## Jr Jn

Eq. 6.74 transforms into

70 Furthermore, if the instantaneous volumetric rate of "heat" generation or removal at a point x, y, z inside ^ is Q(x,y, z,t), then the total amount of heat/flow produced per unit time is

71 Finally, we define the specific heat of a solid c as the amount of heat required to raise a unit mass by one degree. Thus if A^ is a temperature change which occurs in a mass m over a time At, then the corresponding amount of heat that was added must have been cmAfy, or

I3 = / pcA^dü J n where p is the density, Note that another expression of I3 is At(Ii + I2).

72 The balance equation, or conservation law states that the energy produced I2 plus the net energy across the boundary Ii must be equal to the energy absorbed I3, thus

but since t and ^ are both arbitrary, then div (DV^) + Q - pc^- = 0 dt

This equation can be rewritten as

 div (D V\$) + Q d\$ = Pcd dx+dy+q x y = pcdt

1. Note the similarity between this last equation, and the equation of equilibrium d 2ux d°xx + d®xy + pb dx x dy pm dffyy + ^ + pby = pm dy dx dt2 d 2uy

2. For steady state problems, the previous equation does not depend on t, and for 2D problems, it reduces to _

3. For steady state isotropic problems, dH + d2l + d2l + Q = 0

dx2 dy2 dz2 k

which is Poisson's equation in 3D.

4. If the heat input Q = 0, then the previous equation reduces to

dx2 dy2 dz2

which is an Elliptic (or Laplace) equation. Solutions of Laplace equations are termed harmonic functions (right hand side is zero) which is why Eq. 6.84 is refered to as the quasi-harmonic equation.

5. If the function depends only on x and t, then we obtain

which is a parabolic (or Heat) equation.

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