ARTICLE pubs.acs.org/IECR
LiquidLiquid Mixing in Coiled Flow Inverter Monisha Mridha Mandal, Palka Aggarwal, and K. D. P. Nigam* Department of Chemical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110 016, India ABSTRACT: The mixing of liquids is a common operation in process industries such as refineries and chemical and pharmaceutical industries, etc. However, the problem of mixing of different liquids has not been rigorously characterized. Therefore, the objective of this paper is to investigate liquidliquid mixing in a novel coiled flow inverter (CFI). The device works on the principle of flow inversion which is achieved by bending a coiled tube to 90° at equidistant length. In the present study, velocity field and scalar concentration distribution of liquids were characterized. The mixing performances and pressure drop in CFI was investigated and compared with that of a straight, coiled tube and helical element mixer (HEM) for a liquid flow range of 98 e Re e 1020. CFI exhibits significant mixing of two liquids with negligible change in pressure drop as compared to a coiled tube as well as a HEM. The present study reveals that CFI is an efficient device for the mixing of two liquids in process industries.
1. INTRODUCTION Mixing in the laminar flow regime is mainly driven by molecular diffusion. Liquid-phase mixing generally influences the heat and mass transfer rates and reactant conversion in any reactor. However, a careful analysis of the data reported in literature shows that very high fluid flow rate is required in order to induce significant mixing in coiled tubes.1,2 It is not possible to narrow the residence time distribution (RTD) beyond a certain limit in coils with fixed curvature ratio. Hence, in order to reduce axial dispersion, many devices such as motionless mixers,39 flow inverters,10 and chaotic configurations1114 have been reported in the past. Static mixers have limitations for very viscous fluids as it can induce prohibitive pressure drop resulting in higher pumping cost. To overcome this limitation a novel concept was introduced to develop an economical and effective alternative named as the coiled flow inverter (CFI).1 The configuration of a CFI is a novel design, which works on the principle of complete flow inversion. The geometrical configuration of a CFI consists of 90° bends at equal intervals of length in coiled tube geometry. This device helps in intensifying the convective transfer processes and provides enhanced transfer area per unit volume of space. Its performance is substantially closer to plug flow. A modified axial dispersion model has been presented to describe the liquid-phase RTD in gasliquid flow under the conditions of both negligible and significant molecular diffusion in a CFI.2 It was observed that the axial dispersion was reduced with an increase in liquid flow rate and number of bends. The reduction in dispersion number was 2.6 times in the CFI having 15 bends as compared to a coiled tube for two phase gasliquid flow under identical process conditions. Further experiments have been carried out to investigate the effect of design parameters such as gas and liquid flow rates, curvature ratio, pitch, and the number of bends on pressure drop for gasliquid flow in the CFI.15 The transition of flow regimes in gasliquid flow was observed at critical Reynolds numbers of 800010000. Pitch had negligible effect on the pressure drop of gasliquid flow in the CFI. The empirical correlations for the friction factor have been reported for the different gasliquid regimes in the CFI. These correlations take into account the r 2011 American Chemical Society
effect of number of bends, curvature ratio, and gas and liquid flow rates. The void fraction of gasliquid flow in a CFI was investigated.16 The gas void fraction decreased with the increase in number of bends. The effect of pitch on gas void fraction was found to be negligible. At a given gas flow rate, the gas holdup decreased with the increase liquid flow rate. An empirical correlation to predict the void fraction for different flow regimes has been developed. Liquidliquid flow exists in chemical process industries. Information about liquid flow development, pressure drop, and mixing efficiency is required to design as well as optimize operating conditions in the industries. Literature survey shows that information on liquidliquid flow is available for a straight tube configuration.1720 However, very limited efforts have been made in the past to explore the hydrodynamics of liquidliquid flow in coiled tubes.21 Therefore, the objective of the present work is to investigate the flow development and distribution of scalar concentration in a CFI with λ = 10 and a pitch of 0.02 m. An attempt is made to study the mixing of two liquids in straight, coiled, and CFI tubes for the flow range of 98 e Re e 1020. The effect of Reynolds number and number of 90° bends in the CFI on the mixing efficiency has been investigated. The pressure drop as well as mixing performance in the CFI was also compared with the existing experimental data of the helical element mixer (HEM).6,7 All the computations were carried out on a SUN FIRE V440 workstation in the Chemical Reaction Engineering laboratory at Indian Institute of Technology, Delhi, India.
2. NUMERICAL MODEL The coiled flow inverter device with circular cross-sectional having diameter, d; coil diameter, D, and pitch, H was considered Special Issue: Ananth Issue Received: February 2, 2011 Accepted: June 1, 2011 Revised: May 24, 2011 Published: June 01, 2011 13230
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g represents gravity, n is the number of phases, B F is a body FmB u dr,k is force, μm is the viscosity of the mixture. (μm = ∑nk = 1Rkμk). B the drift velocity for secondary phase k. The last term denotes the net rate of momentum transfer per unit volume by the action of drift velocity. The drift velocity for secondary phase can be up B u m where B u p is velocity of secondary expressed as B u dr,p = B phase. The energy equation for the mixture can be expressed as ∂ n ðRk Fk Ek Þ þ r 3 ∂t k ¼ 1
∑
for the present study. The details of the geometry considered for computation has been shown in Figure 1.22 2.1. Governing Equations. The governing equations for mass, momentum, and scalar transport in the CFI were solved with the control volume finite difference method (CVFDM) using commercial CFD code Fluent 6.3.23 In the present study, the mixture model was used to model the liquidliquid flow in the tube. This model is used to study flows where the phases move at different velocities. It works for the case where phases are interpenetrating. This model has been previously used to simulate mixing of liquids in different configurations.8,9 The mixture model approach is used which assumes homogeneous flow with variable volume fraction of each phase. The summed up momentum equation of the phases with phase averaged physical properties is solved. Unlike the Eulerian model, where the conservation equations are coupled via interphase interactions terms, in the mixture model, the mixture continuity, momentum equation, and energy equation are solved along with additional transport equations for the volume fraction of secondary phases. In the present study, the governing continuity equation may be written as ð1Þ
where Fm is the mixture density where Fm = ∑nk = 1RkFk, Rk is the volume fraction of phase k, B u m is the mass-averaged velocity where B u m = (∑nk = 1RkFkuk)/(Fm), m_ represents mass transfer, t represents time. In the case of secondary phase, the volume fraction equation for secondary phase p can be expressed as ∂ ðRp Fp Þ þ r 3 ðRp Fp uBm Þ ¼ r 3 ½Rp Fp uBdr, p ∂t
∂ ðF u Þ þ r 3 ðFm uBm uBm Þ ∂t m Bm ¼ rP þ r 3 ½μm ðr uBm þ r uBTm Þ þ Fm gB þ B F þr3ð
∑ RkFk uBdr, k uBdr, k Þ
k¼1
∂Fm Ck þ r 3 ðFm uBm Ck Γkm rCk Þ ¼ Skm k ¼ 1, ::::, N ð5Þ ∂t where Γkm = ∑ RlΓkl and Skm = ∑lSkl are the mixture diffusivity and source term for transport variable Ck. The mesh of the geometry was built in GAMBIT software. It was then computed in FLUENT 6.3 software. Segregated solver was used to model the flow of liquids. Liquids with constant velocity were employed at the inlet. No-slip boundary condition and the zero derivative conditions for the scalars were treated on the tube wall. Flow was considered as fully developed at the outlet. The scalar transport technique was used to compute the mixing characteristics of liquids. Different scalar concentrations were employed in the two halves of the tube inlet. The interface for initializing the scalar concentration was perpendicular to the direction of the secondary flow. Second-order upwind scheme was used to model the convection term in the governing equations. The coupling between velocity and pressure was resolved using SIMPLE algorithm. The computation was considered converged when the residual summed over all the computational nodes at nth iteration, ∑ Rnϕ, satisfied the following m 8 criterion: ∑ Rnϕ/∑ Rm ϕ e 10 , where Rϕ denotes the maximum residual value of ϕ variable after m iterations, ϕ applied for p, ui, and for scalars. The mixing performance of the geometry was measured in terms of coefficient of variation (COV). It is represents the standard deviation of concentration to the mean concentration of liquids. !0:5 Z ðCavg Ci ÞÞ2 dA
ð2Þ
The momentum equation for the mixture can be obtained by summing the individual momentum equations for all phases. It can be expressed as
n
ð4Þ
where Ek is the sensible enthalpy for phase k, keff is the effective conductivity; keff was calculated as ∑Rkkk where Rk is the volume fraction of any phase k and kk is the conductivity of phase k. The term on the right-hand side of equation represents energy transfer due to conduction. The flow of incompressible fluids was assumed in the given two-phase system. The transport equation for an arbitrary scalar k is
Figure 1. Coiled flow inverter.
∂ ðF Þ þ rðFm uBm Þ ¼ m_ ∂t m
n
∑ ðRk Bv k ðFk Ek þ pÞÞ ¼ r 3 ðkef f rTÞ k¼1
COV ¼ where Cavg ¼
ð3Þ
where r(FmB u mB u m) represents convection term, 3P, represents T pressure, r 3 [μm(ru Bm þ ru Bm )] represents viscous forces,
Cavg Z 1 A Ci dA A 0
ð6Þ
ð7Þ
Cavg is the flow weighted average value of the scalar concentration over the cross-sectional area. A systematic grid sensitivity investigation was performed. Grid-sensitivity tests were carried out with three grids consisting of 625 2050, 625 3100, 690 3100 (cross-section x axial). The pressure drop values calculated for the three grids is shown 13231
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Table 2. Properties of Liquids
cell density,
pressure drop
cells/mm3
(100 Pa/m)
625 2050
parameter
inlet 1
inlet 2
density (kg/m3)
780
872
1.75
viscosity (kg/(m s))
0.007
0.069
625 3100
1.69
diffusion coefficient (m2/s)
10 108
10 108
690 3100
1.69
Figure 2. Comparison between CFD prediction and experiment values of pressure drop at different water volume fraction of oilwater flowing in straight tube with D = 0.055 m, L = 8 m.
in Table 1. It was observed that the 625 3100 grid was necessary to obtain grid independent results. Hence, geometry with 625 3100 grids was used in the present study because it produced data with good accuracy and also saved computation time.
3. RESULT AND DISCUSSION
Figure 3. Velocity contours of liquids flowing at v = 2 m/s at different axial distance in straight tube, coiled tube, and CFI with one, two, and three bends having d = 0.01 m.
3.1. Comparison of Numerical Predictions with Experimental Results. There is lack of quantitative analysis for
liquidliquid mixing in coiled tube. Hence, to check the accuracy and reliability of the computation technique, computations were first validated with the experimental data set reported in the literature19 for liquidliquid flow in straight tube. CFD simulations were carried out to calculate the pressure drop of two-phase flow of oil and water in a 0.055 m diameter, 8 m long straight tube. The oil had a density of 790 kg/m3 and dynamic viscosity of 0.0016 kg/(m s) at 25 °C. Figure 2 shows the comparison between the existing experimental values and predicted values of present CFD study for different water volume fraction ranging from 0.2 to 0.75. The maximum deviation between the CFD predictions and the experimental data was within (2.5%. 3.2. Development of Velocity Contours. The computations were further carried out for an industrially important system of two crude oils, named Arab Mix and Mangla crude, flowing in straight, coiled, and CFI tubes of equal length (L = 5.34 m) and tube diameter (d = 0.01 m). The pitch (H) and curvature ratio (λ) of the tubes considered for the coiled tube and the CFI were 0.02 m and 10, respectively. Table 2 presents the properties of liquids used in the present study. The study was carried out for average Reynolds numbers varying from 98 to 1020 and the number of 90° bends in CFI being from 1 to 3. Figure 3 shows the development of velocity profile at different axial length for straight, coiled and CFI tube of equal length and tube diameter. It can be seen from the figure that the velocity contours were fully developed for the straight tube within length equivalent to first bend (i.e., L = 1.33 m). There was no change in contours with the increase in axial length. However, the velocity contours in coiled tube as well as CFI became asymmetrical as
Figure 4. Distribution of scalar concentrations of liquids flowing at v = 2 m/s at different axial distances in straight, coiled, and CFI tubes having d = 0.01 m.
the axial length was increased. The unbalanced centrifugal force on the fluid flow due to the curvature of the coil shifted the liquid with maximum velocity toward the outer wall of the coil. The flow was fully developed in coiled tube at axial length of 1.33 m which is also length of CFI equivalent to one bend. No further 13232
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Figure 5. Effect of Reynolds number on relative coefficient of variation for straight, coiled, CFI tube, and HEM.
Figure 6. Effect of number of bends on coefficient of variation of two phase liquids flowing in straight, coiled, CFI tube, and SMX static mixer.
modification of contours was observed with further increase in axial length. It was further found that the velocity contours in CFI was inverted to 90° at second bend (L = 2.67 m). This was due to change in the direction of fluid flowing with an introduction of a 90° bend. The contours were again rotated to 90° due to the rotation of the plane of vortex at third bend (L = 4.01 m). 3.3. Mixing Performance. The scalar concentrations of 0 and 1 were set in the two halves of the inlet of the tube. The initial concentration was prescribed perpendicular to the direction of the secondary flow. Figure 4 represents the distribution of scalar concentration of liquids at different axial lengths in straight tube, coiled tube, and CFI with one, two, and three 90° bends having d = 0.01 m. The red and blue color denotes the different scalar concentrations of two liquids. It was observed that the streamlines of scalar concentrations were straight in the straight tube. The two liquids came out from the straight tube exactly as they entered except at the interface where the mixing takes place due to molecular diffusion. There was no convective mixing in either the tangential or radial directions. This shows that mixing of liquids was not significant in case of straight tube. However, in the case of the coiled tube, the mixing in the coiled tube was enhanced due to the presence of Dean vortices. These vortices mix two liquids through advection. It was also observed that the CFI displays a significant increase in uniformity of concentration contours as compared to the straight tube and the coiled tube having equivalent length. The figure clearly shows that the concentrations were almost uniform within 3 bends (L = 4.01 m). This was due to increase in radial mixing of the liquids after introduction of each bend. 3.3.1. Effect of Reynolds Number. COV values computed using eq 6 at the outlet of different geometries were normalized with a COV0 value at the inlet. Normalized COV represents the ratio of standard deviation of concentration to the mean concentration of the unmixed fluid at the injection stage. Figure 5 shows the value of normalized COV with varying Reynolds number for straight, helical coil tube, and CFI of equal lengths. It can be observed from the figure that there was no significant change in normalized COV of liquids flowing in the straight tube with an increase in Reynolds number. However, the COV value of liquids decreased with increase in Reynolds number in coiled as well as CFI. The mixing efficiency increased because of an increase in intensity of secondary flows. However, the normalized COV value of liquids flowing in the CFI was found to be nearly 1626 times lower than that of the coil tube having equal length. This was due to the increase in radial mixing of liquids owing to the fluid flow inversion after the 90o bend in the CFI.
The mixing performance of the CFI was also compared with the existing experimental data available for the HEM.7 It was also observed that the COV value for the CFI was found to be 5 to 8 times lower than that for an equivalent length of motionless mixer such as HEM having 18 elements over the range of 98 e Re e 1020. This shows that the CFI performance is superior as compared to a motionless mixer under identical process conditions 3.3.2. Effect of Number of Bends. Figure 6 represents the effect of number of bends on COV of liquids flowing at Re = 490 in straight, coiled, and CFI tube having d = 0.01 m. The figure shows that there was no substantial variation in mixing performance with an increase in length of straight tube. It was observed for the CFI having one bend and the coiled tube having equivalent length that the COV value of liquids was nearly the same. Nevertheless, the mixing efficiency increased with the introduction of bends in the CFI as compared to that of the straight tube and coiled tube of equal lengths. This shows that the mixing of the two liquids increased with an increase in the number of bends. The figure shows that significant mixing was taking place in the CFI within three bends. The length of CFI is not effectively utilized for mixing after the third bend. This observation agrees with the uniformity of scalar concentration shown in Figure 3. COV values for an SMX static mixer46 were calculated for an equivalent length of CFI from the following equation: bL ð8Þ COV ¼ a exp d Here a and b are adjustable constants and are predicted from laminar flow experimental results of the SMX static mixer with liquid viscosity ratio greater than 1. The values of the exponents in eq 8 were a ≈ 15 and b ≈ 0.505 for the SMX static mixer in laminar flow.5,6 Figure 6 shows that COV values for the static mixer are significantly higher with respect to coiled and CFI tube having length an equivalent one bend. The COV values decrease with an increase in mixer length. Nevertheless, the COV value is still nearly 4 times higher for the static mixer as compared to that for the CFI at the outlet (n = 4). 3.4. Friction Factor in CFI. The multiphase flow studies in coiled tubes mostly use the correlations based on the Lockhart Martinelli parameter.24 Studies show that the pressure drop for two-phase gasliquid flow through coiled tubes satisfies the LockhartMartinelli correlation.2527 In the present study, the friction factor was computed from the pressure drop in different geometries. The details for calculation have been reported in our previous papers.28 The friction factor values for different configurations 13233
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Figure 7. Effect of friction factor on Reynolds number for different configurations.
Figure 8. Effect of product of COV and friction factor for different configurations.
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tube, coiled tube, and CFI of equivalent length. Figure 8 shows the variation of product of COV and friction factor against Reynolds number. It was observed that the product of COV and friction factor in the coiled tube was nearly 14 to 26 times higher than the CFI. The values were found to be 26 to 35 times higher in HEM than in CFI for Reynolds numbers varying from 981020. The performance of coiled tube, HEM, and CFI with respect to straight tube was analyzed in terms of figure of merit. Figure of merit represents the ratio of the unmixedness of liquid in a system to the increase in pumping power by the system. Figure 9 shows the ratio of the figure of merit in coiled tube, HEM, and CFI to that of the straight tube. The figure shows that unmixedness in the CFI is nearly 1825 times lower than that in the coiled tube and nearly 24 times lower than that in the HEM.
4. CONCLUSION In the present study, the physics of flow of two miscible liquids was examined in a complex flow generated in CFI geometry. It was observed that the mixing performance in the CFI increased with increase in Reynolds number as well as number of bends. This was further substantiated by velocity and scalar concentration contours of two liquids. The product of COV and friction factor, a new parameter, has been defined to quantify the mixing of two liquids in flow systems. It was found that the enhancement of mixing efficiency in the CFI as compared to that of coiled tube and HEM is higher than the increase in pressure drop of the liquids. It was observed that the CFI offers higher mixing efficiency as compared to a coiled tube and motionless mixers (HEM) of equivalent length. Hence, it may be concluded that the CFI is a more efficient motionless mixer with reasonably lower pumping cost as compared to conventional static mixer. ’ AUTHOR INFORMATION Corresponding Author
*Tel: þ91-11-26591020. E-mail:
[email protected].
Figure 9. Figure of merit in different configurations as compared to that of straight tube.
were plotted against Reynolds number as shown in Figure 7. The figure shows that the friction factor is least in the case of the straight tube. It is interesting to observe that there was no significant difference between the friction factor in the coiled tube and the CFI with three bends over the Reynolds number range studied in the present study. Similar observations were reported in the literature for single phase flow.29 The experimental data for the friction factor in HEM6 has been compared with that of the CFI. The friction factors in HEM were found to 3.36 times higher than that of CFI. To assess the suitability of a given mixer for the homogenization of two liquids, it is essential to assess the combined effect of mixing performance as well as power consumed by the mixers. Hence, efforts were made to investigate the variation of product of COV and friction factor with Reynolds number for straight
’ NOTATIONS A = cross-sectional area (m2) d = internal diameter of tube (m) D = coil diameter (m) g = gravity (m2/s) H = dimensionless pitch, H = p/d L = length (m) Re = Reynolds number p = pitch (m) P = pressure (N/m2) Rc = coil radius (m) u = velocity, m/s x = spatial position in x-direction, m y = spatial position in y-direction, m Greek symbols
r = volume fraction kη = curvature of free surface λ = curvature ratio (D/d) σ = surface tension (N/m) μ = viscosity (kg/(m 3 s)) F = density of fluid (kg/m3) 13234
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