Effects of Boundary Conditions on Thermal Response of a Cellulose Acetate Layer Using Hottel’s Zonal Method

Energy can transfer internally by radiation in addition to conduction in translucent polymers. Since radiant propagation is very rapid, it can provide energy within the layer more quickly than diffusion by heat conduction. Thus, the transient thermal response of a layer for combined radiative and conduction may be extremely different from that of conduction alone. In this paper, the behavior of a heat conducting, absorbing, and emitting layer of Cellulose Acetate layer is investigated during the transient interval when both conductive and radiative heat transfer are considered. Specifically, the effects of boundary conditions on the response of the layer are considered here. These boundary conditions include both conductive boundary conditions, such as convection coefficient and convective fluid temperature, and radiation boundary conditions, like radiation surrounding temperature and specular reflectivity. To this end, the governing differential equations including the equation of radiative heat transfer within the material coupled to the transient energy equation with radiative terms are presented. The solution procedure is based on nodal analysis and Hottel’s zonal method extended by the ray tracing method. The transient energy equation including the radiative internal energy source is solved using a time marching finite difference procedure with variable space and time increments.     

Optimization of Production Process of a Semitransparent Layer Containing Foam by Considering Radiation and Using Finite Difference Method

Regarding the importance of radiation in contemporary technology of the world, this paper presents a finite difference solution for geometric optimization of a radiative enclosure. In this regard, produced foams during glass melting process are one of the important factors for optimization of melting process and final product quality. On one hand, these foam causes prevention of some part of radiation energy from combustion to glass melt and also leads to increase in fuel and decrease in furnace output. On the other hand, they put high damages to final glass products. For designing a radiant enclosure, the radiative two-flux equation is coupled with the transient energy equation and radiation transfer equation (RTE).     

Film Boiling Heat Transfer around a Sphere in Forced Convection

Film boiling heat transfer around a sphere moving downward in a liquid, is analyzed in this study. The general solution obtained by the theory proposed, developed under certain reasonable assumptions, shows that the Nusselt number for predicting the boiling heat transfer coefficient depends on the Reynolds number, the Prandtl number, the liquid/vapor viscosity ratio, the kinematic viscosity of liquid, the absolute size of sphere, and on another dimensionless number C_p(π_d—π₁)Δi.

Results obtained numerically from the general solution are applied as examples to boiling heat transfer for water and sodium, from which it is possible to estimate the heat transfer coefficients for a sphere with boiling. The effect of radiative heat transfer is also examined.     

Approximate methods for exit distribution problems in inhomogeneous slabs

A generalization of the Chandrasekhar H-equation is solved by a class of approximate methods. The exit distribution problem for radiative transfer, either multigroup or one group, may be solved in terms of the solution to this equation.     

Transient Heat Transfer Analysis of a Layer by Considering the Effect of Radiation

The transient heat transfer analysis of a layer has been studied much less than the steady state. However, transient temperature distribution resulted from including radiation and conduction simultaneously, is significantly different from those obtained by considering conduction alone. In order to include the effect of radiation heat transfer, we must insert the gradient of radiative flux in the energy equation. For this purpose, a variety of multi-flux methods have been suggested. A simplified procedure is the two-flux method, which is the one used in the present paper. This paper is focused on one-dimensional transient heat transfer of a layer using Finite Difference Method. To this end, a computer implementation has been written, which is based on implicit finite difference formulation and is capable of considering effect of radiation. By subjecting the layer to various boundary conditions, the effect of variation of refractive index and scattering on transient temperature distribution is discussed.     

Radiative transfer in plane inhomogeneous media with exponentially varying albedo

Accurate numerical results for the exit distributions and the global reflection and transmission coefficients relevant to radiative transfer in a stratified medium with exponentially varying albedo are obtained and compared to previous results. The semi-analytical solution of the linear transport equation is rigorously performed on the basis of a simple projectional method.     

Existence theory for a one-dimensional problem arising from the boundary layer analysis of radiative flows

We consider a simplified system of equations which models the transfer of energy with conductive, convective and radiative effects inside a convex region filled with a compressible fluid whose velocity field is known. The asymptotic analysis for positive but small distance from an optically thick medium leads to a one-dimensional system of differential equation which couples the temperature and the radiative intensity. We show that this system obeys a conservation law and this feature is explored in order to reduce the problem to a single one-dimension transport equation with anisotropic scattering. This equation admits a formulation in terms of integral operators in a suitable function space which allows us to establish the existence of a solution and infer its behavior far from the boundary. We also provide numerical simulations and comparison with the theoretical results which we have shown in order to validate our methodology.     

Existence theory for a one-dimensional problem arising from the boundary layer analysis of radiative flows

We consider a simplified system of equations which models the transfer of energy with conductive, convective and radiative effects inside a convex region filled with a compressible fluid whose velocity field is known. The asymptotic analysis for positive but small distance from an optically thick medium leads to a one-dimensional system of differential equation which couples the temperature and the radiative intensity. We show that this system obeys a conservation law and this feature is explored in order to reduce the problem to a single one-dimension transport equation with anisotropic scattering. This equation admits a formulation in terms of integral operators in a suitable function space which allows us to establish the existence of a solution and infer its behavior far from the boundary. We also provide numerical simulations and comparison with the theoretical results which we have shown in order to validate our methodology.     

Tritium Isotope Separation by CO2 Laser Irradiation at Dry Ice Temperature

A computational method based on boundary element method (BEM) has been developed to analyze transient temperature distribution in a three-dimensional solid having non-linear boundary condition (for example boiling or thermal radiation). To obtain high numerical accuracy, the heat transfer coefficient which depends on the surface temperature was approximated as a linear combination of interpolation functions with respect to time and analytical time integration of the non-linear term included in boundary integral equation was made.

To investigate feasibility of the present method, it was applied to calculations of the temperature distribution in an infinite flat plate cooled on one side. The heat transfer co-efficient of cooling surface was in proportion to n-th power of the surface temperature. Results are (1) the method has given a stable solution for non-linear boundary value problems such as conventional BEM could not solve, (2) it has given an accurate solution for large time increment except for first time step, and (3) an optimal relaxation factor decreases with strength of non-linearity of the heat transfer coefficient.

Applying the method to thermal analysis of limiter and divertor plate installed in a fusion test facility and fusion reactor respectively, availability of the method has been confirmed.     

Heat conduction analyses on rewetting front propagation during transients beyond anticipated operational occurrences for BWRs

Our previous study investigated the rewetting behavior of dryout fuel surface during transients beyond anticipated operational occurrences for BWRs, which indicated the rewetting velocity was significantly affected by the precursory cooling defined as cooling immediately before rewetting. This study further investigated the previous experiments by conducting additional experimental and numerical heat conduction analyses to characterize the precursory cooling. For the characterization, the precursory cooling was first defined quantitatively based on evaluated heat transfer rates; the rewetting velocity was investigated as a function of the cladding temperature immediately before the onset of the precursory cooling. The results indicated that the propagation velocity appeared to be limited by the maximum heat transfer rate near the rewetting front. This limitation was consistent with results of the heat conduction analysis using heat transfer models for the precursory cooling expressed as a function of distance from the rewetting front, the maximum wetting temperature, and the heat transfer coefficients in the wetted region. This paper also discusses uncertainties in the evaluation of transient heat flux from the measured surface temperature, and technical issues requiring further investigation.     

An analysis of rewetting of a nuclear fuel rod in water reactor emergency core cooling

An exact solution of the quasi-steady two-dimensional conduction equation for the rewetting of a nuclear fuel rod in water reactor emergency core cooling is obtained for a cylindrical rod geometry. The analysis adopts the conventional model of two heat transfer regions: zero heat transfer coefficient over the dry surface and a constant heat transfer coefficient over the wetted surface. Both the wet front velocity and the temperature distribution in the rod are computed. The present solution is valid over the whole range of Biot number.     

Transient pool boiling heat transfer during rapid cooling under saturated water condition

The transient pool boiling heat transfer during rapid cooling was investigated under the saturated water condition. The quenching method was applied, and the vertical rodlet was used as the test specimen. The effects of material property, test specimen size, groove structure, and chromium coating were studied. The small heat capacity and small volume-to-area ratio (VAR) made the quenching duration shorter. In the groove-structured test specimen, the vapor film on the structured portion collapsed earlier than on the plain portion, which could be due to the small local VAR of the structured portion. It caused the groove-structured test specimen to exhibit a shorter quenching duration than the plain test specimen, though both test specimens had almost the same averaged VAR. This implies that the quenching duration and position of the vapor film rupture can be actively controlled by the surface structure. The influence of chromium coating on the quenching duration was preliminarily tested using niobium for accident tolerant fuel (ATF) application in a nuclear light water reactor. The chromium-coated niobium test specimen exhibited a somewhat longer quenching duration than the non-coated one.     

高温气冷堆堆芯主动冷却过程动态分析

高温气冷堆紧急停堆后需要快速冷却堆芯,使其达到重新启动条件,制定合理的冷却方案对于减少电厂运行成本和保护设备安全具有重要意义。本文建立了冷却系统的数学模型,对冷却过程中关键设备的传热传质过程进行了动态数值模拟。首先分析了德国高温气冷堆采用的直接冷却方案,结果表明,此方案无法避免对设备形成冷冲击或热冲击,风险性较大。进而提出了适用于我国高温气冷堆的新方案,新方案包括4个步骤：蒸汽发生器排水-卸压-预冷-冷却堆芯。动态分析表明,新方案成功地避免了冷/热冲击,大幅提高了安全性,冷却时间也在可接受范围内。     

Transient natural convection cooling of a vertical circular cylinder

A numerical solution is presented for transient natural convection cooling of a finite vertical circular cylinder standing on a semi-infinite horizontal base. The method used is the quasi-finite element approach introduced by Mansour et al. in the determination of the steady state temperature field in metal cutting. The method is able to solve heat transfer problems in a more economical way (memory and computing time requirements) than the classical finite element method and the finite difference techniques.     

SENHOR-IV: A computer code for small pipe-break analysis of pressure-tube type reactors

A computer program SENHOR-IV was developed which describes blowdown phenomena associated with a small pipe-break accident in pressure-tube type reactors. Thermal-hydraulic transients of single-phase and two-phase flow in a primary cooling system, which is composed of the pressure tubes, a steam drum, downcomers, a lower header and pipings connecting these components, were calculated from the conservation equations of mass, momentum and energy by assuming pressure propagation and flow rate distribution to be quasi-steady and by applying the method of characteristics to enthalpy transport. The void propagation velocity in two-phase flow was given from Smith's equation for void-quality relationship to the program. Calculation of a flow transient, which has an exact solution, with use of this program showed small deviations from the exact solution. Predicted transients of pressure and water level in the steam drum indicated a good agreement with those observed in a full scale test facility at O-arai Engineering Center.     

Transient experiments on fast reactor core thermal-hydraulics and its numerical analysis: Inter-subassembly heat transfer and inter-wrapper flow under natural circulation conditions

Sodium experiments were conducted on core thermal-hydraulics simulating a scram transient of a large scale fast breeder reactor using the test facility PLANDTL-DHX with seven fuel subassemblies. The influence of inter-subassembly heat transfer on temperature distribution in the subassembly was revealed via measurements. The flow in the gap between neighboring subassemblies called inter-wrapper flow (IWF) was also studied in relation to its capability of cooling the subassemblies. A computational model is presented for predicting the transient without IWF. The multi-dimensional numerical analysis model employs an empirical correlation to simulate mixing effects between adjacent subchannels. It was shown that the present computational method could evaluate the transient behavior of thermal-hydraulics in the subassemblies accurately from forced to natural circulation accompanied by inter-subassembly heat transfer and flow redistribution in the subassembly. The cooling effects of IWF on the fuel subassemblies were found in spite of natural circulation flow reduction in the primary loop attributable to temperature decreases in the upper non-heated section in the core. The inter-wrapper flow can effectively cool the core under extreme conditions of low flow rates through the core.     

Porous media approach in thermohydraulic analysis of high temperature reactors in pressurized/depressurized cooldown: An improvement

The current study aims at introducing a 2D and fast-running code for the issues pertinent to design, analysis and safety in modular high temperature reactors. While the porous media approach is only applied to pebble bed type, the analysis in this paper covers both pebble bed and prismatic reactor.A time-dependent mass equation along with energy conservation equation for the cooling gas and a time-dependent energy conservation equation for the solid was solved. Appropriate series of constitutive equations (e.g. heat transfer coefficient, effective heat conductivity of solid, heat transfer coefficient, pressure drop etc.) has been recruited as well. In addition a finite-volume method is employed for spatial discretization. The SIMPLET algorithm has been used to solve the velocity and pressure linked to the momentum equation. The method of SIMPLET for natural convection is lot more advantageous over the SIMPLE method and will improve the results. Our developed code utilizes advantages of both Zehner and Schlünder and Kasperek and Vortmeyer models which lead to better results. In addition, in Thermo Hydraulic Porous Program (THPP), the Rhie-Chow technique is also used to correct the velocity components while dealing with the discretization problem of the pressure gradient. In the codes developed so far, staggered grids is usually used in computations. However, here we have adopted most of the advantages of Rhie-Chow technique in precision and computational cost.Making use of some simplified assumptions made by the benchmark definition, the core has been modeled in form of 2D-geometry. The calculations below deal with the loss of cooling accidents with or without depressurization. Having compared 2D results of THPP, the well-established thermal-hydraulics codes of THERMIX (Banaschek, 1983) and TH3D (Hossain, 2008) to simulate pebble bed and block fuel elements, it becomes clear that regarding the transient behavior during a depressurized loss of coolant accident, there exists a good agreement. Besides, there were detected more considerable differences between the results of the two codes regarding the pressurized loss of cooling accident. The program code shall be generally applicable for modular High Temperature Reactors (HTRs) e.g. pebble fuel and block fuel elements.     

A two-component equilibrium-diffusion limit

The equilibrium-diffusion limit of the radiative transfer equations is characterized by a medium that is optically thick and diffusive for photons of all frequencies. In reality, this condition is almost never met because the transport medium tends to be optically thin for photons of sufficiently high frequency. Motivated by this fact, we derive a new asymptotic limit of the radiative transfer equations that is characterized by two photon components: one for which the medium is optically thick and diffusive, and the other for which the medium is optically thin. In this limit, the leading-order material temperature satisfies a time-dependent diffusion equation, and the leading-order radiation intensity for the optically thick photons is given by the Planck function evaluated at the leading-order material temperature, but the radiation intensity for the optically thin photons is zero through first order. The O(ϵ²) radiation intensity for the optically thin photons satisfies a quasi steady-state transport equation with zero interaction terms and a Planck emission term that depends upon the leading-order material temperature. We also discuss alternative scalings associated with the two-component limit that are characterized by a stronger coupling between the material and the optically thin component.     

A two-component equilibrium-diffusion limit

The equilibrium-diffusion limit of the radiative transfer equations is characterized by a medium that is optically thick and diffusive for photons of all frequencies. In reality, this condition is almost never met because the transport medium tends to be optically thin for photons of sufficiently high frequency. Motivated by this fact, we derive a new asymptotic limit of the radiative transfer equations that is characterized by two photon components: one for which the medium is optically thick and diffusive, and the other for which the medium is extremely optically thin. In this limit, the leading-order material temperature satisfies a time-dependent diffusion equation, and the leading-order radiation intensity for the optically thick photons is given by the Planck function evaluated at the leading-order material temperature, but the radiation intensity for the optically thin photons is zero through first order. The O(ϵ²) radiation intensity for the optically thin photons satisfies a quasi steady-state transport equation with zero interaction terms and a Planck emission term that depends upon the leading-order material temperature. We also discuss alternative scalings associated with the two-component limit that are characterized by stronger coupling between the material and the optically thin component.     

The effect of retarding torque during a flow transient for Tehran Research Reactor

The primary cooling system of the Tehran Research Reactor (TRR) has been analysed for a possible flow transient phenomenon caused by power cut-off. All the components of the TRR primary cooling loop that offer resistance to the coolant flow are physically modelled. Differential equations of motion for the coolant in the primary piping of the TRR and for the rotating parts of the centrifugal pump are then derived. The equation of flow motion is solved simultaneously with momentum conservation equation of the rotating parts of the pump which predicts the TRR pump speed during the flow transient. Electrical and mechanical losses are measured for the TRR three-phase induction motor in order to calculate the motor retarding torque during the event. The results of the present study are compared with the other similar primary loop results. The present model shows good agreement with the existing experimental and theoretical studies.