The practical applications of the stochastic processes theory are multiple. This fact is a consequence of the capacity of this theory that can predict the future of a dynamic system by use of its history and its actual state. Between the most famous applications we remember:- the analysis of all movement sorts starting with those from atomic and
molecular level [4.62-4.62] and finishing with those that describe the highsystem evolution such as atmospheric system [4.63-4.64]
- the analysis of dynamic links for networks with stations where the service time is stochastic distributed (computers networks, Internet networks, etc);
- the analysis of virtual experiences given with a stochastic model [4.65]]- the analysis of capitals and finances movement [4.66-4.67];- the analysis and development of all games sorts [4.68- the optimization and the control of all sorts of dynamic systems [4.68]
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Flow description by stochastic modeling
Flow description by stochastic modeling
Each transformation or phenomena results from one or many
elementary steps or processes. The equilibrium state results from
similar but contrary transport fluxes (Em Bratu, 1974)
The mathematical theory of complete connected chains [4.12-4.16]
could be described with this condensed statement:
The state of a system at time n is a random variable An with values
in a finite space (measurable) (A A ). The state evolution at time n+1
results from the appearance of a Bn+1 result which is also a random
variable with values in a finite space (measurable) (B B). The
appearance of a result signaling the state evolution could be
represented considering a u application of AxB in A and introducing
the following statement: An+1=u(An, Bn+1) for all n>=0 . The Bn+1
probability distribution conditioned by Bn, An, Bn-1, An-1,…….B1, A1,
A0, and symbolized as (P(Bn+1/Bn,An,…)), depends only on the state
An. The group [(A,A),(B,B), u:AxB→A ,P] defines a random system
with complete connections".
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Flow description by stochastic modeling
Process components and Connection processI) If one elementary particle is participating to a transport phenomena process within a medium with
random characteristics like granular, porous solid etc., the medium will cause random velocity
changes of the particle. In this case the transport process concerning the local velocity are the
named "process components", whereas the transport process changes given by the random
properties of the medium gives the named "connection process
II) The transport phenomena occurs when the displacement of the carriers through different ways
("process components") and the passage from one way to another one is realized by a random
commutation process ("connection process")
III) -A particle (molecule, group of molecules, turbulent group etc) evolves in a medium which
produces its transformation, it means that the process exchange characterizes the particle
evolution. The process occurring before and after the transformation are the "process
components", whereas the transformation itself represents the stochastic evolution "connection
process"
IV) The elementary particles passes randomly from one compartment to another one, the
compartments exchange process form the "connection process" whereas the transformation
realized in each compartment represents the "process components"
V) When a phenomena results on different structures formation, the passage from one structure to
another one is made randomly, in this case the structure formation is the "process component"
and the passing steps are the "connection process"
Flow description by stochastic modeling
Upper current
Upper region
Lower region
Pumping region
H
h
E1
E2
S1
S2
S3
Va
Vi
Vm
D
Mechanical mixing of a liquid media
Some process factors
- the turn rate or frequency of the stirrer.
- the configuration and the distribution of the
stirring paddles in the apparatus.
- the physicochemical properties of the medium.
- the position of entry and the exit of the currents in
the apparatus tank.
Basic topology for mixing with mechanical stirrer
2)(
4
2m
aV
hHD
V
224
2m
sum
iV
VVV
hD
V
)5
d
5
DdD(
60
bV
22
m
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Flow description by stochastic modeling
Mechanical mixing of a liquid media
Upper current
Upper region
Lower region
Pumping region
H
h
E1
E
2
S
1
S2
S3
V
a
Vi
V
m
D
Basic topology for mixing with mechanical stirrer
Identification of detailed topology
i) The start conditions are established;
ii) Va, Vm, Vi are calculated with given relations;
iii)If Va>Vi then r=Va/Vi ; for opposite case r=Vi/Va ;
iv) The number of cells is chosen in the smaller region;
ni=nch=1 (frequently but this is a relative consideration)
if the lower region is the smaller one or na=nch=1 when
the upper region is the smaller one;
v)If h/H=0.5 we can consider ni=na=nch ; if h/H>0.5
which is equivalent to Vi>Va we can write na=nch and
ni=r*nch ; however, if h/H<0.5 and Va>Vi we can
consider ni=nch and na=r*nch
),( 5.10 2 fnbdQ
21 QQQ
5.0H/h or 5.0H/h for hH
h
Q
Q
2
1
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Flow description by stochastic modeling
Mechanical mixing of a liquid media
Upper current
Upper region
Lower region
Pumping region
H
h
E1
E
2
S
1
S2
S3
V
a
Vi
V
m
D
Basic topology for mixing with mechanical stirrer
- The system consists on N-1 cells of ideal mixing
each one with its known volume;
- The cells are rejoined by known currents;
- As notations N corresponds to the system exit, Vk
is the volume of k cell and Qkj is the current (flow
rate ) from the k cell to the j cell.
E(n)=[e0 (n), e1 (n);e2 (n)… ek (n)… eN -1 (n),eN (n) ]
N
1kk ,...2,1,0n 1)n(e
Markov conexion: The probability of the particle presence within the k cell after n+1
time (thus τ=n Δτ) is given only by its probability of presence in the j cell after time n
and by the passage probability from the j to the k cell noted like pjk
jk
N
1jjk p)n(e)1n(e
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Flow description by stochastic modeling
Mechanical mixing of a liquid media
Upper current
Upper region
Lower region
Pumping region
H
h
E1
E
2
S
1
S2
S3
V
a
Vi
V
m
D
Basic topology for mixing with mechanical stirrer
For j=1,N and k=1,N the probability pjk is noted as a matrix P
called the stochastic matrix of the process or "the stochastic one“.
NN3N2N1N
N1N12N11N
N2232221
N1131211
p.ppp
p..pp
.....
p.ppp
p.ppp
P
1pN
1j
ij
1NN
p
P*)0(E)1(E
P*)1(E)2(E
P*)n(E)1n(E
Flow description by stochastic modeling
Mechanical mixing of a liquid media
Upper current
Upper region
Lower region
Pumping region
H
h
E1
E
2
S
1
S2
S3
V
a
Vi
V
m
D
Basic topology for mixing with mechanical stirrer
Which is the information produced with the
assistance of this type stochastic model?
)V
Q
exp(pi
N
ji,1iji
ii
)]V
Q
exp(1[
Q
Qp
i
N
ji,1i
ji
N
ji,1i
ji
ji
ij
1-calculate the system reaction to one disturbance impulse
n ou )n(e)n(e)(F0n
1NN
2- to make a precise estimation for the mean residence time and for
residence time variance around of mean residence time:
))(e1(0n
Nm
2
0nN
0nN
2 ))n(e1(n))(e1(2
3- to appreciate the evolution in time of the function (l) that shows
the intensity of stirring for our topologic cell assembly:
)(e1
)n(e)n(
N
1N
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Flow description by stochastic modeling
Mechanical mixing of a liquid media: Numeric exemplification
In an elliptic based cylindrical apparatus with D=1 m, H=1 m there is a solution stirred with a 6 paddles stirrer
which has a d=0.4 m and b=0.1 m. The mixer is placed in the tank in a position where the ration h/H=0.2 (see
figure 4.1). It can work at n=0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5 turns/sec. One reactant with the same physical
properties that the solution is fed close to the liquid surface. The mixture obtained is evacuated by a
connection on the apparatus base. The entry and the exit flows are identical Qex = 0.0048 m3/s. The question
here is to obtain the parameters dependences characterizing this mixing case according to the number of the
stirrer turns.
V1 V2 V3 V4
V5
6
7
Q+Qex
Q2
V1 V2 V3 V4
V5
V6
Qe
x
Q+Qex
Q2+Qex
Q2
Qex
Q1
Q1+Qex
Q+Qex
Topology of the numerical application.
Topology identification
4/18.0/2.0H/h1
H/hr
- Because h/H<0.5 we can affirm that the stirrer is placed
towards the base of the tank and then with nch = 1.
- So we have ni = 1, na = nch / r=4
- The tank contains 6 cells: one in the lower region, one in
the mixture region and four in the higher
)V
QQexp(p
1
ex1
11 )V
QQexp(p
2
ex122
)V
QQexp(p
3
ex133
)V
QQexp(p
4
ex1
44
)V
QQexp(p
5
ex55
)V
QQexp(p
6
ex266
)V
QQexp(1p
1
ex112
)V
QQexp(1p
2
ex123 )
V
QQexp(1p
3
ex134 )
V
QQexp(1p
4
ex145
))V
QQexp(1(
Qp
5
ex
ex
1
51
))V
QQexp(1(
QQp
5
ex
ex
ex256
))V
QQexp(1(
Qp
6
ex2
ex2
265
))V
QQexp(1(
Qp
6
ex2
ex2
ex
67
,
,
,
,
,
p77=1
V1 V2 V3 V4
V5
6
7
Q+Qex
Q2
V1 V2 V3 V4
V5
V6
Qe
x
Q+Qex
Q2+Qex
Q2
Qex
Q1
Q1+Qex
Q+Qex
Topology of the numerical application.
Flow description by stochastic modeling
Mechanical mixing of a liquid media: Numeric exemplification
Non zero transition probalilities
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media- Deep bed filtration
- for the pores with radius between 10-3-10-5 m it is valid the theory of Poiseuille flow; so the mean force for fluid flow between two planes is expressed by the pressure difference; this can be a consequence of differential actions of external and capillary or/and gravitational forces
-for the pores having mean radius between 10-6-10-
9m we explain the porous body flow by the Knudsen theory; a sort of diffusion coefficient that strongly depends on species dimension shows that this flow has a separation behaviour (two or more species moving inside of porous body present different displacement velocities)
- for the pores or better for the molecular windows, where we have the characteristic dimension smaller than 10-9 m, the movement of one species inside of porous solid occurs due to the molecular interactions between the species and the network of porous body; here for description of species displacement it is used the theory of the molecular dynamics
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media- Deep bed filtration
An interesting method for the analysis of the species motion inside the porous body can be based on the observation that this motion occurs as a result of two or more elementary evolutions that are randomly connected. It is the stochastic way for analysis of specie motion inside the porous body.
Liquid motion inside the porous media.
Between the classic and stochastic methods that are used for the analysis of liquid flow inside the porous media we have strong liaisons. These liaisons are given by the relations between the parameters of stochastic and classic models of this flow type. We show here that the analysis of this flow type by using of stochastic way explains some parameters of deterministic models such as: parameters contained by the Darcy law ; parameters that appear inside the porous medium flow continuity equation ; parameters used by the models that explain the flow mechanism inside the porous body.
Apparently the parameters of stochastic models are very much different from those of classic (deterministic) models where the permeability, the porosity, the pore radius, the tortuosity coefficient of pore, the specific surface, and the effective species diffusion coefficient represents the most used parameters for porous
media characterisation.
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media
xx
Figure Fluid motion inside of the uniform porous body
The liquid motion can be described by the motion of liquid element for direction +x that occurs with the probability p respectively by the opposite motion ( -x displacement) where q gives it evolution probability. Here represents the length portion of pore without contact with the nearby pores
=
The probability to
have the fluid
element at time
to x position
The probability that
shows the fluid
element at time
to xx position with
an evolution along of
x for the following
time
+
The probability that
shows the fluid
element at time
to xx position
with an evolution along
of x for the following
time (4.260)
),xx(qP),xx(pP),x(P
A Taylor expansion of ),xx(P and ),xx(P are used in last relation at their right ter
2
22
x
),x(P
2
x
x
),x(Px)qp(
),x(P
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media
x x
The multiplication x
)qp( , that has a velocity dimension
The ratio 2
x 2
has the dimension of one diffusion coefficient (L2T
-1); it is recognized as dispersion
coefficient (D)
It is not difficult to observe that by this simple stochastic model of liquid flow inside the porous body we obtain that the net flow velocity (w) and the dispersion coefficient (D), as model’s parameters, are determined by the porous body structure.
2
2
x
),x(PD
x
),x(Pw
),x(P
More experimental measurements for flow characterization inside the porous body show that the dispersion coefficient has not constant value; for more body structures it frequently presents a time or diffusing species concentration dependency.
More complexes models has been produced for characterization of these situations. One of the first models that intends to produce a response to this problem is recognized as models of motion with states having multiple velocities
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media
x x
With this model the liquid element evolves inside of porous body with random motions having the velocities
m,..1i,vi
.
These random jumps of velocity from a state to other can be explained by random changes of the flow pore
section or by random changes of the flow rate that go in each pore from each pull road coupling of pores from
the porous body
The completion of this description is given by the consideration that accepts a classical states connection. So
here the elementary states connection becomes a Markov type:ij
*
ijijapp .
For this stochastic description the probabilities balance gives the relation:
1,...mi , )vx(Pp),x(Pij
m
1jjii
),x(P),x(Px
),x(Pv
),x(Pj
m
ij,1jjii
m
ij,1jij
i
i
i
Relation (4.265) shows that the time evolution of the fraction of the fluid particles that attain at time the
position x with the velocity i
v is determined by the following particles types :a) the particles having the
specified velocity i
v that leave the position x; b) the particles having the specified velocity i
v that arrive at
position x ; the particles that arrive at position x and change their velocity from j
v to i
v .
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media
x x
For the particular case where we have two evolution states for the fluid velocity ( vv1
, vv2
) the
general model (4.265) comes to the relations set (4.266). Here the consideration 2112
shows that we
consider the case of the isotropic porous body.
),x(P),x(Px
),x(Pv
),x(P21
11
),x(P),x(Px
),x(Pv
),x(P12
22
),x(P),x(P),x(P21
2
22
2
2
x
),x(P
2
v),x(P
2
1),x(PHyperbolic model (new in chemical engineering)
2
22
x
),x(P
2
v),x(PParabolic model (clasic in chemical engineering)
0for x 0
0for x 0)0,x(P
0for x 0
0for x 1)0,x(P
,
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media
x x
The solution for the couple of the model equation and the above specified conditions is obtained by using the Crank observation about the solution existence of an unsteady one dimensional diffusion equation when for a partly finite medium we have a response to a unitary impulse start
0 x
x
impimpimp d),(Pd),(Pd),)x(P),x(P
The particularization of this last expression to the specified problem contains the following observations: a)
,x(P is normalized having values inside of the interval [0, 1]; b) ,x(P is symmetric with respect to the
plane 0x . So we can write:
x
0
imp 0for x d),(P2
1),x(P
0
x
imp 0for x d),(P2
1),x(P
The results for the probabilities ),x(P are given by the following relations:
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media
d)v2
exp(v22
1)P(x, ; 0x
2
2x
02 )
v2
x(erf
2
1
2
1dze
1
2
1
2
z
v2
x
0
2
2
2v2
x(erf
2
1
2
1)P(x, ; 0x Parabolic model
dv
1I
v1
1
v1I
v2
e
2
1)P(x, ; 0x
x
022
2
1
22
222
2
0
vx0for xvv
Ixv
xv2xv
vIe
2
1
1n
222
n
2
n
222
0
vfor x 0),x(P Hyperbolic model
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Liquid motion inside the porous media P(x,τ)
x/[v2τ/α]0.
5
-2 -1 0 2 1
0.
5
1
increase
P(x,τ)
x/[v2τ/α]0.5
-2 -1 0 2 1
0.5
1
increase
½ exp(- )
Differences between parabolic (up presented) and
hyperbolic (below presented) ),x(P evolution
This discussion, about the deviation of the stochastic model to the parabolic or to the hyperbolic model for the species transport inside the porous body, shows that this problem presents two solutions; the first solution ( frequently used) to the parabolic model and the second one based on the hyperbolic model. For some practical methods for porous body behaviour characterization such as liquid or gaseous permeation methods, the data interpretation by the hyperbolic model can be of interest
2
v
2
)x(limD
22
0,0x
For the porous body with constitutive elements of height dimension such as the fixed packed bed, where the
characteristic dimension is that of the packed element (diameter d of the packed body), the frequency of the
velocity change is d/v (after each passing over a packed body the local fluid velocity v changes its
direction)
Now if we use this value of in the dispersion coefficient we obtain the famous relation 2D/)vd(Pe .
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Deep bed filtration
The deep bed filtration clarifies some suspensions with small content in solid by their flowing into length granular bed formed by fine particles. During the flowing inside the granular bed, between one particle from the suspension and one particle from the granular bed there appear interaction forces that determine the fixing of the particle from the suspension to the particle of the bed. This elementary process occurs in more points placed between suspension input and its bed exit. The quantity of the solid retained at one local section of flow cannot exceed the quantity determined by the granular bed open spaces hold-up.When the retained quantity comes near to the quantity determined by the bed open spaces we affirm that the bed is cloggedAfter clogging we regenerate the granular bed by liquid fluidization when the retained solid goes out from the bed;
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Deep bed filtration
i
Micro-particle moving
around of the
deposition bed particle
Micro-
particle
input
Flow direction
Porous bed particle
External liquid
boundary layer
Retained
particle
Deposition
trajectory
Non-
deposition
trajectory
Figure 4.34 Micro-particle retention by one element of the fixed bed structure
- the inertial force, the gravitational force, the diffusion force, the laminar flow force, the electrostatic force, the Van der Waals, the hydrodinamic adhesion force
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Deep bed filtration
The filtration coefficient model (Mint model)
This coefficient noted as )c,(ss0
depends on its initial value (0) and on the local concentration of the
retained solid around the bed deposition elements (ss
c ). It is defined as the fraction of the solid retained from
the suspension in an elementary length of the granular bed
dx
1
c
dc)c,(
vs
vs
ss0
ss
f
ss
vs
vsc
w
1c
G
A
dx
dc
0 ; cwc
vs0f
ss
0 ; ccwc
ssvs0f
ss
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Deep bed filtration
ss
f
vs0
vs cw
cx
c
00
2
ss
f
vsvs c
w
c
x
c
0x
cc
x
cvsvs
0
vs
2
0c 0x 0vs
0vvscc 0x 0
1nn
1n
0
0
0v
vs )exp(T)!1n(
)x()xexp(
c
c
)!2n(
)(TT
2n
1nn
)exp(T1
1i
2/1
0i
2/i
0
0
0v
vs ])x[(Ix
)x((expc
c
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Dintr-un experiment bazat pe teoria Mint s-a stabilit că pentru un strat filtrant valoarea iniţială
a coeficientului de filtrare este 0 = 4 m-1
, iar valoarea coeficientului de detaşare este =
0,057 h-1
. Pentru un filtru lung de 0,1 m realizat în totalitate din acelaşi material, calculaţi
calitatea filtrantului, ca procentaj din valoarea iniţială, la începutul operaţiei şi după 6 ore de
lucru.
Species displacement and transfer in a porous media- Deep bed filtration
Numerical application
At process start: vsvs cx
c0
)exp( 0
0
Lc
c
vs
vs
Cu datele numerice 0 = 4 m-1
, L = 0,1 se obţine %6767,0)1,04exp(0vs
vs
c
c.
n ( 0L)n-1 (n-1)! ( )n-2 (n-2)! Tn exp(- )
1 1 1 1 - - - 1,41 0,715 0,670
2 0,4 1 0,4 1 1 1 0,41 0,715 0,079
3 0,16 2 0,08 0,344 1 0,344 0,066 0,715 0,003
4 0,064 6 0,0107 0,117 2 0,058 0,008 0,715 0,000
)!1(
10
n
Ln
)!2(
)( 2
n
nAfter six hours:
%2,75752,00vs
vs
c
c
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Deep bed filtration
Numerical application
In an experimental investigation of a deep bed filtration has been used the retention of Fe(OH)3 gel form water in
fixed bed of sand particles. For a fixed bed with H0 = 20 cm and operating with wf = 2,5 m/h at 30 0 C and c0 = 6.75
mg/l, depending of particle diameter, the following results has obtained: Obtain a dependence of Mint parameters on
sand particle diameter.
D
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.05
0.25
0.65
0.85
0.95
0.99
1.0
1.0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.05
0.25
0.65
0.85
0.95
0.99
1.0
1.0
1.0
1.0
1.0
1.0
0
0
0
0
0
0
0
0
0
0.05
0.25
0.65
0.85
0.95
0.99
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0
0
0
0
0
0
0.05
0.25
0.65
0.85
0.95
0.99
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0
0
0
0
0.05
0.25
0.65
0.85
0.95
0.99
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0
0
0.05
0.25
0.65
0.85
0.95
0.99
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
dp 0.2 0.3 0.4 0.5 0.6 0.7( )
1
1
00
0
)exp()!1(
)()exp(
n
n
n
v
vs Tn
xx
c
c
)!2(
)( 2
1n
TTn
nn)exp(1T
ftd1 D1
D0
x 0.2
f 0 exp 0 x exp( ) 0 x exp( ) 1( )
F 0
1
24
i
f 0 iftd1
i
2
0 50 0.2
R Minimize F 0
R36.049
0.214
Species displacement and transfer in a porous media- Deep bed filtration
Stochastic modeling
A deep bed stochastic model [4.5] identifies for micro-particle evolution in the
filtration bed two elementary processes:
a) a type I process that considers for micro-particles their motion with the
velocity vv1
; this velocity is induced by the surrounding flowing fluid; this
process type physically corresponds with the non deposition of the micro-particle;
b) a type II process that shows the possibility of the micro-particle to take
the deposition way; from the viewpoint of the motion the velocity of this process
is 0v2
.
The stochastic model accepts a Markov type connection between its two elementary states. So with 12
we
define the transition probability from the type I process to type II process. The transition probability from a
type II process to a type I process is21
. By ),x(P1
and ),x(P2
we note the probability to locate the
micro-particle at the position x and time with a type I evolution respectively with a type II evolution
),x(P),x(Px
),x(Pv
),x(P221112
11
),x(P),x(P),x(P
112221
2
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
0vvsc/),x(c ),x(P),x(P),x(P
21
Species displacement and transfer in a porous media- Deep bed filtration
Stochastic modeling
0),x(P
)(x
),x(Pv
x
),x(Pv
),x(P122121
2
2
2
By considering the combined variable 2/vxz we eliminate the mixed partly differential term from the
equation
z
),2/vz(P)(
2
v
z
),2/vz(P
4
v),2/vz(P12212
22
2
2
0),2/vz(P
)(1221
0)P(z,x z , 0 x, 0
0P)P(z, 0z , 0 x, 0
The important values of the jumping frequencies from one state to other characterize, as specific, the common
deep bed filtration. This observation permits the transformation of the above-presented hyperbolic model into
the parabolic model
0
),2
vz(P
z
),2
vz(P
2
v
z
),2
vz(P
)(4
v
1221
1221
2
2
1221
2
0)P(z,x z , 0 x, 0
0P)P(z, 0z , 0 x, 0
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Species displacement and transfer in a porous media- Deep bed filtration
Stochastic modeling
1221
2
1221
12
01221
12
v
vx
erf12
1
P
)P(x, , 0
vx
1221
2
1221
12
01221
12
v
vx
erf12
1
P
)P(x, , 0
vx
Now we go on and complete the validation of discussed model. As it is known only the experimental
investigation can validate or invalidate one process model. For this concrete case we appeal to the
experimental data for the filtration of a dilute Fe(OH)3 suspension (no more than 0.1 g Fe(OH)3 /l ) in a sand
bed with various heights and constitution particle diameters. The considered experiments report the
measurements at constant filtrate flow rate and consist in the time evolution of the Fe(OH)3 concentration at
the bed exit when at their input the solid concentration remains unchanged
Turbidity(%)%)%
2 4 6 8 10
H1
H2
0
100
Time (h)
The response curves at deep bed filtration
),0(P/),H(Pc/),H(cvovs
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Deep bed filtration
factorFactor value
Stochastic model parameters
12 21
H [cm]
t=200C
Gv=20cc/min
dg=0.5-0.31mm
C0=6.75mg/lFe(OH)3
2 1.14 326
3 0.592 458.9
5 0.262 420.22
6 17.66 9611.41
t [0C]
H=6cm
Gv=20cc/min
dg=0.5-0.31mm
C0=6.75mg/lFe(OH)3
20 17.95 9805.1
30 2.6 2748.14
35 22.71 1347.5
40 4.337 3028
Gv[cm3/min]
H=6cm
t=300C
dg=0.5-0.31mm
C0=6.75mg/lFe(OH)3
20 3.054 3020.66
30 13.34 6698.96
40 62.08 22605.24
50 118.2 42908.68
C0[mg/l] Fe(OH)3
H=6cm
t=300C
dg=0.5-0.31mm
Gv=50cm3/min
6.75 111.7 40493
13.49 82.7 28999.5
26.98 34.18 10294.41
dg [mm]
H=6cm
C0=6.75mg/lFe(OH)3
Tf=300
Gv=50cm3/min
0.31-0.2 754.98 355456.89
0.5-031 110.65 39773
0.63-0.5 22.409 8795.98
0.85-0.63 23.82 6449.82
The process factors influence on the stochastic model’s parameters
Species displacement and transfer in a porous media- Deep bed filtration
Stochastic modeling
dg28.256C872.2Gv003.375.67012
dg10023.1C1268Gv111110081.3 5
0
4
21
Curgeri in sisteme multifazice , Lectia 5, 14.11.2011
Firstly it is observable that assumption of height values for
2112 and parameters is excellently covered by the experimentally
starting data. Secondly, we find out that all process factors
influence the stochastic model’s parameters values.