Page 222 - Introduction to Continuum Mechanics

Rate of Heat Flow into an Element by Conduction 207

Making use of Eq. (4.12.13), Eq. (ii) becomes

4.13 Rate of Heat Flow Into an Element by Conduction
Let q be a vector whose magnitude gives the rate of heat flow across a unit area by
conduction and whose direction gives the direction of heat flow, then the net heat flow by
conduction Q c into a differential element can be computed as follows:

Referring to the infinitesimal rectangular parallelepiped of Fig. 4.10, the rate at which heat
flows into the element across the face with e^ as its outward normal is
e
X an
X
K~*l" l)jc +dx,x ,X 3^ 2^ 3 d tnat across the face with -ej as its outward normal is
e
x
x
[Or i)je jc ,* d ?d 3- Thus, the net rate of heat inflow across the pair effaces is given by
where #/ = q " e/. Similarly, the net rate of heat inflow across the other two pairs of faces is

so that the total net rate of heat inflow by conduction is

Example 4.13.1
Using the Fourier heat conduction law q = -/cV@, where V© is the temperature gradient
and K is the coefficient of thermal conductivity, find the equation governing the steady-state
distribution of temperature.
Solution. From Eq. (4.13.1), we have, per unit volume, the net rate of heat inflow is given
by