AT was assumed to approach zero becomes invalid. A digital and analog computer solution of the stiff-equations has been carried out, and attempts are being made satisfactorily to correlate the natural convection data combining dimensional analysis with the computer solutions of the equations of change employing a film theory model.
V
= specific volume of gas
8
= = = =
E
coefficient of volume expansion, (1 1 V ) (dV,’aT), absolute viscosity ’ kinematic viscosity degree of advancement of reaction
literature Cited Nomenclature
A
=
CP
=
D
=
g
= =
H, H,.
=
k
=
.YGr,
=
(aV,u’)m =
circumferential area of wire isobaric specific heat capacity wire diameter acceleration owing to gravity specific enthalpy of bulk gas (equi1ibriu.m) specific enthalpy a t Lvire temperature (equilibrium o r frozen) thermal conductivitv dimensionless Grashof number with properties a t T,, (D3q 8.17) 2 (Q .4)0 ( H E - H,)(kc ~ =) dimensionless ~ Nusselt number with k and C, evaluated a t H, H,/
2
.V%
Q
AT
T1 TlL
+
dimensionless Prandtl number, ( C p p k ) = , with properties a t T, = heat flux = temperature driving force ( T , - T,) = bulk temperature of gas = tempeiature of wire =
(1) Brian, P. L. T., A.2.Ch.E. J . 831 (1963). (2) Brokaw, R. S.,J . Chem. Phys. 35, 1569 (1961). (3) Collis, D. C., LVilliams, H. J., “Free Convection of Heat from Fine \Vires.” Aero. Res. Lab. of Australia, Aero Note 140, 1954. (4) Fan, S. S. T.. Mason, D. M., Am. Rocket SOC. 32, 899 (1962). (5) Fan, S. S. T., Mason) D. M., J . Chem. Eng. Data 7, 183 (1962). (6) S. S. T.. Rozsa. R. B., Mason, D. M., Chem. Eng. Sci. 18, 73/ (1963). ( 7 ) Mc.%dams, I V . H., “Heat Transmission,” 2nd ed., Chap. 11, D. 237. McGraw-Hill. S e w York. 1942. Richardson. J L , Boynton, F p . Eng, K y , Mason, D M , ChPm En! J c i 13, 130 (1961) (9) Kosser, \$ A , \Vise, H , J Chem Phys 24, 493 (1956).
9”.
G O R D O N R BOPPl D A V I D M MASON
Stanford C‘ntz)erszty Stanford. Calif
Present address, University of Idaho, Moscow, Idaho RECEIVED for review March 30, 1964 ACCEPTED December 18, 1964
COM MUN I CATION
RISIN’G VELOCITY OF A S W A R M OF SPHERICAL BUBBLES An expression is proposed relating the velocity of rise of a swarm of spherical bubbles to the velocity of a single bubble. The analysis, based on a cellular spherical model, is restricted to the range of high but subcritical Reynolds numbers.
velocity of sedimentation of multiparticle systems has been the object of numerous investigations, both experimental and theoretical. Some authors (2. 3. 7 7 : 72) have treated the problem theoretically on the basis of a “cellular” model: Each particle is supposed to occupy, at any given time, the center of a geometrically regular “cell” of liquid, all particles and cells being equal in size and physical properties. All theoretical analyses based on a cellular model are relative to the motion of spherical particles in the low range of Reynolds numbers (creeping flo\v) ; particularly successful seems to be the result of Happel ( 3 ) .who assumed a “free surface‘’ condition on the external boundary of the assumed spherical cell. Investigation of the motion of a swarm of bubbles may also be afforded using a spherical cellular model; the Reynolds number, defined on the bubble diameter, is supposed to be high but subcritical (1 and 3 represent, respectively, the axial, radial, and angular directions in steady axial flow of a viscoelastic fluid in a concentric annulus, the radial component of the equations of motion, integrated between the bounding radii, is
122(RO) -
dRi) = -
(P22
- P83)dh
7
(1)
+
Here, T~~ = -p J i j P,, and other notation is clear. Therefore: by measuring the total radial stress a t the inner and outer walls at any point along the flow as a function of shear rate, values of the desired quantity, P22 - P33, as a function of shear rate, can be inferred. VOL. 4
NO. 2
M A Y 1965
225
