Efficient inverse heat conduction modeling for parameter estimation and defect detection in multi-layer structures with internal heating

Solution to the 1-D unsteady heat conduction equation with internal Joule heat generation for thermoelectric devices

CConvective instabilities caused by a nonuniform temperature gradient due to vertical throughflow and internal heat generation were investigated in an anisotropic porous layer. The boundaries are taken to be either impermeable or porous and are perfect conductors of heat. The Forchheimer-extended Darcy model is used to describe the flow in the porous medium. The resulting eigenvalue problem is solved numerically by the Galerkin method with a modified external Rayleigh number as the eigenvalue. Both anisotropic parameters (i.e., effective thermal diffusivity, η, and permeability, ξ) appear through their ratio η/ξ only. We found that an increase in η/ξ increases the stability of the system. Furthermore, we observed that in the absence of internal heat generation, throughflow stabilizes when the boundaries are symmetric and destabilizes when they are asymmetric. However, if an internal heat source exists, throughflow destabilizes the system irrespective of the boundary types considered. A more precise control of the buoyancy-driven instability may be achieved by tuning the anisotropy parameters and internal heat source strength. Copyright © 2002 by Begell House, Inc.

In this paper, numerical simulations for the natural convection in a ferrofluid-filled corrugated cavity with internal heat generation under the influence of a magnetic dipole source were performed. The cavity is heated from below and cooled from above while vertical side walls are assumed to be adiabatic. A magnetic dipole source was located under the bottom heated wall. The governing equations were solved by Galerkin weighted residual finite-element formulation. The influence of external Rayleigh number (between 104 and 5 × 105), internal Rayleigh number (between 104 and 5 × 106), magnetic dipole strength (between 0 and 4), horizontal (between 0.2 and 0.8) and vertical (between −5 and −2) locations of the magnetic dipole source on fluid flow, and heat transfer are numerically investigated. It was observed that depending on heating mechanism (the external or internal heating), the presence of corrugation of the bottom wall either enhances or deteriorates the absolute value of the averaged heat transfer. The strength and locations of the magnetic dipole source affect the distribution of the flow and thermal patterns within the cavity for both flat and corrugated wall cavity. The net effect of the complicated interaction of the internal heating, external heating, and ferroconvection of magnetic source results in heat transfer enhancement with increasing values of magnetic dipole strength. Wall corrugation causes more enhancement of averaged heat transfer and this is more pronounced for low values of vertical location of magnetic source.

Convectively cooled solidification in phase change materials in different configurations subject to internal heat generation: Quasi-steady analysis

Axisymmetric thermoelastic stress perturbations in a fuel can due to localized variations in internal heat generation and boundary conditions

The effectiveness of internal heating for enhancing the drying of molded ceramics is evaluated by both modeling and experiments. In the theoretical analysis, three dimensional drying-induced strain-stress are modeled, and the numerical solutions show that the internal heating generates lower internal stress than continuous convective heating or intermittent convective heating. Microwave drying is examined experimentally to study the effect of internal heating on the drying behavior of a wet sample of a kaolin slab. The drying behavior is compared among three modes: microwave heating, hot air heating and radiation heating. The transient behavior of temperatures in microwave drying is quite different from conventional drying by external heating. In particular, the temperature of the slab drops once in the progress of drying. This phenomenon cannot be predicted adequately by a simple model of one-dimensional heat conduction and moisture diffusion accompanied with an internal heat generation rate given as a linear function of the moisture content. It should be noted that the temperature behavior takes place due to the combined interactions with internal evaporation of moisture by rise in internal vapor pressure and shift of impedance or interference in the applicator. Microwave heating with a constant power above 100 W results in sample breakage due to the internal vapor pressure. However, if the power is dynamically controlled so as to maintain the temperature less than the boiling point of water, the drying succeeds without any crack generation until completion with a significantly faster drying rate than drying in convective heating or in the oven.

In this study, heat transfer in a longitudinal rectangular fin with temperature-dependent thermal properties and internal heat generation is revisited. The advanced heat transfer models have been used to study the effects of thermo-geometric parameters, coefficient of heat transfer, and thermal conductivity parameters on the temperature distribution, heat transfer, and thermal performance of the longitudinal rectangular fin. It is shown that its governing nonlinear differential equation with proper boundary conditions is exactly solvable. With this aim, we reduce the order of the differential equation to first and then convert it into a total differential equation by multiplying by a convenient integrating factor. A full discussion and exact analytical solution in the implicit form is given for further physical interpretation and it is proved that three possible cases may occur: there is no solution to the problem, the solution is unique, or the solutions are dual, depending on the values of the parameters of the model.

This research presents a comprehensive thermal performance evaluation of a heat-generating fluid within a magnetohydrodynamic (MHD)-influenced wavy trapezoidal cavity containing a centrally positioned heated solid square block. The study systematically explores the effects of varying Rayleigh number (Ra), Hartmann number (Ha), and internal heat generation coefficient (Δ) on natural convection, flow structure, and thermal transport inside the cavity. Numerical simulations were carried out for a wide range of parameters: Ra (10³–10⁵), Ha (0, 10, 20), and Δ (0, 5, 10). Streamline analysis demonstrates that as Δ increases, the flow circulation becomes more vigorous and buoyancy-driven, resulting in intensified and asymmetric vortex formations around the heated block. However, increasing Ha exerts a damping influence, suppressing fluid motion and limiting the strength of recirculating vortices. Isotherm analysis reveals that heat generation causes tighter and more distorted temperature contours, indicating higher thermal gradients and stronger convective effects. In contrast, a higher Hartmann number smooths out the isotherms, indicating reduced thermal mixing and dominant conduction. The average Nusselt number decreases consistently with increasing Ha and Δ, illustrating that magnetic suppression and internal heat generation reduce the wall heat transfer rate. The highest thermal performance is observed for low Ha and Δ, while the lowest is found at high Ha and Δ combinations. The study concludes that while internal heat generation enhances internal fluid circulation, it lowers overall thermal efficiency under stronger magnetic field conditions. These findings offer insights into optimizing thermal management systems where internal heating and electromagnetic fields are present.

Chapter 7 - Convective flows with internal heat generation in porous media

This study presents a comprehensive numerical investigation of radiative heat transfer and flow dynamics involving Maxwell and Casson nanofluids flowing past a vertically aligned plate, incorporating the effects of internal heat generation. These non-Newtonian fluids exhibiting memory and yield stress behaviors are examined to understand their influence on velocity, temperature, and concentration profiles under coupled thermal and solutal gradients. The governing partial differential equations representing momentum, energy, and mass conservation are transformed into coupled nonlinear ordinary differential equations using similarity variables. These are solved using the fourth-order Runge-Kutta method with a MATLAB-based shooting technique. Key dimensionless parameters such as the Prandtl number, Grashof number, Biot number, heat source parameter, Nusselt number, Sherwood number, and skin friction coefficient are analyzed. The results show that internal heat generation significantly increases fluid temperature and velocity while reducing nanoparticle concentration and thermal transfer efficiency. Furthermore, elevated Biot numbers and thermal radiation broaden the thermal boundary layer, adversely affecting heat transfer. These findings have practical implications for engineering applications such as electronic cooling systems, chemical reactors, biomedical devices, and thermal insulation design. The insights gained enable improved modeling and thermal regulation in systems requiring efficient heat and mass transfer under non-Newtonian fluid behavior and internal heat sources.

Using a scale analysis approach, we model phase change (melting) for pure materials which generate internal heat for small Stefan numbers (approximately one). The analysis considers conduction in the solid phase and natural convection, driven by internal heat generation, in the liquid regime. The model is applied for a constant surface temperature boundary condition where the melting temperature is greater than the surface temperature in a cylindrical geometry. We show the time scales in which conduction and convection heat transfer dominate.

Investigation of heat transfer and pressure drop in a porous media with internal heat generation

Experimental study of temperature distribution in laminar tube flow of a fluid with internal heat generation

We investigated heat and mass transfer on water based nanofluid due to the combined effects of homogeneous–heterogeneous reactions, an external magnetic field and internal heat generation. The flow is generated by the movement of a linearly stretched surface, and the nanofluid contains nanoparticles of copper and gold. Exact solutions of the transformed model equations were obtained in terms of hypergeometric functions. To gain more insights regarding subtle impact of fluid and material parameters on the heat and mass transfer characteristics, and the fluid properties, the equations were further solved numerically using the matlab bvp4c solver. The similarities and differences in the behavior, including the heat and mass transfer characteristics, of the copper–water and gold–water nanofluids with respect to changes in the flow parameters were investigated. Finally, we obtained the numerical values of the skin friction and heat transfer coefficients.

ABSTRACTIn this research work, we developed a mathematical model of heat transfer for different profiles of fins. This paper investigates heat transfer dynamics in continuously moving fins (rectangular, trapezoidal and concave parabolic), focusing on temperature‐dependent thermal conductivity, heat transfer coefficients, internal heat generation and emissivity that varies with temperature and wavelength. The different values of the heat transfer coefficient capture various types of convection, nucleate boiling, condensation, and radiation effects, while treating thermal conductivity as a linear function of temperature. This problem is converted into a dimensionless form, and we adopted the Legendre wavelet collocation method (LWCM) to get the solution of the fin problem for various profiles. An exact solution in specific cases shows congruence up to seven to eight decimal places with LWCM results. Moreover, our analysis explores the effect of numerous non‐dimensional parameters such as thermal conductivity parameter , Peclet number , surface emissivity parameter B, convention‐conduction parameter , radiation‐conduction parameter , internal heat generation , D fin taper ratio on the temperature profile and fin efficiency were studied in detail. As , , and B, D increases in magnitude, the temperature inside the fin decreases, while higher values of Peclet number (), A, m, and Q cause lower heat transfer rate inside the fin. The results provide significant insights into the complex interplay of thermal processes across different fin geometries, emphasizing the importance of these dependencies for accurate modeling and optimization in thermal management systems.