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

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∂ T k =−h( T −T∞) ∂ r at r = R (3.76) Initial condition T=T0 at t=0 (3.77) Dimensionless parameters defined as T −T θ = T −T 0 ∞ ∞ , hR B = k , αt τ = 2 R 32 , r x = R Using equations (3.78) the equation (3.74-3.77) can be written as ( x ) Initial condition Boundary condition SOLUTION PROCEDURE The guess temperature profile is assumed as gR 2 0 G = k T T∞ , ( 0 − ) (3.78) ∂θ 1 ∂ ⎛ ∂θ ⎞ = ⎜x⎟+ G ∂τx ∂x⎝ ∂x ⎠ (3.79) θ ,0 = 1 (3.80) ∂ θ = 0 ∂ x at x = 0 (3.81) ∂ θ = −Bθ ∂ x at x = 1 (3.82) ( ) ( ) ( ) 2 θp= a τ + a τ x+ a τ x 0 1 2 Differentiating the above equation with respect to x we get Applying first boundary condition we have (3.83) ∂θ p = a1+ 2a2x ∂ x (3.84) a1+ 2a2x= 0 (3.85)
Thus, a 1 = 0 (3.86) Applying second boundary condition we have Thus, a + 2a =−Bθ 1 2 a 2 Bθ =− 2 We can also write the second boundary condition as Using equation (3.86), (3.83) and (3.88-3.89) we have Average temperature is expressed as Substituting the value of θ and m we have Integrating the equation (3.91) we have 33 (3.87) (3.88) ∂ θ =− B( a0 + a1+ a2) ∂ x (3.89) ⎛ B ⎞ a0 = θ ⎜1+ ⎟ ⎝ 2 ⎠ (3.90) ( ) 1 1 θ = + ∫ θ m m x dx 0 2 3 ( 0 1 2 ) 1 ∫ 0 (3.91) θ = 2 ax+ ax + ax dx Substituting the value of 0 a , 1 a and 2 a we have ⎛a0a1 a2 ⎞ θ = 2⎜ + + ⎟ ⎝ 2 3 4 ⎠ (3.92) ⎛ B ⎞ θ = θ⎜1+ ⎟ ⎝ 4 ⎠ (3.93)
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: We consider the heat conduction in
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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