A heat exchanger is a device used to transfer thermal energy between two or more fluids, at different temperatures in thermal contact. They are widely used in aerospace, chemical industries, power plants, refineries, HVAC refrigeration, and in many industries. The optimal design and efficient operation of the heat exchanger and heat transfer network plays an important role in industry in improving efficiencies and to reduce production cost and energy consumption. In this paper, significance of shape of inner pipe of double pipe heat exchanger was analyzed with respect to triangular, hexagonal and octagonal shaped inner pipes. The performance of double pipe heat exchangers was investigated with and without dent pattern using CFD analysis in ANSYS and efficient heat transfer results are identified from CFD outputs. On basis of literature review, few factors influencing the efficiency of heat exchanger and method to improve the efficiency are discussed.

As a perturbation method, averaging is a mathematical tool for dynamic analysis of time-periodic and space-periodic dynamical systems, including those governed by partial differential equations. The control design procedure presented in this work uses averaging techniques, the well-developed linear control strategies, and finite element methods. The controller is designed based on the linear averaged dynamics of a time- or space-periodic system. The controller is then used for trajectory tracking or stabilization of the periodic system. The applicability and performance of the suggested method depend on different physical parameters of the periodic system and the control parameters of the controller. The effects of these parameters are discussed in this work. Numerical simulations show acceptable performance of the proposed control design strategy for two linear and nonlinear time- and space-periodic systems, namely, the one-dimensional heat equation and the Chafee-Infante equation with periodic coefficients. / M.S. / Dynamic analysis and control of dynamical systems with varying parameters is a challenging task. It is always of great help if one can perform the analyses for an approximate system with constant parameters and use the results to study and control the original system with varying parameters. Averaging is a mathematical tool that is used to approximate a system with periodic parameters with a ``simpler'' system with constant parameters. In this research averaging is used for design of controllers for systems with periodic parameters. First, an approximate system with constant parameters, called the averaged system, is determined. The averaged system is used for design of a controller which can be then be used for the original system with periodic parameters.

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Termisk modellring av hårdhet genom beräkning och simulering av den värmepåverkade zonen i en lasersvetsad stålplatta är en omfattande process. Dock är analysen viktig då mikrostrukturella fastransformationer förorsakade av svetsningen kan ge oönskade hårdhetsnivåer av den värmepåverkade zonen jämfört med hårdeheten i basmaterialet. I denna avhandling har analytiska ekvationer implementerats och testats för validitet mot simuleringar gjorda av andra författare och mot experimentella värden.Eftersom termisk modellering av svetsar är ett omfattande område var avhandlingen tvungen att smalnas av för att göra analysen mer fokuserad. Begränsningar gjordes för den matematiska modelleringen genom att endast titta på två-dimensionellt värmeflöde i svetsade plattor där endast den analytiska lösningen är av intresse. Arbetet har också inriktats mot stål då detta material är vida använt över hela världen. Då lasersvetsning är en snabb och kostnadseffektiv process så är hårdhetsanalysen av största vikt. Avhandlingen är uppdelad i tre övergripande delar; den första är att ta fram och förstå arbetet som gjorts inom termisk modellering av svetsar, alltså förstå matematiken bakom problemet. Modelleringen är till för att producera diagram parametrar från en termisk cykel, för att kunna fortgå med korrekt hårdhets analys. För det andra så sätts den matematiska modelleringen på prov i ett antal situationer som var och en simulerar olika förutsättningar. Detta gjordes i ett grafiskt användargränssnitt av ren bekvämlighet. Detta gör att ingenjörer lätt kan implementera olika egenskaper för materialet och få fram diagram och kurvor.Sist, ett liknande grafisk användargränssnitt för att simulera hårdheten i valfri punkt i den värmepåverkade zonen programmerades och därigenom implementerades ekvationerna som denna avhandling handlar om i grund och botten. En teoretisk bakgrund till fasomvandlingen är också inkluderad som förklaring till grundproblemet med oönskad hårdhet i den värmepåverkade zonen i lasersvetsat stål.Huvudslutsatser i avhandlingen:•Matematisk modellering av värmeöverföring i svetsar genomförd av Rosenthal är fortfarande applicerbar på modern lasersvetsningsapparatur. •Den empiriska modellen från Ion et al. (1984) är ej applicerbar med godkänt resultat för hårdhetsuppskattning.•Ekvationerna från Ion (2005) är statistiskt godkända för att simulera hårdhet.•Den analytiska lösningen är överlägsen den numeriska när det gäller snabb och enkel implementering för att simulera termiska cykler och hårdhet, medan den numeriska lösningen kan ta i beaktning mera avancerade egenskaper.•Förvärming av stålet innan svetsning kan vara mycket fördelaktigt för hårdheten i den värme-påverkade zonen, speciellt vid högre kolekvivalent. / Thermal modelling of hardness in the heat-affected zone (HAZ) in a laser welded steel plate is a cumbersome process both in calculation and simulation. The analysis is however important as the microstructural phase transformations induced by welding may cause unwanted hardness levels in the HAZ compared with that of the parent material. In this thesis analytical equations have been implemented and checked for validity against simulations made by other authors and against experimental values.With such a large field as thermal modelling, the thesis had to be narrowed down to make the analysis more subject focused. Limitations made were for mathematical modelling only looking at a two-dimensional heat flow in welded plates; in this thesis only the analytical solution to the heat flow is considered. The work was also directed towards steel; such a material as used largely all over the globe. As laser welding is a fast and cost-effective process, an analysis of hardness is of great importance. Work was divided into three overlapping parts; the first was to derive and understand the work done in the field of thermal modelling of welds, thus understanding the mathematics behind the basic problem. This modelling provides a number of curves and parameters from a thermal cycle, thus enabling one to do the hardness analysis correctly. Secondly, this mathematical modelling was applied to a number of cases, simulating different circumstances. This was done using self-programmed Graphical User Interfaces (GUI) for convenience. This enables engineers to easily plug in the materials and processing properties and thus simulate the required parameters and curves for further analysis.Lastly, a GUI for simulating the hardness of any point in the HAZ was programmed and used, thus implementing and validating the equations. A theoretical introduction of the phases induced in the HAZ is also included, in order of understanding the problems of unwanted hardness in the HAZ of laser-welded steel.Main conclusions of this thesis:•Mathematical modelling of heat transfer in welds by Rosenthal (1946) is still applicable for modern laser welding apparatus.•The empirical model presented by Ion et al. (1984) is not applicable with experimental results of hardness in the HAZ of the steels investigated here.•Equations by Ion (2005) are accurate for simulating the hardness.•The analytical solutions investigated are superior to numerical solutions with regard to quick, simple simulations of thermal cycles and hardness. Numerical solutions allows for more advanced modelling, which can be lengthy.•Preheating the steel prior to welding is favourable in reducing hardness levels, especially with steel of higher carbon equivalent.

Numerical models of PDE systems can involve very large matrix equations, but feedback controllers for these systems must be computable in real time to be implemented on physical systems. Classical control design methods produce controllers of the same order as the numerical models. Therefore, reduced order control design is vital for practical controllers. The main contribution of this research is a method of control order reduction that uses a newly developed low order basis. The low order basis is obtained by applying Proper Orthogonal Decomposition (POD) to a set of functional gains, and is referred to as the functional gain POD basis. Low order controllers resulting from the functional gain POD basis are compared with low order controllers resulting from more commonly used time snapshot POD bases, with the two dimensional heat equation as a test problem. The functional gain POD basis avoids subjective criteria associated with the time snapshot POD basis and provides an equally effective low order controller with larger stability radii. An efficient and effective methodology is introduced for using a low order basis in reduced order compensator design. This method combines "design-then-reduce" and "reduce-then-design" philosophies. The desirable qualities of the resulting reduced order compensator are verified by application to Burgers' equation in numerical experiments. / Ph. D.

A unified and systematic study of one-dimensional heat conduction based on thermal relaxation is presented. Thermal relaxation is introduced through the constitutive equation (modified Fourier's law) which relates this heat flux and temperature. The resulting temperature and flux field equations become hyperbolic rather than the usual classical parabolic equations encountered in heat conduction. In this formulation, heat propagates at a finite speed and removes one of the anomalies associated to parabolic heat conduction, i.e., heat propagating at an infinite speed. In situations involving very short times, high heat fluxes, and cryogenic temperatures, a more exact constitutive relation must be introduced to preserve a finite speed to a thermal disturbance. The general one-dimensional temperature and flux formulations for the three standard orthogonal coordinate systems are presented. The general solution, in the temperature domain, is developed by the finite integral transform technique. The basic physics and mathematics are demonstrated by reviewing Taitel's problem. Then attention is turned to the effects of radially dependent systems, such as the case of a cylinder and sphere. Various thermal disturbances are studied showing the unusual physics associated with dissipative wave equations. The flux formulation is shown to be a viable alternative domain to develop the flux distribution. Once the flux distribution has been established, the temperature distribution may be obtained through the conservation of energy. Linear one-dimensional composite regions are then investigated in detail. The general temperature and flux formulations are developed for the three standard orthogonal coordinate systems. The general solution for the flux and temperature distributions are obtained in the flux domain using a generalized integral transform technique. Additional features associated with hyperbolic heat conduction are displayed through examples with various thermal disturbances. A generalized expression for temperature dependent thermal conductivity is introduced and incorporated into the one-dimensional hyperbolic heat equation. An approximate analytical solution is obtained and compared with a standard numerical method. Finally, recommendations for future analytical and experimental investigations are suggested. / Ph. D.

銷售力一直是一個企業關切的主要議題，借助Vidale-Wolfe數學模型，我們提出一個銷售力數學模型。藉由熱傳導方程，刻畫由資訊交流產生的自身銷售力。在資訊交流及商品促銷下產生的銷售力，可經由非齊次熱傳導方程描繪。然而，我們無法以單一非齊次熱傳導方程描繪所有情況，因此，模型建立與問題解決須於不同情況下逐一地討論。 透過充分的數據，銷售力是可以被預估的；另外，我們也可以利用此模型，對於行銷策略加以評估。 異於以往大部分的研究，此模型加入了空間上的概念，對於傳導現象而言，這是相當重要的。 / Sale intensity is always one of the major subjects that business is concerned about. We propose a mathematical model based on the concept given by Vidale-Wolfe to characterize the behavior of sale intensity. Using the sense of diffusion in heat equation, we could characterize the behavior of sale intensity starting from the spontaneous sale intensity caused by the circulating of information. The behavior of changing on sale intensity under the effect of diffusing by the circulating of information and the promoting activities can be generally modeled as nonhomogeneous heat equations. However, because of the great difference between cases, the problem formulating and model solving cannot be generally modeled as one certain nonhomogeneous heat equation and are restricted to be discussed case by case.% The further sale intensity is predictable possibly with sufficient data, but without sufficient data, we can also use the model to appraise the spontaneous sale intensity and the benefit of each advertising strategy in practical. Different from most previous relevant studies, the model supports the studies of sale intensity diffusing over geographic regions, which is especially of significance in spontaneous sale intensity.

Cette thèse a pour objet l'étude de la contrôlabilité à zéro de systèmes d'équations aux dérivées partielles paraboliques, que l'on rencontre en physique, chimie ou en biologie. En chimie ou en biologie, ces systèmes modélisent l'évolution au cours du temps d'une concentration chimique ou de la densité d'une population (de bactéries, de cellules). Le but de la contrôlabilité à zéro est d'amener la solution du système à l'état nul à un temps donné T, en agissant sur le système à l'aide d'un contrôle. Nous recherchons donc un contrôle, de norme minimale, tel que la solution associée y vérifie y(T)=O dans le domaine Omega considéré. Les problèmes de contrôlabilité à zéro considérés dans cette thèse sont de trois types. Dans un premier temps, nous nous intéressons à la contrôlabilité à zéro avec un nombre fini de contraintes sur la dérivée normale de l'état, pour l'équation de la chaleur semi-linéaire. Puis, nous analysons la contrôlabilité simultanée à zéro avec contrainte sur le contrôle, pour un système linéaire de deux équations paraboliques couplées. Notre dernière étude concerne la contrôlabilité à zéro d'un système non linéaire de deux équations paraboliques couplées. Nous abordons ces problèmes de contrôlabilité principalement à l'aide d'inégalités de Carleman. En effet, l'étude des problèmes de contrôlabilité à zéro, et plus généralement de contrôlabilité exacte, peut se ramener à l'établissement d'inégalités d'observabilité pour le problème adjoint, conséquences d'inégalités de Carleman. Nous construisons le contrôle optimal en utilisant la méthode variationnelle et nous le caractérisons par un système d'optimalité / This thesis is devoted to the study of the null controllability of systems of parabolic partial differential equations, which we meet in physics, chemistry or in biology. In chemistry or in biology, the se systems model the evolution in time of a chemical concentration or the density of a population (of bacteria, cells). The aim of nu Il controllability is to lead the solution of the system to zero in a given time T, by acting on the system with a control. Thus we are looking for a control, of minimal norm, such as the associated solution y satisfies y(T)=O in the domain Omega under concern. We consider three types of null controllability problems in this thesis. At first, we are interested in the null controllability with afinite number of constraints on the normal derivative of the state, for the serni-Iinear heat equation. Then, we analyze the simultaneous null controllability with constraint on the control, for a linear system of two coupled parabolic equations. Our last study deals with the null controllability ofa non linear system oftwo coupled parabolic equations. We approach these controllability problems mainly by means of Carleman's inequalities. Indeed, the study of null controllability problems, and more generally exact controllability problems, is equivalent to obtain observability inequalities for the adjoint problem, consequences of Carleman's inequalities. We build the optimal controlusing the variationnal method and we characterize it by an optimality system

Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-25Bitstream added on 2014-06-13T20:48:30Z : No. of bitstreams: 1 rosa_ra_me_sjrp.pdf: 528158 bytes, checksum: 87eb91b0d9f48ee60092159a596eccf5 (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho vamos estudar a existência e unicidade de solução para equações da forma { u + Au = f(t,u) u(t0)= u0 ∈ X, (I) onde X é um espaço de Banach, A : D(A) ⊂ X → X é um operador linear, f é uma função não linear conhecida, u0 ∈ X é um dado inical conhecido e u : I ⊂ R → X é uma função desconhecida e t0 ∈ I. Faremos este estudo usando a Teoria dos Semigrupos de Operadores Lineares. Para melhor entendimento do estudo das equações (I), faremos duas aplicações. A primeira tratando de um modelo (linear) de divisão celular e a segunda, do modelo (não linear) de condução do calor. / In this work we will study the existence and uniqueness of the solutions for the following equation { u + Au = f(t,u) u(t0)= u0 ∈ X, (I) where X is a Banach space, A : D(A) ⊂ X → X is a linear operator, f is a nonlinear function, u : I ⊂ R → X is unknown function. In this study we will use the theory of semigroup of linear operators. For a best understanding of the study of equations (I), we will do two applications. The first one, is a (linear) model of cellular division and the second one, is about the (nonlinear) model od conduction of the heat.

No abstract available. / Thesis (Ph.D.)-University of Natal, Durban, 1986.