Approximate solution of non-linear transient heat conduction in cylindrical geometry

The applicability of the Heat Balance Integral method (HBI) using hybrid profiles (HP _n ) for heat diffusion problems in cylindrical geometry is studied. Three test problems are taken to validate the method, namely, TP1 — long hollow cylinder, inner boundary is supplied with constant heat flux and the outer boundary is at infinity, TP2 — the same geometry with the inner boundary at constant temperature, TP3 — the same geometry with convection at the boundary. It is found that HBI in combination with HP _n gives much better solutions as compared to the HBI-polynomial (P _n ) combinations. This method is applied to non-linear cases like temperature dependent thermal conductivity, boiling or condensation at the boundary. For assessment of accuracy Root Mean Square Residual (E _rms) is defined. The residual minimization technique is used to select the parameter in the HP _n profile. It is shown that this procedure yields solutions which are in excellent agreement with the available exact solutions in the case of the three test problems. In the case of nonlinear problems, the value ofE _rms which is comparable to the values in the test cases, shows that the HBI-HP _n is a very good combination for all problems.In dieser Studie wurde die Verwendbarkeit des Wärmebilanz-Integrationsverfahrens (HBI) unter Benutzung von hybriden Profilen (HP _n ) für Wärmeausbreitungsprobleme in zylindrischen Geometrien untersucht. Um das Verfahren zu veranschaulichen wurden drei Testprobleme genommen; wobei für alle drei Fälle die gleiche Geometrie benützt wurde, und zwar ein langer, ausgehöhlter Zylinder. Beim TP-1 ist die innere Grenze mit konstantem Wärmestrom versorgt und die äußere Grenze ist im Unendlichen angenommen worden. Beim TP-2 herrscht an der inneren Grenze eine konstante Temperatur. Beim TP-3 herrscht Konvektion an den Grenzen. Hierbei wurde festgestellt, daß HBI in Verbindung mit HP _n sehr viel bessere Lösungen liefert, als die HBI-Polynom-Kombinationen (P _n ). Dieses Verfahren ist für nicht lineare Fälle, wie temperaturabhängige Wärmeleitfähigkeit, Sieden oder Kondensation and den Grenzen verwendet worden. Für die Abschätzung der Genauigkeit ist das restliche quadratische Mittel definiert worden. Die Residuen-Minimisierungstechnik ist für die Auswahl des Parameters \ benützt worden. Es ist gezeigt worden, daß dieses Verfahren Lösungen liefert, welche mit den benützten genauen Lösungen der drei Testfälle hervorragend übereinstimmen. Im Fall der nicht linearen Probleme zeigt der Wert vonE _rms, der mit den Testfällen vergleichbar ist, daß die HBI-HP _n eine sehr gute Kombination für alle Probleme ist.     

Nonstationary heat conduction in a half-space with an infinite number of cylindrical heat sources

The problem of nonstationary heat conduction is treated for a half-space containing an infinite number of cylindrical heat sources, and a boundary condition of the first kind at the surfaces. It is assumed that the radii of the sources are small in comparison with their spacing and the ordinate of the center.     

Transient thermal cracking associated with non-classical heat conduction in cylindrical coordinate system

This paper studies the fracture behavior of a thermoelastic cylinder subjected to a sudden temperature change on its outer surface within the framework of non-classical heat conduction.The heat conduction equation is solved by separation of variable technique.Closed form solution for the temperature field and the associated thermal stress are established.The critical parameter governing the level of the transient thermal stress is identified.Exact expression for the transient stress intensity factor is obtained for a crack in the cylinder.The difference between the non-classical solutions and the classical solution are discussed.It is found that the traditional classical heat conduction considerably underestimates the transient thermal stress and thermal stress intensity factor.     

变几何域传热的表面热流反演方法

变几何域的表面热流反演是一类特殊的热传导逆问题,在再入飞行器烧蚀型防热材料的表面热流反演中具有工程实用价值.本文首先对变几何域传热的正问题计算方法进行了校核验证,然后建立了求解变几何域表面热流反演问题的顺序函数法和共轭梯度法;给出了这两种反演方法的基本思想和算法推导,并针对典型算例进行了仿真.结果表明:两种反演方法都能计算出较好的反演结果,并且算法受测量噪声的影响较小,具有较好的鲁棒性;反演算法能适应不同的几何域变化函数,但几何域变化量的测量误差在表面热流的反演结果中会有较为直接的反映.     

Prediction of transient heat transfer coefficients in thick-walled pressure components of cylindrical geometry

This paper presents a procedure for determining the transient heat transfer coefficient in cylindrical, thick-walled pressure parts. From theoretical considerations, the temperatures can be predicted at discrete locations throughout the wall, when input data such as thermocouple responses are known at one or several interior locations.Special emphasis is placed on the dynamic response of the thermometer, which measures the temperature of the inside fluid, to enable exact determination of both heat transfer coefficient and fluid temperature. The transient response of a thermocouple in a convectional thermowell (pocket) is described by the first-order convective heat transfer model in which the rate of thermoelement temperature change is proportional to the instantaneous difference between the thermoelement and fluid temperatures.     

Immersed boundary conditions method for heat conduction problems in slots with time‐dependent geometry

A grid‐less, fully implicit, spectrally accurate algorithm for solving three‐dimensional, both stationary and time‐dependent, heat conduction problems in slots formed by either fixed or time‐dependent boundaries has been developed. The algorithm is based on the concept of immersed boundary conditions (IBC), where the physical domain is immersed within the computational domain and the boundary conditions take the form of internal constraints. The IBC method avoids the need to construct adaptive, time‐dependent grids resulting in the reduction of the required computational resources and, at the same time, maintaining accurate information about the location of the boundaries. The algorithm is spectrally accurate in space and capable of delivering first‐, second‐, third‐ and fourth‐order accuracy in time. Given a potentially large size of the resultant linear algebraic system, various methods that take advantage of the special structure of the coefficient matrix have been explored in search for an efficient solver, including a specialized direct solver as well as serial and parallel iterative solvers. The specialized direct solver has been found to be the most efficient from the viewpoints of the speed of the computations and the memory requirements. Copyright © 2010 John Wiley & Sons, Ltd.     

An alternating direction implicit method for magnetohydrodynamic heat transfer in cylindrical geometry with discontinuity in wall temperature

The temperature distribution in the magnetohydrodynamic axial flow in a circular pipe has been found by using an alternating direction implicit method which has been suitably modified for r-θ-z geometry. First the temperature distribution for from the discontinuity, which ceases to depend upon the axial coordinate, has been found. This has been used to determine the results on both sides of the discontinuity. It is found that the temperature falls as the Hartmann number is increased, and convection dominates for large values of the Peclet number. The effect of the Hartmann number is more pronounced when Peclet number is large.     

非均质材料等效瞬态热传导时域分析

基于时域自适应算法,结合均匀化技术和有限元方法,从而提出一种瞬态温度场求解模型,用来预测非均质材料等效性能并评估宏观温度场的等效行为.在整个计算中,通过自适应算法保证计算精度和稳定性,避免时间段尺寸变化可能引起的计算误差.在数值算例中,等效分析的结果与利用ANSYS求解的非均质有限元解比较,从计算效率和计算精度综合效果而言,结果是令人满意的.     

Whole time domain solution of inverse heat conduction problem in multi-layer media

The unknown surface conditions in composite media is estimated by minimizing the nonlinear least squares error between the computed and measured temperatures over the whole time domain. This approach shown to be stable, efficient and accurate. The unknown surface conditions are assumed to have an abrupt change at unknown time. A sensitivity analysis is conducted to learn more insight into the nature of difficulties that can be encountered in the estimation of the parameters associated with the inverse problem. The stability and accuracy of the method is demonstrated by several numerical examples which provide very strict test conditions. Received on 7 March 1997     

Periodic cavitation shedding in a cylindrical orifice

Cavitation structures in a large-scale (D = 8.25 mm), plain orifice style nozzle within a unique experimental rig are investigated using high-speed visualisation and digital image processing techniques. Refractive index matching with an acrylic nozzle is achieved using aqueous sodium iodide for the test fluid. Cavitation collapse length, unsteady shedding frequency and spray angles are measured for cavitation conditions from incipient to supercavitation for a range of Reynolds numbers, for a fixed L/D ratio of 4.85. Periodic cavitation shedding was shown to occur with frequencies between 500 and 2,000 Hz for conditions in which cavitation occupied less than 30% of the nozzle length. A discontinuity in collapse length was shown to occur once the cavitation exceeded this length, coinciding with a loss of periodic shedding. A mechanism for this behaviour is discussed. Peak spray angles of approximately θ ≈ 14° were recorded for supercavitation conditions indicating the positive influence of cavitation bubble collapse on the jet atomisation process.     

A shear flow instability in cylindrical geometry

Conclusion The results show, that this model experiment is useful for studying the interaction between two parallel shear layers in a rotating system. The structures of the observed instabilities exhibit some similarities with the wake flow behind cylindrical obstacles. Additional investigations in analyzing the dynamical behaviour of the shear instabilities are necessary to get a deeper insight into the physical mechanism. Up to that it is an open question, whether the physical effects described by Meiburg (1987), play an important role in these shear layer instabilities.     

Perturbation solution to heat conduction in melting or solidification with heat generation

The Stefan problem involving a source term is considered in this technical note. As an example, planar solidification with time-dependent heat generation in a semi-infinite plane is solved by use of a perturbation technique. The perturbation solution is validated by reducing the problem to the case without heat generation whose exact solution is available. An application to the case with constant heat generation is presented, for which a closed-form solution is obtained. The effects of heat generation and Stefan number on the evolution of solidification are examined using the perturbation solution.     

Non-fourier heat conduction in a plane slab with prescribed boundary heat flux

The non-stationary heat conduction in an infinitely wide plane slab with a prescribed boundary heat flux is studied. An arbitrary time dependent boundary heat flux is considered and a non-vanishing thermal relaxation time is assumed. The temperature and the heat flux density distributions are determined analytically by employing Cattaneo-Vernotte's constitutive equation for the heat flux density. It is proved that the temperature and the heat flux density distributions can be incompatible with the hypothesis of local thermodynamic equilibrium. A comparison with the solution which would be obtained by means of Fourier's law is performed by considering the limit of a vanishing thermal relaxation time.     

Thermoelastic waves in helical strands with Maxwell–Cattaneo heat conduction

Harmonic thermoelastic waves in helical strands with Maxwell–Cattaneo heat conduction are investigated analytically and numerically. The corresponding dispersion relation is a sixth-order algebraic equation, governed by six non-dimensional parameters: two thermoelastic coupling constants, one chirality parameter, the ratio between extensional and torsional moduli, the Fourier number, and the dimensionless thermal relaxation. The behavior of the solutions is discussed from two perspectives with an asymptotic-numerical approach:(1) the effect of thermal relaxation on the elastic wave celerities, and(2) the effect of thermoelastic coupling on the thermal wave celerities. With small wavenumbers, the adiabatic solution for Fourier helical strands is recovered.However, with large wavenumbers, the solutions behave differently depending on the thermal relaxation and chirality. Due to thermoelastic coupling, the thermal wave celerity deviates from the classical result of the speed of second sound.     

Hyperbolic heat conduction in the semi-infinite body with a time-dependent laser heat source

 The Cattaneo hyperbolic and classical parabolic models of heat conduction in the laser irradiated materials are compared. Laser heating is modelled as an internal heat source, whose capacity is given by g(x,t)= I(t)(1−R)μexp(−μx). Analytical solution for the one-dimensional, semi-infinite body with the insulated boundary is obtained using Laplace transforms and the discussion of solutions for different time characteristics of the heat source capacity (constant, instantaneous, exponential, pulsed and periodic) is presented. Received on 18 May 1999     

Non-Fourier heat conduction in a finite hollow cylinder with periodic surface heat flux

Analytical solution of the non-Fourier axisymmetric temperature field within a finite hollow cylinder exposed to a periodic boundary heat flux is investigated. The problem studied considering the Cattaneo–Vernotte (CV) constitutive heat flux relation. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The standard method of separation of variables is used for solving the problem with time-independent boundary conditions, and the Duhamel integral is used for applying the time dependency. The solution is applied for the special cases of harmonic uniform heat flux and an exponentially pulsed heat flux with Gaussian distribution in outer surface for modeling a laser pulse, and their respective non-Fourier thermal behavior is studied.     

柱状交界面Richtmyer-Meshkov不稳定性的数值研究

采用基于有限体积法的二阶Godunov格式模拟了柱形密度交界面在同轴激波加速下的演化过程。得到了以下初步结果:在参数相同的情形下,内聚激波比中心爆炸波对界面的扰动更危险;内聚激波从轻质流体进入重质流体与从重质流体进入轻质流体相比,界面有更快的增长。周向波数大小对界面增长率有很大的影响,在计算的参数下,n=8～12有最大增长率,大于和小于这个范围的周向波长,增长率均明显减弱。     

Periodic heat transfer in the fins with variable thermal parameters

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