Gas reali

Gas reali

Il modello di gas ideale si adatta per i gas a bassa densit. Quanto bassa? Di quanto si discosta dal gas ideale un gas ad una data pressione e temperatura?. Per rispondere a questi due quesiti si introduce un fattore di comprimibilit Z definito dalla:

Z=1 per i gas ideali e la deviazione dallunit misura lo scostamento rispetto al modello ideale. Considerando ad esempio lazoto, Z varia al variare della pressione e della temperatura come segue:

Il diagramma evidenzia alcuni fatti salienti:

a) che Z(1 per p( 0 indipendentemente da T (comportamento ideale) e che alla temperatura ambiente e per pressioni fino a circa 10 MPa il fattore di comprimibilit Z(1. Questo consente di trattare in tale intervallo lN2, e anche laria che composta per 4/5 di azoto, come un gas ideale;

b) labbassamento di T al di sotto di 300 K a pressione costante (per esempio a 4 MPa) comporta un aumento della densit rispetto al comportamento ideale (Z1) in quanto le forze per distanze molecolari molto piccole diventano repulsive;

Per tutti gli altri gas Z mostra andamenti qualitativamente simili per cui possibile costruire sperimentalmente ed utilizzare con buona approssimazione un unico diagramma generalizzato considerando i parametri riferiti alle condizioni critiche (parametri ridotti): pr = p/pc e Tr = T/Tc.

Si osserva che per gas rarefatti Z(1 indipendentemente da Tr, e che lo stesso vale per Tr > 2 purch pr non superi 4(5. Unequazione di stato largamente usata quella proposta da Lee-Kesler:

dove

b10,118119

b20,265728

b30,15479

b40,030323

c10,023674

c20,018698

c40,042724

d11,55E-05

d26,24E-05

0,65392

0,060167

c) Osservando la zona a bassa pressione fino a pr ( 0.1 la pendenza delle isoterme cresce fino a circa Tr=5 per poi diminuire per temperature decrescenti. Per Tr ( 2.5 lisoterma (temperatura di Boyle) diventa orizzontale (Z=1) e il gas si comporta esattamente come un gas ideale per basse pressioni ma finite.

d) Lo scostamento di Z dallunit mette in evidenza la deviazione rispetto al comportamento ideale come mostra il seguente esempio:

Esempio 1

Si calcoli per il propano (C3H8) il volume specifico a 7 MPa e 150C utilizzando il diagramma di Z generalizzato e lo si confronti con quello che si otterrebbe nel caso di gas ideale

pc = 4.25 MPa

Tc = 369.8 K

R = 0.1886 kJ/kgK

pr = 7/4.25 = 1.647Tr = 423/369.8 = 1.144Dal diagramma (o meglio usando la correlazione di Lee-Kesler da cui tratto) si ottiene Z = 0.522 (contrassegnato dal rombo) da cui si ottiene:

B-0,25275

C0,007328

D7,01E-05

SostanzaPropano

Dati

Pc4250kPa

Tc369,9K

R0,1886kJ/kgK

T423K

P7000kPa

Parametri ridotti

Pr1,65

Tr1,14

Ricorsione per il calcolo di v'r

v'rZPr

0,5000,6061,39

0,4210,5601,52

0,3890,5391,59

0,3740,5301,62

0,3680,5261,63

0,3650,5241,64

0,3640,5231,64

0,3630,5231,65

0,3630,5221,65

0,3630,5221,65

Z0,522

v = ZRT/p = 0.522x0.1886x423/7000 = 0.00596 m3/kg

vid =RT/p = v/Z = v/0.522 = 0.0114 m3/kg.

Esercizio 1

Un recipiente della capacit di 0.25 m3 viene mantenuto a 2 MPa e 17C. Calcolare la massa contenuta nel recipiente nei casi di aria, butano e propano.

GasTc (K)pc (kPa)R (kJ/kgK)

Aria132.537700.287

Butano425.238000.143

Propano369.942500.1886

1) Z=0.992 ; m = 6.05 kg2) Z=0.092 ; m = 132.2 kg3) Z=0.0774 ; m = 118.1 kg

Esercizio 2

Calcolare il volume specifico del refrigerante R134a (il tetrafluoroetano, CF3CH2F, un refrigerante di nuova generazione a bassissimo potenziale distruttivo dellozono stratosferico) a 100C e 3 MPa allo stato di vapore surriscaldato: (a) nellipotesi di gas ideale, (b) usando il fattore di compressibilit Z. Confrontare i risultati con il valore pi attendibile disponibile in letteratura (v=0.00665 m3/kg).

Tc = 374.2 K pc = 4.06 MPa M = 102.03 kg/kmol

Esercizio 3

Una massa di 2 kg di acetilene (C2H2) mantenuta alla pressione di 4.3 MPa in un recipiente a pareti rigide della capacit di 0.045 m3. Calcolare con procedimento ricorsivo la temperatura.

Tc = 308.3 K pc = 6140 kPa M = 26.038 kg/kmol

_1145076835.unknown

_1233988706.unknown

_1233988838.unknown

_1233988567.unknown

_1113544076.unknown

_1113544084.unknown

_1113571403.xlsGrafico1

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0.26808126570.5951480170.8906088781.16760321021.247165584

0.23446305490.56791391560.86369638761.15414433461.238960335

0.22007543780.55055852480.84325973921.14214791171.2312420916

0.21647015670.53977287940.82762635281.1314078641.2239702988

0.2189027570.53346984690.81562464211.12175379151.217108703

0.22471271890.5303037290.8064158311.11304368561.2106248014

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0.25051374340.53206888450.79015151311.09147416371.1931611186

0.26019195930.53495128630.78733169271.08551649021.1879229342

0.27005212410.53854138490.78541273051.08006111061.1829419555

0.2800064070.54267414430.78422940711.07505369111.1782003779

0.29000001720.54722292370.78364947961.0704470961.1736819705

0.29999592010.55208875590.78356607691.06620026071.1693719071

0.30996669040.55719312470.78389210851.06227726811.1652566199

0.35876891130.5844557590.78955002361.04653718481.1471999694

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0.51824697280.68254023280.82827646141.01666061881.1014578924

0.54851611410.70171083760.83723341791.01306737481.0939862241

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0.64133927570.76116230.86657076211.00464218281.0723416731

0.65916333520.772688210.87248634561.00341396731.0683574517

0.6753646570.78319533560.87793552271.00239651761.0647758935

0.69014388310.79280563340.88296483381.00154836021.0615400973

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0.71609676380.80974069580.89192802811.00023875171.0559265297

0.72754268680.81723351910.8959332130.99973248311.0534774465

0.73811826680.82416970350.89966175240.99930288211.0512286513

0.81162810140.87273300910.92629899480.99730779491.0359591809

0.85305831380.90037534120.94185062030.99693450161.0276325083

0.89801354210.93059236250.95915164320.99713949431.0188403844

0.92192376740.94675821230.96853108850.99750902651.0142760986

0.93675650270.95681958370.97441087470.99783003971.0114868676

0.94685476340.96368382570.9784405350.99808907421.009607394

0.95417286120.96866552820.98137421810.99829739421.0082555639

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0.96757074110.97780175210.98677430110.99872387861.005803656

0.97045055870.97976820710.98793993140.99882319771.0052806209

0.97286065360.98141463970.98891676840.99890840431.004844007

0.97490727680.98281328730.98974723770.9989822561.0044740405

0.97666688250.98401616880.99046194020.999046851.0041565502

0.97819588850.98506169520.99108350330.9991038051.0038811152

0.97953683630.98597884630.99162901850.99915438771.0036399014

0.98072240950.98678989740.99211163690.99919960241.0034269062

0.98177813320.98751225230.99254164270.99924025461.003237453

0.98272423180.98815970590.9929271940.99927699771.003067844

0.98357693710.98874333340.99327484530.99931036661.0029151174

0.98434942720.98927212930.99358992290.99934080351.0027768726

0.98505251240.98975347380.99387679990.9993686771.0026511434

0.98569514390.99019347960.99413910.99939429671.0025363042

0.98628479720.99059725260.99437985160.99941792431.002430999

0.98682776420.99096909130.99460160520.99943978281.0023340882

0.98732937820.99131263990.99480652430.99946006311.0022446066

0.98779419010.99163100870.99499645550.99947892981.0021617316

0.98822610670.9919268680.99517298520.99949652531.0020847577

0.98862850060.99220252340.99533748350.99951297361.0020130763

0.98900429870.99245997630.99549113970.99952838321.0019461598

0.98935605330.99270097180.99563499150.99954284921.0018835484

0.98968600050.99292703920.99576994840.99955645571.0018248396

0.9899961070.99313952370.99589681050.99956927681.0017696797

0.99028811030.99333961390.9960162850.99958137851.0017177563

0.99056355070.99352836350.99612899880.99959281981.0016687927

0.99082379850.993706710.99623551010.9996036531.0016225428

0.99107007690.99387549030.99633631710.99961392541.0015787872

0.99130348120.99403545420.9964318660.99962367941.0015373294

0.99152499530.99418727470.99652255780.99963295321.0014979931

0.99173550520.99433155820.99660875380.99964178131.0014606194

0.99193581080.99446885250.996690780.99965019521.0014250651

0.99212663680.99459965340.99676893190.99965822321.0013912004

0.99230864030.99472441080.99684347770.99966589141.0013589078

0.99248241950.99484353420.99691466130.99967322331.0013280803

0.99264851950.99495739680.99698270530.99968024051.0012986203

0.99280743830.99506633950.99704781270.9996869631.0012704389

0.99295963170.99517067450.99711016970.99969340881.0012434546

0.99310551790.02478880920.9971699470.99969959471.0012175927

Tr= 1

Tr= 1,2

Tr= 1,5

Tr= 2,5

Tr= 5

spot

Pressione ridotta, Pr

Fattore di Compressibilit, Z

Fattore di compressibilit - Modello Lee-Kesler(fluidi semplici)

Grafico

b10.1181193SostanzaPropano

b20.265728

b30.15479Dati

b40.030323Pc4250kPa

c10.0236744Tc369.8K

c20.0186984R0.1886kJ/kgK

c30T423K

c40.042724P7000kPa

d11.55E-05

d26.24E-05

b0.65392Parametri ridotti

g0.060167Pr1.647

Tr1.144

B-0.2527521716

C0.0073276664

D7.01E-05Ricorsione per il calcolo di v'r

v'rZPr

0.5000.6061.39

0.4210.5601.52

0.3890.5401.59

0.3750.5311.62

0.3690.5261.63

0.3660.5251.64

0.3640.5241.64

0.3640.5231.65

0.3630.5231.65

0.3630.5231.65

Z0.523

Metodo Bisezione

v'rZ

inf0.500-0.114Z0.523

sup0.2000.193vr0.0059623988m3/kg

media0.350.010vid0.0113968286m3/kg

inf0.5-0.114

sup0.350.010

media0.425-0.049

inf0.425-0.049

sup0.350.010

media0.3875-0.019

inf0.3875-0.019

sup0.350.010

media0.36875-0.004

inf0.36875-0.004

sup0.350.010

media0.3593750.003

inf0.36875-0.004

sup0.3593750.003

media0.3640625-0.001

inf0.3640625-0.001

sup0.3593750.003

media0.361718750.001

inf0.3640625-0.001

sup0.361718750.001

media0.3628906250.000

inf0.3640625-0.001

sup0.3628906250.000

media0.3634765625-0.000

inf0.3634765625-0.000

sup0.3628906250.000

media0.36318359380.000

inf0.3634765625-0.000

sup0.36318359380.000

media0.3633300781-0.000

Grafico

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1.13650785790.97866550571.13484420341.24207358481.2858583314

0.70294104360.809430441.0413923521.21957634691.2751879723

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0.2189027570.53346984690.81562464211.12175379151.217108703

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0.29000001720.54722292370.78364947961.0704470961.1736819705

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0.35876891130.5844557590.78955002361.04653718481.1471999694

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0.69014388310.79280563340.88296483381.00154836021.0615400973

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0.71609676380.80974069580.89192802811.00023875171.0559265297

0.72754268680.81723351910.8959332130.99973248311.0534774465

0.73811826680.82416970350.89966175240.99930288211.0512286513

0.81162810140.87273300910.92629899480.99730779491.0359591809

0.85305831380.90037534120.94185062030.99693450161.0276325083

0.89801354210.93059236250.95915164320.99713949431.0188403844

0.92192376740.94675821230.96853108850.99750902651.0142760986

0.93675650270.95681958370.97441087470.99783003971.0114868676

0.94685476340.96368382570.9784405350.99808907421.009607394

0.95417286120.96866552820.98137421810.99829739421.0082555639

0.95971979660.97244559530.98360537490.9984668741.0072367722

0.96406907280.97541194080.98535930680.99860676531.006441557

0.96757074110.97780175210.98677430110.99872387861.005803656

0.97045055870.97976820710.98793993140.99882319771.0052806209

0.97286065360.98141463970.98891676840.99890840431.004844007

0.97490727680.98281328730.98974723770.9989822561.0044740405

0.97666688250.98401616880.99046194020.999046851.0041565502

0.97819588850.98506169520.99108350330.9991038051.0038811152

0.97953683630.98597884630.99162901850.99915438771.0036399014

0.98072240950.98678989740.99211163690.99919960241.0034269062

0.98177813320.98751225230.99254164270.99924025461.003237453

0.98272423180.98815970590.9929271940.99927699771.003067844

0.98357693710.98874333340.99327484530.99931036661.0029151174

0.98434942720.98927212930.99358992290.99934080351.0027768726

0.98505251240.98975347380.99387679990.9993686771.0026511434

0.98569514390.99019347960.99413910.99939429671.0025363042

0.98628479720.99059725260.99437985160.99941792431.002430999

0.98682776420.99096909130.99460160520.99943978281.0023340882

0.98732937820.99131263990.99480652430.99946006311.0022446066

0.98779419010.99163100870.99499645550.99947892981.0021617316

0.98822610670.9919268680.99517298520.99949652531.0020847577

0.98862850060.99220252340.99533748350.99951297361.0020130763

0.98900429870.99245997630.99549113970.99952838321.0019461598

0.98935605330.99270097180.99563499150.99954284921.0018835484

0.98968600050.99292703920.99576994840.99955645571.0018248396

0.9899961070.99313952370.99589681050.99956927681.0017696797

0.99028811030.99333961390.9960162850.99958137851.0017177563

0.99056355070.99352836350.99612899880.99959281981.0016687927

0.99082379850.993706710.99623551010.9996036531.0016225428

0.99107007690.99387549030.99633631710.99961392541.0015787872

0.99130348120.99403545420.9964318660.99962367941.0015373294

0.99152499530.99418727470.99652255780.99963295321.0014979931

0.99173550520.99433155820.99660875380.99964178131.0014606194

0.99193581080.99446885250.996690780.99965019521.0014250651

0.99212663680.99459965340.99676893190.99965822321.0013912004

0.99230864030.99472441080.99684347770.99966589141.0013589078

0.99248241950.99484353420.99691466130.99967322331.0013280803

0.99264851950.99495739680.99698270530.99968024051.0012986203

0.99280743830.99506633950.99704781270.9996869631.0012704389

0.99295963170.99517067450.99711016970.99969340881.0012434546

0.99310551790.02478880920.9971699470.99969959471.0012175927

Tr= 1

Tr= 1,2

Tr= 1,5

Tr= 2,5

Tr= 5

spot

Pressione ridotta, Pr

Fattore di Compressibilit, Z

Fattore di compressibilit - Modello Lee-Kesler(fluidi semplici)

Calcoli

b10.1181193Tr=v'rZPrTr=v'rZPrTr=v'rZPrTr=v'rZPrTr=v'rZPrTr=v'rZPrTr=v'rZPrTr=v'rZPr

b20.2657280.50.10001.4617.310.70.1111.3898.760.90.121.49711.2310.121.92416.031.20.141.25510.761.50.161.26911.902.50.251.26812.6850.381.29717.07

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_1072960604.unknown