(17) Ibid., 35, 1 1 (1963). (18) Kelley. H. J . , in “Optimization Techniques \vith Applications to hero-spare Systems.” G . Leitmann. ed.. Academic Press. h-ew York, 1962. (19) L.W, E. s..I N D . F,NG. C Y E \ t . F U S D . A M E S T A L S 3, 373 (1964). York University, 1945-46; supplement by H. Rubin and M. (20) O+ztoreli. M. S . .SI.4.M J . Control 1, 290 (1963). Kruskal. 1949-50. (21) Pontryagin. I>. S.. Boltyanskii. V. A . Gamkrelidze. R. V., Mishchenko. 1:. F.. .‘The Mathematical Theory of Optimal (8) Courant, R.. Hilbert. D., “Methods of Mathematical Physics,” Vol. I, Interscience. New York, 1953. Processes.” \Viley. New York. 1962. (9) Denham. \V. F . . Bryson. A. E., A.I.A..4. J . 2, 25 (1964). (22) Sarrus, F . , Compt. Rrnd. 25, 726 (1848). (23) Tompkins. C . B.. i n .‘Modern Mathematics for the Engi(10) Denn, M. M.. Aris, R., A.I.Ch.E. J . , in press. (1 1) Denn: M. M . . Aris. R.. IND.ENG. CHEM.FUNDAMENTALS, neer,” E. F. Beckenbach. ed.. 1st series, McGraiv-Hill. Ke\v 4, 7 (1965). York, 1956. (24) \Tilde. D. J., “Optimum Seeking Methods.” Prentice-Hall. (1 2) Dreyfus, S. F,., “Variational Problems \vith State Variable Inequality Constraints.” RAND Rept. P-2605-1 (.\ugust 1963). Englewood Cliffs, N. J., 1964. (13) Fletcher, R., Powell, M. J . D., Computw J . 6 , 163 (1963). RECEIVED for review August 26. 1964 (14) Ho. Y.-C., Brentani, P. B., SI‘4.M J . Control 1 , 319 (1963). ACCEPTED December 28. I964 (15) Horn. F., Z. Elektrochpm. 65, 244 (1961). This is thr second article of a three-part series. (16) Horn, F., Troltenier, U.: Chem. Ing. Trch. 32, 382 (1960). (4) Bryson, A. E., Denham, I T . F.. Dreyfus, S. E.: A . I . A . A . J . 1, 2544 (1963). (5) Cauchy, X., Compt. Rend. 25, 536 (1847). (6) Courant, R.. Bull. Am. Math. SOC.49, 1 (1943). (7) Courant. R.. .‘Lectures on Calculus of Variations.“ Ne\\
CO M MUN ICAT ION
EXPERIMENTAL NATURAL CONVECTION HEAT TRANSFER FROM WIRES T O T H E NONEQUILIBRIUM CHEMICALLY REACT1NG SYSTEM: NO2-NO-02 Natural convection heat transfer measurements to the nonequilibrium gaseous N02-NO-02 system are presented. Electrically heated horizontal platinum wires from 1 to 10 mils in diameter were immersed in the gas maintained at bulk temperatures from 200” to 600” C. Temperature driving forces of the order of hundreds of degrees are required to cause the increase in heat transfer owing to energy of reaction. The problem of correlating heat transfer rates in nonequilibrium reacting systems i s presented.
PROBLEM of heat transfer in nonequilibrium systems has been theoretically treated for the case where the temperature driving force is small enough that reaction rate constants may be considered to be irdependent of temperature ( 7 . 2 ) . HoLvever. in practical heat transfer devices this condition does not often hold. For the pure conduction case, an analogcomputer technique is available for solving the equations of change (6) for a large temperature driving force. T h e system: THE
2NOn
2N0
+
0 2
(1)
presents a convenient model for study of heat transfer in nonequilibrium systems in that data for thermodynamic and transport properties (5) as well as for chemical kinetics ( 9 ) are available and thermodynamically the reaction goes to rompletion below about l O O O @ C.. a reasonably low temperature for rxperimental study. Experimental
S a t u r a l convection heat transfer rates were measured from horizontal platinum \vires to the NOr-NO-O:! gas system a i bulk temperatures from 2OO@ to GOO@ C. \Vires of 1. 2. 5. and 10 mils diameter were sealed into a quartz vessel 6 inches long and 3 inches in diameter. T h e evacuated vessel \vas loadrd \vith purified NO? gas ( N 2 0 a - N O ? equilibrium mixture) c.;timatrd to contain less than 0.1 \vt, Yc irnpirity. T h r quartz vessel \vas immersed in a constant temperatlire furnace 222
l&EC FUNDAMENTALS
maintained to lvithin 0.j70 of the control temperature. and equilibrium in accordance Ivith Equation 1 was allo\ved to be established. T h e platinum \vires were electrically heated, the temperature of the \vires being measured by standard resistance thermometry techniques and the energy dissipated from the \vires being determined electrically. Results and Discussion
I n equilibrium reacting systems, thermal properties-such as the thermal conductivity. specific heat capacity. and thermal expansion coefficient-are nonmonotonic functions of temperature and are large compared \vith the thermal properties of nonreacting systems o\ving to the diffusion of species transporting the energy of reaction. I n nonequilibrium systems the maximum value of thermal properties of the equilibrium state \vi11 be realized when the ratio of the diffusion time for the given geometry to the reaction time is large; the minimum or so-called frozen-equilibrium values of thermal properties occur \vhen the ratio of thr diffusion time to the reaction time (the nainkohler number) is small ( 7 , 7.6). It is interesting for the small-\vire geometry in the NOaN O - 0 , system that rrlatively large temperature driving forcrs (large wirr temperatures) are required to reveal the effcct of chrmical reaction on heat transfer. Figure 1 shows. from Fan (5). the equilibrium degree of advancement of the reaction of Equation 1 as a function of 3 7 for several bulk
/
0.6b
I
/
04
Figure 1 . Equilibrium degree of advancement of the reaction 2N02 $ 2 N 0 O2 as a function of temperature a t 1 atm.
+
temperatures a t 1 a t m . For reactions of this type stoichiometry a t equilibrium, the maximum in the thermal conductivity occurs around a value of te of 0.5 ( 4 ) . Based on a simple film model, the maximum heat flux density would be expected to occur when the fluid in the vicinity of the solid body is near this state. Experimental d a t a for turbulent heat transfer in the equilibrium ND-NO:! gas system in internal and external flows confirm this behavior. I n a nonequilibrium system, a higher film temperature is needed than for equilibrium systems to reveal the reaction effect on heat transfer. For natural convection from fine uires to the N02-NO-02 gas system a remarkably high degree of overheating is required for the reaction effect to appear. Figure 2 summarizes data for heat transfer from a 5-mil wire with bulk gas temperatures ranging from 200' to GOO' C., the reaction effect being evidenced by the inflection point in the curves. Data for air with a bulk temperature of 400' C. are included for comparison. At a
bulk temperature of 200' C . , a maximum reacting effect in an equilibrium system would be expected at a AT of about 220' C. (Figure 1). whereas no effect occurs a t a value of A T as high as 500' C. Similarly. at bulk temperatures of 350°, 400°, and 500' C., the effect for an equilibrium system would appear a t ATof 70°, 20', and near 0' C . ; whereas for this nonequilibrium system the effect appears at A T ' S of about 450, 400, and 200' C.. respectively. This behavior indicates that the ratio of diffusion time to reaction time tends to be small near the state where the maximum equilibrium value of thermal conductivity occurs. For engineering purposes, it would be desirable to be able to correlate the data of Figure 2 in a conventional manner of a Nusselt number us. Grashof-Prandtl number plot. I n Figure 3, data at T , = 400' C. for 2-, 5-, and IO-mil wires have been plotted for two extreme conditions : assuming frozen-equilibrium a t the state of T , and assuming complete equilibrium throughout the film. Data for air with several wire diameters as well as conventional correlations (3: 7) are included for comparison. I n the Nusselt number, specific enthalpy has been used as a driving force instead of temperature the advantage of which has been previously discussed (8). Figure .3 shows that no satisfactory correlation is obtained over the complete range of A T's studied. T h e equilibrium assumption gives too low values of (.Vxu')m (owing to too large a value of H , being assumed) except when .V,, (and thus T,) becomes large enough for high reaction rates to occur and equilibrium to be approached. Conversely, with the frozen-equilibrium assumption, the correlation is good until NGr(and T u ) becomes large enough that the equilibrium state is approached in the film; the data points then deviate in a direction above the correlation curves. To obtain a good correlation over the whole range of dissociation the state of the film must be predicted and the effective thermal conductivity of the film obtained from solutions of the equations of change as a function of the ratio of diffusion to reaction times. Conclusions
Because of the very large values of AT required to show the reacting effect in this system a linearized theory (2) where
L
A
l
-2
-3
-4
~O~~o~NarNPr)m I
I
200
300 61(.K or .C)
Figure 2. Natural convection heat transfer from a 5-mil wire to the N02-NO-02 gas system a t various bulk temperatures 0 ( 2 0 0 ' C.) CD (350' C.) 0 (400' C.) (600' C.) A Air (400' C.) (500' C.)
V
.
Figure 3. Correlation of natural convection heat transfer to the N02-NO-02 gas system, bulk temperature of 400" C. 2-mil 5-mil 1 0-mil 2, 5, 1 0-mil
A-frozen equilibrium 0,V-frozen equilibrium &frozen equilibrium .-Air Data
- _ _ (7) _
VOL. 4
A-equilibrium 3-equilibrium @-equilibrium
-(3) NO. 2
M A Y
1965
223
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 d a t a 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
