Liquid-Phase Residence Time Distribution for Two-Phase Flow in

Aug 8, 2007 - The range of Dean numbers for the gas and the liquid phase was varied from 235 to 1180 ... of single-phase hydrodynamic and heat transfe...
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Ind. Eng. Chem. Res. 2008, 47, 3630-3638

Liquid-Phase Residence Time Distribution for Two-Phase Flow in Coiled Flow Inverter Subhashini Vashisth and K. D. P. Nigam* Department of Chemical Engineering, Indian Institute of Technology Delhi, New Delhi-110016, India

The coiled flow inverter (CFI) is an innovative device, which has potential for the intensification of processes currently carried out in conventional mixers. Step response experiments were carried out in a CFI to study liquid-phase residence time distribution (RTD) for gas-liquid flow under the conditions of both negligible and significant molecular diffusion using Newtonian fluids. A total of 16 CFIs of different curvature ratios ranging from 6.7 to 20, dimensionless pitch from 1 to 2.5, and number of bends from 1 to 15 were investigated. The range of Dean numbers for the gas and the liquid phase was varied from 235 to 1180 and 3.16 to 1075, respectively. The reduction in dispersion number is nearly 2.6 times for two-phase flow in CFI with 15 number of bends as compared to a straight helix under identical process conditions. A modified axial dispersion model is proposed to describe the RTD. The efficiency of the device is characterized by a mixing criterion. 1. Introduction Mixing operations are essential in the process industries ranging from classical mixing of miscible fluids in single phase flow to complex multiphase reaction systems for which reaction rate, yield, and selectivity are highly dependent upon mixing performance. The consequences of improper mixing may lead to nonreproducible processing conditions and lowered product quality, resulting in the need for more elaborate downstream purification processes and increased waste disposal costs. Numerous mixer designs have been proposed to overcome the above problems. They can be classified as active and passive mixers. Active mixers derive their energy from external source and produce excellent mixing but are often difficult to fabricate and maintain. Passive mixers, which utilize flow energy, are attractive because of their ease of operation and manufacture. Motionless mixers have become standard equipment in the process industries. Their use in continuous processes is an attractive alternative to conventional agitation since similar and sometimes better performance can be achieved at lower cost. The prototypical design of a static mixer is a series of identical, motionless inserts that are called elements, to redistribute fluid in the directions transverse to the main flow, i.e., in the radial and tangential directions.1-8 Many studies aiming at providing an efficient method of mixing by reducing axial dispersion in helical coils,9-15 chaotic configurations,16-18 and serpentine geometries19 have been successfully investigated in past. The state-of-the-art review on the extensive work carried out on single-phase mixing performance of Newtonian and nonNewtonian fluids in straight tube and coiled tube was complied by Saxena and Nigam.14 These configurations have shown promising results in reducing the axial dispersion as compared to the straight tubes. In spite of the advantages of the above devices, they have their own limitations. Static mixers induce prohibitive pressure drop for very viscous fluids. Similarly in helical coils and chaotic geometries, the fluids with very long molecular chains can be damaged by high shear stresses. They also require a very high Dean number to induce significant mixing in the cross-sectional plane. For the last 10 years, industry has been facing the challenges to reduce the energy demands by increasing the process * To whom correspondence should be addressed. Tel.: 91-1126591020. Fax: 91-11-26591020. E-mail: [email protected].

efficiency and at the same time maintaining the environmental norms. This has introduced the concept of process intensification. It focuses on the miniaturization of unit operations to accomplish reduction in energy use, capital expenditures, plant profile (height), and plant footprint (areas) while sustaining the environmental regulations and safety considerations. The coiled flow inverter (CFI) is one such innovative device, which has potential for the intensification of processes currently carried out in conventional motionless mixers and helical coils with much higher efficiency. CFI exploits the concept of effective utilization of centrifugal force and complete flow inversion by bending of helical coils. One unit of CFI consists of several consecutive 90° bends inserted within the helical coils with equal space before and after the bend. By shifting the plane of curvature from one bend to the next, one can induce a class of trajectories in one bend, then deform it to another type in the next bend, and so on as can be seen in Figure 1. Such chaotic cross-sectional movement has been found to enhance the advection of passive scalars and therefore improve the efficiency, leading to homogenization in the fluid volume. Mechanisms that inhibits the axial dispersion in CFI are as follows: (1) Flatter Velocity Profile. Nonuniform velocity profiles cause molecules of different cross-sectional positions to experience different velocities and hence promotes the axial dispersion. In CFI, flatter velocity profiles and uniform thermal and concentration gradients are observed even in laminar flow conditions unlike straight tubes. (2) Bending of Coils. Random mixing at a cross-sectional plane due to coiled geometry and 90° bends inhibits the axial dispersion. This is achieved at low velocities in CFI whereas in straight tubes it is achieved at high Reynolds number. (3) Secondary Flow and Multiple Flow Inversion. Multiple flow inversions due to bending of coils and secondary flow continuously changes the position of molecules in a crosssectional plane, which inhibits axial mixing. This mechanism is not isotropic and, hence, does not increase axial mixing like in the case of random mixing. The details of the optimal configuration of the above device can be found in previous works.20-22 Experimental investigation of single-phase hydrodynamic and heat transfer studies carried out in our laboratory as well as on pilot scale in CFI showed

10.1021/ie070447h CCC: $40.75 © 2008 American Chemical Society Published on Web 08/08/2007

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Figure 1. Working principle of coiled flow inverter.

considerable narrowing of residence time distribution22 and 2530% improvement in Nusselt number23-26 as compared to helical coils. Hence, good knowledge of the extent of axial mixing in the liquid phase is essential for the modeling, design, and optimization of flow reactors. However, only a few studies have been reported on the RTD of two-phase flow systems in complex geometries.27-33 Saxena et al.28 studied liquid-phase RTD for an air-water system through helical coils with λ ranging from 11 to 60.7. They investigated both upward and downward flow and proposed a model to describe RTD. Salengke and Sastry29,30 reported that the mean and the standard deviation of the normalized residence time of particles decreased as either particle size or flow rate was increased in a U-bend geometry. They presented empirical correlations between the means and the standard deviations of the normalized residence time and particle Froude number, particle concentration, and the ratio of bend radius of curvature to tube diameter. Grabowski and Ramaswamy31 investigated the RTD of individual food particles in the curved section (180°) of a holding tube of an aseptic processing simulator. Particle size and shape and viscosity of the carrier liquid influenced the velocity of the food particle in the curved section, but only the shape effect was similar to that in the straight part of the tube. A study of residence times of multiple spherical particles suspended in aqueous solutions of carboxymethylcellulose (CMC) during pseudoplastic flow through a commercial size transparent holding tube system was presented by Sandeep and Zuritz.32 In their another contribution, they studied residence times of multiple particles during the flow of a non-Newtonian suspension through a conventional holding tube assembly and compared it with that in a helical holding tube assembly of identical length and tube diameter.33 They found that the ratio of mean to minimum residence time was within 1.05 to 1.11 in

the helical holding tube and 1.06 to 1.16 in the conventional holding tube for the range of parameters studied. This work attempts to experimentally investigate the liquidphase RTD in CFI. Liquid-phase mixing generally influences the heat and mass transfer rates and reactant conversion in any reactor. Hence, this paper aims to characterize the performance of CFI as a mixer for gas-liquid system by varying geometrical parameters, operating conditions, and fluid properties. A modified axial dispersion model has been proposed to describe the liquid-phase RTD through CFI. A mixing criterion is defined that takes into account both the decrease in axial dispersion and increase in pressure drop in CFI. Our work seeks to exploit the ability of the CFI to improve the conversion and selectivity by improved mixing, residence time characteristics, and thermal control. It is shown that CFI achieves process intensification by improved chemical performance as well as by substantial reduction in reactor size requirements. CFI has enormous potential areas of application such as inline mixers (for liquid-liquid and gas-liquid systems), reactors, heat exchangers, biosensors, and membranes.34 2. Experimental Setup and Methodology Step response experiments were carried out to measure the RTD of the liquid phase in two-phase flow through a coiled flow inverter (CFI). A schematic diagram of the experimental setup is shown in Figure 2. CFI was prepared by using thickwalled, transparent PVC tubing. The transparent wall of the PVC tubing facilitated visual observation of the flow patterns. The PVC tubes were wound around a square-shaped frame made up of cylindrical rods. The internal diameter of tube (d) was varied from 0.005 to 0.015 m, and coil diameter (Dc), from 0.08 to 0.2 m. The angle at the bend was 90° with equal space before and after the bend. The curvature ratio (λ ) Dc/d) of the coil was maintained constant and was varied for different CFIs from

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Figure 2. Schematic of experimental setup. Table 1. Geometrical Consideration of Coiled Flow Invertera sample

D (m)

D (m)

H

λ

NBend

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.005 0.01 0.015 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

0.1 0.1 0.1 0.08 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

1 1 1 1 1 1.5 2 2.5 1 1 1 1 1 1 1 1

20 10 6.7 8 20 10 10 10 10 10 10 10 10 10 10 10

5 5 5 5 5 5 5 5 1 3 7 15 1 3 7 15

a

Length of the tubing ) 27 m. Angle at the bend ) 90°.

6.7 to 20. The complete unit consisted of square-shaped subunits joined together keeping the bend angle constant at 90°. The coils were fixed and carefully tightened with clamps to avoid deformation of tubes. The tubes were wound in close-packed fashion, and the dimensionless pitch (H ) p/d) was varied from 1 to 2.5. The details of the CFI geometry are reported in Table 1. The setup consisted of two reservoirs A1 and A2 containing a water/aqueous solution of diethylene glycol (DEG) and tracer solutions, respectively, and air was drawn from a compressor. The flow rates of air and liquid from these two reservoirs were measured and controlled using calibrated rotameters (Scientific Devices, Pune, India). The reservoirs A1 and A2 and air supply were connected to the CFI through a three-way valve (V1). The minimum possible distance was kept between the three-way valve and the coil. The physical properties of the fluids were taken at ambient temperature and pressure conditions. Congo red dye was used

as the tracer with a concentration of 0.05 kg/m3. Once the flow became steady, the flow rate of the liquid was recorded. Next, the air was introduced into the test section through a small mixer. A straight length of about 1 m was provided before the test section to avoid the fluctuations in the line. A step input technique was used for tracer injection. The tracer concentration was continuously measured at the outlet of CFI by using a UV spectrometer (UV-120-02) at a wavelength of 497 µm (Shinjuku, Tokyo, Japan). A linear relationship between optical density and tracer concentration was experimentally verified that allowed the direct evaluation of dimensionless concentration at the outlet using the optical density measurements. The pressure drop across the CFI was measured with a differential pressure gauge (DPG) (ARR Solution, Hirlekar; DPG 1 range, 0-1 bar; least count, 0.001; DPG 2 range, 1-5 bar; least count, 0.001). The fluid mixture moved through the test section and was separated at the gas-liquid separator. A quick shut-off and displacement method was used to measure the gas hold-up at atmospheric pressure by adjusting the height of the overflow device. 3. Model Consideration The basic property of RTD, ∫0∞|1 - F(θ)| dθ ) 1, should be satisfied to have reliable information on RTD. In this study, the mean residence time (th) for the liquid phase was estimated such that the above stated condition is satisfied. The hold-up was measured experimentally, and those obtained using the mean residence times compared well. This provided a good check for the correctness of measured RTDs. Therefore, the experimental curves (C versus t) were normalized using this value of ht (θ ) t/th) and the final concentration of the dye solution Co (F ) C/Co) to obtain the residence time distribution (F versus θ). For the laminar flow conditions in this study, different flow regimes were encountered such as stratified flow (VSG ) 0.85-

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2.55 m/s; VSL ) 0.001-0.07 m/s), stratified-wavy flow (VSG ) 3-6.37 m/s; VSL ) 0.09-0.13 m/s), and plug flow (VSG ) 0.85-5.31 m/s; VSL ) 0.17-0.72 m/s). It is reported that the liquid velocity profile would be different from the extended velocity profile when the flow deviates from stratified to plug regime.35,36 Hence, many workers incorporated interfacial stresses to account for these differences in velocity.36-39 The RTD for two-phase flow can be modeled in two parts as the following: (1) The fraction of liquid moving with higher velocity and having strong turbulence due to the proximity of the gas phase can be modeled using the axial dispersion model (Fdis). (2) The fraction of liquid moving slower mainly consists of the viscous sublayer. For this part (tail portion), the axial dispersion model is inappropriate. It can be modeled using single-phase laminar flow RTD through straight tube (FN). The way the F curve asymptotically approaches unity is the characteristic of laminar flow in a boundary sublayer which is the same for both straight and coiled tubes.40 Hence, the expression for the RTD for two-phase flow can be given as

F ) Fdisψ + FN(1 - ψ)

(1)

θN )

θ θaV,N

(8)

θdis )

θ θav,dis

(9)

By using the above equations, the F versus θ curve is generated. The three parameters that the model uses are the dispersion number (D/(VSLL)), the fraction of the flow modeled by the dispersion model (ψ), and the relative shift in the velocities of the two fractions (VL). Estimation of Model Parameter. The model was applied to all the 104 experimental F-curves, and the best-fit parameters (VL, ψ, and D/(VSLL)) were estimated:

VL ) 0.1912 - 0.0003xNDe,SGNDe,SL

235 e NDe,SG e 1180 (10) (mean error, 3.4%; maximum error, 7.5%).

ψ ) 0.97 + 0.0122l - 0.0383

( ) NDe,SG NDe,SL

[

]

(1 - θdis) 1 ; (VSLL/D) > 16 (2) Fdis ) 1 - erf 2 x(D/VSLL)θdis 0.25 ; θN > 0.5 θN2

(3)

The complete description of RTD for the liquid phase using eqs 2 and 3 further requires knowledge about the fraction of fluid and the relative shift in their average velocities. Two parameters used in the model to take care of these two aspects are ψ, the fraction of liquid with a distribution of residence time described by eq 2, and VL, the slip velocity between the two fractions, defined as

VL )

Vdis - VN V

θav,dis ) 1 - R

235 e NDe,SG e 1180 (11) (mean error, 3.1%; maximum error, 7.0%). The experimentally obtained values of the dispersion number for two-phase flow were correlated to the design parameter RA:

RA ) volume of the largest arm of bent coil/total volume of the helix (12) This parameter characterizes the performance of CFI. The lower the value of RA, the narrower is the RTD. The volume of the largest arm used in defining RA takes care of the NBend as well as spacing among bends. For a given odd NBend, the lowest value of RA would occur when NBend are equispaced for which RA can be given as

(4)

The shift of the two curves may be defined as

RA )

(5)

D VSLL

(1 - ψ)(1 - R) θav,N ) (1 - R - ψ)

(1 - VL - ψVL)

) 0.0056

RA0.14 NDe,SL0.04

(13)

; 126 e NDe,SL e 1075

235 e NDe,SG e 1180; RA < 0.5 (6)

R can be determined from the implicit eq 7

VL(1 - ψ)

1 NBend + 1

Hence,

and

R)

0.02

126 e NDe,SL e 1075

for each value of θ, where

FN ) 1 -

126 e NDe,SL e 1075

(7)

The derivation of eq 7 is given in Saxena et al.28 The dimensionless residence time for each fraction can be defined as

(14)

(mean error, 1.74%; maximum error, 5.2%). The step response curve (F-curve) for two-phase flow using the above equations can be computed as follows: (1) Calculate the curvature ratio as coil diameter/tube diameter. (2) Calculate the Dean number (NDe ) NRe/xλ) for the gas and the liquid phase using superficial velocity and tube diameter. (3) Estimate the VL, ψ, and D/(VSLL) using eqs 10, 11, and 14. (4) Use eqs 8 and 9 to evaluate θN and θdis for a given θ.

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Figure 3. Step response curves for different values of liquid-phase Dean number, NDe,SL (NDe,SG ) 235, NBend ) 5, λ ) 10, H ) 1).

(5) Evaluate Fdis and FN from eqs 2 and 3, respectively. (6) Use eq 1 to estimate F for the given θ.

Figure 4. Step response curves for different values of gas-phase Dean number, NDe,SG (NDe,SL ) 126, NBend ) 5, λ ) 10, H ) 1).

4. Results and Discussion 4.1. RTD with Significant Molecular Diffusion. The effect of different geometrical parameters and operating variables on axial dispersion in CFI was studied using air and water as the flowing media and Congo red dye as tracer with a concentration of 0.05 kg/m3, under the condition of significant molecular diffusion. The liquid velocity was varied from 0.04 to 0.34 m/s (corresponding to liquid-phase Dean number, NDe,SL from 126 to 1075) and gas velocity from 1.06 to 5.31 m/s (which corresponds to gas-phase Dean number, NDe,SG from 235 to 1180). A total of 104 experiments for the air-water system were conducted. Typical response curves are given for NDe,SL ) 126 and NDe,SG ) 235. Similar trends were observed for the other liquid and gas flow rates range covered in this study and hence not given to avoid the redundancy. Different flow patterns such as stratified, stratified-wavy, and plug flow regimes were observed depending upon the combination of gas and liquid flow rates. 4.1.1. Effect of Dean Number. Figure 3. shows the influence of liquid-phase Dean number, NDe,SL, on the liquid-phase RTD for CFI having λ ) 10, NBend ) 5, and H ) 1. It can be seen from the figure that as the NDe,SL is increased, RTD gets narrower. At NDe,SL ) 126, the first element of tracer appears at dimensionless time, θ ) 0.757, whereas when NDe,SL is increased to 1075, θ ) 0.866, which clearly shows the narrowing of RTD. This is due to the fact that as NDe,SL increases, the intensity of secondary flow is higher which inhibits axial mixing; hence, back-mixing is reduced. This implies that the flow in CFI tends toward ideal plug flow conditions. The proposed model was tested for all the 104 experimental RTD curves. The criterion employed for the best fit was to minimize the area between the theoretical and the experimental F-curves, Σ|Fexp - Ftheo|(∆θ), which is also equal to twice the fraction of fluid that is assigned an incorrect residence time by the model. In this study, the value of min Σ|Fexp - Ftheo|(∆θ) for all the 104 measured RTDs was less than 0.064. This means that e3.2% of the liquid phase has been assigned an incorrect residence time by the model. A similar approach was also used for fitting the dispersion model to two-phase flow in coiled tube.28 A typical model fit to the experimental data is also shown in Figure 3. There is reasonably good agreement between the model and experimental data. The response curves, which were not found

Figure 5. Step response curves for different numbers of bends, NBend (NDe,SL ) 126, NDe,SG ) 235, λ ) 10, H ) 1).

to be smooth, were smoothed using the smoothness criteria that the first and second moments should remain same. Figure 4. shows the influence of gas-phase Dean number, NDe,SG, on the RTD. It was observed that the RTD gets narrower with increase in NDe,SG. It was found that nearly 1.3 times reduction in dispersion number takes place in CFI with 5 number of bends when NDe,SL was increased from 126 to 1075. When the NDe,SG was increased 5-fold, the reduction in dispersion number was 1.08 times. 4.1.2. Effect of Number of Bends on Liquid-Phase RTD. Step response experiments were carried out in 16 different CFIs with NBend ) 1, 3, 5, 7, and 15 and a straight helix. As the NBend is increased, multiple flow inversion takes place. The fluid elements continuously change their positions, which inhibits axial dispersion. Hence, the RTD gets narrower as NBend is increased (Figure 5). The performance of plug flow reactor can also be seen in Figure 5 for which θ ) 1. When the performance was compared with that of the straight helix, it was seen that the dimensionless time at which the first tracer element appear was θ ) 0.657, whereas, for a CFI with NBend )15, θ was found to be 0.868. Comparison of D/(VSLL) observed for two-phase flow in CFI was made against a straight helix (Figure 6). It is interesting to mention that under identical process conditions about 2.6 times reduction in dispersion number is achieved in CFI with 15 number of bends as compared to a straight helix. Hence, it can be said that introduction of bends in a straight

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Figure 6. Comparison of dispersion number for two-phase flow in CFI to that in a straight helix (NDe,SG ) 235, λ ) 10, H ) 1). Figure 8. Effect of Dean number on diffusion-free RTD using DEG (NBend ) 5, λ ) 10, H ) 1, NDe,SG ) 235).

Figure 9. Effect of NBend on diffusion-free RTD using DEG (λ ) 10, H ) 1, NDe,SL ) 126, NDe,SG ) 235). Table 2. Physical Properties of the Working Fluids (n ) 1)

Figure 7. Step response curves for (a) different curvature ratio, λ, and (b) different pitch, H (NDe,SL ) 126, NDe,SG ) 235).

helix improves the performance of the reactor by considerably narrowing the RTD. 4.1.3. Effect of Curvature Ratio and Pitch on Liquid-Phase RTD. Experiments were carried out for different values of curvature ratios (λ ) 6.7, 8, 10, 20) and dimensionless coil pitch, H ()p/d ) 1, 1.5, 2, 2.5), for the combination of gas and liquid flow rates. It was found that RTD of the liquid phase is dependent upon the curvature ratio. As the value of λ was reduced from 20 to 6.7, the RTD was found to approach closer toward a plug flow reactor as can be seen from Figure 7a. This clearly indicates that a CFI with smaller λ favors the uniformity in processing two-phase mixtures. Similarly, when the coil pitch was increased from 1 to 2.5, it led to the broadening of RTD (Figure 7b).

fluid

density (kg/m3)

consistency index K ((kg sn-2)/m)

molecular diffusivity, Dm (× 1010 m2/s)

water 80% DEG pure DEG

1000 1104 1104

0.001 0.0134 0.0273

5 0.414 0.27

4.2. RTD with Negligible Molecular Diffusion. Step response experiments were carried out in CFI under the influence of negligible molecular diffusion using air-pure DEG and air-80% aqueous solution of DEG as flowing media. The geometrical parameters of CFI were kept the same as previous case. The experiments were carried out in the range of 3.16 < NDe,SL < 772 and 235 < NDe,SG < 950. The molecular diffusion coefficient of Congo red dye in air-aqueous DEG solution was experimentally determined by measuring the extent of dispersion in straight tube. The values of molecular diffusion coefficient for different concentration ratios are mentioned in Table 2. A total of 156 experiments were conducted. 4.2.1. Test for Negligible Molecular Diffusion. A test for the absence of molecular diffusion was carried out. RTDs were measured experimentally with air-pure DEG and air-80% aqueous solution of DEG under flow conditions identical with those in the same coil. Despite the 2-fold variation in molecular diffusion coefficients, identical RTD was obtained in both the cases, which confirmed the absence of molecular diffusional effects. Similar experiments were carried out earlier in our laboratory.12

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Figure 10. Model constants versus torsion parameter (NDe,SL ) 126, NDe,SG ) 235, NBend ) 5, λ ) 10, H ) 1).

4.2.2. Effect of Dean Number on Liquid-Phase RTD. Effect of liquid-phase Dean number, NDe,SL, on diffusion-free RTD in CFI was studied. Figure 8. shows the gradual narrowing of RTD with increase in NDe,SL. At higher NDe,SL interchange of velocities among the fluid elements causes narrowing of RTD. At lower values of NDe,SL, mixing will take place only among fluid elements, which have enough secondary momentum before as well as after the bend. The dimensionless time, θmin, for twophase flow in CFI can be correlated as a function of NDe,SL as

symmetrical to the x-axis (i.e., Dean’s velocity profiles), and for T > 0, there exist two asymmetric counter rotating vortices. They reported that the effect of torsion on secondary flow is so dominant that the two counter-rotating vortices becomes one vortex when T > 1/24. The RTD for two-phase flow in CFI over the range of 1 × 10-5 < T < 0.015 (i.e. p/R < π NDe/500) is shown in Figure 10. Nauman’s three-parameter model6 was fitted to the experimentally obtained RTDs, viz.

F(θ) ) 0; θ < θmin

θmin ) 0.66 + 0.0002NDe,SL; 3.16 < NDe,SL < 772; 235 < NDe,SG < 950 (15) 4.2.3. Effect of Number of Bends and Curvature Ratio on Liquid-Phase RTD. Experiments were carried out to examine the effect of number of bends on diffusion-free RTD. For a fixed volume and diameter of CFI, the requirement of some minimum number of turns is necessary for developing secondary flow in each arm to put a restriction on the minimum number of bends. It was observed that, for NBend ) 1 and 3, there was no significant difference in the RTD. This may be due to the fact that the secondary flow is not fully developed. As the number of bends is increased multiple flow inversion takes place at the bends. Mixing of fluid elements of different ages takes place that inhibits axial dispersion. At NBend ) 5, a considerable narrowing of RTD is seen in Figure 9. For CFI having NBend ) 15, the dimensionless time, θ, was found to be 0.874 whereas it is 0.737 for NBend ) 1. The variation of θmin as a function of NBend can be correlated as

θmin ) 0.66 + 0.0102NBend 3.16 < NDe,SL < 772 235 < NDe,SG < 950 (16) A considerable reduction in axial dispersion was observed when λ was decreased from 20 to 6.7. It is evident that bent coils are more effective device in reducing the spread of residence time as compared to a straight tube and straight helix. 4.2.4. Effect of Torsion on Liquid-Phase RTD. The effect of torsion parameter (T ) p/2πRNDe) on liquid-phase RTD in CFI was examined. Saxena and Nigam11 reported that, for single-phase flow at T ) 0.0, RTD is identical with that reported by Nauman1 and as the value of T increases RTD approaches straight tube RTD. At T ) 0 the secondary flow pattern is

F(θ) ) 1 - A1θB+1 -

(17a)

A2 ; θ g θmin θ2

(17b)

Constants A1 and A2 in the model can be related to B and θmin satisfying the basic properties of RTD, i.e.

∫0∞f(θ) dθ ) ∫0∞θf(θ) dθ ) 1.0 where

(18)

)

(19)

A2 ) θmin2(1 - A1θminB+1)

(20)

A1 )

(

(B + 2)(2θmin - 1) θminB+2(B + 2)

The value of T employed for this study covers a wide range of practical applications for which the helical coils is used in industry. Variation of θmin with torsion parameter is shown in Figure 10, which can be correlated as

θmin ) 0.816 - 73.37T + 1598T2; 0 < T < 0.015

(21)

(mean error ) 2.7%; maximum error ) 5%). The values of -B (over the range of 0 < T < 0.015) were also plotted against T in Figure 10, which is correlated as

-B ) 3.882 - 1.544β + 1.4793β2 - 0.57β3 where

β ) 102T (mean error ) 1.2%; maximum error ) 3.2%).

(22)

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with increase in Dean number and number of bends. Experimental results verified that geometric parameters such as curvature ratio and pitch effect the RTD. Compared to a straight helix, it was found that nearly 2.6 times reduction in dispersion number takes place in CFI, with NBend ) 15. A modified axial dispersion model has been proposed to describe the liquid-phase RTD in CFI. A mixing criterion, Cmix, is proposed that takes into account both the mixing characteristics and pressure drop in CFI. The enhancement in mixing performance was nearly 3.5 times that of a straight helix. Superiority of the proposed device has been established on the basis of its performance substantially closer to that of an ideal plug flow, low initial and operating costs, compactness, and ease of fabrication. Nomenclature Figure 11. Comparison of the mixing performance of CFI and straight helix for two-phase flow.

It is interesting to note that as the value of T increases, RTD approaches toward a straight helix RTD for two-phase flow. It was observed that the effect of coil pitch on RTD can be ignored if p/R < πNDe/500. 5. Mixing Performance in CFI The mixing performance of CFI was characterized and compared with a straight helix. It was observed that nearly 2.6 times reduction in axial dispersion takes place in CFI (NBend )15) as compared to a straight helix for two-phase system. The relative increase in pressure drop due to complete flow inversion at bends in CFI should also be taken into consideration before characterizing the mixing performance. The mixing performance criterion, Cmix, considered is defined as

Cmix )

ν DfTP

(23)

where D is the axial dispersion coefficient and fTP is the twophase friction factor. Cmix takes into account the decrease in axial mixing to the relative increase in pressure drop. Figure 11 shows the mixing performance of CFI having NBend ) 1 and 15 for different liquid-phase Dean number. The mixing performance for two-phase flow in straight helix28 is also shown in Figure 10. Cmix decreases as the NDe,SL is increased, showing that pressure drop at higher flow rates is dominant over the decrease in axial dispersion. Also, it can be seen that Cmix is higher for CFI (NBend ) 1) by nearly a factor of 1.6 compared to that of a straight helix. When the number of bends was increased from 1 to 15, the enhancement in mixing performance was nearly 3.5 times that of a straight helix. Hence, it may be concluded that the mixing performance of CFI is much better than that of a straight helix. 6. Conclusion The coiled flow inverter (CFI) is an innovative device, which has potential for the intensification of processes currently carried out in conventional motionless mixers and helical coils. Step response experiments were carried out to study liquidphase RTD under the conditions of both negligible and significant molecular diffusion in CFI for the first time. Secondary flow due to coiled structure and multiple flow inversions because of bends reduces the axial dispersion

A1, A2, B ) model constants, eq 17 C ) instantaneous concentration of the tracer in the output, kg/m3 Co ) final concentration of the tracer in the output, kg/m3 Cmix ) mixing performance criterion; Cmix ≡ ν/(DfTP) d ) internal diameter of tube, m Dc ) coil diameter, m D ) dispersion coefficient, m2/s D/(VSLL) ) dispersion number as described in dispersion model fTP ) two-phase frictional pressure drop F ) dimensionless output concentration for overall flow H ) dimensionless pitch; H ≡ p/d NBend ) number of bends p ) pitch, m RA ) model parameter, eq 12 L ) tube length, m VSG ) superficial gas velocity, m/s VSL ) superficial liquid velocity, m/s t ) instantaneous time of measurement of the output tracer concentration, s jt ) average residence time of the liquid phase, s T ) torsion; T ≡ p/(2πRNDe) VL ) slip velocity parameter R ) coil radius Dimensionless Numbers NDe ) Dean number; (NDe ≡ NRe/(λ)0.5 NRe ) Reynolds number; NRe ≡ dVF/µ Greek Symbols µ ) viscosity, kg/(m s) ν ) kinematic viscosity, m2/s F ) density of fluid, kg/m3 θ ) dimensionless time; θ ≡ t/th θmin ) residence time for the fastest moving fluid element, dimensionless θav ) mean residence time defined by eq 5 and 6, dimensionless ψ ) model parameter representing the fraction of fluid flow governed by eq 1 λ ) curvature ratio; λ ≡ Dc/d Subscripts L ) liquid phase G ) gas phase st ) straight tube CFI ) coiled flow inverter dis ) refers to fraction of liquid described by eq 2 N ) refers to fraction of liquid described by eq 3

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exp ) experimental value theo ) model predicted value Acknowledgment S.V. expresses her gratitude toward the Ministry of Chemical and Fertilizers, Government of India, and The Industrial Research and Development Unit, IIT, Delhi, for providing the High Value Research Scholarship. Literature Cited (1) Nauman, E. B. The residence time distribution for ideal laminar flow in helically coiled tubes. Chem. Eng. Sci. 1977, 32, 287. (2) Nauman, E. B. Invited review: Residence time distributions and micromixing. Chem. Eng. Commun. 1981, 8, 53. (3) Nauman, E. B. Reactions and residence time distributions in motionless mixers. Can. J. Chem. Eng. 1982, 60, 136. (4) Nauman, E. B. On residence time and trajectory calculations in motionless mixers. Chem. Eng. J. 1991, 47, 141. (5) Nauman, E. B.; Kothari, D.; Nigam, K. D. P. Static mixers to promote axial mixing. Chem. Eng. Res. Des. 2002, 80, 1. (6) Nigam, K. D. P.; Nauman, E. B. Residence time distribution of power law fluids in motionless mixers. Can. J. Chem. Eng. 1985, 63, 519. (7) Nigam, K. D. P.; Vasudeva, K. Residence time distribution in static mixer. Can. J. Chem. Eng. 1980, 58, 543. (8) Thakur, R. K.; Vial, Ch.; Nigam, K. D. P.; Nauman, E. B. Static mixers in process industries-A review. Trans. Inst. Chem. Eng. 2003, 81 (A), 787. (9) Saxena, A. K.; Nigam, K. P. D. On RTD for laminar flow in helical coils. Chem. Eng. Sci. 1979, 34, 425. (10) Saxena, A. K.; Nigam, K. P. D. Axial dispersion in laminar flow of polymer solutions through coiled tubes. J. App. Polym. Sci. 1981, 26, 3475. (11) Saxena, A. K.; Nigam, K. P. D. Effect of coil pitch and crosssectional ellipticity on RTD for diffusion-free laminar flow in coiled tubes. Chem. Eng. Commun. 1983, 23, 1097. (12) Saxena, A. K.; Nigam, K. P. D. Laminar dispersion in helically coiled tubes of square cross section. Can. J. Chem. Eng. 1983, 61, 53. (13) Saxena, A. K.; Nigam, K. M.; Nigam, K. P. D. RTD for diffusionfree laminar flow of non-Newtonian fluids through coiled tubes. Can. J. Chem. Eng. 1983, 61, 50. (14) Nigam, K. P. D.; Saxena, A. K. Residence Time Distribution in Straight and Curve Tubes. In Encyclopaedia of Fluid Mechanics; Cheremisinoff, N. P., Ed.; Gulf Publishing: Houston, TX, 1986; Vol. 1, Chapter 22, p 675. (15) Koutsky, J. A.; Adler, R. J. Minimization of axial dispersion by use of the secondary flow in helical tubes. Can. J. Chem. Eng. 1964, 42, 239. (16) Castelain, C.; Mokrani, A.; Legentilhomme, P.; Peerhossaini, H. Residence time distribution in twisted pipe flows: helically coiled and chaotic systems. Exp. Fluids 1997, 22, 359. (17) Castelain, C.; Berger, D.; Legentilhomme, P.; Mokrani, A.; Peerhossaini, H. Experimental and numerical characterization of mixing in a spatially chaotic flow by means of residence time distribution measurements. Int. J. Heat Mass Transfer 2000, 43, 3687. (18) Castelain, C.; Legentilhomme, P. Residence Time Distribution of a purely viscous non-Newtonian fluid in helically coiled or spatially chaotic flows. Chem. Eng. J. 2006, 120, 181. (19) Kaufman, A. D.; Kissinger, P. T. Extra-column band spreading concerns in post-column photolysis reactors for microbore liquid chromatography. Curr. Sep. 1998, 17 (1), 9.

(20) Nigam, K. D. P. (Indian Institute of Technology, Delhi). Indian Patent 159/DEL/2005 and Design Patent 198236, 2005. (21) Nigam, K. D. P. (Indian Institute of Technology, Delhi). U.S. Patent 11/146,913, 2005. (22) Saxena, A. K.; Nigam, K. D. P. Coiled configuration for flow inversion and its effect of residence time distribution. AIChE J. 1984, 30 (3), 363. (23) Kumar, V.; Nigam, K. D. P. Numerical simulation of steady state flow fields in coiled flow inverter. Int. J. Heat Mass Transfer 2005, 48, 4811. (24) Kumar, V.; Nigam, K. D. P. Mixing in curved tubes. Chem. Eng. Sci. 2006, 61, 5742. (25) Kumar, V.; Mridha, M.; Gupta, A. K.; Nigam, K. D. P. Coiled flow inverter as a heat exchanger. Chem. Eng. Sci. 2007, 62, 2386. (26) Mridha, M.; Nigam, K. D. P. Numerical study of turbulent forced convection in coiled flow inverter. Chem. Eng. Process. 2007, in press. (DOI: 10.1016/j.cep.2007.02.026.) (27) Waiz, S.; Cedillo, B. M.; Jambunathan, S.; Hohnholt, S. G.; Dasgupta, P. K.; Wolcott, D. K. Dispersion in open tubular reactors of various geometries. Anal. Chim. Acta 2001, 428, 163. (28) Saxena, A. K.; Nigam, K. P. D.; Schumpe, A.; Deckwer, W. D. Liquid Phase Residence time Distribution for Two Phase Flow in Coiled Tubes. Can. J. Chem. Eng. 1996, 74, 553. (29) Salengke, S.; Sastry, S. K. Residence time distribution of cylindrical particles in a curved section of a holding tube: The effect of particle size and flow rate. J. Food Process. Eng. 1995, 18, 363. (30) Salengke, S.; Sastry, S. K. Residence time distribution of cylindrical particles in a curved section of a holding tube: The effect of particle concentration and bend radius of curvature. J. Food Eng. 1996, 27, 159. (31) Grabowski, S.; Ramaswamy, H. S. Bend-effects on the residence time distribution of solid food particles in a holding tube. Can. Agric. Eng. 1998, 40, 121. (32) Sandeep, K. P.; Zuritz, C. A. Residence times of multiple particles in non-Newtonian holding tube flow: Effect of process parameters and development of dimensionless correlations. J. Food Eng. 1995, 25, 31. (33) Sandeep, K. P.; Zuritz, C. A.; Puri, V. M. Residence time distribution of particles during two-phase non-Newtonian flow in conventional as compared with helical holding tubes. J. Food Sci. 1997, 62, 647. (34) Nigam, K. D. P. Potential areas of application of coiled flow Inverter in process industries. Internal Report; Indian Institute of Technology (IIT), Delhi, 2006. (35) Aggarwal, S. S.; Gregory, A. G.; Govier, G. W. An analysis of horizontal stratified two-phase flow in pipes. Can. J. Chem. Eng. 1973, 51, 280. (36) Ghorai, S.; Suri, V.; Nigam, K. D. P. Numerical modeling of threephase stratified flow in pipes. Chem. Eng. Sci. 2005, 60, 6637. (37) Russel, T. W. F.; Etchells, A. W.; Jensen, R. H.; Arruda, P. J. Pressure drop and hold up in stratified gas-liquid flow. AIChE J. 1974, 20, 664. (38) Cheremisinoff, N. P.; Davis, E. J. Stratified turbulent-turbulent gas liquid flow. AIChE J. 1979, 25, 48. (39) 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. (40) Nauman, E. B.; Buffham, B. A. Mixing in Continuous Flow Systems; Wiley: New York, 1983.

ReceiVed for reView March 27, 2007 ReVised manuscript receiVed June 22, 2007 Accepted June 22, 2007 IE070447H