43
1-D Steady-State Heat Conduction
b. The general solution of the above modified Bessel’s equation
is y = C 1 I n (cx) + C 2 K n (cx), where C 1 and C 2 are constants,
and I n and K n are the modified Bessel’s functions of the first
and second kinds of order n, respectively. Assuming that
the heat transfer on the outer surface of the fin is negligible
[i.e., dT/dr = 0 at r = r o ], solve the governing equation to
obtain the steady 1-D temperature distribution in the annular
fin, T(r).
c. You may also determine the steady 1-D temperature distribution in the annular fin numerically using the finite difference
method. Give the finite-difference equations for the nodes at
r = r i and r = r o and for a typical interior node. Please rearrange the equations to give expressions for the temperatures
at the nodes.
2.14. An engineer has suggested that a triangular fin would be more
effective than a circular fin for a new natural convection heat
exchanger. The fins are very long and manufactured from Al 2024T6. The triangular fin has an equilateral cross section with a base
dimension of 1.0 cm and the second fin has a circular cross section with a 0.955 cm diameter. If the base temperature of the fin is
maintained at 400 ◦ C, which fin will transfer more heat and which
fin has a greater effectiveness?
[h = 25 W 2
/m K, k 2024−T6 = 177 W/mK, T ∞ = 25 ◦ C]
q
1/2
fin = (hPkA c )
· θ b ε fin = q fin /hA c θ b
2.15. A thin conical pin fin is attached to a hot base plate at T b . The
cooling air has temperature T ∞ and convection heat transfer coefficient h. Determine analytically the temperature profile in the pin
fin. Also, determine the heat flux through the pin fin base.
2.16. Solve temperature profile for the annulus fin geometry as shown
in Figure 2.9b, assume that the fin tip is exposed to convection
fluid with same given temperature and convection heat transfer
coefficient.
2.17. Solve the temperature profile for the annulus fin geometry as
shown in Figure 2.9b, assume that the fin tip is fixed at a given
temperature between the fin base and the convection fluid.
2.18. Determine the solutions shown in Equations 2.36 and 2.37.
2.19. Determine the solutions shown in Equations 2.38 and 2.39.
2.20. Determine the solutions shown in Equations 2.40 and 2.41.
References
1. W. Rohsenow and H. Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Inc.,
Englewood Cliffs, NJ, 1961.
1-D Steady-State Heat Conduction
b. The general solution of the above modified Bessel’s equation
is y = C 1 I n (cx) + C 2 K n (cx), where C 1 and C 2 are constants,
and I n and K n are the modified Bessel’s functions of the first
and second kinds of order n, respectively. Assuming that
the heat transfer on the outer surface of the fin is negligible
[i.e., dT/dr = 0 at r = r o ], solve the governing equation to
obtain the steady 1-D temperature distribution in the annular
fin, T(r).
c. You may also determine the steady 1-D temperature distribution in the annular fin numerically using the finite difference
method. Give the finite-difference equations for the nodes at
r = r i and r = r o and for a typical interior node. Please rearrange the equations to give expressions for the temperatures
at the nodes.
2.14. An engineer has suggested that a triangular fin would be more
effective than a circular fin for a new natural convection heat
exchanger. The fins are very long and manufactured from Al 2024T6. The triangular fin has an equilateral cross section with a base
dimension of 1.0 cm and the second fin has a circular cross section with a 0.955 cm diameter. If the base temperature of the fin is
maintained at 400 ◦ C, which fin will transfer more heat and which
fin has a greater effectiveness?
[h = 25 W 2
/m K, k 2024−T6 = 177 W/mK, T ∞ = 25 ◦ C]
q
1/2
fin = (hPkA c )
· θ b ε fin = q fin /hA c θ b
2.15. A thin conical pin fin is attached to a hot base plate at T b . The
cooling air has temperature T ∞ and convection heat transfer coefficient h. Determine analytically the temperature profile in the pin
fin. Also, determine the heat flux through the pin fin base.
2.16. Solve temperature profile for the annulus fin geometry as shown
in Figure 2.9b, assume that the fin tip is exposed to convection
fluid with same given temperature and convection heat transfer
coefficient.
2.17. Solve the temperature profile for the annulus fin geometry as
shown in Figure 2.9b, assume that the fin tip is fixed at a given
temperature between the fin base and the convection fluid.
2.18. Determine the solutions shown in Equations 2.36 and 2.37.
2.19. Determine the solutions shown in Equations 2.38 and 2.39.
2.20. Determine the solutions shown in Equations 2.40 and 2.41.
References
1. W. Rohsenow and H. Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Inc.,
Englewood Cliffs, NJ, 1961.