CHEE317Lect02-01132010toprint

Lecture 2

•  Understand the origins and meaning of Fick’s first law. •  Understand the concepts of mass transfer velocity,

equimolar counter diffusion and unimolar diffusion. •  Understand how to estimate diffusion coefficients. •  Reading: Chapter 3

•  Fick’s first law of diffusion (“fundamental” or “mechanistic”): –  Diffusion is a spontaneous phenomenon when a

system contains two or more components whose concentrations vary from point to point, there is a natural tendency of mass to be transferred in the direction that eliminates (or minimises) concentration differences within that system:

•  This is an extension of Thomas Graham's work – his law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

Fick’s First « Law » of Diffusion

Adolf Eugen Fick (1829-1901)

€

JA,z = −DABdcA

dz

What is mass transport by diffusion?

•  For an active diffusion to occur, the temperature should be high enough to overcome energy barriers to atomic motion.

http://people.virginia.edu/~lz2n/mse209/Chapter5.pdf

Fundamentals of Mass Transfer and Diffusion Fickian Diffusion

•  Let’s consider a unit volume that contains a mixture of “n” different components and write the following definitions:

–  Mass concentration (density): (1)

–  Mass fraction of component A: (2)

–  Where

–  Total molar concentration: (3)

–  Molar concentration of comp. A: (4)

–  Mole fractions (e.g., component A) : »  In liquids (or solids): xA=CA/C »  In gases: yA=CA/C

•  We can also write these same equations for all spatial dimensions (assuming….?):

Fundamentals of Mass Transfer and Diffusion Flux – Fick’s Law

•  In terms of the mass diffusion flux in one dimension, we can write (for constant density):

Fundamentals of Mass Transfer and Diffusion Flux – Fick’s Law

•  What is important is to differentiate the diffusion of species “i” from the movement the same species due to bulk movement (e.g. flow in a pipe).

€

Ni = xiN+ Diffusion Flux (0)

Fundamentals of Mass Transfer and Diffusion Velocities

•  Can define the molar average velocity as the total molar flux, N, divided by the total molar concentration, c:

€

vm =Nc

=NA + NB

c(1)

•  For species “i":

€

vi =Ni

ci

(2)

•  Combine these 2:

€

vm = xAvA + xBvB (3)

Fundamentals of Mass Transfer and Diffusion Velocities

•  Now, do the same thing, but just with the diffusion flux to get the species average diffusion velocity:

€

vi,D =Ji

ci

= vi − vm

€

vi = vm + vi,D

€

vi =Ni

ci•  Substitute:

€

Ni = civm + civi,D

•  For a binary system:

€

NA = xAN− cDAB(dxA

dz)

(4)

Fundamentals of Mass Transfer and Diffusion

•  In our binary system we have:

•  and by analogy

€

NA = xAN− cDAB(dxA

dz)

€

NB = xBN− cDBA(dxB

dz)

Fundamentals of Mass Transfer and Diffusion Binary diffusion – limiting cases

•  From our flux equations, it can be seen that there are two limiting cases that describe mass transfer by diffusion: –  Equimolar counter diffusion: fluxes are

equal, but opposite in direction.

–  Unimolecular diffusion: one component diffuses through a quiescent 2nd component.

Fundamentals of Mass Transfer and Diffusion Steady State Equimolar Counter Diffusion

•  Flux of one gaseous component is equal to but in the opposite direction of the second gaseous component

•  For steady-state, no reaction, in the z-direction:

•  the molar flux is

€

ddz

NA,z = 0

€

NA,z = −cDABdxA

dz+ xA NA,z + NB,z( )

€

N = NA,z + NB,z = 0

A B

z1 z2


•  In equimolar counterdiffusion, NA,z = -NB,z

Integrated at z = z1, cA = cA1 and at z = z2, cA = cA2 to:

•  Or in terms of partial pressure,

€

NA,z =DAB

(z2 − z1)(cA1

− cA 2)

€

NA,z =DAB

RT(z2 − z1)(pA1

−pA 2)


•  The concentration profile is described by

•  Integrated twice with boundary conditions at z = z1, cA = cA1and at z = z2, cA = cA2 to yields a linear concentration profile:

€

ddz

NA,z =d2cA

dz2= 0

€

cA − cA1

cA1− cA 2

=z − z1z1 − z2

Fundamentals of Mass Transfer and Diffusion Steady State Unimolecular Diffusion

•  Flux of one gaseous component is non-zero, the other flux is negligible.

€

NA,z = −cDABdxA

dz+ xANA,z•  the molar flux of A is

€

NA,z =−cDAB

(1− xA )dxA

dz

Fundamentals of Mass Transfer and Diffusion Steady State Unimolecular Diffusion

•  Rearrange and integrate flux of A:

€

dzz1

z

∫ =−cDAB

NA,z

dxA

(1− xA )XA ,1

xA

∫

€

NA,z =cDAB

(z − z1)ln 1− xA

1− xA1

€

xA =1− (1− xA1)expNA (z − z1)

cDAB

Detailed in Seader and Henley page 70. Work through examples 3.1 and 3.2 and the tutorial of this week.

Fundamentals of Mass Transfer and Diffusion Gas Phase Diffusion Coefficients (section 3.2)

•  Have been treating D as a known parameter. •  It is relatively complex:

•  Huge range of values. –  Gases ≈ 10-1 cm2/s –  Liquids ≈ 10-5 cm2/s –  Polymers and Glass ≈ 10-8 cm2/s –  Solids ≈ 10-30 cm2/s

Fundamentals of Mass Transfer and Diffusion Gas Phase Diffusion Coefficients

•  Calculating diffusion coefficients from first principles is easier for gases than liquids. –  Based on Boltzman’s kinetic theory of gases, theorem of

corresponding states and a suitable description of inter-molecular energy potential function (Lennard-Jones potential)…

•  In gases diffusivity proportional to the average molecular velocity times the mean free path (distance travelled until collision):

Gas Phase Diffusion Coeff.

Why so much higher?

Fundamentals of Mass Transfer and Diffusion Gas Phase Diffusion Coefficients – Chapman Enskog

•  Rigorous extensions using molecular sizes, Lennard-Jones interaction potentials etc, lead to Chapman-Enskog equation:

Effective collision diameter (Å) σAB = (σA+ σB)/2

•  Collision Integral •  Tabulated as a function of kBT/εAB •  εAB = (εAεB)0.5

[=] atm [=] cm2/s

[=] K

•  N.B. This is just one possible equation. There are many other “models” for calculating DAB, but they all use essentially the same parameters.



•  Example: Calculate the diffusivity of benzene in air at 100°C and 2 atm (a) using Chapman-Enskog and (b) by extrapolation from tabulated value at 1 atm and 0°C.

(a) From our tables (previous page): ε/k σ M Benzene 412.3 5.349 78.1 Air 78.6 3.711 29


•  Example continued

(b) From our tables (previous page) at 0°C and 1 atm: DAB=0.299 ft2/h = 0.0772 cm2/s

Fundamentals of Mass Transfer and Diffusion Diffusion in Small Pores

•  When diffusion occurs in small pores, and pore diameter is less than mean free path, physical constraints of the pores will influence the rate of diffusion.

•  This is know as Knudsen diffusion. •  For a cylindrical pore (T in K; M = mol wt; r = radius in cm; D[=]cm2/s):

•  For general pore size:

Fundamentals of Mass Transfer and Diffusion Liquid Phase Diffusion Coefficients

•  The theory of diffusion for liquid phase systems is not nearly as advanced as for gas phase.

•  DAB for liquids ≈104-105timessmallerthatforgas.–  Meanfreepathmuchsmaller(typicallylessthanamoleculardiameter).–  Muchgreaterdensitiesmeansthatfluxisroughlythesameorderofmagnitude.

•  Forlargesphericalmoleculesinverydilutesolutions,Stokes-Einsteinequationgives:

Molecular radius Viscosity


•  The S-E equation gives functionality, but underestimates real diffusivities (because the “drag” on the molecules is less than postulated by S-E): –  S-E says D varies as V-⅓ (i.e. r-1) –  In reality D varies as V-0.6

•  Another empirical expression is the Wilke-Chang (for small molecules):

ψB = 2.6 for water = 1.9 for methanol = 1.5 for ethanol

= 1 for benzene = 1 for heptane


•  Example: Find the diffusivity of toluene in benzene and benzene in toluene at 110°C. The physical properties are:

MW TNBP VA µ at 110°C

Benzene 78.1 80.1 96.5 0.24 Toluene 92.1 110.6 118.3 0.26

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CHEE317Lect02-01132010toprint