This theoretical paper determines the effect of the propagation velocity of heat on the temperature and heat-flux distribution in a semi-infinite body due to a step change in temperature at the surface. The solution yields a maximum but finite heat flux under the conditions of a step change. This is contrary to the infinite value predicted by the error function solution to the Fourier transient conduction equation. In addition, assuming convection is conduction limited, an upper limit for convective heat transfer coefficients is postulated.

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Analytical solutions to one-dimensional melting-freezing problems in which the densities of the two phases are constant, but different from each other, are limited to a few special cases, nearly all of which involve a constant wall temperature boundary condition. Much of this work is reviewed by Carslaw and Jaeger (1). Various approximative schemes have been employed. One of the more common is the pseudo-steady state assumption, which implies that the motion of the phase boundary is so slow that the diffusion process is essentially at steady state (2, 3 ) . Another approach uses a polynomial approximation for the temperature distribution, in a manner similar to the wellknown Pohlhausen integral method for laminar boundary layers ( 4 ) . Because of the inherent nonlinearity due to the moving boundary, no general solution for arbitrary boundary conditions exists. Zener ( 2 ) and also Kirkaldy ( 3 ) employed a dimensionless distance X = x/S, where x is the distance from the wall, and 8 is the thickness of the new phase, in discussing the pseudo-steady state approximation. The same variable is used here, but the authors now convert the differential equation and associated boundary conditions into integral equation form in order to establish upper and lower bounds for the rate of phase change with arbitrary monotonic wall temperature or wall flux. STATEMENT OF THE PROBLEMConsider a one-dimensional geometry in which the original phase (liquid or solid) at the fusion temperature is initially in contact with a flat plate, taken at x = 0. At time t = 0 a specified temperature T , or heat flux F o is applied at the plate, initiating the melting or freezing process. The thickness of the newly formed layer S ( t ) is an unknown function which represents the desired solution. Since the original phase remains at the fusion temperature Ts, heat conduction in the new phase only has to be considered. The physical properties of this phase, including density, are assumed to be temperature independent. The heat equation for this phase may then be written:Suppose now the wall temperature is a monotonic, but otherwise arbitrary, function of time. For definiteness consider the melting process. The initial and boundary conditions are then T ( 0 , t ) = T w ( t ) ( 2 ) (4)The last equation expresses an energy conservation re-

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