Acta Mechanica Sinica (2015) 31(1):45 52 DOI 10.1007/s10409-015-0015-7 RESEARCH PAPER Effect of the size and pressure on the modified viscosity of water in microchannels Zhao-Miao Liu Yan Pang Received: 28 April 2014 / Revised: 15 July 2014 / Accepted: 1 September 2014 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2015 Abstract The phenomenon that flow resistances are higher in micro scale flow than in normal flow is clarified through the liquid viscosity. Based on the experimental results of deionized water flow in fused silica microtubes with the inner radii of 2.5 µm, 5 µm, 7.5 µm, and 10 µm, respectively, the relationship between water flow velocity and pressure gradient along the axis of tube is analyzed, which gradually becomes nonlinear as the radius of the microtube decreases. From the correlation, a viscosity model of water flow derived from the radius of microtube and the pressure gradient is proposed. The flow results modified by the viscosity model are in accordance with those of experiments, which are verified by numerical simulation software and the Hagen Poiseuille equation. The experimental water flow velocity in a fused silica microtube with diameter of 5.03 µm, which has not been used in the fitting and derivation of the viscosity model, is proved to be comsistent with the viscosity model, showing a rather good match with a relative difference of 5.56%. Keywords Microtube Viscosity model Pressure gradient Hagen Poiseulle equation 1 Introduction The sustained developments of microfluidics devices have made the basic investigations of microscale flow interested The project was supported by the National Natural Science Foundation of China (11072011 and 11002011) and the Doctoral Fund of Innovation of BJUT. Z.-M. Liu ( ) Y. Pang College of Mechanical Engineering & Applied Electronics Technology, Beijing University of Technology, 100124 Beijing, China e-mail: lzm@bjut.edu.cn by huge number of researchers in various fields. Wide applications of the microfluidics devices, miniaturization and integration of the micro total analysis system demand higher quality, more reliable and predictable performances of corresponding microchannels. As the development of the fabrication technology, the mechanism of flow and heat transfer performances of the microflow has proved to be a significant issue in the microfluidics. There have been a variety of researches on phenomena and influencing factors in the microfluidics [1 5], which show that some conclusions and empirical formulas derived in normal channels are not suitable for microscale flows. From the experiments of deionized water flowing in smooth fused silica microtubes with diameters of 50 µm, 76 µm, 101 µm, and 150 µm in adiabatic condition, the difference of friction factor between experimental value and theoretical one has been indicated by Parlak et al. [6] and was ascribed, under smaller diameter and higher Reynolds number, to the increase of viscous heating, which corresponds to a reduction of the microtube s diameter. Qu et al. [7] investigated the liquid flow in trapezoidal fused silica microchannels with hydraulic diameters of 51 µm and 169 µm and concluded that the experimental friction factor was larger than the theoretical value in the laminar flow. Li et al. [8] found that the friction factor of deionized water in stainless steel microtubes, whose diameters were 128.8 µm, 136.5 µm, and 179.8 µm with the relative roughness of 3% 4% (about 5.5 µm), was 10% 25% higher than the classical theoretical value. Cui et al. [9] experimentally studied the flow behavior of deionized water, isopropanol and carbon tetrachloride in microtubes of 3 10 µm diameters with Reynolds number of 0.1 24. The results showed that the friction factors of water in 3 µm and 5 µm microtubes were lower than those in the normal flow because of the boundary slip, while the results of isopropanol and carbon tetrachloride were contrary to that of water, that is, the experimental friction factor was larger and the difference grew as the pressure dropped.
46 Z.-M. Liu, Y. Pang Various reasons lead to the difference between the microflow and the normal flow, including viscous dissipation [6, 10], roughness of the inner wall [1, 7, 11], boundary slip [9] and other interactions on the solid-liquid interface [12]. The viscosity change incurred would directly affect the behavior of microscale flows. Wen [13] studied the viscosity of water in microscale flow and the hydrostatic boundary layer near a solid and proposed that the stronger interaction between the solid molecular and liquid molecular led to larger viscosity of liquid near the wall, making the liquid viscosity in microscale flow higher than that in normal flow. Nonino et al. [14, 15] investigated the liquid viscosity variation due to the temperature by comparing the numerical simulation results using different models, and indicated that the change of viscosity had significant and non-negligible effects on the microscale flow. Qu et al. [7, 11] studied the flow and heat transfer performances of water in silica microtubes and trapezoidal microchannels. In terms of the concept of apparent viscosity, the roughness viscosity was introduced into the liquid viscosity to correct the numerical results, which turned out to be particularly matched with the experimental results. Xu et al. [16] showed that the feature scale had influence on the liquid viscosity in such a way that the water viscosity in a sub-micron channel would be smaller than that in a conventional size channel. With the reduction of the flow feature size, the liquid viscosity characteristics which are applicable in the common size flow appear to be unsuitable for the micro size flow. The liquid flow behavior has significant influence on the design of microfluidics devices, so accurate predications of the microflow characteristics are in urgent need. This paper analyzes the difference of flows in microscale and conventional tubes to investigate liquid viscosity based on the experimental results of deionized water flowing in the fused silica microtubes with radii of 2.5 µm, 5 µm, 7.5 µm, and 10 µm. A viscosity model of water flow in microtube derived from the radius of microtube and pressure gradient is proposed. The viscosity model is verified by the experimental water flow velocity in a fused silica microtube with diameter of 5.03 µm. 2 Mathematical formulation and data selection The analysis and viscosity model derivation are made from the experimental data concerning driving pressure and corresponding flow rates by referring to the Hagen Poiseuille equation. 2.1 Laminar flow formulations in tubes For laminar flows of incompressible viscous fluid driven by the axial pressure drops in a tube, the Navier Stokes equation can be simplified as dp dz = µ1 d r dr ( r dv dr ), (1) where, p is the axial pressure, v is the axial velocity, µ is the dynamic viscosity of fluid, z is the axial coordinate, r is the radial coordinate. The Hagen Poiseuille equation of developed flows in an equal cross-section straight tube is Q HP = πr4 8µL, (2) where Q HP is flow rate, R is the tube s radius, is the pressure drop between the inlet and outlet, L is the tube s length. The velocity could then be shown as v HP = Q HP A = R2 8µL, (3) where v HP is the average velocity, A is the tube s crosssection area. Let Q exp to be the flow rate measured in the experiment, the velocity can thus be written as v exp = Q exp A. (4) 2.2 Data selection Experimental results of flow velocity and corresponding driving pressure of several kinds of microtubes of the same material, whose geometry parameters are known, are needed to derive the viscosity model related with the geometry of microtube. Li [17] researched the deionized water flowing in fused silica microtubes with radii of 2.5 µm, 5 µm, 7.5 µm, and 10 µm and length of 8 mm under various driving pressure [17], but his data can not be adopted here since the concept of liquid static area near the boundary, which was considered to reduce the effective diameter of microtubes, was involved in the procedure of data reduction. The new experimental velocity is calculated from the relationship of flow rate and driving pressure in Ref. [17] with Eq. (4), in which the theoretical average velocity can be derived from Eq. (3). In this way, the correlation between relative velocity ratios and pressure gradients will be achieved for further analysis. 3 Data reduction and analysis The variation of liquid viscosity has significant influences on the resistance of liquid flowing in the microchannel [18]. Hence, to understand the viscosity of liquids flowing in microtubes, the flow resistance should be analyzed, which can be indicated by the velocity under the same driving pressure. In such a case, the viscosity model derivation could start with the change of velocity. 3.1 The velocity under different pressure gradient As presented in Hagen Poiseuille equation, the average velocity v HP increases linearly with the driving pressure gradient /L when other conditions are identical in the conventional channel. The comparison between velocities from Eqs. (3) and (4) for flows in microtubes with radii of 2.5 10 µm under different pressure gradients are shown in Fig. 1. The differences between the theoretical velocity and exper-
Effect of the size and pressure on the modified viscosity of water in microchannels 47 imental one gradually increase with the reduction of microtube radii. For each microtube, the difference grows with the increase of the axial pressure gradient. The relative difference of the velocity from the Hagen Poiseuille equation and the experiments is increasing with the reduction of the microtube s radius. As shown in Fig. 1a, the relationship between velocity and pressure gradient has so small deviation that it can be presented with the linear function, while there is too obvious difference for the case with radius of 2.5 µm, as shown in Fig. 1d. The variation of the deviation under different diameters indicates that the Hagen Poiseuille equation could not describe the flow accurately when the microtube is small (which is around 2.5 10 µm in this investigation.). Fig. 1 Variation of velocity with pressure gradient for different tubes. a R = 10 µm; b R = 7.5 µm; c R = 5 µm; d R = 2.5 µm 3.2 The relative velocity ratio under different pressure gradient The deviation of experimental and theoretical velocities is affected by the diameter and driving pressure gradient (as shown in Fig. 1). To go further, the concept of relative velocity ratio is defined as S v = v HP v exp v HP. (5) Figure 2 depicts the variation of relative velocity ratio Sv defined in Eq. (5) with pressure gradient in microtubes of different radii. For a given microtube, the relative velocity ratio decreases with increasing pressure gradient, and its rate of reduction also diminishes. Since the velocity is quite small under a small driving pressure, the tiny difference between theoretical and experimental velocities could cause relatively large velocity ratio. There is a critical value of pressure gradient beyond which the Sv stabilizes gradually, but the stable value of Sv is inversely proportional to the radius of microtube. Under the same pressure gradient, smaller radius achieves higher velocity ratio. Fig. 2 Variation of Sv with pressure gradient The changing rate of Sv also has relation with the radius. When R = 10 µm, Sv drops sharply in the pressure gradient range of 0 1 MPa, and then remains rather stable. For a smaller tube, the reduction rate of Sv could be a little slower. When R = 2.5 µm, the range that Sv decreases quite gently is 0 15 MPa/m beyond which Sv almost will not change. Furthermore, the critical value of the pressure gradi-
48 Z.-M. Liu, Y. Pang ent for steady Sv is related with the radius of microtubes. The smaller the microtube s radius, the harder the Sv to achieve the steady value, which means that Sv could not be stabilized unless for a higher pressure gradient. 4 Derivation of the viscosity model From the above analysis, the conclusion that Hagen Poiseuille equation is not suitable for the flow in microchannels has been confirmed. The experimental velocity is lower than that predicted by the conventional theory. In addition, some researches indicated that the viscosity was larger than the physical viscosity in microscale flows [19] since the reduction of feature size changed the dominative elements which affected the flow. Hence, the decrease of the velocity can be explained from the point of a modified viscosity model, and a new viscosity model can be proposed. 4.1 The correlations of velocity, relative velocity ratio with pressure gradient Because of the deviation of linear approximation for the correlation between velocity and driving pressure gradient, as shown in Fig. 1, the variation of velocities with square root of pressure gradient is calculated and shown in Fig. 3, which approaches a parabola. Based on this assumption, there could be a relation as v exp = α L β L, where α and β are empirical coefficients. Table 1 Fitting coefficients of Eq. (6) Radius/µm A B 10 100 0.001 2 7.5 150 0.002 0 5 270 0.003 5 2.5 500 0.010 0 The plots of Eq. (6) are shown in Fig. 4, which can completely imply the value of Sv for each microtube and also the trend under a reasonable deviation. In these four cases, Eq. (6) exhibits a more evident gap for the microtube with radius of 10 µm than those for other smaller tubes, which means that the applicability of the relation should have dependence on the size of the tube, which is also indicated in the velocity and pressure gradient correlation (Fig. 1). The expression of experimental velocity which includes Sv can be derived from Eqs. (3) and (5) is derived in the following form v exp = (1 S v) R2 8µ L. (7) From the derived (7), since Sv varies with the pressure gradient, the correlation between experimental velocity and driving pressure gradient is nonlinear, which is consistent with the velocity and pressure gradient relationship (Fig. 1). Substituting Eq. (6) into Eq. (7) and simplifying the result yields v exp = (1 B) R2 8µ L A R2 8µ L. (8) Setting α = (1 B) R2 8µ, β = A R2, we can rewrite Eq. (8) in 8µ the following form Fig. 3 Variation of velocity with the square root of pressure gradient To verify the parabolic relationship, the correlation of Sv with pressure gradient is employed. From the inversely proportional feature and the steady value of Sv, the correlation could be fitted as A S v = + B, (6) /L where A and B are coefficients of fitness, which vary for varied microtubes, as shown in Table 1. v exp = α L β L. (9) Equation (9) indicates a parabolic relationship between velocity and driving pressure gradient of deionized water flowing in the straight microtube, and the final shape is determined by the radius of microtube and fitting coefficients A, B. According to the value of A, B, there should be some regularity with the radius in their variety. A, B can be fitted as A = 0.001 25, R B = 4 10 11 R 1.5. (10) Plots of Eq. (10) are shown in Fig. 5, which can accurately describe the change of A and B. Substituting Eq. (10) into Eq. (8), the fitting expression of experimental velocity is shown as
Effect of the size and pressure on the modified viscosity of water in microchannels 49 Fig. 4 Fitting plots of Sv and pressure gradient for different tubes. a R = 10 µm; b R = 7.5 µm; c R = 5 µm; d R = 2.5 µm Fig. 5 Relationship between A, B and the radius v exp = (1 4 ) 10 11 R 2 R 1.5 8µ L 0.001 25 R R 2 8µ L. (11) The velocity predicted by Eq. (11) has a nonlinear relationship with the square root of pressure gradient as shown in the experimental velocity plot (Fig. 3) and depends on the radius of the channels. Equation (11) can properly describe the results of experiments in all microtubes involved in the present study (shown in Fig. 6). 4.2 Effects of radius and pressure gradient on viscosity To explain the difference between microscale and conventional flow more reasonably and endue proper physical meaning to Eq. (9), the concept of modified viscosity is employed for analysis. Compared to the Hagen Poiseuille equation, the velocity equation (7) can be written as v exp = 1 R 2 µ/(1 S v) 8 L. (12) Hence, the deionized water s modified viscosity in fused silica microtubes with radius of 2.5 10 µm can be given in the following form µ µ app = (1 S v), which can be transformed into
50 Z.-M. Liu, Y. Pang Fig. 6 Comparison of the experimental velocity and the fitted velocity for different tubes. a R = 10 µm; b R = 7.5 µm; c R = 5 µm; d R = 2.5 µm µ µ app = 0.001 25 1 R /L 4. (13) 10 11 R 1.5 Equation (13) is the expression of the modified viscosity model, depicting that the smaller the radius of microtube, the larger the difference between the modified viscosity and the physical viscosity. The velocity reduced for water flow under this hydrophilic wall condition could be reduced by the resistant force between liquid and solid wall, which will be relatively small when the shear stress is growing with the pressure difference. In contrast, the interaction between liquid and wall will be increased by the reduced tube diameters. The larger the resistance force, the higher the modified viscosity from the model of Eq. (13). The modified viscosity gradually approaches the physical viscosity with the increase of the radius and the driving pressure gradient, which is consistent with the physical phenomenon. 5 Verification of the viscosity model As the properties of the wall (Wettability, et al.) will affect the liquid flow [20 22], this modified viscosity is supposed to apply to the water flow in fused silica tubes with radii of 2.5 10 µm. Numerical simulations are used to verify if a result, which is consistent with the original experimental one, could be achieved by employing the modified viscosity model. Hence, the validity of the model and its derivate process could be proved, while the adaptability can be verified by the experimental result of deionized water flowing in the fused silica tube with diameter of 5.03 µm which has not been used in the fitting and derivation. 5.1 Validity of the modified viscosity model The simulated geometry models are the same as used in the experiment of Ref. [17], and for water viscosity we employed the model described in Sect. 3.2. Fixed pressure condition is used for inlet/outlet in the simulations. The simulation results and the experimental are compared in Fig. 7, which shows a small difference between these two kinds of velocities. The numerical velocity using the modified viscosity model is more accurate in predicting the experimental velocity than the Hagen Poiseuille equation, as shown in Fig. 1. It could be known from the comparison that the viscosity model can properly predict the water flow in the microtubes, which implies that the deduction of the model could be correct in the radius range of 2.5 10 µm.
Effect of the size and pressure on the modified viscosity of water in microchannels 51 Fig. 7 Verification of the viscosity model s validity for different tubes. a R = 10 µm; b R = 7.5 µm; c R = 5 µm; d R = 2.5 µm 5.2 Adaptability of the modified viscosity model Wang et al. [12] experimentally researched the flow of deionized water in fused silica microtubes with inner diameter of 5.03 µm, which is not used in the fitting and derivation of the modified viscosity. For density 0.995 68 g/ml and viscosity 0.866 5 MPa s, the relation of velocity and driving pressure gradient could be calculated from the plot of flow rate vs. Reynolds number given in Ref. [12]. Comparing the numerical velocity with experiments and the result of Hagen Poiseulle equation in Fig. 8, it is shown that the experimental velocity is lower than the theoretical one from Hagen Poiseulle equation but similar with the velocity deduced from the modified viscosity model. The modified viscosity model can predict the velocity quite well, and they almost coincide in the pressure gradient range of 6 8 MPa/m. The maximum velocity deviation of the model compared from the experimental result is less than 0.6 mm/s. Expect for the low pressure cases whose experimental velocity are too slow to obtain small relative deviation (which is still less than 33.2%), the modified viscosity model achieves an rather good match with the experiment under a relative difference of 5.56%. For the Hagen Poiseulle equation, the situation differs from that of viscosity model. The velocity from the Hagen Poiseulle equation can be 55% higher than the experimental value for the low pressure cases. Even in higher pressure condition, the relative difference is still beyond 9.6%. Fig. 8 Verification of the modified viscosity model s adaptability 6 Conclusions Based on the experiment results of deionized water flow in fused silica microtubes with inner radii of 2.5 µm, 5 µm, 7.5 µm, and 10 µm, a viscosity model of water flow in microtube is proposed and verified, which is affected by the radius of microtube and the pressure gradient. (1) The relative velocity ratio decreases with increasing pressure gradient, and its rate of reduction also diminishes. Under the same pressure gradient, the smaller the radius, the higher the velocity ratio. There is a parabolic relationship between velocity and driving pressure gra-
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