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Convection in the General Property Balance

This text focuses on developing the full equations of motion based on balance principles, using control volume analysis with inputs, generation, output, and accumulation of conserved properties. It covers the mathematical formulation and application of these principles in the context of convection, diffusion, and generation. Examples from heat transfer, mass transfer, and momentum transfer are discussed. Special cases with assumptions of zero generation and diffusion are explored, leading to simplified property balance equations.

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Convection in the General Property Balance

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  1. Convection in the General Property Balance Development of the full equations of motion

  2. Based on application of the balance: Control volume analysis: Input + Generation = Output + Accumulation For a conserved property yand corresponding flux Y

  3. 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

  4. Input + Generation = Output + Accumulation Generation: Accumulation: Input: Output:

  5. Rearrange the balance: Accumulation = Output – Input + Generation Next … Focus on terms for [Output – Input]

  6. In the x-direction we can write: [Output – Input] =

  7. [Output – Input] summary: x - direction: y – direction: z – direction:

  8. Accumulation = [Output – Input] + Generation Cancel out the dV terms:

  9. Recall that the flux, Y, is a vector: Short-hand notation … the divergence relation:

  10. A final form for our property balance: To solve this equation, we need to know Y in terms ofy

  11. 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

  12. We need the divergence (derivative) of Y

  13. The general property balance, with becomes Accumulation Convection 1 Generation Diffusion Convection 2

  14. Some examples: Heat transfer, = CpT and we obtain Mass transfer, = Aor CA (mass or moles respectively) and we obtain

  15. Momentum transfer,  = U and we obtain Components for each coordinate direction

  16. An important special case for the general balance: Assume generation and diffusion are zero:

  17. If conserved property is total mass per unit volume, , With constant , /t = 0 and, Hence the property balance for this case becomes,

  18. And in this case (constant ), our original property balance becomes: Divergence of the velocity field is zero

  19. 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”

  20. Explain developing region for this problem!

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