# General Heat Conduction Equation

Consider a small control volume V = 8x8y8z (Fig. 4.27), where the inner heat generation is Q’g" (T) (heat production/volume) and the heat conductivity is A(T). The material is assumed to be homogeneous and iso­tropic, and the internal heat generation and thermal conductivity are

Functions of temperature.

The heat flow to the control volume through area 8y8z at x is

 81. (4.174) Dx 8QX = -8y5zA(T) The outgoing heat at the point % + 8x is

 ( ‘T1 A Xm&x
 (TdT + JL 3% dx
 8QX = —8y8z
 8t (4.175)

Similar formulas can be derived for the other directions. The change of inter­nal energy inside the control volume during time St is

8U = pcp8x8y8z^8t (4.176)

And the heat generation inside the control volume is

SQ* = Qg" (T) 8x8y8z8t. (4.177)

From the first law of thermodynamics,

8Q* + SQy + SQ2 + 8Qg = 8Q* + gx + 8Qy+Sy + + 81/ . (4.178)

Substituting Eq. (4.175) and the formulas for other directions into Eq. (4.178) gives

4.3 HEAT AND MASS TRANSFER

&

 5Q,, Y &QX+Sx

FIGURE 4.27 Control volume.

 +Ty
 A(T)g
 A (T)
 + dz
 51 Dt
 + Qg" (T) = pcp D_ Dx (4.179!

It is normal to assume that the thermal conductivity is constant; hence Eq. (4.179) gives

 (4.180) Ц+Ц+Ц+Цl = 151

Or

 (4.181) ЈЈ = aV2T + H, dt

Where

DT/dt is the derivative of temperature as a function of time; in a steady- state case it is equal to zero a = A/pcp, the heat conductivity or thermal diffusivity H = <P’"/C’" = heat generation inside a material; for example, for Joule’s heat or a nuclear reaction, <&"’ = heat generation/volume and C’" = Cpp = heat capacity/volume For Cartesian coordinates

Dx dy~ dz~

For cylindrical coordinates

V2T=^T+iaJi+i^T+a2T

Dr2 r dr r2d<t>2 dz2

And for spherical coordinates

 ■ ,dT V2T=i dr+ i a

 R2sm24>d(b- R dr2 r2sin tpd1!’

See Fig. 4.28.  FIGURE 4.28 Cylindrical and spherical coordinates.