2754
Ind. Eng. Chem. Res. 1992, 31, 2754-2759
Slow Power Law Fluid Flow Relative to an Array of Infinite Cylinders Anubhav Tripathi and Rajendra P. Chhabra* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India
The creeping flow of incompressible power law fluids past arrays of infinite cylinders has been investigated theoretically. The intercylinder interactions have been modeled using a free surface cell model. While, for the parallel flow, the momentum equation has been integrated numerically to obtain the estimates of drag coefficient for the assemblage, the well-known variational principles have been used to analyze the cross-flow configuration. T h e resulting upper and lower bounds on drag coefficient are very close to each other under most conditions of practical interest. Preliminary comparisons with the scant data available in the literature are satisfactory.
Introduction The steady flow of incompressible fluids relative to an array of cylinders represents an idealization of many industrially important processes. For instance, this configuration has been used extensively to simulate flow in multiparticle assemblages such as fibrous packed and fluidized beds and in shell and tube type heat exchangers (Happel, 1959; Kuwabara, 1959; Ganoulis et al., 1989; Chmielewski et al., 1990). Consequently, a vast body of literature has accumulated on the incompressible flow of Newtonian fluids past an array of cylinders, albeit most of it pertains to the so-called creeping flow regime. Only one set of results, as far as is known to us, is available in the intermediate Reynolds number regime (LeClair and Hamielec, 1970). On the basis of extensive comparisons reported in the literature (Happel, 1959; LeClair and Hamielec, 19701, it is reasonable to conclude that the theoretical predictions based on flow past an array of cylinders provide a satisfactory method of calculating the pressure drop in fibrous packed beds and for banks of tubes as encountered in shell and tube heat exchangers. Most of the work in this area for Newtonian fluids has been reviewed by Drummond and Tahir (1984). In contrast to this, much less is known about the analogous problem involving the flow of non-Newtonian fluids. As far as is known to us, no prior theoretical results are available for non-Newtonian fluids, albeit scant experimental results have been reported in the literature. However, in most of these studies, the flow normal to banks of tubes has been used to elucidate the importance of viscoelastic effects in porous media flows (Vossoughi and Seyer, 1974; Barboza et al., 1979; Chmielewski et al., 1990) and no predictive expressions have been presented in any of these studies. Adams and Bell (1968) and Prakash et al. (1987), on the other hand, have investigated the flow of purely viscous fluids normal to a bank of tubes arranged in a variety of configurations. It is thus safe to conclude that very little information is available on the flow of non-Newtonian fluids relative to an array of cylinders. This work aims to bridge this gap in our existing knowledge. While it is readily recognized that the actual flow field both in a porous medium and in a shell and tube heat exchanger is quite complex, it can be argued that the flow can be approximated by a weighted average of the flows normal and parallel to an array of cylinders. Both cases have been examined in the present study. Extensive theoretical estimates of drag coefficient for the flow of power law fluids parallel and normal to arrays of cylinders have been obtained for a range of values of voidage and the flow behavior index. The paper is concluded by
* To whom correspondence should be addressed.
presenting comparisonswith the appropriate experimental data available in the literature.
Problem Statement and Idealization Let us consider the creeping, steady, and incompressible flow of a power law fluid parallel and normal to an array of infiiite cylinders, as shown schematically in Figure la. In their compact form, the equations of continuity and of motion can be written as follows:
v.v = 0 -vp
(1)
+ V.p(VV + (VV)T)= 0
(2) where p is the nongravitational pressure. The rheological equation of state for a fluid obeying the power law behavior is given by p = m(2n)(n-')/2 (3)
Though the deficiencies of this simple fluid model, especially its inability to predict realistic values of viscosity at low shear rates, are well recognized, experience has shown that its use is adequate for describing macroscopic flow phenomena in a variety of geometries (e.g., see Bird et al., 1987); hence it will be used in this work. It is readily recognized that, in addition to the field equations, a mathematical description of the interparticle interactions is also needed to define the problem completely. Among the various approaches available currently, the spherical and cylindrical free surface cell models (Happel, 1958,1959) have been moderately successful in predicting the so-called macroscopic fluid flow phenomena in multiparticle systems, e.g., pressure drop in fixed and fluidized beds (Chhabra and Manjunath, 1991; Jaiswal et al., 1992) and relative motion between ensembles of fluid particles and quiescent non-Newtonian media (Gummalam et al., 1988; Jarzebski and Malinowski, 1986). The range of applications shows the extremely versatile nature of this approach. Hence, in the present investigation, the intercylinder interactions will be modeled using the free surface cylindrical cell model (Happel, 1959). Detailed descriptions of cell models together with their merits and limitations are available in the literature (Happel and Brenner, 1965). The free surface cell model envisions each cylinder (of radius R) to be surrounded by a hypothetical cylindrical envelope of fluid of radius R, such that the voidage of each cell is equal to the mean voidage of the whole array. Furthermore, Happel (1958, 1959) proposed the cell boundary to be frictionless, thereby emphasizing the noninteracting nature of cells. This approach thus converts a difficult many-body problem into a conceptually much simpler one-body equivalent. Such an idealization is shown in Figure lb, together with a cylindrical coordinate system employed in this work.
0888-5885f 92/2631-2154$03,OOf 0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992 2756 for prechosen values of n and e, it was evaluated numerically using a five-point Gauss-Legendre scheme. Normal Flow In this case, the flow is axisymmetric and two-dimensional since V, = 0. The admissible boundary conditions for this configuration, as postulated by Happel (1959), are written as follows: atr=R
v, = 0 v, = 0
(loa) (lob)
at r = R, = R(l - t)-lJ2 Vr
Figure 1. (a, left) Schematic representationof flow. (b, right) Free surface cell idealization.
Parallel Flow In the absence of end effects there is only one nonzero component of velocity, namely, V,, and the momentum equation simplifies to (4)
The physically realistic boundary conditions for this flow configuration are that of no slip ar r = R and zero shear stress at r = R,, i.e., at r = R: v, = 0 (54 a t r = R, = R ( l - e)-lJ2: dV,/dr = 0
(5b)
= -Vo
COS
0
(10c)
=0 (W Owing to the power law relation between the stress and the rate of deformation tensor, the resulting governing equations are highly nonlinear and are not amenable to an analytical or exact solution. Thus, the variational principles have been employed here to obtain upper and lower bounds on the drag coefficient for arrays of cylinders. For the creeping, steady, and incompressible flow of fluids, Slattery (1972) has presented the following two variational principles: Tre
and
H,=
-p,* +
JV.([T - pba*.n) d S (12)
dV
t
Integration of eq 4 subject to the boundary condition 5b yields
Further analytical integration of eq 6 is not possible, and hence numerical integration was carried out to obtain velocity profiles, followed by yet another integration to evaluate the average velocity as a function of of the rheological characteristics of liquid (n, m) and the voidage of system (R, or e). Often the experimental results in this field are reported in the form of dimensionless variables such as drag coefficient and Reynolds number. I t is thus customary to introduce a drag coefficient by equating the pressure drop gradient to the drag force per unit volume of the cell, i.e.,
where the quantities with an asterisk in eq 11are obtained from a trial velocity profile which satisfies the equation of continuity and the known boundary conditions on S,, the latter being that part of the bounding surface S on which the velocity is explicitly known. Likewise, the quantities with an asterisk in eq 12 are evaluated using a trial stress profile that satisfies Cauchy's first law and the prescribed boundary conditions for stress on St.Slattery (1972) further showed that for the single-phase flows 1
J, 2 JiE dV 2 H,
(13)
For power law fluids,the rate of energy dissipation per unit volume is related to the work functions E and E, via the following inequality. (n l)E L tr.(vVV) L (n + 1)E, (14)
+
where E and E, are given as (Slattery, 1972)
(7) with FD
= C~(1/2pV0~)27fRL
With these definitions, the second integration of eq 6 leads to the following expression: C&e =
2 (1 - e ) n + l
I-n
(8)
where
Though the integral I, in eq 9, can be evaluatd analytically for a few values of n such as 1, f 3, f 2, and f 4, in this work,
Now, inequalities 13 and 14 can be combined to yield (n + 1)J,2 l p - ( r - V V )dV 1 (n + 1)H,
(17)
Clearly, the left- and right-hand parte of the inequality provide the upper and lower bounds, respectively, on the rate of energy dissipation. Upper Bound Calculations The following trial stream function (dimensionless) is used to evaluate the upper bound on drag for the bank of cylinders:
2756 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992
sin
e
)
(18)
For u = 1, eq 18 reduces to the stream function for Newtonian fluids (Happel, 1959). In cylindrical coordinates, the two velocity components V,+ and VO+are related to the stream function 3+as
The application of the four boundary conditions embodied in eq 10 yields the following four relations: A-'/,B + C + D = 0 (204 3.4
+ YzB + u C - D
=0
2 0 ( 1 - 4 3 / 2 = 0 (204 A ( l - E)-' - /&{ln(l - e)
+ 1) + C(1 - t)(l-u)/z +
D(l - t) = -1 (20d)
For a suitable choice of the undetermined constants, eq Besides, the forms of r,,, TOR, etc. assumed here are such that the trace of the extra stress tensor is zero. Some of the unknown constants appearing in trial stress profile can be evaluated using the standard procedure, namely, substituting eq 26 in the r and e components of the momentum balance, eliminating pressure by cross-differentiation method, and, finally, equating the coefficients of similar terms. This procedure yields Al = 1. Another constant can be evaluated using the zero shear condition at r+ = R,+ = (1 - $1/2 as
+ H(l -
-
G(l -
=0
(27)
Now a combination of eqs 12, 13, 17, and 22 yields the lower bound on (C$e) as follows: (C&e) I (C&e)L = max 2" 7r
The second invariant of the rate of deformation tensor is given by II = d,,2 + d o t 2dr02 (21)
(26b)
(r+)3
26 reduces to the corresponding Newtonian expressions.
El(1 -
+ 1/2(u - 1)2C(1- e)(z-0)/2 +
2A(1 -
+ r+
El(r+)A1 - - - sin 8
3
(28)
where Iz is given by
+
The expressions for d,,, dss, and drRin terms of V , and V , are available in standard texts on transport phenomena, e.g., Bird et al. (1960). From a macroscopic mechanical energy balance, one can write V,,FD = tr(7VV) dV (22)
1 V
The surface integral in eq 11 is identically zero because of the no-slip condition at r+ = 1 and zero shear stress at r+ = R,+. Now a combination of eq 22 with eqs 17 and 15 yields C$e I(C$e)" = min (2"11) (23) where Il
=
L2TJ(l-f)-l'*
(2W)(n+1)/2r+ dr+ de
(24)
For the trial stream function chosen here, the second inV:) variant of the rate of deformation tensor (n+= m2/ is given by
The trial stream function has five undetermined constants (A, B, C, D , a), four of which are specified via eq 20, and the right-hand side of eq 23 is minimized with respect to the remaining one parameter. Powell's method was used to obtain the upper bound on (C&e) for a range of values of voidage and the flow behavior index, n. Lower Bound Calculations Based on the known extra stress profile for Newtonian fluids (Happel, 1959), the following modified (nondimensional) form has been used here to evaluate the lower bound on drag coefficient for arrays of cylinders:
II,+ = (T,,+)'
+
TO^+)^
+ 2(7,O+)'
(30)
The trial stress profile has five unknown constants (Al, El, F, G, H), out of which two are already specified, i.e., Al = 1 and one via eq 27, and the right-hand side of eq 28 is maximized with respect to the remaining three constants for a range of values of n and e. Results and Discussion The numerical results are expressed in the form of a loss coefficient A = C&e. Theoretical estimates of A for both parallel and normal flows have been computed over wide ranges of conditions: 0.2 I n I 1 and 0.4 It I0.9. However, before embarking upon the presentation of the new results for power law fluids, it is desirable and instructive first to establish the accuracy and reliability of the numerical scheme used and of the results obtained in this study. Validation of Results. For the flow of Newtonian fluids, analytical expressions for drag on arrays of cylinders in both configurations are available in the literature (Happel, 1959) which can be rearranged as follows: parallel flow 16 (31) A,, = 4(1 - t) - (1 - 3 - 2 ln(1 - c) normal flow =
I*
(32) 1+(1-
The present numerical results for n = 1 are in complete agreement with the predictions of eqs 31 and 32 over the complete voidage range (0.4 5 t 5 0.9); the maximum divergence being less than 0.2 5%. Admittedly, it is not possible to establish the accuracy of the corresponding
Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992 2767 Table I. Calculated Values of Loss Coefficient for Parallel and Normal Flow" n=l
AP
Au AL
n = 0.8
AP
AU AL
n = 0.6
AP
AU AL
n = 0.4
AP
n = 0.2
AL AP
AU
AU AL a
e = 0.4
e = 0.5
e = 0.6
c = 0.7
259.53 397.81 397.46 126.46 184.44 183.33 61.07 86.24 83.93 29.00 40.83 37.88 13.24 19.60 16.56
117.40 171.83 171.71 64.30 90.84 90.31 34.88 48.50 47.19 18.60 26.20 24.35 9.52 14.32 12.21
58.7 83.29 83.24 35.75 49.66 49.36 21.55 29.89 29.09 12.75 18.18 16.94 7.21 11.17 9.69
30.89 43.35 43.34 20.85 29.02 28.85 13.93 19.59 19.09 9.10 13.35 12.48 5.73 9.17 7.94
0.8 16.76 23.313 23.310 10.30 17.59 17.49 8.01 13.38 13.04 6.08 10.248 9.86 4.37 7.91 7.26
c
c = 0.9
7.22 12.101 12.098 6.36 10.55 10.49 5.52 9.27 9.03 4.64 8.21 7.86 3.65 7.32 6.63
Ap = parallel flow; A" and AL = upper and lower bounds, respectively, for normal flow.
results for power law fluids, but based on the aforementioned comparison for n = 1,these are believed to be numerically accurate within fl% Finally, it will suffice to add here that the present predictions for normal flow are about 20% lower than the experimental values of pressure drop for fibrous beds in the range 0.618 Ie I 0.92 as reported by Kyan et al. (1970) for Newtonian media. Power Law Results. Table I provides a summary of the results obtained in this study. As expected, the value of loss coefficient is always smaller in parallel flow than in normal flow and this can solely be attributed to the bending of streamlines in the latter case, thereby resulting in greater viscous dissipation in this case. It is interesting to note that the ratio AI1/A1does not deviate appreciably from about 1.5. For power law fluids, the value of the loss coefficient reduces below its Newtonian value for all values of voidage and in both confiiations, namely, parallel and normal flow conditions. The smaller the value of n,the greater the reduction in A. This decrease in the value of loss coefficient is particularly pronounced in the case of densely packed arrays. For instance, at e = 0.4,the value of A drops from 397.8 to 39.36 (average of upper and lower bounds), an order of magnitude change, as the value of n decreases from n = 1to n = 0.4; the corresponding change 8. This is prefor t = 0.9 is only from A = 12 to A sumably so due to the steeper velocity gradients (hence lower effective viscosity) prevailing in densely packed arrays of cylinders. For a given value of e, the loss coefficient decreases rapidly with the increasing extent of shear-thinning behavior. Finally, since the location of the exact solution enclosed by the upper and lower bounds is not known, the use of the arithmetic average of the two bounds is recommended. This is justifiable in the present case as the two bounds are rarely seen to differ from each other by more than 10% except for highly shear-thinning fluids (n = 0.2) in which case the maximum divergence of about 18% is observed for e = 0.4. By analogy with the creeping flow of power law fluids through packed beds of spherical beads (Christopher and Middleman, 1965; Kemblowski and Michniewicz, 1979), it is customary to intxoduce a generalized Reynolds number which may result in further consolidation of results for the different values of power law index encompassed in the present investigation. On the basis of their own experimental results as well as those of Adams and Bell (19681, Prakash et al. (1987) introduced the following definition of a generalized Reynolds number for flow across a bank of tubes: Re'= Re("-) " 12(1t2 - e) l-n (33) 2n + 1 1-t
.
-
1
} (L)
It is also convenient to introduce a friction factor defined as follows: (34)
It can readily be shown that f = 2CDe3. In the viscous flow regime, one would expect the friction factor to show an inverse dependence on the generalized Reynolds number, that is, fRe' = Q (35) where Q is expected to be a function of voidage but independent of the flow behavior index. The values of Q based on the present theoretical predictions, both in parallel and in normal configuration, are summarized in Table 11. An examination of Table I1 shows that, for a fixed value of voidage, the value of Q shows much weaker dependence on the flow behavior index than the loss coefficient A. For instance, for t = 0.4,the value of Cl, decreases only by about 20% as opposed to A,, which drops by a factor of 20 when n decreases from 1 to 0.2. However, at the other extreme oft = 0.9, the corresponding variation in SI, is only marginally smaller than that in AL. The value of Cl,,displays qualitatively similar dependence on the flow behavior index. Likewise, for a fixed value of n, the variation in Q with voidage is much smaller as compared to the corresponding change in A.
Comparison with Experiments As mentioned earlier, of the five experimental studies available in the literature, only the works of Adams and Bell (1968)and Prakash et al. (1987) fall within the scope of the present study. These investigators have reported pressure drop measurements for the flow of (carboxymethy1)cellulose solutions flowing normal to a bank of tubes arranged on triangular and square pitch mounted in an in-line and staggered manner. On the basis of their results for triangular pitch and those of Adams and Bell (1968) for staggered tube arrangement on square pitch, Prakash et al. (1987) presented an empirical correlation which in the creeping flow region reduces to f = 130/Re' (36) From the dimensions of tube banks given in these papers, the values of voidage are estimated to be about 0.6 for the data of Prakash et al. (1987) and 0.58 for those of Adams and Bell (1968). Furthermore, eq 36 is based on the data gleaned with experimental fluids displaying power law index in the range 0.56 In I1. For these conditions, the present theory predicts a mean value of 0, = 81.12 which is about 37% lower than the experimental value of
2758 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992
Table 11. Values of Qa = 0.4
e
= 0.5
t
e
= 0.6
c
= 0.7
= 0.8
t
e
= 0.9
~~~
n=l
55.36 84.827 54.17 78.77 52.94 73.76 51.63 70.06 50.10 68.4
Q li
n, A.
n = 0.8
Q II
Q,
n = 0.6
Q /I
QL
n = 0.4
li
0,
n = 0.2
Q 1I
a,
averwe values
58.7 85.897 56.94 80.21 55.15 75.65 53.23 72.34 51.06 71.15
70.65 99.14 66.65 92.49 62.75 87.12 58.55 83.10 54.53 81.42
63.4 90.0 60.8 84.2 58.17 79.61 55.39 76.28 52.19 75.49
80.81 119.35 70.45 109.73 64.43 101.79 61.72 96.74 58.50 94.36
105.27
. 176.42 94.09 155.63 83.53 138.46 72.81 126.08 61.49 117.51
I
52*84
1'
75416
1'
a
-2.74 +2*52
55.02
4.76 +9*65 77.05
+3.68 -3,96 +8.84 -5.go
+5.4 -5.8
58
+8.88
81.12
-5.63
62.6
f8.1
67.18
+12.6 -8.68
83-44
88.65
+'OU5 -7.23
104.4
+14"
142.8
k21.8 +33.6 -25.3
Values of 0, are baaed on the mean of upper and lower bounds.
0
w
L
methy1)cellulose solutions. Figure 2 contrasts the experimental and predicted values of (fRe)in the range 0.74 5 n I0.94 and 0.39 It I0.75. The agreement is again seen to be about as satisfactory as can be expected in this type of work.
0.845
-
a
2
22-
$5..
v
18 14
-
5 E 6 0.75
r
10
14
18
22
26
30
3L
38
(+Re)cxperirnental
Figure 2. Comparison between theory and predictions for flow in fluidized beds.
130 in eq 36. However, it is appropriate to mention here that a closer inspection of the experimental results of Prakash et al. (1987) shows that eq 36 has a tendency to overpredict experimental values in the Re < 1 region. In view of this fact coupled with the highly idealized nature of the cell model employed herein, the agreement between the theory and experiments is regarded to be satisfactory and acceptable. In the same paper, Prakash et al. (1987) also reported a value of Q = 60-70 for the in-line square pitch tube banks employed by Adams and Bell (1968). The voidage for this arrangement is estimated to be about -0.49-0.5. For these conditions, the predicted value of Q , is about 77, which is in excellent agreement with the aformentioned experimental value. It is worthwhile to reiterate here that, within the range of narrow experimental conditions (0.56 5 n I1and -0.5 Ie I 0 . 6 ) , the present approximate theory predicts only a slight variation in the value of Q , and therefore it is not at all surprising that a single value of 0 adequately describes the experimental data (Prakash et al., 1987). Also, one would intuitively expect the present theoretical predictions to be more realistic for high-porosity systems, but unfortunately, no experimental results on flow across such tube banks are presently available to substantiate this assertion. Finally, the present predictions for the normal flow configuration are also compared with the pressure drop measurements for both fixed and fluidized beds of spherical particles as reported recently by Srinivas and Chhabra (1991). In this study, beds of glass beads of three different sizes (3.5, 6.3, and 15.8 mm) were fluidized using several (carboxy-
Conclusions In this work, theoretical estimates of drag coefficient (or friction factor) for creeping flow of power law fluids relative to an array of infinite cylinders have been obtained. As expected, the resistance to flow is lower in parallel flow than that encounted in the normal flow confiiation. The upper and lower bounds on the loss coefficient in cross flow are very close to each other under most conditions of interest whence the use of their mean value is suggested. The theoretical predictions show satisfactory agreement with the scant data available on flow past banks of tubes and in fluidized beds. However, more experimental work, especially in fibrous beds with high values of voidage, is required to thoroughly validate the predictions made herein.
Nomenclature A , B , C, D = unknown constants, eq 18 AI, E,, F,G, H = unknown constants, eq 26 CD = drag coefficient d = diameter of cylinder, m d,,, des,dro = components of rate of deformation tensor, sY1 E, E, = work functions, eqs 15 and 16 Pa/s f = friction factor F D = drag force, N H,= function, eq 12, N.m/s J, = function, eq 11, N.m/s L = length of cylinders, m m = power law consistency index, Pa s" n = power law index n = unit vector p = pressure, Pa r = cylindrical coordinate, m R = radius of cylinder, m R , = cell radius, m Re = Reynolds number (pVo2-"(2R)"/m) S = surface T = extra stress tensor, Pa V = velocity vector, m/s Vo = superficial velocity, m/s v = volume of flow domain, m3 Greek Symbols = voidage A = loss coefficient p = viscosity, Pa s t
Ind. Eng. Chem. Res. 1992,31, 275S2764
ll = second invariant of rate of deformation tensor, s-2 = unknown parameter, eq 18 = potential 9 = stream function Q = = fRe'
Superscripts
+ = dimensionless quantity
* = quantity evaluated from trial profiles Subscripts r, B = r and 8 components 11 = parallel flow i = cross flow
Literature Cited Adams, D.; Bell, K. J. Fluid friction and heat transfer for flow of sodium carboxymethyl cellulose solutions across banks of tubes. Chem. Eng. h o g . Symp. Ser. 1968,64 (No. 82), 133. Barboza, M.; Rangel, C.; Mena, B. Viscoelastic effects in flow through porous media. J . Appl. Polym. Sci. 1979,23, 281. Bud, R. B.;Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. Bud, R. B.; Armstrong, R. C.; Hassager, 0. Dynamics of Polymeric Liquids Vol. I: Fluid Mechanics; Wiley: New York, 1987. Chhabra, R. P.; Manjunath, M.Flow of non-Newtonian power law liauids through packed and fluidized beds. Chem. Eng. Commun. 1991,106,33: Chmielewski. C.: Pettv. C. A.: Javaraman. K. Cross flow of Elastic Liquids t d r o k h G a y s of 'Cyfinders. J. Non-Newtonian Fluid Mech. 1990,35,309. Christopher, R. H.; Middleman, S. Power law flow through a packed tube. Ind. Eng. Chem. Fundam. 1965,4,422. Drummond, J. E.; Tahir, M. I. Laminar Viscous Flow Through Regular Arrays of Parallel Solid Cylinders. Int. J. Multiphase Flow 1984,10,515. Ganoulis, J.; Brunn, P.; Durst, F.; Holweg, J.; Wunderlich, A. Laser Measurements and Computations of Viscous Flows Through Cylinders. J . Hydraul. Eng. 1989,115,1223. Gummalam, S.;Narayan, K. A.; Chhabra, R. P. Rise Velocity of a Swarm of Spherical Bubbles through a Non-Newtonian Fluid:
2769
Effect of Zero Shear Viscosity. Int. J . Multiphase Flow 1988,14, 361. Happel, J. Viscous Flow in Multi-particle Systems: Slow Motion of fluids Relative to Beds of Spherical Particles. AIChE J . l958,4, 197. Happel, J. Viscous Flow Relative to Arrays of Cylinders. AIChE J. 1959,5, 174. Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Prentice Halk Englewood Cliffs, NJ, 1965. Jaiswal, A. K.; Sundararajan, T.; Chhabra, R. P. Simulation of Non-Newtonian fluid flow through fixed and fluidized beds of spherical particles. Numer. Heat Transfer 1992,21,275. Jarzebski, A. B.; Malinowski, J. J. Drag and Mass Transfer in Multiple Drop Slow Motion in a Power Law Fluid. Chem. Eng. Sci. 1986,41,2569. Kemblowski, Z.; Michniewicz, M. A new look at the laminar flow of power law fluids through granular beds. Rheol. Acta 1979,18,730. Kuwabara, S . The Forces Experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Numbers. J. Phys. SOC.Jpn. 1959,14,596. Kyan, C. P.; Wasan, D. T.; Kintner, R. C. Flow of Single Phase Fluids through Fibrous Beds. Ind. Eng. Chem. Fundam. 1970,9, 596. LeClair, B.P.; Hamielec, A. E. Viscous Flow Through Particle Aasemblagea at Intermediate Reynolds Numbers: Steady State Solutions for Flow Through Assemblages of Cylinders. Znd. Eng. Chem. Fundam. 1970,9,608. Prakash, 0.;Gupta, S. N.: Mishra, P. Newtonian and Inelastic Non-Newtonian Flow Across Tube Banks. Ind. Eng. Chem. Res. 1987,26,1365. Slattery, J. C. Mass,Momentum and Energy Transfer in Continua; McGraw-Hilk New York, 1972. Srinivas, B. K.; Chhabra, R. P. An Experimental Study of NonNewtonian Fluid Flow in Fluidized Beds: Minimum Fluidization Velocity and Bed Expansion Characteristics. Chem. Eng. Process. 1991,29,121. Vossoughi, S.;Seyer, F. A. Pressure Drop for Flow of Polymer Solution in a Model Porous Medium. Can. J. Chem. Eng. 1974,52, 666.
Received for review April 6,1992 Revised manuscript received August 20, 1992 Accepted September 10,1992
Environmentally Durable Elastomer Materials for Windshield Wiper Blades Ares N. Theodore,* Marsha A. Samus, and Paul C. Killgoar, Jr. Research Staff, Polymer Science Department, Ford Motor Company, Dearborn, Michigan 48121
The primary cause for the poor performance of current windshield wiper blades is the inherently poor resistance of natural rubber to attack by ozone, oxygen, and sunlight causing degradation of the elastomer. EPDM (ethylene-propylene terpolymer) is a low-cost elastomer with excellent resistance to attack by these environmental agents. A significant deterrent to using EPDM has been its high rubber-to-glass friction. Unlike natural rubber, the surface cannot be easily chlorinated to lower the friction. It was found that the friction of the EPDM loaded with graphite could be controlled and the other physical properties needed in a windshield wiper blade maintained at the same time. Critical parameters identified in this work are the size and loading of the graphite and the size of the carbon black. The correct choice of EPDM and cure system is important in mnnimiaing the physical properties of the compound. Prototype wiper blades based on EPDM exhibit superior environmental resistance, frictional properties, compression set, and tensile set when compared to production natural rubber windshield wiper blades.
Introduction Developing an environmentally stable, durable elastomeric windshield wiper blade is important to provide added quality to the automobile and consequently increase customer satisfaction. Windshield wiper blades should have good wipe quality under all environmental conditions, not make noise during operation, and be resistant to envi-
ronmental attack.' Factors which affect wipe quality are the sharpness of the blade edge, the ability of the blade to conform to the glass surface, the ease with which the blade flips from side to side during operation, and the level of glass to rubber friction. Noise is believed to be caused by excessive friction between the rubber and the glass. This is most noticeable during light rain and is charac-
0888-5885/92/ 2631-2759$03.O0/0 0 1992 American Chemical Society
