Inverse estimation of spatially and temporally varying heating boundary conditions of a two-dimensional object

In many dynamic heat transfer situations, the temperature at the heated boundary is not directly measurable and can be obtained by solving an inverse heat conduction problem (IHCP) based on measured temperature or/and heat flux at the accessible boundary. In this study, IHCP in a two-dimensional rectangular object is solved by using the conjugate gradient method (CGM) with temperature and heat flux measured at the boundary opposite to the heated boundary. The inverse problem is formulated in such a way that the heat flux at heated boundary is chosen as the unknown function to be recovered, and the temperature at the heated boundary is computed as a byproduct of the IHCP solution. The measurement data, i.e., the temperature and heat flux at the opposite boundary, are obtained by numerically solving a direct problem where the heated boundary of the object is subjected to spatially and temporally varying heat flux. The robustness of the formulated IHCP algorithm is tested for different profiles of heat fluxes along with different random errors of the measured heat flux at the opposite boundary. The effects of the uncertainties of the thermophysical properties and back-surface temperature measurement on inverse solutions are also examined.     

A heat-flux based “building block” approach for solving heat conduction problems

A numerical approximation of the Green’s function equation based on a heat-flux formulation is given. It is derived by assuming as a functional form of the surface heat flux a stepwise variation with space and time. The obtained approximation is very important in investigation of the inverse heat conduction problems (IHCPs) because it gives a convenient expression for the temperature in terms of the heat flux components. Additionally, it is very important for the unsteady surface element (USE) method which is a modern boundary discretization method. Green’s function approximate solution equation (GFASE) also creates ‘naturally’ fixed groups or modules of work elements called “building blocks” that may be added together to obtain space and time values of temperature. In the current case, they are subject to a partial heating by an applied surface heat flux. The “building block” solution can be derived by using the various analytical and numerical approaches available in heat conduction literature though the exact analysis is preferable, as discussed in the text. Poorly-convergent series deriving from Green’s functions approach are replaced by closed-form algebraic solutions.     

Iterative algorithms for solving inverse problems of heat conduction in multiply connected domains

The presented paper displays a method of solving the inverse problems of heat transfer in multi-connected regions, consisting in iterative solving of convergent series of the direct problems. For known temperature and flux values at the outer boundary of the region the temperature and flux values at the inner boundaries are sought (the cauchy problem for the Laplace equation). In case of such a formulation of the problem, the solution does not always exist, one of the conditions is met in the mean-square sense, providing the optimization criterion. The idea of the process consists in solving the direct problem in which the boundary condition is subject to iterative changes so as to attain minimum of the optimization criterion (the square functional). Two algorithms have been formulated. In the first of them the heat flux at the inner boundaries of the region, while in the other the temperature were subject to changes. Convergence of both the algorithms have been compared.The numerical calculation has been made for selected examples, for which an analytical solution is known. The effect of random disturbance of the boundary conditions on the solution obtained with iterative algorithms has been checked. Moreover, a function was defined, serving as convergence measure of the solution of the inverse problem solved with the algorithms proposed in the paper. The properties of the function give evidence that it tends to the value exceeding unity.     

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.     

Estimation of dimensionless parameters of Luikov's system for heat and mass transfer in capillary porous media

This work deals with the solution of inverse problems of parameter estimation involving heat and mass transfer in capillary porous media, as described by the linear one-dimensional Luikov's equations. Our main objective is to use the D-optimum criterion to design the experiment with respect to the magnitude of the applied heat flux, heating and final experimental times, as well as the number and locations of sensors. The present parameter estimation problem is solved with Levenberg–Marquardt's method of minimization of the ordinary least-squares norm, by using simulated temperature data containing random errors. Moisture content measured data is not considered available for the inverse analysis in order to avoid quite involved measurement techniques. We show that accurate estimates can be obtained for Luikov, Kossovitch and Biot numbers by using only temperature measurements in the inverse analysis. Also, the experimental time can be reduced if the body is heated during part of the total experimental time.     

Fast Identification Method of Total Normal Surface Absorptances Using Inverse Techniques

This paper proposes a simple and fast method to identify the normal absorptance of various surfaces submitted to a radiation source, using inverse techniques. The method consists of imposing during a lap of a few seconds a radiative flux on the front face of a sample whose absorptance is to be identified. The time-dependent temperature on the rear face is measured, and the procedure of inversion is implemented to give a time function of absorbed flux. Only one time–temperature function is measured using a current type K thermocouple. The normal absorptance of the front face is obtained by comparing the time heat flux function of the source and the identified absorbed heat flux function. This method can be quickly and efficiently adopted for many practical applications without the need to use optical devices, which give accurate measurement but at substantial cost. The inverse technique using a conjugate gradient method of minimization with adjoint problem is implemented to estimate the absorbed heat flux. In order to achieve good values of radiative absorptances, reliable knowledge of thermal diffusivities and adequately manufactured samples are required.     

Comparison of three sequential function specification algorithms for the inverse heat conduction problem

Three algorithms for implementing the sequential function specification method of estimating boundary heat flux in the inverse heat conduction problem are compared. They differ from one another in the type of piecewise function used to describe the heat flux and the assumed variation of heat flux over future time. The results of the comparison show that the algorithm that makes use of linear piecewise function for the heat flux and assumes linearly varying heat flux over future time performs slightly better than the other two algorithms.     

Control of the boundary heat flux during the heating process of a solid material

In this paper we apply the conjugate gradient method to solve the inverse problem of determining a time-dependent boundary heat flux in order to achieve a given temperature distribution at the final time. The derivation of sensitivity and adjoint equations in conjunction with the conjugate gradient algorithm are given in detail. The zeroth-order Tikhonov regularization is introduced to stabilize the inverse solution. Solutions by finite differences are obtained for various heat flux profiles. It is found that the time-dependent heat flux may be predicted only for a non-dimensional time of the order of 0.1 while the control problem can be satisfactorily solved for an arbitrary period of time.     

INVERSE HEAT CONDUCTION PROBLEM OF SIMULTANEOUSLY ESTIMATING SPATIALLY VARYING THERMAL CONDUCTIVITY AND HEAT CAPACITY PER UNIT VOLUME

An inverse heat conduction method for simultaneously estimating spatially varying thermal conductivity and heat capacity per unit volume under the conditions of a flash method type of experiment is developed. The unknown thermal properties are assumed to vary only in the space dimension normal to the slab sample and are modeled with piecewise linear representations. Lacking in the literature are specific requirements that must be satisfied by the number of measurements in the spatial domain in order to ensure uniqueness of the inverse solution. We prepared a series of numerical experiments to provide a better understanding of this issue. Multiple temperature sensors are shown to be necessary to determine spatially varying properties. The effectiveness of the method is illustrated through simulated experimental applications of the method.     

INVERSE FORCED CONVECTION PROBLEM OF SIMULTANEOUS ESTIMATION OF TWO BOUNDARY HEAT FLUXES IN IRREGULARLY SHAPED CHANNELS

This article deals with the use of the conjugate gradient method of function estimation for the simultaneous identification of two unknown boundary heat fluxes in channels with laminar flows. The irregularly shaped channel in the physical domain is transformed into a parallel plate channel in the computational domain by using an elliptic scheme of numerical grid generation. The direct problem, as well as the auxiliary problems and the gradient equations, required for the solution of the inverse problem with the conjugate gradient method are formulated in terms of generalized boundary-fitted coordinates. Therefore, the solution approach presented here can be readily applied to forced convection boundary inverse problems in channels of any shape. Direct and auxiliary problems are solved with finite volumes. The numerical solution for the direct problem is validated by comparing the results obtained here with benchmark solutions for smoothly expanding channels. Simulated temperature measurements containing random errors are used in the inverse analysis for strict cases involving functional forms with discontinuities and sharp corners for the unknown functions. The estimation of three different types of inverse problems are addressed in the paper: (i) time-dependent heat fluxes; (ii) spatially dependent heat fluxes; and (iii) time and spatially dependent heat fluxes.     

Analysis and solution of the ill-posed inverse heat conduction problem

The inverse conduction problem arises when experimental measurements are taken in the interior of a body, and it is desired to calculate temperature and heat flux values on the surface. The problem is shown to be ill-posed, as the solution exhibits unstable dependence on the given data functions. A special solution procedure is developed for the one-dimensional case which replaces the heat conduction equation with an approximating hyperbolic equation. If viewed from a new perspective, where the roles of the spatial and time variables are interchanged, then an initial value problem for the damped wave equation is obtained. Since the formulation is well-posed, both analytic and numerical solution procedures are readily available. Sample calculations confirm that this approach produces consistent, reliable results for both linear and nonlinear problems.     

Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.     

Online estimation of multicomponent heat flux using a system identification technique

In this work system identification techniques are used to map the two-dimensional heat flux into the temperatures through a linear model supported by theoretical and numerical results. The basis of this analysis is a discrete version of the Burggraf Method saying a single component heat flux is a linear combination of the temperatures around the time of its occurrence. Taking the same approach, a linear model (i.e. a linear artificial neural network (ANN)) is employed to estimate a multicomponent heat flux as a linear function of the temperatures. A known heat flux is imposed to the direct model, then the history of heat flux-temperature data are fit to the linear mathematical model (i.e. a linear ANN) using system identification techniques. The achieved model estimates the heat flux based on a series of past and future temperatures and the estimated heat flux components are in a good agreement with the exact ones. Finally, the effect of some important factors on the results is investigated. The proposed solution to inverse heat conduction problems does not need thermophysical and geometrical parameters of the system and is robust against noises. It merely needs some series of heat flux-temperature data from solution of a reliable direct numerical model or experiment.     

An analytical solution for two-dimensional inverse heat conduction problems using Laplace transform

An analytical method has been developed for two-dimensional inverse heat conduction problems by using the Laplace transform technique. The inverse solutions are obtained under two simple boundary conditions in a finite rectangular body, with one and two unknowns, respectively. The method first approximates the temperature changes measured in the body with a half polynomial power series of time and Fourier series of eigenfunction. The expressions for the surface temperature and heat flux are explicitly obtained in a form of power series of time and Fourier series. The verifications for two representative testing cases have shown that the predicted surface temperature distribution is in good agreement with the prescribed surface condition, as well as the surface heat flux.     

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.     

Transient surface heating rates from a nickel film sensor using inverse analysis

Convective surface heat transfer measurements play an important role in many industrial, environmental and aerodynamic problems. In most of the cases, the flow is unsteady which results in temperature variation in the body. The surface heating rates are then predicted from the measured temperature histories by suitable heat transfer modeling. In this paper, the temperature history obtained from a nickel film sensor during a flight test is considered to study the effect of sensor thickness on surface heat flux measurements during the flight measurement. Inverse methods using analytical solutions as well as control volume approximations are used to infer the surface heat flux. The experimental temperature data are discretized using cubic-spline method to obtain the closed form solution which is used for inverse analysis. The results are compared with that of standard bench mark results with thin film gauge analysis based on semi-infinite one dimensional medium. No significant change in surface heat flux is observed between inverse and thin film analysis. However, when the thickness of nickel film is increased by 100 times during numerical simulation of inverse method, it is seen that peak surface heat flux increases by 20%.     

Non-iteration estimation of thermal conductivity using finite volume method

The finite volume approach is developed for the inverse estimation of thermal conductivity in one-dimensional domain. The differential governing equation of heat conduction is converted to a system of linear equations in matrix form using the temperature data and heat generation at the discrete grid points as well as surface heat flux. The unknown thermal conductivities are obtained by solving the system equations directly. The features of the present method are that no prior information about the functional form of the thermal conductivity is required and no iterations in the calculation process are needed. The accuracy and robust of the present method are verified by comparing examples of inverse estimation of spatially and temperature-dependent thermal conductivities with the exact solutions.     

Stability and heat transfer characteristics of unsteady condensing and evaporating films

The stability and heat transfer characteristics of an unsteady condensing and evaporating n-pentane film on the underside of a cooled, flat, horizontal plate was studied experimentally. Unsteady conditions were produced by varying the system pressure in a cyclic fashion. The film was imaged using a double-pass shadowgraph system, and an embedded heat flux sensor measured the spatially averaged heat flux. Surface conditions were obtained using an inverse method. Images and data point collection were synchronized to permit direct correlation between thermal data and film behavior. The heat flux was affected by the Rayleigh–Taylor instability after an initial rise due to condensate formation. Hysteresis was observed in the heat flux over each pressure variation cycle, where the heat flux during condensation varied differently with the degree of subcooling than during evaporation. An additional study examined the film stability of non-condensing, growing films with mass addition but without thermal effects. Experiments showed that the film thickness at the point of first droplet break-off increased with increased pumping rate.     

Design and Development of a Test Method for Analyzing Protective Performance of Gloves Exposed to Radiant Heat Based on Computational Fluid Dynamics Analysis

Gloves are one of the most important items in the protective ensemble as they protect hand which is the most frequently burned body part. At present, no international standard or standardized test method is available for analyzing thermal protective performance of gloves exposed to purely radiant heat exposures. In the present work, a systematic approach for developing test setup for radiant heat exposure is proposed using computational fluid dynamics (CFD). A coupled CFD–radiation heat transfer model is first developed and validated with available experimental data. Effects of horizontal and vertical orientations of radiant heaters are then analyzed. Based on this study, an optimized configuration of the experimental setup which results in uniform heat flux distribution throughout the hand under lower level of radiant heat exposure is proposed. Furthermore, effects of different heater temperatures on heat flux distribution are studied. It is found that the proposed configuration works satisfactorily for other higher and lower heat flux levels as well.