Post on 11-Feb-2018
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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