Post-accident Heat Removal Research Articles

A Lagrangian approach to ex-vessel corium spreading over ceramic and concrete substrates using moving particle hydrodynamics

Understanding the spreading of molten core materials (corium) is vital for achieving post-accident heat removal and maintaining the integrity of the reactor containment during a severe nuclear accident. The spreading of highly viscous corium over sacrificial concrete and ablation-resistant ceramic substrates, consistent with recent VE-U9 experiments at the VULCANO facility, is investigated using a Lagrangian moving particle hydrodynamics (MPH) method. The spreading dynamics are coupled with a heat transfer model to account for the increase in melt viscosity due to solidification. Three alternative boundary conditions are considered at the melt-substrate interface to mimic the influence of low viscosity concrete ablation products on the spreading dynamics. Simulation of spreading under no-slip conditions closely resembled the spreading behavior observed over the inert ceramic substrate, advancing with a crawling motion with a steep profile in the proximity of the melt front. Spreading terminated due to a combination of thermal radiation from the free surface and the heat transfer to the substrate, leading to the formation of a continuous crust. Introduction of slip boundary models, to mimic lubrication of the interface by a film of low-viscosity concrete decomposition products, enabled the prediction of the gliding flow observed over the sacrificial concrete substrate, characterized by a shallow melt topology near the spreading front. The simulation results demonstrate excellent potential for Lagrangian models such as MPH to predict complex spreading dynamics in the presence of both melt solidification and lubrication at the substrate.

Enhancement in heat transfer from heat generating core debris using a passive approach in fast reactors: 3-D CFD analysis

The article presents the results of three-dimensional numerical analysis to assess the natural convection heat transfer in pool-type sodium-cooled fast reactors during post-accident heat removal. To improve the coolability of the destroyed core, suitable modifications in core catcher design are suggested. The equations for conservation of mass, momentum and energy are solved in their transient form using finite volume based discretization in (r,θ,z) system. Turbulence is modeled using the k−ω SST model based on sensitivity analysis of different turbulence models and validation exercises. A comparative analysis of 3-D and 2-D simulations is carried out. Various feasible geometrical arrangements for core collection trays are analyzed in detail for better coolability of the core debris by natural convection. Multiple debris collection trays with and without cooling pipes are studied towards achieving coolability of core debris in case of core disruptive accident. Results show that better safety margins are achieved by using 3-D as compared to 2-D simulation. The incorporation of cooling pipes in distributed heat sources on multiple trays leads to enhanced coolability of the debris within the vessel. Maximum temperatures on the core catcher/collection trays and in the core debris (with the selected core catcher design) are 860 K and 932 K respectively. These are substantially below the acceptable thermal design limits. Furthermore, detailed insight of the flow field confirms the turbulence in the lower plenum, partly due to impingement of passive jets from multiple cooling pipes on core debris and partly due to flow bypass through the slots at upper tray. It is established that the use of passive pipes and multi-tray concept for core catcher can successfully accommodate the whole core debris arising from core disruptive accidents within the main vessel without exceeding the safe thermal design limits.

Impact of side injection on heat removal from truncated conical heat-generating porous bed: thermal non-equilibrium approach

Effective removal of heat from the heat-generating porous bed, particularly in a confined space, is essential due to its implications on thermal management and system safety. The safety aspect becomes particularly important during post-accident heat removal from decay heat-generating debris in nuclear reactors, as well as thermal management of self-igniting coal stockpiles. In spite of detailed research on thermal convection through porous media, only a handful of studies have considered mixed convective heat transport involving heat-generating porous bed along with external fluid injection. The situation, however, demands an in-depth analysis due to the associated practical implications. The present work addresses one cooling method providing side injection of cold fluid and assuming a typical conical heat-generating porous bed located centrally within a fluid-filled cylindrical enclosure. The study is carried out in a general way to suit other applications. The dimensionless governing equations, along with the boundary conditions in a two-dimensional coordinate system, are solved numerically assuming laminar and incompressible flow along with the Boussinesq approximation. The analysis is carried utilizing the local thermal non-equilibrium model within the porous bed. Injection of cold fluid can markedly affect the convective heat transport rate. Porous media permeability considerably influences the flow mechanism. The major findings of this study can be very useful in improving the management of thermal energy removal from self-igniting coal stockpiles, grain storages, porous debris, etc. Heat transport intensification from heat-generating porous bed is analyzed by injecting coolant through the sidewall and considering liquid water as the working medium. The impacts of pertinent parameters on the thermal convection characteristics are illustrated using the average Nusselt number and energy flux vectors.

Modelling and capability of severe accident simulation code, AZORES to analyze In-Vessel Retention for a loop-type sodium-cooled fast reactor

Research and development of the physical models and codes of safety analyses have been promoted for loop-type Sodium-cooled Fast Reactors (SFRs). A severe accident code, AZORES has been developing for the evaluation of molten corium behavior and integrity of reactor and containment vessels. AZORES enables the simulation of plant equipment behavior under severe accident conditions. In recent years, much attention regarding the mitigation of severe accident progression has been attracted to In-Vessel Retention (IVR) of SFRs, which has relatively less existing experimental and analysis results than that of the Light Water Reactor (LWR). This paper describes the status of development for main physical models and preliminary verification for IVR simulation mainly with AZORES. For a verification process, the simulation has been conducted for representative Post-Accident Material Relocation (PAMR)/Post-Accident Heat Removal (PAHR) phases subsequent to an Anticipated Transients Without Scram (ATWS) to evaluate the impact of initial conditions on severe accident progression. From the study, it was found that the mass of molten corium and capability of decay heat removal were key parameters for the attainment of IVR. Furthermore, this paper describes the results of Computational Fluid Dynamics (CFD) simulations which have been conducted for molten corium relocation in a control rod assembly and for corium coolability with sodium coolant on the core catcher for the purpose of enhancing the related AZORES physical models.

Numerical simulation of Post Accident Heat Removal in a typical pool-type SFR under severe core relocation scenario

Under a hypothetical core meltdown and relocation scenario in Sodium cooled Fast Reactor (SFR), there may be a chance for reactor main vessel failure if the core debris are allowed to settle at the bottom of this vessel. To avoid this, an in-vessel core catcher assembly is installed in pool type SFRs just above the bottom wall of the main vessel. The function of this assembly is to collect, retain and passively cool the core debris by assisting natural convection decay heat removal from these debris. In this work, the passive decay heat removal from the core debris settled on single-tray-type design of core catcher has been numerically simulated under different amounts of core relocation using a validated 2-D axisymmetric CFD model. The maximum temperature experienced by the debris bed, core catcher plate and the welded joint between the core support structure and the main vessel (i.e., triple point) are evaluated in order to estimate the amount of core relocation that can be safely handled by the primary containment system under Post Accident Heat Removal (PAHR) scenario. The numerical study revealed that the single-tray-type design of core catcher can potentially assist the passive cooling even under the settling of debris material with heat generation equivalent to 60% of the full core on the core catcher. In addition to this, the effect of the size of opening created on grid plate during core relocation on Post Accident Heat Removal (PAHR) is studied for different amounts of core relocation. From this numerical study, the thermal capability of single-tray type in-vessel core catcher of SFR is assessed. The predictions are useful in determining the safe thermal margins available under core relocation scenarios and extending the functionality of single-tray-type core catcher for handling Whole Core Meltdown Accident (WCA) scenario in future SFRs.

Investigation of molten fuel coolant interaction phenomena using real time X-ray imaging of simulated woods metal-water system

Abstract In liquid metal fast breeder reactors, postulated failures of the plant protection system may lead to serious unprotected accidental consequences. Unprotected transients are generically categorized as transient overpower accidents and transient under cooling accidents. In both cases, core meltdown may occur and this can lead to a molten fuel coolant interaction (MFCI). The understanding of MFCI phenomena is essential for study of debris coolability and characteristics during post-accident heat removal. Sodium is used as coolant in liquid metal fast breeder reactors. Viewing inside sodium at elevated temperature is impossible because of its opaqueness. In the present study, a methodology to depict MFCI phenomena using a flat panel detector based imaging system (i.e., real time radiography) is brought out using a woods metal-water experimental facility which simulates the UO 2 -Na interaction. The developed imaging system can capture attributes of the MFCI process like jet breakup length, jet front velocity, fragmented particle size, and a profile of the debris bed using digital image processing methods like image filtering, segmentation, and edge detection. This paper describes the MFCI process and developed imaging methodology to capture MFCI attributes which are directly related to the safe aspects of a sodium fast reactor.

Experimental simulation of fragmentation and stratification of core debris on the core catcher of a fast breeder reactor

The complex and coupled phenomena of two simultaneous molten metal jets fragmenting inside a quiescent liquid pool and settling on a collector plate are experimentally analysed in the context of safety analysis of a fast breeder reactor (FBR) in the post accident heat removal phase. Following a hypothetical core melt down accident in a FBR, a major portion of molten nuclear fuel and clad/structural material which are collectively termed as ‘corium’ undergoes fragmentation in the bulk coolant sodium in the lower plenum of the reactor main vessel and settles on the core catcher plate. The coolability of this decay heat generating debris bed is dependent on the particle size distribution and its layering i.e., stratification.Experiments have been conducted with two immiscible molten metals of different densities poured inside a coolant medium to understand their fragmentation behaviour and to assess the possibility of formation of a stratified debris bed. Molten aluminium and lead have been used as simulants in place of molten stainless steel and nuclear fuel to facilitate easy handling. This paper summarizes the major findings from these experiments. The fragmentation of the two molten metals are explained in the light of relevant dimensionless numbers such as Reynolds number and Weber Number. The mass median diameter of the fragmented debris is predicted from nonlinear stability analysis of slender jets for lead jet and using Rayleigh's classical theory of jet breakup for aluminium jet. The agreement of the predicted values with the experimental results is good. These experiments indicate that complete segregation of the lighter component above the denser component does not happen. However, there is segregation observed in the radial direction to a certain extent in the case of rib-partitioned collector plate pointing to the significance of the geometry of the collector plate in settling and segregation aspects. It is observed that fragmentation phenomenon with its interplaying governing forces, interferes with the settling aspects and seems to suppress stratification that arises due to density differences between the molten melts. Moreover, the thermophysical properties of the metals play a crucial role in the fragmentation and settling behaviour as exemplified by the relevant dimensionless numbers.

An investigation of kinetic reaction and decomposition of sodium uranate

For the management of severe accidents of sodium-cooled fast breeder reactor, the coolability of the fuel debris bed on a core support plate is a key concern during the post-accident heat removal phase. In an air ingress scenario, the reactions between the fuel and highly oxidized sodium are likely to form sodium uranoplutonate. This would negatively influence the coolability of the fuel debris bed due to a lowering of the thermal conductivity and density. This study has focused on the formation kinetics of sodium uranate from UO2 and liquid sodium including oxygen at a high concentration. In this paper, the experiments on reaction initiation temperatures, reaction rates, and the decomposition of sodium uranate are reported.

Investigation of fragmentation phenomena and debris bed formation during core meltdown accident in SFR using simulated experiments

The event of a severe core melt down accident, resulting in the relocation of the active core is analyzed as a part of the nuclear reactor safety research in order to ensure safe removal of decay heat. Molten Fuel Coolant Interaction (MFCI) and debris bed configuration on the core catcher plate assumes importance in assessing the post accident heat removal capability. The key factors affecting the coolability of the debris bed are the bed porosity, morphology of the fragmented particles, degree of spreading/heaping of the debris on the core catcher and the fraction of lump formed. A well defined debris bed is helpful in fixing a prototypical source term for the PAHR studies. Towards this, a series of experiments on fragmentation kinetics and subsequent debris bed formation is conducted with molten Wood's metal (an alloy of Bi 50%, Pb 25%, Sn 12.5% and Cd 12.5% with melting point of 346K) in water simulant system. The experiments are carried out using 2kg, 5kg and 20kg melt inventories. The particle size distribution obtained for the fragmented debris is fit using an Upper Limit Log Normal (ULLN) distribution. The dependence of particle size distribution on initial melt temperature and interaction height is quantified by correlating them to the key parameters i.e. shape factor and location factor in the ULLN expression. Morphology of the debris particles is investigated to understand the fragmentation mechanisms involved. Three major mechanisms of fragmentation are identified namely melt entrainment mechanism, boundary layer stripping and hydrodynamic breakup due to capillary forces. Finally, an approach to quantify the stratification and spreading behaviour of debris on the collector tray is presented by analysing the particle size and mass percentage in different radial zones of the collector tray.

Numerical simulation of passive heat removal under severe core meltdown scenario in a sodium cooled fast reactor

A sequence of highly unlikely events leading to significant meltdown of the Sodium cooled Fast Reactor (SFR) core can cause the failure of reactor vessel if the molten fuel debris settles at the bottom of the reactor main vessel. To prevent this, pool type SFRs are usually provided with an in-vessel core catcher above the bottom wall of the main vessel. The core catcher should collect, retain and passively cool these debris by facilitating decay heat removal by natural convection. In the present work, the heat removal capability of the existing single tray core catcher design has been evaluated numerically by analyzing the transient development of natural convection loops inside SFR pool. A 1-D heat diffusion model and a simplified 2-D axi-symmetric CFD model are developed for the same. Maximum temperature of the core catcher plate evaluated for different core meltdown scenarios using these models showed that there is much higher heat removal potential for single tray in-vessel SFR core catcher compared to the design basis case of melting of 7 subassemblies under total instantaneous blockage of a subassembly. The study also revealed that the side pool of cold sodium plays a significant role in decay heat removal. The maximum debris bed temperature attained during the initial hours of PAHR does not depend much on when the Decay Heat Exchanger (DHX) gets operational, and it substantiates the inherent safety of the system. The present study paves the way for better understanding of the thermal hydraulics of Post Accident Heat Removal (PAHR) towards extending the functionality of core catcher for handling Whole Core Meltdown Accident (WCA) scenarios in future SFRs.

ICONE23-1950 NUMERICAL STUDY ON INFLUENCE OF OHNESORGE NUMBER AND REYNOLDS NUMBER ON THE JET BREAKUP BEHAVIOR USING THE LATTICE BOLTZMANN METHOD

In a core disruptive accident of a sodium-cooled fast reactor, the considerable amount of fuel in the core region may melt, and be discharged into the coolant like a jet. Hence, it is necessary to understand the jet breakup behavior from the viewpoint of the post-accident heat removal. In order to clarify the mechanism of jet breakup caused by the hydrodynamic interaction, we simulated the injection of a jet into a coolant using the lattice Boltzmann method for two-phase fluid which can represent the spontaneous interfacial behavior. In the simulation, we changed the Reynolds number of the jet and the viscosity ratio between the jet and the coolant, and observed their influence to the interfacial behavior. By observing the interfacial behavior, we found that the wave formation and fragmentation at the side of the jet were suppressed when the Reynolds number becomes small, and cognized the importance of considering the influence of the viscosity of a coolant to the jet behavior.

The Phenomenology of Packed Beds in Heavy Liquid Metal Fast Reactors during Postaccident Heat Removal: The Self-Removal Feedback Mechanism

The phenomenology for a particular behavior of packed beds in heavy liquid metal (HLM) fast reactors during postaccident heat removal is proposed. Because of the similar densities of the fuel and the HLM, an inherent passive safety self-removal feedback mechanism due to buoyancy forces is developed, which propels the packed bed away from the wall, thus preventing temperatures that can jeopardize the vessel’s structural integrity and also reducing the recriticality potential by limiting the allowable bed depth. This identified mechanism will have somewhat compensatory tendencies in the self-leveling behavior of debris beds, which are crucial for sodium-cooled reactors, but unfortunately, it is not operative for HLMs because of the absence of boiling of the coolant. By means of a simplified geometrical model, a preliminary analysis of the potentiality of the phenomenon has been performed.

Post-accident heat removal: Numerical and experimental simulation

Post-accident heat removal (PAHR) after severe accident in nuclear reactors plays a vital role to ensure public safety by effectively cooling fragmented core debris within the primary boundary. Design and implementation of multiple safety features demand thorough understanding of the sequences involved in accident progression. The key factors influencing PAHR are molten fuel coolant interaction and grid plate melt through, followed by relocation of fragmented fuel and structural materials settling on the core catcher in the form of debris below the core. The present work is focused on experimental and numerical investigation of melt fragmentation and settlement, morphological characteristics of debris, and finally heat removal from the debris bed via dedicated decay heat removal system. Series of dedicated experimental facilities have been setup in stages and preliminary trials are being conducted to generate the database for development and validation of numerical models, to be used for safety analysis of reactor systems. This paper also discusses preliminary experimental results and comparisons of data with numerical findings.

20812 溶融物のジェットブレークアップ挙動に及ぼす微粒化の影響(086-4【熱流体工学のフロンティア(4)】)

For the safety design of a Fast Breeder Reactor (FBR), Post Accident Heat Removal (PAHR) is required when a hypothetical Core Disruptive Accident (CDA) occurs. In PAHR, it is necessary to understand the interaction between molten core material and coolant m order to estimate whether the molten material jet is completely solidified by the coolant. The objective of the present study is to clarify the dominant factors of the fragmentation behavior on the jet breakup. In this study, we inject molten material and transparent fluid into coolant at free fall speed. The jet breakup and fragmentation behavior of the molten material and the transparent liquid are observed with a high speed video camera. In addition, we simulate liquid jet using the lattice Boltzmann method m order to investigate the jet breakup and the fragmentation behavior, which are difficult to be visualized or measured in the experiment.

ICONE19-43407 Experrimental Study on Influence of Flow Structure on Jet Surface Fragmentation

Fast Breeder Reactor (FBR) is designed with safety in mind. However, there is billion to one possibility that a hypothetical Core Disruptive Accident (CDA) occurs. When CDA occurs, the Post Accident Heat Removal (PAHR) must be achieved. In the PAHR, the molten material is required to be fragmented and solidified in sodium coolant. In order to estimate whether the molten material jet is completely solidified in sodium coolant or not, it is important to estimate jet breakup length, which is the distance from the location where the jet penetrates into coolant to the location where the jet column disappears. Although, the jet breakup length is influenced with fragmentation behavior, the correlation between the jet breakup length and fragmentation behavior are not clear yet. Therefore, it is strongly required to clarify the mechanism of the fragmentation behavior on the jet surface. The objective of the present study is to clarify the flow structure surrounding the jet surface injected into liquid on fragmentation experimentally. Tap water and Fluorinert^<TM> (FC-3283) are used as simulated coolant and molten material. FC-3283 is transparent and color less liquid and its density is higher than water. FC-3283 is injected into water pool, and the jet breakup and the fragmentation behavior of the jet are observed by using a high speed video camera. In order to clarify the surrounding flow structure of the jet, velocity measurement with Particle Image Velocimetry (PIV) is conducted. We also identify the position of the interface with the back lighting technique simultaneously. The interfacial wave behavior on the jet surface is observed. The interface of the jet is deformed and interfacial wave is generated on the jet surface. Also, the velocity of surrounding flow has high values near the interface of the jet. The deformation of the jet surface and the flow structure of the interface are discussed.

Analyses of loads on reactor pressure vessel internals in a pressurized water reactor due to a loss-of-coolant accident considering fluid-structure interaction

Abstract During a postulated loss-of-coolant accident, significant structural loadings on the reactor pressure vessel and its internals can be exerted due to the non-uniform and time-dependent pressure differences. The integrity of the internals must be secured, so as to ensure reactor shutdown and post-accident heat removal. Two developed computer codes to be discussed in this paper are used for the analysis of loads on the reactor pressure vessel internals. In the analyses the fluid-structure interaction effects are taken into account in the annular space between the pressure vessel and the core support barrel as well as in the upper plenum. This is a more realistic approach with respect to peak loadings and structural response frequencies.

Assessment of creep-fatigue damage at main vessel triple point of Prototype Fast Breeder Reactor after core disruptive accident

The main vessel (MV) carries all the major reactor components including core and liquid sodium filled to 12.4 m height. The reactor core along with the inner vessel is placed on the grid plate, which is supported on the core support structure (CSS). The CSS is supported on the MV through a support shell, welded to the dished end at the point where crown and knuckle joins. The CSS support shell to MV joint is called as triple point and damage at this point is of concern. The core catcher (CC) is placed below the CSS and is also supported on the CSS support shell. The triple point is subjected to a dead load of 920 t. During the unlikely event of core disruptive accident (CDA) a high dynamic pressure will act on the core catcher (CC), which in turn exerts a high load on the triple point. Also the hot molten core debris will come to the CC, which is close to the MV dished end. This will result in increase of temperature in the cold pool, which will cause significant creep damage at triple point during post accident heat removal phase. As the triple point is the only junction, which is supporting the reactor core and CC through CSS support shell, its failure can cause serious consequences to the integrity of MV. Hence, triple point is analysed to ensure its integrity during the transient pressure loading and subsequent post accident heat removal (PAHR) phase of CDA. The stress intensities during transient pressure are found to be within the limits. The subsequent creep damage during post accident heat removal (PAHR) phase is also assessed. Based on the apportioned creep damage of 0.4 for category 4 events, the triple point can be kept at a temperature of 923 K (650°C) or 973 K (700°C) effectively for a period of 1.2 × 104 h (500 days) or 662 h (27.6 days) respectively which are acceptable with comfortable margin. More details are presented in the paper.

B210 溶融物ジェットと冷却材の相互作用に及ぼす界面固化の影響(OS8 熱流動),動力エネルギーシステム部門20周年,次の20年への新展開)

For the safety design of the Fast Breeder Reactor (FBR), the Post Accident Heat Removal (PAHR) is required when a hypothetical Core Disruptive Accident (CDA) occurs. To clarify the interaction between molten material and coolant such as jet breakup behavior during the CDA for the FBR, the dominant mechanism of the jet breakup behavior is required. The objective of the present study is to clarify the dominant factor of the jet breakup length and size distribution of the fragments generated through the jet breakup behavior in the condition of surface solidification as simulation of phenomena in a real FBR system. A jet breakup behavior as interaction of simulated molten core material and coolant is observed with high-speed video camera. The solidified fragments are gathered and classified in size, and the each size is measured. The median diameter of the generated fragments is estimated from the mass distribution of the fragments. It is suggested that form of fragments is different by temperature conditions.

E205 液中ジェット界面における内部流動と周囲流動との相互作用(高速炉基盤技術,OS8 軽水炉・新型炉・核燃料サイクル)

For the safety design of the Fast Breeder Reactor (FBR), it is strongly required that the Post Accident Heat Removal (PAHR) is achieved after a hypothetical Core Disruptive Accident (CDA). In the PAHR, it is important that the molten core material is fragmented to be solidified by the sodium coolant and it is necessary to evaluate the jet breakup length. Although there are many previous studies on the jet breakup length, the tendency of jet breakup length is different for the previous studies. To estimate jet breakup length, it is necessary to clarify the influence of fragmentation behavior on jet surface and on jet breakup behavior in coolant, In present study, experiments are conducted to obtain interfacial fragmentation behavior by injecting transparent fluid in water. Internal and external velocity distributions of the jet are measured by PPV technique. Reynolds stress, turbulent energy and shear stress are evaluated from the velocity data. As the results, Reynolds stress and turbulent energy become large in the vicinity of jet surface. Vortex around interfacial wave is observed. Local shear stress acts on the interfacial wave. Mechanism of fragmentation and interaction of jet internal flow and surrounding flow through interface are discussed.

Natural convection fluid flow and heat transfer in porous media

Natural convection and heat transfer of fluid flow are studied numerically inside a rectangular cavity with inclination filled with a porous medium. The mass and momentum equations are given by the Darcy equations coupled with the thermal energy equation through the unsteady Boussinesq approximation. The two-dimensional restriction in terms of the stream function and vorticity variables is considered. The study is analyzed in terms of several values of the parameters that determine the evolution of the flow: the Rayleigh Number, the aspect ratio of the cavity and the angle of inclination. The mass and momentum equations in natural convection fluid flow in a porous medium are given by the Darcy equations coupled with the thermal energy equation through the unsteady Boussinesq approximation to deal with an incompressible structure. In this work the dimensionless problem is formulated in terms of the stream function and vorticity variables; then, the computation of the pressure is avoided and the incompressibility condition is satisfied automatically. Regarding the numerical method, once a convenient second order time discretization is performed, a nonlinear elliptic system is obtained which is solved through a fixed point iterative process. The iterative process leads to the solution of uncoupled, well-conditioned, symmetric linear elliptic problems for which very efficient solvers exist regardless of the space discretization. The study of natural convection and heat transfer of fluid flow in a porous medium has important technological applications: storage and preservation of grains and cereals; solar energy collectors; filter systems; transport of radioactive wastes through the soil; and postaccident heat removal in nuclear reactors. Our numerical study is carried out on tilted rectangular cavities. The study is realized through the parameters that influence directly the behavior and evolution of the flow: the Rayleigh Number Ra, the aspect ratio of the cavity A, and the angle of inclination . We mention below two categories of research in connection with natural convection problems that arise when opposing walls of a cavity are subjected to a temperature gradient and where the other set of walls is insulated—the subject of the present work.