Diffusion

What is Engineering

What do these processes have in common?

1) Hydrogen embrittlement of pressure vessels in nuclearpower plants

2) Flow of electrons through conductors

3) Dispersion of pollutants from smoke stacks

4) Transdermal drug delivery

5) Influenza epidemics

6) Chemical reactions

7) Absorption of oxygen into the bloodstream

They all depend on

Diffusion (conduction)

What is diffusion? The transport of material--atomsor molecules--by random motion

What is conduction? The transport of heat or electronsby random motion.

Brownian motion causes the ink particles to move erraticallyin all directions. A concentration of ink particles will

disperse. DIFUS.HTM

Place a drop of ink into a glass of water. What happens?

Because there are more ways for the particles to drift apartthan there are for the particles to drift closer together.

Why does random motion cause spreading of a concentrationof particles?

We can also explain the spreading of a concentration by entropy.

The second law of thermodynamics says that systems tend towards maximum entropy – or maximum disorder.

Area of high concentration and low/zero concentration is an ordered state and the mixed state is the disordered state!

Other examples?

Why do metal cooking spoons have plastic handles?

Other examples?

What happens if someone across the room sprays perfume?

Perfume diffusion simulation

After adding milk and sugar, why do we stir our coffee?

Diffusion is slow!

Agitation (or stirring) can move fluids much larger distances in the same amount of time, which can accelerate the diffusion process.

Temperature Diffusivity

(°C) (cm2/s)

CO2-N20 0 0.096

Ar-O2 20 0.2

Ethanol(5%)-Water 25 1.13E-05

Water(13%)-Butanol 30 1.24E-05

H2-Ni 85 1.16E-08

Al-Cu 20 1.30E-30

(gas)

(liquid)

(solid)

Values for Diffusivity D

Greater the diffusivity, greater the flux!

In each of these examples, molecules (or heat) are moving down a gradient!

(From an area of high concentration to an area of low concentration)

dz

dcDJ i

i Fick’s Law:

Ji is called the flux. It has units of ))(( 2 tl

diffused material of amount

D is called the diffusion coefficient. It has units oft

l 2

Do our definitions of flux make sense?

N2

CO2

(constant T & P)

time

C(*)

capillary area time

removed gas of amount flux) dioxide carbon(

• If double area of capillary, expect the amount of gas transported to double.

• Want flux independent of apparatus – normalize by area.

lengthcapillary

difference ionconcentrat dioxide carbon flux) dioxide carbon(

• Flux is proportional to the concentration gradient –steeper the gradient, more material transported.

• Flux is inversely proportional to capillary length –increasing the distance to travel will decrease the flux.

2lengthtime

massJ

dx

dcDJ i

i

Steady diffusion across a thin film

Now let’s use our diffusion equation to predict the concentration profile of a material diffusing across a thin film!

If we are at steady-state (the concentration profile has no time dependence, or in other words, there is no accumulation of i in the film), we have a linear concentration profile.

Well-mixed dilute solution with concentration ci,l

Well-mixed dilute solution with concentration ci,0

Thin film

ci,0

ci,l

l

Concentration-dependent diffusion

z=0 z=zc z=l

ci,0

ci,c

ci,l

D1

D2

Which diffusivity is greater? How do you know?

Consider two neighboring thin films with a separation at ci,c:

Unsteady state diffusion

Back to a drop of ink in a glass of water…

If consider diffusion in the z-direction only:

How does the concentration profile change with time?

(add ink drop – all ink located at z = 0)

z=0

t = 0

t

z

A measure of the spread due to diffusion is the diffusion length Ld = (4Dt)0.5,

where D is the diffusivity coefficient and t is time. Note: for small time, spreading is quick, but for long times it slows down. That’s why youstir your coffee after adding cream. Diffusion doesn’t work fast enoughover long distances.

Heat Transfer

Occurs by three means:

1. Conduction:

• Occurs between two static objects

• Heat flows from the hotter to the cooler object

• For example, holding a cup of hot coffee

2. Convection:

• Transport of heat via a fluid medium

• Currents caused by hot air rising, fan circulating air

3. Radiation:

• Transport of energy as electromagnetic waves; the

receiving body absorbs the waves and is warmed

• For example, warmth of a fire

Heat moves down a temperature gradient!

(From an area of high temperature to an area of low temperature)

dz

dTkqz Fourier’s Law:

qz is called the heat flux. It has units of ))(( 2 tl

energy

k is called the thermal conductivity. It has units of))()(( Ttl

energy

α is called the thermal diffusivity. It is defined as)ˆ)(( pC

k

and has units oft

l 2

T k

(°C) (cal/cm s C)

H2 27 4.23E-04

O2 27 6.35E-05

Benzene 23 3.78E-04

Water 60 1.56E-03

Steel 100 9.08E-01

Wood -- 9.00E-05

(gas)

(liquid)

(solid)

Thermal Conductivity Values

Greater the thermal conductivity, greater the heat flux!

Consider a two-paneled door:

metalwood

What will the steady-state temperature profile look like? Why?

Tc

TH

Heat Conduction

z

kmetal > kwood

Here’s a heat-conducting bar with a fixed temperature T at each end:T(t,0)=0; T(t,100)=100. 2k1 = k2 .

z=0 z=100

T(t,0)=0 T(t,100)=100

κ1 κ2

At steady-state:

21

21 .kinkin dz

dCkconst

dz

dCk

Therefore, the ratios of the temperature gradients in each sectionmust equal the inverse ratios of the k’s.

(Constant flux)

Gradient transport summary

1. Momentum transfer—Newton’s Law

flux of x-momentum in z direction tm

zx

xd v

dz

( ), vx is velocity

in x-direction, is density, m is viscosity.

2. Heat transfer—Fourier’s Law

heat flux in z-direction q

A

d c T

dzz p a

( ); a is thermal diffusivity,

is density, cp is heat capacity, T is thermal energy (heat).

3. Mass transfer—Fick’s Law

mass flux of A in z -direction J Ddc

dzA z AB

A ; D is molecular

diffusivity of A in B, C A is the concentration of A.

Heat conduction

Diffusion processes

Diffusion-limited aggregation