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22
Analytical Heat Transfer
Fin cross-section area
T b
q x
x
q
q x
dx
∂
+ ∂x
,
h T ∞
hA s (T−T ∞ )
Hot wall
q f
Cold fluid
Fin
x
FIGURE 2.7
One-dimensional conduction through thin fins with uniform cross-section area.
at a given fin geometry and working conditions. We assume that heat conduction through the fin is 1-D steady state because the fin is thin. The temperature
gradient in the other two dimensions is neglected. We will begin with the constant cross-sectional area fins and then consider the variable cross-sectional
area fins. The following is the energy balance of a small control volume of
the fin with heat conduction through the fin and heat dissipation into cooling
fluid, as shown in Figure 2.7. The result of temperature distributions through
fins of different materials can be seen from Figure 2.8.
dq x
q x − q x +
dx − hA s (T − T ∞ ) = 0
(2.29)
dx
where A s = P dx, and P is perimeter of the fin, and q x is from Fourier’s
Conduction Law shown in Equation 1.10.
d
dT
−
−kA c
dx − hP dx(T − T ∞ ) = 0
(2.30)
dx
dx
x
T b
T ∞
Plastic
Steel
Aluminum
Copper
FIGURE 2.8
Temperature distributions through fins of different materials.