NUMERICAL COMPUTATION OF THE NATURAL CONVECTION FLOW ABOUT A HORIZONTAL CYLINDER USING SPLINES

The present work is devoted to the numerical study of laminar natural convection flow from a heated horizontal cylinder under diverse surface boundary conditions using the spline fractional step method. A general formulation to treat mixed boundary conditions using the spline approximation has been presented. Numerical solutions have been obtained by solving the Navier-Stokes and energy equations. The results for the isothermal boundary condition as well as for the uniform heat flux are in good agreement with published experimental data and with other solutions presently available in the literature. Some new computations at very high Rayleigh numbers indicate the existence of attached separation vortices in the downstream plume region, the appearance of these vortices being dependent on the values of the Biot number. All results were computed on a personal computer using unequally spaced grids that provided good results with a minimum number of computational points. The numerical scheme presented here appears to be sufficiently versatile to permit computation of a large range of problems     

On the analysis of an elliptical Gaussian flux image and its equivalent circular Gaussian flux images

For a continuous elliptical Gaussian flux image over the infinite X-Y plane, the parameter relationships between the elliptical Gaussian flux image and the equivalent circular Gaussian flux images are clearly discussed in both mathematical and graphical ways in this paper with respect to the radial power distribution around the image centre (peak flux location in image plane). This paper presents six typical methods (SIGMA-2Mean, SIGMA-1Mean, SIGMA-0Mean in Class One, SIGMA-RPeak, SIGMA-RMean, SIGMA-SqrRMean in Class Two) to give the equivalent circular Gaussian flux images to the elliptical Gaussian flux image, tries to link these circular Gaussian fitting methods to the relevant solar engineering applications, and makes some assessment comments on the elliptical/circular Gaussian modelling in solar mirror optics. By comparing the approximation performance among these six typical fitting methods, it reveals the reason for the 90% intercept over-estimation phenomenon of Francisco J. Collado’s one-point circular Gaussian fitting practice relative to the experimental flux image. The detailed algorithm for automatically finding out the major/minor axes and the image centre of the digital elliptical flux image is also provided in this paper. SIGMA-2Mean and SIGMA-SqrRMean are equivalent for an elliptical flux image, but they are applied in different ways to figure out either the reasonable intercept factor of the experimental flux image on the physical target plane respecting the aperture region of interest, or the power spillage over the limited experimental target plane. At last, this paper introduces the interpolation reconstruction of an elliptical Gaussian flux image over a rectangular domain just based on the boundary pixel values, so it is quite useful for solar engineering, such as fast simulation of a flux image concentrated by a mirror, and also instant approximation of the flux density over the receiver aperture by the linear array of radiometers around the receiver aperture, when the central receiver system is in the normal working state.     

SIMULTANEOUS ESTIMATION OF INLET TEMPERATURE AND WALL HEAT FLUX IN TURBULENT CIRCULAR PIPE FLOW

An inverse heat convection problem is solved for simultaneous estimation of unknown inlet temperature and wall heat flux in a thermally developing, hydrodynamically developed turbulent flow in a circular pipe based on temperature measurements obtained at several different locations in the stream. The direct problem of turbulent forced convection is solved with a finite difference method with appropriate algebraic turbulence modelling. Although we seek for two unknown functions, we formulate the inverse problem as one of parameter estimation through the representation of the unknown inlet temperature profile and the wall heat flux distribution by one-dimensional finite element interpolation. Nodal values of the inlet temperature and the wall heat flux at chosen positions are determined as unknown parameters through the Levenberg–Marquardt algorithm for minimization procedure. Numerical results for several testing cases with different magnitudes of measurement errors are examined by using simulated experimental data. The effects of the number and the locations of the temperature measurement points are discussed.     

Incorporating boundary conditions in the heat conduction model

This note introduces a mathematical derivation of the heat conduction model that incorporates boundary conditions. In particular, in the present approach boundary conditions are derived in parallel to the heat equation, while in the standard approach to heat conduction modelling they are appended at a later stage. Because of its peculiar mathematical formulation, this method allows modelling heat sources or sinks placed on the boundary. Furthermore, it is shown that when such heat sources depend linearly on the surface temperature and the heat flux, each of their points can be described as a point source emitting a heat wave directed into an infinitesimal volume in the neighbourhood of the surface.     

A variational theory for frictional flow of fluids in inhomogeneous porous systems

For nonlinear steady paths of a fluid in an inhomogeneous isotropic porous medium a Fermat-like principle of minimum time is formulated which shows that the fluid streamlines are curved by a location dependent hydraulic conductivity. The principle describes an optimal nature of nonlinear paths in steady Darcy’s flows of fluids. An expression for the total resistance of the path leads to a basic analytical formula for an optimal shape of a steady trajectory. In the physical space an optimal curved path ensures the maximum flux or shortest transition time of the fluid through the porous medium. A sort of “law of bending” holds for the frictional fluid flux in Lagrange coordinates. This law shows that – by minimizing the total resistance – a ray spanned between two given points takes the shape assuring that its relatively large part resides in the region of lower flow resistance (a ‘rarer’ region of the medium). Analogies and dissimilarities with other systems (e.g. optical or thermal ones) are also discussed.     

Characterization of Flux Rotation and of the Ensuing Core Losses in the Stator of Induction Motors

A literature survey reveals that the study of rotational power losses in stator cores has been conventionally performed by means of finite-element analysis. This paper proposes an alternative characterization of rotational power losses in the tooth roots - a region that has been long known as a seat of flux rotation - relating some geometric dimensions to the values of flux density of interest to compute the rotational losses over the region. A simplified computation of maximum and minimum flux densities is proposed based on the trajectories of flux at singular time intervals. This enables the calculation using analytical expressions. For that reason, the proposed approach improves the power losses analysis in early design stages, showing the impact of projected geometries on the total loss estimate. Moreover, it consequently provides a tool to include the rotational losses estimate within optimization-oriented iterative searches.     

Modelling of magnetic saturation in smooth air-gap synchronousmachines

Main flux saturation in d-q axis representation of synchronous machines is at present modelled by selecting either all the winding currents or all the winding flux linkages as state-space variables. However, these two available models are just a tiny portion of the complete set of models that can be obtained by selecting other combinations of state-space variables. This paper presents a general procedure for main flux saturation modelling in smooth air-gap synchronous machines, that is applicable for most selections of state-space variables. The method relies on the concept of `generalised flux' and `generalised inductance', that has been successfully applied in modelling of saturated single-cage and double-cage induction machines. The concept is extended to saturation modelling in smooth air-gap synchronous machines. A number of models, that result from the application of the method for different selections of state-space variables, are presented in detail     

Extension of the compensated distortion method to the critical heat flux modelling in rectangular inclined channel

Parameters of critical heat flux modelling in inclined rectangular channels for high-pressure convective two-phase flows have been derived using the compensated distortion method of Ahmad [Int. J. Heat Mass Transfer 16 (1973) 641]. Fréon R12 was used as the modelling fluid. A specific expression for the modelling parameter has been obtained by introducing a new dimensionless parameter that accounts for the effect of inclination on critical heat flux. This parameter is built by balancing the inertial force acting on a particle against the viscous effect corrected by a transverse term of gravity. In the same time, a dimensionless correlation of critical heat flux is proposed, which describes available data with an r.m.s. error of 13.1%.     

汽轮机空心静叶缝隙抽吸性能研究

采用Fluent软件对某600MW汽轮机末级空心静叶栅内的蒸汽流场与缝隙抽吸性能进行了三维数值计算,讨论了缝隙结构参数对抽吸性能的影响。结果表明:随着缝隙位置从叶片前缘向尾缘的移动,静叶背弧上缝隙的抽汽量急剧减少;内弧上缝隙的抽汽量先略微增大后减小,在缝隙相对位置ψ=0.723 4时达到最大值;随着缝隙角度的增大,缝隙抽汽量先增大后减小,在缝隙角度α=45°时达到最大值,约占叶栅通道总蒸汽量的1.8%;随着缝隙宽度的增加,缝隙抽汽量先减小后增大,在缝隙宽度为2mm时达到最小;缝隙抽吸能力与缝隙出口处的腔室大小、空腔内形成的涡核位置和涡半径以及抽吸工质的流线的长短有关。     

Main flux saturation modelling in double-cage and deep-barinduction machines

The available models of saturated double-cage and deep-bar induction machines are the current state-space model and the flux state-space model, where state-space variables are selected either as stator current and currents of both rotor cages, or stator flux linkage and flux linkages of both rotor cages. This paper presents a number of models of saturated double-cage (deep-bar) induction machines where alternative sets of state-spate variables are selected. The method of main flux saturation modelling relies on recently introduced concept of `generalised flux space vector', which has originally been developed for modelling of saturated single-cage induction machines. The procedure and the novel models are verified by experimental study and simulation of self-excitation process in a double-cage induction generator     

A unified approach to main flux saturation modelling in D-Q axismodels of induction machines

Main flux saturation is most frequently modelled by selecting either stator and rotor d-q axis currents or stator and rotor d-q axis flux linkages as state-space variables. This paper attempts to unify main flux saturation modelling in d-q axis models of induction machines by presenting a general method of saturation modelling. Selection of state-space variables in the saturated machine model is arbitrary and appropriate models in terms of different state-space variables result by application of the method. A couple of models, obtainable with different selection of state-space variables, are presented. The cross-saturation effect is explicitly present in all the models, except for the one with stator and rotor flux linkage d-q axis components as state-space variables. The models are verified by simulation and experimental investigation of induction generator self-excitation     

Analytical method of two‐dimensional inverse heat conduction problem using Laplace transformation: Effect of number of measurement points

An analytical method has been developed for the inverse problem of two‐dimensional heat conduction using the Laplace transform technique. The inverse problem is solved for only two unknown surface conditions and the other surfaces are insulated in a finite rectangular body. In actual temperature measurement, the number of points in a solid is usually limited so that the number of temperature measurements required to approximate the temperature change in the solid becomes too small to obtain an approximate function using a half polynomial power series of time and the Fourier series of the eigenfunction. In order to compensate for this lack of measurement points, the third‐order Spline method is recommended for interpolating unknown temperatures at locations between measurement points. Eight points are recommended as the minimum number of temperature measurement points. The calculated results for a number of representative cases indicate that the surface temperature and the surface heat flux can be predicted well, as revealed by comparison with the given surface condition. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 618–629, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10116     

Effect of GDL permeability on water and thermal management in PEMFCs—II. Clamping force

In proton exchange membrane fuel cells (PEMFCs), the permeability of the gas diffusion layer (GDL) differs between the in-plane and through plane directions, and the overall permeability in the shoulder region is typically lower than that in the channel region due to the clamping force applied through the bipolar plates. Here, we conducted a numerical investigation of GDLs with different isotropic or anisotropic permeabilities in the channel and shoulder regions for PEMFCs. A three-dimensional, non-isothermal model was employed with a single straight channel. We found that the water and thermal management in PEMFCs depend on the permeability characteristics of the GDL, especially in the shoulder region. Moreover, the ohmic loss and cathode overpotential varied depending on the difference in isotropic permeability between the channel and shoulder regions. In the study on GDLs with anisotropic permeabilities in the channel and shoulder regions, however, we found that variations in the anisotropic permeabilities in the channel and shoulder regions, had little effect on the cathode overpotential at the shoulder region, but caused significant changes in ohmic loss as the ohmic loss depended on water and thermal management. In addition, we found that the cell temperature was much higher in GDLs with low anisotropic permeabilities due to hindering of the water removal process.     

A fitting algorithm for solving inverse problems of heat conduction

The paper presents an algorithm for solving inverse problems of heat transfer. The method is based on iterative solving of direct and adjoint model equations with the aim to minimize a fitting functional. An optimal choice of the step length along the descent direction is proposed. The algorithm has been used for the treatment of a steady-state problem of heat transfer in a region with holes. The temperature and the heat flux density were known on the outer boundary of the region, whereas these values on the boundaries of the holes are to be determined. The idea of the algorithm consist in solving of Neumann problems where the heat flux on the outer boundary is prescribed, whereas the heat flux on the inner boundary is guessed. The guess is being improved iteratively to minimize the mean quadratic deviation of the solution on the outer boundary from the given distribution.The results obtained show that the algorithm provides smooth, non-oscillating, and stable solutions to inverse problems of heat transfer, that is, it avoids disadvantages inherent in other computational methods for such problems. The ill-conditioning of inverse problems in the Hadamard sense is exhibited in that a very quick convergence of the fitting functional to its minimum does not imply a comparable rate of convergence of the recovered temperature on the inner boundary to the true distribution.The considered method can easily be extended to nonlinear problems.Numerical calculation has been carried out with the FE program Felics developed at the Chair of Mathematical Modelling of the Technical University of Munich.     

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.     

Reluctance distribution modelling of saturated salient polesynchronous machines

This model was developed in response to the demand for fast, simple, and reasonably accurate methods for modelling heavily saturated synchronous machines. This paper outlines the development and performance of a new air gap flux density model for saturated salient pole synchronous machines. The model has four main components and its performance is demonstrated by comparing its computed flux density distribution and rotor torque with measured results for static conditions. The model gives reasonable results considering its simplicity     

机械蒸汽压缩机模型及其与风力发电机耦合模拟研究

机械蒸汽压缩海水淡化是一种很有前景的消纳风能的方式,其中的压缩机是这种能源利用方式的核心部件。建立了离心式蒸汽压缩机数学模型,研究了压缩机性能参数间的关系,重点探讨了压缩机输入功率及吸气蒸汽参数与压缩机流量、压比和转速间的关系,并通过耦合风力发电机模型,研究了定压比条件下,压缩机流量和转速在风力发电机随机功率变化时的响应曲线。结果表明,风力发电机驱动的机械蒸汽压缩机需要辅助能源来保证在较小的风力发电机输入功率情况下,压缩机能稳定运行在非喘振区。对于额定功率为160 k W的压缩机,在压比2.4条件下的最小输入功率为50 k W。     

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.     

数值模拟涡轴发动机燃烧室流场

本文运用贴体坐标系统对涡轴发动机燃烧室流场进行了数值模拟。采用ＴＴＭ的非正交贴体坐标网格来处理燃烧室的曲线壁面边界。并把ＳＴＭＬＥ算法应用用到曲线坐标下求解各守恒方程，紊流粘度是用双方程ｋ－ε模型来估算，紊流燃烧模型采用按简单的化学反应系统假设的Ａｒｒｈｅｎｉｕｓ－ＥＢＵ模型，采用热流法辐射模型来估算辐射通量和燃烧室壁温及出口温度分布，计算结果令人满意。本文提供计算方法可供新型燃烧室设计方案选择用     

A 2-D INVERSE METHOD FOR SIMULTANEOUS ESTIMATION OF THE INLET TEMPERATURE AND WALL HEAT FLUX IN A LAMINAR CIRCULAR DUCT FLOW

A two-dimensional inverse analysis is presented for the estimation of the inlet temperature of the fluid flow and wall heat flux in a thermally developing hydrodynamically developed laminar flow in a duct. The inverse analysis is based on the temperature reading located at a stream inside the duct at several different points. At the beginning of the study, finite difference methods are employed to discretize the problem, and then a linear inverse model is constructed to identify the unknown conditions. The present approach is to rearrange the matrix forms of the differential governing equation and estimate the inlet temperature of the fluid and unknown surface conditions of the duct. The linear least squares method is adopted to find the solution. The advantage of applying this method in inverse analysis is that no prior information is needed on the functional form of the unknown quantities, no initial guess is required, and the number of iterations in the calculation process is limited to one. The effects of sensor position, magnitude of measurement error, and number of measurements on the accuracy of estimates are examined. The results show that the preferred position of the sensor is closer to the inlet region and only few measuring points are sufficient to estimate the wall heat flux and inlet temperatures of the fluid when the measurement errors are neglected. When the measurement errors an considered, more measuring points are needed in order to increase the congruence of the estimated results to exact solutions.