Laminar mixed convection boundary layers induced by a linearly stretching permeable surface

The boundary layer flow on a linearly moving permeable vertical surface is studied when the buoyancy force assists or opposes the flow. Similarity and local similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, injection parameter d, and the mixed convection parameter λ=Gr_x/Re_x², which determine the velocity and temperature distributions, the heat transfer coefficient, and the shear stress at the surface are studied. The heat transfer coefficient increases as λ assisting the flow for all d for uniformly or linearly heated surface and as Pr increases it becomes almost independent of λ. However, as the temperature inversely proportional to the distance up the surface, the buoyancy has no effects on the heat transfer coefficient. Critical buoyancy parameter values are obtained for vanished shear stress and for predominate natural convection. Critical values are also presented for predominate buoyancy shear stress at the surface for assisting or opposing flow. A closed form analytical solution is also presented as a special case of the energy equation.     

Study of Variable Turbulent Prandtl Number Model for Heat Transfer to Supercritical Fluids in Vertical Tubes

In order to improve the prediction performance of the numerical simulations for heat transfer of supercritical pressure fluids, a variable turbulent Prandtl number (Pr_t) model for vertical upward flow at supercritical pressures was developed in this study. The effects of Pr_t on the numerical simulation were analyzed, especially for the heat transfer deterioration conditions. Based on the analyses, the turbulent Prandtl number was modeled as a function of the turbulent viscosity ratio and molecular Prandtl number. The model was evaluated using experimental heat transfer data of CO₂, water and Freon. The wall temperatures, including the heat transfer deterioration cases, were more accurately predicted by this model than by traditional numerical calculations with a constant Pr_t. By analyzing the predicted results with and without the variable Pr_t model, it was found that the predicted velocity distribution and turbulent mixing characteristics with the variable Pr_t model are quite different from that predicted by a constant Pr_t. When heat transfer deterioration occurs, the radial velocity profile deviates from the log-law profile and the restrained turbulent mixing then leads to the deteriorated heat transfer.     

Unsteady mixed convection flow from a moving vertical plate in a parallel free stream: Influence of heat generation or absorption

An unsteady mixed convection flow over a moving vertical plate in a parallel free stream is considered to investigate the combined effects of buoyancy force and thermal diffusion in presence of heat generation or absorption. The unsteadiness is introduced by the time dependent free stream velocity as well as by the moving plate velocity. The governing boundary layer equations are transformed into a non-dimensional form by a special group of non-similar transformations. The resulting system of coupled non-linear partial differential equations is solved by an implicit finite difference scheme in combination with the quasi-linearization technique. Computations are performed and numerical results are displayed graphically to illustrate the influence of the buoyancy (mixed convection) parameter, Prandtl number, the ratio of free stream velocity to the composite reference velocity and heat generation or absorption parameter on the velocity and temperature profiles. The numerical results for the local skin-friction coefficient and local Nusselt number are also presented. Present results are compared with previously published work and are found to be in excellent agreement. It is found that in presence of buoyancy force (λ > 0), the velocity profile exhibits velocity overshoot 80% more for lower Prandtl number (Pr = 0.7) as compared to the magnitude of the velocity overshoot for higher Prandtl number (Pr = 7.0).     

Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface

The present study deals with new similarity solution of steady mixed convection boundary layer flow over a permeable surface for convective boundary condition. It has been shown that a self similar solution is possible when the mass transfer velocity at the surface of the plate varies like x^−1/2, where x is the distance from the leading edge of the solid surface. Two point boundary value problem governed by non-linear coupled ordinary differential equations have been solved numerically using implicit finite difference scheme in combination with the quasi-linearization technique. It is interesting to note that dual solutions exist for buoyancy assisting flow, besides that usually reported in literature for buoyancy opposing flow. Further, the buoyancy assisting force causes considerable overshoot in the velocity profile and the Prandtl number strongly affects the thermal boundary layer thickness including the surface heat transfer rate.     

Free convective flow of heat generating/absorbing fluid between vertical porous plates with periodic heat input

This study investigates the free convective flow of heat generating/absorbing fluid between vertical parallel porous plates due to periodic heating of the porous plates. The analysis is performed by considering fully developed flow and steady-periodic regime. The momentum and energy equations, which arise from the definition of velocity and temperature, are written in dimensionless form. Separating the temperature and velocity fields into steady and periodic parts, the resulting second order differential equations are solved to obtain the expressions for velocity, temperature, skin friction and the rate of heat transfer. The effects of various flow parameters such as the suction/injection (s), heat source/sink (δ), Strouhal (St) and Prandtl (Pr) numbers on the skin friction coefficient, rate of heat transfer, velocity and temperature profiles are discussed with the aid of line graphs and contour maps.     

Large eddy simulation of stably stratified turbulent open channel flows with low- to high-Prandtl number

Large eddy simulation of thermally stratified turbulent open channel flows with low- to high-Prandtl number is performed. The three-dimensional filtered Navier-Stokes and energy equations under the Boussinesq approximation are numerically solved using a fractional-step method. Dynamic subgrid-scale (SGS) models for the turbulent SGS stress and heat flux are employed to close the governing equations. The objective of this study is to reveal the effects of both the Prandtl number (Pr) and Richardson (Ri_τ) number on the characteristics of turbulent flow, heat transfer, and large-scale motions in weakly stratified turbulence. The stably stratified turbulent open channel flows are calculated for Pr from 0.1 up to 100, Ri_τ from 0 to 20, and the Reynolds number (Re_τ) 180 based on the wall friction velocity and the channel height. To elucidate the turbulent flow and heat transfer behaviors, some typical quantities, including the mean velocity, temperature and their fluctuations, turbulent heat fluxes, and the structures of the velocity and temperature fluctuations, are analyzed.     

Prandtl number effect on cooling performance of a heated cylinder in an enclosure filled with nanofluid

Natural convection heat transfer from a heated cylinder contained in a square enclosure filled with water–Cu nanofluid is investigated numerically. The main objective of this study is to explore the influence of pertinent parameters such as Prandtl number (Pr) and diameter (D) of the heated body on the flow and heat transfer performance of nanofluids while Rayleigh number (Ra) and the solid particle volume fraction (?) of nanoparticle are considered fixed. The results obtained from finite element method clearly indicate that heat transfer augmentation is possible using highly viscous nanofluid resulting in the compactness of many industrial devices.     

Buoyancy effects on laminar heat transfer in the thermal entrance region of horizontal rectangular channels with uniform wall heat flux for large prandtl number fluid

The Graetz problem for fully developed laminar flow in horizontal rectangular channels with uniform wall heat flux is extended by including buoyancy effects in the analysis for the case of large Prandtl number fluid. A general formulation valid for all Prandtl numbers is presented and the limiting case of large Prandtl number is approached by a numerical method. The typical developments of temperature profile, wall temperature and secondary flow in the thermal entrance region are presented for the case of square channel γ = 1. Local Nusselt number variations are presented for the aspect ratios γ = 0.2, 0.5, 1, 2 and 5 with Rayleigh number as parameter. Due to entry and secondary flow effects, a minimum Nusselt number occurs at some distance from the entrance, depending on the magnitude of Rayleigh number. This behavior is similar to that observed in the thermal entrance region where the transition from laminar to turbulent flow occurs. The effect of Rayleigh number is seen to decrease the thermal entrance length, and the Graetz solution, neglecting buoyancy effects, is found to be applicable only when Rayleigh number is less than about 10³. A study of the practical implications of large Prandtl number on heat transfer results for hydrodynamically and thermally fully developed case reveals that the present heat transfer results are valid for Prandtl number ranging from order 10 to infinity.     

Numerical investigation of pulsating flow around a discrete heater in a channel

In this study, the forced convection heat transfer around a discrete heater located in a channel subjected to laminar pulsating air flow is numerically investigated. Simulations are conducted for six different frequencies and three different amplitudes, while the Reynolds number (Re = 125) and Prandtl number (Pr = 0.71) remain constant for all cases. The impact of the important governing parameters such as the Womersley number (Wo) and the amplitude of flow pulsation (A_o) on heat transfer rate from discrete heaters is examined in detail. The instant velocity and temperature profiles are obtained to determine of the role of dimensionless parameters for pulsating flow. The numerical results show that thermal transport from the heater is greatly affected by the frequency and amplitude of the flow pulsation. The results given are dimensionless parameters.     

Shape design for a cylinder with uniform temperature distribution on the outer surface by inverse heat transfer method

Performance of the inverse heat transfer method in application to the shape design for the heat convection problems has been evaluated. The approach is constructed by combining curvilinear grid generation scheme, direct problem solver, conjugate gradient optimization method, and redistribution method. Shape design for the outer surface profile of a solid medium in a crossflow that contains a heating element and features an isothermal outer surface has been carried out. Practical cases under different combinations of the dominant physical parameters, including Reynolds number (Re), thermal conductivity ratio (k_f/k_s), desired outer surface temperature (θ_d), and Prandtl number (Pr), are studied to evaluate the effects of the physical parameters on the shape design.     

Mixed convection about a rotating sphereConvection mixte autour d'une sphere tournanteMischkonvektion an einer rotierenden kugelCмeшaHHaя кoHвeкция oкoлo вpaщaющeйCя Cфepы

The paper presents a theoretical analysis of flow and heat transfer characteristics of the effects of buoyancy force on laminar boundary layer over a rotating sphere in forced flow under two kinds of heating conditions: uniform wall temperature and uniform surface heat flux. By applying appropriate coordinate transformations and using Merk's types of series, the governing momentum and energy equations are reduced to a set of coupled ordinary differential equations, which depend on wedge, rotation and buoyancy parameters. Numerical computations are carried out for Prandtl numbers 0.7,1.0 and for various values of buoyancy and rotation parameters. For aiding flow, it is found that both the friction factor and the local Nusselt number increase with increasing buoyancy force. The local free stream velocity increases with buoyancy which, in turn, affects the friction coefficient and Nusselt number. The coupling between rotation and buoyancy results in increased overshooting of the velocity profiles in the vicinity of the rotating sphere. For an equivalent buoyancy effect, heating by uniform surface heat flux yields larger local Nusselt number than heating by uniform wall temperature. The ratio Nu_UHF/Nu_UWT is higher for the rotating sphere (as compared to a nonrotating case) and further the ratio increases as the sphere spins faster. The effect of free stream, rotation and buoyancy on the eruption of flow is examined and also a suggestion for further investigation is made.     

Flows and heat transfer of the transition to an unsteady state in a finned cavity for different Prandtl numbers

Flows and heat transfer of the transition to an unsteady state in a finned cavity are studied for Prandtl numbers (Pr) from 0.1 to 100 and Rayleigh numbers (Ra) from 10⁷ to 10¹⁰. Transient flows are described in the finned cavity. Critical Rayleigh numbers of the transition to an unsteady state are obtained for different Prandtl numbers and the relation between two dimensionless parameters is given. The spectral analysis is applied for the oscillations of unsteady flows and the dominant frequency dependent on governing parameters is presented. Heat transfer of the transition to an unsteady flow is quantified and the corresponding relations dependent on the Prandtl number and Rayleigh number are gained. It is demonstrated that the flow rate and the Nusselt number of the finned cavity significantly increase due to the presentence of the fin, which depend on the Prandtl number and the Rayleigh number.     

An exact analysis of radiative heat transfer and unsteady MHD convective flow of a second‐grade fluid with ramped wall motion and temperature

The influence of simultaneously applied ramped boundary conditions on unsteady magnetohydrodynamic natural convective motion of a second‐grade fluid is investigated and analyzed in this study. The motion of the fluid is considered near an infinite upright plate that is nested in a porous medium subject to nonlinear thermal radiation effects. The Laplace transformation technique is utilized to acquire the exact solutions of momentum and energy equations. To effectively examine the rate of heat transfer and shear stress, the Nusselt number and skin friction coefficient are also established. The outcomes of mathematical computations are elucidated through tables and figures to highlight some physical aspects of the problem. Some limiting models of the present problem are also deduced and presented. On comparison, it is observed that the fluid exhibits lower temperature and velocity profiles under ramped boundary conditions. It is also found that wall shear stress can be controlled by choosing large values of the magnetic parameter (M) and Prandtl number (Pr). In addition, the heat transfer rate specifies inverse trends for growing values of radiation parameter (Nr) and Prandtl number (Pr), while it increases rapidly under a ramped surface condition and decreases slowly under a constant surface condition.     

CO2 based natural circulation loops: New correlations for friction and heat transfer

Results obtained from CFD analysis of 3-dimensional natural circulation loops (NCL) that employ carbon dioxide and water as loop fluids are presented for various isothermal wall temperatures of source and sink in the range of 278–341 K. Such a temperature range would be useful in various heat transfer applications of NCL, e.g. air conditioning, solar collectors, extraction of geothermal energy, etc. For the same wall temperature and geometrical parameters, comparison is made between CO₂ and water in terms of heat transfer rate. Water is considered at atmospheric pressure whereas CO₂ is either in subcritical (liquid) or supercritical state. Liquid CO₂ exhibits very high heat transfer rate, approximately seven times higher than water, whereas performance of supercritical CO₂ depends on the operating pressure and temperature. Effect of loop operating pressure on the system performance is also investigated. Results show that near pseudo-critical region, CO₂ yields very high heat transfer rate, approximately seven times higher than water. Results also show that, due to the presence of bends and local buoyancy effects, fluid parameters such as local velocity and temperature vary in all three dimensions. Validation of simulation results against experimental results reported in the literature with respect to modified Grashof number (Gr_m) and Reynolds number (Re) exhibit good agreement. Additionally, new correlations are proposed for Re in terms of Gr_m, friction factor (f) in terms of Re, and Nusselt number (Nu) in terms of Re and Prandtl number (Pr).     

Visualization of heat flow due to natural convection within triangular cavities using Bejan’s heatline concept

Natural convection and flow circulation within a cavity has received significant attention in recent times. The wide range of applicability of flow inside a cavity (food processing industries, molten metal industries, etc.) requires thorough understanding for cost efficient processes. This paper is based on comprehensive analysis of heat flow pattern using Bejan’s heatline concept. The key parameters for our study are the Prandtl number, Rayleigh number and Nusselt number. The values of Prandtl number (0.015, 0.026, 0.7 and 1000) have been chosen based on wide range of applicability. The Rayleigh number has been varied from 10² to 10⁵. Interesting results were obtained. For low Rayleigh number, it is found that the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction. But as Ra increases, flow slowly becomes convection dominant. It is also observed that multiple secondary circulations are formed for fluids with low Pr whereas these features are absent in higher Pr fluids. Multiple circulation cells for smaller Pr also correspond multiple cells of heatlines which illustrate less thermal transport from hot wall. On the other hand, the dense heatlines at bottom wall display enhanced heat transport for larger Pr. Further, local heat transfer (Nu_l, Nu_t) are explained based on heatlines. The comprehensive analysis is concluded with the average Nusselt number plots. A correlation for average heat transfer rate and Ra has been developed and the range of Rayleigh number is also found, to depict the conduction dominant heat transfer.     

Effect of suction/injection on the flow of a micropolar fluid past a continuously moving plate in the presence of radiation

An analysis is carried out to study the effect of suction and injection on the flow and heat transfer characteristics for a continuous moving plate in a micropolar fluid in the presence of radiation. The boundary layer equations are transformed to non-linear ordinary differential equations. Numerical results are presented for the distribution of velocity, microrotation and temperature profiles within the boundary layer. The effects of varying the Prandtl number, Pr, the radiation parameter, N and porosity parameter, F_w, are determined.     

Impingement heat transfer of a single thermal plume on the upper wall

We present numerical calculations of the generation, growth and impingement of a thermal plume in a two-dimensional buoyancy induced flow. Numerical values are obtained for the aspect ratios H/W=1/4, 3/8, 1/2, the Grashof numbers Gr=10⁴, 10⁵, and the Prandtl number Pr=170. Impingement heat transfer on the upper wall is evaluated at various times. Numerical results show that before a thermal plume impinges on the upper heated wall, the thermal conduction layer, which is the stable stratification, near the upper wall becomes thinner and the local heat transfer peaks. The local Nusselt number approaches the steady condition after the impingement of a thermal plume. Additionally, under certain conditions the stream function takes a symmetrical form of two ellipses.     

Heat and mass transfer in a separated flow region for high Prandtl and Schmidt numbers under pulsatile conditions

Heat and mass transfer phenomena were studied in the sudden expansion region of a pipe under steady and pulsatile conditions. The Prandtl number was varied from 100 to 12 000 and the flow was characterized for both uniform and parabolic entrance velocity profiles. A uniform velocity profile was used for pulsatile flow. It was found that heat transfer in the recirculation region was maximal near the area where wall shear was minimal. Blunting of the inlet profile caused the point of maximum heat transfer to move upstream. There was a nonlinear effect of Prandtl number on heat transfer which plateaued for Pr > 10³. The wall shear rate in the separation zone varied markedly with pulsatile flows, but the wall heat transfer remained relatively constant. The time-averaged pulsatile heat transfer at the wall was approximately the same as with steady flow with the mean Reynolds number. However, the isotherms within the pulsatile flow were markedly different from steady flow. The results demonstrate the complexity of separation flows and identify characteristic regions of high and low heat/mass transfer for high Prandtl/Schmidt pulsatile flow.     

Effect of pressure stress work and viscous dissipation in some natural convection flows

A regular two-parameter perturbation analysis is presented here to study the effects of both viscous dissipation and pressure stress on natural convection flows. Four different vertical flows have been analyzed, those adjacent to an isothermal surface and uniform heat flux surface, a plane plume and flow generated from a horizontal line energy source on a vertical adiabatic surface, or wall-plume. For high gravity levels, stress work effects may be important for gases at very low temperatures, and for high Prandtl number liquids. Significant changes in heat transfer and flow quantities are observed even at moderate values of the perturbation parameters. For the entire range of Prandtl number values considered, the viscous dissipation term is seen to inhibit heat transfer from the surface for heated upward flows. The pressure term enhances heat transfer from the surface for lower Prandtl numbers. However, this effect is seen to reverse at Pr = 100, for both the isothermal and uniform flux surfaces. It is observed that viscous dissipation effects on heat transfer are much smaller than those due to the pressure stress, for many practical circumstances.     

Enhancement of heat transfer in turbulent channel flow over dimpled surface

A systematic numerical investigation of heat transfer in turbulent channel flow over dimpled surface is conducted. Both symmetric (or spherical) and asymmetric dimple with different depth ratios (h/D) and skewness (Dx and Dz) are considered for a series of Reynolds numbers Re_2H (based on bulk velocity and full channel height) between 4000 and 6000 while Prandtl number Pr is fixed at 0.7. It is found that the optimum dimple configuration for enhancing heat transfer measured in terms of the volume goodness factor is obtained for the case of asymmetric dimple with a depth ratio of h/D = 15% and stream-wise skewness of Dx = 15%. The heat transfer capacity in terms of Nusselt number is significantly increased, while the associated pressure loss is kept almost to the same level as the symmetric dimple with the same depth ratio. The present study also suggests that the heat transfer enhancement is closely related to ejection with counter-rotating flow, intensified secondary flow and vortex structures at the downstream rim of asymmetric dimple. All these findings suggest that a carefully designed asymmetric dimpled surface presents a viable means of enhancing heat transfer compared to the symmetric dimple.