Influence of longitudinal conduction in the matrix on effectiveness of rotary heat regenerator used in air-conditioning

The paper presents an efficient analytical method of solving the problem of a rotary heat regenerator taking into account longitudinal heat conduction in the matrix. The small parameter method, Laplace transform as well as one of the spline functions have been applied for approximation of an initial condition in the reversion time. In the application part, the solution for a model in analysis of an influence of longitudinal conduction in the matrix on effectiveness of rotary heat regenerator in a wide range of dimensionless parameters as well as for the particular matrix applied in air-conditioning was used.

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An analytical model of a matrix heat exchanger with longitudinal heat conduction in the matrix in application for the exchanger dynamics research

The paper presents an analytical method for the solution of the problem of a fixed matrix heat exchanger with axial heat conduction within the matrix. The small parameter method and Laplace transform have been applied. A general solution has been obtained for the unsteady state in the form of function series, using single and double convolutions of functions, as well as a particular solution for both the uniform and non-uniform initial temperature of the matrix and for an arbitrary function of the fluid temperature at the inlet. Particular solutions have been used in the study of the matrix dynamics in determining dynamic characteristics for the standard input signals in the form of: Dirac pulse, Heaviside function and the function of sinusoidal variable temperature of the fluid at the inlet. The results obtained both illustrate and enable the assessment of the effect of axial heat conduction in the matrix on the dynamic properties of the heat exchanger.

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On Fourier's law of heat conduction

A linear theory of fluid is considered in which the gradients of density, internal energy and velocity are among the constitutive variables. Thus the heat flux may be a linear combination of the gradients of density and internal energy. It is proved that this linear combination may be written as the gradient of temperature so that Fourier's law of heat conduction holds.     

General integral solution of the regenerator transient test equations for zero longitudinal conduction

Since the pioneer work of Anzelius (1) Nusselt (2), Schumann (3), and Hausen (4), a variety of theoretical solutions for thermal response of an initially isothermal matrix subject to specific forms of inlet fluid temperature disturbances have been formulated. This paper provides a single general solution accommodating any inlet temperature disturbance, which is suitable for determining the heat transfer performance of certain matrix geometries near ambient temperature.     

Effects of longitudinal heat conduction of a vertical thin plate in a natural convective cooling process

This paper analyzes the cooling process of a vertical thin plate caused by a free convective flow, taking into account the effects of both longitudinal and transversal heat conduction in the plate. Due to the finite thermal conductivity of the plate, a longitudinal temperature gradient arises within it, which prevents any similarity solution in the boundary layer, changing the mathematical character of the problem from parabolic to elliptic, for large values of the Rayleigh number. The energy balance equations are reduced to a system of three differential equations with two parameters: the Prandtl number and a non-dimensional plate thermal conductivity . In order to obtain the evolution of the temperature of the plate as a function of time and position, the coupled balance equations are integrated numerically for several values of the parameters, including the cases of very good and poor conducting plates. The results obtained, are compared with an asymptotic analysis based on the multiple scales technique carried out for the case of a very good conducting plate. There is at the beginning a fast transient in non-dimensional time scale of order ^–1 followed by a slow non-dimensional time scale of order unity, which gives the evolution of the cooling process. Good agreement is achieved even for values of the conduction parameter of order unity. The asymptotic solution allows us to give closed form analytical solution for the plate temperature evolution in time and space. The overall thermal energy of the plate decreases faster for smaller values of .

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Einfluß der Längswärmeleitung in einer senkrechten dünnen Platte auf den Kühlvorgang unter natürlicher Konvektion

Zusammenfassung In dieser Arbeit wird die Abkühlung einer senkrechten dünnen Platte unter freier Konvektion untersucht, wobei die Einflüsse von Längs- und Querwärmeleitung in der Platte Berücksichtigung finden. Aufgrund der endlichen Wärmeleitfähigkeit der Platte bildet sich darin ein Temperaturgradient aus, der Ähnlichkeitslösungen für die Grenzschichtströmung nicht zuläßt, da der mathematische Charakter des Problems für große Werte der Rayleigh-Zahl vom parabolischen in den elliptischen Typ übergeht. Die Energiebilanzgleichungen reduzieren sich auf ein System von drei Differentialgleichungen mit zwei Parametern: die Prandtl-Zahl und eine dimensionslose Temperaturleitfähigkeit des Plattenmaterials. Um die Entwicklung des Temperaturfeldes in der Platte als Funktion von Zeit und Ort verfolgen zu können, werden die gekoppelten Bilanzgleichungen für mehrer Werte der Parameter — einschließlich der Fälle sehr guter und sehr schlechter Wärmeleitfähigkeit — numerisch integriert. Die gefundenen Ergebnisse lassen sich für den Fall der sehr gut leitenden Platte mit den Ergebnissen einer asymptotischen Untersuchung vergleichen. Zu Beginn — in einem dimensionslosen Zeitbereich der Größenordnung ^–1 — zeigt sich ein sehr rasches Übergangsverhalten, gefolgt von einem Zeitbereich der Größenordnung Eins, in dem der eigentliche Kühlungsprozeß abläuft. Selbst für Werte des Leitfähigkeitsparameters der Größenordnung Eins zeigt sich gute Übereinstimmung. Die asymptotische Lösung gibt die Entwicklung des Temperaturfeldes in der Platte nach Zeit und Ort in geschlossener Form wieder. Für kleinere Werte des Parameters nimmt die in der Platte gespeicherte Gesamtenergie schneller ab.

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Nomenclature C _p fluid's specific heat - C _w plate specific heat - f non-dimensional stream function introduced in Eq. (12) - g function introduced in Eq. (19) - G ₀ non-dimensional heat flux given in Eq. (32) - h plate half-thickness - L plate length - Pr Prandtl number - Ra Rayleigh number defined in Eq. (1) - Re Reynolds number - s non-dimensional strained time introduced in Eq. (17) - t time - t _c characteristic time given byt _c= _w C _w h L Ra ^–1/4/ - T temperature - x cartesian lengthwise coordinate - y cartesian transversal coordinate - z non-dimensional transversal coordinate defined in Eq. (3) Greek symbols non-dimensional parameter defined in Eq. (5) - fluid's thermal expansion coefficient - non-dimensional transversal coordinate given in Eq. (11) - _w plate conductivity - fluid's conductivity - non-dimensional transversal coordinate introduced in Eq. (19) - similarity independent variable defined in Eq. (59) - dynamic viscosity - kinematic viscosity - _w plate density - non-dimensional fast time defined in Eq. (17) - non-dimensional time defined in Eq. (3) - _i time adjusted constants - non-dimensional fluid temperature defined in Eq. (11) - _w non-dimensional plate temperature defined in Eq. (3) - x non-dimensional lengthwise coordinate defined in Eq. (3) - non-dimensional self similar independent variable introduced in Eq. (64)     

Transient heat conduction in a solid slab using multiple-scale analysis

In this paper we study the unsteady heat conduction due to a sudden temperature step in the external surfaces of a solid slab. In order to estimate the temperature profile in the solid, we applied the multiple-scale perturbation technique by identifying the “early” and “late” transient regimes for small values of the Biot number, Bi. In this sense, we have re-visited the classical lumped method, incorporating this particular case as an asymptotic limit, which is fully described by the “late” regime for small values of Bi. Once the temperature distribution is analytically predicted, this solution is compared against the exact solution and with other analytical results obtained by using regular perturbation techniques, for different values of the Biot number Bi. Observing a good agreement between the corresponding comparisons, we obtain a very simple and useful formula to predict the nondimensional temperature of the solid slab.     

Rational design of thermoelastic damping in microresonators with phase-lagging heat conduction law

The design of thermoelastic damping (TED) affected by the phase-lagging non-Fourier heat conduction effects becomes significant but challenging for enlarging the quality factor of widely-used microresonators operating in extreme situations, including ultra-high excitation frequency and ultra-low working temperature. However, there does not exist a rational method for designing the TED in the framework of non-Fourier heat conduction law. This work, therefore, proposes a design framework to achieve low thermoelastic dissipation of microresonators governed by the phase-lagging heat conduction law. The equation of motion and the heat conduction equation for phase-lagging TED microresonators are derived first, and then the non-Fourier TED design problem is proposed. A topology optimization-based rational design method is used to resolve the design problem. What is more, a two-dimensional (2D) plain-strain-based finite element method (FEM) is developed as a solver for the topology optimization process. Based on the suggested rational design technique, numerical instances with various phase lags are investigated. The results show that the proposed design method can remarkably reduce the dissipation of microresonators by tailoring their substructures.     

Approximate analysis for transient heat conduction with radiation in a slab

In an attempt to minimize the numerical computations associated with the solution of transient heat conduction with radiation in a slab, a perturbation type of analysis is being applied to the temperature field and radiation heat flux simultaneously. The resulting partial differential equations for the perturbation functions for the temperature are solved in explicit forms by use of the energy integral methods, while the radiation heat flux is determined by an appropriate scheme of approximating the temperature distribution in the slab. Included in the analysis are the effects of the parameters: the optical thickness, the ratio of conduction transport to radiation and the wall emissivity. It is found that, in a wide range of these governing parameters, the results compare very favorably with those obtained by the numerical solution of the formulated integro-differential equation. With the present analysis, the temperature, conduction and radiation heat fluxes can be predicted without resorting to lengthy numerical analysis.     

热传导问题灵敏度分析的伴随法

在热传导灵敏度分析的直接法的研究基础上，进一步探讨了稳态和瞬态热传导问题灵敏度分析的伴随法.推导了伴随法的计算列式，对于瞬态热传导问题，研究了瞬态约束处理的关键点方法，并提出伴随方程的精细积分解法。算例表明，稳态问题灵敏度计算，伴随法与直接法的结果是一致的；瞬态问题灵敏度计算，两种方法的精度相当。     

INTERVAL ANALYSIS OF FUZZY-RANDOM HEAT CONDUCTION IN COMPOSITE TUBES

IntroductionThere are a lot of uncertainties such as fuzziness and random in the design andmanufacture in engineering.In generally speaking,methods to deal with the fuzzy-randomproblems are first to transfer the fuzzy set to real number set,and then the p…     

The combined effects of wall longitudinal heat conduction and inlet fluid flow maldistribution in crossflow plate-fin heat exchangers

An analysis of a crossflow plate-fin compact heat exchanger, accounting for the combined effect of two-dimensional longitudinal heat conduction through the exchanger wall and nonuniform inlet fluid flow distribution on both hot and cold fluid sides is carried out using a finite element method. Using the fluid flow maldistribution models, the exchanger effectiveness and its deterioration due to the combined effects of longitudinal heat conduction and flow nonuniformity are calculated for various design and operating conditions of the exchanger. It was found that the performance deteriorations are quite significant in some typical applications due to the combined effects of wall longitudinal heat conduction and inlet fluid flow nonuniformity on crossflow plate-fin heat exchanger. Received on 20 May 1999     

Analysis of coupled conduction and radiation heat transfer in presence of participating medium- using a hybrid method

The current study addresses the mathematical modeling aspects of coupled conductive and radiative heat transfer in presence of absorbing, emitting and isotropic scattering gray medium within two-dimensional square enclosure. The walls of the enclosure are considered to be opaque, diffuse and gray. The enclosure comprised of isothermal vertical walls and insulated horizontal walls. A new hybrid method where the concepts of modified differential approximation employed by blending discrete ordinate method and spherical harmonics method, has been developed for modeling the radiative transport equation. The finite volume method has been adopted as the numerical technique. The effect of various influencing parameters i.e., radiation-conduction parameter, surface emissivity, single scattering albedo and optical thickness has been illustrated. The compatibility of the method with regard to solving coupled conduction and radiation has also been addressed.     

半无限介质瞬态热传导的同伦分析法

In the current work, transient heat conduction in a semi-infinite medium is considered for its many applications in various heat fields. Here, the homotopy analysis method （HAM） is applied to solve this problem and analytical results are compared with those of the exact and integral methods results. The results show that the HAM can give much better approximations than the other approximate methods： Changes in heat fluxes and profiles of temperature are obtained at different times and positions for copper, iron and aluminum.     

Thermal convection in a cylindrical enclosure with intense internal heat generation in the presence of a longitudinal magnetic field

Steady convection in a gaseous medium with intense heat generation is considered without making use of the Boussinesq approximation. The effect on convection of the thermal boundary conditions at the walls of the enclosure is investigated, together with the influence of a longitudinal magnetic field which is effective when the medium is strongly heated and becomes electrically conducting.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 11–18, May–June, 1992.The authors are grateful to the participants in the G. A. Lyubimov seminar for discussing their work.     

Thermodynamic analysis of the one-dimensional heat conduction with inhomogeneities in the heat flux direction

Starting from the results of Li, Prigogine and others about the one-dimensional heat conduction with constant temperature boundary conditions, the aim of this paper is to study, according to the methods and the purposes of the generalized thermodynamics, the more general case of one-dimensional heat conduction in systems, whose conductivity is function of both temperature and the coordinate in the heat flux direction and presents a finite number of discontinuities.

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Thermodynamische Analyse für eindimensionale Wärmeströmung mit Ungleichartigkeiten in der Wärmeflußleitung

Zusammenfassung Ausgehend von den Ergebnissen von Li, Prigogine und anderen, für eindimensionale Wärmeströmung mit konstanten Temperaturen an den Grenzen, versucht diese Arbeit, gemäß den Methoden und den Zielen der verallgemeinerten Thermodynamik, den allgemeineren Fall der eindimensionalen Wärmeströmung mit Ungleichartigkeiten zu examinieren.

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Nomenclature c volumetric specific heat - G discriminating parameter [G (t) =k ₂ (t)t ² orG _i (t)=k_i(t) t², when the separation of the variables for thermal conductivity can be done, or in general: G(x, t)=k(x,t)t ²] - J heat transfer rate (generalized flux) [W] - J ₀ heat flux [W/m²] - k thermal conductivity - k ₁ component of thermal conductivity depending upon the coordinate in the heat flux direction - k ₂ component of thermal conductivity depending upon temperature - k _i component of the thermal conductivity of a homogeneous layer (i) depending upon temperature - L ₁,L ₂ extreme coordinates - Lip Lipschitz's function - P entropy production rate - (P(T))_min temperature distribution in a system corresponding to the minimum of entropy production rate - p ₁ thermokinetic potential - (P ₁ (T))_min temperature distribution in a system corresponding to the minimum of thermokinetic potential - P ₂ generalized force potential - Q differential form - dQ total differential form - ¯R set of all real numbers - S(x) area of the isothermal surface corresponding to the coordinatex - T system temperature distribution - t absolute temperature - ¯T set of all the possible system temperature distributions - x symbol of Cartesian product - x coordinate in the heat flux direction - X local generalized force - B,Y ₀,Y ₀ ^*,Z sets (defined in this paper) - function representing the time evolution of the temperature distribution in a system - time - ₀ reference time interval     

Meshless method for nonlinear heat conduction analysis of nano-composites

Carbon nanotubes (CNTs) may become ideal reinforcing materials for high-performance nano-composites due their exceptional properties. Still, much work is needed to be done before the potentials of CNT based composites can be fully realized. The evaluation of effective material properties of nano-composites is one of many difficult tasks. Simulations using continuum mechanics approach can play a significant role in the analysis of these composites. In the present work, nonlinear heat conduction analysis of CNT based composites has been carried out using continuum mechanics approach. Element free Galerkin method has been applied as a numerical tool. Thermal conductivities of nanotube and polymer matrix are assumed to vary quadratically with temperature. Picard and quasi-linearization schemes have been utilized to obtain the solution of a system of nonlinear equations. Cylindrical representative volume element has been used to evaluate the thermal properties of nano-composites. Present simulations show that the temperature dependent matrix thermal conductivity has a significant effect on the equivalent thermal conductivity of the composite, whereas temperature dependent nanotube thermal conductivity has a small effect on the equivalent thermal conductivity of the composite. The results obtained by Picard method have been found almost similar with those obtained by quasi-linearization approach.     

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.