Mean Temperature Profiles Research Articles

Heat transfer enhancement in turbulent dual jet and its thermal characterizations using large eddy simulation

Heat transmission is the prime aim of any thermal device that requires effective cooling. This requirement becomes significantly more vital for a turbulent jet, which is encountered quite often in industrial and engineering purposes. The present study investigates thermal and turbulence characteristics of a turbulent dual jet using Large Eddy Simulation (LES). The bottom surface is maintained at isothermal and isoflux boundary conditions and the flow Re is 7500. The profiles of Reynolds shear-stress, turbulence kinetic energy, wall friction coefficient, r.m.s. temperature, mean Nusselt number, bottom wall temperature, and turbulent heat fluxes are investigated and scaled using outer, thermal and inner parameters. Variations in Nusselt number and bottom wall temperature are also studied. A significant rise in Nusselt number is observed nearer to the jet exit which enriches the cooling rate. Overall, the finding shows that compared to the wall jet and parallel flowing offset jet, the inflection point in the mean temperature profile of the present dual jet scaled with the thermal variable and outer variable is higher. Further, it is also observed that the thermal transmission spreading out of the wall is quite efficient in the low-pressure area of the dual jet. Near-wall eddies and the development of roller-like structures are seen. The interaction occurring through the inner and outer layers increases the momentum transfer along with turbulent heat-flux and a large amount of heat is lost due to turbulent flow.

Direct numerical simulation of one-sided forced thermal convection in plane channels

We carry out direct numerical simulations (DNS) of turbulent flow and heat transfer in pressure-driven plane channels, by considering cases with heating on both walls, as well as asymmetric heating limited to one of the channel walls. Friction Reynolds numbers up to ${Re}_{\tau } \approx 2000$ are considered, and Prandtl numbers from ${Pr}=0.025$ to ${Pr} = 4$ , the temperature field being regarded as a passive scalar. Whereas cases with symmetric heating show close similarity between the temperature and the streamwise velocity fields, with turbulent structures confined to either half of the channel, in the presence of one-sided heating the temperature field exhibits larger regions with coherent fluctuations extending beyond the channel centreline. Validity of the logarithmic law for the mean temperature is confirmed, as well as universality of the associated von Kármán constant, which we estimate to be $k_{\theta } = 0.459$ . Deviations from the logarithmic behaviour are much clearer in cases with one-sided heating, which feature a wide outer region with parabolic mean temperature profile. The DNS data are exploited to construct a predictive formula for the heat transfer coefficient as a function of both Reynolds and Prandtl number. We find that the reduction of the thermal efficiency in the one-sided case is approximately $20\,\%$ at unit Prandtl number; however, it can become much more significant at low Prandtl number.

Characterization of Non-Neutral Urban Canopy Wind Profile Using CFD Simulations—A Data-Driven Approach

Horizontally averaged wind profiles inside the urban canopy are used in many studies and numerical models. The existing analytical models are only applicable to a small range of aspect ratios and mostly neutral atmospheric conditions due to their underlying assumptions. In this study, a surrogate model for predicting horizontally averaged wind profiles in the street canyons of an idealized urban canopy for a wide range of urban morphologies and thermal forcing scenarios is developed with the help of machine learning techniques and computational fluid dynamics (CFD) simulation data. The influence of urban morphological parameters, atmospheric stability and wind conditions on the urban canopy wind flow is modeled using machine learning algorithms applied on CFD simulation results. The numerical model is validated using wind-tunnel data. Steady-state Reynolds averaged Navier–Stokes (RANS) simulations with a standard k-ϵ turbulence model for 252 different simulation conditions are performed on an idealized building geometry that consists of a regular array of 4 × 4 cubes. The simulation results are averaged horizontally to obtain the mean velocity and temperature profiles. Surrogate models are developed using the simulation outputs as training examples and the best model is chosen by comparing the performance of different machine learning models. The surrogate artificial neural network (ANN) model of this study outperforms the current state-of-the-art models in the prediction of horizontally averaged mean wind profile inside the urban canopy. The mean error (ME) and root-mean-square error (RMSE) of the discrete point prediction ANN model of this study are 0.016 m/s and 0.060 m/s, respectively, which is significantly lower compared to the best of the legacy models for which the errors are 0.048 m/s and 0.387 m/s, respectively.

Strong dissimilarity between heat and momentum transfer in rotating Couette flows

Rotation influences flows and transport processes in many engineering applications, however, even in canonical flow cases, the effects of rotation are not fully understood. Here, we present the results of direct numerical simulations of heat transfer plane Couette and Taylor-Couette flows subject to rotation about the spanwise and axial axis, respectively. Temperature is a passive scalar since buoyancy is neglected. The Reynolds number Re and the rotation rate RΩ are systematically varied to thoroughly investigate the influence of rotation on heat and momentum transfer and the Reynolds analogy. We find that with increasing anti-cyclonic rotation, the Nusselt numbers for the momentum transfer Num and heat transfer Nuh both increase at first before declining and approaching unity at rapid rotation rates when the flow becomes fully laminar. The Reynolds analogy factor RA=Nuh/Num is near unity for non-rotating Couette flows, but it grows significantly with increasing rotation rate. RA reaches a maximum of approximately 2 at low Re up to 6 and 8 near RΩ=1 at higher Re in plane Couette and Taylor-Couette flow, respectively. The simulations thus show that the Reynolds analogy between heat and momentum transfer breaks down and that the heat transfer can become much faster than moment transfer when plane Couette and Taylor-Couette flows are subject to anti-cyclonic rotation. This happens at low Re as well as higher Re when the flows are fully turbulent. The turbulent Prandtl becomes much smaller than unity and the mean velocity and temperature profiles differ when the Reynolds analogy breaks down. We also present empirical models for Num and RA, which agree reasonably well to very well with the data within a limited RΩ range.

Single-Mode Solutions for Convection and Double-Diffusive Convection in Porous Media

This work employs single-mode equations to study convection and double-diffusive convection in a porous medium where the Darcy law provides large-scale damping. We first consider thermal convection with salinity as a passive scalar. The single-mode solutions resembling steady convection rolls reproduce the qualitative behavior of root-mean-square and mean temperature profiles of time-dependent states at high Rayleigh numbers from direct numerical simulations (DNS). We also show that the single-mode solutions are consistent with the heat-exchanger model that describes well the mean temperature gradient in the interior. The Nusselt number predicted from the single-mode solutions exhibits a scaling law with Rayleigh number close to that followed by exact 2D steady convection rolls, although large aspect ratio DNS results indicate a faster increase. However, the single-mode solutions at a high wavenumber predict Nusselt numbers close to the DNS results in narrow domains. We also employ the single-mode equations to analyze the influence of active salinity, introducing a salinity contribution to the buoyancy, but with a smaller diffusivity than the temperature. The single-mode solutions are able to capture the stabilizing effect of an imposed salinity gradient and describe the standing and traveling wave behaviors observed in DNS. The Sherwood numbers obtained from single-mode solutions show a scaling law with the Lewis number that is close to the DNS computations with passive or active salinity. This work demonstrates that single-mode solutions can be successfully applied to this system whenever periodic or no-flux boundary conditions apply in the horizontal.

Artificial neural network-based subgrid-scale models for LES of compressible turbulent channel flow

Fully connected neural networks (FCNNs) have been developed for the closure of subgrid-scale (SGS) stress and SGS heat flux in large-eddy simulations of compressible turbulent channel flow. The FCNN-based SGS model trained using data with Mach number Ma=3.0 and Reynolds number Re=3000 was applied to situations with different Mach numbers and Reynolds numbers. The input variables of the neural network model were the filtered velocity gradients and temperature gradients at a single spatial grid point. The a priori test showed that the FCNN model had a correlation coefficient larger than 0.91 and a relative error smaller than 0.43, with much better reconstructions of SGS unclosed terms than the dynamic Smagorinsky model (DSM). In a posteriori test, the behavior of the FCNN model was marginally better than that of the DSM in predicting the mean velocity profiles, mean temperature profiles, turbulent intensities, total Reynolds stress, total Reynolds heat flux, and mean SGS flux of kinetic energy, and outperformed the Smagorinsky model.

Transitional fluctuations of thermal boundary layer in horizontal convection

Fluctuations of thermal boundary layer (BL) in horizontal convection (HC) strongly affect the global heat transport through the HC sample and predominately determine the flow state. Here, we present a systematic experimental study of vertical profiles of mean temperature and temperature variance in the thermal BL of a HC flow that are heated and cooled from the bottom. Three rectangular HC samples filled with de-ionised water are used in the experiment. They have the same aspect ratios Γ≡L:W:H=10:1:1 but with different L (L, W and H are, respectively the length, width and height of the sample). The Rayleigh number Ra varies in the range 2×1010≲Ra≲1.3×1013, and the Prandtl number Pr is in the range 3.9≲Pr≲6.5. For all Ra in the studied range, we find that the cold thermal BL above the cooling plate remains laminar and the vertical profiles of measured mean temperature follow a universal scaling form predicted for laminar BL by Yan, Shishkina and He (J. Fluid Mech., vol. 915, 2021, R5). Above the heated plate, on the contrary, temperature fluctuations in the hot thermal BL undergo several transitional changes as Ra increases. For Ra≲6×1010, the thermal BL has a stable double-layer structure agreeing with the predicted scaling form for laminar BL and the temperature fluctuations are nearly zero. In the range 6×1010≲Ra≲1.1×1011, temperature fluctuations inside the BL have intermittent decayed oscillations, which results in deviations of the measured mean temperature profiles from the predicted scaling form. For 1.1×1011≲Ra≲1012, the temperature oscillations inside the BL become continuous and stable. Correspondingly, an intermediate layer emerges in the thermal BL, where the mean temperature profile has a sinusoidal variation on the distance z above the heated plate. For 1012≲Ra≲1.3×1013, the temperature fluctuations in the intermediate layer become irregular and intermittent, indicating a growing mixing intensity as Ra increases. When Ra exceeds 7×1012, the vertical distribution of mean temperature in the intermediate layer is uniform because of strong mixing. The temperature-variance vertical profiles follow a power-law function σT2(z)∼z−γ with the exponent γ=0.66, which agrees with the Priestley prediction for thermal convection over a heated horizontal surface. Our results thus reveal a series of transitional changes of temperature fluctuations, from laminar to chaotic states, in the hot thermal BL. They predominately account for the enhancement of global heat transport in the HC flow observed by Reiter and Shishkina (J. Fluid Mech., vol. 892, 2020, R1).

Turbulent premixed hydrogen/air flame-wall interaction with heterogeneous surface reactions

A premixed H2/air flame impinging onto a flat surface in statistically stationary state is studied for both reactive and inert wall cases to gain insights into the effects of the heterogeneous surface reactions. Direct numerical simulation (DNS) results with detailed gas-phase chemistry and surface adsorption and desorption mechanisms indicate differences in the flame front topology and near-wall flame dynamics between these two cases. In the reactive surface case, gas-phase free radicals are inclined to be adsorbed with much reduced near-wall concentration. Consequently, the gas-phase heat release rate (HRR) close to the wall decreases as well because of the low availability of free radicals. However, extra heat released from the reactive surface partially compensates for such difference. Moreover, wall reactions will intensify the turbulent fluctuations of the wall temperature and wall heat flux, while for these two cases the mean temperature profile along the flame propagating direction remains similar.

Three-Dimensional Analytical Solutions for Acoustic Transverse Modes in a Cylindrical Duct with Axial Temperature Gradient and Non-Zero Mach Number

Cylindrical ducts with axial mean temperature gradient and mean flows are typical elements in rocket engines, can combustors, and afterburners. Accurate analytical solutions for the acoustic waves of the longitudinal and transverse modes within these ducts can significantly improve the performance of low order acoustic network models for analyses of acoustic behaviours and combustion instabilities in these kinds of ducts. Here, we derive an acoustic wave equation as a function of pressure perturbation based on the linearised Euler equations (LEEs), and the modified WKB approximation method is applied to derive analytical solutions based on very few assumptions. The eigenvalue system is built based on the proposed solutions and applied to predict the resonant frequencies and growth rate for transverse modes. Validations of the proposed solutions are performed by comparing them to the numerical results directly calculated from the LEEs. Good agreements are found between analytical reconstruction and numerical results of three-dimensional transverse modes. The system with both mean temperature profile and mean flow presents a larger absolute value of the growth rate than the condition of either uniform mean temperature or no mean flow.

Heat transfer and behavior of the Reynolds stress in Mach 6 boundary layer transition induced by first-mode oblique waves

This paper investigates a Mach 6 oblique breakdown via direct numerical simulation in conjunction with stability and quadrant analyses. Particular emphasis is placed on, first, the heat transfer and mean flow distortion in the near-wall and outer transitional boundary layer, and, second, the flow events that are responsible for the production of the Reynolds stress. The energy budget reveals that enhancement of viscous dissipation due to mean flow distortion dominates the heat transfer overshoot, while the dissipation due to fluctuations is lesser but not negligible. Downstream of the location of the peak mean heat flux, the wall temperature gradient (non-dimensionalized by the freestream temperature and local boundary layer thickness) varies little, owing to the occurrence of breakdown and the establishment of self-similarity. Renormalized by the boundary layer thickness, a new correlation of the Stanton number shows no overshoot or difference between the original overshoot region and the turbulent region, which indicates the possibility of similarity once breakdown has occurred. In the outer region, enhanced advective heat exchange strongly reshapes the mean temperature profile. Because of successive modal growth and nonlinear saturation, the contributions of the primary oblique mode, streak mode, and a superharmonic to the outer advective heat transfer are found to compete near the location of the peak heat flux. From the perspective of fluid motions, quadrant analysis highlights the evenly and broadly distributed joint probability density function (PDF) of the fluctuating velocities during transition, which results in overproduction of the Reynolds stress, while the PDF is concentrated around zero in the turbulent region. The flow event Q2 (ejection) overtakes Q4 (sweep) in the outer boundary layer of the transitional region, mainly owing to the primary mode, while the two events become attenuated and nearly achieve balance when transition is complete.

Experimental thermohydraulic characterisation of flow boiling and condensation in additive manufactured plate-fin heat exchanger

In this paper, an experimental thermohydraulic characterisation of flow boiling and condensation in additive manufactured plate-fin heat exchanger was conducted. An original experimental setup was developed allowing the operation of the heat exchanger in both boiling and condensation modes. This heat exchanger mainly consists of two identical rectangular cross-section channels, each of hydraulic diameter 3 mm, with n-pentane and water used as refrigerant and heat transfer fluid, respectively. An experimental metrology, based on the infrared thermography, was implemented in order to determine the temperature distribution of the heat exchanger external walls. Based on this temperature distribution, the mean temperature profiles of the n-pentane and water were determined, allowing the identification of three main flow zones: the subcooled liquid zone, the two-phase (boiling/condensation) zone and the superheated vapour zone. These mean temperature profiles were then coupled to a developed thermal model in order to determine the local thermal parameters, such as heat flux, vapour quality and heat transfer coefficient. The total pressure drop in the two-phase zone was finally determined. All the obtained results were analysed and compared with data available in the literature.

Explosive breakup and evolution of the thermal boundary layer around a pulse-heated microwire in sub- and supercritical CO2

This paper reports our experimental findings aimed to understand the importance of compressibility in fluid flow and heat transfer. A platinum microwire of diameter 50 μm was immersed in a pressure vessel filled with CO2 at different thermodynamic states around the critical point. The microwire was heated by an electric pulse resulting in a temperature rise of about 667 K during 0.35 ms. The snapshots of CO2 and the temporal profiles of mean temperature of the microwire were recorded. An explosive breakup of the thermal boundary layer is identified, manifested by a radial spreading fluid layer with a “fluffy” boundary. Since buoyancy can only drive upward motions, such a phenomenon is closely related to compressibility, as a result of complex interactions between thermoacoustic waves and large-density-gradient interfaces. This phenomenon is also responsible for the efficient cooling observed in the first 10 ms because expansion is a cooling process and can also help to evacuate high-temperature fluid. Afterward, the flow exhibits various buoyancy-driven patterns depending on the existence and intensity of surface tension: garland-like cluster, unstable gas column, or normal bubble, followed by a continuously thinning thermal boundary layer. Both the classic and the newly revised thermodynamic phase diagrams are employed and compared in this paper, suggesting the latter is proper and informative.

Modular Method for Estimation of Velocity and Temperature Profiles in High-Speed Boundary Layers

The fluid property variation caused by viscous heating affects the mean velocity and Reynolds stresses in compressible turbulent boundary layers, and as a result also affects the resulting skin friction and wall heat transfer. We present a method based on the work of Huang et al. (AIAA Journal, Vol. 31, No. 9, 1993, pp. 1600–1604) that estimates the mean velocity and temperature profiles, and therefore also the friction and heat transfer coefficients, for a given Mach number, Reynolds number, and wall thermal condition. The method is modular in the sense that it works with multiple variable-property scaling formulas (or “transformations”), velocity–temperature relationships, viscosity–temperature relationships, and equations-of-state. The method is validated against multiple Direct Numerical Simulation (DNS) cases and was found to predict the skin friction and wall heat flux to within 8 and 11% compared to the current state-of-the-art (the Van Driest II method), which is within 16 and 13% for the same data.

Spectral analysis on dissimilarity between turbulent momentum and heat transfers in plane Couette turbulence

Nonlinear interactions between different scales in turbulence result in both interscale and spatial transport of turbulent energy, and their role in the turbulent heat transfer mechanism is also of practical importance from an engineering viewpoint. In this study, we investigate a turbulent plane Couette flow with passive-scalar heat transfer at the Prandtl number of 0.71 to discuss the similarity/difference between scale interactions in velocity and temperature fields. The constant-temperature-difference boundary condition is used so that the mean velocity and temperature profiles are similar, and then, the roles of interscale and spatial transports are compared for the spectral transport budgets of turbulent energies and temperature-related statistics. We show that turbulent heat transfer occurs at relatively small streamwise length scales compared to momentum transfer, although molecular diffusion is more significant in the temperature field as the Prandtl number is less than 1. Detailed analysis on the transport budgets of temperature-related spectra shows that scale interactions in the temperature field supply more energy to small scales than those in the velocity field. This significant temperature cascade causes more energetic temperature fluctuation at small scales, resulting in the spectral dissimilarity between turbulent heat and momentum transfers.

Impact of Riverine Fresh Water on Indian Summer Monsoon: Coupling a Runoff Routing Model to a Global Seasonal Forecast Model

Rivers form an essential component of the earth system, with ~36,000 km3 of riverine freshwater being dumped into the global oceans every year. The role of rivers in controlling the sea-surface salinity and ensuing air-sea interactions in the Bay of Bengal (BoB) is well-known from observational studies; however, attempts to include rivers in coupled models used for seasonal prediction have been limited. This study reports the benefits of river routing in coupled models over prescribing observational river discharge and the impact on the Indian Summer Monsoon (ISM) simulation. Seasonal hindcasts are carried out using a state-of-the-art global coupled ocean-atmosphere-land-sea ice model, Climate Forecast System version 2, coupled to a runoff routing model. It is demonstrated that such a coupling leads to a better representation of the upper ocean stratification in northern BoB, causes mixed layer warming during July-August, and imparts a significant inter-annual variability to the mixed layer heat budget. The rainfall-runoff coupled feedback associated with ISM is captured better, and remote teleconnections with the equatorial Pacific are enhanced. Improved seasonal mean temperature and salinity profiles in the northern BoB lead to the formation of a thicker barrier layer, which is closely tied to the freshwater from rivers. These processes result in an overall enhancement of the ISM rainfall simulation skill, which stems from scale interactions between the sub-seasonal and seasonal variability of ISM. A significant community effort is required to reduce biases in land-surface processes to improve streamflow simulations, along with better parameterization of mixing of river water with the ocean.

Heat transport and temperature boundary-layer profiles in closed turbulent Rayleigh–Bénard convection with slippery conducting surfaces

We report direct numerical simulations (DNS) of the Nusselt number$Nu$, the vertical profiles of mean temperature$\varTheta (z)$and temperature variance$\varOmega (z)$across the thermal boundary layer (BL) in closed turbulent Rayleigh–Bénard convection (RBC) with slippery conducting surfaces ($z$is the vertical distance from the bottom surface). The DNS study was conducted in three RBC samples: a three-dimensional cuboid with length$L = H$and width$W = H/4$($H$is the sample height), and two-dimensional rectangles with aspect ratios$\varGamma \equiv L/H = 1$and$10$. The slip length$b$for top and bottom plates varied from$0$to$\infty$. The Rayleigh numbers$Ra$were in the range$10^{6} \leqslant Ra \leqslant 10^{10}$and the Prandtl number$Pr$was fixed at$4.3$. As$b$increases, the normalised$Nu/Nu_0$($Nu_0$is the global heat transport for$b = 0$) from the three samples for different$Ra$and$\varGamma$can be well described by the same function$Nu/Nu_0 = N_0 \tanh (b/\lambda _0) + 1$, with$N_0 = 0.8 \pm 0.03$. Here$\lambda _0 \equiv L/(2Nu_0)$is the thermal boundary layer thickness for$b = 0$. Considering the BL fluctuations for$Pr>1$, one can derive solutions of temperature profiles$\varTheta (z)$and$\varOmega (z)$near the thermal BL for$b \geqslant 0$. When$b=0$, the solutions are equivalent to those reported by Shishkinaet al.(Phys. Rev. Lett., vol. 114, 2015, 114302) and Wanget al.(Phys. Rev. Fluids, vol. 1, 2016, 082301(R)), respectively, for no-slip plates. For$b > 0$, the derived solutions are in excellent agreement with our DNS data for slippery plates.

A Blasius boundary layer study for liquids and gasses with variable thermo-physical characteristics and radiative heat flux

A numerical study is presented to discuss the temperature dependent viscosity and the thermo-physical properties near a heated flat surface. Temperature dependent viscosity for liquids show decrease in mean velocity profile and enhancement in mean temperature profile, whereas for gasses, it shows reverse effect. Moreover, the thermo-physical properties are treated as variable as they affect the flow phenomena. The basic flow can be modeled in terms of the scaling transformation for the large Reynolds number. Using similarity solutions, the leading order equations are converted into a set of ordinary differential equations which have been solved by fifth order Runga-Kutta integration method and Newton Raphson method. All of the obtained results are investigated for the base flow, concentration and temperature profiles and physical interpretation has been provided in the form figures and tables. The impact of viscosity for gases show an increase behavior in velocity profile but for liquids opposite impact is noticed. The variable conductivity and mass diffusivity parameters rise the temperature and concentration profiles. The temperature profile increases for increasing values of radiation parameter. Comparison has been made with past literature and the results show good agreement.

Central mean temperature scaling in compressible turbulent channel flows with symmetric isothermal boundaries

Originating from the generalized Reynolds analogy theory, an empirical scaling for the central mean temperature in compressible turbulent channel flow with symmetric isothermal boundaries is proposed. The empirical scaling is quite accurate and most of the relative errors are below 1.5% as assessed by available direct numerical simulation data at various Reynolds and Mach numbers. The mean temperature profile can be quantitatively obtained through the mean velocity with empirical scaling.

Heat transport in Rayleigh-Bénard convection with linear marginality.

Recent direct numerical simulations (DNS) and computations of exact steady solutions suggest that the heat transport in Rayleigh-Bénard convection (RBC) exhibits the classical [Formula: see text] scaling as the Rayleigh number [Formula: see text] with Prandtl number unity, consistent with Malkus-Howard's marginally stable boundary layer theory. Here, we construct conditional upper and lower bounds for heat transport in two-dimensional RBC subject to a physically motivated marginal linear-stability constraint. The upper estimate is derived using the Constantin-Doering-Hopf (CDH) variational framework for RBC with stress-free boundary conditions, while the lower estimate is developed for both stress-free and no-slip boundary conditions. The resulting optimization problems are solved numerically using a time-stepping algorithm. Our results indicate that the upper heat-flux estimate follows the same [Formula: see text] scaling as the rigorous CDH upper bound for the two-dimensional stress-free case, indicating that the linear-stability constraint fails to modify the boundary-layer thickness of the mean temperature profile. By contrast, the lower estimate successfully captures the [Formula: see text] scaling for both the stress-free and no-slip cases. These estimates are tested using marginally-stable equilibrium solutions obtained under the quasi-linear approximation, steady roll solutions and DNS data. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

A Comparative Study of Models for Heat Transfer in Bidisperse Gas–Solid Systems via CFD–DEM Simulations

In this study, flow and heat transfers in bidisperse gas–solid systems were numerically investigated using the computational fluid dynamics–discrete element method (CFD–DEM). Three different models to close the gas–solid heat transfer coefficient for each species of bidisperse systems were compared in the simulations. The effect of the particle diameter ratio and particle number ratio between large and small particles on the particle mean temperature and temperature distribution of each species were systematically investigated. The simulation results show that differences in the particle mean temperature and temperature distribution profiles exist among the three heat transfer models at a higher particle number ratio. The differences between the effects of three heat transfer models on heat transfer properties in bidisperse systems with particle diameter ratios of up to 4 are marginal when the particle number ratio between small and large particles is 1.