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%.     

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.     

Laminar forced convection with viscous dissipation in a concentric annular ductConvection laminaire forcée avec dissipation visqueuse dans un conduit annulaire concentrique

Forced convection heat transfer in fully developed flows of viscous dissipating fluids in concentric annular ducts is analyzed analytically. Special attention has been paid to the effect of the viscous dissipation. Two different cases of the thermal boundary conditions are considered: uniform heat flux at the outer wall and adiabatic inner wall (Case A) and uniform heat flux at the inner wall and adiabatic outer wall (Case B). Solutions for the velocity and temperature distributions and the Nusselt number are obtained for different values of the aspect ratio and the Brinkman number. The present analytical results for the case without the viscous dissipation effect are compared with those available in the literature and an excellent agreement is observed. To cite this article: M. Avc?, O. Ayd?n, C. R. Mecanique 334 (2006).     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

Approche analytique de la convection forcée des fluides de Bingham dans un tubeAnalytical approach for forced convection of Bingham fluids in a tube

The asymptotic behaviour of laminar forced convection for Bingham fluid in a circular tube subjected to an axially varying wall heat flux is studied analytically. The effect of viscous dissipation is taken into account while the axial heat conduction in the fluid is considered as negligible. The asymptotic temperature profile and the asymptotic Nusselt number are determined for various axial wall heat flux distributions which yield a thermally developed region. The results obtained show a diminution in asymptotic Nusselt number when the Brinkman number and the dimensionless radius of the plug flow region increase. Comparisons with the results in the literature for Newtonian fluids show the validity of the present analysis. To cite this article: R. Khatyr et al., C. R. Mecanique 330 (2002) 69–75.     

Influence of the variable thermophysical properties on the turbulent buoyancy-driven airflow inside open square cavities

The effects of the air variable properties (density, viscosity and thermal conductivity) on the buoyancy-driven flows established in open square cavities are investigated, as well as the influence of the stated boundary conditions at open edges and the employed differencing scheme. Two-dimensional, laminar, transitional and turbulent simulations are obtained, considering both uniform wall temperature and uniform heat flux heating conditions. In transitional and turbulent cases, the low-Reynolds k − ω turbulence model is employed. The average Nusselt number and the dimensionless mass-flow rate have been obtained for a wide and not yet covered range of the Rayleigh number varying from 10³ to 10¹⁶. The results obtained taking into account variable properties effects are compared with those calculated assuming constant properties and the Boussinesq approximation. For uniform heat flux heating, a correlation for the critical heating parameter above which the burnout phenomenon can be obtained is presented, not reported in previous works. The effects of variable properties on the flow patterns are analyzed.     

Turbulent boundary layer on a mildly curved convex surface

Extensive single point turbulence measurements made in the boundary layer on a mildly curved heated convex wall show that the turbulence heat fluxes and Stanton number are more sensitive to a change in wall curvature than the Reynolds stresses and skinfriction coefficient, and that downstream, as the flow adjusts to new curved conditions, the St/c _f ratio of Reynolds analogy is appreciably lower than in plane wall flow for the same conditions. Details of the turbulence structure in unheated flow have been documented in an earlier paper; temperature field measurements now described comprise mean temperature distributions, the streamwise variation of wall heat flux, profiles of the temperature variance, transverse and streamwise heat fluxes, and triple correlations. Turbulent diffusion of heat flux is drastically reduced even by mild curvature; changes in the heat fluxes are of the same order as changes in the shear stress, that is, an order of magnitude greater than the ratio of boundary layer thickness to wall radius of curvature. The data include plane flow measurements taken in a developed boundary layer upstream of a change in wall curvature.     

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.     

Forced convection heat transfer of Giesekus viscoelastic fluid in pipes and channels

A theoretical solution is presented for the convective heat transfer of Giesekus viscoelastic fluid in pipes and channels, under fully developed thermal and hydrodynamic flow conditions, for an imposed constant heat flux at the wall. The fluid properties are taken as constant and axial conduction is negligible. The effect of Weissenberg number (We), mobility parameter (α) and Brinkman number (Br) on the temperature profile and Nusselt number are investigated. The results emphasize the significant effect of viscous dissipation and fluid elasticity on the Nusselt number in all circumstances. For wall cooling and the Brinkman number exceeds a critical value (Br ₁), the heat generated by viscous dissipation overcomes the heat removed at the wall and fluid heats up longitudinally. Fluid elasticity shifts this critical Brinkman number to higher values.     

Laminar natural convection in inclined enclosures bounded by a solid wall

Laminar natural convection has been studied in enclosures bounded by a solid wall with its outer boundary at constant temperature while the opposing side has a constant heat flux. Two-dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation are solved using a finite difference method. The numerical procedure adopted is based on the SIMPLER algorithm. Various parameters were: Rayleigh number (from 10³ to 10⁶), dimensionless conductivity of bounding wall (from 1 to 10) and dimensionless wall width (from 0.15 to 0.5), aspect ratio (from 0.5 to 1) and the inclination angle (from 30^° to 180^°). The results are reduced in terms of the normalized Nusselt number as a function of the Rayleigh number, and other dimensionless parameters. The isotherms and streamlines are produced for various Rayleigh numbers and geometrical conditions. It is found that the heat transfer is an increasing function of the Rayleigh number, wall to fluid conductivity ratio, enclosure aspect ratio and a decreasing function of the wall thickness. It passes from a maximum for the inclination angle of about 80^°.     

Transient natural convection between concentric and vertically eccentric spheres with mixed boundary conditions

Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with constant heat flux on the inner wall and a specified isothermal temperature on the outer wall. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. The alternating direction implicit method and the successive over-relaxation techniques are applied to solve the finite difference form of governing equations. A physical model is introduced which accounts for the effects of fluid buoyancy as well as eccentricity of the outer sphere. Transient solutions of the entire flow field are obtained for a range of modified Rayleigh number (10³<Ra?<5×10⁵), for a Prandtl number of 0.7 and a radius ratio of 2.0, with the outer sphere near the top and bottom of the inner sphere (ε=±0.625). Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the modified Rayleigh number and the eccentricity of the spherical annulus. The results of average Nusselt numbers are also compared with the results obtained for flow between two isothermal spheres.     

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%.     

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     

Laminar thermally developing flow in rectangular channels and parallel plates: uniform heat flux

Numerical simulations were conducted for thermally developing laminar flow in rectangular channels with aspect ratios ranging from 1 to 100, and for parallel plates. The simulations were for laminar, thermally developing flow with H1 boundary conditions: uniform heat flux along the length of the channel and constant temperature around the perimeter. In the limit as the non-dimensional length, x* = x/(D _h RePr), goes to zero, the Nusselt number is dependent on x* to the negative exponent m. As the non-dimensional length goes to infinity the Nusselt number approaches fully developed values that are independent of x*. General correlations for the local and mean heat transfer coefficients are presented that use an asymptotic blending function to transition between these limiting cases. The discrepancy between the correlation and the numerical results is less than 2.5 % for all aspect ratios. The correlations presented are applicable to all aspect ratios and all non-dimensional lengths, and decrease the discrepancy relative to existing correlations.     

Statistical analysis of instantaneous turbulent heat transfer in circular pipe flows

Turbulent heat transfer in circular pipe flow with constant heat flux on the wall is investigated numerically via Large Eddy Simulations for frictional Reynolds number Re _τ  = 180 and for Prandtl numbers in the range 0.1 ≤ Pr ≤ 1.0. In our simulations we employ a second-order finite difference scheme, combined with a projection method for the pressure, on a collocated grid in cylindrical coordinates. The predicted statistical properties of the velocity and temperature fields show good agreement with available data from direct numerical simulations. Further, we study the local thermal flow structures for different Prandtl numbers. As expected, our simulations predict that by reducing the Prandtl number, the range of variations in the local heat transfer and the Nusselt number decrease. Moreover, the thermal flow structures smear in the flow and become larger in size with less sharpness, especially in the vicinity of the wall. In order to characterize the local instantaneous heat transfer, probability density functions (PDFs) for the instantaneous Nusselt number are derived for different Prandtl number. Also, it is shown that these PDFs are actually scaled by the square root of the Prandtl number, so that a single PDF can be employed for all Prandtl numbers. The curve fits of the PDFs are presented in two forms of log-normal and skewed Gaussian distributions.     

An experimental study of diabatic spiral vortex flow

In spiral vortex flow, between concentric cylinders with the inner cylinder rotating and the outer stationary, the addition of a thermal gradient across the gap is a known complicating factor. The present diabatic study for narrow and wide gaps (radius ratios N=0.955 and N=0.8), with a heated outer and adiabatic inner cylinder, was undertaken to investigate this problem. The heat transfer characteristics and the modes of transition have been investigated together with the relationship between them. Using standard on-line digital computer techniques, the onset of vortex flow and its higher transitions have been shown to cause a sharp increase in Nusselt number. At higher Taylor numbers, of the order of 10⁶, a marked change in the Nusselt number occurs with the onset of the transition to periodic turbulent vortex flow. Outer wall heating is seen to affect the modes of transition. Diabatic critical Taylor numbers are much higher than those for adiabatic conditions and are found to depend on the close approach of the vortices to the outer wall     

A numerical method for forced convection in porous and homogeneous fluid domains coupled at interface by stress jump

A numerical method was developed for flows involving an interface between a homogeneous fluid and a porous medium. It is based on the finite volume method with body‐fitted and multi‐block grids. The Brinkman–Forcheimmer extended model was used to govern the flow in the porous medium region. At its interface, the flow boundary condition imposed is a shear stress jump, which includes the inertial effect, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The forced convection through a porous insert over a backward‐facing step is investigated. The results are presented with flow configurations for different Darcy numbers, 10^?2 to 10^?5, porosity from 0.2 to 0.8, Reynolds number from 10 to 800, and the ratio of insert length to channel height from 0.1 to 0.3. The heat transfer is improved by using porous insert. To enhance the heat transfer with minimal frictional losses, it is preferable to have a medium length of insert with medium Darcy number, and larger Reynolds number. The interfacial stress jump coefficients β and β₁ were varied from ?1 to 1, and within this range the average and local lower‐wall Nusselt numbers are not sensitive to the parameters. Copyright © 2007 John Wiley & Sons, Ltd.     

Enhanced heat transfer performances of molten salt mixed convection in a vertical annular duct

The mixed convection heat transfer of upward molten salt flow in a vertical annular duct is experimentally and numerically studied. The heat transfer performances of mixed convection are measured under Reynolds number 2,500–12,000 and inlet temperature 300–400 °C, and Nusselt number of molten salt flow with cooled inner wall monotonically increases with buoyancy number. The mixed convection is further simulated by low-Reynolds number k-ε model and variable properties, and the heat transfer tendency from numerical results agrees with that from experiments. At low Reynolds number, the natural convection plays more important role in the mixed convection. As the buoyancy number rises, the thickness of flow boundary layer near the inner wall increases, while the effective thermal conductivity remarkably rises, so the enhanced heat transfer of mixed convection is mainly affected by the effective thermal conductivity due to turbulent diffusion.     

3D Numerical study of the effect of aspect ratio on mixed convection air flow in upward solar air heater

A 3D Numerical study of mixed convection air flow in upward solar air heater with large spanwise aspect ratio (A = 10 to 40) was performed using CFD commercial code Fluent 14.5 (ANSYS). The main objective of this study is to investigate the channel height's effect (aspect ratio) on flow pattern and heat transfer in upward solar air heater in the particular case of low Re and high aspect ratio. The bottom plate (absorber) was submitted to Constant Heat Flux (CHF) in the range of 200 to 1000 W/m² and Reynolds number was varied from 50 to 1000. Our results are in concordance with most of authors conclusions about Poiseuille–Rayleigh–Benard flows. In mixed convection, increasing heat flux enhances heat transfer unlike forced convection flows. Simulation results of flow visualizations and Nusselt number calculations have shown that depending on Ri*, the velocity and temperature distributions in SAH vary greatly with the channel's height. The obtained results were different from previous studies. Indeed, our investigation of channel's height was achieved for the same heat flux but different Grashof numbers. For low channel's heights (high aspect ratio), increasing heat flux has not a significant effect but for higher channel's heights, an augmentation of heat flux enhances buoyancy effects in the flow and causes high turbulence. Also, increasing Reynolds number in low channel's heights (high A), can enhance substantially heat transfer. For higher channel's heights (low A), increasing Reynolds number decreases Ri* and thus buoyancy forces. Heat transfer is reduced and so Nusselt number. The obtained results may be very useful for engineers in designing and testing solar collectors.     

Fully Developed Forced Convection Heat Transfer in a Porous Channel with Asymmetric Heat Flux Boundary Conditions

An analytical study is performed on steady, laminar, and fully developed forced convection heat transfer in a parallel plate channel with asymmetric uniform heat flux boundary conditions. The channel is filled with a saturated porous medium, and the lower and upper walls are subjected to different uniform heat fluxes. The dimensionless form of the Darcy–Brinkman momentum equation is solved to determine the dimensionless velocity profile, while the dimensionless energy equation is solved to obtain temperature profile for a hydrodynamically and thermally fully developed flow in the channel. Nusselt numbers for the lower and upper walls and an overall Nusselt number are defined. Analytical expressions for determination of the Nusselt numbers and critical heat flux ratio, at which singularities are observed for individual Nusselt numbers, are obtained. Based on the values of critical heat flux ratio and Darcy number, a diagram is provided to determine the direction of heat transfer between the lower or upper walls while the fluid is flowing in the channel.