# 21-July. Basic Concepts 2. Heat Flow

The amount of thermal energy (J) that flows across a wall per unit time (s), i.e the heat power flowing through a wall, is the Heat Flow $$\Phi$$ (W).

$$\Phi$$ is proportional to the surface A (m $$^2$$) of the wall (wider walls conduct higher heat powers) and to the temperature difference $$\Delta$$T (K) between the two surfaces of the wall. The constant of proportionality is C, the Thermal Conduttance.

$$\Phi=CA\,\Delta T$$

From the above relation we can get the units of the Thermal Conduttance: W/(m $$^2$$ K).

From the units we can get the meaning of C: it tells how many Watts, per unit of temperature difference, are flowing across a wall of surface A due to the thermal gradient between the two sides.

UNITS

The physical meaning of the Thermal conductivity $$\lambda$$

The Flow Density q (Fourier’sLaw) and the Flow $$\Phi$$ are simply related. To get the the flow surface density q we have to divide the flow $$\Phi$$ by the area A of the wall.

$$q=\frac{\Phi}{A}$$

Substituting the following expressions into the equation written above and only considering the modulus of the quantities (i.e. neglecting the minus sign)

$$q=-\lambda \,\frac{\Delta T}{\Delta x}$$ ,

$$\Phi=CA\,\Delta T$$ ,

we get

$$\lambda \frac{\Delta T}{\Delta x}=C\,\Delta T$$ ,

and therefore

$$\lambda=C\,\Delta x$$.

This equation tells us the meaning of the Thermal Conductivity. In order to reach the message we have to remember that when we multiply C by $$\Delta x$$, since C is inversely proportional to $$\Delta x$$, we are de facto eliminating the thickness which is equivalent to consider a unit thickness! Now, looking at the formula that gives $$\lambda$$ we may read:

” $$\lambda$$ is the heat that flows across the wall per unit time, per unit of temperature difference, per unit area (C) and per unit thickness ($$\Delta x$$)”.

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