Download presentation

Presentation is loading. Please wait.

Published byMax Hedley Modified over 3 years ago

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

2
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.

3
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

4
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

5
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

6
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

7
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

8
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

9
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 → ∞

10
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

11
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:

12
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.

13
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)

14
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 )

15
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

16
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)

17
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

18
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.

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

20
Prof. R. Shanthini 05 March 2013 20

21
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.

Similar presentations

OK

Research Paper. Chapter 7: DOPANT DIFFUSION DOPANT DIFFUSION Introduction Introduction Basic Concepts Basic Concepts –Dopant solid solubility –Macroscopic.

Research Paper. Chapter 7: DOPANT DIFFUSION DOPANT DIFFUSION Introduction Introduction Basic Concepts Basic Concepts –Dopant solid solubility –Macroscopic.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on beer lambert law graph Best ppt on motivation Ppt on employee engagement strategy Download ppt on indus valley civilization location Ppt on collection framework in java Ppt on social networking free download Download ppt on cybercrime and security Ppt on hong kong airport Ppt on product advertising letter Ppt on bluetooth based smart sensor networks conferences