Temperature wave

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The basement is considered to be a half-infinite and homogeneous medium, with \(K\) its thermal conductivity, \(\rho\) its density, \(c\) its mass thermal capacity and located in the half-space \(x>0\).

We suppose that the temperature of the ground (\(x = 0\)) is subjected to sinusoidal variations :

\({T_s}(t) = {T_0} + {\theta _0}\cos \omega t\)

Question

Determine the temperature \(T(x,t)\) at the depth \(x\) (use the complex notations ) in stationary mode.

Indice

  • Solve the heat equation using the complex numbers method.

  • This exercise deals with the equivalent of the skin effect in the domain of electromagnetism.

Solution

The heat equation is :

\(D\frac{{\partial ^2 T(x,t)}}{{\partial x^2 }} = \frac{{\partial T(x,t)}}{{\partial t}}\)

We use the complex numbers method and we have :

\(\underline T (x,t)=T_0+\theta_m exp (i(\omega t - \underline k x))\)

The heat equation leads to :

\(-D \underline k ^2=i\omega \)

That is to say :

\(\underline k ^2= e^{-\frac{\pi}{2}}\omega/D \)

Therefore :

\(\underline k = \pm e^{-\frac{\pi}{4}}\sqrt{\omega /D}=\)\( \pm \sqrt{\omega /2D}\;(1-i)\)

We note :(skin thickness)

\(\delta = \sqrt {\frac{2D}{\omega}}\)

We go back to real numbers, and we keep only the solution that does not diverge in the infinite :

\(T(x,t) = T_0 + \theta _0 \;e^{ - \frac{x}{{\delta }}} \cos \left( {\omega t - \frac{x}{{\delta }}} \right)\)

Question

Calculate the velocity of the thermal wave that has been obtained.

Solution

The velocity of the wave is :

\(v = \delta\omega = \sqrt {2D\omega }\)

Question

We consider daily temperature changes, the one on the floor varying between \(0°C\) at night and \(16°C\) during the day.

From which depth are the temperature changes less than \(1°C\) ?

Calculate \(v\).

We give :

\(D = \frac{K}{{\rho c}} = {6.10^{ - 7}}\;{m^2}.{s^{ - 1}}\)

Let us consider annual temperature changes, from\( – 10°C\) to \(26°C\). Answer the same questions.

Solution

First case : \(x = 26,7 \;cm\) and \(v = 80,7\; cm.s^{–1}\).

second case : \(x = 7,1\; m\) and \(v = 4,2 \;cm.s^{-1}\) :

The temperature in an inter cave is fresh in summer and mild in winter.

Indeed, at a \(4,2\; m\) depth, the evolution of the temperature is the same than outside \(100\) days late.