12
Analytical Heat Transfer
T s
y
T ∞ , h
Insulated
q" s
0
a x
FIGURE 1.12
Heat conduction in 2-D system with various boundary conditions.
In addition, in real engineering applications, it is not easy to determine the
precise convection BC shown in Figures 1.10 and 1.12. These require detailed
knowledge of complex convection heat transfer to be discussed in Chapters 6
through 10.
PROBLEMS
1.1 Derive Equation 1.18.
1.2 Derive Equation 1.19.
1.3 a. Write the differential equation that expresses transient heat
conduction in 3-D (x, y, z coordinates) with constant heat
generation and constant conductivity.
b. Simplify the differential equation in (a) to show steady-state
conduction in one dimension, assuming constant conductivity.
c. If the BCs are: T = T 1 . . . at . . . x = x 1 , T = T 2 . . . at . . . x = x 2 ,
solve the second-order differential equation to yield a temperature distribution (T). Express the answer (T) in terms of (T 1 ,
T 2 , x, x 1 , x 2 ).
d. Using the Fourier law and the results from (c), develop an
expression for the heat rate per unit area, assuming constant
conductivity. Express your answer in terms of (k, T 1 , T 2 , x 1 , x 2 ).
Reference
1. F. Incropera and D. Dewitt, Fundamentals of Heat and Mass Transfer, Fifth Edition,
John Wiley & Sons, New York, NY, 2002.
Analytical Heat Transfer
T s
y
T ∞ , h
Insulated
q" s
0
a x
FIGURE 1.12
Heat conduction in 2-D system with various boundary conditions.
In addition, in real engineering applications, it is not easy to determine the
precise convection BC shown in Figures 1.10 and 1.12. These require detailed
knowledge of complex convection heat transfer to be discussed in Chapters 6
through 10.
PROBLEMS
1.1 Derive Equation 1.18.
1.2 Derive Equation 1.19.
1.3 a. Write the differential equation that expresses transient heat
conduction in 3-D (x, y, z coordinates) with constant heat
generation and constant conductivity.
b. Simplify the differential equation in (a) to show steady-state
conduction in one dimension, assuming constant conductivity.
c. If the BCs are: T = T 1 . . . at . . . x = x 1 , T = T 2 . . . at . . . x = x 2 ,
solve the second-order differential equation to yield a temperature distribution (T). Express the answer (T) in terms of (T 1 ,
T 2 , x, x 1 , x 2 ).
d. Using the Fourier law and the results from (c), develop an
expression for the heat rate per unit area, assuming constant
conductivity. Express your answer in terms of (k, T 1 , T 2 , x 1 , x 2 ).
Reference
1. F. Incropera and D. Dewitt, Fundamentals of Heat and Mass Transfer, Fifth Edition,
John Wiley & Sons, New York, NY, 2002.