Creeping Flow of Viscoelastic Fluid through a Packed Bed - American

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Creeping Flow of Viscoelastic Fluid through a Packed Bed Amit Sobti and Ravinder K. Wanchoo* Dr. SSB University Institute of Chemical Engineering and Technology, Panjab University, Chandigarh, India-160014 S Supporting Information *

ABSTRACT: Experimental results on investigation of elastic effects in creeping flow of viscoelastic fluids through a packed bed of spherical particles are presented. A simple form of the corrective Weissenberg number (We) function is proposed to extend the use of the Ergun equation, applicable to fixed bed flow of purely viscous fluids, for the prediction of friction factor in creeping flow of viscoelastic fluids. In order to quantify the effect of fluid elasticity on frictional pressure drop, aqueous solutions of polyacrylamide in the concentration range of 0.25%−0.6% (wt/vol) were used as test fluids and an extensive pressure drop-flow rate data were generated. Friction factor or pressure drop exhibited asymptotic behavior with variation in We, depicting the onset of viscoelastic effects at critical We values with a magnitude of 0.1 and the dominance of elastic effects over inertial effects up to We = 1.1. The percentage drag enhancement, as a function of fluid relaxation time, in the applicable range of We, is also reported. The proposed correlation will be useful in the field of enhanced oil recovery.

1. INTRODUCTION Understanding of hydrodynamic phenomenon for the flow of non-Newtonian fluids through packed beds is of significant importance to enhanced oil recovery processes. Excess pressure drop has been observed by various investigators in the case of flow of viscoelastic fluids through porous media, compared to that of purely viscous fluids under similar flow conditions.1 The increased flow resistance of viscoelastic fluids has been attributed to enhanced normal stresses generated by these fluids when subjected to elongational flow fields,2 with elongation rates that exceed certain “onset” conditions.3 The importance of viscoelastic effects in determining excess pressure drop in the case of flow through porous media has been studied by various investigators4−12 using different complex fluids flowing through porous media; however, there is significant discrepancy13 in the reported critical values of the Deborah number (De), from which viscoelastic effects begin to dominate. Sadowski4 conducted experiments with viscoelastic fluids, characterized using a three-parameter Ellis model, flowing through porous media, and has reported 0.1 as the critical value of the Ellis number (El), beyond which the observed experimental data deviate from the modified Darcy’s law, depicting the onset of viscoelastic regime. Marshall and Metzner5 studied the dependence of viscoelastic effects on the De value of the flow process. Their experimental results reveal that the threshold value of the De value at which viscoelastic effects first found to be measurable, ranges somewhere between 0.05 and 0.06. It is worth noting that the importance of viscoelastic effects was investigated under no inertial conditions. However, Gaitonde and Middleman6 carried out a similar analysis and found no evidence of such a departure, even at De values as high as 1.2. Siskovic et al.7 focused on this disagreement in the role of viscoelasticity on non-Newtonian flow in porous media. These authors have investigated the viscoelastic flow behavior of molten polymers, whose relaxation times are several orders of magnitude higher than the fluids used by Marshall and Metzner,5 through porous media. The authors concluded that the critical De value, where viscoelastic effects begin to assume © 2014 American Chemical Society

significance, is 0.19 or even higher and not 0.05, as reported earlier by Marshall and Metzner.5 Critical De values, as reported by various investigators, are given in Table 1. Literature shows large variation1,13 in the Table 1. Critical Values of the Deborah Number (De)/Weissenberg Number (We)/Ellis Number (El), as Reported by Various Investigators, Representing the Onset of Viscoelastic Effects investigator(s) Bendova et al.1

Kozicki2 Sadowski4 Marshal and Metzner5 Gaitonde and Middleman6 Siskovic et al.7 Vossoughi and Seyer9

Kemblowski and Dziubinski10 Kemblowski and Michniewicz11 Tiu et al.12

observation investigations are based on the assumptions that excess pressure drop for the flow of viscoelastic fluid reach 10% of the corresponding purely viscous fluids at critical De = 1 critical De = 0.5 critical El = 0.1 critical De ≈ 0.05−0.06 no viscoelastic effects were observed critical De ≥ 0.19 quantitative agreement is crude; however, the order of magnitude of De is the same as that observed by Marshal and Metzner5 critical De ≈ 0.052, as observed from their experimental data of loss coefficient versus De plot critical De = 0.07 critical We ≈ 0.022 as observed from their experimental data of loss coefficient versus We plot

magnitude of the critical De values, characterizing the onset of viscoelastic region. The commonly used representation of De (time of relaxation/time of observation) is expressed as

De =

λt (l / v )

Received: Revised: Accepted: Published: 14508

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June 9, 2014 August 8, 2014 August 21, 2014 August 21, 2014 dx.doi.org/10.1021/ie502321a | Ind. Eng. Chem. Res. 2014, 53, 14508−14518

Industrial & Engineering Chemistry Research

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Table 2. Physical and Rheological Properties of Test Fluids test fluid PAA PAA PAA PAA PAA PAA

(0.25 wt %/vol %) (0.3 wt %/vol %) (0.35 wt %/vol %) (0.4 wt %/vol %) (0.5 wt %/vol %) (0.6 wt %/vol %)

temperature (°C)

K (Pa sn)

n

λt (s)

ρf (kg m−3)

30 35 30 30 30 20

0.130 0.164 0.231 0.309 0.549 1.097

0.585 0.553 0.516 0.484 0.427 0.372

0.659 0.887 1.376 1.939 3.755 8.466

997.78 997.88 998.31 998.6 999.73 1001

Figure 1. Experimental setup.

where v and l are the characteristic velocity and linear dimension of the system, respectively. Evaluating the fluid elasticity using different methods13 such as shear viscosity data, primary normal stress difference data, die swell measurements, and dilute solution theories lead to divergent values of λt, which could be the reason for the large deviation in the magnitude of the critical De. For any flow with constant stretch history (i.e., slowly changing or essentially steady flow), the duration of observation is infinite and, hence, the De value alone is insufficient to fully characterize viscoelastic effects.14 An alternate way to characterize the viscoelastic flow behavior is the Weissenberg number (We), which describes the nonlinearity of the rheological response15 involving characteristic rate of deformation (U/Dp) and not the time of observation. The definition of We, as proposed by White,16 is expressed as We =

λt U Dp

Table 3. Column Details parameter diameter of column, Dc (mm) diameter of packing (spherical glass beads), Dp(mm) Dc/Dp bed porosity, ε sphericity of glass beads, φ bed permeability, k (m2) concentrations (wt %/vol %)

value 94.68 4.38

94.68 5.76

21.62 0.345 1 1.22 × 10−8 0.3%, 0.4%, 0.5%, 0.6%, EG

16.44 0.361 1 2.54 × 10−8 0.25%, 0.35%, EG

A wealth of theoretical,18−23 numerical,24−27 and experimental28−34 information, in the form of empirical, semiempirical correlations and numerical/theoretical models, is available on this subject, which has been reviewed by various investigators.13,35−37 Reviews show that much of the research efforts have been directed at evaluating the frictional pressure drop across the fixed bed of particles for a given flow rate, fluid properties, and bed geometrical parameters. However, further complications are added when considering the flow of viscoelastic fluids through porous media due to coexistence of shear and extensional components in the flow path. So far, no general methodology has been developed to address all of the cases of non-Newtonian flow.36 In general, four main models, viz. continuum models, capillary bundle models,

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Investigations have also revealed that a polymer solution may exhibit a drag-reducing effect when flowing through plain tube and drag enhancement may be observed when the same fluid flows through a tube fitted with a porous packing. The dual behavior17 is due to viscometric flow in plain tube and extensional flow in packed tubes. 14509

dx.doi.org/10.1021/ie502321a | Ind. Eng. Chem. Res. 2014, 53, 14508−14518


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Figure 2. Pressure drop per unit bed length (ΔP/L) versus superficial velocity (U0) for (a) Dc/Dp = 21.62 and (b) Dc/Dp = 16.44.

Figure 3. Friction factor ( f mod) versus modified Reynolds number (Remod). 14510



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Figure 4. Loss coefficient ( f mod × Remod) versus Weissenberg number (We), depicting the onset of viscoelastic effects and the dominance of inertial effects over elastic effects.

Figure 5. Pressure drop per unit bed length (ΔP/L) vs We, showing the asymptotic behavior.

others observed no viscoelastic effects, even with the similar polymeric solutions.5,6 There is a lack of experimental data in the literature on pressure drop in the case of viscoelastic fluids flowing through packed beds; this requires further attention. Present work aims to fulfill some of the above shortcomings.

numerical methods, and pore scale network modeling, are currently being used by various investigators to address the problems of porous media flow. These approaches have recently been reviewed by Sochi.36,37 More recently, many authors38,39 have represented flow through porous media in terms of microfluidic systems to assess the usefulness of the microchannels to imitate porous media in a controlled and simplified manner. The authors have assessed the validity of these porous media analogues for viscoelastic fluids and concluded that symmetric and asymmetric microchannels are suitable for studying the flow of viscoelastic fluids at low De values and above the critical De value, respectively. From the literature cited above, it is clear that the information available on the flow of the viscoelastic fluids through porous media is incomplete and somewhat inconclusive. The critical De values, as reported in the literature characterizing the onset of viscoelastic effects, are also in variance. Furthermore, anomalous flow behavior is also reported in case of viscoelastic fluids. Some authors have reported drag enhancement while

2. EXPERIMENTAL SECTION 2.1. Preparation and Characterization of Test Fluids. In the present investigation, aqueous solutions of polyacrylamide (PAA) in the concentration range of 0.25% (wt/vol) to 0.6% (wt/vol) were used as test fluids. PAA having viscosityaverage molecular weight of 8 × 106 was procured from National Chemical, Vadodara, India. Concentration levels were chosen to attain significant viscoelasticity. A batch of 25 L of each test fluid was prepared by slowly adding required amount of polyacrylamide in tap water under ambient conditions. The mixture was gently stirred for half an hour and kept for swelling at least for 24 h to ensure complete hydration of the polymer molecules. The swelled solution was then stirred for 24 h to 14511



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Figure 6. Percentage drag enhancement versus modified Reynolds number (Remod).

Figure 7. Percentage drag enhancement versus fluid elasticity (λt) within viscoelastic regime (0.1 ≤ We ≤ 1.1).

ensure complete homogeneity. A trace amount of formaldehyde was also added, to avoid bacterial degradation of the test fluids. The test fluids were subjected to steady as well as oscillatory tests on a dynamic rheometer (Model MCR-102 (with cone and plate geometry, CP 50-1), Anton Paar GmbH, Graz, Austria). The steady shear viscosity curves of the test fluids exhibit shear thinning behavior, which can adequately be approximated by a two-parameter power-law model over the range of shear rate relevant to the packed-bed experiments. The observed experimental data were fitted to a two-parameter power-law model that can be expressed as

μa = Kγ ṅ − 1

Frequency sweep tests were conducted within the linear viscoelastic region for all test fluids at required temperature to check for the presence of fluid elasticity. The observed experimental data (G′, storage modulus (given in units of Pa), G″, and loss modulus (also given in units of Pa) versus angular velocity (ω, rad s−1) were fitted to the well-known four-element Maxwell model, which is expressed as

3n + 1 ⎛ 8Uo ⎞⎛ ki ⎞ ⎜ ⎟⎜ ⎟ 4n ⎝ εDh ⎠⎝ 2 ⎠

∑

1 + (λiω)2

i=1 4

(3)

G″ =

∑ i=1

The shear-dependent viscosity (μa) was based on the wall shear rate (γ̇), as defined by Tiu et al.12 and Bansal et al.40 and expressed as γ̇ =

ηλ ω2 i i

4

G′ =

(5)

ηω i 1 + (λiω)2

(6)

The values of fluid relaxation time (λt), characterizing the fluid elasticity, were estimated using the following expression:42 4

(4)

λt =

where ki is 4.17, based on the Blake−Kozeny equation.41 14512

∑i = 1 λiηi 4

∑i = 1 ηi

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Figure 8. (a) Excess friction factor versus We. (b) Excess pressure drop versus We.

The observed physical and rheological properties of the test fluids under study are given in Table 2. 2.2. Pressure Drop Measurements. A schematic diagram of experimental setup used for the measurement of pressure drop across the packed bed for the flow of Newtonian and nonNewtonian fluids is given in Figure 1. A glass column randomly packed with spherical glass beads of uniform size (4.38 mm or 5.76 mm) was used in the present study. Test fluid was circulated from the tank through the column against gravity to avoid channeling and to ensure the absence of dead volume. The feed tank was maintained at a constant air pressure of 1.75 kgf/cm2, with the help of pressure regulators. This helps in maintaining the constant volumetric flow rate of the test fluid for a particular valve opening. The pressure drop across the bed was measured at different flow rates using a U-tube manometer filled with carbon tetrachloride (CCl4). The friction factor (f mod) and modified Reynolds number (Remod) were calculated using the following equations: fmod =

Remod =

μa (1 − ε)M n

(9)

where M is the wall correction factor , which can be expressed as ⎡ ⎤ 2Dp M = ⎢1 + ⎥ 3Dc (1 − ε) ⎦ ⎣

(10)

Bed permeability was estimated using the well-known Darcy’s law, which can be expressed as

k=

U0μ ΔP /L

(11)

For this purpose, ethylene glycol (EG) was used as the test fluid and the observed pressure drop-velocity data were fitted to eq 11 (for Remod ≤ 1) to determine the bed permeability. The porosity of the bed was determined using volume displacement method. Furthermore, the obtained bed permeability

ΔPDpε 3 ρf U0 2L(1 − ε)M

Dpρf U0

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Figure 9. Residual plot.

above the data for a Newtonian fluid with almost identical or higher viscosity. This excess pressure drop indicates the presence of enhanced normal stresses due to extensional flow. It is also observed that the pressure drop per unit bed length increases continuously with the increase in superficial velocity. For a fixed value of superficial velocity, the pressure drop increases with increase in fluid elasticity. Similar trend was observed in Figure 2b with Dc/Dp = 16.44. Figure 3 represents the dimensionless plot of friction factor (f mod) versus the modified Reynolds number (Remod) for all the concentrations (0.25−0.6 (wt %/vol %)) of viscoelastic fluids under study. The observed friction factor data, under creeping flow conditions, lie well above the Ergun equation46 (eq 13), suggesting the phenomenon of drag enhancement in these fluids. 3.2. Onset of Viscoelastic and Inertial Effects. To further investigate the phenomenon of drag enhancement, the observed loss coefficient data (f mod × Remod) are expressed as a function of We in Figure 4. The data clearly show an asymptotic behavior, demarcates the onset of viscoelastic effects from purely viscous flow and apparently the dominance of inertial effects over viscoelastic effects at higher Reynolds numbers. The critical We value at which the observed loss coefficient deviates remarkably from the baseline (i.e., f mod × Remod = 150 (applicable for purely viscous fluids)) is ∼0.1, depicting the onset of viscoelastic region and indicating the start of the dominance of elastic forces. The results are in good agreement with those reported earlier by Sadowski4 and, more recently, by Sobti et al.47 It is further observed that the loss coefficient increases linearly for a particular range of We values (0.1 ≤ We ≤ 1.1) beyond which the inertial effects could dominate the elastic effects at higher Reynolds numbers. Figure 5 represents the observed pressure drop data per unit bed length, as a function of We. It is observed that the pressure drop data collapse on a single line for all the concentrations of viscoelastic fluids and the pressure drop increases gradually up to a We value of 1.1. This suggests that a more suitable form of corrective We function, which takes into account the asymptotic behavior, must be incorporated while developing a unified correlation for the prediction of the friction factor, in the case of flow of viscoelastic fluids through a packed bed.

was also used to determine the bed porosity using the Blake− Kozeny equation,8 which is expressed as k=

Dp 2ε 3 150(1 − ε)2

(12)

The results obtained were in agreement with those observed using the volume displacement method. The details of the column and bed geometrical parameters are given in Table 3. Knowing the fluid relaxation time (λt, as determined using eq 7), fluid velocity (U), and geometrical parameters (ε, Dp), the Weissenberg number (We) was calculated using eq 2. An uncertainty analysis was carried out to evaluate the significance of scatter on repeated trials. The uncertainty in the experimental results was determined using the method of Kline and McClintock43 and Saha.44,45 The maximum uncertainty in estimating the friction factor ( f mod) and modified Reynolds number (Remod) under the worst conditions is 11% and 4%, respectively. However, the average uncertainty in the reported experimental data is 6% and 2% for f mod and Remod, respectively. The details of uncertainty analysis for reported experimental data are given in Table S1 in the Supporting Information.

3. RESULTS AND DISCUSSION The packed bed was first calibrated using Newtonian fluids (ethylene glycol (EG) and water). The observed pressure drop data were well-represented by the Ergun equation,46 which can be expressed as fmod =

150 + 1.75 Remod

(13)

3.1. Pressure Drop-Flow Rate Behavior for Viscoelastic Fluids. The observed pressure drop data, as a function of superficial velocity for various concentrations of viscoelastic fluid tested under two different bed geometries (Dc/Dp = 21.62 and 16.44), are represented in Figures 2a and 2b, respectively, along with the data on Newtonian fluids (EG). Each experiment was replicated with constant rheological and physical properties. It is clear from Figure 2a that the observed pressure drop data for a viscoelastic fluid concentration of 0.3% lie well 14514



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Figure 10. Parity between experimental friction factor data ( fexp) and predicted friction factor data ( fcal) ((a) low fexp range and (b) high fexp range).

3.3. Drag Enhancement. To quantify the magnitude of excess pressure drop or friction factor, percentage drag enhancement was plotted as a function of Remod and is represented in Figure 6. It is observed that the percentage drag enhancement increases with increase in Reynolds number for a fixed concentration. The data were further represented in a more realistic form in Figure 7 to see the effect of fluid elasticity (within the viscoelastic regime, i.e., 0.1 ≤ We ≤ 1.1) on percentage drag enhancement for a fixed Reynolds number. It is observed that the percentage drag enhancement increases as the fluid elasticity increases for a constant Reynolds number. Increases in drag as high as 73.37% were observed for a PAA concentration of 0.5% (wt/vol) at Remod = 0.03. 3.4. Correlation Development for Viscoelastic Fluids. Keeping in view the sigmoidal response of the packed bed to viscoelastic fluid, as reported in section 3.2, the following empirical correlation is proposed to represent the observed experimental data for creeping flow conditions (Remod < 1): ⎛ 150 ⎞ fve = ⎜ ⎟f (We) ⎝ Remod ⎠

where ⎡ (C We)C3 ⎤ 2 ⎥ f (We) = 1 + C1⎢ ⎣ 1 + (C2We)C3 ⎦

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The proposed model follows the boundary condition f ve → f vi for We → 0 and asymptotically takes care of the dominance of inertial forces over fluid elastic forces at higher Reynolds numbers. The observed data (f ve versus We) were fitted to eq 14, and the model constants were obtained using nonlinear regression techniques. The values of constants C1, C2, and C3 thus obtained are 0.67, 5.79, and 1.37, respectively. The uncertainty of proposed model has been determined for 95% confidence48 to assess the significance of newly reported results. The uncertainty analysis obtained using nonlinear regression techniques49 is presented in Table S2 in the Supporting Information. The 95% confidence limits for the proposed model are represented in Figure 8a. The proposed model has been evaluated for the fluids with significant elastic character (0.887 ≤ λt ≤ 8.47).

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Figure 11. Friction factor (f mod) versus modified Reynolds number (Remod) for non-Newtonian inelastic fluids, showing a comparison with existing data.

Figure 12. Comparison of proposed model with existing experimental data on viscoelastic flow through porous media.

The model constants (C1, C2, and C3), evaluated using the similar approach, are found to be 0.67, 5.71, and 1.39, respectively. Figure 8b represents the model described by eq 16, along with the experimental pressure drop data for the flow of viscoelastic fluids through a packed bed. The model equations can safely be expressed as

To further validate the proposed model, a residual plot has been constructed and is represented in Figure 9. The residual values are randomly distributed above and below the zero line, which depict the acceptability of the proposed model (eq 14). Figures 10a and 10b represent the parity between the observed friction factor data and model predicted values. It is observed that the model explains the experimental data well, to within a reasonable accuracy of ±10% of the parity. Another correlation used to directly predict the pressure drop per unit bed length has been proposed as ⎛ ΔP ⎞ ⎛ ΔP ⎞ ⎛ ΔP ⎞ ⎜ ⎟ ⎟ ⎜ =⎜ (We)⎟ ⎝ L ⎠ ve ⎝ L ⎠er ⎝ L ⎠

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⎛ ΔP ⎞ ⎛ ΔP ⎞ ⎡ (5.71We)1.39 ⎤ ⎜ ⎟ ⎟ ⎢1 + 0.67 =⎜ ⎥ ⎝ L ⎠ ve ⎝ L ⎠er ⎣ 1 + (5.71We)1.39 ⎦

(19)

and

(16)

where ⎡ (C We)C3 ⎤ ⎛ ΔP ⎞ 2 ⎜ ⎥ (We)⎟ = 1 + C1⎢ ⎝ L ⎠ ⎣ 1 + (C2We)C3 ⎦

⎛ 150 ⎞⎡ (5.79We)1.37 ⎤ fve = ⎜ ⎥ ⎟⎢1 + 0.67 1 + (5.79We)1.37 ⎦ ⎝ Remod ⎠⎣

where

We = (17) 14516

λt U0 εDp

(20)



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3.5. Newtonian and Non-Newtonian Inelastic Fluids. Figure 11 represents the experimental friction factor data on non-Newtonian inelastic and Newtonian fluids such as carboxymethyl cellulose (CMC) and ethylene glycol (EG), respectively, as a function of Remod, along with the proposed model (eq 14) for We → 0. The data for non-Newtonian inelastic fluids have been taken from the recent investigations of Chhabra and Srinivas,50 Kaur et al.,51 and Kaur and Wanchoo.52 It is observed that the proposed model shows reasonable agreement with the existing experimental data for non-Newtonian inelastic fluids. 3.6. Comparison of Proposed Correlation with Available Experimental Data on Viscoelastic Fluids. The model predictions using eq 14 were compared with the existing experimental data, on flow of viscoelastic fluid (Natrosol 250H and aqueous solutions of polyacrylamide) through porous media, of Sadowski4 and Sobti et al.47 respectively. It is evident from Figure 12 that the proposed model shows excellent agreement with experimental data of Sadowski4 and Sobti et al.47

against letter number SR/FTP/ETA-88/2011 (Sanction No. SERB/F/3488/2012-2013) and Panjab University Chandigarh is gratefully acknowledged.

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4. CONCLUSIONS Extensive set of experimental data on viscoelastic fluids (aqueous solutions of polyacrylamide) flowing through porous media have been generated. The data clearly show an asymptotic behavior, and it indicates the onset of viscoelastic effects from purely viscous flow (up to We ≤ 1.1) and the dominance of inertial effects over fluid elastic effect at higher Reynolds numbers. The critical value of the Weissenberg number (We) from which the viscoelastic effects start to dominate is 0.1, which agrees well with the earlier predictions of Sadowski.4 An increase in flow resistance is observed for the fluids with significant elastic character within the viscoelastic regime (i.e., 0.1 ≤ We ≤ 1.1). Based on the observed experimental data, a simple form of corrective Weissenberg number function (f(We) or ΔP/L(We)) is proposed to extend the use of the Ergun equation, applicable to fixed bed flow of purely viscous fluids, for the prediction of friction factor or pressure drop under creeping flow conditions of viscoelastic fluids. The model predicts the data well, to within a reasonable accuracy of ±10% of the parity. Furthermore, the proposed model also explains the experimental data on non-Newtonian inelastic fluids reasonably well, as reported earlier by various investigators. Hence, the proposed model can be treated as a unified correlation that can be used to predict the frictional pressure drop for the flow of Newtonian, as well as non-Newtonian, inelastic and viscoelastic fluids.

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Greek Letters

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ε = porosity of the packed bed γ̇ = shear rate, as determined by eq 4, s−1 ηi = dynamic viscosity in Maxwell model, Pa s λi = fluid relaxation time in Maxwell model, s λt = fluid relaxation time, as determined by eq 7, s μ = viscosity of Newtonian fluid, Pa s μa = apparent viscosity of the non-Newtonian fluid, Pa s ρf = density of test fluid, kg m−3

REFERENCES

(1) Bendova, H.; Siska, B.; Machac, I. Pressure Drop Excess in the Flow of Viscoelastic Liquids through Fixed Beds of Particles. Chem. Eng. Process. 2009, 48, 29−37. (2) Kozicki, W. Flow of FENE Fluid in Packed Beds or Porous Media. Can. J. Chem. Eng. 2002, 80, 818−829. (3) Haas, R; Durst, F. Viscoelastic Flow of Dilute Polymer Solutions in Regularly Packed Beds. Rheol. Acta 1982, 21 (4−5), 566−571. (4) Sadowski, T. J. Non-Newtonian Flow through Porous Media. II. Experimental. Trans. Soc. Rheol. 1965, 9 (2), 251−271. (5) Marshall, R. J.; Metzner, A. B. Flow of Viscoelastic Fluids through Porous Media. Ind. Eng. Chem. Fundam. 1967, 6 (3), 393−400. (6) Gaitonde, N. Y.; Middleman, S. Flow of Viscoelastic Fluids through Porous Media. Ind. Eng. Chem. Fundam. 1967, 6 (1), 145− 147. (7) Siskovic, N.; Gregory, D. R.; Griskey, R. G. Viscoelastic Behavior of Molten Polymers in Porous Media. AlChE J. 1971, 17 (2), 281− 285. (8) Christopher, R. H.; Middleman, S. Power-Law Flow through a Packed Tube. Ind. Eng. Chem. Fundam. 1965, 4 (4), 422−426. (9) Vossoughi, S.; Seyer, F. A. Pressure Drop for Flow of Polymer Solution in a Model Porous Medium. Can. J. Chem. Eng. 1974, 52, 666−669.

ASSOCIATED CONTENT

S Supporting Information *

Uncertainties in the reported experimental data (Table S1), and uncertainty analysis for the proposed model (eq 14) (Table S2) are provided as Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org/.

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NOMENCLATURE Dc = column diameter, m De = Deborah number Dh = hydraulic diameter; Dh = 4εDp/[6(1 − ε) + 4(Dp/Dc)], m Dp = diameter of spherical particle, m fcal = calculated friction factor fexp = experimental friction factor f mod = friction factor, as determined by eq 8 f ve = friction factor for viscoelastic fluids, as determined by eq 14 f vi = friction factor for non-Newtonian inelastic fluids, We → 0 K = flow consistency index, Pa sn k = bed permeability, m2, as determined by eq 11 L = length of the packed bed, m n = flow behavior index Rm = manometer reading, m of CCl4 Remod = modified Reynolds number U = actual fluid velocity in a packed bed; U = U0/ε, m s−1 U0 = superficial fluid velocity, m s−1 We = Weissenberg number, as determined by eq 20 ΔP = pressure drop across the packed bed, N m−2 (ΔP/L)ve = pressure drop per unit bed length for viscoelastic fluids, Pa m−1 (ΔP/L)er = pressure drop per unit bed length predicted by the Ergun equation, Pa m−1

AUTHOR INFORMATION

Corresponding Author

*Tel.: +91-1722534933. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support received from Science and Engineering Research Board (SERB), Government of India, New Delhi 14517



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Creeping Flow of Viscoelastic Fluid through a Packed Bed - American

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