Solid Temperature Distributions Research Articles

Mathematical simulation on coupled flow, heat, and solute transport in slab continuous casting process

A three-dimensional comprehensively coupled model has been developed to describe the transport phenomena, including fluid flow, heat transfer, solidification, and solute redistribution in the continuous casting process. The continuous casting process is considered as a solidification process in a multicomponent solid-liquid phase system. The porous media theory is used to model the blockage of fluid flow by columnar dendrites in the mushy zone. The relation between flow pattern and the shape of the solid shell is demonstrated. Double diffusive convection caused by thermal and concentration gradients is considered. The change in the liquidus temperature with liquid concentration is also considered. The formation mechanism of macrosegregation is investigated. Calculated solid shell thickness and temperature distribution in liquid core are compared with the measured quantities for validating the model.

Modeling of Solid Oxide Heat Exchanger Integrated Stacks and Simulation at High Fuel Utilization

This work provides an evaluation of the behavior of a planar circular solid oxide fuel cell stack with an integrated air preheater fed with hydrogen at the anode and air at the cathode under conditions of high fuel utilization. This evaluation is based on a simulation model, whose hypotheses and equations are presented and discussed, together with model validation. Indeed, it has been shown that the simulation results compare favorably with the experimental data of temperature distribution and I‐V characteristics at low fuel utilization. Having ascertained the reliability of the model, the simulation results obtained at high fuel utilization are presented and discussed. I‐V curves show an almost linear behavior up to fuel utilization factors above 80%. For higher utilizations, the slopes of the curves increase due to an increase of anodic polarization resistance, due to both activation effects and diffusion limitations in the electrode regions underneath the ribs of the gas distributor; on the contrary, diffusion limitations in the regions underneath the gas channels have been demonstrated to be negligible. The results for temperature distributions at high fuel utilization are discussed as well, indicating that the benefit of solid oxide fuel cells with an integrated air preheater, i.e., an almost even distribution of solid temperature on the cell plane, is still valid at high hydrogen utilization factors.

Fluid Temperature and Film Coefficient Prediction and Measurement in Mechanical Face Seals—-Numerical Results

Based on a computational fluid dynamic (CFD) code, an axisymmetric numerical model was developed to predict fluid and solid temperature distributions for an arbitrary mechanical face seal geometry and operating conditions. The effects of varying coolant source location and rates were investigated. Also, various seal chamber configurations were examined. For dead-ended seals, the CFD model shows how heat is transferred away from the seal by turbulent diffusion. Predictions of the local heat transfer coefficients using the present procedure were made and compared to available heat transfer coefficient correlations in the literature. The predictions are also compared to seal fluid and seat temperature measurements. Presented at the 53rd Annual Meeting in Detroit, Michigan May 17–21, 1998

Conjugate natural convection in porous media

This paper reviews conduction-convection conjugated natural convection from plates or bodies in a fluid-saturated porous medium. Heating condition at constant temperature is specified on the surface of a conductive solid body positioned away from the solid-porous interface. The solid-porous boundary temperature is obtained as a part of the solution of temperature field over the domain of interest. Several different configurations including slender bodies, rectangular slabs, horizontal cylinders and spheres are discussed. Associated convective flows and heat transfer characteristics are illustrated. A compact expression for the average Nusselt number is derived from a simple one-dimensional analysis. Analytical solutions for time dependent processes and steady temperature distributions in conductive solids are also included

The viscous stagnation-flow solidification problem

The viscous stagnation-flow solidification problem for a pure substance is investigated by means of quasi-steady, instantaneous-similarity, as well as finite-difference methods. The liquid velocity field, the solid and liquid temperature distributions, the solid-liquid interface location, as well as its growth rate, are obtained using all three methods and comparisons of the solutions are made. The instantaneous-similarity solution at a sufficiently small time is used as the initial field to start the finite-difference calculation. All three methods show the existence of an asymptotic limit of the solidification-front position. The effect on this limit of several important dimensionless parameters, including the Prandtl number, is presented.

Simulation of weld pool flow and capillary interface shapes associated with the plasma arc welding process

A three-dimensional finite element model of the plasma arc welding process is presented. The model allows calculation of the weld pool's approximate capillary and solid-liquid phase boundaries, the weld pool's three-dimensional flow and temperature fields, and the solid phase temperature distribution. Taking into account thermocapillary, jet shear, and buoyancy forces within the pool, it is found that under moderate Reynolds number conditions pool flow is driven primarily by thermocapillary and jet shear forces.

Infrared thermography of dropwise evaporated cooling

An infrared thermographic technique is developed to obtain the transient solid surface temperature distribution in the neighborhood of an evaporating droplet. This technique is nonintrusive and is not affected by the time response of the measuring device (i.e., thermocouple). The entire surface is monitored at any instant of time, and information on the area influenced by the evaporative cooling process is easily derived. A detailed description of the image processing based data reduction is provided. A water droplet in the range of 10–50 μL is deposited on an opaque glasslike material (Macor) that has an initial surface temperature between 100 and 165°C. The evaporative cooling process is fully documented, and these new findings are contrasted with the published literature to gain a better understanding of the phenomena involved.

Transient behavior of an active thermal protectic system

The transient behavior of an active thermal protection system consisting of a semi-infinite layer of a porous medium with gas injection is considered. During the early stages of heating, thermal penetration depth, surface temperature and surface heat flux do not depend on the gas injection velocity. As the heating time increases, the thermal penetration depth of the active thermal protection system reaches a constant value for each gas injection velocity, and the surface heat flux vanishes keeping the back face at a low temperature. However, the thermal penetration depth of conventional insulation without gas injection never reaches a constant value. Transient solid temperature distributions in the layer for two different surface boundary conditions are almost identical, except in the vicinity of the front surface.

Determining Transient Temperature Distribution in Solids from Measurements of the Surface Temperature.

The paper is concerned with a method that enables one to determine transient temperature distribution in a heat conducting body from measurements of the surface temperature. It means that an initial condition is not necessary. The solution can be obtained using analytical or numerical methods. It is presented in terms of a series containing the derivatives of the surface temperature with respect to time. In order to calculate the derivatives the regularization method is employed. Some test cases are included. They prove that the proposed method can produce stable results for a wide range of problems except for rapid processes of heat conduction. It means that the method can be employed when, e.g. one estimates the temperature distribution within blooms or ingots during heating or cooling under moderate conditions.

Thermal characteristics of packed bed storage system

The thermal behaviour of a packed bed storage system charged with hot air is modelled using two partial differential equations representing the energy conservation in the air and solid phases constituting the bed. These two equations are coupled through the heat exchange process between the two phases. A fully implicit numerical scheme based on forward, upwind and central differencing for the time, first and second space derivatives, respectively, is used to solve the modelling equations. Marching technique is used for the air equation and a tri-diagonal matrix solver is employed to solve the solid equation. The solution yields the thermal structure of the bed, namely the air and solid temperature distribution inside the bed at any particular time, and the variation of total energy stored in the bed with time. The effect of bed length, solid diameter and void fraction on the thermal characteristics of the packed bed is studied. Further, the performance of the bed under variable inlet air temperature and mass flow rate is investigated.

Transient analytical temperature distributions in cylindrical packed beds volumetrically heated by radiogenic decay energy

Analytical solutions are presented for the problem of the transient distribution of fluid and solid phase temperatures in a packed, porous, cylindrical particle bed with constant thermophysical properties. The packed particle bed is volumetrically heated by radiogenic decay energy from fission products. Flowing through the particle bed by forced convection is a single-phase fluid, either subcooled liquid or superheated vapor. The dynamic response of the packed bed is for low Reynolds numbers. In this case the transient will develop through the packed bed slowly enough for interphase heat transfer to keep the fluid and solid phase temperatures from having large differences. The two-dimensional, time-dependent Modified Dispersion-Concentric Model (D-C model) is used in the analysis of this problem. The D-C model energy equations are solved using Green's function. The mathematical solution characteristics for the transient fluid and solid phase temperature distributions are presented for three different volumetric heat generation terms: two-dimensional, time-dependent; simplified two-dimensional, time-dependent; and two-dimensional, time-independent. Using the two time-dependent volumetric heat generation terms, a comparison is presented for the transient fluid and solid phase temperatures and the radioactive decay heat power coming from the fission products in the particles.

Analytical solution for the transient thermal response of volumetrically heated fixed porous nuclear debris beds

An analytical solution is presented for the problem of the transient distribution of fluid and solid temperatures in a volumetrically heated fixed porous nuclear debris bed through which a single-phase fluid, either subcooled liquid or superheated vapor, is flowing. The one-dimensional, time-dependent Modified Dispersion-Concentric Model (D-C Model) is used in the analysis of this problem. The method of solution is based on transformations that reduce the equations and then the application of Green's function in a straightforward manner. The mathematical solution characteristics for the transient fluid and solid temperature distributions are analyzed for subcooled liquid flow and for superheated vapor flow. Also, the transient fluid and solid phase temperature distributions for infinite time are compared with steady-state experimental data.

Measurement technique for determining the temperature distribution in a transparent solid using holographic interferometry

Temperature distributions in transparent solids have been determined by measurements of changes in the refractive index of the sample using a holographic interferometry technique. The steady-state temperature distributions within two samples, fused silica and polymethyl methacrylate (PMMA), were measured to demonstrate the technique. Various errors in the measured temperature distribution caused by refraction and heat losses from the sample are discussed and estimated.

A SINGLE PARTICLE MODEL FOR PYROLYSIS OF OIL SHALE

ABSTRACT In many heterogeneous pyrolysis reactions the porous solid undergoes geometrical changes due to the consumption of solid reactant which decrease the compressive strength of the solid matrix. In this study a volume expansion model is proposed for pyrolysis of a single shale particle. This model not only takes into account the structural changes and the intraparticle gradients, but also the functional dependencies of various parameters on the variation of solid reactant conversion and internal temperature distribution. Effects of various parameters which are expected to be encountered in an in-situ retorting process were investigated. Primary emphasis was placed on determining the manner in which transport processes affect the yield of pyrolysis product. The results indicate that Intraparticle heat transfer is a dominant factor in the pyrolysis of oil shale and that volume expansion favors the decomposition rate.

Reduction methods for nonlinear steady‐state thermal analysis

AbstractHybrid analysis techniques based on the combined use of finite elements and the classical Bubnov–Galerkin approximation are presented for predicting nonlinear steady‐state temperature distributions in structures and solids. In these hybrid techniques the modelling versatility of the finite element method is preserved and a substantial reduction in the number of degrees‐of‐freedom is achieved by expressing the vector of nodal temperatures as a linear combination of a small number of global‐temperature modes, or basis vectors. The Bubnov–Galerkin technique is then used to compute the coefficients of the linear combination (i.e. the amplitudes of the global–temperature modes).The basis vectors chosen are the path derivatives commonly used in perturbation techniques, namely, the derivatives of the nodal–temperature vector with respect to a preselected control (or path) parameter(s). The vectors are generated by using the finite element model of the initial discretization. Also, the performance of alternate sets of basis vectors is investigated. In the alternate sets, only a few path derivatives are generated, and they are augmented by a constant vector representing a uniform temperature rise (or drop), and by reciprocal vectors with nonzero components equal to the reciprocals of the nonzero components of the path derivatives. A problem‐adaptive computational algorithm is presented for efficient evaluation of global approximation vectors and generation of the reduced system of equations and for monitoring the accuracy of the reduced system of equations.The potential of the proposed reduction methods for the solution of large‐scale, nonlinear steady‐state thermal problems is also discussed. The effectiveness of these methods is demonstrated by means of four numerical examples, including conduction, convection and radiation modes of heat transfer.This study shows that the use of the uniform‐temperature mode and the path derivatives as global approximation vectors significantly increases the accuracy of the solutions obtained by reduction methods, thereby enhancing the effectiveness of these methods for the solution of large‐scale, nonlinear thermal problems.

Two-dimensional material response of a transpiration-cooled system in a radiative/convective environment

The two-dimensional unsteady material response of a transpiration-cooled system in a radiative/convective heating environment is analyzed numerically with the use of the strongly implicit procedure (SIP). The internal mass flow rate, internal pressure, and in-depth solid and fluid temperature distributions are obtained for various matrix porosities and coolant mass injection rates in a CO2 /silica transpiration-cooled system. The results for application on Saturn entry vehicles provide a feasible idea for a thermal protection system in a severe radiative/convective heating environment. The feasibility of this analysis is tentatively verified by comparison with the experimental results of a radiation simulation of one-dimensional, relatively low-incident radiative heating.

Hybrid perturbation/Bubnov-Galerkin technique for nonlinear thermal analysis

A two step hybrid analysis technique to predict the nonlinear steady state temperature distribution in structures and solids is presented. The technique is based on the regular perturbation expansion and the classical Bubnov-Galerkin approximation. The functions are obtained by using the regular perturbation method. These functions are selected as coordinate functions and the classical Bubnov-Galerkin technique is used to compute their amplitudes. The potential of the proposed hybrid technique for the solution of nonlinear thermal problems is discussed. The effectiveness of this technique is demonstrated by the effects of conduction, convection, and radiation modes of heat transfer. It is indicated that the hybrid technique overcomes the two major drawbacks of the classical techniques: (1) the requirement of using a small parameter in the regular perturbation method; and (2) the arbitrariness in the choice of the coordinate functions in the Bubnov-Galerkin technique. The proposed technique extends the range of applicability of the regular perturbation method and enhances the effectiveness of the Bubnov-Galerkin technique.

Response of thermal storage units to periodic operating conditions

This paper aims to provide an exact solution to the problem of sensible heat transfer and storage in fluid-solid systems. Periodically fluctuating operating conditions are considered in detail, and a simple solution procedure is proposed in order to obtain the fluid and solid temperature distributions for any periodic thermal energy supply. Some significant examples are solved explicitly, and the relevant results are reported and briefly discussed in order to examine directly the storage unit's capability of smoothing out time fluctuations of the power source.

Some potential applications of distributed parameter systems theory to metallurgical heating and casting operations

Two recent computer control studies are discussed with regard to their possible applications in the metals processing industry. The first study involves the on-line estimation and control of temperature profiles for a steel ingot in a furnace. The second study involves the real time estimation of solid crust thickness and temperature distribution in casting operations. The potential applications of these results to the control of soaking pits, slab reheating furnaces and continuous casting operations are discussed.

Combustion of wood charcoal

The dynamics of burning of wood charcoal in an air stream is examined both experimentally and theoretically. To simplify the theory, an experimental arrangement approximating a one dimensional phenomenon was adopted. The theory includes conduction in the solid, chemical reactions and heat release at the surface, and heat and mass transfer in the gas boundary layer above the surface. The molar CO/CO2 ratio is measured. The theory predicts surface temperature, solid temperature distribution and burning rate within experimental error. An effective reaction rate formula is developed.