V.sub.f.sup.2]) and media ([K.sub.M][V.sub.M]) term.
[DELTA]P = [K.sub.G][V.sub.F.sup.2] + [K.sub.M][V.sub.M] (1)
Empirical and computational fluid dynamics (CFD) approaches have been attempted by Chen et al. (1996), Rivers and Murphy (2000), Caesar and Schroth (2002), Del Fabbro et al. (2002), and Tronville and Sala (2003) to determine the constants. Although each method produces accurate results, the models are only applicable to the specific filters studied and lack predictive capabilities due to the heavy reliance on empirical data. The contributions of the pleat tips and filter housing, mentioned by Raber (1982), are often neglected in the models.
The research objective is the development of an accurate model for use as an analytical design tool capable of predicting initial pressure drop performance of pleated filter units based solely on the thickness and permeability of the media utilized. The effects of a filter's geometry and packaging are quantified in a manner that can be universally applied to various pleated filter designs of depths less than 89 mm (3.5 in.). The model is composed entirely of algebraic equations to allow for quick optimization and for prediction calculations to improve the utility.
The approach used to construct the model is similar to Idelchik's (1994) method used to compute pressure drop in an electrostatic filter. In this method, the total pressure drop of a filter is modeled as a summation of smaller, component resistances. The individual components of a pleated filter were deduced from the proposed airflow pathway introduced by Raber (1982). Each component's influence on the total filter resistance was then formulated through the use of Forchheimer-extended Darcy's Law, Bernoulli's equation, and the equation of continuity.
Since the components of a pleat filter interact with one another, the modeling approach could not simply dissect and quantify the exact pressure drop influence of each component. The model is therefore an empirical determination of the relative influence of each term while in the presence of all other terms. This was accomplished by systematically changing design variables, methodically assessing the net increase in the total filter resistance, and then contributing that influence to the appropriate varying term.
Forchheimer-Extended Darcy's Law
In a particulate air filter, the high operational velocities (Reynolds number > 20) often result in nonlinear deviations from Darcy's Law for flow through the media (Rivers and Murphy 2000; Chen et al. 1996). Rivers and Murphy concluded that the deviations in filtration media were the product of fiber compression due to the air's inertial force compressing the media's fibers together at higher operational velocities. The compression changes the internal void volume and tortuosity of the media, leading to higher superficial velocities, decreased permeability, and a nonlinear increase in total resistance.
A practical method to account for the nonDarcian behavior is the addition of a second-order term to Darcy's Law (Scheidegger 1974). Equation 2 is known as a Forchheimer-extended Darcy's law. The A is equivalent to the Darcy's Law constant ([mu]L/[kappa]). The B accounts for the nonlinear deviation due to inertial effects.
[DELTA]P = A[V.sub.M] + B[V.sub.M.sup.2] (2)
Various theoretical equations exist that attempt to relate the physical significance of the second media constant, but these theories require extensive knowledge of the media's fiber dimensions and packing densities (Rivers and Murphy 2000). The research presented by Rivers and Murphy demonstrates the complexity and difficulty in accurately modeling media performance with these theories. Since the primary objective of the research is to identify and determine the resistances created by the geometric design parameters, and not the media, it is preferable to model the media constants using a quick, empirical approach that will not introduce as much theoretical error.
The mechanical energy balance is a summation of kinetic, potential, mechanical, compressive, and viscous energy terms (Bird et al. 2001). Bernoulli's equation is a reduced version of the mechanical energy balance that assumes incompressible, steady-state flow while maintaining a control volume with stationary, solid boundaries. Bernoulli's equation can be further simplified by eliminating elevation change within the control volume, operating at turbulent conditions, and removing all mechanical work. When applied between two points, the following equation results:
[DELTA]P = [P.sub.1] - [P.sub.2] = [1/2][rho]([V.sub.2.sup.2] - [V.sub.1.sup.2]) + L[upsilon] (3)
L[upsilon] = [1/2][rho]K[V.sub.i.sup.2] (4)
The viscous loss term Lv accounts for the change of mechanical energy into heat due to viscous forces (Bird et al. 2001). It is determined by simultaneously solving the mechanical energy and momentum balance or through experimental measurements. The K values, also referred to as coefficients of friction, are functions of either geometry, Reynolds number, or both. The V term is a reference velocity on which the coefficient is based. This reference velocity is an arbitrary choice, yet it must be either the entrance or exit velocity into the control volume. K values have previously been computed for flow systems such as a sudden contraction [K.sub.C], sudden expansion [K.sub.E], and flow through a perforated plate [K.sub.G] (Idelchik 1994).
[K.sub.C] = 0.5 x [([1 - [A.sub.Free]]/[A.sub.Total]).sup.0.75] (5)
[K.sub.E] = [([1 - [A.sub.Free]]/[A.sub.Total]).sup.2] (6)
[K.sub.G] = [([1.707 - [A.sub.Free]]/[A.sub.Total])[([A.sub.Free]/[A.sub.Total]).sup.-2] (7)
Flow Pattern in a Pleated Filter
Air travels through seven regions of varying cross-sectional flow area in a pleated filter (Figure 1). It is assumed that a uniform flow profile exists in the upstream duct before the filter. A typical pleated filter employs a grating to increase the structure integrity of the filter and the pleats. The flow is contracted by the grating, resulting in an increased velocity. The air expands back out after the grating, yet is quickly contracted for a second time as it is channeled around the pleat tips and into the pleats.
[FIGURE 1 OMITTED]
Once inside the filter's pleats, the airflow begins to split and changes directions to allow entrance into the media at an angle perpendicular to the media's surface. The air expanses out onto the media's surface area after the direction change. The proposed flow pattern through the filter's pleats is very similar to the flow in a converging or diverging wye.
The fourth region is the media's accessible surface area. The area does not include the small portion of media that will be pinched shut in the pleat tips. After flowing through the media, the airflow then follows an identical yet reversed path out of the filter system back into the downstream duct. A general schematic for parameter identification is presented. The nomenclature can be found at the end of the paper.
Pressure Drop Assessment of Flow System
The total pressure drop through a pleated filter was to be modeled as a summation of individual resistances. The individual resistances were formulated by applying Bernoulli's equation or the Forchheimer-extended Darcy's Law to the seven proposed flow areas outline above. The result is the following system of equations:
* across front grating: [DELTA][P.sub.1] = [1/2][rho][([V.sub.2.sup.2] - [V.sub.1.sup.2]) + [K.sub.G][V.sub.1.sup.2]]
* flow from grating to pleat inlet: [DELTA][P.sub.2] = [1/2][rho][([V.sub.3.sup.2] - [V.sub.2.sup.2]) + [K.sub.C][V.sub.3.sup.2]]
* flow from pleat inlet to media surface: [DELTA][P.sub.3] = [1/2][rho][([V.sub.4.sup.2] - [V.sub.3.sup.2]) + [K.sub.P1][V.sub.3.sup.2]]
* flow through media: [DELTA][P.sub.4] = [AV.sub.4] + [BV.sub.4.sup.2]
* flow from media surface to pleat outlet: [DELTA][P.sub.5] = [1/2][rho][([V.sub.5.sup.2] - [V.sub.4.sup.2]) + [K.sub.P2][V.sub.5.sup.2]]
* expansion from pleat outlet into grating: [DELTA][P.sub.6] = [1/2][rho][([V.sub.6.sup.2] - [V.sub.5.sup.2]) + [K.sub.E][V.sub.5.sup.2]]
* across back grating: [DELTA][P.sub.7] = [1/2][rho][([V.sub.7.sup.2] - [V.sub.6.sup.2]) + [K.sub.G][V.sub.7.sup.2]]
[DELTA][P.sub.F] = [SIGMA][DELTA][P.sub.i] = [DELTA][P.sub.1] + [DELTA][P.sub.2] + [DELTA][P.sub.3] + [DELTA][P.sub.4] + [DELTA][P.sub.5] + [DELTA][P.sub.6] + [DELTA][P.sub.7] (8)
[K.sub.C], [K.sub.E], and [K.sub.G] are previously published coefficients of friction computed by Equations 5, 6, and 7. [K.sub.P1] and [K.sub.P2] are the friction coefficients for flow in the upstream and downstream pleats. A new coefficient's formula can be identified by simultaneously solving the mechanical energy balance and the momentum balance. Figure 2 is the pleat control volume use to solve the balances for the downstream pleat coefficient.
[FIGURE 2 OMITTED]
[d[GAMMA]/dt] = [[V.sub.1][W.sub.1] + [P.sub.1][A.sub.1]][U.sub.i] - [[V.sub.2][W.sub.2] + [P.sub.2][A.sub.2]][U.sub.i] + [F.sub.s[right arrow]f] + mg
Force of Fluid on the Solid
Forces in y-Direction
Fy = [1/2]([V.sub.4][W.sub.4A] + [P.sub.4][1/2][A.sub.4])sin([gamma]) + [1/2]([V.sub.4][W.sub.4B] + [P.sub.4][1/2][A.sub.4])sin(-[gamma]) - 0 = 0
Forces in x-Direction
Fx = [1/2]([V.sub.4][W.sub.4A] + [P.sub.4][1/2][A.sub.4])cos([gamma]) + [1/2]([V.sub.4][W.sub.4B] + [P.sub.4][1/2][A.sub.4])cos(-[gamma]) - ([V.sub.5][W.sub.5] + [P.sub.5][A.sub.5])cos([gamma]) = sin([beta]) = 0.5[A.sub.5]/0.5[A.sub.4]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Mechanical Energy Balance
[1/2]([V.sub.5.sup.2] - [V.sub.4.sup.2]) + (1/[rho])([P.sub.5] - [P.sub.4]) + L[upsilon] = 0 L[upsilon] = [1/2]([V.sub.4.sup.2] - [V.sub.5.sup.2]) - (1/[rho])([P.sub.5] - [P.sub.4])
Substituting in Momentum Balance Solution
L[upsilon] = [1/2]([V.sub.4.sup.2] - [V.sub.5.sup.2]) - (1/[rho])[[rho]([V.sub.4.sup.2] - [V.sub.5.sup.2])] L[upsilon] = [1/2]([V.sub.5.sup.2] - [V.sub.4.sup.2])[V.sub.4.sup.2] = [[V.sub.5][A.sub.5]/[A.sub.4]).sup.2] L[upsilon] = [1/2][V.sub.5.sup.2][1 - [[([A.sub.5]/[A.sub.4])].sup.2]]
[K.sub.P2] = [1 - [([A.sub.5]/[A.sub.4]).sup.2]] where the reference velocity is [V.sub.5]
In order to solve the balances, two erroneous assumptions had to be made to simplify the equations. Although the force of the fluid on the surface ([F.sub.f[right arrow]s]) cannot be readily quantified, it is incorrect to simply neglect the term. Second, assuming the airflow will enter and exit the media at a strictly perpendicular angle is doubtful. Both solutions do indicate that the coefficient is a function of the pleating ratio (ratio of [A.sub.5]/[A.sub.4] or [A.sub.3]/[A.sub.4]).
An empirical approach was taken to determine the pleat coefficients' formulas, since they could not be determined by the analytical manner employed. The individual contributions of [K.sub.P1] and [K.sub.P2] could not be experimentally separated and analyzed due to the upstream and downstream pleat symmetry. The two coefficients were combined into a single coefficient, since they share identical geometries and experience the same velocities. The newly formed pleat coefficient [K.sub.P] was then substituted in their place. The series can be reduced by replacing all downstream velocities with their reciprocal upstream velocities. The seven terms can be summed and rearranged into the following model:
[DELTA][P.sub.F] = [1/2][rho][(2[K.sub.G])[V.sub.1.sup.2] + ([K.sub.C] + [K.sub.E] + [K.sub.P])[V.sub.3.sup.2]] + [AV.sub.4] + [BV.sub.4.sup.2] (9)
EQUIPMENT AND PROCEDURE
Media Test Rig
The media constants were determined using a 25.4 mm (1 in.) diameter duct (Figure 3) supplied by house air at 6.9 x [10.sup.5] Pa (100 psi). A significantly large duct length-to-diameter ratio of 48 to 1 was used to create a uniform, upstream velocity profile. A media sample was held in place by two plates tightened together by nut and bolt assemblies. A 305 mm (6 in.) long outlet section was located downstream of the media sample to prevent additional flow resistance from a sudden expansion out of the tube.
[FIGURE 3 OMITTED]
Airflow to the rig was controlled by two rotameters that produced a maximum face velocity of 2.48 m/s (490 fpm) within the duct. The rotameters were determined to be accurately calibrated by a volumetric displacement test. Pressure drop measurements were obtained with an Invensys IDP10-T differential pressure transmitter connected to a pressure tap located 50 mm (2 in.) upstream and 125 mm (5 in.) downstream of the media sample.
Media Data Acquisition
Three pressure drop tests were performed on each of the five media types employed. Media samples were obtained by disassembling manufactured filters that consisted of thermally bonded, polyolefin fibers manufactured by Kimberly-Clark. Through each test, multiple data points over the rotameters' velocity range were obtained. A datum point consisted of setting the rotameter to a flow rate and measuring the corresponding pressure drop. A control run with no media sample detected a small amount (0.75 Pa at 2.48 m/s) of background resistance. The resistance corresponded well to the Darcy-Weisbach equation for the whole flow range examined, and the additional resistance was subtracted from the total measured media resistance.
Filter Test Rig
Air was supplied to the filtration test rig (Figure 4) by a Dayton unit with a 15 in. impellor powered by a 3 Hp Hitachi motor. The motor was controlled by a Hitachi SJ200 inverter with a range of 0 to 60 Hz at 0.1 Hz increments. At the blower outlet, a pressure tap coupled with an Omega PX 154 010DI pressure transducer monitored resistance across the blower.
[FIGURE 4 OMITTED]
The baffles, flow straighteners, and main duct created a uniform airflow into the filtration section. The main duct, filter box, and outlet duct all possessed cross sections of 491 mm x 491 mm (19.5 in. x 19.5 in.) to reduce any flow disturbances between the segments. The filter box held and sealed the filters into place with a metal frame. The frame had a height of 3 mm; thus, it fit behind the filter's housing without interfering with the pressure measurements. The top of the filter box contained a window to verify that the pleat integrity remained intact during the experiments. Pressure drop across the filtration section was monitored by a Dwyer Mark II monometer and an Invensys IPO10 differential pressure transmitter. The equipment was connected upstream into the duct by 3 mm (1/10th in.) pressure taps located 250 mm (10 in.) before the filter test box. The manometer's downstream connection was left open to the room's atmosphere. The transmitter's outlet was linked to pressure taps located 150 mm (6 in.) before the end of the outlet section. The outlet section prevented an increased pressure drop due to sudden expansion out into the room. A National Institute of Standards and Technology-calibrated Extech #451104 vane-anemometer monitored exit velocity and temperature at the outlet of the test rig. Velocity measurements were conducted by a nine-point grid procedure outlined in ASHRAE Standard 52.2, Section 5.2 (ASHRAE 2007).
Air to the blower was drawn from the room. All tests were performed in a approximately 20[degrees]C (68[degrees]F) environment and at an elevation of 215 meters (705 ft) above sea level; therefore, the density of air was assumed to possess a constant value of 1.16 kg/[m.sup.3] (0.0725 lb/[ft.sup.3]) throughout the experiments.
Filter Data Acquisition
Experimental data were collected by setting the inverter to the desired frequency and recording the corresponding values from the pressure transducer, manometer, differential pressure transmitter, and vane-anemometer. The process was repeated at 15 uniformly spaced data points between 5 and 40 Hz. The 15-point test was duplicated for each filter. After the data were collected for a filter, a pressure drop curve was created by plotting face velocity versus pressure drop. The face velocity was calculated from the blower curve using the frequency and resistance across the blower. The value was verified by the vane-anemometer measurements. Pressure drop was measured by the pressure cell and cross-checked by the manometer. A regression line was fitted to each pressure drop curve for use in data analysis and to eliminate individual data discrepancies. The small background resistance of the system was removed from the regression line by use of the Darcy-Weisbach equation.
DETERMINATION OF MODEL PARAMETERS
The objective of the experimental program was to verify the validity of utilizing previously published coefficients to model particular aspects of the filter design, as well as to empirically determine a new coefficient for friction encountered in the pleat. Since the media constants will be unique and vary with the media used in the filter, the approach began by measuring the media constants (A and B) and thickness for all materials utilized in the research. The previously published coefficient [K.sub.G], [K.sub.C], and [K.sub.E] were then shown to be applicable. The pleat coefficient for a single filter was empirically determined from empirical [DELTA][P.sub.F] versus face velocity data. This technique was based on Rivers and Murphy's approach to identifying the constants N and [K.sub.G] from their model. A more universal coefficient was developed by determining [K.sub.P] for a multitude of filter designs.
Figure 5 presents the resistance versus face velocity results obtained from flat media samples. The individual points on the graph represent the experimentally collected data. The fitted lines correspond to second-order polynomials; thus, the use of Equation 2 is preferred to a linear Darcy's Law. The fitted constants and R-squared values are presented in Table 1.
Table 1. Media Summary Media Thickness, A, Pa * s/m B, Pa * [s.sup.2] [R.sup.2], mm /[m.sup.2] dimensionless #1 0.5 8.8 20.6 0.998 #2 0.5 11.6 24.6 0.998 #3 1.6 16.7 47.9 0.998 #4 1.0 18.2 81.1 0.999 #5 1.1 29.5 164.4 0.999
[FIGURE 5 OMITTED]
The Grating Coefficient of Friction ([K.sub.G])
The Handbook of Hydraulic Resistance (Idelchik 1994) computes the coefficient of friction for fluid flowing through a shaped, perforated plate using Equation 7. In order to verify that the filter housing could be modeled by the same formula, the frame was altered and the corresponding measured pressure deviation was compared to the calculated deviation.
The filter utilized in the grating experiments consisted of a #2 media with 22 pleats per filter and dimensions of 491 mm x 491 mm x 21 mm (19.5 in. x 19.5 in. x 0.85 in.). The filter's normal housing was composed of a diamond grid that blocked 34.5% of the flow area. Additional grating was uniformly added to the filter's front to increase the blocked flow area from 34.5% to 59.4%. Subsequently, the filter's grating was removed resulting in only 16.0% of the flow area being blocked. The graph of the pressure drops versus face velocity was plotted in Figure 6 for each grating configuration. The markers represented the observed data and the solid lines were Excel-fitted regression lines.
[FIGURE 6 OMITTED]
Since the same filter was utilized in all three tests, the observed pressure difference between the curves was generated solely by the grating modification. The additional resistance created by the extra grating can be quantified using the low blockage filter curve as a reference.
This observed increase in flow resistance between the normal blockage and the low blockage filter was then compared to the expected pressure increase computed using Equation 4 with the friction coefficients determined by Equation 7. Below, Equation 7 was used to calculate the friction coefficients for the following blockage profiles:
For Normal Blockage: [K.sub.G] = [(1.707 - 0.655)/[(0.655).sup.-2]] = 2.50
For Low Blockage: [K.sub.G] = [(1.707 - 0.840)/[(0.840).sup.-2]] = 1.23
For High Blockage: [K.sub.G] = [(1.707 - 0.406)/[(0.406).sup.-2]] = 7.89
This calculation was repeated for the difference between the high blockage and low blockage filter. The results of the measured differences and calculated differences are shown in Figure 7. Since the computed and observed differences overlap, Equation 4 in conjunction with Equation 7 could be used to adequately predict the pressure loss created by the filter's housing. Similar results were obtained after modifying the grating on the back of the filter as well as when filter types S and T (see Table 2) were subjected to the experiment. Although the grating resistance was observed to be independent of pleat count and permeability within the parameter space explored, caution should be taken when applying these equations on lower permeability or higher-pleated systems such as minipleat panels. The unique design of such filters could cause the flow field around the grating to be influenced by the pleats; thus, the use of Equation 7 will no longer be valid.
Table 2. Summary of Pleat Coefficients Filter Pleats Height, Width, Depth, Media Beta, rad mm (in.) mm (in.) mm (in.) 491 491 21 A 22 (19.5) (19.5) (0.85) # 2 0.521 491 491 44 B 16 (19.5) (19.5) (1.75) # 2 0.348 491 491 21 C 14 (19.5) (19.5) (0.85) # 1 0.819 491 491 21 D 19 (19.5) (19.5) (0.85) # 1 0.604 491 491 21 E 23 (19.5) (19.5) (0.85) # 1 0.499 491 491 21 F 28 (19.5) (19.5) (0.85) # 1 0.410 491 491 21 G 32 (19.5) (19.5) (0.85) # 1 0.358 491 491 21 H 37 (19.5) (19.5) (0.85) # 1 0.310 491 491 21 I 42 (19.5) (19.5) (0.85) # 1 0.273 491 491 21 J 47 (19.5) (19.5) (0.85) # 1 0.244 491 491 21 K 55 (19.5) (19.5) (0.85) # 1 0.209 491 491 44 L 19 (19.5) (19.5) (1.75) # 1 0.293 491 491 44 M 34 (19.5) (19.5) (1.75) # 1 0.164 491 491 88 N 19 (19.5) (19.5) (3.5) # 1 0.147 491 491 21 O 19 (19.5) (19.5) (0.85) # 3 0.604 491 491 21 P 32 (19.5) (19.5) (0.85) # 3 0.358 491 491 44 Q 19 (19.5) (19.5) (1.75) # 3 0.293 491 491 44 R 32 (19.5) (19.5) (1.75) # 3 0.174 491 491 44 S 56 (19.5) (19.5) (1.75) # 5 0.099 491 491 88 T 12 (19.5) (19.5) (3.5) # 4 0.232 Filter Kp, dimensionless [R.sub.2], dimensionless A 2.083 0.9761 B 1.893 0.9498 C 1.329 0.9991 D 1.791 0.9998 E 2.501 0.9998 F 3.335 0.9995 G 4.104 0.9998 H 4.960 0.9997 I 5.573 0.9999 J 6.442 0.9998 K 7.802 0.9997 L 2.311 0.9794 M 5.567 0.9974 N 3.506 0.9992 O 1.837 0.9932 P 3.938 0.9737 Q 2.466 0.9612 R 5.116 0.9979 S 10.066 0.9906 T 1.881 0.9850
[FIGURE 7 OMITTED]
The Pleat Tip Assumption
The contraction and expansion into and out of the pleats was assumed to be accurately modeled by Equation 4 using friction coefficients obtained by Equation 5 and Equation 6. For the friction coefficients calculations,
[A.sub.Total] = [F.sub.W]x[F.sub.H]
[A.sub.Free] = ([F.sub.W] - [P.sub.C] x [P.sub.T]) x [F.sub.H].
It would be exceedingly difficult to experimentally alter a pleat tip and analyze the resulting contribution to resistance without inadvertently affecting other resistances. The assumption that a pleat tip acts as a wall is based on Darcy's Law. When a media is pleated, the porous material is folded on top of itself creating a pleat tip of increased thickness and/or decreased permeability. Either an increase in media thickness or a decrease in permeability will result in a path of greater flow resistance according to Darcy's Law. Airflow through the pleat tip is therefore assumed to be blocked due to this heightened resistance and will be channeled around the tips and into the pleats. This assumption was also previously incorporated in the research by Caesar and Schroth (2002) and stated by Raber (1982).
The Pleat Coefficient of Friction ([K.sub.P])
The pleat coefficient for a specific filter was determined by obtaining [DELTA][P.sub.F] versus face velocity data over a range of velocities. The model was rearranged into the following linear form:
[DELTA][P.sub.F] - [1/2][rho][(2[K.sub.G])[V.sub.1.sup.2] + ([K.sub.C] + [K.sub.E])[V.sub.3.sup.2]] + B[V.sub.4] + A[V.sub.4.sup.2] = [1/2][rho][K.sub.p][V.sub.3.sup.2] (10)
[V.sub.i] values were computed from the face velocity, using the equation of continuity. The coefficients [K.sub.G], [K.sub.C], [K.sub.E], A, and B were tabulated by methods previously discussed. The pleat coefficient of friction for the filter was empirically determined by subtracting these known flow resistances from the experimentally measured total pressure drop. The resulting difference was then plotted opposite the reference velocity term ([1/2][rho][V.sub.3.sup.2]), and the pleat coefficient was inferred from the slope.
Figure 8 graphically displays this methodology for a 491 mm x 491 mm x 21 mm (19.5 in. x 19.5 in. x 0.85in.) #1 media filter with 42 pleats. The solid line is a least-squared regression fitted to the experimentally measured pressure drop data. The dashed line is a modeled compilation of the known flow resistances due to the flow through the media, blockage created by the filter grating, and the channeling due to pleat tip contraction and expansion. The hyphenated line represents the observed difference between the least-squared regression and the modeled line, and it is equivalent to the left-hand side of Equation 10. As seen in Figure 8B, a linear line resulted when plotting the observed difference versus ([1/2][rho][V.sub.3.sup.2]) for the filter. The slope of the line equated to the pleat coefficient of friction.
[FIGURE 8 OMITTED]
The resulting coefficient was only valid for a filter with an identical geometry. In order to acquire a more universal coefficient for the model, the pleat coefficient needed to be determined for a wide range of pleat counts, filter depths, media thicknesses, and media permeabilities. Twenty filter variations, manufactured by Quality Filters, Inc. in Robertsdale, Alabama, were used to determine the pleat coefficient. The pleat coefficient for each filter was determined by the same method outlined above. An inventory of the filters, geometric parameters, media type, observed pleat coefficients, and R-squared fit are presented in Table 2.
The formula for the pleat coefficient of friction should be based on Reynolds number, dimensionless geometric ratios, or both. Reynolds number has a prominent effect on the coefficient only when laminar flow is present, yet the flow was almost always turbulent for the test conditions encountered. The Kp coefficient was therefore determined solely on geometric configuration.
The partial solution to the momentum and mechanical energy balances indicated that the pleat coefficient should be related to the function [([P.sub.L]/[P.sub.O]).sup.2]. Figure 9 shows each experimentally-determined coefficient versus the function. As clearly seen, the function did not have a direct correlation to the observed coefficients. This is to be expected due to simplifying assumptions that were made in order to simultaneously solve the mechanical and momentum balances. A general power law trend was visualized between the function and the pleat coefficients. The dashed lines were power law functions with the generic formula y = [mx.sup.[2/3]]. To eliminate the power law fit, the function was rewrote as y = [([P.sub.L]/[P.sub.O]).sup.z] with a scaling exponent (Z) of 4/3 ([x.sup.2] x [x.sup.[2/3]] [approximately equal to] [x.sup.[4/3]]).
[FIGURE 9 OMITTED]
Figure 9 indicates that a second, supplementary term to account for the depth of the filter is needed to determine the pleat coefficient. As seen in Figure 9, the coefficients predicted for the 2 and 4 in. filters by the ratio function [([P.sub.L]/[P.sub.O]).sup.2] are much higher than the empirically determined values. It was hypothesized that the pleating ratio was related to the resistance created by the turn and separation of air in the pleat, but this ratio not fully account for the area available to make this maneuver. Due to the increased spacing within the 2 and 4 in. filters, the airflow was allowed to gradually slow and expand, which reduced friction between air molecules and in turn led to a lower pleat coefficient. A second term was introduced based on the Darcy-Weisbach theory for flow through a duct. The pleat coefficient of friction was accurately modeled with the addition of this dimensionless term and empirically fit scaling factor (0.11) to account for pleat spacing. The friction coefficient was formulated using [V.sub.3] as the reference velocity.
Kp = 0.11[([P.sub.L]/[P.sub.O]).sup.[4/3]]([F.sub.HD]/[F.sub.D]) (11)
[F.sub.HD] = (2[F.sub.H][F.sub.D])/([F.sub.H] + [F.sub.D]) (12)
Since the manner used to formulate the coefficient attributes all remaining resistances to the pleats, it is possible that additional influences are being accounted for by the term. This includes potential deviations created by the additive method used to factor in the media's resistance and the assessment of the pleat tip's influence on the overall filter resistance. To evaluate the assumption that pleat tips affect the overall pressure drop, the same analysis was performed without factoring in the pleat tip blockage. This was accomplished by removing the ([K.sub.C] + [K.sub.E])[V.sub.3.sup.2] term from Equation 10. A linear trend was still observed; however, the R-squared value decreased to 0.87. Of particular note, the filters with the largest pleat tip blockage (Types P, R, and S) possessed a higher-than-average coefficient, indicating that additional resistance effects are being absorbed into the pleat coefficient. This supported the use of a separate term for the pleat tips.
The plot in Figure 11 features the model that compares the calculated values of all 600 observed pressure drop data points compiled from the 20 various filter types. A least-squared regression line shows a one-to-one correspondence. The dashed lines represents [+ or -]5% from the regression line.
[FIGURE 11 OMITTED]
Pleating "U" Curve
Due to the trade-off of media-induced pressure loss for viscous-induced pressure losses, a pleated filter will experience a minimal pressure drop corresponding to an optimal pleat count. Previous research by Chen et al. (1996), Del Fabbro et al. (2002), Tronville and Sala (2003), and Caesar and Schroth (2002) each presented plots of pressure drop versus pleat counts that demonstrate this "U" pleating shaped curve. Chen et al. (1996) labeled the lower pleat count region to the left of the optimal number as the media-dominated regime. The graph is said to be in the viscosity-dominated regime when a filter possesses more than the optimal number of pleats.
[FIGURE 10 OMITTED]
A conventional pleating "U" curve was generated by modeling a 491 x 491 x 21 mm (19.5 x 19.5 x 0.85 in.) filter with FM1 media constants and thickness. The graph, Figure 12, was computed by holding the velocity constant at 2.54 m/s (500 fpm) and varying the pleat count from 12 to 60 pleats per filter. The model predictions were plotted as lines, while the black circles represent the observed total pressure drop of filter types C through K at 2.54 m/s (500 fpm). The modeled resistances due to the pleat contraction and expansion for this filter were left off the graph for clarity, because their contribution was very small (< 0.5 Pa). The observed flow resistances were fitted with error bars signaling [+ or -]5% of their value. The total modeled resistance fell within the experimental data error bars.
[FIGURE 12 OMITTED]
The modeled results of Figure 12 support the previously published general trends regarding pleated filters. The resistance versus pleat count graph clearly indicates a lowest obtainable resistance (LOR) corresponding to an optimal pleat count. The LOR occurs due to the tradeoff of media resistances for viscous resistances as the pleat count is increased. The graph also partly corroborates Chen's assertion that pleat tip blockage can be ignored; however, FM1 is a thin media (~0.5 mm) and the same claim cannot be made for all media types.
A novel feature of the model is the inclusion of a distinct term for the housing losses. Previous research usually ignores the housing effects, or its influence is masked because housing losses are simply combined in with the geometric losses. This has a two-fold disadvantage from a filter design perspective. First, the housing resistance is wrongfully attributed to other geometric design parameters such as pleat height or pleat pitch. This artificially augments the actual influence of these geometric parameters, leading to errors in design estimates. Second, the nature of the housing resistance acts in a different manner than the other geometric losses. Within the parameter range explored, the structural pressure drop essentially serves as a fixed resistance and its influence does not change with pleat count. All other geometric resistances increase with pleat count. A small increase in their resistance due to the incorporation of the grating losses becomes further skewed as pleat count varies. By identifying and separating the grating contribution, the model provides a better understanding of the individual resistances, allowing enhanced analysis, improved design, and increased performance.
One such design improvement would be the elimination of the structural housing. If the same filter could be adequately constructed without the housing, the total pressure drop at the optimal pleat count would be reduced by 30%. However, the structural stability of the filter might become compromised due to the elimination of the grating. This is especially true since most filters are loaded with dirt until a final pressure drop of 249 Pa is obtained. A simple solution is the addition of a wire mesh to the front and back to improve stability. Since the wire is primarily an open void, the net resistance effects would be similar to a small increase in the media constants. A second, similar design improvement would be to eliminate just the front grating. The back grating, in conjunction with the wire mesh, should be sufficient to support the filter media. The removal of the front grating would net a 15% reduction in the overall initial resistance.
An accurate model can also identify various design strategies to minimize material costs, minimize energy consumption, or maximize the useful life of an adsorbent-entrapped filter while maintaining an acceptable initial pressure drop. Figure 12 indicates the presence of a semiflat valley between 27 and 47 pleats that highlights these design goals. In this valley, the initial pressure drop hovers around a starting resistance of 62 Pa. At the low pleat count end, a filter with 27 pleats can be constructed that will perform at an adequate pressure drop, without incurring a higher production cost due to increased material costs. This is especially useful for a filter employing expensive adsorbent or catalyst materials. The most energy efficient filter can be manufactured by increasing pleat count to the LOR of 36 pleats; however, it should be noted that the overall energy consumption is greatly affected if these gas phase filters become loaded with debris. The high end of the valley offers a filter with the largest available filtration area and adsorbent loading without sufficiently increasing the initial pressure drop. The ability to locate and work within this valley demonstrates the utility of an accurate pressure drop model to a filter designer.
Location of the Optimal Pleat Count
Although it can be used as a general heuristic, the optimal pleat count does not simply exist where the media and geometric resistances are equal. For example, the optimal pleat count in Figure 12 was 36 pleats, yet the media and pleat resistances were equal at 38 pleats. The lowest obtainable resistance and the optimal pleat count actually occurred when the total pressure drop's rate of change with respect to pleat count is zero. Since Equation 9 was composed of polynomials, the model can be broken down into individual terms, and the first derivative with respect to pleat count can be readily computed.
[[partial derivative][DELTA][P.sub.F]/[partial derivative][P.sub.C]] = [[partial derivative]([rho][K.sub.G][V.sub.1.sup.2])/[partial derivative][P.sub.C]] + [[partial derivative]([1/2][rho][K.sub.C][V.sub.3.sup.2])/[partial derivative][P.sub.C]] + [[partial derivative]([1/2][rho][K.sub.E][V.sub.3.sup.2])/[partial derivative][P.sub.C]] + [[partial derivative]([1/2][rho][K.sub.P][V.sub.3.sup.2])/[partial derivative][P.sub.C]] + [[partial derivative](A[V.sub.4])/[partial derivative][P.sub.C]] + [[partial derivative](B[V.sub.4.sup.2])/[partial derivative][P.sub.C]] (13)
[[partial derivative][DELTA][P.sub.F]/[partial derivative][P.sub.C]] = Team 1 + Team 2 + Team 3 + Team 4 + Team 5 + Team 6 (14)
The grating contribution's (Term 1) first derivative was zero because it is not a function of pleat count. The first derivative of the viscous/geometric effects (Terms 2, 3, and 4) were always positive, while the media resistances (Terms 5 and 6) had continuously negative first derivatives with respect to pleat count. Equation 14 could be rearranged to give Equation 15 since the left-hand side was equal to zero at the optimal pleat count.
-[[1/2][rho][[[partial derivative]([rho][K.sub.C][V.sub.3.sup.2] + [K.sub.E][V.sub.3.sup.2] + [K.sub.P][V.sub.3.sup.2])]/[partial derivative][P.sub.C]] = [[[partial derivative](A[V.sub.4] + [BV.sub.4.sup.2])]/[partial derivative][P.sub.C]] (15)
Figure 13 is a graphical representation created by taking the first derivatives of the model as shown by Equation 13. Each derivative was calculated while holding the filter dimensions, face velocity, and media properties constant and changing the pleat count between 12 and 70 pleats. The modeling parameters used to create Figure 12 (FM1 media constants, 2.54 m/s face velocity, and 508 x 508 x 25 mm dimensions) were the same as those used to create Figure 13. Figure 13 clearly indicates the balance between viscous and media-dominated resistances.
[FIGURE 13 OMITTED]
A rigorous model was developed based on the physical properties of the media and pleated filter design. The model's pressure drop estimations for 20 different single filters were within [+ or -] 5% of the observed resistance value. Since the model does not employ nontransferable parameters, accurate pressure drop predications can be made solely on empirical data obtained for media thickness and permeability. The model agrees with previously published research in the field.
Model parameters such as pressure drop, pleat height, and pleat count directly influence operational cost, available media area, and performance. A model encompassing these influential design parameters and the effects of their variation can serve as a design tool for filtration units. At a fixed-flow velocity, the model can be used to locate the lowest obtainable resistance and corresponding optimal pleat count for a given media. In HVAC systems where air velocity and allowable pressure drop are fixed, the model can predict the maximum filtration area and sorbent load. It can further serve as a design tool for media construction to back-calculated preferred media properties with respect to permeability versus thickness to achieve a desired operational condition. The end benefits to an adsorbent filtration design would be an increase in available sorbent, decrease in material cost, or a reduction of operational energy costs.
This research was performed under U.S. Army TACOM LCMC (contract #W56HZV-05-C-0686).
A = media constant, Pa * s/m
[A.sub.Free] = free area, [m.sup.2]
[A.sub.i] = area at point i, [m.sup.2]
[D.sub.W] = duct width, m
[F.sub.HD] = filter hydraulic diameter, m
[F.sub.W] = filter width, m
g = gravitational constant, m/[s.sup.2]
[K.sub.G] = geometric constant, dimensionless
L = porous media length, m
[P.sub.C] = pleat count, dimensionless
[P.sub.L] = pleat length, m
[P.sub.T] = pleat tip, m
[V.sub.F] = face velocity (~[V.sub.1]), m/s
[DELTA][P.sub.i] = generic pressure drop, Pa
[beta] = pleat pitch, rad
[mu] = viscosity, Pa*s
B = media constant, Pa*[s.sup.2]/[m.sup.2]
[A.sub.Total] = total area, [m.sup.2]
[F.sub.D] = filter depth, m
[F.sub.H] = filter height, m
Ki = coefficient of friction, dimensionless
[K.sub.M] = media constant, Pa*s/m
[L.sub.v] = viscous losses, Pa
m = mass, kg
[P.sub.i] = pressure at point i, Pa
[P.sub.D] = pleat depth, m
[P.sub.O] = pleat opening, m
Vi = velocity at point I, m/s
[V.sub.M] = media velocity (~[V.sub.4]), m/s
[W.sub.i] = mass flow at point i, kg/s
[DELTA][P.sub.F] = filter pressure drop, Pa
[kappa] = media permeability, [m.sup.-2]
[rho] = density, kg/[m.sup.3]
[gamma] = pleating angle, rad
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Ryan A. Sothen
Bruce J. Tatarchuk, PhD
Received March 25, 2008; accepted July 31,2008
Ryan A. Sothen is a graduate research assistant in the Department of Chemical Engineering at Auburn University, Auburn, AL. Bruce J. Tatarchuk is a professor of chemical engineering at Auburn University and is the director for the Center of Microfibrous Materials Manufacturing Center, Auburn, AL.
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|Author:||Sothen, Ryan A.; Tatarchuk, Bruce J.|
|Publication:||HVAC & R Research|
|Date:||Nov 1, 2008|
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