Download Numerical Analysis for Heat Conduction

1
5. NUMERICAL ANALYSIS FOR HEAT CONDUCTION
1. Why Numerical Methods
› applicable to complex geometries and boundary conditions,
and variable thermal properties
› the advanced mathematical skills are not required
› easily fit to practical “what-if” optimization problems
› the advance in computing technologies
2. Steady-State Heat Conduction
- Numerical Formulation for Interior Nodes
› 1D heat conduction
E element
q cond ,left  q cond ,right  G element 
0
t
where:

T
T
T
T
kA m 1 m  kA m 1 m  g m Ax  0
x
x
g
Tm 1  2Tm  Tm 1  m x 2  0
k
or
m = 1,2,…,M-1
2
› 2D and 3D heat conduction
Consider the discretized element (xy1) around the node
(m,n):
E element
q cond ,left  q cond ,top  q cond ,right  q cond ,bottom  G element 
0
t
where:
Assuming x=y, we can obtain after substituting into the
above energy balance equation:
Tm1,n  Tm1,n  Tm,n1  Tm,n1  4Tm 
g m,n
k
x 2  0 m = 1,2,…,M-1
n = 1,2,…,N-1
3
- Boundary Conditions
Take an internal corner with surface convection as a sample:
qcond ,left  qcond ,top  qcond ,right  qcond ,bottom  qconv  G element  0
T
T
where: qcond ,left  k (y  1) m1,n m,n
x
Tm,n1  Tm,n
q cond ,top  k (x  1)
y
Tm1,n  Tm,n
y
q cond ,right  k (  1)
2
x
Tm,n1  Tm,n
x
q cond ,bottom  k (  1)
2
y
y
x
qconv  h(  1)(T  Tm,n )  h(  1)(T  Tm,n )
2
2
3
G element  g mVelement  g m,n (  x  y  1)
4
Assuming x=y, we can obtain after substituting into the
above energy balance equation:
2(Tm,n1  Tm1,n )  Tm,n1  Tm1,n  2
hx
hx
3 q (x) 2
T  2(3 
)Tm,n 
0
k
k
2 k
4
- Solution Methods
› Matrix Inversion Method/Gaussian Method
Noting nodal equations are linear algebraic equations, they
can be written as:
a11T1  a12T2    a1N TN  C1
a 21T1  a 22T2    a 2 N TN  C 2




a N 1T1  a N 2T2    a NN TN  C N
or
AT   C 
where:
a11

A= a21
 

a N 1
a12  a1N 
a22  a2 N 





a N 2  a NN 
T1 
C1 
 
 
T = T2  C = C 2 
 
 
 
 
TN 
C N 
By inverting the matrix A, the solution matrix is:
T   A1 C 
where:
b11 b12  b1N 


 A1 = b21 b22  b 2 N 
 

 


bN 1 bN 2 bNN 
› Gauss-Seidel Method
1. Re-order the nodal equations as much as possible so that
|aii| > |aij|, where i  j
2. Express the Ti ' s in explicit form with the rest of nodal
temp.
Ti( k ) 
N aij ( k 1 )
Ci i 1 aij ( k )
  Tj  
Tj
aii j 1aii
j i 1 aii
3. Assume the initial Ti( o )' s for each node reasonably
5
4. Calculate the new Ti( k )' s by substituting the assumed
temperatures
5. Repeat the above procedure to calculate the new Ti( k 1 )' s
with the most recent estimates of Ti( k )' s until the
convergence criterion is satisfied
Ti( k )  Ti( k 1 )  
3. Transient Heat Conduction
- FD Discretization
Space discretization:
Time discretization:
t
= incremental time step
T p = nodal temperature at the beginning of the time step
T p 1 = nodal temperature at the end of the time step
- Explicit FD Method
Consider a 2D inner node (m,n) at a time step (tp+1 –tp):
6
Eelement
p
p
 q  G element 
t
all sides
where:
q ( m1,n)( m,n)  k (y  1)
q ( m,n1)( m,n)  k (x  1)
q ( m1,n )( m,n )  k (y  1)
q ( m,n 1)( m,n )  k (x  1)
Tmp1,n  Tmp,n
x
 Tmp,n
Tmp,n 1
y
Tmp1,n  Tmp,n
x
Tm,n 1  Tm,n
y
G element  g mVelement  g m ,n (x  y  1)
Substituting them into the energy equation leads to:
Tmp1,n  Tmp,n
Tmp,n1  Tmp,n
Tmp1,n  Tmp,n
ky
 kx
 ky
x
y
x
C p x  y (T mp,n1  T mp,n )
Tmp,n1  Tmp,n
 kx
 g m ,n (x  y ) 
y
t
If x  y and Fo  t / (x) 2 , it can be further simplified as:
Tmp,n1

Fo(Tmp1,n
 Tmp1,n
 Tmp,n1
g m ,n x 2
 Fo
k
m = 1, 2, …, M-1
n = 1, 2, …, N-1
 Tmp,n1 )  (1  4 Fo)Tmp,n
Note: see Table 5.2 for other transient FD nodal equations
including the nodes subject to the boundary conditions
7
› Solution Technique
Noting that Tmp,n1 is explicitly expressed as Tmp,n ’s, the solution
technique is:
1) write the FD equation at each node,
2) starting at p  0 , we know all Tm0,n from initial conditions,
3) at t1  t , the temperature change commences,
4) solve the FD equations for Tm1 ,n at all the nodes,
5) renew the Tmp, n with the most current temperature at the
node,
6) keep track of the time t p1  t p  t
7) repeat the above process in next time steps until the final
time is reached.
› Stability Criterion
Note that if t is too large or if x is too small, the Tmp,n
coefficient would become negative, which may cause the
solution oscillating and being physically impossible.

t must be so chosen that the coefficients of Tmp,n  0 !
For the above 2D interior nodes, the stability criterion is:
8
- Implicit FD Method
Instead of FD formulation at tp, we replace the spatial
derivatives at tp+1,
In similar way, we can write the energy balance FD equation:
ky
Tmp11,n  Tmp,n1
x
 kx
Tmp,n11  Tmp,n1
y
 ky
Tmp11,n  Tmp,n1
x
C p x  y(Tmp,n1  Tmp,n )
Tmp,n11  Tmp,n1
 kx
 g m ,n (x  y) 
y
t
If x  y and Fo  t / (x) 2 , it can be further simplified as:
(1  4 Fo)Tmp,n1
 Fo(Tmp11,n
 Tmp11,n
 Tmp,n11
 Tmp,n11 )
 Tmp,n
g m ,n x 2
 Fo
k
a) See Table 5.2 for other implicit FD equations
b) The above equations are implicit, thus, must be solved
simultaneously, e.g., using Gauss/Seidal method
c) The solution of the implicit FD method is unconditionally
stable for any chosen x and t.
9
4. Some Remarks
- Irregular Boundaries
Approximate it with a series of simple volume elements.
- Controlling the Numerical Errors
› Types of numerical errors
Discretization error Round-off error
› Controlling the numerical errors
1) Discretization error: check the solution accuracy so that
it is grid-independent (e.g, x x/2, x/4, ,).
2) Round-off error: check the solution accuracy with more
significant digits.
3) Reduce the grid size in regions of importance.
10
- In summary
› Overlay the system with a rectangular grid
› Place nodes at the intersections of the gridlines
› Associate an incremental area (volume) with each node
› Write energy balance on each area in terms of nodal T
› Solve the resultant simultaneous equations for each of the
nodal temperatures