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.     

An inverse radiation boundary design problem for an enclosure filled with an emitting,absorbing, and scattering media

This work is an inverse radiative design problem in which the objective is to determine the spatial distribution of heat source strengths which produces a desired temperature and heat flux distribution on the design surface. The furnace whose walls are diffuse-grey is assumed to be filled with an absorbing, emitting, and scattering medium. The function to be minimized is the sum of squares of the differences between the desired and calculated radiative heat fluxes at the design surface. Radiative heat flux calculations are accomplished by means of the Modified Discrete Transfer Method MDTM using the correction factors suggested by Coelho and Carvalho [P.J. Coelho, M.G. Carvalho, Conservative formulation of the discrete transfer method, ASME J. Heat Transfer, 119 (1997) 118–128.] and Cumber [P.S. Cumber, Improvements to the discrete transfer method of calculating radiative heat transfer, Int. J. Heat Mass Transfer, 38 (12) (1995) 2251–2258.]. For inverse design calculations the Conjugate Gradient Method CGM is employed, in which the sensitivity coefficients are defined and used as needed by the algorithm. Our investigation shows that the presented algorithm is able to estimate heater strengths accurately.     

Temperature and heat flux estimation from sampled transient sensor measurements

Laplace transform is used to solve the problem of heat conduction over a finite slab. The temperature and heat flux on the two surfaces of a slab are related by the transfer functions. These relationships can be used to calculate the front surface heat input (temperature and heat flux) from the back surface measurements (temperature and/or heat flux) when the front surface measurements are not feasible to obtain. This paper demonstrates that the front surface inputs can be obtained from the sensor data without resorting to inverse Laplace transform. Through Hadamard Factorization Theorem, the transfer functions are represented as infinite products of simple polynomials. Consequently, the relationships between the front and back surfaces are translated to the time-domain without inverse Laplace transforms. These time-domain relationships are used to obtain approximate solutions through iterative procedures. We select a numerical method that can smooth the data to filter out noise and at the same time obtain the time derivatives of the data. The smoothed data and time derivatives are then used to calculate the front surface inputs.     

Inverse Boundary Design Problems in Enclosures With Non-Gray Media

In this work, the inverse analysis is applied to radiative heat transfer boundary design problems with non-gray media. The objective of the inverse problem is to find the power of the heaters on the heater surface that produces the desired output, that is, temperature and heat flux distribution over the design surface. The inverse problem is formulated as an optimization problem for minimization of an objective function, which is defined by the sum of the squared difference between estimated and desired heat flux distributions over the design surface. The non-gray optimization problem is solved using the conjugate gradient method, which is a gradient-based optimization method. The spectral line weighted-sum-of-gray-gases model (SLW) is used to account for non-gray gas radiation properties. The radiative transfer equation is solved by the discrete ordinates method combined with two models for simulation of non-gray media. Enclosures with diffuse and gray walls are considered. Radiation is assumed the dominant mode of heat transfer. Example problems including homogeneous/nonhomogeneous, isothermal/nonisothermal media are considered. The results obtained using the SLW model and the gray model are compared.     

Online Temperature Prediction Using a Branch Eigenmode Reduced Model Applied to Cutting Process

In this article, we propose a method to estimate the temperature field in the hottest zone of a cutting tool. Since temperature measurements are not possible in this zone, an inverse method using a branch modal reduction is implemented. The reduced model is used in an inverse problem to identify the heat flux density generated by the frictional forces. Knowing the interface heat flux, the direct problem is solved to compute the temperature field in the tool. The analysis of the results shows that this method enables supervision of the temperature field at the workpiece-tool contact area in real time.     

Real-time solution of heat conduction in a finite slab for inverse analysis

Laplace transform is used to solve the problem of heat conduction over a finite slab. The transfer functions relating the temperature and heat flux on the front and back surfaces of the finite slab are developed. Although there are many competing methods for constructing the inverse Laplace transform, we use polynomial approximation of the transfer function. Therefore, transient solutions for given boundary conditions are easily obtained using SIMULINK. This process is much simpler than other numerical solution methods for the heat equation. Most importantly, our method of solution allows us to obtain, in real-time, the front surface temperature and heat flux based on the thermodynamic measurements on the back surface. We also demonstrate the feasibility of reconstructing the front surface temperature when sensor noise is incorporated to the back surface measurements.     

燃烧室内三维温度场的辐射反问题

本文提出了一种在介质辐射特性已知的条件下,由壁面入射辐射热流的测量值反演燃烧室内三维温度场的方法。该方法是在辐射传递方程离散坐标近似的基础上,用求目标函数极小值的共轭梯度法进行反演计算。通过对吸收系数、散射不对称因子、反照率、壁面黑度和燃烧室大小尺寸等参数对反演精度影响的分析,结果表明,即使存在随机测量误差,这些参数对温度场反演精度的影响也不大,本文所提出的方法可较精确地反演燃烧室内三维温度场。     

Simultaneous estimation of four parameters in a combined-mode heat transfer in a 2D porous matrix with heat generation

This article reports a study on simultaneous estimation of four parameters for combined-mode conduction and radiation heat transfer in a 2D rectangular porous matrix with a localized volumetric heat generation source. Air flows at uniform velocity through the conducting and radiating porous matrix. In the heat generation zone, and its downstream, the gas temperature is higher than that of the solid, and in the upstream the reverse situation occurs. This temperature difference between gas and the solid results in heat transfer by convection between the two phases, and the analysis thus requires consideration of separate energy equations for the two phases. The solid being involved radiatively, the volumetric radiative source term, in the form of the divergence of radiative heat flux, appears only in the solid-phase energy equation. The two equations are coupled through the convective heat transfer term. Four parameters—scattering albedo, emissivity, solid conductivity, and heat transfer coefficient—are simultaneously estimated based on the solid and gas temperature distributions, and convective and radiative heat fluxes at the outer surface of the porous matrix. In both direct and inverse approaches, the energy equations are solved using the finite volume method. For a test case, determining the genetic algorithm is much more time-consuming than the global search algorithm; in other cases, parameter estimations are done using the global search algorithm. Parameters are found to be estimated accurately.     

Coupled radiation‐convection heat transfer of high‐temperature participating medium in heated/cooled tubes

The coupled radiation‐convection heat transfer of high‐temperature participating medium in heated/cooled tubes is investigated numerically. The medium flows in a laminar and fully developed state with a Poiseuille velocity distribution, but the thermal status is developing. By the discrete ordinate method, the nonlinear integrodifferential radiative transfer equation in a cylindrical coordinate form is solved to give the radiative source term in the energy equation of coupled heat transfer. The energy equation is solved by the control volume method. The local Nusselt number and wall heat flux of convection as well as the total wall heat flux are employed to evaluate the influence of radiation heat transfer on convection. The analysis shows that the radiation heat transfer weakens the convection effect, promotes the temperature development, and significantly shortens the tube length with obvious heated/cooled effect. There is an obvious difference between the coupled heat transfer in a heated tube and that in a cooled tube, even though the medium properties are kept constant. The wall emissivity, the medium thermal conductivity and scattering albedo have significant influences on the coupled heat transfer, but the effect of medium scattering phase function is small. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(1): 64–72, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10137     

Application of inverse solution in two‐dimensional heat conduction problem using Laplace transformation

An inverse solution has been explicitly derived for two‐dimensional heat conduction in cylindrical coordinates using the Laplace transformation. The applicability of the inverse solution is checked using the numerical temperatures with a normal random error calculated from the corresponding direct solution. For a gradual temperature change in a solid, the surface heat flux and temperature can be satisfactorily predicted, while for a rapid change in the temperature this method needs the help of a time partition method, in which the entire measurement time is split into several partitions. The solution with the time partitions is found to make an improvement in the prediction of the surface heat flux and temperature. It is found that the solution can be applied to experimental data, leading to good prediction. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 602–617, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10115     

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.     

Modélisation par éléments finis du chauffage infrarouge des membranes thermoplastiques semi-transparentes

This paper reports on a simplified approach for analysing the temperature evolution in a semi-transparent thin membrane of the amorphous polyethylene terephtalate type (PET) exposed to a radiative source. It is based on a 3D finite elements method. The thermophysical properties of the PET are assumed independent of temperature while the internal radiative intensity absorption is taken as one-dimensional and is governed by the Beer–Lambert law. To avoid the difficult problem of computing the shape factors, a semi-analytical approach is used. Finally, the numerical simulations have allowed to validate the analytical and experimental results.In the first place, we have written the energy conservation equation in absence of convection (based on the first law of thermodynamics) (1.a), the radiative source term (1.b) and the boundary conditions (2). As for the finite elements method, the Galerkin approach is used for the formulation of the 3D heat transfer equation (3) and the diagonalisation of the heat capacity matrix, “lumped matrix”, is adopted [M.A. Dokainish, K. Subbaraj, A survey of direct time-integration methods in computational structural dynamics, Comput. Struct. 32 (6) (1989) 1371–1386]. A single step implicit time integration scheme is used for the computation [M.A. Dokainish, K. Subbaraj, A survey of direct time-integration methods in computational structural dynamics, Comput. Struct. 32 (6) (1989) 1371–1386]. After recalling the classical expressions for the radiative flux divergence (5) and the flux itself (6), we have rewritten the radiative source term for an homogeneous medium as a function of spectral intensity (7) and given the integro-differential equation of radiative transfer [R. Siegel, J.R. Howel, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 1992], Eq. (8). The spectral relations linking coefficients of extinction, absorption, scattering and scattering albedo are given by Eqs. (9) and (10), while Eq. (11) expresses the boundary conditions.In the second place, on the basis of the non-scattering behaviour of amorphous PET [K. Esser, E. Haberstroh, U. Hüsgen, D. Weinand, Infrared radiation in the processing of plastics: Precise adjustment-the key to productivity, Adv. Polymer Technol. 7 (2) (1987) 89–128; M.D. Shelby, Effects of infrared lamp temperature and other variables on the reheat rate of PET, in: Proceedings of ANTEC'91 Conference, 1991, pp. 1420–1424; G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249], we have assumed that extinction and absorption coefficients are equal, the corresponding albedo being zero. The radiative transfer equation can thus be rewritten in a reduced form (12). Moreover, for the temperature range prevailing in the processes of PET thermoforming and preforms blowing, temperatures of radiation sources are generally much higher than those used for the forming of these thermoplastic media. Under these conditions, the cold medium hypothesis is used [Y. Le Maoult, F.M. Schmidt, V. Laborde, M. El Hafi, P. Lebaudy, Measurement and calculation of perform infrared heating: a first approach, in: Proceeedings of the Fourth International Workshop on Advanced Infrared, 1997, pp. 321–331], which allows to express transmission of spectral intensity across the material as a function of position and direction (13), and to write the radiative flux divergence in a simplified form as well (14). For one-dimensional radiation, solution of the radiative transfer equation [R. Siegel, J.R. Howel, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 1992] is given by Eq. (15), which combines with Eq. (14) to yield the spectral flux generated inside the semi-transparent medium, Eq. (16). In case of radiation propagation in the same direction as the normal to the polymeric membrane surface, radiation intensity can then be expressed by the Beer–Lambert law (17). This permits to deduce the expression of spectral flux transmitted across the PET membrane thickness (18).Infrared radiation intercepted by the membraneFor determining the infrared radiation intercepted by the membrane surface, the medium between the radiation source and the thermoplastic material is assumed to be completely transparent to radiation. It is also assumed that the surface of the source is diffuse. This allows to infer, from spectral expressions of the energy emitted by the source (Eqs. (19) and (20)) and of the energy received by the semi-transparent membrane (Eq. (21)), a relation for radiation intercepted by the surface at all wavelengths (Eqs. (22) and (23)). For an isotropically radiating source (Lambertian source) when considering an average emissivity of the heating source, Eq. (24), total radiation intercepted by the surface is given by Eq. (25).Infrared radiation absorbed by the membraneFor the determination of infrared radiation absorbed by the polymeric membrane, first, based on the principle of energy conservation, spectral relation (27) which links reflectivity, absorptivity and transmissivity is recalled, as well as relation (28) associating spectral absorption and transmission coefficients following the Beer–Lambert law. Consequently, this allows to find the spectral energy absorbed by the semi-transparent medium (29) with the aid of Eq. (25). Concentrating on wavelength bands that constitute the transmission spectrum of the medium, Eq. (29) is then replaced by Eq. (30) where average values of reflectivity (31) and transmissivity (32) are considered. As a result, the volumetric total energy absorbed by the medium, the flux divergence, is then given by Eqs. (33) and (34). Furthermore, considering that reflectivity is low for polymeric materials (generally lower than 5%), the latter is neglected leading to Eq. (35) instead of Eq. (34). Fig. 1 illustrates a typical transmissivity curve for PET [G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249].Shape factorIn order to take into account arbitrary shapes of sources and preforms, a semi-analytical approach is used for the computation of shape factors [G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249]. The definition of shape factor is recalled in Eq. (36) and Fig. 2. An equivalent form based on the contour principle [R. Rammohan, Efficient evaluation of diffuse view factors for radiation, Int. J. Heat Mass Transfer. 39 (1996) 1281–1286] is given by Eq. (37), while the semi-analytical formula [F. Erchiqui, N.G. Dituba, Analyse comparative des méthodes de calcul des facteurs de formes pour des surfaces à contours rectilignes, Internat. J. Thermal Sci. 46 (2007) 284–293] is expressed by Eq. (39). Validation of this semi-analytical approach is obtained by comparison with the analytical solution [H.C. Hottel, Radiant heat transmission between surfaces separated by non-absorbing media, Trans. ASME 53 (1931) 265–273, FSP-53-196; A. Feingold, Radiant-interchange configuration factors between various selected plane surfaces, Proc. Roy. Soc. London Ser. A 292 (1996) 51–60; J.R. Ehlert, T.F. Smith, View factors for perpendicular and parallel rectangular plates, J. Thermophys. Heat Trans. 7 (1993) 173–174] as well as with results yielded by three different techniques, i.e. the area-integration method, the Gauss quadratic technique and the contour method. Table 1 summarizes the numerical results obtained in each case and gives the relative error.Analytical validation of the reheatingRegarding the numerical validation, we have considered a PET semi infinite medium subjected to a uniform incident radiative flux density. The boundaries of the semi transparent medium are taken as adiabatic. Thermophysical properties and geometrics are given in Table 2. The one-dimensional Laplace equation which governs the radiation heat transfer is solved analytically with the aid of Laplace transforms. The temperature evolution within the depth of the semi transparent medium is obtained by analytical solution [A.B. De Vriendt, in: G. Morin, (Ed.), La transmission de la chaleur, vol. 1, Chicoutimi, Québec, 1984], Eq. (40). Fig. 3 shows a comparison between the numerical solution obtained by the 3D finite elements method and the analytical one. It is seen that, for the three cases, the relative error between numerical and analytical results stays lower than 0.1%.Numerical modeling of infrared heatingAn amorphous PET sheet is considered. The face of the membrane is a square of side 20 cm with a thickness of 1.5 mm. The lateral walls are assumed adiabatic. For modeling with the 3D finite elements method, the sheet is meshed with identical hexahedra comprising eight nodes. The thermophysical properties used for this study are given in [S. Monteix, F. Schmidt, Y. Le Maoult, R Ben Yedder, R.W. Diraddo, D. Laroche, Experimental study and numerical simulation of perform or sheet exposed to infrared radiative heating, Journal of Materials Processing Technology 119 (2001) 90–97] and the reheating time is 35 seconds. The heat transfer coefficient h on front and rear faces is 10 W m^?2 K^?1.Heat flux received by the polymer during heatingTo estimate the incident heat flux distribution on the surface of the polymer, we first calculate the shape factors between each pair of emitting/receiving surfaces. Then the flux distribution is obtained with Eq. (25). Fig. 4 illustrates this distribution.Validation against experimental resultsFigs. 5 and 6 compare the numerical results obtained by the simulation (via MEF 3D) with those obtained experimentally. Fig. 5 displays the temperature distribution on the front and rear surfaces.ResultsFigs. 7–10 illustrate the numerical results obtained for the temporal temperature evolution, during the first five seconds of heating, of the upper receiving face (surface directly exposed to the infrared source), the central plane and the lower face of the PET membrane, respectively.Figs. 11 and 12 give the temperature evolution at four positions on upper and lower faces of the membrane: 0 cm (edge), 2.5 cm, 5 cm and 10 cm (middle). It is observed that the trend is nearly linear during the period of infrared heating.On Figs. 13 and 14 is displayed the temperature evolution in the center of the membrane along its thickness during periods 0–4 and 5–35 seconds.     

Inverse shape design for heat conduction problems via the ball spine algorithm

ABSTRACT

In this article, a novel iterative physical-based method is introduced for solving inverse heat conduction problems. The method extends the ball spine algorithm concept, originally developed for inverse fluid flow problems, to inverse heat conduction problems by employing a subtle physical-sense rule. The inverse problem is described as a heat source embedded within a solid medium with known temperature distribution. The object is to find a body configuration satisfying a prescribed heat flux originated from a heat source along the outer surface. Performance of the proposed method is evaluated by solving many 2-D inverse heat conduction problems in which known heat flux distribution along the unknown surface is directly related to the Biot number and surface temperature distribution arbitrarily determined by the user. Results show that the proposed method has a truly low computational cost accompanied with a high convergence rate.     

An Inverse Analysis for Parameter Estimation Applied to a Non-Fourier Conduction–Radiation Problem

Retrieval of parameters in a non-Fourier conduction and radiation heat transfer problem is reported. The direct problem is formulated using the lattice Boltzmann method (LBM) and the finite-volume method (FVM). The divergence of radiative heat flux is computed using the FVM, and the LBM formulation is employed to obtain the temperature field. In the inverse method, this temperature field is taken as exact. Simultaneous estimation of parameters, namely, the extinction coefficient and the conduction–radiation parameter, is done by minimizing the objective function. The genetic algorithm (GA) is used for this purpose. The accuracies of the estimated parameters are studied for the effects of measurement errors and genetic parameters such as the crossover and mutation probabilities, the population size, and the number of generations. The LBM-FVM in combination with GA has been found to provide a correct estimate of parameters.     

Determination of temperature distributions on the rake face of cutting tools using a remote method

Cutting temperature is a major factor in controlling the tool wear, surface quality, and chip formation mechanics. To understand the exact temperature rise in the tool-chip interface has been recognized as an important study in achieving the best cutting performance. For the above reason, an inverse estimation of the heating history on the rake face of cutting tool is presented in this paper. The first stage of the analysis is to solve a heat transfer model of cutting tool with three-dimensional boundary element method. This is a direct heat transfer problem. Furthermore, the inverse heat transfer technique of sequential estimation is employed to simulate the time histories of heat fluxes at the rake face based on the measured temperature responses on the tool measuring point. The present model can be used not only to determine the part of heat flux conducting into the cutting tool but also the temperature contours on the tool-chip interface.     

燃烧产物辐射特性的误差对炉内辐射传热计算精度的影响

本文用离散坐标法对含吸收散射性介质矩形空腔内的３维辐射传递过程进行了模拟，并编写了相应的数值计算程序。利用该程序分析了介质的吸收系数、散射系数、相函数、光谱特性及壁面灰渣沉积层黑度的不确定性对矩形燃烧室内烟气温度及热流计算精度的影响。结果表明计算精度很大程度上取决于燃烧产物辐射特性的取值精度，特别是壁面灰渣沉积层黑度的取值精度。在煤粉燃烧室中，介质的散射不宜忽略。     

Thermal response of a flat heat pipe sandwich structure to a localized heat flux

The temperature distribution across a flat heat pipe sandwich structure, subjected to an intense localized thermal flux has been investigated both experimentally and computationally. The aluminum sandwich structure consisted of a pair of aluminum alloy face sheets, a truncated square honeycomb (cruciform) core, a nickel metal foam wick and distilled water as the working fluid. Heat was applied via a propane torch to the evaporator side of the flat heat pipe, while the condenser side was cooled via natural convective and radiative heat transfer. A novel method was developed to estimate experimentally, the heat flux distribution of the torch on the evaporator side. This heat flux distribution was modeled using a probability function and validated against the experimental data. Applying the estimated heat flux distribution as the surface boundary condition, a finite volume analysis was performed for the wall, wick and vapor core regions of the flat heat pipe to obtain the field variables in these domains. The results were found to agree well with the experimental data indicating the thermal spreading effect of the flat heat pipe.     

Neural Computation to Estimate Heat Flux in an Atmospheric Plasma Spray Process

Temperature is a key parameter in the thermal spray process and is a consequence of the heat flux experienced by the workpiece. This paper deals with the estimation of the heat flux transmitted to a workpiece from an atmospheric plasma spray torch during the preheating process often implemented in thermal spraying. An inverse heat conduction problem solution using a conjugate gradient method was considered to determine the heat flux starting from a known temperature distribution. Results from the later method were used to train an artificial neural network to discover correlations between selected processing parameters and heat flux.     

Nongray inverse boundary design problems in enclosures at radiative equilibrium

In this paper, an inverse analysis is used to find an appropriate heat flux distribution over the heater surface of radiant enclosures, filled with nongray media at radiative equilibrium from the knowledge of desired (prespecified) temperature and heat flux distributions over the given design surface. Regular and irregular 2D enclosures filled with nongray combustive gas products are considered. Radiation is considered the dominant mode of heat transfer and the medium temperature is obtained from the energy equation. To evaluate the nongray behavior of the participating gases properly, the spectral‐line weighted‐sum‐of‐gray‐gases (SLW) model with updated correlations is used. The dependence of absorption coefficients and the weights of the SLW model on the temperature of the medium makes the inverse problem nonlinear and difficult to handle. Here, the inverse problem is formulated as an optimization problem and the Levenberg‐Marquardt method has been used to solve it. The finite volume method is exploited for the discretization of the energy equation and the spatial discretization of the radiative transfer equation (RTE). The discrete ordinates method (TN quadrature) is used for the angular discretization of RTE. Five test cases, including homogeneous and inhomogeneous media, are investigated to prove the ability of the present methodology for achieving the desired conditions.