a d derivation
TRANSCRIPT
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DERIVATION OF BASICTRANSPORT
EQUATION
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Definitions
[M]
[L]
[T]
Basic dimensions
Mass
Length
Time
Concentration
Mass per unit
volume
[M·L-3]
Mass Flow
Rate
Mass per unittime
[M·T-1]
Flux
Mass flow
rate through unit area
[M·L-2 ·T-1]
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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
x
Positive x direction
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The Transport Equation
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Δ
−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Δ
=
⎥⎥⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎢
⎣
⎡
Δ
t in volume
control the
leaving Mass
t in volume
control the
entering Mass
t interval
timeain
volumecontrol
theinmass
of Change
21J AJ A
t
CV ⋅−⋅=
∂
∂
The mass balance for this case can be
written in the
following form
Equation 1
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The Transport Equation
A closer look to Equation 1
21 J AJ A
t
CV ⋅−⋅=
∂
∂⋅
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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The Transport Equation
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 Transport Equation
The flux is changing in x direction with gradient of
A.J1
x
A.J2
xJ
∂∂
Therefore
Δxx
JJJ
12
⋅
∂
∂+= Equation 4
Positive x direction
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The Transport Equation
ΔxxJJJ 12 ⋅
∂∂+= Equation 4
Equation 3
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ ⎟ ⎠
⎞⎜⎝
⎛ Δ⋅
∂
∂+−⋅=
∂
∂x
x
JJJ
V
A
t
C
11
Equation 5
( )21 JJV A
t
C
−⋅=∂
∂
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The Transport Equation
⎟⎟
⎠
⎞⎜⎜
⎝
⎛ ⎟
⎠
⎞⎜
⎝
⎛ Δ⋅
∂
∂+−⋅=
∂
∂x
x
JJJ
V
A
t
C11 Equation 5
Rearrangements
x
1
V
Ax
A
V
Δ
=⇒Δ=
⎟ ⎠
⎞⎜⎝
⎛ Δ⋅
∂
∂−−⋅
Δ=
∂
∂x
x
JJJ
x
1
t
C
11
Equation 7
Equation 6
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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
x
Particle
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 Ais the cross section area between
the control volumes [L2].
Equation 11
Δx
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Δx
I II
x
Particle
AxCQ ⋅Δ⋅=
t
AxC
t
Q
Δ
⋅Δ⋅=
Δ
Number of
particles passing
from I to II in t
Number of
particles
passing from I
to II in unit time
C
t
xJ
t A
Q ADV ⋅
Δ
Δ==
Δ⋅
Number of particles passing from I to II
in unit time per unit area = FLUX
Ct
x
Ct
x
limJ 0t ADV ⋅∂
∂
=⎟ ⎠
⎞
⎜⎝
⎛
⋅Δ
Δ
= →Δ
Division by cross-section area:
Division by time:
Equation 12
Advective
flux
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The Advective Flux
Ct
x
J ADV ⋅∂
∂= 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.
Δx
I II
xParticle
Δx
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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
xParticle
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 directionsare equal (50%) for all particles.
Therefore, there are two components of the dispersive
mass transfer, one from the control volume I tocontrol volume II and the second from the control
volume II to control volume I
I II
x
q1
q2
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I II
x
q1
q2
AxC5.0q 11 ⋅Δ⋅⋅=
AxC5.0q 22 ⋅Δ⋅⋅=
21 q-qQ =
( )21 CC Ax5.0Q −⋅Δ⋅=
Equation 13
Equation 14
Equation 15
Equation 16
50 % probability
Δx Δx
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Number of particles passing from I to II 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
CC Ax5.0Q −⋅⋅Δ⋅=
( )
t
CC Ax5.0
t
Q 21
Δ
−⋅⋅Δ⋅=
Δ
Δ
xx
C
CC 12 ⋅
∂
∂+=
t
xxCCC Ax5.0
t
Q11
Δ
⎟⎟ ⎠ ⎞⎜⎜
⎝ ⎛ ⎟
⎠ ⎞⎜
⎝ ⎛ Δ⋅
∂∂+−⋅⋅Δ⋅
=Δ
t
xx
CCC Ax5.0
t
Q11
Δ
⎟ ⎠
⎞⎜⎝
⎛ Δ⋅
∂
∂−−⋅⋅Δ⋅
=Δ
t
xx
C Ax5.0
t
Q
Δ
Δ⋅∂
∂⋅Δ⋅−
=
Δ
t
x
x
Cx5.0
Jt A
Q DISPΔ
Δ⋅
∂
∂⋅Δ⋅−
==Δ⋅Divide
by
Area
I II
x
Particle
Equation 16
Equation 17
Equation 18
Equation 19
Equation 20
Equation 21
Equation 22
Divideby time
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t
xx
Cx5.0
J
t A
QDISP
Δ
Δ⋅∂
∂⋅Δ⋅−
==
Δ⋅
( )x
C
t
x5.0J
2
DISP∂∂⋅
ΔΔ⋅−=
( )t
x5.0D
2
ΔΔ⋅=
[ ][ ] ( ) [ ]
[ ]
( ) [ ] [ ][ ]
[ ]1-222
22 TLT
L
t
x5.0D
T t
LxLx
5.0
⋅=⎥⎦
⎤⎢⎣
⎡ ⋅→
Δ
Δ⋅=⇒
⎪⎭
⎪⎬
⎫
→Δ
→Δ⇒→Δ
→
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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THE ADVECTION-DISPERSION
EQUATION FOR A
CONSERVATIVE MATERIAL
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The Advection-Dispersion Equation
J
xt
C
∂
∂−=
∂
∂Equation 8
Ct
xJadvection ⋅
∂
∂= Equation 12
x
CDJdispersion
∂
∂⋅−= Equation 25
dispersionadvection JJJ +=
General
transport
equation
Advective
flux
Dispersive
flux
Equation 26
( )dispersionadvection JJxtC +∂∂
−=∂
∂Equation 27
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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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The Advection-Dispersion Equation
Equation 29⎟ ⎠
⎞⎜⎝
⎛
∂
∂⋅−
∂
∂−⎟
⎠
⎞⎜⎝
⎛ ⋅
∂
∂
∂
∂−=
∂
∂
x
CD
xC
t
x
xt
C
Velocity u in
x direction
x
Cu
∂
∂⋅ 2
2
x
CD
∂
∂⋅−
2
2
x
C
Dx
C
ut
C
∂
∂
⋅+∂
∂
⋅−=∂
∂Equation 30
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Dimensional Analysis of the
Advection-Dispersion Equation
2
2
xCD
xCu
tC
∂∂⋅+
∂∂⋅−=
∂∂ Equation 30
Concentration
over time[M·L-3 ·T-1]
Velocity times
concentrationover 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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The Advection-Dispersion Equation
∑= ⎟⎟ ⎠ ⎞
⎜⎜⎝
⎛
∂
∂⋅+∂
∂⋅−=∂
∂ 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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THE ADVECTION-DISPERSION
EQUATION FOR A NON
CONSERVATIVE MATERIAL
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The Advection-Dispersion Equation
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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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
∂
∂⋅−⋅+
∂
∂⋅+
∂
∂⋅−
∂
∂⋅+∂
∂⋅−∂
∂⋅+∂
∂⋅−=∂
∂
∑ υ
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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
±
∂
∂
⋅−⋅+∂
∂
⋅+∂
∂
⋅−
∂∂⋅+
∂∂⋅−
∂∂⋅+
∂∂⋅−=
∂∂
∑ υ
Sedimentation in z
direction
• External loads
• Interaction with bottom
• Other sources and sinks
Di i l A l i f
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Dimensional Analysis of
Components
∑ ⋅=⎟ ⎠
⎞⎜
⎝
⎛
∂
∂Ck
t
C
kineticsreaction
sinksandsourcesexternal
t
C
external
±=⎟
⎠
⎞⎜
⎝
⎛
∂
∂
z
C
t
Cionsedimentat
ionsedimentat ∂∂⋅−=⎟
⎠ ⎞⎜
⎝ ⎛ ∂∂
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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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
±
∂∂⋅−⋅+
∂
∂⋅+∂
∂⋅−
∂
∂⋅+
∂
∂⋅−
∂
∂⋅+
∂
∂⋅−=
∂
∂
∑ υ