Estimation of heat flux on the surface of an initially hot cylinder cooled by a laminar confined impinging jet

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space-and time-dependent heat flux at the surface of an initially hot cylinder cooled by a laminar confined slot impinging jet from the knowledge of temperature measurements taken on the cylinder’s surface. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. The results show that an excellent estimation on the space-and time-dependent heat flux can be obtained even the distributions of thermal properties inside the cylinder is unknown.     

Estimation of Heat Flux and Thermal Stresses in Functionally Graded Hollow Circular Cylinders

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux at the inner surface of a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Estimation of heat flux and temperature distributions in a composite strip and homogeneous foundation

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux and temperature distributions for the system composed of a multi-layer composite strip and semi-infinite foundation, from the knowledge of temperature measurements taken within the strip. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the time-dependent heat flux can be obtained for the test case considered in this study.     

Inverse Thermo-Mechanical Analysis of a Functionally Graded Fin

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent base heat flux of a functionally graded fin from the knowledge of temperature measurements taken within the fin. Subsequently, the distributions of temperature and thermal stresses in the fin can be determined as well. It is assumed that no prior information is available on the functional form of the unknown base heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors and measurement location upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Estimation of boundary conditions in the presence of unknown moving boundary caused by ablation

Ablative materials can sustain very high temperatures in which surface thermochemical processes are significant enough to cause surface recession. Existence of moving boundary over a wide range of temperatures, temperature-dependent thermophysical properties of ablators, and no prior knowledge about the location of the moving surface augment the difficulty for predicting the exposed heat flux at the receding surface of ablators. In this paper, the conjugate gradient method is proposed to estimate the unknown surface recession and time-varying net surface heat flux for these kinds of problems. The first order Tikhonov regularization is employed to stabilize the inverse solution. Considering the complicated phenomena that are taking place, it is shown via simulated experiment that unknown quantities can be obtained with reasonable accuracy using this method despite existing noises in the measurement data.     

Application of Sequential Function Specification Method in Heat Flux Monitoring of Receding Solid Surfaces

Measurement of heat flux at the receding surface of materials is extremely difficult due to the presence of harsh working conditions. Under these situations, the solution of inverse heat conduction problem can be found as an efficient means of obtaining heat flux in the moving interface. This paper reports the results of applying the sequential function specification method to the problem of estimating the effect of severe thermal environments on the ablative structures. It is assumed that special sensing instruments are used to record the temperatures and the surface recession in real time of the simulated experiment. Due to high temperature regions near the ablating surface, the temperature sensors may burn out. A numerical experiment with simulated noisy measurements is used to analyze the effect of presence and/or absence of the lost sensors on the outcomes of the proposed procedure. The results introduce an efficient technique for designing heat flux gauges that work with temperature, as well as recession sensors.     

An iterative regularization method in estimating the transient heat-transfer rate on the surface of the insulation layer of a double circular pipe

In this study, a conjugate gradient method based inverse algorithm is applied to estimate the unknown space- and time-dependent heat-transfer rate on the surface of the insulation layer of a double circular pipe heat exchanger using temperature measurements. It is assumed that no prior information is available on the functional form of the unknown heat-transfer rate; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the space- and time-dependent heat-transfer rate can be obtained for the test case considered in this study.     

An inverse problem in estimating the base heat flux of an annular fin based on the hyperbolic model of heat conduction

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent base heat flux of an annular fin from the knowledge of temperature measurements taken within the fin. The inverse solutions will be justified based on the numerical experiments in which two specific cases to determine the unknown base heat flux are examined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux can be obtained for the test cases considered in this study.     

Estimation of the Boundary Heat Flux in Grinding via the Conjugate Gradient Method

The unknown boundary surface heat flux in workpieces during grinding is estimated by the application of inverse heat transfer analysis. The conjugate gradient method of function estimation is used for the minimization procedure. Simulated temperature measurements are used in the inverse analysis for typical practical cases, in order to show that results more accurate than those available in the literature are obtained with the present solution approach. Actual experimental data are also used in the computations to estimate the surface heat flux.     

A Comparison of Quasi One-Dimensional and Two-Dimensional Ablation Models for Subliming Ablators

A method to couple a computational fluid dynamic analysis (CFD) with the material response code (ablation code) for a reentry vehicle is presented here. The geometry under consideration is a sphere cylinder geometry that resembles the YES-2 reentry capsule for which the Mach number is 23. The flow is considered to be steady and laminar, and the effect of radiation is ignored. To take into account the dissociative effect, a simplified model based on the use of a modified specific heat has been adopted following Stella et al. The aerothermodynamic analysis is done using Fluent 6.2. In the ablation analysis, both quasi one-dimensional and two-dimensional approaches have been considered. The heat fluxes determined using the CFD analysis are given as input to the ablation program to calculate the recession rate. The change of the flow field due to the geometry change caused by burn-off of the ablative material has been taken into account after every 500s, as the recession rate is seen to be of the order of micro-meters per second. The prime objective of this work is to bring out a methodology to combine CFD results with in-house codes and to give a comparison between quasi one-dimensional ablation model with two-dimensional ablation model for cylindrical geometries. The two-dimensional analysis gives results similar to those obtained by a quasi one-dimensional ablation, and hence the latter is adequate for problems of this kind.     

Fourier analysis of conjugate gradient method applied to inverse heat conduction problems

The convergence and regularization mechanism of the conjugate gradient algorithm applied to inverse heat conduction problems are studied within the context of a Fourier analysis, for a square enclosure subjected to an unknown time-varying heat flux on one side, and to known boundary conditions on the remaining sides. Analytic solutions are derived for the Fourier components of the unknown flux over a given time interval. The convergence rate of the algorithm is thereby shown to depend essentially on the time frequency of the data. Numerical solutions are also presented to describe in details the convergence process and solution regularization power of the conjugate gradient method, when the unknown heat flux contains many frequency components and the measurement data are noisy. It is found that an unknown time-dependent heat flux may be satisfactorily recovered using a single sensor even when the temperature field becomes two-dimensional, and that the sensor should be placed in a symmetric manner for better results.     

Inverse problem of estimating time-dependent heat generation in a frictional heated strip and foundation

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent frictional heat generation for the tribosystem consisting of a semi-infinite foundation and a plane-parallel strip sliding over its surface, from the knowledge of temperature measurements taken within the foundation. It is assumed that no prior information is available on the functional form of the unknown heat generation; hence the procedure is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the time-dependent heat generation can be obtained for the test case considered in this study. The current methodology can be applied to the prediction of heat generation in engineering problems involving sliding-contact elements.     

Particle Swarm Optimization-based algorithms for solving inverse heat conduction problems of estimating surface heat flux

An inverse analysis of estimating a time-dependent surface heat flux for a three-dimensional heat conduction problem is presented. A global optimization method known as Particle Swarm Optimization (PSO) is employed to estimate the unknown heat flux at the inner surface of a crystal tube from the knowledge of temperature measurements obtained at the external surface. Three modifications of the PSO-based algorithm, PSO with constriction factor, PSO with time-varying acceleration of the cognitive and social coefficients, and PSO with mutation are carried out to implement the optimization process of the inverse analysis. The results show that the PSO with mutation algorithm is significantly better than other PSO-based algorithms because it can overcome the drawback of trapping in the local optimum points and obtain better inverse solutions. The effects of measurement errors, number of dimensionalities, and number of generations on the inverse solutions are also investigated.     

An Integral Approximate Solution to Ablation of a Two-Layer Composite with a Temporal Gaussian Heat Flux

Ablation is the most common approach for thermal management for reentry of the spacecraft to the atmosphere. An analytical solution of the ablation of a two-layer composite, which includes an ablative layer and a nonablative substrate, subject to a Gaussian heat flux is presented in this paper. The problem is divided into five stages and the temperature distributions in both layers in the five stages are obtained using an integral approximate method. The locations of ablation interface, thermal penetration depth, and ablation rate are obtained and the effects of Stefan number, subcooling parameter, thickness of the ablative material, and ratio of thermal diffusivities between two materials are investigated.     

Inverse Boundary Design Problem of Turbulent Forced Convection Between Parallel Plates With Surface Radiation Exchange

The inverse methodology is employed to estimate the unknown heat flux distribution over the heater surface of a channel formed by two parallel plates with forced convection and surface radiation exchange, from the knowledge of the desired temperature and heat flux distributions over a given design surface. The energy and radiative transfer equations are solved by the finite-volume method and the net radiation method, respectively. The conjugate gradient method is used for minimization of an objective function, which is expressed by the sum of square residuals between estimated and desired heat fluxes over the design surface. The performance and accuracy of the present method for solving inverse problems are evaluated by some numerical experiments.     

Estimation of spray cooling characteristics on a hot surface using the hybrid inverse scheme

A hybrid technique of the Laplace transform and finite-difference methods in conjunction with the least-squares method and experimental temperature data inside the test material is applied to investigate the spray cooling of a hot surface. In this study, the unknown surface temperature and heat flux will be predicted. Their functional forms are unknown a priori in the present study and are assumed to be the functions of time before performing the inverse calculation. The whole time domain is divided into several analysis sub-time intervals and then these unknown estimates on each analysis interval can be predicted. In order to validate the accuracy of the present inverse method, comparisons between the present estimates and previous estimated results are made. The results show that the present estimates of the unknown temperature at various measurement locations agree with the previous estimated results and experimental temperature data. However, the present estimates of the unknown surface heat flux deviate from the previous estimated results for larger times.     

Flame extinction properties of solids obtained from limiting oxygen index tests

Extinction theory is used in this work to determine from the limiting oxygen index test (LOI), widely used in industry, flame extinction properties such as the critical mass flux at extinction and its dependence on oxygen concentration. A method for characterizing the chemistry related to extinction, namely activation energy and pre-exponential factor, for an assumed one-step global Arrhenius reaction rate is also presented. In previous work, we showed how the LOI is related to key flammability properties of a material as measured by thermogravimetric analysis, differential scanning calorimetry, and cone calorimetry. In this work, even though the chemical kinetics for practical materials are unknown, we use results from a slightly modified LOI apparatus to derive the material extinction properties, including chemistry effects, by assuming a one-step global Arrhenius reaction rate for extinction. The extinction of flames on a solid surface depends on the flow field through both the heat transferred to the solid and the straining of the flame near stoichiometric conditions. The heat transferred to the solid is determined using a convective heat transfer coefficient, because flame radiation near the surface is small at extinction, and the strain rate affecting the flame reactions is expressed by a characteristic flow time. These flow parameters depend on the local flow conditions. For a counterflow situation, these parameters have been identified in our previous work, whereas for other flow situations, they can be determined from the specific flow field. The ratio of the flow time to the chemical time is used to determine a modified Damköhler number, together with the activation and flame temperatures. The heat transfer coefficient is used to determine the dimensionless mass flux at extinction, which is a function of the Damköhler number.     

Simultaneous estimation of heat-transfer rate and coolant fluid velocity in a transpiration cooling process

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat-transfer rate on the heated surface and coolant fluid velocity in a transpiration cooling process using temperature measurements of the porous medium and coolant fluid. It is assumed that no prior information is available on the functional forms of the unknown heat-transfer rate and coolant fluid velocity; hence the procedure is classified as the function estimation inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the time-dependent heat-transfer rate and coolant fluid velocity can be obtained for the test case considered in this study.     

Analytical Solution of Coupled Laminar Heat-Mass Transfer in a Tube with Uniform Heat Flux

Analytical solution is obtained of coupled laminar heat-mass transfer in a tube with uniform heat flux. This corresponds to the case when a layer of sublimable material is coated on the inner surface of a tube with its outer surface heated by uniform heat flux and this coated material will sublime as gas flows throught the tube.     

Estimation of heat generation at the interface of cylindrical bars during friction process

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat generation at the interface of cylindrical bars during friction process from the knowledge of temperature measurements taken within the bar. It is assumed that no prior information is available on the functional form of the unknown heat generation; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat generation can be obtained for the test case considered in this study. The current methodology can be applied to the prediction of heat generation in continuous-drive friction welding or in breaking systems.