porous media transport phenomena (civan/transport phenomena) || suspended particulate transport in...

CHAPTER 10

SUSPENDED PARTICULATE TRANSPORT IN POROUS MEDIA

10.1 INTRODUCTION

Migration and retention of particulate matter during fl ow of particulate suspensions in porous media, such as encountered in petroleum, geothermal, and groundwater reservoirs, are infl uenced strongly in a complicated manner by various transport and retention mechanisms, and temperature variation during injection of fl uids at tem-peratures different from those of porous media, such as for purposes of secondary oil recovery, acid stimulation, and carbon sequestration (Civan, 2010c ). * Description of particulate behavior by phenomenological modeling including dispersive trans-port and nonisothermal conditions is required for generalized handling of various applications of practical importance.

However, whether the particle dispersion phenomenon is of practical impor-tance depends on particular applications and appears to be a matter of continuing debate (Unice and Logan, 2000 ; Altoe et al., 2006 ; Lomin é and Oger, 2009 ). Logan et al. (1997) point out that particle dispersion is different from chemical dispersion. They indicate, for example, that the bromide tracer dispersion is a chemical disper-sion issue and the bacteria dispersion is a particle dispersion issue. Logan et al. (1997) explain that particle dispersion involves other factors such as the sticking phenomenon, slow desorption, and heterogeneity of particles and porous media grains. Lomin é and Oger (2009) state, “ Dispersion of particles results from collisions with the porous matrix and with other moving particles. ” Lomin é and Oger (2009) determined that the value of the dispersion coeffi cient increases with the particle concentration.

Auset and Keller (2004) observed by means of experiments conducted using micromodels that the magnitude of particle dispersion depends primarily on their

Porous Media Transport Phenomena, First Edition. Faruk Civan.© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

353

* Parts of this chapter have been reproduced with modifi cations from Civan, F. 2010c. Non - isothermal permeability impairment by fi nes migration and deposition in porous media including dispersive trans-port. Transport in Porous Media, 85(1), pp. 233 – 258, with permission from Springer.

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354 CHAPTER 10 SUSPENDED PARTICULATE TRANSPORT IN POROUS MEDIA

preferential paths and velocities. Smaller particles tend to move along longer com-plicated tortuous paths and therefore are slower, and larger particles tend to move along shorter straighter paths and therefore are faster. Hence, the dispersivity and dispersion coeffi cient of particles in porous media increase with decreasing particle size and vice versa, depending on the pore structure and the pore channel - to - particle size ratio.

Temperature variation affects the governing particulate transport and rate processes in porous media in very complicated ways. The particulate and pore surface interactions are strongly affected by temperature (Schembre and Kovscek, 2005 ; Civan, 2007a,b, 2008a,c, 2010c ). Particles tend to deposit more preferentially over the pore surface at lower temperatures than higher temperatures. Hence, colder temperature conditions are favorable for more pore surface attachment and retention of fi ne particles (Civan, 2007b ). Conversely, at suffi ciently high temperatures, pore surface conditions become more suitable for particle detachment, and therefore fi ne particles are less likely to deposit over the pore surface but rather migrate toward the pore throats and form particle bridges nevertheless only under favorable condi-tions (Pandya et al., 1998 ; Civan, 2007a , and Tran et al., 2009 ).

The balance between the effects of the pore surface retention and pore - throat plugging conditions varies with the prevailing temperature, fi ne particle suspension, and porous matrix conditions, and determines the severity of permeability impair-ment. In general, pore - throat plugging renders more severe permeability damage than pore surface retention. Such conditions are implicitly included in the value of the fi lter coeffi cient estimated using the correlations developed by Tufenkji and Elimelech (2004) and Chang et al. (2009) . Such effects are important in various applications, such as water injection into hot subsurface formations for extraction of geothermal energy and injection of steam with or without mixing with light hydrocarbons into heavy oil reservoirs for enhanced oil recovery.

Particulate processes occurring in porous media should be expressed in terms of the interstitial fl uid velocity rather than the pore velocity, volumetric fl ux, or fl ow rate because the fl uid forces acting upon particles in porous media are determined by the actual fl uid velocity in tortuous preferential fl uid fl ow paths, referred to as the interstitial fl uid velocity (Dupuit, 1863 ; Civan, 2007b ). Fine particles present in tortuous fl ow paths are drifted more effectively along with the fl owing fl uid because the fl uid moves faster in tortuous paths than in straight paths (Civan, 2007b, 2008d ).

The consequences of injecting particulate suspensions into porous media, such as porous media clogging, permeability reduction, and effl uent solution particle concentration, depend on the essential characteristics of the particles, carrier fl uids, and porous media, and the prevailing conditions, such as temperature, pressure, wettability, and electrolytic and potentiometric activity. Information on values of the parameters involved in various dimensionless groups are necessary, for example, for the calculation of the fi lter coeffi cient using the empirical correlation given by Tufenkji and Elimelech (2004) and Chang et al. (2009) .

In this chapter, fi rst, a phenomenological model is presented for deep - bed fi ltration by considering temperature variation and particle transport by advection and dispersion according to Civan (2010c) . Migration and retention of particles in porous media are formulated by theoretically modeling the relevant processes under

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10.2 DEEP-BED FILTRATION UNDER NONISOTHERMAL CONDITIONS 355

both isothermal and nonisothermal conditions and with and without inclusion of the dispersive transport. The solutions generated with and without the inclusion of the dispersion effect under isothermal and nonisothermal conditions indicate the effect of dispersion and temperature on particulate processes occurring in porous media. Obviously, the transverse dispersion can play an important role in multidimensional cases depending on the ratio of the transverse - to - longitudinal dispersivity. Thus, the dispersion in various directions is taken into account by means of the directional elements of the dispersion tensor. Signifi cant effects of temperature variation and particulate transport by advection and dispersion on particle concentration in the fl owing suspension of particles, particle retention, and permeability variation during fl ow are demonstrated by examples under isothermal and nonisothermal conditions.

Second, a formulation of compressible fi lter cake formation undergoing small particle packing in between large particles is presented according to Tien et al. (1997) . Filter cake is treated as a special porous media that forms over a fi lter used for the separation of particles from a suspension of particles or slurry. Larger par-ticles are assumed to construct the essential porous structure of the fi lter cake. Smaller particles are assumed to be trapped and packed inside the pore space created by the larger particles.

The particle retention rate and the variation of porous matrix properties, such as porosity and permeability, are formulated by considering the temperature and particle retention effects. The nonisothermal and dispersion effects are considered, which may be important under certain conditions for the accurate description of transport and retention of colloidal and fi ne particles during fl ow through porous formations. It is demonstrated that temperature variation has a signifi cant effect on particulate transport through porous media because it affects the fi lter coeffi cient, porous matrix thermal deformation, and pore - throat constriction. Comparison of the results obtained with and without the dispersion effect included indicates that trans-port by dispersion has some effect on suspended fi nes migration and retention, and on the resulting permeability variation in porous media. Dispersion causes the spreading of the effect of the particulate phenomenon on permeability reduction over a long range from the injection port. However, such effect is more pronounced over the region near the injection side of porous media when the dispersion mechanism of particle transport is ignored.

10.2 DEEP - BED FILTRATION UNDER NONISOTHERMAL CONDITIONS

The essential constituents of the overall model, some of which have been adopted with and without modifi cations after Civan (2008a, 2010c) , are described in the following.

Consider a porous medium whose initial and instantaneous effective (intercon-nected) porosities are denoted by φ φi and . Let u and ρ represent the volumetric fl ux and density of the fl owing fl uid containing fi ne particles. ρ p is the particle material mass density. w and σ denote the mass and volume fractions of particles present in

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the fl owing fl uid medium. ε is the volume fraction of particles deposited in bulk porous media. t and x denote the time and distance from the injection point.

10.2.1 Concentration of Fine Particles Migrating within the Carrier Fluid

The mass fraction of fi ne particles w present in a carrier fl uid containing such sus-pended particles can be expressed in terms of the volume fraction of particles σ as the following:

wp=

ρρ

σ. (10.1)

For suffi ciently low concentrations of fi ne particles, the density of the fl owing fl uid ρ (carrier fl uid, i.e., water, plus suspended particles) may be assumed to be the same as the carrier fl uid density ρ w , that is, when σ � 1 0. , then ρ ρ≅ w.

The retention of suspended particles reduces the instantaneous porosity of porous media according to (see Civan, 2007a for a discussion about particle packing effi ciency)

φ φ ε= −i . (10.2)

The momentum balance of fl uid fl owing through porous media is given by Darcy ’ s law as

u K g= − ⋅ ∇ + ⋅∇( )1

μρp z , (10.3)

where K is the permeability tensor, g is the gravitational acceleration vector, z is the Cartesian distance covered in the gravitational acceleration direction, and p and μ are the pressure and dynamic viscosity, respectively, of the fl owing suspension of particles.

The total mass balance equation for the fl owing fl uid (carrier fl uid plus sus-pended particles) is given by

∂( )∂

+∇⋅( ) = −ρφ

ρt

mu � . (10.4)

The suspended particle mass balance equation is given by

∂( )

∂+∇⋅( ) + ∇⋅ = −

ρφρ

w

tw mu j � . (10.5)

Combining Eqs. (10.4) and (10.5) yields the suspended particle mass balance equation for the fl owing fl uid as

ρ φ ∂∂

+ ⋅∇⎛⎝⎜

⎞⎠⎟ +∇⋅ = − −( )w

tw w mu j 1 � , (10.6)

where �m is the mass rate of particles deposited per unit bulk volume of porous media from the fl owing phase and j denotes the dispersive mass fl ux vector of suspended

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particles in the fl owing phase. Note that Eq. (10.6) is applicable regardless of whether the density and porosity are variable or constant.

The dispersive mass fl ux of particles is given by

j D= − ⋅∇φρ w, (10.7)

where D is the coeffi cient of dispersion of suspended particles migrating in the fl owing phase assumed to be a linear function of the interstitial pore fl uid velocity according to

D = va ⋅ , (10.8)

where α is the longitudinal dispersivity and v is the interstitial pore fl uid velocity, given by (Dupuit, 1863 )

vu

=⋅tφ

, (10.9)

where t denotes the tortuosity of the preferential fl ow paths in porous media. Thus, the formulation presented here allows for the modeling of fi nes migration and reten-tion effects also without including the dispersion effect simply by setting a = 0. Note that Eq. (10.7) is more rigorous than the analogous equation given by Altoe et al. (2006) because here, the effect of tortuous fl ow paths is accounted for.

The particle capture (retention) rate �m is assumed proportional to the total suspended particle fl ux modifying the equation given by Iwasaki (1937) as the following:

∂( )

∂= ≡

∂∂

= ⋅ρ εp

tm

m

tk� J n, (10.10)

where k is referred to as the fi ltration rate coeffi cient or fi lter coeffi cient discussed later in the following, n denotes the normal unit vector, and J is the total mass fl ux of suspended particles due to the combined effects of bulk fl ow (advection) and dispersion, given by

J u j= +ρw . (10.11)

The preceding equations, Eqs. (10.6) – (10.11) , can be combined readily to obtain the following equation of convenience:

∂∂

+⋅ − ⋅( ) − ∇⋅ ⋅( )⎡

⎣⎢⎤⎦⎥⋅∇ +

⋅= ⋅ ⋅∇

w

t

kw

kw w

t a ta t a

vv

vv

1 11

2

1

22

τ ρφρφ

τ,, (10.12)

where, considering Eq. (10.1) , a new parameter k 1 is defi ned as

k w k kp1 1 1= −( ) = −⎛

⎝⎜⎞⎠⎟

ρρ

σ . (10.13)

For convenience in the numerical applications presented later, consider con-stant density and fl ow for the fl owing fl uid and neglect the effect of the small amount of fi ne particle retention on the variation of porosity (but not the porosity itself) ( ε φ φ ε φ≅ = − ∂ ≅0 0and i , ). Thus, Eq. (10.12) simplifi es as the following:

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∂∂

+⋅ − ⋅( ) ⋅∇ +

⋅= ⋅ ⋅∇

w

t

kw

kw w

t a t t av v

v1 1

2

1

22

τ τ. (10.14)

The variation of density and porosity with time and space by gradual thermal expansion is negligible relative to their absolute values. But the variation of viscosity with temperature and its effect on the fi lter coeffi cient value are more pronounced and therefore are considered in the present formulation.

The conditions of solution are described by the following. The initial condition throughout the porous media is given by

w w to= =, .0 (10.15)

The boundary condition at the injection side is given by

u J nρw w w w t( ) = = − ⋅ ⋅∇ ⋅ >in inor a t , .0 (10.16)

The boundary condition at the outlet boundary side is given by

∇ ⋅ = >w tn 0 0, , (10.17)

where L denotes the length of the spatial region, the subscript “ in ” represents the inlet condition, and the superscript o represents the initial condition.

Note that Eqs. (10.14) – (10.17) can be expressed in volume fraction of particles in the fl owing fl uid, by means of Eq. (10.1) , as

∂∂

+ ⋅∇ + = ⋅∇σ σ σ σt

a b c 2 . (10.18)


σ σ= =o t, .0 (10.19)


σ σ σin t= − ⋅∇ ⋅ >d n, .0 (10.20)


∇ ⋅ = >σ n 0 0, ,t (10.21)

where

av

bv

c = v d ==⋅ − ⋅( ) =

⋅⋅ ⋅

t a t t a a t1 1

2

1

2

k k

τ τ, , , . (10.22)

Obviously, the values of the parameters a and b vary as a result of variation of the value of the parameter k 1 according to Eq. (10.13) by the variation of the fi lter coeffi cient k and the volume fraction of particles σ in the fl owing fl uid with changing local conditions. Therefore, the numerical solution is obtained iteratively at a time attained after each fi nite time increment until convergence starting with a value of σ known at the previous time, which is the initial value prescribed at the beginning (see Exercise 4).

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10.2.2 Concentration of Fine Particles Deposited inside the Pores of the Porous Matrix

The volume fraction of particles deposited in porous media can be calculated by combining Eqs. (10.7) – (10.11) as the following:

ρ ε ρφτp

t

kw w

∂∂

=⋅

− ⋅ ⋅∇( )t a tv2

. (10.23)

Eq. (10.23) can be expressed in volume fraction of suspended particles in the fl owing fl uid, by means of Eq. (10.1) , as

∂∂

=⋅

− ⋅ ⋅∇( )ε φτ

σ σt

kt a tv2

, (10.24)

where the values of the porosity φ and the fi lter coeffi cient k vary with changing local conditions.

The following initial condition is imposed throughout the porous media:

ε ε= =o t, ,0 (10.25)

where the superscript o represents the initial condition.

10.2.3 Variation of Temperature in the System of Porous Matrix and Flowing Fluid

Variation of temperature as a result of injecting a fl uid at a temperature different from the porous media temperature will affect the physical properties of materials involved, the fi ltration rate coeffi cient, permeability, and porosity, and hence the particle retention rate.

For purposes of the present application, a simplifi ed form of the energy equa-tion for the fl uid fl owing through porous media is used here. Similar to the exercise demonstrated earlier, fi rst, we simplify the general equation of energy balance by means of the total mass balance equation (the equation of continuity given by Eq. 10.4 ). Then, we neglect the heat removed from the fl owing fl uid medium by the small amount of deposited particles, the kinetic energy and the viscous effects of the rather slow moving fl uid, and the substantial derivatives of the slowly varying pressure and density. Finally, we drop the gravity term for horizontal fl ow.

Thus, the temperature T of the fl owing fl uid is described with reasonable accuracy by

ρ φ φcT

tT T q

∂∂

+ ⋅∇⎛⎝⎜

⎞⎠⎟ = ∇⋅ ⋅∇( ) +u k �. (10.26)


T T to= =, .0 (10.27)


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ρ ρ φc T c T T tu n u n( ) ⋅ = − ⋅∇( )⋅ >in k , .0 (10.28)

The boundary condition at the effl uent side is given by

∇ ⋅ = >T tn 0 0, , (10.29)

where �q denotes the amount of heat added to the fl uid, the subscript “ in ” represents the inlet condition, and the superscript o represents the initial condition.

The thermal expansion of the porous matrix is neglected here only for the rate of matrix displacement ( u m = 0) but is considered for reduction of perme-ability by pore - throat constriction by thermal expansion of porous media grains. Hence, the temperature T m of the porous matrix is described by the following equation:

ρ φ φm mm

m m m m mcT

tT T q1 1−( ) ∂

∂+ ⋅∇⎡

⎣⎢⎤⎦⎥= ∇⋅ −( ) ⋅∇[ ]+u k � . (10.30)

The subscript m indicates properties for the porous matrix and �qm denotes the amount of heat added to the porous matrix.


T T tm mo= =, .0 (10.31)


∇ ⋅ = >T tm n 0 0, . (10.32)


∇ ⋅ = >T tm n 0 0, . (10.33)

The interface heat exchange rate between the fl owing fl uid and the porous matrix across the pore surface is given by

� �q q A h T Tm m m= − = −( ), (10.34)

where A m is the matrix pore surface available per unit bulk volume and h is the fi lm heat transfer coeffi cient.

If fl uid fl ows suffi ciently slowly through porous media, such as typically encountered in subsurface reservoirs, then it is reasonable to assume a thermal equilibrium between the fl owing fl uid system and the porous matrix. Thus, Eqs. (10.26) and (10.30) can be added together and then rearranged as

∂∂

+−∇⋅ + −( )[ ]

+ −( )[ ]⎛⎝⎜

⎞⎠⎟⋅∇ =

+ −( )T

t

c

c cTm

m m

muρ φ φφρ φ ρ

φ φk k k k1

1

1[[ ]+ −( )[ ]

⋅∇φρ φ ρc c

Tm m1

2 . (10.35)


T T to= =, .0 (10.36)


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ρ ρ φc T c T T tu n u n( ) ⋅ = − ⋅∇( )⋅ >in k , .0 (10.37)


∇ ⋅ = >T tn 0 0, . (10.38)

If all the physical properties are assumed constant, neglecting the small effect of gradual temperature variation in liquid and solid media, then Eq. (10.35) can be written as

∂∂

+ ⋅∇ = ⋅∇T

tT Ta b1 1

2 . (10.39)


T T to= =, .0 (10.40)


T T T tin = − ⋅∇ ⋅ >c n1 0, . (10.41)


∇ ⋅ = >T tn 0 0, , (10.42)

where

au

b cu

1 1 1 21

1

1=

+ −( )=

+ −( )+ −( )

=⋅ρ

φρ φ ρφ φ

φρ φ ρφ

ρc

c c c c u cm m

m

m m

, ,k k k

.. (10.43)

10.2.4 Initial Filter Coeffi cient

The initial fi lter coeffi cient k o is defi ned as the value of the fi lter coeffi cient deter-mined prior to particle retention, given by (Tien, 1989 ; Elimelech et al., 1995 )

kD

og

o=−( )3

2

11

φβ η , (10.44)

where ηodenotes the single - collector contact effi ciency (a physical effect) represent-ing the rate of collisions between porous media grains and suspended fi ne particles migrating within the fl owing pore fl uid, and the sticking coeffi cient β 1 denotes the particle attachment effi ciency (a chemical effect) representing the fraction of particle collisions leading to successful particle attachment to porous media grains (Tufenkji and Elimelech, 2005 ).

The value of ηo is denoted by ηoS in the absence of electrostatic repulsive force (case of high ionic strength colloidal suspension) and is given by the following cor-relation (Tufenkji and Elimelech, 2004 ):

ηoS s R Pe vdW s R AA N N N A N N= +− −2 4 0 551 3 0 081 0 715 0 052 1 675 0 12. ./ . . . . . 55 0 24 1 11 0 0530 22+ −. .. . .N N NR G vdW (10.45)

The correction factor αCORR required in the presence of electrostatic repulsive force is given by (Chang et al., 2009 )

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αηη

α α αCORR CORR= = ( )[ ] = +( )⎡⎣⎢

⎤⎦⎥

− −o

oSC C B Texp ln exp ln ln

1

2 (10.46)

The correlation of αC C− is given by Chang and Chan (2008) for submicroparticles:

αC C DL E E Lo s RN N N N A N−

− −= +0 024 3 1760 2 196910 423

288 5 1 3 0 08. .. . . . / . 11 0 715 2 687

3 041 0 514 0 125 0 20 222

N N

A N N N N

Pe Lo

s R Pe Lo R

−

− −+ +

. .

. . . .. 44 1 11N NG Lo. .

(10.47)

The correlation of αB T− is given by Bai and Tien (1999) for small and large particles:

αB T DL E E LoN N N N−−= 0 002527 1 3 0352

10 3121

25111 7031. .. . . . (10.48)

The expressions of the dimensionless numbers facilitated in these correlations are given in the following. These include the following parameters: H denotes the Hamaker constant, κ Edenotes the reciprocal electric double - layer thickness, υois the dielectric constant, ξpis the zeta potential of suspended particles, ξg is the zeta poten-tial of porous media grains, k B is the Boltzmann constant, ρp and ρw denote the density of the particles and the carrier fl uid (water for example), respectively, g is the gravitational acceleration coeffi cient, and D∞ denotes the bulk diffusivity. The interstitial velocity v is given by Eq. (10.9) . R p and D p denote the mean radius and diameter, respectively, of the suspended particles. D g is the mean diameter of the porous media grains. Other variables were defi ned in the preceding sections.

The aspect ratio (particle - to - grain diameter ratio) N R is given by

ND

DR

p

g

= . (10.49)

The attraction number N A is given by

NH

R uA

p

=12 2πμ

, (10.50)

where u denotes the magnitude of the volumetric fl ux vector u . The electric double - layer force parameter N DL is given by

N RDL E p=κ . (10.51)

The fi rst electrokinetic parameter N E1 is given by

NR

k TE

o p p g

B1

2 2

4=

+( )υ ξ ξ. (10.52)

The second electrokinetic parameter N E2 is given by

NEp g

p g

2 2

2

1=

( )+ ( )

ξ ξξ ξ

. (10.53)

The gravity number N G is given by

NR g

uG

p p w=−( )2

9

2 ρ ρμ

. (10.54)

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The London force parameter N Lo is given by

NH

k TLo

B

=6

. (10.55)

The Peclet number N Pe is given by

NuD

DPe

g=∞

. (10.56)

The van der Waals number N vdW is given by

NH

k TvdW

B

= . (10.57)

The Stokes – Einstein equation of bulk molecular diffusivity D∞is given by

Dk T

DB

p∞ =

3πμ. (10.58)

The parameter A s is given by

As =−( )

− + −= −( )2 1

2 3 3 21

5

5 6

1 3γγ γ γ

γ φ, . (10.59)

10.2.5 Filter Coeffi cient Dependence on Particle Retention Mechanisms and Temperature Variation

Particles can deposit over the pore surface and/or behind the pore throats restricted for particle transport by means of the particle bridges formed across the pore throats. When particles form bridges across the pore throats but allow some fl ow of the carrier fl uid through the gaps present between the bridging particles, then the pore - bridging particles act as a fi lter for the approaching particles leading to the capture and retention of particles behind the bridged pore throats. Then, the fi lter coeffi cient assumes a value depending on the degree of plugging restriction. When all the pore throats are clogged and sealed completely, then the value of the fi lter coeffi cient becomes zero because the completely plugged pore throats do not allow the fl ow of the carrier fl uid and the fl ow paths involving such pore throats are left out of service.

The dependency of the fi ltration rate coeffi cient k o on temperature T alone, that is, when ε = 0, can be expressed by a Vogel – Tammann – Fulcher (VTF) - type equation, given by (Civan, 2008b )

k T

k

A

T TA

E

Ro

c

k

ckk

k( ) =−

⎛⎝⎜

⎞⎠⎟

≡ −exp , , (10.60)

where T is the absolute temperature, T ck is a characteristic - limit absolute temperature, k c is a pre - exponential coeffi cient, R is the universal gas constant, and E k is the activation energy. Note that the temperature difference ( T – T ck ) has the same value

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regardless of whether the ordinary ( ° C) or absolute (K) temperatures are used. The pre - exponential rate coeffi cient k c represents the high - temperature limit value of the fi ltration rate coeffi cient.

Thus, the temperature dependency factor G ( T ) of the fi lter coeffi cient can be derived from Eq. (10.60) as

k T

k TG T A

T T T To

o ikk

ck ik ck

( )( )

= ( ) ≅−

−−

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

exp ,1 1

(10.61)

where the subscript i denotes a reference condition and T ik is a reference absolute temperature.

The effect of temperature on the fi lter coeffi cient is profound because particle detachment force is higher at higher temperatures and therefore particle retention over the pore surface is less probable (Civan, 2007b ). Consequently, particles are more likely to move toward and to deposit behind the pore throats when the condi-tions are favorable for pore - throat bridging. Hence, rapid particle retention and accumulation occurs behind the bridged pore throats, rendering a pore - fi lling phenomenon.

On the other hand, the particle retention dependency factor F ( ε ) of the fi lter coeffi cient can be expressed as (Ives, 1967 )

k T

k TF

b

o M

m

o

m

o

mεε ε

εε

φεφ

,,

( )( )

= ( ) ≅ −⎛⎝⎜

⎞⎠⎟ +⎛

⎝⎜⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

1 1 11 2 3

εε φ εM o ok T k T≤ ( ) ≡ =( ), , ,0

(10.62)

where εM is the maximum amount of retention at which limit condition the fi lter coeffi cient becomes zero, and therefore the particle retention phenomenon ceases; b is an empirical constant; and m 1 , m 2 , and m 3 are empirically determined exponents of intensity. In the following, the exponent values of m 2 = m 3 = 0 are assumed (Ives, 1987 ).

Thus, the dependency of the fi ltration coeffi cient on temperature and fi nes retention can be expressed based on the method of separation of variables as

k T

k

k T

k T

k T

k TF G T

i o

o

o ik

ε εε

εε

, ,( ) = ( )( )

⎡⎣⎢

⎤⎦⎥

( )( )

⎡⎣⎢

⎤⎦⎥= ( ) ( )

= −1MM

m

kck ik ck

i o ik

k

AT T T T

k k T⎛⎝⎜

⎞⎠⎟ −

−−

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

≡ ( )exp , .1 1

(10.63)

Note in general that ε εM M T= ( ) and φ φo o T= ( ). For example, expressing the grain volume V g by V Vg b= −( )1 φ and assuming that the bulk volume V b is constant for confi ned porous material, the porous media grain volume expansion coeffi cient c g is given by (Civan, 2008c )

cV

V

T Tg

g

g=∂∂

=−( )

∂ −( )∂

1 1

1

1

φφ

. (10.64)

Thus, the variation of porosity with temperature can be expressed by (Gupta and Civan, 1994b )

c10.indd 364c10.indd 364 5/27/2011 12:34:30 PM5/27/2011 12:34:30 PM


φ φ≅ − −( ) −( )[ ]1 1 o g oc T Texp . (10.65)

The thermal expansion coeffi cient of minerals is rather small. For example, the coeffi cient of thermal expansion for quartz and calcite minerals is only in the order of 10 − 5 K − 1 (Gupta and Civan, 1994b ). It is reasonable to neglect the effect of temperature on porosity because the thermal expansion coeffi cient is very small and porosity reduction occurs by pore volume reduction. However, as described in the next section, the effect of temperature on permeability cannot be neglected because permeability reduction occurs primarily by pore - throat constriction. Even a small increase in temperature can cause suffi cient grain expansion to choke the pore - throat openings and to reduce the permeability substantially.

10.2.6 Permeability Alteration by Particle Retention and Thermal Deformation

When externally confi ned porous materials are heated, the pore throats are con-stricted by thermal expansion of porous matrix grains acting like valves to reduce the fl ow through the preferential hydraulic fl ow paths. Applying the method of sepa-ration of variables as demonstrated in the previous section, the total permeability variation can be expressed by the product of the permeability variations resulting from particle retention and temperature variation.

The dependency of permeability K o on temperature T alone, that is, when ε = 0, can be expressed by a VTF - type equation, given by (Civan, 2008c )

K T

K

A

T TA

E

Ro

c

K

cKK

K( ) =−

⎛⎝⎜

⎞⎠⎟

≡ −exp , , (10.66)

where T is the absolute temperature, T cK is the absolute characteristic temperature, K o is the permeability of porous media, R is the universal gas constant, and E K is the activation energy. The pre - exponential permeability coeffi cient K c repre-sents the high - temperature limit value of permeability. Civan (2008c) has demon-strated the validity of this equation by successfully correlating a number of experimental data.

Thus, the temperature dependency factor can be derived from Eq. (10.66) as

K T

K TA

T T T To

o iKK

cK iK cK

( )( )

=−

−−

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

exp .1 1

(10.67)

The subscript i denotes the reference condition and T iK is a reference absolute temperature.

To describe the reduction of permeability by particle retention, consider Civan ’ s power - law fl ow unit equation of permeability based on a bundle of leaky - tube model of tortuous preferential fl ow paths involving cross - fl ow between them depending on the pore interconnectivity in porous media (Civan, 2001 ):

K

φφφ

ββ

=−

⎛⎝⎜

⎞⎠⎟

≤ < ∞Γ1

0, , (10.68)

c10.indd 365c10.indd 365 5/27/2011 12:34:30 PM5/27/2011 12:34:30 PM


where the pore interconnectivity parameter Γ is given by

Γ = + < >φ υυ e e, , ,0 0 (10.69)

where the parameter e represents the effect of the threshold condition, corresponding to the minimum porosity below which permeability vanishes, and β and υ are empiri-cally determined parameters, which can be related to the fractal parameters of pore structure. Thus, using Eqs. (10.68) and (10.69) , the particle retention dependency factor can be formulated as

K T

K T

e

eo o o o

oφ φφ

φφ

φφ

φφ

φ

υ

υ

β,

,( )( )

=++

⎛⎝⎜

⎞⎠⎟

−−

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

=

2 21

1

φφ ε β υ φo o oe K T K T− ≤ < ∞ < > ( ) ≡ ( ), , , , , .0 0 0

(10.70)

For convenience, Eq. (10.70) can be approximated reasonably upon substitu-tion of Eq. (10.2) as

K T

K Tm K T K T

o o

m

K o

Kε εφ

β υ ε,

, , , .( )( )

≅ −⎛⎝⎜

⎞⎠⎟

= + +( ) ( ) ≡ =( )1 1 2 0 (10.71)

Thus, the dependency of permeability on temperature and fi nes retention can be expressed based on the method of separation of variables as

K T

KA

T T T TK T

i o

m

KcK iK cK

i

Kε εφ

,exp ,

( ) = −⎛⎝⎜

⎞⎠⎟ −

−−

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

11 1 (( ) ≡ ( )K To iK . (10.72)

10.2.7 Applications

The numerical solution of the above - given differential equations was obtained for one - dimensional particulate transport in a porous core plug using a fi nite difference scheme according to Civan (2009, 2010c) . The details of this approach are described in Exercise 4.

The representative parameter values employed in the following numerical simulation studies are R p = 1.5E - 06 m, D g = 4.6E - 04 m, ρp = 1050 3kg/m , ρ = 1020 3kg/m , ρm = 1600 3kg/m , c = 4186 J/(kg - K), c m = 820 J/(kg - K), κ = 0 65. ( )W/ m-K , κm = 2 0. ( )W/ m-K , ξp = − −1 0 03. E V, ξg = −3 0 03. E- V, κ E = +2 2 08 1. E /m, υo = 8 9 11. E- , φ = 0 2. , τ = 1 41. , L = 0.1 m, H = 1.0E - 20 J, k B = 1.38E - 23 m 2 kg/s 2 /K, Tcμ = − °100 C, Aμ = 370 K, μc = 0 045. cp, T ik = 25 ° C, T ck = − 200 ° C, A k = − 251 K, k c = 41.2 m − 1 , m k = 2, β1 1 0= . , T iK = 50 ° C, T cK = − 101 o C, A K = 500 K, m K = 5, εM = 0 17 3 3. m /m , u = 1.0E - 04 m/s, T o = 50 ° C, σ o = 0, εo = 0, T in = 80 ° C, and σ in E- m /m= 5 0 3 3 3. .

The effect of temperature on the carrier fl uid (water) viscosity was correlated using the water viscosity data of Bett and Cappi (1965) at 1 atm by the VTF equation as (Civan, 2008b )

μμ

μ

μμ

μT A

T TA

E

Rc c

( ) =−

⎛⎝⎜

⎞⎠⎟

≡ −exp , . (10.73)

c10.indd 366c10.indd 366 5/27/2011 12:34:30 PM5/27/2011 12:34:30 PM


The best estimate values of T c μ , A μ , and μ c reported earlier were determined by the least squares linear regression of their data.

The temperature dependency of the fi lter coeffi cient was obtained also by using a VTF - type equation (Eq. 10.63 ) by correlating the fi lter coeffi cient values predicted by means of the correlations given by Tufenkji and Elimelech (2004) and Chang et al. (2009) as described previously using the best estimate parameter values as reported earlier. Eq. (10.73) was used for the calculation of viscosity in predicting the fi lter coeffi cient using these correlations.

The numerical calculations of the variation of the temperature, suspended particle volume fraction, deposited particle volume fraction, and permeability reduc-tion profi les along the porous medium were carried out by Civan (2010c) with and without the dispersion effect included under isothermal ( T = 50 ° C) and nonisother-mal (50 ° C ≤ T ≤ 80 ° C) conditions by means of the numerical solution scheme described in Exercise 4 using Δx = 0 001. -m grid - point spacing and Δt = 5-s time increments. The predictions of the above - given phenomenological model were cal-culated at various times after starting the injection of the fl uid containing suspended fi ne particles.

Figure 10.1 shows the typical nonisothermal temperature profi les obtained at various times (minutes). As can be seen, the temperature effect propagates with time (minutes) from the injection port to the effl uent port. A comparison of the results presented in Figures 10.2 – 10.4 indicate that the thermal and dispersion factors affect the particle volume fraction in the fl owing suspension, volume fraction of the bulk porous media occupied by the deposited particles, and permeability impairment in porous media.

Figure 10.2 shows the suspended particle volume fraction profi les at various times (minutes) with and without the dispersion effect included under isothermal and elevated nonisothermal conditions. When the dispersion effect is ignored, the particle migration occurs with sharp progressing fronts as indicated in Figure 10.2 a,c.

Figure 10.1 Nonisothermal temperature profi les at various times (minutes) (after Civan, 2010c ; reprinted with permission from Springer) .

045

50

55

60

65

70

75

80

85

0.02 0.04 0.06

Distance from injection side, x (m)

Tem

pera

ture

, T(°

C)

0.08

0.5

2.04.07.0

10.015.045.0

0.1

c10.indd 367c10.indd 367 5/27/2011 12:34:30 PM5/27/2011 12:34:30 PM


When the dispersion effect is considered, the particle migration spreads over a long range as indicated in Figure 10.2 b,d. Figure 10.2 a,b obtained under isothermal con-ditions indicate higher suspended particle concentrations than those indicated by the corresponding Figure 10.2 c,d obtained under elevated nonisothermal conditions.

Figure 10.3 shows the deposited particle volume fraction profi les at various times (minutes) with and without the dispersion effect included under isothermal and elevated nonisothermal conditions. When the dispersion effect is ignored, the particle retention is more pronounced over the region near the injection port as indicated in Figure 10.3 a,c. When the dispersion effect is considered, the particle retention spreads over a long range as indicated in Figure 10.3 b,d. But the effect is small because of low suspended particle concentration in the injected fl uid. Figure 10.3 a,b obtained under isothermal conditions indicate lower deposited particle con-centrations than those indicated by the corresponding Figure 10.3 c,d obtained under

Figure 10.2 Isothermal suspended particle volume fraction profi les at various times (minutes) (a) without and (b) with the dispersion effect included. Nonisothermal suspended particle volume fraction profi les at various times (minutes) (c) without and (d) with the dispersion effect included (after Civan, 2010c ; reprinted with permission from Springer) .

00

0.001

0.002

0.003

0.004

0.005

0.02 0.04 0.06Distance from injection side, x (m)

(a) (b)

(c) (d)

Sus

pend

ed p

artic

al v

olum

e fr

actio

n,s

0.08 0.1

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

00

0.001

0.002

0.003

0.004

0.005


Sus

pend

ed p

artic

al v

olum

e fr

actio

n,s

0.08 0.1

00

0.001

0.002

0.003

0.004

0.005


Sus

pend

ed p

artic

al v

olum

e fr

actio

n,s

0.08 0.1 00

0.001

0.002

0.003

0.004

0.005

0.02 0.04 0.06Distance from injection side, x (m)S

uspe

nded

par

tical

vol

ume

frac

tion,s

0.08 0.1

c10.indd 368c10.indd 368 5/27/2011 12:34:31 PM5/27/2011 12:34:31 PM


elevated nonisothermal conditions. More retention occurs behind constricted and bridged pore throats at elevated temperatures.

Figure 10.4 shows the permeability reduction profi les at various times (minutes) with and without the dispersion effect included under isothermal and elevated non-isothermal conditions. When the dispersion effect is ignored, the permeability impairment occurs more severely over the region near the injection port as indicated in Figure 10.4 a,c. When the dispersion effect is considered, the permeability impair-ment spreads over a long range as indicated in Figure 10.4 b,d. But the effect is small because of low suspended particle concentration in the injected fl uid. Figure 10.4 a,b obtained under isothermal conditions indicate less permeability impairment com-pared with those indicated by the corresponding Figure 10.4 c,d obtained under elevated nonisothermal conditions.

Figure 10.3 Isothermal deposited particle volume fraction profi les at various times (minutes) (a) without and (b) with the dispersion effect included. Nonisothermal deposited particle volume fraction profi les at various times (minutes) (c) without and (d) with the dispersion effect included (after Civan, 2010c ; reprinted with permission from Springer) .

00

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014


Dep

osite

d pa

rtic

le v

olum

e fr

actio

n,e

0.08 0.1 0 0.02 0.04 0.06Distance from injection side, x (m)

Dep

osite

d pa

rtic

le v

olum

e fr

actio

n,e

0.08 0.1

0 0.02 0.04 0.06Distance from injection side, x (m)

Dep

osite

d pa

rtic

le v

olum

e fr

actio

n,e


Dep

osite

d pa

rtic

le v

olum

e fr

actio

n,e

0.08 0.1

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

(a) (b)

(c) (d)

c10.indd 369c10.indd 369 5/27/2011 12:34:31 PM5/27/2011 12:34:31 PM


10.3 CAKE FILTRATION OVER AN EFFECTIVE FILTER

Filter cake is a special porous media formed over a fi lter facilitated for separation of particles from a suspension of particles or slurry. Smaller particles are deposited in between the larger particles, forming a compressible and porous fi lter cake.

Detailed formulations of incompressible and compressible cake fi ltration pro-cesses are presented elsewhere by Civan (1998a,b) . However, the Tien et al. (1997) model for one - dimensional linear fl ow is reviewed here for instructional purposes. Their formulation is expanded to explain the steps leading to model construction. The formulation of this model is presented in a manner consistent with the symbolism used in this chapter. A schematic description of the problem is depicted in Figure 10.5 . A slurry of particles of various sizes is fed into a fi lter. The fi lter is assumed impermeable for particle invasion and, therefore, the fi ltrate is a particle - free carrier fl uid.

Tien et al. (1997) classifi ed the various particles in the slurry fed into the fi lter into two groups as large and small particles, denoted by indices 1 and 2, respectively.

Figure 10.4 Isothermal permeability reduction profi les at various times (minutes) (a) without and (b) with the dispersion effect included. Nonisothermal permeability reduction profi les at various times (minutes) (c) without and (d) with the dispersion effect included (after Civan, 2010c ; reprinted with permission from Springer) .

Per

mea

bilit

y re

duct

ion

ratio

, K/K

°

00.4

0.5

0.6

0.7

0.8

0.9

1

Per

mea

bilit

y re

duct

ion

ratio

, K/K

°

0.4

0.5

0.6

0.7

0.8

0.9

1



0.08 0.1

Per

mea

bilit

y re

duct

ion

ratio

, K/K

°

00.4

0.5

0.6

0.7

0.8

0.9

1

Per

mea

bilit

y re

duct

ion

ratio

, K/K

°

0.4

0.5

0.6

0.7

0.8

0.9

1



0.08 0.1

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

0.52.04.07.010.015.045.0

(a) (b)

(c) (d)

c10.indd 370c10.indd 370 5/27/2011 12:34:31 PM5/27/2011 12:34:31 PM

10.3 CAKE FILTRATION OVER AN EFFECTIVE FILTER 371

They based this classifi cation on Tien ’ s (1989) particle retention criteria, which requires the ratio of the small to large particle diameters to be d p 2 / d p 1 < 0.1. In the following presentation, the fi lter cake and the fl owing suspension of particles are denoted by the indices s and f . The carrier phase (liquid) is denoted by l and the particles by p . It is considered that large particles cannot enter the fi lter cake and, therefore, are deposited over the fi lter cake, whereas the small particles can penetrate the fi lter cake and deposit within the porous cake matrix.

The volume fractions (or concentrations) of the large and small particles are denoted by σ p 1 and σ p 2 . Thus, the total particle volume fraction in the slurry is given by

σ σ σp p p= +1 2. (10.74)

The volume fraction of all the particles (small plus large) forming the fi lter cake can be expressed in terms of the cake porosity, ε φf c≡ , as

ε εs f= −1 . (10.75)

Because the fi lter cake is formed from the large particles plus the deposited small particles, it is also true that

ε ε εs s s= +1 2, (10.76)

where εs1 and εs2 denote the volume fractions of the large and small particles in the fi lter cake, respectively.

The superfi cial (or macroscopic) velocities of the fl owing suspension of par-ticles and the particles forming the cake are indicated by u f and u s , respectively. The fl uid pressure and the compressive stress of the particles of the fi lter cake generated by the fl uid drag forces are represented by p and p s , respectively. Therefore, neglect-ing the inertial effects, the effective pressure at the cake – slurry interface can be expressed as (Tiller and Crump, 1985 )

p p po s= + . (10.77)

Next consider the general mass balance equation given in terms of the mass concentration, cij, of a species i in a phase j , as

Figure 10.5 Cake fi ltration and the coordinate system used in the analysis (prepared by the author with modifi cations after Tien et al., 1997 ; © 1997 AIChE, reprinted by permission from the American Institute of Chemical Engineers) .

Cake F

ilter SlurryFiltrate

L(t)

x

c10.indd 371c10.indd 371 5/27/2011 12:34:31 PM5/27/2011 12:34:31 PM


∂( )

∂+

∂( )∂

=∂∂

∂∂

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ +

εε ρ

ρj ij ij j

j j ijij

jij

c

t

c u

x xD

x

cR , (10.78)

where ε j is the volume fraction of phase j in the bulk of the fi lter cake, �uj is its

superfi cial velocity, ρ j is its density, �Dij is the coeffi cient of dispersion of species i

in phase j , and Rij is the mass rate of addition of species i to phase j . The mass concentration, cij, can be expressed in terms of the volume fraction

(or concentration), σ ij, as

cij i ij= ρ σ . (10.79)

Thus, invoking Eq. (10.79) , the mass balance equation given by Eq. (10.78) can be expressed in terms of the volume fraction as

∂( )

∂+

∂( )∂

=∂∂

∂∂

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

ε ρ σ ρ σε ρ

ρ σρ

j i ij i ij jj j ij

i ij

jt

u

x xD

x++ Rij. (10.80)

Eqs. (10.78) and (10.80) were adopted from Civan (2007a) with modifi cations for the present one - dimensional case in the x - direction.

When the system is assumed incompressible and the dispersion term is neglected following Tien et al. (1997) , Eq. (10.80) simplifi es as

∂( )

∂+

∂∂

=ε σj ij ij

ijt

u

xN , (10.81)

where

u uij ij j= σ (10.82)

is the volumetric fl ux of species i in phase j and

N Rij ij i= ρ (10.83)

is the volumetric rate of generation of species i in phase j . Thus, the volumetric balance equation of the carrier fl uid in the suspension

can be expressed according to Eq. (10.81) for i l≡ and j f≡ . For a diluted suspen-sion of small particles, the carrier fl uid volume fraction is nearly σ lf ≅ 1 0. , and ε εf s= −1 and Nlf = 0. Then, Eq. (10.81) can be written in one dimension, according to Tien et al. (1997) , as

∂∂

=∂∂

εs lf

t

u

x. (10.84)

For the particles of the cake, i p≡ , j s≡ , and σ ps = 1 0. (thus, u ups s= ). Thus, Eq. (10.81) becomes

∂∂

+∂∂

= ≡εs s

pst

u

xN N , (10.85)

or elimination of ∂ ∂εs t between Eqs. (10.84) and (10.85) leads to

c10.indd 372c10.indd 372 5/27/2011 12:34:31 PM5/27/2011 12:34:31 PM


∂∂

+∂∂

= ≡u

x

u

xN Nlf s

ps . (10.86)

For small particles contained in the fi lter cake, Eq. (10.81) becomes

∂∂

( ) + ∂∂

= ≡t

u

xN Ns p s

p sp sε σ 2

22 . (10.87)

The volume fraction of small particles in the cake is given by

ε ε σs s p s2 2= (10.88)

and the volume fl ux is given by

u u up s s p s ss

s2 2

2= =σ εε

. (10.89)

Therefore, substituting Eq. (10.89) into Eq. (10.87) yields

∂∂

+∂∂

⎛⎝⎜

⎞⎠⎟ = ≡

ε εε

ss

sp s

t xu N N2 2

22 . (10.90)

For small particles of the suspension fl owing through the fi lter cake, substitut-ing ε εf s= −1 into Eq. (10.81) yields

∂∂

−( )[ ]+ ∂∂

= = −t

u

xN Ns p f

p fp f1 2

22ε σ , (10.91)

or eliminating N between Eqs. (10.86) and (10.91) yields

∂∂

−( )[ ]+ ∂∂

+∂∂

+∂∂

=t

u

x

u

x

u

xs p f

p f lf s1 022ε σ . (10.92)

The superfi cial velocity, ufr, of the fl owing suspension relative to that of the compressing fi lter cake, us, is given by Darcy ’ s law as

u u uk p

xfr f

s

ss= −

−= −

∂∂

1 εε μ

, (10.93)

where k is the permeability of the cake and μ is the viscosity of the fl owing suspension.

The rate of small particle retention within the fi lter cake is assumed propor-tional to the small particle volume fl ux through the cake as

N ufr p f= λ σ 2 , (10.94)

where λ is the retention rate constant. Next, Tien et al. (1997) algebraically manipulate the preceding equations to

derive a set of workable equations suitable for numerical solution. Tien et al. (1997) integrate Eq. (10.86) over a distance, x , from the fi lter surface

in the fi lter cake as

c10.indd 373c10.indd 373 5/27/2011 12:34:31 PM5/27/2011 12:34:31 PM


u u u u Ndxlf s x lf s x

o

x

+( ) = +( ) += ∫0. (10.95)

Because the solids are not permitted to enter the fi lter, us x= =0 0 and Eq. (10.95) simplifi es as

u u u Ndxlf s x lf x

o

x

+( ) = += ∫0 , (10.96)

in which ulf x=0 is the superfi cial velocity of the fi ltrate (carrier fl uid) entering the fi lter, expressed by Darcy ’ s law as

uk p p

Llf x

m x x L

m

m

== ==

−0

0

μ, (10.97)

where km and Lm denote the permeability and thickness of the fi lter. Thus, substituting Eq. (10.77) and p x Lm= = 0 gage pressure into Eq. (10.97)

yields an equation for the fi ltrate invasion volumetric fl ux into the fi lter as

uk p p

Llf x

m o s x

m=

==−

00

μ. (10.98)

Eliminating the fi lter cake superfi cial velocity, us, between Eqs. (10.93) and (10.96) and substituting u uf lf≅ for a dilute suspension leads to

uk p

xu Ndxlf s s lf x s

o

x

= −∂∂

+ −( ) + −( )= ∫εμ

ε ε1 10 . (10.99)

Then, invoking Eq. (10.99) into Eq. (10.96) results in

uk p

xu Ndxs s s lf x s

o

x

=∂∂

+ += ∫εμ

ε ε0 . (10.100)

A substitution of Eq. (10.100) into Eq. (10.85) yields the following differential equation for the volume fraction of the particles forming the cake as

∂∂

+ +⎡

⎣⎢⎢

⎤

⎦⎥⎥∂∂

+∂∂

∂∂

⎡⎣⎢

⎤⎦⎥= −( )

= ∫ε ε εμ

εsl x

o

x

ss s

tu Ndx

x x

k p

xNf 0

1 .. (10.101)

The boundary conditions at the slurry – cake interface are given by

p x L ts s so= = = ( )0, , .ε ε (10.102)

Based on Shirato et al. (1987) and Tien (1989) , Tien et al. (1997) facilitated the following power - law constitutive relationships for the variation of the volume fraction of particles in a compressible cake by pressure by

εε λ

βs

so

sp= +⎛⎝⎜

⎞⎠⎟1 (10.103)

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and for the fl ow effi ciency factor due to the retention of small particles by

γα εα=

+1

1 1 22

s

. (10.104)

Therefore, the permeability variation is empirically expressed as

k

k

po

ss= +⎛

⎝⎜⎞⎠⎟ +( )

−−

1 1 1 22 1

λα ε

δα , (10.105)

or substituting Eq. (10.103) into Eq. (10.105) ,

k

ko

s

so s= ⎛

⎝⎜⎞⎠⎟

+( )−

−εε

α εδ β

α1 1 22 1

. (10.106)

Eq. (10.103) can be rearranged as

pss

so

= ⎛⎝⎜

⎞⎠⎟

−⎡

⎣⎢

⎤

⎦⎥λ ε

ε

β1

1 . (10.107)

Next, Tien et al. (1997) derive an equation to determine the fi lter cake thick-ness. For this purpose, consider the jump mass balance of species i in a phase l at the interface of the slurry and the cake given by

c u u c u u Rij j j ij j j ij− − − + + +−( ) = −( ) +σ σ σ , (10.108)

where − and + indicate the slurry and cake sides of the interface, cij is the mass concentration of species i in the j th phase, Rij

σ is the amount of the species i deposited over the cake surface from the j th phase, and the superfi cial velocities of the interface for phase j at the slurry and cake sides are given, respectively, by

u dL dtj jσ ε− −= (10.109)

and

u dL dtj jσ ε+ += , (10.110)

where ε j denotes the volume fraction of the j th phase. For convenience, Eq. (10.108) can also be written in an alternative form as

m m c u c u Rij ij ij j ij j ij− + − − + +− = − +σ σ σ , (10.111)

where

m c uij ij j= (10.112)

is the mass fl ux of species i in the j th phase. Dividing Eq. (10.108) by the density of species i leads to the following

expression:

σ σ σ σσ σ σij j ij j ij j ij j iju u u u N− − + + − − + +− = − + , (10.113)

where σ ij is the volume concentration (or fraction) of species i in the j th phase and

N Rij ij iσ σ ρ≡ (10.114)

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denotes the volume of species i deposited over the cake surface from the j th phase. Again, for convenience, Eq. (10.113) can be written as

u u u u Nij ij ij j ij j ij− + − − + +− = − +σ σσ σ σ , (10.115)

where

u uij ij j= σ (10.116)

denotes the volume fl ux of species i in the j th phase. The application of Eqs. (10.109) , (10.110) , and (10.115) to the liquid in the

fl owing suspension for i l j f Nlf f f s≡ ≡ = = = −− + +, , , . ,σ ε ε ε0 1 0 1 , where εs+ is the

volume fraction of the solids in the cake; σ σlf p− −= −1 , and σ σlf p

+ += −1 2 (because only the small particles can penetrate the cake) yields

dL

dt

u ulf lf

p p s

=−

−( ) − −( ) −( )− +

− + +1 1 12σ σ ε. (10.117)

The application of Eqs. (10.109) , (10.110) , and (10.115) to the particles (small plus large) for i p j f f f s pf p≡ ≡ = = − =− + + − −, , . , ,ε ε ε σ σ1 0 1 and σ σpf p

+ += 2 (because only the small particles can penetrate the cake), and N Npf p f

σ σ≡ 1 (because only the large particles deposit over the cake surface) yields

dL

dt

u u Npf pf p f

p p s

=− −− −( )

− +

− + +1

2 1

σ

σ σ ε. (10.118)

Similarly, for the large particles of the cake for i p j s s≡ ≡ =−1 0, , ε ,

ε ε σs s p s+ + −= =, 1 0, and up s p s1 10 1 0− += =, .σ ,

dL

dt

u Np s p s

s

=++

+1 1

σ

ε. (10.119)

Assuming there are less small particles than the large particles, Tien et al. (1997) neglected the small particles in the slurry, that is, σ σ σ σ ε εp p p s s

olf lou u2 1

00− − − + −≅ ≅ ≡ ≡ ≡, , , , and neglected the effect of the small par-ticles in the suspension fl owing through the fi lter cake, that is, σ p2 0+ ≅ . Consequently, Eq. (10.117) simplifi es as

dL

dt

u ulf lo

oso

≅−−

+

σ ε, (10.120)

and the following expression can be obtained by the elimination of N N Np f p s1 1σ σ= ≡

between Eqs. (10.118) and (10.119) , and substituting u up s s1 ≅ (neglecting small particles) and u up f so1 ≡ yields

dL

dt

u us so

so o

=−−

+

ε σ. (10.121)

Therefore, equating Eqs. (10.120) and (10.121) and considering Eq. (10.96) yields

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u u u Ndxlf s lf x

o

L

+ +=+ = + ∫0

. (10.122)

Next, Tien et al. (1997) apply Darcy ’ s law given by Eq. (10.93) at the fi lter surface x us x=( ) =( )=0 00with and at the interface of the slurry and the cake x L=( ) (with u ulf lf L

+ ≡ + and u ulf lf f= σ , where σ lf ≅ 1 0. in the suspension fl owing through the cake) to obtain the following respective expressions:

uk p

xlf x

x=

=

= −∂∂

⎛⎝⎜

⎞⎠⎟0

0μ (10.123)

and

u u uk p

xs s x L

so

so lf x L

x L

+= =

=

≡ =−

+ ⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥+ +

+

εε μ

∂∂1

. (10.124)

Thus, substituting Eqs. (10.123) and (10.124) into Eq. (10.122) and rearrang-ing gives

uk p

x

k p

xNdxlf x L s

o

L

so

so

o

= = − ⎛⎝⎜

⎞⎠⎟

− −( )⎛⎝⎜⎞⎠⎟

+ −( )+

εμ

∂∂

εμ

∂∂

ε1 10

LL

∫ . (10.125)

Assuming that the particles and the carrier liquid move at the same velocity in the slurry, Tien et al. (1997) write

u u up f sf

o

o lf11

− − −≅ =−σσ

(10.126)

Thus, substituting Eqs. (10.123) and (10.126) into Eq. (10.122) yields:

u uk p

xc Ndxlo lf

o

o

o

o

L

≡ = − −( ) ∂∂

⎛⎝⎜

⎞⎠⎟

+ −( )− ∫1 1σμ

(10.127)

Then, substituting Eqs. (10.125) and (10.127) into Eq. (10.120) results in the following cake growth expression:

dL

dt

k p

x

k p

xNdxs

o

so o

L Oo

L

=−

⎛⎝⎜

⎞⎠⎟

− ⎛⎝⎜

⎞⎠⎟

++ ∫ε

ε σ μ∂∂ μ

∂∂

, (10.128)

subject to the condition that the cake thickness is initially zero:

L t= =0 0, . (10.129)

The fi lter cake problem is simulated by numerically solving Eqs. (10.77) , (10.90) , (10.91) , (10.93) , (10.94) , (10.98) – (10.101) , (10.106) , (10.107) , and (10.128) simultaneously.

For illustration purposes, Tien et al. (1997) obtained the numerical solutions of the above - mentioned model for typical constant rate and constant pressure fi ltra-tion cases by using the parameter values given as the following: εs

o = 0 27. , δ = 0 49. , β = 0.09, p a = 1200 Pa, k o = 3.5 × 10 − 15 m 2 , μ = 0.001 Pa · s, p o = 9.0 × 10 5 Pa,

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q lm = 2.0 × 10 − 5 m 3 /m 2 · s, c o = 0.2, t o = 0.1 s, R = 100 m − 1 , ′ =β 5 0. , α 1 = 30, α 2 = 1.0, n 2 o = 0.05, and λ o = 0, 10, and 100. The predicted cake thickness and applied slurry pressure for constant rate fi ltration are shown in Figures 10.6 and 10.7 , respectively. As can be seen, the small particle retention rate constant signifi cantly affects the applied slurry pressure, while the fi lter cake thickness is not appreciable affected. The predicted cake thickness and cumulative fi ltrate volume are given in Figures 10.8 and 10.9 , respectively, for constant pressure fi ltration. It is observed that the small particle retention rate constant signifi cantly affects both the cake thickness and the cumulative fi ltrate volume.

Figure 10.6 Predicted cake thickness versus time. Constant rate fi ltration with n 2 = 0.05 and three different values of λo (after Tien et al., 1997 ; © 1997 AIChE, reprinted by permission from the American Institute of Chemical Engineers) .

Filtration time (s)

00.000

0.005

0.010

Cak

e th

ickn

ess

(m) 0.015

0.020

200 400 600 800

1

2

3

1000

1 l0 = 0 m–1

2 l0 = 10 m–1

3 l0 = 100 m–1

Figure 10.7 Predicted pressure requirement versus time. Constant rate fi ltration with n 2 = 0.05 and three different values of λo (after Tien et al., 1997 ; © 1997 AIChE, reprinted by permission from the American Institute of Chemical Engineers) .

1 l0 = 0 m–1

2 l0 = 10 m–1

3 l0 = 100 m–1

Filtration Time (s)

0

App

lied

Pre

ssur

e, P

0 (P

a)

8E+06

6E+06

4E+06

2E+06

0E+06200 400 600 800

1

2

3

1000

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10.4 EXERCISES 379

10.4 EXERCISES

1. Derive the equations for the one - dimensional case involving laboratory core tests by expressing the equations of the nonisothermal deep - bed fi ltration model presented in this chapter according to Civan (2010c) .

Figure 10.8 Predicted cake thickness versus time. Constant pressure fi ltration with n 2 = 0.05 and three different values of λo (after Tien et al., 1997 ; © 1997 AIChE, reprinted by permission from the American Institute of Chemical Engineers) .

Filtration Time (s)

00.000

0.005

0.010

Cak

e T

hick

ness

(m

)

0.015

0.020

0.025

1 l0 = 0 m–1

2 l0 = 10 m–1

3 l0 = 100 m–1

200 400 600 800

12

3

1000

Figure 10.9 Total fi ltration volume versus. time. Constant pressure fi ltration with n 2 = 0.05 and three different values of λo (after Tien et al., 1997 ; © 1997 AIChE, reprinted by permission of the American Institute of Chemical Engineers) .

1 l0 = 0 m–1

2 l0 = 10 m–1

3 l0 = 100 m–1

Filtration Time (s)

0

0

0.005

0.01

Tot

al F

iltra

te V

olum

e, Q

Tol (

m3 /m

2 )

0.015

0.020

0.025

200 400 600 800

1

2

3

1000

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2. Applying the numerical solution scheme presented in the following according to Civan (2010c) , carry out parametric sensitivity studies by varying the values of the parameters of the one - dimensional nonisothermal deep - bed fi ltration model in a core plug. The special numerical scheme facilitated here accomplishes the solution by means of the second - order accurate discretization of both the differential equations and their associ-ated boundary conditions. Civan (2010c) obtained the numerical solution of the differ-ential equations presented here by using Δx = 0 001. -m grid - point spacing and Δt = 5-s time increments. Carry out numerical solutions to investigate the effect of the values of Δx and Δt on the quality of numerical results. Consider the following general transport equation of the transient - state, advection, dis-persion, and source/sink type:

∂∂

+∂∂

+ =∂∂

f

ta

f

xbf c

f

x

2

2. (10.130)

The initial and boundary conditions are specifi ed as the following:

f f x L to= ≤ ≤ =, , ,0 0 (10.131)

f f df

xx tin = −

∂∂

= >, , ,0 0 (10.132)

and

∂∂

= = >f

xx L t0 0, , . (10.133)

A fully implicit solution of the above partial differential equation, fi rst - order accurate in time and second - order accurate in space, is obtained by applying the fi nite difference method for discretization of Eqs. (10.130) – (10.133) as the following (Civan, 2008a, 2010c ):

f f

ta

f f

xb f c

f f f

xii

nin

iin

in

i in

iin

in

in−

+−

+ =− +−

+ − + −1

1 1 1 1

22

2

Δ Δ Δ, == >1 2 3 0, , , , , .… N n (10.134)

The initial and boundary conditions are discretized as the following:

f f i nin

io= = =, , , , , ,1 2 3 0… (10.135)

f f df f

xi nin i

ni

in

in

= −−

= >+ −1 1

21 0

Δ, , , (10.136)

and

f f

xi N ni

nin

+ −−= = >1 1

20 0

Δ, , . (10.137)

Rearranging Eqs. (10.134) – (10.137) yields the following algebraic analog equations, respectively.

The partial differential equation is represented by the following expression for the interior points:

− + − = = − >− +−A f f A f A f i N ni i

nin

i in

i in1 2 3 2 3 1 01 1

1, , , , , ,… (10.138)

where

c10.indd 380c10.indd 380 5/27/2011 12:34:33 PM5/27/2011 12:34:33 PM

A

e

c

x

aA

e

c

x

a

Ax

tee

ii

i ii

i

i i

ii

i

11

22

1

2

3

= +⎛⎝⎜

⎞⎠⎟ = −⎛

⎝⎜⎞⎠⎟

= =

Δ ΔΔΔ

, ,

,ΔΔΔ

ΔΔ

x

txb

c

xi

i+ +2

. (10.139)

The initial discrete point values are given by

f f i N nin

io= = =, , , , , , .1 2 3 0… (10.140)

The fi ctitious point values are expressed by the following equations:

f fx

df f i ni

nin

iin

in− += − −( ) = >1 12

1 0Δ

, , (10.141)

and

f f i N nin

in

+ −= = >1 1 0, , . (10.142)

Applying Eq. (10.141) to Eq. (10.138) , the partial differential equation is represented by the following expression for the inlet boundary point:

f B f B f B i n

ExA

dB

A Ain

i in

i in

i

ii

ii

i

− = + = >

= + =+

+−1 2 3 1 0

1 12 1

11 2

11 , ,

,Δ ii

ii

i

ii

i

i iEB

A

EB

xA f

d E12

3

13

2 1

1, , .= =

Δ in (10.143)

Applying Eq. (10.142) to Eq. (10.138) , the partial differential equation is represented by the following expression for the outlet boundary point:

− + = = >

= +−

−C f f A f i N n

C A Ai i

nin

i in

i i i

1 3 0

1 1 21

1, ,

. (10.144)

Eq. (10.24) is discretized as the following and is solved explicitly:

ε ε φ

τσ ατ σi

nin

in

in

t

vk

xi N n

−= −⎛

⎝⎜⎞⎠⎟ = >

−1

21 2 3 0

Δ ΔΔ , , , , , ,… and (10.145)

where

Δσ σ σ σin

in

in

in i n= − + − = >+ +3 4 1 01 2, ,and (10.146)

Δσ σ σin

in

in i N n= − = − >+ −1 1 2 3 1 0, , , , ,… and (10.147)

and

Δσ σ σ σin

in

in

in i N n= − + = >− −3 4 01 2, .and (10.148)

When the parameters a , b , c , and d are dependent upon the function value f , then the above - described numerical solution procedure is iterated until convergence starting with their values evaluated at the previous time, which is the initial time at the beginning.

3. Show that the application of Eqs. (10.109) , (10.110) , and (10.115) to the large particles depositing over the cake surface for i p≡ 1, j f≡ , ε f

− = 1 0. , ε εf s+ += −1 , σ σp f p1 1

− −= , and σ p f1 0+ = , and qp f1 0+ = yields the following expression:

dL

dt

u Np f p f

p

=−−

−1 1

1

σ

σ. (10.149)

10.4 EXERCISES 381

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4. Show that the application of Eqs. (10.109) , (10.110) , and (10.115) leads to the following expression for the small particles for i p≡ 2, j f≡ , ε f

− = 1 0. , ε εf s+ += −1 , σ σp f p2 2

− −= , σ σp f p2 2

+ += , and N p f2 0σ = :

dL

dt

u up f p f

p p s

=−

− −( )− +

− + +2 2

2 2 1σ σ ε. (10.150)

5. Tien et al. (1997) used the applied pressure versus time given in Figure 10.7 and the cumulative fi ltrate volume versus time given in Figure 10.9 as a substitute for experi-mental data in order to illustrate the method of estimating the model parameters from experimental data. They determined the parameters of the model to minimize the differ-ence between the measured fi lter cake thickness data and those predicted by the above - mentioned model. For this purpose, Tien et al. (1997) defi ne an objective function as

J f fim

ip

i

N

= −( )∑ 2, (10.151)

in which i N= 1 2, , ,… denote the data points, and fim and fi

p are the measured and predicted values, respectively, of a measurable quantity, such as the slurry application pressure necessary to maintain a constant fi ltration rate or the cumulative fi ltrate volume for constant pressure slurry applications, used here. The best estimates of the model parameters obtained by an optimization method to minimize the objective function given by Eq. (10.151) were determined to be very close to the assumed parameter values given earlier, which were used to generate the numerical solutions, substituted for experimental data. Carry out a similar exercise to determine the best estimate values of the parameters.

c10.indd 382c10.indd 382 5/27/2011 12:34:33 PM5/27/2011 12:34:33 PM

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