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Equation of Continuity

Equation of Continuity. Equation of Continuity. differential control volume:. Differential Mass Balance. mass balance:. Differential Equation of Continuity. divergence of mass velocity vector (  v ). Partial differentiation:. Differential Equation of Continuity. Rearranging:.

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Equation of Continuity

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  1. Equation of Continuity

  2. Equation of Continuity differential control volume:

  3. Differential Mass Balance mass balance:

  4. Differential Equation of Continuity divergence of mass velocity vector (v) Partial differentiation:

  5. Differential Equation of Continuity Rearranging: substantial time derivative If fluid is incompressible:

  6. Equation of Continuity or Conservation of mass for pure liquid flow

  7. Equation of Continuity Applying the conservation of mass to the volume element * May also be expressed in terms of moles

  8. Equation of Continuity

  9. Equation of Continuity

  10. Equation of Continuity Dividing by and letting approach zero,

  11. Equation of Continuity In vector notation, But form the Table 7.5-1 (Geankoplis) and

  12. Equation of Continuity Substituting

  13. Equation of Continuity Dividing both sides by MWA

  14. Equation of Continuity Recall: 1. Fick’s Law 2. Total molar flux of A

  15. Equation of Continuity Substituting NA and Fick’s law and writing for all 3 directions,

  16. Equation of Continuity Two equivalent forms of equation of continuity

  17. Equation of Continuity

  18. Equation of Continuity Special cases of the equation of continuity 1. Equation for constant c and DAB, At constant P and T, c= P/RT for gases, and substituting

  19. Equation of Continuity Special cases of the equation of continuity 2. Equimolarcounterdiffusion of gases, At constant P , with no reaction, c = constant, vM = 0, DAB = constant and RA=0 Fick’s 2nd Law of diffusion

  20. Equation of Continuity Special cases of the equation of continuity 3. For constant ρ and DAB (liquids), Starting with the vector notation of the mass balance We substitute and

  21. Equation of Continuity Example 1. Estimate the effect of chemical reaction on the rate of gas absorption in an agitated tank. Consider a system in which the dissolved gas A undergoes an irreversible first order reaction with the liquid B; that is A disappears within the liquid phase at a rate proportional to the local concentration of A. What assumptions can be made?

  22. Equation of Continuity Gas A dissolves in liquid Band diffuses into the liquid phase An irreversible 1st order homogeneous reaction takes place A + B  AB Assumption: AB is negligible in the solution (pseudobinary assumption)

  23. Equation of Continuity Assuming concentration of A is small, then total c is almost constant and Expanding the equation and taking c inside the space derivative, Assuming steady-state,

  24. Equation of Continuity Assuming that diffusion is along the z-direction only, We can write that since A is disappearing by an irreversible, 1st order reaction

  25. Equation of Continuity Rearranging, Looks familiar? How to solve this ODE?

  26. Equation of Continuity Example 2. A hollow sphere with permeable solid walls has its inner and outer surfaces maintained at a constant concentration CA1 and CA0 respectively. Develop the expression for the concentration profile for a component A in the wall at steady-state conditions. What is the flux at each surface?

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