flow in porous media [compatibility mode]

8/13/2019 Flow in Porous Media [Compatibility Mode]

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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


9/19

=

=> 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 +=


17/19

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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18/19

+

+

+=

+

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.

flow in porous media [compatibility mode]

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