Heat Transfer Methods tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả các lĩnh vực kinh t...
Thermal Aspects of Solar Air Collector 629 [2] Rene Tchinda, A review of the mathematical models for predicting solar air heaters systems, Renewable and Sustainable Energy Reviews 13 (2009) 1734–1759. [3] Perrot, Pierre, A to Z of Thermodynamics, Oxford University Press, Oxford, 1998. [4] Rant, Z., Exergy, a new word for technical available work, Forschung auf dem Gebiete des Ingenieurwesens 22, (1956), pp. 36–37. [5] Gibbs, J. W. ,A method of geometrical representation of thermodynamic properties of substances by means of surfaces: reprinted in Gibbs, Collected Works, ed. W. R. Longley and R. G. Van Name, Transactions of the Connecticut Academy of Arts and Sciences, 2, (1931), pp. 382–404 . [6] Moran, M. J. and Shapiro, H. N., Fundamentals of Engineering Thermodynamics, 6th Edition, 2007. [7] Van Wylen, G.J., Thermodynamics, Wiley, New York, 1991. [8] Wark, J. K., Advanced Thermodynamics for Engineers, McGraw-Hill, New York, 1995. [9] Bejan, A., Advanced Engineering Thermodynamics, 2nd Edition, Wiley, 1997. [10] Saravan , M. Saravan, R and Renganarayanan, S. , Energy and Exergy Analysis of Counter flow Wet Cooling Towers, Thermal Science, 12, (2008), 2, pp. 69-78. [11] Bejan, A., Kearney, D. W., and Kreith, F., Second Law Analysis and Synthesis of Solar Collector Systems, Journal of Solar Energy Engineering, 103, (1981), pp. 23-28. [12] Bejan, A. , Entropy Generation Minimization, New York, CRC press, 1996. [13] Londono-Hurtado, A. and Rivera-Alvarez, A., Maximization of Exergy Output From Volumetric Absorption Solar Collectors, Journal of Solar Energy Engineering , 125, (2003) ,1 , pp. 83-86. [14] Luminosu, I and Fara, L., Thermodynamic analysis of an air solar collector, International Journal of Exergy, 2, (2005), 4, pp. 385-408. [15] Altfeld, K,. Leiner, W., Fiebig, M., Second law optimization of flat-plate solar air heaters Part I: The concept of net exergy flow and the modeling of solar air heaters, Solar Energy 41, (1988), 2, pp. 127-132. [16] Altfeld, K., Leiner, W., Fiebig, M., Second law optimization of flat-plate solar air heaters Part 2: Results of optimization and analysis of sensibility to variations of operating conditions, Solar Energy, 41, (1988),4 , pp. 309-317. [17] Torres-Reyes, E., Navarrete-Gonzàlez, J. J., Zaleta-Aguilar, A., Cervantes-de Gortari, J. G., Optimal process of solar to thermal energy conversion and design of irreversible flat-plate solar collectors, Energy 28, (2003), pp. 99–113. [18] Kurtbas, I., Durmuş, A., Efficiency and exergy analysis of a new solar air heater, Renewable Energy, 29, (2004), pp. 1489-1501. [19] Choudhury C, Chauhan PM, Garg HP. Design curves for conventional solar air heaters. Renewable energy 1995;6(7):739–49. [20] Ong KS. Thermal performance of solar air heaters: mathematical model and solution procedure. Solar Energy 1995;55(2):93–109. [21] Hegazy AA. Thermohydraulic performance of heating solar collectors with variable width, flat absorber plates. Energy Conversion and Management 2000;41:1361–78. [22] Al-Kamil MT, Al-Ghareeb AA. Effect of thermal radiation inside solar air heaters. Energy Conversion and Management 1997;38(14):1451–8. [23] Garg HP, Datta G, Bhargava K. Some studies on the flow passage dimension for solar air testing collector. Energy Conversion and Management 1984;24(3):181–4. Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 630 [24] Forson FK, Nazha MAA, et Rajakaruna H. Experimental and simulation studies on a single pass, double duct solar air heater. Energy Conversion and Management 2003;44:1209–27. [25] Ho CD, Yeh HM, Wang RC. Heat-transfer enhancement in double-pass flatplate solar air heaters with recycle. Energy 2005;30:2796–817. Heat Transfer Methods Heat Transfer Methods Bởi: OpenStaxCollege Equally as interesting as the effects of heat transfer on a system are the methods by which this occurs Whenever there is a temperature difference, heat transfer occurs Heat transfer may occur rapidly, such as through a cooking pan, or slowly, such as through the walls of a picnic ice chest We can control rates of heat transfer by choosing materials (such as thick wool clothing for the winter), controlling air movement (such as the use of weather stripping around doors), or by choice of color (such as a white roof to reflect summer sunlight) So many processes involve heat transfer, so that it is hard to imagine a situation where no heat transfer occurs Yet every process involving heat transfer takes place by only three methods: Conduction is heat transfer through stationary matter by physical contact (The matter is stationary on a macroscopic scale—we know there is thermal motion of the atoms and molecules at any temperature above absolute zero.) Heat transferred between the electric burner of a stove and the bottom of a pan is transferred by conduction Convection is the heat transfer by the macroscopic movement of a fluid This type of transfer takes place in a forced-air furnace and in weather systems, for example Heat transfer by radiation occurs when microwaves, infrared radiation, visible light, or another form of electromagnetic radiation is emitted or absorbed An obvious example is the warming of the Earth by the Sun A less obvious example is thermal radiation from the human body 1/3 Heat Transfer Methods In a fireplace, heat transfer occurs by all three methods: conduction, convection, and radiation Radiation is responsible for most of the heat transferred into the room Heat transfer also occurs through conduction into the room, but at a much slower rate Heat transfer by convection also occurs through cold air entering the room around windows and hot air leaving the room by rising up the chimney We examine these methods in some detail in the three following modules Each method has unique and interesting characteristics, but all three have one thing in common: they transfer heat solely because of a temperature difference [link] Check Your Understanding Name an example from daily life (different from the text) for each mechanism of heat transfer Conduction: Heat transfers into your hands as you hold a hot cup of coffee Convection: Heat transfers as the barista “steams” cold milk to make hot cocoa Radiation: Reheating a cold cup of coffee in a microwave oven Summary • Heat is transferred by three different methods: conduction, convection, and radiation Conceptual Questions What are the main methods of heat transfer from the hot core of Earth to its surface? From Earth’s surface to outer space? 2/3 Heat Transfer Methods When our bodies get too warm, they respond by sweating and increasing blood circulation to the surface to transfer thermal energy away from the core What effect will this have on a person in a 40.0ºC hot tub? [link] shows a cut-away drawing of a thermos bottle (also known as a Dewar flask), which is a device designed specifically to slow down all forms of heat transfer Explain the functions of the various parts, such as the vacuum, the silvering of the walls, the thin-walled long glass neck, the rubber support, the air layer, and the stopper The construction of a thermos bottle is designed to inhibit all methods of heat transfer 3/3 HEAT TRANSFER ͳ MATHEMATICAL MODELLING, NUMERICAL METHODS AND INFORMATION TECHNOLOGY Edited by Aziz Belmiloudi Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Edited by Aziz Belmiloudi Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Iva Lipovic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright Zadiraka Evgenii, 2010. Used under license from Shutterstock.com First published February, 2011 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology, Edited by Aziz Belmiloudi p. cm. ISBN 978-953-307-550-1 free online editions of InTech Books and Journals can be found at www.intechopen.com Part 1 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Part 2 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Preface IX Inverse, Stabilization and Optimization Problems 1 Optimum Fin Profile under Dry and Wet Surface Conditions 3 Balaram Kundu and Somchai Wongwises Thermal Therapy: Stabilization and Identification 33 Aziz Belmiloudi Direct and Inverse Heat Transfer Problems in Dynamics of Plate Fin and Tube Heat Exchangers 77 Dawid Taler Radiative Heat Transfer and Effective Transport Coefficients 101 Thomas Christen, Frank Kassubek, and Rudolf Gati Numerical Methods and Calculations 127 Finite Volume Method Analysis of Heat Transfer in Multi-Block Grid During Solidification 129 Eliseu Monteiro, Regina Almeida and Abel Rouboa Lattice Boltzmann Numerical Approach to Predict Macroscale Thermal Fluid Flow Problem 151 Nor Azwadi Che Sidik and Syahrullail Samion Efficient Simulation of Transient Heat Transfer Problems in Civil Engineering 165 Sebastian Bindick, Benjamin Ahrenholz, Manfred Krafczyk Applications of Nonstandard Finite Difference Methods to Nonlinear Heat Transfer Problems 185 Alaeddin Malek Contents Contents VI Fast BEM Based Methods for Heat Transfer Simulation 209 Jure Ravnik and Leopold Škerget Aerodynamic Heating at Hypersonic Speed 233 Andrey B. Gorshkov Thermoelastic Stresses in FG-Cylinders 253 Mohammad Azadi and Mahboobeh Azadi Experimentally Validated Numerical Modeling of Heat Transfer in Granular Flow in Rotating Vessels 271 Bodhisattwa Chaudhuri, Fernando J. Muzzio and M. Silvina Tomassone Heat Transfer in Mini/Micro Systems 303 Introduction Optimum Fin Profile under Dry and Wet Surface Conditions 29 optimum fins under the volume constraint is less than the surrounding temperature. A significant change in optimum design variables has been noticed with the design constants such as fin volume and surface conditions. In order to reduce the complexcity of the optimum profile fins under different surface conditions, the constraint fin length can be selected suitably with the constraint fin volume. 0.00 0.01 0.02 0.03 0.04 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 A U=0.0001 L=0.05 γ =100% γ =70% θ X Fully dry surface Fully wet surface 0.00 0.01 0.02 0.03 0.04 0.05 0.000 0.001 0.002 0.003 0.004 0.005 B U=0.0001 L=0.05 Y X Fully wet (γ = 100%) Fully wet (γ = 70%) Fully dry Fig. 9. Variation of temperature and fin profile in a longitudinal fin as a function of length for both volume and length constraints: A. Temperature distribution; and B. Fin profile 5. Acknowledgement The authors would like to thank King Mongkut’s University of Technology Thonburi (KMUTT), the Thailand Research Fund, the Office of Higher Education Commission and the National Research University Project for the financial support. 6. Nomenclatures a constant determined from the conditions of humid air at the fin base and fin tip b slop of a saturation line in the psychometric chart, K – 1 C non-dimensional integration constant used in Eq. (84) C p specific heat of humid air, -1 -1 J k g K F functional defined in Eqs. (10), (28), (46), (62), (80) and (96) h convective heat transfer coefficient, 21 W m - K − h m mass transfer coefficient, 21 kg m S h fg latent heat of condensation, 1 J kg - k thermal conductivity of the fin material, 11 W m - K − l fin length, m Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 30 l 0 wet length in partially wet fins, m L dimensionless fin length, hl k L 0 dimensionless wet length in partially wet fins, 0 hl k Le Lewis number q heat transfer rate through a fin, W Q dimensionless heat transfer rate r i base radius for annular fins, m R i dimensionless base radius, i hr k T temperature, K U dimensionless fin volume, see Eqs. (9), (27), (45), (61), (79), (91a) and (95) V fin volume (volume per unit width for longitudinal fins), m 3 x, y coordinates, see Figs. 1 and 2, m X, Y dimensionless coordinates, hx kand hy k , respectively y 0 semi-thickness of a fin at which dry and wet parts separated, m Y 0 dimensionless thickness, 0 hy k Z 1 , Z 2 dimensionless parameters defined in Eqs. (104a) and (104b), respectively Greek Letters α parameter defined in Eqs. (20), (40), (57) and (74a) λ Lagrange multiplier ω specific humidity of air, kg w. v. / kg. d. a. ξ Latent heat parameter φ dimensionless temperature, p θ θ + 0 φ dimensionless temperature at the fin base, 1 p θ + θ dimensionless fin temperature, ( ) ( ) aab TTTT−− p θ dimensionless temperature parameter, see Eq. (5) γ Relative humidity Subscripts a ambient b base d dewpoint opt optimum t tip 7. References Chilton, T.H. & Colburn, A.P. (1934). Mass transfer (absorption) coefficients–prediction from data on heat transfer and fluid friction. Ind. Eng. Chem., Vol. 26, 1183. Duffin, R. J. (1959). A variational problem relating to cooling fins with heat generation. Q. Appl. Math., Vol. 10, 19-29. Guceri, S. & Maday, C. J. (1975). A least weight circular cooling fin. ASME J. Eng. Ind., Vol. 97, 1190-1193. Hanin, L. & Campo, A. (2003). A new minimum volume straight cooling fin taking into account the length of arc. Int. J. Heat Mass Transfer, Vol. 46, Thermal Therap y: Stabilization and Identification 37 Consequently the gradient of J at point (X,Y), in weak sense, is ∂ J ∂X (X, Y)= e (ϕ)V a ˜ u + αn 1 p + a δ(γu + δp − m obs ) n 2 ξ − ˜ Φ , ∂ J ∂Y (X, Y)= ⎛ ⎝ E(ϕ,U 1 , ˜ u) − βm 1 ϕ −(r(ϕ) ˜ u + βm 2 η) −( ˜ u + βm 3 π) ⎞ ⎠ . (72) We can now give the first-order optimality conditions for the robust control problem as follows. The optimal solution (X ∗ ,Y ∗ ) is characterized by (for all (X,Y) ∈U ad ×V ad ) ∂ J ∂X (X ∗ ,Y ∗ ).( X −X ∗ )= Q (e(ϕ ∗ )V ∗ a ˜ u ∗ + αn 1 p ∗ + aδ(M ∗ −m obs ))(p − p ∗ ) dxdt + Σ r (n 2 ξ ∗ − ˜ Φ ∗ )(ξ − ξ ∗ ) dΓdt ≥ 0 ∂ J ∂Y (X ∗ ,Y ∗ ).(Y−Y ∗ )= Ω (E(ϕ ∗ ,U ∗ 1 , ˜ u ∗ ) − βm 1 ϕ ∗ )(ϕ − ϕ ∗ ) dx − Q (r(ϕ ∗ ) ˜ u ∗ + βm 2 η ∗ )(η − η ∗ ) dxdt − Σ ( ˜ u ∗ + βm 3 π ∗ )(π −π ∗ ) dΓdt ≤ 0 where (u ∗ ,θ ∗ ,Ψ ∗ )=F(X ∗ ,Y ∗ ),U ∗ 1 = u ∗ + U, Θ ∗ 1 = Θ + θ ∗ , Φ ∗ 1 = Ψ ∗ + Φ,P ∗ 1 = p ∗ + P, V ∗ a = u ∗ − w a and G ∗ 1 = η ∗ − g, M ∗ (x, t)=γu ∗ + δp ∗ and ( ˜ u ∗ , ˜ θ ∗ , ˜ Φ ∗ )=F ⊥ (X ∗ ,Y ∗ ) is the solution of the adjoint problem (65),(66),(67). Remark 11 We can apply easily our stochastic r obust control approach developed in the section 8 to the problem of coagulation process analyzed in the present section. To help the interested reader with the transition from theory to implementation, we also discuss some optimization strategies in order to solve the robust control problems, by using the adjoint model. 10. Minimax optimization algorithms and conclusion We present algorithms where the descent direction is calculated by using the adjoint variables, particularly by choosing an admissible step size. The descent method is formulated in terms of the continuous variable such is independent of a specific discretization. The methods are valid for the continuous as well as random processes. 10.1 Gradient algorithm The gradient algorithm for the resolution of treated saddle point problems is given by: for k=1, , (iteration index) we denote by (X k ,Y k ) the numerical approximation of the control-disturbance at the kth iteration of the algorithm. (Step1) Initialization: (X 0 ,Y 0 ) (given initial guess). (Step2) Resolution of the direct problem where the source term is (X k ,Y k ),givesF(X k ,Y k ). 69 Thermal Therapy: Stabilization and Identification 38 Heat Tr ansfer (Step3) Resolution of the adjoint problem (based on (X k ,Y k ,F(X k ,Y k )),givesF ⊥ (X k ,Y k ), (Step4) Gradient of J at (X k ,Y k ): (GJ) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c k de f = ∂J ∂X (X k ,Y k ), d k de f = ∂J ∂Y (X k ,Y k ), G k =(c k ,d k ). (Step5) Determine X k+1 : X k+1 = X k −γ k c k , (Step6) Determine Y k+1 : X k+1 = Y k + δ k d k , where 0 < m ≤γ k ,δ ≤ M are the sequences of step lengths. (Step7) If the gradient is sufficiently small: end; else set k : = k + 1andgoto(Step2). Optimal Solution: (X, Y)=(X k ,Y k ). The convergence of the algorithm depends on the second Fr´echet derivative of J (i.e. m, M depend on the second Fr´echet derivative of J) see e.g. (Ciarlet, 1989). In order to obtain an algorithm which is numerically efficient, the best choice of γ k ,δ k will be the result of a line minimization and maximization algorithm, respectively. Otherwise, at each iteration step k of the previous algorithm, we solve the one-dimensional optimization problem of the parameters γ k and δ k : γ k = min λ>0 J(X k −λc k ,Y k ), δ k = min λ>0 J(X k ,Y k + λd k ), (73) To derive an approximation for a pair (γ k ,δ k ) we can use a purely heuristic approach, for example, by taking γ k = min (1,c k −1 ∞ ) and δ k = min (1,d k −1 ∞ ) or by using the linearization of F(X k −λc k ,Y k ) at X k and F(X k ,Y k −λd k ) at Y k by F(X k −λc k ,Y k ) ≈F(X k ,Y k ) −λ ∂ F ∂X (X k ,Y k ).c k , F(X k ,Y k + λd k ) ≈F(X k ,Y k ) −λ ∂ F ∂Y (X k ,Y k ).d k , where ∂ F ∂X (X k ,Y k ).c k = F (X k ,Y k ).(c k ,0) and ∂ F ∂Y (X k ,Y k ).d k = F Radiative Heat Transfer and Effective Transport Coefficients 9 matter. In the sequel we will discuss a few practically relevant closure methods. We will then argue that the preferred closure is given by an entropy production principle. For clarity we will consider the two-moment example; generalization to an arbitrary number of moments is straight-forward. The appropriate number of moments is influenced by the geometry and the optical density of the matter. For symmetric geometries, like plane, cylindrical, or spherical symmetry, less moments are needed than for complex arrangements with shadowing corners, slits, and the same. For optically dense matter, the photons behave diffusive, which can be modelled well by a low number of moments, as will be discussed below. For transparent media, beams, or even several beams that might cross and interpenetrate, may occur, which makes higher order or multipole moments necessary. 4.1 Two-moment example The unknowns are P E , P F ,andΠ, which may be functions of the two moments E and F.For convenience, we will write P E = κ (eff) E (E (eq) − E) , (17) P F = −κ (eff) F F , (18) where we introduced the effective absorption coefficients κ (eff) E and κ (eff) F that are generally functions of E and F. Because the second rank tensor Π depends only on the scalar E and the vector F, by symmetry reason it can be written in the form Π nm = E 1 −χ 2 δ nm + 3χ −1 2 F n F m F 2 , (19) where the variable Eddington factor (VEF) χ is a function of E and F and where δ kl (= 0ifk = l and δ kl = 1ifk = l) is the Kronecker delta. Assuming that the underlying matter is isotropic, κ (eff) E , κ (eff) F ,andχ can be expressed as functions of E and v = F E (20) with F =| F |. Obviously it holds 0 ≤ v ≤ 1, with v = 1 corresponding to a fully directed radiation beam (free streaming limit). According to Pomraning (1982), the additional E-dependence of suggested or derived VEFs often appears via an effective E-dependent single scattering albedo, which equals, e.g. for gray matter, (κE (eq) + σE)/(κ + σ)E. The task of a closure is to determine the effective transport coefficients, i.e., effective mean absorption coefficients κ (eff) E , κ (eff) F ,andtheVEFχ as functions of E and F (or v). This task is of high relevance in various scientific fields, from terrestrial atmosphere physics and astrophysics to engineering plasma physics. 4.2 Exact limits and interpolations In limit cases of strongly opaque and strongly transparent matter, analytical expressions for the effective absorption coefficients are often used, which can be determined in principle from basic gas properties (see, e.g., AbuRomia & Tien (1967) and Fuss & Hamins (2002)). In an optically dense medium radiation behaves diffusive and isotropic, and is near equilibrium with respect to LTE-matter. The effective absorption coefficients are given by the so-called 109 Radiative Heat Transfer and Effective Transport Coefficients 10 Heat Transfer Rosseland average or Rosseland mean (cf. Siegel & Howell (1992)) κ (eff) E = κ ν Ro := ∞ 0 dνν 4 ∂ ν n (eq) ν ∞ 0 dνν 4 κ −1 ν ∂ ν n (eq) ν , (21) where ∂ ν denotes differentiation with respect to frequency, and κ (eff) F = κ ν + σ ν Ro . (22) The Rosseland mean is an average of inverse rates, i.e., of scattering times, and must thus be associated with consecutive processes. A hand-waving explanation is based on the strong mixing between different frequency modes by the many absorption-emission processes in the optically dense medium due to the short photon mean free path. Isotropy of Π implies for the Eddington factor χ = 1/3. Indeed, because ∑ Π kk = E, one has then Π kl = δ kl E/3. With these stipulations, Eqs. (15) and (16) are completely defined and can be solved. In a strongly .. .Heat Transfer Methods In a fireplace, heat transfer occurs by all three methods: conduction, convection, and radiation Radiation is responsible for most of the heat transferred into... mechanism of heat transfer Conduction: Heat transfers into your hands as you hold a hot cup of coffee Convection: Heat transfers as the barista “steams” cold milk to make hot cocoa Radiation: Reheating... microwave oven Summary • Heat is transferred by three different methods: conduction, convection, and radiation Conceptual Questions What are the main methods of heat transfer from the hot core