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

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∂θ ∂ ⎛ ∂θ ⎞ x dx = ⎜x⎟dx ∂τ∂x⎝ ∂x ⎠ 1 m 1 m 0 0 ∫ ∫ Simplifying the above equation we may write Considering the average temperature we may write ∫ 1 0 ∂θ dx = −Bθ ∂τ Substituting the value of θ at equation (3.105) we have Integrating the equation (3.106) we may write as 38 (3.119) ∂ θ = −Bθ ∂ τ (3.120) ∂ θ 3Bθ =− ∂ τ B + 3 (3.121) ∂ θ 3B = − ∂τ θ B + 3 1 1 ∫ ∫ 0 0 Thus by simplifying the above equation we may write θ = exp( −Pτ) Or we may write Where (3.122) ⎛ 3B ⎞ θ = exp⎜− τ ⎟ ⎝ B + 3 ⎠ (3.123) (3.124) 3B P = B + 3 (3.125) Several profiles have been considered for the analysis. The corresponding modified Biot number, P, has been deduced for the analysis and is shown in Table 4.2.
3.7 TRAINSIENT HEAT CONDUCTION IN CYLINDER WITH DIFFERENT PROFILES Fig 3.6: Schematic of cylinder At the previous section we have assumed different profiles for getting the solution for average temperature in terms of time and Biot number for a slab geometry. A cylindrical geometry is also considered for analysis. Heat conduction equation expressed for cylindrical geometry is Boundary conditions are ∂T 1 ∂ ⎛ ∂T ⎞ = α ⎜r⎟ ∂t r ∂r⎝ ∂r ⎠ (3.126) ∂ T = 0 ∂ r at r = 0 (3.127) ∂ T k =−h( T −T∞) ∂ r at r = R (3.128) And initial condition T=T0 at t=0 (3.129) In the derivation of Equation (3.126), it is assumed that thermal conductivity is independent of temperature. If not, temperature dependence must be applied, but the same procedure can be followed. Dimensionless parameters defined as 39
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 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: Differentiating the above equation
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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