Heat Source Term Research Articles

Radiative Rayleigh-Taylor instability and the structure of clouds in planetary atmospheres

Aims.Clouds are expected to form in a broad range of conditions in the atmosphere of exoplanets given the variety of possible condensible species. This diversity, however, might lead to very different small-scale dynamics depending on radiative transfer in various thermal conditions. Here, we aim to provide some insight into these dynamical regimes.Methods.We performed an analytical linear stability analysis of a compositional discontinuity with a heating source term that depends on a given composition. We also performed idealized two-dimensional simulations of an opacity discontinuity in a stratified medium, using theARKcode. We used a two-stream gray model for radiative transfer and explored the brown-dwarf and Earth-like regimes.Results.We revealed the existence of a radiative Rayleigh-Taylor instability (RRTI, hereafter, which is a particular case of diabatic Rayleigh-Taylor instability) when an opacity discontinuity is present in a stratified medium. This instability is similar in nature to diabatic convection and relies only on buoyancy with radiative transfer heating and cooling. When the temperature is decreasing with height in the atmosphere, a lower-opacity medium on top of a higher-opacity medium is shown to be dynamically unstable, whereas a higher-opacity medium on top of a lower-opacity medium is stable. This stability-instability behavior is reversed if the temperature is increasing with height.Conclusions.The existence of a RRTI could have important implications for the stability of the cloud cover with regard to a wide range of planetary atmospheres. In our Solar System, it could help explain the formation of mammatus cloud in Earth atmospheres and the existence of the Venus cloud deck. Likewise, it suggests that stable and large-scale cloud covers could be ubiquitous in strongly irradiated exoplanets, but might be more patchy in low-irradiated or isolated objects such as brown dwarfs and directly imaged exoplanets.

A numerical study of space‐fractional three‐phase‐lag bioheat transfer model during thermal therapy

AbstractPrediction of accurate temperature profile and control of temperature are two important factors to treat the cancerous cells effectively in the living tissue during thermal therapy. To deal with this, we have generated a space‐fractional mathematical model of bioheat transfer within the living skin tissue by using three‐phase‐lag constitutive relation. The present model with Dirichlet's boundary condition is investigated in the companionship of the metabolic and external heat source during the thermal therapy. Exponentially decaying (with position) laser heat source is used as an external heating source term. To find out the solution, the model is converted to an initial value problem by using a fractional backward finite difference scheme and then Runge–Kutta (4,5) scheme is applied for determining the temperature profile within the living biological skin tissue. The effect on the dimensionless temperature profile during thermal therapy due to variations in the value of parameters like an order of space‐fractional derivative, relaxation times, blood perfusion coefficient, rate of thermal conductivity, and the external laser heat source incorporated in the current bioheat transfer model is discussed graphically. Results obtained from the study are beneficial in the field of medicine, especially for oncologists.

A lattice Boltzmann model for reactive mixtures.

A new lattice Boltzmann model for reactive ideal gas mixtures is presented. The model is an extension to reactive flows of the recently proposed multi-component lattice Boltzmann model for compressible ideal gas mixtures with Stefan–Maxwell diffusion for species interaction. First, the kinetic model for the Stefan–Maxwell diffusion is enhanced to accommodate a source term accounting for the change in the mixture composition due to chemical reaction. Second, by including the heat of formation in the energy equation, the thermodynamic consistency of the underlying compressible lattice Boltzmann model for momentum and energy allows a realization of the energy and temperature change due to chemical reactions. This obviates the need for ad-hoc modelling with source terms for temperature or heat. Both parts remain consistently coupled through mixture composition, momentum, pressure, energy and enthalpy. The proposed model uses the standard three-dimensional lattices and is validated with a set of benchmarks including laminar burning speed in the hydrogen–air mixture and circular expanding premixed flame.This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.

Weighted-sum-of-gray-gases models for non-gray thermal radiation of hydrocarbon fuel vapors, CH4, CO and soot

Using the high-resolution experimental spectral absorption coefficients of six fuel gases and the line by line absorption spectra of CO and CH4 based on HITRAN and HITEMP spectral databases, in this paper, novel coefficients for weighted-sum-of-gray-gases (WSGG) model are presented for Heptane, Methane, Methanol, MMA, Propane, Propylene, Toluene, and CO. Moreover, for soot, the spectral absorption coefficients were calculated assuming Rayleigh regime implementing the complex index of refraction obtained from the correlations of Chang and Charalampopoulos. The presented WSGG models were coupled with those of literature for CO2 and H2O by means of the superposition method. The models were first validated in several one-dimensional benchmarks representing various levels of inhomogeneous conditions in temperature, gas concentration and soot loading. Then, the WSGG models were employed in solving a three-dimensional case representing a Heptane pool fire. Using the time averaged 3-D CFD profiles, the WSGG models solved the spectral radiative heat transfer exhibiting excellent agreement with the results of line by line calculations in terms of radiative heat flux and radiative heat source. Moreover, the emissivity charts were provided comparing the emissivity calculated by LBL calculations with those of the new WSGG models.

The coupled heat Maxwell equations with temperature-dependent permittivity

We consider the heat equation coupled with the Maxwell system, the Ampère-Maxwell equation being coupled to the heat equation by the permittivity, which depends on the temperature due to thermal agitation, and the heat equation being coupled to the Maxwell system by the volumic heat source term. Our purpose is to establish the existence of a local-in-time solution to this coupled problem. Firstly, fixing the temperature distribution, we study the resulting Maxwell system, a nonautonomous system due to the dependence of the permittivity on the temperature and consequently on time, by using the theory of evolution systems. Next, we return to our coupled problem, introducing a fixed-point problem in the closed convex set K(0;R):={z∈B¯(0;R);z(0)=0} of the Banach space C1([0,T];C1(Ω¯)) and proving that the hypotheses of Schauder's theorem are verified for R sufficiently large. The construction of the fixed-point problem is nontrivial as we need K(0;R) to be stable.

Identification of the unknown heat source terms in a 2D parabolic equation

The objective of this paper is to reconstruct the unknown time-dependent heat source terms numerically, for the first time, in a two-dimensional parabolic equation in the rectangular domain with initial and Neumann boundary conditions supplemented by the temperature data as over-determination conditions. Although, the problem is ill-posed (in the sense of Hadamard) but has a unique solution. We apply the forward time central space finite difference scheme along with the Tikhonov regularization to find a stable and accurate numerical solution. The MATLAB subroutine lsqnonlin is used to solve the resulting nonlinear minimization problem. The obtained results show that accurate and stable solutions are achieved. Computational efficiency of the method is investigated by small values of CPU-time.

New exact solutions for microscale gas flows

We present a number of exact solutions to the linearised Grad equations for non-equilibrium rarefied gas flows and heat flows. The solutions include the flow and pressure fields associated to a point force placed in a rarefied gas flow close to a no-slip boundary and the temperature field for a point heat source placed in a heat flow close to a temperature jump boundary. We also derive the solution of the unsteady Grad equations in one dimension with a time-dependent point heat source term and the Grad analogue of the rotlet, a well-known singularity of Stokes flow which corresponds to a point torque.

A review of Stirling-engine-based combined heat and power technology

Small- and micro-scale combined heat and power (CHP) technologies offer great potential for reducing energy costs and CO2 emissions in residential and small commercial buildings. Among various CHP technologies, Stirling engines, particularly free-piston ones, show great promise in residential applications because of their remarkable advantages of low emissions, maintenance, noise, and vibration, theoretically high thermal-to-electrical efficiency, and flexibility of fuel sources. This paper presents a comprehensive review of Stirling-engine CHP in terms of different heat sources, tri-generation systems, status of commercial development, techno-economic issues, and challenges and future trends. The techno-economic assessment indicates that the relatively low on-site operational efficiencies and considerable investment costs are the main issues that hinder the further development of Stirling-engine CHP systems, and suggestions for future development are made accordingly. For successful commercial dissemination of Stirling-engine CHP, it is essential to improve the on-site electrical efficiency, reduce the capital cost of systems, and develop renewable-energy-powered and free-piston Stirling-engine-based CHP systems against the background of grid decarbonization and carbon neutrality.

Influence of soot on radiative heat transfer in bubbling fluidized bed combustors

The effect of soot particles and aggregates on radiative heat transfer in bubbling fluidized bed combustors is investigated. For this purpose, a soot radiative property model based on Rayleigh scattering theory is coupled with in-house developed 1-D gray and spectral radiation codes and a 3-D spectral radiation code based on method of lines solution of discrete ordinates method for the application of (i) a 1-D slab problem involving combustion gases and soot and (ii) freeboard of a 0.3 MWt atmospheric bubbling fluidized bed combustion test rig including combustion gases, fly ash particles and soot. The predictions of the 1-D radiation code reveal that (i) Planck-mean soot radiative properties provide reasonably accurate radiative heat transfer predictions (ii) soot particle diameter and number of soot particles in an aggregate do not considerably influence the net wall heat fluxes and source terms as absorption coefficient of soot is not significantly affected by those parameters and is orders of magnitude higher than scattering coefficient. Application of 3-D radiation model to the test rig shows that soot significantly contributes to radiative heat transfer in the freeboards of bubbling fluidized bed combustors. Soot has a noticeable impact on radiation even at very low volume fractions.

An analytic parametric study of rounded laser pulse heating

A theoretical modeling is presented for the laser pulse heating process in cylindrical two-dimensional materials. The laser beam irradiates a local rounded regional strip of the surface and the exposed laser energy is absorbed by the surface. The governing energy equation is due to the Fourier heat conduction law with the laser heat source term incorporated. Making the energy equation non-dimensional leads to three main physical parameters together with three more auxiliary parameters. The dimensionless equation with these parameters are then solved analytically by means of integral transformations suitable to the problem geometry donated with proper insulated boundary constraints in combination with a uniform initial temperature condition. The present model is validated by comparing it with the dimensional counterparts available in the open literature. The rigorous convergent solutions are shown with and without the initial temperature. Through a detailed parametric analysis, the effects of laser irradiation on the material energy absorption of the laser energy, on the temperature rise of the surface, on the heat distribution through the material penetration, and on the radial and temporal laser-beam/material interaction time bandwidth are easily understood from the presented model and analytical solutions.

Thermo-Fluid Design Simulation of Nd3+ POCl3 Transverse Flow Liquid Laser Cavity

Abstract Computational fluid dynamics (CFD) based thermal design of a transverse flow optical cavity is carried out for 1 kW Nd3+ POCl3 liquid laser source to investigate temperature and velocity distribution in the optical pumping region of the cavity. Temperature gradient and turbulence both affect the refractive index of the liquid gain medium, which results in optical path difference (OPD), divergence and hence, poorer quality of the laser beam. The main purpose of this design is to achieve uniform flow and least temperature gradient in the optical pumping region so that the optical path difference can be minimized and a good beam quality can be achieved. CFD model has been developed for carrying out thermo-fluid simulations for this thermal system and based on these simulations, an optimum geometry of inlet ports along with their position from optical pumping region have been proposed. A user defined function (UDF) is incorporated for the input of spatially varying heat source term in each cell of the optical pumping region of the cavity. Variations in refractive index and optical path difference are estimated from the temperature data using another UDF. Simulation reveals that mass flowrate between 1.5 kg/s and 2.0 kg/s maintains the optical homogeneity of gain medium. Preliminary experiments have been carried out to demonstrate the effect of flowrate on the beam divergence and thereby exhibiting the importance of present simulation work.

Optimal control of hyperthermia thermal damage based on tumor configuration

Improvement and optimization of hyperthermia as one of the most effective and conventional cancer treatments have been tackled by many researchers in recent years. The main benchmark for hyperthermia treatment is to elevate the cancerous tissue temperature to a specified level to initiate damage or ablation and simultaneously to prevent any irreversible thermal damage to the surrounding healthy cells. Therefore, the heat source pattern for hyperthermia treatment should be adjusted according to the distribution of cancerous cells. To do this, in the present study the fractional bioheat equation with unknown heat source term is implemented. Two different hyperthermia scenarios with single and dual tumors are considered and the appropriate damage level at both healthy and cancerous tissues is planned according to the Arrhenius model. Moreover, the optimal control criterion is set to simultaneously produce the goal temperature level and minimize the thermal power consumption. The mentioned task is accomplished by the definition of a priority coefficient as a weight function which determines whether the establishment of desired thermal field or the reduction of applied thermal power is more important. Based on the results, the optimal control approach can precisely determine the heat source distribution for single and dual tumor cases (with 1.03 and 0.13% deviation error, respectively). Whereas, by shifting the priority to the minimization of applied energy, up to 23.1 and 38.7% deviation is observed in formation of the desired temperature field (while the overall cost function is minimized). In order to investigate the numerical results, a computational method based on the Legendre cardinal functions is proposed. The main advantage of the proposed method is that it transforms solving such fractional models into solving systems of algebraic equations which greatly simplify the problem.

Analytical modelling of PCM supercooling including recalescence for complete and partial heating/cooling cycles

Abstract Phase change material (PCM) experiencing supercooling and phase change hysteresis are widely reported in the literature. However, only few studies model analytically such PCM, and they mostly focus on either supercooling or phase change hysteresis, but rarely on both phenomena. Moreover, partial phase change cycles, with an incomplete melting or solidification, are rarely considered even if these processes occur frequently in latent heat thermal energy storage (LHTES) systems. The objective of this study is to model analytically the thermal behaviour of a PCM experiencing supercooling and phase change hysteresis, for complete and partial phase change cycles. The developed method, based on a heat source term, enables to model the recalescence process during the solidification. Currently rarely considered, the influence of the cooling rate on the supercooling degree and the recalescence process is evaluated with a phenomenological approach. For partial cycles, the different behavior between heating and cooling is modelled with the hysteresis model “curve scale” already validated in literature. The experimental validation is carried out on a PCM brick sample, monitoring both the heat flux and the PCM temperature, which enables to characterize a greater mass of PCM compared to direct scanning calorimeter (DSC) analysis. The selected PCM undergoing supercooling during solidification is PEG6000, a polymer suitable for domestic hot water (DHW) storage. Results show a good agreement between experimental and numerical results for complete heating and cooling cycles. The behavior laws used to model the solidification with the supercooling and recalescence processes are validated for the cooling rate range tested. The modelling is also satisfactory concerning the experiments on partial solidification for different temperature plateaus. However, the model fails to correctly represent the thermal behavior for a cooling process after a partial melting of the PCM. To conclude, the developed model enables to represent accurately the thermal behaviour of a PCM experiencing supercooling and phase change hysteresis for most of the phase change processes studied. Investigations on the thermal conditions influencing the crystalline structure and the effect on the phase change dynamic are suggested to improve supercooling modelling. The developed model needs to be validated for PCM having a higher supercooling degree, such as sugar alcohols or salt hydrates.

Thermo-magnetic-fluid dynamics analysis of an ONAN distribution transformer cooled with mineral oil and biodegradable esters

This work introduces a coupled electromagnetic, thermal and fluid flow analysis of an oil-natural air-natural distribution transformer in order to study the changes in the heat dissipation performance when a biodegradable ester is used to cool the device instead of mineral oil. The transformer has a rated power of 315kVA and a voltage ratio of 13.2kV / 0.4kV. The heat losses in the magnetic core and the windings are computed with the ANSYS® Maxwell software and they are transferred as volume heat source terms to compute the heat conduction. The natural convection of the fluid flow is taken into account using a temperature-dependent density. The heat conduction through the solid walls and radiators panels are also considered. The thermo–hydraulic problem is solved with the software Code_Saturne. Data from experimental tests carried out with mineral oil are used to validate the numerical simulations. Equivalent and anisotropic thermal conductivities in the core and the windings are calculated both with a semi-analytic procedure and finite element simulations to simplify the heat conduction model in the active parts. It is found that, after reaching a steady state, the transformer cooled with the ester shows a temperature difference between specific locations at the top and the bottom higher than that cooled with mineral oil. The magnetic core and the windings also work hotter when ester is used. Finally, the analysis of the flow through the cooling ducts of the windings confirms that the oil velocity is, on average, 25% higher than the ester one.

Evolution of microstructures in radiation fields using a coupled binary-collision Monte Carlo phase field approach

The simulation of radiation effects in materials broadly falls into two categories. In the limit of short time and length scales lies the modeling of primary radiation damage, such as point defect creation, energy deposition, and ballistic mixing. This is followed by the modeling at longer time scales of thermally activated microstructure evolution and defect reactions, such as recombination, clustering, and coarsening. The binary collision Monte Carlo method is an established, numerically efficient method for the computation of primary radiation damage. Conversely, the phase-field method is a state-of-the-art technique for modeling microstructure evolution on longer time and length scales. We present a concurrent coupling of these two methods, overcoming the difference between the discrete object Monte Carlo paradigm for primary radiation damage and the continuum field variable approach for microstructure evolution. The coupling is bidirectional, in which the microstructure evolution in the MOOSE finite-element framework provides the spatial scattering data set for the charged particle transport and receives point defect, mass transport, and heat source terms from the simulated collision cascades that contribute to the field variable evolution. The concurrent coupling scheme is implemented in the code Magpie and demonstrated by investigating patterning for an irradiated immiscible binary model alloy. The results from the coupled binary collision Monte Carlo/phase-field simulations reproduce the results of analytical models for phase separation, and phase mixing and patterning, supporting the approach and indicating its utility for modeling real-world materials systems.

Measurement of heat generation rate and heat sources of pouch type Li-ion cells

Fundamental understanding of heat generation of lithium-ion batteries during operations is crucial to the cost-effective and efficient design of a thermal management system (TMS) in electric vehicles. Accurate characterization of the heat requires a calorimeter that meets inherent dynamics of the heat with accuracy. Therefore, we have developed an isothermal calorimeter using the thermoelectric assemblies (TEAs) along with the temperature control and Kalman filter, which is used to measure the heat generation rate (HGR) of large format pouch type lithium-ion batteries as an example. The measured HGR is a function of charge and discharge rates, state-of-charge (SOC), and temperatures. Analysis has shown that both the HGR and the ratio of the total heat generation to the total energy stored or released (pheat) of the tested cells increase as the current increases or the temperature decreases. In addition, a new calorimetric method is developed which enables a simultaneous determination of the entropy coefficient and internal resistance in the reversible and irreversible heat source terms in the frequency domain. The method can reduce the testing time about 92% compared with those by the conventional potentiometric method and electrochemical impedance spectroscopy (EIS) analysis, respectively, and the determined parameters are in good accordance with the reference values. Moreover, the calculated reversible and irreversible heat sources terms are compared with the experimental measurement and matched each other.

Numerical solutions of non-gray gases and particles radiative transfer in three-dimensional combustion system using DRESOR and SNBCK

The radiative intensity emitted by non-gray gases and particles can be captured by the charge-coupled device (CCD) camera with more than a thousand pixels in the combustion diagnostic. A solution to efficiently provide radiative intensity with high directional resolution in non-gray gases and particles medium is essential for the analysis of combustion system. In this paper, a three-dimensional enclosure filled with non-gray gases and particles was investigated and two real combustion cases were simulated to estimate the accuracy of the DRESOR method. The radiative heat flux, source term and radiative intensity with different directional resolutions were calculated by the DRESOR and DOM methods using SNBCK model. The results showed that the radiative heat flux and source were in good agreement with the reference results, and similar radiative intensities with low directional resolution were obtained by the DRESOR and DOM methods. For the non-isothermal case, the relative difference of maximum intensity between the results calculated with 288 and 13106 directions was 54.23% which was obviously larger than those in isothermal cases. A higher directional resolution was needed for the non-isothermal case. When the high directional resolution scheme was selected, the computational efficiency of the DREOSR method was significantly higher than that of the DOM method. The time consumed by the DOM method was about seven times of that consumed by the DRESOR method when the discrete directions were 800. The DRESOR method and SNBCK model which could obtain radiative intensity with high directional resolution efficiently for non-gray gases and particles medium, was more suitable for the analysis of three-dimensional combustion system.

Analysis of transient uncoupled thermoelastic problems involving moving point heat sources using the method of fundamental solutions

The method of fundamental solutions (MFS) is very effective for analysis of problems with moving domain loads. In this work, the MFS is formulated for solution of two dimensional transient uncoupled thermoelastic problems involving moving point heat sources. At first, the time-dependent temperature field is obtained by solving the transient heat conduction equation involving a moving heat source term. Then, at each time point, the temperature field is used in the constitutive and equilibrium equations to determine the displacement field. The intensity and location of the point heat source can be arbitrary functions of time. The particular solutions for temperature and stress are described by simple and closed-form expressions and they are used without considering any internal points. Two numerical example problems are provided to demonstrate the efficiency of the presented formulation. The obtained numerical results show that the presented MFS is very efficient and useful. Unlike the finite element method, only a small number of collocation and source points are sufficient to achieve very accurate results in the proposed MFS.

A weighted-sum-of-gray soot-fractal-aggregates model for nongray heat radiation in the high temperature gas-soot mixture

Soot, a product of insufficient combustion, is usually in the form of aggregate. The multi-scattering of soot fractal aggregates has been proved to play an important role in studying the soot radiative properties, which is rarely considered in predicting the radiative heat transfer in combustion flame. In the present study, based on the Rayleigh-Debey-Gans-Fractal-Aggregate (RDG-FA) theory, a model, named Weighted Sum of Gray Soot fractal Aggregates (WSGSA), used to calculate the radiative properties of nongray soot fractal aggregates in gas-soot mixture is developed by combining the features of full-spectrum k-distribution (FSK) model and Weighted-Sum-of-Gray-Gases (WSGG) model. Four gray soot fractal aggregates obtained from reordering in the form of the k-distribution at Gauss-Legendre integral points are selected to represent the nongray soot fractal aggregate, while corresponding weighting factors are obtained by fitting the radiation data obtained by the line-by-line (LBL) model with soot radiation data. Then, the WSGSA model is systematically validated by studying the radiative heat transfer properties in a one-dimensional plane-parallel slab system only with soot aggregates. The maximum relative discrepancies of emittance under different temperatures and different path lengths are less than 12%, while the maximum relative discrepancies of radiative heat source term and radiative heat flux are both less than 11% with path length less than 3 m. Moreover, combined with the WSGG model, the performance of the WSGSA model is further validated by predicting the radiative heat transfer properties in the mixture consisted of gases and soot aggregates, and acceptable results are also obtained. All the results reveal that the developed model can be applied to predict the radiative heat transfer in the mixture containing gases and soot aggregates, even in the case of significant changes in temperature and species concentration.

Semi-analytical source (SAS) method for 3-D transient heat conduction problems with moving heat source of arbitrary shape

In this study, the semi-analytical source method, which has recently developed by the authors, is implemented for a 3-D fully-transient heat conduction problem with a moving heat source. The method utilizes the exact Green’s function for a diffusion problem with a piecewise constant heat source meaning that the heat source term is defined as the superposition of piece-wise constant contributions in each time interval and in each spatial interval. This approach allows the modeling of any arbitrary spatial distribution of heating with time varying power. Moreover, the method is not limited to straight-line motion of the heat source, and can include internal heating as well as surface heating. One important aspect of the method is that spatial discretization is required only on the path of the heating source and at the observation locations of interest, consequently the discretization of the entire domain is not required as in the case of fully-numerical methods. To verify the semi-analytical source method, an experimental setup was constructed and experiments were conducted with a fiber laser, and satisfactory agreement is achieved. Several case studies are also demonstrated with a Gaussian heat source. The semi-analytical source method is particularly well-suited for parallel computing. To explore this aspect, the parallelization of the method is explored using the Message Passing Interface (MPI) and domain decomposition with up to 800 processors on Stampede2. The parallelization results reveal that semi-analytical method is very suitable for parallel computation. For a strong scaling, the method shows an ideal linear scaling with increasing number of processors with a proper load balance. The weak scaling reveals that the parallelization performance exponentially increases with the increasing time domain due to convolution nature of the method in time.