Heat Chap04-047 - Chapter 4 Transient Heat Conduction 4-47...

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Chapter 4 Transient Heat Conduction 4-47 A hot dog is dropped into boiling water, and temperature measurements are taken at certain time intervals. The thermal diffusivity and thermal conductivity of the hot dog and the convection heat transfer coefficient are to be determined. Assumptions 1 Heat conduction in the hot dog is one-dimensional since it is long and it has thermal symmetry about the center line. 2 The thermal properties of the hot dog are constant. 3 The heat transfer coefficient is constant and uniform over the entire surface. 4 The Fourier number is τ > 0.2 so that the one-term approximate solutions (or the transient temperature charts) are applicable (this assumption will be verified). Properties The properties of hot dog available are given to be ρ = 980 kg/m 3 and C p = 3900 J/kg. ° C. Analysis ( a ) From Fig. 4-14b we have 15 . 0 1 1 17 . 0 94 59 94 88 = = = = = - - = - - o o o o o hr k Bi r r r r T T T T The Fourier number is determined from Fig. 4-14a to be 20 . 0 47 . 0 94 20 94 59 15 . 0 1 2 = = = - - = - - = = o i o o r t T T T T hr k Bi α τ The thermal diffusivity of the hot dog is determined to be /s m 10 2.017 2 7 - × = = = α → = α s 120 m) 011 . 0 )( 2 . 0 ( 2 . 0 20 . 0 2 2 2 t r r t o o ( b ) The thermal conductivity of the hot dog is determined from C W/m. 0.771 ° = ° × = αρ = - C) J/kg. )(3900 kg/m /s)(980 m 10 017 . 2 ( 3 2 7 p C k ( c ) From part ( a ) we have 15 . 0 1 = = o hr k Bi . Then, m 0.00165 m) 011 . 0 )( 15 . 0 ( 15 . 0 0 = = = r h k Therefore, the heat transfer coefficient is C . W/m 467 2 ° = ° = → = m 0.00165 C W/m. 771 . 0 00165 . 0 h h k 4-34 Water 94 ° C Hot dog
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Chapter 4 Transient Heat Conduction 4-48 Using the data and the answers given in Prob. 4-43, the center and the surface temperatures of the hot dog 4 min after the start of the cooking and the amount of heat transferred to the hot dog are to be determined. Assumptions 1 Heat conduction in the hot dog is one-dimensional since it is long and it has thermal symmetry about the center line. 2 The thermal properties of the hot dog are constant. 3 The heat transfer coefficient is constant and uniform over the entire surface. 4 The Fourier number is τ > 0.2 so that the one-term approximate solutions (or the transient temperature charts) are applicable (this assumption will be verified). Properties The properties of hot dog and the convection heat transfer coefficient are given or obtained in P4-47 to be k = 0.771 W/m. ° C, ρ = 980 kg/m 3 , C p = 3900 J/kg. ° C, α = 2.017 × 10 -7 m 2 /s, and h = 467 W/m 2 . ° C. Analysis The Biot number is 66 . 6 ) C W/m. 771 . 0 ( ) m 011 . 0 )( C . W/m 467 ( 2 = ° ° = = k hr Bi o The constants λ 1 1 and A corresponding to this Biot number are, from Table 4-1, 5357 . 1 and 0785 . 2 1 1 = = A The Fourier number is 2 . 0 4001 . 0 m) 011 . 0 ( s/min) 60 min /s)(4 m 10 017 . 2 ( 2 2 7 2 = × × = = - L t α τ Then the temperature at the center of the hot dog is determined to be C 73.8 ° = → = - - = = = - - = - - 0 0 ) 4001 . 0 ( ) 0785 . 2 ( 1 0 , 2727 . 0 94 20 94 2727 . 0 ) 5357 . 1 ( 2 2 1 T T e e A T T T T i cyl o θ From Table 4-2 we read J 0 =0.2194 corresponding to the constant 1 =2.0785. Then the temperature at the surface of the hot dog becomes C 89.6 ° = → = - - = = = - - - - ) , ( 05982 . 0
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Heat Chap04-047 - Chapter 4 Transient Heat Conduction 4-47...

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