transport eqn derivn Lecture.pdf

7/29/2019 transport eqn derivn Lecture.pdf

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1

DERIVATION OF BASICTRANSPORT

EQUATION


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2

Definitions

[M]

[L]

[T]

Basic dimensions

Mass

Length

Time

Concentration

Mass per unit

volume

[ML-3]

Mass Flow

Rate

Mass per unittime

[MT-1]

Flux

Mass flow rate through unit area[ML-2 T-1]


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3

The Transport Equation

Mass balance for a control volume where

the transport occurs only in one direction(say x-direction)

Massentering

the control

volume

Massleaving the

control

volume

xPositive x direction


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4


=

tinvolume

controlthe

leavingMass

tinvolume

controlthe

enteringMass

tinterval

timeain

volumecontrol

theinmass

ofChange

21JAJA

t

CV =

The mass balance for this case can be

written in the following form

Equation 1


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5


A closer look to Equation 1

21 JAJA

t

CV =

Volume [L3]

Concentration

over time

[ML-3 T-1]

Area

[L2]

Flux

[ML-2 T-1] Area

[L2]

Flux

[ML-2 T-1]

[L3

][ML-3

T-1

] = [MT-1

] [L2

][ML-2

T-1

] = [MT-1

]Mass over time Mass over time


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6


Change of mass in unit volume (divide all

sides of Equation 1 by the volume)

21 JV

A

JV

A

t

C

=

Equation 2

Rearrangements

( )21 JJV

A

t

C=

Equation 3


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The flux is changing in x direction with gradient of

A.J1

x

A.J2

xJ

Therefore

xx

JJJ

12

+= Equation 4

Positive x direction


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xxJJJ 12

+= Equation 4

Equation 3

+=

x

x

JJJ

V

A

t

C

11

Equation 5

( )21 JJVA

t

C

=


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9


+=

x

x

JJJ

V

A

t

C11 Equation 5

Rearrangements

x

1

V

Ax

A

V

==

=

x

x

JJJ

x

1

t

C

11

Equation 7

Equation 6


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10


Rearrangements

=

xx

JJJ

x

1

t

C11 Equation 7

x

J

t

C

=

Finally, the most general transport equation

in x direction is:

Equation 8


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11


We are living in a 3 dimensional space,

where the same rules for the general massbalance and transport are valid in all

dimensions. Therefore

=

=

3

1i

i

i

Jxt

CEquation 9

x1

= x

x2

= y

x3

= z

+

+

=

zyx JzJyJxt

CEquation 10


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12


The transport equation is derived for aconservative tracer (material)

The control volume is constant as the timeprogresses

The flux (J) can be anything (flows,dispersion, etc.)


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13

The Advective Flux


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14

The advective flux can be analyzed with the simple

conceptual model, which includes two control

volumes. Advection occurs only towards one

direction in a time interval.

x

I I I

x

Particle


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x

I I I

x

Particle

x is defined as the distance, which a particle canpass in a time interval oft.

The assumption is

that the particles move on the direction of

positive x only.


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x

I II

x

Particle

The number of particles (analogous to mass) moving from

control volume I to control volume II in the time interval t

can be calculated using the Equation below, where

AxCQ =

where Q is the number of particles (analogous to mass)passing from volume I to control volume II in the time interval

t

[M], C is the concentration of any material dissolved in

water in control volume I [ML-3

], x is the distance [L] and Ais the cross section area between the control volumes [L2].

Equation 11

x


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17

x

I I I

x

Particle

AxCQ =

t

AxC

t

Q

=

Number of

particles passing

from I to I I in t

Number of

particles

passing from I

to I I in unit time

C

t

xJ

tA

QADV

==

Number of particles passing from I to I I

in unit time per unit area = FLUX

Ct

x

Ct

x

limJ 0tADV

=

=

Division by cross-section area:

Division by time:

Equation 12

Advective

flux


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The Advective Flux

Ct

x

JADV

= Equation 12


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The Dispersive Flux


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The dispersive flux can be analyzed with thesimple conceptual model too. This conceptual

model also includes two control volumes.Dispersion occurs towards both directions in atime interval.

x

I I I

xParticle

x


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x is defined as the distance, which a particle canpass in a time interval oft. The assumption is

that the

particles move on positive and negative x directions. In

this case there are two directions, which particles

can move in the time interval oft.

x

I I I

xParticle

x


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22

Another assumption is that a particle does not change its

direction during the time interval of t

and that the

probability to move to positive and negative x directionsare equal (50%) for all particles.

Therefore, there are two components of the dispersive

mass transfer, one from the control volume I tocontrol volume I I and the second from the control

volume I I to control volume I

I I I

x

q1

q2


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23

I I I

x

q1

q2

AxC5.0q 11 =

AxC5.0q 22 =

21 q-qQ =

( )21 CCAx5.0Q =

Equation 13

Equation 14

Equation 15

Equation 16

50 % probability

x x


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Number of particles passing from I to I I in t

Number of particles passing from I to

II in unit time

Number of particles passing from I toII in unit time per unit area = FLUX

( )21

CCAx5.0Q =

( )

t

CCAx5.0

t

Q 21

=

xx

C

CC 12

+=

t

xxCCCAx5.0

t

Q11

+

=

t

xx

CCCAx5.0

t

Q11

=

t

xx

CAx5.0

t

Q

=

t

x

x

Cx5.0

JtA

Q DISP

==Divide

by

Area

I I I

x

Particle

Equation 16

Equation 17

Equation 18

Equation 19

Equation 20

Equation 21

Equation 22

Divideby time


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25

t

xx

Cx5.0

J

tA

QDISP

==

( )x

C

t

x5.0J

2

DISP

=

( )t

x5.0D

2

=

[ ][ ] ( ) [ ]

[ ]

( ) [ ] [ ][ ]

[ ]1-222

22 TLT

L

t

x5.0D

Tt

LxLx

5.0

=

=

x

CDJ

DISP

=

Equation 22

Equation 25

Equation 23

Equation 24


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Dispersion

Molecular diffusion Turbulent diffusion Longitudinal dispersion

GENEGENERALLYRALLY

Molecular diffusion


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Ranges of the

Dispersion

Coefficient

(D)
http://en.wikipedia.org/wiki/Image:Figure_2_4.jpg


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The Dispersive Flux

Equation 25

x

CDJ

DISP

=


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29

THE ADVECTION-DISPERSION

EQUATION FOR A

CONSERVATIVE MATERIAL


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The Advection-Dispersion Equation

( )dispersionadvection JJxt

C+

=

Equation 27

dispersionadvection Jx

Jxt

C

=

Equation 28

Ct

x

Jadvection

= x

C

DJdispersion

=

=

x

CDxCt

x

xt

CEquation 29


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33

Dimensional Analysis of the

Advection-Dispersion Equation

2

2

xCD

xCu

tC

+

=

Equation 30

Concentration

over time[ML-3 T-1]

Velocity times

concentrationover space

[LT-1][ML-3L-1]

= [ML-3 T-1]

[L2T-1][ML-3L-2]

= [ML-3 T-1]


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34


=

+

=

3

1i2

i

2

i

i

ixCD

xCu

tC

Equation 31

We are living in a 3 dimensional space, where the

same rules for the general mass balance and

transport are valid in all dimensions. Therefore

2

2

z2

2

y2

2

x

z

CD

z

Cw

y

CD

y

Cv

x

CD

x

Cu

t

C

+

+

+

=

Equation 32

x1

= x, u1

= u, D1

= Dx

x2

= y, u2

= v, D2

= Dy

x3

= z, u3

= w, D3

= Dz


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35

THE ADVECTION-DISPERSION

EQUATION FOR A NON

CONSERVATIVE MATERIAL


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for non conservative materials

Equation 32

+

+

+

+

=

Ckz

CD

z

Cw

y

CD

y

Cv

x

CD

x

Cu

t

C

2

2

z

2

2

y2

2

x

Equation 33

2

2

z2

2

y2

2

xz

CDz

Cwy

CDy

Cvx

CDx

Cut

C

+

+

+

=


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37

The Transport Equation for non

conservative materials withsedimentation

Equation 34

+

+

+

+

=

Ckz

CDz

Cw

yCD

yCv

xCD

xCu

tC

2

2

z

2

2

y2

2

x

Equation 33

z

C

Ckz

C

Dz

C

w

yCD

yCv

xCD

xCu

tC

ionsedimentat2

2

z

2

2

y2

2

x

+

+

+

+

=


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38

Transport Equation with all

Components

Equation 35

sinksandsourcesexternal

z

C

Ckz

C

Dz

C

w

yCD

yCv

xCD

xCu

tC

ionsedimentat2

2

z

2

2

y2

2

x

+

+

+

+

=

Sedimentation in z

direction

External loads

Interaction with bottom

Other sources and sinks

Di i l A l i f


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39

Dimensional Analysis of

Components

=

Ck

t

C

kineticsreaction

sinksandsourcesexternal

t

C

external

=

z

C

t

Cionsedimentat

ionsedimentat =

[ML-3][T-1]=[ML-3 T-1]

[LT-1

][ML-3

L-1

]=[ML-3

T-1

]

Must be given in [ML-3 T-1]


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40

Advection-Dispersion Equation with

all components

Equation 35sinksandsourcesexternal

zCCk

z

CD

z

Cw

y

C

Dy

C

vx

C

Dx

C

ut

C

ionsedimentat

2

2

z

2

2

y2

2

x

+

+

+

+

=

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transport eqn derivn Lecture.pdf