Heat transfer and diffusion of particles
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A detailed solution is then proposed to you.
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The basement is considered to be a half-infinite and homogeneous medium, with its thermal conductivity, its density, its mass thermal capacity and located in the half-space .
We suppose that the temperature of the ground ( ) is subjected to sinusoidal variations :
Determine the temperature at the depth (use the complex notations ) in stationary mode.
The heat equation is :
We use the complex numbers method and we have :
The heat equation leads to :
That is to say :
We note :(skin thickness)
We go back to real numbers, and we keep only the solution that does not diverge in the infinite :
Calculate the velocity of the thermal wave that has been obtained.
The velocity of the wave is :
We consider daily temperature changes, the one on the floor varying between at night and during the day.
From which depth are the temperature changes less than ?
We give :
Let us consider annual temperature changes, from to . Answer the same questions.
First case : and .
second case : and :
The temperature in an inter cave is fresh in summer and mild in winter.
Indeed, at a depth, the evolution of the temperature is the same than outside days late.