Chapter 7: 17, 20, 24, 25, 32, 35, 37, 40, 47, 66 and 79.

Transcription

1 hapter 7: 17, 0,, 5,, 5, 7, 0, 7, 66 and A power tranitor mounted on the wall diipate 0.18 W. he urface temperature of the tranitor i to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. Any heat tranfer from the bae urface i diregarded. he local atmopheric preure i 1 atm. 5 propertie are evaluated at 0. Propertie he propertie of air at 1 atm and the anticipated film temperature of 0 7 K are (able A1 k 0.01 W/m β K m / K Analyi he olution of thi problem require a trial-and-error approach ince the determination of the Rayleigh number and thu the Nuelt number depend on the urface temperature which i unknown. We tart the olution proce by gueing the urface temperature to be 150 for the evaluation of h. We will check the accuracy of thi gue later and repeat the calculation if neceary. he tranitor loe heat through it cylindrical urface a well a it top urface. For convenience, we take the heat tranfer coefficient at the top urface of the tranitor to be the ame a that of it ide urface. (he alternative i to treat the top urface a a vertical plate, but thi will double the amount of calculation without providing much improvement in accuracy ince the area of the top urface i much maller and it i circular in hape intead of being rectangular. he characteritic length in thi cae i the outer diameter of the tranitor, D m. hen, and gβ ( Nu (9.81 m/ 0.87Ra 1/ 6 ( K ( (150 5 K(0.00 m [ 1 ( / Pr ] 8 / 7 9 / ( / k 0.01 W/m. h Nu ( W/m m A πdl + πd ha( 0.15 W / m π (0.00 m(0.005 m + π (0.00 m + εaσ ( (15.8 W/m. ( m ( (0.1( m (5.67 urr / 9 /16 [ ] 0.87(78 / 1/ 6 [( (5 + 7 K ] 8 / 7 ( m.0 Power tranitor, 0.18 W D 0. cm ε which i lower that 0.18 W. herefore, the urface temperature mut be higher. By trying other temperature, the urface temperature that reult in a total heat tranfer rate of 0.18 W i determined to be 17 Dicuion he film temperature in thi cae i ( + / (17+5/ 99.5, which i very cloe to the value of 0 ued in the evaluation of the propertie. herefore, there i no need to repeat calculation in thi cae.

2 7-0 Water i boiling in a pan that i placed on top of a tove. he rate of heat lo from the cylindrical ide urface of the pan by natural convection and radiation and the ratio of heat lot from the ide urface of the pan to that by the evaporation of water are to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he local atmopheric preure i 1 atm. Propertie he propertie of air at 1 atm and the film temperature of ( + / (98+5/ K are (able A1 k W/m m / K. 5 K Analyi (a he characteritic length in thi cae i the height of the pan, L 01. m. hen, gβ( ( 9. 8 m/ ( K ( 98 5 K ( 01. m 6 ( (. 191 m / We can treat thi vertical cylinder a a vertical plate ince 5L 5(. 01 5L 007. < 0.5 and thu D 1 6 Gr / ( 717. / / Gr / herefore, 1 / 6 1 / Nu ( k W/m. h Nu ( W/m. 01. m A πdl π(. 0 5 m (. 01 m m and Q ha( ( 7. W/m. ( m ( W (b he radiation heat lo from the pan i Qrad εaσ( urr (. (. ( m 67 W/m.K ( K ( 5+ 7 K 56.1 W (c he heat lo by the evaporation of water i ( / 600 kg / ( 57 kj / kg 15. kw 15 W mh if 5 Vapor kg/h Water 0 Pan 98 ε 0.95 hen the ratio of the heat lot from the ide urface of the pan to that by the evaporation of water then become f % 15

3 7- A printed circuit board (PB i placed in a room. he average temperature of the hot urface of the board i to be determined for different orientation. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he local atmopheric preure i 1 atm. he heat lo from the back urface of the board i negligible. Propertie he propertie of air at 1 atm and the anticipated film temperature of ( + / (5+0/ K are k W/m. 16. m / K 055. K Analyi he olution of thi problem require a trial-and-error approach ince the determination of the Rayleigh number and thu the Nuelt number depend on the urface temperature which i unknown (a Vertical PB. We tart the olution proce by gueing the urface temperature to be 5 for the evaluation of the propertie and h. We will check the accuracy of thi gue later and repeat the calculation if neceary. he characteritic length in thi cae i the height of the PB, L 0. m. hen, gβ( ( 9. 8 m/ ( K ( 5 0 K ( 0. m 7 ( (. 16 m / 1 / 7 1 / Nu ( k W/m. h Nu ( W/m. 0. m A (. 015 m (. 0 m 00. m Heat lo by both natural convection and radiation heat can be expreed a Q ha( + εaσ( urr 8 W (. 51 W / m. (. 0 0 m ( K + (.( m (. 567 ( K L 0. m Inulation PB, 8 W 0 8 It olution i 19 K 6 which i ufficiently cloe to the aumed value of 5 for the evaluation of the propertie and h. (b Horizontal, hot urface facing up Again we aume the urface temperature to be 5 and ue the propertie evaluated above. he characteritic length in thi cae i A (. 00m (. 015m 009. m. P 0 (. m m hen, gβ( ( 9. 8 m/ ( K ( 5 0 K ( m 5 ( (. 16 m / 1 / 5 1 / Nu ( k W/m. h Nu ( W/m m Prob. 7- (continued Heat lo by both natural convection and radiation heat can be expreed a

4 ha( + εaσ ( 8 W (6.8 W/m. (0.0 m [ It olution i urr (0 + 7 K] + (0.8(0.0 m (5.67 [ (0 + 7 K ] 15 K which i ufficiently cloe to the aumed value of 5 in the evaluation of the propertie and h. (c Horizontal, hot urface facing down hi time we expect the urface temperature to be higher, and aume the urface temperature to be 50. We will check thi aumption after obtaining reult and repeat calculation with a better aumption, if neceary. he propertie of air at the film temperature of f K are k W/m m / K 08 K he characteritic length in thi cae i, from part (b, 0.09 m. hen, gβ( ( 9. 8 m/ ( K (50 0 K ( m (. 167 m / 1 / 1 / Nu 0. 7Ra 0. 7( 191, k W/m. h Nu ( W/m m onidering both natural convection and radiation heat loe ha( + εaσ ( (.5 W/m. (0.0 m [ It olution i urr (0 + 7 K] + (0.8(0.0 m (5.67 K 50 which i identical to the aumed value. herefore, there i no need to repeat calculation. [ ( , 879 (0 + 7 K ] 8 W

5 7-5 Aborber plate whoe back ide i heavily inulated i placed horizontally outdoor. Solar radiation i incident on the plate. he equilibrium temperature of the plate i to be determined for two cae. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he local atmopheric preure i 1 atm. Propertie he propertie of air at 1 atm and the anticipated film 700 W/m temperature of ( + / (115+5/ 70 K are k W/m. Aborber plate 198. m / α ε L 1. m K K Inulation Analyi he olution of thi problem require a trial-and-error approach ince the determination of the Rayleigh number and thu the Nuelt number depend on the urface temperature which i unknown. We tart the olution proce by gueing the urface temperature to be 115 for the evaluation of the propertie and h. We will check the accuracy of thi gue later and repeat the calculation if neceary. A (. 1 m ( 08. m he characteritic length in thi cae i 0. m. hen, P 1 (. m m gβ( ( 9. 8 m/ ( K ( K ( 0. m (. 198 m / 1 / 7 1 / Nu ( k W/m. h Nu ( W/m. 0. m A (. 08 m (. 1 m 096. m In teady operation, the heat gain by the plate by aborption of olar radiation mut be equal to the heat lo by natural convection and radiation. herefore, (. ( Q qa (. α W/m 096 m 585W ha( + εaσ ( urr 585 W (5.9 W/m. (0.96 m [ (5 + 7 K] + (0.09(0.96 m (5.67 It olution i 88 K 115 ( [ 7 (5 + 7 K which i identical to the aumed value. herefore there i no need to repeat calculation. If the aborber plate i made of ordinary aluminum which ha a olar aborptivity of 0.8 and an emiivity of 0.07, the rate of olar gain become (. ( (. Q αqa W/m 096m 188. W Again noting that in teady operation the heat gain by the plate by aborption of olar radiation mut be equal to the heat lo by natural convection and radiation, and uing the convection coefficient determined above for convenience (actually, we hould calculate the new h uing data at a lower temperature, and iterating if neceary for better accuracy, ha( + εaσ ( 188. W (5.9 W/m. (0.96 m [ It olution i 9 K 56 urr (5 + 7 K] + (0.07(0.96 m (5.67 [ (5 + 7 K Dicuion If we recalculated the h uing propertie at 5, we would obtain h 5.9 W/m. and 61. ] ]

6 7- An inulated electric wire i expoed to calm air. he temperature at the interface of the wire and the platic inulation i to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he local atmopheric preure i 1 atm. Propertie he propertie of air at 1 atm and the anticipated film temperature of ( + / (+0/ 7 K are (able A1 k W/m m / K K Analyi he olution of thi problem require a trial-and-error approach ince the determination of the Rayleigh number and thu the Nuelt number depend on the urface temperature which i unknown. We tart the olution proce by gueing the urface temperature to be for the evaluation of the propertie and h. We will check the accuracy of thi gue later and repeat the calculation if neceary. he characteritic length in thi cae i the outer diameter of the inulated wire D ( + mm m. hen, gβ( ( 9. 8 m/ ( K ( 0 K ( m (. 0 7 (. 167 m / 1/ Ra 0.87( Nu k W/m. h Nu ( W/m m A πdl π( m(1 m 0. 6 m [ 1 ( / Pr ] 8 / 7 9 / ( / 0.7 1/ 6 9 /16 [ ] 8 / he rate of heat generation, and thu the rate of heat tranfer i Q VI ( 8 V( A 80 W onidering both natural convection and radiation, the total rate of heat lo can be expreed a ha( + εaσ ( 80 W (8.77 W/m. (0.6 m [.9 ( 6] + (0.9(0.6 m (5.67 urr W/m.K [ (0 + 7 K It olution i 6 K 5 which i cloe to the aumed value of. o improve the reult, we repeat the calculation for a urface temperature of 50. It give 5. herefore, we can take urface temperature to be (50+5/ 51. hen the temperature at the interface of the wire and the platic cover in teady operation become πkl ln( / ( ln( / ( ln( / Q D D Q D D1 80 W 6 i i πkl π(. 015 W/m. ( 1 m Reitance heater ] L 1 m ε 0.9 D 6 mm

7 7-5 A circuit board containing quare chip i mounted on a vertical wall in a room. he urface temperature of the chip i to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he local atmopheric preure i 1 atm. he heat tranfer from the back ide of the circuit board i negligible. Propertie he propertie of air at 1 atm and the anticipated film temperature of ( + / (5+5/ 0 0 K are (able A1 k W/m m / K 0 K Analyi he olution of thi problem require a trial-and-error approach ince the determination of the Rayleigh number and thu the Nuelt number depend on the urface temperature which i unknown. We tart the olution proce by gueing the urface temperature to be 5 for the evaluation of the propertie and h. We will check the accuracy of thi gue later and repeat the calculation if neceary. he characteritic length in thi cae i the height of the board, L 0. m. hen, gβ( ( 98. m/ ( 000. K ( 5 5K ( 0. m 7 ( (. 160 m / 1 / 7 1 / Nu (. 1. k W/m. h Nu ( W/m. 0. m A (. 0m 009. m onidering both natural convection and radiation, the total rate of heat lo can be expreed a ha( + εaσ ( ( W (.61 W/m. (0.09 m ( + (0.7(0.09 m (5.67 urr 5 W/m.K [ L 0 cm (5 + 7 K It olution i 06. K. which i ufficiently cloe to the aumed value in the evaluation of propertie and h. herefore, there i no need to repeat calculation by reevaluating the propertie and h at the new film temperature. ] PB, ε W 5 urr 5

8 7-7 It i propoed that the ide urface of a cubic indutrial furnace be inulated for $550 in order to reduce the heat lo by 90 percent. he thickne of the inulation and the payback period of the inulation to pay for itelf from the energy it ave are to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he local atmopheric preure i 1 atm. Propertie he propertie of air at 1 atm and the film temperature of ( + / (1+0/ 70 K are (able A1 k W/m m / K K Analyi he characteritic length in thi cae i the height of the furnace, L m. hen, gβ( ( 9. 8 m/ ( K ( 1 0 K ( m ( (. 199 m / 1/ 1/ Nu 0.1Ra 0.1(.6 19 k 0.09 W/m. h Nu (19.66 W/m. m A ( m 16 m hen the heat lo by combined natural convection and radiation become ha( + εaσ ( (.66 W/m. (16 m (1 5 + (0.7(16 m (5.67 urr W/m.K [(1 + 7 K (0 + 7 K 1. W 1. kw Noting that inulation will reduce the heat loe by 90%, the rate of heat lo after inulation will be aved 0.9 no inulation kw 1.9 kw ( kw 1. kw lo no inulation he furnace operate continuouly and thu 8760 h. hen the amount of energy and money the inulation will ave become 1.9 kj/ 1 therm Energy aved aved t ( /yr 9 therm/yr ,500 kj Money aved (Energy aved(unit cot of energy (9 therm($0.55 / therm $719 herefore, the money aved by inulation will pay for the cot of $550 in $550/($719/yr0.0 yr 7 day. m m Furnace 1 ε 0.7 ] Hot gae 0

9 Prob. 7-7 (continued Inulation will lower the outer urface temperature, the Rayleigh and Nuelt number, and thu the convection heat tranfer coefficient. For the evaluation of the heat tranfer coefficient, we aume the urface temperature in thi cae to be 50. he propertie of air at the film temperature of f 5 98 K are k W/m m / K 98 K hen, gβ( ( 9. 8 m/ ( K (50 0 K ( m ( (. 156 m / 1/ 1/ Nu 0.1Ra 0.1(1.5 9 k W/m. h Nu (9. W/m. m A ( m( + t inul m he total rate of heat lo from the outer urface of the inulated furnace by convection and radiation become conv + rad ha( + εaσ ( urr 18 W (. W/m. A( 98 + (0.7 A(5.67 W/m.K [ (98 K ] In teady operation, the heat lot by the ide urface of the pipe mut be equal to the heat lot from the expoed urface of the inulation by convection and radiation, which mut be equal to the heat conducted through the inulation. herefore, ( ]K Q Q ka urnace [(1 + 7 inulation 18 W (. 008 W / m. A t t in Solving the two equation above by trial-and error (or better yet, an equation olver give 09. K 6. and t inul 0.0 m or. cm. he accuracy can be improved omewhat by reevaluating the propertie and h at the new film temperature of (6.+5/ 0.7. inul

10 7-0 he equilibrium temperature of a light gla bulb in a room i to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he local atmopheric preure i 1 atm. he light bulb i approximated a an 8-cm-diameter phere. Propertie he olution of thi problem require a trial-and-error approach ince the determination of the Rayleigh number and thu the Nuelt number depend on the urface temperature which i unknown. We tart the olution proce by gueing the urface temperature to be 170 for the evaluation of the propertie and h. We will check the accuracy of thi gue later and repeat the calculation if neceary. he propertie of air at 1 atm and the anticipated film temperature of ( + / (170+5/ K are (able A1 k 0.00 W/m β 1 m K / K 5 Lamp 60 W ε 0.9 Analyi he characteritic length in thi cae i D 0.08 m. hen, gβ ( hen.97 Nu Ra (9.8 m/ 1/ + ( K (.16 (170 5 K(0.08 m 9 /16 9 [ 1+ ( 0.69 / Pr ] / 1+ ( 0.69 / π (0.08 m m m 0.589(.97 / 9 /16 [ ] 6 1/ k 0.00 W/m. h Nu ( W/m m A πd / (0.705 onidering both natural convection and radiation, the total rate of heat lo can be written a ha( + εaσ ( ( W (7.9 W/m. (0.001m ( + (0.9(0.001 m (5.67 urr 5 W/m.K [( + 7 (5 + 7 K D 8 cm It olution i 169 which i ufficiently cloe to the value aumed in the evaluation of propertie and h. herefore, there i no need to repeat calculation. ] Light, 6 W

11 7-7 wo urface of a pherical encloure are maintained at pecified temperature. he rate of heat tranfer through the encloure i to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he air preure in the encluure i 1 atm. Propertie he propertie of air at 1 atm and the average temperature of ( 1 + / (50+75/ 1.5 K are (able A1 k W/m m / K 1. 5 K Analyi he characteritic length in thi cae i determined from D D cm. hen, and gβ ( 1 Nu 0.8Ra 0.6 (9.8 m/ 0.8(7.09 (0.000 K ( A πd1d π(. 015 m (. 0 5 m m (50 75 K(0.05 m.8 knua 1 (. (. (. ( W/m m W 005. m m / 5 cm D 5 cm 75 K ( D 1 15 cm 1 50 K 5

12 7-66 An electronic box i cooled internally by a fan blowing air into the encloure. he fraction of the heat lot from the outer urface of the electronic box i to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. Heat tranfer from the bae urface i diregarded. he preure of air inide the encloure i 1 atm. Propertie he propertie of air at 1 atm and the film temperature of ( + / (+15/ K are (able A1 k W/m m / K 015. K Analyi Heat lo from the horizontal top urface: A (. 05m he characteritic length in thi cae i 015. m. hen, P [( 05. m + ( 05. m] gβ ( (9.81 m/ ( K ( 5 K(0.15 m (1.57 m / 1 / 6 1 / Nu ( k W/m. h Nu ( W/m m (. 05m 05. m A top and top hatop ( ( 8. W/m. ( 05. m ( W Heat lo from vertical ide urface: he characteritic length in thi cae i the height of the box 015. m. hen, gβ ( 1/ (9.81 m/ 6 1/ ( K (1.57 ( 5 K(0.15 m m / 5 ( (0.71. Nu 0.59Ra 0.59(..8 k W/m. h Nu ( W/m m A ide 015 (. m (. 05m 0. m and Qide haide ( ( 96. W/m. ( 0. m ( 5 8. W he radiation heat lo i rad εaσ ( urr (0.85( m (5.67 W/m.K [( + 7 K (5 + 7 K ] 0. W hen the fraction of the heat lo from the outer urface of the box i determined to be ( W f % 180 W 15 cm 180 W ε cm cm

13 7-79 A group of 5 tranitor are cooled by attaching them to a quare aluminum plate and mounting the plate on the wall of a room. he required ize of the plate to limit the urface temperature to 50 i to be determined. Aumption 1 Steady operating condition exit. i an ideal ga with contant propertie. he local atmopheric preure i 1 atm. Any heat tranfer from the back ide of the plate i negligible. Propertie he propertie of air at 1 atm and the film temperature of ( + / (50+0/ 0 1 K are (able A1 k W/m m / K 1 K Analyi he Rayleigh number can be determined in term of the characteritic length (length of the plate to be gβ( ( 9. 8 m/ ( K (50 0 K( L 9 ( L (. 170 m / We aume the length of the plate go be ufficiently le than 1 m o that Ra < 9 and thu we can determine the Nu number from Eq It give 1/ 9 1/ Nu 0.59Ra 0.59(1.5 L 116.9L h conv k W/m. Nu (116.9L L A L / /.156L 1/ W/m. Noting that both the urface and urrounding temperature are known, the rate of convection and radiation heat tranfer are determined to be 1/ 7 / ha( (.156L L ( L W conv 8 rad εaσ ( ky (0.9 L (5.67 W/m.K [( (0 + 7 ]K 15. L he rate of total heat tranfer i expreed a total conv + rad 7 / 5 (1.5 W 6.1L + 15.L W Solving for L, the length of the plate i determined to be L 0.0 m Dicuion Note that L < 1 m, and therefore the aumption of Ra < 9 i verified. hat i, L 1.5 ( < Room 0 Plate L L ranitor, W ε