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Ind. Eng. Chem. Fundam. 1981, 20, 181-180
Axial Dispersion in a Segmented Gas-Liquid Flow Henrlk Pedersen’ Department of Chemical and Biochemiccl Engineering, College of Engineering, Rutgers-lhe New Brunswkk, New Jersey 08902
State Unlversliy of New Jersey,
Csaba Horvdth Department of C h e m h l Engineering, Yak Universliy, New Haven, connecticUt 06520
Axial dispersion of a tracer in liquld flow segmented by gas bubbles is investigated by numerical calculations using a lumped parameter model. Slow mixing In the liquid segments is shown to be the major factor in determining the magnitude of axial dlspersbn, measured by the second central moment of the tracer concentratlon distrlbvtlon. Experimental results obtained at a variety of tracer inputs substantiate the model predictions. Additknally, convolutbn and deconvolution of concentratlon profiles at the output are discussed.
Introduction Two-phase flow characterized by two different phases moving alternately through a tube is usually referred to as slug flow. In this work, the term segmented flow is used for a particular type of slug flow comprised of equidistant regions of the two phases flowing through a tube. Segmented flow can be generated for a gas-liquid system in a certain regime of gas and liquid flow rates that is essentially determined by the surface tension and viscosity of the liquid phase (Wallis, 1969). On the other hand, liquid flow segmented by solid spheres (Vrentas et al., 1978) is expected to be stable, independent of the liquid phase properties. Our study deals with gas-liquid segmented flow with particular regard to the use of this type of flow in automated analytical systems, so-called continuous-flow analyzers. Two characteristics of segmented flow have particular appeal in a number of applications. First, radial heat or mass transfer is augmented; second, axial dispersion is attenuated when segmented flow occurs instead of homogeneous laminar flow. Thus,Horvith et d. (1973a) have demonstrated the increase in radial mass transfer in laminar flow through tubes when the liquid stream was segmented by gas bubbles. The effect has been exploited for the use of tubular wall reactors with immobilized enzymes in continuous-flow analytical systems (Horvith and Pedersen, 1977). Vrentas et al. (1978) have shown how radial heat transfer is augmented in a liquid-solid segmented flow. The experimental results were used to substantiate the predictions of an earlier theoretical analysis by Duda and Vrentas (1971b). Skeggs (1957) introduced the use of gas-liquid segmented flow in order to reduce axial dispersion of analybs and their derivatives in continuous-flow analytical systems. The speed and efficiency of commercial continuous-flow analyzers such as Technicon SMAC system (Snyder, 1976) are still determined largely by the magnitude of axial dispersion. In the most widely used continuous-flow analyzers, a solution of the analyte (sample) and some “inter-sample wash” fluid are alternately aspirated into a gas-segmented liquid carrier stream. As the analyte is carried through the system, in most cases it undergoes reaction and backmixing. The product of the reaction, the concentration of which is usually monitored at the outlet, is also subject to axial dispersion. Consequently, the recorded 01964313/81/1020-0181$01.25/0
signal of the colorimeter or other concentration monitoring device may not show baseline separation of the tracings for closely spaced samples due to interference of the product concentration distributions at the outlet. Under proper operating conditions, however, tracings engendered by each sample show at maximum concentration a small flat portion representing that product concentration which would be obtained at continuous feed of the analyte, i.e., at steady state. The maximum permissible sampling frequency at which the signal for each sample yields a “flat” is determined by the magnitude of axial dispersion in the system. We would exped, therefore, that a more detailed knowledge of the factors affecting axial dispersion in such analytical intrumenta will facilitate the design of continuous-flow analysis for sampling rates significantly higher than those recently available. Furthermore, the analysis of such phenomena in segmented flow may facilitate the application of this type of flow field to other systems, such as tubular polymerization reactors where axial dispersion is of great importance (Ray, 1972). Theory Numerous studies have been conducted to characterize and predict axial dispersion in a gas-liquid segmented flow under conditions that are pertinent to continuous-flow analyzers (Thiers et al., 1971; Begg, 1972; Snyder and Adler, 1976a,b). These models, however, have generally assumed infinitely fast radial mass transport or included the effect of radial mass transport a posteriori. In all cases, Poisson or Gaussian distribution functions were used to describe axial dispersion. By and large, all previous investigations on segmented flow have been concerned with either the enhancement of radial transport or the characterization of axial dispersion, separately. Our model is based on a two-region description of segmented flow and relates the magnitude of axial dispersion to mass transport resistances in a gas-liquid segmented flow with nonreactive tracers. Consequently, it allows a simultaneous treatment of the two phenomena having the greatest practical significance. Axial dispersion in a gas-liquid segmented flow is a result of relatively slow mass transfer between the moving liquid segments and the quasistatic liquid film at the tube wall. In addition to molecular diffusion, secondary convective flow due to recirculation in the liquid segment and centrifugally driven flow due to coiling of the tube, often 0 1981 American Chemical Society
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m = n = 2/3 (Bretherton, 1961). Equations 1 can be made dimensionless by introducing the quantities x ( t ; i ) = Cb(t;i)/cb(@O) y(t;i) = C f ( t ; i ) / C b ( O ; O )
I
I
and
= u t / l = dimensionless time or distance /3 = l / d , = aspect ratio c = 1 / ( 1 + 1,) = dimensionless segment length a = l/df = aspect ratio of the film segment St = h / u = Stanton number 7
(b)
(0)
Figure 1. Schematic illustration of streamlines in a segmented flow (a) and in a coiled tube with homogeneous laminar flow (b).
: F]
-m+r]-D -D-/dT]Va
II( I
FLM REGION
...
2
,-I
I
Figure 2. Graphical illustration of the segmented flow model. Concentration distribution among the bulk segments is measured as a function of time.
done in practice, will contribute to the radial mass transport. Figure 1shows a schematic illustration of these secondary flow patterns. As the concentration and velocity fields in this type of flow are rather complex (Duda and Vrentas, 1971a,b; Gross and Aroesty, 1972), it would be a formidable task to develop a sufficiently detailed microscopic description of the system for the evaluation of axial dispersion. Therefore, we use a lumped parameter model that comprises the main features of segmented flow as illustrated schematically in Figure 2. Furthermore, for the sake of convenience we shall view segmented flow from a lagrangian reference frame that is stationary with respect to the bulk liquid segments. The tracer is assumed to have a t time t a uniform concentration of Cb(t;i)and cXt;i) in the ith liquid segment and the corresponding film region, respectively. The rate of transfer between the two regions is characterized by a mass transfer coefficient, h, and the transfer area is given by A . The liquid film region is modeled as a series of stirred tanks with a forward flow rate given by the product of the average liquid velocity, u, and the cross-sectional area of the film annulus, S. A mass balance on the ith segment gives
where I and 1, are the liquid and air segment lengths, respectively. The appropriate dimensionless mass balances when i 1 0 are for the bulk liquid segment dx(7;i) (34 d7 = -4/3St[x(7;i) - y(r;i)] and for the film segment
with y(r;-l) = 0. The equations are solved assuming that the tracer is initially distributed only in segment i = 0 x(@i) = y(0;i) = 6(i) (4) where 6(i)is the delta function. Solutions to eq 3 and 4 are obtained numerically for different values of the parameters @, t, a, and St. The Stanton number can be estimated for a segmented gas-liquid flow from an earlier analysis by Horviith et al. (1973b). Profiles for any arbitrary initial distribution can be found by an appropriate convolution procedure. The tracer profiles obtained from numerical solutions are used to evaluate the moments of the distribution, mj, given by mj = C i j x ( 7 ; i ) (i, j I 0 ) (5) i
As seen, tracer present in the liquid film region is not considered in the calculation of the moments. The mean, q, and variance of the distribution, 2, are defined by the zeroth, mo,first, ml, and second, m2,moments as follows q = m1/mo (6)
mz/mo- (ml/moI2 (7) The magnitude of axial dispersion in the system is measured by the value of 2. The units of q and D are in terms of a dimensionless segment number that can be related to actual distance via the segment length, 1. Experimental Section The experimental apparatus used to measure tracer dispersion is shown in Figure 3. A Technicon (Tarrytown, N.Y.) peristaltic pump was used to introduce air bubbles at regular intervals into a flowing liquid stream containing 0.1% (v/v) Brij 35 surfactant (Sigma, St. Louis, Mo.) in distilled water. The segmented carrier stream was passed through a four-port rotary injection valve (Hamilton, Reno, Nev.) and then through a 500 cm long, 0.13 cm i.d. Tygon coil. The coil diameter was 15 cm. The temperature of the apparatus was maintained at 30 "C. A transverse flow-through detector supplied by Technicon was used to u2 =
dc,(t;i) V f T = -hA(c,(t;i) - cb(t;i))- uS{cf(t;i) - cf(t;i- l)] (1b) where vb and Vf are the volumes of the bulk liquid segment and the underlying liquid film, respectively. The thickness of the annular liquid film dfcan be evaluated from the expression (Fairbrother and Stubbs, 1935; Taylor, 1961; Bretherton, 1961; Concus, 1970)
where dt is the tube diameter, p is the viscosity, and y is the surface tension of the liquid. For values of the bracketed dimensionless group between loa and typical of the analytical systems considered in this work,
Id.Eng. Chem. Fundam., Vol. 20, No. 3, 1981 183
’ RECORDER
MATING 4 PCRT INJECTION VALVE
/
mIFIER
I
THERMOSTATED UWL WASTE
I
t
I I mllmn
Figure 3. Flow sheet of the apparatus used for the measurement of axial dispersion.
i.... I
1
I
0
5
D
I IS
SEGMENT NUMBER,
Table I. Parameter Values Used in This Study for Characterizing Axial Dispersion in Slug Flow. The Conditions Correspond to Those in Technicon’s SMAC Analyzer at a Sampling Frequency of 150 per Hour symbol
parameter
magnitude
df dt h
inner diameter of tube liquid film thicknessa mass transfer coefficient b
0.1 cm 5x cm 10-3to 10-2 cm/s 1.0 cm 0.3 cm 10-4to 10-1
I 1, St U &
P E
length of liquid segment length of air bubble Stanton number for mass transfer mean fluid velocity reciprocal film thickness aspect ratio for segment fraction of the tube space occupied by the liquid
Calculated from eq 2. Horv&h et al. (1973b).
-
- --.-.-............. I
20
I
Figure 4. Tracer profiles at 7 = 100 calculated for the ideal case St = 0 (solid line) and for St = 0.002 (broken line). Parameter values are given in Table I. The hatched area represents the initial (T = 0)tracer input.
-
In the limit St a,the bulk and film concentrations are equal and the solution to eq 4 yields a Poisson distribution (Carslaw and Jaeger, 1948) qie-
x(7;i) = q=-
Z!
(8)
4P 7
ff
0.7 cmls 2 x 103 10.0 0.75
* Estimated from the data of
monitor tracer concentration in the individual liquid segments at the outlet of the coil. A 1-mg/mL dye solution of Evans Blue (Sigma, St. Louis, Mo.) was introduced into single liquid segments at appropriate intervals by rotating the injection valve with the flow stopped. The volume inside the valve is sufficiently small that only a part of the liquid segment occupies the valve space during injection. After injection, visual examination of the liquid segment containing the dye confiied that it was evenly distributed throughout the segment before entering the dispersion tube. Absorbance profiles were recorded on a stripchart recorder (Honeywell, Fort Washington, Pa.) for latter analysis. At least ten replicate runs were made for all experiments and in each case peak heights of the individual segments were obtained from the recorded output. Data were normalized to correspond to a unit steady-state concentration. In this way, variations in the amount of injected dye were accounted for. Results and Discussion Equations 3 and 4 have been solved numerically for parameter values that are representative of the usual operating conditions with Technicon’s SMA analytical system and listed in Table I. The film thickness was estimated from eq 2, which yields the lower theoretical limit. The appropriate viscosity and surface tension data were obtained from Snyder and Adler (1976a). The balance of radial and axial mass transfer is given by the Stanton number, St, so that the magnitude of tracer dispersion is expected to depend on St. Large values of St imply that the transfer of material between bulk liquid and film regions is rapid relative to the convective flow.
Equation 8 yields a lower bound for axial dispersion in any gas-liquid segmented flow system and the tracer profiles so obtained correspond to the so-calledideal model discussed by Thiers et al. (1971) as well as by Snyder and Adler (1976a). Thus, their models represent a limiting case for our model when the rate of radial mass transfer is infinitely high. For Poisson distributions, q represents both the mean and the variance. Therefore, eq 9 shows how the mean and variance of the tracer distribution depend on the parameters when the system is ideal. For Stanton numbers having finite values, tracer profiles have to be calculated numerically. The concentration profile calculated for St = 0.002 is depicted in Figure 4 along with that obtained for the ideal model (St = a) at 7 = 100. In both cases, the tracer was initially in segment 0 only. It is apparent from a comparison of the two profiles that slower radial mass transfer yields greater dispersion that is smallest for the ideal model. The mean and variance of the distribution can be calculated from eq 5-7, and this is shown in Figure 5 as a function of 7 with St as the parameter. The ratio a2/q is related to the “plate height” contribution to the dispersion resulting from slow mixing (Snyder and Adler, 1976a,b). A constant value is obtained at large 7 values and it is seen that the transient part of the curve becomes less pronounced as St increases. The ideal model corresponds to a2/q equal unity. Instead of being present only in one segment, the tracer occupies a series of subsequent segments, each having the same tracer concentration, in the input of automated analyzers with segmented flow. However, in such cases the output can be readily generated by convoluting the solution obtained with the tracer originally in a single segment. Accordingly, tracer concentrations in segment i, x * ( T ; ~ ) ,are found by the expression k-1
x*(7;i) = C x ( ~ -j); ; i j 5 i, i 1 0
(8)
j=O
where k is the number of segments covered by the initial tracer input and the summation is carried out for j’ Ii.
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Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981
sc’
t
xo
(7;i) =
I
e
I
1
1
4
4-
i
2 a@)x*(7;m)
s \
P-1
m = i - 0, - l)(k + s); m,i 1 0 (9) where the tracer concentration in sample p is a @ ) and s is the number of wash segments, i.e., segments between subsequent inputs containing no tracer. The concentration profilexo(7;i)isdisplayed and recorded in the practice of continuous-flow analysis, where one is interested in determining a@). In analytical instruments the analyte (tracer) is injected into a sufficiently large number of segments so that eq 8 closely approximates the zeroth moment for a finite value of i and the number of wash segments is large enough to obtain a “flat” portion for the output profiles of each sample. With these conditions, x*(7;i) equals unity for a particular i and xO(7;i) equals a@) for a sequence of i values, according to eq 9. The maximum sampling rate is that at which, for each output profile, a flat portion is discernible. It is determined by the maximum number of sample and wash segments for which the above two conditions hold and is dependent on the magnitude of axial dispersion. This situation is illustrated in Figure 6 for large values of the S t number. Three tracer concentrations having relative values 1:42 are used and the number of segments initially containing tracer (k)is varied from 5 to 25. The number of wash segments (s) is five in each case. It is seen that only large values of k yield “flats” and allow the “steady state” output concentmtion to be read directly from the tracer curves. Axial dispersion causes the disappearance of “flats” at small k values and also prevents a baseline separation of the sequential sample outputs. In practice, it would be desirable to operate the system at small k values in order to increase sample throughput. Besides appropriate design to minimize axial dispersion, a suitable deconvolution procedure might be used to evaluate input concentrationsfrom output profiles without “flats”provided sensitivity is sufficiently high to measure output concentrations in individual liquid segments accurately and the system is linear. The effect of axial dispersion on the tracer output curves
\
o
,:
m
,
,
,
m
KO
120
Lp-__-_-
40
w
SEGMENT NUMBER.
I
Figure 6. Tracer output profiles for sequential inputs of three samples having different concentrations in the ratio 2:kl. As seen, the number of segments for the initial tracer distribution is varied and “flats” are lost as the number of initial sample segments is reduced. The St number is large so that 2 = q. The number of wash segments between all sample inputs is 5.
is shown in Figure 7 for three tracer samples each distributed initially in five segments and separated by five wash segments. The relative concentrations for the tracer inputs are again 1:4:2. With increasing axial dispersion, as measured by the variance, the original “square-wave” concentration profiles are more and more distorted 80 that the evaluation of the “steady state” concentrations becomes more and more difficult. Since the variance increases with time, cf. eq 9, Figure 7 also illustrates the development of the tracer profiles in a continuous-flow system with increasing length of the conduit. Experimenta have been carried out and compared with the predictions of the mathematical model. Tracer was originally introduced into a single liquid segment or a series of segments, which moved through a 500 cm long tube, and the concentration profiles were monitored at the outlet. Experimental results and results calculated from the model with St = (ideal case) and St = 5 X lo4 are depicted in Figures 8 and 9. The values of the other parameters are given in Table 11. The mean and variance of the experimental results were evaluated by using eq 5-7 and Q)
Ind. Eng. Chem. Fundam., Vol. 20, 32
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No. 3, 1981 185
I
r
0
SEGMENT N M B E R ,
I
1
Figure 7. Effect of the dispersion on tracer profiles at large St numbers. Each tracing encompasses three samples which initially occupy five segments, have the concentration ratio 241 and are separated by five “wash” segments. Table 11. Experimental Results Obtained by Using the Apparatus Depicted in Figure 3. (The Mean and Variance Were Evaluated by Using Eq 5-7. Experimental Results Were Correlated with Calculated Values by Using the Model with the Stated Parameter Values for Both the Ideal and Nonideal Cases, and the Correlation Coefficients are Listed.) correlation coefficient varinontracer pattern (0123456789) mean, q ance, o 2 ideala 0.999 1000000000 1.262 1.511 0,999 1100000000 1.255 1.465 0.999 1110000000 1.261 1.506 1100110000 1.315 1.529 0.998 1010100000 1.310 1.683 0.999 1001001000 1.301 1.580 0.999 0.999 1111111111 1.285 1.720
I
0
1
I
1
2
1
3
1
4
I
I
I
S
1
7
I
I
SEGMENT MlMBER, i Figure 8. Theoretical concentration profile (solid line) and experimental data (solid circles) for the output of tracer initially distributed in a single segment. The ideal model predictions (St = =) are shown by open circles. Parameter values used for the calculation of theoretical curves are given in Table 11.
idealb
0.941 0.982 0.988 0.970 0.978 0.944 0.997
a Parameter values are: 7 = 100,p = 40,E = 0.901,(Y = Theoretical mean and variance 1.25 x lo4,St = 5 x are 1.277 and 1.655,respectively. ti Parameter values as above except St = -. Theoretical mean and variance are both 1.30.
are listed in Table I1 for different initial tracer patterns, indicated by 1 for a segment containing tracer and 0 for a segment without tracer. The means and variances given in Table I1 each represent the difference between the final and initial values. All experimental results show that u2 > q ; therefore, St < and slow radial transport affects the magnitude of axial dispersion. Comparison of the calculated profiles (solid line) with experimental data (solid circles) in Figures 8 and 9 shows that the model is able to predict experimental results accurately. The same conclusion can be drawn by comparing the values of the respective correlation Coefficients in Table 11. On the other hand, the “ideal” model (St = m) gives a poor representation of the physical reality as seen from the deviation of the ideal model (open circles) from the experimental values in Figures 8 and 9, as well as from the relatively low correlation coefficients in Table 11. Summary The preceding discussion has demonstrated how the variance and mean of the concentration distribution at the outlet can be calculated for a tracer in a segmented flow
SEMHT HM%R,i
Figure 9. Predicted (solid line) and experimental (solid circles) tracer concentrations at the outlet under the same conditions given in Figure 8 except that the tracer is initially distributed over 10 segments.
by using a lumped parameter model similar to that employed in chemical engineering for the treatment of axial dispersion. The results are in good agreement with experimental data. In the model used here the distribution of tracer concentration in the individual segments is calculated rather than its value in a particular effluent segment. The approach facilitates the evaluation of axial dispersion and calculation of output profiles encountered in continuous-flow analytical systems. Recently, small bore enzymic wall reactors have found wide employment in continuous-flow analyzers (Lebn et al., 1976,1977). When the tracer undergoes chemical reaction at the tube wall,the axial dispersion of the product is likely to be greater than that of the unreacted tracer.
Ind. Eng. Chem. Fundam. 1981, 20, 186-195
180
Preliminary investigations (Horvlth and Pedersen, 1977) suggest that the method developed here can be used for the quantitative treatment of axial dispersion of tracers subject to chemical transformation in automated analyzers with segmented flow.
Superscripts * = convoluted tracer profile 0 = tracer profile for sequential samples Literature Cited
Nomenclature a = tracer concentration in sequential sampling A = transfer area in liquid slugs; cf. eq l a c = concentration dt = tube diameter df = film thickness h = mass transfer coefficient i, j, k , m, n = segment number 1,1, = liquid and air segment lengths m = moment p = sample number in sequential sampling p = normalized first moment s = number of wash segments between sample inputs
S = cross sectional area of annular liquid film St = Stanton number for mass transfer t = time u = average velocity of bulk liquid slug V = volume 1:
j = order of the moment
= dimensionless bulk concentration
y = dimensionless film concentration
Greek Symbols CY = l/dr B = lldt y = surface tension c = 1/(1 + la) 7 = ut11 a2 = variance 1.1 = viscosity
Begg, R. D. Anal. Chem. 1972, 44, 631. Brethetton, F. P. J . FkrMMech. 1961, 70, 166. Carslaw, H. S.; Jaeger, J. C. “Operatknal Methods In Applied MWmaW, 2nd ed.; Oxford Unlvedty: Cambrldgs, 1948; pp 299-303. Concus, P. J. J. phys. Chem. 1970, 74, 1818. Duda, J. L.; Vrentas, J. S. J. F M M e c h . 1971a, 45,247. Duda, J. L.; Vrentas, J. S. J. FbidMech. 197lb. 45,261. Falrbrother, F.: Stubbs, A. E. J . Chem. Soc. 19821, 1 , 527. Gross, J. F.; Aroesty, J. -1972, 8 , 225. HorvHth, Cs; Pedecsen, H. In Advances in Automated Analysis, Technicon International Congress 1976”, Vd. 1; Mediad: Tarrytown, NY, 1977: pp 86-95. HorvHth. Cs;Sardl, A.: Solomon, B. A. physkl. (2”.phys. 1978a, 4, 125. HorvHth, Cs; Solomon, 8. A.; Ewsser, J-M. Ind. Eng. Chem. Fundam. 197Sb, 12, 431. Ldn, L. P.; Narayan, S.; Dellenbach, R.; HOrvBth, Cs. CUn. Chem., 1976, 22, 1017. L&, L. P.; Sansur, M.;Snyder, L. R.; ~orvHth,Cs. Clin. Chem., 1977, 23, 1556. Ray, W. H., Jr. J . Mecromd. Scl. Rev. 1972, C8, 1. Skeggs, L. T., Jr. Am. J. CNn. PaW. 1957, 28, 311. Snyder, L. R. J. chrometq. 1976, 125, 287. Snyder, L. R.; Adler, H. J. Anal. Chem. 1976a, 48, 1017. Snyder, L. R.; Adler, H. J. Anal. Chem. 1976b, 48,1922. Taylor, G. I. J. FbidMech. 1961, 70, 161. Theirs, R. E.: Reed, A. H.; Delander, K. C h . Chem. 1971, 17, 42. Vrentas, J. S.; Duda, J. L.; Lehmkuhl, G. D. Id.Eng. Chem. Fundam. 1978, 17, 39. Wallis, Q. B. “OneDlmenslonalTwo-Phase Flow”; McGrawHIII: New York, 1969; pp 282-314.
Receiued for review December 11, 1978 Resubmitted June 23, 1980 Accepted May 4,1981
Subscripts
This work was supported by Grants No. GM 20993 and CA 28037 from the National Institute of Health, US. Public Health Service,
b = bulk f = film
DHEW.
Mass and Momentum Transfer to Newtonian and Non-Newtonian Fluids in Fixed and Fluidized Beds S. Kumar and S. N. Upadhyay’ Lbpatiment of Chemical Engineering d Technobgy, InstlMe of Technology, Banaras Hindu Unkerslty, Varanasi22 7005, Mia
Mass transfer and pressure drop measurements during flow of Newtonian and nonHewtonian fluids through fixed and fluidized beds of uniformly sized cylindrical pellets and spheres have been made. Demlneralized water and 1.O % aqueous carboxymethylcellulose (CMC)solution have been used as the fluids. Measurements have covered a particle Reynolds number range from 0.0387 to 6000 and Schmidt numbers from 816 to 71 871. It has been shown that with appropriate choice of 8 viscosity, the results for both Newtonian and non-Newtonian fluids can be expressed by corretatlons based on the capillary tube bundle model.
Introduction
Heat, mass, and momentum transfer characteristics of Newtonian fluids flowing through fixed and fluidized particulate systems have been extensively studied and the
* To whom corrmpondence &odd be addressed at the Department of Energy Engineering, College of Engineering, University of Illinois at Chicago Circle, Box 4348,Chicago,IL 60680. 01964313/81/1020-0186$01.25/0
available information is adequately reviewed by many workers (Barker, 1965; Kunii and Levenspiel, 1969; Leva, 1959; Pandey et al., 1978; Upadhyay and Tripathi, 1975a). The study of flow of non+htc”an fluids through P a ticulate systems has gained momentum recently due to its applications in many areas (Savins, 1969). For example, a knowledge of non-Newtonian flow mechanism through porous media is needed in diversified fields such as ceramic engineering, filtration of polymer solutions and slurries, 0 1981 Amerlcan Chemical Society
