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egion radius. At a spherical geometry to have one-dimensional analysis a uniform condition is applied to each concentric surface which bounds the region. 1.4.2 Steady and unsteady analysis Steady state analysis A steady-state thermal analysis predicts the effects of steady thermal loads on a system. A system is said to attain steady state when variation of various parameters namely, temperature, pressure and density vary with time. A steady-state analysis also can be considered the last step of a transient thermal analysis. We can use steady-state thermal analysis to determine temperatures, thermal gradients, heat flow rates, and heat fluxes in an object which do not vary over time. A steady-state thermal analysis may be either linear, by assuming constant material properties or can be nonlinear case, with material properties varying with temperature. The thermal properties of most material do vary with temperature, so the analysis becomes nonlinear. Furthermore, by considering radiation effects system also become nonlinear. Unsteady state analysis Before a steady state condition is reached, certain amount of time is elapsed after the heat transfer process is initiated to allow the transient conditions to disappear. For instance while determining the rate of heat flow through wall, we do not consider the period during which the furnace starts up and the temperature of the interior, as well as those of the walls, gradually increase. We usually assume that this period of transition has passed and that steady-state condition has been established. In the temperature distribution in an electrically heated wire, we usually neglect warming upperiod. Yet we know that when we turn on a toaster, it takes some time before the resistance wires attain maximum temperature, although heat generation starts instantaneously when the current begins to flow. Another type of unsteady-heat-flow problem involves with periodic variations of temperature and heat flow. Periodic heat flow occurs in internal-combustion engines, air-conditioning, instrumentation, and process control. For example the temperature inside stone buildings remains quite higher for several hours after sunset. In the morning, even 4
though the atmosphere has already become warm, the air inside the buildings will remain comfortably cool for several hours. The reason for this phenomenon is the existence of a time lag before temperature equilibrium between the inside of the building and the outdoor temperature. Another typical example is the periodic heat flow through the walls of engines where temperature increases only during a portion of their cycle of operation. When the engine warms up and operates in the steady state, the temperature at any point in the wall undergoes cycle variation with time. While the engine is warming up, a transient heat-flow phenomenon is considered on the cyclic variations. 1.4.3 One Dimensional unsteady analysis In case of unsteady analysis the temperature field depends upon time. Depending on conditions the analysis can be one-dimensional, two dimensional or three dimensional. One dimensional unsteady heat transfer is found at a solid fuel rocket nozzles, in reentry heat shields, in reactor components, and in combustion devices. The consideration may relate to temperature limitation of materials, to heat transfer characteristics, or to the thermal stressing of materials, which may accompany changing temperature distributions. 1.5 DESCRIPTION OF ANALYTICAL METHOD AND NUMERICAL METHOD In general, we employ either an analytical method or numerical method to solve steady or transient conduction equation valid for various dimensions (1D/2D). Numerical technique generally used is finite difference, finite element, relaxation method etc. The most of the practical two dimensional heat problems involving irregular geometries is solved by numerical techniques. The main advantage of numerical methods is it can be applied to any twodimensional shape irrespective of its complexity or boundary condition. The numerical analysis, due to widespread use of digital computers these days, is the primary method of solving complex heat transfer problems. The heat conduction problems depending upon the various parameters can be obtained through analytical solution. An analytical method uses Laplace equation for solving the heat conduction problems. Heat balance integral method, hermite-type approximation method, polynomial approximation method, wiener–Hopf Technique are few examples of analytical method. 5
Page 1 and 2: ANALYSIS OF TRANSIENT HEAT CONDUCTI
Page 3 and 4: National Institute of Technology Ro
Page 5 and 6: CONTENTS Abstract i List of Figures
Page 7 and 8: ABSTRACT Present work deals with th
Page 9 and 10: Fig4.6 Average dimensionless temper
Page 11 and 12: NOMENCLTURE B Biot Number k Thermal
Page 13 and 14: 1.1 GENERAL BACKGROUND CHAPTER 1 IN
Page 15: direction. When no single and domin
Page 19 and 20: higher. When Biot number is more th
Page 21 and 22: 1.8 OBJECTIVE OF PRESENT WORK 1. An
Page 23 and 24: 2.1 INTRODUCTION CHAPTER 2 LITERATU
Page 25 and 26: A. G. Ostrogorsky [5] has used Lapl
Page 27 and 28: linear polynomial pieces which are
Page 29 and 30: error. In this case the approximate
Page 31 and 32: systems based on the flux formulati
Page 33 and 34: CHAPTER 3 THEORTICAL ANALYSIS OF CO
Page 35 and 36: equation. This can then be converte
Page 37 and 38: Based on the analysis a closed form
Page 39 and 40: a + 2a =−Bθ 1 2 Subtracting the
Page 41 and 42: ∂T 1 ∂ ⎛ m ∂T ⎞ = α r G
Page 43 and 44: Thus, a 2 Bθ =− 2 We can also wr
Page 45 and 46: We consider the heat conduction in
Page 47 and 48: Thus, a 1 = 0 (3.86) Applying secon
Page 49 and 50: θ = − τU e + V U ⎛ 2B ⎞ U =
Page 51 and 52: Differentiating the above equation
Page 53 and 54: 3.7 TRAINSIENT HEAT CONDUCTION IN C
Page 55 and 56: Thus a + 2a =−Bθ 1 2 a 2 Bθ =
Page 57 and 58: CHAPTER 4 RESULT AND DISCUSSION
Page 59 and 60: Dimensionless temperature (θ) 1000
Page 61 and 62: 4.2 HEAT GENERATION FOR BOTH SLAB A
Page 63 and 64: Dimensionless temperature (θ) 1.1
Page 65 and 66: Dimensionless temperature 1.0 0.8 0
Page 67 and 68:
Table 4.2 Comparison of modified Bi
Page 69 and 70:
5.1 CONCLUSIONS CHAPTER 5 CONCLUSIO
Page 71 and 72:
CHAPTER 6 REFRENCES
Page 73 and 74:
12. O¨. Ercan Ataer, “An approxi
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