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is one-dimensional since there is thermal symmetry about the centerline and no significant variation in the axial direction. 3 Thermal properties are constant. Properties The thermal conductivity of cast iron is given to be k = 52 W/m\u22c5°C. Analysis Using water properties at room temperature, the mass flow rate of water and rate of heat transfer from the water are determined to be [ ] W13,296=C)6770)(CJ/kg. kg/s)(4180 06.1( kg/s 06.1m4/(0.03)m/s) )(1.5kg/m 1000( 223 °\u2212°=\u394= ==== TcmQ VAm p cc && && \u3c0\u3c1\u3c1V The thermal resistances for convection in the pipe and the pipe itself are C/W 001768.0 m)]15((0.03)C)[. W/m400( 11 C/W 000031.0 )m 15(C) W/m.52(2 )5.1/75.1ln( 2 )/ln( 22iconv, 12 pipe °=°== °=°= = \u3c0 \u3c0 \u3c0 ii Ah R kL rr R Rconv ,i Rpipe Rcombined ,o T\u221e1 T\u221e2 Using arithmetic mean temperature (70+67)/2 = 68.5°C for water, the heat transfer can be expressed as o aveaveave Ah RR TT RRR TT R TT Q combined pipeiconv, 2,1, ocombined,pipeiconv, 2,1, total 2,1, 1++ \u2212=++ \u2212=\u2212= \u221e\u221e\u221e\u221e\u221e\u221e& Substituting, 2 combined )]m(0.035)(15[ 1+C/W) 001768.0(+C/W) 000031.0( C)155.68( W296,13 \u3c0h°° °\u2212= Solving for the combined heat transfer coefficient gives C. W/m272.5 2 °=combinedh PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission. 3-106 3-163 An 10-m long section of a steam pipe exposed to the ambient is to be insulated to reduce the heat loss through that section of the pipe by 90 percent. The amount of heat loss from the steam in 10 h and the amount of saved per year by insulating the steam pipe. Tair =8°C Ts =82°C Steam pipe Assumptions 1 Heat transfer through the pipe is steady and one-dimensional. 2 Thermal conductivities are constant. 3 The furnace operates continuously. 4 The given heat transfer coefficients accounts for the radiation effects. 5 The temperatures of the pipe surface and the surroundings are representative of annual average during operating hours. 6 The plant operates 110 days a year. Analysis The rate of heat transfer for the uninsulated case is 2 o m 77.3m) 10(m) 12.0( === \u3c0\u3c0 LDA o W9764C)882)(m 77.3)(C. W/m35()( 22 =°\u2212°=\u2212= airso TThAQ& The amount of heat loss during a 10-hour period is day)(per )s 3600kJ/s)(10 764.9( kJ 103.515 5×=×=\u394= tQQ & The steam generator has an efficiency of 85%, and steam heating is used for 110 days a year. Then the amount of natural gas consumed per year and its cost are therms/yr2.431days/yr) 110( kJ 105,500 therm1 0.85 kJ 10515.3used Fuel 5 =\u239f\u239f\u23a0 \u239e\u239c\u239c\u239d \u239b×= $517.4/yr=erm))($1.20/th therms/yr2.431( fuel) ofcost fuel)(Unit ofAmount (fuel ofCost = = Then the money saved by reducing the heat loss by 90% by insulation becomes $466=$517.4/yr0.9fuel) of(Cost 0.9=savedMoney ×=× 3-164 A multilayer circuit board dissipating 27 W of heat consists of 4 layers of copper and 3 layers of epoxy glass sandwiched together. The circuit board is attached to a heat sink from both ends maintained at 35°C. The magnitude and location of the maximum temperature that occurs in the board is to be determined. Assumptions 1 Steady operating conditions exist. 2 Heat transfer can be approximated as being one- dimensional. 3 Thermal conductivities are constant. 4 Heat is generated uniformly in the epoxy layers of the board. 5 Heat transfer from the top and bottom surfaces of the board is negligible. 6 The thermal contact resistances at the copper-epoxy interfaces are negligible. Properties The thermal conductivities are given to be k = 386 W/m\u22c5°C for copper layers and k = 0.26 W/m\u22c5°C for epoxy glass boards. Analysis The effective conductivity of the multilayer circuit board is first determined to be C W/m.48.58 m)0015.0(3)0002.0(4[ CW/)00117.03088.0()()( C W/00117.0)]m 0015.0)(C W/m.26.0[(3)( C W/3088.0)]m 0002.0)(C W/m.386[(4)( epoxycopper epoxycopper epoxy copper °=+ °+=+ += °=°= °=°= tt ktkt k kt kt eff Copper Epoxy The maximum temperature will occur at the midplane of the board that is the farthest to the heat sink. Its value is C56.8°=°+°=+== \u2212= =+= )m 000954.0)(C W/m.48.58( )m 2/18.0)( W2/27( C35 )( m 000954.0)]0015.0(3)0002.0(4[18.0 2 eff 21max 21 eff 2 Ak LQTTT TT L Ak Q A & & PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission. 3-107 3-165 The plumbing system of a house involves some section of a plastic pipe exposed to the ambient air. The pipe is initially filled with stationary water at 0°C. It is to be determined if the water in the pipe will completely freeze during a cold night. Assumptions 1 Heat transfer is transient, but can be treated as steady since the water temperature remains constant during freezing. 2 Heat transfer is one-dimensional since there is thermal symmetry about the centerline and no variation in the axial direction. 3 Thermal properties of water are constant. 4 The water in the pipe is stationary, and its initial temperature is 0°C. 5 The convection resistance inside the pipe is negligible so that the inner surface temperature of the pipe is 0°C. Properties The thermal conductivity of the pipe is given to be k = 0.16 W/m\u22c5°C. The density and latent heat of fusion of water at 0°C are \u3c1 = 1000 kg/m3 and hif = 333.7 kJ/kg (Table A-9). Analysis We assume the inner surface of the pipe to be at 0°C at all times. The thermal resistances involved and the rate of heat transfer are Water pipe Tair = -5°C Soil C/W 0258.16631.03627.0 C/W 6631.0 m)] 5.0)(m 024.0(C)[. W/m40( 11 C/W 3627.0 )m 5.0(C) W/m.16.0(2 )1/2.1ln( 2 )/ln( oconv,pipetotal 2oconv, 12 pipe °=+=+= °=°== °=°== RRR Ah R kL rr R o \u3c0 \u3c0\u3c0 W874.4 C/W 1.0258 C)]5(0[ total 21 =° °\u2212\u2212=\u2212= \u221e R TT Q s& The total amount of heat lost by the water during a 14-h period that night is kJ 7.245)s 3600J/s)(14 874.4( =×=\u394= tQQ & The amount of heat required to freeze the water in the pipe completely is kg 157.0)m 5.0()m 01.0()kg/m 1000( 232 ==== \u3c0\u3c1\u3c0\u3c1 Lrm V kJ 4.52kJ/kg) 7.333kg)( 157.0( === fgmhQ The water in the pipe will freeze completely that night since the amount heat loss is greater than the amount it takes to freeze the water completely . )4.527.245( > PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission. 3-108 3-166 The plumbing system of a house involves some section of a plastic pipe exposed to the ambient air. The pipe is initially filled with stationary water at 0°C. It is to be determined if the water in the pipe will completely freeze during a cold night. Assumptions 1 Heat transfer is transient, but can be treated as steady since the water temperature remains constant during freezing. 2 Heat transfer is one-dimensional since there is thermal symmetry about the centerline and no variation in the axial direction. 3 Thermal properties of water are constant. 4 The water in the pipe is stationary, and its initial temperature is 0°C. 5 The convection resistance inside the pipe is negligible so that the inner surface temperature of the pipe is 0°C. Properties The thermal conductivity of the pipe is given to be k = 0.16 W/m\u22c5°C. The density and latent heat of fusion of water at 0°C are \u3c1 = 1000 kg/m3 and hif = 333.7 kJ/kg (Table A-9). Analysis We assume the inner surface of the pipe to be at 0°C at all times. The thermal resistances involved and the rate of heat transfer are C/W 0153.36526.23627.0 C/W 6526.2 m)] 5.0)(m 024.0(C)[. W/m10( 11 C/W 3627.0 )m 5.0(C) W/m.16.0(2 )1/2.1ln(