This work examines the steady two-dimensional mixed convection boundary layer flow of non-Newtonian Carreau fluid embedded in a porous medium. The impermeable wedge is at rest over which the momentum and thermal boundary layers form due to motion of Carreau fluid with a large Reynolds number. We consider local thermal non-equilibrium for which the temperature of the solid porous medium is different from that of fluid phase, and hence, a single heat-transport equation is replaced by a two-temperature model. The governed equations for flow and heat transfer are converted into a system of ordinary differential equations using a similarity approach. It is observed that local thermal non-equilibrium effects are dominant for small interphase heat transfer rate and porosity scaled conductivity parameters. It is shown that the temperature at any location of the solid porous medium is always higher than that of fluid phase. When these parameters are increased gradually the local thermal equilibrium phase is recovered at which the temperatures of the fluid and solid are identical at each pore. Similar trend is noticed for both shear-thinning and shear-thickening fluids. The results further show that heat exchange between the fluid and solid porous medium is similar to both assisted and opposed flows and Carreau fluid. The velocity and temperature fields for the various increasing fluid index, Grashof number and permeability show that the thickness of the momentum and thermal boundary layer is thinner.

In this study, a numerical investigation on mixed convection inside a trapezoidal cavity with a pair of rotating cylinders has been conducted. Three different power-law fluid indexes (n = 1.4, 1.0, and 0.6) have been considered to model different sets of non-Newtonian fluids. Four separate cases are considered dependent on the rotation orientation of the cylinders within the cavity. In the first two cases, the cylinders rotate in the same direction, i.e., both counter-clockwise (CCW), and both clockwise (CW), whereas, in the other two cases, cylinders rotate in opposite directions (CW-CCW and CCW-CW). Simulations have been carried out over a broad range of Reynolds number (from 0.5 to 500) and angular speeds (a dimensionless value from 0 to 10). The average Nusselt number values at the isothermal hot inclined cavity surface are determined to evaluate heat transfer performance in various circumstances. Streamlines and isotherm contours are also plotted for better understandings of the effects of different cases for various parameters on thermal and fluid flow fields. It is found that the Nusselt number varies non-linearly with different angular speeds of the cylinders. The combined effect of the mixing induced by cylinder rotation and viscosity characteristics of the fluid dictates the heat transfer in the system. Predictions from the numerical investigation provide insights onto the sets of key parametric configuration that have dominant influence on the thermal performance of lid driven cavity with double rotating cylinders.

Physics-informed neural networks (PINNs) have gained popularity across different engineering fields due to their effectiveness in solving realistic problems with noisy data and often partially missing physics. In PINNs, automatic differentiation is leveraged to evaluate differential operators without discretization errors, and a multitask learning problem is defined in order to simultaneously fit observed data while respecting the underlying governing laws of physics. Here, we present applications of PINNs to various prototype heat transfer problems, targeting in particular realistic conditions not readily tackled with traditional computational methods. To this end, we first consider forced and mixed convection with unknown thermal boundary conditions on the heated surfaces and aim to obtain the temperature and velocity fields everywhere in the domain, including the boundaries, given some sparse temperature measurements. We also consider the prototype Stefan problem for two-phase flow, aiming to infer the moving interface, the velocity and temperature fields everywhere as well as the different conductivities of a solid and a liquid phase, given a few temperature measurements inside the domain. Finally, we present some realistic industrial applications related to power electronics to highlight the practicality of PINNs as well as the effective use of neural networks in solving general heat transfer problems of industrial complexity. Taken together, the results presented herein demonstrate that PINNs not only can solve ill-posed problems, which are beyond the reach of traditional computational methods, but they can also bridge the gap between computational and experimental heat transfer.

This paper studies the local thermal nonequilibrium (LTNE) model for two-dimensional mixed convection boundary-layer flow over a wedge, which is embedded in a porous medium in the presence of radiation and viscous dissipation. It is considered that the temperature of the fluid and solid phases is not identical; hence, we require two energy equations: one for each phase. The motion of the mainstream and wedge is approximated by the power of distance from the leading boundary layer. The flow and heat transfer in the LTNE phase is governed by the coupled partial differential equations, which are then reduced to nonlinear ordinary differential equations via suitable similarity transformations. Numerical simulations show that when the interphase rate of heat transfer is large, the system attains the local thermal equilibrium (LTE) state and so is for porosity scaled conductivity. When LTNE is strong, the fluid phase reacts faster to the mainstream temperature than the corresponding solid phase. The state of LTE rather depends on radiation and viscous dissipation of the model. Further, numerical solutions successfully predicted the upper and lower branch solutions when the velocity ratio is varied. To assess which of these solutions is practically realizable, an asymptotic analysis on unsteady perturbations for a large time leading to linear stability needs to be performed. This shows that the upper branch solutions are always stable and practically realizable. The physical dynamics behind these results are discussed in detail.

An objective function combining the first and second laws of thermodynamics has been employed to delineate the thermodynamic performance on mixed convection around a vertical hollow, rotating cylinder within the laminar range with the variation of Rayleigh number (10 4 ≤ Ra ≤ 10 8 ), Reynolds number (Re D < 2100), and aspect ratio (1 ≤ L/D ≤ 20). Entropy generation in the system is predominantly triggered by heat transfer in comparison to fluid friction. The irreversibility incurred progressively increases with an increase in Ra and Re D . The variation pattern of ( I / Q ) Rotation / ( I / Q ) Non − Rotation has been demonstrated to find out the optimized regime where heat transfer is maximum within the laminar range. The contribution of fluid friction irreversibility toward total irreversibility rises abruptly with an increase in Re D for all cases of L/D and Ra. To demonstrate this study's thermodynamic characteristics, the static temperature contours as well as the contours of entropy generation have been represented pictorially. The estimation of cooling time has been reported by using the method of lumped capacitance.

Numerical investigations are performed on natural and mixed convection around stationary and rotating vertical heated hollow cylinder with negligible wall thickness suspended in the air. The fluid flow and heat transfer characterization around the hollow cylinder are obtained by varying the following parameters, namely, Rayleigh number (Ra), Reynolds number (Re D ), and cylindrical aspect ratio (L/D). The heat transfer quantities are estimated by varying the Rayleigh number (Ra) from 10 4 to 10 8 and aspect ratio (L/D) ranging from 1 to 20. Steady mixed convection with active rotation of hollow vertical cylinder is further studied by varying the Reynolds number (Re D ) from 0 to 2100. The velocity vectors and temperature contours are shown in order to understand the fluid flow and heat transfer around the vertical hollow cylinder for both rotating and nonrotating cases. The surface average Nusselt number trends are presented for various instances of Ra, Re D, and L/D and found out that the higher rate of heat loss from the cylinder wall occurs at high Ra, low L/D (short cylinder) and high Re D .

Mixed convection heat transfer in a channel filled with porous medium and containing an isolated heat source at the bottom wall is studied in this work. The porous medium is assumed to be made of circular cylinders and is placed only on the heater surface. Three different configurations of porous medium are considered in this study. Pore-scale numerical simulations are carried out using the exact geometry of porous medium. The same configuration is also investigated using the volume-averaged approximation. The temperature distribution of the heater surface obtained from the pore-scale numerical simulation is compared with the results obtained from the volume-averaged numerical simulation. Parametric studies are carried out by varying the material of the cylinders, the porosity, and the height of the porous medium. The effects of Grashof number and Reynolds number of the flow are also studied as part of this investigation. The results obtained from the pore-scale numerical simulations show that the presence of the porous medium leads to reduction in heat transfer, while the results obtained from the volume-averaged numerical simulations show an enhancement of heat transfer due to the presence of the porous medium on the heater surface. However, the pore-scale numerical simulation results show that the heat transfer enhancement is only possible if the channel height is completely filled with the selected porous medium.

In this work, experimental and numerical results were obtained to analyze the effect of surface thermal radiation on heat transfer by mixed convection in a ventilated cavity. Experimental temperature profiles were obtained at six different depths and heights consisting of 14 thermocouples each. Five turbulence models were evaluated against experimental data. The radiative heat transfer model was solved with the discrete ordinate method. The effect of thermal radiation on experimental heat transfer coefficients is significant; it increases between 87% (Re = 30,372 and Ra = 3.04 × 10 11 ) and 110% (Re = 6021 and Ra = 2.27 × 10 11 ), when the emissivity of the walls increases from 0.03 to 0.98.

Mixed convection heat transfer in a vertically oriented air-cooled square enclosure is simulated numerically using the finite volume method. The vertical left wall being heated and all others are considered insulated. The effects of six opposite and staggered inlet/outlet openings' locations over the top and bottom walls are investigated, with different sizes of the opening height (0.05 ≤ d / H ≤ 0.3). The objective is to figure out the better size and location of inlet and outlet to acquire more efficient cooling system in the enclosure by maximizing the heat removal rate. This is conducted over ranges of the governing parameters, which are the Richardson number (0 ≤ Ri ≤ 30) and the Reynolds number (50 ≤ Re ≤ 250). The results show that the level of heat transfer enhancement increases with increasing the opening height, with an optimal size of d/H = 0.25 for obtaining maximum heat removal, for all Richardson and Reynolds numbers. The results also indicate that the higher heat dissipation occurs when the cold air is injected vertically near the hot wall and exits the enclosure from the opposite or staggered outlet, for all Richardson and Reynolds numbers.

The mixed convection in an open trapezoidal lid-driven cavity connected with a channel is investigated in the present paper. Four different cases were considered depending on the movement of the cavity sidewalls. For case I, the left sidewall moves downward; for case II, the left sidewall moves downward and the right one moves upward; while for case III, only the right sidewall moves upward. A comparative case (case 0) is accounted when both sidewalls are assumed stationary. The base of the cavity is subjected to a localized heat source of constant temperature T h . The effects of Richardson number Ri and Reynolds number ratio Re r on the flow and thermal fields have been investigated. The results indicated that for cases I and II, the average Nusselt number increases with the increase of the Richardson number and Reynolds number ratio. Moreover, it was found that the maximum average Nusselt number occurs with case I. When the lid-driven speed is three times that of the inlet airflow velocity, the augmentations of the average Nusselt number compared with stationary walls are 163%, 158%, and 96% for cases I, II, and III, respectively.

This work addresses the magnetic and radiation effects on the fully developed mixed convective flow in a vertical channel occupied by a porous medium with the thermal nonequilibrium state. The assumption that the fluid is electrically conducted is taken into account and permitted by a uniform transversal magnetic field while the temperature of the wall is changing linearly with the direction of the fluid flow. The spectral collocation technique is used for the numerical solution, whereas the analytical solution is governed for the special case when the drag force F * and the ratio of porosity-scaled thermal conductivity γ are zero. It is observed that, in the buoyancy assisted case, the fluid flow for Ra < 10 2 , ( Nu f ) increased near the wall with increasing the Hartmann number (M). Beyond this when Ra ≥ 10 2 , ( Nu f ) is decreased with increasing M. It is also perceived that there exists an interval [ 0 , H 0 ] in which ( Nu f ) increases with increasing M as well as increasing radiation parameter R d , furthermore beyond the value of H 0 , Nu f decreasing asymptotically. While for the buoyancy opposed case, the flow separation and inflection point appear in the velocity profile for different values of M, further both the flow separation and inflection point are dying out as M increases. Overall, for the both cases, the magnetic and radiation parameters are stabilizing the flow in the system.

The present comment concerns some doubtful results included in the above paper.

A numerical study on the mixed convection of Al 2 O 3 –water nanofluid in a lid-driven inclined square enclosure partially heated from below is performed based on Buongiorno's two phase model. The velocity of the nanoparticles relative to the base fluid is considered due to thermophoresis and Brownian diffusion. The thermophysical properties of the nanofluid are assumed to be dependent on temperature as well as the nanoparticle volume fraction. A control volume method over a staggered grid arrangement is used to discretize the governing equations. The discretized equations of two-dimensional continuity, momentum, energy, and volume fraction are solved through a pressure-correction-based semi-implicit method for pressure linked equations (SIMPLE) algorithm. The effects of relevant parameters such as nanoparticle diameter (25 nm ≤ d p ≤ 90 nm), Richardson number ( 0.1 ≤ Ri ≤ 5 ), nanoparticle bulk volume fraction (0 ≤ φ b ≤ 0.05) on the mixed convection of the nanofluid is studied by considering the inclination angle of the enclosure to vary between 0 deg and 60 deg. The entropy generation as well as the Bejan number is evaluated to illustrate the thermodynamic optimization of the mixed convection. Both the heat transfer and entropy generation are higher in the nanofluid compared to the clear fluid and the rate of increment in entropy generation remains lower than the rate by which the heat transfer is augmented in the nanofluid. We find that due to the presence of the Brownian diffusion and thermophoresis in the nonhomogeneous model, a higher heat transfer is yielded as compared to the homogeneous model. The discrepancy between the homogeneous and nonhomogeneous models is significant when the mixed convection is dominated by the shear force. When the mixed convection is dominated by the thermal buoyancy, an increase in positive inclination angle of the enclosure creates a significant increment in the heat transfer.

In this work, a solution technique is proposed by synergistically combining asymptotics and computational fluid dynamics to ascertain a problem of laminar mixed convection heat transfer in a vertical channel. First, numerical simulation is carried out on a vertical channel that consists of an aluminum heater plate assembly at the center of the channel. The numerical model is treated as a conjugate heat transfer problem, and the concept of perturbation and blending is incorporated wherein the limiting solution of natural and forced convection is obtained in terms of average Nusselt number. These correlations are then blended to find a unified composite correlation that work very well for extreme limits of mixed convection. The Richardson number is chosen as an independent variable in the present analysis; as a result, the Nusselt number correlation is cogent for the mixed convection region. Upon performing the numerical simulations, the results of the mixed convection are then compared with experimental results available in the literature for the purpose of validation of the numerical solution. The results of the present work emphasize that, with minimum computational fluid dynamics (CFD) solutions, one can obtain a reasonably good composite correlation for the Nusselt number for mixed convection and also a substantial reduction of computations is possible ensuing an asymptotically flawless solution.

A numerical study of magneto-hydrodynamic mixed convection in a cavity has been conducted to investigate the influence of magnetic field on integrated flux of thermal energy, linear momentum, and kinetic energy. Shear force through lid motion establishes the forced convection effect and buoyancy force due to differential heating of the moving lid and the stationary interface ensures the natural convection phenomenon. Additionally, conduction through the solid slab with prescribed temperature at the outer surface attached to the cavity induces conjugate heat transfer. Further, the top and bottom walls throughout the domain are kept insulated and a uniform horizontal magnetic field is applied on the interface toward left. Fluid flow and heat transfer characteristics are examined for a range of Hartmann number (Ha): 0, 10, 50, and 120 at fixed values of Reynolds number, Grashof number, and Prandtl number of 300, 9 × 10 4 and 0.71, respectively. The result shows that the transport of heat in the near wall regions of the fluid domain is primarily governed by diffusion, whereas advection appears stronger in the central region of the cavity. Increase in magnetic field strength from Ha = 0 to 120 gradually suppresses the recirculating structure of the flow signifying a reduction in advective strength as depicted by the decrease in the value of total integrated heat flux from 25.15 × 10 - 3 to 6.0 × 10 - 3 . The reduction in heat flux, momentum fluxes, and kinetic energy fluxes with increase in magnetic field has been well correlated in the range of 0 ≤ Ha ≤ 120 .

This paper aims to investigate the entropy generation in slip flow due to double rotating disks. Heat equation is formulated by considering effects of viscous dissipation, Joule heating and nonlinear thermal radiation. Brownian motion and thermophoresis effects of nanofluid are also discussed. Applied magnetic field is considered to be time dependent. Homogeneous–heterogeneous reactions are also studied. Von Karman transformations are used. Homotopy analysis method is implemented on system of equations for convergent series solutions. Influence of various flow parameters on entropy, Bejan number, velocity, temperature, Nusselt number, and skin friction is discussed through graphs and tables. Axial velocity decays for higher nonlinear mixed convection variable of temperature and velocity slip parameter. Temperature rises for larger thermal slip parameter and thermophoresis parameter. Entropy and Bejan number are increasing for higher estimation of homogeneous reaction parameter and diffusion parameters.

This analysis studies the impact of an inclined magnetic field, hall current, and thermal radiation on fully developed electrically conducting mixed convection flow through a channel. The governing equations are nondimensionalized. The resulting system of nonlinear ordinary differential equations is solved utilizing spectral quasi-linearization method. Impact of all the pertaining flow parameters of this study on all the dimensionless profiles was calculated and presented through plots. Also, the nature of the physical parameters was calculated and presented in table form. This study clearly exhibits that the inclined magnetic field influences the fluid flow remarkably.

High pressure/high temperature forced and mixed convection experiments have been performed with helium and nitrogen at temperatures and pressures up to 893 K and 64 bar, respectively. The test section had a 16.8 mm ID flow channel in a 108 mm OD graphite column. Flow regimes included turbulent, transitional, and laminar flows with the inlet Reynolds numbers ranging from 1500 to 15,000. Due to strong heating, the local Reynolds number decreased by up to 50% over the 2.7 m test section. In addition, heat transfer degradation and flow laminarization caused by intense heating led to Nusselt numbers 20–50% lower than the values given by the modified Dittus–Boelter and modified Gnielinski correlations. Flow laminarization criteria were considered based on a dimensionless acceleration parameter (K v ) and buoyancy parameter (B o * ). Upward turbulent flows displayed higher wall temperatures than downward flows, due to the impact of flow laminarization which is not expected to affect buoyancy-opposed flows. Laminar Reynolds number flows presented an opposite behavior due to the enhancement of heat transfer for buoyancy-aided flows. At low Reynolds numbers, downward flows displayed higher and lower wall temperatures in the upstream and downstream regions, respectively, than the upward flow cases. In the entrance region of downward flows, convection heat transfer was reduced due to buoyancy leading to higher wall temperatures, while in the downstream region, buoyancy-induced mixing caused higher convection heat transfer and lower wall temperatures.

In this study, convective heat transfer in a discretely heated parallel-plate vertical channel which simulates an IC package is investigated experimentally and numerically. Both natural and mixed convection cases are considered. The primary focus of the study is on determining optimum relative lengths of the heat sources in order to reduce the hot spot temperature and to maximize heat transfer from the sources to air. Various values of the length ratio and the modified Grashof number (for the natural convection case)/the Richardson number (for the mixed convection case) are examined. Conductive and radiative heat transfer is included in the analysis while air is used as the working fluid. Surface temperatures of the heat sources and the channel walls are measured in the experimental study. The numerical studies are performed using a commercial CFD code, ANSYS fluent . The variations of surface temperature, hot spot temperature, Nusselt number, and global conductance of the system are obtained for varying values of the working parameters. From the experimental studies, it is showed that the use of identical heat sources reduces the overall cooling performance both in natural and mixed convection. However, relatively decreasing heat sources lengths provides better cooling performance.

Two-dimensional computational fluid dynamics simulations of mixed convection heat transfer through aluminum metal foams partially filled in a vertical channel are carried out numerically. The objective of the present study is to quantify the effect of metal foam thickness on the fluid flow characteristics and the thermal performance in a partially filled vertical channel with metal foams for a fluid velocity range of 0.05–3 m/s. The numerical computations are performed for metal foam filled with 40%, 70%, and 100% by volume in the vertical channel for four different pores per inch (PPIs) of 10, 20, 30, and 45 with porosity values varying from 0.90 to 0.95. To envisage the characteristics of fluid flow and heat transfer, two different models, namely, Darcy Extended Forchheirmer (DEF) and Local thermal non-equilibrium, have been incorporated for the metal foam region. The numerical results are compared with experimental and analytical results available in the literature for the purpose of validation. The results of the parametric studies on vertical channel show that the Nusselt number increases with the increase of partial filling of metal foams. The thermal performance of the metal foams is reported in terms of Colburn j and performance factors.