3-42. Chapter 15 Steady Heat Conduction. Heat Conduction in Cylinders and Spheres

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1 Chapter 5 Steady Heat Cnductn Heat Cnductn n Cylnders and Spheres 3-64C When the dameter f cylnder s very small cmpared t ts length, t can be treated as an ndefntely lng cylnder. Cylndrcal rds can als be treated as beng nfntely lng when dealng wth heat transfer at lcatns far frm the tp r bttm surfaces. Hwever, t s nt prper t use ths mdel when fndng temperatures near the bttm and the tp f the cylnder. 3-65C Heat transfer n ths shrt cylnder s ne-dmensnal snce there wll be n heat transfer n the axal and tangental drectns. 3-66C N. In steady-peratn the temperature f a sld cylnder r sphere des nt change n radal drectn (unless there s heat generatn. 3-4

2 Chapter 5 Steady Heat Cnductn 3-67 sphercal cntaner flled wth ced water s subjected t cnvectn and radatn heat transfer at ts uter surface. The rate f heat transfer and the amunt f ce that melts per day are t be determned. ssumptns Heat transfer s steady snce the specfed thermal cndtns at the bundares d nt change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the mdpnt. 3 Thermal cnductvty s cnstant. Prpertes The thermal cnductvty f steel s gven t be k = 5 W/m C. The heat f fusn f water at atm s h f kj / kg. The uter surface f the tank s black and thus ts emssvty s =. nalyss (a The nner and the uter surface areas f sphere are D ( 5 m m D (5.03m 79.49m We assume the uter surface temperature T t be 5 C after cmparng cnvectn heat transfer ceffcents at the nner and the uter surfaces f the tank. Wth ths assumptn, the radatn heat transfer ceffcent can be determned frm h rad ( T T (5.670 surr 8 ( T W/m T.K 4 surr The ndvdual thermal resstances are [(73 5 K (73 30K ](73 30K(73 5 K] 5.570W/m.K T T rad T cnv, cnv, rad eqv ttal C/W h (80W/m. C(78.54 m r r (.55.5 m sphere C/W 4kr r 4 (5W/m. C(.55m(.5 m 0.006C/W h (0 W/m. C(79.49 m 0.006C/W hrad (5.57W/m. C(79.54 m eqv C/W cnv, cnv, rad Then the steady rate f heat transfer t the ced water becmes T T (30 0 C Q 30,58W C/W ttal eqv C/W (b The ttal amunt f heat transfer durng a 4-hur perd and the amunt f ce that wll melt durng ths perd are 6 Q Q t (30.58kJ/s(4 3600s.640 kj m ce Q h f kj 798kg kj/kg Check: The uter surface temperature f the tank s Q Q hcnvrad ( T Ts Ts T h 30,58W 30C 5.3 C ( W/m. C(79.54m cnvrad whch s very clse t the assumed temperature f 5C fr the uter surface temperature used n the evaluatn f the radatn heat transfer ceffcent. Therefre, there s n need t repeat the calculatns. 3-43

3 Chapter 5 Steady Heat Cnductn 3-68 steam ppe cvered wth 3-cm thck glass wl nsulatn s subjected t cnvectn n ts surfaces. The rate f heat transfer per unt length and the temperature drps acrss the ppe and the nsulatn are t be determned. ssumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the center lne and n varatn n the axal drectn. 3 Thermal cnductvtes are cnstant. 4 The thermal cntact resstance at the nterface s neglgble. Prpertes The thermal cnductvtes are gven t be k = 5 W/m C fr steel and k = W/m C fr glass wl nsulatn nalyss The nner and the uter surface areas f the nsulated ppe per unt length are D L ( m ( m 057. m D L ( m ( m m T The ndvdual thermal resstances are ttal nsulatn ppe 0.08C/W (80 W/m. C(0.57m ln( r / r ln(.75/ C/W k L (5 W/m. C(m ln( r3 / r ln(5.75/ C/W k L (0.038W/m. C(m 0.847C/W (5 W/m. C(0.36m C/W Then the steady rate f heat lss frm the steam per m. ppe length becmes ( Q T T 30 5 C C / W 93.9 W ttal The temperature drps acrss the ppe and the nsulatn are T T ppe nsulatn Q ( W(0.000 C / W 0.095C ppe Q ( W(3.089 C / W 90C nsulatn T 3-44

4 Chapter 5 Steady Heat Cnductn 3-69 "GIVEN" T_nfnty_=30 "[C]" T_nfnty_=5 "[C]" k_steel=5 "[W/m-C]" D_=0.05 "[m]" D_=0.055 "[m]" r_=d_/ r_=d_/ "t_ns=3 [cm], parameter t be vared" k_ns=0.038 "[W/m-C]" h_=5 "[W/m^-C]" h_=80 "[W/m^-C]" L= "[m]" "NLYSIS" _=p*d_*l _=p*(d_+*t_ns*cnvert(cm, m*l _cnv_=/(h_* ppe=ln(r_/r_/(*p*k_steel*l _ns=ln(r_3/r_/(*p*k_ns*l r_3=r_+t_ns*cnvert(cm, m "t_ns s n cm" _cnv_=/(h_* ttal=_cnv_+_ppe+_ns+_cnv_ Q_dt=(T_nfnty_-T_nfnty_/_ttal DELTT_ppe=Q_dt*_ppe DELTT_ns=Q_dt*_ns Tns [cm] Q [W] Tns [C]

5 Chapter 5 Steady Heat Cnductn Q [W] Tns [C] t ns [cm] 3-46

6 Chapter 5 Steady Heat Cnductn m lng sectn f a steam ppe passes thrugh an pen space at 5C. The rate f heat lss frm the steam ppe, the annual cst f ths heat lss, and the thckness f fberglass nsulatn needed t save 90 percent f the heat lst are t be determned. ssumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the center lne and n varatn n the axal drectn. 3 Thermal cnductvty s cnstant. 4 The thermal cntact resstance at the nterface s neglgble. 5 The ppe temperature remans cnstant at abut 50C wth r wthut nsulatn. 6 The cmbned heat transfer ceffcent n the uter surface remans cnstant even after the ppe s nsulated. Prpertes The thermal cnductvty f fberglass nsulatn s gven t be k = W/m C. nalyss (a The rate f heat lss frm the steam ppe s DL ( 0.m(50m 5.7m Q bare ( T T s ar (b The amunt f heat lss per year s (0W/m. C(5.7m (505 C = 4,4W Q Q t (4.4kJ/s( s/yr kj/yr The amunt f gas cnsumptn frm the natural gas furnace that has an effcency f 75% s Q gas kj/yr therm 6,903therms/yr ,500kJ The annual cst f ths energy lst s Energy cst (Energy used(unt cst f energy = (6,903therms/yr($0.5/ therm $8790/yr (c In rder t save 90% f the heat lss and thus t reduce t t 0.4,4 = 44 W, the thckness f nsulatn needed s determned frm Q nsulated Ts T ar nsulatn Ts Tar ln( r / r kl Substtutng and slvng fr r, we get (505 C 44W r 0.069m ln( r / 0.05 (0W/m. C[(r (50m] (0.035W/m. C(50m Then the thckness f nsulatn becmes t nsulatn r r cm T s nsulatn T ar 3-47

7 Chapter 5 Steady Heat Cnductn 3-7 n electrc ht water tank s made f tw cncentrc cylndrcal metal sheets wth fam nsulatn n between. The fractn f the ht water cst that s due t the heat lss frm the tank and the payback perd f the d-t-yurself nsulatn kt are t be determned. ssumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the center lne and n varatn n the axal drectn. 3 Thermal cnductvtes are cnstant. 4 The thermal resstances f the water tank and the uter thn sheet metal shell are neglgble. 5 Heat lss frm the tp and bttm surfaces s neglgble. Prpertes The thermal cnductvtes are gven t be k = 0.03 W/m C fr fam nsulatn and k = W/m C fr fber glass nsulatn nalyss We cnsder nly the sde surfaces f the water heater fr smplcty, and dsregard the tp and bttm surfaces (t wll make dfference f abut 0 percent. The ndvdual thermal resstances are D L ( m ( m. 89 m fam ttal 0.09C/W ( W/m. C(.89m ln( r / r ln(3/ C/W kl (0.03W/m. C( m fam C/W The rate f heat lss frm the ht water tank s (55 Q T w T 7 C 70 W 0.40 C / W ttal The amunt and cst f heat lss per year are Q Q t ( kw(365 4 h / yr 63. kwh / yr Cst f Energy (munt f energy(unt cst = (63. kwh($0.08/ kwh $ $ f % $80 If 3 cm thck fber glass nsulatn s used t wrap the entre tank, the ndvdual resstances becmes D L ( 0.5m( m 3.67m h fberglass 0.06 ( W/m. C(3.67m fam ttal C/W ln( r / r ln(3/ C/W k L (0.03W/m. C( m ln( r3 / r ln(6/ C/W k L (0.035W/m. C( m fam fberglass C/W The rate f heat lss frm the ht water heater n ths case s Tw T (55 7 C Q 4.4W ttal 0.676C/W The energy savng s savng = = 8.58 W The tme necessary fr ths addtnal nsulatn t pay fr ts cst f $30 s then determned t be Cst (0.0858kW(Tmeperd($0.08/ kwh $30 Then, Tmeperd 3,hurs 547days.5 years T w fam T w fam T fberglass T 3-48

8 Chapter 5 Steady Heat Cnductn 3-7 "GIVEN" L= "[m]" D_=0.40 "[m]" D_=0.46 "[m]" r_=d_/ r_=d_/ "T_w=55 [C], parameter t be vared" T_nfnty_=7 "[C]" h_=50 "[W/m^-C]" h_= "[W/m^-C]" k_ns=0.03 "[W/m-C]" Prce_electrc=0.08 "[$/kwh]" Cst_heatng=80 "[$/year]" "NLYSIS" _=p*d_*l _=p*d_*l _cnv_=/(h_* ns=ln(r_/r_/(*p*k_ns*l _cnv_=/(h_* ttal=_cnv_+_ns+_cnv_ Q_dt=(T_w-T_nfnty_/_ttal Q=(Q_dt*Cnvert(W, kw*tme tme=365*4 "[h/year]" Cst_HeatLss=Q*Prce_electrc f_heatlss=cst_heatlss/cst_heatng*cnvert(, % Tw [C] fheatlss [%]

9 Chapter 5 Steady Heat Cnductn fheatlss [%] T w [C] 3-50

10 Chapter 5 Steady Heat Cnductn 3-73 cld alumnum canned drnk that s ntally at a unfrm temperature f 3 C s brught nt a rm ar at 5 C. The tme t wll take fr the average temperature f the drnk t rse t 0C wth and wthut rubber nsulatn s t be determned. ssumptns The drnk s at a unfrm temperature at all tmes. The thermal resstance f the can and the nternal cnvectn resstance are neglgble s that the can s at the same temperature as the drnk nsde. 3 Heat transfer s ne-dmensnal snce there s thermal symmetry abut the centerlne and n varatn n the axal drectn. 4 Thermal prpertes are cnstant. 5 The thermal cntact resstance at the nterface s neglgble. Prpertes The thermal cnductvty f rubber nsulatn s gven t be k = 0.3 W/m C. Fr the drnk, we use the prpertes f water at rm temperature, = 000 kg/m 3 and C p = 480 J/kg.C. nalyss Ths s a transent heat cnductn, and the rate f heat transfer wll decrease as the drnk warms up and the temperature dfference between the drnk and the surrundngs decreases. Hwever, we can slve ths prblem apprxmately by assumng a cnstant average temperature f (3+0/ = 6.5C durng the prcess. Then the average rate f heat transfer nt the drnk s Q D ( m D L ( m ( 05. m m 4 4 bare, ave h ( Tar Tcan, ave (0W/m. C(0.09m (5 6.5 C = 5.40W The amunt f heat that must be suppled t the drnk t rase ts temperature t 0 C m V r L ( 000 kg / m ( m ( 05. m kg Q mc T ( kg(480 J / kg(0-3 C 0, 39 J p 3 Then the tme requred fr ths much heat transfer t take place s Q 0, 39 J t 9 s 3.9 mn Q 5.4 J / s We nw repeat calculatns after wrappng the can wth -cm thck rubber nsulatn, except the tp surface. The rate f heat transfer frm the tp surface s ( ( (. Q T T 0 W / m. C[ 0 03 m ]( C = 0.5 W tp, ave tp ar can, ave Heat transfer thrugh the nsulated sde surface s Q D L ( m ( 05. m m nsulatn, sde sde T ttal ar 3.83C/W (0 W/m. C(0.034m ln( r / r ln(4 / 3.88C/W kl (0.3 W/m. C(0.5m T cnv, can,ave nsulatn C/W (5 6.5 C 3.08 W 6.00C/W The rat f bttm t the sde surface areas s ( r / ( rl r / ( L 3/ ( Therefre, the effect f heat transfer thrugh the bttm surface can be accunted fr apprxmately by ncreasng the heat transfer frm the sde surface by %. Then, Q Q Q W nsulated sde bttm tp Then the tme f heatng becmes Q 0, 39 J t 60 s 43.4 mn Q 3.97 J / s T can s nsulatn T ar 3-5

11 Chapter 5 Steady Heat Cnductn 3-74 cld alumnum canned drnk that s ntally at a unfrm temperature f 3C s brught nt a rm ar at 5C. The tme t wll take fr the average temperature f the drnk t rse t 0C wth and wthut rubber nsulatn s t be determned. ssumptns The drnk s at a unfrm temperature at all tmes. The thermal resstance f the can and the nternal cnvectn resstance are neglgble s that the can s at the same temperature as the drnk nsde. 3 Heat transfer s ne-dmensnal snce there s thermal symmetry abut the centerlne and n varatn n the axal drectn. 4 Thermal prpertes are cnstant. 5 The thermal cntact resstance at the nterface s t be cnsdered. Prpertes The thermal cnductvty f rubber nsulatn s gven t be k = 0.3 W/m C. Fr the drnk, we use the prpertes f water at rm temperature, = 000 kg/m 3 and C p = 480 J/kg.C. nalyss Ths s a transent heat cnductn, and the rate f heat transfer wll decrease as the drnk warms up and the temperature dfference between the drnk and the surrundngs decreases. Hwever, we can slve ths prblem apprxmately by assumng a cnstant average temperature f (3+0/ = 6.5C durng the prcess. Then the average rate f heat transfer nt the drnk s D ( m D L ( m ( 05. m m 4 4 Q ( T T ( 0 W / m. C ( m ( C = 5.40 W bare, ave ar can, ave The amunt f heat that must be suppled t the drnk t rase ts temperature t 0 C s 3 m V r L ( 000 kg / m ( m ( 05. m kg Q mc T ( kg(480 J / kg(0-3 C 0, 39 J p Then the tme requred fr ths much heat transfer t take place s Q 0, 39 J t 9 s 3.9 mn Q 5.4 J / s We nw repeat calculatns after wrappng the can wth -cm thck rubber nsulatn, except the tp surface. The rate f heat transfer frm the tp surface s ( ( (. Q T T 0 W / m. C[ 0 03 m ]( C = 0.5 W tp, ave tp ar can, ave Heat transfer thrugh the nsulated sde surface s D L ( m ( 05. m m nsulatn, sde cntact ttal Q sde 3.83C/W (0 W/m. C(0.034m ln( r / r ln(4 / 3.88C/W kl (0.3W/m. C(0.5m m. C/W C/W (0.06m(0.5m Tar T nsulatn cnv, can,ave cntact C/W (5 6.5 C 3.08 W 6.004C/W The rat f bttm t the sde surface areas s ( r /(rl r /(L 3/( Therefre, the effect f heat transfer thrugh the bttm surface can be accunted fr apprxmately by ncreasng the heat transfer frm the sde surface by %. Then, Q nsulated Qsdebttm Qtp W Then the tme f heatng becmes Q 0, 39 J t 60 s 43.4 mn Q 3.97 J / s Dscussn The thermal cntact resstance dd nt have any effect n heat transfer. T can cntact nsulatn Tar 3-5

12 Chapter 5 Steady Heat Cnductn 3-75E steam ppe cvered wth -n thck fberglass nsulatn s subjected t cnvectn n ts surfaces. The rate f heat lss frm the steam per unt length and the errr nvlved n neglectng the thermal resstance f the steel ppe n calculatns are t be determned. ssumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the center lne and n varatn n the axal drectn. 3 Thermal cnductvtes are cnstant. 4 The thermal cntact resstance at the nterface s neglgble. Prpertes The thermal cnductvtes are gven t be k = 8.7 Btu/hft F fr steel and k = 0.00 Btu/hft F fr fberglass nsulatn. nalyss The nner and uter surface areas f the nsulated ppe are D L ( 35. / ft ( ft ft D L (8 / ft ( ft. 094 ft The ndvdual resstances are ttal nsulatn ppe 0.036h F/Btu (30 Btu/h.ft. F(0.96ft ln( r / r ln( / h F/Btu k L (8.7 Btu/h.ft. F(ft ln( r3 / r ln(4 / 5.56h F/Btu k L (0.00Btu/h.ft. F(ft 0.096h F/Btu (5 Btu/h.ft. F(.094ft h F/Btu Then the steady rate f heat lss frm the steam per ft. ppe length becmes T Q T ttal ( F 69.9Btu/h 5.65hF/Btu If the thermal resstance f the steel ppe s neglected, the new value f ttal thermal resstance wll be ttal h F/Btu Then the percentage errr nvlved n calculatns becmes errr% ( h F / Btu % 565. h F / Btu whch s nsgnfcant. T ppe nsulatn T 3-53

13 Chapter 5 Steady Heat Cnductn 3-76 Ht water s flwng thrugh a 3-m sectn f a cast rn ppe. The ppe s expsed t cld ar and surfaces n the basement. The rate f heat lss frm the ht water and the average velcty f the water n the ppe as t passes thrugh the basement are t be determned. ssumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the center lne and n varatn n the axal drectn. 3 Thermal prpertes are cnstant. Prpertes The thermal cnductvty and emssvty f cast rn are gven t be k = 5 W/m C and = 0.7. nalyss The ndvdual resstances are D L ( m ( 5 m 885. m D L ( m ( 5 m. 68 m ppe C/W (0 W/m. C(.885m ln( r / r ln(.3 / C/W k L (5 W/m. C(5m The uter surface temperature f the ppe wll be smewhat belw the water temperature. ssumng the uter surface temperature f the ppe t be 80 C (we wll check ths assumptn later, the radatn heat transfer ceffcent s determned t be h rad ( T T surr (0.7(5.670 ( T 8 T W/m surr.k 4 [(353K (83K ]( W/m.K Snce the surrundng medum and surfaces are at the same temperature, the radatn and cnvectn heat transfer ceffcents can be added and the result can be taken as the cmbned heat transfer ceffcent. Then, h cmbned ttal h h rad h cmbned cnv, ppe W/m. C 0.09C/W (0.67W/m. C(.68m C/W The rate f heat lss frm the ht water ppe then becmes T Q T ttal (900 C 97W 0.073C/W Fr a temperature drp f 3C, the mass flw rate f water and the average velcty f water must be Q mc p m V T c Q m C T m V c p 97J/s 0.33kg/s (480J/kg. C(3 C 0.33kg/s 3 (0.04m (000kg/m 4 Dscussn The uter surface temperature f the ppe s T Q T s ppe (90Ts C 97W T ( C/W 0.86 m/s s = 77C whch s very clse t the value assumed fr the surface temperature n the evaluatn f the radatn resstance. Therefre, there s n need t repeat the calculatns. T ppe T 3-54

14 Chapter 5 Steady Heat Cnductn 3-77 Ht water s flwng thrugh a 5 m sectn f a cpper ppe. The ppe s expsed t cld ar and surfaces n the basement. The rate f heat lss frm the ht water and the average velcty f the water n the ppe as t passes thrugh the basement are t be determned. ssumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the centerlne and n varatn n the axal drectn. 3 Thermal prpertes are cnstant. Prpertes The thermal cnductvty and emssvty f cpper are gven t be k = 386 W/m C and = 0.7. nalyss The ndvdual resstances are D L ( m ( 5 m 885. m D L ( m ( 5 m. 68 m ppe C/W (0W/m. C(.885m ln( r / r ln(.3 / C/W kl (386W/m. C(5m The uter surface temperature f the ppe wll be smewhat belw the water temperature. ssumng the uter surface temperature f the ppe t be 80 C (we wll check ths assumptn later, the radatn heat transfer ceffcent s determned t be h rad ( T T surr (0.7(5.670 ( T 8 T W/m surr.k 4 [(353K (83K ]( W/m.K Snce the surrundng medum and surfaces are at the same temperature, the radatn and cnvectn heat transfer ceffcents can be added and the result can be taken as the cmbned heat transfer ceffcent. Then, h cmbned ttal h rad h cnv, W/m. C 0.09C/W (0.67 W/m. C(.68m ppe The rate f heat lss frm the ht tank water then becmes T Q T ttal C/W (90 0 C 930W 0.073C/W Fr a temperature drp f 3C, the mass flw rate f water and the average velcty f water must be Q mc p m V T c Q m C T m V c p 930J/s (480J/kg. C( kg/s 3 (0.04 m (000kg/m 4 Dscussn The uter surface temperature f the ppe s T Q T s ppe (90Ts C 930W T ( C/W 0.34 kg/s C 0.86 m/s s = 77C whch s very clse t the value assumed fr the surface temperature n the evaluatn f the radatn resstance. Therefre, there s n need t repeat the calculatns. T ppe T 3-55

15 Chapter 5 Steady Heat Cnductn 3-78E Steam extng the turbne f a steam pwer plant at 00F s t be cndensed n a large cndenser by clng water flwng thrugh cpper tubes. Fr specfed heat transfer ceffcents, the length f the tube requred t cndense steam at a rate f 400 lbm/h s t be determned. ssumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the center lne and n varatn n the axal drectn. 3 Thermal prpertes are cnstant. 4 Heat transfer ceffcents are cnstant and unfrm ver the surfaces. Prpertes The thermal cnductvty f cpper tube s gven t be k = 3 Btu/hft F. The heat f vaprzatn f water at 00F s gven t be 037 Btu/lbm. nalyss The ndvdual resstances are D L ( 0. 4 / ft ( ft 005. ft D L ( 0. 6 / ft ( ft 057. ft ppe ttal ln( r / r ln(3 / hF/Btu kl (3Btu/h.ft. F(ft 0.7hF/Btu (35Btu/h.ft. F(0.05ft hF/Btu (500Btu/h.ft. F(0.57ft ppe The heat transfer rate per ft length f the tube s T Q T ttal hF/Btu (00 70 F 08.44Btu/h F/Btu The ttal rate f heat transfer requred t cndense steam at a rate f 400 lbm/h and the length f the tube requred s determned t be Q ttal mh Q Tube length Q ttal (0lbm/h(037 Btu/lbm 4,440 Btu/h 4, ft T ppe T 3-56

16 Chapter 5 Steady Heat Cnductn 3-79E Steam extng the turbne f a steam pwer plant at 00F s t be cndensed n a large cndenser by clng water flwng thrugh cpper tubes. Fr specfed heat transfer ceffcents and 0.0-n thck scale buld up n the nner surface, the length f the tube requred t cndense steam at a rate f 400 lbm/h s t be determned. ssumptns Heat transfer s steady snce there s n ndcatn f any change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the centerlne and n varatn n the axal drectn. 3 Thermal prpertes are cnstant. 4 Heat transfer ceffcents are cnstant and unfrm ver the surfaces. Prpertes The thermal cnductvtes are gven t be k = 3 Btu/hft F fr cpper tube and be k = 0.5 Btu/hft F fr the mneral depst. The heat f vaprzatn f water at 00F s gven t be 037 Btu/lbm. nalyss When a 0.0-n thck layer f depst frms n the nner surface f the ppe, the nner dameter f the ppe wll reduce frm 0.4 n t 0.38 n. The ndvdual thermal resstances are depst ppr T T D L ( 0. 4 / ft ( ft 005. ft D L ( 0. 6 / ft ( ft 057. ft ppe depst ttal ln( r / r ln(3 / hF/Btu kl (3Btu/h.ft. F(ft ln( r / r k L 0.7hF/Btu (35Btu/h.ft. F(0.05ft dep ln(0. / h. F/Btu (0.5 Btu/h.ft. F(ft hF/Btu (500Btu/h.ft. F(0.57ft ppe depst The heat transfer rate per ft length f the tube s T Q T ttal (00 70 F 0.40Btu/h 0.998F/Btu hF/Btu The ttal rate f heat transfer requred t cndense steam at a rate f 400 lbm/h and the length f the tube requred can be determned t be Q ttal mh Q Tube length Q ttal (0lbm/h(037 Btu/lbm 4,440 Btu/h 4, ft 3-57

17 Chapter 5 Steady Heat Cnductn 3-80E "GIVEN" T_nfnty_=00 "[F]" T_nfnty_=70 "[F]" k_ppe=3 "[Btu/h-ft-F], parameter t be vared" D_=0.4 "[n]" "D_=0.6 [n], parameter t be vared" r_=d_/ r_=d_/ h_=037 "[Btu/lbm]" h_=500 "[Btu/h-ft^-F]" h_=35 "[Btu/h-ft^-F]" m_dt=0 "[lbm/h]" "NLYSIS" L= "[ft], fr ft length f the tube" _=p*(d_*cnvert(n, ft*l _=p*(d_*cnvert(n, ft*l _cnv_=/(h_* ppe=ln(r_/r_/(*p*k_ppe*l _cnv_=/(h_* ttal=_cnv_+_ppe+_cnv_ Q_dt=(T_nfnty_-T_nfnty_/_ttal Q_dt_ttal=m_dt*h_ L_tube=Q_dt_ttal/Q_dt kppe [Btu/h.ft.F] Ltube [ft] 3-58

18 Chapter 5 Steady Heat Cnductn D[n] Ltube [ft] 3-59

19 Chapter 5 Steady Heat Cnductn Ltube [ft] k ppe [Btu/h-ft-F] Ltube [ft] D [n] 3-60

20 Chapter 5 Steady Heat Cnductn m dameter sphercal tank flled wth lqud ntrgen at atm and -96C s expsed t cnvectn and radatn wth the surrundng ar and surfaces. The rate f evapratn f lqud ntrgen n the tank as a result f the heat gan frm the surrundngs fr the cases f n nsulatn, 5-cm thck fberglass nsulatn, and -cm thck supernsulatn are t be determned. ssumptns Heat transfer s steady snce the specfed thermal cndtns at the bundares d nt change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the mdpnt. 3 The cmbned heat transfer ceffcent s cnstant and unfrm ver the entre surface. 4 The temperature f the thn-shelled sphercal tank s sad t be nearly equal t the temperature f the ntrgen nsde, and thus thermal resstance f the tank and the nternal cnvectn resstance are neglgble. Prpertes The heat f vaprzatn and densty f lqud ntrgen at atm are gven t be 98 kj/kg and 80 kg/m 3, respectvely. The thermal cnductvtes are gven t be k = W/m C fr fberglass nsulatn and k = W/m C fr super nsulatn. nalyss (a The heat transfer rate and the rate f evapratn f the lqud wthut nsulatn are D ( 3 m 8. 7 m 0.000C/W (35 W/m. C(8.7 m Ts T [5 ( 96] C Q 08,90W 0.000C/W Q 08.90kJ/s Q mh m.055kg/s h 98kJ/kg (b The heat transfer rate and the rate f evapratn f the lqud wth a 5-cm thck layer f fberglass nsulatn are D ( 3. m 309. m C/W (35 W/m. C(30.9m nsulatn ttal 4 kr r r r (.55.5 m C/W 4 (0.035W/m. C(.55m(.5 m nsulatn C/W Ts T [5 ( 96] C Q 433W ttal C/W Q 4.33kJ/s Q mh m 0.04kg/s h 98 kj/kg (c The heat transfer rate and the rate f evapratn f the lqud wth -cm thck layer f supernsulatn s D ( 304. m m C/W (35 W/m. C(9.03m nsulatn r r 4 kr r (.5.5 m 3.96C/W 4 ( W/m. C(.5m(.5 m ttal nsulatn C/W [ ( ] Q T s T 5 96 C 5. W ttal 3.96 C / W. Q mh m Q 0 05 kj / s kg / s h 98 kj / kg T s T s T s nsulatn nsulatn T T T 3-6

21 Chapter 5 Steady Heat Cnductn m dameter sphercal tank flled wth lqud xygen at atm and -83C s expsed t cnvectn and radatn wth the surrundng ar and surfaces. The rate f evapratn f lqud xygen n the tank as a result f the heat gan frm the surrundngs fr the cases f n nsulatn, 5-cm thck fberglass nsulatn, and -cm thck supernsulatn are t be determned. ssumptns Heat transfer s steady snce the specfed thermal cndtns at the bundares d nt change wth tme. Heat transfer s ne-dmensnal snce there s thermal symmetry abut the mdpnt. 3 The cmbned heat transfer ceffcent s cnstant and unfrm ver the entre surface. 4 The temperature f the thn-shelled sphercal tank s sad t be nearly equal t the temperature f the xygen nsde, and thus thermal resstance f the tank and the nternal cnvectn resstance are neglgble. Prpertes The heat f vaprzatn and densty f lqud xygen at atm are gven t be 3 kj/kg and 40 kg/m 3, respectvely. The thermal cnductvtes are gven t be k = W/m C fr fberglass nsulatn and k = W/m C fr super nsulatn. nalyss (a The heat transfer rate and the rate f evapratn f the lqud wthut nsulatn are D ( 3 m 8. 7 m 0.000C/W (35 W/m. C(8.7 m Ts T [5 ( 83] C Q 96,040W 0.000C/W Q kJ/s Q mh m 0.90kg/s h 3kJ/kg (b The heat transfer rate and the rate f evapratn f the lqud wth a 5-cm thck layer f fberglass nsulatn are D ( 3. m 309. m C/W (35 W/m. C(30.9m nsulatn r r 4 kr r (.55.5 m C/W 4 (0.035W/m. C(.55m(.5 m ttal nsulatn C/W [ ( ] Q T s T 5 83 C 3976 W ttal C / W. Q mh m Q 3976 kj / s kg / s h 3 kj / kg (c The heat transfer rate and the rate f evapratn f the lqud wth a -cm supernsulatn s D ( 304. m m C/W (35 W/m. C(9.03m nsulatn r r 4 kr r (.5.5 m 3.96C/W 4 ( W/m. C(.5m(.5 m ttal nsulatn C/W [ ( ] Q T s T 5 83 C 48. W ttal 3.96 C / W. Q mh m Q kj / s kg / s h 3 kj / kg T s T s T s nsulatn nsulatn T T T 3-6

Chapter 3, Solution 1C.

Chapter 3, Solution 1C. COSMOS: Cmplete Onlne Slutns Manual Organzatn System Chapter 3, Slutn C. (a If the lateral surfaces f the rd are nsulated, the heat transfer surface area f the cylndrcal rd s the bttm r the tp surface