1 Molecular Diffusion

Notion of concentrationMolecular diffusion, Fick’s LawMass balanceTransport analogies; salt-gradient solar pondsSimple solutionsRandom walk analogy to diffusionExamples of sources and sinks

Motivation

Molecular diffusion is often negligible in environmental problemsExceptions: near interfaces, boundariesResponsible for removing gradients at smallest scalesAnalytical framework for turbulent and dispersive transport

Concentration

Contaminant => mixtureCarrier fluid (B) and contaminant/tracer (A)If dissolved, then solvent and soluteIf suspended, then continuous and dispersed phase

Concentration (c or ρA) commonly based on mass/volume (e.g. mg/l); also

mol/vol (chemical reactions)Mass fraction (salinity): ρA/ρ

Note: 1 mg/l ~ 1 mg/kg (water) = 1ppm

Molecular Diffusion

One-way flux (M/L2-T)

Net flux

z

AAwρ=

zww

AmA

AAA

∂∂−=

(2)

ml

wA

−=

/)( 21

ρρρ

l

DAB

AABA DJ ρ∇−=r

(1)

c or ρΑ

Ficks Law and Diffusivities

AABA DJ ρ∇−=r

kz

jy

ix

rrr)()()()(

∂∂

+∂∂

+∂∂

=∇

DAB is isotropic and essentially uniform(temperature dependent), but depends on A, B

Table 1.1 summarizes some values of DAB

Roughly: Dair ~ 10-1 cm2/s; Dwater ~ 10-5 cm2/s

Diffusivities, cont’d

Diffusivities often expressed through Schmidt no. Sc = ν/D

Roughly: νair ~ 10-1 cm2/s; νwater ~ 10-2 cm2/s

Scair ~ 1

Scwater ~ 103

Also: Prandlt no. Pr = ν/κ (κ = thermal cond.)

Add advection; total flux of A is:

AABAA DqN ρρ ∇−=rr

macroscopic velocity vector

Conservation of Mass

x

Like a bank account except expressed as rates:

(rate of) change in account = (rate of) (inflow – out) +/-(rate of) prod/consumption

Example for x-direction

z

y

zyxxNinnet

zyxxNNzyNout

zyNin

A

AxAxxA

xA

∆∆⎥⎦

⎤⎢⎣

⎡∆⎟

⎠⎞

⎜⎝⎛

∂∂

−=

∆∆⎥⎦

⎤⎢⎣

⎡∆⎟

⎠⎞

⎜⎝⎛

∂∂

+=∆∆=

∆∆=

∆+ )()(

)(∆x

∆y

(NA)x+∆x(NA)x

∆z

Conservation of Mass, cont’d

x

y

zAccount balance

zyxA ∆∆∆= ρRate of change of account balance

zyxt A ∆∆∆

∂∂

= )(ρ

Rate of production

zyxrA ∆∆∆=

Conservation of Mass, cont’d

x

y

zSum all terms (incl. advection in 3D)

Flux divergence (dot product of two vectors is scalar)

For carrier fluid B

BBB rNt

=⋅∇+∂

∂ rρ

AAA

zAyAxAA

rNt

Nz

Ny

Nxt

=⋅∇+∂

∂∂∂

+∂∂

+∂∂

+∂

∂

rρ

ρ )()()(

Conservation of Mass, mixture

0

0

0)(

=⋅∇

≅∇≅∂∂

=⋅∇+∂∂

=+

−=

qt

qt

qNN

rr

BA

BA

r

r

rrr

ρρ

ρρρ

Conservation of total mass

Liquids are nearly incompressible

Divergence = 0; Continuity

ρA +ρB =ρ

Conservation of Mass, contaminant

∇∇=∂∂

+∇∇=∇⋅+∂∂

+∇∇+∇=∇⋅+⋅∇+∂∂

+∇⋅∇=⋅∇+∂∂

=

cDtc

rcDcqtc

rcDcDcqqctc

rcDqctc

cA

2

2

2 ))((

)()(

r

rr

r

ρ

Conservative form of mass cons.

0 0N.C. form

If => Ficks Law of Diffusion0== rqr

drop subscript A

Heuristic interpretation of Advection and diffusion

c

t-1 tu∆t

∆c

AdvectionFlux ~ negative gradient

U

x

cDiffusionDifference in fluxes (divergence) ~ curvature

t-1

t

J Jx

Analogs

ρυ

κ

mrqqqtq

TtT

cDtc

rrrr

r

+∇=∇⋅+∂∂

∇=∂∂

∇=∂∂

2

2

2

)(

Ficks Law (mass transfer)

Fourier’s Law (heat transfer)

Newton’s Law (mom. Transfer)

Air Waterν/D Sc ~1 ~103

ν/κ Pr ~0.7 ~8

D~κ∼ν D<<κ<ν

Example: Salt Gradient Solar Ponds (WE 1-1)like El Paso Solar Pond

S T

UCZ

BCZ

GZ

~3m

dense brine

Solar Pond, diffusive salt flux to UCZ

S T

UCZ

BCZ

GZ

oooS 50=

oooS 250=

c = ρS

cUCZ = (1033 Kg/m3)(50x10-3 Kg/Kg) = 52 Kg/m3

cBCZ = (1165 Kg/m3)(250x10-3 Kg/Kg) = 291 Kg/m3

Area = 10,000 m2

Z1=0.3mZ2=1.8m

dzDdcJs /==(2x10-9)(239 Kg/m3)/1.2m = 4.0x10-7 Kg/m2-s

= 344 Kg/day

Solar Pond: diffusive thermal flux to UCZ

T

φsn = 250 W/m2T1 = 30oC

φs(z) = φsn(1-β)e-ηz

T2 = 80oC

z1

z2

β= fraction of φsn absorbed at surfaceη = extinction coefficientCp = heat capacity (4180 J/KgoC);

κ = thermal diffusivity (1.5x10-7 m2/s)212

2

2

*

)1(0

czceT

edzTdC

z

zsnp

++−

=

−+=

−

−

η

η

ηφ

φβηκρ

κρφβηφ psn C/)1(* −=

c1, c2 from T=T1 at z1, T=T2 at z2

Solar Pond: thermal flux

T

φsn = 250 W/m2T1 = 30oC

φs(z) = φsn(1-β)e-ηz

T2 = 80oC

z1

z2

2)/( zzpt dzdTCJ == κρ

⎥⎦

⎤⎢⎣

⎡−

−+−+

−−

=−−

−

)()()1(

1212

1212

2

zzeee

zzTTC

zzz

snp ηφβκρ

ηηη

z1 = 0.3m; z2 = 1.5m, β=0.5, η = 0.6 => Jt = 7 W/m2

Compare with (1-0.5)(250)exp(-0.6*1.5) = 51 W/m2

reaching BCZ (~13% lost)

Rankine Cycle Heat Engine

Figure by MIT OCW.

GFeed pump

Cooling water

Condenser

EvaporatorBrine

Turbine generator

Solar Pond: total energy extraction

T

φsn = 250 W/m2T1 = 30oC

φs(z) = φsn(1-β)e-ηz

T2 = 80oC

z1

z2

Energy Flux at surface 250 W/m2 100

Energy Flux reaching BCZ 51 20

Energy Flux extracted 34 14

Electricity extracted (theoretical) 4.8 2

Electricity extracted (net actual) 2.4 1 [24 KWe for 1 ha]

% of φsn

Carnot efficiencyηc = (T2-T1)/(T2+273)

Simple Solutions

Inst. injection of mass M

Ax

00)()(

0)(:

0:

2

2

===

==

±∞==∂∂

=∂∂

+

tatcwithtxAMr

tatxAMcic

xatcbcxcD

tc

δδ

δ alternative

Simple Solutions, cont’d

Ax

Inst. injection of mass M

Solution by similarity transform (Crank, 1975) or inspection

DABM

McdxA

etBc Dt

x

π2

42/1

2

=

=

=

∫∞

∞−

− Dtx

eDtA

Mtxc 4

2

2),(

−=

πAdd a current

Dtutx

eDtA

Mtxc 4

2)(

2),(

−−

=π

Gaussian Solution

t

c

x

σx

x

Spatial Momentsinterpretation

∫

∫

∫

∞

∞−

∞

∞−

∞

∞−

=

=

=

dxcxm

cxdxm

dxtxcmo

22

1

),(

Dtmm

mm

utmmx

AmM

ox

oc

o

22

2

122

1

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

==

=

σ

Mass; indep of t

Center of mass

Plume variance

Spatial Moments, cont’dRelationship of moments to equation parameters

DtAMAMDt

mm

DtAM

dtxeDtA

Mm

ox

Dtx

2/

/2

2

2

22

242

2

===

=

=−∞

∞−∫

σ

π

Without current, odd moments are 0

Spatial Moments, cont’dRewrite in terms of σ

Plume dilutes by spreading:

In 1-D, c ~ t-1/2 ~ σx-1

In 3-D, c ~ t-3/2 ~ σ-3

2

222

222

2)(

32/3

4)(

2/3

)2(

)(8),(

σ

σπ

πzyx

Dtzyx

eM

eDtMtxc

++−

++−

=

=

2

2

2

2

4

2

2),(

σ

σπ

πx

x

Dtx

eAM

eDtA

Mtxc

−

−

=

=

or in 3-D (isotropic)

Moment generating equation

∫∞

∞−

=∂∂

=∂∂ cdxxm

xcD

tc i

i2

2

Approach 1: moments of c(x,t) => σ2 = m2/mo = 2Dt

Approach 2: moments of ge => moment generation eq.

dxtermeachxi∫∞

∞−

)(

Moment generating eq., cont’d

∫∞

∞−

=∂∂

=∂∂ cdxxm

xcD

tc i

i2

2

o

o

DmdxDccxDdxxcxD

xcDxdx

xcDx

dtdmdxcx

tdxtcx

xcDdx

xcD

dtdmdxc

tdxtc

22)(22

0

22

22

222

2

2

=+−=∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

=∂∂

=∂∂

=∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

=∂∂

=∂∂

=∂∂

∫∫∫

∫∫

∫

∫∫

∞

∞−

∞∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−0th

moment

0

00

2nd

moment

Moment generating eq., cont’d

∫∞

∞−

=∂∂

=∂∂ cdxxm

xcD

tc i

i2

2

0th

moment0=

dtdmo => mo = const = M/A

oo

o

Dmdtdm

Dmdtdm

2

2

2

2

=

=

σ2nd

moment=> dσ2/dt = 2D or σ2 = 2Dt

How fast is molecular diffusion?

Creating linear salinity distribution from initial step profile

Assume 80 cm tank; 40 2cm steps

z

S

Time to diffuse: σ2 = 2Dt

t = σ2/2D ~(2cm)2/(2)(1.3x10-5 cm2/s)

= 1.5x105s ~ 2 days

If thermal diffusion (100 x faster), t < 1 hr

Table 1-1

slow!

Spatially distributed sourcesc

co

t

0

00000

0

2

2

=>==<=

−∞==∞==

∂∂

=∂∂

tatxforctatxforccic

xatccxatcbc

xcD

tc

o

o or c = co/2 at x = 0

x

Spatially distributed sources

0 xx

dξ

ξco

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−==

==

−∞

−−

∫ Dtxerfc

cDtxerf

ce

Dtdc

txc

eDtdc

eDtA

dMtdc

ooDt

x

o

DtoDt

2221

22),(

22),(

4

44

2

22

ξ

ξξ

πξ

πξ

πξ

Error Function 22 α

π−e

erf(ω)erfc(ω)

αω

∫

∫∞

−

−

=

=

ω

α

ωα

απ

ω

απ

ω

deerfc

deerf

2

2

2)(

2)(0

)(1)(1)(0)0(

xerfxerfcerferf

−==∞=

Example: DO in Fish aquarium (WE 1-4)

2(c2-c1)

z

c

t

C1 C2 2C2 – C1

( ) ( )Dt/zccc)t,z(c 2erfc221

121 −+=

( )Dt/zcccc

2erfc12

1 =−−

t=0: c=c1=8 mg/l (csat at 27oC)

t>0: c(0)=c2=10 mg/l (csat at 16oC)

Evaluate at z =15 cm, D = 2x10-5 cm2/s (Table 1-1)

t z/(4Dt)0.5 erfc[(z/(4Dt)0.5]

1hr 28 0

1d 5.7 10-15

1 mo 1.0 0.15

Again, very slow!

Diffusion as correlated movements

x=0 at t=0 ∫=t

dttutx0

')'()(

p(x,t)

x

Analogy between p(x) and c(x); ergotic assumption

For many particles, both distributions become Normal (Gaussian) through Central Limit Theorem

Statistics of velocity

)()(

)()(

)0()()()0()()(

.

0

2

2

ττ

τττ

Rtututu

uuuututu

constu

u

=−

−==−

=

= mean velocity

variance

auto correlation

auto co-variance

R(τ)

1

τ0

Statistics of position

∫

∫∫

∫∫

∫∫

∫

===

=−=

=⎥⎦

⎤⎢⎣

⎡==

===

=

t

tt

tt

tt

t

dRudtd

dtxdD

dRtudtttRtudttxd

dttututudttudtdxtx

dttdx

tx

dttudttux

dttutx

0

222

0

2

0

22

00

2

2

00

0

)(22

)()(2')'()(2)(

')'()(2)(')'(2)(2)(

)(

0)(')'(

')'()(

ττσ

ττ

increases with time, as follows

[D] = [V2T]

Earlier, [VL] or

Taylor’s Theorem (1921); classic

mAwD l=t

D2

2σ= [L2/T]

Random Walk (WE 1-3)Special case:

Walker’s position at time t=N∆t

Probability distribution

UorUtu −=)(

∑ ∆=N

tux1

( ) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +

= NNNNNp

21

22

!,

!! χχχ

0

0.125

0.25

0.375

0.5

-3 -1 1 3

-3 -1 1 3

3/8

2/8

1/8

χ = x/Ut

direction changes randomly after ∆t

Bernoulli Distribution

Approaches Gaussian for large N

Example for N=3

Statistics of position

( )( )[ ]

t∆/x∆tt∆tUUt∆N

uuuuuuut∆t∆uσ

2222

N21Ni212

2N

1i

2

===

++++=⎟⎠

⎞⎜⎝

⎛= ∑

=

KKK

( ) ∑=

==N

1i0)t(utx

txtU

tD

∆∆

=∆

==222

222σ [L2/T ]

Alternatively, derive D from Taylor’s Theorem

Examples of Sources and Sinks (r terms)

1st order0th order2nd orderCoupled reactionsMixed order

1st Order

c/coExample: radioactive decay 1

0t1/2

0.50.37

k-1

0.1kto ecc

kcdtdc

−=

−=

/ tt90

Half life e-folding time

Linearity => 1st O decay multiplies simple sol’n by e-kt; e.g.

Also very convenient in particle tracking models

0th OrderS/SoExample: silica uptake

by diatoms (high diatom conc)

1

BtSS

BdtdS

o −=

−=

0 tSo/BS=substrate (silica) concentration

B=rate (depends on diatom population, but assume large)

2nd Orderc/coExample: particle-

particle collisions/reactions; flocculant settling)

1

co

oo Btccc

Bcdtdc

+=

−=

11

2

0 t

Behavior depends on co; slower than e-kt.

Can be confused with multiple species undergoing 1st order removal

Coupled Reactions

Example: Nitrogen oxidation

2233

2231122

1121

NKdtdN

NKNKdtdN

NKdtdN

=

−=

−= N1 = NH3-N

N2 = NO2-N

N3 = NO3-N

If N’s are measured as molar quantities, or atomic mass, then successive K’s are equal and opposite

Mixed Order—Saturation Kinetics(Menod kinetics)

Example: algal uptake of nutrients—focus on algae

oSSkS

dtdc

+=

c = algal concentrationS = substrate concentrationSo = half-saturation const

rrmax

Rmax/2

So S

kdtdc

SkSkS

dtdc

o

≅

=≅ ' (1st Order)

(0th Order)

S << So =>

S >> So =>

1 Wrap-upMolecular diffusivities

D is “small” ~ 1x10-5 cm2/s for waterInst. point source solutions are Gaussian; other solutions built from

Spatial and temporal integration, coordinate translation, linear source/sink terms

ττ

σ

∫∞

=

=

=

0

2

2

2

)(

2

dRuD

dtdD

wD ml Molecular motion; Eulerian frame

Method of moments

Molecular motion; Lagrangian frame