The document discusses heat transfer analysis through thermal conduction and convection. It provides the governing equations for 1D transient and steady-state heat conduction using the finite element method. Weak formulations are presented for both cases. It also discusses formulations for linear elements and presents the element equations. Time integration methods for transient problems like explicit and implicit Euler are introduced. Stability criteria like the Courant–Friedrichs–Lewy condition for the time step is also covered. An example transient heat conduction problem is given at the end.

1. 110
UNIT - 4
Heat transfer analysis

2. ThermalConvection
Newton’s Lawof Cooling
q  h(Ts T)
h: convective heat transfer coefficient (W m2
Co
)
111

3. 112
Thermal Conduction in1-D
Boundary conditions:
Dirichlet BC:
Natural BC:
Mixed BC:

4. Weak Formulation of 1-D HeatConduction
(SteadyState Analysis)
• GoverningEquationof 1-D Heat Conduction -----

d (x)A(x)
dT(x)   AQ(x)  0
dx  dx 
 
0<x<L
• WeightedIntegral Formulation -----
• WeakFormfrom Integration-by-Parts -----
0
L
dx
 d  dT(x)  
  AQ(x)dx
 
 
0  w(x) 
dx (x)A(x)
L
dw  dT  
113

L
dT 
   0
0  
0   dx  A
dx   wAQdx  w A
dx 

5. Formulation for 1-D LinearElement
Let T(x)  T11(x)T22(x)
f2
1
x1
2
x2
x
T1 T2
f1
2 1
1 2
l l
x  x
,
x  x
 ( x)  ( x) 
x2
x1
1T1
2T2
2
2
1
1
x
x
, f (x)  A
T
f (x)  A
T
114

6. Formulation for 1-D LinearElement
Letw(x)= i (x), i =1, 2
2 x2
i j
j i i 2 2 i 1 1
j1
x2 
 d d 
  ( x )f 
dx   AQdx (x ) f
 dx dx 
 
x 
 1  1
0  T  A
 
x
2
Qi i (x2 ) f2 i (x1) f1
 KijTj
j1
 



   
 11
1
1
K22 T2
K12
K12 T1 
K
 2  Q2
 f   
f Q
1
115
1
2
x2 x2
i j
ij i i 1
x1 x2
x
dT dT
dx dx
 d d 
where K  A dx, Q   AQdx, f  A , f  A
 dx dx 
 
 
x

7. Element Equations of 1-D LinearElement
 




 
 1
1
1 T2 
1 1T1 
L 1
 2  Q2
Q
 f   
f A
 
i i 1
dT dT
dx xx1 xx2
where Q   AQ dx, f  A , f2   A
dx
x2

x1
f2
1
x1
2
x2
T1
x
T2
116
f1

8. 1-D Heat Conduction -Example
t1 t2 t3
0
T 200o
C T 50o
C
x
1 2 3
Acomposite wall consists of three materials, asshown in the figure below.
Theinside wall temperature is 200oCand the outside air temperature is50oC
with aconvection coefficient of h =10 W(m2.K). Find the temperature along
the composite wall.
1  70W m  K , 2  40W m  K , 3  20W m  K 
t1  2cm, t2  2.5cm, t3  4cm
117

9. Thermal Conduction andConvection-
Fin
Objective: to enhanceheattransfer
dx
t
x
w
loss
Q 
2h(T T)dx  w  2h(T T)dx t

2h(T T)w  t
Ac dx Ac
Governing equation for 1-D heat transfer in thin fin
c c
d  A
dT   A Q 0
dx  dx 
 
c
118
 c
d  A
dT   Ph T  T  A Q  0
dx  dx 
 
P  2w  t
where

10. Fin- WeakFormulation
(SteadyStateAnalysis)
• GoverningEquationof 1-D Heat Conduction -----


d (x)A(x)
dT(x)   PhT T  AQ 0
dx  dx 
 
0<x<L
• WeightedIntegral Formulation -----
• WeakFormfrom Integration-by-Parts -----
0
L
dx
 d  dT(x)  
  Ph(T T)  AQ(x)dx
 
 
0  w(x) 
dx (x)A(x)
dx
119
dx dx

L
dw  dT   
L
dT 
0  A  wPh(T T
 )  wAQ dx  w A
   

   0
0  


11. Formulation for 1-D LinearElement
Letw(x)= i (x), i =1, 2
2
dx dx
d d
j1

x2
 
iAQ  PhTdx
A i j
 Ph dx 
i j 

 x 
 1  1
i (x2 ) f2 i (x1) f1
0  T 
j 
x2


x
2
i (x1) f1
Qi i (x2 ) f2
 KijTj
j1
 



 

 
 11
1
1
K22 T2
K12
K12 T1 
K
 2  Q2
 f  
f Q
1 1
1 2
x2
ij i i
i j
dx dx
dx dx
d d
A i 
xx1 xx2
x2
 
where K   Ph dx, Q   AQ  PhT dx,
j 
x  
f   A
dT
, f  A
dT
 
x
120

12. Element Equations of 1-D LinearElement
1 1
f Q  1
    1 T1 

A  1

Phl 2
 f          
 2  Q2  1  6 1
 L 1 2T2 
f2
1
x=0
2
x=L
T1
x
T2
f1
x2
121
dT dT
xx1 xx2
f1   A
dx
, f2   A
dx
where Qi  i AQ  PhT dx,
x1

13. 122
Time-Dependent Problems

14. 123
Time-DependentProblems
In general,
Keyquestion: How to choose approximatefunctions?
Two approaches:
ux,t
ux,t u jj x,t
ux,t u j tj x

15. Model Problem I– Transient Heat Conduction
Weakform:
 
t x

x

c
u

 a
u   f x,t
1 1
1
a )Q2w(x2 )





x2

 wf dx Q w(x
u
t
 cw
w u
x x
0  
x
2
124
1
x2
x1
dx


Q  a
du
dx


Q  a
du  ;

16. let:
)
1 1 2 2
1
a 




x2

 wf dx Q w(x ) Q w(x
u
t
 cw
w u
x x
0  
x
Transient HeatConduction
n
ux,t u j tj x
j1
and w i x
KuMu 
F

x2
x1
ij
x x
K  a i j
dx
 
Mij
x2
 cijdx
x1
x2
Fi  i fdx  Qi
x1
ODE!
125

17. TimeApproximation – First OrderODE
dt
126
0  t  T 0
a
du
 bu  f t u0u
Forward difference approximation - explicit
Backward difference approximation - implicit
k k k
a
k1
u  u
 t
f bu 
k k k
k1
u  u 
t
f bu 
a bt

18. StabilityRequirment
max
127
2
1 2
cri
t  t 
K M uQ
where
Note: Onemust usethe samediscretization for solving
the eigenvalueproblem.

19. Transient Heat Conduction -Example
0
128
u

2
u
 0  x 1
t
u
1,t 0
t x2
u0,t 0
ux,01.0
t  0

20. Transient Heat Conduction -Example
129

21. Transient Heat Conduction -Example
130

22. Transient Heat Conduction -Example
131

23. Transient Heat Conduction -Example
132

24. Transient Heat Conduction -Example
133