PPT Diffusion Logan

8/10/2019 PPT Diffusion Logan

1/23

DIFFUSION

Bruce E. Logan

Department of Civil & Environmental Engineering

The Pennsylvania State University

Email: [email protected]

http://www.engr.psu.edu/ce/enve/logan.htm


2/23

What are the mechanisms for

chemical motion? Advection

Bulk transport by imposed flow. Examples: currentin a stream, flow in a pipe.

Convection Transport due to fluid instability. Examples: air

rising over a hot road. Diffusion- molecular

Scattering of particles (molecules) by randommotion due to thermal energy

Diffusion- turbulent Scattering due to fluid turbuence. Also called

eddy diffusion. This type of diffusion is much

faster than molecular diffusion


3/23

Diffusion & Dispersion

Diffusion is a method by which a chemical is

dispersed.

Dispersion is the spreading out of a

chemical that can be caused by different

mechanisms

Dont confuse a molecular diffusion

coefficient with a dispersion coefficient (more

on dispersion will come later in the course).


4/23

Chemical Flux

Molecules move in a

random direction based on

thermal energy

][2

tL

moles

Area

rateflowTotalFlux

Is there a net flux?


5/23

Chemical Flux

Net flux occurs when

molecules move in a direction

where there are no molecules

to balance their motion back in

the opposite direction.

][2

tL

moles

Area

rateflowTotalFlux


6/23

Chemical Flux






][2

tL

moles

Area

rateflowTotalFlux

Direction of net chemical flux


7/23

Chemical Flux






][2

tL

moles

Area

rateflowTotalFlux

Direction of net chemical flux


8/23

We often show chemical flux graphically

as concentration versus time

t=0

t=2

t=4

t=1

Conc.

Distance

t=3


9/23

Ficks First Law

Fick recognized that there must be a difference in

concentration to drive the net diffusion of a chemical, and

formulated the law:

dz

dxDcj CCwwzCw ,

cw= molar density of water

dxC/dz= molar gradient of C in z-

direction

jCw,z= molar flux of C in z-direction

D= Diffusion constant (fitted

parameter)


10/23

Ficks First Law



formulated the law:

dz

dxDcj CCwwzCw ,



direction



parameter)Note negative sign

(flux in directionopposite to gradient)

Gradient in + z-direction

Flux in - direction

Distance (z)

Conc.


11/23

Ficks First Law



formulated the law:

dz

dcDj CwCwzCw ,



direction



parameter)

dz

dxDcj CCwwzCw ,

Under isothermal, isobaric conditions,

this can be simplified to:


12/23

Flux in different directions:

dy

dxDcj CCwwyCw ,

dr

dxDcj CCwwrCw ,

dx

dxDcj CCwwxCw ,

dz

dxDcj CCwwzCw ,


13/23

Typical values of Diffusion Coefficient

DCw = 10-10 [cm2/s]Solid

DCw = 10-5 [cm2/s]Liquid

DCa = 10-1 [cm2/s]Gas


14/23

EXAMPLE CALCULATION

A jar of phenol contaminates a room with onlyone cylindrical vent (10 cm diameter, 20 cmdeep)

Neglecting advection, what is the rate ofphenol loss through the vent if the roomconcentration of phenol in air is 0.05%?

Assume: constant temperature of 20oC; alinear concentration gradient in the vent;Dpa=10

-1 cm2/s.


15/23

Solution

3

32521

,

10

)20(

)79)(/101.2)(/10(

cm

L

cm

cmLmolscmW zPa

)0(

)0(

2

1,

,z

cD

z

cD

dz

dcD

dz

dyDcj PaPa

PaPa

PaPa

CPaazPa

L

mol

L

molecyc aPPa

5

1, 101.2)

1.24

)(0005.0(

222 79)10(44

cmcmdA

2

1,

,,:z

AcDAjWRate PaPazPazPa

WPa,z = 8.210-9 mol/s


16/23

Ficks Second Law

What is the effect of time on the flux? Or how does the

flux change over time?

distance

gradientconc.concinchange

t

2

2

)/(zcD

zzcD

tc

zzcD

tc

)/(


17/23

Ficks Second Law

constantdz

dcCwand we have that

0

t

cCwAt steady state,

For a chemical in water:2

2

z

cD

t

c CwCw

Cw

so this becomes2

2

0 z

c

Dt

c CwCw

Cw

This tells us that at steady

state, the flux is constant, i.e. dx

dxDcj CCwwxCw ,

Ficks

First Law


18/23

How to calculate Diffusion Coefficients?

General Approaches

Tabulated values- best approach

Correlations- many can exist

Experimental- can be time consuming

Air Correlations for DCa Kinetic theory of gases: many limitations, such as

binary mixture.

Hirshfelder correlation: requires a lot of constants

Fuller correlation: best approach


19/23

FULLER CORRELATION: Diffusivities from Structure

Chemical C in gas g

Where: [these units must be used]

DCa= diffusion coefficient [cm2/s]

T= temperature [K]

P= pressure [atm]

MC= molecular weight of chemical [g/mol]

Mg= molecular weight of gas [g/mol]

VC,d= atomic diffusion volume (from formula and

tabulated values) [cm3]

2/1

23/1

,

3/1

,

375.1 11)(

10

gCdgdC

CgMMVVP

TD

For chemicals

in air:2/1

23/1

,

375.1

0345.01

)73.2(

10

CdC

CaMVP

TD


20/23

Table 3.1: Logan, Environmental Transport processes, p. 63.


21/23

How to calculate Diffusion Coefficients?

Water: more on this later

Solids Less information available

Chemical in porous medium is in liquid phase:

hindered diffusion

Chemical adsorbed to soil is in solid phase:

surface diffusion

For hindered and surface diffusion, need to know

diffusion constant in water


22/23

Examples: Diffusion constants in/on solids

surCwhCwsurhCw

DKDD

,

,,

*)1(

Where:

DCw,h+sur= overall surface diffusion coefficient

DCw,sur= surface diffusion coefficient

K*= dimensionless adsorption partition coefficient

Surface diffusion

hCwpmCw DD ,,

Where:

DCw,pm= diffusion coefficient in porous medium

DCw,h= diffusion coefficient in a single pore

= porosity; typically 0.3

= tortuosity factor; typically 3

Hindered diffusion


23/23

Diffusion constants: summary

Tabulated values always best

For air, simple correlations should besufficient for level of our calculations

In a porous medium, porosity and tortuosity

factors probably more important than correctvalue for molecular diffusion in air.

For water, situation is much more complex

more on that next.

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PPT Diffusion Logan