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Convection in the General Property Balance. Development of the full equations of motion. Based on application of the balance:. Control volume analysis:. Input + Generation = Output + Accumulation. For a conserved property y and corresponding flux Y.
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Convection in the General Property Balance Development of the full equations of motion
Based on application of the balance: Control volume analysis: Input + Generation = Output + Accumulation For a conserved property yand corresponding flux Y
Consider a control volume in Cartesian coordinates: dV = dx dy dz Property transport entering or leaving each face of the form, A where A is an area element
Input + Generation = Output + Accumulation Generation: Accumulation: Input: Output:
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 ofy
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
The general property balance, with becomes Accumulation Convection 1 Generation Diffusion Convection 2
Some examples: Heat transfer, = CpT and we obtain Mass transfer, = Aor CA (mass or moles respectively) and we obtain
Momentum transfer, = U 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, , With constant , /t = 0 and, Hence the property balance for this case becomes,
And in this case (constant ), our original property balance becomes: Divergence of the velocity field is zero
Cases with constant lead to 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 describes the temperature field, T(x, y, z), given boundary conditions at specified edges of a Cartesian “box”
