Ind. Eng. Chem. Res. 1987,26, 1248-1254
1248
n = number of species in the composition vector n = number of species in cluster p f = (i, j ) entry in the hth step-response matrix dh= hth step-response matrix R E response matrix R, R1,etc. = bar indicates some columns have been averaged x ( t ) = species composition vector at time t x i = zth species in x x ( h ) = composition vector at time th = h.7 xo = initial composition z ( t ) ,z (h) = lumped composition vectors Greek Symbol 7 = fixed sampling-time interval
Literature Cited
Coxson, P. G.; Bischoff, K. B., submitted for publication in Ind. Eng. Chem. Res. 1987. Golub, G. H.; Van Loan, C. F. Matrix Computations; Johns Hopkins University Press: Baltimore, 1983. Gross, B.; Jacob, S. M.; Nace, D. M.; Voltz, S. E. US Patent 3960707, 1976. Jacob, S. M.; Gross, B.; Voltz, S. E.; Weekman, V. W. AIChE J. 1976, 22, 701-713. Kuo, J. C. W.; Wei, J. Ind. Eng. Chem. Fundam. 1969,8,124-133. Nace, D. M.; Voltz, S. E.; Weekman, V. W. Ind. Eng. Chem. Process Des. Deu. 1971, 10, 530-541. Ozawa, Y. Ind. Eng. Chem. Fundam. 1973, 12, 191-196. Weekman, V. W., Jr. AZChE Monogr. Ser. 1979, 75(11), 3-29. Wei, J.; Kuo, J. C. W. Znd. Eng. Chem. Fundam. 1969,8,114-123. Wishart, C. CLUSTAN User Manual, 3rd ed.; University of Edinburgh Edinburgh, Jan 1978; Inter-University/Research Council Report 47. Received for reuiew December 26, 1985 Accepted October 8, 1986
Anderberg, M. R. Cluster Analysis for Applications; Academic: New York, 1973.
Momentum and Heat Transfer in Non-Newtonian Fluids Flowing through Coiled Tubes Yoshinori Kawase and Murray Moo-Young* Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI
A simple approach to momentum and heat transfer in inelastic non-Newtonian fluids flowing through coiled tubes is discussed. The proposed approach is based on the integral momentum boundary layer method and is applicable to both laminar and turbulent flows. T h e predicted values of the friction factor and the Nusselt number are compared with experimental data and other previous correlations. Satisfactory agreement is found. Coiled tubes have been widely used as heat exchangers and reactors because of more efficient heat transfer, reduced backmixing, and longer average residence time per unit floor space as compared with straight tubes. Therefore, the Newtonian fluid flow in coiled tubes has been extensively investigated. A flow in a coiled tube develops secondary flow caused by the difference in centrifugal force which is exerted on elements of fluid moving with different axial velocities. The fluid in the central part of the coiled tube is driven toward the outer wall by the centrifugal force and flows inward along the wall by a pressure gradient. The resultant secondary flow appears as twin counterrotating vortexes in the cross section of the tube and increases heat-transfer rates in addition to the rate of momentum transfer. Dean (1927, 1928) showed that a single dynamic similarity parameter, De R e ( d / D ) ' f 2 ,can characterize Newtonian flow in a coiled tube. The use of tubular coiled reactors for continuous fermentation and polymerization reactions has considerable advantages, and most fermentation broths and polymerization reaction mixtures are viscous and non-Newtonian in nature. In coiled tube blood oxygenators, the secondary flow which is relatively gentle can be expected to improve the gas-transfer rate. Despite a variety of practical applications, relatively little attention has been paid to the flow of non-Newtonian fluids in coiled tubes. Particularly, little theoretical work has been published dealing with the flow of inelastic non-Newtonian fluids. The pseudoplasticity flattens the primary velocity profile and enhances the secondary circulation flow close to the tube wall due to the reduction of the apparent viscosity in the high shear rate region. Mashelkar and Devarajan (1976a, 1977) analyzed the laminar and turbulent flows of a power-law fluid in coiled
tubes by using a boundary layer approximation. In spite of the fact that they applied the momentum integral approach, they had to solve the governing equations numerically and present numerical results in a form of a formula suitable for use in engineering design. Later, Shenoy et al. (1980) extended this approach to the turbulent flow of mildly viscoelastic fluids. Although Mashelkar and Devarajan (1976a) stated that their result for laminar Newtonian fluid flows is in excellent agreement with Ito's (1956) solution for Newtonian fluids, as pointed out by Hsu and Patankar (1982), their correlation for n = 1 cannot describe the laminar Newtonian flow data. It should be noted, furthermore, that in their paper (1977) there is confusion in the definition of the friction factor. Recently Hsu and Patankar (1982) solved numerically the governing equations for the momentum in the laminar flow of a power-law fluid. Very little information is available on heat transfer in non-Newtonianfluids flowing through coiled tubes. Oliver and Asghar (1976) measured heat transfer to laminar pseudoplastic liquids through coiled tubes. Numerical solutions for the laminar power-law fluid flow heat transfer in coiled ttibes under the condition of constant wall temperature were presented by Hsu and Patankar (1982). Incidentally, it is to be mentioned that the Nusselt number ratio in their work should be the ratio of their solutions to the corresponding solutions of the modified Graetz problem under the condition of constant wall temperature instead of those under constant wall heat flux. As shown in the measurement of the velocity distributions by Ranade and Ulbrecht (1983), the existing numerical solutions are not in satisfactory agreement with the data of the secondary flow. Moreover, it is conceivable that the results of numerical calculations are difficult to
0888-588518712626-1248$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1249
1
secormry tiow field
,
Figure 1. Secondary flow field in a coiled tube.
use in practical design. For the purpose of design, simple and reliable correlations for momentum and heat transfer are definitely desirable. Since the flow in a coiled tube is in actuality three dimensional and nonviscometric, some approximations have to be introduced to obtain simple correlations. In this paper, we discuss a simple approach for momentum and heat transfer in inelastic non-Newtonian fluids flowing through coiled tubes under the condition of fully developed velocity and temperature fields. The utility of the proposed model is tested by using the available correlations and experimental data. Analysis We consider an inelastic non-Newtonian flow behavior described by a power-law model. It has been shown that in the region of high Dean numbers (De > lo2) which is of practical importance (Dravid et al., 1971), the secondary flow field consists of a central inviscid core and a thin boundary layer adjacent to the wall (see Figure 1). It may be reasonable, therefore, that the transport processes in a coiled tube are controlled by the thin boundary layer at high Dean numbers, and a boundary layer analysis is applied to this problem. On the whole, the axial velocity dominates the flow in the coiled tube, but it becomes comparable to the angular velocity in the boundary layer. We use the following procedure to solve the transfer equations. In formulating the momentum and heat-transfer equations, the secondary flow is not considered and the expressions for the shear stress and the Nusselt number, zero-order solutions, are obtained as functions of the boundary layer thickness. The substitution of the expression for the boundary layer thickness, in which the secondary flow is taken into account, into the zero-order solutions yields first-order solutions. The velocity distribution in the boundary layer formed in the coiled tube is different from that in the case of a steady parallel flow over a flat plate where the velocity increases monotonically from zero at the wall to the free stream velocity. As shown in Figure 1, the fluid in the central part is driven toward the outer wall by the centrifugal force (A B), enters the boundary layers, and flows back along the wall toward the inner wall by the pressure gradient (C D). Therefore,
-
the velocity in the boundary layer increases from zero at the wall to the maximum and decreases to zero at the edge of the boundary layer (Mori and Nakayama, 1965; Mashelkar and Devarajan, 1976a). In other words, this velocity distribution in the boundary layer is caused by the flow in the central part which moves opposite to that in the boundary layer. We assume that at high Dean numbers, the transport processes in a coiled tube are dominated by the flow in the region where the velocity increases from zero at the wall (u = 0) to the maximum (u = u,) rather than the flow in the whole boundary layer (Figure 1))and this hypothetical boundary layer is characterized as follows. The boundary layer in the cross section of a tube due to the secondary flow is closed. Therefore, its thickness will increase over part of its length and decrease over the remainder. However, it may be approximated by the layer which increases in thickness over a path of length of order d under the action of an external stream velocity of order u, (Batchelor, 1960). Furthermore, the boundary layer thickness is very thin compared with the tube diameter so that the curvature of the boundary layer is negligible. Consequently, the boundary layer along the tube wall in which the axial flow of the average axial velocity u,,is accompanied by the secondary flow can be replaced by that over a flat plate in which the free stream velocity is u, and mathematical description of the boundary layer flow in the coiled tube is greatly simplified. The thickness of the hypothetical boundary layer appearing in the cross-sectional plane is represented by the characteristic thickness of that formed on the flat plate which is estimated by the force balance in the boundary layer. (i) Laminar Momentum and Heat Transfer. The laminar flow in the coiled tube is approximated by the hypothetical boundary layer described above. Its thickness is estimated as follows. An expression for the thickness of the boundary layer on a flat plate at the characteristic length 7 d / 2 may be written as (Skelland, 1967)
where A =
E(1+ i)n n)(
From the balance in the boundary layer between the driving centrifugal force, u 0 / ( D / 2 ) , and the retarding viscous force per unit mass, (K/p)u,a,n,/bl+n,we have
where C is a proportionality constant and its value is discussed later. Substituting eq 2 for u,, into eq 1gives an expression for the characteristic thickness of the boundary layer due to the centrifugal force appearing in the cross section of a tube
The average thickness of the boundary layer formed in the coiled tube is evaluated by eq 3. The boundary layer flow is characterized by this boundary layer thickness equation which is obtained by taking into account the influence of
1250 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 &--Present
eq ?
work
on the boundary layer thickness is first assumed to be negligible. When we substitute an expression for the boundary layer thickness into the resulting equation for the Nusselt number, the effect of the secondary flow is considered. The local Nusselt number for laminar power-law fluid flow over a flat d a t e is given as (Kawase and Ulbrecht, 1984)
C=042
r-
i i
-
Nu, = 1.5x/
eMishra and Guptqil9791 oCllier and Asghar('975)
ASebar and McLaugnlin(l%.1
15' IC
1c'
102
lo4
where
[-I
Le
(K/p)2/(l+n)X(1-n)/(l+n)
Figure 2. Friction factor in coiled tubes for laminar Newtonian fluids.
secondary flow. The shear stress a t the wall in the boundary layer of thickness 6 may be written as (Skelland, 1967)
Pr*, =
umax
l3(n-l)l/(l+n)
ff!
Substituting eq 2 and 3 into the zero-order solution, eq 8, and replacing x by the characteristic length ( a d / 2 ) ,we have
(4) This zero-order solution for the momentum equation is obtained by not taking into account of the secondary flow. Considering the relation of the friction factor for fully developed flow in a tube, f = - -70 1
Nu = 0.944De1/2Pr1/3 (5)
ZPUO
substituting eq 3 into eq 4, and rearranging, we have the first-order solution
where fs
=
(10)
(ii) Turbulent Momentum and Heat Transfer. As well as the analysis for laminar flow, the turbulent flow in a coiled tube is replaced by the hypothetical turbulent boundary layer flow, and the same procedure is used to determine the boundary layer thickness. The Characteristicthickness of turbulent power-law fluid boundary layer may be given by (Skelland, 1976)
where p is a function of the flow index (Dodge and Metzner, 19591,
16 Regen
-
The friction factor ratio fc/fs is a function of Degenand dlD. In the model proposed by Mashelkar and Devarajan (1976a), the dependency of d l D on fJfSis neglected. When the fluid is Newtonian (n = l),eq 6 reduces to
fcf s = 0.0925C-'/4De'/2
For n = 1, eq 9 reduces to
n= and
'
2 - ,8(2 - n) 2 - p(2 - n) + pn) 2 - 2p + 38n
= 2(1- p
(7)
We assume that the value of C is independent of the non-Newtonian flow behavior, and from comparison of eq 7 with the Newtonian flow data of Seban and McLaughlin (1963), Oliver and Asghar (1975), and Mishra and Gupta (1979) and the empirical correlations of Prandtl(l949) and White (19291, it is determined as 0.42 (Figure 2). Since the boundary layer approximation introduced in the present analysis is applicable at high Dean numbers, eq 7 for C = 0.42 is, as expected, not in good agreement with the data and the correlations for De < lo2. Laminar heat transfer in a coiled tube is also discussed using the results for the laminar boundary layer on a flat plate as well as the momentum transfer. The heat transfer in a coiled tube is replaced by that in a boundary layer of thickness 6 given by eq 3. In solving the heat-transfer equation, the effect of the secondary flow
40.817)2-O(2-n) 21+Bn
The force balance in the turbulent boundary layer between the centrifugal force and the viscous force gives
The constant C'is determined using Newtonian data as well as the case of a laminar flow. Manipulation leads the following relation between 6 and uo X
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1251 Present work
eq 17 6-0.0057 @d/D-0.1
~dlD~o.oo~
- - -whlle(1932)
-
Mqshelkar
urd Devsrajan(l9771
I
Y
Mqshelkqr qnd Devwajan(1976a)
oMishra
Id-
.%ban
and Guptq(1979)@=0148
-
and McLauqhlin(1963)d/0-00096 I
I
I
[-I
De gen
I
Figure 4. Friction factor correlations for laminar power-law fluids in coiled tubes.
mined as 0.0057. The Colburn analogy between momentum and heat transfer leads to The approach of von Karman (1946) is applied to obtain the zero-order solution for the friction factor in a coiled tube. The modification of the well-known Prandtl oneseventh power law gives
where an effective viscosity pe is defined as c
The values for a are functions of the flow index n and are given in the paper of Dodge and Metzner (1959). The friction factor for a turbulent boundary layer on a flat plate in power-law fluid flows may be written as
(~158~l-48nU01+68n-10~ l+Sn-5i3)l/(~n+l)(
P
$p
[5,Pn(z-n)/ ~ ( B n + l ) ]
When n = 1, eq 18 becomes In the derivation of this equation, the boundary-layer thickness 6 is used in place of y, and uo/u at r = 0 is assumed to be independent on the flow index n and 0.8 (Schlichting, 1979). Substitution of eq 13 into the zero-order solution, eq 15, yields the first-order solution
where CY
Regen@
9 )1
2 0.05
f,
Discussion (i) Laminar Momentum and Heat Transfer. Figure 4 compares the present model for laminar power-law fluid flows with the theoretical results of Mashelkar and Devarajan (1976a) and Hsu and Patankar (1982). It can be seen that the pseudoplasticity decreases with an increase of pressure loss caused by the curvature of the tube. The present results given as the average of the values for d/D = 0.1 and 0.001 agree well with Hsu and Patankar's numerical solutions at high Dean numbers (Degen> 200), but Mashelkar and Devarajan's results lie somewhat below the present model except for the region of Degen< 250. The following fact should be mentioned. The numerical results of Mashelkar and Devarajan (1976a) are given in the form
fB
For n = 1,eq 16 reduces to the result for Newtonian fluids
- = 0.662C'-0.025(Re(
(19)
f. = (0.05668 - 0.5899n + 0.2734n2)De0.232+0~122n (20)
f, = -
fc
Nu = 0.0298Re0.8Pr1/3
d
-0.075
(5)
(17)
From the comparison of eq 17 with the available correlation (White, 1932; Ito, 1959; Mashelkar and Devarajan, 1977) and experimental data (Seban and McLaughlin, 1963; Rogers and Mayhew, 1964; Mishra and Gupta, 1979) (Figure 3) for Newtonian fluids, the value of C'is deter-
For n = 1, this equation becomes
fc - = 0.2503De0.354
(21)
f B
As pointed out by Hsu and Patankar (19821, this correlation is not in good agreement with the Newtonian data. The present model is compared with the data for n = 0.8 by Gupta and Mishra (19751, Mashelkar and Devarajan (1976b),and Mujawar and b o (1978) in Figure 5. On the whole, the agreement is satisfactory.
1252 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 10
I
2.0,
I
I
I
I
Present work(@=O.l35-0.0188)
I
I
I
I
1
I
I
Gupt4 qnd Mishra(1975) d/D-C10369,0.076 Wjawar and R90(1978! @=0.0198-0.0695
un.082
en-0754 Mashelkor and Devarajan(1976b)6a-00188~0.135
'01
1I'
lo4
De F [-I Figure 5. Comparison of the present model with friction factor data for laminar power-law fluids. 1 c2
5
I
I
Corrdet!ons ___ Present work
L
0.50
02
04
06
io
08
12
14
n [-I Figure 8. Effect of non-Newtonian fluid behavior on the laminar heat transfer in a coiled tube.
- - .HSU _ and Fbtenkar(1982)
Kalbqnd Seaderl19721
-
-
- Orevideta1.(1971! --Mor1 ~akayama(i9671
,.
Experimental data *Msrl and N4k~ya"X19651R.07
L
I
'x
1
ii?
ASebQn Wd MCLWghlln(1963)PrulOO
10
5
I
I
102
10'
10'
[-I
De
Figure 6. Laminar heat transfer in coiled tubes for Newtonian fluids. '
'
I
---
-Mashelkar and
I
,
I
I
I
-_ Present work -
- - -hsu and 4tankar(%321
I 102 Degen
I 10'
5
[-I
Figure 7. Heat-transfer correlations for laminar power-law fluids in coiled tubes.
The applicability of the present model is restricted to the high Dean number region. There is the restriction on the value of the flow index, too. The present results may be valid for n 2 0.5 as well as Mashelkar and Devarajan's (1976a) analysis. As shown in Figure 6, eq 10 for heat transfer in Newtonian fluids agrees well with the numerical solution of Hsu and Patankar (1982), the correlations of Mori and Nakayama (1965), Dravid et al. (1971),Kalb and Seader (19721, and Tarbell and Samuels (1973), and the experimental data of Mori and Nakayama (1965) for Pr = 1. However, the correlations of Mori and Nakayama (1965) and Dravid et al. (1971) and the experimental data of
(60 < De < 2000) (22a)
(4 < De
< 60) (22b)
It can be seen in Figure 8 that this correlation predicts the enhancement of the heat-transfer rate due to pseu-
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1253 I
1
1
work _ _ - - Seben and McLPughlin(l'j63) Pr-4.3 @ d/Ds0.00¶6 @)d/D-0.059 -Present
For further discussion, more experimental data are necessary particularly in nowNewtonian fluids which are often encountered in biochemical and biomedical industries.
Rogers and Mqyhew(i9641 Pr-lO', d / D = O . O l
-
.... .... .~ratt(1947)Pr-lo',
d/b=0.01
Nomenclature A = function defined by eq 1
doplasticity as well as the present model. More experimental data are required to discuss this problem in detail. (ii) Turbulent Momentum and Heat Transfer. Figure 9 compares the present model, eq 16, with the model proposed by Mashelkar and Devarajan (1977) which correlates their own data (n = 0.93-0.75) very well. Good agreement can be found. The pZn/(pn+ 1)power dependence on the similarity variable ( R e ,n(d/D)2)in the correlation for fJf, is consistent with tLe analysis of Mashelkar and Devarajan (1977) but the power of d / D in eq 16 is slightly smaller than that in their analysis. It should be emphasized that Mashelkar and Devarajan's (1977) analysis based on the momentum integration is numerical and the results are not given in a form of a formula unlike their result for laminar flows (Mashelkar and Devarajan, 1976a). It can be seen that the pseudoplasticity slightly decreases the friction factor for turbulent flows in coiled tubes, and the influence of curvature on pressure loss in turbulent flows is weaker than that in laminar flows. The present model for turbulent flows is applicable a t high Dean numbers as well as that for laminar flows. However, a t present the upper and lower Dean number limits of its applicability are difficult to specify. It should be noted that the Dodge and Metzner (1959) correlation used in this analysis is applicable in the range of 5 X lo3 < Regen< lo5, and this limit is also effective in the present analysis. Unfortunately, no experimental data for turbulent heat transfer in non-Newtonian fluids flowing through coiled tubes are available. It can be seen in Figure 10 that eq 19 for turbulent Newtonian flows in coiled tubes is in reasonable agreement with the empirical correlations proposed by Pratt (19471, Seban and McLaughlin (1963), and Rogers and Mayhew (1964).
Conclusions The simple model for momentum and heat transfer in inelastic non-Newtonian fluids flowing through coiled tubes is developed on the basis of the momentum integral method. The results given in a closed form are in reasonably good agreement with the available data. It is concluded, therefore, that the present model provides satisfactory predictions of friction factor and Nusselt number at high Dean numbers and is useful in engineering design.
C = constant in eq 2 C ' = constant in eq 12 D = diameter of the coil helix De = Dean number, R e ( d / D ) 1 / 2 Degen= generalized Dean number, Re,e,(d/D)'/2 d = tube diameter f = friction factor f, = friction factor in a coiled tube f, = friction factor in a straight tube GZ = Graetz number, ( 7 ~ / 4 ) ( R e P r ) ( d / L ) g = acceleration due to gravity h = heat-transfer coefficient j, = modified j factor K = consistency index in a power-law model k = thermal conductivity L = tube length N u = Nusselt number, h d / k Nu, = Nusselt number based on x , h x / k n = flow index in a power-law model Pr = Prandtl number ( p / p ) / a Pr+ = Prandtl number (K/p)(u0/d)"-l/a pr* = Prandtl number, (K/p)Z/(ln)d(l-n)/(l+n)u [3(n-l)]/(l+n) 0 /a! Pr*, = Prandtl number based on x , ( K / p)2/(l+n)x(l-n)/(l+n) [3(n-l)I/(l+n) 4" /a Re = Reynolds number, pduo/p y1 Regen= generalized Reynolds number, pdnuO2-"/ r = radial distance u = velocity uo = average axial velocity umaX = velocity component along the wall at the edge of the hypothetical boundary layer u* = friction velocity x = distance along the wall y = distance perpendicular to the wall Greek Symbols a , /3 = functions of the flow index a = thermal diffusivity y = function of consistency index, P-lgK y 1 = function of consistency index, K8"-l((3n 1)/4nJ yB = y based on fluid bulk temperature conditions yw = y based on tube wall temperature conditions b = boundary layer thickness ,u = viscosity p e = effective viscosity defined by eq 18 p = density T,, = shear stress at the wall $, 52 = functions defined by eq 11
+
Literature Cited Batchelor, G. K. J . Fluid Mech. 1960, 7, 416. Dean, W. R. Phil. Mag. 1927, 4, 208. Dean, W. R. Phil. Mag. 1928,5, 673. Dodge, D. W.; Metzner, A. B. AZChE J. 1959,5, 189. Dravid, A. N.; Smith, K. A,; Merrill, E. W.; Brian, P. L. T. AIChE J. 1971, 17, 1114. Gupta, S.N.; Mishra, P. Indian J. Technol. 1975, 13, 245. Hsu, C.-F.; Patankar, S.V. AIChE J. 1982, 28, 610. Ito, H. Mem. Inst. High Speed Mech. Tohoku Uniu. ( J p n . ) 1956,6, 54. Ito, H. J . Basic Eng. 1959, 81, 123. Kalb, C . E.; Seader, J. D. Znt. J. Heat Mass Transfer 1972,15,801. Kawase, Y.; Ulbrecht, J. J. Znt. Commun. Heat Mass Transfer 1984, 11, 143. Mashelkar, R. A.; Devarajan, G. V. Trans. Inst. Chem. Eng. 1976a, 54, 100. Mashelkar, R. A.; Devarajan, G. V. Trans. Inst. Chem. Eng. 1976b, 54, 108.
Ind. Eng. Chem. Res. 1987, 26, 1254-1259
1254
Mashelkar, R. A.; Devarajan, G. V. Trans. Inst. Chem. Eng. 1977, 55, 29. Mishra, P.; Gupta, S. N. Ind. Eng. Chem. Process Des. Deu. 1979, 18, 137. Mori, Y.; Nakayama, W. Znt. J . Heat Mass Transfer 1965, 8, 67. Mujawar, B. A.; Rao, M. R. Znd. Eng. Chem. Process Des. Deu. 1978, 17, 22. Oliver, D. R.; Asghar, S. M. Trans. Znst. Chem. Eng. 1975,53, 181. Oliver, D. R.; Asghar, S. M. Trans. Znst. Chem. Eng. 1976,54, 218. Prandtl, L. In Boundary-Layer Theory; 7th ed.; Schlichting, H. Ed.; McGraw-Hill: New York, 1979; p 627. Pratt, N. H. Trans. Znst. Chem. Eng. 1947, 25, 163. Ranade, V. R.; Ulbrecht, J. J. Chem. Eng. Commun. 1983,20, 253. Rogers, G. F. C.; Mayhew, Y. R. Int. J . Heat Mass Transfer 1964, 7, 1207.
Schlichting, H. Boundary-Layer Theory, 7th ed.; McGraw-Hill: New York, 1979; p 600. Seban, R. A.; McLaughlin, E. F. Znt. J . Heat Mass Transfer 1963, 6, 387. Shenoy, A. V.; Renade, V. R.; Ulbrecht, J. J. Chem. Eng. Commun. 1980, 15, 269. Skelland, A. H. P. Non-Newtonian Flou and Heat Transfer; Wiley: New York, 1967; p 276. Tarbell, J. M.; Samuels, M. R. Chem. Eng. J . 1973, 5, 117. von Karman, T. NACA T N 1092, 1946. White, C. W. Proc. R. Sac. London, Ser. A 1929, A123, 645. White, C. W. Trans. Znst. Chem. Eng. 1932, 10, 66.
Received for review January 21, 1986 Accepted December 31, 1986
COMMUNICATIONS Improved Apparatus for Measurement of Specific Surface Areas of Powders An account is given of a new electronic permeameter designed for measurement of the specific surface areas of powders by observing gas flow rates and pressure drops across beds of the solid particles. The gas permeametry results were examined by using the Carman-Arne11 equation, a two-term equation consisting of a viscous flow term (the Carman-Kozeny expression) and a slip flow term (the Knudsen expression). The results obtained from this new flow apparatus are reproducible. Regression analysis produced high correlations between gas flow rate and mean pressures which are statistically valid at a significance level of 0.1%. The estimated fractional standard errors of the specific surface were only a few percent. The equation (Carman and Arnell, 1948) relating the transport of a gas through a compacted bed of powder under nonturbulent flow conditions to the surface area of the powder is made up of two terms
Rather than solve this equation directly for Sv,the calculation can be carried out by evaluating two separate terms and combining them to give Sv.
Equation 2 and the first term in eq 1 apply to viscous (or laminar) flow~andgive the contribution to the surface area of the powder due to this streamline flow. This equation was first developed by Kozeny (1927) and later modified by Carman (1938), who adopted a value of 5 for the constant of proportionality. The second term applies to molecular (or Knudsen) flow and relates to a surface which includes surface cracks and fissures as well as the voids between the particles of which the bed is composed.
0888-5885/87/2626-1254$01.50/0
The combined equation was proposed by Carman and Arnell (1948), who introduced a “slip” coefficient in the second term. This coefficient corrects for the nature of the molecule-surface interaction: A fraction of the molecules striking the surface wall will be specularly reflected; i.e., if the capillary walls are smooth, molecules striking them at any angle rebound at the same angle with the Same average velocity and with the component of velocity perpendicular to the wall reversed. The surfaces of packed beds of powder are not smooth, and molecules striking them rebound in any direction, i.e., diffuse reflection. For any powder, as the inlet pressure falls, the contribution of the viscous flow term to measured surface area will decrease and the contribution of the molecular flow term will increase. Further, the finer the powder, the greater the contribution of the slip flow term in the combined equation. Combining eq 1-3 gives the quadratic in S v having the solution
These equations have to be evaluated. Instrumentations in Gas Permeametry A large number of gas permeameters have been designed for measuring the flow rate and pressure drops across packed beds. They can be classified into two basic types according to the way in which flow resistance is measured. With constant-flow-rate permeameters, gas is passed through the bed of powder at a constant volume rate of 0 1987 American Chemical Society
