A brief introduction to Numerical methods. Application of Numerical Methods (Finite Difference) in Heat Transfer.

1. Application of Numerical Methods
(Finite Difference) In
Heat Transfer
Presented By:
Shivshambhu Kumar
Roll No.-15MPE15
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2. Introduction:
 The evolution of numerical methods, especially Finite
Difference methods for solving ordinary and partial
differential equations, started approximately with the
beginning of 20th
century (1)[1].
 Many problems in engineering and science can be
formulated in terms of differential equations. A differential
equation is an equation involving a relation between an
unknown function and one or more of its derivatives.
 Equations involving derivatives of only one independent
variable are called ordinary differential equations and may
be classified as either initial-value problems (IVP) or
boundary-value problems (BVP)[2].
2

3. Heat Equation:
The one-dimensional heat equation is
2
2
,0 , 0x L t
x x
φ φ
α
∂ ∂
= < > >
∂ ∂
3

4. Numerical Methods for Unsteady Heat Transfer
2 2
2 2
1 T T T
t x yα
∂ ∂ ∂
= +
∂ ∂ ∂
Unsteady heat transfer equation, no generation, constant k, two-
dimensional in Cartesian coordinate:
To discretize the Laplacian operator into system of finite difference
equations using nodal network. For the unsteady problem, the temperature
variation with time needs to be discretized too. To be consistent with the
notation from the book, we choose to analyze the time variation in small
time increment ∆t, such that the real time t=p∆t. The time differentiation
can be approximated as:
1
, ,
,
........................( )
while m & n correspond to nodal location
such that x=m x, and y=n y.
P P
m n m n
m n
T TT
A
t t
+
−∂
≈
∂ ∆
∆ ∆
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5. The Finite-Difference Method
• An approximate method for determining temperatures at discrete (nodal) points
of the physical system and at discrete times during the transient process.
• Procedure:
─ Represent the physical system by a nodal network, with an m, n notation used
to designate the location of discrete points in the network,
─ Use the energy balance method to obtain a finite-difference equation for
each node of unknown temperature.
─ Solve the resulting set of equations for the nodal temperatures at
t = ∆t, 2∆t, 3∆t, …, until steady-state is reached.
and discretize the
problem in time by designating a time increment ∆t and expressing the time
as t = p∆t, where p assumes integer values, (p = 0, 1, 2,…).
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6. The Explicit Method of Solution
• All other terms in the energy balance are evaluated at the preceding time
corresponding to p. Equation (A) is then termed a forward-difference
approximation.
• Example: Two-dimensional conduction
for an interior node with ∆x=∆y.
( ) ( )1
, ,1, 1, , 1 , 1 1 4p p p p p p
m n m nm n m n m n m nT Fo T T T T Fo T+
+ − + −= + + + + −
( )2
finite-difference form o Fourf ier number
t
Fo
x
α∆
= →
∆
• Unknown nodal temperatures at the new time, t = (p+1)∆t, are determined
exclusively by known nodal temperatures at the preceding time, t = p∆t, hence
the term explicit solution.
6

7. 1
, ,..............................
0
p p
m n m nT AT
A
+
= +
≥
Hence, for the two-dimensional interior node:
( )1 4 0Fo− ≥
1
4
Fo ≤
( )2
4
x
t
α
∆
∆ ≤
For a finite-difference equation of the
form,
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8. Marching Solution
• Transient temperature distribution is determined by a marching solution,
beginning with known initial conditions.
1 ∆t -- -- -- ……………
--
Known
2 2∆t -- -- -- ……………
--
3 3∆t -- -- -- ……………
--
. .
. .
. .
. .
. .
. .
Steady-state -- -- -- -- …………….
--
p t T1 T2 T3……………….. TN
0 0 T1,i T2,i T3,i………………. TN,i
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9. Finite Difference Equations
From the nodal network to the left, the heat
equation can be written in finite difference
form:
( )
( )
1
, , 1, 1, , , 1 , 1 ,
2 2
2
1
, 1, 1, , 1 , 1 ,
2 21
( ) ( )
t
Assume x= y and the discretized Fourier number Fo=
x
(1 4 )
This is the
P P P P P P P P
m n m n m n m n m n m n m n m n
P P P P P P
m n m n m n m n m n m n
T T T T T T T T
t x y
T Fo T T T T Fo T
α
α
+
+ − + −
+
+ − + −
− + − + −
= +
∆ ∆ ∆
∆
∆ ∆
∆
= + + + + −
expl , finite difference equation for a 2-D,
unsteady heat transfer equation.
The temperature at time p+1 is explicitly expressed as a
function of neighboring temperatures at an earlier time p
icit
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10. Nodal Equations
Nodal equation can be written by using different points in the
given problem, there is a stability criterion for each nodal
configuration. This criterion has to be satisfied for the finite
difference solution to be stable. Otherwise, the solution may be
diverging and never reach the final solution.
For example, Fo≤1/4. That is, α∆t/(∆x)2
≤1/4 and ∆t≤(1/4α)(∆x)2
.
Therefore, the time increment has to be small enough in order to
maintain stability of the solution.
This criterion can also be interpreted as that we should require the
coefficient for TP
m,n in the finite difference equation be greater
than or equal to zero.
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11. Finite Difference Solution:
Steps used to solve the finite difference equation:
1.First, by specifying initial conditions for all points inside the
nodal network. That is to specify values for all temperature at
time level p=0.
2. Important: check stability criterion for each points.
3.From the explicit equation, we can determine all temperature at
the next time level p+1=0+1=1. The following transient response
can then be determined by marching out in time p+2, p+3, and so
on.
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12. Example
Example: A flat plate at an initial temperature of 100 deg. is suddenly
immersed into a cold temperature bath of 0 deg. Use the unsteady finite
difference equation to determine the transient response of the temperature
of the plate.
1
2 3
x
L(thickness)=0.02 m, k=10 W/m.K, α=10×10-6
m2
/s,
h=1000 W/m2
.K, Ti=100°C, T∞=0°C, ∆x=0.01 m
Bi=(h∆x)/k=1, Fo=(α∆t)/(∆x)2
=0.1
There are three nodal points: 1 interior and two
exterior points: For node 2, it satisfies the case 1
configuration in table.
1
2 1 3 2 2 2 1 3 2
1 3 2
( ) (1 4 ) ( ) (1 2 )
0.1( ) 0.8
1
Stability criterion: 1-2Fo 0 or Fo=0.1 ,it is satisfied
2
P P P P P P P P P
P P P
T Fo T T T T Fo T Fo T T Fo T
T T T
+
= + + + + − = + + −
= + +
≥ ≤
12

13. 1
1 2 1 1 1
2 1 2 1
1
3 2 3
For nodes 1 & 3, they are consistent with the case 3 in table.
Node 1: (2 2 ) (1 4 2 )
(2 2 ) (1 2 2 ) 0.2 0.6
Node 3: 0.2 0.6
Stability cr
P P P P P
P P P P
P P P
T Fo T T T BiT Fo BiFo T
Fo T BiT Fo BiFo T T T
T T T
+
∞
∞
+
= + + + + − −
= + + − − = +
= +
1
1 2 1
1
2 1 3 2
1
3 2 3
1
iterion: (1-2Fo-2BiFo) 0, (1 ) 0.2 and it is satisfied
2
System of equations
0.2 0.6
0.1( ) 0.8
0.2 0.6
P P P
P P P P
P P P
Fo Bi
T T T
T T T T
T T T
+
+
+
≥ ≥ + =
= +
= + +
= +
Use initial condition, T = T = T = 100,1
0
2
0
3
0
T T T
T T T T
T T T
1
1
2
0
1
0
2
1
1
0
3
0
2
0
3
1
2
0
3
0
0 2 0 6 80
01 0 8 100
0 2 0 6 80
= + =
= + + =
= + =
. .
. ( ) .
. .
Marching in time, T = T = 80, T = 100
, and so on
1
1
3
1
2
1
T T T
T T T T
T T T
1
2
2
1
1
1
2
2
1
1
3
1
2
1
3
2
2
1
3
1
0 2 0 6 0 2 100 0 6 68
01 0 8 01 80 0 8 100 96
0 2 0 6 0 2 100 0 6 68
= + = + =
= + + = + + =
= + = + =
. . . ( ) . (80)
. ( ) . . (80 ) . ( )
. . . ( ) . (80)
Example
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14. References:
1. John C, Dale Anderson, Richard Pletcher. Computational Fluid Mechanics and Heat
Transfer. Second Edition. USA : Taylors&Francis, 1997.
2. Davis, Mark E.. Numerical methods and modeling for chemical. New York : John Wiley &
Sons, 2001
3. Introduction to Heat Transfer by S.K SOM Page 177-192
4. http://nptel.ac.in/
5. https://en.wikipedia.org
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