INTROP E N G - R O B I N S O N E Q U A T I O N O F S T A T E :
Z F A C T O R
PurposeThis workbook provides a working example of the z factor,
enthalpy and entropy calculations for a multi-component fluid using
the Peng-Robinson Equation of State (EOS). This is the same
calculation written into the zfactor Excel add-in but provided in
this format to help users who are less familiar with code to
understand the calculation approach.LiabilityNo warrantees are made
with respect to the accuracy or applicability of the calculations
in this spreadsheet. The onous is on the user to verify that any
results obtained are correct and appropriate for the work being
carrying out.CopyrightThis spreadsheet is the intellectual property
of the author, Andrew Hooks. You are free to use it and distribute
it however, you may not make it available for download from any
website without prior written consent and you must not remove or
obscure any notices regarding authorship.ContactEmail:For other
tools, visit: www.firstprincipleseng.wordpress.com
zFactorP E N G - R O B I N S O N E Q U A T I O N O F S T A T E
: Z F A C T O R Stream
conditionsConstantsTemperature310.92KR8.31451kPa.m3/(kmol.K)Pressure689.47kPaAz
factor0.9767-Enthalpy-80019kJ/kgmolEntropy171kJ/(kgmol.K)Constants
and derived propertiesCompositionComponent constantsBinary
interaction parameters (from Unisim Design)Component derived
propertiesIdeal gas heat capacitiesdH(ideal)
Andrew.Hooks: dH idealCp = dH/dT = A + B*T + C*T^2 + D*T^3 + ...dH
= Integral{Cp.dT} from T1 to T2 = A*(T2-T1) + B/2*(T2^2-T1^2) +
C/3*(T2^3-T1^3) + ...dS(ideal)
Andrew.Hooks: dS idealdS = Integral{dQrev/T}dQrev = dH at constant
pressure, and dH = Cp.dTCp = A + B*T + C*T^2 + D*T^3 + ...Cp/T =
A/T + B + C*T + D*T^2 + ...dS = Integral{Cp/T.dT} from T1 to T2 =
A*[LN(T2)-LN(T1)] + B*(T2-T1) + C/2*(T2^2 - T1^2) + D/3*(T2^3-T1^3)
+ ...Reference H,
SIDMolFrnMolWeightCritTempCritPresAccFactorNitrogenCO2MethaneEthanePropanei-Butanen-Butanei-Pentanen-Pentanekappa
Andrew.Hooks: kappaw>0.49= 0.379642 + (1.48503 - (0.164423 -
1.016666 * w) * w) * ww0.49= 0.379642 + (1.48503 - (0.164423 -
1.016666 * w) * w) * ww 00.9767**r = 0-If there are multiple roots
we use fugacity (related to Gibbs Free Energy) to determine which
is the stable root (the root with the lowest fugacity is the-stable
root). For a single component fluid the transition between z(v) and
z(l) denotes the vapour-liquid phase change. However, for a
multi-componentr < 0-fluid the phase transition occurs over a
range of P,T (lighter components vapourising first etc) and the
change from z(v) to z(l) does not correspond to the -phase
transition boundary (a dew/bubble point calculation is required for
this which considers the fugacities of the individual components).
In fact for a -multi-component fluid the z values in the region
either side of the phase transition will be "suspect" as we are
likely to be in the two phase region.z factor can only meaningfully
be calculated for a single phase fluid. If two phases exist then a
flash calc must be performed to determine the compositionDetermine
stable rootof the separate liquid and vapour phases and z
calculated for each of them.z(vap)0.9767In addition note that even
though there may be 2 or 3 roots this does not imply that we are in
the two phase region - this may or may not be the
case.z(liq)0.9767Fugacity(vap)0.9769Fugacity(liq)0.9769z0.9767-z(add-in)UNAVAIL.EthalpyEthalpy
and entropy are state properties which means that their value at a
given T, P is independent of the path taken to get thereWe define a
reference enthalpy at a given P, T then calculate the change in
enthalpy to the requested P, T in two steps - first an ideal step
(no change in P), then a departure function to account for
non-ideality at high pressureDifferent literature/software uses
different reference values which isn't really important since we
are normally interested in the change in enthalpyHYSYS and UNISIM
use the Heat of formation at 25C as the reference enthalpy and this
is also adopted here to make it easy for users to carry out their
own validation if desiredReference enthalpyArbitary reference value
(since practical calculations are interested in change in
enthalpy)Reference T298.15KReference P101.325kPaAH
reference-80385kJ/kgmolH(ref) = SUM[xi.dH(formation)]dH ideal
(T.ref --> T)Change in enthalpy from reference T to requested T
(at P=1 bara therefore "ideal" change in enthalpy)dH
ideal556kJ/kgmoldH (ideal) = SUM[xi.dHideal]H departure (P=1bara
--> P)Change in enthalpy from (T requested, P=1bara) to (T
requested, P=P requested). "Departure
function"Kappa0.431-Tc224Kalpha0.852-H departure-191kJ/kgmolHd = (z
- 1 - LN((z + (1 + Sqr(2)) * B) / (z + (1 - Sqr(2)) * B)) * A / (B
* Sqr(8)) * (1 + K * Sqr(Tr) / Sqr(alpha))) * GAS_CONST *
Temp"Real" enthalpyHreal-80019kJ/kgmolEnthalpy = Href + dHideal +
HdHreal (add-in)UNAVAIL.kJ/kgmol EntropyS
reference179kJ/(kgmol.K)S(ref) = SUM[xi.dS(formation)]dS
ideal1.83kJ/(kgmol.K)dS (ideal) = SUM[xi.dSideal]dS
mixing6.28kJ/(kgmol.K)Enthalpy of mixing is zero for an ideal fluid
but entropy of mixing is not, dS(mix) = -R*SUM[xi.LN(xi)]S
departure-0.42kJ/(kgmol.K)Sd = GAS_CONST * LN(z - B) - LN((z + (1 +
Sqr(2)) * B) / (z + (1 - Sqr(2)) * B)) * A * GAS_CONST / (B *
Sqr(8)) * (K * Sqr(Tr) / Sqr(alpha))S depart.
(ref.)0.00kJ/(kgmol.K)Sd(ref) is ignored since it requires
recalculation of z at reference P,T and is generally very small. It
is calculated in the zfactor add-inS real171kJ/(kgmol.K)Entropy =
Sref + dSideal + dSmix - GAS_CONST * LN(Pres / Pref) + Sd - Sd_refS
real (add-in)UNAVAIL.kJ/(kgmol.K)z factor add-inThe zfactor add-in
expands on the above calculations - adding Cp-real and Cv-real,
Isenthalpic and Isentropic temperature/pressure change etc. which
allowus to model real world processes (e.g. compression or
expansion across a valve or turbo-expander) These calculations are
the same as those carried out above but need to be solved multiple
times, or as an iteration, and are therefore well suited for code,
e.g.* Cp-real = dH/dT as dT approaches 0 requires two enthalpy
calculations* Isenthalpic temperature rise (compression) requies an
iteration to find the temperature (at the target pressure) that
corresponds to dS=0
Used for Enthalpy/Entropy calculation only
zFactorChartP E N G - R O B I N S O N E Q U A T I O N O F S T
A T E : Z F A C T O R C H A R TTable created using the 'Scenario
Tool' Add-in - available from
https://firstprincipleseng.wordpress.com/category/excel/Re-run
scenarios to update results for a change in
compositionPressureTemperaturezkPaAK-INP:
'[zfactor-spreadsheet-version.xlsx]zFactor'!$C$5INP:
'[zfactor-spreadsheet-version.xlsx]zFactor'!$C$4OUT:
'[zfactor-spreadsheet-version.xlsx]zFactor'!$C$6100248.150.9952000.9913000.9864000.9815000.9776000.9727000.9678000.9629000.95810000.95320000.90530000.85740000.80850000.76060000.71270000.66780000.62790000.594100000.570110000.556120000.551130000.551140000.556150000.564160000.574170000.585180000.598190000.612200000.627210000.642220000.657230000.673240000.689250000.705260000.721270000.737280000.753290000.770300000.786310000.803320000.819330000.836340000.852350000.869360000.886370000.902380000.919390000.935400000.952100273.150.9962000.9933000.9894000.9865000.9826000.9797000.9758000.9729000.96910000.96520000.93030000.89640000.86350000.83160000.80070000.77180000.74590000.722100000.703110000.688120000.677130000.669140000.666150000.665160000.667170000.671180000.677190000.685200000.694210000.703220000.714230000.725240000.737250000.749260000.762270000.775280000.788290000.801300000.815310000.828320000.842330000.856340000.870350000.884360000.898370000.913380000.927390000.941400000.955100298.150.9972000.9953000.9924000.9895000.9876000.9847000.9818000.9799000.97610000.97420000.94830000.92340000.89950000.87660000.85570000.83580000.81790000.801100000.787110000.776120000.766130000.759140000.754150000.751160000.750170000.751180000.754190000.757200000.762210000.768220000.775230000.782240000.791250000.800260000.809270000.819280000.829290000.840300000.850310000.862320000.873330000.884340000.896350000.908360000.920370000.932380000.944390000.956400000.969100323.150.9982000.9963000.9944000.9925000.9906000.9887000.9868000.9849000.98210000.98020000.96130000.94240000.92450000.90860000.89370000.87880000.86690000.854100000.844110000.836120000.829130000.823140000.819150000.816160000.815170000.815180000.816190000.818200000.821210000.825220000.830230000.835240000.841250000.848260000.855270000.862280000.870290000.879300000.888310000.897320000.906330000.915340000.925350000.935360000.945370000.955380000.966390000.976400000.987T=-25'C100200300400500600700800900100020003000400050006000700080009000100001100012000130001400015000160001700018000190002000021000220002300024000250002600027000280002900030000310003200033000340003500036000370003800039000400000.995327927047317210.990648932999079750.985963020703241040.981270196703364910.976570471499109920.971863859823293370.967150380936580680.962430058940922710.957702923112919490.952969008258372390.905269456076601140.856989511471352160.808345072701727620.759787807191708580.712184042632622690.667053307815112960.626703603396105760.593836451735467510.57037846262167080.556445534569872780.550604094614454920.55091622188580680.55566664148505640.563567812405520120.573713221524428270.585475823357194280.598420349024643580.612240789335638190.62671851762303210.64169469876575980.657052059471872620.672702705520343060.68857985675911060.704632148043407460.720819634974010910.7371109501242280.753481248041344530.769910699427890030.786383373359761380.802886397525467980.819409320283754680.835943621026433160.852482330774705250.86901973558879320.88555114281325020.902072695444291030.918581223671816050.935074125372895470.95154926932415562T=0'C100200300400500600700800900100020003000400050006000700080009000100001100012000130001400015000160001700018000190002000021000220002300024000250002600027000280002900030000310003200033000340003500036000370003800039000400000.996491291832512620.992984701035564580.989480348119574190.985978357469063040.982478857456715150.97898198056015850.975487863481471210.971996647269407490.968508477444335550.965023504125865020.930386162101841330.896273079233149920.862932728153378960.830691929632158650.799967290014502060.77126505181993610.745157696208033740.722226758231061970.702973336681720130.687720354456706540.676547663793350740.669290928727783840.665600266819907360.665025456453005370.667092946535202460.671357348869265680.677427114531291560.684971912284125570.69371977641204330.703449794342223540.713983661292222020.72517771278964060.736916053811162760.749104903666385710.761668058097547670.774543296658543050.787679556544565230.801034713691923580.81457383916002390.828267825340776250.842092299429647580.856026760207985740.870053888856126330.884158995878267180.898329574937346440.912554941058049020.926825935737723090.941134685386324050.95547440248792748T=25'C100200300400500600700800900100020003000400050006000700080009000100001100012000130001400015000160001700018000190002000021000220002300024000250002600027000280002900030000310003200033000340003500036000370003800039000400000.997332332564637050.994670394347128670.992014298889921430.989364161670212240.986720100120154540.984082233646520480.981450683649772060.978825573542481230.976207028767050340.97359517681266650.947874302910618690.92297863005569880.899070786094029680.8763337509862430.854967129781390560.835179840599411080.817178753201400990.801153727245462740.787260808927873420.775606596419205820.766237271493719560.759134988372450750.754222356353750010.75137352087007270.750428870239295160.751210172204796470.753533738447390470.757220401935481680.762102126293916050.768025704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Pressure [kPa(A)]
z factor [-]
https://firstprincipleseng.wordpress.com/category/excel/