USING INVERSE MODELING TO ESTIMATE THE INCIDENT HEAT FLUX REQUIRED TO ACHIEVE TEMPERATURE UNIFORMITY ACROSS A CIRCULAR DISK

This article presents a systematic inverse modeling analysis for estimating the incident heat flux required to achieve temperature uniformity across a circular disk during thermal processing. A one-dimensional thermal model, temperature-dependent thermal properties of silicon, and a future-time algorithm of inverse heat transfer method are used. Vertical and lateral edge-heat compensations on the perimeter are discussed. The required edge-heat compensations for maintaining uniform temperature across 100-mm-diameter (0.6-mm-thick), 150-mm-diameter (0.675-mm-thick), 200-mm-diameter (0.725-mm-thick), and 300-mm-diameter (0.775-mm-thick) silicon disks are evaluated intuitively using inverse modeling. Our numerical results show that temperature uniformity can be efficiently achieved using inverse modeling. The resulting maximum temperature differences in our present study were only 0.279, 0.583, 0.989, and 0.178 ° C across 100-, 150-, 200-, and 300-mm-diameter disks, respectively.     

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.     

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.     

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.     

SOLUTION OF THE INVERSE PROBLEMS IN HEAT TRANSFER AND THERMAL STRESS ANALYSIS

The stable method of soloing inverse problems in heat transfer and inverse problems concerning thermal stress control in elements of complex shapes is presented. Based on the discrete form of Duhamel's integral and on so-called future time steps, a simple and accurate method of solving one- and multi-dimensional inverse problems is developed. Two examples illustrating the application of the described method are presented.     

A SERIAL SOLUTION FOR THE 2-D INVERSE HEAT CONDUCTION PROBLEM FOR ESTIMATING MULTIPLE HEAT FLUX COMPONENTS

A serial algorithm for the inverse heat conduction problem (IHCP) has been developed to estimate the individual flux components, one by one, at the unknown boundary, based on the function specification method. The sensitivity coefficient defined in this algorithm brings out the influence of the heat flux components independent of each other. The objective function minimizes the difference in the measured temperature and the contribution of the individual flux component to the thermal field at the sensor location. The serial algorithm developed here could be used with data from both overspecified and underspecified sensors with respect to the number of flux components. The method was tested for delineating independent heat fluxes at the boundary of a two-dimensional solid for both space- and time-varying heat fluxes. Simulated thermal histories obtained from direct solution were used as inputs for the inverse problem for characterizing the new algorithm.

Three types of analyses were done on the results of the IHCP, focused on (1) the convergence of error in estimated temperatures at the different sensor locations, (2) overall error in estimated temperatures for the whole domain, and (3) the total heat energy transferred across the boundary. It is shown that the optimum configuration of independent unknown fluxes is given by the one with minimum energy estimates across the boundary, for both cases.     

高温太阳能吸热管在非均匀热流下的温度研究及结构优化

对腔式管状集热器在非均匀受热条件下进行温度场和热应力场的模拟.研究发现周向温差的不同造成吸热管的热变形分布不同,且吸热管后半段热变形最大;同时,太阳能辐射量随时间变化导致热流密度和吸热管壁面上的周向温差都在发生变化.鉴于此,分别从腔式集热器吸热管的结构、工作介质物性等方面入手以解决非均匀热流带来的形变问题.通过改变吸热...     

Extending thermal response test assessments with inverse numerical modeling of temperature profiles measured in ground heat exchangers

Thermal response tests conducted to assess the subsurface thermal conductivity for the design of geothermal heat pumps are most commonly limited to a single test per borefield, although the subsurface properties can spatially vary. The test radius of influence is additionally restricted to 1–2 m, even though the thermal conductivity assessment is used to design the complete borefield of a system covering at least tens of squared meters. This work objective was therefore to develop a method to extend the subsurface thermal conductivity assessment obtained from a thermal response test to another ground heat exchanger located on the same site by analyzing temperature profiles in equilibrium with the subsurface. The measured temperature profiles are reproduced with inverse numerical simulations of conductive heat transfer to assess the site basal heat flow, at the location of the thermal response test, and evaluate the subsurface thermal conductivity, beyond the thermal response test. Paleoclimatic temperature changes and topography at surface were considered in the model that was validated by comparing the thermal conductivity estimate obtained from the optimization process to that of a conventional thermal response test.     

分层岩土中地埋管换热器传热解析模型与分析

为准确预测地埋管换热器在分层岩土中的传热特征,采用分离变量法和格林函数法,基于单个圆环热源基本传热单元问题的解答,建立考虑岩土结构分层和横观各项同性特征的地埋管传热解析模型。该模型适用于工程中常见的垂直钻孔和桩基埋管换热器分层传热问题,具有较好的普适性。以2层岩土为例,利用模型解答对分层岩土中地埋管的传热特征以及分层参数对其影响规律进行研究。结果表明：均匀介质假设计算误差随作用时间的增加而逐渐增大,在靠近热源处误差更加明显,预测地埋管长时间温度响应时,应采用分层传热模型;在临界区域范围内,可用均质假设模型预测地埋管的传热特性,均质等效热物性参数取为对应岩土分层的热物性参数值;分层岩土导热系数对地埋管传热性能影响较大,岩土平衡温度随分层导热系数比的增大而明显降低;地埋管长度和直径的比值对地埋管传热性能有所影响,岩土平衡温度随长径比的增大而升高,且其影响程度随分层导热系数比的减小而增强。     

METHODOLOGY TO GENERATE ACCURATE SOLUTIONS FOR VERIFICATION IN TRANSIENT THREE-DIMENSIONAL HEAT CONDUCTION

This article describes the development of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat conduction problems. Verification is a process to ensure that a computer code is free of errors and accurately solves the mathematical equations. The exact solutions presented in this article can have any combination of boundary conditions of specified temperature, prescribed heat flux, or imposed convection coefficient and ambient temperature on the surfaces of the parallelepiped. Additionally, spatially uniform nonzero initial condition and internal energy generation are treated. The methodology to obtain the analytical solutions and sample calculations are presented.     

NUMERICAL CALCULATIONS FOR COMBINED CONDUCTION AND RADIATION TRANSIENT HEAT TRANSFER IN A SEMITRANSPARENT MEDIUM

A numerical computer code was developed for calculating the combined conduction and radiation transient heat transfer in cylindrical, semitransparent materials that have temperature-dependent thermal properties. The radiative component is combined with the equation of conduction heat transfer by adding it as a heat source. The finite element method (FEM) was used for calculating the radiative component and for solving the temperature field in the medium. Very good agreement was observed between results obtained by using our code and those that exist in the literature for several steady-state cases. The advantage of the code is due to the fact that it incorporates temperature-dependent properties; thus it leads to more realistic and accurate results. The code was applied to calculate the cooling path of a large cylindrical sapphire boule while using varying, transient, temperature-dependent, combined heat transfer coefficients.     

THERMOELASTIC PROBLEM OF CURVILINEAR CRACK WITH A POINT HEAT SOURCE

Boundary value problems for a circular-arc crack embedded in an infinite medium due to a point heat source are formulated and solved in closed form. Based on the Hilbert problem formulation and a special technique of contour integration, exact solutions of a semicircular crack are obtained in an explicit form. It is found that the thermal stresses or temperature giadient near the tips of a curved crack always possess the characteristic inverse square-root singularity in terms of the radial distance away from the crack tip under the application of a heat source. The simultaneous existence of mode-I and mode-II stress intensity factors are shown in this article to be dependent on the strength of a heat source, heat conductivity, as well as thermal and elastic isotropy. The nonnegative mode-I stress intensity factor is found to be present in this article for the application of the heat sink, which validates the fully open crack assumption.     

U型管桩埋换热器稳态传热模拟研究

采用能量平衡的方法建立了土壤层内U型管桩埋换热器稳态传热模型，并以天津市一地源热泵实际工程为背景，模拟计算了管脚热影响因子、土壤导热系数等对U型管桩埋换热器的传热特性的影响。     

NUMERICAL SOLUTION OF 2-D NONLINEAR INVERSE HEAT CONDUCTION PROBLEMS USING FINITE-ELEMENT TECHNIQUES

A general method is presented for solving different classes of nonlinear inverse heat conduction problems (IHCP) for two-dimensional, arbitrarily shaped bodies. It is based on the systematic use of a finite-element library. It is shown that, following this approach, the conjugate gradient method can be easily implemented. The method offers a very wide field of practical applications in inverse thermal analysis, while reducing very significantly the amount of work which remains specific for each particular IHPC. Two numerical experiments illustrate the influence of data errors and the iterative regularization principle.     

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.     

同轴地埋管换热器岩土热响应试验研究

岩土热物理性质是影响地源热泵系统设计和运营的关键因素,对位于武汉市洪山区的2口不同深度的同轴地埋管换热孔分别进行48 h的热响应试验,并对同轴地埋管换热器内外管之间环形空间中的平均流体温度进行测试.根据同轴地埋管换热器的几何特性,以简便实用的方式测量同轴地埋管换热器环状空间传热流体的平均温度,结合同轴地埋管换热器钻孔热...     

Inverse problem of transpiration cooling for estimating wall heat flux by LTNE model and CGM method

In this work, a transient inverse problem of transpiration cooling is investigated in detail. The heat flux on the wall to be cooled is estimated by single point temperature measurement. The local thermal non-equilibrium (LTNE) model is utilized to describe the energy conservation of transpiration cooling process, and the conjugate gradient method (CGM) is extended to solve the inverse problem. The accuracy of the solutions of the inverse problem is examined through three given heat fluxes with given measurement errors. The examination shows that with the LTNE model and CGM, satisfactory solutions can be obtained. The influences of the variation in thermal properties, compressibility and the location of sensor on the accuracy of the solutions are analyzed. The analysis indicates that the variation in thermal properties and compressibility should be considered when a large temperature gradient exists, and the sensor location should be as close as possible to the hot wall. The inverse solutions obtained by the measurements of solid and fluid temperatures are compared. Through the comparison, it is found that using the solid temperature measurement as the input of the inverse problem is better than using the fluid temperature measurement.     

Explicit boundary heat flux reconstruction employing temperature measurements regularized via truncated eigenfunction expansions

The present work is aimed at developing and validating an explicit inverse problem solution for boundary heat flux reconstruction in thermally thin plates, supposing a large amount of temperature measurements is available, both in space and time, such as typically obtained by modern infrared thermography systems with fine spatial resolution and high frequency. In order to handle the magnification of the measurement errors when applying the derived explicit inversion formula, a regularized representation is proposed for the measured temperatures, consisting of truncated eigenfunction expansions in which only the most important modes, based on the discrepancy principle, are employed. Two applications are considered, the first one consists of a spatially uniform but time varying heat flux with fast transitions, and the second one consists of a spatially varying heat flux concentrated in two small spots. Good results are obtained with very little computational effort, indicating the feasibility of the proposed approach. This work adds to the available standard inverse problems tools, either as an alternative approach or as a companion in obtaining a starting approximate solution for iterative methods.     

变温热源地热热泵系统的可用能分析

地热能作为一种新能源已成为许多国家的研究重点，如何提高地热能的利用率是其重要的研究方向，而热泵又是其中一项关键技术。本文针对地热热泵系统的变温热源，从减小可用能损失的角度进行了详尽的理论分析。首先，对该地热热泵系统中的各个主要部件进行了可用能分析，发现和能损失情况均与循环工质的性能有关；其次，在实际工况下比较了纯工质和非共沸混合工质的可用能损失情况；最终提出了几种有可能应用于实际地热热泵工况下的循环工质。     

REVERSE TIME TLM MODELING OF THERMAL PROBLEMS DESCRIBED BY THE HYPERBOLIC HEAT CONDUCTION EQUATION

A method for reverse-time transmission line matrix (TLM) modeling of thermal diffusion problems described by the hyperbolic heat conduction equation is presented. The method involves pulse scattering governed by the inverse of the usual scattering matrix. Consideration is given to the accuracy of the reverse time method by comparison with the forward scatter. The numerical behavior of the reverse-time algorithm is investigated, and related to the amplification factor of the inverse scattering matrix.