64
Analytical Heat Transfer
Remarks
In this chapter, we have introduced a very powerful mathematical tool, the
method of separation of variables, to solve typical 2-D heat conduction problems with various thermal BCs. In the undergraduate-level heat transfer, we
normally employ the finite-difference energy balance method to solve the
2-D heat conduction problems with various thermal BCs. The finite-difference
numerical methods and solutions will be discussed in Chapter 5. Here we
are more focused on the analytical methods and solutions for various 2-D
heat conduction problems. In general, all kinds of 2-D heat conduction problems with various BCs can be solved analytically by using superposition of
separation of variables.
The most important thing for applying separation of variable is that you
have to set up your problem where only one nonhomogeneous BC is allowed.
If you have more than one nonhomogeneous BC, you have to employ the
superposition principle to split into two or three subproblems in order to
use separation of variables. The problems become more complicated if you
work with 3-D heat conductions with heat generation and with complex BCs,
but they are still workable. However, if you are interested in solving for the
2-D and 3-D cylindrical coordinate systems and for the 2-D and 3-D spherical
coordinate systems with complex BCs, they are beyond the intermediate-level
heat transfer, you need to look at the advanced heat conduction textbook for
solutions.
Another popular method is using the finite-difference method to be discussed in the later chapter. It is particularly true when you deal with
complicate BCs such as convection. In real-life engineering applications, the
convection heat transfer coefficients normally are varied along the solid surface. This will cause additional complexity for the separation of variables
because we normally assume the uniform convection BCs to simplify the problem. This will not cause any complexity at all by using the finite-difference
numerical method.
PROBLEMS
3.1. A long rectangular bar 0 ≤ x ≤ a, 0 ≤ y ≤ b, and a, b << L, the
bar length, is heated at y = o 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 h.
a. Write down, step by step, a solution method and associated
BCs, which can be used to determine the bar steady-state
temperature distributions.
b. Sketch the heat flows and the isothermal profiles in the
rectangular bar.
3.2. An infinitely long rod of square cross section (LXL) floats in a
fluid. The heat transfer coefficient between the rod and the fluid
Analytical Heat Transfer
Remarks
In this chapter, we have introduced a very powerful mathematical tool, the
method of separation of variables, to solve typical 2-D heat conduction problems with various thermal BCs. In the undergraduate-level heat transfer, we
normally employ the finite-difference energy balance method to solve the
2-D heat conduction problems with various thermal BCs. The finite-difference
numerical methods and solutions will be discussed in Chapter 5. Here we
are more focused on the analytical methods and solutions for various 2-D
heat conduction problems. In general, all kinds of 2-D heat conduction problems with various BCs can be solved analytically by using superposition of
separation of variables.
The most important thing for applying separation of variable is that you
have to set up your problem where only one nonhomogeneous BC is allowed.
If you have more than one nonhomogeneous BC, you have to employ the
superposition principle to split into two or three subproblems in order to
use separation of variables. The problems become more complicated if you
work with 3-D heat conductions with heat generation and with complex BCs,
but they are still workable. However, if you are interested in solving for the
2-D and 3-D cylindrical coordinate systems and for the 2-D and 3-D spherical
coordinate systems with complex BCs, they are beyond the intermediate-level
heat transfer, you need to look at the advanced heat conduction textbook for
solutions.
Another popular method is using the finite-difference method to be discussed in the later chapter. It is particularly true when you deal with
complicate BCs such as convection. In real-life engineering applications, the
convection heat transfer coefficients normally are varied along the solid surface. This will cause additional complexity for the separation of variables
because we normally assume the uniform convection BCs to simplify the problem. This will not cause any complexity at all by using the finite-difference
numerical method.
PROBLEMS
3.1. A long rectangular bar 0 ≤ x ≤ a, 0 ≤ y ≤ b, and a, b << L, the
bar length, is heated at y = o 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 h.
a. Write down, step by step, a solution method and associated
BCs, which can be used to determine the bar steady-state
temperature distributions.
b. Sketch the heat flows and the isothermal profiles in the
rectangular bar.
3.2. An infinitely long rod of square cross section (LXL) floats in a
fluid. The heat transfer coefficient between the rod and the fluid