Holdup Characteristics of Gas−Non-Newtonian Liquid Flow through

Ind. Eng. Chem. Res. 2006, 45, 7287-7292

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Holdup Characteristics of Gas-Non-Newtonian Liquid Flow through Helical Coils in Vertical Orientation A. B. Biswas and S. K. Das* Department of Chemical Engineering, UniVersity of Calcutta, 92 A. P. C. Road, Kolkata 700 009, India

Experimental investigations have been carried out to evaluate the holdup for gas-non-Newtonian liquid flow through helical coils in vertical orientation. The experiments performed using 36 different helical coils and 4 different concentrations of sodium salt of carboxymethyl cellulose (SCMC) as non-Newtonian liquids. The intermittent flow regime was observed in the experiments. The effects of gas and liquid flow rate, coil diameter, helix angle, and liquid properties on the holdup were illustrated. An attempt has been made to fit the experimental holdup data by the Lockhart and Martinelli (Chem. Eng. Prog. 1949, 45, 39-48) correlation and the modified Lockhart and Martinelli correlation as presented by different authors. In another approach, a correlation was developed to predict the holdup as functions of the physical and dynamic variables of the system. Statistical analysis of the correlation suggests that they are of acceptable accuracy. 1. Introduction Helical coils are extensively used in compact heat exchangers, heat-exchanger networks, heating or cooling coils in the piping systems, intake in aircrafts, fluid amplifiers, coil steam generators, refrigerators, nuclear reactors, thermosyphones, and other heat transfer equipment involving phase change, in chemical plants as well as in the food, drug, and cryogenic industries. Either single-phase or two-phase flow can occur in helical coils, depending on specific applications. A literature survey indicated that numerous publications could be found dealing with flow phenomenon and the pressure drop in single-phase flow through helical coils.2-4 However, two-phase flow in helical coils has rarely been investigated5-16 as compared to single-phase flow studies. Mandal and Das12,13 reported the extensive literature survey for gas-Newtonian liquid flow through helical coils. It was observed from the literature that gas-Newtonian liquid flow through helical coils showed similar flow patterns as obtained in the case of two-phase gas-Newtonian liquid flow in a straight horizontal tube, but with a smaller liquid holdup and higher pressure drop. Mujawar and Raja Rao17 used the Lockhart and Martinelli1 approach to analyze the experimental data of gas-non-Newtonian liquid two-phase flow in helical coils. Bandaru and Chhabra18 reported the pressure drop for gasnon-Newtonian liquid flow through helical coils and concluded that the Lockhart and Martinelli1 technique is sufficient to correlate their experimental data. A literature review shows that most of the experimental studies reported in the literature are based on either Newtonian liquid flow or gas-Newtonian liquid two-phase flow through coils. Compared with Newtonian fluids, the flow of nonNewtonian fluids has remained very much unstudied, either theoretically or experimentally. However, accurate prediction of frictional pressure losses and holdup, which are the main hydrodynamic parameters for gas-non-Newtonian liquid twophase flow through coiled tubing, has remained a challenge in hydraulics design. This is mainly due to the lack of adequate experimental data, correlations, and proper understanding of the complex flow phenomena. So the systematic investigations on gas-non-Newtonian liquid flow through helical coils are necessary for a better understanding of the flow and to generate * Corresponding author. E-mail: [email protected].

experimental data for design purposes. Therefore, the aim of the present work is to investigate the characteristics of holdup of gas-non-Newtonian liquid flow through helical coils in vertical orientation. 2. Experimental Section The schematic diagram of the experimental apparatus is shown in Figure 1. The experimental apparatus consisted of an air-supply system, a liquid storage tank, centrifugal pumps, a test section, control and measuring systems for flow rate, pressure drop, and holdup, and other accessories. The details of the experimental setup and procedure are reported by Mandal and Das.13 The test liquids were prepared by dissolving the required amount of sodium salt of carboxymethyl cellulose (SCMC) (Loba Cheme Pvt. Ltd., Mumbai, India) in tap water and stirring until a homogeneous solution was obtained. Adding trace amounts of formalin prevented biological degradation. The content of the tank was kept at a constant temperature by circulating water through a copper coil. Detailed dimensions of the coils used in the experiments are given in Table 1. Four different concentrations, 0.2, 0.4, 0.6, and 0.8 kg/m3, of SCMC in water were used as the non-Newtonian liquids and atmospheric air was used as the other fluid. The SCMC solution is a time-independent pseudoplastic fluid, and the Ostwald deWaele model or the power-law model describes its rheological behavior. For a power-law fluid, calculations are generally carried out on the basis of the effective viscosity, µeff, which is given by

µeff ) 8n′-1D1-n′Vln′-1K′

(1)

The values of n′, K′, and other physical properties of the liquids are shown in Table 2. The rheological properties of the SCMC solutions were measured by means of a pipeline viscometer. Surface tension and density were measured using a DuNouy ring tensiometer and specific gravity bottles, respectively. The experimental system has been verified by experiments of single-phase flow through helical coils by Mandal.19 The results were compared with correlations available in the literature.20-24

10.1021/ie060420i CCC: $33.50 © 2006 American Chemical Society Published on Web 09/15/2006

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Figure 1. Schematic diagram of the experimental setup: (A) compressor, (B) oil filter, (C) gas cylinder, (D) gas regulator, (E) solution tank, (H) dryer, (M) manometer, (P) pump, (S) separator, (T) T-mixer, (HC) helical coil, (HE) heat exchanger, (LC) level controller, (RG1-RG2) gas rotameters, (RL1-RL2) liquid rotameters, (ST) stirrer, (SV1-SV2) solenoid valves, (T1-T2) thermometers, and (V1-V14) valves.

Figure 2. Variation of liquid holdup with gas flow rate at constant liquid flow rate.

Table 1. Dimension of Vertical Helical Coils tube diameter Dt, m

coil diameter Dc, m

angle, deg

0.009 33 0.009 33 0.009 33 0.009 7 0.009 7 0.009 7 0.012 0 0.012 0 0.012 0

0.176 2 0.216 2 0.266 2 0.176 7 0.216 7 0.266 7 0.133 0 0.178 0 0.212 0

0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12

Table 2. Physical Properties of the Liquids concentration, kg/m3

flow behavior index n′

consistency index K′, Nsn′/m2

density rl, kg/m3

surface tension σl, N/m

0.2 0.4 0.6 0.8

0.901 3 0.744 3 0.660 5 0.601 5

0.014 2 0.122 2 0.341 6 0.711 2

1 001.69 1 002.13 1 002.87 1 003.83

0.078 34 0.080 03 0.081 42 0.083 20

Figure 3. Variation of liquid holdup with gas flow rate at constant liquid flow rate and coil diameter.

The gas and liquid flow rates used in the experiments were in the range of 0.44 to 42.03 × 10-5 m3/s and 3.33 to 15.00 × 10-5 m3/s, respectively. The intermediate flow regime was observed under the experimental conditions. The liquid and the gas used in the experiment were at normal room temperature, 30 ( 2 °C. 3. Results and Discussion Effect of Gas Flow Rate on the Holdup. Figure 2 shows the effect of the gas flow rate on the liquid holdup for various liquid flow rates. It is clear that, as the gas flow rate increases, the liquid holdup decreases for a given liquid flow rate. Another feature of these curves is that the liquid holdup increases with increasing liquid flow rate at a constant gas flow rate. Effect of Coil Diameter on the Holdup. Figure 3 shows the effect of the coil diameter on the liquid holdup. The main feature of these curves is that the liquid holdup increases with increasing coil diameter at constant liquid flow rate, gas flow rate, and helix angle. This result can be explained by the introduction of slip effect in two-phase flow conditions. Because the liquid density is >600× higher than the gas density and the overall gas flow rate is much higher than the liquid flow

Figure 4. Variation of liquid holdup with gas flow rate at constant liquid flow rate and helix angle.

rate, the centrifugal forces acting on the liquid phase are much higher than those acting on the gas phase at any particular coil diameter. The liquid is accelerated because of the slip existing between the gas and liquid phases. As the coil diameter decreases, the slip increases, i.e., the liquid is accelerated to a greater extent; hence, the liquid holdup decreases. Effect of Helix Angle on the Holdup. Figure 4 shows that the two-phase liquid holdup decreases with increasing gas flow rate at constant liquid flow rate and coil diameter but is

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Figure 5. Variation of liquid holdup with gas flow rate at constant liquid flow rate for different system.

independent of the helix angle. Similar results were obtained by Banerjee et al.6, Xin et al.,11 and Mandal and Das13 for gasNewtonian liquid flow systems. Effect of Liquid Concentration on the Holdup. Figure 5 shows the liquid holdup as a function of gas flow rate obtained with the four different concentrations of SCMC solution. It is clear from the graph that the liquid holdup decreases with increasing liquid viscosity at constant liquid flow rate and helix angle. The gas has a retarding effect as the viscosity of the liquid increases, and also the slip is expected to be higher in a viscous liquid. But the retarding effect of the gas is more pronounced, and hence, the liquid holdup decreases with increasing liquid viscosity at constant liquid flow rate. Analysis of the Holdup. Lockhart and Martinelli1 proposed a graphical correlation for the analysis of the liquid holdup data for horizontal gas-Newtonian liquid systems. They presented a graphical relation between the liquid holdup, Rl, and the parameter X, defined by the following equation

X2 )

∆Pfl/L ∆Pfg/L

(2)

Initially the experiments on the single phase non-Newtonian liquid flow through coils have been carried out by Mandal,19 and it was observed that the Mishra and Gupta22 equation wellcorrelated the experimental data for both Newtonian and nonNewtonian liquid flow. The Mishra and Gupta22 equation for the laminar flow is as follows,

fcl - 1 ) 0.033(log Del)4.0 fst

(3)

Researchers in the field of gas-non-Newtonian two-phase horizontal flow have tried to extend the applicability of this graphical correlation. The experimental liquid holdup data obtained in the present case and the parameter, X, calculated from standard equations, have been plotted in Figure 6. It may be observed that the average liquid holdup in the present case is lower, and the Lockhart and Martinelli correlation is unable to predict the experimental data of liquid holdup properly because of the non-Newtonian characteristics of the liquid. In the Zuber and Findlay25 method, the actual gas velocity in two-phase flow, i.e.,

Vg ) Vg/Rg

(4)

Figure 6. Comparison of the experimental liquid holdup data with the Lockhart and Martinelli correlation.

Figure 7. Zuber and Findlay correlation.

is plotted against the superficial mixture velocity Vm, yielding the linear relationship, and is represented by

V g ) C oV m + V d

(5)

A typical plot is show in Figure 7. The values of the distribution parameter, Co, and the weighted average drift velocity, Vd, can thus be obtained. Once Co and Vd are known, Rg can be predicted by using eqs 5 and 4. Table 3 shows the values of Co and Vd for different non-Newtonian fluids used in the present study as well as values from literature for the vertical system. The higher value of Co is due to the fact that the gas experiences an expansion and the velocity profile in the slug liquid changes continuously. The negative value of Vd is due to the fact that the gas flows with higher velocity than the liquid in the slug. The values of Co and Vd vary with the liquid system. Though Rg can be predicted from the equations above, it will apply only for a particular liquid solution, and for generalization, Co and Vd would have to be expressed as a function of different parameters. Chandrakar’s26 study used improved gas-mixing devices, and hence, the values Co and Vd were varied with the mixing devices. Das et al.27 studied the gas-non-Newtonian liquid flow through vertical pipeline, whereas Clark and

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Table 3. Values of the Zuber-Findlay Parameters

investigator

avg drift velocity Vd, m/s

distribution parameter Co

system

-0.026 16 -0.045 03 -0.008 1 -0.210 7

present work

air-0.2 SCMC kg/m3 air-0.4 SCMC kg/m3 air-0.6 SCMC kg/m3 air-0.8 SCMC kg/m3

1.177 5 1.188 3 1.178 4 1.195 8

Chandrakar26

air-0.66 CMC kg/m3 air-1.50 CMC kg/m3 air-2.00 CMC kg/m3 air-2.5 CMC kg/m3

1.23-1.51 1.43-1.65 1.57-1.84 1.60-2.00

0.16-0.29 0.26-0.33 0.20-0.31 0.18-0.28

Das et al.27

air-0.50 CMC kg/m3 air-0.67 CMC kg/m3 air-0.83 CMC kg/m3 air-1.00 CMC kg/m3

1.40 1.47 1.50 1.67

0.53 0.52 0.50 0.62

1.07

0.25

Clark and Flemmer28 air-water

Flemmer28 reported the Zuber and Findlay parameters for airwater vertical flow. Eisenberg and Weinberger29 suggested modifications in the Lockhart and Martinelli1 parameters φL and X for annular twophase gas-non-Newtonian liquid flow through pipes considering the shear-thinning behavior of the non-Newtonian liquid. In the two-phase flow, the average shear rate in the liquid phase will be higher than that in the single-phase flow at the same flow rate of the liquid phase. To account for this effect of lower effective viscosity, a correction factor has been suggested in the estimation of the single-phase frictional pressure drop by Eisenberg and Weinberger. The corrected value of the frictional pressure drop for single-phase liquid flow is given by

∆P′fl )

(µeff)wtp (µeff)wl

∆Pfl

(6)

An expression for the correction factor (µeff)wtp/(µeff)wl has been given by Eisenberg and Weinberger on the basis of the assumptions of axial symmetry, constant location of the gasliquid interface, and rectilinear flow of liquid phase in the annular flow regime and is expressed as

(µeff)wtp

)

(µeff)wl

[

n′ - 1 ′ ′ 2n′ (3n′ + 1) 1 - Rg (1 - Rg(3n +1)/2n ) 3n′ + 1

[

]]

n′-1

(7)

They modified the Lockhart and Martinelli correlation1 as

φLEW )

(

)

∆Pftp/L ∆P′ft/L

0.5

( )

Figure 9. Comparison of the experimental liquid holdup with the Farooqi and Richardson correlation.

in Figure 8. It may be seen that the experimental liquid holdup data do not agree well with the Eisenberg and Weinberger correlation. Farooqi and Richardson30 modified the Lockhart and Martinelli1 parameter, X, to analyze their holdup data for gasnon-Newtonian liquid two-phase flow in horizontal pipeline. Their interest was aimed at utilizing the original Lockhart and Martinelli1 correlation for gas-Newtonian fluids in the case of shear-thinning fluids. They modified the parameter, X, as

(8) XFR ) X

and

∆P′ft/L XEW ) ∆Pfg/L

Figure 8. Comparison of the experimental liquid holdup data with the Eisenberg and Weinberger correlation.

0.5

(9)

Since the correction factor for ∆P′ft, as proposed by eq 6, has been developed for the annular two-phase flow of gasnon-Newtonian liquid, it will not be applicable in the present case. However, the present data have been analyzed by this method because this method incorporates the change of effective viscosity in the two-phase gas-non-Newtonian liquid flow. φLEW and XEW calculated from the present data have been plotted

( ) Vl VLC

1-n′

(10)

where VLC is the critical value of liquid superficial velocity when laminar flow ceases to exist, i.e., this value is obtained at a Reynolds number equal to 2000. Khatib and Richardson31 observed that their holdup data for vertical gas-shear-thinning liquid agrees well with the above-modified parameter. Figure 9 shows the liquid holdup versus XFR plot, but it failed to represent the data. The reason for this failure may be due to the action of continuous centrifugal forces along the flow path. It is extremely difficult to approach a theoretical analysis of a completed system where bubble separation, coalescence, and

Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7291 Table 4. Range of Variables Investigated for Vertical Helical Coil minimum e variable e maximum 70 e Rel e 2200 40 e Reg e 3000 0.783 180 × 10-7 e Npl e 0.380 82 × 10-3 0.035 042 e Dt/Dc e 0.090 226

of 1.98 for 1085 degrees of freedom at a 0.05 probability level and 95% confidence range. The ranges of variables investigated are shown in Table 4. 4. Conclusion

Figure 10. Correlation plot of the gas holdup.

centrifugal action are taking place throughout the flow field; the method of similitude has been used for the analysis. The tube roughness factor, k, is neglected because the investigation is carried out in a smooth poly(vinyl chloride) (PVC) tube. Here, the parameters influencing the liquid holdup are the physical and dynamic variables of the system. So, the liquid holdup will be the a function of the following parameters,

Rl ) F(Vl,Vg,Fl,Fg,µeff,µg,σl,Dt,Dc,g)

(11)

The influence of all these parameter on Rl is very complex; hence, by the use of dimensional analysis, this functional relationship can be reduced to an equivalent functional relationship as

Rl ) F(Rel,Reg,Npl,Dt/Dc)

(12)

The liquid property group, Npl [)µeff4g/Flσl3 ) (Wel2/Rel4Frl)], signifies some complex balance between viscous, surface tension, and gravitational forces. Equation 12 must follow the following limiting condition

Reg ) 0 at Rl ) 1 To accommodate the boundary condition, eq 12 is modified as

-ln(Rl) ) F(Rel,Reg,Npl,Dt/Dc)

(13)

-ln(1 - Rl) ) F(Rel,Reg,Npl,Dt/Dc)

(14)

Rg ) 1 - exp(F(Rel,Reg,Npl,Dt/Dc))

(15)

or

or

On the basis of eq 15, the multiple linear regression analysis for the liquid holdup data in helical coils was carried out, which yielded the following correlation

Re0.67(0.016 N -0.162(0.011 Rg ) 1 - exp(-0.094Re0.658(0.038 l g pl (Dt/Dc)0.188(0.048) (16) The values of the Rg as predicted by eq 16 have been plotted against the experimental values as shown in Figure 10. The variance of estimate and correlation coefficient of the above equation are 0.0047 and 0.9530, respectively, for a t value32

The two-phase holdup was measured for gas-non-Newtonian liquid (SCMC) flow through 36 helical coils in vertical orientation. Effects of the gas flow rate, liquid flow rate, concentrations of SCMC, and coil diameter on the liquid holdup have been critically examined. The effect of helix angle (0°-12°) on the liquid holdup was found to be negligible. The analysis of the experimental holdup data exhibits large deviations with the Lockhart and Martinelli1 or modified Lockhart and Martinelli correlation due to the non-Newtonian characteristics of the liquid and also the continuous application of centrifugal forces. An empirical correlation was developed to calculate the holdup. A detailed statistical analysis has shown that the correlations are of acceptable accuracy. Acknowledgment The authors would like to acknowledge the financial support in the form of a research grant from the University Grant Commission, Government of India, No. F 14-8/2003(SR) dated March 27, 2003. A.B.B. wishes to thanks the Council of Scientific and Industrial Research (CSIR), Government of India, for the Senior Research Fellowship, Award No. 9/28 (574)/2002EMR-I dated August 27, 2002. Nomenclature Co ) distribution parameter D ) diameter, m De ) Dean number, dimensionless f ) friction factor, dimensionless F ) function, dimensionless Fr ) Froude number, dimensionless g ) acceleration due to gravity, m/s2 k ) roughness height, m K, K′ ) consistency index (Nsn′/m2) L ) length, m n, n′ ) flow behavior index, dimensionless Npl ) liquid property group (µl4g/Fl σl3), dimensionless ∆P ) pressure drop, N/m Q ) flow rate, m3/s r ) radius of the tube, m Re ) Reynolds number, VDF/µeff, dimensionless V ) velocity, m/s V h ) actual velocity, m/s Vd ) weighted average drift velocity, m/s VLC ) critical velocity at Rel ) 2000, m/s We ) Weber number, dimensionless X ) Lockhart and Martinelli parameter, dimensionless Greek Letters R ) holdup, dimensionless φ ) two-phase multiplier, dimensionless µ ) viscosity, Pa‚s

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F ) density, kg/m3 σ ) surface tension, N/m τ ) shear stress, N/m2 Subscripts c ) coil g ) gas l ) liquid t ) tube w ) wall cl ) coil liquid EW ) Eisenberg and Weinberger fl ) frictional liquid fg ) frictional gas st ) straight tp ) two-phase wl ) wall liquid eff ) effective LEW ) liquid Eisenberg and Weinberger wtp ) wall two-phase Literature Cited (1) Lockhart, R. W.; Martinelli, R. C. Proposed correlation of data for isothermal two-phase two components flow in pipes. Chem. Eng. Prog. 1949, 45, 39-48. (2) Berger, S. A.; Talbot, L. Flow in curved pipes. Annu. ReV. Fluid Mech. 1983, 15, 461-512. (3) Das, S. K. Water flow through helical coils in turbulent condition. In Multiphase reactor and polymerization system hydrodynamics; Cheremisinoff, N. P., Ed.; Advances in Engineering Fluid Mechanics Series; Gulf Publishing Company: Houston, TX, 1996; pp 379-403. (4) Ali, S. Pressure drop correlations for flow through regular helical coil tubes. Fluid Dyn. Res. 2001, 28, 295-310. (5) Ripple, G. R.; Eidt, C. M.; Jordon, H. B. Two-phase flow in a coiled tubes. Ind. Eng. Chem. Process Des. DeV. 1966, 5, 32-39. (6) Banerjee, S.; Rhodes, E.; Scott, D. S. Studies on concurrent gasliquid flow in helically coiled tubes. I. Flow patterns, pressure drop and holdup. Can. J. Chem. Eng. 1969, 47, 445-453. (7) Akagawa, K.; Sakanguch, T.; Veda, M. Study on gas-liquid twophase flow in helically coiled tubes. Bull. Jpn. Soc. Mech. Eng. 1971, 14, 564-571. (8) Kasturi, G.; Stepanek, J. B. Two-phase flow. I. Pressure drop and void fraction measurements in concurrent gas-liquid flow in a coil. Chem. Eng. Sci. 1972, 27, 1871-1880. (9) Kasturi, G.; Stepanek, J. B. Two-phase flow. II. Parameters for void fraction and pressure drop correlations. Chem. Eng. Sci. 1972, 27, 18811891. (10) Saxena, A. K.; Schumpe, A.; Nigam, K. D. P.; Deckwer, W. D. Flow regimes, hold-up and pressure drop for two-phase flow in helical coils. Can. J. Chem. Eng. 1990, 68, 553-559. (11) Xin, R. C.; Awwad, A.; Dong, Z. F.; Ebadian, M. A.; Soliman, H. M. An investigation and comparative study of the pressure drop in airwater two-phase flow in vertical helicoidal pipes. Int. J. Heat Mass Transfer 1996, 39, 735-743.

(12) Mandal, S. N.; Das, S. K. Gas-Newtonian liquid flow through helical coils in horizontal orientation. Can. J. Chem. Eng. 2002, 80, 979983. (13) Mandal, S. N.; Das, S. K. Gas-liquid flow through helical coils in vertical orientation. Ind. Eng. Chem. Res. 2003, 42, 3487-3494. (14) Guo, L.; Feng, Z.; Chen, X. An experimental investigation of the frictional pressure drop of steam-water two-phase flow in helical coils. Int. J. Heat Mass Transfer 2001, 44, 2601-2610. (15) Murai, Y.; Sakai, K.; Toda, S.; Yoshikawa, S.; Tamayama, K.; Ishikawa, M.; Yamamoto, F. Local pressure fluctuation of gas-liquid twophase flow in helical tubes. Proceedings of the 15th International Symposium on Transport Phenomena, Bangkok, Thailand, May 9-13, 2004; pp 1-6. (16) Murai, Y.; Yoshikawa, S.; Toda, M.-I.; Ishikawa, M.-A.; Yamamoto, F. Structure of air-water two-phase flow in helically coiled tubes. Nucl. Eng. Des. 2006, 236, 94-106. (17) Mujawar, B. A.; Raja Rao, M. Gas-non-Newtonian liquid twophase flow in helical coils. Ind. Eng. Chem. Process Des. DeV. 1981, 20, 391-397. (18) Krishna Bandaru, S. V. S. R.; Chhabra, R. P. Pressure drop for single and two-phase flow of non-Newtonian liquids in helical coils. Can. J. Chem. Eng. 2002, 80, 315-321. (19) Mandal, S. N. Studies on the flow of liquid through helical coils. M.Tech. Thesis, University of Calcutta, India, 1996. (20) Ito, H. Laminar flow in curved pipes. Z. Angew. Math. Mech. 1969, 49, 653-663. (21) Srinivasan, P. S.; Nandapukar, S. S.; Holland, F. A. Friction factors for coils. Trans. Inst. Chem. Eng. 1970, 48, T156-T161. (22) Mishra, P.; Gupta, S. N. Momentum transfer in curved pipes. 2. Non-Newtonian fluids. Ind. Eng. Chem. Proc. Des. DeV. 1979, 18, 130-137. (23) Liu, S.; Masliyah, J. H. Axially invariant laminar flow in helical pipes with a finite pitch. J. Fluid Mech. 1993, 251, 315-353. (24) Ito, H. Friction factors for turbulent flow in curved pipes. J. Basic Eng. 1959, D81, 123-134. (25) Zuber, N.; Findlay, J. A. Average volumetric concentration in twophase flow system. J. Heat Transfer 1965, C87, 453-468. (26) Chandrakar, S. K. Gas dispersion in non-Newtonian liquid-jet ejector and two-phase co-current vertical flow. Ph.D. Thesis, Indian Institute of Technology, Kharagpur, India, 1985. (27) Das, S. K.; Biswas, M. N.; Mitra, A. K. Holdup for two phase flow gas-non-Newtonian liquid mixtures in horizontal and vertical pipes. Can. J. Chem. Eng. 1992, 70, 431-437. (28) Clark, N. N.; Flemmer, R. L. Predicting the holdup in two-phase bubble upflow and downflow using the Zuber and Finlay drift flux model. AIChE J. 1985, 31, 500-503. (29) Eisenberg, F. G.; Weinberger, C. B. Annular two-phase flow of gases and non-Newtonian liquids. AIChE J. 1979, 25, 240-246. (30) Farooqi, S. I.; Richardson, J. F. Horizontal flow of air and liquid (Newtonian and non-Newtonian) in a smooth pipe. Part I: A correlation for average holdup. Trans. Inst. Chem. Eng. 1982, 60, 292-305. (31) Khatib, Z.; Richardson, J. F. Vertical co-current flow of air and shear thinning suspension of kaoline. Chem. Eng. Res. Des. 1984, 62, 139154. (32) Volk, V. Applied statistics for engineers; McGraw-Hill Book Company: New York, 1958; p 345.

ReceiVed for reView April 4, 2006 ReVised manuscript receiVed July 24, 2006 Accepted August 16, 2006 IE060420I