Solute segregation temperature fields

These measures of solute segregation are closely related to the spatial and temporal patterns of the flow in the melt. Most of the theories that will be discussed are appropriate for laminar convection of varying strength and spatial structure. Intense laminar convection is rarely seen in the low-Prandtl-number melts typical of semiconductor materials. Instead, nonlinear flow transitions usually lead to time-periodic and chaotic fluctuations in the velocity and temperature fields and induce melting and accelerated crystal growth on the typically short time scale (order of 1 s) of the fluctuations. 

In equations 5-8, the variables and symbols are defined as follows p0 is reference mass density, v is dimensional velocity field vector, p is dimensional pressure field vector, x is Newtonian viscosity of the melt, g is acceleration due to gravity, T is dimensional temperature, tT is the reference temperature, c is dimensional concentration, c0 is far-field level of concentration, e, is a unit vector in the direction of the z axis, Fb is a dimensional applied body force field, V is the gradient operator, v(x, t) is the velocity vector field, p(x, t) is the pressure field, jl is the fluid viscosity, am is the thermal diffiisivity of the melt, and D is the solute diffiisivity in the melt. The vector Fb is a body force imposed on the melt in addition to gravity. The body force caused by an imposed magnetic field B(x, t) is the Lorentz force, Fb = ac(v X v X B). The effect of this field on convection and segregation is discussed in a later section. 

Formation kinetics will depend upon the concentrations present in the vapor phase, properties of the encapsulated elements or compounds (such as carbon solubility and carbide formation), buffer gas conditions that controls cooling rates and the possibility of independent carbon condensation (polyaromatic carbon), and other process variables such as temperature and pressure. Catalytic effects can be present not only in the segregation of carbon in solution but also in the kinetic pathways of independent carbon species formed. Magnetic and electric fields can interact in these processes through the presence of paramagnetic (free radicals) and polar species. 

A high fidelity model of a fuel cell may include many dependent variables such as concentration, temperature, velocity, potential, pressure, and other fields described in the model. For this reasOTi, a segregation of the solution procedure is required by solving a smaller set of equations fuUy coupled and then iterating over all possible smaller sets in one nonlinear iteration. To obtain faster convergence, variables such as the potential in the electrodes and in the electrolyte should then be solved in the same set, due to their tight coupling and presence in the... 

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