Heat Transfer

There are three modes of heat transfer: conduction, convection and radiation. These are illustrated below for heat transfer through a plate on the left, air or fluid then a remote surface on the right:

Note that the temperature decreases from the left face of the plate to the right, then there is an elevated temperature of the air very close to the plate which drops to ambient away from the plate.

In general one dimensional rate of heat transfer can be defined in terms of thermal resistivity and temperature difference: $q = \frac{1}{R} Δ T$

Conduction

Heat transfer by conduction through a plane wall for the steady state case can be simplified from Fourier's law (1822) to:

q_{c o n d} = \frac{k}{L} A (T_{1} - T_{2})

Or for heat transfer through a hollow cylinder per unit length:

q_{c o n d} = \frac{2 π k (T_{1} - T_{2})}{ln \frac{ϕ_{2}}{ϕ_{1}}}

This same equation can be represented as a a thermal resistance for heat conduction through a plane wall:

R_{c o n d} = \frac{L}{k A}

Convection

The steady state heat transfer rate by convection is given by Newton's law of cooling:

q_{c o n v} = h A (T_{2} - T_{\infty})

This can also be expressed as a thermal resistance for convection:

R_{c o n v} = \frac{1}{h A}

Radiation

TODO

Series Addition

Thermal resistivity in series can be added as follows:

R = R_{1} + R_{2} + \dots + R_{n}

Parallel Addition

Thermal restistance over parallel paths are combined:

\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{n}}

Symbols Used

$q$	is the heat transfer rate (W);
$A$	is the cross-sectional area (m²);
$k$	is the material's conductivity coefficient (W/m.K);
$h$	is the convective heat transfer coefficient (W/m².K);
$L$	is the distance between the left and right faces of the plate (m);
$T_{1}$	temperature on the left hand side of the plate (K or °C);
$T_{2}$	temperature on the right hand side of the plate (K or °C);
$T_{3}$	temperature of the remote surface (K or °C);
$T_{\infty}$	is the temperature of the air/fluid (K or °C);
$ϕ_{1}$	is the hollow cylinder inside radius;
$ϕ_{2}$	is the hollow cylinder outside radius;