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