COMPARISON OF VARIOUS CONJUGATE GRADIENT METHODS FOR INVERSE HEAT TRANSFER PROBLEMS

Many inverse heat transfer problems can be solved efficiently through the minimization of a performance function utilizing the conjugate gradient method. The gradient of the performance function needed in the minimization procedure of the conjugate gradient method is obtained by employing either the adjoint variable method or the direct differentiation method. In the present study we consider an inverse problem of estimating time-varying strength of a heat source in a two-dimensional heat conduction system, and compare the adjoint variable method and the direct differentiation method in terms of accuracy and computational efficiency, and finally suggest a new method that exploits the advantageous aspects of both methods while avoiding the shortcomings of them.     

Inverse solution for a biochemical heat source in a porous medium in the presence of natural convection

The inverse problem approach by conjugate gradient with adjoint equations is adapted to the context of a non-isothermal bioprocess, controlled by internal heat generation from microbial oxidation, as could be found in a composting reactor for instance, to determine the heat source from internal temperature measurements. The volumetric heat source is assumed proportional to the rate of consumption of a substrate by a biomass, as described by a Monod model. Computations are performed for Rayleigh numbers equal to 0.25 and 25, for a representative biochemical reaction under typical boundary conditions, for constant and temperature-dependent model parameters. The influence of noisy input data is also considered. It is found that good solutions can be obtained when heat release and diffusion occur over very different time scales. The variation of the model parameters with temperature must be taken into account, but single sensor solutions are possible at relatively small Rayleigh numbers when convection is present.     

An inverse radiation problem of estimating heat-transfer coefficient in participating media

In the radiant cooler, where the hot gas from the pulverized coal gasifier or combustor is cooled to generate steam, the wall heat-transfer coefficient varies due to ash deposition. In the present work, we investigate an inverse radiation problem of estimating the heat-transfer coefficient from temperature measurement in the radiant cooler. The inverse radiation problem is solved through the minimization of a performance function, which is expressed by the sum of square residuals between calculated and observed temperature, utilizing the conjugate gradient method. The gradient of the performance function is evaluated by means of the improved adjoint variable method, which resolves the difficulty associated with the singularity of the adjoint equation through its inherent regularization property. The effects of the number of measurement points and measurement noise on the accuracy of estimation are also investigated.     

A GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR COMBINED FREE AND FORCED CONVECTION FLOW ABOUT INCLINED SURFACES IN POROUS MEDIA

In this article, we apply the differential transform method (DTM) to obtain approximate analytical solutions of combined free and forced (mixed) convection about inclined surfaces (or wedges) in a saturated porous medium. Both aiding and opposing flows are considered. It is found that the parameter mixed convection from inclined surfaces in porous media is Gr/Re, where Gr is the local Grashof number and Re is the local Reynolds number. DTM solutions are obtained for mixed convection from an isothermal vertical flat plate as well as an inclined plate with constant heat flux having an inclination of 45°. Temperature and velocity profiles for these two cases at different values of Gr/Re are presented. The similarity transformations are applied to reduce the governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. DTM is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Thereafter a Padé approximant is applied to the solutions to increase the convergence of the given series. Excellent correlation between DTM-Padé and numerical quadrature (shooting) solutions is achieved. The DTM-Padé simulation is shown to be a robust benchmarking tool providing an excellent means of validation of numerical methods. The study has applications in geothermal energy systems, chemical engineering filtration systems, and packed beds.     

MHD MIXED-CONVECTION INTERACTION WITH THERMAL RADIATION AND nTH ORDER CHEMICAL REACTION PAST A VERTICAL POROUS PLATE EMBEDDED IN A POROUS MEDIUM

This article investigates the hydromagnetic mixed convection heat and mass transfer flow of an incompressible Boussinesq fluid past a vertical porous plate with constant heat flux in the presence of radiative heat transfer in an optically thin environment, viscous dissipation, and an nth order homogeneous chemical reaction between the fluid and the diffusing species. The dimensionless governing equations for this investigation are solved numerically by the fourth-order Runge-Kutta integration scheme along with shooting technique. Numerical data for the local skin-friction coefficient, the plate surface temperature, and the local Sherwood number have been tabulated for various values of parametric conditions. Graphical results for velocity, temperature, and concentration profiles based on the numerical solutions are presented and discussed.     

NATURAL CONVECTION HEAT AND MASS TRANSFER FROM A VERTICAL FLAT PLATE WITH VARIABLE WALL TEMPERATURE AND CONCENTRATION TO POWER-LAW FLUIDS WITH YIELD STRESS IN A POROUS MEDIUM

The problem of natural convection coupled heat and mass transfer of non-Newtonian power law fluids with yield stress from a vertical flat plate in a fluid-saturated porous medium is analyzed under boundary layer approximations. Similarity solutions are obtained for the general case of power law variations of the wall temperature and concentration. Velocity, temperature, and concentration profiles as well as local heat and mass transfer rates are presented and discussed for different values of the rheological parameters of a power-law fluid, the Lewis number, and the thermal and concentration buoyancy ratio.     

THERMAL CONVECTIVE INSTABILITY IN TRANSLUCENT POROUS MEDIA WITH RADIATIVE HEAT TRANSFER

The onset of thermal convection in a translucent porous layer is considered. Attention is focused on the effect of radiative heat transfer on the critical Rayleigh-Darcy number and the convection cell shape. If we consider the contribution of radiative heat transfer, the basic temperature profile is non-linear and the thermal convective instability is influenced by the ratio of conduction to radiation heat flux, the temperatures at the boundary surfaces, and radiative parameters such as wall emissivity, scattering albedo and extinction coefficient as well as the usual Rayleigh-Darcy number. Effects of these parameters on the onset of convective instability are investigated with the help of linear stability theory employing the Darcy's law and the radiative transport equation simplified by the P₁ approximation. The increased effective thermal conductivity due lo the radiation inhibits the onset of convection and causes increased critical Rayleigh-Darcy number and decreased convection cell size. The results of the present work may be exploited to find out the optimal diameter of aerogel pellets and the air pressure in the double pane window filled with the translucent silica aerogel granules to suppress natural convection.     

基于复变量求导法的共轭梯度法及其在热传导反问题边界条件辨识中的应用

为了利用共轭梯度法的计算精度高和收敛速度快的优点, 避免传统共轭梯度法在求解非线性热传导反问题中的微分处理、复杂的推导过程等问题, 给出一种改进的共轭梯度法, 即将复变量求导法引入传统的共轭梯度法, 准确计算了各灵敏度系数, 进而对瞬态非线性热传导反问题进行求解, 并对边界条件进行辨识。算例验证了本文方法的有效性与精度。与传统共轭梯度法相比, 在处理非线性问题方面, 本文方法具有操作简单和精度高的优点。     

Natural convection on a vertical plate with variable heat flux in supercritical fluids

The present article investigates laminar free convection with uniform or non-uniform prescribed surface heat flux over a vertical flat plate in supercritical fluid, numerically. A new equation for thermal expansion coefficient in a supercritical fluid is derived based on Redlich–Kwong equation of state as a function of pressure, temperature and the compressibility factor. Calculated values of thermal expansion coefficient have been compared with the experimental results which show better accuracy in comparison with van der Waals ones. The governing systems of partial differential equations were solved numerically using the finite difference method. The local Nusselt number was calculated and plotted as a function of the local Rayleigh number. Using supercritical fluids in constant heat flux free convection, decrease wall temperature in comparison with under critical fluids. It was observed that positive and negative slopes of surface heat flux distribution increases and decreases the heat transfer coefficient, respectively.     

MOMENTUM INTEGRAL METHOD FOR FORCED CONVECTION IN THERMAL NONEQUILIBRIUM POWER-LAW FLUID-SATURATED POROUS MEDIA

Forced convection heat transfer for power-law fluid flow in porous media was studied analytically. The analytical solutions were obtained based on the Brinkman-extended Darcy model for fluid flow and the two-equation model for forced convection heat transfer. As a closed-form exact velocity profile is unobtainable for the general power-law index, an approximate velocity profile based on the parabolic model is proposed by subscribing to the momentum boundary layer integral method. Heat transfer analysis is based on the two-equation model by considering local thermal nonequilibrium between fluid and solid phases and constant heat flux boundary conditions. The velocity and temperature distributions obtained based on the parabolic model were verified to be reasonably accurate and improvement is justified compared to the linear model. The expression for the overall Nusselt number was derived based on the proposed parabolic model. The effects of the governing parameters of engineering importance such as Darcy number, power-law index, nondimensional interfacial heat transfer coefficient, and effective thermal conductivity ratio on the convective heat transfer characteristics of non-Newtonian fluids in porous media are analyzed and discussed.     

Influence of natural convection on the heat explosion in porous media

The interaction between natural convection and the heat explosion in porous media is studied. The model consists of a nonlinear heat equation coupled with the Darcy equation for the motion of an incompressible fluid in a porous medium. Numerical simulations are performed using the alternate direction finite difference method and the fast Fourier transform method. A complex behavior of solutions is observed, including periodic and aperiodic oscillations and an oscillating heat explosion. It is shown that convection can decrease the risk of the explosion due to additional mixing and heat loss, but it can also facilitate the explosion due to temperature oscillations arising as a result of instability of stationary convective regimes.     

Mixed convection flow of a power-law fluid in horizontal ducts

The laminar mixed convection heat transfer of a non-Newtonian fluid modelled by a power-law constitutive equation is studied. The equations of motion subject to the Boussinesq approximation and an axially uniform heat flux and peripherally uniform wall temperature condition are discretized by finite difference approximation and are solved by the successive relaxation method. Dual solutions with two-cell and four-cell patterns have been observed in certain regions of the parameter space. The secondary flow caused by the buoyancy force has a shear thinning effect for pseudoplastic fluids. The Nusselt number, however, always increases with increasing Grashof number.     

NATURAL CONVECTION HEAT TRANSFER IN A POROUS LAYER WITH MULTIPLE PARTITIONS

Laminar natural convection heat transfer in inclined porous layers with multiple diathermal partitions is studied analytically and numerically. On the basis of the Darcy-Oberbeck-Boussinesq equations, the problem is solved analytically in the limit of a thin layer with constant flux boundary conditions. The relationship between overall heat transfer and the number of partitions N, their relative positions Nu and the angle of inclination φ of the system is determined

The critical Darcy-Rayleigh number R_c for the onset of convection in a bottom-heated horizontal system is predicted. The results are compared with limiting cases of the problem and are found to be in agreement. The influence of a thermal barrier which is sandwiched between two porous layers is also discussed. A numerical study of the same phenomenon, obtained by solving the complete system of governing equations, is conducted. A good agreement is found between the analytical predictions and the numerical simulation.     

SLIP FLOW AND HEAT TRANSFER OF A SECOND-GRADE FLUID IN A POROUS MEDIUM OVER A STRETCHING SHEET WITH POWER-LAW SURFACE TEMPERATURE OR HEAT FLUX

This article deals with the study of boundary layer flow of a second-grade fluid in a porous medium past a stretching sheet and heat transfer characteristics with power-law surface temperature or heat flux. The flow in the boundary layer is considered to be generated solely by the linear stretching of the boundary sheet adjacent to a porous medium, and boundary wall slip condition is assumed. In the energy equation effects of viscous dissipation, work done due to deformation and internal heat generation/absorption is taken into account. Closed form solutions are obtained for this problem.     

Thermal Diffusion and MHD Effects on Combined Free‐Forced Convection and Mass Transfer of a Viscous Fluid Flow Through a Porous Medium with Heat Generation

The problem of thermal diffusion and magnetic field effects on combined free‐forced convection and mass transfer flow past a vertical porous flat plate, in the presence of heat generation is studied numerically. The governing momentum, energy and concentration equations are converted into a system of nonlinear ordinary differential equations by means of similarity transformations. The resulting system of coupled nonlinear ordinary differential equations is solved numerically by using the Shooting method. Numerical results are presented for velocity, temperature and concentration profiles within the boundary layer for different parameters of the problem including suction parameter, heat generation parameter, Soret number, Dufour number, magnetic parameter, etc. In addition, the effects of the pertinent parameters on the skin friction and the rates of heat and mass transfer are discussed numerically and illustrated graphically.     

格子Boltzmann方法模拟高Darcy数多孔介质内融化传热过程

采用Brinkmann-Forchheimer-Darcy方程描述多孔介质内流动过程,并通过选择合适的平衡态分布函数及非线性源项形式构建出表征融化相变温度场的分布函数演化方程。应用格子Boltzmann模型模拟了方腔内无多孔介质以及填充多孔介质自然对流融化传热过程,模拟结果与其他文献结果吻合较好,模型的正确性得到了一定的验证。模拟结果还表明液相的自然对流对融化传热过程有着重要的影响,并且在高Rayleigh 数或高Darcy 数情况下作用更强。在高Darcy 数情况下由于非Darcy效应的存在,增大Rayleigh 数以强化融化传热的效应要大于提高同样倍数Darcy 数的效应。     

Computation of the gradient and sensitivity coefficients in sum of squares minimization problems with differential equation models

In parameter estimation problems where the system model consists of differential equations, methods for minimizing a sum of squares of residuals objective function require derivatives of the residuals with respect to the parameters being estimated (sensitivity coefficients) or the gradient of the objective function (depending on the numerical optimization method). This paper considers two methods for generating such derivatives: (1) the adjoint equation — gradient formula; and (2) complimentary sensitivity coefficient differential equations. Particular attention is given to the consistency between the method used to solve the model equations and the proper formulation of the additional equations required by the two methods. Two example problems illustrate computational experience using a modified quasi-Newton method with the adjoint method used to generate gradients and applying a modified Gauss-Newton approach with the sensitivity coefficient equations to calculate both the Gauss-Newton matrix and the objective function gradient. Results indicate the superiority of the sensitivity coefficient approach. When comparing the computational effort required by the two methods and the results from the simple examples, it appears that the use of complimentary sensitivity coefficient equations is much more efficient than using only the gradient of the sum of squares function.     

有内热源的多孔层中自然对流的稳定性分析

采用局部非热平衡模型, 通过数值法和Garlerkin近似法, 分析存在均匀内热源和边界浓度梯度时, 有效热导率比、流体和固相间的传热系数、浓度梯度的大小以及内热源在流体与固相内的分布情况对水平多孔层中临界内热源Rayleigh数的影响, 来研究相关参数对自然对流的稳定性的影响, 并得到临界内热源Rayleigh数的表达式。结果表明, 浓度Rayleigh数的增加可以促进自然对流的形成;内热源为正时, 自然对流的形成区域主要位于上半区域;内热源为负时, 自然对流的形成区域位于下半区域, 内热源总是促进自然对流的发生;有效热导率比、流体和固相间的内部传热系数、内热源在流体与固相内的分布情况相互耦合, 影响自然对流的稳定性, 这种影响取决于各参数的范围。     

INTERACTION OF THERMAL RADIATION IN THE LAMINAR PIPE FLOW OF OPTICALLY THIN GASES

The influence of thermal radiation on laminar forced convection of a gray gas in a pipe flow is studied semi-analytically. The gas is considered as an absorbing-emitting medium and the thin gas model approximation is used to describe the radiative heat flux in the energy equation. Invoking the method of lines (MOL), the nonlinear boundary value problem is reduced to an initial value problem which is eventually solved by a standard Runge-Kutta integration scheme. Numerical results are presented graphically for a selected group of thermal parameters encompassing the thin gas behavior. Additionally, it is found that limiting cases namely: laminar plug flow and laminar parabolic flow, both in the absence of radiation define an envelope for the curves of bulk temperature and total Nusselt number describing the combined thermal process for a thin gas in laminar motion. In general, the comparisons reveal that these asymptotic solutions are valid when the entrance-to-wall temperature ratio is less than 2.     

Transient forced convection in laminar channel flow with stepwise variations of wall temperature

Transient laminar forced convection of Newtonian fluids inside circular ducts and parallel-plate channels subjected to stepwise variations of wall temperature is considered. Approximate analytical solutions are developed by making use of the generalized integral transform and the classical Laplace transform techniques. Higher order solutions are given for the case of a step change in wall temperature. A second-order accurate finite-difference solution is developed to assess the accuracy of the available approximate solutions. The Sign-Count method is used to solve the resulting eigenvalue problems and determine as many eigenvalues and eigenfunctions as needed. Results are presented for the local Nusselt number, average fluid temperature and the wall heat flux over a wide range of the dimensionless axial coordinate.