Applications of Dirichlet boundary problem
PMiKNoM seminars - class 6
Mass transfer (diffusion)
Derivation of second Fick's Law
- Mass balance equation reads:
$$ \frac{\partial c_i}{\partial t} = - \frac{\partial J_i}{\partial x} $$where `c_i` is the concentration of the `i`-th ion/molecule, `J_i` corresponds to its flux; `x` and `t` denote space and time, respectively.
- First Fick's Law reads:
$$ J_i = - D_i \frac{\partial c_i}{\partial x} $$where `D_i` is the diffusion coefficient of the `i`-th ion/molecule.
- After inserting the first Fick's Law into the mass balance equation one obtains:
$$ \frac{\partial c_i}{\partial t} = - \frac{\partial (- D_i \frac{\partial c_i}{\partial x}))}{\partial x} $$
- Assuming that the diffusion coefficient is independent of space, i.e. `D_i = const`, we obtain the second Fick's Law:
$$ \frac{\partial c_i}{\partial t} = D_i \frac{\partial^2 c_i}{\partial x^2} $$
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:
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:
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:
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.