3. Numerical methods in Heat Conductions
Finite Difference techniques
The analytical solution, which is considered an exact solution, of engineering heat
transfer problems is not available for all situations due to several reasons, e.g. complicated
geometries, unusual boundary conditions …etc. One is seeking approximate solutions for such
problems. Finite difference technique is a numerical technique for differential equations and
according to this technique the solution of these equations is obtained at specified discrete
position rather than the case of analytical solution, where the solution is a continuous function
valid everywhere of domain. According to this technique, the differential equation transformed
to an equivalent set of finite difference equation (set of linear algebraic equations).
3.1 Nodal mesh
Fig. (1) Two dimensional conduction, (a) Nodal network, (b) Finite difference
approximation.
The domain of interest is covered with a mesh; each node of this mesh has an identifier. For x-y
domain, every node has (m, n) identifier according to its x- and y-position. The distance (step
size) between two successive nodes in x-direction is ∆x and in corresponding y-direction is ∆y.
The first and second derivative of the temperature T at general interior node (m,n), can be
approximated as shown in Fig. (1):
𝜕𝑇
𝑇𝑚+1,𝑛 − 𝑇𝑚,𝑛
|
≈
𝜕𝑥 𝑚+1/2,𝑛
∆𝑥
𝜕𝑇
𝑇𝑚,𝑛 − 𝑇𝑚−1,𝑛
|
≈
𝜕𝑥 𝑚−1/2,𝑛
∆𝑥
1
𝜕 2𝑇
𝑇𝑚+1,𝑛 + 𝑇𝑚−1,𝑛 − 2𝑇𝑚,𝑛
|
≈
2
(∆𝑥)2
𝜕𝑥 𝑚,𝑛
In the same manner, the second derivative in ydirection is given by:
𝜕 2𝑇
𝑇𝑚,𝑛+1 + 𝑇𝑚,𝑛−1 − 2𝑇𝑚,𝑛
|
≈
(∆𝑥)2
𝜕𝑦 2 𝑚,𝑛
The nodes of mesh lie at the boundaries of the
domain have different finite difference approximation
depending upon the kind of such boundary:
Specified temperature boundary:
Simply, each node lies at this boundary has a
temperature equal to this specified value as shown in
Fig. (2).
Fig. (2) Finite difference formulation of
specified temperature boundary conditions on
both surfaces of a plane wall.
For other expected boundary conditions, energy balance of volume element of the considered
node is carried out to determine its temperature as shown in figure:
∑ 𝑄̇ + 𝐺̇𝑒𝑙𝑒𝑚𝑒𝑛𝑡 = 0
𝑎𝑙𝑙 𝑠𝑖𝑑𝑒𝑠
1. Specified heat flux boundary condition
As shown in figure (3), one can carry out the energy
balance as follows;
𝑞𝑜̇ 𝐴 + 𝑘𝐴
𝑇1 − 𝑇0
+ 𝑔0̇ (𝐴 ∆𝑥⁄2) = 0
∆𝑥
In case of specified heat flux boundary with no heat
generation (𝑔0̇ = 0), the finite difference equation
of this boundary takes the form:
𝑞𝑜̇ 𝐴 + 𝑘𝐴
𝑇1 − 𝑇0
=0
∆𝑥
And in case of insulated boundary ( 𝑞𝑜̇ = 0), the
finite difference equation takes the form:
2
Fig. (3) Finite difference formulation
of the left boundary node of a plane.
𝑘𝐴
𝑇1 − 𝑇0
+ 𝑔0̇ (𝐴 ∆𝑥⁄2) = 0
∆𝑥
Moreover for insulated boundary with no heat generation (𝑞𝑜̇ = 0 & 𝑔0̇ = 0)
𝑘𝐴
𝑇1 −𝑇0
∆𝑥
=0
2. Convection boundary condition
ℎ𝐴(𝑇∞ − 𝑇0 ) + 𝑘𝐴
𝑇1 − 𝑇0
+ 𝑔0̇ (𝐴 ∆𝑥⁄2) = 0
∆𝑥
3. Radiation boundary condition
4
𝜀𝜎𝐴(𝑇𝑠𝑢𝑟𝑟
− 𝑇04 ) + 𝑘𝐴
𝑇1 − 𝑇0
+ 𝑔0̇ (𝐴 ∆𝑥⁄2) = 0
∆𝑥
4. Combined Convection and radiation boundary
condition
As it is clear from figure (4):
Fig.(4) Finite difference formulation of
combined convection and radiation
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ℎ𝐴(𝑇∞ − 𝑇0 ) + 𝜀𝜎𝐴(𝑇𝑠𝑢𝑟𝑟
− 𝑇04 ) + 𝑘𝐴
𝑇1 − 𝑇0
+ 𝑔0̇ (𝐴 ∆𝑥⁄2) = 0
∆𝑥
Or
ℎ𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 𝐴(𝑇∞ − 𝑇0 ) + 𝑘𝐴
𝑇1 − 𝑇0
+ 𝑔0̇ (𝐴 ∆𝑥⁄2) = 0
∆𝑥
5. Combined convection, radiation and heat flux boundary condition
4
𝑞𝑜̇ 𝐴 + ℎ𝐴(𝑇∞ − 𝑇0 ) + 𝜀𝜎𝐴(𝑇𝑠𝑢𝑟𝑟
− 𝑇04 ) + 𝑘𝐴
𝑇1 − 𝑇0
+ 𝑔0̇ (𝐴 ∆𝑥 ⁄2) = 0
∆𝑥
6. Interface boundary condition
Two different solid media A and B are assumed to be in perfect contact, and thus at the same
temperature at the interface at node m (as shown in figure (5)):
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Fig. (6) Finite difference formulation of
boundary node is obtained by applying
energy balance on its volume element.
Fig.(5) Finite difference formulation of
the interface boundary condition for two
mediums A and B.
For the condition of two dimensional problems the development of finite difference formulation
of boundary nodes is similar to that of one dimensional case, taking in consideration that heat
transfers in x- and y-directions as shown in figure (6).
∑ 𝑄̇ + 𝐺̇𝑒𝑙𝑒𝑚𝑒𝑛𝑡 = 0
𝑎𝑙𝑙 𝑠𝑖𝑑𝑒𝑠
Figure 5 fig 5-16 C
Figure 6 fig 5-16 C
3.2 Finite difference form of the steady heat conduction equation
Taking in account the one dimensional steady heat conduction problems with heat generation 𝑔̇ ,
the differential equation describing this case is as follows:
𝜕 2 𝑇 𝑔̇
+ =0
𝜕𝑥 2 𝑘
Substitution in the foregoing equation with approximate second derivative of T for a general
point (m) lie on the nodal mesh, yields to:
𝑇𝑚−1 + 𝑇𝑚+1 − 2𝑇𝑚 𝑔̇ 𝑚
+
=0
∆𝑥 2
𝑘
For m=1,2,3……M-1
Where M is the total number of nodes. This recursive relation represents a set of linear algebraic
equations of M-1 unknown. This set can be solved by Gauss-seidel method.
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