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Equation of Continuity

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differential control volume:

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Differential Mass Balance mass balance:

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

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

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Equation of Continuity or Conservation of mass for pure liquid flow

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

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Equation of Continuity

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In vector notation, But form the Table 7.5-1 (Geankoplis) and

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Equation of Continuity

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Dividing both sides by MW A

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

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Equation of Continuity Substituting N A and Fick’s law and writing for all 3 directions,

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Equation of Continuity Two equivalent forms of equation of continuity

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Equation of Continuity

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Special cases of the equation of continuity 1. Equation for constant c and D AB,

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Equation of Continuity Special cases of the equation of continuity 2. Equimolar counterdiffusion of gases, At constant P, with no reaction, c = constant, v M = 0, D AB = constant and R A =0 Fick’s 2 nd Law of diffusion

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

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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?

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Equation of Continuity 1.Gas A dissolves in liquid B and diffuses into the liquid phase 2.An irreversible 1 st order homogeneous reaction takes place A + B AB Assumption: AB is negligible in the solution (pseudobinary assumption)

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

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Equation of Continuity Assuming that diffusion is along the z-direction only,

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

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Equation of Continuity A hollow sphere with permeable solid walls has its inner and outer surfaces maintained at a constant concentration C A1 and C A0 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? Example 2.

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