Convection in the General Property Balance

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					Convection in the General
   Property Balance

Development of the full equations of motion
Control volume analysis:
 Based on application of the balance:

      Input + Generation =
     Output + Accumulation
For a conserved property y and corresponding flux Y
  Consider a control volume in Cartesian coordinates:
                                 Y zA ) 2
                                 (
dV = dx dy dz
                       dy                        Y yA ) 2
                                                 (


      Y xA ) 1
      (                                          Y xA ) 2
                                                 (
                  dz
                                                   z
      Y yA ) 1
      (                     dx                              y

                                                            x
                                 Y zA ) 1
                                 (

Property transport entering or leaving each face of the form, Y A
                  where A is an area element
                                                               Y zA ) 2
                                                               (
Input + Generation =
                                                                          Y yA ) 2
                                                                          (
Output + Accumulation                                dy


                                     Y xA ) 1
                                     (                                    Y xA ) 2
                                                                          (
                                                dz
Generation:                                                                 z
                                     Y yA ) 1             dx
        y G dV  y G dx dy dz
                                     (                                               y

                                                                                     x
                                                               Y zA ) 1
                                                               (

                   y        y 
Accumulation:          dV       dx dy dz
                   t        t 

Input:      Yx 1 dy dz  Yy 1 dx dz  Yz 1 dx dy

 Output:     Yx 2 dy dz  Yy 2 dx dz  Yz 2 dx dy
Rearrange the balance:

Accumulation = Output – Input + Generation


Next …

Focus on terms for [Output – Input]
                                                                 Y zA ) 2
                                                                 (
In the x-direction we can write:
                                                       dy                   Y yA ) 2
                                                                            (


                                       Y xA ) 1
                                       (                                    Y xA ) 2
                                                                            (
                                                  dz
                                                                              z
                                       Y yA ) 1
                                       (                    dx                         y

                                                                                       x
                                                                 Y zA ) 1
                                                                 (



  [Output – Input] =       Yx 2 - Yx 1  dy dz  - Yx 1 - Yx 2  dy dz

                Yx              Y              Y
                  x dy dz  -    dx dy dz  -    dV
                 x              x              x
[Output – Input] summary:


                                Y
          x - direction:    -      dV
                                x

                              Y
          y – direction:    -    dV
                              y

                              Y
          z – direction:    -    dV
                              z
Accumulation = [Output – Input] + Generation

        y    Y  Y  Y 
    dV          x   y   z  dV  y G dV
            -                 
        t                    

Cancel out the dV terms:

        y   Y Y Y
        t     x   y   z   yG
           -                 
                              
  Recall that the flux, Y, is a vector:

         Y i Yx  jYy  k Yz

Short-hand notation … the divergence relation:

              Yx  Yy  Yz
       Y          
              x   y   z
 A final form for our property balance:


           y
               -   Y   y G
           t

To solve this equation, we need to know Y in terms of y
  In engineering practice, we do this by
splitting the flux up into two components:

           Y  Yconv  Ydiff
     Yconv is a convective component, and

          Ydiff is a diffusive component


 Yconv  y U               Ydiff  -  y
    - where U is the local convective velocity
           y
               -   Y   y G
           t
  We need the divergence (derivative) of Y

           Y    YDiff  YConv 

        Y  -    y   y U 

  Y  -    y   y   U   U   y
          The general property balance,

                          y
                              -   Y   y G
                          t
       with

            Y  -    y   y   U   U   y

       becomes

   y
       U   y  y G     y  - y   U 
   t
Accumulation   Convection 1    Generation   Diffusion   Convection 2
   y
       U   y  y G     y  - y   U 
   t
 Some examples:
   Heat transfer, y =  Cp T and we obtain
   CPT 
             U   CPT   y G      CPT   - CPT   U 
     t
Mass transfer, y = A or CA (mass or moles respectively) and we obtain

          A
               U    A  y G     D A  - A   U 
          t
          CA
               U    CA  y G     D CA  - CA   U 
          t
  Momentum transfer, y = U and we obtain

 ρU x 
           U   ρU x   ψG    ρU x  - ρU x   U 
   t
 
 ρU y     U   ρU   ψ    ρU  - ρU   U 
                        y     G           y        y
  t

 ρU z 
           U   ρU z   ψG    ρU z  - ρU z   U 
   t

          Components for each coordinate direction
An important special case for the general balance:
   y
       U   y  y G     y  - y   U 
   t

Assume generation and diffusion are zero:

                ψ
                      ψ U   0
                t
 If conserved property is total mass per unit volume, ,

                 
                    +    U = 0
                 t
           With constant , /t = 0 and,

         U = U    +    U =    U

Hence the property balance for this case becomes,
                    U = 0

                 or   U = 0
And in this case (constant ), our original property balance

    y
        U   y  y G     y  -y   U 
    t

becomes:

           y
               U   y  y G     y 
           t

         Divergence of the velocity field is zero
Cases with constant  lead to

                                   2y  2y  2y 
        y      y    
                                   x2   y2   z 2 
                                                       
                                                      

The dot product, , operating on a scalar is given the
symbol 2 and is called the Laplacian operator

   e.g. the steady state conduction equation
                    2 T 2 T 2 T
                                     0
                    x 2
                           y 2
                                  z 2


describes the temperature field, T(x, y, z), given boundary
conditions at specified edges of a Cartesian “box”
                                           V

                                               x

                                       U
                             i i
                           L qu d
                                 i
                           F a lln g
                           Fm il
                                                           i
                                                   V e r t ca l
                                                        l
                                                    P a te




                                           y


Explain developing region for this problem!

				
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posted:4/8/2010
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