Literature Cited
( I ) Babbit, J. D., Can. J . Res. A28, 449 (1950). (2) Barrer, R . M., Barrie, J. ‘4.: Proc. ROY.S ~ C . (London) A2137 250 (1952). (3) Calston Research Society, Proceedings of ‘l’erith Symposium, p. 1’8, Butterworths, London, 1958. (4) Carman, P. C.. Kaal, F. A . , Proc. Roy. Soc. (London) A209, 38 (1951). (5) Chu, Ctiiek. Hougen, 0. A , , Chem. En?. Sci.17, 167 (1962). (6) de Boer, J. H., Adcan. Catalysis, 8, 18 (1959). (7) Gilliland, F.. K., Baddour, R. F., Russell, J. I,., A.1.Ch.E. J . 4, 90 (1958). (8) Masamune, S., Smith, J. M., Zbid.,to he published. (9) Miller, 11. N., Kirk, R. S., Ibid., 8, 183 (1962).
(10) Patterson, D., Trans. Faraifay Soc. 49, 802 (1953). (11) Ross, J. \V., Good. R.J., J . Phys. Cizem. 60, 1167 (1956). (12) ‘I‘hiele, E. \V., Ind. En?. Ciiem. 31, 916 (1939). (13) T o d i n s o n , R. H., Flood? E. ,\., Con. .J. R P J .B26, 38 (1948). (14) \.~eisz,p. R.,~ch\+\.artr, )\,B., J . Cntn/ysis 1 , 399 (1062). (15) \Vheeler, A , 4 n i ~ n n Cdolyszr . 3, 250 [1951). J. 11. KR.\SUK Clniz,ersity of Ruenos A r e s Ruenos Aires, Argpntina
J. M. SMITH University of California ~ ~Calif. ~ i ~
, RECEIVED for review April 5, 1964 ACCEPTEDSeptember 28, 1964
COM MUN ICATION
ABSENCE OF CONCENTRATION GRADIENTS I N SLURRIES S E T T L I N G A T HIGH REYNOLDS N U M B E R S In settling slurries of uniform spheres, concentration gradients can form if the final packing exceeds a critical value which i s a function of Reynolds number and sphere-cylinder diameter ratio.
it has been shown that concentration gradients occur in the settling of closely sized spheres (3, 6, 8) and that the width and position of the gradient can be predicted from a flux plot based on initial settling rates (3, 6 ) . As the previous studies have dealt with settling a t very low Reynolds numbers, it may be of interest to determine a theoretical relationship between Reynolds number and the occurrence of concentration gradients. Coulson. Richardson, and Zaki (2, 5, 9 ) have shown that ECENTLY
u =
By equating the second derivative to zero, the inflectiori point is found to be
O n a plot of S’against E , c y is set equal to that final packing voidage \vhich is ~ i i s thigh enough to eliminate a concentration gradient. ‘[he slope of the line from e o is then equal to the slope of the curve a t the inflection point--Le.,
U,?
where
0 < Re! 0.2
< 0.2
< Re! < 1
n
=
4.6 f 20
n = (4.4
d
-
D
+ 18
g)
3
(2b)
1
< Re: < 200
n
=
200
< R e ! < 500
n
=
4.4
(2d)
> 500
n
=
2.4
(2e)
Re!
(
4.4 f 18 ~- Re!+.’
= En
-
cis” ~- -
- (n
+ 1)
en
+l
(3)
en
(4)
rhus
de
n‘n-l
from Equation 6 yields after a
Figure 1 is the theoretical relationship bet\veen (1 - eo) and R e ! based on Equations 2 (with d,’D = 0) and 9 . For regular close packing, the solids fraction is 0.74. Thus, there is theoretically no possibility of forming concentration gradients above a Reynolds number of 40 if d / D = 0. Because of the wall effect ( 7 ) , the value for. close random packing of uniform spheres in a cylinder varies with d / D . For d / D less than 0.05, the effect is slight (7), and (1 - e , ) may be taken as 0.61 ( 7 ) . I n this ca4e, a concentration gradient should al~vaysform if Re! < 1. Rearrangement of Equations 2c and 9 with e o = e, = 0.39 yields
Re!O.i and
et
e y = (n+1> n - 1
(2c)
Their results a t low R e ! are supported by other investigations (4, 7). T h e dimensionless solids flux is given by
S’ = s / u , p ,
Substitution of the value of some simplification
1.017
+ 4.16 & L l
(10)
As shown in Figure 2, concentration gradients form more readily \vith increaqing d , D .
In practice, a very slight concentration gradient cannot be 106
l&EC
FUNDAMENTALS
L
I
I
I
I
I
1
I
0.75 u”
0.55
l i L u + b 10
0.I
1000
100
d /D
Re‘
Figure 1 . gradients
Figure 2. Maximum diameter ratio for no concentration gradients
Maximum final packing for no concentration
distinguished from a n end effect. Hence the value of (1 - em) must be appreciably greater than (1 - eo) for experimental confirmation. For 67-micron glass spheres in water, the results Wrre represmtetl by a n empirical equation (7) \vhich. in dimensionless form. is S ’ p S u , = 0.863349 (1 - e) - 3.51204 (1 - e)*
4.13967 (1 --
+ 0.28735 (1
-
+
= settling velocity of solids, cm./sec.
Stokes velocity. cm.isec. voidage? dimensionless = kinematic viscosity, stokes = solids density, grams/cc. = =
e u
pa
SUBSCRIPTS = value which just fails to permit formation of gradient
8z
-
2.30162 (1 -
u u,
= =
m
e)‘
value at inflection point final value
(11)
[Each term of the last equation in Shannon, Stroupe, a n d Tory’s T a b l e I11 (7) should be: divided by the Stokes velocity uo = 0.392 c m . per second to give the dimensionless form (above) which is plotted in their Figure 9.1 Setting the slope of the line from (1 - e o ) equal to the slope of the Slp,u, us. (1 - e) curve a t the inflection point (1 - e) = 0.339, it is easily found that (1 - e,) is 0.56. Both this value a n d that of 0.59 from Figure 1 (for R e ! = 0.3) are appreciably below the final solids fraction of 0.64, so a concentration gradient was readily observcd (3. 6 ) . Nomenclature
= sphere diameter, c m . = inside diameter of cylinder, cm. n = exponent, dimensionless R e ! = modified Reynolds n u m b e r = uod/u3 dimensionless solids flux, gramslsq. cm.-sec. S’ = solids flux, dimensionless d
literature Cited
(1) Benenati, R. F., Brosilow, C. B., A.I.CI2.E. J . 8, 359 (1962). (2) Coulson. J . M., Richardson, J. F., “Chemical Engineering,” Vol. 2, Chap. 15, Pergamon Press, London, 1955. (3) De Haas, R. D., bl. S. thesis, Purdue University, 1962. (4) Hanratty, T. J., Bandukwala, A , , A.2.Ch.E. J . 3, 293 (1957). (5) Richardson, J. F., Zaki, FV. N., Trans. Inst. Chem. Engrs. 32, 35 (19 5 4) . (6) S h a n n o n ,~ . P. T.. De Haas. R.D.. Stroum. E. P.. Torv. ,, E. M.. IND.ENG.CHEM.’FUSDAUENTALS 3 , 2 5 0 (1964). (7) Shannon, P. T., Stroupe, E., Tory, E. M., Zbid.,2, 203 (1963). (8) Verhoeiren, J., B. Eng. thesis, McMaster University, 1963. (9) Zaki, F$‘. N.,J . Imp. Coll. Chem. Eng. SOC. 7, 119 (1953). \
~
,
~~~
I
~~~
E L M E R M. TORY
Brookhaven ,Vational Laboratory Cyton, L.I., ‘Y.Y.
D
s =
RECEIVED for review October 1, 1964 ACCEPTED October 6, 1964 Work performed under the auspices of the U. S. Atomic Energy Commission.
COM MUNICATION
M E L T I N G A T A POROUS PLATE W I T H SUCTION An exact similarity solution i s given for the steady-state melting rate at the surface o f a heated porous plate, the melt being removed by suction.
ANUMBER of investigations, of both theoretical a n d experimental nature, have been performed recently of condensation or evaporation a t a porous solid surface? with removal of the neivly formed phase by suction ( 7 , 4? 7-9). I n each case a n approximate soliution was sought, based upon various perturbation schemes. O n the other h a n d , various exact. or self-similar, solutions of the Stefan (melting-freezing) problem
exist (2, 3, 6 ) for which, in the absence of inertial effects, the uniqueness of the solution, for given thermal initial a n d boundary conditions, is (correctly) assumed. I n this brief note we call attention to a n exact Stefan solution a t a porous surface with suction? which is nonunique unless the liquid pressure is specified. T h e problem can be very simply stated. Consider a solid of VOL. 4
NO.
1
FEBRUARY
1965
107
