what do these processes have in common? 1) hydrogen embrittlement of pressure vessels in nuclear

Engineering Innovation--Engineering Innovation--DiffusionDiffusion

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.



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

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

This is NOT diffusion. How canYou tell?



Why does random motion cause spreading of a concentrationof particles?

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

DIFUS.HTM


What is Engineering?In one dimension. . .

NjNj-1 Nj+1

j-1 j j+1

Net change of particles in box j per time step is

ΔNj = Nj-1 Pa - Nj Pa +Nj+1Pb - Nj Pb

ΔNj = (Nj+1- Nj) Pb - (Nj - Nj-1)Pa

Let δNa = Nj - Nj-1 δNb = Nj+1 - Nj

Then ΔNj = Pb δNb - Pa δNa = δ(PδN)

Pa Pb

Δ is a change in timeδ is a change in space


What is Engineering?If the P’s are constant, i.e., the probabilities are the samefrom box to box, then

ΔNj = P δ2(N)

In three dimensions and in the continuous limit, this equationbecomes the diffusion equation where C is concentrationand κ is the diffusivity of the medium.

Cz

C

y

C

x

C

t

C 22

2

2

2

2

2

or, C is temperature and κ is thermal conductivity


What is Engineering?Consider diffusion in only one dimension. Then we have

2

2

x

C

t

C

Consider now the condition of “steady-state”, i.e., concentration C no longer changes with time. Then,

002

2

2

2

dx

Cd

x

C

t

C

This can be integrated to .constdx

dC


What is Engineering?What can one learn from this equation?

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

X=0 X=100C(t,0)=0 C(t,100)=100

κ1 κ2

At steady-state: 21

21 .kinkin dx

dCkconst

dx

dCk

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



2. Heat transfer—Fourier’s Law

heat flux in z-direction q

A

d c T

dzz p

( ); 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

dzAz ABA ; D is molecular

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

1. Momentum transfer—Newton’s Law

flux of x-momentum in z direction

zx

xdv

dz

( ), vx is velocity

in x-direction, is density, is viscosity.

Gradient transport



Heat conductionConduction-1D

Diffusion processesDiffusion-2D

Diffusion-limited aggregation

Setup: ρgolf ball = 1.15 ρsalt water = 1.13

Conc(sat) =1.20 Dsalt = 1.4 x 10-5 cm2/sec

Initial condition: Dry salt at bottom of cylinder.Drop in ball. Add water.

What happens? How long does it take?

what do these processes have in common? 1) hydrogen embrittlement of pressure vessels in nuclear

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