J
0
1
0
1
I
I 1 (x)
I 0 (x)
J 0 (x)
J 1 (x)
x
x
x
0
Y
Y 0 (x)
K 0 (x)
K 1 (x)
Y 1 (x)
K
x
0
k A
r 3
r 2
r 1
k B
R″ tc,A , R″ tc,B
q″ h , T h
h o , T ∞,o
h i , T ∞,i
q″ h , (2πr 2 )
1
ln(r 2 /r 1 )
ln(r 3 /r 2 )
1
R″ tc,B , R″ tc,A
h i 2πr 1 2πk B
2πk A h o 2πr 3
q′ i
q′ o
T ∞,o
T ∞,i
T h
31
1-D Steady-State Heat Conduction
FIGURE 2.10
The characteristics of Bessel functions.
Examples
2.1. A composite cylindrical wall (Figure 2.11) is composed of two materials of
thermal conductivity k A and k B , which are separated by a very thin, electric
resistance heater for which interfacial contact resistances with material A and
B R ""
tc,B , respectively. Liquid pumped through the tube is at a tempertc,A , R ""
ature T ∞,i and provides a convection coefficient h i at the inner surface of
FIGURE 2.11
Thermal circuit of a composite cylindrical wall and all resistance.
0
1
0
1
I
I 1 (x)
I 0 (x)
J 0 (x)
J 1 (x)
x
x
x
0
Y
Y 0 (x)
K 0 (x)
K 1 (x)
Y 1 (x)
K
x
0
k A
r 3
r 2
r 1
k B
R″ tc,A , R″ tc,B
q″ h , T h
h o , T ∞,o
h i , T ∞,i
q″ h , (2πr 2 )
1
ln(r 2 /r 1 )
ln(r 3 /r 2 )
1
R″ tc,B , R″ tc,A
h i 2πr 1 2πk B
2πk A h o 2πr 3
q′ i
q′ o
T ∞,o
T ∞,i
T h
31
1-D Steady-State Heat Conduction
FIGURE 2.10
The characteristics of Bessel functions.
Examples
2.1. A composite cylindrical wall (Figure 2.11) is composed of two materials of
thermal conductivity k A and k B , which are separated by a very thin, electric
resistance heater for which interfacial contact resistances with material A and
B R ""
tc,B , respectively. Liquid pumped through the tube is at a tempertc,A , R ""
ature T ∞,i and provides a convection coefficient h i at the inner surface of
FIGURE 2.11
Thermal circuit of a composite cylindrical wall and all resistance.