Thermal Solver Research Articles

MultIHeaTS: A Fast and Stable Thermal Solver for Multilayered Planetary Surfaces

A fully implicit scheme is proposed for solving the heat equation in 1D heterogeneous media, available as a computationally efficient open-source Python code. The algorithm uses finite differences on an irregular grid and is unconditionally stable due to the implicit formulation. The thermal solver is validated against a stiff analytical solution, demonstrating its robustness in handling stiff initial conditions. Its general applicability for heterogeneous cases is demonstrated through its use in a planetary surface scenario with nonlinear boundary conditions induced by blackbody thermal emission. MultIHeaTS's advantageous stability allows for computation times up to 100 times faster than Spencer’s explicit solver, making it ideal for simulating processes on large timescales. This solver is used to compare the thermal signatures of homogeneous and bilayer profiles on Europa. Results show that homogeneous materials cannot reproduce the thermal signature observed in bilayer profiles, emphasizing the need for multilayer solvers. In order to optimize the scientific return of a space mission, we propose a strategy made of three local time observations that is enough to identify bilayer media, for instance, for the next missions to the Jovian system. A second application of the solver is the estimation of the temperature profile of Europa’s near surface (first 10s m) throughout a 1 million yr simulation with varying orbital parameters. The probability distribution of temperature through depth is obtained. Among its various applications, MultIHeaTS serves as the core thermal solver in a multiphysics simulation model detailed in the companion article by C. Mergny & F. Schmidt.

LunaIcy: Exploring Europa’s Icy Surface Microstructure through Multiphysics Simulations

Abstract A multiphysics simulation model incorporating a sintering model coupled with the MultIHeaTS thermal solver was developed to study the evolution of icy moons’ microstructure. The sintering process is highly dependent on temperature, and this study represents the first attempt in planetary science to examine the coupled interaction between heat transfer and sintering. Our approach to ice sintering is based upon the literature while offering a refined description of the matter exchange between grains, bonds, and the pore space. By running the numerical framework, we simulate the evolution of ice microstructure on Galilean satellites, specifically tracking the changes in the ice grain and bond radii over time. LunaIcy, our multiphysics model, was applied to study the evolution of Europa’s ice microstructure over 1 million yr along its orbit, with a parameter exploration to investigate the diverse configurations of the icy surface. Our results indicate that effective sintering can take place in regions where daily temperatures briefly surpass 115 K, even during short intervals of the day. Such sintering could not have been detected without the diurnal thermal coupling of LunaIcy due to the cold daily mean temperature. In these regions, sintering occurs within timescales shorter than Europa’s ice crust age, suggesting that, in present times, their surface is made of an interconnected ice structure.

Virtual body-fitted grid-based immersed boundary method for simulation of thermal flows with Dirichlet and Neumann boundary conditions

In this work, a virtual body-fitted grid-based immersed boundary method (IBM) combined with the thermal lattice Boltzmann flux solver (TLBFS) is proposed to efficiently simulate thermal flows with Dirichlet and Neumann boundary conditions. The method involves generating a virtual body-fitted grid along the immersed boundary and implementing the boundary conditions using a fractional step technique. In the prediction step, the finite-volume TLBFS is used to obtain the predicted flow field on the Eulerian mesh without considering the immersed boundary, while in the correction step, the correction process of the flow field is conducted on the virtual grid. Thus, the effect of the boundary is transferred to the fluid domain through the virtual grid indirectly. For Dirichlet boundary conditions, such as the no-slip condition and the specified temperature condition, the velocity and temperature corrections are considered unknowns and solved through a matrix system formed from enforcing the boundary conditions. As for the Neumann boundary condition, such as the specified heat flux condition, the temperatures at virtual layers inside the immersed body are regarded as unknowns and computed from those at virtual layers outside the wall according to the quadratic distribution. Several benchmark cases including natural, forced, and mixed convection problems are well simulated to validate the method. The numerical results are in good agreement with reference data, demonstrating the capacity of the present method to simulate thermal flows with Dirichlet and Neumann boundary conditions while requiring fewer Lagrangian points and achieving higher computational efficiency.

Finite-temperature minimally entangled typical thermal states impurity solver

Finite-temperature minimally entangled typical thermal states impurity solver

New thermal solver for mitigating surface temperature instability in laser-induced heating

An accepted approach to computing laser-induced peak surface temperature is to employ the enthalpy formulation of the transient heat conduction equation [Grigoropoulos et al., Adv. Heat Transfer 28, 75–144 (1996); Sawyer et al., J. Laser Appl. 29, 022212 (2017)]. This approach is generally implemented using an explicit numerical scheme to solve the thermal transport equation. While it offers the advantage of modeling the solid-melt phase transition automatically, the approach results in instability-like behavior in the computed surface temperature. When laser-induced ablation becomes significant, the heating rate in the surface cell becomes unrealistically large. This results in spikes in the computed peak surface temperature due to large errors in calculating the heating rate. In this paper, we present a new approach, which we refer to as the Moving Frame Solver, that employs a moving-coordinate frame of reference, located at the receding evaporating surface. We also use an analytical representation for the phase transition region of the enthalpy-temperature relationship. The Moving Frame Solver combined with an implicit scheme leads to a stable solution without surface temperature, pressure, or velocity spikes. In other words, any instability in these computed parameters due to use of an explicit scheme (such as Dufort–Frankel) has been eliminated. Details of the new thermal solver and example calculations are presented. Numerical experiments suggest that the surface cell size needs to be small, ∼0.1 μm, to obtain a highly accurate solution with a typical metal such as aluminum. Using the Moving Frame Solver with a refined grid near the surface, but coarse elsewhere, enables accurate and stable surface temperature computation.

GO-MELT: GPU-optimized multilevel execution of LPBF thermal simulations

Computationally modeling the laser powder bed fusion process can be expensive despite the fact that the laser covers a small fraction of the entire domain. Hence, we developed GO-MELT, a multilevel approach inspired by the variational multiscale method. GO-MELT calculates the full temperature solution using three coupled thermal solvers, each solving one of three overlapping domains, subsequently referred to as levels. Level 1’s solver obtains a global, coarse-scale solution. Level 2’s solver computes a meso-scale thermal field spanning the melt pool region. Level 3’s solver obtains a fine-scale temperature field spanning the laser’s immediate region. Coarser solutions provide boundary conditions for finer solvers. Finer solutions compute subgrid scales that influence coarser solvers through additional source terms. Being independently meshed, Levels 2 and 3 can track a laser along its tool path without remeshing. Moreover, fixed-sized, structured meshes allow for GPU acceleration using Google’s JAX library with just-in-time compilation. Five case studies were conducted to demonstrate proper convergence of GO-MELT, to show that GO-MELT can reach the same accuracy as a uniform mesh with fewer degrees of freedom, and to quantify the computational load from each level’s solver. A simulated production run averaged 1.64ms per time step after taking 9.36h to complete 20.5 million time steps, which is approximately 678× faster than a uniform mesh solver with identical resolution and GPU acceleration. Future work will implement different solvers into GO-MELT to improve fidelity and speed.

Development of a Fourier‐expansion based differential quadrature method with lattice Boltzmann flux solvers: Application to incompressible isothermal and thermal flows

AbstractThis paper presents a high‐order Fourier‐expansion based differential quadrature method with isothermal and thermal lattice Boltzmann flux solvers (LBFS‐FDQ and TLBFS‐FDQ) for simulating incompressible flows. The numerical solution in the present method is approximated via trigonometric basis. Therefore, both periodic and non‐periodic boundary conditions can be handled straightforwardly without the special treatments as required by polynomial‐based differential quadrature methods. The incorporation of LBFS/TLBFS enables the present methods to efficiently simulated various types of flow problems on considerably coarse grids with spectral accuracy. The high‐order accuracy, efficiency and competitiveness of the proposed method are comprehensively demonstrated through a wide selection of isothermal and thermal flow benchmarks.

Evaluation of acoustic-thermal simulations of invivo magnetic resonance guided focused ultrasound ablative therapy.

To evaluate numerical simulations of focused ultrasound (FUS) with a rabbit model, comparing simulated heating characteristics with magnetic resonance temperature imaging (MRTI) data collected during invivo treatment. A rabbit model was treated with FUS sonications in the biceps femoris with 3D MRTI collected. Acoustic and thermal properties of the rabbit muscle were determined experimentally. Numerical models of the rabbits were created, and tissue-type-specific properties were assigned. FUS simulations were performed using both the hybrid angular spectrum (HAS) method and k-Wave. Simulated power deposition patterns were converted to temperature maps using a Pennes' bioheat equation-based thermal solver. Agreement of pressure between the simulation techniques and temperature between the simulation and experimental heating was evaluated. Contributions of scattering and absorption attenuation were considered. Simulated peak pressures derived using the HAS method exceeded the simulated peak pressures from k-Wave by 1.6 ± 2.7%. The location and FWHM of the peak pressure calculated from HAS and k-Wave showed good agreement. When muscle acoustic absorption value in the simulations was adjusted to approximately 54% of the measured attenuation, the average root-mean-squared error between simulated and experimental spatial-average temperature profiles was 0.046 ± 0.019 °C/W. Mean distance between simulated and experimental COTMs was 3.25 ± 1.37 mm. Transverse FWHMs of simulated sonications were smaller than in invivo sonications. Longitudinal FWHMs were similar. Presented results demonstrate agreement between HAS and k-Wave simulations and that FUS simulations can accurately predict focal position and heating for invivo applications in soft tissue.

An in silico testbed for fast and accurate MR labeling of orthopedic implants

One limitation on the ability to monitor health in older adults using magnetic resonance (MR) imaging is the presence of implants, where the prevalence of implantable devices (orthopedic, cardiac, neuromodulation) increases in the population, as does the pervasiveness of conditions requiring MRI studies for diagnosis (musculoskeletal diseases, infections, or cancer). The present study describes a novel multiphysics implant modeling testbed using the following approaches with two examples: (1) an in silico human model based on the widely available Visible Human Project (VHP) cryo-section dataset; (2) a finite element method (FEM) modeling software workbench from Ansys (Electronics Desktop/Mechanical) to model MR radio frequency (RF) coils and the temperature rise modeling in heterogeneous media. The in silico VHP-Female model (250 parts with an additional 40 components specifically characterizing embedded implants and resultant surrounding tissues) corresponds to a 60-year-old female with a body mass index of 36. The testbed includes the FEM-compatible in silico human model, an implant embedding procedure, a generic parameterizable MRI RF birdcage two-port coil model, a workflow for computing heat sources on the implant surface and in adjacent tissues, and a thermal FEM solver directly linked to the MR coil simulator to determine implant heating based on an MR imaging study protocol. The primary target is MR labeling of large orthopedic implants. The testbed has very recently been approved by the US Food and Drug Administration (FDA) as a medical device development tool for 1.5 T orthopedic implant examinations.

Assessing the impact of wet and dry flue gas recycling on heat transfer in oxygen-fired circulating fluidized bed furnaces

Oxygen-driven circulating fluidized bed combustion (oxy-CFB) has been suggested as a potential technique for facilitating carbon capture in large-scale power generation. Precise modeling of radiative heat transfer is vital for forecasting the thermal performance of such processes. This research examines the impact of wet and dry flue gas recycling on the thermal behavior of a large-scale oxy-CFB furnace, utilizing a combined modeling strategy that merges a three-dimensional furnace process model with a thermal radiation solver. The semi-empirical furnace model operates in steady-state conditions, accounting for fluid dynamics, reactions, and heat transfer phenomena. The radiation solver, based on the zone method, employs novel correlations for radiative heat exchange between optically thick zones. The radiation solver employs a weighted sum of the gray gases model, developed specifically for oxy-combustion, and geometric optics to determine the radiative characteristics of combustion gases and particles, respectively.This research covers various process conditions and boiler loads to evaluate the influence of wet versus dry flue gas recycling on heat transfer and temperature profiles. Model results were successfully compared against small-scale oxygen-fired CFB tests and large-scale measurements in a supercritical air-fired CFB. Utilizing an external radiation solver allows for a more precise representation of radiative heat transfer and the influence of varying process conditions. The insights from this research can aid in the design of large-scale oxy-CFB facilities, and the demonstrated modeling approach can be employed for additional model development, such as integrating the external radiation solver with other modeling tools.

Diffuse interface method for solid composite propellant ignition and regression

Solid Composite Propellants (SCPs) are extensively used in the field of propulsion for their chemical and mechanical stability in long-term storage, and for having simple production and operation processes. Computational modeling enables cost reduction, increased efficiency, and greater coverage of the configuration space in the SCP design process. However, accurate and efficient SCP modeling presents a number of numerical challenges. A primary obstacle in modeling these systems is capturing the complex evolving interface. Recently, it has been shown that the phase-field method has a strong ability to model the combustion behavior of SCPs, implicitly capturing the topological evolution at a relatively low computational cost. Initial phase-field methods show promise in their ability to do predictive regression modeling but require a number of approximations and heuristic modeling methods. This work presents a new formulation for the phase field regression model that combines a thermal solver with an Arrhenius rate law to model interface regression using a comprehensive and physics-based approach. This improves the capability of previous methods by increasing the number of kinematic forces that are accounted for, and allowing the study of thermal diffusivity in the system. It also enables the integration of a fully coupled solid-fluid interface. To demonstrate the efficacy of the model, it is applied and calibrated to a homogeneous monopropellant (ammonium perchlorate). The model is validated for a range of temperatures, with a reasonable quantitative match to experimental data. It was also demonstrated that the model recovered the relationship between ignition time and heat flux, with no fitting required. Overall, the method showed great capability of reproducing experimental data by matching temperature, burn rate, and thermal diffusivity profiles. Statement of SignificanceA new diffuse interface method is developed for calculating ignition and regression rates in SCPs. While this builds on previous work in diffuse interface modeling of SCPs, it is novel in its coupling to thermal transport, as well as its development of (and coupling to) a approximation for gas phase heat flux. The model is shown to reproduce experimental regression rate data with reasonable accuracy. Moreover, it quantitatively captures ignition behavior with no additional modifications to the model. The model is implemented in a high performance computational framework that is able to resolve large, three-dimensional mesostuctures.

Immersed boundary method based fluid–structure–thermal interaction solver with conjugate heat transfer

Immersed boundary method based fluid–structure–thermal interaction solver with conjugate heat transfer

An implicit immersed boundary method for Robin boundary condition

In this work, a novel boundary condition-enforced immersed boundary method (IBM) for simulating complex thermal–fluid–structure interaction (TFSI) problems with Robin boundary conditions is proposed. In this work, two auxiliary layers of Lagrangian points are introduced and placed within the inner and outer parts of the immersed object to enable the simultaneous evaluation of the temperature and its gradient on the solid surface. The mutual thermal interactions between the immersed object and the surrounding flow are taken into account through the temperature corrections on the Lagrangian points by accurately enforcing the Robin boundary condition on the solid boundary. Hence, a system of linear equations can be formulated to compute the temperature corrections. Subsequently, the temperature corrections are biasedly distributed to the Eulerian points located in the inner auxiliary layer to eliminate the diffusion generated by the smooth delta function on the temperature field outside the solid object. The proposed IBM integrated with the reconstructed thermal lattice Boltzmann flux solver (RTLBFS) is extensively and critically assessed with classical benchmark cases ranging from two-dimensional stationary problems to three-dimensional problems involving moving boundaries to demonstrate the robustness and accuracy of the proposed method for Robin boundary conditions. Results from the present work are extensively corroborated with previous works that adopt different approaches for imposing the Robin boundary condition. It shows that the proposed method not only provides a much more accurate representation of the Robin boundary condition but also introduces a simple implementation for the complex Robin boundary condition within the diffuse interface IBM framework.

Importance, influence and limits of CFD radiation modeling for containment atmosphere simulations

In the appearance of steam, the CFD modeling of thermal-hydraulic phenomena in reactor containments during various phases of an accident necessarily requires consideration of radiative heat transfer. These radiation phenomena involve (i) energy transfer within the gas mixture and (ii) between the gas and the surrounding structures which have a much higher thermal inertia. Preliminary calculations performed for the present experiments in the OECD/NEA HYMERES-2 project with the CFD code containmentFoam using a Monte Carlo thermal radiation solver with non-gray gas modeling of steam absorption, showed that radiant heat transfer is important even with a very low amount of steam (≈ 2%). Therefore, the test matrix was tailored to the two opposite extremes: Either gas compositions with low steam content, where radiative heat transfer can be neglected, or gas mixtures containing larger amounts of steam, so that radiative heat transfer plays a dominant role. For the two selected experiments within the H2P2 series and the corresponding CFD calculations, a vessel with a diameter of 4 m and a height of 8 m was preconditioned with different mixtures of air/steam at room and elevated temperatures. This was followed by the build-up of a stable helium stratification in the upper part of the vessel. Helium was then injected from the top at a higher mass flow rate, which (a) compresses the gas atmosphere, (b) resembles piston compression and (c) leads to a height-dependent and transient increase in gas temperature below this helium stratification, influenced by thermal radiation effects — or their absence. These experiments and the associated CFD calculations were developed to isolate the thermal radiation phenomena as much as possible from convective and diffusive effects. They are the first of their kind to be conducted in a large-scale thermal-hydraulic facility. It is demonstrated (a) that neglecting thermal radiation for the high steam content case, the temperature rise is significantly over-predicted by a factor of two compared to the application of the suggested Monte Carlo thermal radiation solver with non-gray gas modeling, (b) the computational effort considering CFD and non-gray gas radiation is feasible with the suggested tailored Monte Carlo solver and (c) that the suggested radiation model outperforms the widespread used P1 radiation model.

Large eddy simulation of a row of impinging jets with upstream crossflow using the lattice Boltzmann method

Large eddy simulations of a row of seven jets emerging from a perforated pipe and impinging on a flat heated plate were carried out using a compressible hybrid thermal lattice Boltzmann solver. The average Reynolds number of the emerging jets was 5000, with an exit-to-plate distance of 3 jet diameters. Two levels of upstream crossflow were simulated: one with weak cross flow, with a velocity ratio of ≈9.7, and one with strong cross flow, with a velocity ratio of ≈2. The flow field and heat transfer statistics were validated against experimental PIV and infrared thermography data from a recent study, showing good agreement. The effect of varying the Mach number on the wall heat transfer was subsequently tested. It was found that an increased Mach number lead to an increased value of the Nusselt number near the stagnation points.

Efficient transient thermal analysis of chiplet heterogeneous integration

Thermal issue has significant effect on the reliability and performance of integrated circuits with the increase of integrated density, especially for 2.5-D and 3-D heterogeneous integration packaging systems. In this paper, a new highly efficient 3-D transient thermal distribution model has been constructed to capture the heat conduction behavior of multiple heat sources for chiplet heterogeneous integration (CHI) by improving the alternating direction implicit finite difference method (ADI-FDM). Firstly, a new floating optimization algorithm (FOA) of coefficient matrix assignment of heat conduction equation and a new boundary treatment utilizing virtual points are proposed to modify the ADI-FDM and improve the model efficiency of thermal analysis. Furthermore, a universal equivalent thermal conductivity model is also built up to describe the design features of through silicon via (TSV), bump and redistribution layer structures, which can enhance the flexibility of FDM-based thermal simulator. Compared with previous finite element analysis results, the present FOA-accelerated thermal model can not only obtain an accurate simulation result of temperature profile, but also give a simulation efficiency of 3.74 times faster than Douglas-Gunn approach for the coefficient matrix assignment. Finally, the effect of power consumption variation and the floorplaning effects of chiplets, TSVs and bumps on the temperature distribution are investigated in detail by utilizing the present FOA-based thermal solver. The simulated results of the present work demonstrate the viability and computational efficiency for temperature field optimization and indicate that the new proposed simulator is helpful in thermal analysis of packaging structures and can be adopted to assist in physical design optimization of 2.5-D CHI or 3-D heterogeneous stacked chips.

An explicit boundary condition-enforced immersed boundary-reconstructed thermal lattice Boltzmann flux solver for thermal–fluid–structure interaction problems with heat flux boundary conditions

For the thermal–fluid–structure interaction (TFSI) problems with moving boundaries, the original implicit boundary condition-enforced immersed boundary method (IBM) by Wang et al. [1] has to generate a large correlation matrix and compute its inversion at each iteration, implying that the virtual memory requirement and computational cost would grow exponentially with the number of Lagrangian points. In this work, we proposed an efficient explicit boundary condition-enforced immersed boundary method for Neumann boundary condition (NBC) to achieve high computational efficiency and maintain similar accuracy as the original implicit IBM [1], which circumvents the needs to assemble a large correlation matrix and inverse it in the original implicit IBM [1] through second-order approximation based on the error analysis using Taylor series expansion. Most importantly, the proposed explicit IBM can efficiently solve practical physics problems with a tremendous amount of Lagrangian points involving Neumann boundary condition. The comparisons of the virtual memory and computational cost between the explicit and implicit IBMs demonstrate that the explicit boundary condition-enforced IBM is not only computational efficient, but also has memory saving performance. The proposed explicit IBM integrated with the reconstructed thermal lattice Boltzmann flux solver (RTLBFS) is validated with some classical benchmarks, and the results show that the proposed explicit IBM can successfully resolve TFSI problems with Neumann boundary condition.

Enhancing SLM Stainless Steel: Model-Based Parameter Optimization

SLM (selective laser melting) has faced challenges in ensuring consistent mechanical properties for various geometries. However, the authors have developed a system that solves this issue by calculating the optimal process parameters for each scan vector on each layer. This approach ensures consistent melt pool geometry, derived from the most robust parameters in conduction welding mode. The results from the single melt-line experiments and the presented thermal conduction solver demonstrate the effectiveness of this approach in achieving a stable melt pool geometry in stainless steel (1.4404 or 316L). Furthermore, a testing regimen has been developed to evaluate the system under different changing boundary conditions, to further demonstrate its usefulness. By utilizing this system, manufacturers will achieve consistent and high-quality mechanical properties which improves cost-effectiveness for small lot sizes, including spare and obsolescent parts.

Reconstruction of interfacial thermal transport mediated by hotspot in silicon-based nano-transistors

Transistors are typical multi-interface nanostructures, and the heat dissipation problem has become the bottleneck restricting size reduction and performance improvement of semiconductor devices. Local high-temperature hotspot generated by self-heating effects seriously affects the stability and reliability of the devices, and the interfaces further hinder the heat conduction. Thermal analysis is complicated by subcontinuum phenomena which reshapes the hotspot region and interface-mediating. There has been less description of device thermal transport mediated by hotspot coupling with the interfaces. In this paper, based on the phonon BTE (pBTE) thermal solver and electronic Monte Carlo (e-MC) simulation, we systematically study the thermal transport behavior of 1-D Si transistor and Si-based heterojunction interfaces under the hotspot effect. Also, the influence of other scattering events is investigated by modifying the relaxation time. Inside hard hotspot, the temperature and heat flux of different phonon modes are decomposed, which shows a strong non-equilibrium effect. The contributions of non-Fourier effects are separated and quantified based on split-flux model. In heterojunction structure, the interfacial thermal resistance further increases the hotspot temperature. Interestingly, on the side of hotspot near the interface, the direction of heat flux is opposite to the direction of the temperature drop, which violates Fourier's law. Through different substrate combinations, the heat dissipation effect of diamond, BN is not excellent due to the low interfacial thermal conductance (ITC), time-dependent BTE is used to study the heating rate. Mode-dependent temperature and heat flux reveal nonequilibrium phonon transport performance. Finally, the wave-resolved spectral heat flux is calculated to explore the predominant thermal transport channel and interface phonon transport characters. Above studies provide useful insights for further improving the accuracy of transistor-level thermal simulations.

Mixed convection in a lid driven square cavity using lattice Boltzmann method: Effects of thermal gradient direction and moving lid length

This numerical study investigates the effect of thermal boundary conditions and moving lid length on the mixed convective heat transfer and flow regime dynamics in a square top-lid driven cavity. The numerical simulations are performed using an in-house developed thermal lattice Boltzmann solver. Here, the two cavity walls are defined using the opposite pairs of hot and cold isothermal walls. The rest walls are set as adiabatic with zero thermal flux conditions such that a thermal gradient is established either in the vertical or horizontal direction. Additionally, we have varied the size of the moving top-lid to couple the inertia effect with the thermal gradient and investigated how the mixed convective heat transfer and flow regime are affected in the square lid-driven cavity. Here, the Richardson number () is varied as 0.1, 1.0, and 10.0 to cover a wide range of mixed convective flow situations in the lid-driven cavity. is fixed at 106 in all the cases. Qualitative results of the flow regime and heat transfer characteristics are presented as isotherms and streamline patterns. The quantitative information is analyzed by the conduction-convection characteristics along the cavity centerlines and on the boundary walls. It is intuitive to say that the rate of heat transfer or Nusselt number will be higher for larger moving lid lengths, but this is achieved at the cost of higher driving force estimated by the line integral of the shear stress at the moving wall. Additionally, we have defined an efficiency parameter that can relate the heat transfer rate (Nuhot ) with the fluid friction (Cf ). Furthermore, η helps identify the effective and energy-efficient way to achieve the optimized heat transfer in the lid-driven cavity in the considered parametric range.