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The fastest capillary penetration of power-law fluids
Chemical Engineering Science 137 (2015) 583–589
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The fastest capillary penetration of power-law fluids Dahua Shou n, Jintu Fan n Department of Fiber Science & Apparel Design, College of Human Ecology, Cornell University, Ithaca, NY 14853, USA
H I G H L I G H T S
This paper studies dynamics of capillary-driven power-law fluids. A model for flow in straight tube and Y-shaped tree network is developed. The minimum penetration time is found in the optimized structure. Unique optimal transport behaviors are analyzed with different power components.
art ic l e i nf o
a b s t r a c t
Article history: Received 7 April 2014 Received in revised form 27 June 2015 Accepted 11 July 2015 Available online 20 July 2015
The engineering and modeling of non-Newtonian power-law fluids (i.e., shear thinning and thickening fluids) in porous media has received wide attention in natural systems, oil recovery, and microfluidic devices. In this work, we theoretically explore the dynamics of power-law fluids in the form of capillary flow confined by a straight tube and a Y-shaped tree network, both of which are basic elements of many advanced materials. The straight tube and the tree network are composed of sub-tubes with different radii and lengths. The proposed model reveals that the evolution of the penetration time to the penetration distance is highly dependent on the viscous and capillary effects. If the viscous resistance is high, the flow is slow. If the capillary pressure increases, the flow accelerates. An interesting question is therefore in what optimal structure is the power-law flow fastest, considering different responses of viscous resistance and capillary force to the structural parameters. Based on optimization of the radius and length distribution of sub-tubes, we find the minimum penetration time of the fluids or the fastest flow in both straight tubes and tree networks under size constraints. The unique optimal transport behaviors of power-law fluids, which are different from those of Newtonian fluids, are analyzed in details with different power components. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Capillary pressure Fast flow Porous media Power-law fluid Tree network
1. Introduction Non-Newtonian power-law fluids have found broad applications in natural and engineering fields, including microfluidic devices (Groisman et al., 2003), blood rheology (Fedosov et al., 2011), oil recovery (Pu et al., 2015), capillary breakup (Zimoch et al., 2013), and droplet ejection (Bartolo et al., 2007). The powerlaw fluid (or the Ostwald–de Waele fluid) in a circular tube is characterized by a modified Hagen–Poiseuille equation (Christop and Middlema, 1965) Q n ¼ ðπ R2 uÞn ¼
π R3
!n
3 þ 1n
R dp ; 2K dx
n
Tel.: þ 1 607 727 9272; fax: þ 1 607 255 1093. E-mail addresses: [email protected], [email protected] (D. Shou), [email protected] (J. Fan). http://dx.doi.org/10.1016/j.ces.2015.07.009 0009-2509/& 2015 Elsevier Ltd. All rights reserved.
ð1Þ
where Q is the volume flow rate, n is the power exponent for power-law fluids, u is the flow velocity, R is the tube radius, p is the pressure drop along x-direction, and K is the flow consistency index. Eq. (1) is computed based on the constitutive equation τ ¼ kγ_ n , where τ and γ_ are shear stress and strain rate of the fluid, respectively. When n ¼ 1, the non-Newtonian fluid power-law fluid becomes Newtonian, and Eq. (1) recovers the Hagen– Poiseuille equation in laminar regime. When n 4 1, the powerlow fluid is known for shear-thickening or dilatant as viscosity increases with shear rate. When n o 1, shear thinning or pseudoplastic behavior occurs due to the decrease in viscosity at higher shear rates. The more generalized model for the non-Newtonian fluid is described by the Herschel–Bulkley model, τ ¼ τ0 þ kγ_ n , where τ0 is the yield stress (Yun et al., 2010). The Herschel–Bulkley model represents the Bingham plastic fluid with n ¼ 1, and becomes the power-law fluid model with τ0 ¼ 0. Based on the fractal geometry of porous media (Cai et al., 2010), the flow behavior of power-low fluids was quantified by fractal dimensions from single tortuous tubes (Yun et al., 2008) to generous porous
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systems with wide pore size distribution (Zhang et al., 2006). Eq. (1) was also extended to model the power-law flow through packed and porous media by introducing the Blake–Kozeny equation as the permeability (Christop and Middlema, 1965). Without using empirical constants, a resistance model of powerlaw flow in packed beds was recently developed by taking into account the inertia force (Tang and Lu, 2014). The power-law fluid through three-dimensional disordered porous media was explored numerically with a broad range of Reynolds number, and an enhancement of permeability was found at intermediate conditions due to the interplay of fluid rheology, disordered geometry, and inertial effects (Morais et al., 2009). The self-driven power-law fluids caused by the capillary force have unique transport properties. The capillary dynamics of the power-law fluids in a uniform circular tube was expressed as a function of n, and the flow was found to be retarded when the liquid became more strongly shear-thinning with n o1 (Turian and Murad, 2005). The impact of fluid rheology and dynamic contact angle on the capillary rise of power-law fluid was explored experimentally and theoretically (Digilov, 2008). The shearthinning fluid under gravity begins to rise faster than the shear thickening counterpart, but they have the same equilibrium height with the Newtonian liquid due to self-retardation (Digilov, 2008). For power-law fluids in conical tubes, asymmetric flow times have been found from different ends of the tubes, and the asymmetry is strengthened by shear-thinning fluid but weakened by the shearthickening fluid (Berli and Urteaga, 2014). A comprehensive model is developed to predict the permeability of power-law fluid flow in fractal-like tree network, and the permeability is found to be a function of the branching diameter ratio, the branching length ratio, the total number of branching levels, and the power exponent of power-law fluids (Wang and Yu, 2011). Control and acceleration of capillary flow is always advantageous for the applications mentioned above. The manipulated capillary flow in non-uniform porous structures, in terms of the evolution of flow distance to time, deviates from the classical Washburn equation based on uniform tubes (Reyssat et al., 2008, 2009; Shou et al., 2014b, 2014c). As well, the fastest capillary flow on the basis of Newtonian fluids has been found in the optimally designed tubes and porous systems (Shou and Fan, 2015; Shou et al., 2014b, 2014c, 2014d). However, to our best of knowledge, less studies concern the development and optimization of the fast capillary-driven power-law fluids. In this context, we aim for finding the minimum penetration time required for a given penetration distance, from a straight tube to a tree network. Both the straight tube and the tree network, which contain sub-tubes with different radii and lengths, are fundamental elements of various applications. More specifically, the straight tubes with varying local radii and lengths are often used in microfluidics devices (Stone et al., 2004) and found in general porous media (Yun et al., 2008), while the tree networks conduct blood in body as vessels (Murray, 1926) and drainage water/oil mixtures in forms of rock fractures (Wang and Yu, 2011). The investigation of transport in tree networks was dated back to 1926, when Murray (1926) found the optimal radius ratio between parent and daughter branches in a cardiovascular system restricted by energy expenditure. Based on the contructal theory (Reis, 2006), the minimization of flow resistance in a simple tree network under the given volume yields the same optimal radius ratio as Murray's law (Bejan et al., 2000), as the volume constraint is equivalent to the constraint of energy expenditure considered by Murray (1926). Even when the number of branch levels increases, the optimal radius ratio stays a constant (Kou et al., 2014). Moreover, the tree network requires a lower pumping power for transport (Chen and Cheng, 2002). In this work, we focus on investigating power-law fluids in the tree network, whereas the
tree network becomes a straight tube when the branching number is unit.
2. Model generation Capillary fluid, either Newtonian or non-Newtonian, moves spontaneously in tiny tubes due to the pressure of cohesion and adhesion. The capillary flow accelerates with the decrease in tube size as capillary force or capillary pressure increases. Conversely, the flow slows down with decreasing tube size, generating higher flow resistance based on Eq. (1). In a uniform tube, the time for a given flow distance is monotonously dependent on the tube size or the microstructure, in analogy to Washburn equation. In the tree network and the straight tube, the sub-tubes have different radii and lengths, whereas the capillary pressure and the viscous resistance have different sensitivity to microstructures of the local sub-tubes at different levels. Therefore, it is possible to find the minimum penetration flow time in the tree network as well as the straight tube with the dynamic competition between capillary and viscous effects. We investigate here the power-law flow through a multi-level Y-shaped tree network and a multi-section straight tube constrained by fixed volume and length. The gravitational force, the liquid evaporation, and the effect of meniscus shape is neglected, and a large aspect ratio of length to radius of the tubes is assumed. The fluids are in the continuum regime, as the tubes concerned in this work are much larger than the size of liquid molecules. A Y-shaped tree network and a parallel tube net with the same volume and length are shown in Fig. 1. The parallel tube net is used as a control sample, as the flow time for a given total penetration distance (i.e., the tube length) is constant in a uniform tube. The influence of the branching angle α is eliminated, since all sub-tubes in the tree network have the same branching angle with the parallel tube net. In analogy to Ref. Chen et al. (2007), the number of parallel tubes is assumed to be equal to that of outlet tubes of the tree network, viz., m. In Fig. 1, m is equal to 2, while the tree network becomes a straight tube composed of two successive sub-tubes when m ¼ 1. The lower sub-tube at the parent branching level is named the first sub-tube, and the upper sub-tubes at the daughter branching level are named the second sub-tubes. We define the radius ratio and the length ratio between the second to the first sub-tubes e¼
R2 R1
ð2Þ
and f¼
h2 ; h1
ð3Þ
Fig. 1. Illustration of a Y-shaped tree network (left) and a parallel tube net (right).
D. Shou, J. Fan / Chemical Engineering Science 137 (2015) 583–589
respectively. Here, R2 and R1 , and h2 and h1 are radii and lengths of the second and first sub-tubes, respectively. From Fig. 1, the total length h0 and the total volume V of the tree network and the parallel tube net are equal, given by h0 ¼ h1 þh2 ¼ h1 ð1 þf Þ
ð4Þ
and V ¼ π R20 h0 ¼ π ðR21 h1 þ mR22 h2 Þ;
ð5Þ
respectively. Here, R0 is the tube radius of the parallel tubes, which is obtained based on Eqs. (2–5)
1 þ me2 f R0 ¼ 1 þf
0:5 R1 :
ð6Þ
When the meniscus is in a circular tube, the capillary pressure Δp driving the power-law fluids is expressed by the Young– Laplace equation (Cai et al., 2014)
Δp ¼
2γ cos θ ; R
ð7Þ
where γ stands for the (liquid–vapor) surface tension of the liquid, and θ is the contact angle between the liquid meniscus and the wall of the tube. The capillary pressure is equal to the loss of hydrostatic pressure of the power-law fluids on the basis of Eq. (1) for a moving distance z !n !n Z z 3 þ 1n 3 þ 1n dx ∂z n z Δp ¼ 2k Qn ¼ 2k π R2 : 3n þ 1 ∂t R3n þ 1 π π 0 R ð8Þ Substituting Eq. (7) into Eq. (8), we have a simplified expression of Eq. (8) as follows: n A dz z ¼ R2 ; ð9Þ R dt R3n þ 1 γ cos θ
where A is a constant given by n . And the flow time required kð3 þ 1nÞ for filling the tube with a length h is obtained by integration of Eq. (9) Z h 1=n þ 1 1 1 h z1=n dz ¼ 1=n : ð10Þ t ¼ 1=n 1 þ 1n 0 RA RA Therefore, the flow time for filling the first sub-tube at the parent branching level is given by t1 ¼
1=n þ 1
h1 : 1=n 1 þ 1n R1 A 1
ð11Þ
When the meniscus moves into the second sub-tubes at the daughter branching level, the flow rate in each sub-tube is Q d ¼ Q =m based on conservation of mass. The capillary pressure is determined by the sub-tube radius at the position of the meniscus
Δp ¼
2γ cos θ ; R2
ð12Þ
which is equal to the total hydrostatic pressure of the power-law fluids with a moving distance z (z 4h1 ) !n " # Z h1 Z z 3 þ 1n dx dx n Δp ¼ 2k ðmQ d Þn þ Q ; ð13Þ d þ1 3n þ 1 π R3n h1 R2 0 1 where Q d is expressed as π R22 ðdz=dtÞ in the second sub-tube. Substituting Eq. (12) into Eq. (13) yields # n "Z h1 Z z A dz dx dx ¼ R22 mn 3n þ 1 þ : ð14Þ 3n þ 1 R2 dt R1 h1 R2 0
585
Integrating Eq. (14) against z leads to the flow time within the meniscus moving in the second sub-tubes #1=n Z h " n 2n þ 1 1 m R2 h1 z h1 t ¼ 1=n þ dz: ð15Þ þ1 Rn2 R3n h1 A 1 Thus the flow time for filling the second sub-tubes is obtained by Eq. (16) as follows: 1=n Z h1 þ h2 1 z h1 n 2n þ 1 t2 ¼ m e h þ dz: ð16Þ 1 en R1 A1=n h1 We also obtain the flow time for the control sample by substituting Eq. (4) and Eq. (6) into Eq. (10), viz. t0 ¼
1=n þ 1
1 1=n
R1 A
h1 ð1 þ f Þ1=n þ 1:5 : 1 þ 1n ð1 þ me2 f Þ0:5
ð17Þ
Then the dimensionless time of the power-law flow in the Yshaped tree network is expressed as the ratio of t 1 þ t 2 to t 0 R 1 þ f n 2n þ 1 z^ 1 1=n 1 m e þ en dz^ t 1 þ t 2 1 þ 1n þ 1 T¼ ¼ ; ð18Þ 1=n þ 1:5 ð1 þ f Þ t0 1 ð1 þ me2 f Þ0:5 1 þ ð nÞ where the dimensionless distance z^ is defined as z=h1 . For the straight tube composed of two sub-tubes with different radii and lengths, the dimensionless time is easily obtained with m ¼ 1, R 1 þ f 2n þ 1 z^ 1 1=n 1 þ 1 e þ en dz^ 1 þ 1n T¼ : ð19Þ 1=n þ 1:5 ð1 þ f Þ
ð1 þ 1nÞð1 þ e2 f Þ0:5
3. Results and discussion In this section, we investigate the power-law flow in the straight tube and the tree network, respectively. The two-level Y-shaped tree network with m ¼ 2, a possible smallest dichotomous structure (see Fig. 1), the two-section straight tube, a special case of the tree network with m ¼ 1, the three-level tree network, and the three-section straight tube are used to represent the present work. The impact of the viscosity on the flow is analyzed with variations in power exponent n. The Newtonian flow with n ¼ 1 is revisited as a comparison to the power-law flow. First, we compare the model for the capillary-driven power-law fluids in a single tube with experimental results (Digilov, 2008) in Fig. 2. Digilov (2008) measured the quasi-steady-state flow of a power law liquid (1 wt% solution of carboxymethyl cellulosesodium salt) in a vertical tube. The tube radius R is 0.148 mm, the power exponent n is 0.78, the flow consistency index K is 0.06 Pa sn , and the surface tension is 0.066 N m 1 . In the initial stage of the capillary flow (i.e., h o6 mm in Fig. 2), the capillary force is dominant and the effect of the gravitational force is negligible. The mean velocity for the penetration distance h ¼ 6 mm in Fig. 2 is around 11.5 m/s, correspond to a contact angle equal to θ ¼ 77o (Digilov, 2008). Based on the model of Eq. (10), the theoretical model agrees satisfactorily with the experimental data as shown in Fig. 2. This model has also been verified by the measurement of capillary-driven blood flow with n ¼ 0:72 using the optical coherence tomography (Cito et al., 2012). Next we analyze the effect of shear rate on the power-law flow in the uniform tube with different values of n. Dividing Eq. (10) by 1 H 1=n þ 1 , RA1=n 1 þ 1n
the dimensionless flow time T versus the dimensionless
penetration distance h=H is obtained 1=n þ 1 h ; T¼ H
ð20Þ
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behavior across the range of shear rates to which the power can be fitted. For instance, Eq. (1) was validated by the experimental results of the power-law fluids, including aqueous solutions of Carboxymethyl cellulose with approximately n ¼ 0:53 and toluene solutions of Polyisobutylene with n ¼ 0:96, which flow within packed tubes (Christop and Middlema, 1965). Based on Eq. (20), we also find a simple but effective approach to determine the power component n by measuring the flow time T a and T b , corresponding to the penetration distance ha and hb , respectively. The expression of n is obtained by modifying Eq. (20), viz.
ln hha b
ð21Þ n¼ ln TT ab ln hha b
Fig. 2. Comparison of the penetration time of the capillary-driven power-law fluids versus the penetration distance in the uniform circular tube with R ¼ 0:148 mm between the present model and experimental results.
Fig. 3. The dimensionless time T versus the dimensionless flow distance h=H in a uniform tube for different values of power exponent n.
where H is a fixed flow distance. Fig. 3 reveals the variations of T with h=H at n ¼ 4; 2; 1; 1=2; 1=4. The shear thinning fluids move faster at the beginning of the capillary dynamics, but they slow down later and finally arrive at the end of the tube at the same T with the shear thickening and the Newtonian fluids, which was previously reported by Berli and Urteaga (2014). This phenomenon is ascribed to the dependence of the viscous resistance on the shear rate. It is pointed out that the apparent viscosity is given by μ ¼ kγ_ n 1 (Wang and Yu, 2011), which varies with the shear rate at different flow distance or time. When capillary flow occurs, the shear rate of the shear thinning fluids (n o1) is great with a relatively high starting velocity. Then the viscosity increases with the decrease in the shear rate, as more fluids penetrate the tube under a fixed capillary pressure and the flow slows. On the contrary, the viscosity of the shear thickening fluids (n 4 1) decreases with time. Hence, a convergence of T for all the fluids is found in Fig. 3, as predicted by Eq. (20). It is noted from the power correlation μ ¼ kγ_ n 1 , an unrealistic infinite viscosity is determined in the case of vanishing shear rate at the beginning of the shear thinning flow. Indeed, the effective viscosity in a real fluid has the upper and lower bounds, determined by the physical chemistry at the molecular level. With this objection, however, Eq. (1) still provides reasonable estimate of the power-law flow
Then we examine the model for the two-section tube with numerical results of Erickson et al. (2002), who simulated the capillary-driven flow with n ¼ 1 in a convergent tube by the finite element approach. The tube contains two sub-tubes connected by an extremely mild contraction. The first sub-tube has radius R1 ¼ 50 μm and h1 ¼ 20 mm length, the second sub-tube has radius R2 ¼ 25 μm, and the length of the mild contraction between the two sub-tubes is 2:86 mm (Erickson et al., 2002). The fluid properties are given as follows, γ ¼ 0:03 N=m, η ¼ 0:001 kg=s m and θ ¼ 30o . It has been found that the effect of the mild contraction on the total penetration time is very small (Shou et al., 2014c). Fig. 4 shows that the present model based on Eq. (11) and Eq. (16) agrees closely with numerical results of Erickson et al. (2002), indicating the accuracy of our model and the negligible effect of the mild contraction. Thus the model can also be applied to the tree networks composed of long sub-tubes, in which the junction effect is less significant (Shou et al., 2014b). The dynamics of capillary flow in single uniform tubes with radii 25 μm and 50 μm are also added for comparison. It is interesting to note that the flow is faster in the two-section tube than in both of the two uniform tubes at 20 mm o h o50 mm. In comparison to the uniform tube with R ¼ 25 μm, the two-section tube has a smaller viscous resistance; in comparison to the uniform tube with R ¼ 50 μm, the two-section tube has a higher capillary pressure. The evolution of time to flow distance for the power-law fluids in the two-section straight tube is shown in Fig. 5. The figure indicates the dependence of the dimensionless time T on the radius ratio e at five different values of n (i.e., n ¼ 4; 2; 1; 1=2; 1=4). In order to highlight the effect of e, the length ratio f is fixed as a constant 1. It is observed that T first
Fig. 4. Comparison of the penetration time of the capillary-driven fluids versus the penetration distance in the two-section circular tube between the present model and numerical results. The dynamics of capillary flow in uniform circular tubes are also added for comparison.
D. Shou, J. Fan / Chemical Engineering Science 137 (2015) 583–589
587
Fig. 5. The dimensionless time T versus the radius ratio e of the two-section straight tube for different values of power exponent n at a fixed length ratio f ¼ 1.
of T reduce with the decrease in n, as expected. Besides, the optimal values of f increase inversely with n. It is interesting to find that T is slightly higher than 1 for n ¼ 4, whereas T of the Newtonian fluids stays smaller than 1 when e o 1 (Shou et al., 2014c). It is expected that the shear thickening fluids generate a higher viscous resistance than the Newtonian fluids under the same capillary pressure, enabling to delay the time required for a given flow distance. The dimensionless time T versus the length ratio f of the twosection straight tube for different n at e ¼ 2 is shown in Fig. 7. In this case, the second sub-tube has larger radius than the first subtube. Similar to Fig. 6, the values of T tend to be 1 when f tends to be 0 or infinite in Fig. 7. When f is in the intermedium regime, the maximum T is found for all values of n. The slower flow is caused by that the first sub-tube with smaller radius has higher flow resistance, whereas afterwards the second sub-tube with larger radius generates smaller capillary pressure for the movement of power-law fluids. The optimal f exists for the slowest flow when the contribution of the capillary and hydrostatic effects is properly allocated in the two sub-tubes. The maximum results of T reduce as the decrease in n, and the optimal f increases inversely with n. Based on Fig. 6 and Fig. 7, the asymmetry of flow times from different ends of the same tube is found, which is more remarkable in the case of shear thinning fluids such as blood. The liquid handling in an asymmetric fashion may be of interest to wound dressings: the designed porous layers composed of the above optimized tube bundles promote drainage ability of inside liquids but inhibit invasion of external liquids. The dynamics of the power-law flow in the two-level Y-shaped tree network is shown in Fig. 8. In the figure, the variations of the dimensionless time T against the radius ratio e at five different values of n (i.e., n ¼ 4; 2; 1; 1=2; 1=4) and f ¼ 1 are demonstrated. We find that the relationship between T and e is characterized by concave curves, indicating the fastest capillary flow in tree networks. Similar to the straight tube, the tree network has extremely large flow resistances with e ¼ 0 or e ¼ inf . And the optimal e close to 0:5 is found with a proper balance between the capillary pressure and the hydrostatic condition for the five values of n. The minimum T and the optimal e enhances with the increase in n. As well, the corresponding results of e to the minimum T decreases with decreasing n. Based on calculation we notice that no convergence of T is found at e ¼ 1 for different n, although the results of T look close to each other. It is reasonable that the Y-shaped tree network does not become a uniform tube at e ¼ 1 as the two-section straight tube. As well, the optimal values of e for tree networks are smaller than those for straight tubes.
Fig. 6. The dimensionless time T versus the length ratio f of the two-section straight tube for different values of power exponent n at a fixed radius ratio e ¼ 0:5.
Fig. 7. The dimensionless time T versus the length ratio f of the two-section straight tube for different values of power exponent n at a fixed radius ratio e ¼ 2.
decreases and then increases with e for different values of n. When e is close to 0 or much greater than 1, the flow resistance becomes extremely large due to the rather tiny second or first sub-tube. However, when the value of e is properly selected, i.e., e o1, the power-law flow becomes faster than in a uniform tube. It is expected that the first sub-tube with larger radius has less flow resistance, whereas the second sub-tube with smaller radius generate higher capillary pressure to drive the power-law fluid. Consequently, the concave curves in Fig. 5 show the existence of the minimum T or the fastest flow, as a result of the optimal dynamic balance between the capillary pressure and the hydrostatic condition. As well, we find that the minimum T decreases with the decrease in n. The corresponding results of e to the minimum T also reduce with decreasing n. When e ¼ 1 all the results of T are equal to 1, as the tube becomes uniform and identical with the control sample. Fig. 6 shows the variations of the dimensionless time T versus the length ratio f of the two-section straight tube at five different values of n (i.e., n ¼ 4; 2; 1; 1=2; 1=4). The radius ratio is fixed at e ¼ 0:5. The shape of all the curves in Fig. 4 is also concave, so the minimum T exists against f . The values of T tend to be 1 when it is close to 0 or infinite, because the two-section tube becomes a uniform tube. When f is in an intermedium region, the minimum T is found for all values of n. This phenomenon is attributed to the optimal distribution of the contribution of capillary pressure and hydrostatic condition in the two sub-tubes. The minimum results
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Fig. 8. The dimensionless time T versus the radius ratio e of the two-level tree network for different values of power exponent n at a fixed length ratio f ¼ 1.
Fig. 9. The dimensionless time T versus the length ratio f of the two-level tree network for different values of power exponent n at a fixed radius ratio e ¼ 0:5.
Fig. 11. The dimensionless time T versus the length ratio f of the three-level tree network for different values of power exponent n at a fixed radius ratio e ¼ 0:5.
optimized structure is greater than both the shear thickening and thinning fluids. As well, all the results of T tend to be 0:707 when f ¼ 0, because the tree network becomes a straight uniform pffiffiffi tube at R1 ¼ 2R0 . On the other hand, with f ¼ inf , the results of T tend to be 1 as the tree network becomes two uniform tubes as the control sample. Moreover, the results of T increase faster in tree networks than in straight tubes when f 4 1, because the flow resistances in the second sub-tubes of tree networks enhance considerably with the rapidly decreasing tube size. This phenomenon indicates that T of tree networks is more sensitive to microstructures than that of straight tubes. Based on Eq. (S13) in the supplementary material, the evolution of the penetration time to the penetration distance for the powerlaw fluids in the three-section straight tube is shown in Fig. 10. Analogous to the two-section tube, the minimum results of T versus the radius ratio e are found when f ¼ 1. The values of the minimum T in Fig. 10 are lower than those in Fig. 3 under the same n, indicating the acceleration of the flow in the three-section tube is more significant than that in the two-section tube. We also investigate the dynamics of the power-law flow in the three-level tree network by Eq. (S12) in the supplementary material. The minimum T is found against f at e ¼ 0:5 in Fig. 11, which is similar with the optimized results for the two-level tree network in Fig. 9. As well, the optimal values of f are much smaller in the three-level tree network than in the three-section tube. It is expected that the fastest flow can also be found for the structures with more sections or levels.
4. Conclusion
Fig. 10. The dimensionless time T versus the radius ratio e of the three-section straight tube for different values of power exponent n at a fixed length ratio f ¼ 1.
Fig. 9 shows the variations of the dimensionless time T versus the length ratio f for the two-level Y-shaped tree network. We assume the radius ratio as a constant, viz., e ¼ 0:5. The minimum T is found for all values of n, and the minimum results of T reduce as the decrease in n. We note that the optimal f that accounts for the minimum penetration time of the Newtonian fluid in the
A theoretical model has been developed to encode the dynamics of power-law fluids in the form of capillary flow confined in a straight tube and a Y-shaped tree-network. The power-law flow in both optimized structures can be accelerated in comparison to in parallel tube nets under a given volume. Based on dynamic competition between viscous and capillary effects, the optimal distribution of sub-tubes is found, leading to the minimum time or fastest flow for the liquid to fill the whole straight tube or tree network. The power-law flow behaviors in optimized structures are observed to deviate from those of Newtonian fluids on the basis of Washburn equation. For power-law fluids, the flow time of the shear thinning fluids with n ¼ 1=4 can be reduced by over 90% comparing with that in parallel tubes. As well, it is shown that the variations of T for tree networks with microstructures are more dramatic than those for straight tubes. So far, we limit our attention to an initial investigation of the fastest capillary-driven
D. Shou, J. Fan / Chemical Engineering Science 137 (2015) 583–589
power-law flow in simple and basic structures. However, it is straightforward to extend the proposed robust framework to more realistic and complex systems such as interconnected networks and porous media under specific constraints. The limitations of present model are also summarized and will be addressed in future work. (1) In this paper, the stationary contact angle (SCA) is assumed during the flow process of the power-law fluids. We will take into account the effect of the dynamic contact angle (DCA) for more practical conditions, although the deviation of the SCA-based model from the DCAbased model is small in a single tube (Digilov, 2008). (2) The present model is conducted by assuming the effect of gravity negligible. It is expected that the fastest flow of the power-law fluids also exits under gravity as that of the Newtonian fluids found by Shou et al. (2014a). Thus we will concern the dependence of the optimal structure on the gravitational influence. (3) We have not investigate the effect of the junction of the straight tube or the tree network on the flow behavior. Numerical simulations have shown that the liquid meniscus suffers from complex deformation when moving through contraction and expansion regions of the tubes (Wiklund and Uesaka, 2013); in particular, the liquid passing through an expansion was pinned near a highcurvature point (Mehrabian et al., 2011), and the meniscus within the wider sub-tube is arrested for a time when the liquid moves into the sub-tubes with different sizes in a tree network (Sadjadi et al., 2015). Although the junction effect is found to be negligible for the long and thin sub-tubes (Shou et al., 2014b), modification of the current dynamic model is required when the short and thick sub-tubes are investigated. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ces.2015.07.009. References Bartolo, D., Boudaoud, A., Narcy, G., Bonn, D., 2007. Dynamics of non-newtonian droplets. Phys. Rev. Lett. 99, 174502. Bejan, A., Rocha, L.A.O., Lorente, S., 2000. Thermodynamic optimization of geometry: T- and Y-shaped constructs of fluid streams. Int. J. Thermal Sci. 39, 949–960. Berli, C.L.A., Urteaga, R., 2014. Asymmetric capillary filling of non-Newtonian power law fluids. Microfluid. Nanofluidics 17, 1079–1084. Cai, J., Yu, B., Zou, M., Mei, M., 2010. Fractal analysis of invasion depth of extraneous fluids in porous media. Chem. Eng. Sci. 65, 5178–5186. Cai, J.C., Perfect, E., Cheng, C.L., Hu, X.Y., 2014. Generalized Modeling of Spontaneous Imbibition Based on Hagen-Poiseuille Flow in Tortuous Capillaries with Variably Shaped Apertures. Langmuir 30, 5142–5151. Chen, J., Yu, B., Xu, P., Li, Y., 2007. Fractal-like tree networks increasing the permeability. Phys. Rev. E 75, 056301. Chen, Y.P., Cheng, P., 2002. Heat transfer and pressure drop in fractal tree-like microchannel nets. Int. J. Heat Mass Transf. 45, 2643–2648.
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Christop, R.H., Middlema, S., 1965. Power-law flow through a packed tube. Ind. Eng. Chem. Fundam. 4, 422–426. Cito, S., Ahn, Y.-C., Pallares, J., Duarte, R.M., Chen, Z., Madou, M., Katakis, I., 2012. Visualization and measurement of capillary-driven blood flow using spectral domain optical coherence tomography. Microfluid. Nanofluidics 13, 227–237. Digilov, R.M., 2008. Capillary Rise of a Non-Newtonian power law liquid: impact of the fluid rheology and dynamic contact angle. Langmuir 24, 13663–13667. Erickson, D., Li, D., Park, C.B., 2002. Numerical simulations of capillary-driven flows in nonuniform cross-sectional capillaries. J. Colloid Interface Sci. 250, 422–430. Fedosov, D.A., Pan, W., Caswell, B., Gompper, G., Karniadakis, G.E., 2011. Predicting human blood viscosity in silico. Proc. Natl. Acad. Sci. USAmerica 108, 11772–11777. Groisman, A., Enzelberger, M., Quake, S.R., 2003. Microfluidic memory and control devices. Science 300, 955–958. Kou, J., Chen, Y., Zhou, X., Lu, H., Wu, F., Fan, J., 2014. Optimal structure of tree-like branching networks for fluid flow. Physica A: Stat. Mech. Appl. 393, 527–534. Mehrabian, H., Gao, P., Feng, J.J., 2011. Wicking flow through microchannels. Phys. Fluids 23, 122108. Morais, A.F., Seybold, H., Herrmann, H.J., Andrade Jr., J.S., 2009. Non-Newtonian fluid flow through three-dimensional disordered porous media. Phys. Rev. Lett. 103, 194502. Murray, C.D., 1926. The physiological principle of minimum work applied to the angle of branching of arteries. J. Gen. Physiol. 9, 835–841. Pu, W.F., Liu, R., Wang, K.Y., Li, K.X., Yan, Z.P., Li, B., Zhao, L., 2015. Water-soluble core-shell hyperbranched polymers for enhanced oil recovery. Ind. Eng. Chem. Res. 54, 798–807. Reis, A.H., 2006. Constructal theory: From engineering to physics, and how flow systems develop shape and structure. Appl. Mech. Rev. 59, 269–282. Reyssat, M., Courbin, L., Reyssat, E., Stone, H.A., 2008. Imbibition in geometries with axial variations. J. Fluid Mech. 615, 335–344. Reyssat, M., Sangne, L.Y., van Nierop, E.A., Stone, H.A., 2009. Imbibition in layered systems of packed beads. EPL 86, 56002. Sadjadi, Z., Jung, M., Seemann, R., Rieger, H., 2015. Meniscus arrest during capillary rise in asymmetric microfluidic pore junctions. Langmuir 31, 2600–2608. Shou, D., Fan, J., 2015. Structural optimization of porous media for fast and controlled capillary flows. Phys. Rev. E 91, 053021. Shou, D., Ye, L., Fan, J., 2014a. The fastest capillary flow under gravity. Appl. Phys. Lett. 104, 231602. Shou, D., Ye, L., Fan, J., 2014b. Treelike networks accelerating capillary flow. Phys. Rev. E 89, 053007. Shou, D., Ye, L., Fan, J., Fu, K., 2014c. Optimal design of porous structures for the fastest liquid absorption. Langmuir 30, 149–155. Shou, D., Ye, L., Fan, J., Fu, K., Mei, M., Wang, H., Chen, Q., 2014d. Geometry-induced asymmetric capillary flow. Langmuir 30, 5448–5454. Stone, H.A., Stroock, A.D., Ajdari, A., 2004. Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech.s 36, 381–411. Tang, G.H., Lu, Y.B., 2014. A resistance model for Newtonian and power-law nonNewtonian fluid. Transport Porous Media 104, 435–449. Turian, R.M., Murad, S., 2005. Capillary flow of power-law non-Newtonian liquids in circular tubes. Chem. Eng. Commun. 192, 575–580. Wang, S.F., Yu, B.M., 2011. Analysis of seepage for power-law fluids in the fractallike tree network. Transport Porous Media 87, 191–206. Wiklund, H.S., Uesaka, T., 2013. Microfluidics of imbibition in random porous media. Phys. Rev. E 87, 023006. Yun, M.-J., Yu, B.-M., Xu, P., Cai, J.-C., 2008. Fractal analysis of power-law fluid in a single capillary. Chin. Phys. Lett. 25, 616–619. Yun, M., Yu, B., Lu, J., Zheng, W., 2010. Fractal analysis of Herschel-Bulkley fluid flow in porous media. Int. J. Heat Mass Transf. 53, 3570–3574. Zhang, B., Yu, B., Wang, H., Yun, M., 2006. A fractal analysis of permeability for power-law fluids in porous media. Fractals-Complex Geom. Patterns Scaling Nat. Soc. 14, 171–177. Zimoch, P.J., McKinley, G.H., Hosoi, A.E., 2013. Capillary breakup of discontinuously rate thickening suspensions. Phys. Rev. Lett. 111, 036001.
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
The fastest capillary penetration of power-law fluids Dahua Shou n, Jintu Fan n Department of Fiber Science & Apparel Design, College of Human Ecology, Cornell University, Ithaca, NY 14853, USA
H I G H L I G H T S
This paper studies dynamics of capillary-driven power-law fluids. A model for flow in straight tube and Y-shaped tree network is developed. The minimum penetration time is found in the optimized structure. Unique optimal transport behaviors are analyzed with different power components.
art ic l e i nf o
a b s t r a c t
Article history: Received 7 April 2014 Received in revised form 27 June 2015 Accepted 11 July 2015 Available online 20 July 2015
The engineering and modeling of non-Newtonian power-law fluids (i.e., shear thinning and thickening fluids) in porous media has received wide attention in natural systems, oil recovery, and microfluidic devices. In this work, we theoretically explore the dynamics of power-law fluids in the form of capillary flow confined by a straight tube and a Y-shaped tree network, both of which are basic elements of many advanced materials. The straight tube and the tree network are composed of sub-tubes with different radii and lengths. The proposed model reveals that the evolution of the penetration time to the penetration distance is highly dependent on the viscous and capillary effects. If the viscous resistance is high, the flow is slow. If the capillary pressure increases, the flow accelerates. An interesting question is therefore in what optimal structure is the power-law flow fastest, considering different responses of viscous resistance and capillary force to the structural parameters. Based on optimization of the radius and length distribution of sub-tubes, we find the minimum penetration time of the fluids or the fastest flow in both straight tubes and tree networks under size constraints. The unique optimal transport behaviors of power-law fluids, which are different from those of Newtonian fluids, are analyzed in details with different power components. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Capillary pressure Fast flow Porous media Power-law fluid Tree network
1. Introduction Non-Newtonian power-law fluids have found broad applications in natural and engineering fields, including microfluidic devices (Groisman et al., 2003), blood rheology (Fedosov et al., 2011), oil recovery (Pu et al., 2015), capillary breakup (Zimoch et al., 2013), and droplet ejection (Bartolo et al., 2007). The powerlaw fluid (or the Ostwald–de Waele fluid) in a circular tube is characterized by a modified Hagen–Poiseuille equation (Christop and Middlema, 1965) Q n ¼ ðπ R2 uÞn ¼
π R3
!n
3 þ 1n
R dp ; 2K dx
n
Tel.: þ 1 607 727 9272; fax: þ 1 607 255 1093. E-mail addresses: [email protected], [email protected] (D. Shou), [email protected] (J. Fan). http://dx.doi.org/10.1016/j.ces.2015.07.009 0009-2509/& 2015 Elsevier Ltd. All rights reserved.
ð1Þ
where Q is the volume flow rate, n is the power exponent for power-law fluids, u is the flow velocity, R is the tube radius, p is the pressure drop along x-direction, and K is the flow consistency index. Eq. (1) is computed based on the constitutive equation τ ¼ kγ_ n , where τ and γ_ are shear stress and strain rate of the fluid, respectively. When n ¼ 1, the non-Newtonian fluid power-law fluid becomes Newtonian, and Eq. (1) recovers the Hagen– Poiseuille equation in laminar regime. When n 4 1, the powerlow fluid is known for shear-thickening or dilatant as viscosity increases with shear rate. When n o 1, shear thinning or pseudoplastic behavior occurs due to the decrease in viscosity at higher shear rates. The more generalized model for the non-Newtonian fluid is described by the Herschel–Bulkley model, τ ¼ τ0 þ kγ_ n , where τ0 is the yield stress (Yun et al., 2010). The Herschel–Bulkley model represents the Bingham plastic fluid with n ¼ 1, and becomes the power-law fluid model with τ0 ¼ 0. Based on the fractal geometry of porous media (Cai et al., 2010), the flow behavior of power-low fluids was quantified by fractal dimensions from single tortuous tubes (Yun et al., 2008) to generous porous
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systems with wide pore size distribution (Zhang et al., 2006). Eq. (1) was also extended to model the power-law flow through packed and porous media by introducing the Blake–Kozeny equation as the permeability (Christop and Middlema, 1965). Without using empirical constants, a resistance model of powerlaw flow in packed beds was recently developed by taking into account the inertia force (Tang and Lu, 2014). The power-law fluid through three-dimensional disordered porous media was explored numerically with a broad range of Reynolds number, and an enhancement of permeability was found at intermediate conditions due to the interplay of fluid rheology, disordered geometry, and inertial effects (Morais et al., 2009). The self-driven power-law fluids caused by the capillary force have unique transport properties. The capillary dynamics of the power-law fluids in a uniform circular tube was expressed as a function of n, and the flow was found to be retarded when the liquid became more strongly shear-thinning with n o1 (Turian and Murad, 2005). The impact of fluid rheology and dynamic contact angle on the capillary rise of power-law fluid was explored experimentally and theoretically (Digilov, 2008). The shearthinning fluid under gravity begins to rise faster than the shear thickening counterpart, but they have the same equilibrium height with the Newtonian liquid due to self-retardation (Digilov, 2008). For power-law fluids in conical tubes, asymmetric flow times have been found from different ends of the tubes, and the asymmetry is strengthened by shear-thinning fluid but weakened by the shearthickening fluid (Berli and Urteaga, 2014). A comprehensive model is developed to predict the permeability of power-law fluid flow in fractal-like tree network, and the permeability is found to be a function of the branching diameter ratio, the branching length ratio, the total number of branching levels, and the power exponent of power-law fluids (Wang and Yu, 2011). Control and acceleration of capillary flow is always advantageous for the applications mentioned above. The manipulated capillary flow in non-uniform porous structures, in terms of the evolution of flow distance to time, deviates from the classical Washburn equation based on uniform tubes (Reyssat et al., 2008, 2009; Shou et al., 2014b, 2014c). As well, the fastest capillary flow on the basis of Newtonian fluids has been found in the optimally designed tubes and porous systems (Shou and Fan, 2015; Shou et al., 2014b, 2014c, 2014d). However, to our best of knowledge, less studies concern the development and optimization of the fast capillary-driven power-law fluids. In this context, we aim for finding the minimum penetration time required for a given penetration distance, from a straight tube to a tree network. Both the straight tube and the tree network, which contain sub-tubes with different radii and lengths, are fundamental elements of various applications. More specifically, the straight tubes with varying local radii and lengths are often used in microfluidics devices (Stone et al., 2004) and found in general porous media (Yun et al., 2008), while the tree networks conduct blood in body as vessels (Murray, 1926) and drainage water/oil mixtures in forms of rock fractures (Wang and Yu, 2011). The investigation of transport in tree networks was dated back to 1926, when Murray (1926) found the optimal radius ratio between parent and daughter branches in a cardiovascular system restricted by energy expenditure. Based on the contructal theory (Reis, 2006), the minimization of flow resistance in a simple tree network under the given volume yields the same optimal radius ratio as Murray's law (Bejan et al., 2000), as the volume constraint is equivalent to the constraint of energy expenditure considered by Murray (1926). Even when the number of branch levels increases, the optimal radius ratio stays a constant (Kou et al., 2014). Moreover, the tree network requires a lower pumping power for transport (Chen and Cheng, 2002). In this work, we focus on investigating power-law fluids in the tree network, whereas the
tree network becomes a straight tube when the branching number is unit.
2. Model generation Capillary fluid, either Newtonian or non-Newtonian, moves spontaneously in tiny tubes due to the pressure of cohesion and adhesion. The capillary flow accelerates with the decrease in tube size as capillary force or capillary pressure increases. Conversely, the flow slows down with decreasing tube size, generating higher flow resistance based on Eq. (1). In a uniform tube, the time for a given flow distance is monotonously dependent on the tube size or the microstructure, in analogy to Washburn equation. In the tree network and the straight tube, the sub-tubes have different radii and lengths, whereas the capillary pressure and the viscous resistance have different sensitivity to microstructures of the local sub-tubes at different levels. Therefore, it is possible to find the minimum penetration flow time in the tree network as well as the straight tube with the dynamic competition between capillary and viscous effects. We investigate here the power-law flow through a multi-level Y-shaped tree network and a multi-section straight tube constrained by fixed volume and length. The gravitational force, the liquid evaporation, and the effect of meniscus shape is neglected, and a large aspect ratio of length to radius of the tubes is assumed. The fluids are in the continuum regime, as the tubes concerned in this work are much larger than the size of liquid molecules. A Y-shaped tree network and a parallel tube net with the same volume and length are shown in Fig. 1. The parallel tube net is used as a control sample, as the flow time for a given total penetration distance (i.e., the tube length) is constant in a uniform tube. The influence of the branching angle α is eliminated, since all sub-tubes in the tree network have the same branching angle with the parallel tube net. In analogy to Ref. Chen et al. (2007), the number of parallel tubes is assumed to be equal to that of outlet tubes of the tree network, viz., m. In Fig. 1, m is equal to 2, while the tree network becomes a straight tube composed of two successive sub-tubes when m ¼ 1. The lower sub-tube at the parent branching level is named the first sub-tube, and the upper sub-tubes at the daughter branching level are named the second sub-tubes. We define the radius ratio and the length ratio between the second to the first sub-tubes e¼
R2 R1
ð2Þ
and f¼
h2 ; h1
ð3Þ
Fig. 1. Illustration of a Y-shaped tree network (left) and a parallel tube net (right).
D. Shou, J. Fan / Chemical Engineering Science 137 (2015) 583–589
respectively. Here, R2 and R1 , and h2 and h1 are radii and lengths of the second and first sub-tubes, respectively. From Fig. 1, the total length h0 and the total volume V of the tree network and the parallel tube net are equal, given by h0 ¼ h1 þh2 ¼ h1 ð1 þf Þ
ð4Þ
and V ¼ π R20 h0 ¼ π ðR21 h1 þ mR22 h2 Þ;
ð5Þ
respectively. Here, R0 is the tube radius of the parallel tubes, which is obtained based on Eqs. (2–5)
1 þ me2 f R0 ¼ 1 þf
0:5 R1 :
ð6Þ
When the meniscus is in a circular tube, the capillary pressure Δp driving the power-law fluids is expressed by the Young– Laplace equation (Cai et al., 2014)
Δp ¼
2γ cos θ ; R
ð7Þ
where γ stands for the (liquid–vapor) surface tension of the liquid, and θ is the contact angle between the liquid meniscus and the wall of the tube. The capillary pressure is equal to the loss of hydrostatic pressure of the power-law fluids on the basis of Eq. (1) for a moving distance z !n !n Z z 3 þ 1n 3 þ 1n dx ∂z n z Δp ¼ 2k Qn ¼ 2k π R2 : 3n þ 1 ∂t R3n þ 1 π π 0 R ð8Þ Substituting Eq. (7) into Eq. (8), we have a simplified expression of Eq. (8) as follows: n A dz z ¼ R2 ; ð9Þ R dt R3n þ 1 γ cos θ
where A is a constant given by n . And the flow time required kð3 þ 1nÞ for filling the tube with a length h is obtained by integration of Eq. (9) Z h 1=n þ 1 1 1 h z1=n dz ¼ 1=n : ð10Þ t ¼ 1=n 1 þ 1n 0 RA RA Therefore, the flow time for filling the first sub-tube at the parent branching level is given by t1 ¼
1=n þ 1
h1 : 1=n 1 þ 1n R1 A 1
ð11Þ
When the meniscus moves into the second sub-tubes at the daughter branching level, the flow rate in each sub-tube is Q d ¼ Q =m based on conservation of mass. The capillary pressure is determined by the sub-tube radius at the position of the meniscus
Δp ¼
2γ cos θ ; R2
ð12Þ
which is equal to the total hydrostatic pressure of the power-law fluids with a moving distance z (z 4h1 ) !n " # Z h1 Z z 3 þ 1n dx dx n Δp ¼ 2k ðmQ d Þn þ Q ; ð13Þ d þ1 3n þ 1 π R3n h1 R2 0 1 where Q d is expressed as π R22 ðdz=dtÞ in the second sub-tube. Substituting Eq. (12) into Eq. (13) yields # n "Z h1 Z z A dz dx dx ¼ R22 mn 3n þ 1 þ : ð14Þ 3n þ 1 R2 dt R1 h1 R2 0
585
Integrating Eq. (14) against z leads to the flow time within the meniscus moving in the second sub-tubes #1=n Z h " n 2n þ 1 1 m R2 h1 z h1 t ¼ 1=n þ dz: ð15Þ þ1 Rn2 R3n h1 A 1 Thus the flow time for filling the second sub-tubes is obtained by Eq. (16) as follows: 1=n Z h1 þ h2 1 z h1 n 2n þ 1 t2 ¼ m e h þ dz: ð16Þ 1 en R1 A1=n h1 We also obtain the flow time for the control sample by substituting Eq. (4) and Eq. (6) into Eq. (10), viz. t0 ¼
1=n þ 1
1 1=n
R1 A
h1 ð1 þ f Þ1=n þ 1:5 : 1 þ 1n ð1 þ me2 f Þ0:5
ð17Þ
Then the dimensionless time of the power-law flow in the Yshaped tree network is expressed as the ratio of t 1 þ t 2 to t 0 R 1 þ f n 2n þ 1 z^ 1 1=n 1 m e þ en dz^ t 1 þ t 2 1 þ 1n þ 1 T¼ ¼ ; ð18Þ 1=n þ 1:5 ð1 þ f Þ t0 1 ð1 þ me2 f Þ0:5 1 þ ð nÞ where the dimensionless distance z^ is defined as z=h1 . For the straight tube composed of two sub-tubes with different radii and lengths, the dimensionless time is easily obtained with m ¼ 1, R 1 þ f 2n þ 1 z^ 1 1=n 1 þ 1 e þ en dz^ 1 þ 1n T¼ : ð19Þ 1=n þ 1:5 ð1 þ f Þ
ð1 þ 1nÞð1 þ e2 f Þ0:5
3. Results and discussion In this section, we investigate the power-law flow in the straight tube and the tree network, respectively. The two-level Y-shaped tree network with m ¼ 2, a possible smallest dichotomous structure (see Fig. 1), the two-section straight tube, a special case of the tree network with m ¼ 1, the three-level tree network, and the three-section straight tube are used to represent the present work. The impact of the viscosity on the flow is analyzed with variations in power exponent n. The Newtonian flow with n ¼ 1 is revisited as a comparison to the power-law flow. First, we compare the model for the capillary-driven power-law fluids in a single tube with experimental results (Digilov, 2008) in Fig. 2. Digilov (2008) measured the quasi-steady-state flow of a power law liquid (1 wt% solution of carboxymethyl cellulosesodium salt) in a vertical tube. The tube radius R is 0.148 mm, the power exponent n is 0.78, the flow consistency index K is 0.06 Pa sn , and the surface tension is 0.066 N m 1 . In the initial stage of the capillary flow (i.e., h o6 mm in Fig. 2), the capillary force is dominant and the effect of the gravitational force is negligible. The mean velocity for the penetration distance h ¼ 6 mm in Fig. 2 is around 11.5 m/s, correspond to a contact angle equal to θ ¼ 77o (Digilov, 2008). Based on the model of Eq. (10), the theoretical model agrees satisfactorily with the experimental data as shown in Fig. 2. This model has also been verified by the measurement of capillary-driven blood flow with n ¼ 0:72 using the optical coherence tomography (Cito et al., 2012). Next we analyze the effect of shear rate on the power-law flow in the uniform tube with different values of n. Dividing Eq. (10) by 1 H 1=n þ 1 , RA1=n 1 þ 1n
the dimensionless flow time T versus the dimensionless
penetration distance h=H is obtained 1=n þ 1 h ; T¼ H
ð20Þ
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behavior across the range of shear rates to which the power can be fitted. For instance, Eq. (1) was validated by the experimental results of the power-law fluids, including aqueous solutions of Carboxymethyl cellulose with approximately n ¼ 0:53 and toluene solutions of Polyisobutylene with n ¼ 0:96, which flow within packed tubes (Christop and Middlema, 1965). Based on Eq. (20), we also find a simple but effective approach to determine the power component n by measuring the flow time T a and T b , corresponding to the penetration distance ha and hb , respectively. The expression of n is obtained by modifying Eq. (20), viz.
ln hha b
ð21Þ n¼ ln TT ab ln hha b
Fig. 2. Comparison of the penetration time of the capillary-driven power-law fluids versus the penetration distance in the uniform circular tube with R ¼ 0:148 mm between the present model and experimental results.
Fig. 3. The dimensionless time T versus the dimensionless flow distance h=H in a uniform tube for different values of power exponent n.
where H is a fixed flow distance. Fig. 3 reveals the variations of T with h=H at n ¼ 4; 2; 1; 1=2; 1=4. The shear thinning fluids move faster at the beginning of the capillary dynamics, but they slow down later and finally arrive at the end of the tube at the same T with the shear thickening and the Newtonian fluids, which was previously reported by Berli and Urteaga (2014). This phenomenon is ascribed to the dependence of the viscous resistance on the shear rate. It is pointed out that the apparent viscosity is given by μ ¼ kγ_ n 1 (Wang and Yu, 2011), which varies with the shear rate at different flow distance or time. When capillary flow occurs, the shear rate of the shear thinning fluids (n o1) is great with a relatively high starting velocity. Then the viscosity increases with the decrease in the shear rate, as more fluids penetrate the tube under a fixed capillary pressure and the flow slows. On the contrary, the viscosity of the shear thickening fluids (n 4 1) decreases with time. Hence, a convergence of T for all the fluids is found in Fig. 3, as predicted by Eq. (20). It is noted from the power correlation μ ¼ kγ_ n 1 , an unrealistic infinite viscosity is determined in the case of vanishing shear rate at the beginning of the shear thinning flow. Indeed, the effective viscosity in a real fluid has the upper and lower bounds, determined by the physical chemistry at the molecular level. With this objection, however, Eq. (1) still provides reasonable estimate of the power-law flow
Then we examine the model for the two-section tube with numerical results of Erickson et al. (2002), who simulated the capillary-driven flow with n ¼ 1 in a convergent tube by the finite element approach. The tube contains two sub-tubes connected by an extremely mild contraction. The first sub-tube has radius R1 ¼ 50 μm and h1 ¼ 20 mm length, the second sub-tube has radius R2 ¼ 25 μm, and the length of the mild contraction between the two sub-tubes is 2:86 mm (Erickson et al., 2002). The fluid properties are given as follows, γ ¼ 0:03 N=m, η ¼ 0:001 kg=s m and θ ¼ 30o . It has been found that the effect of the mild contraction on the total penetration time is very small (Shou et al., 2014c). Fig. 4 shows that the present model based on Eq. (11) and Eq. (16) agrees closely with numerical results of Erickson et al. (2002), indicating the accuracy of our model and the negligible effect of the mild contraction. Thus the model can also be applied to the tree networks composed of long sub-tubes, in which the junction effect is less significant (Shou et al., 2014b). The dynamics of capillary flow in single uniform tubes with radii 25 μm and 50 μm are also added for comparison. It is interesting to note that the flow is faster in the two-section tube than in both of the two uniform tubes at 20 mm o h o50 mm. In comparison to the uniform tube with R ¼ 25 μm, the two-section tube has a smaller viscous resistance; in comparison to the uniform tube with R ¼ 50 μm, the two-section tube has a higher capillary pressure. The evolution of time to flow distance for the power-law fluids in the two-section straight tube is shown in Fig. 5. The figure indicates the dependence of the dimensionless time T on the radius ratio e at five different values of n (i.e., n ¼ 4; 2; 1; 1=2; 1=4). In order to highlight the effect of e, the length ratio f is fixed as a constant 1. It is observed that T first
Fig. 4. Comparison of the penetration time of the capillary-driven fluids versus the penetration distance in the two-section circular tube between the present model and numerical results. The dynamics of capillary flow in uniform circular tubes are also added for comparison.
D. Shou, J. Fan / Chemical Engineering Science 137 (2015) 583–589
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Fig. 5. The dimensionless time T versus the radius ratio e of the two-section straight tube for different values of power exponent n at a fixed length ratio f ¼ 1.
of T reduce with the decrease in n, as expected. Besides, the optimal values of f increase inversely with n. It is interesting to find that T is slightly higher than 1 for n ¼ 4, whereas T of the Newtonian fluids stays smaller than 1 when e o 1 (Shou et al., 2014c). It is expected that the shear thickening fluids generate a higher viscous resistance than the Newtonian fluids under the same capillary pressure, enabling to delay the time required for a given flow distance. The dimensionless time T versus the length ratio f of the twosection straight tube for different n at e ¼ 2 is shown in Fig. 7. In this case, the second sub-tube has larger radius than the first subtube. Similar to Fig. 6, the values of T tend to be 1 when f tends to be 0 or infinite in Fig. 7. When f is in the intermedium regime, the maximum T is found for all values of n. The slower flow is caused by that the first sub-tube with smaller radius has higher flow resistance, whereas afterwards the second sub-tube with larger radius generates smaller capillary pressure for the movement of power-law fluids. The optimal f exists for the slowest flow when the contribution of the capillary and hydrostatic effects is properly allocated in the two sub-tubes. The maximum results of T reduce as the decrease in n, and the optimal f increases inversely with n. Based on Fig. 6 and Fig. 7, the asymmetry of flow times from different ends of the same tube is found, which is more remarkable in the case of shear thinning fluids such as blood. The liquid handling in an asymmetric fashion may be of interest to wound dressings: the designed porous layers composed of the above optimized tube bundles promote drainage ability of inside liquids but inhibit invasion of external liquids. The dynamics of the power-law flow in the two-level Y-shaped tree network is shown in Fig. 8. In the figure, the variations of the dimensionless time T against the radius ratio e at five different values of n (i.e., n ¼ 4; 2; 1; 1=2; 1=4) and f ¼ 1 are demonstrated. We find that the relationship between T and e is characterized by concave curves, indicating the fastest capillary flow in tree networks. Similar to the straight tube, the tree network has extremely large flow resistances with e ¼ 0 or e ¼ inf . And the optimal e close to 0:5 is found with a proper balance between the capillary pressure and the hydrostatic condition for the five values of n. The minimum T and the optimal e enhances with the increase in n. As well, the corresponding results of e to the minimum T decreases with decreasing n. Based on calculation we notice that no convergence of T is found at e ¼ 1 for different n, although the results of T look close to each other. It is reasonable that the Y-shaped tree network does not become a uniform tube at e ¼ 1 as the two-section straight tube. As well, the optimal values of e for tree networks are smaller than those for straight tubes.
Fig. 6. The dimensionless time T versus the length ratio f of the two-section straight tube for different values of power exponent n at a fixed radius ratio e ¼ 0:5.
Fig. 7. The dimensionless time T versus the length ratio f of the two-section straight tube for different values of power exponent n at a fixed radius ratio e ¼ 2.
decreases and then increases with e for different values of n. When e is close to 0 or much greater than 1, the flow resistance becomes extremely large due to the rather tiny second or first sub-tube. However, when the value of e is properly selected, i.e., e o1, the power-law flow becomes faster than in a uniform tube. It is expected that the first sub-tube with larger radius has less flow resistance, whereas the second sub-tube with smaller radius generate higher capillary pressure to drive the power-law fluid. Consequently, the concave curves in Fig. 5 show the existence of the minimum T or the fastest flow, as a result of the optimal dynamic balance between the capillary pressure and the hydrostatic condition. As well, we find that the minimum T decreases with the decrease in n. The corresponding results of e to the minimum T also reduce with decreasing n. When e ¼ 1 all the results of T are equal to 1, as the tube becomes uniform and identical with the control sample. Fig. 6 shows the variations of the dimensionless time T versus the length ratio f of the two-section straight tube at five different values of n (i.e., n ¼ 4; 2; 1; 1=2; 1=4). The radius ratio is fixed at e ¼ 0:5. The shape of all the curves in Fig. 4 is also concave, so the minimum T exists against f . The values of T tend to be 1 when it is close to 0 or infinite, because the two-section tube becomes a uniform tube. When f is in an intermedium region, the minimum T is found for all values of n. This phenomenon is attributed to the optimal distribution of the contribution of capillary pressure and hydrostatic condition in the two sub-tubes. The minimum results
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Fig. 8. The dimensionless time T versus the radius ratio e of the two-level tree network for different values of power exponent n at a fixed length ratio f ¼ 1.
Fig. 9. The dimensionless time T versus the length ratio f of the two-level tree network for different values of power exponent n at a fixed radius ratio e ¼ 0:5.
Fig. 11. The dimensionless time T versus the length ratio f of the three-level tree network for different values of power exponent n at a fixed radius ratio e ¼ 0:5.
optimized structure is greater than both the shear thickening and thinning fluids. As well, all the results of T tend to be 0:707 when f ¼ 0, because the tree network becomes a straight uniform pffiffiffi tube at R1 ¼ 2R0 . On the other hand, with f ¼ inf , the results of T tend to be 1 as the tree network becomes two uniform tubes as the control sample. Moreover, the results of T increase faster in tree networks than in straight tubes when f 4 1, because the flow resistances in the second sub-tubes of tree networks enhance considerably with the rapidly decreasing tube size. This phenomenon indicates that T of tree networks is more sensitive to microstructures than that of straight tubes. Based on Eq. (S13) in the supplementary material, the evolution of the penetration time to the penetration distance for the powerlaw fluids in the three-section straight tube is shown in Fig. 10. Analogous to the two-section tube, the minimum results of T versus the radius ratio e are found when f ¼ 1. The values of the minimum T in Fig. 10 are lower than those in Fig. 3 under the same n, indicating the acceleration of the flow in the three-section tube is more significant than that in the two-section tube. We also investigate the dynamics of the power-law flow in the three-level tree network by Eq. (S12) in the supplementary material. The minimum T is found against f at e ¼ 0:5 in Fig. 11, which is similar with the optimized results for the two-level tree network in Fig. 9. As well, the optimal values of f are much smaller in the three-level tree network than in the three-section tube. It is expected that the fastest flow can also be found for the structures with more sections or levels.
4. Conclusion
Fig. 10. The dimensionless time T versus the radius ratio e of the three-section straight tube for different values of power exponent n at a fixed length ratio f ¼ 1.
Fig. 9 shows the variations of the dimensionless time T versus the length ratio f for the two-level Y-shaped tree network. We assume the radius ratio as a constant, viz., e ¼ 0:5. The minimum T is found for all values of n, and the minimum results of T reduce as the decrease in n. We note that the optimal f that accounts for the minimum penetration time of the Newtonian fluid in the
A theoretical model has been developed to encode the dynamics of power-law fluids in the form of capillary flow confined in a straight tube and a Y-shaped tree-network. The power-law flow in both optimized structures can be accelerated in comparison to in parallel tube nets under a given volume. Based on dynamic competition between viscous and capillary effects, the optimal distribution of sub-tubes is found, leading to the minimum time or fastest flow for the liquid to fill the whole straight tube or tree network. The power-law flow behaviors in optimized structures are observed to deviate from those of Newtonian fluids on the basis of Washburn equation. For power-law fluids, the flow time of the shear thinning fluids with n ¼ 1=4 can be reduced by over 90% comparing with that in parallel tubes. As well, it is shown that the variations of T for tree networks with microstructures are more dramatic than those for straight tubes. So far, we limit our attention to an initial investigation of the fastest capillary-driven
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power-law flow in simple and basic structures. However, it is straightforward to extend the proposed robust framework to more realistic and complex systems such as interconnected networks and porous media under specific constraints. The limitations of present model are also summarized and will be addressed in future work. (1) In this paper, the stationary contact angle (SCA) is assumed during the flow process of the power-law fluids. We will take into account the effect of the dynamic contact angle (DCA) for more practical conditions, although the deviation of the SCA-based model from the DCAbased model is small in a single tube (Digilov, 2008). (2) The present model is conducted by assuming the effect of gravity negligible. It is expected that the fastest flow of the power-law fluids also exits under gravity as that of the Newtonian fluids found by Shou et al. (2014a). Thus we will concern the dependence of the optimal structure on the gravitational influence. (3) We have not investigate the effect of the junction of the straight tube or the tree network on the flow behavior. Numerical simulations have shown that the liquid meniscus suffers from complex deformation when moving through contraction and expansion regions of the tubes (Wiklund and Uesaka, 2013); in particular, the liquid passing through an expansion was pinned near a highcurvature point (Mehrabian et al., 2011), and the meniscus within the wider sub-tube is arrested for a time when the liquid moves into the sub-tubes with different sizes in a tree network (Sadjadi et al., 2015). Although the junction effect is found to be negligible for the long and thin sub-tubes (Shou et al., 2014b), modification of the current dynamic model is required when the short and thick sub-tubes are investigated. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ces.2015.07.009. References Bartolo, D., Boudaoud, A., Narcy, G., Bonn, D., 2007. Dynamics of non-newtonian droplets. Phys. Rev. Lett. 99, 174502. Bejan, A., Rocha, L.A.O., Lorente, S., 2000. Thermodynamic optimization of geometry: T- and Y-shaped constructs of fluid streams. Int. J. Thermal Sci. 39, 949–960. Berli, C.L.A., Urteaga, R., 2014. Asymmetric capillary filling of non-Newtonian power law fluids. Microfluid. Nanofluidics 17, 1079–1084. Cai, J., Yu, B., Zou, M., Mei, M., 2010. Fractal analysis of invasion depth of extraneous fluids in porous media. Chem. Eng. Sci. 65, 5178–5186. Cai, J.C., Perfect, E., Cheng, C.L., Hu, X.Y., 2014. Generalized Modeling of Spontaneous Imbibition Based on Hagen-Poiseuille Flow in Tortuous Capillaries with Variably Shaped Apertures. Langmuir 30, 5142–5151. Chen, J., Yu, B., Xu, P., Li, Y., 2007. Fractal-like tree networks increasing the permeability. Phys. Rev. E 75, 056301. Chen, Y.P., Cheng, P., 2002. Heat transfer and pressure drop in fractal tree-like microchannel nets. Int. J. Heat Mass Transf. 45, 2643–2648.
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