...
Outputs
...
Inputs
...
| Time lapse after the temperature step from up to |
| Spatial coordinate along the transversal direction to constant temperature plane |
| |
| Thermal diffusivity of the surroundings |
Equations
...
Driving equation | Initial conditions | Boundary conditions |
---|
LaTeX Math Block |
---|
| \frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2} |
|
LaTeX Math Block |
---|
| T(t=0, z) = T_G(z) |
|
LaTeX Math Block |
---|
| T(t, z=0) = T_f = {\rm const}, \quad T(t, z \rightarrow \infty) = T_G(z) |
|
Solution
...
LaTeX Math Block |
---|
| T(t,z) = T_f + (T_G(z) - T_f) \cdot \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi |
|
Approximations
...
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \zeta = \frac%7Bz%7D%7B4 a t%7D \sim 0 |
---|
|
|
LaTeX Math Block |
---|
| T(t,z) = T_f + (T_G(z) - T_f) \cdot \Bigg[ 1- \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg) \Bigg] |
|
See also
Heat flow equation for Semispace Linear Conduction:
...
Physics / Fluid Dynamics / Linear Fluid Flow / Temperature Flat Source Solution @model
References
...
...