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:

 dV = dx dy dz

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




Rearrange the balance:

Accumulation = Output – Input + Generation

Next …

Focus on terms for [Output – Input]
In the x-direction we can write:

  [Output – Input] =
[Output – Input] summary:

          x - direction:

          y – direction:

          z – direction:
Accumulation = [Output – Input] + Generation

Cancel out the dV terms:
  Recall that the flux, Y, is a vector:

Short-hand notation … the divergence relation:
 A final form for our property balance:

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:

     Yconv is a convective component, and

          Ydiff is a diffusive component

    - where U is the local convective velocity
We need the divergence (derivative) of Y
          The general property balance,



Accumulation   Convection 1   Generation   Diffusion   Convection 2
 Some examples:
   Heat transfer, y = r Cp T and we obtain

Mass transfer, y = rA or CA (mass or moles respectively) and we obtain
Momentum transfer, y = rU and we obtain

    Components for each coordinate direction
An important special case for the general balance:

Assume generation and diffusion are zero:
 If conserved property is total mass per unit volume, r,

           With constant r, ¶r/¶t = 0 and,

Hence the property balance for this case becomes,
And in this case (constant r), our original property balance


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

The dot product, Ñ·Ñ, operating on a scalar is given the
symbol Ñ2 and is called the Laplacian operator

   e.g. the steady state conduction equation

describes the temperature field, T(x, y, z), given boundary
conditions at specified edges of a Cartesian “box”
Explain developing region for this problem!

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