This document summarizes key concepts in Chapter 12 on temperature distribution with more than one independent variable. It discusses unsteady heat conduction in solids, including heating of a finite slab and cooling of a sphere in contact with a fluid. It also covers steady heat conduction in laminar incompressible flow, including complete and asymptotic solutions, as well as boundary layer theory for non-isothermal flow. Dimensionless variables, differential equations, boundary conditions, and solution methods like separation of variables and Laplace transforms are presented.

1. Ch t 12 T t di t ib tiChapter 12. Temperature distribution
with more than one independent variablep
U t d h t d ti i lid• Unsteady heat conduction in solids
• Steady heat conduction in laminar,y ,
incompressible flow
• Steady potential flow of heat in solids• Steady potential flow of heat in solids
• Boundary layer theory for non-isothermal
flow
1
Unsteady heat conduction in solidsUnsteady heat conduction in solids
• Energy equation
T
 Tk
t
T
Cˆ p 



• For constant k
t
• For constant k
T
T 2



t
2
Heating of a finite slabHeating of a finite slab
I i i l T• Initial temperature = To
• T1 wall temperature at t = 01 p
• Dimensionless variables
1 tyTT 


 2
01 bbTT



2b
3
Heating of a finite slabHeating of a finite slab
• Differential equationq
2

2





• Boundary conditions

10atIC 
2b
01at21BC 
4

2. Solvingg
• Separation of variables
      gf
• Substitution into equation
      gf,
• Substitution into equation
2
22
f1gd1f1gd1  2
22
c
f
f
1
d
gd
g
1f
f
1
d
gd
g
1










• Separation into two differential equationsp q
fc
f
gc
gd 2
2
2



5
fcgc
d 2




Solvingg
• Integrating
   ccosCcsinBfcexpAg 2
• Temperature distribution


TT
TT1
    
 TT
22n
01
 



 





 




 





 







b
y
2
1
ncos
b
t
2
1
nexp
1
1
2 2
22
0n
n
6
  




  b2b2
2
n0n
7 8

3. 9
U t d h t d ti i lidUnsteady heat conduction in solids
• Cooling of a sphere in contact with a well-
stirred fluidstirred fluid
• Fluid temperature, To
• Sphere temperature, T1
• Insulated tank• Insulated tank
• Determine T=T(t)
10
Cooling of a sphere in contact with ag p
well-stirred fluid
i i l i bl• Dimensionless variables
TT
 etemperatursolid
TT
TT
),(
o1
s1
s 



 fl id
TT
)( f1
o1

  etemperaturfluid
TT
TT
)(
o1
f1
f 


   time
t
coordinateradial
r s



11
   time
R
coordinateradial
R 2

Equations and BCsEquations and BCs
• Solid Fluid
 3d 














 s2
2
s 1
t 1
sf
B
3
d
d





 t
00t 
1Bd 

1
0,0at s

 1,0at f 
,1at fs 
12finite,0at s 

4. Applying Laplace transform method
• Solid Fluid



 


 s21
p s3
1p




 


 2sp
1
f
B
1p


,1at fs 
finite,0at s 
13
Solution at the Laplace plan (x and p)p p ( p)
• Solid CC 21
pcosh
C
psinh
C 21
s 




0C2BCgsinU 2 
• Fluid
 
 









p3ptanhpBp3
ptanhp11
3
p
1
f
   p3ptanhpBp3p
14
Solution at the plane x and tSolution at the plane x and t
• Temperature of the fluid in function of timeTemperature of the fluid in function of time
 1 pN  fff VCˆ 
 
1
f
pD
pN
L31






 
spss
fpff
VCˆ
VC
B



 2
kbexp
B6
B 
 

p
 
 1k
2
k
2
k
f
bBB19
p
B6
B1 


 

 
 
kb3
btanofrootsareb 
15
 
 2
k
kk
Bb3
btanofrootsareb


16

5. Steady heat conduction in laminary
incompressible flow
• Laminar tube flow with constant heat
flux at the wall.
– Complete solution by the method of
separation of variables (Ex 12 2-1)separation of variables (Ex 12.2-1)
– Asymptotic solution for short distance
down the tube by the method ofdown the tube by the method of
combination of variable. (Ex 12.2-2)
A i l i f l di– Asymptotic solution for large distances
down of tube. (§10.8)
17
Complete solutionComplete solution
• Using the dimensionless variables (§ 10 8) a• Using the dimensionless variables (§ 10.8) a
solution is proposed
  


 



1
1 2
  


 




1 2
            ,,, d
18
Complete solutionp
• The boundary conditions for the dampingy p g
function is

01at00at dd







 00at d 

• Assumption
 0,0at d  
      ZX,d
19
Complete solutionp
• Separation of variablesSeparation of variables
  0X1c
XdXd1
Zc
Zd 22




   0X1c
dd
Zc
d



 




        

 k
2
kk XcexpB,,
    

1 2
1k
    
    
 


1 22
1
0
2
k
k
d1X0,
B
20
     
1
0
22
k
k
d1X

6. Asymptotic solution for the entrance region
• Assumptions
– No curvature effects, flat surface. Using y=R-r
– Fluid is a semi-infinite medium
– Linear velocity profile
 RPP 2–
 
 
L2
RPP
v
R
y
vyv
2
L0
00z



2
2
o
y
T
z
T
R
y
v






• Can be solved by combination of variables
yzR 
21
• Can be solved by combination of variables
Boundary layer theory for non-isothermal
flow
C ti it • Continuity
• Motion 0
y
v
x
v yx






 




 
TT
vvvv x
2
exx
yx 
 












 TTg
yx
v
y
v
x
v x2
xe
e
x
y
x
x
• Energy
2
x
2
2
yxp
vT
k
T
v
T
vCˆ 






















22
2yxp
yyyx  

 

Von Karman momentum and energy
balances
M t• Momentum

   dyvv
x
v
dyvvv
dx
d
y
v
0
xe
e
xex
0
x









 dyTTg
xdxy 000y





• Energy
 dyTTg
0
x 
 dyTTvCˆ
d
dT
k xp 


 

23
  y
dxy
xp
00y


 
 24

7. Heat transfer in laminar forced convection.
Von Karman integral method
• Velocity and temperature profiles
43
  xyfor
yy
2
y
2
v
v
43
x






















yyyTT
v
43

   
 xyfor
yy
2
y
2
TT
TT
TTTo
o

























 xyfor1
TT
and1
v ox
TTTo



25
 xyfor1
TT
and1
v o
ox




Results
• Assuming constant  xT

g
ratio and with
 
 x
xT


1
  


 

x1260
x  




v37
x
  




x4
x  




v
180
1
140
3
15
2
x
43
T
26
18014015
lResults
• Equation for 
1653 37132 1653
Pr
315
37
180
1
140
3
15
2 

• Solving• Solving
  3
1
Pr

  
27