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Unsteady State Conduction

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					          HEAT TRANSFER
                (ME 421)

              Spring 2012

                Chapter5

           Kazem M. Osaily
            kazemosaily@hotmail.com


Department of Mechanical Engineering PPU
           Unsteady State Conduction
           (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

				
DOCUMENT INFO
Description: Unsteady State Conduction (Transient Heat Transfer)