Stefan condition

This type of relationship, accounting for the flux into and out of the interface, is generally known as a Stefan condition [1]. The Stefan condition introduces a new variable, the interface position, and one new equation. 

To deal with this more complex problem, we follow Sekerka et al. [2] and Sekerka and Wang [3] and first establish a general analysis that allows for these changes of volume. The previous melting problem was solved by first obtaining independent solutions to the diffusion equation in each phase and then coupling them via the Stefan flux condition at the interface, similar approach can be employed for the present problem. To accomplish this, it is necessary to identify suitable frames for analyzing the diffusion in each phase and then to find the relations between them necessary to construct the Stefan condition. 

Stefan Condition at an a/(3 Interface. Consider the interface between the moving a and (3 phases shown in Fig. 20.26. The a and 3 phases (along with their l -frames) will be bodily displaced with respect to each other, and the Stefan condition can... 

Stefan Condition at a Free Surface. Commonly, a component (i.e., B) with a high vapor pressure is diffused into the free surface of a (3 phase. Component A has a much lower vapor pressure and does not evaporate from the surface. The Stefan condition at the free surface is then... 

First, solutions of the diffusion equation in the a and j3 phases (in the F - and F -frame, respectively) are found that match the boundary and initial conditions and then the Stefan condition is invoked. The solutions are of the error-function type and are given by... 

In this example, the equilibrium concentrations maintained at the interface are functions of the interface temperature, which in turn is a function of time. In addition, the velocity of the interface, v (i.e., rate of solidification), depends simultaneously upon the mass diffusion rates and the rates of heat conduction in the two phases, as may be seen by examining the two Stefan conditions that apply at the interface. For the mass flow the condition is... 

A complete general solution therefore requires solving for temperature and concentration fields in both phases which satisfy all boundary conditions as well as the two coupled Stefan conditions. Solving this problem is a challenging task [4, 5] however, an analysis of the concentration spike under certain simplifying conditions when v is known is given in Section 22.1.1. 

Next, t)r is determined in the usual way by invoking the Stefan condition at the interface, which has the form... 

As a first approximation, it is assumed that the radial flux entering the edge is constant over the cylindrical interface. The Stefan condition, integrated over the interface, is then... 

The Stefan conditions at interfaces 1 and 2 are given by Eqs. 20.12 and 20.17, respectively. Substituting the appropriate relationships from those given above into these two equations then yields equations that can be solved simultaneously for Ai and A2 ... 

We now determine rjx by solving for c% and invoking the Stefan condition at the interface. The diffusion equation was scaled and integrated in Cartesian coordinates in Section 4.2,2 with the solution given by Eq. 4,28. When this solution is matched to the present boundary conditions,... 

The velocity at any point uy(6) is determined by the rate of melting at the interface (Fig. 6.67), which is obtained from the Stefan condition or heat balance between conduction and the rate of melting at that interface,... 

The coordinate xs has its origin at the external surface when melting started, and it is stationary. In addition to the boundary conditions just given, we can write a heat balance for the interface (this is occasionally referred to as the Stefan condition). 

From here we get the Stefan condition. This is the equation for pit), p < 0 ... 

This expression allows to exclude A from (7) and the Stefan condition. As a result we obtain the system of differential equations p = f p,B), B = f2(p,B),... 

It is considered that the fluid front moves with the same speed v/, as the fluid particles v(x) at the front do (Stefan condition)... 

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