An analytical permeability model for power-law fluids in porous fibrous media with consideration of electric double layer

International Journal of Heat and Mass Transfer 91 (2015) 255–263

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An analytical permeability model for power-law fluids in porous fibrous media with consideration of electric double layer Q.Y. Zhu a, Y.J. Zhuang a, H.Z. Yu a,b,⇑ a b

School of Engineering, Sun Yat-sen University, Guangzhou 510275, China China Earthquake Networks Center, Beijing 100045, China

a r t i c l e

i n f o

Article history: Received 27 March 2015 Received in revised form 28 July 2015 Accepted 30 July 2015 Available online 8 August 2015 Keywords: Permeability Power-law fluids Porous fibrous media Fractal model Electric double layer (EDL)

a b s t r a c t This work studies the permeability of power-law fluids in porous fibrous media with electrokinetic effects. By solving the linearized Poisson–Boltzmann and Navier–Stokes equations, we get the analytical solutions of pressure driven flow of power-law fluids in a microcapillary with electric double layer (EDL). The flow rate in a single capillary, combining with the fractal model of pore distribution of fractures in naturally fractured porous media, deduces the total flow rate. Then the analytical result of effective permeability for power-law fluids with EDL effects is derived as a function of the porosity, the flow behavior index and a dimensionless number derived from the solid surface zeta potential and maximum pore radius. The present results show that the EDL effects as well as other variable parameters may greatly influence the effective permeability of the power-law fluids in porous fibrous media: the larger the porosity, the higher the effective permeability; the larger the maximum pore radius, the higher the effective permeability; the higher the solid surface zeta potential, the more the EDL effects; the more the EDL effects, the lower the effective permeability. Comparing the effective permeability produced by different flow behavior indexes, we further illustrate that the EDL has virtually no effects when the flow behavior index is great than 1, moderate effects when equal to 1, and very significant effects when less than 1. Therefore the EDL effects may provide strong constraints on evaluation of the effective permeability of the shear thinning fluids rather than the shear thickening and Newtonian fluids. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Porous media has been broadly applied to heat exchangers, medical science, and other aspects [1,2]. One of the most important measures of characterizing the transport properties in porous media is the permeability. Analytical expressions for calculating permeability of the porous media are the goal of many researchers in the field of the flow through porous media [3,4]. For examples, a Fast Fourier Transform (FFT) based method was presented to compute the dynamic permeability of periodic porous media by Nguyen [5]. The Monte Carlo technique was proposed to develop a probability model for radial permeability in fractured porous media [6]. In fact, the determination of the permeability depends strongly on the porosity and pore radius as well as geometrical formation factors [7,8]. Due to the random distribution of the fibers in porous fibrous media, the pores in porous media are non-uniform in size; causing a great difference between the traditional models and the ⇑ Corresponding author at: School of Engineering, Sun Yat-sen University, Guangzhou 510275, China. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.127 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

real cases exists. Fortunately, many researchers have found that the pore size distribution in porous media can be described well by using the fractal geometry [9,10], promoting the studies of this aspect. Xu [11] used the fractal geometry theory to investigate the permeability of the fractal-like tree network by parallel and series models. Zhu [12] developed a fractal model to analyze the effects of porosity of porous fibrous media on heat and mass transfer. Miao [13] proposed a fractal model to analyze the permeability of the rocks with shear fractures which widely exist in nature such as oil/gas reservoirs. However, for the analysis of the pressure driven flow in porous fibrous media, the electrokinetic effects should be included. Considerable researches of the electroviscous effects (EDL effects) on microchannel flow have been reported in recent years. Zhao [14] presented a comprehensive review of electrokinetics flow of non-Newtonian fluids. Power-law non-Newtonian flow in microchannels combined with electroviscous effect was numerically simulated by Tang [15]. Vakili [16] also studied the electroosmotic flow of power-law fluids in rectangular microchannels by using the numerical method. Quite recently, Zhu [17] provided a fourth-order compact difference method to discuss the periodical

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Nomenclature A0 A DF d Ez z E e F 1 G 2 G Jc Js K k kB L m N Nt n1 n p z P Q q Rc Rf R0 R  R Rmin Rmax

cross-sectional flow area (m2) the total cross-sectional area of a unit cell (m2) fractal dimension Euclidean dimension electric field vector (V/m) dimensionless electric field the elementary charge (C) body force vector a dimensionless parameter a dimensionless parameter conducting current density (A/m2) streaming current density (A/m2) permeability tensor of the porous materials dimensionless electroosmotic radius Boltzmann constant length scale dynamic viscosity of power-law fluids (N m2 sn) number of pores or capillaries in material total number of pores or capillaries in material the bulk ionic concentration (m3) flow behavior index pressure in the capillary (N/m2) dimensionless pressure gradient total flow rate of liquid moisture through the material (m3/s) flow rate of liquid moisture thought a single microchannel (m3/s) mean pore radius (m) radius of fibers (m) capillary radius in material (m) radial coordinate of a material (m) dimensionless radial coordinate of a material minimum pore radius in material (m) maximum pore radius in material (m)

flow of power-law fluids through a rectangular microchannel with electroviscous effects. These results indicate that the EDL effect can be an important factor to assess the pressure driven flow in porous media [18–20]. Nowadays, the electrokinetic phenomena in porous media have widely been applied to the micropump, chemical engineering, medical science and other fields [21,22]. Above studies of fluid flow in porous media are focused on the Newtonian fluids. Actually, complex fluids (e.g. polymer solutions and colloids) with long-chain molecules exhibiting obvious non-Newtonian characteristics (e.g. changes in viscosity, memory effect, yield stress and hysteresis fluid properties) are also manipulated in medical and hygiene applications (e.g. blood). Many studies basically follow the works in this regard. Chen [23] used the lattice Poisson–Boltzmann method (LPBM) to explore the effects of the porous medium structure on electro-osmotic permeability of a power-law fluid. Turcio [24] applied the Bautista–Manero–P uig (BMP) model to analyze the effective permeability in fractal porous media. The power-law fluids were numerically simulated in 3-D fibrous structures to study the effects of fibers orientations on the permeability of fibrous media by Emami [25]. Presently, however, the analytical study related to the EDL effects on permeability of power-law fluids in porous fibrous media has not been reported. And, few analytical studies, which can be incorporated into the fractal model of pore distribution in naturally fractured porous media, have been carried out to investigate the power-law fluids flow in a cylindrical microcapillary with electrokinetic phenomena.

T U V V  v ðRÞ vz  v ðRÞ mean z

absolute temperature (K) volume-averaged superficial fluid velocity (m/s) flow velocity vector maximum velocity of capillary without consideration of EDL dimensionless velocity velocity of the fluid (m/s) mean velocity of the liquid (m/s) z-coordinate

Greek symbols h contact angle w local electrical potential (V) W dimensionless local electrical potential ww wall zeta potential (V) q mass density (kg m3) qe the net charge density per unit volume (C/m3) e the relative dielectric permittivity e0 the permittivity of vacuum (C V1 m1) j Debye–Huckel parameter (m1) v valence number of the ion f0 zeta potential (V) k electrical conductivity of the electrolyte solution (S/m) / porosity of the fibrous materials /i micro-porosity /eff effective porosity H dimensionless velocity variable defined in Eq. (28) c_ strain rate tensor c_ the magnitude of strain rate tensor c_ w shear rate for power-law fluids in a single capillary s stress tensor l the apparent viscosity for power-law fluids x a modified coefficient d ratio of Rmin to Rmax

In this paper, we experiment an analytical model, by using the fractal technique of pore distribution in porous fibrous media, to investigate the effective permeability of power-law fluids with EDL effects. We investigate the variations of the effective permeability with the porosity, solid surface zeta potential, and maximum pore radius. Meanwhile, we study the effects of EDL on the evaluation of the effective permeability. To show the validity of the model, the experimental data given by Kostornov [26] and Labrecque [27] are tested. In addition, the relationship between the EDL effect and the flow behavior index is also discussed.

Fig. 1. Schematic diagram of the capillaries. R0 denotes radius of the capillary.

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2. Mathematical model In our study, the porous fibrous media is regarded as a large number of yarns woven out of fibers. A yarn is made up of a cluster of twisted fibers. The capillaries, which are interconnected, are formed by these pores in yarns woven into the porous fibrous media. Fig. 1 shows a capillary in the porous fibrous media. Because of the low Reynolds number in the porous fibrous media, the steady flow of an electrolyte solution is considered through a microchannel that comprises a cluster of fibers.

In our study, we assume that the electrical potential (w) of the electrical double layer (EDL) is small enough, and the Debye–Huckel approximation can therefore be used to linearize the Poisson–Boltzmann equation (as Wu [18] did in their studies of electroosmotic flow). Thus, the electric potential of the EDL can be governed by Possion–Boltzmann equation

ð1Þ

where e is the relative dielectric permittivity and e0 is the permittivity of vacuum. The volumetric net charge density qe can be expressed with Debye–Huckel linearization approximation

qe ¼ j2 ee0 w;

ð2Þ 1=2

where j ¼ ð2v2 e2 n1 =ee0 kB TÞ is Debye–Huckel parameter, v is the valence number of the ion, e is the elementary charge, n1 is the bulk ionic concentration at the neutral condition, kB is the Boltzmann constant and T is the absolute temperature, respectively. The boundary conditions are

 dw ¼ 0; dR R¼0

wjR¼R0 ¼ ww :

ð3Þ

coshðjRÞ ; coshðjR0 Þ

 n   n  n nþ1 1 d  dv nþ1  1E  z Þ1 G z W 2   2   R ðP n n dR R dR ¼ 0:

ð8Þ

@V q þ V  rV @t



¼ rp þ r  s þ F;

 ¼ v ðRÞ

n þ 1  1n ðPz Þ n

where q is the mass density, V is the flow velocity vector, t is the time, P is the pressure, s ¼ 2lðc_ Þc_ is the stress tensor, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðc_ Þ ¼ m  ðc_ Þn1 is the viscosity of power-law fluids, c_ ¼ 12 ðc_ : c_ Þ is the magnitude of the strain rate tensor c_ , m is the fluid dynamic viscosity constant with dimension ½Nm2 sn , n is the flow behavior index, F is body force vector. n < 1 and n > 1 correspond to the case of shear thinning fluids and shear thickening fluids, respectively. The components of V satisfy v z ¼ v z ðRÞ and v R ¼ v h ¼ 0 in the cylindrical coordinate, hence the porous media with the microcapillaries can be considered as 1D problem. If assume that the electrolyte solution is a steady incompressible fluid, and the effect of gravity can be neglected, Eq. (6) can be written in the simplified form as

"

#  n1 dp 1 d dv z dv z þ qe Ez ¼ 0: mR   þ dz R dR dR dR

ð7Þ

Z

1



 R

   0 n  z ÞR  0  G1 Ez Ww sinhðkR Þ dR 0 : ðP k coshðkÞ 1

ð10Þ

In order to obtain the velocity distribution of power-law fluids, we use the following approximation [28] 0  E  G 1 z Ww sinhðkR Þ k coshðkÞ

0 ðPz ÞR

¼ 1:

ð11Þ

Combining with the method of Taylor series, the analytical solution of Eq. (10) can be obtained,

 ¼ v ðRÞ



 nþ1  1   nþ1  Fðn; kÞ  Fðn; kRÞ n 1R  2 ðPz Þ G 1 Ww Ez nþ1 n n k coshðkÞ

þ

 1  n2  2  2 2 2 Hðn; kÞ  Hðn; kRÞ ðPz Þ G1 Ww Ez : nþ1 2 2n3 n k cosh ðkÞ

ð12Þ

 The mean velocity v ðRÞ mean of power-law fluids can be expressed as:

Z

1

  v ðRÞd  R  R

0 n þ 1 n þ 1  1  z ¼2  2 ðPz Þ G1 Ww E 6n þ 2 n

ð4Þ

ð6Þ

ð9Þ

From Eq. (8), we can get the solution of velocity:

2

k Fðn; kÞ  2½F 1 ðn; kÞ  F 1 ðn; 0Þ 3nþ1 n

2k

coshðkÞ

k Hðn; kÞ  2½H1 ðn; kÞ  H1 ðn; 0Þ 3nþ1 n

2k

1  n2  2  2 2 2 ðP z Þ G1 Ww Ez 2n3 )

þ

2

ð5Þ



¼ 0:

R¼0

where ww is the wall zeta potential on the surfaces of fibers. The present work concerns electrokinetic force as a source term in the momentum equation. The modified Navier–Stokes equations can be expressed as

r  V ¼ 0;

 dv    dR

v jR¼1 ¼ 0; 

 v ðRÞ mean ¼ 2

The solution of Eq. (2) can be derived as:

w ¼ ww

the induced streaming potential, V denotes the maximum velocity of the capillary without consideration of EDL, f0 is a reference electrical potential, Eq. (7) can be rewritten as

The boundary conditions are

2.1. Electrokinetic flow of power-law fluids in a microcapillary

  1 d dw q ¼ e ; R R dR dR ee0

 ¼ R=R0 , k ¼ jR0 , W ¼ vew=kB T, If we define the normalized variables R  1 nþ1 2 v en f R E R dp v 1 n 1 dp n v ¼ Vz , Pz ¼ qV02 dz, Ez ¼ zf0 0 , G 1 ¼ qV 2 0 , V ¼ nþ1  2m R0n , where Ez is dz

¼ H;

2

ð13Þ

cosh ðkÞ

where the auxiliary functions Fðn; xÞ, F 1 ðn; xÞ, Hðn; xÞ and H1 ðn; xÞ are displayed in Appendix A and the dimensionless number H denotes   ðRÞ the mean velocity v mean . The fluid flow in the micro-channel causes an electric current, called the streaming current J s , in the direction of fluid flow. The streaming potential associated with the streaming current in turns generates an electric current, called the conducting current J c , to flow in the opposite direction to the pressure driven liquid flow. When the ions move in the liquid, they exert a force on the liquid molecules, thus generating a viscous effect, which is called the electroviscous effect. Specifically,

J s ¼ 2p

Z

R0

0

Rv ðRÞqe dR ¼ 4vzn1 V  pR20

 0 =R0  pR2 ; J c ¼ EðRÞkS ¼ kEf 0

Z

1

 v ðRÞ  WdR;  R

ð14Þ

0

ð15Þ

where k is the electric conductivity of the flow. In the steady flow we have

J c þ J s ¼ 0:

ð16Þ

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Substituting Eqs. (14) and (15) into Eq. (16), the streaming potential can be expressed as:

2  ¼G E

Z

1

  v ðRÞ  WdR;  R

ð17Þ

0

 2 ¼ 4vzn1 R0 V=ðkf0 Þ is a dimensionless parameter. where G Combining Eq. (17) with Eq. (12), the streaming potential satisfies the quadratic equation

2  ð1 þ bn ÞE z þ cn ¼ 0; an E z

bn ¼

nþ1 n

3nþ1 n

where the auxiliary functions F 2 ðn; xÞ, F 3 ðn; xÞ, F 4 ðn; xÞ and H2 ðn; xÞ are listed in Appendix A. Then we have

z ¼ E

n–1

:

n¼1

2an

cn 1þbn

:

In this section, we establish the governing equations in fractal porous fibrous media. For the fractal objects, the measure M(L) is related to the length scale L, through a scaling law in the form of MðLÞ  LDF [29], where DF is the fractal dimension of an object, M(L) can be the length of a line, the area of a surface, the volume of a cube, and the mass of an object, and L is the length scale. The number of pores is related to the pore radius through a scaling law in the form of [30]

ð20Þ

where Rmax is maximum pore radius, R0 is the pore radius. If 1 < DF < 2, it is in the two-dimensional space, and if 2 < DF < 3, it is in the three-dimensional space. Then, we can obtain that ðDF þ1Þ

F dN ¼ DF RDmax R0

dR0 :

ð21Þ

The total number of pores, from the minimum radius Rmin to the maximum radius Rmax , can be derived from Eq. (20):

Nt ðL P Rmin Þ ¼

Z

1

f ðR0 ÞdR0 ¼

 D Rmax F : Rmin

dN ðD þ1Þ F ¼ DF RDmin R0 F dR0 ¼ f ðR0 ÞdR0 ; Nt



Rmin Rmax

DF ð24Þ

:

Rmax

f ðR0 ÞdR0 ¼ 1:

ð25Þ

Eq. (25) shows that Eq. (24) holds if and only if [30]

DF

¼ 0:

ð26Þ

ln / ; ln d

d ¼ Rmin =Rmax ;

ð27Þ

where d is the Euclidean dimension, which can be 2 and 3, denoting the two- and three- dimension spaces, respectively. When DF = d, the spaces are totally occupied by pores, i.e. all the pores are merged into a large one to fill the space. In general, Rmin =Rmax < 0:01 in porous fibrous media and Eq. (26) holds approximately.  ¼ vVz , the On the other hand, from the normalized variable of v mean velocity v ðRÞmean of power-law fluids can be expressed as

 1 n 1 dp n nþ1  R0n  H; nþ1 2m dz

ð28Þ

 where H denotes the dimensionless mean velocity v ðRÞ mean (see Eq. (13)), describing the electric resistant effects on the velocity of power-law fluids in the pores with the radius of R0 . Then the flow rate through a single capillary with the radius of R0 in the vertical direction can be deduced as

   n  1 dp1n  3nþ1   qðR0 Þ ¼ x  jv ðRÞmean j  pR20 ¼ xp  HR0 n ;  n þ 1 2m dz

ð29Þ

where x, a modified coefficient, is the determined by the internal geometry and the structure of the porous fibrous media. For a simple porous fibrous media structure, x is determined mainly by the effective angle h (x ¼ sin h), the angle between the capillaries and the surface of porous fibrous media. Using Eqs. (24) and (29), the total flow rate in the porous fibrous media can be expressed as

Q¼

Z

Rmax

qðR0 Þ  f ðR0 Þ  Nt dR0 Rmin

  1   1  nþ3  i xp 2  lnln /d Rmax 1þ1þln / 1 dp n h  n n ln d : ¼  H 1  d   ln / 1  n þ 1 2m dz þ 1 þ ln d n

ð30Þ

If define the area of total pores as

ð22Þ A0 ðR0 Þ ¼

Z

Rmax

Rmin

ð23Þ

F where f ðR0 Þ ¼ DF RDmin R0 F , representing the probability density function of the pore distribution, which should be f ðR0 Þ P 0. The integration of Eq. (23) is

ðD þ1Þ

dR0 ¼ 1 

Rmin

1

If divide Eq. (21) by Eq. (22), then



Z

 v ðRÞmean ¼ V  v ðRÞ mean ¼

2.2. Effective permeability in fractal porous fibrous media

 D Rmax F ; R0

ðDF þ1Þ

ð19Þ

z in Eqs. (12) and If we replace the induced streaming potential E (13) by Eq. (19), the complete results of velocity can be obtained.

NðL P R0 Þ ¼

F DF RDmin R0

According to the probability theory, the probability density function should satisfy the following relationship:

DF ¼ d 

coshðkÞ

8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < ð1þbn Þ ð1þbn Þ2 4an cn

Rmax

Eq. (26) should be a criterion if the porous fibrous media can be characterized by fractal theory and technique. The unified relation between the fractal dimension DF and the porosity of the porous fibrous media / has been given by [31]

½F 3 ðn; kÞ  F 3 ðn; 0Þ  ½F 4 ðn; kÞ  F 4 ðn; 0Þ k

Z

Rmin

1

Rmin Rmax

nþ1   z Þ1 W2 G1 G2 ðP w n2 Fðn; kÞ½F 3 ðn; kÞ  F 3 ðn; 0Þ  ½F 2 ðn; kÞ  F 2 ðn; 0Þ  3nþ1 2 k n cosh ðkÞ k

f ðR0 ÞdR0 ¼



1  n2  2   z Þ2 W3 G G2 ðP w 2n3 1 Hðn; kÞ½F 3 ðn; kÞ  F 3 ðn; 0Þ  ½H2 ðn; kÞ  H2 ðn; 0Þ  3nþ1 3 k n cosh ðkÞ

 2 Ww cn ¼ G

1

ð18Þ

where

an ¼

Z





pR20 pR2max ð1  /Þ 2 ln d f ðR0 ÞNt dR0 ¼  1 ; ð31Þ ln / x x

the total cross sectional area of a unit cell perpendicular to the flow direction can be

A¼

  A0 ðR0 Þ pR2max ð1  /Þ 2 ln d ¼ 1 : / ln / x/

Thus, the superficial velocity can be obtained by

ð32Þ

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U¼

Q A

   n  1 dp1n h i ln / 1     H 1  dnþ1þ ln d : / n þ 1  2m dz ð1  /Þ ln d 1n þ 1 þ ln ln d 1

¼

nþ1 x2 /Rmax ln /

ð33Þ In terms of the reference [32], the shear rate for power-law fluids in a single capillary is

 1 1 dp n 1n c_ w ¼  R0 ; 2m dz

ð34Þ

where c_ w is the shear rate, n is the flow behavior index. The total shear rate in the porous fibrous media is therefore expressed as

c_ ¼ 

Z

Rmax

c_ w dNðR0 Þ

Rmin

"  1  1D # 1 n 1 dp n nRmax Rmin n F : DF 1 ¼  2m dz 1  nDF Rmax

ð35Þ

Then the apparent viscosity for power-law fluids can be defined as

l ¼ mc_ n1 ( " 1  1D #)n1 1 1 dp n nRnmax Rmin n F  DF 1 : 2m dz 1  nDF Rmax

¼m

Eq. (40) indicates that the effective permeability of the porous fibrous media with EDL effects is a function of the porosity /, the dimensionless number H, and the flow behavior index n. According to the definitions of Eqs. (12) and (28), the dimensionless number H is determined by the solid surface zeta potential and maximum pore radius.

3. Results and discussion To get the velocity of power-law fluids in a cylindrical microcapillary, we used an approximate scheme to calculate the induced streaming potential by solving the P–B equation and modified momentum equation. The dimensionless velocity with consideration of EDL is then expressed as a dimensionless electric resistance number. This number is consisted of the invariants, such as the liquid dielectric constant, liquid dynamic viscosity, electric conductivity, etc, and the variable parameters, such as the porosity, solid surface zeta potential, and square of the maximum pore radius. In following examples, we will discuss the influences of such variable parameters on the pressure driven flow of power-law fluids in porous fibrous media. Fig. 2 presents the variations of the dimensionless velocity with the dimensionless radius the flow behavior indexes of 0.9, 1.0, and 1.1. Detailed parameters used in the calculations are as follows: relative dielectric permittivity e ¼ 80, permittivity of vacuum

ð36Þ

e0 ¼ 8:85  1012 F=m, absolute temperature T ¼ 293 K, ratio of valence number of the ion vþ : v ¼ 1 : 1, elementary charge

According to the Debye–Huckel linear approximation, with the low electrical potential of w, the apparent viscosity without EDL effects can also be used for the case with EDL effects [17,28]. Based on the Darcy’s law, the expression for the effective permeability of the porous fibrous media with EDL is obtained as follow

e ¼ 1:6  1019 C, Boltzmann constant kB ¼ 1:38  1023 J=K, fluid

K¼ ¼

lU Pz

x2 R2max / ln /

   ln / ln / n1 1 1  1  dnþ1þ ln d 1  dn2þ ln d

2ð1  /Þ ln d    n1  n  n ln d nð2 ln d  ln /Þ   H : nþ1 ln d þ n ln d þ n ln / ln d  nð2 ln d  ln /Þ ð37Þ

Moreover, the mean pore radius Rc can be calculated by

Rc ¼

Z

Rmax

R0 f ðR0 ÞdR0 ¼

Rmin

¼

Z

Rmax

Rmin

F F DF RDmin RD 0 dR0

  ln d R 1  /ln /1 : ln / min

/ 2  ln ln d

ð38Þ

1  ln d

If the mean pore radius is approximated by the function of porosity and fiber radius Rf [33]

Rc ¼ 

p0:5 2



1þ

 Rf ; 2 ln /

p

ep=2 6 / 6 1:

ð39Þ

Substituting Eqs. (38) and (39) into Eq. (37), the dimensionless effective permeability is written as:

K R2f

¼

x2 p/ ln /

" / 2  ln ln d

ln d 1 ln /

1/

#  2  1þ

p

2

/ 2 ln / 8ð1  /Þd2 ln d 1  ln ln d    n1  n  n ln d nð2 ln d  ln /Þ   H nþ1 ln d þ n ln d þ n ln / ln d  nð2 ln d  ln /Þ    ln / n1 1þ1þln / 1  1  dn ln d 1  dn2þ ln d :

ð40Þ

density

q ¼ 103 kg=m3 ,

fluid

dynamic

viscosity

constant

m ¼ 9  104 Nm2 sn , pore radius R0 ¼ 10  106 m, wall zeta potential ww ¼ 70 mV , pressure difference dp=dz ¼ 20 kPa, electric conductivity of the flow k ¼ 1:2639  107 S=m, bulk ionic concentration n1 ¼ 6:022  1020 =m3 , constant d ¼ 0:01. Comparing the dimensionless velocities produced with and without EDL effects, it is clear that both curves have yielded the same evolution. The velocity at centreline is about 1, and then begins to decrease with increase of the dimensionless radius, and  = 1 (at the surface of the fibers). More approaches zero when, R detailed comparison between the results obtained with and without EDL effects for the different flow behavior indexes, however, does show some noticeable differences. They are: (1) Although the variation of velocity produced with and without EDL effects show the same evolution process, the one produced without EDL looks more prominent than the others produced with EDL. The corresponding values at centreline are 1.0 vs 0.91, 1.0 vs 0.95, and 1.0 vs 0.99 for the flow behavior indexes of 0.9, 1.0, and 1.1 (Fig. 2a–c). Such contrasts make the EDL effects stand out more clearly in the entire process. (2) The shear thinning fluids (n < 1.0) are more sensitive to the resistant effects of EDL than the shear thickening (n > 1.0) and the Newtonian (n = 1.0) fluids. All the velocities produced with EDL are less than that produced without EDL. However, the differences are different. For example, the differences of the centreline velocity for the flow behavior indexes of 0.9, 1.0, and 1.1 are 9%, 5%, and 1%. Thereby, the flow behavior index n may play an important role in the calculation. In Fig. 3, the effective permeability, both with and without EDL, is plotted as a function of porosity. The dimensionless effective permeability of power-law fluids is obtained by using the fractal technique of pore distribution in porous fibrous media. Note that

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power-law fluids with the porosity: the larger the porosity, the higher the effective permeability (Fig. 4). As shown in Fig. 4, the fact that the permeability of the fluids without EDL is obviously higher than the permeability of the fluids with EDL further suggest the effects of EDL on the evaluation of the permeability. Moreover, we explore the change of the permeability associated with the flow behavior index. Once again, the EDL

Fig. 2. Distribution of the dimensionless velocity with the dimensionless radius of R. (a) n = 0.9; (b) n = 1.0; (c) n = 1.1. The solid and dashed lines denote respectively the velocities produced with and without EDL effects.

the dimensionless effective permeability of Newtonian fluids, when n = 1.0, as a function of porosity for fibrous media without EDL effects can be supported by previous studies of Zhu [34]. The analytical expressions have been developed as a function of porosity, dimensionless local averaging net charge density and dimensionless electric resistance number. The results can also be supported by the experimental data retrieved by Kostornov and Shevchuk [26], Labrecque [27], and Wheat [35], which illustrated the variation of the dimensionless effective permeability of

Fig. 3. Dimensionless effective permeability as a function of porosity. (a) n = 0.9; (b) n = 1.0; (c) n = 1.1. All the used parameters are the same as for Fig. 2. The solid and dashed lines represent the effective permeability derived with and without EDL effects.

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261

Fig. 4. Dimensionless effective permeability of Newtonian fluids as a function of porosity without EDL effects derived by different researchers. All the used parameters are the same as for Fig. 2. The inset indicates sources of the curves.

exhibits more significant effects on permeability of the fluids with the smaller behavior index. If we merge the evolutions of the dimensionless effective permeability with EDL for different flow behavior indexes into a figure (Fig. 5), we find that the change of the flow behavior index from 1.1 to 0.9 can almost decrease one order of magnitude of the effective permeability. Finally, we show the relationship between the dimensionless effective permeability of power-law fluids and the zeta potential for two different maximum radius and two porosities in Fig. 6, and some interesting results are listed as follows: (1) The maximum radius as well as the porosity greatly influences the dimensionless effective permeability for power-law fluids: the larger the porosity, the higher the effective permeability; the larger the maximum pore radius, the higher the effective permeability. Here, the maximum rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1/i radius Rmax is given by Ref. [36], Rmax ¼ R20 2 1/ 1 ,

Fig. 6. Relationship between dimensionless effective permeability of power-law fluids and the zeta potential. All the parameters are the same as for Fig. 2 except some specific parameters used: (a) / ¼ 0:5; (b) / ¼ 0:8; Rmax = 5 and 50 lm.

eff

where /i and /eff are respectively the micro and effective porosity, with relationship of /i ¼ 0:342/eff [37]. (2) The larger the absolute value of the zeta potential is, the larger the effects of EDL on the power-law fluids flows are, and the lower the effective permeability is. Significant changes of

Fig. 5. Dimensionless effective permeability as a function of porosity with EDL effects for the flow behavior indexes of 0.9, 1.0, and 1.1.

the dimensionless effective permeability associated with the zeta potential are found when Rmax ¼ 5 lm, and are not found when Rmax ¼ 50 lm. The selection of the Rmax value has been used by studies of Zhu [34], which illustrates that these values are suitable for this calculation. (3) Similar to our previous discussions, the effective permeability of shear thinning fluids is more sensitive to the EDL effects than that of Newtonian and shear thickening fluids. This conclusion can easily be derived from the remarkable differences between the dimensionless effective permeability derived with different flow behavior indexes. Most solid surfaces carry electrostatic charges, i.e., an electrical surface potential. The solid surface interacting with electrolyte is charged being responsible for the rearrangement of ions in the neighborhood of the charged surface where the so-called electric double layer (EDL) can be formed as a result. When a liquid is forced by pressure gradient, the migration of counter-ions in EDL along with the opposite direction of the flow creates a streaming potential which produces an electrical resistance hindering the pressure-driven flow in return. The results we show above are possibly such examples, whose high effects of EDL on the permeability of power-law fluids are detected by the significant differences between the curves produced with and without EDL (Figs. 2–6). On the other hand, for the fluids with smaller flow behavior index,

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such as the shear thinning fluids of n = 0.9 in Figs. 2–6, higher streaming potential would tend to be induced. Correspondingly, the resistant effect on the pressure-driven flow is more, and the effective permeability is lower. Knowing this unique feature of the EDL effects, we might use that for tuning the permeability of various kinds of non-Newtonian fluids in the practices of petroleum engineering, soil, groundwater and other aspects. 4. Conclusions We devise an analytical model to solve the effective permeability in porous fibers with consideration of EDL effects. Although more data testing is needed to further verify the model, it is ready to be employed in the real world for quantitative evaluation of the effective permeability of power-law fluids in the porous fibrous media with EDL effects, should the porosity, solid surface zeta potential, and square of the maximum pore radius be detected. The technique is general and can be applied to both the Newtonian and non-Newtonian fluids. Through our application examples, we know that the resistant effects of EDL have great influence on the on the evaluation of the effective permeability. More importantly, we find that the resistant effects of EDL depend strongly on the flow behavior index. If all the invariants and variable parameters are known, the approach presented in this paper allows us to systematically assess the effective permeability of power-law fluids in the porous fibrous media with EDL effects. And, even if such a priori information is not available, one could still turn down the effective permeability by using the shear thinning fluids with the smaller flow behavior indexes.

Z x x 1 1 e þ e dx x2þn coshðxÞdx ¼ x2þn 2      1 1 1 1 C 3 þ ; x þ ð1Þ3n C 3 þ ; x ¼ 2 n n

F 4 ðn; xÞ ¼

Z

Z

1

2

x2þn sinh ðxÞdx "    # 1 1 nx1þn 1 1 1=n ¼  ð2Þ C 1 þ ; 2x þ 21=n C 1 þ ; 2x 2 1 þ n n n

Hðn; xÞ ¼

H1 ðn;xÞ ¼ ¼

Z

Hðn;xÞxdx

1 n2 1þ1n x 2 n2  1      1 1 1 1=n 2 21=n  ð2Þ x C 1 þ ;2x  ð2Þ C 1 þ ; 2x 4 n n      1 1=n 2 1 1 21=n þ 2 x C 1 þ ;2x  2 C 1 þ ;2x 4 n n

Z

Hðn;xÞx coshðxÞdx ¼ Hðn; xÞxsinhðxÞ  Hðn; xÞcoshðxÞ        1 1 1 1 1 1  3n C ; 3x  3C ; x þ ð1Þn 3C ; x 8 n n n       1 1 1 1n 11n ð3Þ C ; 3x  3 C 1 þ ; 3x þ C 1 þ ; x n n n     1 1 11n 11n þð1Þ C 1 þ ;x  ð3Þ C 1 þ ; 3x n n

H2 ðn;xÞ ¼

Conflict of interest References None declared. Acknowledgement We would like to acknowledge the Grant support from the Chinese NSFC (No. 91230114). Appendix A In the present analysis, the incomplete Gamma function Cða; nÞ involved in the following auxiliary functions can be efficiently solved by Matlab.

Fðn; xÞ ¼

Z

1

x1þn sinhðxÞdx ¼

     1 1 1 C ; x  ð1Þ1=n C ; x 2 n n

     1 2 1 1 x C ;x  C 2 þ ;x Fðn; xÞxdx ¼ 4 n n     1 1 1=n 2 21=n x C ; x þ ð1Þ C 2 þ ; x ð1Þ n n

F 1 ðn; xÞ ¼

Z

Z

Fðn; xÞx coshðxÞdx ¼ Fðn; xÞx sinhðxÞ  Fðn; xÞ coshðxÞ      1 1 1 1 1  21n C 1 þ ; 2x þ ð2Þ1n C 1 þ ; 2x 4 n n     2n 1þ1n 1 1 1n 1n x  2 C ; 2x þ ð2Þ C ; 2x  nþ1 n n

F 2 ðn; xÞ ¼

F 3 ðn; xÞ ¼

Z

x coshðxÞdx ¼

Z

¼ x sinhðxÞ  coshðxÞ

xd sinhðxÞ ¼ x sinhðxÞ 

Z

sinhðxÞdx

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