Pressure Losses in Bends during Two-Phase GasNewtonian Liquid

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Ind. Eng. Chem. Res. 2001, 40, 2340-2351

Pressure Losses in Bends during Two-Phase Gas-Newtonian Liquid Flow S. N. Mandal† and S. K. Das* Department of Chemical Engineering, University of Calcutta, 92 A.P.C. Road, Calcutta 700009, India

Experimental investigations have been carried out to evaluate the two-phase pressure drop across different types of bends in the horizontal plane for gas-Newtonian liquid flow. Four different Newtonian liquids were used for the experiments. Correlations have been developed to predict the two-phase friction factor as a function of various physical and dynamic variables of the system. Statistical analysis of the correlation suggests that the correlations are of acceptable accuracy. Introduction

Table 1. Dimensions of Bends

Bends are an integral part of any pipeline transport processes, and the flow patterns developed are more complex than those of straight tubes. Fluid motion in a bend is not parallel to the curved axis of the bend. As the flow enters into the bend, the centrifugal force acts outward from the center of curvature on the fluid particles. The slower moving fluid particles move along paths whose radii of curvatures are smaller than those of the faster moving particles. This leads to the onset of secondary flow such that fluid nearer the wall moves toward the inner wall while fluid far from it flows to the outer wall. Das1,2 discussed in detail the singlephase flow through curved geometry and piping components. Two-phase flow in a straight pipe is complex, involving a number of flow regimes. However, two-phase flow in a bend is always in the developing stage3 and is more difficult to analyze than that for a straight pipe. The curvature generates a centrifugal force and causes the denser phase (i.e., liquid) to move away from the center of curvature, while the air flows toward the center of curvature. Separation of phases in this way is likely to give rise to significant slip between the phases. Studies on the bend for a two-phase flow are relatively few in numbers. Sekoguchi et al.4 have investigated the airwater flow through a 90° bend and analyzed the twophase pressure drop across the bend data using parameters φlb and Xb, which are similar to the LockhartMartinelli5 parameters. Maddock et al.3 studied the flow structure during two-phase flow through different types of bends (30-90°) and concluded that the flow was in the developing region within the bend. Chisholm6,7 developed equations for pressure drop prediction based on a two-phase multiplier for 90° and 180° bends. Experimental observations of the flow structure and pressure drop have been presented by Hoang and Davis8 for air-water froth flow in the entrance of 180° bends. They observed that the overall loss coefficients were substantially larger than those in single-phase flow, particularly for bends with a larger radius of centerline curvature, and the flow structure was almost stratified. Norstebo9 reported the two-phase pressure drop in pipe fittings, the 90° bend, and the return bend in the * To whom all correspondence should be addressed. E-mail: [email protected]. † Present address: Technical Teachers’ Training Institute (ER), Block FC, Sector III, Salt Lake City, Calcutta 700 091, India. E-mail: sailen•[email protected].

angle R (deg)

radius of centerline curvature Rc (m)

linear length of the bend portion Lb (m)

45 90 135 180

0.1195 0.0505 0.0645 0.1060

0.13 0.08 0.13 0.33

Table 2. Physical Properties of the Test Liquids

liquid used

density Fl (kg/m3)

viscosity µl (kN‚s/m2)

surface tension σl (kN/m)

995.67 996.37

0.85 0.84

71.23 50.00

1067.95

2.00

63.38

1098.20

2.91

68.40

water 1% amyl alcohol-water solution (% by volume) 30% glycerin-water solution (% by volume) 42% glycerin-water solution (% by volume)

refrigeration plant. He analyzed the pressure drop data across the pipe fittings by correlating φlb and Xb in a manner similar to that of the Lockhart-Martinelli correlation. He also compared the experimental pressure drop data with the Chisholm7 method and found a +110% deviation. Subbu et al.10 studied the air-water flow through different U-bends, and they developed an empirical correlation for the prediction of the two-phase pressure drop. Das et al.11 studied gas-non-Newtonian liquid flow through bends and developed an empirical equation to calculate the two-phase friction factor. Even today the data or equations for pressure loss in twophase gas-Newtonian liquid flow through bends are meager, and the present study is an attempt to generate experimental data on pressure drop with respect to certain finite bends in the horizontal plane. Experimental Section The schematic diagram of the experimental apparatus incorporating a 180° bend is shown in Figure 1. For other bends the upstream straight portion was identical, but the downstream portion and the gas-liquid separator were shifted as per bend angle. The experimental setup consisted of a liquid storage tank (0.45 m3), an air supply system, a test section, a gas-liquid separator, control and measuring systems for the flow rates, pressure drops, and other accessories. The test section consisted of a horizontal upstream straight tube of 4.5 m length, a bend portion, and a horizontal downstream straight tube of 3 m length. The internal diameter of

10.1021/ie0003988 CCC: $20.00 © 2001 American Chemical Society Published on Web 04/14/2001

Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2341

Figure 1. Schematic diagram of the experimental setup for the 180° bend.

Figure 2. Typical static pressure distribution curve for a Newtonian liquid (45° bend).

the tubes and the bends was 0.019 m. The reason for having long horizontal upstream and downstream portions before and after the bend was to achieve fully developed flow conditions to facilitate the measurement of pressure drop across the bend. The bend portion of the test section was connected to the upstream and downstream portions with the help of flanges. The entrance and exit lengths were 2.0 and 1.4 m, respectively, which were more than 50 pipe diameters to ensure fully developed flow. Before the test section, a 0.5 m length Perspex tube of the same diameter was incorporated in the system to visualize the flow pattern. The rest of the test section was fabricated from mild

steel. The test section was fitted horizontally with the help of a leveling gauge. It was provided with pressure taps (piezometric ring) at different points in the upstream and downstream sections of the pipe and in bends. Four different types of bends have been used, and their radii of curvature are given in Table 1. The bends were specially manufactured in order to ensure uniform internal diameter, constant curvature, and roundness. Four liquids were used for the experiment, and their physical properties are given in Table 2. The gas and liquid flow rates used in the experiments were in the ranges of 0.42 × 10-4-5.97 × 10-4 and 2.00 × 10-4-

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Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001

Figure 3. Typical static pressure distribution curve for a Newtonian liquid (90° bend).

Figure 4. Typical static pressure distribution curve for a gas-Newtonian liquid (45° bend).

5.975 × 10-4 m3/s, respectively. Experiments were repeated a number of times to ensure reproducibility of the data. The observed flow pattern was intermittent at the inlet of the bend. The temperature of the liquid and gas used in the experiments was maintained at 30 ( 2 °C, i.e., ambient temperature. Results and Discussion Evaluation of the Two-Phase Pressure Drop across the Bend. The detailed technique of the evalu-

ation of the pressure drop across the bend has been described by Subbu et al.10 The pressure drop due to the bend was obtained from the difference between the static pressure of the upstream fully developed flow region and the static pressure of the downstream fully developed flow region across the bend. Typical static pressure distribution curves are shown in Figures 2 and 3 for single-phase Newtonian liquid flow through bends and Figures 4-7 for gas-Newtonian liquid flow through bends. It may be noted that for gas-Newtonian liquid




flow through bends the slope of the static pressure variation in the upstream fully developed flow and the downstream fully developed flow may not be the same because of the modification of the flow structure caused by the bubble migration, coalescence, etc., for the different flow systems. Single-Phase Flow. Effect of Liquid Characteristics on the Pressure Drop across the Bend. Figure 8 shows the pressure drop across the bend as a function of the liquid velocity. It is clear from the graph

that as the viscosity of the liquid increases, the pressure drop across the bend increases. Analysis of the Experimental Pressure Drop Data. The pipe fitting is characterized by a loss coefficient, KL, and the frictional energy loss, hf, is evaluated from

hf )

∆Pb V12 ) KL F1 2

(1)

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as

Table 3. Comparison of the KL Values loss coefficient, KL type of bend

exptl

McCabe et al.12

Perry et al.13

45° 90° 135° 180°

0.86 1.39 1.05 1.32

0.4 0.9

0.2 0.45

2.2

1.5

Figure 9 shows the frictional energy loss (∆Pb/Fl) versus velocity head (Vl2/2) data for different bends. The measured KL values for different bends are compared with the literature values12,13 (Table 3). The main drawback of this method is that it depends on the Reynolds number (Kittredge and Rowley,14 Ito,15 and Hooper,16 whereas Sookprasong et al.17 reported that it was not very sensitive to the Reynolds number in the range of 1 × 104-2.2 × 105). Initially, the pressure drops were measured first in a straight horizontal tube. Experimental values of Fanning’s friction factor were found to be within +5% with the Blasius equation for a smooth straight pipe in turbulent flow conditions

fh ) 0.079Rel-0.25

fb ) Dt∆Pb/2Vl FlLb

(3)

Parameters influencing the friction factor are the physical and operating variables of the system. The physical variables include the radius of the tube, the radius of curvature of the bend, and physical properties of the fluid, while the operating variable is the flow rate of the fluid. So, the functional relationship may be written

(4)

To extend the applicability of eq 4 to all of the different bends in the horizontal plane, an angle factor R/π, defined as the ratio of the angle of the bend to that of the 180° bend, has been introduced in the functional relationship as

fb ) F(Rel,Rt/Rc,R/π)

(5)

In order that the type of relationship obtained in eq 5 should also be able to predict the friction factor for the straight tube, i.e., Rc f ∝, the above equation has been modified to the following form:

fb - fh ) F(Rel,Rt/Rc,R/π)

(6)

On the basis of eq 6 the multivariable linear regression analysis of the experimental data yielded the correlation

fb - fh ) 10.2075Rel-0.2754(0.0673(Rt/Rc)1.6847(0.1277 × (R/π)-0.7573(0.0859 (7)

(2)

The analysis of the experimental pressure drop data across the bend is carried out by means of the friction factor. The friction factor, fb, for a bend is calculated by the well-known Fanning friction factor equation 2

fb ) F(Rel,Rt/Rc)

for

3000 < Rel < 50 000 0.08 e Rt/Rc e 0.19 45° e R e 180° The values of fb - fh predicted by eq 7 have been plotted against the experimental values as shown in Figure 10. The correlation coefficient and the variance of estimate are 0.9221 and 0.0750, respectively, for a t value18 of 1.98 for 152 degrees of freedom at 0.05 probability level and 95% confidence range.


Figure 8. Variation of the pressure drop across the bend with liquid flow rate.

Streamwise Pressure Loss due to the Bend. Observations of pressure were made in the long upstream and long downstream portion of the bend in order to obtain the overall pressure drop across the bend. The static pressure starts to deviate from steady value within 15 pipe diameter for the 45°, 90°, and 135° bends and within 20 pipe diameter for the 180° bend upstream of the inlet of the bend, depending on the flow rate. In the downstream of the bend, the pressure recovery lengths were found to be within 25 pipe diameter for the 45°, 90°, and 135° bends and within 25 pipe diameter for the 180° bend, depending on the flow rate. Ito15 reported that the pressure drop starts to deviate from the developing flow 5-10 pipe diameters upstream of the inlet and gradually approaches the developing flow in the downstream straight section about 40-50 pipe diameter from the bend exit, whereas Ohadi et al.19 showed that the developing flow reached within 20 pipe diameter from the bend outlet. In the upstream and downstream of the fully developed flow region of the bend, it was observed that friction factors were found to be in very good agreement with the Blasius equation (2). Hence, the presence of the bend did not affect the fully developed friction factors. Similar results were also observed by Ohadi et al.19

Two-Phase Gas-Newtonian Liquid Flow through the Bends. Effect of the Air Flow Rate on the TwoPhase Pressure Drop. Figure 11 shows the typical pressure drop across the 45° and 90° bends as a function of the air flow rate. As the air flow rate increases, the two-phase pressure drop across the bend gradually increases. Figure 12 shows the typical pressure drop across the 45° and 135° bends as a function of the air flow rate and liquid properties as a parameter. It is clear from the graph that for a constant liquid flow rate the two-phase pressure drop across the bend is higher for highly viscous liquids. An increase in the viscosity has a retarding effect on the liquid phase, and slip is expected to be higher in viscous liquid. Hence, the twophase drop increases. Surface tension also has a pronounced effect on the two-phase pressure drop. Slight foaming has been observed in the case of air-1% amyl alcohol-water solution flow through the bends versus no foaming in the air-water two-phase flow. It reduces the slip between the phases and also creates a tendency toward a retarding effect on the gas phase. Because the gas-phase pressure drop is very small in comparison to that of the liquid phase, the net effect is a decrease in the two-phase pressure drop. Hence, the two-phase pressure drop slightly decreases in comparison to the air-water system.

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Figure 9. hf versus Vl2/2 for different bends.

Analysis of the Two-Phase Frictional Pressure Drop. In the horizontal pipeline, the total pressure drop is the frictional pressure drop because the gravitational and hydrostatic head components are both absent. Hence, in the present case

∆Ptpb ) ∆Pftpb

(8)

In gas-liquid two-phase flow, the phenomena of momentum transfer between the phases, the wall friction, shear at the phase interface and the interaction of centrifugal, viscous, and surface tension forces, and coalescence of the phases cannot be specified quantitatively.11,20 So, a theoretical analysis may not be possible and will lead to the development of an empirical correlation. Literature review suggests that two-phase gas-Newtonian pressure drop data analysis in horizontal flow has been carried out by the LockhartMartinelli5 method and other methods as reported by Mandal and Das.21

Lockhart-Martinelli5 Correlation. The two-phase multiplier for the bend can be defined as

φlb2 )

∆Pftpb/Lb ∆Pflb/Lb

(9)

Xb2 )

∆Pflb/Lb ∆Pfgb/Lb

(10)

Figure 13 shows the φlb versus Xb plots for different bends along with the Lockhart-Martinelli line. It may be seen from the plots that there is unacceptable deviation of the experimental data from that of the Lockhart-Martinelli line as Lockhart-Martinelli plots were originally developed for gas-Newtonian liquid flow through horizontal pipeline. Chisholm6,7 Correlation. Chisholm6,7 suggested the two-phase multiplier, φlb, for different bends for airwater flow as

φlb2 ) [1 + (F1/Fg - 1)(Bx(1 - x) + x2)]

(11)


Friction Factor Method. The analysis of the experimental data, i.e., total pressure drop across the bend, is best correlated by the friction factor method. Govier et al.22 developed an expression for the total pressure drop as follows:

( )

∆Ptp 1 + Rm ∆Pf 1 ) + FlgL 1 + Rv 1 + Rv FlgL

(14)

1

In eq 14, the first term on the right-hand side represents the hydrostatic head component and the second term represents the irreversibility component. In the case of horizontal two-phase flow, the hydrostatic head component is negligible and the above equation reduces to

( )

∆Pf ∆Ptp 1 ) FlgL 1 + Rv FlgL

(15)

l

For the horizontal two-phase flow in the bends, the total pressure drop is equal to the frictional pressure drop only; thus, the above equation can be written as

( )

∆Pfb ∆Pftpb 1 ) FlgLb 1 + Rv FlgLb

Figure 10. Correlation plot for Newtonian liquid flow through bends.

For the 90° bend,

(16)

1

or

B)

2.2 KL(2 + Rc/2Rt)

(12)

For the 180° bend,

B ) 0.5(1 + B90)

(13)

The deviation between the measured pressure drop across the bend (90° and 180°) and that estimated from the Chisholm6,7 equation is high (Table 4) and hence unacceptable.

( ) ∆Pfb FlgLb

) (1 + Rv)

1

( ) ∆Pftpb FlgLb

(17)

where (∆Pfb/FlgLb)1 is the frictional pressure drop per unit length across the bend due to single-phase flow of liquid. This can be expressed as

( ) ∆Pfb FlgLb

1

)

2f1V12 gD

(18)

In a similar manner, a two-phase friction factor ftplb,

Figure 11. Variation of the two-phase pressure drop across the bend with gas flow rate.

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Figure 12. Variation of the two-phase pressure drop across the bend with gas flow rate at constant liquid flow rate.

based on the liquid superficial velocity, may be defined as

( ) ∆Pf FlgLb

2ftplbV12 ) gD 1

(19)

Substituting in eq 17 gives

∆Pftpb 2ftplbV12 ) (1 + Rv) gD FlgD

(20)

or

ftplb ) (1 + Rv)

( )( ) ∆Pftpb gD FlgLb 2V 2 1

(21)

(22)

The liquid property group (Npl ) µl4g/F1σl3 ) Wel3/Rel3Frl) signifies some complex balance between viscous, surface tension, and gravitational forces. For the individual bend, the ratio of Rt/Rc is constant. Hence, for the individual bend, the above equation reduces to

ftplb ) F(Reg,Rel,Npl)

ftplb ) 46.3227Rel-0.9626(0.1255Reg0.6996(0.0581Npl0.0438(0.0431 (24) The values of ftplb predicted by eq 24 have been plotted against the experimental values as shown in Figure 14. The variance of estimate and correlation coefficient of the above equation are 6.4224 × 10-2 and 0.9381, respectively, for a t value18 of 1.98 for 225 degrees of freedom at 0.05 probability level and 95% confidence range. For the 90° bend,

ftplb )

Correlation for ftplb. The values of ftplb have been calculated by the above equation using experimental data on the two-phase pressure drop across the bend. Parameters influencing the two-phase friction factor are the physical and operating variables of the system. The physical variables include the radius of the tube, Rt, the radius of the curvature of the bend, Rc, and the physical properties of the gas and liquids, while the operating variables are the flow rates of gas and liquid. Dimensional analysis yields the following dimensionless parameters: (i) geometric parameter (Rt/Rc), (ii) dynamic parameter (Reg and Rel), and (iii) a physical property parameter (Npl). The functional relationship is

ftplb ) F(Reg,Rel,Npl,Rt/Rc)

For the 45° bend,

(23)

59.521Rel-0.5754(0.1661Reg0.5908(0.0804Npl0.1585(0.0565 (25) The values of ftplb predicted by eq 25 have been plotted against the experimental values as shown in Figure 14. The variance of estimate and correlation coefficient of the above equation are 0.1224 and 0.8876, respectively, for a t value18 of 1.98 for 246 degrees of freedom at 0.05 probability level and 95% confidence range. For the 135° bend,

ftplb ) 0.4211Rel-0.8595(0.1089Reg0.6440(0.0519Npl-0.1299(0.0408 (26) The values of ftplb predicted by eq 26 have been plotted against the experimental values as shown in Figure 14. The variance of estimate and correlation coefficient of the above equation are 5.5780 × 10-2 and 0.8985, respectively, for a t value18 of 1.98 for 212 degrees of freedom at 0.05 probability level and 95% confidence range.


Figure 13. Comparison of the two-phase pressure drop across the bend data with the Lockhart-Martinelli correlation. Table 4. Comparison between Predicted Values from the Chisholm6,7 Correlation and the Measured Values of the Frictional Pressure Drop for the 90° and 180° Bends system

RE (%)

AE (Pa/m)

90° Bend air-water air-1% amyl alcohol-water solution air-30% glycerin-water solution air-42% glycerin-water solution

30.393 29.921 79.026 61.402

0.908 93 × 104 0.580 16 × 104 0.606 79 × 105 0.241 74 × 105

180° Bend air-water air-1% amyl alcohol-water solution air-30% glycerin-water solution air-42% glycerin-water solution

42.290 41.301 36.596 45.053

0.382 28 × 104 0.259 61 × 104 0.181 94 × 104 0.262 97 × 104

For the 180° bend,

ftplb ) 0.0683Rel-0.6249(0.0623Reg0.7565(0.0338Npl-0.0433(0.0229 (27) The values of ftplb predicted by eq 27 have been plotted against the experimental values as shown in Figure 14. The variance of estimate and correlation coefficient of the above equation are 1.65 × 10-2 and 0.9642, respectively, for a t value18 of 1.98 for 200 degrees of freedom at 0.05 probability level and 95% confidence range. To develop a single generalized correlation for all of the different bends in the horizontal plane, an angle

factor, R/π, defined as a ratio of the angle of the bend to that of the 180° bend, has been introduced in the above functional relationship:

ftplb ) F(Reg,Rel,Npl,Rt/Rc,R/π)

(28)

This expression should also be valid for two-phase gas-Newtonian liquid flow in the horizontal tube; when the bend becomes straight, i.e., Rc f R, the two-phase friction factor for the bend, ftplb, should become equal to the two-phase friction factor for the horizontal tube, ftplh. This limiting condition may be incorporated by slightly modifying eq 28 as

ftplb - ftplh ) F(Reg,Rel,Npl,Rt/Rc,R/π)

(29)

where F being such a function in that it will become 0 as Rc f R, i.e., Rt/Rcf 0. Initially, the gas-Newtonian liquid flow through the horizontal pipeline was carried out by Mandal and Das,21 and they developed the following correlation for a two-phase friction factor:

ftplh ) 6.0816 × 10-3Rel-1.0600Reg1.1045Npl-0.2029 (30) On the basis of eq 29, the multiple linear regression analysis of the two-phase friction factor data for the

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Figure 14. Correlation plot ftplb for an individual bend. Table 5. Range of Variables Investigated 5000 < Rel < 50 000 180 < Reg < 2500 0.14 × 10-10 < Npl < 0.20 × 10-8 0.08 < Rt/Rc < 0.19 0.25 < R/π < 1.0

bend yielded the following correlation:

ftplb - ftplh ) 392.61Reg0.5087(0.0443Rel-0.5642(0.0901 × Npl0.1216(0.0320(Rt/Rc)1.8590(0.0728(R/π)-0.9660(0.5070 (31) The values of ftplb - ftplh predicted by eq 31 have been plotted against the experimental values as shown in Figure 15. The variance of estimate and correlation coefficient of the above equation are 0.1451 and 0.9256, respectively, for a t value18 of 1.98 for 893 degrees of freedom at 0.05 probability level and 95% confidence range. The range of variables investigated for eq 31 has been shown in Table 5. Streamwise Pressure Loss due to the Bend. Observations of pressure were made in the long upstream and long downstream portions of the bend in order to obtain the overall pressure drop across the bend. The static pressure starts to deviate from steady value within 30 pipe diameter for the 45°, 90°, 135°, and 180° bends upstream of the inlet of the bend, depending on the flow rate. In the downstream of the bend, the

Figure 15. Generalized correlation plot of ftplb - ftplh for all bends.

pressure recovery lengths were found to be within 35 pipe diameter for the 45°, 90°, 135°, and 180° bends, depending on the flow rate.


Conclusions The two-phase pressure drops across the bends have been measured for four different types of bends in the horizontal plane. Four different Newtonian liquids are used for the experiments. Experiments were carried out in a single tube diameter pipeline with varying radius of curvature and angle of bends. Correlations have been proposed to predict the two-phase friction factor for gasNewtonian liquid flow through the bends. Nomenclature AE ) average absolute error ) 1 N ∑ (∆Pflpb)expt - (∆Pflpb)cal , Pa/m N 1 B ) constant, eqs 12 and 13 D ) diameter, m f ) friction factor F ) function Fr ) Froude number g ) acceleration due to gravity, m/s2 KL ) loss coefficient L ) length, m Npl ) liquid property group ∆P ) pressure drop, Pa R ) radius, m RE ) average relative error ) 1 N (∆Pflpb)expt - (∆Pflpb)cal ∑ × 100, % N 1 (∆Pflpb)expt Rm ) input gas-liquid mass ratio RV ) input gas-liquid volumetric flow ratio Re ) Reynolds number We ) Weber number X ) Lockhart-Martinelli parameter x ) mass fraction

|

|]

[ |

|]

[

Greek Letters R ) bend angle, deg µ ) viscosity, Pa‚s F ) density, kg/m3 σ ) surface tension, N/m φ ) two-phase multiplier Subscripts b ) bend c ) curvature g ) gas h ) horizontal l ) liquid t ) tube lb ) liquid for the bend flb ) frictional liquid for the bend fgb ) frictional gas for the bend tpb ) two phase for the bend ftpb ) frictional two phase for the bend tplb ) two phase based on liquid for the bend tplh ) two phase based on liquid for the horizontal tube

Cheremisinoff, N. P., Ed.; Gulf Publishing Co.: Houston, TX, 1996; Chapter 13, p 379. (2) Das, S. K. Non-Newtonian Liquid Flow through Globe and Gate Valves. In Multiphase Reactor and Polymerization System Hydrodynamics Advances in Engineering Fluid Mechanics Series; Cheremisinoff, N. P., Ed.; Gulf Publishing Co.: Houston, TX, 1996; Chapter 17, p 487. (3) Maddock, C.; Lacey, P. M. C.; Patrick, N. A. The Structure of Two-Phase flow in a Curved Pipe. Multiphase Flow System Symposium, Glasgow, England, 1974; Paper J2. (4) Sekoguchi, K.; Sato, Y.; Kariyasaki, A. The Influences of Mixtures, Bend and Exit sections on Horizontal Two-Phase flow. Proceedings of the International Symposium on Research of Cocurrent Gas-Liquid Flow, Waterloo, Canada, 1968; p 109. (5) Lockhart, R. W.; Martinelli, R. C. Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes. Chem. Eng. Prog. 1949, 45, 39. (6) Chisholm, D. Prediction of Pressure Drop at Pipe Fittings during Two-phase Flow. Proceedings of the 13th International Congress on Refrigeration; Washington, 27 Aug.-3 Sept., 1971; International Institute of Refrigeration: Paris, 1971; Vol. 2, p 781. (7) Chisholm, D. Two-phase Flow in Bends. Int. J. Multiphase Flow 1980, 6, 363. (8) Hoang, K.; Davis, M. R. Flow Structure and pressure Loss for Two-phase flow in Return Bends. Trans. ASME J. Fluids Eng. 1984, 106, 1. (9) Norstebo, A. Pressure Drop in Bends And Valves in TwoPhase Refrigerant flow. Proceedings of the 2nd International Conference on Multiphase Flow; London, England, 19-21 June, 1985; BHRA, The Fluid Engineering Centre: Cranfield, Bedford, England, p 82. (10) Subbu, S. K.; Das, S. K.; Biswas, M. N.; Mitra, A. K. Pressure Drop in U-Bends for Air-Water Flow. Int. J. Eng. Fluid Mech. 1990, 3, 239. (11) Das, S. K.; Biswas, M. N.; Mitra, A. K. Friction Factor for Gas-non-Newtonian Liquid Flow in Horizontal Bends. Can. J. Chem. Eng. 1991, 69, 179. (12) McCabe, W. L.; Smith, J. C.; Harriott, P. Unit Operations of Chemical Engineering; McGraw-Hill: New York, 1993. (13) Perry, R. H.; Green, D. W.; Maloney, J. O. Perry’s Chemical Engineers’ Handbook; McGraw-Hill: New York, 1984. (14) Kittredge, C. P.; Rowley, D. S. Resistance Coefficient for Laminar and Turbulent Flow through one-half-inch Valves and Fittings. Trans. ASME 1957, 79, 1759. (15) Ito, H. Pressure Losses in Smooth Pipe Bends. Trans. ASME J. Basic Eng. 1960, 82, 131. (16) Hooper, W. B. The Two-K Method Predicts Head Losses in Pipe Fittings. Chem. Engg. 1981, 88, 96. (17) Sookprasong, P.; Bill J. P.; Schnlidt, Z. Two Phase Flow in Piping Component. J. Energy Res. Technol. 1986, 108, 197. (18) Volk, V. Applied Statistics for Engineers; McGraw-Hill Book Co.: New York, 1958; p 345. (19) Ohadi, M. M.; Sparrow Walavalkar, E. M. A.; Ansari, A. L. Pressure Drop Characteristics for Turbulent Flow in a Straight Circular Tube situated Downstream of a Bend. J. Heat Transfer 1990, 33, 583. (20) Friedel, L. Pressure Drop during Gas/Vapour Liquid Flow in Pipes. Int. Chem. Eng. 1980, 20, 352. (21) Mandal, S. N.; Das, S. K. Gas-Newtonian Liquid Flow through Horizontal Tube. Ind. Chem. Eng. 1998, 40, 241. (22) Goveir, G. W.; Radford, B. A.; Dunn, J. S. C. The Upward Vertical flow of Air-Water Mixtures I. Effect of Air and Water Rates on Flow Pattern, Holdup and Pressure Drop. Can. J. Chem. Eng. 1957, 35, 58.

Literature Cited (1) Das, S. K. Water Flow through Helical Coils in Turbulent Condition. In Multiphase Reactor and Polymerization System Hydrodynamics Advances in Engineering Fluid Mechanics Series;

Received for review April 7, 2000 Accepted February 28, 2001 IE0003988

Pressure Losses in Bends during Two-Phase GasNewtonian Liquid

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