Dimensionless Numbers for Dissolution

Dimensionless Numbers for Dissolution
Modified Sherwood number
Sherwood number
Stanton number
Schmidt number
Lewis number
Reynolds number
Grashöf number
Rainer Helmig, Eric Bullinger für 100-online
λl2
Sh =
D
Kc l
Sh =
D
Kc
St =
v
v
Sc =
D
α
Le =
D
lv
Re =
v
l3g∆ρ/ρ
Gr =
ν2
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0
mass transfer rate
diffusion rate
mass transfer rate
diffusion rate
mass transfer rate
flow rate
diffusivity of momentum
diffusivity of mass
diffusivity of energy
diffusivity of mass
inertial forces
viscous forces
buoyancy forces
viscous forces
1
Mass transfer in the three–phase system water–NAPL–gas:
phase (n)
components: c (contaminant),
w (water), a (air)
NAPL
n
ss
ol
di
n
n
tio
sa
en
ut
io
di
io
at
riz
ss
ol
o
ap
ut
io
n
ev
nd
co
evaporization
phase (w)
components: w (water),
c (contaminant), a (air)
GROUNDWATER
GAS
condensation
phase (g)
components: a (air),
w (water), c (contaminant)
Mass transfer in the three–phase system water–NAPL–gas:
phase (n)
components: c (contaminant),
w (water), a (air)
NAPL
n
ss
ol
di
n
n
tio
sa
en
ut
io
di
io
at
riz
ss
ol
o
ap
ut
io
n
ev
nd
co
evaporization
phase (w)
components: w (water),
c (contaminant), a (air)
GROUNDWATER
GAS
condensation
phase (g)
components: a (air),
w (water), c (contaminant)
The thermodynamic phase equilibrium partitioning coefficients (K–factors)
k
ωαk = Kα:β
ωβk
are given by e.g. Henry’s law.
Rainer Helmig, Eric Bullinger für 100-online
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2
Interphase Mass Flux
General form of the interphase mass flux relationship:
J = Kc(Csat − C)
for the NAPL-water interface:
J=
1 dm
s
= Kc(Cw
− Cw )
Acs dt
dividing through by the REV and rearranging:
1 dm dC
Acs
s
s
=
=
Kc(Cw
− Cw ) = Kca(Cw
− Cw )
V dt
dt
V
common to use a lumped mass transfer rate coefficient:
λ = Kc · a
Rainer Helmig, Eric Bullinger für 100-online
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3
Hyperbolic case - Buckley–Leverett problem
pc ≡ 0, g = 0 leads in the 1D case to
the hyperbolic saturation equation
∂Sw
∂
+
(f (Sw )) = 0
∂t
∂x
f has S–shape
solutions have a shock, rarefaction
wave, or both
Hyperbolic case - Buckley–Leverett problem
rarefaction wave:
250 days
500 days
750 days
1000 days
1
Saturation w
pc ≡ 0, g = 0 leads in the 1D case to
the hyperbolic saturation equation
Linear Relative Permeabilities (mun/muw=2)
0.8
0.6
0.4
0.2
0
∂Sw
∂
+
(f (Sw )) = 0
∂t
∂x
0
50
100
150
200
250
300
x [m]
rarefaction wave and shock:
f has S–shape
Brooks-Corey, lambda=2, mun/muw=100
250 days
500 days
750 days
solutions have a shock, rarefaction
wave, or both
Saturation w
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
x [m]
Rainer Helmig, Eric Bullinger für 100-online
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4