Transport processes Emission – [mass/time] pollutants released into the environment Imission –...

Transport processesTransport processes

bull Emission ndash [masstime]pollutants released into the environment

bull Imission ndash [massvolume]amount of pollutants received by a living

being at a given location Also the state of the environment with respect to the pollutant

bull TransmissionEverything between emission and

immission


What is a transport processbull Describes the fate of any material in the

environmentbull The continuous steps of transportation

reactions and interactions which happen to the pollutants

bull It determines the concentration-distribution of the observed material in time and in space

bull Transmission


Transport processes includebull Physical transportation due to the

movement of the mediumbull Advectionbull Diffusion and dispersion

bull Physical chemical biochemical conversion processes

bull Settlingbull Ad- and desorptionbull Reactionsbull Degradation decomposition


What is our aimbull Defend our values

bull Decrease the immission by controlling the emission ndash EFFORT MONEY

bull Describe transport processes

Transport processesTransport processesImportant conceptsbull State variable concentration density temperaturehellip

bull Conservative material no reactions no settling

bull Non-conservative material opposite realistic

bull Flow processes 3D u(xyzt) h(xyzt) V(xyzt)hellip

bull Steady state dudt=0 dCdt=0hellip

bull Homogeneous distribution ndash totally mixed reactor

Transport processesTransport processesThis presentation deals only with transport

processes in surface waters So the transporting medium is water

Classical water quality state variables arebull Dissolved oxygen (DO)bull Organic material - biochemical oxygen

demand (BOD)bull Nutrients ndash N- P-forms (NO3-N PO4-Phellip)

bull Suspended solids (SS)bull Algae

Mass balanceMass balance

General mass balance equationbull Expresses the conversation of massbull Differential equations

IN ndash OUT + SOURCES ndash SINKS = CHANGE

VIN

(1)

OUT

(2)

Controlling surface

We work with mass fluxes [gs]

Mass balanceMass balanceBy solving the general mass balance equation

we can describe the transport processes for a given substance

Solutionbull Analyticalbull Numerical

Discretizationbull Temporalbull Spatial

Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other

Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA

Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE

Assumptionsbull Steady statebull Conservative material

Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0

Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D

vs

A ~ B middot H [m2]

H

Simplified case IISimplified case II

Q

(1) (2)x

Q

IN

OUT

LOSS settled matter

C(x) is linear (assumption)

Av

v

vs

If x = O C = Co

Exponential decrease

Simplified case IISimplified case IIThe calculation

C0 concentration under the inlet

Determination of C0 value

Q

E = q middot c emission

Cbg background concentration

1D ndash Complete mixing (two water mixing with each other)

Increment

Dilution ratio

E


The solution

Transmission coefficient

Dilution Sedimentation Conservative matter


Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC

Concentration at distance L from the emission

Flow velocity

ms10275

15HB

Qv

Maximum possible C0 value

Maximum allowed waste water concentration

Allowed maximum concentration

C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc


Necessary reduction of emission

gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa

The general situationThe general situation

bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D

General transport equation

Advection-Dispersion Equation (ADE)

Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of

the observed state variablebull Itrsquos solution is the concentration-distribution over

timebull Differential equation

Change in concentration with time

Change due to advection

=Change due to diffusion or dispersion

Change due to conversion processes

+ +

Molecular diffusionMolecular diffusion

DIFFUSION

CONVECTION

v

Molecular Molecular diffusiondiffusion

FickrsquosFickrsquos LawLaw

c1 c2x

bull c1 c2 separated tanks

bull open valve

Mass flux (kgs)

D - molecular diffusioncoefficient [m2s]

bull equalization and mixing (Brown-movement)

through area unit

bull depends on temperature

IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion

Mass balance Mass balance equationequation

x direction

IN conv + diff

dxdy

dzOUT conv + diff


Convection Diffusion

If D(x) = const in x direction

Convection - Diffusion 1D Equation

The other directions are similar

Convection transfering Diffusion spreading


x y z directions (3D)

D ndash molecular diffusion coefficient

material specific water 10-4 cm2s

space independent

slow mixing and equalization

Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity

Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration

Mass balance equationMass balance equation

bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)

because of surface roughness (friction) rarr causes intensive mixing

v(c)

t

v deviation pulsation

T Time step of the turbulence

averagev~

0

bull Natural streams always turbulent

We usually can use the average only

TurbulenceTurbulence

Convection v middot c [ kgm2 middot s ]

0

0

Turbulent transport Turbulent transport descriptiondescription

v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D

Depends on the direction

Function of the velocity space

X direction (the other are similar)

Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter

Turbulent diffusionTurbulent diffusion

Dx = D + Dtx Dy = D + Dty Dz = D + Dtz

Convection

Diffusion

Turbulent diffusion

Stochastic fluctuations of the flow velocity (pulsations)

Diffusive process in mathematical sense (~ Fickrsquos Law)

Convective transport (temporal differences)

Temporal averaging (T)

3D transport equation in turbulent flow3D transport equation in turbulent flow

Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion

Depth averaged flow velocity v

0v

H

After the expansion of the convectiv part (v middot c) in x direction

Depth Integration (3D2D)

Turbulent dispersion (~ diffusion)

z

x

Velocity depth profile

Dx Dy

turbulent dispersion coefficients in 2D equation

Depth averaging (H)

11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))

Dx turbulent dispersion coefficient in 1D equation

Cross-section averaging (A)

2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)

Turbulent dispersionTurbulent dispersion

Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)

v

In the 2D and 1D equations only

Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)

Dx = Ddx + Dtx + D Dy

= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D

Dx Dy

gtgt Dx Dx gtgt Dx

In case of more irregular channel and higher depth or larger cross-section area its value is bigger

Also present in laminar flow

Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients

10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff

Vertical turbulent diff

High depth Shallow depth

Cross disp (2D)

Longitudinal disp (2D)

Horizontal turbulent diff


Evaluation of dispersion coefficients measuring

Measuring with tracing matter (eg paint)

Inverse calculation from the measured concentration

2D Cross dispersion coefficient (Fischer)

Dy = dy u R (m2s)

dy - dimensionless cross dispersion coefficient

Straight regular channel dy 015

Moderately winding channel dy 02 ndash 06

Hard winding irregular channel dy gt 06 (1-2)

R ndash hydraulic radius (m)

u - shearing velocity at the channel bottom (ms)

u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)

Velocity space turbulent pulsation unequal distribution

Estimation of dispersion coefficients empirical forms

TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS

bull There are sources and losses in the flow space

bull Physical chemical biochemical transformations take place

bull Non-conservative pollutants reaction kinetics ( R(C) )

bull They are taken into account in a linear way

dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)

cαxc

DAxx

c)v(Atc)(A

xx

1D equation in this case

ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION

Permanent mixing of the pollutants

Polluntant-wave transmission

Estimation of the geometric and hydraulic parameters

Exacter calculations based on measurments using numerical methods

The analitical solutions can be obtained in simpler cases only (approximate calculations)

Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING

The pollutant emission is steady in time

Permanent river discharge (low flow condition)

Constant flow velocity flow depth and dispersion coef

2D equation negligible vertical changes (shallow river)

Convection transferring

Dispersion spreading

Initial condition M0 (x0 y0) - emission

Boundary cond cy = 0 at the bank

)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2

INLET AT THE MAIN FLOW PATH

Longitudinal according to x-frac12 function

Cross direction Gauss-distribution

cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234

Bb at value 01 cmax 1522bBBrand width

B ~ Bb2

1 0270 BD

vL

y

x

First distance of the mixing

M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x

INLET AT THE RIVER BANK

)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M

INLET NEAR THE RIVER BANK

)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1

IMPACTS OF THE RIVER BANKS (total river section)

Boundary condition reflection theory (infinite series)

Complete mixing the change along the cross-section is less than 10

L2 ~ 3L1 second distance of the mixing

Inlet at a point in y0 distance from the bank

xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2

More inlets or diffuser-row theory of the superposition

Separated calculations for each inlet point and addition

M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx

TRANSMISSION OF NON-PERMANENT EMISSIONS

Suddenly jerky pollutant-waves

In time intensively changing emissions

2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6

Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)

Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)

CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x

1D equation ndash narrow and shallow rivers

2 tDA

GCmax

x At the fixed time moment (xvx)

x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i

TIME-CHANGING EMISSIONS

][ skgM i

tti=1 i=n

Dividing into discrete units with constant values

Superposition

Gi ~ Mi Δt t - (i-1) Δt ge 0

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What is a transport processbull Describes the fate of any material in the

environmentbull The continuous steps of transportation

reactions and interactions which happen to the pollutants

bull It determines the concentration-distribution of the observed material in time and in space

bull Transmission
























VIN

(1)

OUT

(2)

Controlling surface












vs

A ~ B middot H [m2]

H


Q

(1) (2)x

Q

IN

OUT

LOSS settled matter


Av

v

vs

If x = O C = Co





Q




Increment

Dilution ratio

E


The solution




Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

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febr08

febr09

febr10

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febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


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VIN

(1)

OUT

(2)

Controlling surface












vs

A ~ B middot H [m2]

H


Q

(1) (2)x

Q

IN

OUT

LOSS settled matter


Av

v

vs

If x = O C = Co





Q




Increment

Dilution ratio

E


The solution




Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

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vs

A ~ B middot H [m2]

H


Q

(1) (2)x

Q

IN

OUT

LOSS settled matter


Av

v

vs

If x = O C = Co





Q




Increment

Dilution ratio

E


The solution




Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52


Q

(1) (2)x

Q

IN

OUT

LOSS settled matter


Av

v

vs

If x = O C = Co





Q




Increment

Dilution ratio

E


The solution




Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

v

vs

If x = O C = Co





Q




Increment

Dilution ratio

E


The solution




Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52



Q




Increment

Dilution ratio

E


The solution




Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

The solution




Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Clim = 15 gm3

concentration limit

Q = 15 m3s

q = 1 m3sc = 300 gm3

Cbg = 10 gm3

L = 5 km

B = 75 m

vS = 03 mh

H = 2 m

EXAMPLE

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Complete mixing

3bg0 gm28125

11530011015

qQ

cqCQC

3lim

33

s0

gm15Cgm22836)10513600

0321

( exp28125

L)vv

H1

( expC L)(xC


Flow velocity

ms10275

15HB

Qv




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52




C (x = L)max = Clim

3

3s

limmax0

gm18474

)10513600

0321

( exp

15

L)vv

H1

( exp

C C

3

bgmax0max

gm145584

11015-11518474

q

CQ-qQCc



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52



gs154416145584)(3001)c(cqΔE max

33

s

sm0051)10513600

0321

( exp115

1

L)vv

H1

( expqQ

1 L)(xa












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52












+ +


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52


DIFFUSION

CONVECTION

v



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52



c1 c2x


bull open valve

Mass flux (kgs)



through area unit


IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

IN conv + diff

dxdy

dzOUT conv + diff

x direction

IN OUT

bull convection

CHANGE

bull diffusion


x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

x direction

IN conv + diff

dxdy

dzOUT conv + diff











space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52










space independent







v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52



v(c)

t



averagev~

0





0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52


0

0


v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

v

turbulent diffusion

molecular diffusion

Dtx Dty Dtz gtgt D







Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52


Convection

Diffusion

Turbulent diffusion








0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52



0v

H




z

x


Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Dx Dy


Depth averaging (H)







v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52



v





Dx Dy

gtgt Dx Dx gtgt Dx




10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52


10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s

Pore water

Mol diff



Cross disp (2D)








Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52





Dy = dy u R (m2s)
















cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52







cαxc

DAxx

c)v(Atc)(A

xx








Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52







Main steps

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

2

2

y

cD

x

cv yx

PERMANENT MIXING









)()()(

cvhy

cvhxt

chyx )()(

y

cDh

yx

cDh

xyx

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

x

yy

v

xD2




cmax

M [kgs]cmax

)4

exp(2

2

xD

yv

xvDh

Mc (x t)y

x

xy

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

x

yb

v

xDB

234


B ~ Bb2

1 0270 BD

vL

y

x


M

1L

Bb

C (x1 y)

x1

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

M

x

yb

v

xDB

2152

21 110 B

D

vL

y

x


)4

exp(2

xD

yv

xvDh

Mc

y

x

xy

cmax

C (x1 y)

x1

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

M


)4

(exp ( xDy

( y-y0 )2-v

xvD2h

Mc x

xy

cmax

))4

+exp ( xDy

( y+y0 )2-vx

y0 C (x1 y)

x1






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52






xvD2h

Mc

xy

)4

exp ( xDy

( y-y0 +2nB)2-vx

)4

+ exp ( xDy

( y+y0 -2nB)2-vx

sumn=infin

n=minusinfin(

)

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

M2



M1 C = C1 + C2

C2

C1

SUPERPOSITION

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

)( cvxt

cx )()(

y

cD

yx

cD

xyx




2D Equation

febr03

febr04

febr05

febr06

febr07

febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


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febr03

febr04

febr05

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febr08

febr09

febr10

febr11

febr12

febr13

0

1

2

3

4

5

6



CIANID(mgl)

DISPERSION-WAVE

)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


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)44

)(exp(

4

22

tD

y

tD

tvx

DDht

Gc

yx

x

xy

tDxx 2 tDyy 2

xcL 34 ycB 34

G [kg]

c2B

c2L

x2=vt2

cmax

x1=vt1

C (t2 x 0)

C (t2 x2 y)

C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


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C

xv

t

Cx 2

2

x

CDx

2

)4

)(exp(

2 tD

tvx

tDA

GC

x

x

x


2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


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2 tDA

GCmax


x

C C (t1x) C (t2x)

xcL 34 tDxx 2

x1 = vx t1 x2 = vx t2

Lc1 Lc2

)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


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)))1((4

)))1(((exp(

))1(((2

2

121 titD

titvx

titDA

tMC

x

xn

i x

i


][ skgM i

tti=1 i=n


Superposition


Slide 15

Slide 16

Slide 17

Slide 18

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Transport processes Emission – [mass/time] pollutants released into the environment Imission –...

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