Steam properties.docx

Steam properties.docx

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Steamproperties

30/10/2014

Steamandsilencers

Summary

2methodscanbeusedtodeterminetheparametersofthesonicneck :

∙AgraphicalmethodwithEXCELfromthesteamtables（saturatedsteam）byusingtheratioxwhichistheratioofthemassofthegaseousphasedividedbythemassofthemixture.

∙Amethodusingrelationshipssimilartothoseofperfectgases,withanisentropiccoefficientkinsteadof

.

TheygivethesameresultsastheBertincalculationofatypicalcase.

Then,concerningthevelocityofthejetdownstreaminfinite（wherethepressureistheatmosphericpressure）:

Wehavetoconsiderthatitisanadiabaticexpansion.

Onecanfindverydetailedsteamtablesonthefollowingsite:

sitehttp:

//webbook.nist.gov/chemistry/fluid.

Tabledesmatières

1Goal3

2Steamtable3

3Caseofstudy4

3.1Personalinterpretation5

4Steam5

4.1Enthalpydiagrams5

4.2Isentropiccurves6

4.3Relationshipbetweenpressureanddensity7

4.4Isentropicexpansionrelationships8

4.4.1Zeunerrelationship8

4.4.2Eulerequation8

4.4.3Soundspeeddefinition8

4.4.4Continuity9

4.4.5Additionallaws9

5Backtothecasestudy10

5.1Usingisentropicrelationships10

5.2CurrentBertincalculation11

5.3Downstream12

1Goal

ExplainthethermodynamicalbehaviourusedtocomputeanddesigntheBertinsilencers.

2Steamtable

Websitewhereonecandownloadveryaccuratedata:

http:

//webbook.nist.gov/chemistry/fluid/

http:

//webbook.nist.gov/cgi/fluid.cgi?

TLow=100&THigh=300&TInc=1&Applet=on&Digits=5&ID=C7732185&Action=Load&Type=SatP&TUnit=C&PUnit=MPa&DUnit=kg%2Fm3&HUnit=kcal%2Fmol&WUnit=m%2Fs&VisUnit=uPa*s&STUnit=N%2Fm&RefState=DEF

3Caseofstudy

Theimportantparametersforthesilencerdesignare :

∙Thetotalsectionnecessaryfortheflowrateattherequiredpressure（sonicneck）

∙Thejetspeedatthedownstreaminfiniteconditionwhichisusedtocalculatetheacousticpowerforafreejet.

Wechooseanumericalapplicationforwhichwehavetheresultsofcalculation:

∙NISteamDumpSilencers-DumpingModuleandAcousticsDimensioning-Réf.:

05483-005-DC002-A

∙Pages5et6,itiswritten:

∙

∙

∙

3.1Personalinterpretation

Asthevalvebehaviourisunknown,wejustconsidertheupstreamconditions（3）ofthevalve

Thenweadmitthattheexpansionisadiabatic（isentropic）for2reasons:

ØAsthesteamvelocityattheneckwillbehigh（soundspeed）weadmitthatthereisnoheatexchangewithenvironment;

ØWehavesimplerelationshipsinthiscase.

Thenwemanagetohaveasonicflowattheneck.

Then,weadmitthatattheoutlet,thepressuredecreasestoreachtheatmosphericpressureinthefarfield.Wehavestillanadiabaticexpansionandwecancalculatethesteamvelocityattheatmosphericpressure.

4Steam

Thesteamisnotaperfectgas.

4.1Enthalpydiagrams

Weusetheenthalpydiagramsversusentropyforsaturatedsteambecausethetransformationisassumedtobeisentropic（adiabatic）.

Above:

Enthalpyforvariousxcoefficients.

Thecurvex=1isthesaturatedsteamcurve.

Onthisgraph:

Theinitialenthalpyis2765kJ/kg.Thenwehaveanisentropicexpansionuptoh1.Thetitrexis0,9.Itmeansthatthecorrespondingsteammixturecontains90%ofgas（inmass）.

4.2Isentropiccurves

Theyareeasytobuildwhentheenthalpydecreasesbecausewecanusethetitrex.

Fortheentropy,wehave:

Where

istheentropyoftheliquidphaseand

theentropyofthegaseousphase.

Itisthesamewithenthalpyandvolume（inm^3/kg）.

Wecaneasilycomputethexcoefficientcorrespondingtotheentropywherex=1whichistheentropyforsaturatedsteam.Forx=1wehave:

Thereforeforeachlineofthetablewecancompute:

Thenwecomputeforeachline:

Thepressureandtemperaturedonotchangewhenxchanges.

NotaBene:

xcannotexceed1.

4.3Relationshipbetweenpressureanddensity

Thankstothesteamtable,wehavenowonanisentropiccurvethevaluesofthedensityandpressureofthemixture.Wecanseethat:

Hereafteraresomeisentropiccoefficientsforsomevaluesoftheentropy:

S（kJ/kg/°K）

k

6,9894

1,1376

6,014

1,1213

5,7739

1,1093

5,5026

1,0909

5,2915

1,0733

4.4Isentropicexpansionrelationships

Inthefollowingequationsweneverassumethatsteamisaperfectgas.

4.4.1Zeunerrelationship

ONLYifthetransformationisadiabatic（isentropic）:

Theenthalpyishanduisthefluidvelocity.

Betweentheinitialstateandanycurrentpoint:

Wederivethisrelationship:

Infact,itisthethermodynamicalequationintheparticularcaseofanadiabatictransformation.

4.4.2Eulerequation

Withoutexternalforceswork :

UsingZeunerequation,wehavealso:

4.4.3Soundspeeddefinition

Thesoundspeedisgivenbythepartialderivativeofpressureversusdensityinisentropicconditions :

4.4.4Continuity

Itistheconservationofthemass,andthusofthemassflowrate:

WhereAisthesection.

Therefore:

4.4.5Additionallaws

Wecanobtainadditionallawswhichareverysimilartothoseforperfectgases.

Using :

Wherekdependsupontheentropy（butisconstantforagivenentropy）weobtainafterboringcalculationsthefollowingrelationships :

ØWhenthedownstreamvelocityisneglected:

Massflowrateforasubsonicflow:

Criticalpressureforasonicflowrate:

Velocityattheneck（soundvelocity） :

Massflowrateforasonicflow:

5Backtothecasestudy

Inlet（i）

DumpingModule（3）

Sonicneck（c）

Outlet（o）

Pi=7.6MPaabs

Ti=292°C

Hi=2765kJ/kg

P3=4.56MPaabs

T3=258°C

Tc

Po=0.1MPaabs

To=145°C

Ho=Hi=2765kJ/kg

Nota:

V3isabout80m/s.Thecondition（V3）²/2<

Upstreamwehave:

Pi=7.6MPaabs

Ti=292°C

Hi=2765kJ/kg

5.1Usingisentropicrelationships

∙Necksection :

Theequation

Gives:

With:

k=

1,1213

cte1=

0,61543621

cte2=

1,05718192

P0=

4,56

Mpa

P0=

4560000

Pa

rho_0=

23,013

kg/m^3

For:

Qm=

90

kg/s

Wefind:

A=

1,3884E-02

m^2

∙Criticalpressure（atthesonicneck）:

Pc/P0=

0,580

P0/Pc=

1,723

Pc=

2,646

Mpa

∙Soundspeed:

Wefind:

c^2=

2,095E+05

c=

457,69

m/s

5.2CurrentBertincalculation

Itgivesthesameresult.

5.3Downstream

Weusetherelationship:

Attheatmosphericpressure:

k=

1,1213

P0=

4,56

Mpa

P0=

4560000

Pa

rho_0=

23,013

kg/m^3

Uj=

1113,56138

m/s