Equations of state - Revision history

Priya.bagde123 at 21:36, 18 June 2013

2013-06-18T21:36:27Z

Jpye: moved Equations OF State to Equations of state

2013-06-18T06:59:47Z

moved Equations OF State to Equations of state

← Older revision		Revision as of 06:59, 18 June 2013
(No difference)

Priya.bagde123 at 04:19, 18 June 2013

2013-06-18T04:19:02Z

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-== '''Van der Waals equation of state''' ==
+where <math>V_m</math> is [[molar volume]]. The substance-specific constants <math>a</math> and <math>b</math> can be calculated from the [[critical properties]] <math>p_c, T_c</math> and <math>V_c</math> (noting that <math>V_c</math> is the molar volume at the critical point) as:
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Priya.bagde123: Created page with "In physics and thermodynamics, an equation of state is a relation between state variables.[1] More specifically, an equation of state is a thermodynamic equation describing th..."

2013-06-18T03:51:32Z

Created page with "In physics and thermodynamics, an equation of state is a relation between state variables.[1] More specifically, an equation of state is a thermodynamic equation describing th..."

New page

In physics and thermodynamics, an equation of state is a relation between state variables.[1] More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars.

== '''Cubic Equations of State''' ==

Cubic equations of state are called such because they can be rewritten as a cubic function of Vm.

All the models below are first modeled as a general, re-usable model and then used in a test-model.
This code defines virial coefficients as well as the model for ideal gas laws:

<source lang=a4c>
REQUIRE "thermodynamics.a4l";

ATOM vdwa REFINES solver_var
DIMENSION M*L^5/Q^2/T^2
DEFAULT 0.1 {kg*m^5/mole^2/sec^2};
lower_bound := 0.0 {kg*m^5/mole^2/sec^2};
upper_bound := 10.0 {kg*m^5/mole^2/sec^2};
nominal := 0.1 {kg*m^5/mole^5/sec^2};
END vdwa;

ATOM virialC REFINES solver_var
DIMENSION L^6/Q^2
DEFAULT 7000.0 {cm^6/mol^2};
lower_bound := 0.0 {cm^6/mol^2};
upper_bound := 1.0 {m^6/mol^2};
nominal := 7000.0 {cm^6/mol^2};
END virialC;

ATOM virialB REFINES solver_var
DIMENSION L^3/Q
DEFAULT 1.0 {m^3/mol};
lower_bound := -1000.00 {m^3/mol};
upper_bound := 1000.0 {m^3/mol};
nominal := 1.0 {m^3/mol};
END virialB;

MODEL idealgas(P WILL_BE pressure;
T WILL_BE temperature;
V WILL_BE molar_volume;);

R IS_A molar_gas_constant;

eq: P*V = R*T;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;
METHOD specify;
P.fixed := TRUE;
T.fixed := TRUE;
V.fixed := FALSE;
END specify;
METHOD values;
P := 1.0 {atm};
T := 298.0 {K};
END values;

END idealgas;

MODEL testidealgas;

ig IS_A idealgas(P,T,V);
P IS_A pressure;
T IS_A temperature;
V IS_A molar_volume;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;
METHOD specify;
P.fixed := TRUE;
T.fixed := TRUE;
V.fixed := FALSE;
RUN ig.specify;
END specify;
METHOD values;
P := 1.0 {atm};
T := 800.0 {K};
RUN ig.values;
END values;
END testidealgas;

</source>

== '''Van der Waals equation of state''' ==

<source lang=a4c>

MODEL vanderwaals(P WILL_BE pressure;
T WILL_BE temperature;
V WILL_BE molar_volume;
Z WILL_BE factor;
data WILL_BE td_component_constants;
);

R IS_A molar_gas_constant;
b IS_A molar_volume;
a IS_A vdwa;
Pc IS_A pressure;
Tc IS_A temperature;

eq: (P*V^2 + a)*(V - b) = R*T*V^2;
eqb: b*8.0*data.Pc = R*data.Tc;
eqa: a*64.0*data.Pc = 27.0*R^2*(data.Tc)^2;
eqc: Z*R*T = P*V;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;
METHOD specify;
P.fixed := TRUE;
T.fixed := TRUE;
V.fixed := FALSE;
Z.fixed := FALSE;
END specify;
METHOD values;
P := 12.0 {atm};
T := 298.15 {K};
V := 0.075 {m^3/mole};
V.nominal := 0.05 {m^3/mole};
V.upper_bound := 100 {m^3/mole};
END values;

END vanderwaals;

MODEL testvanderwaals;

c1 IS_A symbol_constant;
vd IS_A vanderwaals(P,T,V,Z,cd.data[c1]);
Tr, Pr IS_A factor;
P IS_A pressure;
T IS_A temperature;
V IS_A molar_volume;
Z IS_A factor;

c1 :== 'ethylene';
cd IS_A components_data([c1],c1);

eqTr: Tr = T/cd.data[c1].Tc;
eqPr: Pr = P/cd.data[c1].Pc;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;

METHOD specify;
P.fixed := TRUE;
V.fixed := FALSE;
T.fixed := TRUE;
Z.fixed := FALSE;
RUN vd.specify;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
RUN vd.values;
END values;

END testvanderwaals;

</source>

<source lang=a4c>

MODEL truncated_virial_equation (P WILL_BE pressure;
T WILL_BE temperature; V WILL_BE molar_volume; Z WILL_BE factor;
data WILL_BE td_component_constants;);

R IS_A molar_gas_constant;
Pc IS_A pressure;
Tc IS_A temperature;
T_degC IS_A factor;
omega IS_A factor;
Tr, Pr IS_A factor;
B IS_A virialB;
C IS_A virialC;

Bhat, B0, B1 IS_A factor;

Pc = data.Pc;
Tc = data.Tc;
omega = data.omega;

eq1: Tr = T/Tc;
eq2: Pr = P/Pc;
eq3: T_degC = T/1{K} - 273.15;

eq4: Bhat = B0 + omega*B1;
eq5: B0*Tr^(1.6) = 0.083*Tr^(1.6) - 0.422;
eq6: B1*Tr^(4.2) = 0.139*Tr^(4.2) - 0.172;

eq7: Z = 1.0 + B0*Pr/Tr + omega*B1*Pr/Tr;

eq8: Z*R*T = P*V;
eq9: B = Bhat*1.0{m^3/mol};

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;
METHOD specify;
P.fixed := TRUE;
T_degC.fixed := FALSE;
T.fixed := TRUE;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
END values;

END truncated_virial_equation;

MODEL testtruncatedvirial;

c1 IS_A symbol_constant;
c1 :== 'acetonitrile';
cd IS_A components_data([c1],c1);
Tr, Pr IS_A factor;
P IS_A pressure;
T IS_A temperature;
V IS_A molar_volume;
Z IS_A factor;
tve IS_A truncated_virial_equation(P,T,V,Z,cd.data[c1]);

eqTr: Tr = T/cd.data[c1].Tc;
eqPr: Pr = P/cd.data[c1].Pc;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;

METHOD specify;
P.fixed := TRUE;
V.fixed := FALSE;
Z.fixed := FALSE;
T.fixed := TRUE;
RUN tve.specify;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
RUN tve.values;
END values;

END testtruncatedvirial;

</source>

<source lang=a4c>

MODEL rk(P WILL_BE pressure;
T WILL_BE temperature;
V WILL_BE molar_volume;
Z WILL_BE factor;
data WILL_BE td_component_constants;
);

R IS_A molar_gas_constant;
Pc IS_A pressure;
Tc IS_A temperature;
T_degC IS_A factor;
alpha, beta, q,Tr, Pr, omega IS_A factor;

PSI, OMEGA, sigma, eps IS_A real_constant;

PSI :== 0.42748;
OMEGA :== 0.08664;
sigma :== 1.0;
eps :== 0.0;

Pc = data.Pc;
Tc = data.Tc;
omega = data.omega;

eq1: Tr = T/Tc;
eq2: Pr = P/Pc;
eq3: T_degC = T/1{K} - 273.15;

eq4: alpha = Tr^(0.5);
eq5: q = PSI*alpha/(OMEGA*Tr);
eq6: beta = OMEGA*Pr/Tr;
eq7: Z = 1.0 + beta - q*beta*(Z - beta)/((Z + eps*beta)*(Z + sigma*beta));
eq8: P*V = Z*R*T;
METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;
METHOD specify;
P.fixed := TRUE;
T_degC.fixed := FALSE;
T.fixed := TRUE;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
Z.lower_bound := 0.0;
Z.upper_bound := 10.0;
Z.nominal := 1.0;
END values;
END rk;

MODEL testrk;

c1 IS_A symbol_constant;
c1 :== 'ethylene';
cd IS_A components_data([c1],c1);
Tr, Pr IS_A factor;
P IS_A pressure;
T IS_A temperature;
V IS_A molar_volume;
Z IS_A factor;
trk IS_A rk(P,T,V,Z,cd.data[c1]);

eqTr: Tr = T/cd.data[c1].Tc;
eqPr: Pr = P/cd.data[c1].Pc;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;

METHOD specify;
P.fixed := TRUE;
V.fixed := FALSE;
Z.fixed := FALSE;
T.fixed := TRUE;
RUN trk.specify;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
RUN trk.values;
END values;

END testrk;

</source>

<source lang=a4c>

MODEL srk(P WILL_BE pressure;
T WILL_BE temperature;
V WILL_BE molar_volume;
Z WILL_BE factor;
data WILL_BE td_component_constants;);

R IS_A molar_gas_constant;
Pc IS_A pressure;
Tc IS_A temperature;
T_degC IS_A factor;
alpha, beta, q,Tr, Pr, omega IS_A factor;

PSI, OMEGA, sigma, eps IS_A real_constant;

PSI :== 0.42748;
OMEGA :== 0.08664;
sigma :== 1.0;
eps :== 0.0;

Pc = data.Pc;
Tc = data.Tc;
omega = data.omega;

eq1: Tr = T/Tc;
eq2: Pr = P/Pc;
eq3: T_degC = T/1{K} - 273.15;

eq4: alpha = (1.0 + (0.480+1.574*omega-0.176*omega^2)*(1.0-Tr^(0.5)))^2;
eq5: q = PSI*alpha/(OMEGA*Tr);
eq6: beta = OMEGA*Pr/Tr;
eq7: Z = 1.0 + beta - q*beta*(Z - beta)/((Z + eps*beta)*(Z + sigma*beta));
eq8: P*V = Z*R*T;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;

METHOD specify;
P.fixed := TRUE;
T.fixed := TRUE;
T_degC.fixed := FALSE;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
Z.lower_bound := 0.0;
Z.upper_bound := 10.0;
Z.nominal := 1.0;
END values;
END srk;

MODEL testsrk;

c1 IS_A symbol_constant;
c1 :== 'ethylene';
cd IS_A components_data([c1],c1);
Tr, Pr IS_A factor;
P IS_A pressure;
T IS_A temperature;
V IS_A molar_volume;
Z IS_A factor;
tsrk IS_A srk(P,T,V,Z,cd.data[c1]);

eqTr: Tr = T/cd.data[c1].Tc;
eqPr: Pr = P/cd.data[c1].Pc;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;

METHOD specify;
P.fixed := TRUE;
V.fixed := FALSE;
Z.fixed := FALSE;
T.fixed := TRUE;
RUN tsrk.specify;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
RUN tsrk.values;
END values;

END testsrk;

</source>

<source lang=a4c>

MODEL pengrobinson(P WILL_BE pressure;
T WILL_BE temperature;
V WILL_BE molar_volume;
Z WILL_BE factor;
data WILL_BE td_component_constants;);

R IS_A molar_gas_constant;
Pc IS_A pressure;
Tc IS_A temperature;
T_degC IS_A factor;
alpha, beta, q,Tr, Pr, omega IS_A factor;

PSI, OMEGA, sigma, eps IS_A real_constant;

PSI :== 0.45724;
OMEGA :== 0.07780;
sigma :== 2.414213562;
eps :== 0.414213562;

Pc = data.Pc;
Tc = data.Tc;
omega = data.omega;

eq1: Tr = T/Tc;
eq2: Pr = P/Pc;
eq3: T_degC = T/1{K} - 273.15;

eq4: alpha = (1.0 + (0.37464+1.54226*omega-0.26992*omega^2)*(1.0-Tr^(0.5)))^2;
eq5: q = PSI*alpha/(OMEGA*Tr);
eq6: beta = OMEGA*Pr/Tr;
eq7: Z = 1.0 + beta - q*beta*(Z - beta)/((Z + eps*beta)*(Z + sigma*beta));
eq8: P*V = Z*R*T;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;

METHOD specify;
P.fixed := TRUE;
T.fixed := TRUE;
T_degC.fixed := FALSE;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
Z.lower_bound := 0.0;
Z.upper_bound := 10.0;
Z.nominal := 0.1;
END values;

END pengrobinson;

MODEL testpengrobinson;

c1 IS_A symbol_constant;
c1 :== 'ethylene';
cd IS_A components_data([c1],c1);
Tr, Pr IS_A factor;
P IS_A pressure;
T IS_A temperature;
V IS_A molar_volume;
Z IS_A factor;
tpeng IS_A pengrobinson(P,T,V,Z,cd.data[c1]);

eqTr: Tr = T/cd.data[c1].Tc;
eqPr: Pr = P/cd.data[c1].Pc;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;

METHOD specify;
P.fixed := TRUE;
V.fixed := FALSE;
Z.fixed := FALSE;
T.fixed := TRUE;
RUN tpeng.specify;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
RUN tpeng.values;
END values;

END testpengrobinson;

</source>

<source lang=a4c>

MODEL vdw(P WILL_BE pressure;
T WILL_BE temperature;
V WILL_BE molar_volume;
Z WILL_BE factor;
data WILL_BE td_component_constants;);

R IS_A molar_gas_constant;
Pc IS_A pressure;
Tc IS_A temperature;
T_degC IS_A factor;
beta, q,Tr, Pr, omega IS_A factor;

alpha, PSI, OMEGA, sigma, eps IS_A real_constant;

PSI :== 27.0/64.0;
OMEGA :== 1.0/8.0;
sigma :== 0.0;
eps :== 0.0;
alpha :== 1.0;

Pc = data.Pc;
Tc = data.Tc;
omega = data.omega;

eq1: Tr = T/Tc;
eq2: Pr = P/Pc;
eq3: T_degC = T/1{K} - 273.15;

eq5: q = PSI*alpha/(OMEGA*Tr);
eq6: beta = OMEGA*Pr/Tr;
eq7: Z = 1.0 + beta - q*beta*(Z - beta)/((Z + eps*beta)*(Z + sigma*beta));
eq8: P*V = Z*R*T;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;
METHOD specify;
P.fixed := TRUE;
T.fixed := TRUE;
T_degC.fixed := FALSE;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
Z.lower_bound := 0.0;
Z.upper_bound := 10.0;
Z.nominal := 0.1;
END values;

END vdw;

MODEL testvdw;

c1 IS_A symbol_constant;
c1 :== 'ethylene';
cd IS_A components_data([c1],c1);
Tr, Pr IS_A factor;
P IS_A pressure;
T IS_A temperature;
V IS_A molar_volume;
Z IS_A factor;
tvdw IS_A vdw(P,T,V,Z,cd.data[c1]);

eqTr: Tr = T/cd.data[c1].Tc;
eqPr: Pr = P/cd.data[c1].Pc;

METHODS
METHOD default_self;
RUN ClearAll;
RUN specify;
RUN values;
END default_self;

METHOD specify;
P.fixed := TRUE;
V.fixed := FALSE;
Z.fixed := FALSE;
T.fixed := TRUE;
RUN tvdw.specify;
END specify;
METHOD values;
P := 12.0 {bar};
T := 298.15 {K};
RUN tvdw.values;
END values;

END testvdw;

</source>

<source lang=a4c>

MODEL testall;

P IS_A pressure;
T IS_A temperature;
Vvanderwaals, Vvdw, Videal, Vtruncvirial, Vrk, Vsrk, Vpeng IS_A molar_volume;
Zvanderwaals, Zvdw, Ztruncvirial, Zrk, Zsrk, Zpeng IS_A factor;

c1 IS_A symbol_constant;
c1 :== 'acetonitrile';
cd IS_A components_data([c1],c1);
Tr, Pr IS_A factor;
ig IS_A idealgas(P,T,Videal);
vd IS_A vanderwaals(P,T,Vvanderwaals,Zvanderwaals,cd.data[c1]);
vdd IS_A vdw(P,T,Vvdw,Zvdw,cd.data[c1]);
tve IS_A truncated_virial_equation(P,T,Vtruncvirial,Ztruncvirial,cd.data[c1]);
trk IS_A rk(P,T,Vrk,Zrk,cd.data[c1]);
tsrk IS_A srk(P,T,Vsrk,Zsrk,cd.data[c1]);
tpeng IS_A pengrobinson(P,T,Vpeng,Zpeng,cd.data[c1]);

eqTr: Tr = T/cd.data[c1].Tc;
eqPr: Pr = P/cd.data[c1].Pc;

METHODS
METHOD default_self;
(* RUN ClearAll; *)
RUN specify;
RUN values;
END default_self;
METHOD specify;
P.fixed := TRUE;
T.fixed := TRUE;
Videal.fixed := FALSE;
Vvanderwaals.fixed := FALSE;
Vvdw.fixed := FALSE;
Vtruncvirial.fixed := FALSE;
Vrk.fixed := FALSE;
Vsrk.fixed := FALSE;
Vpeng.fixed := FALSE;
Zvanderwaals.fixed := FALSE;
Zvdw.fixed := FALSE;
Ztruncvirial.fixed := FALSE;
Zrk.fixed := FALSE;
Zsrk.fixed := FALSE;
Zpeng.fixed := FALSE;
RUN ig.specify;
RUN vd.specify;
RUN vdd.specify;
RUN tve.specify;
RUN trk.specify;
RUN tsrk.specify;
RUN tpeng.specify;
END specify;
METHOD values;
P := 306.12 {atm};
T := 561 {K};
RUN ig.values;
RUN vd.values;
RUN vdd.values;
RUN tve.values;
RUN trk.values;
RUN tsrk.values;
RUN tpeng.values;
END values;

END testall;
</source>