Assessment of mixed convection heat transfer in a flow through an induced tube

Transcription

1 International Invention Journal of Engineering Science and Technology (ISSN: ) Vol. 2(2) pp , August, 2015 Available online Copyright 2015 International Invention Journals Full Length Research Paper Assessment of mixed convection heat transfer in a flow through an induced tube Taiwo O. Oni *1 and Manosh C. Paul 2 1 Mechanical Engineering Department, Faculty of Engineering, Ekiti State University, P.M.B. 5363, Ado-Ekiti, Nigeria. 2 Systems, Power and Energy Research Division, School of Engineering, University of Glasgow, Glasgow G12 8QQ, UK. Abstract This research article focuses an assessment of mixed convection heat transfer in a flow through an induced tube. The assessment was numerically carried out to examine the influence of flow orientation on heat transfer and flow characteristics of mixed convection for laminar, transition and turbulent flows through a tube induced with an alternate-axis triangular cut twisted tape. Water with Prandtl number of 5.83 was the working fluid, flow orientation of 15 o θ 90 o was considered, and the wall of the tube was subjected to uniform heat flux. Fluent software was used and the RNG κ ε was employed for the numerical simulation of the turbulent flow while the transitional variant of the Shear-Stress Transport (SST)κ ω was applied for the transition flow. The heat transfer was presented in terms of the dimensionless quantity known as Nusselt number and it was discovered that the Nusselt number and the friction factor for the mixed convection flow (15 o θ 90 o ) were higher than those of the forced convection flow (0 o ). Also, the maximum Nusselt number and the friction factor are obtained at a flow orientation of 90 o. Keywords: Assessment, mixed convection, flow orientation, heat transfer, Nusselt number. INTRODUCTION In heat transfer, cases frequently occur in which both natural and forced convection mechanisms participate simultaneously. This occurrence is referred to as mixed or combined convection (Bergles and Simonds, 1971). In natural convection, the motion of fluid is due to variation of gravitational body force associated with variation of fluid density. In forced convection, fluid motion is caused by an externally applied force (Holman, 2009). The contribution of each of the natural and forced convections depends on parameters such as the flow orientation, geometry of the arrangement, flow regime, nature of working fluid, magnitude of the temperature driving force for heat transfer, and boundary condition (Joye et al., 1989; Patil and Vijay-Babu, 2012; Tanaka et al., 1987). *Corresponding author tooni1610@yahoo.com; Tel: 234(0) Mixed convection heat transfer in tubes is utilized in a wide variety of many industrial engineering applications such as cooling of electronic equipment, solar collector heat exchangers, pipe lines used for transporting oil, boilers, compact heat exchangers, cooling of rotating parts such as rotor blades of gas turbine, etc. (Aung, 1987; Mohammed, 2008; Yousef and Tarasuk, 1982). The importance of mixed convection heat transfer in tubes has incited a large number of research activities with a view to optimizing and improving the performance of various industrial application (Ozsunar et al., 2001). The investigations on mixed convection for laminar, transition and turbulent flow through an induced tube in an inclined orientation are rare in the literature. Majority of the literature reviews on mixed convection give attention to the horizontal and vertical orientations of plain tubes and ducts with air as working fluid. As early as 1954, mixed convection heat transfer in a short vertical tube was analysed(eckert and Diaguila, 1954). Another early work on mixed convection heat transfer

2 18 Int. Inv. J. Eng. Sci Tech. was done in 1966 in which an analytical investigation of the influence of flow orientation on laminar mixed convection in an upward circular tube was carried out and the result indicated that the heat transfer coefficients increased up to a maximum at an optimum flow orientation and the heat transfer decreased thereafter (Iqbal and Stachiewicz, 1966). Mixed convection for laminar flow of water through a horizontal circular duct was investigated both experimentally and numerically under a uniform wall heat flux by(piva et al., 1995). The results implied that the pressure drop and the heat transfer were found to increase up to 22% and 150% respectively as a result of the buoyancy effect created by the mixed convection. Lin and Lin (1996) discovered in an experimental study of buoyancy-driven flow, flow transition and the accompanying heat transfer process in a mixed convection of air through an inclined rectangular duct that the presence of the buoyancy increased the heat transfer. In an experiment on the laminar flow of mixed convection of water under uniform wall temperature conditions, Patil and Vijay-Babu (2012) investigated the importance of non-dimensional parameters like Prandlt number, Richardson number and Reynolds number in natural and mixed convection regions. They deduced that the intensity of natural convection and magnitude of Richardson number decreased as the Reynolds number increased. In addition, the average Nusselt number in the mixed convection was found to be higher than that in forced convection. For the first time, Abdelmeguid and Spalding (1979) applied a two-equation model to perform numerical predictions of turbulent flow and heat transfer in a horizontal, vertical and inclined pipes under the effects of buoyancy and with the pipe wall subjected to uniform heat flux. The results revealed that for Grashof number less than critical Grashof number, the effect of buoyancy on turbulence is modest whereas for Grashof number greater than critical Grashof number, both the averaged flow and the heat transfer are affected by the buoyancy. The problem of mixed convection flow in a horizontal cylinder rotating in air was numerically and experimentally conducted by Farouk and Ball (1985) and the mean Nusselt number of the rotating cylinder was found to be higher than that of the stationary cylinder. Nguyen et al. (2004) numerically explored the laminar mixed convection flow in a vertical tube subjected to a uniform heat flux and the results obtained indicated that the heat transfer increased due to the buoyancy effect. Notwithstanding its importance in engineering applications, the mixed convection heat transfer for laminar, transition and turbulent flows in an induced tube subjected to an inclined orientation has virtually been neglected. The present study numerically assesses laminar, transition and turbulent mixed convection in a tube induced with alternate-axes triangular cut twisted tape with water as working fluid. Importance is attached to the effects of flow orientation (15 o θ 90 o ) on the heat transfer and friction factor. Comparison is also made between the results obtained for the mixed convection and that forced convection (0 o ). Flow model The geometry of the model used for the numerical assessment is shown in Figure 1. It is inclined at an angle to the horizontal and this angle is varied to obtain different flow orientations (θ). The model has been sliced so that the tape inside its tube can be seen. The tube has an internal diameter of 19mm and a length of 1000mm. The alternate-axis twisted tape was obtained by cutting the twist lengths of a tape without alternate-axis and arranged each plane at 90 o relative to the adjacent plane. In addition, there are triangular cuts on the tape.the configuration of the tape showing its width (w), pitch (y) and thickness (δ) of 18mm, 54mm and 1mm respectively is displayed in Figure 2. Mathematical modelling Governing equations The heat transfer and fluid flow are governed by the Navier-Stokes and energy transport equations (Fluent, 2006; Versteeg and Malalasekera, 2007). The working fluid (water) is assumed to be Newtonian, incompressible and steady. A constant formulation for properties is used for the fluid except for the density which is taken as a function of the local-to-reference temperature difference, approximated (Yan, 1994) as ρ = ρ 0 1 β T T 0 (1) The conservation equations of momentum, mass and energy are written as (a) the mass conservation equation: u x v y w z = 0 (2) (c) the momentum conservation equations are x- momentum equation: ρ u ρuu t x ρuv y ρuw z p x μ 2 u x 2 2 u y 2 2 u z 2 gcosθ(ρ ρ 0) = (3)

3 Oni and Paul 19 y x z Figure 1. Flow model inclined at an orientation. Figure 2.Twisted tape. y-momentum equation: ρ v ρvu t x ρvv y = p ρvw z y μ 2 v x 2 2 v y 2 2 v z 2 gsinθ(ρ ρ 0 ) z-momentum equation: ρ w ρwu ρwv ρww t x y z = p z μ 2 w x 2 2 w y 2 2 w z 2 With the Boussinesq model, the buoyancy term in the equations (3) and (4)is approximated (Fluent, 2006) as ρ ρ 0 g ρ 0 β T T 0 g (6) (d) (e) x the energy equation: T t k T ρc p x ut x y vt y k T ρc p y wt z z = k T ρc p z (4) (5) (7) In the equations above, u, v and w are the velocity components, ρ is the density, ρ 0 is the reference density, T 0 is the reference temperature, T is the fluid temperature, p is the pressure, μ is the dynamic viscosity, g is the gravitational acceleration, θ is the angle of inclination and β is the thermal expansion coefficient of the fluid. The above conservation equations (2) - (5) and (7) are solved directly for laminar flow but for turbulent flow the terms μ and k are substituted by μ eff and k eff (Baskaya et al., 1999; Davidson, 1990) respectively and expressed as κ 2 (8) μ eff = μ μ t = μ ρc μ μ t ε k eff = k k t = k C (9) p Pr t where μ eff, k eff, μ t, k t,pr t and C μ are the effective dynamic viscosity, effective thermal conductivity, turbulent dynamic viscosity, turbulent thermal conductivity, turbulent Prandtl number and model constant respectively. The solution variables (velocity, temperature and pressure) in the turbulent flow represent the time-averaged quantities but not the

4 20 Int. Inv. J. Eng. Sci Tech. instantaneous quantities. RNG κ ε model Renormalization Group (RNG) κ ε model is a twoequation model (Yakhot and Orszag, 1986)based on model transport equations for the turbulence kinetic energy (κ) and its dissipation rate (ε). The transport equations and turbulent viscosity for the RNG κ ε models (Fluent, 2006) are t ρκ ρκu x i i κ (10) = α x κ μ eff G j x k G b j ρε t ρε = x i α ε μ eff ε x j x j ε (11) C 1ε κ G k C 3ε G b C 2ε ρ ε2 κ R ε where G κ is the generation of turbulence kinetic energy due to the mean velocity gradients; G b is the generation of turbulence kinetic energy due to buoyancy;α κ and α ε are the inverse Prandtl numbers for κ and ε respectively; R ε is a term which is related to the mean strain and turbulence quantities; C 1ε = 1.42, C 2ε = 1.68, σ κ = and σ ε = are model constants (Fluent, 2006). Shear-Stress Transport (SST) κ-ω model The Shear-Stress Transport (SST) κ ω modelis based on transport equation of turbulence kinetic energy (κ) and specific dissipation rate (ω)(menter, 1994). Since transitional flow is not fully turbulent, the transitional variant of the SST κ ω is adopted for this research. The transport equations of the SST κ ω model are given (Abraham et al., 2009; Fluent, 2006) as t ρκ x i t ρω x i ρκu i = ρωu i x j ( μ μ t σ κ ) κ x j G k β 1 κω = μ μ t ω G x j σ ω x ω β 2 ω 2 j 1 κ ω 2(1 F 1 )σ ω,1 ω x j x j The turbulent viscosity (μ t ) for the SST κ ω model is presented as (12) (13) μ t = ρκ ω Where G ω is the generation of specific dissipation rate, the quantities σ κ and σ ω are the turbulent Prandtl numbers for κand ωrespectively, F 1 and F 2 are the blending functions, S is the absolute value of shear strain rate. β 1 = 0.072,β 2 = 0.072,σ ω,1 = 1.168, a 1 = 0.31are the model constants. Boundary conditions The tube wall is subjected to a no-slip condition, i.e. u i = 0 and a uniform heat flux condition was imposed on its surface. At the pipe inlet, a velocity (v) derived by v = Re. μ ρ. D (where Re is the Reynolds number), temperature T i = 301K and diameter D = 0.019m were set. The tube wall is subjected to a uniform heat flux (q) expressed as k T = q. For the turbulent flow, the x D 2 intensity I = 0.16Re 1 8 was stated. At the outlet of the tube, a zero gauge pressure was specified. An enhanced wall treatment was adopted for the simulation of the turbulent flow because it will not significantly reduce the accuracy for wall-function meshes. A value of wall plus in the range 1 y 5 was obtained. This is in agreement with the enhanced wall treatment law (Fluent, 2006). Numerical techniques and grid independence test A pre-processor software known as Gambit was used to form the computational domain. The Gambit software was also used to mesh and set up boundary conditions for the domain. The above-mentioned governing partial differential equations with their appropriate boundary conditions were then discretized with second order upwind scheme, and hence solved by the finite volume solver in fluent software. Other flow quantities were extrapolated from the interior domain by the solver in fluent software (Fluent, 2006) In order to know the grid which gives acceptable numerical accuracy, analyses of the grid independence were carried out on the domain at a flow orientation of 60 o with respect to the horizontal. Four different grids with cells , , and were used. For the laminar mixed convection flow, the temperature and velocity across the cross-section at the exit of the domain for Re = 1820 were extracted and displayed in figure 3. In figure 3(a), there is a discrepancy of 1.9%, 2.8% and 2.8% between the values of temperature in the cells , and respectively and that in the cell max 1 α, S.F 2 a 1 ω (14)

5 Oni and Paul 21 Figure2.Twisted tape. 342 T (K) 335 (a) cell_ cell_ cell_ cell_ v (m/s) cell_ cell_ cell_ cell_ r (m) r (m) Figure 3. Grid independence test of laminar flow for (a) temperature and velocity across the crosssection at the exit of the tube for flow orientation of 60 o at Re = Figure 3 shows that there is a discrepancy of 2.9%, 5.8% and 5.8% in comparing the values of velocity in the cell , and respectively with that in the cell Considering solution precision and time for convergence to be reached, the grids with cell was adopted. For the numerical solutions of the transition flow of the mixed convection, grid dependence study was carried out on the SST κ-ω model for Re = The data of temperature and velocity at the exit of the domain were extracted and the results are shown in figure 4. The results in figure 4(a) indicate that the grids with cells and produce a less satisfactory result since the difference in the value of the temperature in each of them and that in grid with cell is noticeable. The difference in the value of the temperature between the grid with cell and that with cell is negligible and can therefore be concluded that they produce a more satisfactory result. The same trend was observed in the values of the velocity in each of the four grids as demonstrated in figure 4. Considering convergence time as well as accuracy of solution, the grid with cell was selected. In the case of the turbulent mixed convection flow, grid independence test was carried out on the RNG κ εturbulent model to findout asuitable grid which will be applicable to resolve the flow inside the domain. To achieve this, the temperature, velocity and turbulent kinetic energy across the cross-section at the exit of the domain for Re = were extracted and presented in figure 5. It is indicated in figure 5(a) that a discrepancy of 3.8% between the value of the temperature in the cells and but comparing the values in the cell with those in the cells and reveals a variation of 0.1%. The same trend in the results of the temperature was observed in the results of the velocity (Figure ) and the turbulent kinetic energy (Figure 5 (c)). By giving consideration to solution precision and convergent time, the grid with cell was chosen. RESULTS AND DISCUSSION The numerical assessment for the mixed convection was performed at various flow orientations 15 o θ 90 o for laminar flow at 830 Re 2000, transition flow at 2150 Re 4650 and turbulent flow at 5000 Re The solutions were obtained with Prandtl number and Grashorf number of 5.83 and respectively. The results and discussion, which are presented below, focus on the effects of the different flow orientations on the heat transfer and flow characteristics of the mixed convection. Turbulent kinetic energy The turbulent kinetic energy of the turbulent flow at Re = for the mixed convection was compared with that of the forced convection and the results are presented in figure 6. The turbulent kinetic energy grew gradually up to the end of the tube. The values of the turbulent kinetic energy in the mixed convection with flow orientation of 15 o (frame b) show a little departure from that of the forced convection (frame a). This means that there was a low buoyancy influence when the flow orientation was 15 o. The maximum turbulent kinetic energy in the tube with forced convection (frame a) was approximately 8%, 14%, 22% and 26% less than those in the tube with the mixed convection at flow orientation

6 22 Int. Inv. J. Eng. Sci Tech. (a) 319 T (K) cell_ cell_ cell_ cell_ v (m/s) cell_ cell_ cell_ cell_ E r (m) Figure 4. Grid independence test of transition flow for (a) temperature and velocity across the crosssection at the exit of the tube for flow orientation of 60 o at Re = E r (m) 362 T (K) 338 (a) cell_ cell_ cell_ cell_ cell_ cell_ v (m/s) cell_ cell_ cell_ cell_ cell_ cell_ (c) k (m 2 /s 2 ) r (m) cell_ cell_ cell_ cell_ cell_ cell_ Figure 5. Grid independence test of turbulent flow for (a) temperature, velocity and (c) turbulent kinetic energy across the cross-section at the exit of the tube for flow orientation of 60 ^o at Re=20000

7 Oni and Paul 23 Figure 6. Turbulent kinetic energy for turbulent flow for Re=20000 for (a) forced convection, and mixed convection for flow orientation of 15 ^o, (c) 30 ^o, (d) 60 ^o, (e) 90 ^o. of 15 o (frame b), 30 o (frame c), 60 o (frame d) and 90 o (frame e) respectively. This indicates that the magnitude of the turbulent kinetic energy of the mixed convection increased as the flow orientation increased. As a result of the disturbances which the cuts imparted on the flow, the turbulent kinetic energy in the vicinity of the cuts on the tape was higher than those in the region that was not in close proximity to the cuts. Temperature distribution in the domain The results for the temperature distribution in the tubes obtained for the laminar, transition and turbulent flows are displayed in figure 7, figure 8 and figure 9 respectively and reveal that the temperature gradually increases along the tube. Due to the buoyancy effect, which caused a density difference in the fluid, the temperature in the tubes under mixed convection (frames b, c, d and e) were higher than that in the tube under forced convection (frame a). The temperature inside the tube under mixed convection increased as the flow orientation increased. The temperature of the tube under forced convection was slightly higher than that in the tube with flow orientation of 15 o under the mixed convection. In addition, the temperature in the tube under mixed convection increased slightly as the flow orientation increased from 15 o to 60 o but increased noticeably as the flow orientation increased from 60 o to 90 o. Outlet temperature and velocity The effects of the flow orientation at different Reynolds numbers on the normalised outlet temperature are shown for the laminar (Figure 10(a)), transition (Figure 10) and turbulent flow (Figure 10(c)). It is seen that the outlet temperature decreased when the Reynolds number increased. Also, the normalised temperature increased slightly as the flow orientation increased. However, in the transition flow the increase in the temperature was more pronounced, particularly when the flow orientation was greater than 23 o. The effects of the flow orientation on the outlet velocity are shown in Figure 11(a),Figure 11 and

8 24 Int. Inv. J. Eng. Sci Tech. Figure 7. Temperature distribution for laminar flow for Re = 1820 for (a) forced convection, and mixed convection for flow orientation of 15 o, (c) 30 o, (d) 60 o, (e) 90 o. Figure 8. Temperature distribution for transition flow for Re = 2150 for (a) forced convection, and mixed convection for flow orientation of 15 o, (c) 30 o, (d) 60 o, (e) 90 o.

9 Oni and Paul 25 Figure 9. Temperature distribution for turbulent flow for Re = for (a) forced convection, and mixed convection for flow orientation of 15 o, (c) 30 o, (d) 60 o, (e) 90 o (T o -T i )/T i (a) Re=830 Re=1150 Re=1820 Re= (T o -T i )/T i Re=2150 Re=3000 Re= θ ( o ) θ ( o ) (c) (T o -T i )/T i Re=5000 Re= Re= θ ( o ) Figure 10. Effect of flow orientation on normalized outlet temperature for various Reynolds numbers of (a) laminar flow, transition flow and (c) turbulent flow.

10 26 Int. Inv. J. Eng. Sci Tech v (m/s) Re=830 Re=1150 Re=1820 Re= (a) v (m/s) Re=2150 Re=3000 Re= θ ( o ) θ ( o ) v (m/s) Re=5000 Re=16000 Re=20000 (c) θ ( o ) Figure 11. Average outlet velocity at various flow orientation for various Reynolds numbers of (a) laminar flow, transition flow and (c) turbulent flow. Figure 11(c) for the laminar, transition and turbulent flows respectively. The velocity increased as the Reynolds number reduced. Moreover, the velocity of flow reduced as the flow orientation increased from 15 o to 90 o due to a drawback to the flow moving toward the end of the tube, and there is a sudden drop at flow orientation of about 23 o. For the laminar flow, Re = 830 produced the highest velocity drop of about 7% at flow orientation of 23 o θ 90 o. In the transition flow, the highest velocity drop of about 8% was obtained for flow orientation of 23 o θ 90 o at Re = In the case of the turbulent flow, the highest velocity drop of about 9% at flow orientation of 23 o θ 90 o was obtained at Re = Evaluation of heat transfer A dimensionless quantity known as Nusselt number is used to present the heat transfer. The average Nusselt number (Nu) is mathematically expressed (Incropera et al., 2007) as Nu = 1 L x dx. D L 0 k (15) Where L, x,d and k are the length of the tube, heat transfer coefficient at a point (x) along the tube, inner diameter of the tube and thermal conductivity of fluid. The Nusselt number for the mixed convection was compared with that of the forced convection, and the results are presented in figure 12. For the laminar flow, shown in figure 12(a), the Nusselt number for the mixed convection with flow orientation of 15 o, 30 o, 60 o and 90 o were 3.8 to 5.8%, 5.1 to 7.0%, 4.8 to 7.45% and 6.4 to 8.9% respectively higher than that in the tube under forced convection. The Nusselt number in the tubes with flow orientation of 15 o, 30 o, 60 o and 90 o for the transition flow, shown in figure 12, were 4.0 to 6.1%, 4.6 to 7.4%, 5.3 to 8.7% and 6.3 to 10.1% respectively higher than that in the tube under forced convection. For the turbulent flow, shown in figure 12(a), the Nusselt number in the tubes with flow orientations of 15 o, 30 o, 60 o and 90 o are 3.3 to 3.9%, 3.5 to 4.4%, 4.0 to 4.9% and 5.5 to 6.4% respectively higher than that in the tube under forced convection. It is evident in the above presentation that the Nusselt number in the tubes under mixed convection was higher than that in the tube under forced convection. In addition, the maximum enhancement in the Nusselt number of the tube was obtained at a flow orientation of 90 o. The heat transfer enhancement in the mixed convection is as a result of the combined forced and natural convections which yields three different mechanisms. These mechanisms are the external force

11 Oni and Paul (a) 155 Nu 55 Forced convection 45 θ = 15⁰ θ = 30⁰ 36 θ = 60⁰ θ = 90⁰ Re Nu 133 Forced convection 110 θ = 15⁰ θ = 30⁰ 88 θ = 60⁰ θ = 90⁰ Re (c) Forced convection Nu θ = 15⁰ θ = 30⁰ θ = 60⁰ θ = 90⁰ Re Figure 12. Effect of flow orientation on Nusselt number for various Reynolds numbers of (a) laminar flow, transition flow and (c) turbulent flow. provided by the forced convection, and the parallel and normal components of the buoyant force derived from the natural convection. The external force from the forced convection reduced the thermal resistance and consequently resulted in an increase in heat transfer. The parallel component of the buoyant force is in the same direction to the main flow in the tube and this led to a decrease in thermal resistance and an increase in the heat transfer. The normal component of the buoyant force disturbed the boundary layer and subsequently increased the heat transfer (Maughan and Incropera, 1987; Wickern, 1991). Friction Factor The effect of different flow orientations at different Reynolds numbers on the friction factor is examined in this section. The Darcy friction factor is mathematically expressed (Incropera et al., 2007) as f = 2D. p u m 2. L. ρ (16) Where f is friction factor, p is pressure drop, u m is mean velocity of fluid and ρis density of the working fluid. As demonstrated in figure 13, the friction factor for the mixed convection increased as the flow orientation increased from 15 o to 90 o. The friction factor for the mixed convection was higher than that of the forced convection. The factor responsible for this is the buoyant force due to the natural convection acting on the flow in addition to the forces due the forced convection that is also acting on the flow. Quantitatively, the friction factor for the laminar flow (Figure 13(a)) in the tubes with flow orientation of 15 o, 30 o, 60 o and 90 o were 4.3%, 5.6%, 6.2% and 9.2% respectively higher that in than the tube under forced convection. The friction factor in the tubes with flow orientation of15 o, 30 o, 60 o and 90 o for the

12 28 Int. Inv. J. Eng. Sci Tech. f Forced convection θ = 15⁰ θ = 30⁰ θ = 60⁰ θ = 90⁰ (a) f Forced convection θ = 15⁰ θ = 30⁰ θ = 60⁰ θ = 90⁰ Re Re Forced convection θ = 15⁰ θ = 30⁰ (c) f θ = 60⁰ θ = 90⁰ Re Figure 13. Effect of flow orientation on friction factor for various Reynolds numbers of (a) laminar flow, transition flow and (c) turbulent flow. transition flow (Figure 13) were 6.6%, 8.4%, 9.8% and 10.8% respectively higher than that in the tube under forced convection. For the turbulent flow (Figure 13 (c)), the friction factor in the tubes with flow orientation of 15 o, 30 o, 60 o and 90 o were up to 10.7%, 13.7%, 15.3% and 18% respectively higher than that in the tube under forced convection. CONCLUSION In this work, an assessment was numerically conducted to examine the influences of flow orientation on heat transfer and flow characteristics of mixed convection for laminar, transition and turbulent flows of water through a tube induced with an alternate-axis triangular cut twisted tape. The numerical results were presented for flow orientations of 15 o θ 90 o, Reynolds number 830 Re 2000 (laminar flow),2150 Re 4650 (transition flow), and 5000 Re 20000(turbulent flow). The results indicated that for the laminar, transition and turbulent flows, both the Nusselt number and the friction factor obtained in the tubes under mixed convection were higher than those in the tube under forced convection. Also, the Nusselt number and the friction factor for the mixed convection increased as the flow orientation increased from 15 o to 90 o. For the flow orientations considered, the maximum Nusselt number and friction factor were obtained at a flow orientation of 90 o while the minimum Nusselt number and friction factor were obtained at a flow orientation of 15 o. At theflow orientation of 90 o, the induced tubes under mixed convection provided a maximum enhancement in the Nusselt number of up to 8.9%, 10.1% and 6.4% for the laminar, transition and turbulent flows respectively over that in the induced tube under forced convection. For the friction, a maximum enhancement of up to 9.2%, 10.8% and 18% for the laminar, transition and turbulent flows respectively over that in the forced convection were obtained in the mixed convection. ACKNOWLEDGMENT The financial support from the Tertiary Education Trust Fund (TETFund) Nigeria and Ekiti State University Ado- Ekiti Nigeria enjoyed by this research is faithfully appreciated. Abbreviations C 1ε, C 2ε, C 3ε, C μ C p Model constant Specific heat capacity at constant

13 Oni and Paul 29 D f pressure (J kg. K) Diameter of tube (m) Darcy friction factor g Gravitational acceleration (m s 2 ) F 1, F 2 Blending function Generation of turbulence kinetic G κ energy due to the mean velocity gradients G ω Generation of specific dissipation rate G b Generation of turbulence kinetic energy due to buoyancy Gr Grashof number Heat transfer coefficient (W m 2. K) I Intensity of turbulence k Thermal conductivity (W m. K) k eff Effective thermal conductivity (W m. K) L Length of tube (m) Nu Nusselt number p Pressure (N m 2 ) Pr Prandtl number Pr t Turbulent Prandtl number Re Reynolds number t Time (s ) T Temperature (K) T i Inlet temperature (K) T o Reference temperature (K) u, v, w Velocity component(m/s ) w Width of twisted tape (m) y Twist ratio of tape Greek symbols α κ, α ε Inverse Prandtl number for κ and ε α Correction factor β 1, β 2 Model constant β Thermal expansion coefficient (K 1 ) δ Thickness of tape (m) ε Dissipation rate of Turbulent kinetic energy (m 2 m 3 ) θ Flow orientation (degree) κ Turbulent kinetic energy (m 2 s 2 ) μ Dynamic viscosity (kg ms ) μ eff Effective dynamic viscosity (kg ms ) μ t Turbulent dynamic viscosity (kg ms ) ρ Density (kg m 3 ) ρ o Reference density (kg m 3 ) σ ε, σ κ, σ ω Turbulent Prandtl numbers for ε, κ, ω ω Specific dissipation rate (K 1 ) REFERENCES Abdelmeguid AM, Spalding DB (1979). "Turbulent flow and heat transfer in pipes with buoyancy effects", J. Fluid Mech., Vol 94, pp Abraham JP, Sparrow EM, Tong JCK (2009). "Heat transfer in all pipe flow regimes: laminar, transitional/intermittent and turbulent", Int. J. Heat Mass Transfer, Vol 52, pp Aung W (1987)."Mixed convection in internal flow", in: Kakac S,Shah RK and Aung W, (Eds.), Handbook of single-phase convective heat transfer, Wiley, New York. Baskaya S, Aktas MK, Onur N (1999). "Numerical simulation of the effects of plate separation and inclination of heat transfer in buoyancy driven open channel", Heat and Mass Transfer, Vol 35, pp Bergles AE, Simonds RR (1971). "Combined forced and free convection for laminar flow in a horizontal tube with uniform heat flux", Int. J. Heat Mass Transfer, Vol 14, No. 12, pp Davidson L (1990). "Calculation of the turbulent buoyancy driven flow in a rectangular cavity using an efficient solver", Num. Heat Transfer, Part A, Vol 18, pp Eckert ERG, Diaguila AJ (1954). "Convective heat transfer for mixed, free and forced flow through tubes", ASME Transac. J. Heat Transfer, Vol 76, pp Farouk B, Ball KS (1985). "Convective flows around a rotating isothermal cylinder", Int. J. Heat Mass Transfer, Vol 28, pp Fluent Provide initial (2006). "Fluent 6.3 User s Guide ", Lebanon. Holman JP (2009). "Heat Transfer", 10th ed., McGraw-Hill, New York. Incropera FP, Dewitt DP, Bergman TL, Lavine AS (2007). "Fundermentals of heat and mass transfer", 6th ed., John Wiley and sons, USA. Iqbal M, Stachiewicz JW (1966). "Influence of tube orientation on combined free and forced laminar convection heat transfer", ASME Transac. J. Heat Transfer, Vol 109, No. 2, pp Joye DD, Bushinsky JP, Saylor PE (1989). "Mixed convection heat transfer at high Grashorf number in a vertical tube", Ind. Eng. Chem. Res., Vol 28, No. 12, pp Lin WL, Lin TF (1996). "Unstable aiding and opposing mixed convection of air in a bottom-heated rectangular duct slightly inclined from the horizontal", ASME Transac. J. Heat Transfer, Vol 118, No. 1, pp Maughan JR, Incropera FP (1987). "Experiments on mixed convection heat transfer for airflow in a horizontal and inclined channel", Int. J. Heat Mass Transfer, Vol 30, No. 7, pp Menter FR (1994). "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications", American Inst. of Aeronautics and Astronautics, Vol 32, No. 8, pp Mohammed HA (2008). "Laminar mixed convection heat transfer in a vertical circular tube under buoyancy-assisted and opposed flows", Energy Conversation and Management, Vol 49, pp Nguyen CT, Roy G, Landry MA, Maiga SEB (2004). "Transient laminar mixed convection flow in a vertical tube under high Grashof number condition", ASME Transac. J. Heat Transfer, Vol 21, No. 3, pp Ozsunar A, Baskaya S, Sivriogl M (2001). "Numerical analysis of Grashof number, Reynolds number and inclination effects on mixed convection heat transfer in rectangular channels", Int.l Comm. Heat and Mass Transfer, Vol 28, No. 7, pp Patil SV, Vijay-Babu PV (2012). "Experimental studies on mixed convection heat transfer in laminar flow through a plain square duct", Heat and Mass Transfer, Vol 48, pp Piva S, Barozzi GS, Collins MW (1995). "Combined convection and wall conduction effects in laminar pipe flow: numerical predictions and experimental validation under uniform wall heating", Heat and Mass Transfer, Vol 30, No. 6, pp Tanaka H, Maruyama S, Hatano S (1987). "Combined forced and natural convection heat transfer for upward flow in a uniformly heated, vertical pipe", Int. J. Heat Mass Transfer, Vol 30, No. 1, pp

14 30 Int. Inv. J. Eng. Sci Tech. Versteeg HK, Malalasekera W (2007). "An introduction to computational fluid dynamics- The finite volume method", 2nd ed., Pearson, England. Wickern G (1991). "Mixed convection from an arbitrarily inclined semiinfinite flat plate - I. The influence of the inclination angle", Int. J. Heat Mass Transfer, Vol 34, No. 8, pp Yakhot V, Orszag SA (1986). "Renormalization Group analysis of turbulence I: Basic theory", J. of Scientific Computing, Vol 1, No. 1, pp Yan WM (1994). "Mixed convection heat and mass transfer in inclined rectangular ducts", Int. J. Heat Mass Transfer, Vol 37, No. 13, pp Yousef WW, Tarasuk JD (1982). "Free convection effects on laminar forced convective heat transfer in a horizontal isothermal tube", ASME Transac. J. Heat Transfer, Vol 104, No. 1, pp How to cite this article: Oni TO, Paul MC (2015). Assessment of mixed convection heat transfer in a flow through an induced tube. Int. Inv. J. Eng. Sci. Tech. 2(2):17-30