A Review on the Potential Applications of Curved Geometries in

Ind. Eng. Chem. Res. 2008, 47, 3291-3337

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A Review on the Potential Applications of Curved Geometries in Process Industry Subhashini Vashisth, Vimal Kumar, and Krishna D. P. Nigam* Department of Chemical Engineering, Indian Institute of Technology, Delhi, India

The potential industrial applications of curved tubes for single- and two-phase flow are reviewed within the context of physics of flow, trends in the development of technology, and its laboratory to industrial-scale commercialization. Comparison of the performance of curved tube configurations demonstrates its edge over the conventional motionless mixers, heat exchangers, and reactors. Alongside, their respective advantages and limitations are also highlighted. Further, a compendium of the available correlations for single- and twophase friction factor and heat- and mass-transfer coefficient in curved tubes has also been presented. Key issues regarding the design parameters governing the performance of the curved tubes for mixing and heatand mass-transfer that impact the research, development, and scale-up or scale-down of such devices are also analyzed. Emerging trends for the development of a new class of curved tubes, namely, inverters and serpentine and chaotic devices are also presented. 1. Introduction Complex fluid dynamics occurring in curved tubes features an invariant issue of research for last few decades. The various types of curved tube geometries can be classified as follows: (a) torus (constant curvature and zero pitch), coiled or helical tube (constant curvature and pitch), serpentine tubes (periodic curved tubes with zero pitch) with bends or elbows, spirals (Archimedian spirals), and twisted tubes are shown in Figure 1. The facts concerning the working principle of curved tubes and reasons for its enhanced performance are well established as mentioned: (a) generation of secondary flow due to unbalanced centrifugal forces; (b) enhanced cross-sectional mixing; (c) reduction in axial dispersion; (d) improved heat-transfer coefficient; (e) improved mass-transfer coefficient. The details of hydrodynamics and heat-transfer characteristics will only be lightly sketched where appropriate since more complete reviews are available in other sources and the references cited therein.1-6 The literature survey suggests that the earliest recognition of the curved geometries (coiled tubes, U-tubes, and elbows) can be found in the work of Thompson,7 Williams et al.,8 Grindley and Gibson,9 and Eustice.10 Since then there are now approximately 5000 U.S. patents and more than 10 000 research articles on curved tube geometries and their applications. Curved tubes are an essential component of nearly all industrial processes, ranging from power production, chemical and food industries, electronics, environment engineering, waste heat recovery, manufacture industry, air-conditioning, refrigeration, and space applications. The use of curved tubes in continuous processes is an attractive alternative to conventional agitation since similar and sometimes better performance can be achieved at lower energy consumption and reduced maintenance requirement because of no moving parts. For heating and mass-transfer applications, they have the 2-fold advantage of increasing the transfer rate due to secondary flow and providing high heatand mass-transfer area per unit volume of space. Curved tubes can provide homogenization of feed streams with a minimum residence time and are available in most materials of construction. Various applications of curved tubes as reported in literature are compiled in Table 1. It can be seen from Table 1 that curved geometries have wide range of applications varying from human organs (lungs, blood vessels, catheter, etc.) to industrial devices (heat exchanger, mixers, microdevices, nuclear reactor, etc.).

The classifications of curved tubes in process industry are based on three broad areas (mixing, heat- and mass-transfer) as can be seen in Figure 2. The industrial processes involving curved tubes include the classical mixing of miscible fluids in single-phase flow, heat- and mass-transfer enhancement, dispersion of gas into a continuous liquid phase, two-phase contacting, and mixing of solids. As a mixer, curved tubes are intended to achieve composition homogeneity in the directions transverse to the predominant flow, e.g., in the radial direction. The masstransfer operations include applications to homogeneous reactions, with separation processes, such as membrane separation process and reactive absorption. Curved tubes have been used for all these applications, including liquid-liquid systems (e.g., liquid-liquid extraction), gas-liquid systems (e.g., absorption), and solid-liquid systems (e.g., pulp slurries). The heat-transfer operation includes traditional thermal homogenization and heat transfer in heat exchangers involving viscous fluids in the laminar (such as polymer solutions, highly exothermic chemical reactions, etc.) and turbulent (e.g., gas-gas heat exchanger, heaters, cooler. etc.) flow regimes. The primary objective of this work is to provide an overview of the perspective on evolution of curved tube technology and introduction to a new class of curved tubes. The paper is divided into four parts. The first part discusses the fundamentals involved in enhancing the mixing performance and local phenomena in curved tubes, to better understand how mixing and heat- and mass-transfer proceed for single-phase flow. In the second part, the chemistry of two-phase flow in curved tube is discussed and reported. Both first and second parts also review the wellestablished models available for pressure drop, heat transfer, mixing, and mass-transfer in the curved tubes. Methods for estimating the performance of curved tubes from experimental data, empirical correlations, and computational fluid dynamics are discussed. The third part discusses the development of a new class of curved chaotic geometries (bent coils) as a course of technological development. The hydrodynamics and heattransfer and mass-transfer performance in curved chaotic configurations have been discussed and compared against curved tube. The last section concludes with the ongoing research and future developments in this field.

10.1021/ie701760h CCC: $40.75 © 2008 American Chemical Society Published on Web 04/30/2008

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Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008

Figure 1. Different types of curved tube geometries: (a) helical coil, (b) bend tube, (c) serpentine rube, (d) spiral, and (e) twisted tubes. (Source: Kumar and Nigam.203)

2. Single-Phase Fluid Flow, Heat- and Mass-Transfer 2.1. Fluid Flow in Curved Tubes. Unlike the flow in straight pipe, fluid motion in a curved pipe is not parallel to the curved axis of the bend, owing to the presence of a secondary motion caused by secondary flow. As the flow enters a curved bend, centrifugal force acts outward from the center of curvature on the fluid elements. Pressure gradients parallel to the axis of symmetry are almost uniform along lines normal to that of the symmetry axis. Because of the no-slip condition at the wall, the axial velocity in the core region is much faster than that near the wall. To maintain the momentum balance between the centrifugal force and pressure gradient, slower moving fluid elements move toward the inner wall of the curved tube. This leads to the onset of a secondary flow such that fluid near the wall moves along upper and lower halves of the torus wall while fluid far from it flows to the outer wall (Figure 3). The coil curvature affects the flow patterns.9 Even slight curvature was observed to modify the critical velocity of the fluid.10 Dean88,89 was the first to develop an analytical solution for fully developed laminar flow in curved tube of circular crosssection. He developed a series solution as a perturbation of the Poiseuille flow in a straight pipe for low values of Dean number (NDe < 17). He reported that the relation between pressure gradient and the rate of flow is not dependent on the curvature to the first approximation. In order to show its dependence, he modified the analysis by including the higher-order terms and was able to show that reduction in flow due to curvature depends on a single variable K (known as Dean number), equal to 2NRe2a/L (where NRe is the Reynolds number, a is the radius of the tube, and L is the radius of curvature of curved tube in Dean’s notation). Various other authors used different definitions

of the Dean number for curved tube studies.1 Tables 2 and 3 present the analysis of work carried out by numerous researchers on fluid flow for laminar and turbulent flow regime over a wide range of Dean numbers. The various studies reported in the Tables 2 and 3 are varied for different flow (low to high Reynolds number) and geometrical parameters (curvature ratio and tube pitch). It was reported that, for low Dean numbers, the axial-velocity profile was parabolic and unaltered from the fully developed straight tube flow. As the Dean number is increased, the maximum velocity began to be skewed toward the outer periphery. Similarly, for low values of curvature ratio, the secondary flow intensity is very high while for high values of curvature ratio the secondary flow intensity is much less. 2.1.1. Flow Transition from Laminar to Turbulent Regime. Secondary flow in a curved tube is the potential reason behind stabilizing the laminar fluid flow, resulting into a higher critical Reynolds number. Table 4 summarizes some of the correlations useful in predicting the critical Reynolds number as a function of curvature ratio (λ ) D/d) in the curved tube. The transitional behavior of the flow in curved tubes was investigated in detail by Sreenivasan and Strykowski142 and Webster and Humphrey.143 Sreenivasan and Strykowski142 observed that the critical Reynolds number of turbulence is different at different locations when the disturbance is measured at a cross-section or along a tube. They also found that the critical Reynolds number, which corresponds to the first appearance of turbulence everywhere at the chosen cross-section, reaches a maximum value and then drops as the curvature increases. Webster and Humphrey143 found low-frequency oscillation in half of the pipe cross-section, while the flow near the outer wall remains steady when the Reynolds number is

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3293 Table 1. Applications of Coiled Tubes Reported in the Literature application coiled membrane blood oxygenators

flow regime laminar flow

authors Weissman and Mockros11 Dorson et al.12 Chang and Mockros13 Chang and Tarbell14 Baurmeister et al.15

reverse osmosis units

laminar flow

Srinivasan and Tien16 Nunge and Adams17 Moulin et al.18

ultrafiltration (membrane separation)

laminar flow

Elmaleh and Ghaffor19,20

chemical reactors

laminar/ Koutsky and Adler22 turbulent flow Seader and Southwick23 Janssen24 Waiz et al.25

lung, blood vessels, and artificial respiration

laminar/Horsfield et al.26 turbulent flow Lighthill27 Pedley 28 Lin and Tarbell29 Gilroy et al.30 Patel and Sirs31 Donaldson and De32 Padnabhan and Jayaraman33 Jain and Jayaraman34 Niimi et al.35 Eckmann and Grotherg36 Sharp et al.37 Sarkar and Jayaraman38 Zhang39 Pontrelli and Tatone40

Liu et al.21

Guan and Martonen41

human lungs and cathetor arteries

Karahalios42 Jayaraman and Tiwari43 Kim et al.44 Krams et al.45 Torii et al.46 Liu47 Dash et al.48 Nunge and Lin49 Payatakes et al.50 Deiber and Schowalker51

flow through porous media

protein separation and emulsification

laminar flow

Leclerc et al.52 Matsuda et. al.53 Kaur and Agrawal54

heat exchanger, heater/cooler, laminar/ Collins55 cryogenic applications turbulent flow Georgiev and Kovatchev56 Prasad et al.57 Mote et al.58-61 Klien et al.62 Chauvet et al.63 Inagaki et al.64 Rindt et al.65,66 Acharya et al.67 Kim et al.68 Prabhajan et al.69,70 Rennie and Raghvan71-73 Naphon and Wongwises74,75 Wongwises and Polsongkram76,77 Naphon78 Xie et al.79 Park et. al.80 evaporator, steam generator

laminar/ Yi et al.81 turbulent flow Jo and Jhung82

rectification and absorption column

laminar flow

Hameed and Muhammed83 Jose et al.84 Akbrinia and Behzadmehr85

nanofluidics

laminar flow

nuclear reactor (IRIS)

turbulent flow Carelli et al.86

between 5060 and 6330. The flow shows a high-frequency turbulent fluctuation for NRe > 6330. Figure 4 shows the critical Reynolds number data reported by various workers. From the various studies reported on critical Reynolds number, the correlation of Srinivasan et al.138 approaches the limit of 2100

Figure 2. Application of curved tubes according to various transport phenomenon.

for straight tube. Later Mishra and Gupta139 incorporated the effect of pitch in the correlation of critical Reynolds number proposed by Ito.135 2.1.2. Development of Flow Fields. Hawthrone92 was the first to give a numerical solution for flow development in curved tubes from straight tube Poiseuille flow solution. Later, Austin and Seader144 experimentally investigated the velocity profiles in entry flow region using laser Doppler velocimetery. They proposed a correlation assuming fully developed flow at the entrance for developing length θ (deg) in curved tubes after which secondary flow is fully developed. They established that, with the increase in Dean number, the developing length also increases. Since then a number of analytical solutions have been proposed by various workers for entry development in the curved tubes.145-150 Agrawal et al.149 measured the developing flow velocities for both axial and cross-flow components using a laser Doppler velocimeter and found an embedded vortex in addition to secondary flow separation near the inner bend of curved tube. Electrochemical limited-current measurements for the local wall shear were performed by Choi et al.150 They found a valley in the circumferential wall shear profile and a region of non-monotonic variation of wall shear with downstream distance, hence suggested that the vortex structure in the entry region is much more complicated than that in the fully developed flow. Later, the data on numerical solution of flow development length in curved tube at various flow rates and geometrical parameters were further appended on by Moulin et al.,18 Guan and Martonen,41 Liu,148 and Zheng et al.153 Table 5 summarizes the analytical, numerical, and experimental studies for flow length development in curved tubes. In most of the studied, it was observe that the flow profiles are almost symmetrical to center point on both horizontal and vertical centerlines near the inlet and at the inlet. With the increase of flow length, the velocity becomes asymmetrical. 2.1.3. Influence of Torsion on Secondary Flow. Since the work of Dean,88.89 a lot of research has been carried out, but little attention was paid toward the flow in a curved tube having finite pitch. The results in this direction obtained by various authors are in apparent conflict. While a curved tube or helical coil of finite pitch is the favored configuration for experimental work, theoretical work has been limited almost entirely to flow in curved tubes with zero pitch, i.e., torus. Mishra and Gupta139 conducted an experimental investigation from which they concluded that the effect of pitch on the friction factor can be eliminated if the coil diameter D is replaced by the diameter of

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Figure 3. Velocity streamlines and Dean vortices in curved tube.

curvature Dc given by

b )] [ (2πD 2

Dc ) D 1 +

(1)

use of Dc instead of D in the Dean number definition results in a new number, referred to as the helical coil number, NHe, defined as,

NHe ) NRe

() d Dc

1/2

) NDe

[ ( )] b 2 1+ 2πD

-1/2

(2)

where d is the tube diameter. According to eq 2, as b increases for the same D, the centrifugal forces decrease for the same NRe, thus weakening the secondary flow field, and ultimately straight tube behavior is approached. Lin and Tarbell29 and Yamamoto et al.154 conducted experiments in curved tubes of finite pitch to evaluate the effect of torsion on the secondary flow. This problem has been discussed theoretically95,98,155-160 and numerically99,132,154,161-163 for different orthogonal and nonorthogonal coordinate systems (Figure 5) and solutions for steady flow in a helical pipe. It was found that the twin vortex in the toroidal tube changes its shape to almost a single vortex as torsion increases at a constant Dean number. The friction factor in the helical tube first decreases from that of a toroidal pipe and then increases with increasing torsion. Torsion has a destabilizing effect on the fluid flow. The critical Reynolds number shows an oscillatory flow structure, which decreases with increasing torsion, has a minimum, and then increases with further increasing torsion.154 It is quite interesting to note that the curvature of a curved tube has a stabilizing effect on the flow whereas the torsion has a reversed effect. The detailed discussion on the effect of torsion on secondary flow in reported in Table 6. It was observed that there is a conflict between the predictions of secondary flow for the finite value of pitch. This may be due to the use of different coordinate systems used by various authors. 2.2. Friction Factor in Curved Tubes. When fluid flows through a curved tube, it experiences a positive pressure gradient arising in the radial direction due to the centrifugal force, which can be written as

( Dd )

∆Pradial ) 2FV2

(3)

The pressure drop in the axial direction can be obtained from the equation for fluid flow through a straight tube after replacing the Fanning friction factor (f) by fc, as

∆Paxial )

4fc

(Ld)FV

2

2

(4)

where fc is the friction factor for a curved tube, F is the fluid density, and V is the average velocity. Equation 4 applies to an empty pipe with f ) 16/NRe for laminar flow and for turbulent flow with f ) 0.079NRe-0.25. 2.2.1. Flow Resistance at Low Reynolds Numbers. The friction factor correlations for a laminar flow regime as proposed by various authors in the literature are complied in Table 7. Due to the lack of global solutions for the whole geometrical range, a number of design correlations were developed for the friction factor (Dean,89 White,90 Ito,127 Trusdell and Adler161) with the applicability of each correlation restricted to certain geometrical or flow conditions. Disagreement often occurred between investigators on the relevant dimensionless groups and form of dependency in the design correlations. Some of the proposed correlations mentioned in the Table 7 have at least one of the following disadvantages: (a) the formulas are only applicable in a limited range of Dean numbers, (b) correlations do not satisfy the boundary conditions fc f f if NDe f 0. Most of the available correlations for laminar flow in curved tubes are for the ratio of the Fanning friction factor for the curved tube to the straight tube under similar process parameters, fc/fs. If fc/fs is visualized as a comparison of the pressure drop in a coiled tube to that in a straight tube, the former value should be used only up to the value of a Reynolds number equal to the critical Reynolds number of flow in straight tubes, i.e., 2100. Beyond this value, the flow in the straight tube ceases to be laminar, whereas in coiled tubes, it persists to be laminar up to much higher Reynolds number. However, the use of the former value of fs up to NRe ) 2100 and then the latter value up to the critical Reynolds number for the coil flow will result in an additional break in the log C vs log NRe curve at NRe ) 2100. This makes the nature of the C vs NRe curves more complex than the fc vs NRe curve, which is continuous. Thus, it may be concluded that a correlation in fc/fs can be developed, at the most, up to NRe ) 2100, whereas a correlation in fc can be developed for the entire range of coil flow. Further, because

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3295 Table 2. Fluid Flow in Curved Tubes under Laminar Flow Condition author

technique

geometry

Grindley and Gibson9 Eustice87

experimental

coiled pipe

experimental

Dean88,89

theoretical

coiled tubes, U-tubes, and elbows torus

White90 Taylor91

experimental experimental

curved tube curved tube

Hawthrone92

theoretical

curved tube

Detra93

theoretical

curved tube

Mori and Nakayama94 Topakoglu95

theoretical and experimental theoretical

curved tube

McConalogue and Srivastava96

numerical (reduced curved tube PDEs to ODEs by Fourier analysis)

parameter range low to high flow rate regime

NDe < 17

low Reynolds number low Reynolds number NDe < 100

curved tube 96< NDe < 605.72

Trusdell and Adler97 numerical method (FDM) Larrain and Bonilla98 theoretical

helical coil

1 < NDe < 280

Austin and Seader99

numerical (overrelaxation technique)

torus (toroidal coordinate system)

5 < λ < 500; 1 < NDe < 1000

Tarbell and Samuels100

numerical technique (ADI scheme)

curved tubes

3 < λ < 30; 20 < NDe < 580

Greenspan101

numerical technique (FDM) analytical

torus

Smith102-104

Collins and Denis105 numerical Zapryanov and numerical Christov106 107 analytical Masliyah Soh and Berger108 Dennis and Ng109 Nandakumar and Masliyah110

numerical (artificial compressibility technique) numerical (reduced PDEs to ODEs by Fourier analysis) numerical solution

remarks Coil curvature affects the flow patterns in a coiled pipe. In turbulent flow regime the frictional resistance to be proportional to the velocity raised to the power 1.25 Observed the secondary flow patterns in coiled tubes, U-tubes and elbows. Solution was applicable for loose coiling approximation for the reasons of analytically tractability Developed a solution for fully developed laminar flow in curved tube of circular cross-section. The relation between pressure gradient and the rate of flow is not depending on the curvature for the first approximation. Solution was valid for very low value of Dean number The flow in a curved tube is much more stable than that in a straight pipe Taylor’s observation for flow in the curved tube was much more correct as compared to the Eustice87 as he was able to predict the flow stabilization even when it was turbulent on entering the straight part of the tube Reported secondary circulation in the curved tube and the solution was applicable only for flow rates Reported the secondary flow in curved tubes. Solution was applicable for low flow rates Theoretically reported the fully developed velocity profile using boundary layer idealization for large curvature and higher values of Dean number Reported an approximation solution for the laminar flow of an incompressible viscous fluid in curved tube for large value of curvature ratio. Solution was applicable for low flow rates For low values of NDe, a symmetrical secondary flow pattern with respect to the vertical plane was revealed, while for the large values of NDe the axial momentum peak was found to be convected from the center of the cross-section toward the outer wall of the curved tube Obtained the axially and secondary velocities in the helically coils of finite pitch (H/2πRc < 0.2) Reported the solution for all Dean number values, after a certain amount of series manipulation and restructuring For low Dean numbers, the axial-velocity profile was parabolic and unaltered from the fully developed straight tube flow. As the Dean number is increased, the maximum velocity began to be skewed toward the outer periphery. Coil pitch was not taken into consideration. Results are not agreeing with the work of Mori and Nakayama94 A separate influence of NRe and λ to be more realistic than NDe alone to characterize the secondary flow in the curved tube

Reported the secondary flow in curved tubes. The solution is only applicable for constant pitch Modified the solution of Hawthrone92 and Detra93 and reported the flow structure near the wall for circular and noncircular geometries. The solutions were applicable only for very low Reynolds number torus 96 < NDe < 5000 Reported the variation of the axial velocity with Dean number curved tubes 7000 < NDe The oscillatory flow in a curved circular tube has been reported < 20 000 torus with Reported four-vortex mode for a cross-section of semicircular shape with the flat semicircular surface forming the outer bend cross-section 1/20 < d/D < 1/7; Secondary flow separation near the inner wall in the developing region of the 108.2 < NDe curved tube was observed. The separation and the magnitude of the secondary < 680.3 flow is highly influenced by the curvature ratio curved tube 96 < NDe e 5000 Found non-uniqueness in their solutions, with a four vortex as well as two vortex type of secondary flow appearing for NDe g 956 torus

(NRe ≈ 1)

curved semicircular pipes curved tube

Yanase et al.111

analytical solution

Lai et al.112

curved tube

Dennis and Riley113

numerical solution (Reynolds stress model) analytical solution

Verma and Ram114

analytical solution

helical coil

curved tube low Reynolds number

Chung et al.,115-117 analytical solution Belfort et al.118,119 and experimental

curved channel

Park et al.120

experimental

curved tubes

Moulin et al.121,122

experimental and numerical

NRe ) 250 and λ ) 6 25 < NRe < 3000

Dey123

theoretical and numerical

torus

57 < NDe < 5 × 105

Eduard and Igor124

theoretical

curved tube

any correlation is only an approximation of the actual relationship valid only for a limited range of parameters, it needs to be simple. Some of the correlations listed above are quite complicated, which might have arisen from the failure of arriving at accurate characterizing groups.

Reported four-vortex solutions for curved semicircular pipes than for curved circular pipes using a bipolar toroidal coordinate system. Their solution was in agreement with the work of Dennis and Ng109 Four-vortex flow in a circular tube is unstable to asymmetric disturbances with respect to the centerline in the cross section Secondary flow is primarily induced by high anisotropy of the cross-stream turbulent normal stresses near the outer bend. The secondary flow appears as a counterrotating vortex pair embedded in a Dean-type secondary motion At high Dean number the flow develops into an inviscid core with a viscous boundary layer at the pipe wall but they could not find a complete solution for the problem The flow of a magnetic fluid through a helical coil for low Reynolds number and reported that torsion does not affect the flow rate for the order of parameter considered in their study Reported Dean vortices in a curved channel with optical and magnetic-resonance imaging methods. Defined a critical Dean number related to the flow rate above which the flow pattern changes and vortices are detected Reported the velocity profiles using a laser photochromic velocimetry method for single values of Reynolds number and curvature ratio Secondary flow do not have any noticeable effect at NDe < 20. It was observed that the two Dean vortices, their respective rotation and the center of rotation of the vortices moves towards the inner wall of the coil as Reynolds numbers increase The secondary boundary layer thickness along the outer pipe wall increases gradually and decreases with NRe. It grows very rapidly near the point of secondary boundary-layer separation Reported theoretical solution for finite curvature and tube pitch low flow rates

Hart et al.177 proposed that the friction factor of a curved tube could better be calculated with the following empirical correlation:

(

fc ) f 1 +

)

0.09NDe1.5 70 + NDe

(5)

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Table 3. Fluid Flow in Curved Tubes under Turbulent Flow Condition author

technique

Eustice10

experimental

Adler125

numerical solution experimental and numerical solution analytical solution analytical solution numerical solution experimental

Barua126 Ito127 Van

Dyke128

Anwer et al.129 Anwer and So130

Boersma and numerical Nieuwstadt131 solution (LES) Huttl and numerical 132 Friedrich solution (DNS) Eduard and numerical Pazanin131-134 solution

geometry coiled tube, U-tube, and elbows torus torus torus torus U-bend curved pipe (rotating drum at the inlet of curved tube) curved tube straight, curved, and helical tube

remarks Even slight curvature can modify the critical velocity that is a common indicator of the transition from laminar to turbulent flow Inferred that the two boundary layers, on the upper and lower halves collide at the innermost point of the cross-section, separate their and forms a reentrant jet that moves outwards through the core The analyses assumed a nonturbulent core where fluid moved towards the outer periphery and a thin boundary layer where fluid moved back to the inner periphery of the tube. There was good agreement for higher values of NDe than at low values The boundary layer solution proposed by him breaks down near the inner bend The asymptotic behavior for large NDe shows a 1/4th power dependence, which is clearly in disagreement with the results of a number of other workers The energy of secondary flow is only dissipated by viscous dissipation, explaining why it takes so long for the flow to return to the expected straight tube profile The superimposed solid-body rotation completely dominated the secondary flow, though strong swirl was used and this may not be the case for weaker swirls. The wall static pressure was lower on the outer wall than the inner wall, which is opposite to normal secondary flow caused by curvature The numerical results obtained using large eddy simulations were in agreement with the experimental results from the literature. This approach for determining secondary flow patterns is feasible Showed that turbulent fluctuations are reduced in curved pipes compared to the straight pipes. The effect of torsion on the axial velocity is much lower than the curvature effect

large coil diameter and Used an explicit formula of the Poisseuile type including a small distortion due to the particular geometry low Dean number of the pipe to predict flow patterns in the helical tubes

Table 4. Correlations for Critical Reynolds Number

This correlation satisfies the boundary conditions fc ) f if NDe f 0 and covers the whole domain, 0 < NRe < NRe,crit. The correlations given by Although White, Adler, Schmidt, and Ito does not consider the effect of pitch. Trusdell and Adler161 and Mishra and Gupta,139 however, investigated the

effect of pitch on friction factor. Trusdell and Adler161 employed helical number in place of the Dean number in order to incorporate the effect of pitch in their empirical correlation for the friction factor. The helical number is given below as

NHe )

Figure 4. Critical Reynolds number versus curvature ratio.

[

2d〈wφ〉 d/D V 1 + (b/2πD)2

]

1/2

(6)

where D is the coil diameter (which equals the radius of curvature only if the pitch, b ) 0) and d is the internal diameter of the tube. The actual radius of curvature of the coil helix is given by D ) 1 + (b/2πD)2; when pitch of the coil, b ) 0, the coil (torus) radius of curvature becomes D. Further, Liu and Masaliyah168 and Yamamoto et al.169 numerically calculated the secondary flow pattern and the friction factor for curved tubes having larger pitch. They reported that the friction factor at a large curvature first increases from that of a toroidal pipe and then decreases to that of a straight pipe with increasing torsion. 2.2.2. Flow Resistance at High Reynolds Numbers. Pressure drop in the coiled tubes for the higher values of Reynolds numberhavebeeninvestigatedbymanyresearchers.90,94,135,136,139,154,180-184 Table 8 summarizes some of the friction factor correlations developed by these investigators. The correlations developed

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3297 Table 5. Entry Flow in Curved Tubes geometry

remarks

Austin and Seader144

authors

experimental

curved tube

Singh145

analytical solution

curved tube

Yao and Berger146

analytical solution

curved tube

Smith103,104

analytical solution

curved tube

Humphery147 Liu148

numerical solution numerical solution

Agrawal et al.149

experimental (LDA technique) experimental (electrochemical limitedcurrent measurements)

curved tube 90° bend (elbow) curved tube

Proposed a correlation for developing length θ (deg) in curved tubes after which secondary flow in fully developed. The correlation proposed by them is good within 10° of measured data. They reported that the increase in Dean number increases the developing length The two different inlet conditions (constant dynamic pressure uniform velocity) affected the development of the flow in the near-entry region; however, it does not significantly affect the flow farther downstream Patched a solution for the flow inside and outside the viscous region at the outer edge of the boundary layer in the radial direction. Solution was in good agreement with the Barua’s fully developed solution and differed from Austin and Seader’s results144 Investigated the transition of a parabolic flow in a straight pipe to a curved one near their junction for circular, triangular, and rectangular cross-sections Developing laminar flow in curved pipes of arbitrary curvature radius is reported Solved the elliptic Navier-Stokes equations in a 90° bend elbow, where fully developed conditions were assumed to hold Measured the developing flow velocities of both axial and cross-flow components and found an embedded vortex in addition to secondary flow separation near the inner bend Found a valley in the circumferential wall shear profile and a region of non-monotonic variation of wall shear with downstream distance. Suggested that the vortex structure in the entry region is much complicated in the fully developed flow. Results were in good agreement with the Singh (1974) predictions in the developing region Found vanishing axial shear at the inner bend at a downstream distance ld ) 0.943δ1/2, where l is the arclength along the centerline of the bend. The singular behavior near the point of vanishing axial shear has been studied Reported that the developing lengths are smaller for curved tubes than straight tubes. The study was limited for torus only The velocity is almost symmetrical to center point on both horizontal and vertical centerlines near the inlet and at the inlet. With the increase of flow length, the velocity becomes asymmetrical Reported the developing length over several cross-sections and there results were in good agreement with the predictions of Austin and Seader99

Choi et al.150

Stewartson et al.151,152 Guan and Martonen41 Zheng et al.153 Moulin et al.18

technique

curved tube

analytical solution (boundary layer calculations) numerical solution numerical solution

coiled tubes

numerical solution

coiled tubes

by White90 and Mishra and Gupta139 appear to be most suitable from a practical point of view. The functional forms of these equations are in agreement with that of Ito’s one-seventh law correlation. Note that the effect of finite pitch can be accounted for in the correlation of Mishra and Gupta139 by using the actual radius of curvature of the helical coil. It also indicates that the increased frictional resistance in a curved tube depends only on the curvature of the coil. The results of Akagawa et al.,185 Mori and Nakiyama,94 Unal et al.,186 Watanbe et al.,187 and Guo et al.188 for curved pipe or helical coiled tubes shows that Ito’s correlation is of high accuracy. Grundmann189 and Hart et al.177 developed a friction factor diagram to predict the friction factor in coiled tubes that is similar to the one for straight pipes (see Figure 6). The friction factor diagram also offers a graphic view of the pertinent parameters and the flow condition (laminar or turbulent) in the helically coiled tube. The friction factor fc is depicted as a function of the Reynolds number in log-log plot, the parameter being the diameter ratio (d/D). When the limit D or D/d approaches infinity, the relation holds good for a straight pipe. Hence, the diagram is valid for hydraulic smooth pipe and coiled tubes. The influence of the diameter ratio (D/d) can be clearly recognized in the friction diagram as well as the shift

of the critical Reynolds number. The values for NRe,crit have been taken from Srinivasan’s critical Reynolds number correlation. 2.2.3. Influence of Orientation on Flow Resistance. The behavior of fluid flow in vertically or horizontally positioned coils is expected to be different. This difference may be pronounced when the centrifugal force is less than or equal to the gravity force, and therefore, the frictional pressure drop may differ as well. On the other hand, gravity influences the secondary flow, which is developed along the circumferential direction of the curved tube and is considered as one of the major variations that increases the frictional pressure drop as compared to the straight tube. Guo et al.181 studied frictional pressure drop in two helical coiled tubes in which one is placed in four helix axial inclinations. They observed that the coil positioned at 45° upward inclination exhibits a little higher frictional pressure drop; however, the difference among four inclined coils is less than 12%. 2.3. Mixing and Mass-Transfer in Curved Tubes. In continuous processes, many investigations are dedicated to the dispersion phenomenon or mass-transfer operations in order to

Figure 5. Coordinate systems for studying flow in coiled tubes: (a) orthogonal cylindrical-toroidal coordinate, (b) nonorthogonal rotating coordinate. (c) orthogonal helical coordinate, and (d) nonorthogonal helical coordinate.

3298


Table 6. Effect of Torsion on Fluid Flow in Curved Tubes authors

coordinate system

Truesdell and toroidal coordinates Adler161 Mishra and Gupta139 Manlapaz and Churchill164,165 Wang156

Germano157 Kao159

Germano158 Xie166

Chen and Fan167 Tuttle160

Huttle132,162 Liu,170 Liu and Masliyah168 Yamamoto et al.169 Liu and Masliyah171

Chen and Jan172 Liu and Masliyah173

method of solution

remarks

numerical (FDM)

Dean number is replaced by helical number for small pitch. Results were valid for NDe e 200

experimental

The effect of pitch is significant only for coils for which the elevation per revolution greater than the radius of the coil The effects of nonorthogonality to be negligible in the limit of small pitch. Compared the results of flow resistance and heat transfer reported in the literature The first-order effect of torsion is found on the secondary flow comparable with the effect of the curvature. The analysis was limited to Reynolds number less than unity

nonorthogonal analytical coordinate system nonorthogonal finite difference coordinate system method ith Gauss-Siedel iteration orthogonal perturbation method The effect of torsion on a helical pipe flow is of second order in , while the effect of curvature is coordinates first order of when ) ba, which was contrary to the results of Wang156 where torsion had a first order effect orthogonal perturbation method Torsion produces a significant influence at the secondary flow pattern if the ratio of curvature coordinates to torsion is of the order of unity. Some deviations were found between the series and the numerical solutions for secondary flow and the axial velocities, due to anomaly of the series solution orthogonal perturbation and The flow in a helical tube depends not only on Dean number but also Reynolds number and the coordinate numerical curvature ratio. The torsion has no first-order effect on the flow system methods helical coordinate series expansion Reported that curvature has no effect on the flow rate within the order of the 2 (pipe radius system and numerical multiplied by curvature), and that torsion exerted a second-order effect on the secondary method vortices. The torsion effect could be great enough to rotate the line separating the two vortices from vertical to horizontal, for very small Reynolds numbers nonorthogonal perturbation The torsion has a second-order effect on the secondary vortices and the torsion effect on the flow coordinate method system orthogonal and Coordinate system determines whether the torsion effect is first or second order. However, nonorthogonal Tuttle160 agreed with Wang’s conclusion that the effects are first order as the method of Wang156 coordinate was the best method to determine the torsion effect systems orthogonal second-order The torsion effects are weaker than the curvature effect and it influence the secondary flow induced coordinate finite volume by pure curvature and leads to an increase in fluctuating kinetics energy and dissipation rate system technique When the torsion is dominant, the flow in helical pipes tends to be the similar as that in a straight pipe. When torsion is small, the developing flow is oscillatory and develops more quickly than in a straight pipe experimental torsion from 0.45 to As the torsion parameter increases the friction factor values deviates from that of a toroidal tube 1.72 curvature and decreases toward a straight tube from 0.01 to 0.1 orthogonal numerical method When the torsion is dominant, the flow in helical pipes tends to be the same as that in a straight coordinate based on pipe. When torsion is small, the developing flow is oscillatory and develops more quickly system separation than in a straight pipe method nonorthogonal double series The flow in a helical pipe is governed by three parameter: Reynolds number, Dean number coordinate expansion and torsion number (NTn ) τosNRe) system method orthogonal numerical method A new model was developed for the secondary flow in coiled tube coordinate based on system separation method

(a) homogenize the residence time distribution (b) to increase mixing in order to enhance the conversion rate in chemical reactors. Typical operations where mass-transfer is the dominant step are falling film evaporation and reaction, total and partial condensation, distillation and absorption in packed columns, liquid-liquid extraction, multiphase reactors, membrane separation, etc. 2.3.1. Axial Dispersion and Residence Time Distribution in Curved Tube. Dispersion theory is concerned with the dispersal of a solute in a flowing liquid owing to the combined action of a nonuniform velocity profile and molecular diffusion. Additional complexities arise when flow instabilities are introduced using Dean vortices, where lateral mixing owing to molecular diffusion is augmented by the convective secondary motion. Many studies aimed at providing an efficient method of mixing by reducing axial dispersion in coiled tubes22,52,190-194 have been successfully investigated in the past. The state-ofthe-art review on the extensive work carried out on single-phase mixing performance of Newtonian and non-Newtonian fluids in straight tube and coiled tube was complied by Saxena and Nigam.194 Coiled tube configuration has shown promising results in reducing the axial dispersion as compared to the straight tubes. In spite of the advantages of curved tubes, they have their own limitations such as that the fluids with very long molecular chains can be damaged by high shear stresses.

Curved tubes are used in chromatographic columns to enhance the radial mixing. Hofmann and Halasz195,196 investigated the radial mixing without considering the axial dispersion in band broadening in squeezed, twisted, and coiled tubes. Band broadening occurs due to the multiple path of an analyte through the column packing. Tijssen197,198 observed the heavy equivalent theoretical plate in a chromatographic column using Dean’s profile, based on dye tracer experiments in a transparent tube. Katz and Scott199 used serpentine tubes as low dispersion connectors. Leclerc et al.52 reported an almost curvatureindependent change in dispersion in coiled tubes of different curvature ratios. They found that dispersion increases as (NRe/ NRe,c)2/3NSc at low fluid velocities and decreases as (NRe,c1/6/ (NRe/NRe,c)4/5NSc0.08) at higher velocities. Grochowicz et al.200 modeled the secondary flow in serpentine and coiled tubes as the measure of radial mixing. They showed that flow characteristics in serpentine tubes result in considerably less band broadening per unit length than in linear tubes of the same inner diameter. They compared the serpentine and helical tubes from the point of view of band broadening. The use of comparatively large column diameters is permissible for the enhancement of native diffusion mass-transfer with a secondary mechanism of radial mixing. According to Kauffman and Kissinger,201 there is no distinct advantage in terms of band spreading between any of the serpentine geometries and the

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3299 Table 7. Friction Factor Correlations under Laminar Flow Conditions in Curved Tubes

3300


Table 8. Friction Factor Correlations under Turbulent Flow Conditions in Curved Tubes

smallest diameter helical coil. However, these geometries reduced the band spreading by up to 44% when compared with a linear tube of the same length. A new compact model of toroidal coil centrifuge was developed by Matsuda et al.53 that enabled an efficient analytical separation achieving over 10 000 theoretical plates. They reported that the partition efficiency can further be improved by the use of a longer or a narrower toroidal coil. Waiz et al.202 experimentally studied and compared the band dispersions observed in straight, coiled, and various types of serpentine and superserpentine geometries of open tubular reactors (OTRs) as a function of flow rate. They reported that the dimensional changes (e.g., grid spacing) within a given geometry can have a very major effect on the observed dispersion. More detailed reviews on axial dispersion in curved tubes can be found in Saxena and Nigam163 and Kumar and Nigam.203 2.3.2. Membrane Separation Process. Curved tubes are extensively in demand for mass-transfer operations such as for separation processes (like gas absorption/stripping, extraction, membrane distillation, etc.). They offer surprisingly high

interfacial area, no flooding at high flow rate, and nondispersive phase contact avoiding entrainment as compared to conventional separation devices. Curved hollow fiber membrane module is regarded as an effective alternative. The investigation on Dean vortices and their application to membrane separation processes has been the subject of several experimental and theoretical studies concerning the improvement of microfiltration (MF), ultrafiltration (UF), and nanofiltration or pervaporation. A coiled tube membrane reactor developed by Hagedorn and Kargi204 is used for the cultivation of mouse-mouse hybridoma cells producing monoclonal antibodies. The cell and antibody concentrations obtained in the membrane reactor were found to be higher than that obtained in a batch spinner flask without a membrane. Belfort and co-workers115-119 experimentally and theoretically studied the Dean flow instabilities to improve the performance of membrane processes for micro-, ultra-, and nanofiltration in a helically coiled and curved slit membrane system because of Dean vortex flow. Coiled modules for ultrafiltration were studied by Manno et al.205 They found that, compared with the conventional straight module, the secondary

Figure 6. Friction factor chart for straight and curved smooth tubes (0 e d/D < 0.2 and 0 e NRe < 2 × 105). The dotted line represents values of fc, where NRe ) NRecrit. (Source: Hart et al.177)

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3301

Figure 7. (a)Variation of the Sherwood number with the Dean number for oxygen transfer in coiled and straight modules. (Source: Moulin et al.121) (b) Variation of the Sherwood number with energy consumption (water, modules 1S-1D, Dia ) 3.2 mm, and 2S-2D Dia ) 2.4 mm).

flow in a coiled module could enhance its membrane permeation by a factor of 5. Curved woven hollow fiber UF membranes are reported to increase mass flux in a manner similar to the results observed for helical fibers but at lower energy consumption.206 Ghogomu et al.207 employed several curved membrane modules of different configurations (straight, helically coiled, twisted and sinusoidal, meander-shaped) in ultrafiltration. Their experiments showed a remarkable enhancement on mass-transfer in curved modules in comparison with conventional straight ones. Mallubhotla et al.208 also used coiled modules in nanofiltration and found that Dean vortices could effectively reduce concentration polarization and increase membrane permeation. Moulin et al.121 have shown an increased mass-transfer coefficient in the range of 2-4 in their oxygenation experiments for a helical hollow fiber module compared with a linear hollow fiber module and demonstrated the effectiveness of using Dean vortices for several applications, such as pervaporation and ultrafiltration of suspensions of yeast and bentonite.122 Figure 7a shows the variation of Sherwood number with Dean number for oxygen transfer in coiled and straight modules. It can been seen from the figure that, for the same Dean number, the Sherwood numbers of coiled modules are between 2 and 4 times greater than straight modules. Variation of Sherwood number with energy consumption for coiled and straight modules can be seen in Figure 7b. Similar improvement factors were obtained

by Liu et al.209 in their membrane extraction experiments for a liquid-liquid system. Some authors investigated the effect of other important geometric parameters of coiled module, such as wind angle, coil diameter, and fiber internal diameter.210,211 Other interesting experimental and theoretical findings from different works available in the literature are presented in Kumar and Nigam.212 2.3.3. Mass-Transfer in Absorption Columns. Mohhammed and Muhammed213 experimentally determined the mass-transfer coefficients for thin films of liquid falling in case of straight and coiled tubes for the absorption of CO2 into liquid films of distilled water, ethyl alcohol (96.25%), or ethylene glycol of 12 or 5.2%. They reported a higher mass-transfer coefficient in the helical liquid film and proposed empirical correlations for the mass-transfer coefficient. A detailed study on the ammoniawater vapor rectification process in absorption systems using a helical coil rectifier was carried out by Seara et al.214 They showed that the vapor mass-transfer coefficient has the most significant effect on the rectifier length (number of turns); while the other heat- and mass-transfer coefficients have no substantial effect. 2.4. Heat Transfer in Curved Tubes. Curved tubes are often used for heat transfer in mixing, storage, and reactor vessel as well as in heat exchangers owing to the advantages of high heattransfer area and higher heat-transfer coefficient. The heat transfer in curved tubes takes place not only by diffusion but also by convection. This convective heat transfer is more or less dominating depending upon the flow conditions and fluid properties. Jeschke215 was the first who experimentally reported the heattransfer data for two coils (D/d ) 6.1 and 18.2) for a turbulent flow (NRe < 150 000) condition and developed an empirical correlation. However, his experimental technique, described in a rather sketchy manner, is suspect in several aspects. Later, Merkel216 revised his empirical correlation and suggested that the average heat transfer in coiled tubes is (1 + 3.5λ) times the heat transfer in straight tubes (Diettus-Bolter correlation). Hawe217 predicted the fully developed temperature profile in curved tubes. Their experimental data showed that temperature profile is markedly different from those obtained in straight tubes and that the local heat-transfer coefficient at the outer wall was greater than that at the inner wall. Berge and Bonilla218 investigated the heat-transfer coefficient for heated air, water, and oil in coils with condensing steam. However, the heattransfer rates reported by them for air were lower than in straight pipes. Since then, extensive work on heat-transfer enhancement in a curved tube has been carried out by various workers. Tables 9 and 10 show the various studies and the empirical correlations reported in the literature for the heat-transfer enhancement in the curved tubes for the different thermal boundary conditions. In spite of the extensive research on heat transfer in a curved tube (Tables 9-11), several points of controversy still remain on the structure of the velocity and temperature fields and development region as well as the form of dependence of these fields on the geometrical (curvature ratio and pitch) parameters. Due to the lack of global solutions for the whole geometrical range, a number of design correlations were developed for the Nusselt number (Tables 9-11), with the applicability of each correlation restricted to certain geometrical or flow conditions. Disagreement often occurred between investigators on the relevant dimensionless groups and form of dependency in the design correlations. For example, the Prandtl dependency in Nusselt number correlations was found to be NPr1/3 by Seban

3302


Table 9. Experimental Studies on Heat Transfer in Coiled Tubes

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3303 Table 10. Numerical and Theoretical Studies on Heat Transfer in Coiled Tubes

3304


Table 10 (Continued)

Table 11. Thermal Development in Curved Tubes authors

technique

geometry

parameter range

numerical

torus

Tarbell and Samuel100 numerical

torus

NDe > 100, λ > 18.66, BC: Φw BC: Tw

torus

BC: Tw

Dravid et al.225

Patankar et Singh and

al.254

Bell226

Patankar et

al.255

numerical

numerical and curved pipe experimental numerical curved pipe

Akiyama and Cheng 256,257

numerical

Janssen and Hoogendoorn229 Manafzadeh et al.258

Lin and Ebadian260

numerical and curved pipe experimental experimental straight and helical tube experimental helical heat exchanger numerical helical pipes (orthogonal helical coordinate system) numerical helical pipes

Lin et al.261

numerical

Lin and Ebadian262

numerical

Li et al.263

numerical

Rindt et al.264

numerical

Kalb and Seader259 Liu and Masliyah168

curved pipe

BC: Φw turbulent flow, BC: Φw NDe > 100, BC: Tw

remarks Observed a cyclic variation in wall temperature and Nusselt number in the developing region and this oscillation decreases with increasing axial position and an asymptotic value is reached for large Graetz number The similar oscillations were observed near the entrance in Nusselt number as observed by Dravid et al.225 The similar oscillations were observed near the entrance in Nusselt number as observed by Dravid et al.225 Wavy behavior of the Nusselt number has been observed in the development of the temperature field in a fully developed forced curved pipe flow Standard k- two-equation turbulence model was used to predicted the turbulent developing fluid in curved pipes for fully developed hydrodynamics results at zero pitch Thermal boundary layer development is not uniform around the circumference. It develops quicker at the inner wall than at the outer wall. For NPr ) 0.1, Nusselt number development along the axial length was similar to that of a straight pipe. While for NPr ) 0.7, 10, and 500 the Nusselt number decrease to a minimum and then increase again and level off to a constant value at a certain distance downstream. The fluctuations were due to numerical instability rather than a physical phenomenon Reported the cyclic variation in Nusselt number in the entrance region

BC: ΦW

Reported the mechanism of Nusselt number oscillations at the junction of a straight and helical tube

BC: Tw

There was a rapid transition from turbulent to laminar flow and that the Nusselt number also obtained the fully developed value well before the two turns of the coil Correlated the thermal entrance length with the fluid Prandtl number. Similar to the thermal developing length, the asymptotic Nusselt number is also affected by curvature ratio and torsion

laminar flow

turbulent flow; Circumferential average Nusselt number development is found to be oscillatory before (k- standard it is fully developed and the oscillation phenomenon enhanced with increase in pitch, two-equation curvature, and NRe turbulent model) helical pipes laminar flow The Nusselt number and the friction factor were oscillatory in the entrance region and when the curvature ratio decreases, the oscillations of both the Nusselt number and the friction factor increases helical pipes turbulent flow; The increased inlet turbulence intensity tended to reduce the velocity gradient at the walls (k- standard of the pipe, although it had negligible effect on the maximum axial velocity. Increasing two-equation the intensity level also increased the intensity of the secondary flow but did not change turbulent model) its pattern. The thermal boundary layer developed quicker with increased inlet turbulence intensity helical pipes laminar flow There was an oscillatory behavior noted for the friction factor and Nusselt number in developing flows helical pipes laminar flow; Nusselt number oscillated along the axial position and attributed the phenomena to BC: Φw circulating secondary flow along the tube wall

and McLaughlin220 and Singh and Bell226 and NPr1/6 by Dravid et al.225 and Janssen and Hoogendoorn,229 while Kalb and Seader240 proposed NPr0.0108 for low Prandtl numbers and Pr0.2 for high Prandtl numbers. Also, the Dean number dependency appeared as NDe0.5 (Dravid et al.225), NDe0.115 at low NPr and NDe0.476 at high NPr,240 or NDe0.75.265 While the Dean number appeared in most heat-transfer correlations, the use of NRe and D/d (Janssen and Hoogendren229) or fc and NRe (Seban and McLaughlin,220 Janssen and Hoogendren229) was suggested instead as correlating parameters under certain flow conditions. These differences exist in spite of the fact that coil pitch “b”

was not a factor in any of the above studies (b/d < 5) and laminar forced convection was the only mode of heat transfer except for Abul-Hamayel and Bell265 and Manlapaz and Churchill,165 who found significant effects of free convection at low Reynolds numbers. Manlapaz and Churchill165 reported the theoretical effect of pitch on heat transfer and concluded from their results that this effect is negligible for b/D < 0.5. They proposed the use of Dc as a correlating parameter for b/D > 0.5 (similar to the approach for friction factors139); however, this proposal was merely a hypothesis without computational support. Manlapaz and


Figure 8. Heat transfer in curved tubes for various thermal boundary conditions under thermal flow conditions.

Figure 9. Grashof number versus Dean number.

Churchill165 generated fairly useful correlations utilizing their own data, and those of previous workers (see Table 12, presented later in this work) for different thermal boundary conditions (TW, ΦW). Their predictions give fairly good agreement between the correlation and most of the available data. Manlapaz and Churchill’s165 work was limited for lower values of Reynolds number. For the higher values of Reynolds number, a comparison between heat-transfer values (NNuNPr-0.4) as a function Reynolds number is reported in Figure 8. It can be seen from the figure that there is lot of variation in the heattransfer values reported by various workers. This may be due to the discrepancies in the measurements during the experiments. There is much less attention paid to radiation effects on heat transfer. Only a few workers dealt with this and only through computational study; no experimental validation has been reported so far. 2.4.1. Thermal Development in Curved Tube. Dravid et al.225 numerically studied the entrance region laminar flow heat transfer for Dean numbers greater than 100 with coil-to-tube radius ratio of 18.66 subjected to the boundary conditions of axially uniform flux with peripherally uniform wall temperature. They observed a cyclic variation in wall temperature and Nusselt number in the developing region. The oscillation decreases with increasing axial position and an asymptotic value is reached for a large Graetz number. The developing region is extremely short, and for design purposes, the asymptotic Nusselt number should be used along the whole length of the coil. The development of a temperature profile in the entrance region has been numerically studied by Tarbell and Samuel100 for the axially uniform wall temperature conditions and by Patankar et al.254 for longitudinal uniform heating and peripheral uniform wall temperature. They also confirm the oscillations in Nusselt number observed by Dravid et al.225 A similar type of wavy behavior of the Nusselt number has been observed in the study of Singh and Bell.226 They numerically and experimentally studied the development of the temperature field in a fully developed forced curved pipe flow, subject to a constant wall heat flux. Patankar et al.,255 using the standard k- two-equation turbulence model, predicted the turbulent developing fluid in curved pipes for fully developed hydrodynamics results at zero pitch. Akiyama and Cheng257 also obtained fluctuating Nusselt number relations with the Graetz number as was observed by Dravid et al.225 However, Akiyama and Cheng256,257 proposed that these fluctuations were due to numerical instability rather than a physical phenomenon. Liu

and Masliyah168 numerically studied the development of laminar flow and heat transfer in helical pipes using an orthogonal helical coordinate system. They correlated the thermal entrance length with the fluid Prandtl number, which is in agreement with the correlation reported by Janssen and Hoogendoorn.229 2.4.2. Effect of Buoyancy Force. Buoyancy effect is predominant in the secondary flow at low Reynolds number, depending upon the physical properties and the difference between the wall and the bulk temperatures. The varying gravitational force due to difference in density between cold and hot fluids causes the motion of fluid in the vertical direction. Morton266 showed that if a temperature distribution is present in a heated straight pipe, the varying gravitational force due to difference in the density causes a motion of fluid elements in the vertical direction. This effect forces secondary flow to form two vertical vortices with a vertical dividing streamline. According to Morton,266 buoyancy forces induce a secondary flow in the case of straight tube, which depends upon the nondimensional parameter NReNRa, the ratio of buoyancy force to viscous force, where NRa is Rayleigh number. The heattransfer rate is enhanced by the secondary flow and is larger than that expected without the buoyancy effect for the same mass flow rate. By using perturbation analysis, Yao and Berger146 showed that the buoyancy effect could indeed be as important as the effect produced by centrifugal force. Prusa and Yao267 numerically solved the combined effects of buoyancy and centrifugal forces in heated curved tubes for intermediate range of Dean number and NReNRa. They provided a flow regime map to indicate the three basic regime (Figure 9). In region I, the centrifugal force is dominant; here the problem can be solved by taking momentum and energy equations uncoupled. In region III, the buoyancy force is dominant; the heat transfer is treated here as that for a straight tube with natural convection. In region II, both forces are dominant, and the full momentum and energy equations have to be solved simultaneously. Lee et al.268 studied the effect of buoyancy on steady curved tubes with a relatively wide range of Prandtl, Grashof, and Reynolds numbers and showed that buoyancy has a significant effect on the heat-transfer performance. They found that buoyancy effects resulted in an increase in the average Nusselt number, as well as modifying of the local Nusselt number distribution. Futagami and Aoyama233 developed an expression to predict the average Nusselt number for situations where both the secondary flow and buoyancy forces are important on the heat-transfer coefficients.

3306


Mixed convective flow in a helically coiled heat exchanger with the constant wall temperature boundary conditions and axially varying wall temperature was investigated by Rindt et al.264,269 They showed that, for certain values of Grashof number, buoyancy effects increase the oscillations, reflecting the fact that when the heat transfer is higher, buoyancy effects becomes more dominant than centrifugal effects. The magnitude of the secondary flow was found to decrease and oscillations in the Nusselt number damp out. This can be attributed to the strong stabilizing effects of stratification in the pipe. Goering et al.270 used two thermal boundary conditions, both of constant heat fluxes, but one was a constant peripheral heat flux while the other had a constant peripheral wall temperature to study the combined effects of curvature and buoyancy on the heat-transfer rates. Li et al.263 numerically investigated turbulent heat transfer in a curved pipe using the control volume finite difference method. They reported that the buoyancy forces strongly affected the secondary flow, as these forces were much stronger than the centrifugal forces near the critical point. There was also an oscillatory behavior noted for the friction factor, similar to what has been seen with the Nusselt number in developing flows. 2.4.3. Influence of Viscosity on Heat Transfer. Another important issue of viscosity effect on heat transfer is addressed by Bergles,271 Andrade and Zaparoli,272 and Kumar et al.273 Bergles271 reported the effects of temperature-dependent viscosity in the curved tube, but they did not consider curvature in their study. Andrade and Zaparoli272 studied the fully developed laminar water flow in a curved duct under heating and cooling conditions with temperature-dependent viscosity while keeping other properties (density, specific heat, thermal conductivity) constant. Kumar et al.273 studied the effect of temperaturedependent properties on fully developed laminar flow in a curved tube under both heating and cooling conditions at various Reynolds numbers ranging from 100 to 400 for water and DEG. It was found that the velocity and temperature profiles were distorted when the effects of temperature-dependent properties were considered. Under cooling conditions, the Nusselt number values were found to be lower for temperature-dependent properties due to the increase of the thermophysical properties at the inner points of the curved tube section. The opposite trends were observed when the heating condition was considered. 2.4.4. Effect of Pulsatile Flow on Heat Transfer. In some instances, it may even be beneficial to induce pulsation in the flow system if enhanced performance is ensured. However, not many reports have been published on unsteady fluid flow for heat transfer in curved tubes. Lyne274 predicted unsteady flow resulting from a sinusoidal pressure gradient. They showed that the secondary flow could be in the opposite direction compared to steady pressure gradients. Zalosh and Nelson275 further reported the unsteady flows by studying pulsating flows in curved tubes for fully developed laminar flows with pressure gradients that oscillated sinusoidally in time. Three different solutions to the problem were proposed. The predictions of Zalosh and Nelson275 were in good agreement with the reversal of the secondary flow as reported by Lyne.274 Patankar et al.254 presented a perturbation analysis for fluid flow and heat transfer. Their result showed that increase in time-average Nusselt number is most evident at high Prandtl number, high excitation relative amplitude, and low excitation frequencies. They reported that the NNu ratio (curved to straight pulsatile) passes through a maximum value at low Reynolds number. Local and peripherally averaged temperature distribution and Nusselt numbers were determined by Rabadi et al.276 for fully developed pulsating laminar flow in a curved tube. They found

that the Nusselt number varied to a large degree both around the periphery of the tube and during cycles. These effects were higher for large Prandtl numbers and low values of the frequency parameter. Specific features of unsteady-state heat transfer and heat carrier mixing in complex-shaped channels formed by bundles of coiled tubes were studied by Ashmantas et al.277 Iyevlev et al.278 experimentally demonstrated the investigation of unsteadystate heat- and mass-transfer processes on bundles made of 127 coiled tubes for Reynolds numbers ranging from 3500 to 17 500. The investigations of unsteady-state heat- and mass-transfer with an increasing thermal load showed that the change in time of the working fluid temperature distributions and accompanying variations of the effective coefficient of diffusion, which characterizes the transfer properties of the flow, were due to the influence of unsteady-state boundary conditions. The observed rearrangement of temperature fields and pronounced enhancement of heat- and mass-transfer at the first instants of time can be attributed to the change in the turbulent flow structure during unsteady-state heating of the bundle. Chung and Hyun279 presented an inappropriate complete numerical solution, time-dependent Navier-Stokes equations for a fully developed pulsating flow in a pipe and the effect of pulsation on the global heat-transfer phenomena. Local Nusselt numbers were developed based on the Womersley number (ratio of transient inertial to viscous forces), which is a function of the pipe radius, the kinematic viscosity, and the frequency of the pulsation. It was found that the strength of the Womersley number affected the distribution of the Nusselt number around the periphery. Guo et al.280 experimentally investigated the effects of pulsation upon transient heat-transfer characteristics for fully developed turbulent flow in a uniformly heated helical coiled tube. They analyzed the secondary flow mechanism and the effect of interaction between the flow oscillation and secondary flow and proposed new correlations for the average and local heat-transfer coefficients both under steady and oscillatory flow conditions. Guo et al.281 showed the outcomes of the interaction between the flow oscillation and secondary flow. They reported that the oscillation heat-transfer coefficients were higher than that of steady flow with corresponding conditions. Oscillation enhanced the single-phase turbulent heat transfer, and the enhancement with increasing time-averaged Reynolds number. Pontrelli and Tatone40 numerically investigated the pulsatile flow in a curved pipe and demonstrated the role of curvature in the wave propagation and in the development of a secondary flow. Recently, Timite et al.282 carried out an experimental and numerical study on alternated Dean pulsated flow that showed an important modification of the secondary flow structure due to the pulsation. For certain control parameters, the secondary flow becomes more complex, in some cases with the appearance of Lyne instability or swirling structures due to the periodic movement. 2.4.5. Heat Transfer in Curved Annulus. There are very limited studies reported on fluid flow and heat-transfer coefficient in the outer duct of the curved devices. Prasad et al.,57 Patil et al.,283 Haraburda,284 and Figueiredo and Raimunda285 have discussed the design procedure for coil-in-shell heat exchangers. In these studies, helically coiled tubes were approximated as a bank of straight tubes for calculating outer heat-transfer coefficients. However, poor circulation is observed in shell regions near the coil in the coil-in-shell heat exchangers. This problem could be avoided by using a coil-in-coil tube configuration.


Kapur et al.286 performed theoretical calculations for steady laminar flow of a Newtonian fluid through a curved annulus. They predicted two vortices fitting the curved annulus similar to that as observed in curved tubes. Topakoglu95 used an approximate solution to determine the secondary flow streamlines for flow between two concentric torus-shaped pipes. They observed four secondary flow vortices, with symmetry about the plane that cut the torus tube through the center. In the two vortices on each side of the central plane, the direction of the secondary flows was opposite. Garimella et al.287 reported average heat-transfer coefficients for laminar and transition flow regimes in a coiled annulus. They found that the heat-transfer coefficients obtained from the coiled annular ducts were higher than those obtained from a straight annulus, especially in the laminar region. Petrakis and Karahalios288-290 investigated the fluid flow and heat transfer in a curved pipe with a solid core and found that the size of the core affects the flow in the annulus with flows approaching parabolic for large cores. The entrance flow into a curved annulus was studied numerically by Choi and Park291 for the case of an incompressible steady laminar flow for various curvature ratios and found that this parameter had a large effect on the secondary flow field. They also stated that, unlike the case of a straight annular duct, the fully developed flow does not necessarily develop earlier when the radius ratio is larger. Further, Xin et al.292 studied the effects of coil geometries and the flow rates of air and water on pressure drop in both annular vertical and horizontal helicoidal pipes for three different diameters of inner and outer tubes. The results showed that the transition from laminar to turbulent flow covers a wide range of Reynolds numbers. 2.4.6. Fluid-to-Fluid Heat Transfer in Curved Tubes. All the above-reported studies for helical coils are confined to two major boundary conditions, constant wall heat flux or constant wall temperature.2,3 However, for industrial applications, constant wall flux conditions do not appear to be physically realistic. This complicates the design of coil-in-coil heat exchangers, where either the heating or cooling is supplied by a secondary fluid, with the two fluids separated by the wall of the coil. Numerical investigations to study the heat-transfer characteristics in a two-turn coil-in-coil helical coil heat exchanger for various tube-to-tube ratios and Dean numbers for laminar flow in both annulus and in-tube were carried out by Prabhanjan et al.69 It was found that the flow in the inner tube at high tubeto-tube ratios was the limiting factor for the overall heat-transfer coefficient. This dependency reduced at the smaller tube-totube ratio, where the influence of the annulus flow was increased. Further, Rennie and Raghavan71 carried out experimental studies to determine the heat-transfer characteristics of a double-pipe helical heat exchanger for both parallel flow and counterflow configurations. heat-transfer coefficients from parallel flow and counterflow systems were similar, as changing the flow direction should have negligible effect on heat transfer. Kumar et al.293 reported experimental as well as numerical studies on the hydrodynamics and heat-transfer characteristics of a tube-in-tube helical heat exchanger at pilot plant scale for the counterflow configuration. They reported that the overall heat-transfer coefficient increases with increase in the innercoiled tube Dean number for a constant flow rate in the annulus region. Similar trends in the variation of overall heat-transfer coefficient were observed for different flow rates in the annulus region for a constant flow rate in the inner-coiled tube. Rennie and Raghavan73 numerically modeled the double-pipe helical heat exchanger for laminar fluid flow and heat-transfer char-

acteristics under different fluid flow rates and tube sizes. They found that the annulus Nusselt number is highly dependent on the Dean number. They observed that the thermal-dependent viscosities have very little effect on the Nusselt number correlations because Newtonian fluids have significant effects on the pressure drop. However, changing the flow rate in the annulus can significantly affect the pressure drop in the inner tube, since the average viscosity of the fluid in the inner tube would change due to the change in the average temperature. Recently, Xie et al.79 carried out experiments and also applied artificial neural network for heat-transfer analysis of shell-andtube heat exchangers with segmental baffles or continuous helical baffles for three different heat exchangers. Naphon78 studied the thermal performance and pressure drop of the coil-in-shell heat exchanger with and without helical crimped fins and reported the effects of the inlet conditions of both working fluids flowing through the coiled heat exchanger on the heat-transfer characteristics. Naphon and Suwagrai237 investigated the heat transfer and flow developments in the horizontal spirally coiled tubes under constant wall temperature boundary condition and compared the performance with the straight tube heat exchanger. They reported that spirally coiled heat exchanger shows higher heat-transfer coefficient as compared to the straight tube heat exchangers. 2.4.7. Non-Newtonian Fluid Flow and Heat Transfer. Knowledge of residence time distribution (RTD), fluid flow, and heat transfer of non-Newtonian fluid in curved tubes is important in the biomedical field, for the design of a flow reactor in a continuous fermentation or polymerization reaction. In particular, the thermal pasteurization of non-Newtonian liquid foods, where the death rate of microorganisms is proportional to the population density of the microorganism in a process that could be modeled as a convective diffusion with a first-order chemical reaction. Most of the studies for fluids in circular curved tubes are confined to Newtonian fluids. Relatively very few attempts have been made to understand the flow phenomenon of non-Newtonian fluids in curved tubes, despite its importance in the field of polymers and biochemical and biomedical areas. The analyses, which were based on the extension of Dean’s analysis, were the work of Jones,294 Yitung et al.,295 and Sharma and Prakash.296 Jones294 was the first who reported the theoretical solution for the flow of a non-Newtonian fluid, visco-inelastic, ReinerRivlin fluid in a coiled tube and developed a successive approximation solution. The flow phenomenon of power law fluids flowing through circular curved tube has been analyzed by Clegg and Power297 for a small range of Dean numbers. Thomas and Walter298,299 reported a successive approximation solution, but using a different equation of state to describe the material behavior. They considered the flow of an elastic viscofluid in a curved pipe of circular298 and elliptical299 crosssections. Thomas and Walter also observed that the presence of elasticity in a fluid increases the volumetric flow through the pipe for a given pressure gradient. The experiments of Barne and Walters300 indicate that the generation of a secondary flow may be sufficient to achieve a drag reduction even in laminar flow. Their experiments in turbulent flow indicate that the turbulent drag reduction in a strongly curved tube is not as significant as in a straight tube under similar conditions. Rajshekhran et al.301,302 experimentally presented the flow of aqueous CMC solutions in coiled tubes with a limited variation in curvature ratio. They proposed a correlation, which has a number of inconsistencies, the most serious one being the fact that the proposed correlation does not agree with

3308


experimental data these authors have presented. Kelkar and Mashelkar303 experimentally analyzed the drag reduction in curved tubes for a single coil. In the laminar regime, they concluded that there is no drag reduction over the range of parameter studied in their work. Oliver and Asghar304 presented experimental results on flow and heat transfer using a power law model for viscoelastic fluid polyacrylamide solution in different coils (λ ) D/d ) 12.7, 16.8, 26.7, 32, and 43) as

NNu ) 1.75(1 + 0.118xNDe)NGz0.33

( ) ()

3n + 1 0.33 γm 0.14 4n γw 60 < NDegen < 2000 (7)

( ) ()

3n + 1 4n 4.0 < NDegen < 60,

NNu ) 1.75(1 + 0.36NDe0.25)NGz0.33

0.33

γm γw NGz > 20 (8) 0.14

where γ is a function of the consistency index in the power law model (γ ) 8n-1gk)). When n ) 1, the above correlation has been verified to represent experimental heat-transfer data for Newtonian fluids as well. Mashelkar and Devrajan305 and Hsu and Patankar306 reported that decreasing n (power law index) tends to flatten the velocity profiles and denote the secondary flow, which was inconsistent with the observation for the flow of a power law fluid through a straight tube. Mujawar and Rao308 presented experimental data on isothermal frictional pressure drop for viscous non-Newtonian fluids. Their data are in satisfactory agreement with that of Mashelkar and Devrajan.309 But the experimental data of Rajshekhran et al.302 appear to be in considerable error; this was also indicated by Hsu and Patankar306 and Mashelkar and Devarjan.309 Singh and Mishra310 have reported the frictional factor for non-Newtonian fluids in curved tubes. Shenoy et al.311 numerically studied the non-Newtonian flow behavior in curved tubes. Their predictions were in good agreement with the results of Thomas and Walter.298 Ranade and Ulbrecht312 have numerically reported RTDs for non-Newtonian helical flow using velocity profiles reported by Rathana.313 In the case of non-Newtonian fluids, the dependence of apparent viscosity on shear rate changes the velocity distribution over the tube cross-section, which in turn affects the RTD. However, a similar study has also been reported by Saxena and Nigam.193 They fitted the numerically computed RTDs for various values of power law index (0.2 e n e 2.0) to Nauman’s model and developed a new expression for RTD. They found that conversion for first- and second-order chemical reactions at the outlet of a coiled tube reactor are predicted using numerically computed RTDs; the volume of reactor required to obtain a given conversion decreases as the value of the power law index decrease. Heat transfer in polymer melt flow (power law fluid) using a temperature-dependent power law model for constant wall temperature was numerically studied by Zavadsky et al.314 They presented the velocity, temperature, and viscosity profiles, bulk temperature, and Nusselt number in the curved tube of a circular cross-section. Curvature effect on the flow pattern of pseudoplastic fluid is similar to that of the Newtonian fluid in curved pipes over the range of parameters NDe e 1000, n ) 1-0.5, and λ ) 10-100. The secondary flow weakens as the power index decreases, but its dependence on Dean number is not so significant. Rao317 experimentally measured turbulent Fanning friction factors and Nusselt numbers for viscous power law nonNewtonian fluids with curvatures in the range of 1/10-1/26, the generalized Reynolds number (N*Re) from 15 000 to 50 000,

and power law exponents (n) in the range of 0.78-1. The results indicated that the turbulent Nusselt number increases with decreasing n. It was also observed that the lower the value of n, the greater the secondary flow. He reported that, other than space saving, helical coils do not offers any advantages over a straight tube for heat exchanger applications involving turbulent flow of Newtonian or power law fluids. He also proposed correlations for heat transfer for water:

NNuc/NNus ) 1 + 1.48

( Dd )

1.15

(9)

Das and Batra318 analyzed the steady-state laminar flow of power law fluids in a circular curved tube using boundary layer approximations. The influence of yield stress on the velocity profiles and frictional resistance in the higher Dean number region was reported. Robertson and Muller319 used a perturbation method to study the steady flow of Oldroyd-B fluids and for Newtonian fluids through curved pipes and curved annuli for creeping flow (NRe ≈ 0) and noncreeping flow with NRe < 25. For creeping flow, both the curved and annular tubes have secondary flows produced due to elasticity similar to those produced by inertia. However, the effects of elasticity and inertia were not found to be additive. Fan et al.320 used finite element computations to study the flow of both viscous and viscoelastic fluids through curved tubes. They quantified the intensity of the secondary flows by calculating two stream functions, one being the secondary volumetric flux per unit work consumption and the other as the secondary volumetric flux per unit axial volumetric flux. They showed that, for both these parameters, the values first increased with the Reynolds number and then decreased, for both the Newtonian and the non-Newtonian fluids. Agrawal and Nigam322 numerically investigated the velocity profiles in the coiled tube. It was observed that, as the pesudoplasticity increases, the viscous boundary layer becomes limited to the outer wall of the pipe and its thickness reduces. Their results were in good agreement with the findings of Mashelkar and Devrajan,305 Hsu and Patankar,306 and Takami et al.315,316 Agrawal and Nigam322 also numerically studied the convective diffusion with first-order reaction in curved circular tubes for both Newtonian and non-Newtonian fluids in the range 1 < NDe < 250, 1 < NSc < 1.4 × 104, 10 < λ < 100, and 0.5 < n < 1. They observed that the variation of bulk concentration along the axial distance decreases with increases in the values of R for particular values of n, NDe, NSc, and λ. It was reported that the performance of coiled chemical reactor lies between the performance of plug and laminar reactors. A comprehensive literature survey on the flow of power law fluids flowing through circular curved tubes has been listed in Tables 12-14. 3. Two-Phase Fluid Flow and Heat Transfer in Curved Tubes Unlike single-phase flow, there are considerable complications in describing and quantifying the nature of two-phase flow. In most cases, the gas phase, which may be flowing with a much greater velocity than the liquid, continuously accelerates the liquid thus involving transfer of energy. A secondary flow is induced in the gas core, which acts through interfacial shear to distribute the liquid around the entire surface up to 90-95% quality.5 In addition to inertia, viscous and pressure forces present in single-phase flow, two-phase flows are also affected by interfacial tension forces, the wetting characteristics of the liquid on the tube wall, and the exchange of momentum between the liquid and vapor phases in the flow. Phase separation occurs

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3309 Table 12. Experimental Studies for the Flow Phenomena of Power Fluids

Table 13. Theoretical Studies for the Flow Phenomena of Power Fluids range of parameters name of investigator

theoretical method

l

n

Rathna313

analytical method (perturbation method)

20-100

0.5-1.5

1-20

Mashelkar and Devrajan305

boundary layer approach

10-100

0.5-1.0

>50

Hsu and Patankar306

numerical finite technique described by Patanker (1980) numerical time marching method

0.5-1.25

1-103

Takami et al.315,316

Kewase and Young321

integral momentum

Agarwal and Nigam322

analytical perturbation method; numerical finite difference

NDe

10-100

0.5-1.0

1-1000

10-100

0.5-1.0

>100

5-100

0.5-1.5

1-500

in all kinds of curved pipes because of the differences in the inherent centrifugal force, gravity force, and secondary flow effect between heavy and light phases. Two-phase flow in coiled tube geometry may lead to inhomogeneous phase distribution, flow reversal, flooding, secondary flow, and film inversion. These phenomena can cause problems such as burnout, corrosion, and tube failure in industrial components, resulting in costly outages, repairs, and early replacement affecting plant reliability and safety. Hence, it gives rise to a more complex and challenging flow phenomenon, which needs to be explored. In spite of vast application

remarks Followed Dean’s constraint, flow rate is independent of curvature ratio Theory is applicable for higher values of Dean number. Theoritical correlation for friction coeffcient (fc) is given in Table 1B Ratio is large The relation fc and NDe can be expressed with single curve which is dependent on n The theory in applicable for both laminar and turbulent flow and heat transfer of power law fluids for high Dean number (NDe > 100) Extended the Dean’s perturbation solution for secondary velocity profiles and heat transfer in coiled tube

of multiphase flow in industrial practices, only limited information is available for gas-liquid, liquid-liquid, and gas-solid two-phase flow system in coiled tubes as compared to singlephase studies. The focus of this section is to characterize the flow characteristics (like flow patterns, pressure drop, holdup, and residence time) and heat transfer for two-phase flow in curved tubes. 3.1. Two-Phase Fluid Flow in Curved Tubes. 3.1.1. GasLiquid Systems. These types of flow systems are encountered in gas-liquid dispersion, continuous fermentation, and polym-

3310


Table 14. Heat-Transfer Results for Power Law Fluidsa Asymptotic Nusselt Number n

NDe

NPr

Agarwal and Nigam322

Hsu and Patankar306

0.5

25

5 15 30 5 15 5 15 30 50 5 15 30 50 5 15 30 50 5 15 30 50

7.25 8.10 10.03 8.30 10.55 9.23 11.50 12.85 15.60 12.05 14.50 16.55 20.22 15.90 22.10 26.80 32.20 20.95 24.12 30.50 37.20

6.64 7.83 9.49 7.59 9.25 8.75 10.35 12.25 14.85 11.25 12.15 14.85 17.10 15.39 21.80 25.08 31.21 20.52 23.60 29.93 30.37

50 0.75

50

100

1.25

75

100

a

Source: Agarwal and Nigam.322

Figure 10. Various flow patterns encountered in coiled tubes.

erization reactions, which are often carried out in coiled tubes. The working fluids adopted by various researchers include steam-water, air-water, SF6-water, and chemical compound (glycerine, butanol, etc.)-water. 3.1.1.1. Flow Patterns, Pressure Drop, Void Fraction, and RTD. The complicate and complex nature of two-phase flow phenomena requires detection, monitoring, and description of flow patterns. The flow patterns observed in coiled tubes involve a combination of horizontal and vertical tubes. When a twophase gas-liquid mixture flows through a coiled tube, the various possible flow patterns encountered are such as bubbly flow, slug, plug, stratified/wavy, and annular flow (Figure 10). In order to predict the two-phase pressure drop and holdup in a coiled tube, the most widely used correlations are based on Lockhart-Martinelli (L-M) parameter.323 Most of the researchers used the L-M method to correlate their experimental data on two-phase frictional pressure drop and holdup while others modified the L-M parameters to fit their experimental data. Ripple et al.324 carried out experimental work on coiled tubes and observed that pressure drop for such systems satisfied the L-M correlation. They found that the in situ liquid void fraction (L) was in good agreement with that predicted by the L-M curve for straight horizontal tubes, at higher valves of X, but it was found that L values were lower at lower values of X. They attributed this difference to the downward flow orientation in the coiled tubes. Further, they also pointed out that the liquid void fraction was influenced by liquid properties rather than gas properties. Similar observation was also made by Owhadi325 that pressure drop for two-phase gas-liquid flow through coiled tubes satisfied the L-M correlation.

In 1969, Banerjee et al.326 investigated the gas-liquid flow through transparent coils with different tube diameters, coil diameters, and helix angles. They concluded that Baker’s plots327 adequately predicted the flow patterns and reported that their experimental values of L agreed with those predicted by the L-M curve within (30%. They modified the L-M correlation to satisfy their experimental data on the two-phase pressure drop and holdup. Another observation made by them was that the helix angle, if small, appears to have no discernible effect on the pressure drop. Boyce et al.328 carried out two-phase pressure drop and flow regime measurements in coiled plastic tubes. Unlike Banerjee et al.,326 they reported that Baker327 flow maps could not predict the flow pattern transition adequately. They observed that the L-M correlation adequately predicted their experimental pressure drop data. Kasturi and Stepanek329 worked with different working fluids such as air-water, air-corn sugar-water, air-glycerol-water, and air-butanol-water to determine pressure drop and void fraction in coiled tubes. They compared the results for pressure drop with L-M and Dukler’s correlations. Voidage results were compared with Hughmark’s correlation. They found that L-M correlation fitted better than Dukler’s correlation but observed a systematic displacement of the curves for the various systems with the L-M plot. The curve for the most viscous corn sugar solution-air system was the lowest in the set of curves and the curve for the butanol-water-air system was highest. The same applied for the Hughmark’s correlation for the void fraction. They intuitively thought that the L-M parameter could be modified to take into account the effects of viscosity and voidage. Hence, in their second contribution,330 they proposed correlations for void fraction and pressure drop in terms of new parameters that could account more fundamentally for the complex behavior of the two-phase flow than the simple correlation in terms of L-M parameters. Akagawa et al.331 also used the Hughmark332 correlation to correlate the mean volume fraction of gas. Mujawar and Rao308 pointed out that if the flow patterns were specified for two-phase flow then the pressure drop could be successfully correlated by the L-M method. Kaji et al.333 observed that the flow regime transition for an airwater two-phase flow was close to the map of Mandhane et al.334 except for the annular-wavy stratified flow boundary, which was close to Baker’s map. In 1984, Rangacharyulu and Davis335 studied pressure drop and holdup for a system of air-liquid co-current upward flow in coiled tubes. Water, glycerol, and isobutyl alcohol were used as working fluids. Based on the modified L-M parameter, they presented a new correlation for the two-phase frictional pressure drop. In another study, Chen and Zhang336 observed that the helix angle has a pronounced effect on the flow pattern transition. They proposed empirical correlations for predicting transitions under different flow conditions. Saxena and Nigam337 conducted experiments to study the flow patterns, holdup, and pressure drop for co-current upward and downward flow in coiled tubes. They found close similarities between the flow patterns in coiled tubes and those of inclined tubes reported by Spedding et al.338 They proposed a new mechanistic approach to correlate the pressured drop in coils instead of using the L-M method. The noteworthy point about their model is that it retains the identity of each phase and separately accounts for the effects of curvature and tube inclination resulting from the torsion of the tube as well. Awaad et al.339 investigated the air-water, two-phase flow in horizontal and vertical coiled tubes, respectively. For horizontal coiled tubes, they found that the superficial velocities of air or water


Figure 11. Comparison of published pressure drop correlations for steamwater two-phase flow.

had significant effect on the pressure drop multiplier, while the helix angle had insignificant effect, and the pipe and coil diameters had a certain effect only at low flow rates. They compared the pressure drop using the L-M parameter and found that no significant difference occurred for the helix angle, but a slight modification was caused due to change in tube and coil diameter. In the case of vertical coiled tubes, Xin et al.340 observed that two-phase pressure drop depends not only on the L-M parameter but also on the flow rates. For small curvature ratio coils, they proposed a two-phase frictional pressure drop equation by modifying the L-M parameter. They found good agreement between the holdup data and the L-M correlation. In line with the observations of Awaad et al.,339 they also noticed that the helix angle has no effect on the holdup but had some effect on the frictional pressure drop. Chen and Guo341 experimentally investigated the three-phase, oil-air-water flow in helical coils to study the effect of flow rate and liquid properties. They reported that the flow characteristics can be classified into more than four flow patterns and also suggested flow regime maps and correlation for pressure drop. Guo et al.181,182 studied the pressure drop of steam-water two-phase flow in two coiled tubes with four different helix angles. They found that the system pressure and mass quality had significant effect on the two-phase pressure drop. They gave a correlation based on the Chen correlation for boiling twophase flow frictional pressure drop. The comparison of the pressure drop correlations of Akagawa et al.,331 Unal et al.,186 Chen and Zhang,336 and Guo et al.188 are shown in Figure 11. It can be seen from the figure that great differences exist among them, and in addition, these correlations are way too sophisticated to be practically used, except that of Guo et al.188 The various pressure drop correlations resulting from the above investigations are listed in Table 15. A set of empirical correlations were given by Mandal and Das342 and Biswas and Das343 for two-phase friction factor and void fraction for Newtonian and non-Newtonian fluid flow in coiled tubes. Murai et al.344 used the backlight imaging tomography to study the effect of centrifugal acceleration on phase distribution and interfacial structure for gas-liquid flow in a coiled tube. They observed that centrifugal force resulting from the curvature of coil generated a wall-clinging liquid layer against gravity, which caused the interfacial area to be enhanced. They photographed the interfacial structure by a high-speed video system with synchronized measurement of local pressure

fluctuations and reported that the flow transition line alters due to centrifugal force acting on the liquid phase in the tube. In particular, the bubbly flow regime is narrowed significantly. Recently, Kumar et al.345 numerically reported the Taylor flow in curved microchannels on gas and liquid slugs. The slug flow development for different inlet conditions and geometries (premixed feed, T-type and Y-type inlets) were studied in the curved microchannels. They reported that, for low curvature ratio (D/d ) 3), the phenomenon of flow reversal and slug freezing takes place due to centrifugal and buoyancy forces. For the similar process conditions, with an increase of curvature ratio to 5 and 10, the phenomenoa of flow reversal and slug freezing observed were very minor. The nonuniformity in the slug formation was observed for low curvature ratio as compared to the higher curvature ratios. Further, the influence of surface tension, viscosity, and wall adhesion was also studied. Two competing mechanisms of dispersion are present in coiled systems: (i) a dislocation, to the outer wall, of the maximum of the velocity profile (asymmetric profile) due to the curvature effects, which increases the spread of residence time distribution; (ii) the secondary flow (Dean-roll cells) generates a transverse mixing that decreases axial dispersion. The relative balance between these two competing actions depends on the Reynolds number, since the effective dispersion first increases and then decreases with increasing Reynolds number. Saxena et al.349 experimentally determined the liquid residence time distribution for two-phase flow in coiled tubes. Upward and downward cocurrent flows were investigated in three coils with curvature ratios ranging from 11 to 60.7. They proposed a model that describes the liquid RTD as combination of two different RTDs applicable for turbulent and laminar liquid flows. The area of studying two-phase RTD in coiled tubes has not been much explored as compared to single-phase flow in spite of its great importance in reactor design. 3.1.2. Gas-Solid Systems. 3.1.2.1. Pressure Drop and Void Fraction. The studies on gas-solid systems performed to date have shown that the performance of various hydraulic systems can be significantly improved by using a coiled tube instead of a straight tube. The main issue of concern in any gas-solid system is the concentration buildup near the wall, e.g., in membrane filtration modules. It turns out that only very limited efforts have been made in the past to understand the fluid mechanics of such flows. Several issues regarding the effect of coil curvature on local multidimensional phenomena governing fluid flow need to be explored in depth. To overcome the limitation of near-wall concentration buildup, two effective methods have been used to date. One is associated with the use of well-ordered Taylor vortices established in a rotating annular filter module,350 and other is the use of multiple Dean vortices resulting from flow around a curved duct.88 Due to the use of rotating parts, Taylor vortices are associated with the difficulty in scaling up, high-energy consumption, and high maintenance costs. On the other hand, no moving parts are needed for Dean vortex-based systems, because the coil tube curvature directly induces secondary flows, so that no moving parts are needed. Hence, they serve as an attractive alternative. Dean vortices can provide a stronger mixing effect than Taylor vortices. The research on gas-solid systems351-354 has shown that the near-wall solid particle buildup in liquid/particle suspension systems can be significantly reduced by taking advantage of Dean vortex-induced secondary flows. U-bend geometries355,356 and helical coiled tubes18,179 are the configurations that have been developed to date that capitalize on the use of Dean vortices. The U-bend tubes, with a zero

3312


Table 15. Pressure Drop Correlations for Two-Phase Flow in Coiled Tubes

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3313 Table 15 (Continued)

3314


Table 15 (Continued)

pitch, involve the curvature-induced centrifugal force only, whereas for coiled tubes, where the pitch of the coiled tube is greater than zero, the secondary flow fields become more complex due to the combined effects of the Coriolis force due to torsion of the tube centerline and the centrifugal force due to the curvature. So far, only limited attempts have been made to model gassolid357,358 and liquid-solid359 flow systems for different applications. Recently, promising results have also been reported for suspended particulate flows in U-bend and coiled tubes by Tiwari et al.353,354 The next important factor of concern in gas-solid flow in coiled tubes is pressure drop. Weinberger and Shu360 reported that the variations of solid pressure drop depends on solids flow rate, helix radius, and loading ratios. In their second paper,361 they determined the transition velocities as a function of bend or helix radius and solids flow rate. They found that the transition velocities decreased with increasing bend radius and solid flow rate when compared with those predicted from modified horizontal flow correlation. Numerical analyses of fluid flow in membrane tubes that have been reported in the literature are primarily for either single-phase liquids18 or for particle/ liquid flows in straight ducts.362 Tiwari et al.353 carried out numerical investigation on coiled tubes and demonstrated that azimuthal vortices may bifurcate at lower Dean numbers. They also discussed the use of vorticity magnitude as a measure of vortex strength and the role that Dean vortices play to mitigate the effect of gravity on particle settling. Their work has direct relevance to synthetic membrane fouling during filtration of particle suspensions.

Numerical and experimental investigations on liquid-solid separation on 2-D curved channel and coiled tube, respectively, were carried out by Gao et al.359 They analyzed that the drag force, pressure gradient, and turbulent dispersion force play an opposite role for separation whereas centrifugal force is the prime driving force for separation, which moves the particles from inner bend to outer bend. Also, the lift force is notable near the wall and helpful to separation, and the virtual mass force can be neglected. Xu et al.363 numerically studied liquidsolid, two-phase turbulent flow in a curved pipe employing a two-fluid model and finite element method. Guo280 carried out numerical simulations to study the particle concentration profiles for various flow situations and particle dynamic analysis performed along particle trajectories. They also investigated the gas-solid, two-phase flows on a pilot-scale separator and made a comparison of the conventional separation techniques. They recommended the use of coiled tube separators due to their lucrative advantages, such as simplicity and compactness in structure, high efficiency, and low pressure loss against the conventional methods. Tiwari et al.364 investigated the mutual interactions between the liquid flow and solid particles in particulate two-phase flows in both the U-bend and helical geometries. They observed that particle inertia causes an increase in the wall shear. They also characterized two effects, which are important for equipment design and optimization in biotechnology and process industries. One was the shift in the particle-settling zone from the bottom of the horizontal or nearly horizontal tube toward the inner bend of the tube. The other, even more important, is a dramatic reduction in peak concentration with increasing Dean number.


Very few experimental efforts have been made to study the gassolid flow in coiled tubes, and the numerical studies scarcely considered the mechanism of particle dynamics. 3.1.2.2. Residence Time Distribution. RTD of particles during continuous processing of solid-liquid mixtures has been of great interest to researchers since it relates to the pharmaceutical and biological industries and aseptic processing of particulate foods using coiled holding tubes. Uniformity of heating in a continuous thermal process increases with narrowing RTD of particles in the holding tube. An important parameter influencing the residence time (RT) of a particle in a continuous flow system is the density of the particle, since it determines the radial location in the tube at which the particle travels. Neutrally buoyant particles travel close to the tube center where the carrier fluid velocity is the highest, and hence, they have a low RT. Particles with a density greater than that of the carrier fluid, however, have a larger RT, since they travel near the tube wall where the carrier fluid velocity is lower.365 Other factors affecting RTD include particle concentration, carrier fluid viscosity, and flow rate. Studies have been conducted to determine the RTD of particles in a straight holding tube as affected by different processing parameters.366,367 The velocity profile approaches that of ideal plug flow, and the RTD of the particles narrows down with increasing flow rate368 and with increasing particle concentration.366 Abdelrahim et al.369 found that increasing the carrier fluid concentration resulted in an increase in the RT of the fastest-moving particle and both particle types appeared to be accelerated upon mixing, the effect being more obvious at high flow rate. From the literature, it is evident that there is very little information on RTD of a heterogeneous mixture of particles. There is a need to better understand the flow behavior of solid-liquid mixtures having more than one type of particle in coiled tubes. The flow of solid-liquid mixtures, containing a heterogeneous mixture of particles, in coiled tubes was investigated.370 The effects of curvature ratio of the coiled tube, flow rate, carrier fluid viscosity, and particle concentration on the RT and RTD were determined. All of the parameters, except the carrier fluid viscosity, have strong effects on the flow behavior of particles. 3.1.3. Liquid-Liquid Systems. Baier et al.371 investigated the problem of extraction of admixtures having extremely low diffusion coefficients, e.g., proteins. The first experimental attempt in this direction was made by Baier et al.371 with twofluid, Taylor-Couette vortical flow in a specially designed bioseparator/bioreactor. Further, the same configuration was studied theoretically and numerically, where the effect of the Taylor vortices and axial through-flow on the mass-transfer rates was investigated.372 The major limitation working with TaylorCouette flow is rapidly rotating cylindrical boundaries. Hence, Chung et al.373 and Mullubhotla et al.374 got motivated to use the Dean vortices arising in coiled tubes. They observed that the concentration buildup, known as concentration polarization, can be significantly reduced if solutions were supplied through curved channels, where the Dean vortices arise. The Dean vortices effectively mix the retained solute and this results in depolarization and a significant increase in the transmembrane flux. Gelfgat et al.375 numerically investigated two-fluid, Dean vortex flow in a coiled tube with vanishing torsion and its effect on the mass-transfer through the liquid-liquid interface of two immiscible fluids. They considered both co-current and countercurrent axial flows in the fluid layers. The liquids got stratified by gravity, with the denser one occupying the lower part of the pipe. They reported that the Dean vortex flow is an effective

tool for mass-transfer enhancement at the liquid-liquid interface. It is also shown that the Dean flow provides a stronger mixing than the Taylor-Couette flow. A lot of literature is available on liquid-liquid flow in straight tubes, but rare efforts have been made to explore its behavior and utilities in coiled tubes. 3.2. Two-Phase Heat-Transfer Characteristics in Curved Tubes. Two-phase flow structure found in helically coiled heattransfer tubes varies with centrifugal acceleration caused due to the curvature ratio. The flow structure inside the heat-transfer tubes due to helicity is one of the fundamental problems needing critical attention for the optimization of compact heat exchangers, evaporators, and condensers used in many industrial applications. It is well-known that the two main mechanisms, nucleate boiling and convective boiling or forced convection evaporation, determine the heat transfer in two-phase forced flow within tube. In the practical industrial applications, convective boiling is often the prevailing heat-transfer mode in heat exchangers, where high heat-transfer rate can be achieved at extremely small wall-liquid temperature differences. Hence, it is of great practical significance to have well-established design recommendations, which would allow this process to be calculated with a high degree of precision. Extensive studies on the flow and heat transfer in helically coiled tubes have been conducted for several decades. So far, it is well-known that the secondary flow due to centrifugal force and Coriolis force in the cross-section of the tube is a significant factor affecting the flow patterns and, consequently, affecting heat transfer in both single- and two-phase flows. The research work for two-phase flow, however, is insufficient, compared to that of single-phase flow. Owhadi et al.325 carried out pioneering research on forced convective boiling heat transfer to water at atmospheric pressure in two coiled tubes of 12.5 mm i.d., and d/D ) 0.05 and 0.024, respectively. It is evident from their results that, over most of the quality region, the prevailing heat-transfer mode is convection and a nucleate boiling component is present at low qualities. They found that the local average boiling heat-transfer coefficient of coiled tube could be predicted by Chen’s correlation376 with an accuracy of (15% over the range tested. Tarasova et al.377 introduced a correction factor to account for the heating effect on two-phase frictional pressure losses. The correction factor is directly proportional to the heat flux and, hence, appears to be valid only for the bubbly flow regime, where the pressure drop increases with increasing heat flux. Similar observations have also been made by Petukhov et al.378 for the same heating effect in helium flow. There is still lack of thermodynamic analysis about the effects of curvature ratio on entropy generation and the further optimal analysis. Kozeki et al.379 conducted a test on heat-transfer and pressure drop characteristics in a helically coiled tube heated by hightemperature water at steam pressures of 0.5-2.1 MPa. They found their two-phase, frictional pressure drop data were larger than those predicted by Martinelli-Nelson’s correlation for a straight tube, the differences increase with the decrease of system pressure and the increase of the flow rate, and two-phase forced convection occupied most portions due to the effect of centrifugal force and secondary flow. The study on forced convection heat transfer to high-quality, two-phase water-steam mixtures in helically coiled tube was taken up by Crain and Bell.380 They correlated the circumferential average heat-transfer coefficients as a function of L-M parameter. Izumi et al.381 suggested using a similar correction factor for boiling Freon-12 flow. Campolunghi et al.382 con-

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ducted full-scale tests for coiled once-through steam generators with subcooled water at inlet and superheated steam at outlet. They correlated the boiling heat-transfer coefficients with heat flux and operation pressure as a dimensional formula. Boom et al.383 observed a slightly different trend of the heat flux effect on two-phase pressure drop in helium flow inside a heated tube. The two-phase pressure drop increases continuously over the full range of quality (i.e., from 0% to 100%). In boiling flow regime, Reddy et al.384 observed a minor effect of surface heating on the two-phase frictional pressure loss. Tarasova et al.377 on the other hand, observed a strong surface-heating effect on the two-phase frictional pressure loss from their high-pressure steam-water data. An inevitable problem met by coiled tubes is that the heattransfer enhancement in the flow fields is always achieved at the expense of an increase in frictional losses. For example, the heat-transfer enhancement method by using alternating axis coils to induce chaotic mixing proposed by Narasimha et al.385 not only increases the heat-transfer rate but also increases significant pressure drop simultaneously. Such conflict raises the question as to what is the optimal tradeoff by selecting the most appropriate geometry and the best flow condition. The design-related concept of efficient energy use386 has become the answer to the question. Bejan387 proposed a systematic methodology of computing entropy generation through heat and fluid flow in heat exchangers, based on which considerable optimal designs of thermal systems have been proposed by various investigators.388-391 During the design work of coiled tubes, it can be found that several important parameters have effective influence on heat and momentum transport phenomena in the devices. Meantime, these parameters also contribute to irreversibility that inherently competes with one another. Nonetheless, in previous investigations relevant to the flow field in coiled tubes, most of them were restricted to the first law of thermodynamics and very rare energy analysis has been addressed. For practical design and application of the steam generators, the prediction of two-phase flow and boiling heat-transfer characteristics is extremely important. The investigation of twophase flow and boiling heat-transfer characteristics in coiled tubes is highly lacking, compared to the investigation of that in straight channels. The methods for analyzing the pressure drop and heat transfer for the straight tube are still used or modified to describe forced convective boiling two-phase flow and heat transfer in coiled tube. Chen’s correlation376 given by eq 10 is widely used to calculate flow boiling heat-transfer coefficient in a straight tube is one of the most famous correlations that is based on the mechanism of nucleate and convective dominated heat transfer. Owhadi et al.325 indicated that Chen’s correlation could successfully be applied to helically coiled tube with acceptable agreement.

kl0.79Cpl0.45Fl0.49 h ) 0.00122 0.5 0.29 0.24 0.24∆Tsat0.24∆psat0.75S + σ µL ilg Fg kl 0.023NRel0.8Prl0.4 F (10) d where S and F factors are calculated using eqs 11 and 12, respectively.

S)

1 1 + 2.53 × 10-6F1.25NRel

(11)

F ) 1.0

(

for

1 e 0.1 Xtt

F ) 2.35 (1/Xtt + 0.213)0.736

(

)

for

(12) 1 > 0.1 Xtt

)

For correlating two-phase heat-transfer coefficient in helically coiled tube, L-M type correlation is another popular method, whichisusedbymostresearchersinthisfield,188,280,325,346,380,381,392-394 some of them adding a nucleate boiling term, such as boiling number, Bo, to combine the effect of nucleate boiling mechanism.393,395 Most of the above correlations are presented in Table 16. Figure 12 shows the comparisons of experimental results of boiling heat-transfer coefficients397,398 with the predicted values of Kozeki’s correlation, Schrock-Grossman’s correlation, and Guo’s correlation. Schrock-Grossman’s correlation is one of the most famous correlations for predicting heat-transfer coefficients at annular forced convective boiling flow regime in straight tubes and it was suggested by Nariai et al.393 that it can also be used for predicting the boiling heat-transfer coefficient for a coiled tube as well. Kozeki’s and Guo’s correlations could not predict the experimental results very well. In these correlations, the heat-transfer coefficients are expressed as the ratio of a two-phase heat-transfer coefficient to the single-phase heattransfer coefficient at the same mass flux when the fluid is completely liquid. Hence, the comparison indicates that the correlations based on water-steam two-phase flow overestimate the heat-transfer enhancement of two-phase heat transfer to single-phase heat transfer for refrigerant flow boiling especially in higher Xtt region. It can be found that there exists some relationship between convective boiling number, NCB, and the ratio of two-phase heat-transfer coefficient and single-phase heat-transfer coefficient, hTP/hlo, which must be taken into account in order to settle down to a better correlation to predict the convective boiling heat transfer. Nariai et al.393 conducted an investigation of thermalhydraulic behavior in a once-through steam generator used for an integrated type nuclear reactor, in which the coiled tube was heated with liquid sodium. Their experimental result indicated that modified Kozeki and Martinelli-Nelson correlations agree with their experimental results of two-phase frictional pressure drop within 30% and the effect of coiled tube on average heattransfer coefficients was small. The Schrock-Grossman correlation could also be applied to a coiled tube with good accuracy at pressures lower than 3.5 MPa. The SchrockGrossman correlation covered the effect of both saturated nucleate boiling and forced convection. In 1981, Unal et al.186 conducted the same experiment in a sodium-heated helically coiled tube and found that the diameter ratio d/D has little influence on the two-phase frictional pressure drop. This conclusion was confirmed by later research of Zhou398 and Guo et al.,188 employing an electrically heated helically coiled tube test section. Based on their own experimental data of pressure drop. Unal et al.,186 Zhou,398 Guo et al.,399 and Bi et al.400 provided a series of empirical correlations developed from L-M turbulent relationship to calculate the steam-water two-phase frictional pressure drop inside vertical or horizontal coils at high pressure. So far, the empirical or semiempirical correlations of different researchers can only be used in a specific range and there are some conflicts among their conclusions. Many researches indicate that the two-phase flow and boiling heat-transfer characteristics in a small channel are different from that in a

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3317 Table 16. Heat-Transfer Correlations for Two-Phase Flow in Coiled Tube

Figure 12. Comparison of experimental boiling heat-transfer results available in the literature.

large channel.401,402 Their results indicate that the boiling heattransfer coefficient is sensitive to both heat flux and mass flux and that convective boiling dominates at lower wall superheat values and nucleate boiling dominates at higher wall superheat

values. Horizontal coiled tube once-through steam generator is favored for space, navigation, and other specific techniques because of its lower gravitation center and higher efficiency both in heat transfer and steam generation.398,403 Kang et al.396 discussed the effects of cooling wall temperature on the condensation pressure drop characteristics of refrigerant HFC-134a in annular-coiled tubes. They also investigated the effects of tube wall temperature on the heat-transfer coefficients and pressure drops. The results showed that the refrigerant side heat-transfer coefficients decreased with increasing mass flux or the cooling water flow Reynolds number. Naphon and Wongwises404 experimentally investigated the heat-transfer characteristics in coiled tubes with water flowing in the tube side and dry air flows outside the tube. They obtained a new experimental correlation of Nusselt number to Dean number to the power index of 0.287, implying heat-transfer enhancement by the internal secondary flow. Guo et al.280 measured the transient convective heat-transfer coefficient of steam-water two-phase flow in a coiled tube and found a correlation to pressure drop-type oscillation. Yu et al.405 presented an experimental study on the condensation heat transfer of HFC-134a flowing inside a coiled tube with cooling water flowing in the annulus. The experiments were performed

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for mass flux in the range of 100-400 kg/m2‚s and the Reynolds number of cooling water in the range of 1500-10 000. They reported that the orientations of the coiled tube had significant effects on the heat-transfer coefficient. In 2003, Louw et al.406 studied the effect of annular contact on the heat-transfer coefficients and pressure drop in a coiled tube-in-tube heat exchanger with hot water in the inner tube and cold water in the annulus. The heat-transfer and pressure drop characteristics of the coiled tube with annular contact were compared with those without annular contact. They observed that annular contact had an insignificant effect on the conduction heat-transfer resistance. The change in flow patterns in the annulus made the heat-transfer coefficient and pressure drop increase substantially. The heat-transfer characteristics of an evaporating two-phase flow in a small coiled tube of 4 mm internal diameter was determined by Yi et al.407 They noted that the pulsation by slug flow was effective for heat-transfer enhancement and greatly improved the critical heat flux. Xin et al.,408 Zhou and Xin,409 and Cui et al.410,411 have experimentally studied the flow boiling heat-transfer characteristics inside both the 3D microfinned straight and coiled tubes. Cui et al.397 developed a new kind of cross-grooved microfinned tube, called a three-dimensional microfinned tube, from which many valuable results have been obtained. Despite the considerable work discussed above, most of which are based on pressurized water boiling experiments, the convective boiling heat transfer in coiled tubes has been scarcely investigated compared with the similar work conducted in straight tubes, and very few heat-transfer coefficient correlations available are to predict the convective boiling heat-transfer process. Although different flow and heat-transfer characteristics exist, the methods for analyzing the pressure drop and heat transfer for the straight tube are still used or modified to describe forced convective boiling two-phase flow and heat transfer in helically coiled tube. 4. Chaotic Configurations (Combination of Coils and Bends) Heat- and mass-transfer in coiled tubes can further be enhanced by inserting some perturbation in the geometry. The idea of generating spatial (Lagrangian) chaotic behavior from a deterministic flow by simple geometrical perturbations has attracted much attention.4,412-414,416,462 The geometrical perturbation induces complex three-dimensional chaotic trajectories in which fluid elements can visit a large number of positions in physical space. Very complex flow paths can be produced in this way, and a fluid element undergoing such flows follows a chaotic path. Such chaotic cross-sectional movement has been found to enhance the advection of passive scalars and therefore to improve the efficiency of the wall heat transfer, leading to homogenization in the fluid volume212,273,416,417 and thus better mixing. Figure 13 shows the various chaotic and serpentine geometries being used by different investigators. These geometries have the potential to create chaotic flows and are used in several applications, such as analytical chemistry, systems of microbiological analysis, “lab on chip”, and cooling systems. Chaotic feature is a simple and regular flow with chaotic trajectories of fluid.418 These mechanisms were subsequently elucidated by various authors.413,418-425 Aref418 achieved chaotic mixing at small Reynolds numbers with appropriate blinkingvortex flow of a two-dimensional temporally periodic model. Ottino et al.421 performed a two-dimensional analysis on periodic mixing with cavity flow and journal-bearing flow at small Reynolds numbers.

4.1. Fluid Flow and Heat Transfer. Jones et al.,413 Guer and Peerhossaini,412 Liu et al.,426 Acharya et al.,427-429 Peerhossaini et al.,430 Duchene et al.,431 and Mokrani et al.417 presented an alternative regime in laminar flow that has dispersive properties close to a turbulent regime using the phenomenon of chaotic advection. In chaotic advection, the fluid-particle trajectories are chaotic, which enhance mixing. Guer and Peerhossaini412 and Peerhossaini et al.430 analyzed two coiled heat exchangers over a range, NDe ) 141-530. They recorded velocity pattern by laser Doppler velocimetry and visualized the flow field with fluorescence. Acharya and Sen427 demonstrated that heat transfer is enhanced through chaotic trajectories. They found that flow in a coil with an alternating axis leads to chaotic trajectories of particles, thus enhancing thermal transfer. Yang et al.432 studied a system with periodically varying curvature, much like the chaotic systems developed above. Their findings showed that changing the amplitude or the wavelength of the curved pipe could increase the heattransfer rates. Amon433 numerically reported vortices and described chaotic phenomena at NRe ) 450 in a wavelike convergent-divergent channel. Accompanying the development of microtechnology, various microfluidic mixers have been described for which the order of Reynolds number is 10 or less.426,434-436 In order to achieve satisfactory mixing efficiency at a small Reynolds number, chaotic mixing in a serpentine channel has attracted much attention. Liu et al.426 utilized a microscale, three-dimensional serpentine channel to induce chaotic mixing in a range of NRe ) 6-70. Castelain et al.437 investigated the mixing of fluids in a three-dimensional serpentine channel, generating an irregular trajectory of fluid particles with a centrifugal force and geometrical perturbation. Castelain et al.438,439 further experimentally studied the RTD of a pseudoplastic fluid in different configurations of helically coiled or chaotic systems for Reynolds numbers varying from 30 to 270. They observed that Peclet number (NPe) is found to increase with Reynolds number whatever the number of bends in the system. The values of the Peclet number are greater for the pseudoplastic fluid, the local change of apparent viscosity affecting the secondary flow, and the apparent viscosity is lower near the wall and higher at the center of the cross-section. The maximum axial velocity is flattened as the flow behavior index is reduced, inducing a decrease of the secondary flow in the central part of the pipe, which reduces the axial dispersion. A method to study fully developed flow and heat transfer in channels with periodically varying shape was first developed by Patankar et al.440 Webb and Ramadhyani441 and Park et al.442 analyzed fully developed flow and heat transfer in periodic geometries. They reported an increase in both the heat-transfer rate and pressure drop as the Reynolds number is increased. Popiel and van der Merwe443 and Popiel and Wojtkowiak444 carried out investigations on geometries with an undulating sinusoidal shape or U-bend configuration to determine the friction factors for a number of geometrical parameters, such as curvature ratio, wavelength, and amplitude over a range of Reynolds numbers spanning laminar and low-Reynolds-number turbulent flow. They reported an interesting observation that when the friction factor is plotted against the Reynolds number, there is either no definite transition from laminar to turbulent flow or a delayed transition relative to that of a straight pipe. This is also reflected in the work of Johnston and Hyanes.445 Their results demonstrate a smooth transition from laminar to turbulent flow in plots of the friction factor and Colburn j-factor versus the Reynolds number. They hypothesized that a smooth


Figure 13. Geometries of chaotic and serpentine channels: (a) straight channel, (b) square wave channel, (c) C-shaped channel, (d) V-shaped channel, and (e) B-shaped channel. (Source: Lasbet et al.459) (g) Discontinuous baffles and (h) snake-like baffle. (Source: Ottino et al.421) (i) Serpentine geometry with 90° bends and (j) serpentine geometry with U-bends. (Source: Wua et al.451)

transition to turbulence occurs due to secondary flows produced in the complex geometries. In 2004, Metwally and Manglik446 also studied the effect of Prandtl and Reynolds numbers, as well as geometrical parameters, on fully developed flow and heat transfer in corrugated or wavy channels with constant wall temperature. They found that the creation of transverse vortices is dependent on the Reynolds number and also on the ratio of the wall wave amplitude to wavelength. Mixing produced by the creation of these vortices promoted further increase in the rate of heat transfer. They took into account the concomitant increase in the rate of heat transfer and pressure drop by using an area goodness factor originally defined in Shah and London447 in order to compare cross-sectional shapes independent of hydraulic diameter. Results indicated that the optimum geometry is dependent on Reynolds number for a given Prandtl number. It is suggested that a less wavy geometry is more desirable at

higher Reynolds numbers. It is known that, for most configurations, an increase in the rate of heat transfer can be achieved by increasing the Reynolds number;448 however, higher waviness could cause excessively large pressure drops at higher Reynolds numbers. Chintada et al.449 numerically studied the laminar flow and heat transfer in square serpentine channels with right-angle turns, which have applications in heat exchangers. It was found that the heat-transfer performance of serpentine channels is higher than that for straight channels for NPr ) 7.0 and is lower for NPr ) 0.7. Flow in serpentine channels has also been investigated by Liu et al.,426 who quantified the mixing obtained in such passages experimentally, for the purpose of designing a passive three-dimensional serpentine micromixer. The serpentine geometries studied by Liu et al.426 snaked in all three coordinate directions and were found to promote mixing when compared with square wave channels (by a factor of 1.6 for Reynolds

3320


numbers up to 70) and straight channels (by a factor of 16 for Reynolds numbers up to 70). The extent of mixing had been determined by measuring color variation in the reaction between mixing streams of a pH indicator (phenolphthalein) and sodium hydroxide, both dissolved in an alcohol. Secondary flows were said to be responsible for stretching and folding the fluid elements increases rates of mixing. Sudarsan and Ugaz450 explored the use of compact spiralshaped flow geometries designed to achieve efficient mixing. They observed that, under appropriate conditions, transverse Dean flows are induced that augment diffusive transport and promote mixing in considerably shorter downstream distances as compared with conventional planar straight channel designs. Wua et al.451 carried out numerical simulations to investigate the refrigerant flow boiling in a horizontal serpentine round tube with the Eulerian multiphase flow model and a phase-change model for the mass-transfer and also conducted experiments to provide validation and data for the simulations. The phase distributions taking place at the bend sections showed competitive influence of buoyancy force and centrifugal force at different operating conditions. In particular, the simulation very well explained the bend effects on flow reconstruction and thermal nonequilibrium release observed in the experiments. A 3-D serpentine geometry with recurring “C-shaped” units has been demonstrated by Liu et al.426 The efficiency of this mixer increases with NRe due to the presence of eddies formed at the channel bends. Kim et al.452 demonstrated a mixer having triangular projections on the channel walls that induce stretching and folding of the fluid segments. A mixer with combined effects of splitting-and-recombining and advection has been realized in a micromixer composed of a series of “F-shaped” mixing elements.452 Other examples of mixing techniques that have recently been introduced involve self-circulation in a mixing chamber that is efficient at high flow rates (NRe g 50).453 Dean effects have been investigated to study mixing in serpentine-like microchannels.454-456 Unfortunately, in serpentine channels that are made up of opposing curved segments, this effect is reversed as the fluids flow from one bend to the other, and this cycle continues along the entire length of the channel. Consequently, the interface between the two fluids simply undulates between the channel walls without achieving appreciable mixing. One way of overcoming this flow reversal problem is to design a channel such that the transverse secondary flows are sustained over longer distances. Howell et al.457 have shown this to be possible in wide (>1 mm) spiral channels, and Vanka et al.458 have studied this effect in a 793 mm wide spiral channel at a flow rate corresponding to NRe ) 6.8. In such channels, the fluid experiences a reduction in the channel radius of curvature as it flows downstream, accompanied by a corresponding increase in the strength of then transverse secondary flow. Figure 14a presents the comparison of the Nusselt number versus the curvilinear coordinate for the straight, square-wave mixer, V-shaped, C-shaped, and B-shaped geometries at NRe ) 100.459 It can be seen from the figure that a periodic variation in Nusselt number takes place in chaotic geometries after the entry region. The mean value remains constant over the computational domain and is much greater than the square and straight channels. The presence of recirculation zones in these geometries contributes to a significant increase in the convective heat transfer over the straight channel. In a straight channel, the thermal mixing is done only by conduction. In the case of chaotic channels, the thermal mixing is done not only by conduction but also by the mixing produced by the recirculation

Figure 14. Comparison of (a) the Nusselt number and (b) the mixing rate between straight tube and different chaotic channels at NRe ) 100. (Source: Lasbet et al.459)

zones. Thus, mixing is more vigorous, when the Reynolds number increases and the mixing rate increases. Figure 14b shows that the mixing rate for C-shaped, V-shaped, and B-shaped geometries is higher than the square and straight channels. The flow in the C-shaped geometry is chaotic, explaining its great effectiveness in mixing. The B-shaped and V-shaped geometries present an interesting alternative to the C-shaped serpentine channels. Coiled flow inverter (CFI) configuration is one of the curved chaotic confirguration design. It works on the principle of complete flow inversion and centrifugal force. The geometrical configuration of CFI consists of 90° bends inserted in coils, with equal space before and after the bend. One such unit has several consecutive 90° bends and coils depending upon the number of bends involved. The details of the optimal configuration of the above device can be found in Nigam460 and Saxena and Nigam.461,462 The introduction of 90° bends in this device creates random mixing in the cross-sectional plane due to helical coils and complete flow inversion because of bending. The flow generated in this device, due to curvature of a stationary surface bounding the flow, changes direction continuously. This results in complex secondary flows in a plane normal to the principal flow direction. By shifting the plane of curvature from one bend to the next, one can induce a class of trajectories in one bend and then deform it to another type in the next bend, and so on. Very complex flow paths can be produced in this way, and a fluid element undergoing such flow follows a chaotic


Figure 15. Working principle of coiled flow inverter.

path. 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 and thus better mixing. Figure 15 displays the development of velocity contours in a curved tube and after the introduction of one and two bends. Kumar and Nigam212,273 numerically studied the threedimensional developing flows and heat transfer in helical tubes and bent coiled configuration (CFI). From the study it was observed that the Dean roll cells generated in the CFI are locally similar to a helical tube after fully developed flow. However, their effects on convective heat transfer between the tube wall and fluid are quite different. In the CFI, the Dean roll-cells smear temperature differences in the tube cross-section and, therefore, render it uniform. It was also observed that the bent coil configuration displays a heat-transfer enhancement of 20%30% in terms of the fully developed Nusselt numbers compared to the straight coil over a range of 25 e NRe e 1200 with little change in the pressure drop. Kumar et al.463 further experimentally reported that heat-transfer enhancement in CFI over a large range of Reynolds numbers (1000-16 000) using water in the tube side of the heat exchanger. The shell side fluids used were either cooling water (from cooling tower) or ambient air (from blower), and it was fitted with baffles to provide high turbulence and avoid channeling in the shell side. It was observed that the efficiency of the heat exchanger at low Reynolds number was near 1 and as the Reynolds number increases the efficiency decreases. It was also observed that at low Reynolds number the heat transfer was 25% higher and at higher Reynolds number it was 12% higher as compared to the coiled tube data reported in the literature. Mridha and Nigam464 numerically investigated turbulent forced convection in CFI for different fluids (air, water, kerosene, ethylene glycol). The gain in heat transfer in CFI for turbulent flow condition as compared to the straight tube for

the same flow rate and boundary conditions is 35%-45% while the increase in pressure drop is 29-30%. Figure 16 shows the performance of CFI for single-phase flow. The effectiveness of CFI can be assessed by the fact that even at a Dean number of 3 the value of dispersion number as low as 0.0013 is obtained under the condition of significant diffusion, and in the case of negligible diffusion, the value of dimensionless time at which the first element of tracer appears at the outlet is as high as 0.85 (see Figure 16a).462 Mridha and Nigam465 numerically investigated the scalar mixing of two miscible fluids in CFI for different process conditions (Dean number, Schmidt number, and number of bends) by using a scalar transport technique. They found significant increase in mixing performance of CFI as compared to regular helical coils at low Dean number. The mixing efficiency increased with the increase in Dean number and number of bends. It can be seen from Figures 16b and c that the friction factor and the Nusselt number are higher in the tube side of the CFI, as compared to the coiled tube and straight tube.463 Vashisth and Nigam466,467 experimentally investigated the pressure drop, void fraction, and flow regime prevalent for a gas-liquid two-phase flow system in CFI. They observed various flow patterns such as stratified, slug, plug, wavy, and churn flow. Figure 17a shows the comparison of two-phase friction factor with that of different geometries such as straight tube and coiled tube for single- and two-phase systems. It was observed that, due to the effect of secondary flow and flow inversion, the two-phase frictional pressure drop in CFI is 2.5-3 fold higher than that of a coiled tubes and straight tube, respectively. Empirical correlation for two-phase friction factor and gas void fraction in CFI was developed that separately

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Figure 16. Performance of coiled flow inverter for single-phase flow: (a) dispersion number versus Dean number; (b) friction factor versus Reynolds number; (c) inner heat-transfer coefficient versus Reynolds number.

accounts for the effect of curvature ratio, number of bends, and gas and liquid flow rates and also retains the identity of each phase. Vashisth and Nigam468 further performed step response experiments under the conditions of both negligible and significant molecular diffusion using gas-Newtonian and gasnon-Newtonian fluids. Reduction in axial dispersion in CFI occurs due to secondary flow and flow inversion with increase in Dean number and number of bends. An interesting observation was made that under identical process conditions 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 coiled tubes. A modified axial dispersion model was proposed to describe the liquid-phase RTD through CFI. In order to characterize the efficiency of the device, a criterion is proposed

that takes into account both the mixing characteristics and pressure drop in CFI (Figure 17b). Superiority of the proposed device has been established on the basis of its performance substantially closer to ideal plug flow, low initial and operating costs, compactness, and ease of fabrication. Correlations for single-phase and two-phase flow in CFI are presented in Table 17. 5. Applications of Coiled Tubes The wide range of applications of coiled tubes can be divided into three major categories: (a) mixing, (b) mass transfer, and (c) heat transfer. Table 18 summarizes the potential areas of applications where curved tube configurations may be applied in the process industries.


Figure 17. Performance of coiled flow inverter for single-phase flow: (a) friction factor versus the Reynolds number and (b) mixing performance.

5.1. Axial Dispersion and Mixing Applications. A reliable device for the routine determination of coal residence time in a long slender coiled tube reactor is in operation in the University of Utah process for coal liquefaction.469-471 The application of curved tubes in a chromatographic column has been reported by many workers.195-198,200,201,472,473 Matsuda et al.53 proposed that the toroidal coil centrifuge can be extremely useful for the analytical separation of various biologically active compounds, which tend to produce emulsification in the high-speed countercurrent chromatography system. An application of curved tubes as flow injection systems and postcolumn reactors has been reported by Leclerc et al.52 for a protein mixing process. Liu et al.426 used a series of short bends for mixing operations. Scho¨nfeld and Hardt474 proposed a micromixer based on curved configuration, in which the mixing time was in the range of milliseconds, thus lending toward the fast, mass-transfer-limited reactions. Jiang et al.475 and Vanka et al.476 proposed a chaotic mixer based on a curved tube for enhancing mixing performance. Coiled tubular reactors find extensive use in postcolumn HPLC reaction detectors as well. Reaction coils and coil holders are commercially available for this purpose.477 Mixers with alternating helices put in a linear conduit has also been shown to efficiently mix viscous fluids.478 Agrawal and Nigam322 proposed the applications of coiled tubes as chemical reactor for first-order chemical reactions. Such mixers are commercially

available. However, the minimum inner diameter of there mixers is 3 mm and this is too large to design a low-dispersion system. Low-dispersion reactors can reportedly be achieved by knitting479,480 or crocheting481 flexible tubing. Selavka et al.482 demonstrated a lower dispersion for the knitted designs when compared to the coiled reactors. Crocheted open tubular poly(tetrafluoroethylene) (PTFE) reactors have been used for photochemical postcolumn reactor applications.481 Crocheted reactors have not been widely used, due to the lack of suitable knitting machines or the necessary skill to do this manually. It has been shown that tubes in a serpentine configuration have superior radial transport properties and give less band broadening per unit length than helical OTRs of the same inner diameter. These “serpentine” OTRs are generally applicable.483 For the serpentine designs, generally a perforated board or a screen/net is used as the framework. This allows greater reproducibility and ease of fabrication. The serpentine reactors are generally made of flexible Teflon PTFE tubing, as it is easier to wind such tubing around the framework, allowing smaller bend radii. Even stainless steel capillary tubes have been wound through steel nets yielding serpentine designs with very low band dispersion.482 However, PTFE tubes find more numerous applications as OTRs due to their flexibility and inertness. 5.2. Mass-Transfer Applications. The coiled tubes are widely used for mass-transfer operations, e.g., membrane separation processes of reverse osmosis, ultra-microfiltration,

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Table 17. Correlations for Single- and Two-Phase Flow in CFI

gas permeation, and chromatography. The mass-transfer improvement is induced by complex hydrodynamic phenomena involving the centrifugal force. The efficiency of Dean vortices in several membrane applications such as water oxygenation,121,351 pervaporation,121,122,484 ultrafiltration, and microfiltration121 have been clearly demonstrated. The application of membrane processes for micro-, ultra-, and nanofiltration in a helically coiled and curved slit membrane system have been reported by Belfort and co-workers.115-119,143,351,352,354,356,373,374 Chung et al.115-117 developed a new spiral reverse osmosis system and quantified the concentration polarization. They showed that the presence of Dean vortices promotes mixing and inhibits the growth of the concentration polarization boundary layer. Ophoff et al.485,486 proposed a new design of helically twisted tubular membrane and reported significant flux improvement for the UF of a dextran solution. Wille et al.487 used single straight and meander-shaped tubular polypropylene membranes for the crossflow MF of polystyrene latex (0.005 wt %) and baker’s yeast suspension (0.3 wt %). Guigui et al.488 used the curved tube ultrafiltration membrane for water and wastewater treatment. Kuakuvi et al.206 and Ghogomu et al.207 used curved woven hollow fiber for UF membranes. Jose et al.489 presented an application of helical coiled tubes in the

rectification process of a single-stage ammonia-water absorption refrigeration system with partial condensation. Mohhammed and Muhammed213 showed that helically coiled tube can be used for the absorption of CO2 into liquid films of distilled water, ethyl alcohol, or ethylene glycol. Besides the application in separation and absorption processes, coiled tubes are also used in the pyrolysis of coal-related aromatic compounds (benzene and its derivatives).490,491 Hagedorn and Kargi204 used a coiled tube membrane reactor for the cultivation of mouse-mouse hybridoma cells for producing monoclonal antibodies as the cell. 5.3. Heat-Transfer Applications. The curved tubes are used in industry either for thermal homogenization or for pure heat transfer in heat exchangers. In undisturbed laminar flow in an empty pipe, thermal diffusion is the only mechanism for heat transfer in the radial direction. A great variety of curved tube configurations have been used to promote radial flow and thus to reduce radial temperature gradients in process fluids. Mori and Nakayama,94 Akiyama and Cheng,256,257 and Kalb and Seader240,241 reported applications of curved tubes to enhance thermal homogenization. Curved tubes are used for thermal homogenizations of polymer flow in practically all equipment used for manufacturing and processing plastics.314

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3325 Table 18. Industrial Processes for the Potential Applications for Curved Tubes process

applications

polymerization

Blending of immiscible polymers (polyamide-6 and low-density polyethylene) Thermal homogenization for film blowing or sheet extrusion (polyethylene, polypropylene, polystyrene, and ABS resins processing) Exothermic polymerization processes Mixing with reaction (polystyrene process, styrene polymerization, polymerization of methyl methacrylate, production of polymerization of methyl methacrylate)

process glues

Epoxy resins

food industry

Mixing of acids, juices, oils, beverages, milk drinks, or sauces in food formulations and melted chocolate Starch slurry cooking Heating and cooling of sugar solutions

mixing of gases

Pre-reactor gas mixing (nitric acid production) Chemical reactions (production of vinyl chloride, ethylene dichloride, styrene, xylene, and maleic anhydride)

environmental applications

Reducing NO emission in combustors Sampling and analysis of contaminants in air in nuclear industry

waste water treatment

Dispersion of flocculating agent Ultrafiltration, ultraflocculation, and turbulent microflotation, disinfection, and dechlorination Dewatering of sludge Sterilization and water treatment with ozone

mixing with reaction pulp and paper

Lactase treatment of whole whey, phenol alkylation

petrochemical

Cracking of heavy and crude oils Amine washing, caustic washing, water washing of organics; extraction of H2S from petroleum fractions using diethanolamine Blending of multicomponent drugs Dispersion of oils Sterilization pH control

extraction

Fractionation of lipids in order to separate squalene from triglycerides and diacylglyceryethers Caffeine from supercritical CO2 with water Kerosene-water Butanol-succinic acid-water Toluene-acetone-water Carbon tetrachloride-propionic acid-water Copper Indium

emulsification

Water-kerosene Water-carbon tetrachloride Concentrated emulsions of epoxy resins Microencapsulation processes

absorption

H2S from natural gas using NaOH solutions, amines, or proprietary solvents CO2 using amines or proprietary solvents Dehydrate gases with glycols Purifying the effluent from an oxychlorination reactor with a NaOH solution in ethylene dichloride production

scrubbing

NH3, HCl, HF, or cyanides with water Chlorine gas and acid gases with NaOH solutions or solvents; noxious organic compounds with various solvents Continuous hydrogenation of vegetable oils

Production of starch ethers Liquefaction of starch Pulp bleaching Stock dilution and consistency control pH control

pharmaceuticals

Production of Amiodarone

refrigeration and cryogenics

Rectifying circuit with mixed refrigerants

The applications of curved tubes in the food processing71-73,492 industry for heating and cooling of highly viscous liquid food, such as pastes, or for products that are sensitive to high shear stresses are very beneficial. Due to high heat transfer, coiled tubes are used in steam power plants; the water to be heated

and boiled into steam is pumped through pipes that are exposed to the hot gases formed in the furnace section of the steam generator. The cooling water circulating through the coiled pipes is heated as the low-pressure steam leaving the turbine is condensed on the outer surface of these pipes. In home heating systems, hot water flows through curved pipes, transferring heat to the living space. Georgiev and Kovatchev56 developed a new type of multichannel low-temperature heat exchanger based on helical coiled tubes, which is mainly designed for liquefiers and refrigerators with small and average capacity. The application of coiled tube heat exchangers have been reported by Sheft et al.493 in the Argonne hydrogen conversion and storage system (HYCSOS), which is a two-hydride concept that operates as a chemical heat pump for the storage and recovery of thermal energy for heating, cooling, and energy conversion. Prasad et al.57 tested the performance of the coiled tube heat exchanger as a waste heat recovery device for a 60 HP gas turbine and found excellent corroboration of the effectiveness-NTU relation between their simulation results with in situ experiments. A typical application of coiled pipe heat exchanger was the extraction of heat from the hot water storage tank due to the high surface area and compact structure.58-61 The application of U-tube and helically coiled tube heat exchanger was explored exemplarily for the process-heat reactor AHTR 500 with central graphite column by a thermohydraulic simulation of a secondary cooler circuit, which is thermally connected with the primary circuit.62 Chauvet et al.63 demonstrated the application of coiled tube in the traditional domestic hot water tank for extracting heat from the tank. Inagaki et al.64 used helically coiled tubes of an intermediate heat exchanger for the HTTR, using a full-size partial model and air as the fluid. The heat exchanger was composed of 54 helically coiled tubes separated into three-layer bundles, surrounding the center pipe. Acharya et al.,67 Prabhanjan,69,70 Naphon and Wongwises,6,74,75 Gupta et al.,494 Guobing and Yufeng,495 and Naphon78 have shown the application of coiled heat exchangers in various industrial applications (Figure 18). Horizontal helically coiled tubing once-through steam generator is favored for space, navigation, and other specific techniques because of its lower gravitation center and higher efficiency both in heat transfer and in steam generation.394 In IRIS design the steam generators, besides steam production, are required to drive the natural circulating, emergency heat removal systems, which provide both the main post-LOCA (loss of coolant accident) depressurization and the reactor core emergency cooling function. Helically coiled tubes are used as once-through steam generators of the IRIS nuclear reactor project.496-498 In industrial applications, for example, the evaporators in refrigeration, forced convective evaporation is the dominant process and high heat-transfer efficiency can be obtained under smaller temperature difference between wall and liquid and compact geometries. Therefore, helical coiled tube finds a very goodapplicationintherefrigerationandcryogenicsprocesses.76,77,397,499-502 Fusion technology requirements gave rise to the recent applications of curved tubes for subcooled flow boiling critical heat flux. Celata et al.503 proposed that the curved tubes will be a very useful device for the heat removal from components such as divertors, plasma limiters, neutral beam calorimeters, ion dump, and first-wall arm or that arc subjected to very high heat loads. Takahashi and Momozaki179 demonstrated the application of a helical heat exchanger in the fusion reactor. Jo and Jhung505 reported that the helically coiled tubes are also used in the nuclear reactors as once-through steam generator, where the

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Figure 18. (a) U-tube heat exchanger,554 (b) U-tube double-pipe heat exchanger,555 and (c) coiled tube reactor.556

tubes are subjected to liquid cross-flow externally and multiphase flow internally for heat removal from the reactor. Coiled tubes are also used as the receiver for a solar energy concentrator, the Crosbyton (Texas) System.231,504-507 5.4. Application of Curved Geometries in Microdevices. Microreactors belong to the most predominant equipment in the developing area of microprocess engineering. The short diffusion length between the components in microstructures results in a short mixing time, which has been shown by various applications in analytical equipment508 and for laboratory applications.475 Most of the microreactors studied today are running in continuous operation under laminar flow conditions.509 Recently, Ookawara et al.,510-512 Scho¨nfeld and Hardt,474 and Yamaguchi et al.456 showed that the curved microchannels can be used in the separation processes. A successful microreactor, i.e., providing a complete reaction with a high selectivity and product yield, is based on an efficient mixing process. More specifically, the time scale of the mixing process in relation to the time scale of the reaction plays the major role, as the mixing process must be finished before the reaction is completed. Jiang et al.475 used a curved microtube in the mixing application and reported that the curved microchannel shows more efficiency as compared to the conventional microchannels. Table 19 shows the applications of curved channels as microreactors.

Table 19. Potential Applications of Chaotic and Serpentine Geometry application

authors

heat transfer parallel lamination mixers

Choi and Anand,513 Lasbet et al.459 Wolfgang et al.,514 Hongmiao,515 Nielsen et al.,516 Karp and Covington,517 Koch et al.,518 Erbacher et al.,519 Ehrfeld et al.,520 Haverkamp et al.,521 Bessoth et al.,522 Mitchell et al.,523 Xiang et al.,524 He et al.,525 Sudarsan and Ugaz,450 Kim et al.,527Lin et al.,526 Kang et al.,528 Wu et al.529 Santiago et al.,530 Branebjerg et al.,531 Liu et al.,426 Beebe et al.,532 Vijayendran et al.,533 Stroock et al.,436 Therriault et al.454 Miyake et al.,534 Bo¨hm et al.535 Volpert et al.,536 Deshmukh et al.,537 Lee et al.,538 Niu and Lee,539 Glasgow and Aubry540 Choi et al.,541 Oddy et al.,542 Ajdari,543 Lee et al.538 Moroney et al.,544 Woias et al.,545 Zhu and Kim,546 Vivek et al.,547 Rife et al.,548 Yasuda,549 Yang et al.,550,551 Liu et al.,552 Jagannathan et al.553

serial lamination mixers laminar jets pulsed junction electrokinetic acoustic vibration

6. Summary and Conclusions Potential process applications of curved tubes for various operations involving mixing and heat- and mass-transfer with notably compact size and improved performance are attractive alternatives to conventional devices. The role of curved tubes is expected to gain significant importance in times to come due


to their advantages such enhanced cross-sectional mixing, reduction in axial dispersion, higher heat- and mass-transfer coefficient, and compactness. The present review was divided into three groups according to the geometrical configurations. The following conclusions can be summarized. (1) The above survey on curVed tubes indicated that numerous experimental and theoretical studies have been carried to understand the flow behavior, mixing, and heat- and masstransfer performances. The outcome of these investigations on curved tubes is reasonably documented for single- and twophase flow systems. (2) Heat- and mass-transfer in coiled tubes can further be enhanced by inserting some perturbation in the geometry. The performance of these chaotic configurations, which consists of combinations of coils and bends, has also been reviewed. In the case of chaotic and serpentine geometries, an extensive work on hydrodynamics is available, but the heat-transfer characteristics need to be further explored. There is a need for generalized correlations for friction factor and Nusselt number, which takes into account the change in geometrical configurations. (3) A new class of chaotic configuration, coiled flow inverter is presented and discussed. It is an innovative device and needs to be further explored extensively for both single- and twophase fluid flow and heat transfer. (4) Although a significant amount of research has been performed on the hydrodynamics and heat transfer for singlephase flow in coiled tubes, the investigations on two-phase flow in coiled tubes and complex geometries still needs to be explored. Several researchers attempted successfully to develop numerical methods for liquid-liquid and gas-solid systems and verified their results with the known experimental results. There is little information on local variables like flow profiles, interfacial phenomena, local shear stress, phase distribution, and entry length development, which needs attention. (5) Keeping in view the increasing emphasis on the effectiveness of curved tubes and chaotic configurations, their potential applications in process industry is ever-increasing. Nomenclature A ) cross-section area, m2 a ) tube radius, m b ) coil pitch, [m] Cp ) specific heat, kJ/(kg‚K) d ) tube diameter, m db ) baffle diameter, m D ) coil diameter, m DAB ) molecular diffusion coefficient, m2/s De ) equivalent diameter, m DS ) shell diameter, m f ) Fanning friction factor for coil () ∆Pc/2FV2‚d/Lt), (-) Ghrc ) geometrical number for regular helical coil k ) thermal conductivity, W/(m‚K) Lt ) length of heat exchanger, m l ) arc length along the centerline of the curved tube, m H ) pitch of coil, m h ) heat-transfer coefficient, W/m2‚K m˘ ) mass flow rate, kg/h n ) power law index, (-) p ) pressure, N/m2 q′′ ) heat flux, W/m2 q ) heat-transfer rate, W Q ) volumetric flow rate, L/h r ) tube radius, m

R ) coiled tube curvature, m Rc ) radius of the coil, m Rmin ) minimum radii of curvature of the beginning of the spiral, m Rmax ) maximum radii of curvature of the end of the spiral, m rh ) hydraulic radius, m T )temperature, K U ) overall heat-transfer coefficient, W/m2‚°C u0, V ) inlet velocity, m/s Greek Letters δ ) inverse of curvature ratio () d/D), (-) λ ) curvature ratio () D/d), (-) µ ) viscosity of the fluid, kg/m‚s F ) density of the fluid, kg/m3 Φ ) wall flux, W/m2 τ ) torsion () p/(R2 + p2)), (-) Dimensionless Numbers NDe ) Dean number () NRe/xλ), (-) NEu ) Euler number () d0.85De0.85/L), (-) NHe Helical number () NRe[(d/D)/1 + (p/πD)2 1/2]), (-) NPr ) Prandtl number () Cpµ/k), (-) NSt ) Stanton number () h/(CpFV), (-) NRe ) Reynolds number () FVd/µ), (-) NRe,crit ) critical Reynolds number, (-) NPr ) Prandtl number () Cp/µk), (-) NPr,w ) turbulent wall Prandtl number, (-) NNu ) Nusselt number () hd/k), (-) NRa ) Rayleigh number () gβ/VR(Ts - T∞)Lt), (-) NSc ) Schmidt number () µ/(FDAB)), (-) NTn ) torsion number () 2τRe), (-) Subscripts and Superscripts 0 ) inlet condition 1 ) first bend 2 ) second bend b ) bulk quantity c ) coiled tube m ) mean value cp ) value with constant property s ) straight tube inner ) inner tube outer ) outer tube i ) inside/inner o ) outside/outer w ) wall condition + ) standard wall coordinates. θ ) local quantity Literature Cited (1) Berger, S. A.; Talbot, L.; Yao, L. S. Flow in curved pipes. Annu. ReV. Fluid Mech. 1983, 15, 461. (2) Nandakumar, K.; Masliyah, J. H. Swirling flow and heat transfer in coiled and twisted pipes. In AdVances in Transport Process, 4; Mujumdar, A. S., Masliyah, R. A., Eds.; Wiley Eastern: New York, 1987. (3) Shah, R. K.; Joshi, S. D. Convective heat transfer in curved ducts, In Handbook of Single-Phase ConVectiVe Heat Transfer; Kakac, S., Shah, R. K., Aung, W., Eds.; Wiley: New York, 1987; Chapter 5, pp 5.3-5.47 and 3.1-3.147. (4) Saxena, A. K.; Nigam, K. D. P. Residence time distribution in straight and cured tubes. Encylopedia of Fluid Mechechanics 1; Cheremishinoff, N. P., Ed.; Gulf Publishing Co.: Houston, TX, 1986; p 675. (5) Young, M. A.; Bell, K. J. Review of two-phase flow and heat transfer phenomena in helically coiled tubes. Am. Inst. Phys. 1991, 1214.

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ReceiVed for reView December 25, 2007 ReVised manuscript receiVed March 25, 2008 Accepted March 26, 2008 IE701760H

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