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LaTeX Math Inline
bodyt

Time lapse after the temperature step from 

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bodyT_e(z=0) =0
  up to 
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bodyT_e(z=0) =T_f

LaTeX Math Inline
bodyz

Spatial coordinate along the transversal direction to constant temperature 

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bodyT_e(z)= T_f
plane 
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bodyz=0

LaTeX Math Inline
bodyz_f

TVDss of the top of the lateral flow unit

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bodyh_f

True vertical thickness of the the lateral flow unit

LaTeX Math Inline
bodyT_f

Boundary temperature at 

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bodyz=0

LaTeX Math Inline
bodya_e

Thermal diffusivity of the surroundings

LaTeX Math Inline
bodyT_G(z)

Geothermal Temperature Profile

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Driving equationInitial conditions Boundary conditions


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\frac{\partial T_e}{\partial t} = a^2a_e^2 \Delta T_e = a^2a_e^2\frac{\partial^2 T_e}{\partial z^2}



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T_e(t=0, z) = T_G(z)



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T_e(t, z_f \leq z \leq z_f + h_f) = T_f = {\rm const}


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T_e(t, z \rightarrow \infty) = T_G(z)


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\mbox{if} \, z < z_f \; \Longrightarrow \;T_e(t,z) = T_f + (T_G(z) - T_f) \cdot \mbox{erf} \left( \frac{z_f-z}{\sqrt{4at4 a_e t}} \right)



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\mbox{if} \, z_f \leq z \leq z_f + h_f  \; \Longrightarrow \; T_e(t,z) = T_f



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\mbox{if} \, z > z_f + h_f  \; \Longrightarrow \; T_e(t,z) = T_f + (T_G(z) - T_f) \cdot \mbox{erf} \left( \frac{z-z_f-h_f}{\sqrt{4at4 a_e t}} \right)


where

LaTeX Math Inline
body--uriencoded--\mbox%7Berf%7D(x)

Error function

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