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```					          HEAT TRANSFER
(ME 421)

Spring 2012

Chapter5

Kazem M. Osaily
kazemosaily@hotmail.com

Department of Mechanical Engineering PPU
(Transient Heat Transfer)
 2 T 1 T

x  2
 t

Initial Condition(I.C.);               x


T  Ti @ t = 0; 0  x  2L
2L
Boundary Conditions (B.Cs.);

T  T1 @ t > 0; x = 0 and x  2L
Changing variable:
  T Ti

 2  1 

x  t
2

New Initial and Boundary Conditions:

  i = Ti - T1 @ t = 0; 0  x  2L
0               @ t > 0; x = 0
0             @ t > 0; x = 2L

Solution Method: Separation of Variable

                 (x,t)  X(x)H(t)
2X 2
 X0          X(x)  C1 cosx  C 2 sin x
x 2

2 t
 2H                 H(t)  Ce
 2 H  0
t 2
2 t
(x,t)  X(x)H(t)  (C1 cosx  C 2 sin x)e


  0 @ t > 0; x = 0     C1  0                             n
      0 @ t > 0; x = 2L      sin(2L)  0                  
2L

                     n 2
n     (
(x,t)   C n sin( x)e
) t
2L

n1       2L                

n
  i @ t = 0; 0  x  2L                        i   C n sin( x)
n1       2L
2L
n
 i  sin( x)dx
2L       4
C n  2L  0
    i   
n      n
  sin 2 ( x)dx
2L
0

n 2
(      ) t
 T  T1 4  e              2L
n
                               sin( x)
 i Ti  T1  n1 n                      2L


Lumped Heat Capacity System
Quenching with uniform temperature.
Condition: It is realistic for small physical size bodies.

Heat transferred to the environment=change of internal energy

dT
hA(T  T )  cV
d

I.C.:                 T  T0 @  = 0

T  T
            e [hA/cV]

T0  T
Validity of the Lumped Capacity Method

hV / A hs
Biot number        
k     k

The lumped capacity method is only valid when:


Biot number  Bi  0.1


Example1
A steel ball (5Cm in diameter) with uniform temperature of 450 oC
is suddenly placed in a 100 oC environment. How much time needed
for the ball to attain a temperature of 150 oC?
( c  0.46 kJ / kg C; k  35W /m C; h =10 W/m C )
0               o             2o

Solution:
4 / 3r 3
10(           )
hV / A         r 2
Biot number                           0.0023  0.1
k           35
T  T
Lumped-Capacity Method;                           e [hA/cV]
T0  T
hA

10(r 2 )                  150 100             4

cV 7800  460  (4 / 3  r 3 )   ;             e 3.34410    

450 100
 3.44 10 4
                          5819 s 1.62 h
Example2(Quenching of a steel plate)
Transient Heat Flow
in Semi-Infinite Solid
T0
 T 1 T
2

x 2
 t

Initial Condition(I.C.); T(x,t)  T(x,0)  Ti

Boundary Conditions (B.Cs.); T(x,t)  T(0,t)  T0 for                         t >0

Using Laplace-Transform:

T(x,t)  T0   2 x / 2                                       x

t        2
                           e          d  erf
Ti  T0                                                  2 t
Solution is shown in Figure 4-4, Page 137 of the book!               T kA(T0  Ti )
q x  kA      
x    t
Constant Heat Flux on Semi-Infinite Solid
(Transient Heat Flow)
 2 T 1 T

x  2
 t

Initial Condition(I.C.); T(x,t)  T(x,0)  Ti
q0      T
Boundary Conditions (B.Cs.);          k         for t > 0
A       x x =0


x2
2q t /                  q 0x           x
T(x,t)  Ti  0          e      4 t
      (1 erf      )
kA                       kA           2 t
Convection Boundary Condition on
Semi-Infinite Solid (Transient Heat Flow)
 2 T 1 T

x  2
 t

Initial Condition(I.C.);              T(x,t)  T(x,0)  Ti

T
Boundary Conditions (B.Cs.);                hA(T  T ) x 0    kA
x x =0

hx h 2 t
T(x,t)  Ti           x                                     x    h t
1 erf       {e        k   k2
[1 erf(            )]}
T  Ti            2 t                                   2 t    k

t   kt
Fo .Bi   2       ;       Fo       
s 2 cs 2

Solution is shown in Figure 4-5, Page 140 of the book!
Infinite surfaces
(suddenly subjected to convection environment)
Definition: Surface whose thickness is smaller than the other
two dimensions.

  T(x,t)  T      or   T(r,t)  T
 0 
 i  Ti  T                                            
i i 0
 0  T0  T
The solution is plotted in Figures 4-7 through 4-12
      Slab
hs
       
Biot Number:       Bi          &             s  ro Sphere
k                         r Cylinder
 o
t    kt
Fourier Number:        Fo  2 
                 s    cs 2

Note:
If the centerline temperature is desired only one chart/graph
is required. (Charts/graphs 4-7 through 4-9)

To determine an off-center temperature two charts/graphs are
required. (Charts/graphs 4-7 through 4-9 and
Charts/graphs 4-10 through 4-12)

**If Bi<0.01 and Fo>0.2 then Chart/graph 4-13 should be used for
Small value of h, convection heat coefficient.

***To evaluate the HEAT LOSS charts/graphs 4-14 through 4-16
should be used. In these graphs Q 0  cV(Ti  T ) .


Multi-Dimensional Systems
Semi-Infinite rectangular bar
Semi-infinite Plate    Infinite rectangular bar
                          
                                                    
                                               i
i                      i

                      

C(X) :Solution for Infinite Cylinder
P(X) :Solution for Infinite Plate
S(X) :Solution for Semi - Infinite Solid
Rectangular parallelepiped                               Short Cylinder
Semi-Infinite Cylinder   
                                                       
                                                 i
i                           i




C() :Solution for Infinite Cylinder
P(X) :Solution for Infinite Plate
S(X) :Solution for Semi - Infinite Solid
Heat Transfer In Multi-Dimensional Systems

Intersection of Two bodies:

 Q        Q   Q    Q 
          1
Q 0 
total Q 0  Q 0   Q 0 
1        2           1


Intersection of Three bodies:
(Semi-infinite rectangular bar; rectangular parallelepiped)

 Q                                                            
 Q   Q    Q   Q    Q   Q  
               1   
1                     1         
Q 0 
total Q 0  Q 0   Q 0  Q 0   Q 0  Q 0  
1        2           1          3           1        2

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Description: Unsteady State Conduction (Transient Heat Transfer)