Most often heat is conducted in two dimensions instead of one dimension
as discussed in Chapter 2. For example, we are interested in determining the
temperature distribution in a 2-D rectangular block with appropriate BCs.
Once the temperature distribution is known, the associated heat transfer rate
can be determined. The following are the steady-state 2-D heat conduction
equations without heat generation and the typical BCs with given surface
temperatures.
∂ 2 T
∂x 2 +
∂ 2 T
∂y 2 = 0
(3.1)
∂ 2 θ
∂x 2 +
∂ 2 θ
∂y 2 = 0, if let θ = T − T 0
(3.2)
Boundary conditions:
x = 0, T = 0 or x = 0, θ = 0 homogeneous BC
x = a, T = 0 or x = a, θ = 0 homogeneous BC
y = 0, T = 0 or y = 0, θ = 0 homogeneous BC
y b, T T s or y b, θ T s T 0 θ s nonhomogeneous BC
=
=
=
= −
=
3
2-D Steady-State Heat Conduction

3.1 Method of Separation of Variables: Given
Temperature BC
Here, we defined a homogeneous BC as T = 0, or ∂T/∂x = 0, ∂T/∂y = 0;
θ = 0, or ∂θ/∂x = 0, ∂θ/∂y = 0, that is, temperature or temperature gradient
at a given boundary surface (in the x- or y-direction) equals 0. In contrast,
we define a nonhomogeneous BC as T = 0, ∂T/∂x = 0, ∂T/∂y = 0; or θ = 0,
∂θ/∂x = 0, ∂θ/∂y = 0, that is, temperature or temperature gradient at a given
boundary surface (in the x- or y-direction) does not equal 0.
Equations 3.1 and 3.2 can be solved by the method of separation of variable.
By means of this, we can separate the temperature from depending on two
directions, T(x, y), to one direction each, T(x) and T(y), respectively. The final
2-D temperature distribution is a product of each 1-D temperature solution,
that is, T(x, y) = T(x) · T(y). The following outlines the method of separation
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