APPLIED THERMODYNAMICS D201. SELF ASSESSMENT SOLUTIONS TUTORIAL 2 SELF ASSESSMENT EXERCISE No. 1
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1 APPIED ERMODYNAMICS D0 SEF ASSESSMEN SOUIONS UORIA SEF ASSESSMEN EXERCISE No. Show how the umeti effiiey of a ideal sigle stage eioatig ai omesso may be eeseted by the equatio ( / Whee is the leaae atio, the delivey essue ad the idutio essue. A eioatig ai omesso followig the ideal yle has a fee ai delivey of 60 dm /s. he leaae atio is 0.0. he ilet is at atmoshei essue of ba. he delivey essue is 7 ba ad the omessio is olytoi with a idex of.. Calulate the followig. i. he ideal umeti effiiey. (8.7% ii. he ideal idiated owe. (.7 kw Swet ume Idued ume - Cleaae ume Swet ume Idued ume - Cleaae ume ( ( / / / / / ( { } 8.7% /. / [ ] kw x0.. xf.a.d W,.
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3 SEF ASSESSMEN EXERCISE No.. A sigle atig stage omesso daws i 8. m /mi of fee ai ad omesses it to 0 ba. he omesso us at 00 ev/mi. he atmoshei oditios ae.0 ba ad o C. hee is a iteoole betwee stages whih ools the ai bak to o C. he olytoi idex fo all omessios is.. he umeti effiiey is 90% fo the low essue stage ad 8% fo the high essue stage. Igoe the effet of the leaae ume. Calulate the followig. he itemediate essue fo miimum idiated wok. (6.6 ba he theoetial idiated owe fo eah stage. (.8 kw he heat ejeted i eah ylide. (6. kw he heat ejeted by the iteoole. (6. kw he swet umes of both stages. (. dm ad. dm What advatage is thee i usig a afte-oole? State the effet o you aswes of ot igoig the leaae ume ad leakages. 8. m /mi 0.66 m /s N ev/s Idued ume 0.66/ 0.08 m /stoke k (0/ b 6.6 ba IP mr( 0. 88( K.0x0 x 0.66 m 0.76 kg/s R 87 x 88 Note that thee is a oflit hee. If the leaae ume is egleted the 90%( ad 0 so 0.08 m /s ad this will give diffeet aswes.. IP x0.76 x 87 x ( kw 0. Aly the SFEE to the omesso h A + P h B + Φ Φ m ( A B + P Φ 0.76 x.00 ( kw Aly the SFEE to the whole system h A + P - Φ Φ h C h A h C Φ.8 6, 6. kw P CYINDER ol/stoke 0.08 m Atual ume/swet olume S 0.08/ m P CYINDER ( m R m ( 8 R ( S 0.000/ m But sie x 0 ( 8 6.6
4 . A sigle atig stage omesso daws i fee ai ad omesses it to 8. ba. he omesso us at 600 ev/mi. he atmoshei oditios ae.0 ba ad o C. he itestage essue is ba ad the iteoole ools the ai bak to 0 o C. he olytoi idex fo all omessios is.8. Due to the effet of wamig fom the ylide walls ad the essue loss i the ilet valve, the essue ad temeatue at the stat of the low essue omessio stoke is 0.96 ba ad o C. he high essue yle may be take as ideal. he leaae ume fo eah stages is % of the swet ume of that stage. he low essue ylide is 00 mm diamete ad the stoke fo both stages is 60 mm. Calulate the followig. he fee ai delivey. he umeti effiiey of the low essue stage. he diamete of the high essue ylide. he idiated owe fo eah stage. OW PRESSURE SAGE N 600 ev/mi D 00 mm 60 mm C % S S π x 0. / x 0.6 S 0.0 m. dm C % x. 0. m Fo a ideal yle idued ume Fid.8.8 x x. dm dm 0.66 dm Chage this to FAD (ume at.0 b ad o C FAD/stoke 0.66 x (88/98 x (0.96/ dm /stoke FAD 9.76 x dm /mi o.89 m /mi Ideal FAD. x dm /mi o m /mi.89/ % If we eglet the wok of the idutio stoke.8.8 W xf.a.d x 0.96 x 0 / IP W N/60 x 600/60 0 W o. kw IG PRESSURE SAGE Idued mass mass of the FAD m /R.0 x 0 x 9.76 x 0 - /(8 x 88 m kg/stoke 0.8 ( J/yle K / (/8. / (A he mass exelled i oess 6 to 7 is kg 6( x0 x ( R6 87 x m.7 dm...(b 7 0.0( (C Put (A ito (C 6 0. x (D Put (D ito (B dm Put 7 ito (C 6 x dm Put ito (A 6.68 dm S m πd / hee D 0.7 m o 70 mm dm 8 IP x N ( x x x.7kW
5 . A stage eioatig ai omesso has a iteoole betwee stages. he idutio ad exulsio fo both stages ae at ostat essue ad temeatue. All the omessios ad exasios ae olytoi. Negletig the effet of the leaae ume show that the itemediate essue, whih gives miimum, idiated wok is M ( ½ Exlai with the aid of a sketh how the delivey temeatue fom both ylides vaies with the itemediate essue as it hages fom to. W W + W whee W is the wok doe i the low essue stage ad i the high essue stage. mr( - mr(6 - W + ( - ( - Sie the assumig the same value of fo eah stage ( ( / / W mr mr ( - ( - Sie ad ad m 6 (-/ ad 6 ( / ( / M 6 W mr ( + mr - ( - M Fo a miimum value of W we diffeetiate with eset to ad equate to zeo. dw ( / / ( / ( / mr M - mr M d M M W is the wok doe If the iteoole etus the ai to the oigial ilet temeatue so that, the equatig to zeo eveals that fo miimum wok M ( ½ It a futhe be show that whe this is the ase, the wok doe by both stages is equal. he elatioshi betwee the temeatues ad essues ae: ( m / -/ 6 ( / m -/ his odues the lot show. 6 (-/
6 .a. Pove that the ideal umeti effiiey of a sigle stage eioatig omesso is - (/- is the essue atio, is the olytoi idex ad the leaae atio. Sketh uves of agaist fo tyial values of ad. b. A two stage eioatig ai omesso woks betwee essue limits of ad 0 ba. he ilet temeatue is oc ad the olytoi idex is.. Iteoolig betwee stages edues the ai temeatue bak to oc. Fid the fee ai delivey ad mass of ai that a be omessed e kw h of wok iut. Fid the atio of the ylide diametes if the isto have the same stoke. Neglet the effet of the leaae ume. Pat (a Deivatio as i Q he sketh is show below. Pat (b (0 x.7 ba Sie the wok i both stages is equal W kwh 00 x 60 x 60.6 x 0 6 J 6.6 x m e kwh 00 x 0 x 0.06 m. kg/kwh R 87 x 88 mr.7x 87 x 88. m /kwh.7 x 0 swet ume of P stage /.7. m /kwh (ofimatio swet ume of P stage πd / d. 0. d/d 0.7 πd / D 0.06
7 SEF ASSESSMEN EXERCISE No.. Show that fo ay omessio oess the oveall effiiey is give by whee is the olytoi effiiey. O Detemie the idex of omessio fo a gas with a adiabati idex of. ad a olytoi effiiey of 0.9. (.6 Detemie the oveall effiiey whe the essue omessio atio is / ad 8/. (0.879 ad Gas aws lus Comessio aws ae / ad C / / Combiig these we have C x C / d d Diffeetiate d C d Next divide by C -/ / d d Substitute C Fo a isetoi oess desigate the fial temeatue as ad the diffeetial as d d' d...( Isetoi Effiiey is defied as ' is et the hage be ifiitesimally small. d' + d ad + d d Suose the omessio is made fom may tiy ste hages. d' d d' Substitute ( ito this. d d d [ l] [ l] Itegate ad is alled the Polytoi Effiiey adiabati oess Substitutio gives ' O d d he oveall effiiey is o olytoi oess Comae ad it follows that. Now ut 0.9 ad x '
8 Now ut O 0. 7 Now ut O A omesso daws i ai at. K temeatue ad 0.6 ba essue. he omessio atio is 6. he olytoi effiiey is Detemie the temeatue afte omessio. ake. Fom the evious questio we have ( 6. x 0 K
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