Boundary Conditions
When a component is subjected to an environment of different carbon concentration, carbon can diffuse into or out from its surface. This mass transfer can be modeled in two ways:
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A prescribed carbon flux (a Neumann condition): Carbon mass transfer is modeled using a convection boundary condition
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A prescribed carbon concentration (a Dirichlet condition): Carbon mass transfer is enabled by means of a prescribed value on the surface
A special case of prescribed carbon flux is the case of zero carbon flux.
Prescribed Carbon Flux
The convective mass transfer of carbon from the environment into a surface with outward normal n is given by
where b is the mass transfer coefficient, and cext is the carbon concentration of the exterior. A natural way to define the carbon concentration of the exterior is to take it as equal to the carbon potential cpot of the surrounding carburizing atmosphere. The mass transfer coefficient can be specified directly, or defined as thermally activated
where b0 is a pre-exponential factor and Qb is an activation energy. In a given situation, these quantities would have to be estimated or measured experimentally.
Prescribed Carbon Concentration
A simple way to model the carbon exchange is to simply prescribe the carbon concentration at the surface of the component to be equal to the carbon concentration of the exterior. This modeling approach avoids having to characterize the carbon mass transfer, but it may instead exaggerate it because the surface will be saturated.