Applications of Dirichlet boundary problem

PMiKNoM seminars - class 6

Mass transfer (diffusion)

Derivation of second Fick's Law
Second Ficks Law in steady state

Steady state (stationary state) represents a situation when the ionic concentrations do not change in time, hence `{\partial c_i} / {\partial t} = 0`. The Fick's Law in steady state takes the form:

$$ D_i \frac{\partial^2 c_i}{\partial x^2} = 0$$

As one my notice it is a special case of the problem solved in class 5. Solving the general problem with `\alpha (x) = -D_i`, `\beta (x) = 0`, `\gamma (x) = 0` and `f(x) = 0` and known concentrations at the boundaries, allows obtaining the steady state concentration profiles.


Heat transfer

Heat equation

The heat equation describes the flow of heat in a homogeneous and isotropic medium:

$$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} $$
where `T` is the temperature of the medium, `\alpha` is known as thermal diffusivity; `x` and `t` denote space and time, respectively.

Temperature profile in steady state

Steady state (stationary state) represents a situation when the temperature does not change in time, hence `{\partial T} / {\partial t} = 0`. The heat equation in steady state takes the form:

$$ \alpha \frac{\partial^2 T}{\partial x^2} = 0$$

As one my notice it is a special case of the problem solved in class 5. Solving the general problem with `\alpha (x) = -\alpha`, `\beta (x) = 0`, `\gamma (x) = 0` and `f(x) = 0` and known temberature at the boundaries, allows obtaining the steady state temperature profiles.

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