Mdule 7: Slved Prblems 1 A tn-walled nentr tube eat exanger f 019-m lengt s t be used t eat denzed water frm 40 t 60 at a flw rate f 5 kg/s te denzed water flws trug te nner tube f 30-mm dameter wle t press water at 95 flws n te annulus frmed wt te uter tube f 60-mm dameter e term pysal prpertes f te fluds are: nsderng a parallel-flw nfguratn f te exanger determne te mnmum flw rate requred fr te t press water Determne te verall eat transfer effent requred fr te ndtns f part a nsderng a unter flw nfguratn determne te mnmum flw rate requred fr te t press water Wat s te effetveness f te exanger fr ts stuatn? Knwn: n-walled nentr tube Parallel flw eat exanger f presrbed dameter and lengt wt press and denzed water Inlet and utlet temperatures and flw rate f desred water Inlet temperature and utlet temperature and flw rate f denzed water Inlet temperature f press water
Fnd: (1) mnmum flw rate requred fr te t press water (b) requred verall eat transfer effent and weter t s pssble t ampls ts eatng and () fr F arrangements mnmum press water flw requred and te effetveness? Semat: Assumptns: (1) Neglgble eat lss t surrundngs () Neglgble knet and ptental energy anges Analyss: (a) frm verall energy balanes q ( m ) ( ) ( m ) ( ) Fr a fxed term (m ) wll be a mnmum wen s a mnmum Wt te parallel flw nfguratn ts requres tat 60 Hene m mn ( m) ( ( ) ) 5kg / s 4181J / kg K(60 40) 4197J / kg K(95 60) 85kg / s (b)frm te rate equatn and te lg mean temperature relatn
q UAΔ lm PF Δlm PF Δ1 - Δ ln Δ 1 Δ And sne Δ 0 Δ lm 0 s tat UA Sne AπDL s fnte U must be extremely large Hene te eatng annt be amplsed wt ts arrangement () Wt te F arrangements ṁ wll be a mnmum wen s a mnmum s requres tat s a mnmum s requres tat s a mnmum s requres tat 40 Hene frm te verall energy balane 5kg / s 4181J / kg K(60 40) K m 181kg / s 4197J / kg K(95 40) K Fr ts ndtn mn w s led frm t ene ε1 mments: Fr te unter flw arrangement te eat exanger must be nfntely lng
An autmble radatr may be vewed as a rss-flw eat exanger wt bt fluds unmxed Water w as flw rate f 005kg/s enters te radatr at 400K and s t leave at 330 K e water s led by ar w enters at 075kg/s and 300K If te verall eat transfer effent s 00W/m K wat s te requred eat transfer surfae area? Knwn: flw rate and nlet temperature fr autmble radatr Overall eat transfer effent Fnd: Area requred t aeve a presrbed utlet temperature Semat: Water 400K m 005kg/s Ar 300K m 075kg/s 330K Assumptns: (1) Neglgble eat lss t surrundngs and knet and ptental energy anges () nstant prpertes Analyss: e requred eat transfer rate s q ( m ) ( ) 005kg / s(409j / kg K )70K 14 73W
Usng te ε-nu metd mn 1045W / K max 7555W / K ene mn / max ( ) 1045W / K(100K ) 1045W and ε q / q max 1473W / 1045W 0700 Frm fgure NU 15 ene A NU( m mn / U ) 15 1045W / K(00W / m K ) 1 58 mments: (1) te ar utlet temperature s + q / 300K + (1473W / 7555W / K ) 319 5K () Usng te LMD appra Δ lm 51 K R079 and P07 Hene frm fg F 095 and A q / FUΔlm (1473W ) /[095(00W / m K )51K ] 151m
3 Saturated water vapr leaves a steam turbne at a flw rate f 15kg/s and a pressure f 051 bars e vapr s t be mpletely ndensed t saturated lqud n a sell-and tube eat exanger w uses ty water as te ld flud e water enters te tn-walled tubes at 17 and s t leave at 57 assumng an verall eat transfer effent f 00W/m K determne te requred eat exanger surfae area and te water flw rate After extended peratn fulng auses te verall eat transfer effent t derease t 100W/m K and t mpletely ndense te vapr tere must be an attendant redutn n te vapr flw rate Fr te same water nlet temperature and flw rate wat s te new vapr flw rate requred fr mplete ndensatn? Knwn: Pressure and ntal flw rate f water vapr Water nlet and utlet temperatures Intal and fnal verall eat transfer effents Fnd: (a) Surfae area fr ntal U and water flw rate (b) Vapur flw rate fr fnal U Semat: Assumptns: (1) Neglgble eat lss t surrundngs () Neglgble wall ndutn resstane
Prpertes: able fr satwater: ( 310K ) : p 4178J / kg K;(p 051 bars) : sa 355K 304kJ/kg t fg Analyss: (a) e requred eat transfer rate s q m fg 6 1 5kg / s(304 10 J / kg) 346 10 6 W And te rrespndng eat apaty rate fr te water s mn q /( 6 ) 348 10 W / 40 K 86400W / K ene ε q /( mn [ 6 ]) 346 10 W / 86400W / K(65K ) 06 sne mn / max 0 NU -ln(1 - ε ) ln(1 06) 097 And A NU( mn / U ) 097(86400W / K / 000W / m K ) 419m m / 86400W / K / 4178J / kg K 07kg / s p (b) usng te fnal verall eat transfer effent fnd Sne / 0 mn max
ε 1 exp( NU ) 1 exp( 0485) 0384 ene q ε mn ( ) 0384(886400W / K )65K 16106W m q / fg 16 10 6 W 6 / 304 10 J / kg 0936kg / s mments: e sgnfant redutn (38%) n ṁ represents a sgnfant lss n turbne pwer Perd leanng f ndenser surfaes suld be emplyed t mnmze te adverse effets f fulng
4 Water at 5 kg/ s t be eated frm 35 t 95 by means f a nentr tube eat exanger Ol at 5kg/ and 10 wt a spef eat f 095 J/kgK s t be used as te t flud If te verall eat transfer effent based n te uter dameter f te nner tube f 550W/m Kdetermne te lengt f te exanger f te uter dameters s 100mm Knwn: nentr tube eat exanger Fnd: Lengt f te exanger Semat: Assumptns: (1) Neglgble eat lss t surrundngs () Neglgble knet and ptental energy anges (3) nstant prpertes Prpertes: able fr Water: _ ( (35 + 95) / 338K ) : p 4188J / kg K Analyss: Frm rate equatn wt A πd L Lq/U D Δ l m e eat rate q an be evaluated frm an energy balane n te ld flud 5kg / q m ( ) 4188J / kg K(95 35) K 15 W 3600s / 705 0
In rder t evaluate Δ l m we need t knw weter te exanger s peratng n F r PF Frm an energy balane n te t flud fnd 5kg / J q / m 10 15705W / 095 901 3600s / kg K Sne < t fllws tat HXer peratn must be F Frm eq fr lg mean temperature dfferene Δ1 Δ (10 95) (901 35) Λl m F 81 5 ln Δ / Δ ) ln(115 / 551) ( 1 Substtutng numeral values te HXer lengt s L 15705W / 550W / m Kπ (010m) 814K 1 1m mments: e ε NU metd uld als be used It wuld be neessary t perfrm te t flud energy balane t determnng F peratn exsted e apaty rate s mn / max 050 Frm eq fr effetveness and frm wt q evaluated frm an energy balane n te t flud ε 10 901 069 10 35
Frm fg fnd NU 15 gvng W W L NU mn / U π D 15 13094 550 π (010m) 1 14m K m K Nte te gd agreement by bt metds
5 nsder a very lng nentr tube eat exanger avng t and ld water nlet temperatures f 85 and 15 e flw rate f te t water s twe tat f te ld water Assumng equvalent t and ld water spefes eats; determne te t water utlet temperature fr te fllwng mdes f peratn (a) unter flw (b) Parallel flw Knwn: A very lng nentr tube eat exanger avng t and ld water nlet temperatures f 85 and 15 respetvely: flw rate f te t water s twe tat f te ld water Fnd: utlet temperatures fr unter flw and parallel flw peratns Semat: Assumptns: (1) equvalent t and ld water spef eats () Neglgble Knet and ptental energy anges (3) N eat lss t surrundngs Analyss: te eat rate fr a nentr tube Heat exanger wt very large surfae area Operatng n te unter flw mde s
) ( mn max q q mbnng te abve relatn and rearrangng fnd mn ) ( ) ( + + Substtutng numeral values + 50 85 15) (85 1 Fr parallel flw peratn te t and ld utlet temperatures wll be equal; tat s Hene ) ( ) ( Settng and rearrangng + + + + 617 1 1 / 15 1 85 1 / mments: Nte tat wle ε 1 fr F peratn fr PF peratn fnd ε q/q max 067
6 A nentr tube eat exanger uses water w s avalable at 15 t l etylene glyl frm 100 t 60 e water and glyl flw rates are ea 05 kg/s Wat are te maxmum pssble eat transfer rate and effetveness f te exanger? W s preferred a parallel flw r unter flw mde f peratn? Knwn: Inlet temperatures and flw rate fr a nentr tube eat exanger Fnd: (a) Maxmum pssble eat transfer rate and effetveness (b) Prffered mde f peratn Semat: Assumptns: (1) Steady-state peratn () Neglgble KE and PE anges (3) Neglgble eat lss t surrundngs (4) Fxed verall eat transfer and effent Prpertes: able: Etylene glyl ( Water( _ 30 ) : 4178J / kg K m p _ n 80 ); Analyss: (a) Usng te ε-nu metd fnd p650j/kgk;
m p (05kg / s)(650j / kg K ) 135W / K mn q ma x mn ( ) (135W / K )(100 15) 113 10 5 W 5 q m p ( ) 05kg / s(650j / kg K )(100 60) 053 10 W 5 5 ε q / qmax 053 10 / 113 10 047 (b) 5 q 053 10 + 15 + 40 4 05kg / s 4178J / kg K m p Sne < a parallel flw mde f peratn s pssble Hwever wt ( mn / max ) ( / m ) 063 m p Frm fg (NU) PF 095 (NU) F 075 Hene (A F /A PF ) (NU) F/ (NU) PF (075/095)079 Beause f te redued sze requrement ene aptal nvestment te unter flw mde f peratn s prffered p