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FCSys.Characteristics.BaseClasses.CharacteristicEOS

Base thermodynamic package with only the p-v-T relations

Information

This package may be used with the assumption of ideal gas or of constant specific volume, although it is more general than that.

Notes regarding the constants:

  • bv: The powers of p/T increase by row. The powers of T increase by column. If n_v[1] == -1, then the rows of bv correspond to 1, B*T, C*T2, D*T3, …, where 1, B*, C*, and D* are the first, second, third, and fourth coefficients in the volume-explicit virial equation of state [Dymond2002, pp. 1–2]. Currently, virial equations of state are supported up to the fourth coefficient (D*). If additional terms are required, review and modify the definition of bp.
  • The defaults for bv and nv represent ideal gas.
Extends from Modelica.Icons.MaterialPropertiesPackage (Icon for package containing property classes).

Package Content

NameDescription
p0=U.barReference pressure (po)
n_v={-1,0}Powers of p/T and T for 1st row and column of bv (nv)
b_v=[1]Coefficients for specific volume as a polynomial in p/T and T (bv)
isCompressible=anyTrue({anyTrue({abs(b_v[i, j]) > Modelica.Constants.small and n_v[1] + i - 1 <> 0 for i in 1:size(b_v, 1)}) for j in 1:size(b_v, 2)})true, if density depends on pressure
hasThermalExpansion=anyTrue({anyTrue({abs(b_v[i, j]) > Modelica.Constants.small and n_v[2] + j - n_v[1] - i <> 0 for i in 1:size(b_v, 1)}) for j in 1:size(b_v, 2)})true, if density depends on temperature
n_p={n_v[1] - size(b_v, 1) + 1,n_v[2] + 1}Powers of v and T for 1st row and column of bp
b_p=if size(b_v, 1) == 1 then b_v .^ (-n_p[1]) else {(if n_v[1] + i == 0 or n_v[1] + i == 1 or size(b_v, 1) == 1 then b_v[i, :] else (if n_v[1] + i == 2 and n_v[1] <= 0 then b_v[i, :] + b_v[i - 1, :] .^ 2 else (if n_v[1] + i == 3 and n_v[1] <= 0 then b_v[i, :] + b_v[i - 2, :] .* (b_v[i - 2, :] .^ 2 + 3*b_v[i - 1, :]) else zeros(size(b_v, 2))))) for i in size(b_v, 1):-1:1}Coefficients of p as a polynomial in v and T
FCSys.Characteristics.BaseClasses.CharacteristicEOS.dp_Tv dp_Tv Derivative of pressure as defined by pT v()
FCSys.Characteristics.BaseClasses.CharacteristicEOS.dv_Tp dv_Tp Derivative of specific volume as defined by vT p()
FCSys.Characteristics.BaseClasses.CharacteristicEOS.p_Tv p_Tv Pressure as a function of temperature and specific volume (pT v())
FCSys.Characteristics.BaseClasses.CharacteristicEOS.v_Tp v_Tp Specific volume as a function of temperature and pressure (vT p())
FCSys.Characteristics.BaseClasses.CharacteristicEOS.beta beta Isothermal compressibility as a function of temperature and pressure (β)

Types and constants

constant Q.PressureAbsolute p0=U.bar 
  "Reference pressure (po)";
constant Integer n_v[2]={-1,0} 
  "Powers of p/T and T for 1st row and column of bv (nv)";
constant Real b_v[:, :]=[1] 
  "Coefficients for specific volume as a polynomial in p/T and T (bv)";
final constant Boolean isCompressible=anyTrue({anyTrue({abs(b_v[i, j]) >
    Modelica.Constants.small and n_v[1] + i - 1 <> 0 for i in 1:size(b_v, 1)}) 
    for j in 1:size(b_v, 2)}) 
  "true, if density depends on pressure";
final constant Boolean hasThermalExpansion=anyTrue({anyTrue({abs(b_v[i, j]) >
    Modelica.Constants.small and n_v[2] + j - n_v[1] - i <> 0 for i in 1:size(
    b_v, 1)}) for j in 1:size(b_v, 2)}) 
  "true, if density depends on temperature";
final constant Integer n_p[2]={n_v[1] - size(b_v, 1) + 1,n_v[2] + 1} 
  "Powers of v and T for 1st row and column of bp";
final constant Real b_p[size(b_v, 1), size(b_v, 2)]=if size(b_v, 1) == 1 then 
    b_v .^ (-n_p[1]) else {(if n_v[1] + i == 0 or n_v[1] + i == 1 or size(b_v, 1)
     == 1 then b_v[i, :] else (if n_v[1] + i == 2 and n_v[1] <= 0 then b_v[i, :]
     + b_v[i - 1, :] .^ 2 else (if n_v[1] + i == 3 and n_v[1] <= 0 then b_v[i,
    :] + b_v[i - 2, :] .* (b_v[i - 2, :] .^ 2 + 3*b_v[i - 1, :]) else zeros(
    size(b_v, 2))))) for i in size(b_v, 1):-1:1} 
  "Coefficients of p as a polynomial in v and T";

FCSys.Characteristics.BaseClasses.CharacteristicEOS.dp_Tv FCSys.Characteristics.BaseClasses.CharacteristicEOS.dp_Tv

Derivative of pressure as defined by pT v()

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

TypeNameDefaultDescription
TemperatureAbsoluteT298.15*U.KTemperature [L2.M/(N.T2)]
VolumeSpecificAbsolutev298.15*U.K/p0Specific volume [L3/N]
TemperaturedT0Derivative of temperature [L2.M/(N.T2)]
VolumeSpecificdv0Derivative of specific volume [L3/N]

Outputs

TypeNameDescription
PressuredpDerivative of pressure [M/(L.T2)]

Modelica definition

function dp_Tv 
  "Derivative of pressure as defined by pT v()"
  import FCSys.Utilities.Polynomial;
  extends Modelica.Icons.Function;

  input Q.TemperatureAbsolute T=298.15*U.K "Temperature";
  input Q.VolumeSpecificAbsolute v=298.15*U.K/p0 "Specific volume";
  input Q.Temperature dT=0 "Derivative of temperature";
  input Q.VolumeSpecific dv=0 "Derivative of specific volume";
  output Q.Pressure dp "Derivative of pressure";

algorithm 
  dp := if isCompressible then Polynomial.f(
    v,
    {Polynomial.f(
      T,
      b_p[i, :] .* {(n_p[1] + i - 1)*T*dv + (n_p[2] + j - 1)*v*dT for j in 1:
        size(b_p, 2)},
      n_p[2] - 1) for i in 1:size(b_p, 1)},
    n_p[1] - 1) else 0;

end dp_Tv;

FCSys.Characteristics.BaseClasses.CharacteristicEOS.dv_Tp FCSys.Characteristics.BaseClasses.CharacteristicEOS.dv_Tp

Derivative of specific volume as defined by vT p()

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

TypeNameDefaultDescription
TemperatureAbsoluteT298.15*U.KTemperature [L2.M/(N.T2)]
PressureAbsolutepp0Pressure [M/(L.T2)]
TemperaturedT0Derivative of temperature [L2.M/(N.T2)]
Pressuredp0Derivative of pressure [M/(L.T2)]

Outputs

TypeNameDescription
VolumeSpecificdvDerivative of specific volume [L3/N]

Modelica definition

function dv_Tp 
  "Derivative of specific volume as defined by vT p()"
  import FCSys.Utilities.Polynomial;
  extends Modelica.Icons.Function;
  input Q.TemperatureAbsolute T=298.15*U.K "Temperature";
  input Q.PressureAbsolute p=p0 "Pressure";
  input Q.Temperature dT=0 "Derivative of temperature";
  input Q.Pressure dp=0 "Derivative of pressure";
  output Q.VolumeSpecific dv "Derivative of specific volume";

algorithm 
  dv := Polynomial.f(
    p,
    {Polynomial.f(
      T,
      b_v[i, :] .* {(n_v[1] + i - 1)*T*dp + (n_v[2] - n_v[1] + j - i)*p*dT for 
        j in 1:size(b_v, 2)},
      n_v[2] - n_v[1] - i) for i in 1:size(b_v, 1)},
    n_v[1] - 1);

end dv_Tp;

FCSys.Characteristics.BaseClasses.CharacteristicEOS.p_Tv FCSys.Characteristics.BaseClasses.CharacteristicEOS.p_Tv

Pressure as a function of temperature and specific volume (pT v())

Information

If the species is incompressible then p(T, v) is undefined, and the function will return a value of zero.

The derivative of this function is dp_Tv().

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

TypeNameDefaultDescription
TemperatureAbsoluteT298.15*U.KTemperature [L2.M/(N.T2)]
VolumeSpecificAbsolutev298.15*U.K/p0Specific volume [L3/N]

Outputs

TypeNameDescription
PressureAbsolutepPressure [M/(L.T2)]

Modelica definition

function p_Tv 
  "Pressure as a function of temperature and specific volume (pT v())"
  annotation(derivative=dp_Tv);
  import FCSys.Utilities.Polynomial;
  extends Modelica.Icons.Function;
  input Q.TemperatureAbsolute T=298.15*U.K "Temperature";
  input Q.VolumeSpecificAbsolute v=298.15*U.K/p0 "Specific volume";
  output Q.PressureAbsolute p "Pressure";

algorithm 
  // assert(isCompressible,
  //  "The pressure is undefined since the material is incompressible.",
  //  AssertionLevel.warning);
  // Note:  This isn't used because it creates an error instead of a warning
  // in Dymola 2014
  p := if isCompressible then Polynomial.f(
    v,
    {Polynomial.f(
      T,
      b_p[i, :],
      n_p[2]) for i in 1:size(b_p, 1)},
    n_p[1]) else p0;
end p_Tv;

FCSys.Characteristics.BaseClasses.CharacteristicEOS.v_Tp FCSys.Characteristics.BaseClasses.CharacteristicEOS.v_Tp

Specific volume as a function of temperature and pressure (vT p())

Information

The derivative of this function is dv_Tp().

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

TypeNameDefaultDescription
TemperatureAbsoluteT298.15*U.KTemperature [L2.M/(N.T2)]
PressureAbsolutepp0Pressure [M/(L.T2)]

Outputs

TypeNameDescription
VolumeSpecificAbsolutevSpecific volume [L3/N]

Modelica definition

function v_Tp 
  "Specific volume as a function of temperature and pressure (vT p())"
  annotation(derivative=dv_Tp);
  import FCSys.Utilities.Polynomial;
  extends Modelica.Icons.Function;
  input Q.TemperatureAbsolute T=298.15*U.K "Temperature";
  input Q.PressureAbsolute p=p0 "Pressure";
  output Q.VolumeSpecificAbsolute v "Specific volume";

algorithm 
  v := Polynomial.f(
    p,
    {Polynomial.f(
      T,
      b_v[i, :],
      n_v[2] - n_v[1] - i + 1) for i in 1:size(b_v, 1)},
    n_v[1]);
end v_Tp;

FCSys.Characteristics.BaseClasses.CharacteristicEOS.beta FCSys.Characteristics.BaseClasses.CharacteristicEOS.beta

Isothermal compressibility as a function of temperature and pressure (β)

Information

For an ideal gas, this function is independent of temperature (although temperature remains as a valid input).

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

TypeNameDefaultDescription
TemperatureAbsoluteT298.15*U.KTemperature [L2.M/(N.T2)]
PressureAbsolutepp0Pressure [M/(L.T2)]

Outputs

TypeNameDescription
PressureReciprocalbetaIsothermal compressibility [L.T2/M]

Modelica definition

function beta 
  "Isothermal compressibility as a function of temperature and pressure (β)"
  extends Modelica.Icons.Function;

  input Q.TemperatureAbsolute T=298.15*U.K "Temperature";
  input Q.PressureAbsolute p=p0 "Pressure";

  output Q.PressureReciprocal beta "Isothermal compressibility";

algorithm 
  beta := -dv_Tp(
    T=T,
    p=p,
    dT=0,
    dp=1)/v_Tp(T, p);
end beta;
Copyright © 2007–2014, Kevin Davies, Hawaii Natural Energy Institute. Last updated Thu Jan 23 09:07:03 2014.