Department f Ciil ngeerg Applied Mechanics McGill Uniersity, Mntreal, Quebec Canada CI 90 THRMODYNAMICS HAT TRANSFR Assignment #4 SOLUTIONS. A 68-kg man whse aerage bdy temperature is 9 C drks L f cld water at C an effrt t cl dwn. Takg the aerage specific heat f the human bdy t be.6 kj/kg C, determe the drp the aerage bdy temperature f this persn under the fluence f this cld water. A man drks ne liter f cld water at C an effrt t cl dwn. The drp the aerage bdy temperature f the persn under the fluence f this cld water is t be determed. Assumptins Thermal prperties f the bdy and water are cnstant. The effect f metablic heat generatin and the heat lss frm the bdy durg that time perid are negligible. Prperties The density f water is ery nearly kg/l, and the specific heat f water at rm temperature is c 4.8 kj/kg C (Table A-). The aerage specific heat f human bdy is gien t be.6 kj/kg. C. Analysis. The mass f the water is m w ρ ( kg/l)( L) kg We take the man and the water as ur system, and disregard any heat and mass transfer and chemical reactins. Of curse these assumptins may be acceptable nly fr ery shrt time perids, such as the time it takes t drk the water. Then the energy balance can be written as 44 Net energy transfer Δsystem 44 Change ternal, ketic, 0 ΔU ΔU + ΔU bdy water r [ mc ( T T )] + [ mc( T T )] 0 bdy water f T f Substitutg ( 68 kg)(.6 kj/kg C)( T 9) C + ( kg)(4.8 kj/kg C)( ) C 0 It gies T f 8.4 C Then T9 8.4 0.6 C Therefre, the aerage bdy temperature f this persn shuld drp ab half a degree celsius.
. The diffuser a jet enge is designed t decrease the ketic energy f the air enterg the enge cmpressr with any wrk r heat teractins. Calculate the elcity at the exit f a diffuser when air at 00 kpa and 0 C enters it with a elcity f 500 m/s and the exit state is 00 kpa and 90 C. The ketic energy f a fluid decreases as it is decelerated an adiabatic diffuser. What happens t this lst ketic energy? Air is decelerated an adiabatic diffuser. The elcity at the exit is t be determed. Assumptins This is a steady-flw prcess sce there is n change with time. Air is an ideal gas with cnstant specific heats. Ptential energy changes are negligible. 4 There are n wrk teractins. 5 The diffuser is adiabatic. Prperties The specific heat f air at the aerage temperature f (0+90)/55 C 8 K is c p.007 kj/kg K (Table A-b). Analysis There is nly ne let and ne exit, and thus m. We take diffuser as the system, which is a cntrl lume sce mass crsses the bundary. The energy balance fr this steady-flw system can be expressed the rate frm as 44 Δsystem 0 Rate f change ternal, ketic, 44444 ( h + / ) ( h + /) h + / h + / Slg fr exit elcity, 0.5 [ + ( h h )] [ + c ( T T )] (500 m/s) 0. m/s p 0.5 000 m /s + (.007 kj/kg K)(0 90)K kj/kg 00 kpa 0 C 500 m/s The lst ketic energy is mstly cnerted t ternal energy as shwn by a rise the fluid temperature. 0.5 AIR 00 kpa 90 C. Refrigerant-4a at 700 kpa and 0 C enters an adiabatic nzzle steadily with a elcity f 0 m/s and leaes at 400 kpa and 0 0 C. Determe a) the exit elcity, and b) the rati f the let t exit area A /A. In the case f a nn-adiabatic nzzle, hw wuld heat transfer affect the fluid elcity at the nzzle exit? R-4a is accelerated a nzzle frm a elcity f 0 m/s. The exit elcity f the refrigerant and the rati f the let-t-exit area f the nzzle are t be determed. Assumptins This is a steady-flw prcess sce there is n change with time. Ptential energy changes are negligible. There are n wrk teractins. 4 The deice is adiabatic and thus heat transfer is negligible. Prperties Frm the refrigerant tables (Table A-) and P 700 kpa 0.0458 m /kg T 0 C h 58.90 kj/kg R-4a
P 400 kpa 0.056796 m /kg T 0 C h 75.07 kj/kg Analysis (a) There is nly ne let and ne exit, and thus. We take nzzle as the system, which is a cntrl lume sce mass crsses the bundary. The energy balance fr this steady-flw system can be expressed the rate frm as Substitutg, It yields 44 Δsystem 0 0 ( h Rate f change ternal, ketic, 0 h 4444 + / ) ( h + /) (sce W 0) ( 75.07 58.90) 409.9 m/s h + kj/kg + ( 0 m/s) kj/kg 000 m /s (b) The rati f the let t exit area is determed frm the cnseratin f mass relatin, A A A A ( 0.0458 m /kg)( 409.9 m/s) ( 0.056796 m /kg)( 0 m/s) 5.65 Heat transfer t the fluid as it flws thrugh a nzzle is desirable sce it will prbably crease the ketic energy f the fluid. Heat transfer frm the fluid will decrease the exit elcity. 4. A steam turbe perates with.6 MPa and 50 C steam at its let and saturated apur at 0 C at its exit. The mass flw rate f the steam is 6 kg/s, and the turbe prduces 9000 kw f pwer. Determe the rate at which heat is lst thrugh the casg f this turbe. Steam expands a turbe whse pwer prductin is 9000 kw. The rate f heat lst frm the turbe is t be determed. Assumptins This is a steady-flw prcess sce there is n change with time. Ketic and ptential energy changes are negligible. Prperties Frm the steam tables (Tables A-6 and A-4) P.6 MPa h 46.0 kj/kg T 50 C T 0 C h 555.6 kj / kg x.6 MPa 50 C 6 kg/s Analysis We take the turbe as the system, which is a cntrl lume sce mass crsses the bundary. Ntg that there is ne let and ne exiti the energy balance fr this steady-flw system can be expressed the rate frm as Heat Turbe 0 C sat. ap.
44 Δsystem 0 Substitutg, Q h Rate f change ternal, ketic, ptential, etc. energies 44444 h + W + ( h h ) W (6 kg/s)(46.0 555.6) kj/kg 9000 kw 446.4 kw 5. Air is cmpressed by an adiabatic cmpressr frm 00 kpa and 0 C t.8 MPa and 400 C. Air enters the cmpressr thrugh a 0.5-m peng with a elcity f 0 m/s. It exits thrugh a 0.08-m peng. Calculate the mass flw rate f air and the required pwer put. Why des the air exit the cmpressr at a higher temperature?.8 MPa 400 C Cmpressr 00 kpa 0 C 0 m/s Air is cmpressed an adiabatic cmpressr. The mass flw rate f the air and the pwer put are t be determed. Assumptins This is a steady-flw prcess sce there is n change with time. The cmpressr is adiabatic. Air is an ideal gas with cnstant specific heats. Prperties The cnstant pressure specific heat f air at the aerage temperature f (0+400)/0 C48 K is c p.06 kj/kg K (Table A-b). The gas cnstant f air is R 0.87 kpa m /kg K (Table A-). Analysis (a) There is nly ne let and ne exit, and thus m. We take the cmpressr as the system, which is a cntrl lume sce mass crsses the bundary. The energy balance fr this steady-flw system can be expressed the rate frm as 44 Δsystem 0 h Rate f change ternal, ketic, ptential, etc. energies Sce 0 W h 44444 + c p ( T T ) + The specific lume f air at the let and the mass flw rate are RT ( 0.87 kpa m /kg K)(0 + 7 K) 0.8409 m P 00 kpa A (0.5 m )(0 m/s) 5.5kg/s 0.8409 m /kg /kg 4
Similarly at the let, RT ( 0.87 kpa m /kg K)(400 + 7 K) 0.07 m P 800 kpa (5.5 kg/s)(0.07 m /kg) 7.77 m/s A 0.08 m (b) Substitutg t the energy balance equatin gies W c p ( T T ) + (7.77 m/s) (0 m/s) (5.5 kg/s) (.06 kj/kg K)(400 0)K + 084 kw /kg kj/kg 000 m /s The air exits the cmpressr at a higher temperature because energy ( the frm f shaft wrk) is beg added t the air. 6. An adiabatic capillary tube is used sme refrigeratin systems t drp the pressure f the refrigerant frm the cndenser leel t the eapratr leel. The R-4a enters the capillary tube as a saturated liquid at 50 0 C, and leaes at - C. Determe the quality f the refrigerant at the let f the eapratr. Refrigerant-4a is thrttled by a capillary tube. The quality f the refrigerant at the exit is t be determed. Assumptins This is a steady-flw prcess sce there is n change with time. Ketic and ptential energy changes are negligible. Heat transfer t r frm the fluid is negligible. 4 There are n wrk teractins led. Analysis There is nly ne let and ne exit, and thus m. We take the thrttlg ale as the system, which is a cntrl lume sce mass crsses the bundary. The energy balance fr this steady-flw system can be expressed the rate frm as 50 C Sat. liquid h h Δ system mh mh 0 sce Q W Δke 0. The let enthalpy f R-4a is, frm the refrigerant tables (Table A-), T 50 C h sat. liquid The exit quality is h f.49 kj/kg - C R-4a T h C h h x h h fg f.49 5.9 0.4 07.8 5
7. A ht-water steam at 80 C enters a mixg chamber with a mass flw rate f 0.5 kg/s where it is mixed with a stream f cld water at 0 C. If it is desired that the mixture leae the chamber at 4 C, determe the mass flw rate f the cld-water stream. Assume all the streams are at a pressure f 50 kpa. When tw streams are mixed a mixg chamber, can the temperature f the exit stream eer be lwer than the temperature f the clder stream? T 80 C m 0.5 kg/s H O (P 50 kpa) T 4 C A ht water stream is mixed with a cld water stream. Fr a specified mixture temperature, the mass flw rate f cld water is t be determed. Assumptins Steady peratg cnditins exist. The mixg chamber is well-sulated s that heat lss t the surrundgs is negligible. Changes the ketic and ptential energies f fluid streams are negligible. 4 Fluid prperties are cnstant. 5 There are n wrk teractins. Prperties Ntg that T < T sat @ 50 kpa 7.4 C, the water all three streams exists as a cmpressed liquid, which can be apprximated as a saturated liquid at the gien temperature. Thus, h h f @ 80 C 5.0 kj/kg h h f @ 0 C 8.95 kj/kg h h f @ 4 C 75.90 kj/kg Analysis We take the mixg chamber as the system, which is a cntrl lume. The mass and energy balances fr this steady-flw system can be expressed the rate frm as Mass balance: nergy balance: Δ system 0 + m 44 0 Δsystem 4444 Rate f change ternal, ketic, 0 h + h h (sce W Δke 0) Cmbg the tw relatins and slg fr m gies m h + h ( + ) h h h h h m Substitutg, the mass flw rate f cld water stream is determed t be ( 5.0 75.90) ( 75.90 8.95) T 0 C m kj/kg kg/s kj/kg ( 0.5 ) 0.865 kg/s When tw streams are mixed a mixg chamber, the temperature f the exit stream can be lwer than the temperature f the clder stream if the mixg chamber is lsg heat t the surrundgs. 6
8. Steam is t be cndensed n the shell side f a heat exchanger at 85 F. Clg water enters the tubes at 60 F at a rate f 8 lbm/s and leaes at 7 F. Assumg the heat exchanger t be well-sulated, determe the rate f heat transfer the heat exchanger and the rate f cndensatin f the stream. Steam is cndensed by clg water a cndenser. The rate f heat transfer the heat exchanger and the rate f cndensatin f steam are t be determed. Assumptins Steady peratg cnditins exist. The heat exchanger is well-sulated s that heat lss t the surrundgs is negligible and thus heat transfer frm the ht fluid is equal t the heat transfer t the cld fluid. Changes the ketic and ptential energies f fluid streams are negligible. 4 Fluid prperties are cnstant. Prperties The specific heat f water is.0 Btu/lbm. F (Table A- ). The enthalpy f aprizatin f water at 85 F is 045. Btu/lbm (Table A-4). Analysis We take the tube-side f the heat exchanger where cld water is flwg as the system, which is a cntrl lume. The energy balance fr this steady-flw system can be expressed the rate frm as 44 0 Δsystem 4444 Rate f change ternal, ketic, + mh mh (sce Δke 0) mc p( T T ) 85 F Then the rate f heat transfer t the cld water this heat exchanger becmes Q [ mc ( T T )] water (8 lbm/s)(.0 Btu/lbm. F)(7 F 60 F) 794 Btu/s p Or ne culd apprximate the cmpressed liquid as a saturated liquid at the same temperature. Q ( h - h ) Table A-4: h h f @ 60F 8.08 Btu/lbm h h f@7f 4.07 Btu/lbm Q 8 lbm/s (4.07-8.08)Btu/lbm 79 Btu/s 0 Steam 85 F 7 F 60 F Water Ntg that heat ga by the water is equal t the heat lss by the cndensg steam, the rate f cndensatin f the steam the heat exchanger is determed frm 794 Btu/s ( mh fg ) steam steam.7 lbm/s h 045. Btu/lbm fg 7