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Prof. R. Shanthini 05 March 2013 1 CP302 Separation Process Principles Mass Transfer - Set 3

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Prof. R. Shanthini 05 March 2013 2 One-dimensional Unsteady-state Diffusion Fick’s First Law of Diffusion is written as follows when C A is only a function of z: (1) J A = - D AB dCAdCA dzdz ∂CA∂CA ∂z∂z (39) Fick’s First Law is written as follows when C A is a function of z as well as of some other variables such as time: Observe the use of ordinary and partial derivatives as appropriate.

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Prof. R. Shanthini 05 March 2013 3 J A, in J A, out ΔCAΔCA z+Δz J A, in = - D AB ∂CA∂CA ∂z∂z at z J A, out = - D AB ∂CA∂CA ∂z∂z at z+Δz z Mass flow of species A into the control volume = J A, in x A x M A where Mass flow of species A out of the control volume = J A, out x A x M A where A: cross-sectional area M A : molecular weight of species A One-dimensional Unsteady-state Diffusion

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Prof. R. Shanthini 05 March 2013 4 J A, in J A, out ΔCAΔCA z+Δzz A: cross-sectional area M A : molecular weight of species A = ∂CA∂CA ∂t∂t Accumulation of species A in the control volume x (A xΔz) x M A Mass balance for species A in the control volume gives, J A, in = J A, out A M A + ∂CA∂CA ∂t∂t (AΔz) M A One-dimensional Unsteady-state Diffusion

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Prof. R. Shanthini 05 March 2013 5 Mass balance can be simplified to - D AB ∂CA∂CA ∂z∂z at z - D AB ∂CA∂CA ∂z∂z at z+Δz = ∂CA∂CA ∂t∂t +ΔzΔz ∂CA∂CA ∂t∂t = D AB (∂C A /∂z) z+Δz (∂C A /∂z) z - ΔzΔz (40) The above can be rearranged to give One-dimensional Unsteady-state Diffusion

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Prof. R. Shanthini 05 March 2013 6 In the limit as Δz goes to 0, equation (40) is reduced to = ∂2CA∂2CA ∂z 2 D AB (41) ∂CA∂CA ∂t which is known as the Fick’s Second Law. Fick’s second law in the above form is applicable strictly for constant D AB and for diffusion in solids, and also in stagnant liquids and gases when the medium is dilute in A. One-dimensional Unsteady-state Diffusion

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Prof. R. Shanthini 05 March 2013 7 = ∂2CA∂2CA ∂z 2 D AB (42) Fick’s second law, applies to one-dimensional unsteady-state diffusion, is given below: ∂CA∂CA ∂t = ∂ ∂r D AB (43) Fick’s second law for one-dimensional diffusion in radial direction only for cylindrical coordinates: ∂CA∂CA ∂tr ∂C A ∂r r = ∂ D AB (44) Fick’s second law for one-dimensional diffusion in radial direction only for spherical coordinates: ∂CA∂CA ∂tr2r2 ∂C A ∂r r2r2 One-dimensional Unsteady-state Diffusion

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Prof. R. Shanthini 05 March 2013 8 = ∂2CA∂2CA ∂x 2 D AB Fick’s second law, applies to three-dimensional unsteady-state diffusion, is given below: ∂CA∂CA ∂t ++ ∂2CA∂2CA ∂z 2 ∂2CA∂2CA ∂y 2 = ∂ ∂r D AB Fick’s second law for three-dimensional diffusion in cylindrical coordinates: ∂CA∂CA ∂t r ∂C A ∂r r = ∂ D AB Fick’s second law for three-dimensional diffusion in spherical coordinates: ∂CA∂CA ∂tr2r2 ∂C A ∂r r2r2 + ∂ ∂θ∂θ ∂C A r ∂θ + ∂ ∂z ∂C A ∂z r (45b) (45a) ∂ ∂θ∂θ ∂C A ∂θ∂θ sinθ + 1 ∂2∂2 ∂2Φ∂2Φ + 1 sin 2 θ CACA (45c) Three-dimensional Unsteady-state Diffusion

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Prof. R. Shanthini 05 March 2013 9 Unsteady-state diffusion in semi-infinite medium z = 0z = ∞ = ∂2CA∂2CA ∂z 2 D AB (42) ∂CA∂CA ∂t z ≥ 0 Initial condition: C A = C A0 at t ≤ 0 and z ≥ 0 Boundary condition: C A = C AS at t ≥ 0 and z = 0 C A = C A0 at t ≥ 0 and z → ∞

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Prof. R. Shanthini 05 March 2013 10 Introducing dimensionless concentration change: = ∂2Y∂2Y ∂z 2 D AB ∂Y∂Y ∂t Y = C A – C A0 C AS – C A0 Use and transform equation (42) to the following: where ∂Y∂Y ∂t = ∂CA∂CA / ∂t C AS – C A0 ∂2Y∂2Y ∂z 2 = ∂2CA∂2CA / ∂z 2 C AS – C A0

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Prof. R. Shanthini 05 March 2013 11 Introducing dimensionless concentration change: Initial condition: C A = C A0 becomes Y = 0 at t ≤ 0 and z ≥ 0 Boundary condition: C A = C AS becomes Y = 1 at t ≥ 0 and z = 0 C A = C A0 becomes Y = 0 at t ≥ 0 and z → ∞ Y = C A – C A0 C AS – C A0 Use and transform the initial and boundary conditions to the following:

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Prof. R. Shanthini 05 March 2013 12 Solving for Y as a function of z and t: Initial condition:Y = 0 at t ≤ 0 and z ≥ 0 Boundary condition: Y = 1 at t ≥ 0 and z = 0 Y = 0 at t ≥ 0 and z → ∞ = ∂2Y∂2Y ∂z 2 D AB ∂Y∂Y ∂t Since the PDE, its initial condition and boundary conditions are all linear in the dependent variable Y, an exact solution exists.

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Prof. R. Shanthini 05 March 2013 13 Non-dimensional concentration change (Y) is given by: Y = C A – C A0 C AS – C A0 = erfc z 2 D AB t (46) where the complimentary error function, erfc, is related to the error function, erf, by erfc(x) = 1 – erf(x) = 1 – 2 π ⌠ ⌡ 0 x exp(-σ 2 ) dσ (47)

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Prof. R. Shanthini 05 March 2013 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 – π x exp(-x 2 )

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Prof. R. Shanthini 05 March 2013 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 D AB = 0.1 cm 2 /s. Solution: Starting from (46) and (47), we get Y = 1 -erf z 2 D AB t Using Y = 0.01, z = 1 m (= 100 cm) and D AB = 0.1 cm 2 /s, we get = 1 - 0.01 = 0.99erf 100 2 0.1 x t 100 2 0.1 x t = 1.8214 t = 2.09 h

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Prof. R. Shanthini 05 March 2013 16 Get back to (46), and determine the equation for the mass flux from it. J A = - D AB ∂CA∂CA ∂z∂z at z D AB / π t J A = exp(-z 2 /4D AB t) (C AS -C A0 ) (48) D AB / π t J A = (C AS -C A0 ) Flux across the interface at z = 0 is at z = 0 (49)

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Prof. R. Shanthini 05 March 2013 17 Exercise: Determine how the dimensionless concentration change (Y) profile changes with time in a semi-infinite medium. Assume D AB = 0.1 cm 2 /s. Work up to 1 m depth of the medium. Solution: Starting from (46) and (47), we get Y = 1 -erf z 2 D AB t

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Prof. R. Shanthini 05 March 2013 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.

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Prof. R. Shanthini 05 March 2013 19 t = 1 min to 3 h

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Prof. R. Shanthini 05 March 2013 20

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Prof. R. Shanthini 05 March 2013 21 Diffusion in semi-infinite medium: In gas: D AB = 0.1 cm 2 /s Time taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is 2.09 h. In liquid: D AB = 10 -5 cm 2 /s Time taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is 2.39 year. In solid: D AB = 10 -9 cm 2 /s Time taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is 239 centuries.

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