Fluid Flow and Heat Transfer in Curved Tubes with Temperature

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Fluid Flow and Heat Transfer in Curved Tubes with Temperature-Dependent Properties Vimal Kumar, Pooja Gupta, and K. D. P. Nigam* Department of Chemical Engineering, Indian Institute of Technology, Delhi-110016, India

Helically coiled tubes find applications in various industrial processes like solar collectors, combustion systems, heat exchangers, and distillation processes because of their simple and effective means of enhancement in heat and mass transfer. Though extensive work is available in the literature on curved tubes, no study is available considering the variation in thermo-physical properties of fluids (density, viscosity, thermal conductivity, and specific heat) with temperature. In the present work, the effect of temperature dependence of fluid properties is examined on both hydrodynamic and thermal performance of the curved tube having finite curvature and pitch, under cooling and heating conditions. The range of Reynolds number studied in the present work is varied from 100 to 400 using water and diethylene glycol as two different fluids. The secondary flow induced due to centrifugal force distorts the velocity and temperature profiles when the effect of temperature-dependent properties is taken into account. The friction factor obtained with variable property assumption under cooling is higher as compared to the constant property results. This is due to the increase in the value of viscosity near the wall of the curved tube that reduces the effect of the secondary flow. The Nusselt number also shows a marked dependence on the properties variation in the coil tube cross section. A new model is also developed in the present study based on the property-ratio technique for both friction factor and heat transfer. The study provides understanding of the fundamentals of energy transportation in curved ducts and will be very helpful in designing coiled tube heat exchangers with temperature-dependent properties. 1. Introduction

n ) -0.14, m ) +0.58 for µw/µm < 1 (heating)

Flow and heat transfer at constant thermo-physical properties in various configurations have been a topic of important fundamental engineering interests during the past decades. Since the thermo-physical properties of most fluids are temperaturedependent, the current practice is to use the constant property solutions for heat transfer problems. In the literature,1-3 the effect of temperature-dependent properties is analyzed by using two techniques: (i) property ratio method and (ii) reference temperature method. These techniques were proposed either for pipes or boundary layer flow problems. Shah and London1 reported the temperature-dependent results in a straight tube of circular cross section for gases and liquids separately, because they involve different property ratio corrections. Harms et al.4 reviewed the temperature-dependent behavior of viscous liquids in a straight tube using different methods and reported that these techniques provide reasonable predictions for many fluids but they are not valid over a large range of temperature and different configurations. Kakac5 reported a correction for the temperaturedependent viscosity effect on Nusselt number and friction factor for a straight tube under laminar flow conditions. The corrections reported by Kakac5 for liquid are

n ) -0.14, m ) +0.50 for µw/µm > 1 (cooling)

() ()

µw Nu St ) ) Nucp Stcp µm f fcp

)

µw µm

Liu and Lee6 numerically studied the characteristics of fluid flow and heat transfer in microchannel heat sinks. They reported the effect of temperature-dependent thermo-physical properties of the fluid on the local temperature, heat flux, and Nusselt number distributions. In the curved tubes, the vortices are generated because of the centrifugal force, which generates a secondary flow field with a circulatory motion pushing the fluid particles toward the outer wall of the curved tube. Therefore, the heat and mass transfer in coiled tubes are higher as compared to the straight tubes in the laminar flow region. Dean7 was the first to report the analytical expression for flow fields in a curved tube, considering constant properties. The extensive reviews of fluid

n

(1)

m

(2)

where the values of exponents n and m for laminar flow through a circular straight tube are * Corresponding author. Tel.: +91-11-26591020. Fax: 91-1126591020. E-mail: [email protected], [email protected].

Figure 1. Schematic geometry and coordinates of the coiled tube.

10.1021/ie0608399 CCC: $37.00 © 2007 American Chemical Society Published on Web 02/09/2007

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3227 Table 1. Coefficients for Temperature-Dependent Properties (Density, Viscosity, Specific Heat, and Thermal Conductivity) for Water properties

A0

A1

A2

A3

A4

A5

A6

µ (kg/m‚s) F (kg/m3) Cp (J/kg‚K) k (W/m‚K)

2.3750 E-16 -8.0964 E-12 2.1684 E-19 -2.6470 E-23

-6.9716 E-13 2.1247 E-08 2.5986 E-05 5.6027 E-20

8.4513 E-10 -2.2889 E-05 -1.0256 E-02 -5.5391 E-06

-5.4171 E-07 1.2948 E-02 -1.9417 E-01 4.6120 E-03

1.9374 E-04 -4.0547 4.4738 E+03 -2.7580 E-01

-3.6701 E-02 6.6544 E+02

2.8838 -4.3598 E+04

flow and heat transfer in helical pipes were reported by Berger et al.8 and Shah and Joshi.9 Bergles10 reported the effects of temperature-dependent viscosity in the curved tube. However, they did not consider the effect of curvature and pitch in their study, and the study was also limited to the heating case only. Andrade and Zaparoli11 studied the fully developed laminar flow in a curved duct under heating and cooling conditions with temperature-dependent viscosity while keeping other properties (density, specific heat, and thermal conductivity) constant for the case of water. They reported that the Nusselt number and the friction factor results show a marked dependence on the viscosity variations in the coil tube cross section. They did not consider the complex phenomena of hydrodynamic and thermal development at various cross sections and Reynolds numbers. They have also not considered the effects of pitch, curvature ratio, and variation of all thermo-physical properties in their study. Though extensive work is reported in the literature for fluid flow and heat transfer in a curved tube, there is still no work reported, to the best of our knowledge, that takes care all the thermo-physical properties variation of viscous fluids with temperature as well as the effects of curvature and pitch. The purpose of this work is to analyze the influence of temperaturedependent thermo-physical properties on fully developed laminar forced convection considering the finite values of curvature and pitch in coil tube. The hydrodynamics and thermal profiles for the flow of water in the coiled tube are also presented for the first time under both heating and cooling conditions for temperature-dependent properties. Friction factor and heat transfer are also predicted for a highly viscous fluid (diethylene glycol) considering temperature-dependent properties. A new model is developed for a curved tube for both heating and cooling conditions. 2. Mathematical Model The geometry considered and the systems of the coordinates are illustrated in Figure 1. The circular pipe has a diameter of

2a and is coiled at a radius Rc, while the distance between the two turns (the pitch) is reported by H. In the present study, the Cartesian coordinate system (x, y, z) is used to represent a coiled tube for the purpose of numerical simulations. The laminar flow and heat transfer develop simultaneously downstream in the helical pipe. The flow is considered to be steady. The temperature-dependent properties (viscosity, density, thermal conductivity, and specific heat) for water are calculated using polynomial functions as follows,

µ(T) ) A0 + A1T + A2T2 + A3T3 + A4T4 + A5T5 + A6T6 (3a) F(T) ) A0 + A1T + A2T2 + A3T3 + A4T4 + A5T5 + A6T6 (3b) k(T) ) A0 + A1T + A2T2 + A3T3 + A4T4

(3c)

Cp(T) ) A0 + A1T + A2T2 + A3T3 + A4T4

(3d)

where T is the temperature and An (n ) 1, 2, 3, 4, 5, and 6) are coefficients determined by a polynomial fitting over the water thermo-physical property data.12 Table 1 shows the values of the coefficients of water thermo-physical properties distribution as a function of temperature resulting from eqs 3a-3d. The differential equations governing the three-dimensional laminar flow in the coiled tube could be written in the master Cartesian coordinate system as

Continuity: ∂ui )0 ∂xi

Figure 2. Heat transfer comparison in present predictions with the results of Andrade and Zaparoli11 under (i) heating and (ii) cooling conditions.

(4)

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Momentum:

[(

)

parameters are used,

]

∂ui ∂uj ∂ µ + - Fujui - δijp + Fgi ) 0 ∂xj ∂xj ∂xi Energy:

[(

)

]

∂T ∂ k - FujCpT + µΦv ) 0 ∂xj ∂xj

(5)

(6)

NRe )

NNu,θ )

q w di

Fu0di µ

, NNu,m )

k(Tw - Tb)

1 usA

(

)

∂ui ∂ui ∂uj 2 ∂ui + - µ δ ∂xj ∂xj ∂xi 3 ∂uj ij

(7)

No-slip boundary condition, ui ) 0, and constant temperature, Tw, are imposed on the wall. At the inlet, the fully developed duct flow velocity profiles and a fixed pressure at the outlet of the coiled tube were employed. The diffusion flux at the outlet for all variables in the exit direction is set to zero. To represent the results and characterize the heat transfer in a coiled tube, the following nondimensional variables and

fθ )

τw 1 FU 2 2 0

, fm )

(8)

∫0A NNu,θ dθ, Tb )

where µΦv is the viscous heating term in the energy equation and Φv is represented by

Φv )

D di

λ)

1 2π

(

1

∫0AusT dA

usA

∫02π fθ dθ

)

(9a)

(9b)

where λ is the curvature ratio and Tb is the bulk temperature; fθ and NNu,θ are the local friction factor and Nusselt number along the circumference of the pipe, respectively; fm and NNu,m are the circumference average friction factor and Nusselt number, respectively; u0 denotes the velocity at the inlet of the tube; and qw denotes the heat flux on the wall.

Figure 3. Velocity fields for temperature-dependent properties in curved tube at Reynolds number of 400 for (i) heating condition, (ii) cooling condition, and (iii) constant properties (heating condition) at (a) φ ) 15°, (b) φ ) 30°, (c) φ ) 60°, and (d) φ ) 180°.

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Figure 4. Velocity fields for temperature-dependent properties in curved tube at (i) heating condition, (ii) cooling condition, and (iii) constant properties (heating condition) at (a) Re ) 100, (b) Re ) 200, (c) Re ) 300, and (d) Re ) 400.

Figure 5. Velocity fields for heating and cooling conditions in curved tube for temperature-dependent properties at (i) horizontal centerline and (ii) vertical centerline.

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Figure 6. Velocity fields for heating and cooling conditions in curved tube for constant properties at (i) horizontal centerline and (ii) vertical centerline.

Figure 7. Temperature fields for temperature-dependent properties in curved tube for Reynolds number of 400 for (i) heating condition, (ii) cooling condition, and (iii) constant properties (heating condition) at (a) φ ) 15°, (b) φ ) 30°, (c) φ ) 60°, and (d) φ ) 180°.

In order to predict the performance of the coiled heat exchanger under constant properties and temperature-dependent properties, the governing equations for hydrodynamics and heat transfer were solved in the master Cartesian coordinate system

with a control volume finite difference method (CVFDM). Considering the large difference of thermo-physical properties between the fluid region and the solid region, the doubleprecision solver is used. The numerical computation is consid-

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Figure 8. Temperature fields for temperature-dependent properties in curved tube at (i) heating condition, (ii) cooling condition, and (iii) constant properties (heating condition) at (a) Re ) 100, (b) Re ) 200, (c) Re ) 300, and (d) Re ) 400.

ered converged when the residual summed over all the computational nodes at the nth iteration was e10-8. The methodology for the numerical simulations and the standardization of its accuracy can be found in Kumar and Nigam.13 The Nusselt number and friction factor values obtained with temperature-dependent viscosity under heating and cooling conditions for water in the present study were compared with the predictions of Andrade and Zaparoli11 while keeping other properties constant. It can be seen from Figure 2 that the present predictions were found to be in good agreement with the results of Andrade and Zaparoli.11 3. Results and Discussion The numerical simulations with constant and variable properties assumptions were carried out for the Reynolds number in the range of 100-400. The temperature considered on the wall was 350 K. For the constant-properties case, the physical properties were calculated at the mean temperature of tube inlet and tube wall. The hydrodynamics and thermal fields were reported for temperature-dependent properties under heating and cooling conditions. In the case of temperature-dependent properties, all the thermo-physical properties (namely, density, viscosity, thermal conductivity, and specific heat) were varied with

temperature. In order to explain the physics of fluid flow and heat transfer in a curved tube under temperature-dependent properties, the study was also carried out for constant properties for better understanding and comparison of the present predictions. It was observed that, under constant properties, the velocity and thermal profiles were not affected by heating and cooling conditions, respectively. 3.1. Velocity Fields. Velocity profiles for temperaturedependent properties are shown in Figure 3 under heating and cooling conditions and for constant properties, respectively. The velocity fields are presented at various axial locations (φ ) 15°, 30°, 60°, and 180°) at Re ) 400. In the case of temperaturedependent properties (parts i and ii of Figure 3), the velocity fields get twisted toward the bottom wall for the heating condition and toward the top for the cooling condition, but in the case of constant properties (Figure 3iii), the velocity profiles move toward the outer wall. Figure 3 shows that, under temperature-dependent properties, the velocity profiles are asymmetric because of the interaction of centrifugal forces and the variation in the properties. From Figure 3, it can also be seen that the flow is not fully developed at φ ) 180° for the temperature-dependent properties, while in the case of constant properties (Figure 3iii), the flow is fully developed at φ ) 180°

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Figure 9. Temperature fields for heating and cooling conditions in curved tube for temperature-dependent properties at (i) horizontal centerline and (ii) vertical centerline.

for Re ) 400. Fully developed velocity fields (φ ) 270°) are also reported at different Reynolds numbers for temperaturedependent properties (parts i and ii of Figure 4) and constant properties (Figure 4iii) at Reynolds numbers ranging from 100 to 400. It can be seen from Figure 4 that the maximum velocity is shifted toward the outer wall from the top wall in the heating case, while in the case of cooling conditions, the maximum velocity shifted toward the outer wall from the bottom wall with the increases in the Reynolds number of the curved tube. Figure 4iii shows that, for constant properties, the velocity fields move toward the outer wall as the Reynolds number increases. It can also be seen from Figure 4 that the velocity fields are not symmetric, which may be due to the pitch in the curved tube. Figures 5 and 6 show the velocity profiles at horizontal and vertical centerlines for temperature-dependent properties and constant properties, respectively, at different Reynolds numbers at φ ) 270°. In Figures 5 and 6, the solid line represents velocity profiles at heating conditions, while the dashed line represents velocity profiles at cooling conditions. Figure 5 shows the same phenomenon as observed in Figure 3: the velocity profiles at horizontal and vertical centerlines are asymmetrical because of the imbalance between centrifugal forces and main flow due to thermo-physical properties variation. It can be seen from Figure 6 that there is no effect of heating and cooling conditions on the velocity profiles in the curved tube at constant properties. 3.2. Temperature Fields. Figure 7 shows the thermal fields at various axial locations for a constant value of Reynolds number equal to 400. The patterns of axial temperature development at various axial distances are clearly related to the fluid mechanics of the system. The maximum thermal fields were toward the top wall for the heating condition and toward the bottom wall for the cooling condition at various locations in the curved tube. In the case of constant properties with the increase of φ, the secondary velocity is enhanced, and the temperature zones shift to the outer side because of centrifugal force (Figure 7iii). It can also be seen from Figure 7iii that, for a very short distance from the tube inlet, the secondary flow effect is negligible so the temperature profiles in the curved tube are similar to that in the straight tube except for the effect of skewed axial velocity profiles, which is to set up a circumferential temperature gradient. The thermal fields are also reported at various Reynolds number for temperature-dependent and constant properties ranging from 100 to 400 in Figure 8 at φ ) 270°. It can be seen from Figure 8 for temperature-dependent properties that

Figure 10. Temperature fields for heating and cooling conditions in curved tube for constant properties at (i) horizontal centerline and (ii) vertical centerline.

the maximum temperature values shifted toward the top for the heating condition and toward the bottom for the cooling condition, while for the constant-properties case (Figure 8iii), the thermal field becomes more skewed toward the outer wall of the curved tube with the increase in the value of Reynolds number. Figure 8 also show that the thermal fields are perpendicular for temperature-dependent properties as compared to the thermal fields in constant properties (Figure 8iii). Figures 9 and 10 show the thermal profiles at horizontal centerline and vertical centerline for temperature-dependent and constant properties, respectively, at φ ) 270° under heating and cooling conditions. The figures show that the temperature profiles below the temperature 350 K were for the heating condition and those above 350 K were for the cooling condition. Figure 9 shows that the thermal fields are asymmetrical both in horizontal and vertical centerline. It can be seen from Figure 10 that the thermal profiles move toward the wall at horizontal centerline as the Reynolds number increases. The unbalanced centrifugal force of the main flow results in the shift of the point of the maximum temperature to the outside of the pipe, forming a steep thermal gradient near the outer wall. 3.3. Thermo-physical Property Distribution. Figure 11 shows the variation of density and viscosity profiles. It can be seen from Figure 11a that, under the heating condition, a minimum value in the temperature profile (Figure 9i) is linked with the maximum in the viscosity and density distributions (Figure 11a(i)). Similarly, under the cooling condition, the temperature exhibits a maximum (Figure 9i) that is related with

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Figure 11. Properties distribution profiles under heating and cooling conditions in curved tube for temperature-dependent properties: (a) density profiles and (b) viscosity profiles.

a minimum in the viscosity and density distributions. The effects of thermal conductivity and specific heat on velocity and thermal profiles were negligible. 3.4. Flow Resistance. In the present work, the friction factor is calculated using the value of wall shear stress for different Reynolds numbers at constant and temperature-dependent properties. Figure 12 shows the variation of friction factor under both heating and cooling of water in a curved tube for constant and temperature-dependent properties. Figure 12a shows that, under the heating condition, the friction factor data for temperature-dependent properties are lower than the constantproperties data. This may be due to the fact that the bulk mean temperature is greater in the case of temperature-dependent properties as compared to the constant-properties case. Therefore, the values of thermo-physical properties are lower in the case of temperature-dependent properties, which reduces the friction factor in comparison with the constant-property case. The opposite phenomenon was observed for cooling conditions; the friction factor value with temperature-dependent properties is greater than that in the constant-properties case, as shown in Figure 12b. 3.5. Heat Transfer Variation for Temperature-Dependent Properties. Figure 13 shows the Nusselt number as a function

of Reynolds number for heating and cooling conditions. It can be seen from parts a and b of Figure 13 that, for constant properties, the heat transfer is higher in the heating than the cooling condition. On comparing heat transfer between the constant and temperature-dependent properties under the cooling condition, the higher heat transfer was observed for the constantproperties case. When the fluid is cooled considering temperature-dependent properties, the bulk mean temperature is lower as compared to the constant-property case. Therefore, the thermo-physical properties values increase in the curved tube section, which reduces the secondary flow effect and, therefore, the heat transfer rate. 3.6. Model Development. It was observed from the velocity and thermal development profiles that viscosity is the property of importance that varies strongly with temperature. Though density also affects the velocity and thermal distributions, its contribution is small as compared to the viscosity. In the present study, the Nusselt number and friction factor were computed for both constant properties and temperature-dependent properties under heating and cooling conditions. The Nusselt number and friction factor results for temperature-dependent properties were then fitted using the property-ratio method, and the exponents n and m obtained in the present study are n ) -0.2

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Figure 12. Variation of friction factor with Reynolds Number for temperature-dependent properties for water and DEG under (a) heating and (b) cooling conditions.

and m ) +0.52, for heating and cooling conditions, respectively. Therefore, eqs 1 and 2 become

() ()

µw Nu St ) ) Nucp Stcp µm f fcp

)

µw µm

-0.2

for heating and cooling

(10)

0.52

for heating and cooling

(11)

The exponent values obtained in the present study are higher than the values reported by Kakac.5 This may be due to the following two reasons: (i) the simulations were carried out in a curved tube, where secondary flow plays a dominant role, and (ii) simulations were carried out under temperaturedependent thermo-physical properties. Figure 12 also shows the friction factor values obtained from the property ratio method using eq 11 with the constantproperties results from the present study. It can be seen from Figure 12a that the friction factor results obtained from eq 11 and the temperature-dependent properties are in good agreement for the cooling condition. However, the differences are more significant in the heating condition, for the predictions from the fitted model (eq 11) and the temperature-dependent properties results. For the higher Reynolds number values, the fitted

Figure 13. Variation of Nusselt number with Reynolds number for temperature-dependent properties for water and DEG under (a) heating and (b) cooling conditions.

model leads to an underestimation of the friction factor values from the temperature-dependent properties. Similarly, the Nusselt number values from the temperaturedependent properties and the fitted model (eq 10) are also reported in Figure 13 for heating and cooling conditions, respectively. Under both heating and cooling conditions, the fitted model exhibits a good agreement with the curved tube results for lower Reynolds number. As the Reynolds number increases, the fitted model provides higher Nusselt number values in comparison with the temperature-dependent properties results. Under the cooling condition, the fitted model values are in good agreement with the temperature-dependent properties results. 3.7. Temperature-Dependent Properties for Highly Viscous Fluid. The temperature-dependent properties study was also carried out for a highly viscous fluid, diethylene glycol (DEG), and the friction factor and heat transfer results were fitted with eqs 10 and 11. The temperature-dependent properties (viscosity, density thermal conductivity, and specific heat) for DEG are reported in Table 2. The friction factor and heat transfer studies for DEG were carried out for both heating and cooling conditions for the Reynolds number ranging from 100-400. Though all the properties for DEG were varied during the simulations, the viscosity is the property of importance, and it affects both velocity and thermal profiles. Figures 12 and 13 also show the friction factor and heat transfer predictions for

Table 2. Coefficients for Temperature-Dependent Properties (Density, Viscosity, Specific Heat, and Thermal Conductivity) for Diethylene Glycol properties

A0

A1

A2

A3

A4

A5

A6

µ (kg/m‚s) F (kg/m3) k (W/m‚K) Cp (J/kg‚K)

1.6855 E-12 -3.0339 E-15 2.20759 E-13 1.73472 E-17

-3.4258 E-09 4.4209 E-12 -1.92843 E-10 1.76220 E-05

2.8934 E-06 -3.2163 E-09 1.13917 E-07 -1.83213 E-02

-1.2999 E-03 1.0574 E-06 2.22313 E-05 8.06397

3.2767 E-01 -4.2020 E-04 1.92320 E-01 1.19160 E+03

-4.3947 E+01 -5.3913 E-01

2.4506 E+03 1.3011 E+03

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constant properties, temperature-dependent properties, and fitted model for DEG. The same phenomenon was observed for DEG under heating and cooling conditions as observed in the case of water. Under constant properties, it can be seen from the figures that the friction factor and heat transfer values are higher for DEG as compared to those for water. Figure 13 shows that, under the heating condition, the temperature-dependent properties show 25% higher heat transfer as compared to constant properties, while under the cooling condition, the heat transfer is 12% less for the temperature- dependent properties than for the constant properties. Beside this, the effect of tube curvature (ranging from 5-20) and pitch (ranging from 0.015-0.03 m) on friction factor and heat transfer was also determined in the coiled tube considering temperature-dependent properties under heating and cooling conditions, respectively. It was observed from the present predictions that the effect of both curvature and pitch on flow fluid and heat transfer under temperature-dependent properties variation is the same as that reported in the literature14-18 for constant properties. The heat transfer and friction factor decreases as tube curvature and pitch increases, while the reverse phenomenon was observed when the values of tube curvature and pitch decreases. 4. Conclusions A fully developed laminar flow in a curved tube with temperature-dependent properties was studied under both heating and cooling conditions for different values of tube curvature and pitch. The heat transfer in the curved tubes is higher as compared to that in the straight tube because of the secondary flow. The velocity and temperature profiles were distorted when the effects of temperature-dependent properties were considered. The distribution of all the thermo-physical properties was reported at various Reynolds numbers ranging from 100-400. It was observed that specific heat and thermal conductivity have negligible effects on the velocity and thermal profiles and density has a small effect, while viscosity plays a dominant role of importance. The Nusselt number and the friction factor results show a marked dependence on the properties variation in the coiled tube. Under cooling conditions, the Nusselt number values considering temperature-dependent properties are lower than the constant-properties results because of the increase of the thermophysical properties at the inner points of the curved tube section. That reduces the secondary flow effects and the heat transfer rate. The opposite trends were observed when the heating condition was considered. The temperature-dependent study was also carried out for a highly viscous fluid like diethylene glycol for both heating and cooling conditions. Under constant properties, it was observed that the friction factor and heat transfer are higher for DEG as compared to that for water. The friction factor and heat transfer values obtained by considering temperature-dependent properties and the correlation proposed were in good agreement for both water and DEG. It was also observed that the heat transfer values obtained by considering temperaturedependent properties under the heating condition for DEG were 25% higher as compared to the values obtained by considering constant properties. The friction factor and heat transfer obtained using temperature-dependent properties were also fitted using the property-ratio method, and a new correlation was proposed. Acknowledgment The authors gratefully acknowledge the Ministry of Chemical and Fertilizers, GOI, India, for funding the project.

Nomenclature An ) coefficient for thermo-physical properties (n ) 1, 2, 3, 4, 5, and 6) a ) tube radius, m Cp ) specific heat, J/kg‚K di ) tube diameter, m D ) coil diameter, m De ) Dean number f ) friction factor k ) thermal conductivity, W/m‚K L ) length of curved tube, m H ) pitch of coil, m n, m ) exponents in eqs 1 and 2 Nu ) Nusselt number p ) pressure Q ) heat transfer rate, W qw ) heat flux, W/m2 Re ) Reynolds number St ) Stanton number u, V, w ) velocity in x, y, and z directions, m/s T ) temperature, K Greek Letters µ ) viscosity of the fluid, kg/m‚s F ) density of the fluid, kg/m3 Subscripts b ) bulk mean value m ) mean value cp ) value with constant property w ) value at the wall Literature Cited (1) Shah, R. K.; London, A. L. Laminar Flow Forced Convection in Ducts. In AdVances in Heat Transfer; Academic Press: New York, 1978. (2) Herwig, H. The effect of variable properties on momentum and heat transfer in a tube with constant heat flux across the wall. Int. J. Heat Mass Transfer 1985, 28, 423. (3) Etemand, S. Gh.; Mujumdar, A. S. Effects of variable viscosity and viscous dissipation on laminar convection heat transfer of a power law fluid in the entrance region of a semi-circular duct. Int. J. Heat Mass Transfer 1995, 38, 2225. (4) Harms, T. M.; Jog, M. A.; Manglik, R. M. Effects of temperaturedependent viscosity variations and boundary conditions on fully developed laminar forced convection in a semicircular duct. ASME J. Heat Transfer 1998, 120, 600. (5) Kakac, S. Handbook of Single-Phase ConVectiVe Heat Transfer; Kakac, S., Shah, R. K., Aung, W., Eds.; John Wiley & Sons, Inc.: New York, 1987; Chapter 18. (6) Liu, D.; Lee, P. S. Numerical investigation of fluid flow and heat transfer in microchannel heat sinks. In ConVection of Heat and Mass; ME 605; ASME: West Lafayette, IN, 2003. (7) Dean, W. R. Note on the motion of fluid in a curved pipe. Philos. Mag. 1927, 4, 208. (8) Berger, S. A.; Talbot, L.; Yao, L. S. Flow in curved pipes. Ann. ReV. Fluid Mech. 1983, 15, 461. (9) 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. (10) Bergles, A. E.; Kakac, S.; Shah, R. K. Low Reynolds Number Flow Heat Exchangers, 1st ed.; Hemisphere Publishing Corp.: New York, 1983; p 451. (11) Andrade, C. R.; Zaparoli, E. L. Effects of temperature-dependent viscosity on fully developed Laminar forced convection in a curved duct. Int. Commun. Heat Mass Transfer 2001, 28, 211. (12) Incropera, F. P.; Dewitt, D. P. Fundamentals of Heat Transfer; John Wiley & Sons, Inc.: New York, 1981. (13) Kumar, V; Nigam, K. D. P. Numerical simulation of steady flow fields in Coiled Flow Inverter. Int. J. Heat Mass Transfer 2005, 48, 4811.

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(14) Roger, G. F. C; Mayhew, Y. R. Heat transfer and pressure losses in helically coiled tubes under turbulent flow. Int. J. Heat Mass Transfer 1964, 7, 1207. (15) Kalb, C. E.; Seader, J. D. Heat and Mass transfer phenomena for viscous flow in curved circular pipe. Int. J. Heat Mass Transfer 1972, 15, 801. (16) Kalb, C. E.; Seader, J. D. Fully developed viscous flow, Heat transfer in curved tubes with uniform wall temperatures. AIChE J. 1974, 20, 340.

(17) Manlapaz, R. L.; Churchill, S. W. Fully developed laminar flow in helically coiled of finite pitch. Chem. Eng. Commun. 1980, 7, 57. (18) Manlapaz, R. L.; Churchill, S. W. Fully developed laminar convection from a helical coil. Chem. Eng. Commun. 1981, 97, 185.

ReceiVed for reView June 30, 2006 ReVised manuscript receiVed November 9, 2006 Accepted November 10, 2006 IE0608399