Finite-Difference Equations and Solutions

Multi-Dimensional Conduction:
Finite-Difference Equations
and
Solutions
Chapter 4
Sections 4.4 and 4.5

The Finite-Difference Method
• An approximate method for determining temperatures at discrete
(nodal) points of the physical system.
• Procedure:
– Represent the physical system by a nodal network.
–Use the energy balance method to obtain a finite-difference
equation for each node of unknown temperature.
– Solve the resulting set of algebraic equations for the
unknown nodal temperatures.

The Nodal Network and Finite-Difference Approximation
• The nodal network identifies discrete points at which the temperature
is to be determined and uses an (m,n) notation to designate their
location.
What is represented by the temperature determined at a nodal
point, as for example, T m,n

• A finite-difference approximation is used to represent temperature
gradients in the domain.
How is the accuracy of the solution affected by construction of
the nodal network
What are the trade-offs between selection of a fine or a coarse mesh

Derivation of the Finite-Difference Equations
- The Energy Balance Method -
• As a convenience that obviates the need to predetermine the direction
of heat flow, assume all heat flows are into the nodal region of
interest, and express all heat rates accordingly.
Hence, the energy balance becomes:
i i
in+ Eg
=
E
0
(4.30)

• Consider application to an interior nodal point (one that exchanges heat by
conduction with four, equidistant nodal points):
4
∑
i=
1
() i →( m, n)
i
( )
q + q ∆x⋅∆y⋅ =
0
where, for example,
q k y
( m−1, n) →( m,
n) = ( ∆ ⋅)
T
−T
m−1, n m,
n
∆x
(4.31)
Is it possible for all heat flows to be into the m,n nodal region
• A summary of finite-difference equations for common nodal regions
is provided in Table 4.2.

Consider an external corner with convection heat transfer.
q q q
1, ,
+
, 1 ,
+
,
= 0
( m− n) →( mn) ( mn− ) →( mn) ( ∞) →( mn)
⎛∆y
⎞T −T ⎛∆x
⎞T −T
k⎜ ⋅ ⎟ + k⎜ ⋅⎟
⎝ 2 ⎠ ∆x
⎝ 2 ⎠ ∆y
m−1, n mn , mn , −1 mn ,
⎛∆x
⎞ ⎛∆y
⎞
+ h⎜ ⋅⎟( T∞
− Tmn
, ) + h⎜ ⋅⎟( T∞
− Tmn
, ) = 0
⎝ 2 ⎠ ⎝ 2 ⎠
or, with ∆ x=∆y,
h∆x
⎛h∆x
⎞
Tm−1, n+ Tmn , −1 + 2 T∞
− 2⎜
+ 1⎟Tmn
, = 0
k ⎝ k ⎠
(4.43)

• Note potential utility of using thermal resistance concepts to express rate
equations. E.g., conduction between adjoining dissimilar materials with
an interfacial contact resistance.
q
( mn , −1 ) →( mn , )
=
T
−T
mn , −1 mn ,
R
tot
R
tot
∆y/2 Rtc
′′
, ∆y/2
= + +
k x x k x
A
( ∆ ⋅) ∆ ⋅ ( ∆ ⋅)
B
(4.46)

Solutions Methods
• Matrix Inversion: Expression of system of N finite-difference
equations for N unknown nodal temperatures as:
[ A] [ T] = [ C]
(4.48)
Coefficient
Matrix (N x N)
Solution Vector
(T 1 ,T 2 , …T N )
Right-hand Side Vector of Constants
(C 1 ,C 2 …C N )
−1
Solution [ T] = [ A] [ C]
(4.49)
Inverse of Coefficient Matrix

• Gauss-Seidel Iteration: Each finite-difference equation is written in
explicit form, such that its unknown nodal temperature appears
alone on the left-hand side:
C a
a
T T T
( k) i−1
( )
N
i
ij k ij ( k −1)
i = − j −
j
aii j= 1aii j= i+
1aii
∑ ∑ (4.51)
where i =1, 2,…, N and k is the level of iteration.
Iteration proceeds until satisfactory convergence is achieved
for all nodes:
i
( k) ( k−1)
T −T ≤ ε
i
• What measures may be taken to insure that the results of a
finite-difference solution provide an accurate prediction of the
temperature field

Example 4.3 A long column in a furnace (no heat generation)

Matrix Inversion

Gauss-Seidel Iteration

Homework 5
• 4.4
• 4.23
• 4.51 (a)