The influence of the continuous phase Pe numbers on thermal wake phenomenon

The reverse of the transfer direction in the unsteady conjugate heat transfer between a spherical particle and a surrounding fluid flow has been analysed. The aspect this work is focused on is the influence of the continuous phase convection on the occurrence and development of this phenomenon. The energy equations are solved by the ADI finite difference method. The range of the Pe numbers investigated is between 0 and 10. The ratios of the thermal conductivity and volume heat capacity between the particle and its ambient flow belong to the interval 0.01–100. It was found that, in creeping flow, the thermal wake occurs at Pe=0.690·10⁻³. Increasing the Pe number up to 1 the dimension of thermal wake increases. For Pe>1, the increase in Pe decreases thermal wake. Received on 13 January 1998     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

Forced convection from a straight elliptical tube

The paper deals with the problem of two-dimensional laminar forced convection heat transfer from a straight isothermal tube of elliptic cross-section placed in a uniform stream. The study is based on numerical solutions of the conservation equations of mass, momentum, and energy which covers the entire flow domain including the wake region. The parameters influencing the heat transfer process are essentially the Reynolds number, Re, the tube geometry represented by its minor to major axis ratio, Ar, and the angle of inclination, λ. The study focuses on the effect of Re, Ar, and λ on the heat transfer process in the range of Re from 20 to 500. The study reveals that the rate of heat transfer reaches its maximum when λ=0^∘ while the minimum occurs when λ=90^∘. The results also show that smaller axis ratio gives higher heat transfer rate when λ=0^∘. The local Nusselt number and surface vorticity distributions are plotted for a number of cases and the effect of vortex shedding on the overall rate of heat transfer is briefly discussed. Received on 20 September, 1997     

Viscosity of concentrated noncolloidal bidisperse suspensions

At the same solid volume fraction (Φ) the relative viscosity (η _r ) of a concentrated noncolloidal bidisperse suspension of hard spherical particles is lower than that of a monodisperse suspension. In this paper a semi-analytical viscosity model of noncolloidal bidisperse suspensions is derived using an integration method. In this model the random loose packing density obtained by computer simulation is taken as the limit of solid volume fraction Φ _m which depends upon both the diameter ratio (λ) of large to small particles and the volume fraction of large particles (ξ=Φ _l /Φ). This model shows that at high solid volume fraction, Φ > 0.40, both λ and ξ significantly influence η _r . For example, at Φ=0.5, it predicts that for monodisperse suspensions η _r =70, while for bidisperse suspensions (λ=2 and ξ=0.7) η _r =40. Comparison shows that, at high solid volume fraction (0.4–0.5), the relative viscosity predicted by this model is in good agreement with that predicted by the work of Shapiro and Probstein (1992) and of Patlazhan (1993), but is higher than that predicted by the work of others. Received: 27 February 2001 Accepted: 25 April 2001     

Theoretical analysis of convective heat transfer enhancement of microencapsulated phase change material slurries

This paper analyzes the convective heat transfer enhancement mechanism of microencapsulated phase change material slurries based on the analogy between convective heat transfer and thermal conduction with thermal sources. The influence of each factor affecting the heat transfer enhancement for laminar flow in a circular tube with constant wall temperature is analyzed using an effective specific heat capacity model. The model is validated with results available in the literature. The analysis and the results clarify the heat transfer enhancement mechanism and the main factors influencing the heat transfer. In addition, the conventional Nusselt number definition of phase change slurries for internal flow is modified to describe the degree of heat transfer enhancement of microencapsulated phase change material slurries. The modification is also consistent evaluation of the convective heat transfer of internal and external flows.c volumetric concentration of microcapsules - c_m mass concentration of microcapsules - c_p specific heat, kJ kg^–1 K^–1 - h_fs phase change material heat of fusion, kJ kg^–1 - h_m* modified convective heat transfer coefficient, W m^–2 K^–1 - k thermal conductivity, W m^–1 K^–1 - k_e effective thermal conductivity of slurry, W m^–1 K^–1 - k_b slurry bulk thermal conductivity, W m^–1 K^–1 - ML dimensionless initial subcooling - Mr dimensionless phase change temperature range - Nu conventional Nusselt number - Nu* improved Nusselt number - q_wⁿ wall heat flux, Wm^–2 - Pe Peclet number - Pr Prandtl number - Re Reynolds number - r radial coordinate, m - r₀ duct radius, m - r₁ dimensionless radial coordinate - Ste Stefan number - T temperature, K - T₁ lower phase change temperature limit, K - T₂ upper phase change temperature limit, K - T_i slurry inlet temperature, K - u axial velocity, m/s - v radial velocity, m/s - x axial coordinate, m - x₁ dimensionless axial coordinate - thermal diffusivity, m²/s - dimensionless temperature - dynamic viscosity, N·s/m² - kinematic viscosity, m²/s - _t width of thermal boundary, m - degree of heat transfer enhancement, = h_m*/(h_m*)_single - b bulk fluid (slurry) - b0 slurry without phase change - l liquid - m mean - s solid - f suspending fluid - p microcapsule particles - w wall - single single-phase fluid     

A THEORY OF EFFECTIVE THERMAL CONDUCTIVITY FOR MATRIX-INCLUSION- MICROCRACK THREE-PHASE HETEROGENEOUS MATERIALS BASED ON MICROMECHANICS

The effective thermal conductivity of matrix-inclusion-microcrack three-phase heterogeneous materials is investigated with a self-consistent micromechanical method (SCM) and a random microstructure finite element method(RMFEM). In the SCM, microcracks are assumed to be randomly distributed and penny-shaped and inclusions to be spherical, the crack effect is accounted for by introducing a crack density parameter, the effective thermal conductivity is derived which relates the macroscopic behavior to the crack density parameter. In the RMFEM, the highly irregular microstructure of the heterogeneous media is accurately described, the interaction among the matrix-inclusion-microcracks is exactly treated, the inclusion shape effect and crack size effect are considered. A Ni/ZrO₂ particulate composite material containing randomly distributed, penny-shaped cracks is examined as an example. The main results obtained are: (1) the effective thermal conductivity is sensitive to the crack density and exhibits essentially a linear relationship with the density parameter; (2) the inclusion shape has a significant effect on the effective thermal conductivity and a polygon-shaped inclusion is more effective in increasing or decreasing the effective thermal conductivity than a sphere-shaped one; and (3) the SCM and RMFEM are compared and the two methods give the same effective property in the case in which the matrix thermal conductivity λ₁ is greater than the inclusion one λ₂. In the inverse case of λ₁ < λ₂, the two methods agree as the inclusion volume fraction and crack density are low and differ as they are high. A reasonable explanation for the agreement and deviation between the two methods in the case of λ₁ < λ₂ is made. This work was supported by the National Natural Science Foundation of China and Chnese “863” High-Tech, Program.     

Lattice Boltzmann simulation of heat transfer and fluid flow in a microchannel with nanofluids

Mathematical modeling is performed to simulate forced convection flow of 47 nm- Al₂O₃/water nanofluids in a microchannel using the lattice Boltzmann method (LBM). Single channel flow and conjugate heat transfer problem are taken into consideration and the heat transfer rate using a nanofluid is examined. Simulations are conducted at low Reynolds numbers (2 ≤ Re ≤ 16). The computed average Nusselt number, which is associated with the thermal conductivity of nanofluid, is in the range of 0.6 ￡ [`(Nu)] ￡ 13 0.6 \le \overline{Nu} \le 13 . Results indicate that the average Nusselt number increases with the increase of Reynolds number and particle volume concentration. The fluid temperature distribution is more uniform with the use of nanofluid than that of pure water. Furthermore, great deviations of computed Nusselt numbers using different models associated with the physical properties of a nanofluid are revealed. The results of LBM agree well with the classical CFD method for predictions of flow and heat transfer in a single channel and a microchannel heat sink concerning the conjugate heat transfer problem, and consequently LBM is robust and promising for practical applications.     

Determination of heat transfer coefficients on the surfaces of plastic tubes

 The paper discusses the statistical steady heat and momentum transfer problem in the inlet section of the plastic tubes. The modified two equation k–ɛ turbulent model utilizing variability of turbulent Prandtl number, Pr_t, was used for the analysis. Considering the thermophysical anisotropy of the tube material, a balance of local temperatures and local heat fluxes on the boundary between the fluid and the tube wall was assumed (conjugate heat transfer problem). The thermal boundary condition on the external surface of the tube (temperature) measured in the experiment was taken into account. The boundary problem described was solved by the control volume method. The values of the parameters of Pr and Re obtained from the experiments were included in the numerical calculations. Based on the results obtained, profiles of mean fluid temperatures, local Nusselt numbers on the internal and external surface of the tube, and profiles of temperatures on the internal surface of the tube and inside of the tube wall were determined. The analysis shows that changes in Pr_tand turbulence intensity, Tu, influence the local values of Nusselt numbers, and it also shows that the results for the local Nusselt numbers inside the tube obtained from numerical calculations are of great accuracy in comparison with results published in the available literature. Received on 11 June 2001     

The influence of the Henry number on the conjugate mass transfer from a sphere

Unsteady conjugate mass transfer between a sphere and a surrounding fluid flow has been analysed. Two sphere models, the sphere with rigid interface and the sphere with mobile interface and internal circulation, have been studied. Creeping flow is assumed. The mass balance equations were solved numerically by the ADI finite difference method. The influence of the parameters that characterize the coupling features of the conjugate mass transfer, diffusivity ratio (Φ_D) and Henry number (H) (0.01?≤?H (Φ_D)?≤?100), is studied at intermediate Pe numbers. The occurrence and evolution of mass wake phenomenon is analysed. In contradiction with the assumptions practiced in classical theories of interface mass transfer, the results obtained in this work show that the Henry number and the diffusivity ratio have a significant and distinct influence on the values of both overall and fractional (internal and external) Sh numbers. Using the additivity relation as an interpolation formula between two asymptotic solutions does not lead to accurate approximations of the numerical results on the entire domain of variation of H and Φ_D. An analogy between conjugate heat and mass transfer was established.     

Steady and transient liquid sphere heating in a convective gas stream

 A finite-difference scheme has been developed to solve the equations governing the laminar forced convection heat transfer around and inside a spherical fluid droplet moving steadily in another immiscible fluid for both steady and transient thermal conditions. For large values of the external flow Reynolds number (Re), results not available in the literature have been obtained for circulating droplets at intermediate and high interior-to-exterior viscosity ratios (μ^*). Detailed results over a wide range of viscosity ratio (μ^*) and for 200≤Re≤1000 are presented for the temperature profiles outside and inside the sphere, Nusselt number, the time required to attain a uniform surface temperature and the time required to reach the steady-state temperature. Results show that convective heating is dependent on the external flow Reynolds number (Re) and the interior-to-exterior viscosity ratio (μ^*) where increasing Re or decreasing μ^* result in increasing heat transfer rate convected to the liquid sphere. Received on 1 March 1999     

Computation of conjugate heat transfer in the turbulent mixed convection regime in a vertical channel with multiple heat sources

This paper presents the results of a comprehensive numerical study to analyze conjugate, turbulent mixed convection heat transfer from a vertical channel with four heat sources, uniformly flush-mounted to one of the channel walls. The results are presented to study the effect of various parameters like thermal conductivity of wall material (k _s), thermal conductivity of flush-mounted discrete heat source (k _c), Reynolds number of fluid flow (Re _s), modified Richardson number (Ri ⁺) and aspect ratio (AR) of the channel. The standard k-ε turbulence model, modified by including buoyancy effects with physical boundary conditions, i.e. without wall functions, has been used for the analysis. Semi-staggered, non-uniform grids are used to discretise the two dimensional governing equations, using finite volume method. A correlation, encompassing a wide range of parameters, is developed for the non-dimensional maximum temperature (T ^*) using the asymptotic computational fluid dynamics (ACFD) technique.     

The optimal dimensions of convective-radiating circular fins

The optimal dimensions of convective-radiating circular fins with variable profile, heat-transfer coefficient and thermal conductivity, as well as internal heat generation are obtained. A profile of the form y=(w/2) [1+(r _o/r) ⁿ ] is studied, while variation of thermal conductivity is of the form k=k _o[1+ɛ((T−T _∞)/ (T _b−T _∞)) ^m ]. The heat-transfer coefficient is assumed to vary according to a power law with distance from the bore, expressed as h=K[(r−r _o)/(r _e−r _o)]^λ. The results for λ=0 to λ=1.9, and −0.4≤ɛ≤0.4, have been expressed by suitable dimensionless parameters. A correlation for the optimal dimensions of a constant and variable profile fins is presented in terms of reduced heat-transfer rate. It is found that a (quadratic) hyperbolic circular fin with n=2 gives an optimum performance. The effect of radiation on the fin performance is found to be considerable for fins operating at higher base temperatures, whereas the effect of variable thermal conductivity on the optimal dimensions is negligible for the variable profile fin. It is also observed, in general, that the optimal fin length and the optimal fin base thickness are greater when compared to constant fin thickness. Received on 22 February 1999     

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.     

Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.     

Combined heat and mass transfer along a vertical moving cylinder with a free stream

The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u _w/(u _w + u _∞), where u _w and u _∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1 it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution. For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction decreases as the relative velocity between the surface and free stream decreases. Received on 17 May 1999     

The developing heat transfer and fluid flow in micro-channel heat sink with viscous heating effect

The numerical modeling of the conjugate heat transfer and fluid flow through the micro-heat sink was presented in the paper, considering the viscous dissipation effect. Three different fluids with temperature dependent fluid viscosity are considered: water, dielectric fluid HFE-7600 and isopropanol. The square shape of the cross-section is considered with D _h  = 50 μm with a channel length L = 50 mm. As most of the reported researches dealt with fully developed fluid flow and constant fluid properties in this paper the thermal and hydro-dynamic developing laminar fluid flow is analyzed. Two different heat transfer conditions are considered: heating and cooling at various Br. The influence of the viscous heating on local Nu and Po is analyzed. It was shown that for a given geometry the local Po and Nu numbers are strongly affected by the viscous heating. Moreover the Po number attains the fully developed value as the external heating is equal with the internal viscous heating.     

Molecular dynamics simulation of annular flow boiling with the modified Lennard-Jones potential function

Molecular dynamics simulation of annular flow boiling in a nanochannel is numerically investigated. In this research, an annular flow model is developed to predict the superheated flow boiling heat transfer characteristics in a nanochannel. To characterize the forced annular boiling flow in a nanochannel, an external driving force F^?_\textext \overrightarrow {F}_{\text{ext}} ranging from 1 to 12 PN (PN = pico newton) is applied along the flow direction to inlet fluid particles during the simulation. Based on an annular flow model analysis, it is found that saturation condition and superheat degree have great influences on the liquid–vapor interface. Also, the results show that due to the relatively strong influence of the surface tension in small channels, the interface between the liquid film and the vapor core is fairly smooth, and the mean velocity along the stream-wise direction does not change anymore. Also, it is found that the heat flux values depend on the boundary conditions. Finally, the Green–Kubo formula is used to calculate the thermal conductivity of liquid Argon. The simulations predict thermal conductivity of liquid Argon quite well.     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

Conjugate heat transfer in the unbounded flow of a viscoelastic fluid past a sphere

This work addresses the conjugate heat transfer of a simplified PTT fluid flowing past an unbounded sphere in the Stokes regime (Re = 0.01). The problem is numerically solved with the finite-volume method assuming axisymmetry, absence of natural convection and constant physical properties. The sphere generates heat at a constant and uniform rate, and the analysis is conducted for a range of Deborah (0 ≤ De ≤ 100), Prandtl (10⁰ ≤ Pr ≤ 10⁵) and Brinkman (0 ≤ Br ≤ 100) numbers, in the presence or absence of thermal contact resistance at the solid–fluid interface and for different conductivity ratios (0.1 ≤ κ ≤ 10). The drag coefficient shows a monotonic decrease with De, whereas the normalized stresses on the sphere surface and in the wake first increase and then decrease with De. A negative wake was observed for the two solvent viscosity ratios tested (β = 0.1 and 0.5), being more intense for the more elastic fluid. In the absence of viscous dissipation, the average Nusselt number starts to decrease with De after an initial increase. Heat transfer enhancement relative to an equivalent Newtonian fluid was observed for the whole range of conditions tested. The dimensionless temperature of the sphere decreases and becomes more homogeneous when its thermal conductivity increases in relation to the conductivity of the fluid, although small changes are observed in the Nusselt number. The thermal contact resistance at the interface increases the average temperature of the sphere, without affecting significantly the shape of the temperature profiles inside the sphere. When viscous dissipation is considered, significant changes are observed in the heat transfer process as Br increases. Overall, a simplified PTT fluid can moderately enhance heat transfer compared to a Newtonian fluid, but increasing De does not necessarily improve heat exchange.     

Heat and mass transfer characteristics of simulated high moisture flue gases

The heat transfer process occurring in a condensing heat exchanger where noncondensible gases are dominant in volume is different from the condensation heat transfer of the water vapor containing small amount of noncondensible gases. In the process the mass transfer due to the vapor condensation contributes an important part to the total heat transfer. In this paper, the Colburn-Hougen method is introduced to analyze the heat and mass transfer process when the water vapor entrained in a gas stream condenses into water on the tube wall. The major influential factors of the convective-condensation heat transfer coefficient are found as follows: the partial pressure of the vapor p _v , the temperature of the outer tube wall T _w , the mixture temperature T _g , Re and Pr. A new dimensionless number Ch, which is defined as condensation factor, has been proposed by dimensional analysis. In order to determine the relevant constants and investigate the convection-condensation heat and mass transfer characteristics of the condensing heat exchanger of a gas fired condensing boiler, a single row plain tube heat exchanger is designed, and experiments have been conducted with vapor-air mixture used to simulate flue gases. The experimental results show that the convection-condensation heat transfer coefficient is 1.52 times higher than that of the forced convection without condensation. Based on the experimental data, the normalized formula for convention-condensation heat transfer coefficient is obtained. A heat transfer area m² - Ch condensation factor - c _p specific heat at constant pressure, J/(kg·K) - G mass flux Kg/(m²·s) - h heat transfer coefficient W/(m²·K) - J J-factor - Nu Nusselt number - pa pressure - Pr Prandtl number - Q heat transfer rate - q heat flux W/m² - r latent heat, kJ/kg - Re Reynolds number - Sc Schmidt number - T temperature, C or K - heat conductivity m W/(m·K) - density, kg·m³ - g gas - h moistened hot air - i interface - v vapor - w water