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.
. B :
4.62 4.62
4.63 4.64
( , , ); 4.65 4.66 4.67 ;
4.68 4.68
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10 3 10 5 ;
;
/
10 6 109
;
(
)
, 10 9 ,
;
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A
.
.
B .
.
: D ; ; .
A ( ) , , , , ,
.
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Figure Fluid motion inside of the uniform porous body
+
( ) .
=
The probability tohave the fluid
element at time
to x position
The probability tha tshows the fluidelemen t at time to xx position withan evolution along of
x+ for the following time
+
The probability thatshows the fluid
element at time to xx + positionwith an evolution alongof 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
2x
x),x(Px
)qp(),x(P
=
+
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The multiplication
x
)qp( , that has a velocity dimension The ratio
2
x 2 has the dimension of one diffusion coefficient (L 2T -1); it is recognized as dispersion
coefficient (D)
( ) (D), , .
2
2
x
),x(PD
x
),x(Pw
),x(P
=
+
; .
.
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With this model the liquid element evolves inside of porous body with random motions having the velocitiesm,..1i,v i = .
These random jumps of velocity from a state to other can be explained by random changes of the flow poresection or by random changes of the flow rate that go in each pore from each pull road coupling of pores fromthe 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*ijij app .
:
1,...mi, )vx(Pp),x(P i jm
1 j jii ==+
=
),x(P),x(Px
),x(Pv
),x(P j
m
i j,1 j jii
m
i j,1 j ij
i
i
i +
=
==
Relation (4.265) shows that the time evolution of the fraction of the fluid particles that attain at time theposition x with the velocity iv is determined by the following particles types :a) the particles having thespecified velocity iv that leave the position x; b) the particles having the specified velocity iv that arrive at
position x ; the particles that arrive at position x and change their velocity from jv to iv .
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For the particular case where we have two evolution states for the fluid velocity ( v v1 += , vv 2 = ) thegeneral model (4.265) comes to the relations set (4.266). Here the consideration == 2112 shows that weconsider 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(P 21 +=
2
22
2
2
x),x(P
2v),x(P
21),x(P
=
+
( )
2
22
x
),x(P
2
v),x(P
=
( )
=0for x 00for x 0
)0,x(Pf
=+ 0for x 00for x 1
)0,x(Pf
,
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C
===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),(P21
),x(P
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=
=> d)v2
exp(v22
1)P(x, ; 0x
2
2x
02 )
v2
x(erf
21
21
dze1
21
2
z
v2
x
0
2
2
=
+=
dv
1I
v1
1v
1Iv2
e21
)P(x, ; 0xx
022
2
1
22
222
2
0
++
=
vx0forxvv
Ixvxv
2xvv
Ie21
1n
222n
2n
2220
pp
= vfor x 0),x(P f
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( ,)
/ 2/ 0.
2 1 0 2 1
0.
1
( ,)
/ 2/ 0.5
2 1 0 2 1
0.5
1
( )
Differences between parabolic (up presented) and
hyperbolic (below presented) ),x(P evolution
,
, ; ( )
.
,
=
= 2
v2
)x(limD
22
0,0x
For the porous body with constitutive elements of height dimension such as the fixed packed bed, where thcharacteristic dimension is that of the packed element (diameter d of the packed body), the frequency of thevelocity change is d / v= (after each passing over a packed body the local fluid velocity v changes itsdirection)
Now if we use this value of in the dispersion coefficient we obtain the famous relation 2D / )vd(Pe == .
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.D ,
.
.
.
A
;
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i
Micro-particle movingaround of t hedeposition bed particle
Micro-particleinput
Flow direction
Porous bed particle
External liquidboundary layer
Retainedparticle
Depositiontrajectory
Non-depositiontrajectory
Figure 4.34 Micro-particle retention by one element of the fixed bed struc ture
, , , , , ,
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( )
This coefficient noted as )c,( s s0 depends on its initial value ( 0 ) and on the local concentration of theretained solid around the bed deposition elements ( ssc ). It is defined as the fraction of the solid retained fromthe suspension in an elementary length of the granular bed
dx1
cdc
)c,(vs
vsss0 =
=
= ss
f
ss
vs
vs cw1c
GA
dxdc
0 ; cwc
vs0f ss ==
0 ; ccwc
ssvs0f ss f=
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ss
f
vs0vs c
wc
xc =
002
=
=
ss
f
vsvs cw
c x
c
0x
ccxc vsvs
0vs
2
=
+
+
0c 0x 0 vs ==
0vvs cc 0x 0 ==
=
=1n
n
1n0
0
0v
vs )exp(T)!1n(
)x()xexp(
cc
)!2n(
)(TT
2n
1nn
=
)exp(T1 =
=
+=
1i
2 / 10i
2 / i
0
0
0v
vs ])x[(Ix
)x((expcc
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Dintr-un experiment bazat pe teoria Mint s-a stabilit c pentru un strat filtrant valoarea ini ial
a coeficientului de filtrare este 0 = 4 m -1 , iar valoarea coeficientului de deta are este =
0,057 h -1 . Pentru un filtru lung de 0,1 m realizat n totalitate din acela i material, calcula i
calitatea filtrantului, ca procentaj din valoarea ini ial , la nceputul opera iei i dup 6 ore delucru.
A : vsvs c
x
c0 =
)exp( 0
0
Lc
c
vs
vs =
Cu datele numerice 0 = 4 m-1
, L = 0,1 se ob ine %6767,0)1,04exp(0 ===vsvs
c
c.
n ( 0L)n-1 (n-1)! ( )n-2 (n-2)! T n 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 L n
)!2()( 2
n
n A :
%2,75752,00
==vs
vs
c
c
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A deep bed stochastic model [4.5] identifies for micro-particle evolution in thefiltration bed two elementary processes:
a) a type I process that considers for micro-particles their motion with thevelocity 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 processis 0v 2 = .
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 atype II process to a type I process is 21 . By ),x(P1 and ),x(P2 we note the probability to locate themicro-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 1122212 +=
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0vvs c / ),x(c ),x(P),x(P),x(P 21 +=
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0),x(P
)(x
),x(Pv
x),x(P
v),x(P
122121
2
2
2
=
++
++
By considering the combined variable 2 / vxz = we eliminate the mixed partly differential term from theequation
+
++
+
+z
),2 / vz(P)(
2v
z),2 / vz(P
4v),2 / vz(P
12212
22
2
20),2 / vz(P)( 1221 =
++
0)P(z,xz , 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 commondeep bed filtration. This observation permits the transformation of the above-presented hyperbolic model intothe parabolic model
0),
2v
z(P
z
),2v
z(P
2
v
z
),2v
z(P
)(4
v
1221
12212
2
1221
2
=
++
+
+
=
+
+
0)P(z,xz , 0 x, 0 ==>=
0P)P(z, 0z , 0 x, 0 =>=>
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+
+
+=
+
1221
2
1221
12
01221
12
v
vx
erf 121
P)P(x,
, 0v
x
Now we go on and complete the validation of discussed model. As it is known only the experimentalinvestigation can validate or invalidate one process model. For this concrete case we appeal to theexperimental data for the filtration of a dilute Fe(OH) 3 suspension (no more than 0.1 g Fe(OH) 3 /l ) in a sandbed with various heights and constitution particle diameters. The considered experiments report themeasurements at constant filtrate flow rate and consist in the time evolution of the Fe(OH) 3 concentration atthe bed exit when at their input the solid concentration remains unchanged
(%)%)%
2 4 6 8 10
1
2
0
100
( )
),0(P / ),H(Pc / ),H(c vovs =
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Deep bed filtrationfactor
Factor value Stochastic model parameters
12 21
H [cm] t=20 0CGv=20cc/mindg=0.5-0.31mmC0=6.75mg/lFe(OH) 3
2 1.14 3263 0.592 458.95 0.262 420.22
6 17.66 9611.41
t [0C] H=6cmGv=20cc/mindg=0.5-0.31mmC0=6.75mg/lFe(OH) 3
20 17.95 9805.130 2.6 2748.1435 22.71 1347.5
40 4.337 3028
G v[cm 3 /min] H=6cmt=30 0Cdg=0.5-0.31mmC0=6.75mg/lFe(OH) 3
20 3.054 3020.6630 13.34 6698.9640 62.08 22605.24
50 118.2 42908.68
C 0[mg/l] Fe(OH) 3 H=6cmt=30 0Cdg=0.5-0.31mmGv=50cm 3 /min
6.75 111.7 4049313.49 82.7 28999.5
26.98 34.18 10294.41
dg [mm] H=6cm
C0=6.75mg/lFe(OH) 3 Tf =30 0 Gv=50cm 3 /min
0.31-0.2 754.98 355456.890.5-031 110.65 39773
0.63-0.5 22.409 8795.980.85-0.63 23.82 6449.82
The process factors influence on the stochastic models parameters
dg28.256C872.2Gv003.375.67 012 +=
dg10023.1C1268Gv111110081.3 504
21 +=
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Firstly it is observable that assumption of height values for2112 and parameters is excellently covered by the experimentally
starting data. Secondly, we find out that all process factors
influence the stochastic models parameters values.