Prof. R. Shanthini 05 March 2013*CP302 Separation Process PrinciplesMass Transfer - Set 3

Prof. R. Shanthini 05 March 2013*One-dimensional Unsteady-state Diffusion Ficks First Law of Diffusion is written as follows when CA is only a function of z:(1)JA = - DABdCAdzJA = - DABCAz(39)Ficks First Law is written as follows when CA 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.

Prof. R. Shanthini 05 March 2013*JA, inJA, outCA z+zJA, in = - DABCAzat zJA, out = - DABCAzat z+zzMass flow of species A into the control volume= JA, in x A x MAwhereMass flow of species A out of the control volume= JA, out x A x MAwhereA: cross-sectional area

MA: molecular weight of species AOne-dimensional Unsteady-state Diffusion

Prof. R. Shanthini 05 March 2013*JA, inJA, outCA z+zzA: cross-sectional area

MA: molecular weight of species A=CAtAccumulation of species A in the control volumex (A xz) x MAMass balance for species A in the control volume gives, JA, in=JA, outA MAA MA+CAt (A z) MAOne-dimensional Unsteady-state Diffusion

Prof. R. Shanthini 05 March 2013*Mass balance can be simplified to - DABCAzat z- DABCAzat z+z=CAt+zCAt=DAB(CA/z)z+z(CA/z)z-z(40)The above can be rearranged to give One-dimensional Unsteady-state Diffusion

Prof. R. Shanthini 05 March 2013*In the limit as z goes to 0, equation (40) is reduced to= 2CA z2DAB(41)CA twhich is known as the Ficks Second Law. Ficks 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. One-dimensional Unsteady-state Diffusion

Prof. R. Shanthini 05 March 2013*= 2CA z2DAB(42)Ficks second law, applies to one-dimensional unsteady-state diffusion, is given below:CA t= rDAB(43)Ficks second law for one-dimensional diffusion in radial direction only for cylindrical coordinates:CA trCA rr= rDAB(44)Ficks second law for one-dimensional diffusion in radial direction only for spherical coordinates:CA tr2CA rr2One-dimensional Unsteady-state Diffusion

Prof. R. Shanthini 05 March 2013*= 2CA x2DABFicks second law, applies to three-dimensional unsteady-state diffusion, is given below:CA t+ + 2CA z22CA y2= rDABFicks second law for three-dimensional diffusion in cylindrical coordinates:CA trCA rr= rDABFicks second law for three-dimensional diffusion in spherical coordinates:CA tr2CA rr2+ CA r + zCA zr(45b)(45a) CA sin+ 1 sin2 2+ 1 sin2CA (45c)Three-dimensional Unsteady-state Diffusion

Prof. R. Shanthini 05 March 2013*Unsteady-state diffusion in semi-infinite mediumz = 0z == 2CA z2DAB(42)CA tz 0Initial condition: CA = CA0 at t 0 and z 0Boundary condition: CA = CAS at t 0 and z = 0CA = CA0 at t 0 and z

Prof. R. Shanthini 05 March 2013*Introducing dimensionless concentration change:= 2Y z2DABY tY = CA CA0CAS CA0Useand transform equation (42) to the following:where Y t= CA / tCAS CA02Y z2= 2CA / z2CAS CA0

Prof. R. Shanthini 05 March 2013*Introducing dimensionless concentration change:Initial condition: CA = CA0 becomes Y = 0 at t 0 and z 0Boundary condition: CA = CAS becomes Y = 1 at t 0 and z = 0CA = CA0 becomes Y = 0 at t 0 and z

Y = CA CA0CAS CA0Useand transform the initial and boundary conditions to the following:

Prof. R. Shanthini 05 March 2013*Solving for Y as a function of z and t:Initial condition: Y = 0 at t 0 and z 0Boundary condition: Y = 1 at t 0 and z = 0Y = 0 at t 0 and z = 2Y z2DABY tSince the PDE, its initial condition and boundary conditions are all linear in the dependent variable Y, an exact solution exists.

Prof. R. Shanthini 05 March 2013*Non-dimensional concentration change (Y) is given by: Y = CA CA0CAS CA0 = erfc z2 DAB t(46)where the complimentary error function, erfc, is related to the error function, erf, by erfc(x) = 1 erf(x) = 1 2exp(-2) d(47)

Prof. R. Shanthini 05 March 2013*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 xexp(-x2)

Prof. R. Shanthini 05 March 2013*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 getY = 1 - erf z2 DAB tUsing Y = 0.01, z = 1 m (= 100 cm) and DAB = 0.1 cm2/s, we get = 1 - 0.01 = 0.99 erf 1002 0.1 x t1002 0.1 x t= 1.8214 t = 2.09 h

Prof. R. Shanthini 05 March 2013*Get back to (46), and determine the equation for the mass flux from it. JA = - DABCAzat z DAB / tJA = exp(-z2/4DABt) (CAS-CA0)(48) DAB / tJA = (CAS-CA0)Flux across the interface at z = 0 is at z = 0(49)

Prof. R. Shanthini 05 March 2013*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 getY = 1 - erf z2 DAB t

Prof. R. Shanthini 05 March 2013*clear allDAB = 0.1;%cm2/st = 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);endplot(z,Y)xlabel('z (cm)')ylabel('Non-dimensional concentration, Y')gridpauseplot(t/3600,Y(100,:))xlabel('t (h)')ylabel('Y at z = 100 cm')gridLet us get the complete profile using MATLAB which has a built-in error function.

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

Prof. R. Shanthini 05 March 2013*

Prof. R. Shanthini 05 March 2013*Diffusion in semi-infinite medium:In gas: DAB = 0.1 cm2/sTime taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is2.09 h. In liquid: DAB = 10-5 cm2/sTime taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is2.39 year. In solid: DAB = 10-9 cm2/sTime taken for the dimensionless concentration change (Y) to reach 0.01 at a depth 1 m is239 centuries.