2.1.1 Conduction through Circular Tube Walls
1-D steady-state heat conduction, without heat generation, in the radial
system shown in Figure 2.3, can be simplified from Equation 1.18
1
r
d
dr
�
r
dT
dr
�
= 0
(2.13)
dT
r dr
= c 1
The general solution of Equation 2.13 is
T(r) = c 1 ln r + c 2
with boundary conditions
at r = r 1, T = T s,1 = c 1 ln r 1 + c 2
at r r 2 , T T s,2 c 1 ln r 2 c 2
=
=
=
+
1-D Steady-State Heat Conduction
15
Similar to electric current, the heat rate through the wall is
T ∞,1 − T ∞,2
T ∞,1 − T ∞,2
q =
=
= UA(T ∞,1 − T ∞,2 ) (2.11)
(1/Ah 1 ) + (L/kA) + (1/Ah 2 )
R tot
Figure 2.2 shows conduction through two plane walls with contact resistance
between them, the total heat resistance becomes
R
""
1
1
L 1
tc
L 2
R tot =
+
+
+
+
(2.12)
Ah 1
k 1 A
A
k 2 A Ah 2
where R "" = (T a − T b /(q/A)) = pre-determined (depends on contact material
tc
surface roughness and contact pressure).
Contact surface
T ∞,1 h 1
T s,1
k 2 , α 2
Hot fluid
Cold fluid
T a
ΔT
k 1 , α 1
T b
T s,2
T ∞,2 h 2
0
L 1
L 2
FIGURE 2.2
Temperature drop due to thermal contact resistance between surface a and surface b.