Derivation of Basic Transport Equation (A. Ertürk)

DERIVATION OF BASIC
TRANSPORT
EQUATION
1

Definitions
Basic dimensions
[M] Mass
[L] Length
[T] Time
Concentration
Mass per unit
volume
[M·L -3 ]
Mass Flow Rate
Mass per unit
time
[M·T -1 ]
Flux
Mass flow rate
through unit area
[M·L -2 ·T -1 ]
2

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
Δx
Positive x direction
3

The Transport Equation
The mass balance for this case can be
written in the following form
⎡ Change of ⎤
⎢
⎥
⎢
mass in the
⎥
⎢control
volume⎥
=
⎢
⎥
⎢ in a time ⎥
⎢
⎥
⎣ interval Δt
⎦
⎡Mass
entering⎤
⎢
⎥
⎢
the control
⎥
⎢⎣
volume in Δt
⎥⎦
−
⎡Mass
leaving⎤
⎢
⎥
⎢
the control
⎥
⎢⎣
volume in Δt⎥⎦
V
∂C
∂t
=
A
⋅ J
−
A
1 J 2
⋅
Equation 1
4

The Transport Equation
A closer look to Equation 1
V
⋅
∂C
∂t
=
A
⋅
J
−
A
1 J 2
⋅
Flux
Flux
Volume [L 3 ]
Concentration
over time
[M·L -3 ·T -1 ]
Area
[L 2 ]
[M·L -2 ·T -1 ]
Area
[L 2 ]
[M·L -2 ·T -1 ]
[L 3 ]·[M·L -3·T -1 ] = [M·T -1 ] [L 2 ]·[M·L -2·T -1 ] = [M·T -1 ]
Mass over time
Mass over time
5

The Transport Equation
Change of mass in unit volume (divide all
sides of Equation 1 by the volume)
∂C
∂t
=
A
V
⋅
J
−
A
1 J 2
V
⋅
Equation 2
Rearrangements
∂C
∂t
A
= ⋅
V
( − )
J 1 J 2
Equation 3
6

The Transport Equation
A.J 1
A.J 2
x
Δx
Positive x direction
The flux is changing in x direction with gradient of
Therefore
∂J
∂
∂J
= J + Δx
Equation 4
∂x
J2 1 ⋅
7

The Transport Equation
∂C
∂t
=
A
V
⋅
( − )
J 1 J 2
Equation 3
∂J
= J + Δx Equation 4
∂x
J2 1 ⋅
∂C
∂t
=
A
V
⋅
⎛
⎜J
⎝
⎛
−⎜J
⎝
+
∂J
∂x
⋅ Δx
⎞⎞
⎟
⎟
⎠⎠
1 1
Equation 5
8

The Transport Equation
Rearrangements
∂C
∂t
=
A
V
⋅
⎛
⎜J
⎝
⎛
−⎜J
⎝
+
∂J
∂x
⋅ Δx
⎞⎞
⎟
⎟
⎠⎠
1 1
Equation 5
V
A
= Δx
⇒
A
V
=
1
Δx
Equation 6
∂C
∂t
=
1
Δx
⋅
⎛
⎜
⎝
J
−
J
−
∂J
∂x
⎞
⋅ Δx⎟
⎠
1 1
Equation 7
9

The Transport Equation
Rearrangements
∂C
∂t
=
1
Δx
⋅
⎛
⎜
⎝
J
−
J
−
∂J
∂x
⎞
⋅ Δx⎟
⎠
1 1
Equation 7
Finally, the most general transport equation
in x direction is:
∂C
∂t
=
−
∂J
∂x
Equation 8
10

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
C
∂t
∂
3
=
−
∑
i= 1 ∂
∂
x
i
J
i
x 1 = x
x 2 = y
x 3 = z
Equation 9
∂C
∂t
=
−
⎛
⎜
⎝
∂
∂x
J
x
+
∂
∂y
J
y
+
∂
∂z
J
z
⎞
⎟
⎠
Equation 10
11

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.)
12

The Advective Flux
13

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
14

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.
Δx
I
II
x
Particle
15

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
Q
=
C
⋅ Δx
⋅
A
Equation 11
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 [L 2 ]. 16

Δx
I
II
Q
=
C
⋅
Δx
⋅
A
Number of
particles passing
from I to II in Δt
x
Division by cross-section area:
Q
A ⋅ Δt
=
J
ADV
=
Δx
Δt
⋅ C
Particle
Division by time:
Q
Δt
=
C
⋅ Δx
⋅
Δt
A
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
J
ADV
=
lim
Δt→0
⎛
⎜
⎝
Δx
Δt
⋅
C
⎞
⎟
⎠
=
∂x
∂t
⋅
C
Advective
flux
17
Equation 12

The Advective Flux
J ADV
∂x
= ⋅ C Equation 12
∂t
18

The Dispersive Flux
19

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
Δx
I
II
x
Particle
20

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
Δx
I
II
x
Particle
21

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
q 1
q 2
I
II
x
22

I
q 1
II
q 2
A
x
q 1
1 = 0.5 ⋅ C ⋅ Δx
⋅
q 2
2 = 0.5 ⋅ C ⋅ Δx
⋅
Q = q1 - q 2
50 % probability
A
Equation 13
Equation 14
Equation 15
Q
=
0.5
⋅
Δx
⋅
A
( − )
C 1 C 2
Equation 16
23

Number of particles passing from I to II in Δt
Q
Δt
= 0.5
Equation 16
⋅ Δx
⋅
A
0.5 ⋅ Δx
⋅ A ⋅
=
Δt
( − )
C 1 C 2
Number of particles passing from I to
II in unit time per unit area = FLUX
⋅
( C − C )
Q 1 2
Equation 17
C = C1
∂C
∂x
2 + ⋅
Equation 18
Q
A ⋅ Δt
=
J
Equation 22
DISP
=
Δx
− 0.5 ⋅ Δx
⋅
Δt
∂C
∂x
⋅ Δx
Divide
by time
Divide
by
Area
Number of particles passing from I to
II in unit time
⎛ ⎛ ∂C
⎞⎞
0.5 ⋅ Δx
⋅ A ⋅ ⎜C1
− ⎜C1
+ ⋅ Δx⎟⎟
Q
⎝ ⎝ ∂x
⎠
=
⎠
Δt
Δt
Equation 19
Q
Δt
Q
Δt
⎛
0.5 ⋅ Δx
⋅ A ⋅ ⎜C1
− C
=
⎝
Δt
Equation 20
=
Equation 21
x
I
− 0.5 ⋅ Δx
⋅ A
Δt
Δx
∂C
∂x
Δx
1
⋅ Δx
II
∂C
−
∂x
⎞
⋅ Δx⎟
⎠
24
Particle

Q
A ⋅ Δt
=
J
DISP
=
−
0.5
⋅
Δx
⋅
Δt
∂C
∂x
⋅
Δx
Equation 22
J
DISP
Equation 23
0.5 ⋅
= −
Δt
D =
( Δx)
2
0.5 ⋅
Δt
Equation 24
( Δx)
∂C
⋅
∂x
2
J DISP
Equation 25
= −D
⋅
∂C
∂x
0.5
Δx
Δt
→
→
→
[ ]
2
[] ( ) [ ]
2
L ⇒ Δx
→ L
[ T]
⎫
⎪
⎬
⎪
⎭
⇒
D
=
0.5⋅
Δt
2
( ) [][ ]
2
Δx
⎡ ⋅ L
→
⎢
⎣
[ T]
⎤
⎥
⎦
=
[ ]
2 -1
L ⋅ T
25

Dispersion
Molecular diffusion Turbulent diffusion Longitudinal dispersion
GENERALLY
Molecular diffusion

Ranges of the
Dispersion
Coefficient (D)
27

The Dispersive Flux
∂C
J DISP
= −D
⋅
∂x
Equation 25
28

THE ADVECTION-DISPERSION
EQUATION FOR A
CONSERVATIVE MATERIAL
29

The Advection-Dispersion Equation
∂C
∂t
=
J advection
J dispersion
−
∂
J
Equation 8
∂x
∂x
= ⋅ C
Equation 12
∂t
∂C
= − D ⋅ Equation 25
∂x
J = J advection
+ J dispersion
∂C
∂t
=
−
∂
∂x
Equation 26
( + )
J advection
J dispersion
General
transport
equation
Advective
flux
Dispersive
flux
Equation 27
30

The Advection-Dispersion Equation
∂C
∂t
=
−
∂
∂x
( + )
J advection
J dispersion
Equation 27
∂C
∂t
=
−
∂
∂x
J
−
∂
∂x
advection
J dispersion
Equation 28
J advection
=
∂x
∂t
⋅
C
J dispersion
= −D
⋅
∂C
∂x
∂C
∂t
=
−
∂
∂x
⎛
⎜
⎝
∂x
∂t
⋅
⎞
C⎟
⎠
−
∂
∂x
⎛
⎜
⎝
−
D
⋅
∂C
∂x
⎞
⎟
⎠
Equation 29
31

The Advection-Dispersion Equation
∂C
∂t
=
−
∂
∂x
⎛
⎜
⎝
∂x
∂t
⋅
⎞
C⎟
⎠
−
∂
∂x
⎛
⎜
⎝
−
D
⋅
∂C
∂x
⎞
⎟
⎠
Equation 29
Velocity u in
x direction
⋅ 2
∂C
u ∂ x
2
∂ C
− D ⋅
∂ x
∂C
∂t
=
−u
⋅
∂C
∂x
+
D
⋅
∂
2
∂x
C
2
Equation 30
32

Dimensional Analysis of the
Advection-Dispersion Equation
∂C
∂t
=
−u
⋅
∂C
∂x
+
D
⋅
∂
2
∂x
C
2
Equation 30
Concentration
over time
[M·L -3 ·T -1 ]
Velocity times
concentration
over space
[L·T -1 ]·[M·L -3·L -1 ]
[L 2·T -1 ]·[M·L -3·L -2 ]
= [M·L -3 ·T -1 ]
= [M·L -3 ·T -1 ]
33

The Advection-Dispersion 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
∂C
=
∂t
3
∑
i=
1
⎛
⎜−u
⎝
Equation 31
i
⋅
∂C
∂x
i
+
D
i
⋅
∂
2
∂x
C
2
i
⎞
⎟
⎠
x 1 = x, u 1 = u, D 1 = D x
x 2 = y, u 2 = v, D 2 = D y
x 3 = z, u 3 = w, D 3 = D z
∂C
∂t
= −u
⋅
∂C
∂x
+
D
x
⋅
∂
2
∂x
C
2
− v
⋅
∂C
∂y
+
D
y
⋅
∂
2
∂y
C
2
−
w
⋅
∂C
∂z
+
D
z
⋅
∂
2
∂z
C
2
Equation 32
34

THE ADVECTION-DISPERSION
EQUATION FOR A NON
CONSERVATIVE MATERIAL
35

36
The Advection-Dispersion Equation
for non conservative materials
Equation 32
+ ∑ ⋅
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
= −
∂
∂
C
k
z
C
D
z
C
w
y
C
D
y
C
v
x
C
D
x
C
u
t
C
2
2
z
2
2
y
2
2
x
Equation 33
2
2
z
2
2
y
2
2
x
z
C
D
z
C
w
y
C
D
y
C
v
x
C
D
x
C
u
t
C
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
= −
∂
∂

37
The Transport Equation for non
conservative materials with
sedimentation
Equation 34
+ ∑ ⋅
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
= −
∂
∂
C
k
z
C
D
z
C
w
y
C
D
y
C
v
x
C
D
x
C
u
t
C
2
2
z
2
2
y
2
2
x
Equation 33
z
C
C
k
z
C
D
z
C
w
y
C
D
y
C
v
x
C
D
x
C
u
t
C
sedimentation
2
2
z
2
2
y
2
2
x
∂
∂
⋅
−
⋅
+
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
= −
∂
∂
∑
υ

Transport Equation with all
Components
∂C
∂t
= −u
⋅
∂C
∂x
+
D
x
⋅
∂
2
∂x
C
− v
+ D
∂C
∂ C
− w ⋅ + Dz
⋅ + ∑k
⋅ C −υ
2
∂z
∂z
± external sources and sinks
2
2
∂C
⋅
∂y
y
⋅
∂
2
∂y
C
2
sedimentation
Equation 35
∂C
⋅
∂z
Sedimentation in z
direction
• External loads
• Interaction with bottom
• Other sources and sinks
38

⎛
⎜
⎝
∂C
⎞
⎟
∂t
⎠
reaction
Dimensional Analysis of
kinetics
[M·L -3 ]·[T -1 ]=[M·L -3 ·T -1 ]
Components
= ∑k
⋅ C
⎛
⎜
⎝
∂C
⎞
⎟
∂t
⎠
sedimentation
= −υ
sedimentation
[L·T -1 ]·[M·L -3·L -1 ]=[M·L -3 ·T -1 ]
⋅
∂C
∂z
⎛
⎜
⎝
∂C
∂t
⎞
⎟
⎠
external
= ±
external
sources
and
sinks
Must be given in [M·L -3 ·T -1 ]
39

40
Advection-Dispersion Equation with
all components
Equation 35
sinks
and
sources
external
z
C
C
k
z
C
D
z
C
w
y
C
D
y
C
v
x
C
D
x
C
u
t
C
sedimentation
2
2
z
2
2
y
2
2
x
±
∂
∂
⋅
−
⋅
+
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
−
∂
∂
⋅
+
∂
∂
⋅
= −
∂
∂
∑
υ