analysis of transient heat conduction in different geometries - ethesis ...

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a difficult one to materialize it physically. The heat flux across the boundaries is specified to be a constant or a function of space and/or time. The mathematical description of this condition may be given in the light of Kirchhoff’s current law; that is the algebraic sum of heat fluxes at a boundary must be equal to zero. Here after the sign is to be assuming positive for the heat flux to the boundary and negative for that from the boundary. Thus remembering the Fourier’s law as heat flux is independent of the actual temperature distribution, and selecting the direction of heat flux conveniently such that it becomes positive. A special case of no heat flux (insulation) from previous one is obtained by inserting heat flux as zero. When the heat transfer across the boundaries, of a continuum cannot be prescribed, it may be assumed to be proportional to the temperature difference between the boundaries and the ambient. Which we may call as Newton’s cooling Law. The importance of radiation relative to convection depends to a large extent, on the temperature level. Radiation increases rapidly with increasing temperature. Even at room temperature, however, for low rates of convection to air, radiation may contribute up to fifty percent of the total heat transfer. Prescribed heat flux involves in any body surrounded by the atmosphere, capable of receiving radiant heat, and near a radiant source (a light bulb or a sun lamp) or exposed to the sun exemplifies the forgoing boundary condition. Interface of two continuums of different conductivities which are called composite walls and insulated tubes have a common boundary, the heat flux across this boundary are elevated from both continua, regardless of the direction of normal. A second condition may be specified along this boundary relating the temperature of the two continua. When two solid continua in contact, one moving relative to other, we say Interface of two continua in relative motion. The friction brake is an important practical case of the forgoing boundary conditions. When part of a continuum has temperature below the temperature at which the continuum changes from one phase to another by virtue of the lubrication or absorption of heat, there exist a moving boundary between the two phase. For problems in this category, the way in which the boundary moves has to be determined together with the temperature variation in the continuum. After formulating the governing equation and boundary conditions, we have converted the parameters to dimensionless values. Based on this application various approximation methods is employed for solutions. 8
1.8 OBJECTIVE OF PRESENT WORK 1. An effort will be made to predict the temperature field in solid by employing a polynomial approximation method. 2. Effort will be made analyze more practical case such as heat generation in solid and specified heat flux at the solid surface is investigated. 3. Effort will be made to obtain new functional parameters that affect the transient heat transfer process. 4. It is tried to consider various geometries for the analysis. 1.9 LAYOUT OF THE REPORT Chapter 2 discuss with the literature review of different types of problems. Chapter 3 deals with the theoretical solution of different heat conduction problems (slab/tube) by employing polynomial approximation method. Chapter 4 reports the result and discussion obtained from the present theoretical analysis. Chapter 5 discuss with the conclusion and scope of future work. 9
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 and 16: direction. When no single and domin
Page 17 and 18: though the atmosphere has already b
Page 19: higher. When Biot number is more th
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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