6
Analytical Heat Transfer
1.2 General Heat Conduction Equations
If the temperature profile inside a solid material T(x, y, z, t) is known, the
heat rate q through the solid can be determined as shown in Figure 1.5. As
far as thermal stress is concerned, it is equally important to predict the temperature profile in some high-temperature applications. In this chapter, the
general heat conduction equations will be derived. The general heat conduction equation can be used to solve various real problems with the appropriate
boundary conditions (BCs) and initial condition. The heat conduction can be
modeled as 1-D, 2-D, or 3-D depending on the nature of the problem:
1-D
T(x) for steady state or T(x, t) for transient problem
2-D
T(x, y) for steady state or T(x, y, t) for transient problem
3-D
T(x, y, z) for steady state or T(x, y, z, t) for transient problem
To determine the temperature profile, the following should be given:
1. Initial condition and BCs
2. Material thermal conductivity k, density ρ, specific heat C p , and
diffusivity α = k/ρC p
1.2.1 Derivations of General Heat Conduction Equations
The general form of the conservation of energy in a small control volume of
solid material is
E in − E out + E g = E st
(1.8)
where E in − E out is the net heat conduction, E g is the heat generation, and E st
is the energy stored in the control volume.
Figure 1.6 shows the conservation of energy in a differential control
volume in a 3-D Cartesian (rectangular) coordinate. If we consider energy
y
x
z
T H
T(x, y, z, t)
k, α
T L
FIGURE 1.5
Heat conduction through a solid medium.
Analytical Heat Transfer
1.2 General Heat Conduction Equations
If the temperature profile inside a solid material T(x, y, z, t) is known, the
heat rate q through the solid can be determined as shown in Figure 1.5. As
far as thermal stress is concerned, it is equally important to predict the temperature profile in some high-temperature applications. In this chapter, the
general heat conduction equations will be derived. The general heat conduction equation can be used to solve various real problems with the appropriate
boundary conditions (BCs) and initial condition. The heat conduction can be
modeled as 1-D, 2-D, or 3-D depending on the nature of the problem:
1-D
T(x) for steady state or T(x, t) for transient problem
2-D
T(x, y) for steady state or T(x, y, t) for transient problem
3-D
T(x, y, z) for steady state or T(x, y, z, t) for transient problem
To determine the temperature profile, the following should be given:
1. Initial condition and BCs
2. Material thermal conductivity k, density ρ, specific heat C p , and
diffusivity α = k/ρC p
1.2.1 Derivations of General Heat Conduction Equations
The general form of the conservation of energy in a small control volume of
solid material is
E in − E out + E g = E st
(1.8)
where E in − E out is the net heat conduction, E g is the heat generation, and E st
is the energy stored in the control volume.
Figure 1.6 shows the conservation of energy in a differential control
volume in a 3-D Cartesian (rectangular) coordinate. If we consider energy
y
x
z
T H
T(x, y, z, t)
k, α
T L
FIGURE 1.5
Heat conduction through a solid medium.