Pulsating Flow of Polymer Solutions Donald W. Sundstrom" and Arthur Kaufman Department of Chemical Engineering, The University of Connecticut, Storrs, Connecticut 06268
The flow of pseudoplastic fluids in a tube under sinusoidal pressure gradients was examined theoretically and experimentally. Theoretical results were obtained by solving the equation of motion numerically using both the Ellis and power law models to characterize the fluid viscosity. Experimental data taken with aqueous polymer solutions confirmed the theoretical approach and demonstrated that the addition of pulsations may increase flow rates significantly. The flow rate enhancement increased with increasing deviation from Newtonian behavior, with increasing pulsation amplitude, and with decreasing values of a frequency parameter. For any given flow rate, however, power requirements were greater for pulsating than for steady-state flow. Dimensionless graphs were developed to aid in estimating flow rates and power requirements for polymer solutions in a pulsating flow process.
The behavior of fluids subjected to a periodic pressure gradient is often classified as either oscillating or pulsating flow. With oscillating flow, the mean value of the periodic pressure gradient is zero so that the fluid moves back and forth with no net flux. In the case of pulsating flow, a periodic pressure gradient with a nonzero mean produces a net flow of fluid. Pulsating flow is encountered in many situations as a result of positive displacement pumps. The use of pulsations has also been of interest in connection with heat transfer, mass transfer, and chemical kinetic processes. The laminar flow of Newtonian fluids under pulsating conditions has been studied extensively by Lambossy (1952), Womersley (1955), Uchida (1956), and others. With pulsating flow of Newtonian liquids, the velocity profile is readily found by superimposing the oscillatory component of velocity upon the steady flow component. The mean flow rate is unaffected by the oscillatory component of the pressure gradient and is therefore the same as the steady-state flow rate produced by the mean pressure gradient. However, the average rate of energy dissipation by friction is increased by the addition of an oscillatory component, resulting in higher power requirements for pulsating flow than steady flow of Newtonian fluids. In most polymer processes, the fluid is non-Newtonian in nature and may also exhibit elastic as well as viscous characteristics during flow. Several studies of oscillating and pulsating flow of viscoelastic fluids have been reported, many of which are reviewed by Harris and Maheshwari (1975). If the time scale of the process is large relative to the relaxation times of the fluid, the elastic effects can usually be neglected. Pulsating flow of inelastic non-Newtonian fluids has received limited attention. Gianetto and Baldi (1970,1972) and Edwards et al. (1972 a, b) made theoretical analyses for pulsating flow of power law fluids in circular tubes under sinusoidal pressure gradients. For pseudoplastic power law fluids, they found that time-averaged flow rates with pulsations were greater than the steady-state flow rates a t the same mean pressure gradient. The power requirements for pulsating flow, however, exceeded steady-state values at corresponding flow rates. Barnes et al. (1969) compared theoretical and experimental flow rates for pulsating flow of aqueous polyacrylamide solutions in a tube. The theoretical analysis was based on a purely viscous fluid, and rheological data from pipe flow experiments were used directly in the model. Both the theoretical and experimental increases in flow rates resulting from pulsations went through a maximum as the mean pressure gradient was increased. Although shapes of the curves were similar, measured flow rate increases were greater than the 320
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
theoretical flow rate increases. For the range of frequencies covered by Barnes et al., however, Edwards et al. predicted that the flow rate increase was a constant value independent of the mean pressure gradient. The discrepancies between the two analyses may result from differences in rheological behavior. Edwards et al. employed a power law model for viscosity whereas Barnes et al. used measured viscosity values. If the viscosity data of Barnes et al. reached a second Newtonian region a t high shear rates, then flow rate enhancement would approach zero. Since no rheological data were presented by Barnes et al., this possible explanation of their flow curves could not be confirmed. Most of the theoretical work on inelastic fluids has used the power law as a rheological model but data are not available to verify the predicted results. In this study, pulsating flow experiments were conducted with polymer solutions to provide data for examining the validity of a theoretical analysis. The theoretical model was then used to obtain results over a range of conditions to illustrate the effect of pulsations on mean flow rates and power requirements, and to provide dimensionless graphs for use in design.
Theory The axial component of the equation of motion for flow of an incompressible fluid in a horizontal tube is
If end effects are negligible, the pressure gradient is independent of axial position and can be expressed as AplL. For pulsating flow, the pressure drop is the sum of a steady component and a sinusoidal component of amplitude A and frequency w Ap =
+ A sin w t = z(l+ X sin u t )
where X is the ratio of amplitude A to mean pressure drop G. The equation of motion then becomes
T o solve the equation of motion, a suitable relationship between shear stress, T , and shear rate, aular, is needed. Many rheological models have been proposed to relate T and aular for inelastic non-Newtonian fluids. The power law is the most widely used model because of its simplicity. T
= K(-au/ar)*
(4)
Although the power law has the flaw of predicting infinite
viscosity a t zero shear rate, it is often adequate in characterizing non-Newtonian fluids for practical situations. The three-constant Ellis model provides greater flexibility than the power law
convenience, the pulsating power will be expressed in dimensionless form by dividing it by the steady-state power requirement, QJp,. Thus, the ratio of pulsating flow to steady-state power requirement at the same mean pressure gradient is obtained from = Bplp = 2
In this model qO is the limiting viscosity as shear rate approaches zero and re is the shesr stress a t which viscosity is equal to '12 of 70. Since the E l k modei reduces LO 2 sxi.e: law relationship at high shear rates, the power law is in effect a special case of the Ellis model. The Ellis model has been used successfully to solve a variety of problems involvicg flow of polymer solutions. The power law and Ellis expressions were selected as rheological models for solving the equation of motion.
Dimensional Analysis Dimensional analysis techniques were applied to the equation of motion to relate flow and power to the minimum number of dimensionless groups. The use of dimensionless equations is more general and efficient since the important combinations of variables are specified rather than individual parameters. For the Ellis model, the following dimensionless groups were used
APS R L 2
7w=--
=
?O
J1 U Y ( 1 +A
sin 270) dY de (10)
Since the power law model provides an explicit expression for shear stress, it can be substituted in the equation of notion to eliminate T. The dimensionless equation of motion for a power law fluid is given by
"-
as
[1+ X sin
2x8
n Since the /3 group is not needed for power law fluids, the reduced pulsating velocity is a function of only the a , A, and n groups. The dimensionless groups have the same form as for the Ellis model fluid but the expressions for u s and vs are
[ 1 + --& (94/Te)Q-']
.Ips =.Ip
The terms qs and u srepresent the effective viscosity and average velocity in the tube a t steady state. Substituting these dimensionless groups into the equation of motion and the Ellis model gives
T
L1
For power law fluids, the ratio of pulsating to steady-state flow rate does not depend upon the level of shear stress in the tube. A t very low and very high values of the dimensionless frequency parameter ( a ) ,approximate analytical solutions can be obtained for the power law case. In general, however, the equation of motion must be solved by numerical techniques. A finite difference procedure was used in this study to solve for averaged reduced velocity over a wide range of values of the dimensionless groups. The numerical solutions were checked against analytical solutions when possible at low and high dimensionless frequencies.
where
n .. '13
Q J p s
+ T a = -41 [ p +-p] 4 a+3
dU ay
These equations must be solved simultaneously since the rheological equation is implicit in T . The average reduced velocity obtained by integrating over radial position and time, is a function of the dimensionless parameters a, @,A, and
u,
a.
The reduced velocity is equivalent to the ratio of average pulsating flow rate to steady-state flow rate at the same mean pressure gradient G.
The time averaged power requirement for pulsating flow is the product of flow rate and pressure drop, QpAp. For
Experimental Section Pulsating flow experiments were carried out in circular tubes using aqueous polymer solutions as the non-Newtonian fluids. A schematic diagram of the apparatus is shown in Figure 1. Sinusoidally varying pressure gradients were established by coupling a sinusoidal signal generator to a proportional electro-pneumatic device upstream of the circular tube. The signal generator (Procedyne Model SG-101) produced a sinusoidally varying voltage of controllable magnitude and frequency. The electro-pneumaticconverter (Leeds and Northrup Model 10970-1) developed an output air pressure proportional to the input voltage from the signal generator. Aqueous polymer solution was supplied to the test section from a pressurized feed tank. A tee between the feed tank and test section led to the electropneumatic converter through an indicator section of transparent tubing. The output air pressure from the electro-pneumatic device acted on the liquid in the indicator section and developed the sinusoidally varying pressure gradient. During pulsating flow experiments, the liquid in the indicator section oscillated back and forth and produced time varying flow rates in the test section. Pulsation amplitudes and frequencies were changed by means of controls on the sinusoidal signal generator. For most of the runs, the amplitude of the pressure wave was adjusted so that it equalled the mean pressure gradient through the test section. The test sections were 36-in. lengths of stainless steel tubing with inside diameters of 0.0905 and 0.123 in. The diameters Ind. Eng. Chem., Process Des. Dev., Vol. 16, NO. 3, 1977
321
t
Air IO-
Generotor
Test Section Tubing 1
IO1
I
0
1,000 1.8
9
I
I
I
I
1.2
Figure 3. Comparison of experimental and theoretical flow rate enhancement ratios for 0.27% Natrosol solution.
Figure 1. Schematic diagram of experimental apparatus.
"E
I
0.4 06 0.8 1.0 FREQUENCY PARAMETER, (I
0.2
1
500
In
E
-
G
0 >.
e
1.4
w
a 3
a
&
I
(L
200
v) v)
2
100
e
-
*) **
\\* \
(j,
Theoretical
a a W
5
50
20 100
1.0 L 0 I
I
I
I
I
500 1,000 5,000 SHEAR R A T E , s e c - '
I
10.000
Figure 2. Rheological data for 0.27% and 0.45% Natrosol 250-HR solutions.
were calculated from steady-state flow data with water as the fluid. Pressure levels were monitored by transducers connected to pressure taps located 6 in. from each end of the test section. The output signals from the transducers were recorded on a two-channel oscillographic instrument (Beckman Type RS Dynograph). Time-averaged flow rates were determined by weighing the effluent liquid from the tubes. Aqueous solutions of a soluble hydroxyethylcellulose polymer (Natrosol250-HR)were used as the non-Newtonian fluids. Natrosol250-HR exhibits appreciable deviations from Newtonian behavior at low concentrations but is only weakly viscoelastic as evidenced by fluid relaxation times near 0.02 s (Biery, 1964). To minimize biological degradation of the Natrosol solutions, 0.5% sodium benzoate was dissolved in all solutions. The viscosity characteristics of different batches with a given polymer concentration could generally be duplicated within 2%. Rheological properties of the polymer solutions were determined by using the apparatus as a steadystate capillary viscometer in which liquid flow rates were measured at several constant levels of applied pressure. The steady-state viscometer data were analyzed by the method of Rabinowitsch (1929) to determine shear stress at the wall as a function of shear rate. The temperature of the liquid in the test section was controlled at 25 "C for all experiments. Results a n d Discussion (a) Viscometry. The shear stress-shear rate results from the steady state experiments are plotted in Figure 2 for the 0.27 and 0.45% Natrosol solutions. Since both solutions are straight lines on logarithmic coordinates, their rheological behavior can be described by the power law model. The slope of the line is the power law index n and the intercept at a shear 322
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
I
I
I
1
0.15 FREQUENCY PARAMETER, 0.10
0.05
0.20 (I
Figure 4. Comparison of experimental and theoretical flow rate enhancement ratios for 0.45% Natrosol solution.
rate of unity is the constant K . The equations describing the data of Figure 2 are 0.27%Natrosol:
T
0.45%Natrosol:
T
(- 3°'65s = 5.18 - ( = 0.954
3°'52g
where 7 is shear stress in dyn/cm2 and du/dr is shear rate in S-1.
(b) Pulsating Flow. For all experiments, the sinusoidal forcing function was adjusted to maintain the minima of the pressure cycles at zero gauge pressure. Thus, the ratio of the amplitude of the sinusoidal component to the steady-state component of pressure gradient was unity for all runs (A = 1). The measurements yielded time-averaged flow rates and mean pressure gradients for frequencies ranging from 0.03 to 1 cycle/s, wall shear stresses ranging from 30 to 240 dyn/cm2, and tube diameters of 0.0905 and 0.123 in. Steady-state flow rates-corresponding to each mean pressure gradient in the pulsating experiments were obtained from the steady-state results. The ratio Q of time-averaged pulsating flow rate to steady-state flow rate at the same pressure gradient is plotted vs. the dimensionless frequency parameter in Figures 3 and 4. Calculated Reynolds numbers showed that all runs were well within the laminar regime. Theoretical ratios of pulsating to steady-state flow rates were evaluated by numerical solution of the equation of motion with a X of unity. Theoretical results for the two polymer solutions, shown as solid lines in Figures 3 and 4, are in general agreement with the experimental data over the range of conditions studied. For a given pressure gradient, both theory and experiment demonstrate that pulsations increase flow rates by about 20% for 0.27% Natrosol solutions and by about 40% for 0.45%Natrosol solution. Since the fluids obeyed the power
1.20k
a = 1.5
W
I-
d
1.4
3
s I. LL
2
10 0
FREQUENCY PARAMETER, Q
5
10
20
15
FREQUENCY PARAMETER, Q
Figure 5. Pulsating to steady-state flow rate ratios for Ellis model fluid with a = 1.5,
Figure 7. Pulsating t o steady-state flow rate ratios for Ellis model fluid with a = 2.5.
POWER LAW n 0.5
1.5 IO IO
1.4
zF 13 W
F
a
12
3
''b 9LL
I I
LL
1.0 0
5
IO
20
15
FREQUENCY PARAMETER,
Q
Figure 6. Pulsating to steady-state flow rate ratios for Ellis model fluid with a = 2.0.
law model (6= m ) , the level of shear stress had a negligible effect on the flow rate ratio. The experimental data points in Figures 3 and 4 represent frequencies up to 1 cycle/s. At higher frequencies, the pressure traces deviated from sinusoidal behavior because of limitations of the electro-pneumatic device. In addition, viscoelastic effects would assume greater importance a t higher frequencies where natural response times of the fluid become comparable to cycle periods. Therefore, higher dimensionless frequencies were not included in this study. ( c ) Flow Rate Graphs. The theoretical and experimental results are in reasonable agreement over the range of conditions studied and show a significant flow rate enhancement with pulsating flow. The theoretical analysis predicts that greater flow rate enhancement is realized a t low dimensionless frequencies and high values of amplitude ratio. Since the experimental amplitude ratio of X = 1represents a high practical value, the experimental data confirms the theoretical predictions in a region of large flow rate enhancement. Although the range of experimental conditions is limited, the agreement is sufficiently encouraging to warrant extending the theory to a wider range of dimensionless groups. Since the Ellis model is a more flexible rheological equation than the power law, the Ellis model was emphasized in the theoretical analysis. As shear stress or rate of strain is decreased, the viscosities of many polymer solutions approach a constant value. The Ellis model contains a zero shear rate viscosity, 70,whereas the power law model predicts an infinite viscosity a t zero shear rate. Thus, the Ellis model provides a better correlation for fluids that exhibit a change from Newtonian to power law behavior over the shear rate range of in-
10 0
5
10
20
15
FREQUENCY PARAMETER,
Q
Figure 8. Effect of amplitude ratio on pulsating to steady-state flow rate ratios for power law model with n = 0.5.
terest. As shown previously, the flow rate ratio for an Ellis fluid is a function of a , 6,A, and a. The effect of these dimensionless groups on flow rate ratio is illustrated in Figures 5 to 8. The flow rate ratio has a maximum value a t low dimensionless frequency and then declines to unity as a approaches zero. Since experimental 01 values were less than 1.2, the measured flow rate ratios were in the low-frequency region near the asymptotic values. For a values exceeding about 20, the deviations of flow rate ratio from unity is small. At high frequencies, the fluid movement cannot respond fast enough to the pressure fluctuations and the velocity is essentially equal to the steady-state value. The entire range of N values shown in Figures 5 to 8 can be encountered easily in practice. For example, if a fluid with qs = 25 CPand p = 1 g/cm" flows in a 1-in. diameter tube under a pulsation frequency of 1 cycle/s, the value of parameter N is 40. The shear stress parameter p ranges from zero for a Newtonian fluid to infinity for a power law fluid. The figures show that the flow rate ratio is unaffected by pulsations for a Newtonian fluid. As 6 increases, the fluid behavior is increasingly non-Newtonian and the flow rate ratio increases correspondingly. For a power law fluid, the flow rate ratio is independent of the level of shear stress. The flow index, a , in the Ellis model exceeds unity for pseudoplastic fluids. In the special case of a power law fluid, a = l h , so that n is less than 1for pseudoplastic fluids. As the deviation of a and n from unity increases, the fluid becomes more non-Newtonian and the flow rate ratio increases. For Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
323
1
2.0
1.5 A = 1.0
0'
1 0
IO-
P-
1.8
Ga I-
16
e5
14
8 a 6
1.2
z W
i:
t
a
c z W
J
a 3
2.0
sa (L
1.5
W
ba
a I 0
5
1 10
I
I
15
20
FREQUENCY PARAMETER, CL
Figure 9. Pulsating to steady-state power requirement ratios for Ellis model fluid with a = 1.5.
example, the asyinptotic flow rate enhancement increases from 1.2 for a = I .5 to 2.5 for a = 3. The results shown in Figures 5 to 7 are for X = 1,where the minimum of the sinusoidal pressure wave is taken as zero and the amplitude of the oscillatory component of pressure gradient is equal to the time averaged component. To illustrate the effect of A on flow rate ratio, results were calculated for a power law fluid with n = %. Figure 8 shows that flow rate ratio decreases with decreasing amplitude ratio. For this particular case, the amount by which Q exceeds unity is proportional to A 2 a t low dimensionless frequencies. Thus, flow rate enhancement increases by a factor of 4 when amplitude ratio is doubled. Since the fluid is non-Newtonian, the flow rate pulsates asymmetrically about the steady-state value corresponding to the mean pressure gradient. For pseudoplastic fluids, the deviation of instantaneous flow rate from steady-state flow rate is greater on the high side of the cycle than on the low side. As a result, the value of time averaged flow rate over a cycle exceeds the steady-state value. In using these graphs, it should be remembered that the results are valid for inelastic fluids obeying a power law or Ellis rheological model. A fluid may be considered inelastic in an unsteady-state process when characteristic time periods for the process are much greater than relaxation times of the fluid. For the system studied, the fluid relaxation times of about 0.02 s (Biery, 1964) were small compared to the shortest cycle time of 1s. Under these conditions, the fluid accommodated rapidly to changes, and elastic effects were probably insignificant a t all frequencies. (d) Power Requirement Graphs. The quantity P is the ratio of time averaged power requirement for pulsating flow to steady-state power requirement a t the same mean pressure gradient. The theoretical results presented in Figures 9 to 11 correspond to the same range of dimensionless groups used in Figures 5 to 7. The power requirement ratio exceeds unity at all conditions for both Newtonian and non-Newtonian fluids. As in the case of flow rate ratio, the power requirement ratio decreases with increasing cy and increases with increasing 8, A, and a . At low dimensionless frequencies, the instantaneous values of flow rate and pressure are largely in phase throughout the cycle. The high value of power requirement ratio a t low a results mainly from this in phase character of flow rate and pressure. As dimensionless frequency increases, the phase angle by which instantaneous flow rate lags pressure becomes larger and reaches about 90' a t a dimensionless frequency of 25. By comparing corresponding flow rate and power requirement figures, the power requirement ratio is seen to exceed 324
3.0
0 = 2.0 A = 1.0
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
I
0
I
5 IO 15 FREQUENCY PARAMETER, a
T
20
Figure 10. Pulsating to steady-state power requirement ratios for Ellis model fluid with a = 2.0.
FREQUENCY PARAMETER, a
Figure 11. Pulsating to steady-state power requirement ratios for Ellis model fluid with a = 2.5.
the flow rate ratio a t any particular set of conditions. For ex-
_ample, - from Figures 6 and 10 a t cy = 0 and @ = a,the ratio of
PIQ is 2.5j1.5. Although pulsating flow produces a flow rate enhancement, the additional power needed to sustain the flow is proportionately larger. For a given flow rate, then, the cost of power is greater for pulsating flow than for steady-state flow. Conclusions 1. Sinusoidal pulsations of pressure gradient about a mean value result in a flow rate enhancement for pseudoplastic fluids flowing in a circular tube, Experimental data taken a t frequencies up to 1 cycleh confirm the validity of the theoretical approach. 2. Dimensionless graphs are developed from the pulsating flow theory to illustrate the effect of variables on flow rate and power required for fluids described by the Ellis model. For the special case of a power law fluid, the theoretical results agree with those of Gianetto and Baldi (1970, 1972) and Edwards et al. (1972a, b). 3. The fractional increase in power required for pulsating flow is greater than the fractional increase in flow rate. For a given time-averaged flow rate, then, the power requirements are larger for pulsating than for steady-state flow. 4. The excess flow rate and power requirement attributable to pulsations increase as the fluid characteristics become more non-Newtonian. For an Ellis model fluid, the flow rate and power ratios increase with incressing flow index a and shear stress parameter 13.
5. The flow rate and power requirement ratios decrease with increasing dimensionless frequency and ultimately approach values of unity. At low dimensionless frequencies, the flow and power ratios reach maximum asymptotic values. 6. If the rheological behavior of a polymer solution or polymer melt can be described by an Ellis or power law model, the dimensionless graphs can be used to estimate flow rates and power requirements for a pulsating flow process.
Nomenclature a = Ellis model parameter A = amplitude of sinusoidal pressure oscillation K = power law model parameter L = lengthoftube n = power law model parameter p = pressure h p = time averaged pressure drop bS= steady state pressure dropP = reduced power requirement, QpAp/QsAps Q p = time-averaged pulsating flow rate Qs = steady-state flow rate, Qp/Qs r = radial position R = radiusoftube t = time T = dimensionless shear stress u = velocity up = time-averaged pulsating velocity u s = steady-state velocity U = dimensionless velocity U = reduced velocity, up/us
Y = dimensionless radial position 2 = axial position Greek Letters a = dimensionless frequency parameter 6 = dimensionless shear stress parameter 70
= Ellis model parameter
qs = effective steady state viscosity 0 = dimensionless time A = dimensionless amplitude
= density stress model parameter T~ = shear stress a t wall o = angular velocity p
7 = shear 7, = Ellis
Literature Cited Barnes. H. A., Townsend, P., Walters, K., Nature (London), 224, 585 (1969). Biery, J. C., AlChEJ., I O , 551 (1964). Edwards, M. F., Nellist, D. A., Wilkinson, W. L., Chem. Eng. Sci., 27, 545 (1972a). Edwards, M. F., Nellist, D. A . , Wilkinson, W. L., Chem. Eng. Sci., 27, 671 ( 1972b). Gianetto, A., Baldi, G., lng. Chim. /tal., 6, 186 (1970). Gianetto, A., Baldi, G., Capra, V., Chem. Eng. Sci., 27, 670 (1972) Harris, J., Maheshwari, PI., Rheol. Acta, 14, 162 (1975). Lambossy, P., Helv. Phys. Acta, 25, 371 (1952). Rabinowitsch, B., 2.Phys. Chem., A145, l(1929). Uchida, S., Z.Angew. Math. Phys., 7, 403 (1956). Womersley, J. R.,J. Physiol., 127, 556 (1955).
Received for review June 7, 1976 Accepted January 18,1977
Mass Transfer Units in Single and Multiple Stage Packed Bed, Cross-Flow Devices Louis J. Thibodeaux," David R. Daner, Arato Kimura, Jerry D. Millican, and Rajesh J. Parikh College of Engineering, University of Arkansas, Fayetteville, Arkansas 7270 1
This paper unifies the design of packed bed cross-flow mass transfer units with the countercurrent tower design procedure by employing the "number of transfer unit" (NTU) technique. The cross-flow equations are solved numerically for the case of linear operating and equilibrium expressions to yield a set of charts for the "number of overall transfer units" (NOU) for a single stage device. The concept of a cross-flow cascade consisting of a series of single-stage cross-flow devices is introduced. The single-stage results are then extended to develop a design procedure for cross-flow cascades. An aDplication of cross-flow devices to coal gas scrubbing is given.
Introduction Single-stage cross-flow gas-liquid contactors have found extensive use in water cooling, stripping, and absorption operations. Cross-flow cooling towers have enjoyed use as water coolers in competition with countercurrent towers mainly because of low pressure drop. Air stripping operations include ammonia and volatile organics removal from wastewater by use of cross-flow cooling tower devices. Absorption of gas in densely packed cross-flowchambers is a common air pollution control technique. Single-stage devices also find primary use in scrubbing particulates from air with gas absorption as a parallel benefit. Design methods and techniques for single-stage cross-flow devices have been developed to meet specific applications.
Baker and Shryock (1961) present a cross-flow cooling tower integration technique by considering incremental volumes that are geometrically similar in shape to the tower cross section. Mechanical integration starts with the volume increment at the top of the air inlet and successively considers each sub-volume down and across the section. The mean driving force was assumed to be the entering potential difference. Wnek and Snow (1972) present a practical design method for two applications: cooling tower, and ammonia stripping towers with simultaneous cooling. They observed that the temperature change of the liquid across the width of the tower is small in comparison to that across its height and this allows the development of an analytical solution of tower dimensions for stripping of ammonia from water. Roessler et al. (1970) have developed a digital computer program for deInd. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977
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