 # Equation of Continuity. differential control volume:

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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: substantial time derivative If fluid is incompressible:

Equation of Continuity or Conservation of mass for pure liquid flow

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

Equation of Continuity

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

Equation of Continuity

Dividing both sides by MW A

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

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

Equation of Continuity Two equivalent forms of equation of continuity

Equation of Continuity

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

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

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

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?

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)

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

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

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

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.