An optimal control algorithm for entrance concurrent flow problems

The conjugate gradient method of minimization is applied successfully in the present optimal control algorithm in determining the optimal boundary control function for a concurrent flow problem based on the desired thermal entry length and fluid temperatures.The validity of the present optimal control analysis is examined by means of numerical experiments. Different desired thermal entry length and fluid temperature distributions are given in three test cases and the corresponding optimal control heat fluxes are determined. The results show that the optimal boundary heat fluxes can be obtained with an arbitrary initial guess within seconds of CPU time on a Pentium III-600 MHz PC.     

Three-Dimensional Transient Optimal Boundary Heating of Functionally Graded Plates

An optimal control procedure for estimating the heat fluxes on the boundaries of functionally graded (FG) thick plates to reach the desired domain temperature distributions in a specified time interval of heating is presented. The conjugate gradient method (CGM) is employed for optimization, and the differential quadrature method as an accurate and numerically efficient method in conjunction with the forward finite-difference method are applied to solve the three-dimensional transient heat transfer, sensitivity, and adjoint problems. The validity of the presented optimal control problem is demonstrated by solving different numerical examples. Results show that excellent estimation on the boundary heat fluxes can be obtained with arbitrary initial guesses of these functions.     

MIXED CONVECTION IN CAVITIES WITH A LOCALLY HEATED LOWER WALL AND MOVING SIDEWALLS

An optimal control algorithm for cryopreservation of cells using ultrarapid freezing technique is applied successfully in the present study to determine the strength of optimal laser heating based on the desired limited temperature distribution of the cell. The validity of this optimal control analysis utilizing the conjugate gradient method of minimization is examined using numerical experiments. Three different heating times are given, and the corresponding optimal control heat fluxes are determined. Results show that the optimal boundary heat fluxes can be obtained with any arbitrary initial guesses within a very short CPU time on a Pentium III 600-MHz PC.     

To the theory of underwater ice evolution,or nonlinear dynamics of “false bottoms”

We present a mathematical model describing evolution of false bottoms often met between an under-ice melt pond and the underlying ocean during summer. The model treats a false bottom as the region of mixed phase (mushy layer) whose coordinates depend on time and determine the phase transition area. As the heat and the salt fluxes in the ocean are strongly influenced by turbulence and the ice meltwater accumulating underneath the ice cover is practically fresh, we use modified boundary conditions for heat and mass fluxes at the interfaces of phase transition. Explicit analytical solutions (thickness of false bottom and growth rates of its boundaries, temperature and salinity distributions, solid phase fraction and ocean-to-ice heat flux) of the nonlinear model under consideration are found. Model predictions are in good agreement with existing experimental data and physical concepts of phenomena under study.     

APPLICATION OF OPTIMAL CONTROL THEORY IN PIPE INSULATION

Optimum insulation thickness of a pipe subjected to convective heat transfer that minimizes the heat loss is studied using the control theory approach and steepest descent method. As a constraint to the problem, the amount of insulation material is assumed to be fixed. A circular pipe through which fluid is transported from one end to the other is considered. Variations of the bulk temperature of the fluid as well as the temperatures of the outer surface of the insulation are evaluated. It is shown that obtaining an optimal thickness variation of insulation that minimizes the heat losses to the ambient using control theory can be done in a systematic manner. The method can be extended easily to more complex and nonlinear problems.     

边界点法在传热问题数值分析中的应用

将一种新的数值分析方法-边界点法应用于传热问题的研究，对无内热源稳态热传导问题，通过传统边界元法将边界积分方程离散化，发现可以不直接求解影响系数矩阵，而是通过对偶关系，由域外虚源构造方程组的特解场形成边界已知和未知温度，热流密度的系数矩阵，而且域内温度和热流密度的求解将不依赖于边界未知参数的求解，对于有内热源的问题，可以将非齐次方程的解转换为齐次方程的解和某一确定解的叠加，对于非线性问题，可以通过基尔霍夫变换，将非线性问题转化为线性问题求解，这种边界点方法不但具有边界元法降维的优势，而且不须求解奇异积分，大大节约了计算时间，计算精度极高，以有内热源非线性稳态热传导问题的实例印证了这种方法的高效性。     

Identifying heat conductivity and source functions for a nonlinear convective-diffusive equation by energetic boundary functional methods

Abstract

In the article, we solve the inverse problems to recover unknown space-time dependent functions of heat conductivity and heat source for a nonlinear convective-diffusive equation, without needing of initial temperature, final time temperature, and internal temperature data. After adopting a homogenization technique, a set of spatial boundary functions are derived, which satisfy the homogeneous boundary conditions. The homogeneous boundary functions and zero element constitute a linear space, and then a new energetic functional is derived in the linear space, which preserves the time-dependent energy. The linear systems and iterative algorithms to recover the unknown parameters with energetic boundary functions as the bases are developed, which are convergent fast at each time marching step. The data required for the recovery of unknown functions are parsimonious, including the boundary data of temperatures and heat fluxes and the boundary data of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the exact solutions with the identified results, which are obtained under large noisy disturbance.     

Explicit full field analytic solutions for two-dimensional heat conduction problems with finite dimensions

This study presents explicit analytical solutions of heat conduction problems for isotropic media with finite dimensions. The geometry configurations considered in this study include composite layer, wedge and circular media. The boundary conditions are assumed to be either thermal isolation or isothermal. The full field analytical solutions of temperature and heat fluxes for the composite layered media subjected to an embedded heat source are derived first by Fourier transform technique in conjunction with the image method. The corresponding problems of composite wedge and circular media are constructed by conformal mapping and the solutions of composite layer media. The explicit full field solutions are expressed in simple closed-forms which can be easily used in engineering applications. The numerical calculations of the temperature and heat fluxes distributions are provided in full field configuration base on the available analytic solutions.     

THE RAY METHOD FOR SOLVING BOUNDARY-VALUE PROBLEMS CONNECTED WITH THE PROPAGATION OFTHERMOELASTIC SHOCK WAVES OF FINITE AMPLITUDE

The ray method for solving dynamic boundary value problems for nonlinear thermo-elastic media, wherein heat propagates with a finite speed, is developed. By the action of initial and boundary conditions, two types of finite amplitude shock wave propagate in such media: quasi-thermal wave (fast wave) and quasi-longitudinal wave (slow wave). Behind the wave fronts the solution for the desired functions is constructed along the rays in terms of power series (ray series) whose coefficients are the discontinuities in various orders of partial derivatives of the functions to be found with respect to time, but a variable value is the time needed for a disturbance to propagate along the ray from the point under consideration up to the wave front; in so doing the power of the variable value corresponds to the order of the partial time-derivative of the desired function.     

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.     

Analysis of the Impact of Nonlinear Heat Transfer Laws on Temperature Distribution in Irradiated Biological Tissues: Mathematical Models and Optimal Controls

In this paper, we consider nonlinear control problems governed by some generalized transient bioheat transfer-type models with the nonlinear Robin boundary conditions. The control estimates the blood perfusion rate, the heat transfer parameter, the distributed energy source terms, and the heat flux due to the evaporation, which affect the effects of thermal physical properties on the transient temperature of biological tissues. The result can be very beneficial for thermal diagnostics in medical practices, for example, for laser surgery, photo and thermotherapy for regional hyperthermia often used in treatment of cancer. First, the mathematical models are introduced and the existence, uniqueness, and regularity of a solution of the state equation are proved as well as the stability and maximum principle under extra assumptions. Afterwards, the optimal control problem is formulated in order to control the online temperature given by radiometric measurement. We prove that an optimal solution exists and obtain necessary optimality conditions. Some strategy for numerical realization based on the adjoint variables are provided.      

Inverse boundary design of square enclosures with natural convection

An optimization technique is applied to design of heat transfer systems in which the natural convection is important. The inverse methodology is employed to estimate the unknown strengths of heaters on the heater surface of a square cavity with free convection from the knowledge of the desired temperature and heat flux distributions over a given design surface. The direct and the sensitivity problems are solved by finite volume method. 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 convection heat transfer problems is evaluated by comparing the results with a benchmark problem and a numerical experiment.     

Optimal number and location of heaters in 2-D radiant enclosures composed of specular and diffuse surfaces using micro-genetic algorithm

In this study, a combinatorial optimization methodology has been presented for determining the optimal number and location of equally powered heaters over some parts of the boundary, called the heater surface, to satisfy the desired heat flux and temperature profiles over the design surface while keeping the total heaters power constant but floating the number of heaters. In a typical enclosure, candidate locations were numerous for placing the heaters. The optimal number and location could be found by checking among all the possible combinations of heater power ranges and locations on the heater surface. The possibility of checking only a small portion of the total search space was increasingly desirable for finding an overall optimal solution. Micro-genetic algorithm was a candidate method which displayed a significant potential in achieving that task. Micro-genetic algorithm was used to minimize an objective function which was expressed by the sum of square errors between estimated and desired heat fluxes on the design surface. Radiation element method by ray emission model (REM²) was used to calculate the radiative heat flux on the design surface. It enabled us to handle the effects of specular surfaces and blockage radiation due to enclosure geometry. The capabilities of this methodology were demonstrated by finding the optimal number and position of heaters in two irregular enclosures. The effects of refractory surface characteristics (i.e., diffuse and/or specular) on the optimal solution have been studied in detail. The results show that the refractory surface characteristics have profound effects on the optimal number and location of heaters.     

Non‐Fourier Boundary Conditions Effects on the Skin Tissue Temperature Response

Thermal wave and dual phase lag bioheat transfer equations are solved analytically in the skin tissue exposed to oscillatory and constant surface heat flux. Comparison between the application of Fourier and non‐Fourier boundary conditions on the skin tissue temperature distributions is studied. The amplitude of temperature responses increases and also the phase shift between the temperature responses and heat flux decreases under the non‐Fourier boundary conditions for the case of an oscillatory surface heat flux. It is supposed the stable temperature cycles in order to estimate the blood perfusion rate via the existing phase shift between the surface heat fluxes and the temperature responses. It is shown that the higher rates of the blood perfusion correspond to lower phase shift between the surface temperature responses and the imposed heat flux.     

Thermal performance of an MHD radiative Oldroyd‐B nanofluid by utilizing generalized models for heat and mass fluxes in the presence of bioconvective gyrotactic microorganisms and variable thermal conductivity

The current paper analyzes the thermal and concentration attributes with the temperature‐dependent mass diffusion coefficient and thermal conductivity for the flow of an Oldroyd‐B nanoliquid over a stretchable configuration using the Buongiorno model under the application of boundary layers. The mechanisms of heat and mass transport are modeled by using the revised definitions of heat and mass fluxes. Mathematical expressions for the conservation laws are transformed into ordinary differential expressions by making the appropriate changes. The resulting complexly structured expressions are handled via an optimal homotopy procedure. The impact of influential variables on the desired solutions is plotted, tabulated, and discussed in detail. Comparative analysis of the thermal wall flux coefficient, concentration flux coefficient, density magnitude of the motile microorganisms, and reduced dimensionless stresses with already published research as a limiting case of this exploration is presented for the validity of the proposed scheme, and an excellent agreement is observed, which confirms the reliability of the homotopic solution.     

Calculating transient wall heat flux from measurements of surface temperature

Wall heat fluxes can be derived from time resolved measurements of the surface temperature. This paper describes an analytical approach to calculate the heat flux from an analytical solution of the one-dimensional transient energy equation with transient boundary conditions using the Laplace transformation. The results are compared to simple test cases for which the heat fluxes are given in literature. The method is used to calculate the heat flux from a fuel spray to a wall at diesel engine conditions.     

Inverse radiation–conduction design problem in a participating concentric cylindrical medium

Inverse conduction–radiation problem for design analysis in a two-dimensional concentric cylindrical absorbing, emitting and isotropically scattering medium has been solved, when the desired boundary conditions are available on the design surface. The finite-volume method was adopted to deal with energy conservation equation including conduction and radiation. The radiative transfer equation was also taken into consideration in direct problem, whereas the Levenberg–Marquardt method was used to solve a set of equations in inverse problem, which are expressed by errors between estimated and desired total heat fluxes on the design surface. The automatic differentiation as well as the Broyden combined update was utilized to reduce computational time in calculating the sensitivity matrix. The results have shown that the desired total heat flux distribution on design surface could be successfully estimated with less computational time using the present inverse procedure developed here.     

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.     

A method for predicting heat and moisture transfer through multilayered walls based on temperature and moisture content gradients

A mathematical formulation applied to a numerically robust solver is presented, showing that moisture content gradients can be used as driving forces for heat and moisture transport calculation through the interface between porous materials with different pore size distribution functions. For comparison purposes, several boundary conditions are tested—in order to gradually increase the discontinuity effects—and a detailed analysis is undertaken for the temperature and moisture content distributions and sensible and latent heat fluxes, when the discontinuity on the moisture content profile is taken or not into account.     

A shape design problem in determining the interfacial surface of two bodies based on the desired system heat flux

A shape design problem (or inverse geometry problem) in determining the geometry of interfacial surface between two conductive bodies in a three-dimensional multiple region domains, based on the desired system heat flux and domain volume, is examined in this study. The design algorithm utilized the Levenberg–Marquardt method (LMM), B-spline surface generation and the commercial software CFD-ACE+. The validity of this shape design analysis is examined using the numerical experiments. Different desired system heat fluxes are considered in the numerical test cases to justify the validity of the present algorithm in solving the three-dimensional shape design problems. Finally, the results show that for the two different cases considered in this work, the maximum increasing in the system heat flux is obtained as 11.3% and 14.1%, respectively. It is also concluded that when the boundary control points of interfacial surface are free to move, maximum system heat flux can be obtained by the present algorithm since it has more degree of freedom in describing the interfacial surface.