Pure Component Equations

Transcription

1 Pure Component Equatons Fttng of Pure Component Equaton Parameters DDBSP Dortmund Data Bank Software Package DDBST Software & Separaton Technology GmbH Mare-Cure-Straße 10 D Oldenburg Tel.: Fax: E-Mal: support@ddbst.com Web:

2 Contents 1 Introducton Lst of Equatons Usng the program Intal Dalog Fle Menu Help Menu Component Selecton Check Data Avalablty Ft Input by Hand Ft Results Plot Understandng the ParameterDDB Data Set Dsplay Workng wth a Parameter Data Set Copy Edt Plot Detals Calculate Ft Archve T c /P c Evaluaton Densty Predcton by Equaton of State Vral Coeffcents Isotherms Ratonale Applcaton Flow Descrpton of the Graphcs Output Mathematcal and Physcal Relatons Dsplay of Compressblty Factor-1 aganst Densty Optmzaton Evaluaton of the Optmzaton Qualty Pressure PmaxB Practcal Tps Gas Constant, Molar Mass, Crtcal Densty All Data Smultaneously Ratonale Problem Descrpton Regresson Short Tutoral Volume Translaton...38 DDB Pure Component Equatons Page 2 of 39

3 1 Introducton PCPEquatonFt fts parameters for a large varety of equatons for pure component propertes. Parameters can be stored n and retreved from a parameter database, they can be plotted, and they can be used for calculatons. PCPEquatonFt normally uses the pure component propertes data bank whch s a part of the Dortmund Data Bank. It can also be used to ft data from other data sources snce tables can be pasted from the clpboard or loaded from fles. DDB Pure Component Equatons Page 3 of 39

4 2 Lst of Equatons Property Lqud Vscosty T [K] η [mpa s] Equaton 1. Andrade A B =e T 2. Vogel =e A B T C A 3. DIPPR 101 B E C lnt DT =e T 4. PPDS 9 =E exp[ A C T 1 T D B 3 C T T D 4 ] 3 Vapor Vscosty T[K] η [mpa s] 5. Extended Andrade η=e A+ B T + CT + DT 2 + ET 3 = A T B 1. DIPPR C T D T 2 2. Polynomal η=a+ B T + CT 2 + DT 3 + ET 4 DDB Pure Component Equatons Page 4 of 39

5 Property Saturated Vapor Pressure T [K] P [kpa] Equaton 1. Antone P=10 A B T C ( Other Unts: T [ C], P [mmhg]) 2. Wagner 2.5,5 P=exp ln P c A 1 T r B 1 T r 1.5 C 1 T r 2.5 D 1 T r 5 3. Wagner 3,6 P=exp ln P c A 1 T r B 1 T r 1.5 C 1 T r 3 D 1 T r 6 A B 4. Cox P=exp[ln T T B C T T B e 1 T ] B T T r T r 5. DIPPR 101 A B E C lnt DT T P=e ( Other Unts: P [Pa]) 6. Extended Antone (Lonza) P=exp( A+ B T +C +DT + ET 2 +F ln(t )) ( Other Unts: P [bar]) 7. Extended Antone (Aspen) P=exp( A+ B +DT +E ln(t )+ F T T +C G) G=1 or G=2 8. Extended Antone (Hysys) P=exp( A+ B +D ln(t )+E T T +C F) F=1 or F=2 9. Rarey2P [ T B P=P atm A T 8 ] B 10 B T b 1 A Xang/Tan P=P c exp(ln T R ( A 1 + A 2 (1 T R ) A 3 (1 T R ) 5.67 )) 11. PVExpanson: P=exp( A+ B T +C ln(t )+DT + E T 2 + F T 2 +G T6 + H T 4) 12. Hoffman/Florn: P=exp( ( A+ B 1 T log 10 (T ) T)) 6 ( Other Unts: P [Pa]) DDB Pure Component Equatons Page 5 of 39

6 Property Saturated Vapor Pressure by EOS T [K] P [kpa] Equaton 1. Mathas-Copeman Constants for EOS = 1 m 1 T r 2 m=c 1 c 2 1 T r c 3 1 T r 2 2. Twu-Bluck-Cunnngham-Coon Constants for EOS =T r c 3 c 2 1 exp c 1 1 T r c 2 c 3 (c 1, c 2, c 3 used n DDB programs) =T r N M 1 exp L 1 T r M N (L, M, N lke orgnal authors) 3. Melhem-San-Goodwn Constants for EOS =exp c 1 1 T r c 2 1 T r 2 4. Stryjek-Vera Constants for EOS κ=κ 0 +κ 1 (1+ T r )(0.7 T r ) α=(1+κ(1 T r )) 2 5. Stryjek-Vera-2 Constants for EOS κ=κ 0 +[κ 1 +κ 2 (κ 3 T r 0.5)(1 T r 0.5)](1+T r 0.5)(0.7 T r ) α=(1+κ(1 T r 0.5 )) 2 6. Schwartzentruber/Renon/Watanasr Constant for EOS P rm = ω ω 2 Lqud Heat Capacty T [K] c p [J/mol K] 2 α=(1+ P rm (1 T r ) (1 T r ) (c 1 +c 2 T r +c 3 T r )) 2 1. Polynomal c P =A BT CT 2 DT 3 ET 4 2. PPDS 15 c p =R A C D 2 E 3 F 4 wth =1 T T c DDB Pure Component Equatons Page 6 of 39

7 Property Ideal Gas Heat Capacty T [K] c p [J/mol K] Equaton 1. Polynomal c P =A BT CT 2 DT 3 ET 4 2. Aly-Lee, DIPPR 107 c p =a 0 a 1 a 2 T snh a 2 T 2 a 3 a 4 T cosh a 4 T 2 3. PPDS 2 C P =R B C B y 2 [1 y 1 D Ey Fy 2 Gy 3 ] wth y= T A T 4. Shomate c P =A B T C T 2 DT 3 E T 2 Sold Heat Capacty T[K] c p [J/mol K] Polynomal c P =A BT CT 2 DT 3 ET 4 DDB Pure Component Equatons Page 7 of 39

8 Property Lqud (Saturated) Densty T [K] ρ [kg/m³] Equaton 1. DIPPR 105 = A B 1 1 T C D 2. Polynomal = A B T CT 2 DT 3 ET 4 3. Tat (pressure-dependent data) P ref =max f T, MPa (Wagner-Equaton) ref = f T kg m 3 (DIPPR 105-Equaton) T reduced =100 T R = T T reduced C=c 0 + c 1 T R B=b 0 + b 1 T R + b 2 T R 2 + b 3 T R 3 + b 4 T R 4 ref = 1 C ln[ B P B P ref ] 4. DIPPR 116 (wth addtonal addend ρ C, the crtcal densty) 2 L = c [ A 0.35 B 5. DIPPR C D 4 3] wth =1 T T c ρ=a+ B τ 3 +C τ 3 + D τ 3 +E τ 3 +F τ 3 3 wth +G τ τ=1 T T c Surface Tenson T [K] σ [N/m] 1. Polynomal = A BT CT 2 DT 3 ET 4 2. Short DIPPR 106 = A 1 T R n wth T R = T T c 3. = A T T C B 4. Full DIPPR 106 = A 1 T r B C T r DT 2 3 r E T r wth T r = T T c 5. PPDS 14 σ= A τ B (1+C τ ) wth τ=1 T T c DDB Pure Component Equatons Page 8 of 39

9 Property Second Vral Coeffcent T [K] B [cm³/mol] Heat of Vaporzaton T [K] H vap [J/mol] Equaton 1. B = A T B T 2. DIPPR 104 B = A B T C T 3 D T 8 E T 9 1. DIPPR 106 H Vap = A 1 T T C B C T T C D T T C 2 E T T C 3 2. Extended Watson H Vap =a c T b d 3. PPDS H Vap =R T c A 3 B 3 C D 2 E 6 wth =1 T T c Lqud Thermal Conductvty T [K] λ [W/m K] Vapor Thermal Conductvty T [K] λ [W/m K] Isothermal Compressblty Thermal Expanson Coeffcent 1. Polynomal = A BT CT 2 DT 3 ET 4 2. PPDS 8 1. PPDS 3 Lnear Interpolaton Lnear Interpolaton 1 = A 1 B 3 C T = r A B C T r T D 2 3 r T r 2 3 D wth =1 T T c wth T r = T T c Meltng Temperature T (Pressure Dependency) Smon-Glatzel Equaton P m =a m T m normal c 1. Delectrc Constants of Lquds, Permttvty T K], ε [.] 1. Polynomal = A BT CT 2 DT 3 ET 4 DDB Pure Component Equatons Page 9 of 39

10 3 Usng the program 3.1 Intal Dalog Fgure 1 Man PCPEquatonFt Dalog The program's start dalog contans three major parts: 1. The components area allows 1. selectng components 2. dsplayng component detals wth the component edtor 3. dsplayng the content of the Dortmund Data Bank for the selected component 4. verfyng f enough data sets or ponts are avalable (ths s only a hnt, snce there mght be further constrants) 2. The lst of equatons. The lst s organzed herarchcally. The methods are summarzed below the property they descrbe. 3. The parameter data set shows the current content of the ParameterDDB. The toolbar buttons are manly short cuts for the Fle and Help menus. DDB Pure Component Equatons Page 10 of 39

11 3.2 Fle Menu Open Component Numbers Fle Ths functon allows loadng a fle wth a lst of DDB component numbers. Such component fles can be created, for example, n the component selecton dalog or n the man Dortmund Data Bank program from search results. The data set numbers are shown n a separate wndow. Fgure 2: Fle menu A clck on a lne sets the component number n the man ft wndow. Count Count shows the number of avalable parameter data sets for the current model. Statstcs Statstcs creates a table wth an overvew over all equatons Fgure 3: Parameter Data Set Count Fgure 4: Statstcs DDB Pure Component Equatons Page 11 of 39

12 Database Detals (Current Equaton) Ths functon creates a table wth all data sets avalable for the current equaton. Database Overvew Ths functons creates a table wth the number of components for expermental data n the Pure Component Propertes part of the Dortmund Data Bank are avalable for the sngle equatons. Fgure 5: Database Detals (Current Equaton) Archve See chapter Ft Archve on page 25. Fgure 6: Database Overvew ParamDBOrganzer Ths functon call the program for managng the parameter data base. Ths program s descrbed n a separate PDF ( ParameterDDBOrganzer.pdf ). Buld ParameterDB Index Ths wll rebuld the component ndex of the parameter data base. Ths s normally done automatcally when needed. Ths functon s only needed f changes outsde PCPEquatonFt have been made. 3.3 Help Menu The help menu contans a button whch brngs ths PDF help up and an About button whch shows some nformaton about the program. Fgure 7: Help menu DDB Pure Component Equatons Page 12 of 39

13 3.4 Component Selecton DDB component numbers can be typed drectly n the component feld. After a Return the component name s added. The buttons The button allow to navgate through the DDB component lst. calls the component selecton dalog whch s descrbed n detals n other documents. 3.5 Check Data Avalablty Fgure 8 Component Selecton Ths button starts a search n the pure component property data bank for expermental data for the currently selected equaton. When ths search s fnshed the Check Data Avalablty s hdden and nformaton about the avalablty of data s shown. DDB Pure Component Equatons Page 13 of 39

14 The nformaton lnes show for how many components the Dortmund Data Bank contans expermental data sets. The example shows the number of components for the Antone equaton (saturated vapor pressures). Clckng on the underlned label ( Components 5028 ) wll open a wndow wth the lst of components. The Data are avalable lne ndcates that there are enough data ponts for the specfc equaton. Ths number s normally set to <number of parameters + 1>. If no data are avalable ths text wll be dsplayed:. The check box should be used n walk-through mode where a lst of components s n work. If checked ths wll avod the dsplay of components wthout expermental data ponts. A detaled descrpton of all component selecton features s avalable n the Component Management documentaton. 3.6 Ft After the component and the equaton has been selected and the program ndcates that enough data ponts are avalable ( ) the Ft button dsplays a model specfc dalog wth almost the same content for the dfferent models. The used example for showng a typcal ft s the Wagner equaton for saturated vapor pressures. DDB Pure Component Equatons Page 14 of 39

15 Fgure 9 Ft Dalog for Wager equaton The dalog dsplays the data source whch s n most cases the pure component propertes data bank. All possble sources are 1. Database 2. Input by hand 3. Readng from fle 4. Calculated data or stored data ponts (here marked as '-') The Append PCP Fle would allow to append data from an external fle. The dalog dsplays the number of avalable data ponts and the number of dfferent references (number of dfferent authors) and repeats the dsplay of the component name. The two buttons besdes the name nvoke the component edtor and the Dortmund Data Bank program. DDB Pure Component Equatons Page 15 of 39

16 The temperature and pressure range are also dsplayed. These lmts are edtable and can be used to cut ponts by ncreasng the lower lmt or decreasng the upper lmt. The knfe button wll actually throw the ponts outsde the gven ranges away. The Edt Data Ponts allows to modfy the data from the data sources. It uses the Input by Hand dalog. The normal bolng pont (T b ), the crtcal data (T c, P c, ρ c ), and the meltng pont (T m ) are read from pure component basc fles (not from the pure component propertes data bank). The lower part of the dalog s model specfc but contans n most cases startng parameters and a selecton for an objectve functon where approprate Input by Hand If ths nput mode s selected a dalog wth a data grd s shown where the user can ether type or paste or load data. Fgure 10 Input by Hand DDB Pure Component Equatons Page 16 of 39

17 3.6.2 Ft Results After pressng the Ft button the ft wll start and present a New Parameters box when t's fnshed: Fgure 11 Ft Result Ths box shows the new parameters, a mean error, the used temperature lmts, the data source and the current date and n some cases addtonally used constants lke n ths example T c and P c. These entres wll be stored n the ParameterDDB f one of the Save buttons wll be pressed Plot For an overvew on the ft qualty PCPEquatonFt provdes several plots. DDB Pure Component Equatons Page 17 of 39

18 Fgure 12 Plot Ft Result The lst of plots slghtly vares from model to model. Always the same s the rubber band drawn from the mouse cursor to the nearest pont. Detaled nformaton of ths pont are dsplayed n the status lne. Addtonally the reference s shown below the tool bar. The dagram lmt can be wdened and narrowed. The Expermental Data button adjusts the dagram so that the expermental data are fllng the chart wndow. Ths s useful n the cases where crtcal data and meltng ponts are shown and the expermental data are avalable only for a smaller range. DDB Pure Component Equatons Page 18 of 39

19 Through a context menu on the plot t s possble to 1. Exclude ponts (ether sngle or by crtera) 2. Include formerly excluded ponts 3. Dsplay data sets shown n the chart (ether sngle or a lst of data sets for the current component or reference) 4. Call the data sets edtor 5. Change the background color Fgure 13 Plot Context Menu Addtonally a complete lst of devatons can be created ( Table of Devatons tool button) and the dagram can be coped to the Wndows clpboard or prnted. DDB Pure Component Equatons Page 19 of 39

20 The Data Ponts tool button used to nclude and exclude data ponts. Fgure 14 Table of Devatons opens a dalog where all data ponts are lsted. Ths dalog can be Fgure 15: Data Ponts Selecton Ths functon has been added because of ponts occupyng exactly the same poston (exactly same data) whch makes t mpossble to select all these ponts by mouse. If ponts have been excluded t s necessary to start a new ft by the Reft button to the ft dalog allowng to store the modfed parameters.. Ths wll return us DDB Pure Component Equatons Page 20 of 39

21 4 Understandng the ParameterDDB Data Set Dsplay Fgure 16 Parameter Data Set The ParameterDDB contans key/value pars. The keys descrbe the values. The grd shows the lst of keys and the values belongng to them. 1. The keys A, B, C, D and so on are the parameters of the equatons. 2. C1 s the DDB component number. Its name can be found n the component edtor. 3. Pc, Tc are crtcal temperature and pressure. Other possble entres are e.g. Tb. 4. EQID s the nternal equaton number. 5. Tmax and Tmn are the upper and lower temperature lmts of the expermental data used. Please regard these values also as valdty range for the equaton. 6. User specfes the person who stored the parameter dataset. 7. DateD, DateM, DateY specfy the date when the dataset has been stored. 8. Error gves the model and ft specfc error. 9. Source specfes the source of the data ponts whch have been used for the ft. 10. Locaton specfes f the parameter set s stored n the publc DDB (0) or n the prvate DDB (1) or, f mssng or another number, some other locaton. 11. AUTOSELECT s necessary f more than one dataset s avalable for a component and a sngle equaton. It specfes the preferred parameter set. 12. SourceFle s gven n some cases and specfes a fle from whch the set has been mported. DDB Pure Component Equatons Page 21 of 39

22 5 Workng wth a Parameter Data Set 5.1 Copy The data set grd wll be coped to the wndows clpboard as t s dsplayed n Fgure 16 (source) and Fgure 17 (destnaton). Fgure 17 Data set pasted n spreadsheet program DDB Pure Component Equatons Page 22 of 39

23 5.2 Edt The edtor s another vew on the parameter data set grd. The grd s now edtable and new values can be typed n the Value column. The Key column s not drectly edtable but new keys ( ) can be added and keys wth empty values wll be removed automatcally when the data set s saved. The Recommended Value check mark should be set f more than one data set s avalable for the same component and equaton and the current data set should be preferred over all others. 5.3 Plot Ths plot shows the stored equaton parameters together wth ponts from the pure component propertes data bank. It's the same plot as used n the ft procedure wth the excepton that some edtng functons are not avalable lke removal of data ponts. 5.4 Detals Ths functon dsplays a more detaled and explanatory vew on the current parameter set. It s part of the ParamDDBOrganzer program. Ths program s descrbed n detal n the separate document ParameterDDBOrganzer.pdf. DDB Pure Component Equatons Page 23 of 39

24 5.5 Calculate Fgure 18 Data set detals Stored parameter sets can be used to calculate the property at arbtrary temperatures. It s ether possble to calculate values n a temperature range where start and end temperature as well as a step wdth can be specfed or sngle values typed n the data grd. DDB Pure Component Equatons Page 24 of 39

25 6 Ft Archve Fgure 19: Calculate propertes wth stored parameters PCPEquatonFt stores a hstory of ftted parameters and used datasets. Ths archve s accessble through the tool bar button. The archve s ntended to be the memory of all fts. It should allow to save the data whch have been used for the ft and to restore them and perform a full re-ft under the same condtons as done orgnally. Ths goal s currently not perfectly acheved. The archve dalog tself (Fgure 20) shows a lst of of parameter sets dentfed by component number and model descrpton separated for the publc and prvate data banks. The detals grd shows the x and y, the reference number and the dataset number and n the Used column a + f the value has been used n the ft or a - f the pont has been excluded. The Reft button creates a ft dalog for the gven equaton and component wth the stored data ponts (Fgure 21). DDB Pure Component Equatons Page 25 of 39

26 Fgure 20 Ft archve Fgure 21 Reft wth archved data DDB Pure Component Equatons Page 26 of 39

27 7 T c /P c Evaluaton PCPEquatonFt allows wth ths functon the evaluaton of expermental pure component crtcal data and saturated vapor pressures together wth calculated and estmated values. For a full nvestgaton t s necessary to have at least a parameter set for a vapor pressure equaton and the Artst program package should also be present snce t s used for dsplayng estmated crtcal data. Fgure 22 Crtcal Data Evaluaton - Plot The Optons page allows selectng vapor pressure equatons from PCPEquatonFt and T c and P c estmaton methods from Artst. Fgure 23 Crtcal Data Evaluaton - Vapor Pressure Equatons DDB Pure Component Equatons Page 27 of 39

28 Fgure 24: Zoomed n for Crtcal Pont The resultng dagram shows all expermental, calculated, and estmated data ponts n a Temperature vs. Pressure plot. Devatons are shown n the same dagram wth ts scale on the dagram's rght sde. The dagram allows swtchng between T vs. P and 1000/T vs. log 10 P and the dsplay of the devatons can be swtched on and off. The mportant pont s the end pont of the vapor pressure curve. The expermental and estmated crtcal T c and P c are shown as horzontal and vertcal lne. The ntersectons gve a hnt where the correct crtcal pont les. DDB Pure Component Equatons Page 28 of 39

29 8 Densty Predcton by Equaton of State Ths dalog Fgure 25: Densty Predcton can be used to calculate lqud and vapor denstes and volumes of pure components by equaton of states. The supported equatons of state are the same whch can be used to regress α functon parameters n the man dalog and the regressed α functon parameters are used also for ths densty calculaton. Input for the calculaton by the equaton of state are temperatures and pressures. The pressure can ether be gven drectly or the saturated vapor pressure can be used. The saturated vapor pressure would be determned by the equaton of state. Fgure 26: Usng saturated vapor pressures DDB Pure Component Equatons Page 29 of 39

30 9 Vral Coeffcents Orgnal Author: Romana Laznckova 9.1 Isotherms Ratonale The sub program ISOTHERM calculates second and thrd vral coeffcents from qualfed sothermal gasphase PVT-data. The program also allows to compare the datasets and judge ther qualty Applcaton Flow The program allows ether to load a pure component propertes fle contanng PVT data or searches the datasets tself after a component has been selected. The data from ths lst are sorted by temperature and data ponts measured at the same temperatures are collected and combned n sotherms. These sotherms are searched for applcable data. For the calculaton of vral coeffcents only data up to ¾ of the crtcal densty are used. Near the crtcal sotherm, at reduced temperatures between T r =0.95 and T r =1.2, only data wth denstes up to ½ of the crtcal densty are used. If an sotherm has at least two data ponts n the specfed range t wll be used to regress the second and thrd vral coeffcents. The vral coeffcents are regressed by an optmzng algorthm whch mnmzes the sum of the squared errors of the compressblty factor. The qualty of the optmzaton can be judged by the absolute and relatve devaton n the compressblty factor and the densty of the regressed vral equaton from the expermental values. The regresson qualty s also characterzed by the numbers square root from the mean squared error of the compressblty factor and the densty. Addtonally the program determnes a maxmum pressure (PmaxB), whch gves a real densty value for a vral equaton made up only wth the second coeffcent B. The results are lsted on screen gvng an overvew over all temperatures. Regresson results are gven for all temperatures where expermental data ponts have been avalable. The expermental datasets are lsted together wth the regressed second and thrd vral coeffcents, the maxmum pressure (PmaxB), the absolute and relatve densty devaton and both characterzaton numbers Descrpton of the Graphcs Output Compressblty Factor 1 The man chart s the dsplay of aganst the densty. The vral equaton buld Densty wth B and C s a straght lne n ths case. The axs ntercept on the y-axs s the second vral coeffcent B and slope of the straght lne s the thrd vral coeffcent C. Ths projecton allows evaluatng the qualty of the optmzaton n a very clear way. For sotherms where B and C have been obtaned a calculated lne s ncluded. There are four other charts whch dsplay dfferences between the expermental values and the correlaton: 1. Absolute devaton n the densty, 2. Relatve devaton n the densty, 3. Absolute devaton n the compressblty factor, 4. Relatve devaton n the compressblty factor aganst the densty. DDB Pure Component Equatons Page 30 of 39

31 The chart also ncludes the crtcal densty Mathematcal and Physcal Relatons Dsplay of Compressblty Factor-1 aganst Densty Ths presentaton s based on the relaton for second vral coeffcent B=lm d 0( z 1 d ) and the thrd vral coeffcent C=lm d 0( ( z 1 d )) d The equaton evolved up to the thrd vral coeffcent z=1+ B d+c d 2 s a straght lne n the presentaton of z 1 ρ =B+C ρ Compressblty Factor 1 Densty aganst densty. Because vral coeffcents are normally shown n molar unts (B [cm 3 *mol -1 ], C[cm 6 *mol -2 ]) and denstes n Compressblty Factor 1 [kg*m 3 ] the ordnate shows n [cm Densty 3 *mol -1 ] and the abscssa shows denstes n [kg*m -3 ]. If the thrd vral coeffcent shall be determned graphcally from ths presentaton t s necessary to convert both unts Optmzaton The optmzaton routne searches for a combnaton of the second and thrd vral coeffcents where the sum of squares of errors of the compressblty factor s mnmal. F= (z zcalc ) 2 =! Mn runs over all expermental data ponts for a specfed sotherm. The compressblty factor z s calculated from the measured temperature T, pressure P, and densty. z = P M ρ R T The vral equaton calculates the compressblty factor zcalc for the expermental densty 2 z calc, =1+ B' ρ +C ' ρ wth B '= B M and C '= C M 2 Equaton 8 DDB Pure Component Equatons Page 31 of 39

32 The mnmum of the objectve functon F=F (B',C ') s determned mathematcally exact. The necessary condton for a mnmum s the exstence of a combnaton of the second and thrd vral coeffcents that the partal dervatons of the objectve functon by B' and C' are zero. F B ' =0 and F C ' =0 These condtons lead to lnear equaton system. P M R T ρ B ' ρ 2 C ' ρ 3 =0 P M ρ R T ρ 2 B' ρ 3 C ' ρ 4 =0 Ths equaton system s solved by the Gauß-Jordan method. The results are the second and thrd vral coeffcents B' and C' n mass unts. These values are converted by equatons (8) nto molar unts. The program dsplays the second vral coeffcent n [cm 3 *mol -1 ] and the thrd n [cm 6 *mol -2 ] Evaluaton of the Optmzaton Qualty The goodness of the optmzaton can be evaluated by the dfference between the expermental values and the calculated values. absolute devaton n the densty ρ ρ calc, relatve devaton n the densty ρ ρ calc, ρ 100. absolute devaton n the compressblty factor z z calc, relatve devaton n the compressblty factor z z calc, z 100. These devatons are determned for all expermental values. Addtonal qualty numbers are square root from the mean squared error of the compressblty factor ( z z calc, ) 2 n and the square root from the mean squared error of the densty (ρ ρ calc, ) 2 n These number are obtaned only from the expermental values used n the optmzaton. n s the number of these values. DDB Pure Component Equatons Page 32 of 39

33 Pressure PmaxB A vral equaton wth only B s quadratc aganst the densty. If the second vral coeffcent s negatve, t depends on the pressure f the quadratc equaton yelds real solutons for the densty. The pressure PmaxB s the maxmum pressure where the equaton wth only B yelds a real soluton. PmaxB= R B 4 B Practcal Tps Ths program only calculates vral coeffcents from measured values n a reasonable range, despte ths statement t s stll necessary to carefully evaluate the results. Expermental values mght be dstrbuted only n a narrow range whch mght lead to an arbtrary result dependng on scatterng. If the denstes are very small the expermental error wll ncrease Gas Constant, Molar Mass, Crtcal Densty Ths program uses the gas constant from the DDB fle STOFF. 9.2 All Data Smultaneously Ratonale R= J K mol. The molar mass and the crtcal densty are taken The smultaneous correlaton can be used for the evaluaton of PVT datasets, especally for non-sothermal data (see prevous chapter 30 Isotherms for sothermal data). Addtonally the program allows to select datasets and nterpolaton between dfferent data. The mplemented vral equaton regresses the second and thrd vral coeffcent and uses a two-parameter temperature relaton. Therefore the correlaton needs at least four ponts n a system Problem Descrpton The correlaton s a three-dmensonal problem. T, P, are lyng on a surface. Ths surface has to be descrbed by the vral equaton wth second and thrd coeffcent and a two-parameter temperature functon.because t s hard to obtan meanngful three-dmensonal graphcal dsplays the program uses a projecton of the P T space to the P (pressure aganst densty) plan. The vral equaton s drawn as a seres of sothermal P=f( ) curves Regresson The objectve functon s F= The vral equaton s ( z z calc, ) 2 Mn wth z= P M ρ R T DDB Pure Component Equatons Page 33 of 39

34 z calc, =1+ B ( ρ M ) +C ( ρ M ) 2 wth M [kg/mol], B [m 3 /mol], C [m 6 *mol -2 ] The two-parameter temperature dependence for the second vral coeffcent B s B = b 1 T + b T Two-parameter temperature dependence for the thrd vral coeffcent C s C = c 1 T c 2 T 10 The exact mathematcal soluton ( F =0, F =0, F =0, F =0 ) leads to the lnear equaton system: b 1 b 2 c 1 c 2 A 11 x 1 + A 12 x 2 + A 13 x 3 +A 14 x 4 =D 1 A 21 x 1 + A 22 x 2 + A 23 x 3 +A 24 x 4 =D 1 A 31 x 1 + A 32 x 2 + A 33 x 3 +A 34 x 4 =D 1 A 41 x 1 +A 42 x 2 +A 43 x 3 +A 44 x 4 =D 1 wth b 1 = x 1, b 2 = x 2, c 1 =x 3, c 2 = x 4 A 11 = 1 M 2 2 ρ T A 12 = 1 M 2 ρ 2 T 1.5 A 13 = 1 M 3 ρ 3 T 1,7 A 14 = 1 M 3 ρ 3 T 10.5 A 21 = 1 M 2 ρ 2 T 1.5 A 22 = 1 M 2 ρ 2 T 2 A 23 = 1 M 3 ρ 3 T 2.2 A 24 = 1 M 3 ρ 3 T 11 7 A 31 = 1 M 3 ρ 3 T 1.7 A 32 = 1 M 3 ρ 3 T 12.2 A 33 = 1 M 4 ρ 4 T 2,4 A 34 = 1 M 4 ρ 4 T 11,2 A 41 = 1 M 3 ρ 3 T 10.5 A 42 = 1 M 3 ρ 3 T 11 A 43 = 1 M 4 ρ 4 T 11.2 A 44 = 1 M 4 ρ 4 T 20 DDB Pure Component Equatons Page 34 of 39

35 D 1 = D 2 = D 3 = ( ( D 4 = ( P R T ρ ) 1.5 T 0.5 M ( P R T ρ 2 T 0.5 M) P M R T ρ 2 ) 2.2 T 1.2 M P M R T ρ 2 ) 11 T 10 M Ths equaton system s solved by the Gauß-Jordan method. The results are b 1 [m 3 mol 1 K 0.5 ] b 2 [m 3 mol 1 K ] c 1 [m 6 mol 2 K 1.2 ] c 2 [m 6 mol 2 K 10 ] On screen the values are multpled by 10 6 for b 1 and b 2, and for c 1 and c 2 (m cm) Short Tutoral Fgure 27: Start Screen Start ScreenFgure 27 shows the start screen of SIMULTAN. The PVT data are ether obtaned from the DDB pure component propertes database f a component s selected or loaded from a PCP nterface fle whch has been created by another program. DDB Pure Component Equatons Page 35 of 39

36 After selectng a component or loadng a fle the program dsplay the ranges n densty, pressure, and temperature and allows here to set new lmts. Fgure 28: Densty Lmts Fgure 29: Pressure Lmts Fgure 30: Temperature Lmts After these dalogs the program mmedately regresses the vral coeffcents and dsplay a result. The result lst gves 1. name of component wth ts molecular weght, 2. number of ponts gven and used, 3. the expermental values ether from fle or database, 4. used temperature, pressure, and densty lmts, 5. the regressed b 1, b 2, c 2, c 3 values, 6. examples f the B and C at 353 K, 7. a table wth expermental and calculated data, 8. error numbers for specfyng the qualty of the regresson. The plot output dsplays sx charts. 1. normal plot (no sotherms) 2. B aganst T 3. C aganst T 4. relatve compressblty factor devaton 5. compressblty factor devaton 6. relatve densty devaton DDB Pure Component Equatons Page 36 of 39

37 7. densty devaton 8. normal plot: P aganst molar densty The plot output has a context menu (see Fgure 31) whch allows to dsplay the expermental data n the database retreval program or all the data comng from a sngle reference or some component detals. Fgure 31 Addtonally t allows to select data from a sngle reference for correlaton. In ths case the program recorrelates b 1, b 2, c 1, c 2 only from ths reference's datasets. The chart contans some addtonal lnes whch are the crtcal densty, 0.5 and 0.75 of the crtcal densty, a zero lne and the crtcal pressure, f the ordnate shows pressure values. DDB Pure Component Equatons Page 37 of 39

38 10 Volume Translaton VTPR uses a volume translaton based on the dfference between the expermental volume and the volume calculated by the Peng-Robnson equaton of state at T=T c *0.7. Ths temperature s normally qute close to the normal bolng pont. PSRK normally does not use a volume translaton for the Redlch-Kwong EOS but t can use such a correcton, n prncple. In ths dalog the volumes calculated by the equatons DIPPR 105, DIPPR 116, and Polynomal are used as source for the expermental volume. The left table shows the calculaton result wth the volume translaton value c n lght green. The rght table shows the already stored values n the parameter data bank. The Dagram page shows the dfferent calculated volume (1/ρ) curves, a vertcal lne at T c *0.7 and expermental values from the pure component property data base. DDB Pure Component Equatons Page 38 of 39

39 DDB Pure Component Equatons Page 39 of 39