A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder

Forced convection heat transfer from an unconfined circular cylinder in the steady cross-flow regime has been studied using a finite volume method (FVM) implemented on a Cartesian grid system in the range as 10 ≤ Re ≤ 45 and 0.7 ≤ Pr ≤ 400. The numerical results are used to develop simple correlations for Nusselt number as a function of the pertinent dimensionless variables. In addition to average Nusselt number, the effects of Re, Pr and thermal boundary conditions on the temperature field near the cylinder and on the local Nusselt number distributions have also been presented to provide further physical insights into the nature of the flow. The rate of heat transfer increases with an increase in the Reynolds and/or Prandtl numbers. The uniform heat flux condition always shows higher value of heat transfer coefficient than the constant wall temperature at the surface of the cylinder for the same Reynolds and Prandtl numbers. The maximum difference between the two values is around 15–20%.     

Semi-analytic solution of the two-dimensional turbulent energy equation in round tubes using an analytic velocity profile and its experimental validation

 A semi-analytic solution of the temperature development of single-phase, turbulent viscous flows inside smooth round tubes is performed. The special feature of the theoretical analysis revolves around two single universal functions of analytic form for the accurate characterization of the turbulent diffusivity of momentum and the turbulent velocity profile in the entire cross-section of a round tube. Using this valuable information that emanates from the analytic solution of the one-dimensional momentum balance equation, the two-dimensional energy balance equation was reformulated into an adjoint system of ordinary differential equations of first–order with constant coefficients. Each equation in the system of differential equations governs the axial variation of the average temperature of a finite volume of fluid of annular shape. Exploiting the linearity of the system of differential equations, an analytic solution of it was obtained via the matrix eigenvalue method with LAPACK, a library of Fortran 77 subroutines for numerical linear algebra. Reliable series have been determined for the axial variation of the two thermal quantities of importance: (a) the time-mean bulk temperature and (b) the local Nusselt number. The semi-analytic nature of the local Nusselt number distribution is advantageous because it may be viewed as an analytic-based correlation equation. Prediction of the local Nusselt numbers for turbulent air flows compare satisfactorily with the comprehensive correlation equations and the abundant experimental data that are accessible from the literature. The air flows are regulated by a wide spectrum of turbulent Reynolds numbers. Received on 4 June 2001 RID="★" ID="★" Current address Mechanical Engineering Dept. The University of Vermont Burlington, VT 05405, USA     

Natural convection heat transfer along vertical rectangular ducts

Experimental investigations have been reported on steady state natural convection from the outer surface of vertical rectangular and square ducts in air. Seven ducts have been used; three of them have a rectangular cross section and the rest have square cross section. The ducts are heated using internal constant heat flux heating elements. The temperatures along the vertical surface and the peripheral directions of the duct wall are measured. Axial (perimeter averaged) heat transfer coefficients along the side of each duct are obtained for laminar and transition to turbulent regimes of natural convection heat transfer. Axial (perimeter averaged) Nusselt numbers are evaluated and correlated using the modified Rayleigh numbers for laminar and transition regime using the vertical axial distance as a characteristic length. Critical values of the modified Rayleigh numbers are obtained for transition to turbulent. Furthermore, total overall averaged Nusselt numbers are correlated with the modified Rayleigh numbers and the area ratio for the laminar regimes. The local axial (perimeter averaged) heat transfer coefficients are observed to decrease in the laminar region and increase in the transition region. Laminar regimes are obtained at the lower half of the ducts and its chance to appear decreases as the heat flux increases.     

Forced convection from a microstructure on a flat plate

In this work, a cooling of a flat microelectronic structure with single-phase forced convection is investigated. The axial conduction, usually neglected in boundary layer theory, is considered here since the length of the heated element is in the same order of magnitude as the thickness of the boundary layer. The microstructure represents a package of chips mounted flush with the surface of the plate, and uniformly heated with a constant heat flux. The differential method is used to reduce the governing partial differential equations to ordinary differential ones, which are solved numerically by the use of a computational code developed by the authors. This code is based on Keller–Box method. The temperature profiles and Nusselt numbers are plotted at several locations on the heated element and are given as functions of the Reynolds number at the beginning of heated microstructure and of the ratio of unheated to heated length. Furthermore, the average Nusselt numbers on the heated length are computed for Prandtl numbers in the range 0.7≤Pr≤7,000. The results are compared to the boundary layer solution of unheated starting length problem. The results will be used as a baseline for successively more complex situations of cooling in electronics.     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

Coupled heat and mass transfer in the entrance region of a circular tube with fully-developed parabolic flow and external convective heating

Coupled heat and mass transfer in the entrance region of a circular tube with fully-developed parabolic flow and external heating is analyzed. The closed form solutions of temperature and concentration distributions, local Nusselt numbers based on the total external heat transfer and convective heat transfer inside the tube, as well as the Sherwood number were obtained. The effects of Biot and Lewis number on heat and mass transfer were investigated.     

Effects of Lewis number on coupled heat and mass transfer in a circular tube subjected to external convective heating

Heat and mass transfer in a circular tube subject to the boundary condition of the third kind is investigated. The closed form of temperature and concentration distributions, the local Nusselt number based on the total external heat transfer and convective heat transfer inside the tube, as well as the Sherwood number were obtained. The effects of Lewis number and Biot number on heat and mass transfer were investigated.     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

Transient free convection flow of a viscoelastic fluid over vertical surface

In this paper, the viscoelsatic boundary layer flow and the heat transfer near a vertical isothermal impermeable surface and in a quiescent fluid are examined. The gov-erning equations are formulated and solved numerically using MackCormak’s technique. The results show excellent agreement with previously published results by a compari-sion. Representative results for the velocity and temperature profiles, boundary layer thicknesses, Nusselt numbers, and local skin friction coefficients are shown graphically for different values of viscoelsatic parameters. In general, it is found that the velocities increase inside the hydrodynamic boundary layers and the temperatures decrease inside the thermal boundary layers for the viscoelsatic fluid as compared with the Newtonian fluid due to favorable tensile stresses. Consequently, the coefficients of friction and heat transfer enhance for higher viscoelsatic parameters.     

Interaction of turbulent natural convection and surface thermal radiation in inclined square enclosures

Two-dimensional numerical studies of flow and temperature fields for turbulent natural convection and surface radiation in inclined differentially heated enclosures are performed. Investigations are carried out over a wide range of Rayleigh numbers from 10⁸ to 10¹², with the angle of inclination varying between 0° and 90°. Turbulence is modeled with a novel variant of the k–ε closure model. The predicted results are validated against experimental and numerical results reported in literature. The effect of the inclination of the enclosure on pure turbulent natural convection and the latter’s interaction with surface radiation are brought out. Profiles of turbulent kinetic energy and effective viscosity are studied to observe the net effect on the intensity of turbulence caused by the interaction of natural convection and surface radiation. The variations of local Nusselt number and average Nusselt number are presented for various inclination angles. Marked change in the convective Nusselt number is found with the orientation of enclosure. Also analyzed is the influence of change in emissivity on the flow and heat transfer. A correlation relevant to practical applications in the form of average Nusselt number, as a function of Rayleigh number, Ra, radiation convection parameter, N _RC and inclination angle of the enclosure, φ is proposed.     

Heat transfer characteristics of an annular turbulent impinging jet with a confined wall measured by thermosensitive liquid crystal

The present paper describes the heat transfer characteristics of an annular turbulent impinging jet with a confined wall. The local temperature distribution on the impingement surface was measured using a thermosensitive liquid crystal sheet and an image processor. The net heat flux was evaluated by considering the heat conduction in the heated substrate and the thermal radiation between an upper confining insulated wall and an impingement surface. Distributions of the temperature and Nusselt number on the impingement surface were captured in two-dimensional maps. Effects of the diameter ratio of the annular nozzle, the space between nozzle and impingement surface and the Reynolds number on radial distributions of the local Nusselt number were examined. Experimental formulas of the local Nusselt number were obtained in power-law expressions of r/r_p for the major and minor flow regions.     

Experimental investigation of heat transfer in vertical upward and downward supercritical CO2 flow in a circular tube

An experimental investigation of turbulent heat transfer in vertical upward and downward supercritical CO₂ flow was conducted in a circular tube with an inner diameter of 4.5 mm. The experiments were performed for bulk fluid temperatures from 29 to 115 °C, pressures from 74.6 to 102.6 bar, local wall heat fluxes from 38 to 234 kW/m², and mass fluxes from 208 to 874 kg/m² s. At a moderate wall heat flux and low mass flux, the wall temperature had a noticeable peak value for vertical upward flow, but increased monotonically along the flow direction without a peak value for downward flow. The ratios of the experimental Nusselt number to the value obtained from a reference correlation were compared with Bo^* and q⁺ distributions to observe the buoyancy and flow-acceleration effects on heat transfer. In the experimental range of this study, the flow acceleration predominantly affected the heat-transfer phenomena. Based on an analysis of the shear-stress distribution in the turbulent boundary layer and the significant variation of the specific heat across the turbulent boundary layer, a new heat-transfer correlation for vertical upward and downward flow of supercritical pressurized fluid was developed; this correlation agreed with various experimental datasets within ±30%.     

Conjugate heat transfer study of incompressible turbulent offset jet flows

In the present case, the conjugate heat transfer involving a turbulent plane offset jet is considered. The bottom wall of the solid block is maintained at an isothermal temperature higher than the jet inlet temperature. The parameters considered are the offset ratio (OR), the conductivity ratio (K), the solid slab thickness (S) and the Prandtl number (Pr). The Reynolds number considered is 15,000 because the flow becomes fully turbulent and then it becomes independent of the Reynolds number. The ranges of parameters considered are: OR = 3, 7 and 11, K = 1–1,000, S = 1–10 and Pr = 0.01–100. High Reynolds number two-equation model (k–ε) has been used for turbulence modeling. Results for the solid–fluid interface temperature, local Nusselt number, local heat flux, average Nusselt number and average heat transfer have been presented and discussed.     

Heat transfer and fluid flow around semi-circular tube in cross flow at different orientations

Fluid flow and heat transfer characteristics around a semi-circular tube in cross flow were experimentally and numerically investigated. Three different tube-flow arrangements were considered. Firstly, the flat surface of the tube was placed parallel to the freestream flow; secondly, the flat surface of the tube was facing the upstream flow and thirdly, the curved surface of the tube was facing the upstream flow. For the second and the third arrangements, different angles of attack were studied. Flow visualization was carried out to illustrate streamlines around the tube and to verify flow patterns obtained from the numerical calculations. It was found that: (1) for any angle of attack, the arrangement of the curved surface facing the flow gave higher average Nusselt number than the arrangement of the flat surface facing the flow and (2) for all tube-flow arrangements, increasing the angle of attack slightly increases the average Nusselt number. Correlations of Nusselt numbers in terms of Reynolds number and angle of attack were deduced from the experimental results for the three arrangements. The comparisons between the experimental data, correlations’ predictions and numerical results showed reasonable agreements.     

Laminar forced convection conjugate heat transfer over a flat plate

Coupled heat transfer between laminar forced convection along and conduction inside a flat plate wall is theoretically studied. The laminar convective boundary layer is analyzed by employing the integral technique. The energy equations for the fluid and the plate wall are combined under the condition of the continuity in the temperature and heat flux at the fluid-solid interface. The analysis results in a simple formal solution. Expressions have been obtained for calculating local Nusselt number, wall heat flux and temperature along the plate, all are functions of the local Brun number, Br _x , which is a measure of the ratio of the thermal resistance of the plate to that of the convective boundary layer. The results indicate that for Br _x ≥0.15, neglecting the plate resistance will results in an error of more than 5% in Nusselt number. Comparison of the present solution with other previous studies has been made. The solution may be of a considerable theoretical and practical interest. Received on 19 August 1998     

Heat transfer from the heated concave wall of a return bend with rectangular cross section

The characteristics of the turbulent heat transfer along the heated concave walls of return bends which have rectangular cross sections with large aspect ratio have been examined for various clearances of the ducts in detail. The experiments are carried out under the condition that the concave walls are heated at constant heat flux while the convex walls are insulated. Water as the working fluid is utilized. Using three kinds of clearance of 9, 34, and 55 mm, the Reynolds number in the turbulent range are varied from 5×10³ to 8×10⁴ with the Prandtl numbers ranging from 4 to 13. As a result it is elucidated that both the mean and the local Nusselt numbers are always greater than those for the straight parallel plates or for the straight duct, respectively. This is attributed to Görtier vortices, which are visualized here. It is also found that the more the clearance increases, the more both the local and the mean Nusselt numbers increase. Correlation equations for the mean and the local Nusselt numbers are determined in the range of parameters covered. Introducing the Richardson number, it appears that the local Nusselt number,Nu _x , may be described as the following equation:Nu _x =447.745 ·Re _x ^1.497 ·De _x ^?1.596 ·F ^0.960 ·Pr ^0.412     

Momentum and heat transfers from fully developed turbulent flow in an eccentric annulus to inner and outer tube walls

An analysis and measurement were made of momentum and heat transfers from fully developed turbulent flow in an eccentric annulus to inner and outer tube walls. The eddy diffusivities of momentum and heat obtained in the turbulent flow in a circular tube were applied to the annular flow in the analysis. The result indicates that the calculated friction factor and average Nusselt numbers on the inner and outer tube walls are in fairly good agreement with the measurements.     

Analysis of viscous dissipation effect on thermal entrance heat transfer in laminar pipe flows with convective boundary conditions

Consideration is given to the influence of viscous dissipation on the thermal entrance region laminar pipe flow heat transfer with convective boundary condition. The Eigenfunction series expansion technique is employed to solve the governing energy equation. The results for axial distributions of dimensionless bulk and wall temperatures, local Nusselt number as well as modified local Nusselt number are presented graphically forNu ₀ =0.1, 2, and 100. The complicated variations of conventional local Nusselt number is due to the inappropriate definition of conventional heat transfer coefficient in this problem. A modified local heat transfer coefficient, based on the difference of bulk fluid temperature and wall temperature, is introduced. Its value can clearly indicate the extent and the direction of heat exchange between the fluid in the pipe and the ambient. The effects of outside Nusselt number are also investigated. Significant viscous dissipation effects have been observed for large Br.     

Average Nusselt number correlation in the thermal entrance region of steady and pulsating turbulent pipe flows

Empirical correlation has been developed for local and average Nusselt numbers in the thermal entrance region of steady and pulsating turbulent air-flows in a pipe. The correlation was based on experimental data obtained from experiment carried out on a pipe heated under uniform heat flux conditions. The rate of flow was periodically varied at frequencies ranging from 1 Hz to 13 Hz while the average Reynolds number varied from 6400 to 42000. Received on 20 October 1998     

Analytical and numerical study of buoyancy-driven convection in a vertical enclosure filled with nanofluids

This paper presents an analytical and numerical study of natural convection of nanofluids contained in a rectangular enclosure subject to uniform heat flux along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number Ra, the Prandtl number Pr, the aspect ratio of the cavity A and the solid volume fraction of nanoparticles, Φ. Three types of nanoparticles are taken into consideration: Cu, Al₂O₃ and TiO₂. Various models are used for calculating the effective viscosity and thermal conductivity of nanofluids. In the first part of the analytical study, a scale analysis is made for the boundary layer regime situation. In the second part, an analytical solution based on the parallel flow approximation is reported for tall enclosures (A ≫ 1). In the boundary layer regime a good agreement is obtained between the predictions of the scale analysis and those of the analytical solution. Solutions for the flow fields, temperature distributions and Nusselt numbers are obtained explicitly in terms of the governing parameters of the problem. A numerical study of the same phenomenon, obtained by solving the complete system of the governing equations, is also conducted. A good agreement is found between the analytical predictions and the numerical simulations.