67
2-D Steady-State Heat Conduction
a. Sketch the steady 2-D temperature distribution in the fin.
b. If you were to determine the steady 2-D temperature distribution in the fin using a finite-difference numerical method, you
would solve a set of algebraic nodal equations simultaneously
for the temperatures at a 2-D array of nodes. Derive the equation for a typical node on one of the surfaces of the fin. Please
do not simplify the equation.
c. Using the method of separation of variables, derive an expression for the steady local temperature in the fin, in terms of the
thermal conductivity of the fin, k, the convective heat transfer
coefficient, h, the half-height of the fin, L, and the base and
fluid temperatures, T b and T ∞ .
Note that
W �
[cos 2
1
(aw)] dw = a[2aW + sin (2aW)
4
]
0
W
and
�
[cos(aw) · cos(bw)] dw = 0, when a = b.
0
3.12. A long rectangular rubber pad of width a = W and height b = 2W
is a component of a spacecraft structure. Its sides and bottom
are bonded to a metal channel at constant temperature T 0 , and
the temperature distribution along the top of the pad can be
approximated as a simple sine curve T = T 0 + T m sin(πx/W).
a. Write the differential equation and BCs needed to solve for the
temperature distribution in the pad.
b. Find the solution for the temperature distribution from the
differential equation and BCs.
3.13. A long rod of right triangular cross section has the horizontal
length “a” at temperature T 1 , the vertical length “b” at temperature T 2 , and the inclined length perfectly insulated. Obtain an
expression for the steady-state temperature distribution T(x, y) in
the long rod of triangular cross section as stated. Assume that the
thermal conductivity of the material of the rod is constant.
3.14. A long rectangular bar 0 ≤ x ≤ a, 0 ≤ y ≤ b, and a, b << L, the
bar length, is heated at y = 0 and y = b, respectively, to a uniform
temperature T o and is insulated at x = 0. The side of x = a loses
heat by convection to a fluid at temperature T ∞ with a convection
coefficient of h ∞ .
a. Write down, step by step, a solution method and the associated BCs, which can be used to determine the bar steady-state
temperature distributions. You do not need to obtain the final
solution of the steady-state temperature distributions.
b. Sketch the heat flows and the isothermal profiles in the
rectangular bar.
2-D Steady-State Heat Conduction
a. Sketch the steady 2-D temperature distribution in the fin.
b. If you were to determine the steady 2-D temperature distribution in the fin using a finite-difference numerical method, you
would solve a set of algebraic nodal equations simultaneously
for the temperatures at a 2-D array of nodes. Derive the equation for a typical node on one of the surfaces of the fin. Please
do not simplify the equation.
c. Using the method of separation of variables, derive an expression for the steady local temperature in the fin, in terms of the
thermal conductivity of the fin, k, the convective heat transfer
coefficient, h, the half-height of the fin, L, and the base and
fluid temperatures, T b and T ∞ .
Note that
W �
[cos 2
1
(aw)] dw = a[2aW + sin (2aW)
4
]
0
W
and
�
[cos(aw) · cos(bw)] dw = 0, when a = b.
0
3.12. A long rectangular rubber pad of width a = W and height b = 2W
is a component of a spacecraft structure. Its sides and bottom
are bonded to a metal channel at constant temperature T 0 , and
the temperature distribution along the top of the pad can be
approximated as a simple sine curve T = T 0 + T m sin(πx/W).
a. Write the differential equation and BCs needed to solve for the
temperature distribution in the pad.
b. Find the solution for the temperature distribution from the
differential equation and BCs.
3.13. A long rod of right triangular cross section has the horizontal
length “a” at temperature T 1 , the vertical length “b” at temperature T 2 , and the inclined length perfectly insulated. Obtain an
expression for the steady-state temperature distribution T(x, y) in
the long rod of triangular cross section as stated. Assume that the
thermal conductivity of the material of the rod is constant.
3.14. A long rectangular bar 0 ≤ x ≤ a, 0 ≤ y ≤ b, and a, b << L, the
bar length, is heated at y = 0 and y = b, respectively, to a uniform
temperature T o and is insulated at x = 0. The side of x = a loses
heat by convection to a fluid at temperature T ∞ with a convection
coefficient of h ∞ .
a. Write down, step by step, a solution method and the associated BCs, which can be used to determine the bar steady-state
temperature distributions. You do not need to obtain the final
solution of the steady-state temperature distributions.
b. Sketch the heat flows and the isothermal profiles in the
rectangular bar.