APPLIED THERMODYNAMICS TUTORIAL 2 GAS COMPRESSORS

Transcription

1 APPLIED HERMODYNAMICS UORIAL GAS COMPRESSORS I order to omlete this tutorial you should be familiar with gas laws ad olytroi gas roesses. You will study the riiles of reiroatig omressors i detail ad some riiles of rotary omressors. O omletio you should be able to the followig. Desribe the workig riiles of reiroatig omressors. Defie ad alulate swet volume. Defie ad alulate volumetri effiiey. Defie ad alulate isothermal effiiey. Defie ad alulate idiated ower. Dtate the beefits of oolig. Calulate the heat rejeted through oolig. Defie ad alulate the iterstage ressures for multile omressors. Defie olytroi effiiey. Let s start by osiderig the geeral use of omressed air. D.J.Du

2 . COMPRESSED AIR. YPES Air is a exasive substae ad dagerous whe used at high ressures. For this reaso, most aliatios are ofied to thigs requirig low ressures (0 bar or lower) but there are idustrial uses for high ressure air u to 00 bar. he ommo soure of the air is the omressor. here are may tyes of omressors with differet workig riiles ad workig oditios. hese are examles. Reiroatig. Slidig vae omressors. Lobe omressors. Helial srews. Cetrifugal. Axial turbie omressors. he futio of all of them is to draw i air from the atmoshere ad rodue air at ressures substatially higher. Usually a storage vessel or reeiver is used with the omressor. he same riiles are alied to the omressio of other gasses. his tutorial is maily about reiroatig omressors. he other tyes are overed briefly. he reiroatig omressor is robably the most versatile of all the tyes ad is oly out erformed by rotary tyes whe large volumes at low ressures are required. For high ressures, the reiroatig omressor is almost uiversal.. AMOSPHERIC APOUR Air ad vaour mixtures are overed i detail i a later tutorial. We should ote, however, the effets it has o the erformae of a air omressor. Atmosheri air otais WAER APOUR mixed with the other gases. Whe the air is ooled to the dew oit, the vaour odeses ito water ad we see rai or fog. he ratio of the mass of water vaour i the air to the mass of the air is alled the ABSOLUE HUMIDIY. he quatity of water that a be absorbed ito the air at a give ressure deeds uo the temerature. he hotter the air, the more water it a evaorate. Whe the air otais the maximum ossible amout of vaour it is at its dew oit ad rai or fog will aear. he air is the said to have 00% humidity. Whe the air otais o water vaour at all (dry air), it has 0% humidity. his refers to RELAIE HUMIDIY. For examle if the air has 0% humidity it meas that it otais 0% of the maximum that it ould otai. here are various ways to determie the humidity of air ad istrumets for doig this are alled HYGROMEERS. he imortae of humidity to air omressors is as follows. Whe air is suked ito the omressor, it brigs with it water vaour. Whe the air is omressed the ressure ad the temerature of the air goes u ad the result is that the omressed air will have a relative humidity of about 00% ad it will be warm. Whe the air leaves the omressor it will ool dow ad the water vaour will odese. Water will the log the omressor, the reeiver ad the ies. D.J.Du

3 Water auses damage to air tools, ruis ait srays ad orrodes ies ad equimet. For this reaso the water must be removed ad the best way is to use a well desiged omressor istallatio ad distributio etwork.. YPICAL COMPRESSOR LAYOU he diagram below shows the layout of a two stage reiroatig omressor tyially for sulyig a worksho. Figure. Idutio box ad sileer o outside of buildig with ourse sree.. Idutio filter.. Low ressure stage.. Iterooler. 5. High ressure stage. 6. Sileer. 7. Drai tra. 8. After ooler 9. Pressure gauge. 0. Air reeiver.. Safety ressure relief valve.. Sto valve. FREE AIR DELIERY Whe a gas suh as air flows i a ie, the mass of the air deeds uo the ressure ad temerature. It would be meaigless to talk about the volume of the air uless the ressure ad temerature are osidered. For this reaso the volume of air is usually stated as FREE AIR DELIERY or FAD. FAD refers to the volume the air would have if let out of the ie ad retured to atmosheri ressure at the same temerature. he FAD is the volume of air draw ito a omressor from the atmoshere. After omressio ad oolig the air is retured to the origial temerature but it is at a higher ressure. Suose atmosheri oditios are aa ad a (the FAD) ad the omressed oditios are, ad. D.J.Du

4 Alyig the gas law we have a aa a a a F. A. D.. CYCLE FOR RECIPROCAING COMPRESSOR. HEOREICAL CYCLE he diagram shows the basi desig of a reiroatig omressor. he isto reiroates drawig i gas, omressig it ad exellig it whe the ressure iside the ylider reahes the same level as the ressure i the delivery ie. Figure If the isto exels all the air ad there is o restritio at the valves, the ressure - volume yle is as show below. Figure Gas is idued from to at the ilet ressure. It is the traed iside the ylider ad omressed aordig the law C. At oit the ressure reahes the same level as that i the delivery ie ad the outlet valve os oe. Air is the exelled at the delivery ressure. he delivery ressure might rise very slightly durig exulsio if the gas is beig omated ito a fixed storage volume. his is how ressure builds u from swith o. D.J.Du

5 . OLUMERIC EFFICIENCY I reality, the isto aot exel all the gas ad a learae volume is eeded betwee the isto ad the ylider head. his meas that a small volume of omressed gas is traed i the ylider at oit. Whe the isto moves away from the ylider head, the omressed gas exads by the law C util the ressure falls to the level of the ilet ressure. At oit the ilet valve oes ad gas is draw i. he volume draw i from to is smaller tha the swet volume beause of this exasio. Figure he volumetri effiiey is defied as Idued olume vol Swet olume his effiiey is made worse if leaks our ast the valves or isto. he learae ratio is defied as Clearae volume/swet volume. Ideally the roess to ad to are isothermal. hat is to say, there is o temerature hage durig idutio ad exulsio. D.J.Du 5

6 WORKED EXAMPLE No. Gas is omressed i a reiroatig omressor from bar to 6 bar. he FAD is dm /s. he learae ratio is he exasio art of the yle follows the law. C. he rak seed is 60 rev/mi. Calulate the swet volume ad the volumetri effiiey. SOLUION Swet olume Clearae volume 0.05 Cosider the exasio from to o the - diagram. bar 6 bar... 6(0.05). (. ) 0. or.%% of F.A.D. 0.0 m/s. Idued volume Idued volume / m/s Crak seed 6 rev/s so the swet volume 0.057/6.6 dm vol Idued olume Swet olume 0.88 vol 8.8 % D.J.Du 6

7 D.J.Du 7 WORKED EXAMPLE No. Show that if the learae ratio of a ideal sigle stage reiroatig omressor is that the volumetri effiiey is give by ) / ( L H vol where L is the ilet ressure ad H the outlet ressure. SOLUION Swet volume Idued volume - Clearae volume ( ) ( ) / / / / / L H vol L H vol L H vol L H vol vol

8 I real omressors the warm ylider auses a slight temerature rise over the idutio from to. he gas is restrited by the valves ad is slightly less tha. he valves also ted to move so the real yle looks more like this. Figure 5 D.J.Du 8

9 WORKED EXAMPLE No. A sigle stage reiroatig omressor rodues a FAD of dm /s at 0 rev/mi. he ilet oditios are bar ad 0 o C. he olytroi idex is. for the omressio ad exasio. he outlet ressure is 8 bar. he learae volume is 0 m. Due to the restritio of the ilet valve ad the warmig effet of the ylider walls, the ressure at the start of omressio is 0.97 bar ad the temerature is 7 o C. Determie the volumetri effiiey. SOLUION Beause the idutio stroke is either at ostat ressure or ostat temerature, we must solve the swet volume by usig the exulsio stroke, whih is assumed to be at ostat ressure ad temerature. he umbers of the yle oits are as before K 0.97 x60 F. A. D. er stroke dm er stroke xaH he omressed volume exelled Exulsio volume 8x dm x 8x dm 0.. hee. Idued volume Swet volume vol % xx. 0.05dm 8x dm 0.06 dm 0.5 dm a H D.J.Du 9

10 D.J.Du 0. INDICAED POWER he idiated work er yle is the area elosed by the - diagram. he easiest way to fid this is by itegratig with reset to the ressure axis. Figure 6 he roesses to ad to are olytroi C. C / -/ C / Cosider the exressio ) ( ) ( ) / ( / / / / / / C d C d Betwee the limits of ad this beomes ( ) Betwee the limits ad this beomes ( ) he idiated work (iut) is the W ( ) ( ) r r W W W Where r is the ressure ratio. Sie this redues to [ ] r W. Where - is the swet volume.

11 If the learae volume is egleted this beomes W r Sie mr W mr r Note that if the roess was isothermal ad the the itegratio would yield W mr l ( / ) m is the mass omressed eah yle ad W is the idiated work er yle. he idiated ower is foud by multilyig W by the strokes er seod. I.P. W x N where N is the shaft seed i Rev/s If the learae volume is egleted, the mass omressed is the mass exelled. I this ase the atual mass flow rate delivered may be used for m ad W beomes the idiated ower.. ISOHERMAL EFFICIENCY he miimum idiated ower is obtaied whe the idex is a miimum. he ideal omressio is hee isothermal with. he isothermal effiiey is defied as (iso) Isothermal work/atual work ( ) l( / ) ( ) Note that i the ideal ase, ad are the ilet ad outlet temeratures. D.J.Du

12 SELF ASSESSMEN EXERCISE No. Show how the volumetri effiiey of a ideal sigle stage reiroatig air omressor may be rereseted by the equatio vol ( / ) H L Where is the learae ratio, H the delivery ressure ad L the idutio ressure. A reiroatig air omressor followig the ideal yle has a free air delivery of 60 dm /s. he learae ratio is he ilet is at atmosheri ressure of bar. he delivery ressure is 7 bar ad the omressio is olytroi with a idex of.. Calulate the followig. i. he ideal volumetri effiiey. (8.7%) ii. he ideal idiated ower. (.7 kw) D.J.Du

13 . MULIPLE COMPRESSOR SAGES. HE EFFEC OF INERCOOLING he advatage of omressig the fluid i stages is that iteroolers may be used ad the overall omressio is earer to beig isothermal. Cosider the - diagram for a two stage omressor. Figure 7 he yle to is a ormal yle oduted betwee L ad M. he air is exelled durig roess to at M ad ostat temerature. he air is the ooled at the itermediate ressure ad this auses a otratio i the volume so that the volume eterig the high ressure stage is 5 ad ot. he high ressure yle is the a ormal yle oduted betwee M ad H. he shaded area of the diagram reresets the work saved by usig the iterooler. he otimal savig is obtaied by hoosig the orret itermediate ressure. his may be foud as follows.. OPIMAL INERSAGE PRESSURE W W + W where W is the work doe i the low ressure stage ad W is the work doe i the high ressure stage. mr( - ) mr(6-5 ) W + ( -) ( -) (-/) (-/) 6 Sie ad the assumig the same value of for eah stage 6 W mr + mr ( -) ( -) ( / ) ( / ) 6 5 D.J.Du

14 Sie dw mr d ( / ) ( / ) ( )/ / ( ) / ( )/ L m M - mr H M 6 H W mr + mr ( -) L ( -) M For a miimum value of W we differetiate with reset to ad equate to zero. M 5 ad 6 5 H ad M If the iterooler returs the air to the origial ilet temerature so that 5, the equatig to zero reveals that for miimum work M ( L H ) ½ It a further be show that whe this is the ase, the work doe by both stages are equal. Whe K stages are used, the same roess reveals that the miimum work is doe whe the ressure ratio for eah stage is ( L / H ) /K L M D.J.Du

15 WORKED EXAMPLE No.5 A sigle atig reiroatig omressor rus at 60 rev/mi ad takes i air at bar ad 5 o C ad omresses it i stages to 6 bar. he free air delivery is m /s. here is a iterooler betwee eah stage, whih returs the air to 5 o C. Eah stage has oe isto with a stoke of 00 mm. Calulate the followig. he ideal iterstage ressure. he ideal idiated ower er stage. he heat rejeted from eah ylider. he heat rejeted from eah iterooler. he isothermal effiiey. he swet volume of eah stage. he bore of eah ylider. Igore leakage ad the effet of the learae volume. he idex of omressio is. for all stages. SOLUION Pressure ratio for eah stage (6/) / Hee the ressure after stage is x bar. he ressure after the seod stage is x 6 bar he fial ressure is 6 x 6 bar. 88 K. m /R x 0 5 x /(87 x 88) kg/s 88() 0./ K he idiated ower for eah stage is the same so it will be alulated for the st. stage. - mr I. P. x sie m is the mass omressed x 87 x. x 88. I. P. 96 Watts. - D.J.Du 5

16 CYLINDER COOLING Cosider the eergy balae over the first stage. Balaig the eergy we have Figure 8 H A + P(i) H B + Φ (out) Φ (out) P(i) - mc( B - A ) Φ (out) x.005 ( ) Φ (out).78 kw (rejeted from eah ylider) INERCOOLER Now osider the Iterooler. No work is doe ad the temerature is ooled from to 5. Φ (out) mc( - 5) x.005 ( ) 7.9 kw ISOHERMAL EFFICIENCY he ideal isothermal ower mrl(/) er stage. P(isothermal) x 87 x 88 l 7.86 kw (iso) 7.86/ % SWEP OLUMES Cosider the first stage. he F.A.D. is m /s. I the ideal ase where the air is draw i at ostat temerature ad ressure from the atmoshere, the FAD is give by FAD Swet olume x Seed ad the seed is 6 rev/s D.J.Du 6

17 Hee S.. (st. Stage) / m S.. Bore Area x Stroke π D / x 0. D 0.7 m. Now osider the seod stage. he air is retured to atmosheri ressure at ilet with a ressure of bar. he volume draw is hee / of the origial FAD. he swet volume of the seod stage is hee 0.009/ m π D / x 0. hee D 0.7 m By the same reasoig the swet volume of the third stage is S(rd stage) 0.009/ m π D / x 0. D m D.J.Du 7

18 SELF ASSESSMEN EXERCISE No.. A sigle atig stage omressor draws i 8.5 m /mi of free air ad omresses it to 0 bar. he omressor rus at 00 rev/mi. he atmosheri oditios are.0 bar ad 5 0 C. here is a iterooler betwee stages whih ools the air bak to 5 o C. he olytroi idex for all omressios is.. he volumetri effiiey is 90% for the low ressure stage ad 85% for the high ressure stage. Igore the effet of the learae volume. Calulate the followig. he itermediate ressure for miimum idiated work. (6.65 bar) he theoretial idiated ower for eah stage. (.85 kw) he heat rejeted i eah ylider. (6. kw) he heat rejeted by the iterooler. (6.5 kw) he swet volumes of both stages. (. dm ad 5. dm ) What advatage is there i usig a after-ooler? State the effet o your aswers of ot igorig the learae volume ad leakages.. A sigle atig stage omressor draws i free air ad omresses it to 8.5 bar. he omressor rus at 600 rev/mi. he atmosheri oditios are.0 bar ad 5 o C. he iterstage ressure is bar ad the iterooler ools the air bak to 0 o C. he olytroi idex for all omressios is.8. Due to the effet of warmig from the ylider walls ad the ressure loss i the ilet valve, the ressure ad temerature at the start of the low ressure omressio stroke is 0.96 bar ad 5 o C. he high ressure yle may be take as ideal. he learae volume for eah stages is % of the swet volume of that stage. he low ressure ylider is 00 mm diameter ad the stroke for both stages is 60 mm. Calulate the followig. he free air delivery. (5.858m /mi) he volumetri effiiey of the low ressure stage. (86. %) he diameter of the high ressure ylider. (7 mm) he idiated ower for eah stage. (.6 kw ad. kw) D.J.Du 8

19 . A stage reiroatig air omressor has a iterooler betwee stages. he idutio ad exulsio for both stages are at ostat ressure ad temerature. All the omressios ad exasios are olytroi. Negletig the effet of the learae volume show that the itermediate ressure, whih gives miimum, idiated work is M ( L H ) ½ Exlai with the aid of a sketh how the delivery temerature from both yliders varies with the itermediate ressure as it hages from L to H..a. Prove that the ideal volumetri effiiey of a sigle stage reiroatig omressor is vol - (r/-) r is the ressure ratio, is the olytroi idex ad the learae ratio. Sketh urves of vol agaist r for tyial values of ad. b. A two stage reiroatig air omressor works betwee ressure limits of ad 0 bar. he ilet temerature is 5oC ad the olytroi idex is.. Iteroolig betwee stages redues the air temerature bak to 5oC. Fid the free air delivery ad mass of air that a be omressed er kw h of work iut. (0.06 m/kw h.7 kg/kw h) Fid the ratio of the ylider diameters if the isto have the same stroke. Neglet the effet of the learae volume. (d/d 0.7) D.J.Du 9

20 . POLYROPIC or SMALL SAGE EFFICIENCY his is a alterative way of aroahig isetroi effiiey. I this method, the omressio is suosed to be made u of may stages, eah raisig the ressure a small amout. he theory alies to ay tye of omressor. For a adiabati gas omressio the law of omressio ad the gas law may be ombied. d C Divide by d d C d d C d d d Sie differetiate ad C For a isetroi omressio, let the fial temerature be desigated ' ad the hage i temerature be '. d ' + d d...() Isetroi effiiey ad + d ' Figure 9 Let the hage be ifiitesimally smallsuh that ' d ' he ratio is desigated d is alled the POLYROPIC EFFICIENCY. ' D.J.Du 0

21 D.J.Du If we thik of the omressio as beig made u of may tiy stes eah with the same value of. l l ' ' d d Itegrate d d d d d d is the startig temerature ad is the fial temerature. he overall effiiey is 0 ' as exeted. with Comarig SUBSIUE PROCESS POLYROPIC PROCESS ADIABAIC ' ' ' ' r r r r r r r r r r r O his theory may be alied to exasios as well as omressios. It may also be alied to exasios i ozzles. I steam work, it is more usual to use the REHEA FACOR, whih is based o the same riile.

22 WORKED EXAMPLE 6 A omressor draws i air at 5 o C ad 0. bar. he air is omressed to.6 bar with a olytroi effiiey of Determie the temerature ad the isetroi effiiey. ake. SOLUION (0.86) 88 ' 88(5.) is ( 5.) K 6.5K D.J.Du

23 SELF ASSESSMEN EXERCISE No.. Show that for ay omressio roess the overall effiiey is give by r O r where is the olytroi effiiey. Determie the idex of omressio for a gas with a adiabati idex of. ad a olytroi effiiey of 0.9. (.65) Determie the overall effiiey whe the ressure omressio ratio is / ad 8/. (0.879 ad 0.866). A omressor draws i air at. K temerature ad 0.65 bar ressure. he omressio ratio is 6. he olytroi effiiey is Determie the temerature after omressio. ake. (05 K) D.J.Du