Ind. Eng. Chem. Res. 2003, 42, 3487-3494
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Gas-Liquid Flow through Helical Coils in Vertical Orientation S. N. Mandal† and S. K. Das* Department of Chemical Engineering, University of Calcutta, 92, A. P. C. Road, Calcutta 700 009, India
Experimental investigations have been carried out to evaluate the two-phase pressure drop and the hold-up for gas-liquid flow through vertical helical coils. The coils were made of thickwalled transparent PVC tubes of diameters 0.01 and 0.013 m. Twenty-four coils were made with different coil diameters and different helix angles (0-12°). Three different liquids and air were used for the experimental studies. Empirical correlations were developed to predict the two-phase friction factor and the liquid hold-up as functions of the physical and dynamic variables of the system. Statistical analysis of the correlations suggests that they are of acceptable accuracy. Introduction Helical coils are extensively used in compact heat exchangers, heat exchanger networks, heating or cooling coils in piping systems, intake in air crafts, fluid amplifiers, coil steam generators, refrigerators, nuclear reactors, thermosiphons, other heat-transfer equipment involving phase changes, and chemical plants, as well as in the food and drug industries. One of the main advantages in the use of helical coiled tubes as chemical reactors or heat exchangers lies in the fact that considerable lengths of tubing can be contained in a spacesaving configuration that can easily be placed in a temperature-controlled environment. The heat- and mass-transfer coefficients in helical coiled tubes are higher than those in straight tubes. When fluid flows through a curved pipe, the presence of curvature generates a centrifugal force that acts at right angles to the main flow, resulting in secondary flow. The strength of the secondary flow depends on the curvature of the surface. A literature survey indicates that numerous studies dealing with flow phenomena and pressure drop in single-phase flow through helical coils have been published. These are well summarized in Berger and Talbot,1 Shah and Joshi,2 and Das.3 Two-phase gas-liquid flow through curved pipes is much more complex in nature. When flow enters the curved region, because of centrifugal action, the heavier phase (i.e., liquid), which is subjected to a larger centrifugal force, moves away from the center of curvature, whereas the gas flows toward the center of the curvature. Separation of phases in this way is likely to give rise to significant slip between the phases. This process is a continuous function of the coil geometry. Despite various applications, the literature on twophase flow through coiled tubes is rather meager. Ripple et al.4 and Owhadi et al.5 observed that the pressure drop for two-phase gas-liquid flow through a helicoidal pipe satisfied the Lockhart-Mertinelli6 correlation. Banerjee et al.7 investigated the gas-liquid flow through transparent coils with different tubes, coil diameters, and helix angles. They concluded that Baker’s8 plots * 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:
[email protected].
adequately predicted the flow patterns. They modified the Lockhart-Martinelli6 correlation to satisfy their experimental data on the two-phase pressure drop and hold-up. They also observed that the helix angle, if small, appears to have no discernible effect on the pressure drop. Boyce et al.9 carried out two-phase pressure drop and flow regime measurements in helically coiled plastic tubes. They observed that the Lockhart-Martinelli6 correlation adequately predicted the experimental pressure drop data. They also reported that Baker8 flow maps could not predict the flow pattern transition adequately. Akagawa et al.10 observed that the two-phase frictional pressure drop for helical coils was 1.1-1.5 times higher than that for straight pipes. The authors proposed empirical equations for the frictional pressure drop and observed that the hold-ups were approximately equal to those in straight tubes. Kasturi and Stepanek11,12 studied gas-liquid two-phase flow through a helical coil and developed correlations for the pressure drop and hold-up in terms of known dimensionless groups. Chen and Zhou13 developed an empirical correlation for calculating the frictional pressure drop of air-water two-phase flow in helical coils. Chen and Zhang14 reported that the helix angle has a pronounced effect on the flow pattern transition and proposed empirical correlations for predicting transitions under different flow conditions. Rangacharylu and Davis15 proposed a new correlation for the two-phase frictional pressure drop using modified LockhartMartinelli6 parameters. Kaji et al.16 observed that the flow regime transition for air-water two-phase flow was close to the map of Mandhane et al.,17 except for the annular-wavy stratified flow boundary, which was close to Baker’s8 map. Saxena et al.18 studied the flow pattern, hold-up, and pressure drop for co-current upward and downward flow in helical coils. They observed close similarities between the flow patterns in helical coils and those of inclined tubes reported by Spedding et al.19 Xin et al.20 measured the two-phase pressure drop and hold-up for the air-water system in vertical helical coils. They observed that the two-phase pressure drop depended on both the Lockhart-Martinelli6 parameter and the flow rates. For small coils, they proposed a twophase frictional pressure drop equation by modifying the Lockhart-Martinelli6 parameter. They found good agreement between the hold-up data and the LockhartMartinelli6 correlation. They observed that the helix
10.1021/ie0200656 CCC: $25.00 © 2003 American Chemical Society Published on Web 05/31/2003
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Figure 1. Schematic diagram of 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; T1-T2, thermometers; V1-V14, valves; LC, level controller; HC, helical coil; HE, heat exchanger; RG1-RG2, gas rotameters; RL1-RL2, liquid rotameters; ST, stirrer; SV1-SV2, solenoid valves.
angle, coil diameter, and pipe diameter had no apparent effect on the hold-up but had some effect on the frictional pressure drop. Hence, it can be concluded that only a few literature reports on gas-liquid two-phase flow through coils are available. The purpose of the present study is to generate experimental data on the two-phase pressure drop and hold-up for gas-Newtonian liquid flow through vertical helical coils. 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 the flow rate, pressure drop, and hold-up; and other accessories. Thick-walled flexible, transparent PVC tubes with internal diameters of 0.010 and 0.013 m were used for experiments. The PVC tubes were wound around hard cylindrical PVC frames of known diameter to form helical coils. The helical coils were fixed and carefully tightened with clamps, to avoid deformation of the tubes. The tubes were wound in close-packed fashion so that the pitch was equal to the outer diameter of the tube and maintained constant for all cases. The entire test section was vertically mounted on a frame to prevent vibration. The use of transparent tubes enabled the observation of the flow pattern in the helical coil. Four helix angles (0°, 4°, 8°, and 12°) were used for the experiments. Detailed dimensions of the coils used in the experiments are given in Table 1. All systems were more than 10 m long. Before the helical coil, a horizontal tube about 1.5 m in length was provided for the mixing of the two-phase flow.
Table 1. Dimensions of Vertical Helical Coils tube diameter Dt (m)
coil diameter Dc (m)
helix angle (deg)
total
turns manometers
0.01 0.01 0.01 0.013 0.013 0.013
0.131 0.185 0.216 0.137 0.191 0.222
0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12 0, 4, 8, 12
20 16 13 19 15 12
8, 8, 8, 8 6, 6, 6, 6 4, 4, 4, 4 7, 7, 7, 7 5, 5, 5, 5 4, 4, 4, 4
A rectangular tank (0.45 m3) was used for the storage of liquid. The liquid was kept at a constant temperature by recirculation of tap water through a copper coil. The liquid was circulated from the tank by means of a centrifugal pump to the test section. Its flow rate was controlled by bypass valves and measured by a set of rotameters (RL1 and RL2) (Transducers and Controls Pvt. Ltd., Hyderabad, India, accuracy of (2%). The liquid was discharged into the separator and returned to the tank. The separator, of diameter 0.31 m and height 0.6 m, was made from stainless steel sheet. The size of the separator was such that it accommodated the complete range of the flow in the experimental study. The liquid level in the separator was always kept below the pipe, as shown in Figure 1. The level of liquid in the separator was controlled with the help of a level controller (LC). Air was drawn from a compressor, and its pressure was reduced to 103 kPa (gauge) prior to injection into the pipeline through a T-entry. The flow rate of the air was measured by a set of rotameters (RG1 and RG2) (Transducers and Controls Pvt. Ltd., Hyderabad, India, accuracy of (2%).
Ind. Eng. Chem. Res., Vol. 42, No. 14, 2003 3489 Table 2. Physical Properties of the Test Liquids
liquid used water 1 vol % amyl alcoholwater solution 30 vol % glycerinwater solution
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
Pressure drop measurements are known to be difficult because of the inherent variable nature of two-phase flow. The upstream pressure tap was mounted after four to six coil turns from the inlet, and the downstream pressure top was located before four to six turns from the outlet to reduce the effect of the upstream and downstream flow, respectively. The pressure taps were adjusted to ensure that they were on the same vertical line. A simple U-tube manometer containing mercury beneath water was used for the measurement of the pressure difference. Arrangements for purging the air bubbles/liquid in the manometer line were also provided. At times, especially with the high air input rates, it was necessary to constrict the manometer lines to reduce fluctuations. The liquid hold-up measurements in the tube were done by the simultaneous shutting of solenoid valves (SV1 and SV2) in the inlet and outlet of the coil, after a steady two-phase flow condition was reached, to trap gas and liquid. The trapped liquid was then removed from the tube into a graduated cylinder and measured. A previously determined wall wettage was added to the amount collected to give the total liquid collection. The wall wettage determination was made by adding a known amount of liquid to the dry tube, then blowing it from the coil, and finally determining the differences between the amounts collected and added. Liquid hold-ups are expressed as the ratio of the amount collected plus the wall wettage loss to the total amount held in the tube for single-phase liquid flow. Three different liquids (water, 1% amyl alcohol water solution, 30% glycerin water solution) and atmospheric air were used as the experimental fluids. The physical properties of the experimental solutions were measured by employing standard techniques. The viscosity, surface tension, and density of the liquid were measured with an Ostwald viscometer, a DuNouy tensiometer, and a specific gravity bottle, respectively. The physical properties of the experimental liquids are given in Table 2. The experimental system has been verified by experiments of single-phase flow through helical coils by Mandal.21 The results were compared with correlations available in the literature. In the actual experiments, first, the liquid flow was maintained at a fixed flow rate through the test section. Once the flow became steady, the flow rate of the liquid and the manometer readings were noted. Next, the air at a pressure of 103 kPa (gauge) was introduced into the test section through a T-mixer. The fluid mixture moved through the test section and was separated in the gas-liquid separator. Under steady-state conditions of two-phase flow, the liquid and air flow rates were noted. The readings of the manometer attached to the tapping were also noted, and the solenoid valves were closed to measure the liquid hold-up. The gas and liquid flow rates used in the experiments were in the range of (0.13-5.25) × 10-4 and (3.65-14.2) × 10-5 m3/s, respectively. The experiments were repeated a number of
times to ensure reproducibility of the data. The flow pattern was intermittent in nature under the experimental conditions. The temperature of the liquid was maintained at 30 ( 2 °C. Results and Discussion Two-Phase Frictional Pressure Drop. The total pressure drop for gas-liquid flow through vertical helical coil of length L () nπDc/cos β), can be expressed as the sum of frictional, ∆Pftp; hydrostatic, ∆Phtp; and accelerational, ∆Patp, components
∆Ptp ) ∆Pftp + ∆Phtp + ∆Patp
(1)
Hughmark and Pressburg22 have indicated that the accelerational component, ∆Patp, is negligible as compared to the total pressure drop in a tube of uniform cross section when no phase change occurs. Hence
∆Ptp ) ∆Pftp + ∆Phtp
(2)
A literature review suggests that the hydrostatic head component, ∆Phtp, can be calculated either by assuming that the gas and liquid form a homogeneous mixture, which gives
∆Phtp )
Ml + Mg hg Ql + Qg
(3)
or by considering the in situ hold-up in the system given by
∆Phtp ) (FlRl +FgRg)hg
(4)
The evaluation of ∆Phtp through eq 3 is relatively straightforward and depends only on the entry flow rate. Therefore, the hydrostatic head component in the present case was estimated from eq 3. The frictional pressure drop was calculated by the following equation
∆Pftp ) ∆Ptp -
Ml + Mg hg Ql + Q g
(5)
Effect of the Gas Flow Rate on the Two-Phase Frictional Pressure Drop. Figure 2 illustrates the effect of the gas flow rate on the two-phase frictional pressure drop per unit length of coil for various liquid flow rates. The main feature of these curves is that the two-phase frictional pressure drop per unit length of coil increases with increasing liquid flow rate. Figure 3 shows the effect of the coil diameter on the two-phase frictional pressure drop per unit length of coil. The main feature of these curves is that the two-phase frictional pressure drop per unit length of coil increases with increasing coil diameter. This result can be explained by the introduction of slip effect in two-phase flow conditions. Because the liquid density is more than 600 times higher than the gas density and the overall gas flow rate is nearly 10 times higher than the liquid flow 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 pressure drop for the liquid phase decreases. The gas-phase pressure drop is very small
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Figure 2. Variation of two-phase frictional pressure drop with gas flow rate at constant liquid flow rate. Figure 4. Variation of two-phase frictional pressure drop with gas flow rate and helix angle.
Figure 3. Variation of two-phase frictional pressure drop with gas flow rate at constant liquid flow rate and coil diameter.
compared to that of the liquid phase, so the net effect is a decrease in the two-phase frictional pressure drop per unit length of coil. Figure 4 shows that the two-phase frictional pressure drop per unit length of coil increases with increasing gas flow rate at constant liquid flow rate and coil diameter, but is independent of the helix angle. Banerjee et al.7 also observed that the two-phase frictional pressure drop was independent of helix angle, whereas Xin et al.20 concluded that the helix angle had some effect on the frictional pressure drop. Figure 5 shows the two-phase frictional pressure drops per unit length of coil as a function of gas flow rate obtained with the three different liquids used in the experiments. It is clear from the graph that the two-
Figure 5. Variation of two-phase frictional pressure drop with gas flow rate for different systems.
phase frictional pressure drop per unit length of coil increases with increasing liquid viscosity at constant liquid flow rate and helix angle. The liquid has a retarding effect as its viscosity increases, and also the slip is expected to be higher in a viscous liquid. Hence, the two-phase pressure drop per unit length of coil increases with increasing liquid viscosity at constant gas flow rate. Surface tension also has a pronounced effect on the two-phase pressure drop per unit length of coil. In the case of the air-1% amyl alcohol water solution, the liquid surface tension decreases, and slightly more foaming is observed than for the air-water two-phase system. This reduces the slip between the phases and creates a tendency to retain the gas phase. Because of
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the continuous changes in the centrifugal forces, the liquid is continuously pushing to the outer wall while the gas phase is at the inside wall. The probable effect of the centrifugal force is greater than the retarding effect of gas phase. Hence, the two-phase frictional pressure drop per unit length of coil is slightly higher at constant gas flow rate for the air-1% amyl alcohol water solution than for the air-water system. Analysis of the Two-Phase Frictional Pressure Drop. Mandal21 carried out extensive experiments on liquid flow through helical coils made of transparent PVC tube. He observed that the Mishra and Gupta23 equations fitted his experimental data within (8% deviation. The well-known Mishra and Gupta23 equations are as follows:
For laminar flow fc ) 1 + 0.033(log DeM)4.0 fst
(6)
where Figure 6. Correlation plot for two-phase friction factor.
DeM ) Re(Dt/Dc)0.5
Table 3. Range of Variables Investigated
For turbulent flow
minimum
fc - fst ) 0.0075(Dt/Dc)0.5
(7)
The Lockhart-Martinelli6 parameters (φl and X) for a coil are defined as
φl2 )
∆Pftp/L ∆Pfl/L
(8)
X2 )
∆Pfl/L ∆Pfg/L
(9)
The single-phase pressure drops for gas and liquid were calculated using the Mishra and Gupta23 equations. It was observed that the present experimental data on two-phase frictional pressure drop deviate (by more than 50%) from the Banerjee et al.7 and Boyce et al.8 correlations. Govier et al.24 developed an expression for the twophase friction factor (ftplc) for vertical gas-liquid flow as
( )( )
ftplc ) (1 + Rv)
∆Pftp gDt FlgL 2V 2 l
(10)
The values of ftplc were calculated by the above equation using the experimental data. Friedel25 pointed out that the determination of the hydrodynamic parameter is not possible by theoretical analysis alone as the phenomena of momentum transfer between the two phases, wall friction, shear at the phase interface, and secondary flow due to centrifugal action cannot be specified quantitatively. Because the physical process of two-phase flow is not clearly understood, the alternative method generally used is dimensional analysis. Therefore, the frictional pressure drop was analyzed in terms of the twophase friction factor as a function of various physical and dynamic variables of the system. Dimensional analysis yields the following functional relationship
ftplc ) F(Rel,Reg,Npl,Dt/Dc)
(11)
variable
0.217 04 × 104 0.105 95 × 103 0.142 31 × 10-10 0.462 96 × 10-1
e e e e
Rel Reg Npl Dt/Dc
maximum e e e e
0.192 56 × 105 0.4053 × 104 0.577 27 × 10-9 0.948 91 × 10-1
The liquid property group (Npl ) µl4g/Fl σl3) signifies some complex balance between viscous, surface tension, and gravitational forces. On the basis of eq 11, a multiple linear regression analysis of the experimental data on the two-phase friction factor in the vertical helical coil for a 0° helix angle was carried out, which yielded the following correlation
ftplc ) 5.885Rel-1.183(0.021 Reg0.952(0.014 × Npl0.022(0.008 (Dt/Dc)-0.282(0.037 (12) The values of ftplc predicted by eq 12 are plotted against the experimental values in Figure 6. The variance of the estimate and the correlation coefficient of the above equation are 2.158 × 10-2 and 0.9855, respectively, for a t value of 1.98 for 1074 degrees of freedom at the 0.05 probability level and the 95% confidence range (Volk26). The ranges of the variables investigated in this study are listed in Table 3. Table 4 presents the average relative error and average absolute error for the frictional pressure drop evaluated by different methods, i.e., the LockhartMartinelli6 method, the Xin et al.20 method, and the friction factor method. It can be seen that the average relative error in the friction factor method is the lowest, i.e., eq 12. Consequently, it can be concluded that the frictional pressure drop correlation by the friction factor method gives the best agreement between the predicted and the measured values. Liquid Hold-up. Effect of the Gas Flow Rate on the Liquid Hold-up. Figure 7 shows the effect of the gas flow rate on the liquid hold-up for various liquid flow rates. It is clear from the graph that, as the gas flow rate increases, the liquid hold-up decreases for a given liquid flow rate.
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Figure 7. Variation of liquid hold-up with gas flow rate at constant liquid flow rate.
Figure 8. Variation of liquid hold-up with gas flow rate and coil diameter.
Table 4. Comparison between Predicted and Measured Values of the Two-Phase Frictional Pressure Drop system used
REa (%)
AEb (Pa/m)
Lockhart-Martinelli6 Method air-water 103.59 air-1% amyl alcohol-water solution 63.09 air-30% glycerin-water solution 28.06
3166.38 1517.70 2337.42
Xin et al.20 Method air-water 265.12 air-1% amyl alcohol-water solution 188.69 air-30% glycerin-water solution 35.42
8491.17 5806.2 1452.79
Present Friction Factor Method (eq 12) air-water 14.82 air-1% amyl alcohol-water solution 11.13 air-30% glycerin-water solution 9.01 a
635.64 843.28 813.85
Average relative error. b Average absolute error.
Figure 8 illustrates the effect of the coil diameter on the liquid hold-up. It is clear from the graph that the liquid hold-up decreases as the coil diameter increases at constant liquid flow rate, gas flow rate, and helix angle. Because the liquid density is much higher than the gas density, the centrifugal forces acting on the liquid phase are much higher at any particular coil diameter. As the coil diameter decreases, the slip between the liquid phase and the gas phase increases; hence, the liquid phase is accelerated more. Again, the number of turns is greater for a smaller-diameter coil. Hence, the flow path of the smaller-diameter coil is greater, i.e., the effect of gravity is greater. The individual effects of slip and gravity act in opposite directions, and the combined effect increases with increasing number of turns. This combined effect is responsible for the slightly more liquid hold-up for the smaller coil diameter. Figure 9 shows the effect of the gas flow rate on the liquid hold-up for different helix angles. It is clear from the graph that the liquid hold-up decreases with incresing gas flow rate at constant liquid flow rate, tube, and coil diameter and is independent of the helix angle.
Figure 9. Variation of liquid hold-up with gas flow rate and helix angle.
Similar results were obtained by Banerjee et al.7 and Xin et al.20 Figure 10 shows the liquid hold-up as a function of the gas flow rate for the different liquids used in the experiments. It can be observed that the liquid hold-up decreases with increasing gas flow rate and increasing viscosity of the liquid at constant liquid flow rate and helix angle. The 30% glycerin-water solution has a higher density than the other liquids tested, so because of centrifugal forces, the liquid moves away from the center, and the slip is expected to be higher. The combined effect of increasing the density and viscosity results in a decrease in the liquid hold-up for this higher-viscosity liquid. The liquid surface tension for the 1% amyl alcohol-water solution is less than that of
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Figure 10. Variation of liquid hold-up with gas flow rate for different systems.
water. In two-phase flow, as the liquid surface tension decreases, the slip between the phases decreases. Consequently, the tendency of the gas phase is to be retained. However, the effect of centrifugal forces is greater than the retarding effect of the gas phase. Hence, the effect of the surface tension on the liquid hold-up is negligible for the air-1% amyl alcohol-water solution and for the air-water system. Analysis of the Liquid Hold-up. It was observed that the present experimental data present a large deviation (high by about 50%) from the Banerjee et al.7 and Xin et al.20 correlations. The experimental liquid hold-up data have been analyzed as a function of various physical and dynamic variables of the system. Dimensional analysis yields the following functional relationship
Rl ) F(Rel, Reg, Npl, Dt/Dc)
(13)
On the basis of eq 13, a multiple linear regression analysis of the experimental data on the hold-up for the helix angle of 0° was carried out, which yielded the following correlation
Rl ) 0.172Rel0.462( 0.008 Reg-0.363(0.005 × Npl-0.007(0.003 (Dt/Dc)0.371( 0.014 (14) The values of the Rl predicted by eq 14 are plotted against the experimental values in Figure 11. The variance of the estimate and the correlation coefficient of the above equation are 2.0725 × 10-3 and 0.9876, respectively, for a t value of 1.98 for 1074 degrees of freedom at the 0.05 probability level and the 95% confidence range (Volk26). The ranges of variables investigated are shown in Table 3. Table 5 presents the average relative error and average absolute error for the liquid hold-up evaluated by the different methods. It can be seen that the average relative error is the lowest for eq 14, i.e., the best agreement is obtained between the predicted and measured values of the liquid hold-up.
Figure 11. Correlation plot for liquid hold-up. Table 5. Comparison between Predicted and Measured Values of the Liquid Hold-up REa (%)
system used Lockhart-Martinelli6
AEb
Method 35.99 33.34 72.17
0.1292 0.1224 0.2370
Present Hold-up Method (eq 14) air-water 3.89 air-1% amyl alcohol-water solution 4.15 air-30% glycerin-water solution 0.72
0.02 0.017 0.0032
air-water air-1% amyl alcohol-water solution air-30% glycerin-water solution
a
Average relative error. b Average absolute error.
Conclusions Experiments have been carried out to measure the two-phase pressure drop and liquid hold-up for different vertical helical coils. The coils were made of thick-walled PVC tubes of diameters 0.01 and 0.013 m. Twenty-four coils were tested at different coil diameters and helix angles (0-12°). Three different liquids and air were used for the experimental studies. The two-phase pressure drop was measured by a U-tube manometer and the liquid hold-up by the displacement technique. The experimental data on the two-phase pressure drop and hold-up were analyzed by different methods available in the literature. Empirical correlations, eqs 12 and 14, were developed to calculate the two-phase friction factor and liquid hold-up using experimental data on the twophase pressure drop and liquid hold-up of coils with a 0° helix angle. Nomenclature AE ) average absolute error for pressure drop, Pa/m
)
1 N
N
∑1 |(∆Pftpc)expt - (∆Pftpc)cal|
AE ) average absolute error liquid hold-up
)
1
N
∑|(Rl)expt - (Rl)cal| N 1
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D ) diameter, m F ) function f ) friction factor g ) acceleration due to gravity, m/s2 h ) vertical height of the manometer tapings, m L ) length, m M ) mass flow rate, kg/s n ) number of coil turns N ) number of data points Npl ) liquid property group (µl4g/Flσl3) ∆P ) pressure drop, Pa Q ) flow rate, m3/s R ) radius, m RE ) average relative error for pressure drop
)
1 N
∑1 | N
(∆Pftpc)expt - (∆Pftpc)cal (∆Pftpc)expt
|
RE ) average relative error for hold-up
)
∑| N 1 1
N
(Rl)expt - (Rl)cal (Rl)expt
|
Rm ) gas/liquid mass flow ratio Rv ) input gas/liquid volumetric flow ratio Re ) Reynolds number, VDF/µ V ) superficial velocity, m/s X ) Lockhart-Martinelli parameter Greek Letters R ) hold-up β ) helix angle, degrees µ ) viscosity, N s/m2 F ) density, kg/m3 σ ) surface tension, N/m φ ) two-phase multiplier Subscripts c ) coil g ) gas l ) liquid t ) tube st ) straight tp ) two-phase fg ) frictional gas fl ) frictional liquid atp ) accelerational two-phase cal ) calculated ftp ) frictional two-phase htp ) hydrostatic two-phase expt ) experimental ftpc ) frictional two-phase for coil tplc ) two-phase based on liquid for coil
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Received for review January 22, 2002 Revised manuscript received April 22, 2003 Accepted April 29, 2003 IE0200656
