Transp Porous Med (2008) 71:331–343 DOI 10.1007/s11242-007-9129-0
A resistance model for flow through porous media Jinsui Wu · Boming Yu · Meijuan Yun
Received: 12 July 2006 / Accepted: 27 March 2007 / Published online: 8 June 2007 © Springer Science+Business Media B.V. 2007
Abstract A new model for resistance of flow through granular porous media is developed based on the average hydraulic radius model and the contracting–expanding channel model. This model is expressed as a function of tortuosity, porosity, ratio of pore diameter to throat diameter, diameter of particles, and fluid properties. The two empirical constants, 150 and 1.75, in the Ergun equation are replaced by two expressions, which are explicitly related to the pore geometry. Every parameter in the proposed model has clear physical meaning. The proposed model is shown to be more fundamental and reasonable than the Ergum equation. The model predictions are in good agreement with the existing experimental data. Keywords
Laminar flow · Porous media · Flow resistance
1 Introduction The widely applied resistance model for flow through porous media (the term “porous media” refers to granular porous media in this work) was proposed by Ergun (1952). This model is called the Ergun equation, which is often used to analyze the resistance/pressure drop for flow through porous media (Vafai and Tien 1981; Demirel and Al-Ali 1997; Demirel and Kahraman 2000; Alazmi and Vafai 2001; Khaled and Vafai 2003; Jiang et al. 2004). The Ergun equation is expressed as P 1 − ε ρvs2 150µ(1 − ε)2 vs + 1.75 3 , = 2 3 L ε Dp DP ε
(1)
J. Wu · B. Yu (B) · M. Yun Department of Physics, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, P. R. China e-mail:
[email protected] J. Wu Department of Physics, North China Institute of Science and Technology, 206 Yanjiao 101601, Langfang, P. R. China
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where P is the pressure, L is the length along the macroscopic pressure gradient in porous media, vs is superficial (volume averaged) velocity (defined by vs = Q/A, where Q is the total flow rate through a cross-section of area A), µ is dynamic viscosity of fluid, ε is porosity of porous media, and D p is average diameter of particles, ρ is density of fluid. Equation 1 is based on the average hydraulic radius (Bird et al. 1960; Sheffield and Metzner 1976), see Appendix A. The first-term on the right-hand side of Eq. 1 is called the Blake–Kozeny 150µ(1−ε)2 vs equation, i.e., P , which represents the viscous energy loss primarily in L = D 2 ε3 P
laminar flow, i.e., when the modified Reynolds number (Re p = (D p ρv/µ)(1 − ε)−1 ) is less than 10 (Bird et al. 1960), the second-term on the right-hand side of Eq. 1 can be neglected. If the first-term on the right-hand side of Eq. 1 is neglected, Eq. 1 is reduced to P L = ρv 2
s 1.75 1−ε , which is called the Burke–Plummer equation, which denotes the kinetic energy ε3 D p loss primarily in turbulent flow, i.e., when the modified Reynolds number is higher than 100 (Bird et al. 1960). The Ergun equation was examined from the point of view of its dependence upon flow rate, properties of fluids, porosity, orientation, size, shape, and particle’s surface. According to Ergun, the orientation of the randomly packed beds is not susceptible to exact mathematical formulation (Ergun 1952), thus in Ergun’s work, effect of orientation was not included. This work focuses on granular porous media, and effects of orientation will not be included in the present analysis. It has been found that the pressure loss as shown by Eq. 1 is obtained by adding the viscous and kinetic energy losses. The two constants 150 and 1.75 in Eq. 1 were obtained by fitting experimental data, and it was shown that using ε 3.6 instead of ε 3 in the Ergun equation would result in a better fit to the experimental data (Macdonald et al. 1979). The Ergun equation has been hotly debated in the area of porous media in the past decades. Hicks (1970) thought that the two coefficients (150 and 1.75) in the Ergun equation are not constants and are a function of Reynolds number. Bradshaw and Myers (1963) found that their measured values for the pressure drop for the bed of Celite cylindrical packing are the half of those calculated by the Ergun’s equation. Handly and Heggs (1968) also found that the Ergun equation is not applicable to predict the pressure drop across irregular beds packed with spheres, cylinders, rings, or planes. MacDonald et al. (1979) obtained that the two coefficients are 180 and 1.8, respectively, for the bed packed with quadrate particles. Nemec and Levec (2005) concluded that the Ergun equation is mainly applicable for spherical particles in the porosity range of 0.35–0.55 (Nemec and Levec 2005; Endo et al. 2002; Hill et al. 2001). Du Plessis (1994) solved the Navier–Stokes equation for flow in the pore space and proposed a model for flow resistance with one adjustable parameter. The solution was very tedious. Du Plessis thought that the porosity of packed beds usually lies in the range from 0.35 to 0.5 for the quadrate particles, and the first coefficient increases from 185 to 250 with porosity increasing, the second coefficient decreases from 2.25 to 1.5 with porosity increasing. According to Du Plessis, the two coefficients are 207 and 1.88 when porosity is 0.44 with the quadrate particles (Du Plessis 1994). It is evident that the Ergun equation is a semi-empirical equation and has two empirical constants 150 and 1.75, which have no physical meaning. The two empirical constants may have different values at different porosities, and the mechanisms behind these constants are still not clear until now. The experiments (Zhang and Yue 2004) showed that the changing trend of resistance of fluid flowing through the contracting–expanding paths/channels accords with the seepage resistance in porous packed beds. Therefore, the pores and capillaries in granular porous media can also be considered as a series of contracting–expanding paths/channels in this work. Researchers usually ignored the irregularity of shapes of the interspaces and assumed the
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interspaces to be circular pipes when using the contracting–expanding path/channel model (Proefschrift and Denys 2003). Niven (2002) analyzed the Ergun equation in terms of a model of pore conduits consisting of alternating expanding and contracting sections and obtained a model for the pressure drop in granular materials. However, his model contains as many as eight parameters to be determined, except fluid properties and fluid velocity. This work attempts to derive a new model for resistance of flow through granular porous media such as sandstones, soil, and packed beds. In these media, pores and particles are assumed to be uniformly distributed, and the medium is isotropic and homogeneous. Therefore, we can use an average-sized cell as a unit cell, which contains only one particle and the influence of microstructural differences of porous media can thus be neglected. In this work, we combine the average hydraulic radius model and the contracting– expanding path/channel model, and assume that the capillaries in porous media are tortuous and have a series of contracting and expanding sections, and then derive a new resistance model for flow through porous media.
2 The new resistance model 2.1 The viscous energy loss According to the superposition principle, the total energy losses can be obtained by adding the viscous energy loss along the flow path and the kinetic energy loss. For the viscous energy loss along the flow path, we assume the one-dimensional flow under the one-dimensional macroscopic pressure drop across a porous medium. The effect of transverse flow is indirectly considered by introducing the tortuosity. The porous media studied in this work are assumed to be comprised of a bundle of tortuous capillaries. According to Du Plessis (1994), at very slow flow, when conditions resemble the classical Darcy flow, Hagen–Poiseuille can be assumed in each pore/capillary. The fluid is assumed to consist of a single fluid phase with constant properties. For laminar flow through a tortuous capillary (neglecting the effect of contracting–expanding because the flow is assumed to be slow), the flow rate is governed by modifying the well-known Hagen–Poiseuille equation (Denn 1980) q=
π P d 4 , 128 L t µ
(2)
where d is the diameter of a capillary, L t is the real length of a tortuous capillary and is related to tortuosity τ by Bear (1972) and Dullien (1979) τ = L t /L ,
(3)
where the tortuosity τ is approximated by Yu and Li (2004) ⎡ τ=
√ √ 1⎢ ⎢1 + 1 1 − ε + 1 − ε ⎣ 2 2
2 −1 + √ 1− 1−ε
√1 1−ε
⎤ 1 4
⎥ ⎥, ⎦
(4)
which is a function of porosity and is based on the assumption that some particles in a porous medium are unrestrictedly overlapped and the others are not. The cross-sectional area of a
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capillary is A = πd 2 /4.
(5)
The average velocity in the pore/capillary (i.e., pore velocity) can be obtained from Eqs. 2 to 5 as v¯ =
q d2 = P1 . A 32µLτ
(6)
From Eq. 6, we can get that P1 8µτ v¯ 32µτ v¯ = , = L d2 r2
(7)
where d = 2r = 4Rh is the diameter of a capillary. Using the “average hydraulic radius” and inserting r = 2Rh (Bird et al. 1960) into Eq. 7 yield 2µτ v¯ P1 = , L Rh2
(8)
where Rh is average hydraulic radius, which is expressed as (Bird et al. 1960; Sheffield and Metzner 1976) (see Appendix A) Rh =
ε DP . 6(1 − ε)
(9)
Inserting (9) into (8) yields 72µτ (1 − ε)2 v¯ P1 = . L D 2p ε 2
(10)
Due to the relation between the average (pore) velocity and the superficial velocity v¯ = vs /ε (Ergun 1952; Bear 1972), inserting v¯ = vs /ε into Eq. 10 results in 72µτ (1 − ε)2 vs P1 = . L D 2p ε 3
(11)
Equation 11 depicts the pressure drop caused by the viscous energy loss along the flow path. It is seen that this equation is similar to the Blake–Kozeny equation, which represents the viscous energy loss. Note that the constant 150 in the Blake–Kozeny equation has no physical meaning and is only an empirical constant and is independent of porosity. Comparing Eq. 11 to the Blake–Kozeny equation, we can find that the constant 150 in the Blake–Kozeny equation is now replaced by 72τ , in which τ is related to porosity. Therefore, Eq. 11 might be more fundamental and reasonable than the Blake–Kozeny equation. Let P1 = avs , L
(12)
where the coefficient a is related to porosity ε, diameter D p of particles, tortuosity τ and fluid property µ, and is expressed by a=
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72µτ (1 − ε)2 . D 2p ε 3
(13)
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Fig. 1 A simple model for porous media
2.2 The kinetic energy loss According to Du Plessis (1994), the Burke–Plummer equation represents an empirical quantification of the Forchheimer effect and is derived in a physically realistic manner. The modeling for high Reynolds number cases is based on the inclusion of local flow recirculation within pores. This is related to the well-known shape drag for flow around an obstacle. In this work, the modeling for high Reynolds numbers is based on flow through pores/capillaries with a series of contracting and expanding sections, also called pore–throat model. In high Reynolds number cases, the effect of contracting and expanding sections on flow resistance is taken into account. It should be noted that the tortuosity and the contracting–expanding are different. A tortuosity capillary means that the capillary is tortuous, and both a tortuous and a straight capillary may be comprised of a series of contracting and expanding cross sections. Figure 1 displays an idealized model for porous media, in which D p is the average diameter of particles, L B F represents the average interspatial distance. The ratio of pore diameter L +D to throat diameter is defined by β = BLFB F p (see Fig. 1), where L B F and L B F + D p are the diameters of throat and pore, respectively. The average length of a pore–throat is approximated to be the average pore diameter d, and the total number of pore–throats is L t /d along the tortuous pathway length L t . Figure 2 shows a schematic structure of a pore–throat, which is the inter-space of the dashed square in Fig.1. The flow is assumed to be one-dimensional from the left to the right in Fig. 2. A1 , A2 , and A3 are the cross-sectional areas. The dashed square in Fig. 1 represents a unit cell. The area of the unit cell is approximated as
A=
D 2p 1−ε
.
(14)
Since porous media discussed in this work are assumed to be isotropic and homogeneous, Eq. 14 is approximately valid. The diameter of a pore is L BF + Dp =
√
A = Dp
1 . 1−ε
(15)
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Fig. 2 Schematic of the pore–throat model
The diameter of a throat is
L BF = Dp
1 −1 . 1−ε
The ratio β of the pore diameter to the throat diameter can be found from √ L BF + Dp A2 A2 = = = 1/(1 − 1 − ε). β= A1 A3 L BF
(16)
(17)
Now, we consider the kinetic energy loss. The kinetic loss of head is expressed by Zhang (1999) h f2 =
v¯ 2 ξ , 2g
where the coefficient for a suddenly expanding pipe is expressed as Zhang (1999) A1 2 ξe = 1 − . A2 So, the kinetic loss of head caused by the suddenly expanding pipe is v¯ 2 A1 2 v¯e2 h fe = 1 − = ξe e , A2 2g 2g
(18)
(19)
(20)
where v¯e is the average velocity through plane A1 (see Fig. 2). For a suddenly contracting pipe, the coefficient is Zhang (1999) A3 . (21) ξc = 0.5 1 − A2 So, the kinetic loss of head caused by the suddenly contracting pipe is v¯ 2 A3 v¯c2 h f c = 0.5 1 − = ξc c , A2 2g 2g
(22)
where v¯c is the average velocity through plane A3 (see Fig. 2). Due to A1 = A3 , see Fig. 1, v¯e = v¯c = v¯ holds for the average velocity through the cross-sections A1 and A3 . Combining Eqs. 20 and 22, we get
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v¯ 2 h f 2 = h fe + h fc = (ξe + ξc ) 2g 2 v¯ 1 5 3 + . − 2 = 2 β4 2β 2g
(23)
The pressure drop across the length of a pore–throat can be obtained from Eq. 23 as p2 = ρgh f 2 =
3 5 1 − 2 + 2 β4 2β
ρ v¯ 2 . 2
(24)
Equation 24 represents the pressure drop across one pore–throat (see Fig. 1), whose length is approximated to be the average pore diameter. Because porous media discussed in this work is assumed to be isotropy, the average pore diameter is approximated to be the average diameter d of capillaries. Thus, the pressure drop across the length of one pore–throat is approximated by 2 2 ρ v¯ L t ρ v¯ τ 3 1 5 3 1 5 p2 Ldt 2 + β 4 − 2β 2 2 d 2 + β 4 − 2β 2 2 P2 = = = . (25) L L L d Since we have assumed that d is related to the average hydraulic radius by d = 2r = 4Rh , Eq. 25 becomes 2 ρ v¯ τ 3 1 5 3τ (1 − ε)ρ v¯ 2 23 + β14 − 2β5 2 2 + β 4 − 2β 2 2 P2 = . (26) = L 4Rh 4ε D p Due to v¯ = vs /ε (Ergun 1952; Bear 1972), Eq. 26 is rewritten as 3τ (1 − ε)ρvs2 ( 23 + P2 = L 4ε 3 D p
1 β4
−
5 ) 2β 2
.
(27)
Equation 27 denotes that the kinetic energy loss, Eq. 27 also shows that the pressure drop decreases with increasing β, this is in accord with the experiment observation (Du Plessis 1994). In Eq. 27, β is determined by Eq. 17, which is related to porosity. Equation 27 is similar to the Burke–Plummer equation, which is the second-term on the right-hand side of Eq. 1. However, the Burke–Plummer equation contains an empirical constant 1.75, whereas there is no empirical constant in Eq. 27, in which every parameter has clear physical meaning. Equation 27 also indicates that the pressure drop is proportional to ρvs2 and tortuosity τ . This is consistent with the physical situation. Therefore, Eq. 27 might be more fundamental and reasonable than the Burke–Plummer equation. Let P2 = bρvs2 , L
(28)
where the coefficient b is related to porosity ε, ratio β of the pore diameter to the throat diameter, diameter D p of particles and tortuosity τ , and is expressed by b=
3τ (1 − ε)
3 1 2 + β4 4ε 3 D P
−
5 2β 2
.
(29)
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Fig. 3 A comparison on the dimensionless flow resistance vs. Re p at Re p < 10 and ε = 0.4
1400 1200 1000
G
800
Eq. (31)
600 Eq. (33)
400 200 0 2
0
4
6 Rep
8
10
The total pressure drop is the sum of the viscous energy loss and kinetic energy losses along the flow paths, i.e. P P1 P2 = + L L L
(30)
3τ (1 − ε)ρvs2 23 + 72µτ (1 − ε)2 vs + = 4ε 3 D p D 2P ε 3
−
1 β4
5 2β 2
.
Equation 30 describes the pressure drop when fluid flows through granular porous media. Equations 11, 27, and 30 can, respectively, be written as the dimensionless forms: G1 =
P1 ρ D 3p ε 3 = 72τ Re p , L µ2 (1 − ε)3
G2 =
P2 ρ D 3p ε 3 L µ2 (1 − ε)3
=
3τ
3 2
+
−
5 2β 2
4
3τ P ρ D 3p ε 3 = 72τ Re p + G= 2 3 L µ (1 − ε)
1 β4
(31)
3 2
+
1 β4
4
−
Re2p , 5 2β 2
(32)
Re2p ,
(33)
where Re p = (D p ρvs /µ)(1 − ε)−1 . Figure 3 compares the dimensionless flow resistances predicted by Eqs. 31 and 33, respectively, vs. the modified Reynolds numbers as Re p < 10. The results show that when the modified Reynolds numbers Re p < 7.8, the difference between the predicted values by the two equations is less than 10%. This means that the pressure drop is mainly determined by the viscous energy loss Eq. 31, and Eq. 31 can be a good approximation to the flow resistance at low Reynolds numbers. Figure 4 compares the dimensionless flow resistances predicted by Eqs. 32 and 33, respectively, vs. the modified Reynolds numbers as Re p > 400. The results show that when the modified Reynolds numbers Re p > 620, the difference between the predicted values by the two equations is less than 10%. This means that at high Reynolds numbers the flow resistance is dominated by the kinetic/local energy loss Eq. 32. However, Eq. 30 is the general form for resistance of flow through porous media. From Eq. 30 it can be seen that the pressure drop for flow at low speed (or at low Reynolds numbers)
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A resistance model for flow through porous media Fig. 4 A comparison on the dimensionless flow resistance vs. Re p at Re p > 400 and ε = 0.4
339
6
4x10
6
G
3x10
Eq. (32)
6
2x10
6
1x10
0 400
Eq. (33)
600
800
1000
1200
1400
1600
3.5
4.0
Rep
Fig. 5 The coefficient a vs. (1 − ε)2 /ε 3
14
Present model Eq. (13) Experiment data
Cofficient a
12 10 8 6 4 1.0
1.5
2.0
2.5 2
1−ε) /ε
3.0
3
)
is mainly determined by the viscous energy loss, which is represented by the first-term on the right-hand side of Eq. 30. In this case, the irregularity of porous shape can be ignored. At high speed (or at high Reynolds numbers) the irregularity of porous shapes has the significant influence on the flow resistance, and the kinetic/local energy loss dominates the pressure drop, which is mainly determined by the second-term on the right-hand side of Eq. 30. Equation 30 is similar to the Ergun equation, which has two empirical constants 150 and 1.75. Whereas, every parameter has the clear physical meaning in Eq. 30.
3 Comparisons and discussions To verify the proposed model, Fig. 5 compares the present model predictions by Eq. 13 with the available experimental data (Ergun 1952) about the coefficient a vs. (1 − ε)2 /ε 3 . It is seen from Fig. 5 that good agreement is found between them. Figure 6 compares the present model predictions by Eq. 29 with the available experimental data (Ergun 1952) about the coefficient b vs. (1 − ε)/ε 3 , and good agreement is again found between them. Figure 7 compares the present model predictions by Eq. 30 with those by the Ergun equation Eq. 1 and experimental data (Wang and Ding 2004), which were measured for beds packed with spherical particles of D p = 10 mm at ε = 0.42. In Eq. 30, the two coefficients
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Fig. 6 The coefficient b vs. (1 − ε)/ε 3
200 Present model Eq. (29)
C o f f ic i en t b
180
Experiment data
160 140 120 100 80 4
3
(1−ε)/ε
3
1000 Ergun equation (1) Equation (30) Experiment data
800 Ppressure drop
Fig. 7 Pressure drop of pure air flow through a bed packed with glass balls of D p = 10 mm at ε = 0.42
7
6
5
600 400 200 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Velocity
3 1 5 72τ = 117.5 and 3τ 4 2 + β 4 − 2β 2 = 1.67 are used in calculations at ε = 0.42. Figure 7 shows that good agreement is obtained among the present model predictions, experimental data and the Ergun equation Eq. 1. Equation 30 can also be expressed in terms of the dimensionless form: D 2p ε 3 P 3τ fv = = 72τ + L µvs (1 − ε)2 4
3 5 1 − 2 + 2 β4 2β
Re p .
(34)
While the Ergun equation (1) can also be written as the dimensionless form, i.e. fv =
D 2p ε 3 P = 150 + 1.75Re p . L µvs (1 − ε)2
(35)
Figure 8 compares the present model predictions by Eq. 34 with those by the Ergun equation Eq. 35 and the experimental data (Yu and Zhang 2002), which were measured for beds packed with spherical particles of D p = 12 mm at ε = 0.364, In Fig. 8, thecalculations of Eq. 34 3 1 5 are based on the two coefficients 72τ = 129.9 and 3τ 4 2 + β 4 − 2β 2 = 1.89 at ε = 0.364. From Fig. 8 it can be found that the model predictions present much better agreement with the experimental data than those by the Ergun equation.
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A resistance model for flow through porous media Fig. 8 A comparison among the present model Eq. 35, Ergun equation (35) and experimental data (Yu and Zhang 2002) for beds packed with ball particles of D p = 12 mm and at ε = 0.364
341 9000 8000 7000
Ergun equation (35) Present equation (34) Experiment data
fv
6000 5000 4000 3000 2000 1000 1500
2000
2500 3000
3500 4000
Rep
Since effects of flow rate, properties of fluids, size, shape, and particle’s surface on the flow resistance have been discussed in detail in the Ergun’s paper, this paper does not intend to repeat such discussions due to the limitation of page length.
4 Conclusions We have derived a new model for resistance of flow through homogeneous and isotropic granular media based on the average hydraulic radius model and the contracting–expanding channel model. This model is expressed as a function of tortuosity, porosity, ratio of pore diameter to throat diameter, diameter of particles, and fluid properties. The two empirical constants 150 and 1.75 in the Ergun equation are replaced by two expressions in the proposed model. Every parameter in the proposed model has clear physical meaning in this paper. The proposed model has been shown to be more fundamental and more reasonable than the Ergun equation. The results show that the model predictions are in good agreement with those from the existing experimental data. The validity of the proposed model for resistance of flow through homogeneous and isotropic granular media is thus verified. Acknowledgements This work was supported by the National Natural Science Foundation of China through grant number 10572052.
Appendix A According to Ergun (1952), Bird et al. (1960) and Du Plessis (1994), the hydraulic radius Rh is related to porosity ε and specific surface area Sv of particles by Rh = =
A Surface of cross-section of cavity Volume cavity = = S Circumference cavity Wall surface cavity
(A1)
ε Porosity (m3 /m 3 ) Volume cavity/m3 = = . 3 Specific wall surface S Wall surface cavity/m v
The specific surface area is, in turn, again a function of ε and particle diameter Sv =
1 6(1 − ε)AL 6(1 − ε) π D 2P . = 3 AL DP π DP
(A2)
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Equation A2 implies that the capillaries along the flow direction are approximated to be cylindrical pores. From Eqs. A1 to A2, we obtain Rh =
εDp . 6(1 − ε)
(A3)
Pr 2 , 8µL
(A4)
The modified Hagen–Poiseuille equation is v¯ =
where r = 2Rh , and inserting it into Eq. A4 results in P Rh2 . 2µL
(A5)
ε3 P D 2p . L 72µ (1 − ε)2
(A6)
P 72µvs (1 − ε)2 . = L D 2p ε3
(A7)
v¯ = ¯ we have From Eqs. A3, A5 and vs = vε, vs = From Eq. A6, we obtain
Assume L to be the length of porous media, and the real length of fluid flow is large than L, using a constant to modify it, we get P µvs (1 − ε)2 . = 72c 2 L Dp ε3
(A8)
From experiments, c = 25/12, then we have P µvs (1 − ε)2 . = 150 2 L Dp ε3
(A9)
This is the Blake–Kozeny equation. For the Burke–Plummer equation, the friction coefficient is P L = 4 f, 2 ρ v¯ /2 d
(A10)
¯ d = 4Rh and where d is the diameter of capillary. Using the relations vs = vε, Rh = ε D p /6/(1 − ε), we obtain P 1 1 21−ε . =6f ρv L D p 2 s ε3
(A11)
From experiments, 6 f = 3.5, inserting it into Eq. A11 yields P ρv 2 1 − ε . = 1.75 s L D p ε3
(A12)
This is the Burke–Plummer equation. Adding the Blake–Kozeny equation and the Burke–Plummer equation, we have Eq. 1.
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