130
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
initial solid velocity, m/s us = dimensionless solid velocity, u,/u,, W = gas mass flow rate, kg/m2-s = solid mass flow rate, kg/m2-s x = coordinate parallel to flow direction R = dimensionless coordinate parallel to flow, X / L us, =
4
Greek Letters e = volumetric fluid concentration el = initial volumetric fluid concentration
z = dimensionless fluid concentration, t / q 0 = the angle between the tube axis and the horizontal line hf =
viscosity of the fluid
pf = density of the fluid, kg/m3 pg = density of the gas, kg/m3 pg, = initial density of the gas, kg/m3 ps = density of the solid, kg/m3
Literature Cited Arastoopour, H., Gidaspow, D., "Two Phase Flow Fundamentals", in "Two Phase Transport and Reactor Safety", S. Kakac and T N. Veziroglu, Ed., Vol. I , pp 133-158, Hemisphere Publishing Corp., 1978. Arastoopour, H., Ph.D. Thesis, Illinois Institute of Technology, Chicago, 1978. Capes, C. E., Nakamura, K., Can. J Chem. Eng., 51, 31-38 (1973). Davidson, J. F., Trans. Inst. Chem. Eng., 39, 230-232 (1961). Deich, M. E.. Danilin. V. S., Sleznev, L. I.. Solomko, V. I., Taiklouri, G. V., Shannon, V. K., High Temp. 12(2), 299-307 (Nov 1974); (translation of Teplofiz. Vys. Temp., 12 (2), 344-53 (1974) by Consultants Bureau, New York). Farbar, L., Ind. Eng. Chem., 41, 1184-1191 (1949). Gidaspow, D.. "Hyperbolic Compressible Two-Phase Flow Equations Based on Stabbnary Principles and the Fick's Law", in "Two phase Transport and Reactor Safety", S. Kakac and T. N. Veziroglu, Ed., Vol. I, pp . . 283-298, Hemisphere Publishing Corp., 1978. Gidaspow, D., "Fluid Particle Systems", in "Two Phase Flow and Heat Transfer", S. Kakac and F. Mayinger, Ed., Vol I, pp 115-128, Hemisphere Publishing CorD.. 1977. Gidaspow, D. Solbrig, C. W., "Transient Two-Phase Flow Models in Energy Production," State of the Art Paper presented at the AIChE 81st National
Meetings, Apr 11-14, 1976; in revised form preprinted for NATO Advanced Study Institute on Two-Phase Flows and Heat Transfer, Aug 16-27, 1976, AS1 Proceedings, Istanbul, Turkey. Gdaspow, D., Round Table Discussion (RT-1-2). "Modeling of Two-Phase Flow", 5th International Heat Transfer Conference, Sept 3, Tokyo, Japan, in "Heat Transfer 1974", Vol. VII, pp 163-168, 1974. Giot, M., Fritte, A., Prog. Heat Mass Transfer, 5, 651-670 (1972). "Handbook of Natural Gas Engineering", C. F. Bonilh et al., Ed., p 303, M&aw-Hill, New York. N.Y., 1959. Hariu, 0. H., Molstad, M. C., Ind. Eng. Chem., 41, 1148-60 (1949). Jackson, R.. "Chapter 3. Fluid Mechanical Theory". pp 65-1 19 in "Fluk3zation", J. F. Davidson and D. Harrison, Ed., Academic Press, New York, N.Y., 1971. Jackson, R., Trans. Inst. Chem. Eng., 41, 13-28 (1963). Knowlton, T. M., Bachovchin, D. M., Fluidication Technoi., 2, 253-82 (1976). Lamb, H., "Hydrodynamics", 6th ed, Chapter 6, Dover Publications, New York, N.Y., 1932. Lax, P. D., Comm. Pure Appi. Math., 11, 175-94 (1958). Lyczkowski, R. W., Gdaspow, D., Solbrig. C. W., Hughes, E. D., Nucl. Sci. Eng., 66, 377-96 (1978). Mehta, N. C., Smith, J. M., Comings, E. W., I d . f n g . Chem., 49,986-92 (1957). Nakamura, K., Capes, C. E., Can. J . Chem. Eng., 51, 39-46 (1973). Pritchett, J. W., Levine, H. B., Blake, T. R., Gary, S. K., "Numerical Model of Gas Fluidized Beds", 69th AIChE Annual Meeting, Chicago, Nov 1976. Richardson, J. F., Zaki, W. N., Trans. Inst. Chem. Eng., 32, 35-53 (1954). Rowe, P. N., Henwood, G. A., Trans. Inst. Chem. Eng., 39, 43-54 (1961). Rudinger, G., Chang, A., fhys. Fluids, 7, 1747-54 (1964). Shook, C. A., Masliyah, J. H., Can. J . Chem. Eng., 52, 228-33 (1974). Soo, S.L., "Fluid Dynamics of Mukiphase Systems", p 279, Blaisdell Publishing Co., Waltham, Mass., 1967. Soo, S. L., Int. J . Muitiphase Flow, 3, 79-82 (1976). Wen, C. Y.. Galli, A. F., "Chapter 16, Dilute Phase Systems", in "Fluidization", J. F. Davidson and D. Harrison, Ed., Academic Press, New York, N.Y., 1971. Yousfi, Y., Gau, G., Chem. Eng. Sci., 29, 1939-46 (1974). Zenz, F. A., Ind. Eng. Chem., 41, 2801-07 (1949).
Received for reuieu March 6 , 1978 Accepted December 7, 1978
Partial support for this study was provided to one of the authors (D.G.)by Department of Energy Contract DOE ET-78-G-01-3381.
Viscous Flow in Rectangular Open Channels Seshadri Srinivasan, Krishna M. Bobba, and Leonard A. Stenger Owens-Corning Fibergias Corporation, Technical Center, Granville, Ohio 43023
An exact solution is obtained to the problem of isothermal creeping flow of liquids in rectangular open channels. This exact solution exhibits excellent convergence and is very convenient to use.
Introduction An interesting approach to the problem of viscous flow of liquids in rectangular open channels was suggested by Johnston (1973), who obtained a numerical solution and applied it for flow in a groove. Details of the numerical solution, however, were not reported. A nondimensional plot of the open channel surface profile was presented. This is difficult to use in the computation of the changes in liquid level, especially when the changes are relatively small compared to the dimensions of the channel. We therefore examined the governing equations and were successful in obtaining a convenient exact solution. The details are presented in the following section. Mathematical Formulation and Solution The mathematical formulation described here is the same as that of Johnston (1973). We present some of the details of the derivation as part of the development of the exact solution. The mathematical formulation consists of the following three steps. 0019-7874/79/1018-0130$01 .OO/O
1. Step One. The first step is to consider isothermal creeping steady incompressible flow in a sloping closed rectangular duct. Figure 1 describes the geometry of the system. Let u , u , and w represent the velocities in the x , y , and z directions. For two-dimensional flow in the x direction, u = w = 0 and u = u(y,z). The x , y , and z equations of motion are aP - pg, = ax
.( $
+
$)
-aP= o aY
aP -
az
= Pi?,
(3)
The symbols are defined in the Nomenclature section. In 0 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
131
13 is now utilized in eq 8. This yields
e5
n=l m=l
Am.( n2T2+
y)
sin (nTV sin
(5) =1
(14) Multiplying both sides of eq 14 by sin ( L a Z l a ) and integrating with respect to Z with the limits 0 to a , the following equation is obtained.
f AL.( n2T2+ T ) E)( sin (nTY) = a [I T
n=l
Figure 1. Sloping rectangular channel.
view of eq 2, P is a t best a function of x and z only. Equation 3 can, therefore, be integrated to give P = (pg,)z + arbitrary f(x) (34 From eq 3a it is seen that aP/ax is a function of x only. Now dh g, = g cos 0 = -g(4) dx Using eq 4 in eq 1, therefore, gives
It is seen that the left-hand side of eq 5 can be a function
-
(-1)Ll
(15) Both sides of eq 15 are now multiplied by sin ( K T Y )and integrated with respect to Y with the limits 0 to 1. The resulting equation is rearranged to give
This completes the solution to 4. It is now convenient to define a quantity denoted by F (flow function) that is related to the volumetric flow rate.
where, Q, the volumetric flow rate, is given by
of x only, while the right hand side can be a function of y and z only, and independent of x since u = u(y,z). Hence, it follows that
a -(P
+ pgh) = constant
(6)
ax A quantity 4 is now defined.
6=-u/[
Using the definition of 4 (eq 7) and the dimensionless quantities (eq 9, 10, and 12) it can be shown that
F =
.“ a ( P + pgh) cc ax
1
sal’4 0
0
(Y,Z) d Y d Z
(7)
It is evident that F is a function of a only. Utilizing eq 13 and 16 and carrying out the integrations in eq 19 yields
(8)
2. Step Two. The problem of steady isothermal flow down a sloping open rectangular channel with a uniform depth of b / 2 is considered in step two. The coordinate system is the same as that shown in Figure 1. For the present case aPlax =O since the fluid is subjected to uniform atmospheric pressure. Following the arguments in step one, it can be shown that the equation of motion reduces to eq 8, where
Utilizing eq 7 , eq 5 may be rearranged to give
a24 atR
a24
-+-=-I
aZ2
where
y = Ya
(9)
and
z =z a
Johnston indicates that he obtained a “numerical solution” of eq 8 “using a computer” without divulging the details or the nature of the numerical scheme. We have derived a n exact solution to eq 8 subject to the conditions d(0,Z) = 6(1,Z) = 4(Y,O) = 4(Y,a) = 0 (11) where
.=[
depth of liquid channel width
1
m
m
n=l m = l
A,, sin (nnY)sin
(22)
(12)
i.e.
for a closed, rectangular channel. In the present instance, a = b/a. We shall now develop an exact solution to eq 8 with the associated boundary conditions. Let
4=
As before, we have 4(0,Z) = 4(1,z) = d Y , O ) = 0 The fourth condition is
(9) (13)
Equation 13 satisfies the conditions of eq 11. Equation
This condition, however, is automatically satisfied by 4 (eq 13 and 16). The results of step 1 (i.e., eq 20) are, therefore, applicable for the sloping, open, rectangular channel with uniform depth with the understanding that Qo = 0.5Q (25)
132
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
0.5
t
a
Figure 3. Flow function, F.
a
Figure 2. Flow function, F.
where Q = volumetric flow rate in closed rectangular channel of width a and depth b. Qo = volumetric flow rate in open rectangular channel of width a and depth b/2. 3. Step Three. Free surface flow (varying depth) in a horizontal rectangular channel is now considered. If the free surface slope (the rate of change of the depth of the fluid with respect to the length of the channel) is small (0.01 or less) then eq 17 is applicable along with eq 20 with Q = 2Q0 and aPlax = 0. Equation 17 for the present case, reduces to c
0.2
0.4
0.6
0.8
1.0
2
A
Figure 4. Dimensionless surface profile. Table I. Convergence’ of Series Representations for = 3. (See Eq 20, 30, and 31)
oi
Also, it must be pointed out that if h is the local depth of the liquid in the open horizontal rectangular channel, the equivalent closed channel problem would require a liquid depth of 2h. Hence, if the “depth ratio” H is defined to be the ratio h l a , then it is evident from the definition of a (eq 12) that h - = H = 0.5a (27) a A new coordinate x * is defined which is in opposite direction to x (i.e., dx* = -dx). This is done to achieve an exact correspondence with Johnston’s paper (1973). Utilizing eq 27, eq 26 may be rearranged to give
where
x = X-*
(29)
a
If x * = 0 (i.e., X = 0) chosen to correspond to h = 0 (i.e., a = 0), eq 28 can be integrated readily. This yields m
=
m
$ + 5 In (L2 + )]
C C BLK[
L=l K = l
L2
L2
Pa2
(30)
where
BLK= 2[1 - (-1)L]2[l - ( - 1 ) K ] 2 / ( ~ 6 L 2 P(31) ) and
This completes the solution. Results and Discussion The flow function F (eq 20) and the dimensionless surface profile ( a vs. dimensionless group X, eq 30) are
computing timeb (CPU s) N
F
h
comp. of F
comp. of h
10 20 30 40 50
0.19714 0.19744 0.19747 0.19748 0.19748
0.24024 0.24050 0.24052 0.24053 0.24053
0.15 0.57 1.29 2.32 3.77
0.19 0.80 1.97 3.45 5.43
Similar convergence behavior was found for other values of a . IBM 3701138.
plotted in Figures 2, 3, and 4. The results are in good agreement with those obtained by Johnston (1973) by a numerical procedure. As was pointed out earlier, the details of the numerical procedure were not furnished. An exact comparison of the results of the present work with those of Johnston (1973) was, therefore, not possible and the comparison is based on the graphical results published in Johnston’s paper. A possible numerical approach to the problem involves a finite difference procedure (Carnahan et al., 1969) with Gauss-Seidel iteration to solve eq 8 followed by a numerical double integration scheme to obtain F for each a. Finally, the solution to the governing differential equation (eq 28) would also require a numerical routine. The exact solution developed in the present paper is certainly much easier to use. Also, the series representations (eq 20 and 30) converge very well. Typical convergence behavior is shown in Table I. From a practical point of view, the change in the level of viscous liquid flowing in a rectangular open channel of length IALI is of interest. This requires the following steps. 1. Compute initial value of X corresponding to the initial liquid level. Let this be denoted by Xo. (Ao is computed from eq 30 corresponding to ao). 2. Compute the final X using Xf = Xo - [AX! (33)
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
where (34)
133
channels. The solution converges well, requires only modest computing time, and is extremely convenient and easy to use in practical calculations. Nomenclature
3. Find afcorresponding to Xf and hence the change in level. Finding af corresponding to Xf requires the solution of the algebraic equation da)= 0 (35) where
(36)
BLKis given in eq 31. Applying the Newton-Raphson scheme, the iterative algorithm becomes aj+1
q(aj) - aj - V'(aj)
j = 0, 1, 2,
...
(37)
with cyo as the starter value. The prime indicates differentiation with respect to a. After carrying out the differentiation and rearranging, the algorithm becomes aj+1 m
[C
- aj m
C BL.K1(a?/K2)+ (L2/K4) In ( L 2 / L 2+ P a l 2 ) -}
L = l K=l
m
O
:
K +(~j~fl)] &I/[ LC= l KC= l ~ B L aj3/(L2
j = 0, 1, 2,
...
(38) According to Johnston (1973), the mathematical formulation is valid if the ratio of the change in liquid height to the length of the channel is less than 0.01. It must be pointed out that for small values of IAXI the graphical results of Johnston (1973) are difficult to use and could lead to considerable error. The exact solution, however, is very convenient to use in the computation. Conclusion An exact solution has been developed for isothermal creeping flow of viscous liquids in rectangular open
ALK= defined by eq 16 a = width of channel BLK
= defined by eq 31
b = depth of channel
F = flow function, eq 17 g = acceleration due to gravity g,, g, = x and z components of g h = height of free surface ho = initial height of fluid H = h/a K , L = summation indices N = number of terms in the series P = fluid pressure Q = volumetric flow rate in closed channel Qo = volumetric flow rate in open channel x = direction of flow (opposite to x * ) X = defined by eq 29 y = coordinate along the width Y = y/a z = coordinate normal to x and y Z = z/a Greek Letters a = defined by eq 1 2 a0 = a corresponding to h af = a at the end of the channel IALI = length of channel IAXJ = change in X 9 = defined by eq 36 0 = slope angle (Figure 1) X = defined by eq 32 Xf = X at the end of the channel Xo = X corresponding to cyo 4 = defined by eq 7 and 21
Literature Cited Johnston, A. K.,Ind. Eng. Chem. Fundam., 12, 482 (1973). Carnahan, E..Luther, A. H., Wilkes. J. O., "Applied Numerical Methods", pp 491-497, Wiiey, New York, N.Y., 1969.
Received for review March 16, 1978 Accepted January 15, 1979
