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DERIVATION OF BASIC TRANSPORT EQUATION
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Definitions
[M][L][T]
Basic dimensionsMassLengthTime
ConcentrationMass per unit volume[M·L-3]
Mass Flow
RateMass per unit time[M·T-1]
FluxMass flow
rate
through unit area[M·L-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)
Mass entering
the control volume
Mass leaving the
control volume
ΔxPositive x direction
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4
The Transport Equation
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Δ−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Δ=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
Δt in volume
control theleaving Mass
t in volumecontrol theentering Mass
t intervaltimea involume control
theinmassofChange
21 JAJAtCV ⋅−⋅=∂∂
The mass balance for this case can be written in the
following form
Equation 1
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5
The Transport Equation
A closer look to Equation 1
21 JAJAtCV ⋅−⋅=∂∂⋅
Volume [L3]
Concentration over time
[M·L-3 ·T-1]
Area
[L2]
Flux
[M·L-2 ·T-1] Area
[L2]
Flux
[M·L-2 ·T-1]
[L3]·[M·L-3·T-1] = [M·T-1] [L2]·[M·L-2·T-1] = [M·T-1]
Mass over time Mass over time
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6
The Transport Equation
Change of mass in unit volume (divide all sides of Equation 1 by the volume)
21 JVAJ
VA
tC
⋅−⋅=∂∂
Equation 2
Rearrangements
( )21 JJVA
tC
−⋅=∂∂ Equation 3
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7
The Transport Equation
The flux is changing in x direction with gradient of
A.J1
Δx
A.J2
xJ∂∂
Therefore
ΔxxJJJ 12 ⋅∂∂
+= Equation 4
Positive x direction
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8
The Transport Equation
ΔxxJJJ 12 ⋅∂∂
+= Equation 4
Equation 3
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ Δ⋅
∂∂
+−⋅=∂∂ x
xJJJ
VA
tC
11 Equation 5
( )21 JJVA
tC
−⋅=∂∂
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9
The Transport Equation
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ Δ⋅
∂∂
+−⋅=∂∂ x
xJJJ
VA
tC
11 Equation 5
Rearrangements
x1
VAx
AV
Δ=⇒Δ=
⎟⎠⎞
⎜⎝⎛ Δ⋅
∂∂
−−⋅Δ
=∂∂ x
xJJJ
x1
tC
11 Equation 7
Equation 6
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10
The Transport Equation
Rearrangements
⎟⎠⎞
⎜⎝⎛ Δ⋅
∂∂
−−⋅Δ
=∂∂ x
xJJJ
x1
tC
11 Equation 7
xJ
tC
∂∂
−=∂∂
Finally, the most general transport equation in x direction is:
Equation 8
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11
The Transport Equation
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
∑= ∂
∂−=
∂∂ 3
1ii
i
Jxt
C Equation 9
x1
= x
x2
= y
x3
= z
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
−=∂∂
zyx Jz
Jy
Jxt
C Equation 10
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The Transport Equation
•
The transport equation is derived for a conservative tracer (material)
•
The control volume is constant as the time progresses
•
The flux (J) can be anything (flows, dispersion, etc.)
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The Advective Flux
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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.
Δx
I II
x Particle
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Δx
I II
x Particle
Δx
is defined as the distance, which a particle can pass in a time interval of Δt.
The assumption is
that the particles move on the direction of positive x only.
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Δx
I II
xParticle
The number of particles (analogous to mass) moving from control volume I to control volume II in the time interval Δt can be calculated using the Equation below, where
AxCQ ⋅Δ⋅=where Q is the number of particles (analogous to mass) passing from volume I to control volume II in the time interval Δt
[M], C is the concentration of any material dissolved in
water in control volume I [M·L-3], Δx
is the distance [L] and A is the cross section area between
the control volumes [L2].
Equation 11
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Δx
I II
xParticle
AxCQ ⋅Δ⋅=
tAxC
tQ
Δ⋅Δ⋅
=Δ
Number of particles passing from I to II in Δt
Number of particles passing from I to II in unit time
CtxJ
tAQ
ADV ⋅ΔΔ
==Δ⋅
Number of particles passing from I to
II in unit time per unit area = FLUX
CtxC
txlimJ
0tADV ⋅∂∂
=⎟⎠⎞
⎜⎝⎛ ⋅ΔΔ
=→Δ
Division by cross-section area:
Division by time:
Equation 12
Advective flux
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The Advective Flux
CtxJADV ⋅∂∂
= Equation 12
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The Dispersive Flux
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The dispersive flux can be analyzed with the simple conceptual model too. This conceptual model also includes two control volumes. Dispersion occurs towards both directions in a time interval.
Δx
I II
x Particle
Δx
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21
Δx
is defined as the distance, which a particle can pass in a time interval of Δt. 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 of Δt. Δx
I II
x Particle
Δx
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Another assumption is that a particle does not change its direction during the time interval of Δt
and that the
probability to move to positive and negative x directions are equal (50%) for all particles. Therefore, there are two components of the dispersive mass transfer, one from the control volume I to control volume II and the second from the control volume II to control volume I
I II
x
q1
q2
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23
I II
x
q1
q2
AxC5.0q 11 ⋅Δ⋅⋅=
AxC5.0q 22 ⋅Δ⋅⋅=
21 q-qQ =
( )21 CCA x5.0Q −⋅Δ⋅=
Equation 13
Equation 14
Equation 15
Equation 16
50 % probability
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Number of particles passing from I to II in Δt
Number of particles passing from I to II in unit time
Number of particles passing from I to II in unit time per unit area = FLUX
( )21 CCAx5.0Q −⋅⋅Δ⋅=
( )t
CCAx5.0t
Q 21
Δ−⋅⋅Δ⋅
=Δ
ΔxxCCC 12 ⋅∂∂
+=t
xxCCCAx5.0
tQ 11
Δ
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ Δ⋅
∂∂
+−⋅⋅Δ⋅=
Δ
t
xxCCCAx5.0
tQ 11
Δ
⎟⎠⎞
⎜⎝⎛ Δ⋅
∂∂
−−⋅⋅Δ⋅=
Δ
t
xxCA x5.0
tQ
Δ
Δ⋅∂∂
⋅Δ⋅−=
Δ
t
xxCx5.0
JtA
QDISP Δ
Δ⋅∂∂⋅Δ⋅−
==Δ⋅
Divide by
Area
Δx
I II
xParticle
Δx
Equation 16
Equation 17
Equation 18
Equation 19
Equation 20
Equation 21
Equation 22
Divide by time
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25
t
xxCx5.0
JtA
QDISP Δ
Δ⋅∂∂⋅Δ⋅−
==Δ⋅
( )xC
tx5.0J
2
DISP ∂∂⋅
ΔΔ⋅
−=
( )t
x5.0D2
ΔΔ⋅
=
[ ][ ] ( ) [ ][ ]
( ) [ ] [ ][ ] [ ]1-2
2222 TL
TL
tx5.0D
T tLxLx
5.0⋅=⎥
⎦
⎤⎢⎣
⎡ ⋅→
ΔΔ⋅
=⇒⎪⎭
⎪⎬
⎫
→Δ→Δ⇒→Δ
→
xCDJDISP ∂∂⋅−=
Equation 22
Equation 25
Equation 23
Equation 24
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Dispersion
Molecular diffusion Turbulent diffusion Longitudinal dispersion
GENEGENERALLYRALLY
Molecular diffusion << Turbulent diffusion << Longitudinal dispeMolecular diffusion << Turbulent diffusion << Longitudinal dispersionrsion
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The Dispersive Flux
Equation 25xCDJDISP ∂∂⋅−=
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THE ADVECTION-DISPERSION EQUATION FOR A
CONSERVATIVE MATERIAL
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30
The Advection-Dispersion Equation
Jxt
C∂∂
−=∂∂ Equation 8
CtxJadvection ⋅∂∂
= Equation 12
xCDJdispersion ∂∂⋅−= Equation 25
dispersionadvection JJJ +=
General transport equation
Advective flux
Dispersive flux
Equation 26
( )dispersionadvection JJxt
C+
∂∂
−=∂∂
Equation 27
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31
The Advection-Dispersion Equation
( )dispersionadvection JJxt
C+
∂∂
−=∂∂
Equation 27
dispersionadvection Jx
Jxt
C∂∂
−∂∂
−=∂∂ Equation 28
CtxJadvection ⋅∂∂
=xCDJdispersion ∂∂⋅−=
⎟⎠⎞
⎜⎝⎛
∂∂⋅−
∂∂
−⎟⎠⎞
⎜⎝⎛ ⋅∂∂
∂∂
−=∂∂
xCD
xC
tx
xtC
Equation 29
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32
The Advection-Dispersion Equation
Equation 29⎟⎠⎞
⎜⎝⎛
∂∂⋅−
∂∂
−⎟⎠⎞
⎜⎝⎛ ⋅∂∂
∂∂
−=∂∂
xCD
xC
tx
xtC
Velocity u in x direction
xCu∂∂⋅ 2
2
xCD
∂∂⋅−
2
2
xCD
xCu
tC
∂∂⋅+
∂∂⋅−=
∂∂
Equation 30
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33
Dimensional Analysis of the Advection-Dispersion Equation
2
2
xCD
xCu
tC
∂∂⋅+
∂∂⋅−=
∂∂
Equation 30
Concentration over time
[M·L-3 ·T-1]
Velocity times concentration over space
[L·T-1]·[M·L-3·L-1]
= [M·L-3 ·T-1]
[L2·T-1]·[M·L-3·L-2]
= [M·L-3 ·T-1]
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34
The Advection-Dispersion Equation
∑=
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅+
∂∂⋅−=
∂∂ 3
1i2
i
2
ii
i xCD
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 zCD
zCw
yCD
yCv
xCD
xCu
tC
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−=
∂∂
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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36
The Advection-Dispersion Equation for non conservative materials
Equation 32
∑ ⋅+∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−=
∂∂
CkzCD
zCw
yCD
yCv
xCD
xCu
tC
2
2
z
2
2
y2
2
x
Equation 33
2
2
z2
2
y2
2
x zCD
zCw
yCD
yCv
xCD
xCu
tC
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−=
∂∂
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37
The Transport Equation for non conservative materials with
sedimentation
Equation 34
∑ ⋅+∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−=
∂∂
CkzCD
zCw
yCD
yCv
xCD
xCu
tC
2
2
z
2
2
y2
2
x
Equation 33
zCCk
zCD
zCw
yCD
yCv
xCD
xCu
tC
ionsedimentat2
2
z
2
2
y2
2
x
∂∂⋅−⋅+
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−=
∂∂
∑ υ
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38
Transport Equation with all Components
Equation 35
sinksandsourcesexternalzCCk
zCD
zCw
yCD
yCv
xCD
xCu
tC
ionsedimentat2
2
z
2
2
y2
2
x
±∂∂⋅−⋅+
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−=
∂∂
∑ υ
Sedimentation in z direction
• External loads
• Interaction with bottom
• Other sources and sinks
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39
Dimensional Analysis of Components
∑ ⋅=⎟⎠⎞
⎜⎝⎛∂∂ Ck
tC
kineticsreaction
sinksandsourcesexternaltC
external
±=⎟⎠⎞
⎜⎝⎛∂∂
zC
tC
ionsedimentationsedimentat ∂
∂⋅−=⎟
⎠⎞
⎜⎝⎛∂∂ υ
[M·L-3]·[T-1]=[M·L-3
·T-1]
[L·T-1]·[M·L-3·L-1]=[M·L-3
·T-1]
Must be given in [M·L-3 ·T-1]
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40
Advection-Dispersion Equation with all components
Equation 35
sinksandsourcesexternalzCCk
zCD
zCw
yCD
yCv
xCD
xCu
tC
ionsedimentat
2
2
z
2
2
y2
2
x
±∂∂⋅−⋅+
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−=
∂∂
∑ υ