h proportiorial controller with a fast-acting feedback loop could be used to provide adequate control of a metastable condition. Best control is obtained by locating the sensing element a t a position where the reaction rate, rather than the heat transfer rate, is predominant.
GREEK (Y
P Y 0 P 7
Nomenclature
A AL
heat transfer area per unit length, sq. ft./ft.
= heat capacity of mixture, B.t.u./mole O F. GP E = energy of aciivation, Arrhenius law F = molal flow rate, moles/sec. = molal velocity, rnoles/sq. ft. sec. G, (-AH)= heat of reaction, B.t.u./mole k = p exp( - E / R T ) = reaction rate constant, Arrhenius law,
moles/cu. ft. sec.
K, L
PT U
UL X
z
FCP
( - A H ) a / c p = (-AH)/cPG, time, sec. density, moles/cu. ft. = feedback loop time constant, sec. = = =
SUBSCRIPTS
= cross-sectional area, sq. ft. =
= l/Gm, reciprocal molal velocity
- -ULAL
= proportional controller gain
total length, ft. frequency factor, Arrhenius law temperature, O R. velocity, ft./sec. over-all heat transfer coefficient, B.t.u./sq. ft. sec. F. = mole fraction of reactant = length, ft.
= = = = =
e
r S
t
exchanger reactor = shell side = tube side
= =
literature Cited
(1) (2) 3) 4)
Ark, R., Can. J . Chem. Eng. 40, 87 (1962). Aris, R., Amundson, N. R., Chem. Eng. Sci. 7, 121 (1958). Denbigh, K. G., Trans. Faraday Sac. 40, 352 (1944). Douglas, J. M., Eagleton, L. C., IND.ENG.CHEM.,FUNDAMENTALS 1, 116 (1962). (5) Foss, A. S., Chem. Eng. Progr., Symp. Ser. 45, No. 25, 47 (1959). (6) Parts, A. G., Australian J . Chem. 11, 251 (1958). (7) van Heerden, C., Chem. Eng. Sci. 8, 133 (1958). (8) van Heerden, C., Znd. Eng. Chem. 45, 1242 (1953).
I
RECEIVED for review October 26, 1961 ACCEPTEDAugust 10, 1962
BUBBLE DYNAMICS A T THE SURFACE OF A N EXPONENTIALLY HEATED PLATE S. G . BANKOFF Chemical Engineering Department, Northwestern University, Evanston, 111.
The consequences of Zuber’s hypothesis, which states that at every instant the latent heat increase of a bubble growing at a heated surface is equal to the superheat energy of the liquid displaced by the bubble, are evaluated for an exponentially heated plate. For a bubble growing in saturated liquid, the radius increases initially as t3” a result in agreement with Zwick’s relationship for a bubble growing in a liquid with exponential volume heat sources. The effect of plate thickness is also discussed.
N AN EARLIER WORK
( I ) , the heat flux from a solid surface
I to the liquid brought into contact with it as the result of the growth and collapse of a hemispherical bubble was estimated as the integral of a sequence of one-dimensional heat flow problems. Zuber (4) proposed that the superheat content of the liquid layer thus being displaced could be equated, a t every instant, to the gain in latent heat content of the bubble in that instant. Essentially, this is a phenomenological theory, in which it is argued, on physical grounds, that the advancing bubble front cannot displace the heated liquid from its position next to the solid surface, but instead sweeps over it, a thin tongue of evaporating liquid separating a portion of the bubble from the solid surface. The existence of such thin films of liquid at the base of a sessile bubble was first noticed by Derjaguin (2) some years ago, as reported by Frenkel (3). The present work explores the consequences of this hypothesis for bubbles growmg at the surface of an exponentially
heated plate. The problem has bearing on the rapidity of shutdown, and hence the safety, of liquid-cooled nuclear reactors during a power excursion. Zuber’s hypothesis has the novel feature that it focuses attention on the heat flow in the solid, with the result that the bubble growth rates are dependent, at least in rhrory, on the thickness of the solid and its physical properties. Heat Flow in Solid
As a preliminary to the application of Zuber’s hypo thesis, we consider the simple one-dimensional heat flow resulting when a metal slab of thickness 1 is brought in contact with a semi-infinite liquid. Neglecting edge effects, the heat conduction equations may be written
VOL. 1
NO. 4
NOVEMBER 1 9 6 2
257
Uniform Exponential Volume Source
where the subscript 1 refers to the liquid region and the subscript 2 refers to the solid slab; and (r. v , x , and t are the thermal diffusivity, temperature, position variable, and time, respectively. Consider first the case where the two regions are initially a t different temperatures. tvith continuity of flux and temperature a t the interface x = 0, and no heat flux a t the back face x = 1. The conditions for this case are expressed by the equations: v i ( . ,
0) =
GI( m , dU2 -
dX
t)
(-z,
=
0 ; u?(x, 0 ) =
t) = 0
ul(O, t ) = D2(0, t ) ; t
Consider now the same problem as above except that we replace the uniform constant heat source by a uniformly distributed exponential heat source. Equation 2 becomes
where Qo is the initial rate of volume heat release, p and cn are the metal density and specific heat. and A is the reciprocal period. From this. one obtains
(3) (4)
>0
(5)
Proceeding as above, one eventually obtains where k is the thermal conductivity, and u is the initial temperature excess of the solid. Denoting the Laplace transform by the bar superscript, Equations 1 and 2 become where Q represents the expression on the right hand side of Equation 10 and where y . x n ’ > and x,” have been defined in Equation 11, and \There where p is the Laplace transform time variable. O n transforming the boundary conditions, one obtains for the transformed metal temperature Bubble Dynamics from an Exponentially Heated Surface
O n expanding the second term on the right hand side in a n infinite series of exponentials and taking the inverse transform, one obtains
Consider now the problem of bubble growing on a n exponentially heated surface. For convenience. we consider the initial temperature of the liquid to be the saturation temperature. If. in accordance with the Zuber hypothesis ( 4 ) , the quenching heat flux may be equated to the rate of latent heat gain of the bubble. we obtain. for a hemispherical bubble
where K’
y = -
- K. +
K,
x,‘
=
2nl - x; xn” = x
+ 21(n + 1 )
(11)
This converges rapidly for small values oft. Another representation may be obtained from Equation 9 in terms of the inversion integral
where L is the latent heat of vaporization, R is the bubble radius, and p i , is the vapor density. The bubble radius is therefore
O n taking the Laplace transform and on employing Equation 15, one obtains
where
(2)
112
P2
=
The integral has a branch point a t Q = 0. over a suitable cor.tour one obtains
which converges rapidly for large values oft. 258
I ~ E CF U N D A M E N T A L S
Making use of the identity Upon evaluating
where K I / d ( x ) represents the modified Bessel function of the second kind of order l / 4 : and also performing some rearrangements to get the identity
,where
1 exp ( b x
+
ollb2t)
erfc
(2e2
4- adz) (23)
we finally obtain the result
where y n = 2nl
in very rapid transients
limiting Solutions
This solution. although complicated, converges rapidly for small values o f t . Simpler solutions are obtained if the thickness for the metal slab is considered to be large. From Equation 15, if 1 approaches infinity
Dz
=
(-VP + P--) Q‘ ( P --
(1
If the initial value of the slab temperature is the same as that of the water ( V = 0), the bubble initially grows as PI2. This is similar to the predictions of Zuber (4)and Zwick ( 5 ) for bubbles growing in a liquid containing uniform exponential heat sources. No data have been published on bubble dynamics from exponentially heated surfaces. Hence the expressions derived herein must be regarded with considerable reserve until subject to verification. One may note that the Zuber hypothesis ignores the heat losses from the polar rrgions of the bubble, and hence would be expected to be principally applicable during the early stages of bubble growth. Coalescence of neighboring bubbles, which occurs almost instantaneously
-1 - Y 2
e
-fzlXzI
)
(k
200
msec
).
Acknowledgment
(25)
O n differentiating with respect to x and substituting into Equation 19, one obtains after taking Laplace transforms
This work was performed for Research Laboratory, RamoWooldridge Corp. (now Space Technology Laboratories), Canoga Park, Calif. The writer wishes to acknowledge stimulating discussions with Kovak Zuber. Literature Cited
This leads to the result
R =
(1
-
-,)k*t”Z LP,
x
Upon expanding the second term in Equation 26 by means of the binomial theorem, one obtains a solution which is useful for small values of time.
(1) Bankoff, S. G., Jet Propulsion Laboratory, Memo No. 30-8 Pasadena, Calif., 1959. (2) Derjaguin, B., Acta Physicochirn. U.R.S.S. 5 , 1 (1936) ; 10, 333 (1937). (3)‘ Frenkel, J., “Kinetic Theory of Liquids,” p. 332, Dover Publications, New York, 1946. (4) Zuber, N., AEC Rept. TID 6338, January 1960. ( 5 ) Zwick, S. A., P h y . Fluids 3, 685 (1960).
RECEIVED for review March 12, 1962 ACCEPTED August 22, 1962
VOL.
1
NO. 4
NOVEMBER 1 9 6 2
259
