# 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**

Heat Flow \( \Phi \) | W | Heat that flows per unit time through a wall |

Thermal Conduttance C | W/ (m \( ^2\) K) |

**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 (\(\Delta x \))”.