Chapter 4

# Thermal resistance

### Fondamental : Definition of the thermal resistance

Definition of the thermal resistance of an insulated cylindrical rod :

We want to determine, in stationnary mode, the temperature in a homogeneous cylindrical rod, with its length, its section and its edges are at temperatures and .

Let's suppose that the lateral surface is insulated.

The heat equation is simply

So :

The thermal flow through the rod is :

We can defined a thermal resistance such as :

That is to say :

We can also define thermal conductance :

### Exemple : Insulation of the walls of a house

• Blockworks, polystyrene, plasterboard and wallpapers : serial thermal resistances

• Likewise with a window added : shunt thermal resistances.

Let's take the case of a double-glazing with a window (surface , its thickness and its conductivity), a gas layer with its conductivity, and second window similar to the first one.

It's the serial association of three heat resistances, hence the total resistance :

Where is the resistance of a window of thickness .

With , it appears that double-glazing allows us to multiply the thermal resistance by and thus to divide by the losses through the windows.

### Complément : Analogy between Fourier's and Ohm's phenomenological laws

The classical Drude model affords to interpret local Ohm's law in the metals :

Let's remember that the expression of the heat resistance of a metal wire, with its length, its section and its conductivity ( its resistivity) :

Obviously, we can make the analogy with the thermal resistance of a one-dimensional straight-lined bar.

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Heat transfers in a cylinder
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Experimental measurement of a thermal conductivity