CP302 Separation Process Principles

1. CP302 Separation Process Principles Mass Transfer - Set 3

2. One-dimensional Unsteady-state Diffusion Fick’s First Law of Diffusion is written as follows when CA is only a function of z: dCA JA = - DAB (1) dz Fick’s First Law is written as follows when CA is a function of z as well as of some other variables such as time: ∂CA JA = - DAB (39) ∂z Observe the use of ordinary and partial derivatives as appropriate.

3. One-dimensional Unsteady-state Diffusion Mass flow of species A into the control volume = JA, in x A x MA JA, out JA, in where ∂CA ΔCA JA, in = - DAB ∂z at z z z+Δz Mass flow of species A out of the control volume = JA, out x A x MA A: cross-sectional area MA: molecular weight of species A where ∂CA JA, out = - DAB ∂z at z+Δz

4. One-dimensional Unsteady-state Diffusion Accumulation of species A in the control volume JA, out ∂CA JA, in = x (A x Δz) x MA ∂t ΔCA Mass balance for species A in the control volume gives, z z+Δz A: cross-sectional area MA: molecular weight of species A = A MA A MA JA, in JA, out ∂CA (A Δz) MA + ∂t

5. One-dimensional Unsteady-state Diffusion Mass balance can be simplified to ∂CA ∂CA ∂CA - DAB - DAB = + Δz ∂t ∂z ∂z at z at z+Δz The above can be rearranged to give ∂CA (∂CA/∂z)z+Δz (∂CA/∂z)z - DAB = ∂t Δz (40)

6. One-dimensional Unsteady-state Diffusion In the limit as Δz goes to 0, equation (40) is reduced to ∂CA ∂2CA DAB (41) = ∂t ∂z2 which is known as the Fick’s Second Law. Fick’s second law in the above form is applicable strictly for constant DAB and for diffusion in solids, and also in stagnant liquids and gases when the medium is dilute in A.

7. One-dimensional Unsteady-state Diffusion Fick’s second law, applies to one-dimensional unsteady-state diffusion, is given below: ∂CA ∂2CA DAB = (42) ∂t ∂z2 Fick’s second lawfor one-dimensional diffusion in radial direction only for cylindrical coordinates: ∂CA DAB ∂ ∂CA r = (43) ∂t r ∂r ∂r Fick’s second lawfor one-dimensional diffusion in radial direction only for spherical coordinates: ∂CA DAB ∂ ∂CA r2 = (44) ∂t r2 ∂r ∂r

8. Three-dimensional Unsteady-state Diffusion Fick’s second law, applies to three-dimensional unsteady-state diffusion, is given below: ∂CA ∂2CA ∂2CA ∂2CA (45a) DAB = + + ∂t ∂x2 ∂y2 ∂z2 Fick’s second lawfor three-dimensional diffusion in cylindrical coordinates: ∂CA DAB ∂ ∂CA ∂ ∂CA ∂ ∂CA r r = + + (45b) r ∂t ∂r ∂r ∂θ r ∂θ ∂z ∂z Fick’s second lawfor three-dimensional diffusion in spherical coordinates: ∂CA DAB ∂ ∂CA 1 ∂ ∂CA 1 ∂2 CA r2 sinθ = + + ∂t r2 ∂r ∂r sinθ ∂θ ∂θ sin2θ ∂2Φ (45c)

9. Unsteady-state diffusion in semi-infinite medium ∂CA ∂2CA DAB = (42) ∂t ∂z2 ∞ z = 0 z ≥ 0 z = Initial condition: CA = CA0 at t ≤ 0 and z ≥ 0 Boundary condition: CA = CAS at t ≥ 0 and z = 0 CA = CA0 at t ≥ 0 and z → ∞

10. Introducing dimensionless concentration change: CA – CA0 Use Y = CAS – CA0 and transform equation (42) to the following: ∂Y ∂2Y DAB = ∂t ∂z2 ∂CA / ∂t ∂Y where = ∂t CAS – CA0 ∂2CA / ∂z2 ∂2Y = ∂z2 CAS – CA0

11. Introducing dimensionless concentration change: CA – CA0 Use Y = CAS – CA0 and transform the initial and boundary conditions to the following: Initial condition: CA = CA0 becomes Y = 0 at t ≤ 0 and z ≥ 0 Boundary condition: CA = CAS becomes Y = 1 at t ≥ 0 and z = 0 CA = CA0 becomes Y = 0 at t ≥ 0 and z → ∞

12. Solving for Y as a function of z and t: ∂Y ∂2Y DAB = ∂t ∂z2 Initial condition: Y = 0 at t ≤ 0 and z ≥ 0 Y = 1 at t ≥ 0 and z = 0 Y = 0 at t ≥ 0 and z → Boundary condition: ∞ Since the PDE, its initial condition and boundary conditions are all linear in the dependent variable Y, an exact solution exists.

13. ⌠ ⌡ x 0 Non-dimensional concentration change (Y) is given by: CA – CA0 z Y = = erfc (46) CAS – CA0 2 DAB t where the complimentary error function, erfc, is related to the error function, erf, by 2 erfc(x) = 1 – erf(x) = 1 – exp(-σ2) dσ  π (47)

14. A little bit about error function: • Error function table is provided (take a look). • Table shows the error function values for x values up to 3.29. • For x > 3.23, error function is unity up to five decimal places. • For x > 4, the following approximation could be used: • erf(x) = 1 – exp(-x2)  π x

15. Example 3.11 of Ref. 2: Determine how long it will take for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m in a semi-infinite medium. Assume DAB = 0.1 cm2/s. Solution: Starting from (46) and (47), we get z Y = 1 - erf 2 DAB t Using Y = 0.01, z = 1 m (= 100 cm) and DAB = 0.1 cm2/s, we get 100 erf = 1 - 0.01 = 0.99 2 0.1 xt 100 = 1.8214 2 0.1 xt t = 2.09 h

16. Get back to (46), and determine the equation for the mass flux from it. ∂CA JA = - DAB ∂z at z JA = exp(-z2/4DABt) (CAS-CA0)  DAB / π t (48) Flux across the interface at z = 0 is JA = (CAS-CA0)  DAB / π t (49) at z = 0

17. Exercise: Determine how the dimensionless concentration change (Y) profile changes with time in a semi-infinite medium. Assume DAB = 0.1 cm2/s. Work up to 1 m depth of the medium. Solution: Starting from (46) and (47), we get z Y = 1 - erf 2 DAB t

18. clear all DAB = 0.1; %cm2/s t = 0; for i = 1:1:180 %in min t(i) = i*60; z = [0:1:100]; %cm x = z/(2 * sqrt(DAB*t(i))); Y(:,i) = 1 - erf(x); end plot(z,Y) xlabel('z (cm)') ylabel('Non-dimensional concentration, Y') grid pause plot(t/3600,Y(100,:)) xlabel('t (h)') ylabel('Y at z = 100 cm') grid Let us get the complete profile using MATLAB which has a built-in error function.

19. t = 1 min to 3 h

21. Diffusion in semi-infinite medium: In gas: DAB = 0.1 cm2/s Time taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is 2.09 h. In liquid: DAB = 10-5 cm2/s Time taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is 2.39 year. In solid: DAB = 10-9 cm2/s Time taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is 239 centuries.