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
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81. (4.174) |
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Dx |
8QX = -8y5zA(T) The outgoing heat at the point % + 8x is
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( ‘T1 A Xm&x |
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(TdT + JL 3% dx |
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8QX = —8y8z |
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8t |
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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
&
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5Q,, |
Y &QX+Sx
FIGURE 4.27 Control volume.
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+Ty |
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A(T)g |
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A (T) |
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+ dz |
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51 Dt |
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+ Qg" (T) = pcp |
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It is normal to assume that the thermal conductivity is constant; hence Eq. (4.179) gives
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(4.180) |
Ц+Ц+Ц+Цl = 151
Dx2 dv2 dz2 A adt
Or
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(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
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■ ,dT |
V2T=i dr+ i a
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R2sm24>d(b- |
R dr2 r2sin tpd1!’
See Fig. 4.28.
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FIGURE 4.28 Cylindrical and spherical coordinates.
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