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Analytical Heat Transfer
(3) x = 0, T = T 1
(4) x = 0, −k(∂T/∂x) = h(T − T ∞ )
x = a, −k(∂T/∂x)
x = a, T = T 1
= h(T − T ∞ )
y = 0, T = T 1
y = 0, T = T 1
y = b, T = T 1
y = b, T = T 1
3.7. Use the principle of superposition to determine the temperature
distribution for 2-D heat conduction with four nonhomogeneous
BCs shown in Figure 3.4.
3.8. Refer to Figure 3.6, and determine the temperature distributions
for 3-D heat conduction with the following BCs:
(1) x = 0, T = T 1
(2) x = 0, T = T 1
x = a, T = T 1
x = a, T = T 1
y = 0, T = T 1
y = 0, T = T 1
y = b, T = T 1
y = b, T = T 1
z = 0, T = T o
z = 0, T = T 1
z = c, T = T 1
z = c, T = T o
3.9. Refer to Equation 3.19, and determine the temperature distributions for 2-D heat conduction with uniform heat generation with
the following BCs:
(1) x = 0, −k(∂T/∂x) = 0
(2) x = 0, T = T o
x = a, −k(∂T/∂x) = h(T − T ∞ )
x = a, T = T o
y = 0, T = T o
y = 0, −k(∂T/∂y) = 0
y = b, T = T o
y = b, −k(∂T/∂y)
= h(T − T ∞ )
3.10. Obtain an expression for the steady-state temperature distribution
T(x, y) in a long square bar of side a. The bar has its two sides and
bottom maintained at temperature T 1 , while the top side is loosing
heat by convection. Consider the surrounding temperature to be
T ∞ , the heat transfer coefficient at the top wall be denoted by h,
and let the thermal conductivity of the bar be equal to k.
a. Sketch the domain and write the governing equation and BCs
for this problem.
b. Define the temperature θ(x, y) = T(x, y) − T 1 , and find a series
solution using separation of variables.
c. Using the BCs write the expression for the coefficients used in
the series solution for θ(x, y), and write an expression.
d. Write an expression for (T(x, y) − T 1 )/(T ∞ − T 1 ).
3.11. Given a very long and wide fin with a height of 2L. The base of
the fin is maintained at a uniform temperature of T b . The top and
bottom surfaces of the fin are exposed to a fluid whose temperature is T ∞ (T ∞ < T b ). The convective heat transfer coefficient
between the fin surfaces and the fluid is h.
Analytical Heat Transfer
(3) x = 0, T = T 1
(4) x = 0, −k(∂T/∂x) = h(T − T ∞ )
x = a, −k(∂T/∂x)
x = a, T = T 1
= h(T − T ∞ )
y = 0, T = T 1
y = 0, T = T 1
y = b, T = T 1
y = b, T = T 1
3.7. Use the principle of superposition to determine the temperature
distribution for 2-D heat conduction with four nonhomogeneous
BCs shown in Figure 3.4.
3.8. Refer to Figure 3.6, and determine the temperature distributions
for 3-D heat conduction with the following BCs:
(1) x = 0, T = T 1
(2) x = 0, T = T 1
x = a, T = T 1
x = a, T = T 1
y = 0, T = T 1
y = 0, T = T 1
y = b, T = T 1
y = b, T = T 1
z = 0, T = T o
z = 0, T = T 1
z = c, T = T 1
z = c, T = T o
3.9. Refer to Equation 3.19, and determine the temperature distributions for 2-D heat conduction with uniform heat generation with
the following BCs:
(1) x = 0, −k(∂T/∂x) = 0
(2) x = 0, T = T o
x = a, −k(∂T/∂x) = h(T − T ∞ )
x = a, T = T o
y = 0, T = T o
y = 0, −k(∂T/∂y) = 0
y = b, T = T o
y = b, −k(∂T/∂y)
= h(T − T ∞ )
3.10. Obtain an expression for the steady-state temperature distribution
T(x, y) in a long square bar of side a. The bar has its two sides and
bottom maintained at temperature T 1 , while the top side is loosing
heat by convection. Consider the surrounding temperature to be
T ∞ , the heat transfer coefficient at the top wall be denoted by h,
and let the thermal conductivity of the bar be equal to k.
a. Sketch the domain and write the governing equation and BCs
for this problem.
b. Define the temperature θ(x, y) = T(x, y) − T 1 , and find a series
solution using separation of variables.
c. Using the BCs write the expression for the coefficients used in
the series solution for θ(x, y), and write an expression.
d. Write an expression for (T(x, y) − T 1 )/(T ∞ − T 1 ).
3.11. Given a very long and wide fin with a height of 2L. The base of
the fin is maintained at a uniform temperature of T b . The top and
bottom surfaces of the fin are exposed to a fluid whose temperature is T ∞ (T ∞ < T b ). The convective heat transfer coefficient
between the fin surfaces and the fluid is h.