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Tutorial_4 - 1 To warm up some milk for a baby a mother...

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1. To warm up some milk for a baby, a mother pours milk into a thin-walled glass whose diameter is 6cm. The height of the milk in the glass is 7cm. She then places the glass into a large pan filled with hot water at 70ºC. The milk is stirred constantly, so that its temperature is uniform at all times. If the heat transfer coefficient between the water and the glass is 120 W/m 2 ·K, determine how long it will take for the milk to warm up from 3ºC to 38ºC. Take the properties of the milk to be the same as those of water. Can the milk in this case be treated as a lumped system? Why? Solution Assumptions 1 The glass container is cylindrical in shape with a radius of r 0 = 3 cm. 2 The thermal properties of the milk are taken to be the same as those of water. 3 Thermal properties of the milk are constant at room temperature. 4 The heat transfer coefficient is constant and uniform over the entire surface. 5 The Biot number in this case is large (much larger than 0.1). However, the lumped system analysis is still applicable since the milk is stirred constantly, so that its temperature remains uniform at all times. Properties The thermal conductivity, density, and specific heat of the milk at 20 ° C are k = 0.598 W/m. ° C, ρ = 998 kg/m 3 , and c p = 4.182 kJ/kg. ° C (Table A-9). Analysis The characteristic length and Biot number for the glass of milk are 1 . 0 > 21 . 4 ) C W/m. 598 . 0 ( ) m 0105 . 0 )( C . W/m 240 ( m 01050 . 0 m) 03 . 0 ( 2 + m) m)(0.07 03 . 0 ( 2 m) 07 . 0 ( m) 03 . 0 ( 2 2 2 2 2 2 2 = ° ° = = = = + = = k hL Bi r L r L r A L c o o o s c π π π π π π V For the reason explained above we can use the lumped system analysis to determine how long it will take for the milk to warm up to 38 ° C: min 2.25 s 135 = = → = → = = ° ° = = = t e e T T T t T L c h c hA b t bt i c p p s ) s 005477 . 0 ( 1 - 3 2 1 - 70 3 70 38 ) ( s 005477 . 0 m) C)(0.0105 J/kg. 4182 )( kg/m (998 C . W/m 240 ρ ρ V Therefore, it will take 135 s to warm the milk from 3 to 38 ° C. Water 70 ° C Milk 3 ° C
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2. A long 35-cm-diameter cylindrical shaft made of stainless steel 304 ( k = 14.9 W/m·K, ρ = 7900 kg/m 3 , c p = 477 J/kg·K, and α = 3.95 x 10 -6 m 2 /s) comes out of an oven at a uniform temperature of 500deg. The shaft is then allowed to cool slowly in a chamber at 150ºC with an average convection heat transfer coefficient of h = 60 W/m 2 ·K. Determine the temperature at the center of the shaft 20min after the start of the cooling process. Also determine the heat transfer per unit length of the shaft during this time period.
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