49
2-D Steady-State Heat Conduction
Therefore, one obtains
(2/nπ) [1 − (−1) n ] T s
C n =
sinh(nπb/a)
Finally, the 2-D temperature distribution follows:
∞
nπx
nπy
T(x, y) =
C n sin
sinh
(3.7)
a
a
n=1
∞ (2/nπ) [1 − (−1) n ] T s
nπx
nπy
T(x, y) =
sin
sinh
(3.8)
sinh(nπb/a)
a
a
n=1
The second-order partial differential equations (PDEs) T(x, y) are split into
two second-order ordinary differential equations (ODEs) T(x) and T(y). The
second-order ODE with two homogeneous BCs is the so-called eigenvalue
equation. The solution of the eigenfunctions, sine and cosine, depend on
the two homogeneous BCs. The eigenvalues, λ, can be determined by one
of the two homogeneous BCs (either both in the x-direction, or both in the
y-direction). The solution of the other second-order ODE is a decay curve
of combining e(x) and e(−x) for an infinite-length problem, or sinh(x) and
cosh(x) for a finite-length problem. The only nonhomogeneous BC will be
used to solve the final unknown coefficient C n . The integrated value C n can
be determined by performing integration of sin, sin-square or cos, cos-square,
depending on the given BCs, by using the characteristics of the orthogonal
functions.
If we let θ = T − T 0 , follow the same procedure, the 2-D temperature
distribution becomes
nπx
nπy
θ(x, y) = θ(x)θ(y) =
C n sin
sinh
(3.9)
a
a
∞ (2/nπ) [1 − (−1) n ] θ s
nπx
nπy
θ(x, y) =
sin
sinh
(3.10)
sinh(nπb/a)
a
a
n=1
3.2 Method of Separation of Variables: Given Heat Flux
and Convection BCs
3.2.1 Given Surface Heat Flux BC
The following shows the similar principal of using the separation of variable
method to solve the 2-D heat conduction problems with one nonhomogeneous
boundary specified as surface heat flux or surface convection condition [2].
As the aforementioned procedure, we need to make all BCs homogeneous
2-D Steady-State Heat Conduction
Therefore, one obtains
(2/nπ) [1 − (−1) n ] T s
C n =
sinh(nπb/a)
Finally, the 2-D temperature distribution follows:
∞
nπx
nπy
T(x, y) =
C n sin
sinh
(3.7)
a
a
n=1
∞ (2/nπ) [1 − (−1) n ] T s
nπx
nπy
T(x, y) =
sin
sinh
(3.8)
sinh(nπb/a)
a
a
n=1
The second-order partial differential equations (PDEs) T(x, y) are split into
two second-order ordinary differential equations (ODEs) T(x) and T(y). The
second-order ODE with two homogeneous BCs is the so-called eigenvalue
equation. The solution of the eigenfunctions, sine and cosine, depend on
the two homogeneous BCs. The eigenvalues, λ, can be determined by one
of the two homogeneous BCs (either both in the x-direction, or both in the
y-direction). The solution of the other second-order ODE is a decay curve
of combining e(x) and e(−x) for an infinite-length problem, or sinh(x) and
cosh(x) for a finite-length problem. The only nonhomogeneous BC will be
used to solve the final unknown coefficient C n . The integrated value C n can
be determined by performing integration of sin, sin-square or cos, cos-square,
depending on the given BCs, by using the characteristics of the orthogonal
functions.
If we let θ = T − T 0 , follow the same procedure, the 2-D temperature
distribution becomes
nπx
nπy
θ(x, y) = θ(x)θ(y) =
C n sin
sinh
(3.9)
a
a
∞ (2/nπ) [1 − (−1) n ] θ s
nπx
nπy
θ(x, y) =
sin
sinh
(3.10)
sinh(nπb/a)
a
a
n=1
3.2 Method of Separation of Variables: Given Heat Flux
and Convection BCs
3.2.1 Given Surface Heat Flux BC
The following shows the similar principal of using the separation of variable
method to solve the 2-D heat conduction problems with one nonhomogeneous
boundary specified as surface heat flux or surface convection condition [2].
As the aforementioned procedure, we need to make all BCs homogeneous