THE UNIVERSITY OF CONNECTICUT SCHOOL OF ENGINEERING 11E MOTION OF A PLANE EVAPORATION FRONT IN A SUPERHEATED LIQUID R. S. BRAND

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1 THE UNIVERSITY OF CONNECTICUT Q SCHOOL OF ENGINEERING q 7 TECHNICAL REPORT NO. 2 11E MOTION OF A PLANE EVAPORATION FRONT IN A SUPERHEATED LIQUID by R. S. BRAND MECHANICAL ENGINEERING DEPARTMENT This research was carried out under the Bureau of Ships Fundamental Hydromechanics Research Program S-RO , administered by the David Taylor Model Basin Contract Nonr 3292 (01) Reproduction in whole or in part is permitted for any purpose of the United States Government DDC STOERS, CONNECTICUT MA' APRIL, 1963 xl'sia

2 The Motion of a Plane Evaporation Front in a Superheated Liquid R. S, Brand Professor of Mechanical Engineering University of Connecticut Storrs, Connecticut ABSTRACT The velocity of the interface between liquid and vapor phases when a slightly superheated liquid changes to vapor is found. The temperature distribution in the liquid ahead of the evaporation front can then be determined by a numerical integration.

3 2 NOMENCLATURE A = length scale factor defined in equation (20) b M slope, dp/dt, of the saturation curve C - speed of sound in the liquid g(t) - position (i.e. x co-ordinate) of interface at time t g'(t) - velocity of interface k - thermal conductivity L - latent heat of efaporation m - mass rate of evaporation per unit area p - pressure t T % time w temperature u - fluid velocity x - space co-ordinate y - x-g(t) = space co-ordinate measured from interface 0 - thermal diffusivity 77 - y/a - dimensionless distance from interface 9 - dimensionless temperature defined in equation (9) f - density of liquid F - density of vapor - 0( t/a 2 - dimensionless time - dimensionless velocity of interface defined in equation (14) SUBSCRIPTS: o, refers to conditions at interface, liquid side a, refers to equilibrium saturation condition,l

4 The Motion of a Plan. Evaporation Front in a Superheated Liquid R. S. Brand Professor of Mechanical Engineering University of Connecticut Storrs, Connecticut INTRODUCTION This study is an attempt to solve the problem of the motion of a plano front in a fluid medium across which a change of phase t Okes place, In particular, we assume that an infinitely long tube is filled with a liquid at a uniform pressure, p(oo), and at a uniform temperature, T(00), slightly above the saturation temperature corresponding to p(oo). The liquid is thus in an unstable state. We assume that at some point in the tube (xxo) some vapor is formed, and that we then have an evaporation front moving into the superheated liquid. As fluid crosses this front it is converted into a vapor at equilibrium saturation conditions. The energy to supply the latent heat of evaporation is assumed to come by conduction through the liquid. Hence the liquid will be cooler at the interface than at infinity. We let x - g(t) be the location of the front and g'(t) its velocity. The determination of this function, g'(t), is the principal object of the investigation.

5 4 ANALYSIS The temperature in the liquid must satisfy the one-dimensional diffusion equation in a moving substance, 'AT + WT bzt S+ C L (1) in the region x-g(t), t2:o, subject to the boundary conditions: xi1)r: - '(2) T@,O = Too)(3) T(0, T(o) (t0) In boundary condition (2), which expresses the fact that the latent heat of evaporation is supplied by conduction through the liquid, the functions g(t) and m(t) are unknown. If we assume that the moving front converts the liquid into stationary vapor, then from which 4o 0 = -- (6) At 212 F, 7.5 x 10 hence it is permissible in considering the effect of motion of the liquid to take 't 0s (7) The curved boundary x - g(t) can be removed by choosing a co-ordinate measured from the interface. That is, letting y - x-g(t), equation (1) becomes 6 r at t [ - (b)

6 The temperature gradient will be large only near the boundary y - 0, and here [u - g'(t9 is small because of (7). Hence the convective term in (8) is neglected. The problem is reformulated in dimensionless quantities by writing.l ' of t T--TOO) A- e TO " )- (0) (9) where T 5 to p(oo). (oo) is the equilibrium saturation temperature corresponding We then have the standard diffusion equation?i fe (10) (the subscripts denote partial differentiation) to be solved in the region 7/ & 0, r2! 0, with the boundary conditions: (- 1o) 0 (1) e(do., 0) o(12) The boundary function v(z) in (13), still to be determined, is the dimensionless form of the velocity of the interface, g'(t), as is seen from (2), (5), and (9). (13) The boundary value problem, equations (10), (11), (12), (13), is most conveniently solved by applying the Fourier cosine transform on the variable 7. Lot the transform be denoted by Aj r 0 r)co's

7 For the boundary oonditions of the present problem the partial differential equation (10) transforms to Of 2 with the solution d" = - i (r) M a e -x M- (S) 1 (15) It can be shown that the inversion of (15) is o(or=-}je $(s)('r-s- O's (16) giving the temperature distribution in terms of the motion of the interface. From (16) we find the temperature at the interface to be, - - -lf d (17) Since thermodynamic equilibrium is established at the evaporation front, the boundary temperature, T (0 ) - T(O,ZV), given by (17), will determine the pressure at the interface p 0 (Z'). If the range of temperature variation is small enough, we can take the relationship between pressure and temperature at saturation to be linear. That is, W p(oo) + b (T(i) We now assume that the evaporation front moving into the originally motionless liquid sends a wave into this liquid in which the fluid velocity and the pressure are related as in an acoustic medium. p(xt) - p(oo) In particular, at x + fcu(x,t) g(t) _ (I) + L A )(19)

8 If we now choose our length scale A to be A - pc K (20) equating (18) and (19) provides the following integral equation for the motion of the interface: 0 r -I-;if 2--s d (21) Equation (21) is solved by an iterative technique in which with " 1, the iteration can be carried to any order with only elementary integrals being encountered. The result is the following power series: *~~~' t1t#..... After some manipulation the series can be recognised as the expansion of r ( ') (22) where erfc denotes the complimentary error function defined by & Z c.rfc (Z) 1- erf (.F) - - 2,I RESULTS Fig. 1 is a graph of equation (22). Fig. 2 shows some typical temperature distributions obtained by numerical evaluation of the integral in equation (16).

9 LU SL 0 L-

10 0 O_ ze 0e Ac

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