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polykin.properties.pvt_polymer¤

This module implements methods to evaluate the PVT behavior of pure polymers.

Flory ¤

Flory equation of state for the specific volume of a polymer.

This EoS implements the following implicit PVT dependence:

\[ \frac{\tilde{P}\tilde{V}}{\tilde{T}} = \frac{\tilde{V}^{1/3}}{\tilde{V}^{1/3}-1}-\frac{1}{\tilde{V}\tilde{T}}\]

where \(\tilde{V}=V/V^*\), \(\tilde{P}=P/P^*\) and \(\tilde{T}=T/T^*\) are, respectively, the reduced volume, reduced pressure and reduced temperature. \(V^*\), \(P^*\) and \(T^*\) are reference quantities that are polymer dependent.

References

  • Caruthers et al. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions. AIChE, 1998.
PARAMETER DESCRIPTION
V0

Reference volume, \(V^*\).

TYPE: float

T0

Reference temperature, \(T^*\).

TYPE: float

P0

Reference pressure, \(P^*\).

TYPE: float

Tmin

Lower temperature bound. Unit = K.

TYPE: float DEFAULT: 0.0

Tmax

Upper temperature bound. Unit = K.

TYPE: float DEFAULT: inf

Pmin

Lower pressure bound. Unit = Pa.

TYPE: float DEFAULT: 0.0

Pmax

Upper pressure bound. Unit = Pa.

TYPE: float DEFAULT: inf

name

Name.

TYPE: str DEFAULT: ''

Source code in src/polykin/properties/pvt_polymer/eos.py 
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class Flory(PolymerPVTEoS):
    r"""Flory equation of state for the specific volume of a polymer.

    This EoS implements the following implicit PVT dependence:

    $$ \frac{\tilde{P}\tilde{V}}{\tilde{T}} =
      \frac{\tilde{V}^{1/3}}{\tilde{V}^{1/3}-1}-\frac{1}{\tilde{V}\tilde{T}}$$

    where $\tilde{V}=V/V^*$, $\tilde{P}=P/P^*$ and $\tilde{T}=T/T^*$ are,
    respectively, the reduced volume, reduced pressure and reduced temperature.
    $V^*$, $P^*$ and $T^*$ are reference quantities that are polymer dependent.

    **References**

    *   Caruthers et al. Handbook of Diffusion and Thermal Properties of
        Polymers and Polymer Solutions. AIChE, 1998.

    Parameters
    ----------
    V0 : float
        Reference volume, $V^*$.
    T0 : float
        Reference temperature, $T^*$.
    P0 : float
        Reference pressure, $P^*$.
    Tmin : float
        Lower temperature bound.
        Unit = K.
    Tmax : float
        Upper temperature bound.
        Unit = K.
    Pmin : float
        Lower pressure bound.
        Unit = Pa.
    Pmax : float
        Upper pressure bound.
        Unit = Pa.
    name : str
        Name.
    """

    @staticmethod
    def equation(v: float,
                 t: float,
                 p: float
                 ) -> tuple[float, float, float]:
        r"""Flory equation of state and its volume derivatives.

        Parameters
        ----------
        v : float
            Reduced volume, $\tilde{V}$.
        t : float
            Reduced temperature, $\tilde{T}$.
        p : float
            Reduced pressure, $\tilde{P}$.

        Returns
        -------
        tuple[float, float, float]
            Equation of state, first derivative, second derivative.
        """
        f = p*v/t - (v**(1/3)/(v**(1/3) - 1) - 1/(v*t))  # =0
        d1f = p/t - 1/(t*v**2) - 1/(3*(v**(1/3) - 1)*v**(2/3)) + \
            1/(3*(v**(1/3) - 1)**2*v**(1/3))
        d2f = (2*(9/t + (v**(4/3) - 2*v**(5/3))/(-1 + v**(1/3))**3))/(9*v**3)
        return (f, d1f, d2f)

V ¤

V(
    T: Union[float, FloatArrayLike],
    P: Union[float, FloatArrayLike],
    Tunit: Literal["C", "K"] = "K",
    Punit: Literal["bar", "MPa", "Pa"] = "Pa",
) -> Union[float, FloatArray]

Evaluate the specific volume, \(\hat{V}\), at given temperature and pressure, including unit conversion and range check.

PARAMETER DESCRIPTION
T

Temperature. Unit = Tunit.

TYPE: float | FloatArrayLike

P

Pressure. Unit = Punit.

TYPE: float | FloatArrayLike

Tunit

Temperature unit.

TYPE: Literal['C', 'K'] DEFAULT: 'K'

Punit

Pressure unit.

TYPE: Literal['bar', 'MPa', 'Pa'] DEFAULT: 'Pa'

RETURNS DESCRIPTION
float | FloatArray

Specific volume. Unit = m³/kg.
Source code in src/polykin/properties/pvt_polymer/base.py 
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def V(self,
      T: Union[float, FloatArrayLike],
      P: Union[float, FloatArrayLike],
      Tunit: Literal['C', 'K'] = 'K',
      Punit: Literal['bar', 'MPa', 'Pa'] = 'Pa'
      ) -> Union[float, FloatArray]:
    r"""Evaluate the specific volume, $\hat{V}$, at given temperature and
    pressure, including unit conversion and range check.

    Parameters
    ----------
    T : float | FloatArrayLike
        Temperature.
        Unit = `Tunit`.
    P : float | FloatArrayLike
        Pressure.
        Unit = `Punit`.
    Tunit : Literal['C', 'K']
        Temperature unit.
    Punit : Literal['bar', 'MPa', 'Pa']
        Pressure unit.

    Returns
    -------
    float | FloatArray
        Specific volume.
        Unit = m³/kg.
    """
    TK = convert_check_temperature(T, Tunit, self.Trange)
    Pa = convert_check_pressure(P, Punit, self.Prange)
    return self.eval(TK, Pa)

alpha ¤

alpha(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Calculate thermal expansion coefficient, \(\alpha\).

\[\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Thermal expansion coefficient, \(\alpha\). Unit = 1/K.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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def alpha(self,
          T: Union[float, FloatArray],
          P: Union[float, FloatArray]
          ) -> Union[float, FloatArray]:
    r"""Calculate thermal expansion coefficient, $\alpha$.

    $$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}$$

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Thermal expansion coefficient, $\alpha$.
        Unit = 1/K.
    """
    dT = 0.5
    V2 = self.eval(T + dT, P)
    V1 = self.eval(T - dT, P)
    return (V2 - V1)/dT/(V1 + V2)

beta ¤

beta(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Calculate isothermal compressibility coefficient, \(\beta\).

\[\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Isothermal compressibility coefficient, \(\beta\). Unit = 1/Pa.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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def beta(self,
         T: Union[float, FloatArray],
         P: Union[float, FloatArray]
         ) -> Union[float, FloatArray]:
    r"""Calculate isothermal compressibility coefficient, $\beta$.

    $$\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Isothermal compressibility coefficient, $\beta$.
        Unit = 1/Pa.
    """
    dP = 1e5
    P2 = P + dP
    P1 = np.max(P - dP, 0)
    V2 = self.eval(T, P2)
    V1 = self.eval(T, P1)
    return -(V2 - V1)/(P2 - P1)/(V1 + V2)*2

equation staticmethod ¤

equation(
    v: float, t: float, p: float
) -> tuple[float, float, float]

Flory equation of state and its volume derivatives.

PARAMETER DESCRIPTION
v

Reduced volume, \(\tilde{V}\).

TYPE: float

t

Reduced temperature, \(\tilde{T}\).

TYPE: float

p

Reduced pressure, \(\tilde{P}\).

TYPE: float

RETURNS DESCRIPTION
tuple[float, float, float]

Equation of state, first derivative, second derivative.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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@staticmethod
def equation(v: float,
             t: float,
             p: float
             ) -> tuple[float, float, float]:
    r"""Flory equation of state and its volume derivatives.

    Parameters
    ----------
    v : float
        Reduced volume, $\tilde{V}$.
    t : float
        Reduced temperature, $\tilde{T}$.
    p : float
        Reduced pressure, $\tilde{P}$.

    Returns
    -------
    tuple[float, float, float]
        Equation of state, first derivative, second derivative.
    """
    f = p*v/t - (v**(1/3)/(v**(1/3) - 1) - 1/(v*t))  # =0
    d1f = p/t - 1/(t*v**2) - 1/(3*(v**(1/3) - 1)*v**(2/3)) + \
        1/(3*(v**(1/3) - 1)**2*v**(1/3))
    d2f = (2*(9/t + (v**(4/3) - 2*v**(5/3))/(-1 + v**(1/3))**3))/(9*v**3)
    return (f, d1f, d2f)

eval ¤

eval(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Evaluate specific volume, \(\hat{V}\), at given SI conditions without unit conversions or checks.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Specific volume. Unit = m³/kg.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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@vectorize
def eval(self,
         T: Union[float, FloatArray],
         P: Union[float, FloatArray]
         ) -> Union[float, FloatArray]:
    r"""Evaluate specific volume, $\hat{V}$, at given SI conditions without
    unit conversions or checks.

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Specific volume.
        Unit = m³/kg.
    """
    t = T/self.T0
    p = P/self.P0
    solution = root_scalar(f=self.equation,
                           args=(t, p),
                           # bracket=[1.1, 1.5],
                           x0=1.05,
                           method='halley',
                           fprime=True,
                           fprime2=True)

    if solution.converged:
        v = solution.root
        V = v*self.V0
    else:
        print(solution.flag)
        V = -1.
    return V

from_database classmethod ¤

from_database(name: str) -> Optional[PolymerPVTEquation]

Construct PolymerPVTEquation with parameters from the database.

PARAMETER DESCRIPTION
name

Polymer code name.

TYPE: str

Source code in src/polykin/properties/pvt_polymer/base.py 
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@classmethod
def from_database(cls,
                  name: str
                  ) -> Optional[PolymerPVTEquation]:
    r"""Construct `PolymerPVTEquation` with parameters from the database.

    Parameters
    ----------
    name : str
        Polymer code name.
    """
    table = load_PVT_parameters(method=cls.__name__)
    try:
        mask = table.index == name
        parameters = table[mask].iloc[0, :].to_dict()
        return cls(**parameters, name=name)
    except IndexError:
        print(
            f"Error: '{name}' does not exist in polymer database.\n"
            f"Valid names are: {table.index.to_list()}")

get_database classmethod ¤

get_database() -> pd.DataFrame

Get database with parameters for the respective PVT equation.

Method Reference
Flory [2] Table 4.1.7 (p. 72-73)
Hartmann-Haque [2] Table 4.1.11 (p. 85-86)
Sanchez-Lacombe [2] Table 4.1.9 (p. 78-79)
Tait [1] Table 3B-1 (p. 41)

References

  1. Danner, Ronald P., and Martin S. High. Handbook of polymer solution thermodynamics. John Wiley & Sons, 2010.
  2. Caruthers et al. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions. AIChE, 1998.
Source code in src/polykin/properties/pvt_polymer/base.py 
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@classmethod
def get_database(cls) -> pd.DataFrame:
    r"""Get database with parameters for the respective PVT equation.

    | Method          | Reference                            |
    | :-----------    | ------------------------------------ |
    | Flory           | [2] Table 4.1.7  (p. 72-73)          |
    | Hartmann-Haque  | [2] Table 4.1.11 (p. 85-86)          |
    | Sanchez-Lacombe | [2] Table 4.1.9  (p. 78-79)          |
    | Tait            | [1] Table 3B-1 (p. 41)               |

    **References**

    1.  Danner, Ronald P., and Martin S. High. Handbook of polymer
        solution thermodynamics. John Wiley & Sons, 2010.
    2.  Caruthers et al. Handbook of Diffusion and Thermal Properties of
        Polymers and Polymer Solutions. AIChE, 1998.
    """
    return load_PVT_parameters(method=cls.__name__)

HartmannHaque ¤

Hartmann-Haque equation of state for the specific volume of a polymer.

This EoS implements the following implicit PVT dependence:

\[ \tilde{P}\tilde{V}^5=\tilde{T}^{3/2}-\ln{\tilde{V}} \]

where \(\tilde{V}=V/V^*\), \(\tilde{P}=P/P^*\) and \(\tilde{T}=T/T^*\) are, respectively, the reduced volume, reduced pressure and reduced temperature. \(V^*\), \(P^*\) and \(T^*\) are reference quantities that are polymer dependent.

References

  • Caruthers et al. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions. AIChE, 1998.
PARAMETER DESCRIPTION
V0

Reference volume, \(V^*\).

TYPE: float

T0

Reference temperature, \(T^*\).

TYPE: float

P0

Reference pressure, \(P^*\).

TYPE: float

Tmin

Lower temperature bound. Unit = K.

TYPE: float DEFAULT: 0.0

Tmax

Upper temperature bound. Unit = K.

TYPE: float DEFAULT: inf

Pmin

Lower pressure bound. Unit = Pa.

TYPE: float DEFAULT: 0.0

Pmax

Upper pressure bound. Unit = Pa.

TYPE: float DEFAULT: inf

name

Name.

TYPE: str DEFAULT: ''

Source code in src/polykin/properties/pvt_polymer/eos.py 
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class HartmannHaque(PolymerPVTEoS):
    r"""Hartmann-Haque equation of state for the specific volume of a polymer.

    This EoS implements the following implicit PVT dependence:

    $$ \tilde{P}\tilde{V}^5=\tilde{T}^{3/2}-\ln{\tilde{V}} $$

    where $\tilde{V}=V/V^*$, $\tilde{P}=P/P^*$ and $\tilde{T}=T/T^*$ are,
    respectively, the reduced volume, reduced pressure and reduced temperature.
    $V^*$, $P^*$ and $T^*$ are reference quantities that are polymer dependent.

    **References**

    *   Caruthers et al. Handbook of Diffusion and Thermal Properties of
        Polymers and Polymer Solutions. AIChE, 1998.

    Parameters
    ----------
    V0 : float
        Reference volume, $V^*$.
    T0 : float
        Reference temperature, $T^*$.
    P0 : float
        Reference pressure, $P^*$.
    Tmin : float
        Lower temperature bound.
        Unit = K.
    Tmax : float
        Upper temperature bound.
        Unit = K.
    Pmin : float
        Lower pressure bound.
        Unit = Pa.
    Pmax : float
        Upper pressure bound.
        Unit = Pa.
    name : str
        Name.
    """

    @staticmethod
    def equation(v: float,
                 t: float,
                 p: float
                 ) -> tuple[float, float, float]:
        """Hartmann-Haque equation of state and its volume derivatives.

        Parameters
        ----------
        v : float
            Reduced volume.
        t : float
            Reduced temperature.
        p : float
            Reduced pressure.

        Returns
        -------
        tuple[float, float, float]
            Equation of state, first derivative, second derivative.
        """
        f = p*v**5 - t**(3/2) + log(v)  # =0
        d1f = 5*p*v**4 + 1/v
        d2f = 20*p*v**3 - 1/v**2
        return (f, d1f, d2f)

V ¤

V(
    T: Union[float, FloatArrayLike],
    P: Union[float, FloatArrayLike],
    Tunit: Literal["C", "K"] = "K",
    Punit: Literal["bar", "MPa", "Pa"] = "Pa",
) -> Union[float, FloatArray]

Evaluate the specific volume, \(\hat{V}\), at given temperature and pressure, including unit conversion and range check.

PARAMETER DESCRIPTION
T

Temperature. Unit = Tunit.

TYPE: float | FloatArrayLike

P

Pressure. Unit = Punit.

TYPE: float | FloatArrayLike

Tunit

Temperature unit.

TYPE: Literal['C', 'K'] DEFAULT: 'K'

Punit

Pressure unit.

TYPE: Literal['bar', 'MPa', 'Pa'] DEFAULT: 'Pa'

RETURNS DESCRIPTION
float | FloatArray

Specific volume. Unit = m³/kg.
Source code in src/polykin/properties/pvt_polymer/base.py 
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def V(self,
      T: Union[float, FloatArrayLike],
      P: Union[float, FloatArrayLike],
      Tunit: Literal['C', 'K'] = 'K',
      Punit: Literal['bar', 'MPa', 'Pa'] = 'Pa'
      ) -> Union[float, FloatArray]:
    r"""Evaluate the specific volume, $\hat{V}$, at given temperature and
    pressure, including unit conversion and range check.

    Parameters
    ----------
    T : float | FloatArrayLike
        Temperature.
        Unit = `Tunit`.
    P : float | FloatArrayLike
        Pressure.
        Unit = `Punit`.
    Tunit : Literal['C', 'K']
        Temperature unit.
    Punit : Literal['bar', 'MPa', 'Pa']
        Pressure unit.

    Returns
    -------
    float | FloatArray
        Specific volume.
        Unit = m³/kg.
    """
    TK = convert_check_temperature(T, Tunit, self.Trange)
    Pa = convert_check_pressure(P, Punit, self.Prange)
    return self.eval(TK, Pa)

alpha ¤

alpha(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Calculate thermal expansion coefficient, \(\alpha\).

\[\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Thermal expansion coefficient, \(\alpha\). Unit = 1/K.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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def alpha(self,
          T: Union[float, FloatArray],
          P: Union[float, FloatArray]
          ) -> Union[float, FloatArray]:
    r"""Calculate thermal expansion coefficient, $\alpha$.

    $$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}$$

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Thermal expansion coefficient, $\alpha$.
        Unit = 1/K.
    """
    dT = 0.5
    V2 = self.eval(T + dT, P)
    V1 = self.eval(T - dT, P)
    return (V2 - V1)/dT/(V1 + V2)

beta ¤

beta(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Calculate isothermal compressibility coefficient, \(\beta\).

\[\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Isothermal compressibility coefficient, \(\beta\). Unit = 1/Pa.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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def beta(self,
         T: Union[float, FloatArray],
         P: Union[float, FloatArray]
         ) -> Union[float, FloatArray]:
    r"""Calculate isothermal compressibility coefficient, $\beta$.

    $$\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Isothermal compressibility coefficient, $\beta$.
        Unit = 1/Pa.
    """
    dP = 1e5
    P2 = P + dP
    P1 = np.max(P - dP, 0)
    V2 = self.eval(T, P2)
    V1 = self.eval(T, P1)
    return -(V2 - V1)/(P2 - P1)/(V1 + V2)*2

equation staticmethod ¤

equation(
    v: float, t: float, p: float
) -> tuple[float, float, float]

Hartmann-Haque equation of state and its volume derivatives.

PARAMETER DESCRIPTION
v

Reduced volume.

TYPE: float

t

Reduced temperature.

TYPE: float

p

Reduced pressure.

TYPE: float

RETURNS DESCRIPTION
tuple[float, float, float]

Equation of state, first derivative, second derivative.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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@staticmethod
def equation(v: float,
             t: float,
             p: float
             ) -> tuple[float, float, float]:
    """Hartmann-Haque equation of state and its volume derivatives.

    Parameters
    ----------
    v : float
        Reduced volume.
    t : float
        Reduced temperature.
    p : float
        Reduced pressure.

    Returns
    -------
    tuple[float, float, float]
        Equation of state, first derivative, second derivative.
    """
    f = p*v**5 - t**(3/2) + log(v)  # =0
    d1f = 5*p*v**4 + 1/v
    d2f = 20*p*v**3 - 1/v**2
    return (f, d1f, d2f)

eval ¤

eval(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Evaluate specific volume, \(\hat{V}\), at given SI conditions without unit conversions or checks.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Specific volume. Unit = m³/kg.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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@vectorize
def eval(self,
         T: Union[float, FloatArray],
         P: Union[float, FloatArray]
         ) -> Union[float, FloatArray]:
    r"""Evaluate specific volume, $\hat{V}$, at given SI conditions without
    unit conversions or checks.

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Specific volume.
        Unit = m³/kg.
    """
    t = T/self.T0
    p = P/self.P0
    solution = root_scalar(f=self.equation,
                           args=(t, p),
                           # bracket=[1.1, 1.5],
                           x0=1.05,
                           method='halley',
                           fprime=True,
                           fprime2=True)

    if solution.converged:
        v = solution.root
        V = v*self.V0
    else:
        print(solution.flag)
        V = -1.
    return V

from_database classmethod ¤

from_database(name: str) -> Optional[PolymerPVTEquation]

Construct PolymerPVTEquation with parameters from the database.

PARAMETER DESCRIPTION
name

Polymer code name.

TYPE: str

Source code in src/polykin/properties/pvt_polymer/base.py 
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@classmethod
def from_database(cls,
                  name: str
                  ) -> Optional[PolymerPVTEquation]:
    r"""Construct `PolymerPVTEquation` with parameters from the database.

    Parameters
    ----------
    name : str
        Polymer code name.
    """
    table = load_PVT_parameters(method=cls.__name__)
    try:
        mask = table.index == name
        parameters = table[mask].iloc[0, :].to_dict()
        return cls(**parameters, name=name)
    except IndexError:
        print(
            f"Error: '{name}' does not exist in polymer database.\n"
            f"Valid names are: {table.index.to_list()}")

get_database classmethod ¤

get_database() -> pd.DataFrame

Get database with parameters for the respective PVT equation.

Method Reference
Flory [2] Table 4.1.7 (p. 72-73)
Hartmann-Haque [2] Table 4.1.11 (p. 85-86)
Sanchez-Lacombe [2] Table 4.1.9 (p. 78-79)
Tait [1] Table 3B-1 (p. 41)

References

  1. Danner, Ronald P., and Martin S. High. Handbook of polymer solution thermodynamics. John Wiley & Sons, 2010.
  2. Caruthers et al. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions. AIChE, 1998.
Source code in src/polykin/properties/pvt_polymer/base.py 
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@classmethod
def get_database(cls) -> pd.DataFrame:
    r"""Get database with parameters for the respective PVT equation.

    | Method          | Reference                            |
    | :-----------    | ------------------------------------ |
    | Flory           | [2] Table 4.1.7  (p. 72-73)          |
    | Hartmann-Haque  | [2] Table 4.1.11 (p. 85-86)          |
    | Sanchez-Lacombe | [2] Table 4.1.9  (p. 78-79)          |
    | Tait            | [1] Table 3B-1 (p. 41)               |

    **References**

    1.  Danner, Ronald P., and Martin S. High. Handbook of polymer
        solution thermodynamics. John Wiley & Sons, 2010.
    2.  Caruthers et al. Handbook of Diffusion and Thermal Properties of
        Polymers and Polymer Solutions. AIChE, 1998.
    """
    return load_PVT_parameters(method=cls.__name__)

SanchezLacombe ¤

Sanchez-Lacombe equation of state for the specific volume of a polymer.

This EoS implements the following implicit PVT dependence:

\[ \frac{1}{\tilde{V}^2} + \tilde{P} + \tilde{T}\left [ \ln\left ( 1-\frac{1}{\tilde{V}} \right ) + \frac{1}{\tilde{V}} \right ]=0 \]

where \(\tilde{V}=V/V^*\), \(\tilde{P}=P/P^*\) and \(\tilde{T}=T/T^*\) are, respectively, the reduced volume, reduced pressure and reduced temperature. \(V^*\), \(P^*\) and \(T^*\) are reference quantities that are polymer dependent.

References

  • Caruthers et al. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions. AIChE, 1998.
PARAMETER DESCRIPTION
V0

Reference volume, \(V^*\).

TYPE: float

T0

Reference temperature, \(T^*\).

TYPE: float

P0

Reference pressure, \(P^*\).

TYPE: float

Tmin

Lower temperature bound. Unit = K.

TYPE: float DEFAULT: 0.0

Tmax

Upper temperature bound. Unit = K.

TYPE: float DEFAULT: inf

Pmin

Lower pressure bound. Unit = Pa.

TYPE: float DEFAULT: 0.0

Pmax

Upper pressure bound. Unit = Pa.

TYPE: float DEFAULT: inf

name

Name.

TYPE: str DEFAULT: ''

Source code in src/polykin/properties/pvt_polymer/eos.py 
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class SanchezLacombe(PolymerPVTEoS):
    r"""Sanchez-Lacombe equation of state for the specific volume of a polymer.

    This EoS implements the following implicit PVT dependence:

    $$ \frac{1}{\tilde{V}^2} + \tilde{P} +
        \tilde{T}\left [ \ln\left ( 1-\frac{1}{\tilde{V}} \right ) +
        \frac{1}{\tilde{V}} \right ]=0 $$

    where $\tilde{V}=V/V^*$, $\tilde{P}=P/P^*$ and $\tilde{T}=T/T^*$ are,
    respectively, the reduced volume, reduced pressure and reduced temperature.
    $V^*$, $P^*$ and $T^*$ are reference quantities that are polymer dependent.

    **References**

    *   Caruthers et al. Handbook of Diffusion and Thermal Properties of
        Polymers and Polymer Solutions. AIChE, 1998.

    Parameters
    ----------
    V0 : float
        Reference volume, $V^*$.
    T0 : float
        Reference temperature, $T^*$.
    P0 : float
        Reference pressure, $P^*$.
    Tmin : float
        Lower temperature bound.
        Unit = K.
    Tmax : float
        Upper temperature bound.
        Unit = K.
    Pmin : float
        Lower pressure bound.
        Unit = Pa.
    Pmax : float
        Upper pressure bound.
        Unit = Pa.
    name : str
        Name.
    """

    @staticmethod
    def equation(v: float,
                 t: float,
                 p: float
                 ) -> tuple[float, float, float]:
        """Sanchez-Lacombe equation of state and its volume derivatives.

        Parameters
        ----------
        v : float
            Reduced volume.
        t : float
            Reduced temperature.
        p : float
            Reduced pressure.

        Returns
        -------
        tuple[float, float, float]
            Equation of state, first derivative, second derivative.
        """
        f = 1/v**2 + p + t*(log(1 - 1/v) + 1/v)  # =0
        d1f = ((t - 2)*v + 2)/((v - 1)*v**3)
        d2f = (-3*(t - 2)*v**2 + 2*(t - 6)*v + 6)/((v - 1)**2*v**4)
        return (f, d1f, d2f)

V ¤

V(
    T: Union[float, FloatArrayLike],
    P: Union[float, FloatArrayLike],
    Tunit: Literal["C", "K"] = "K",
    Punit: Literal["bar", "MPa", "Pa"] = "Pa",
) -> Union[float, FloatArray]

Evaluate the specific volume, \(\hat{V}\), at given temperature and pressure, including unit conversion and range check.

PARAMETER DESCRIPTION
T

Temperature. Unit = Tunit.

TYPE: float | FloatArrayLike

P

Pressure. Unit = Punit.

TYPE: float | FloatArrayLike

Tunit

Temperature unit.

TYPE: Literal['C', 'K'] DEFAULT: 'K'

Punit

Pressure unit.

TYPE: Literal['bar', 'MPa', 'Pa'] DEFAULT: 'Pa'

RETURNS DESCRIPTION
float | FloatArray

Specific volume. Unit = m³/kg.
Source code in src/polykin/properties/pvt_polymer/base.py 
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def V(self,
      T: Union[float, FloatArrayLike],
      P: Union[float, FloatArrayLike],
      Tunit: Literal['C', 'K'] = 'K',
      Punit: Literal['bar', 'MPa', 'Pa'] = 'Pa'
      ) -> Union[float, FloatArray]:
    r"""Evaluate the specific volume, $\hat{V}$, at given temperature and
    pressure, including unit conversion and range check.

    Parameters
    ----------
    T : float | FloatArrayLike
        Temperature.
        Unit = `Tunit`.
    P : float | FloatArrayLike
        Pressure.
        Unit = `Punit`.
    Tunit : Literal['C', 'K']
        Temperature unit.
    Punit : Literal['bar', 'MPa', 'Pa']
        Pressure unit.

    Returns
    -------
    float | FloatArray
        Specific volume.
        Unit = m³/kg.
    """
    TK = convert_check_temperature(T, Tunit, self.Trange)
    Pa = convert_check_pressure(P, Punit, self.Prange)
    return self.eval(TK, Pa)

alpha ¤

alpha(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Calculate thermal expansion coefficient, \(\alpha\).

\[\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Thermal expansion coefficient, \(\alpha\). Unit = 1/K.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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def alpha(self,
          T: Union[float, FloatArray],
          P: Union[float, FloatArray]
          ) -> Union[float, FloatArray]:
    r"""Calculate thermal expansion coefficient, $\alpha$.

    $$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}$$

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Thermal expansion coefficient, $\alpha$.
        Unit = 1/K.
    """
    dT = 0.5
    V2 = self.eval(T + dT, P)
    V1 = self.eval(T - dT, P)
    return (V2 - V1)/dT/(V1 + V2)

beta ¤

beta(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Calculate isothermal compressibility coefficient, \(\beta\).

\[\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Isothermal compressibility coefficient, \(\beta\). Unit = 1/Pa.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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def beta(self,
         T: Union[float, FloatArray],
         P: Union[float, FloatArray]
         ) -> Union[float, FloatArray]:
    r"""Calculate isothermal compressibility coefficient, $\beta$.

    $$\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Isothermal compressibility coefficient, $\beta$.
        Unit = 1/Pa.
    """
    dP = 1e5
    P2 = P + dP
    P1 = np.max(P - dP, 0)
    V2 = self.eval(T, P2)
    V1 = self.eval(T, P1)
    return -(V2 - V1)/(P2 - P1)/(V1 + V2)*2

equation staticmethod ¤

equation(
    v: float, t: float, p: float
) -> tuple[float, float, float]

Sanchez-Lacombe equation of state and its volume derivatives.

PARAMETER DESCRIPTION
v

Reduced volume.

TYPE: float

t

Reduced temperature.

TYPE: float

p

Reduced pressure.

TYPE: float

RETURNS DESCRIPTION
tuple[float, float, float]

Equation of state, first derivative, second derivative.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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@staticmethod
def equation(v: float,
             t: float,
             p: float
             ) -> tuple[float, float, float]:
    """Sanchez-Lacombe equation of state and its volume derivatives.

    Parameters
    ----------
    v : float
        Reduced volume.
    t : float
        Reduced temperature.
    p : float
        Reduced pressure.

    Returns
    -------
    tuple[float, float, float]
        Equation of state, first derivative, second derivative.
    """
    f = 1/v**2 + p + t*(log(1 - 1/v) + 1/v)  # =0
    d1f = ((t - 2)*v + 2)/((v - 1)*v**3)
    d2f = (-3*(t - 2)*v**2 + 2*(t - 6)*v + 6)/((v - 1)**2*v**4)
    return (f, d1f, d2f)

eval ¤

eval(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Evaluate specific volume, \(\hat{V}\), at given SI conditions without unit conversions or checks.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Specific volume. Unit = m³/kg.
Source code in src/polykin/properties/pvt_polymer/eos.py 
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@vectorize
def eval(self,
         T: Union[float, FloatArray],
         P: Union[float, FloatArray]
         ) -> Union[float, FloatArray]:
    r"""Evaluate specific volume, $\hat{V}$, at given SI conditions without
    unit conversions or checks.

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Specific volume.
        Unit = m³/kg.
    """
    t = T/self.T0
    p = P/self.P0
    solution = root_scalar(f=self.equation,
                           args=(t, p),
                           # bracket=[1.1, 1.5],
                           x0=1.05,
                           method='halley',
                           fprime=True,
                           fprime2=True)

    if solution.converged:
        v = solution.root
        V = v*self.V0
    else:
        print(solution.flag)
        V = -1.
    return V

from_database classmethod ¤

from_database(name: str) -> Optional[PolymerPVTEquation]

Construct PolymerPVTEquation with parameters from the database.

PARAMETER DESCRIPTION
name

Polymer code name.

TYPE: str

Source code in src/polykin/properties/pvt_polymer/base.py 
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@classmethod
def from_database(cls,
                  name: str
                  ) -> Optional[PolymerPVTEquation]:
    r"""Construct `PolymerPVTEquation` with parameters from the database.

    Parameters
    ----------
    name : str
        Polymer code name.
    """
    table = load_PVT_parameters(method=cls.__name__)
    try:
        mask = table.index == name
        parameters = table[mask].iloc[0, :].to_dict()
        return cls(**parameters, name=name)
    except IndexError:
        print(
            f"Error: '{name}' does not exist in polymer database.\n"
            f"Valid names are: {table.index.to_list()}")

get_database classmethod ¤

get_database() -> pd.DataFrame

Get database with parameters for the respective PVT equation.

Method Reference
Flory [2] Table 4.1.7 (p. 72-73)
Hartmann-Haque [2] Table 4.1.11 (p. 85-86)
Sanchez-Lacombe [2] Table 4.1.9 (p. 78-79)
Tait [1] Table 3B-1 (p. 41)

References

  1. Danner, Ronald P., and Martin S. High. Handbook of polymer solution thermodynamics. John Wiley & Sons, 2010.
  2. Caruthers et al. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions. AIChE, 1998.
Source code in src/polykin/properties/pvt_polymer/base.py 
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@classmethod
def get_database(cls) -> pd.DataFrame:
    r"""Get database with parameters for the respective PVT equation.

    | Method          | Reference                            |
    | :-----------    | ------------------------------------ |
    | Flory           | [2] Table 4.1.7  (p. 72-73)          |
    | Hartmann-Haque  | [2] Table 4.1.11 (p. 85-86)          |
    | Sanchez-Lacombe | [2] Table 4.1.9  (p. 78-79)          |
    | Tait            | [1] Table 3B-1 (p. 41)               |

    **References**

    1.  Danner, Ronald P., and Martin S. High. Handbook of polymer
        solution thermodynamics. John Wiley & Sons, 2010.
    2.  Caruthers et al. Handbook of Diffusion and Thermal Properties of
        Polymers and Polymer Solutions. AIChE, 1998.
    """
    return load_PVT_parameters(method=cls.__name__)

Tait ¤

Tait equation of state for the specific volume of a liquid.

This EoS implements the following explicit PVT dependence:

\[\hat{V}(T,P)=\hat{V}(T,0)\left[1-C\ln\left(\frac{P}{B(T)}\right)\right]\]

with:

\[ \begin{gather*} \hat{V}(T,0) = A_0 + A_1(T - 273.15) + A_2(T - 273.15)^2 \\ B(T) = B_0\exp\left [-B_1(T - 273.15)\right] \end{gather*} \]

where \(A_i\) and \(B_i\) are constant parameters, \(T\) is the absolute temperature, and \(P\) is the pressure.

References

  • Danner, Ronald P., and Martin S. High. Handbook of polymer solution thermodynamics. John Wiley & Sons, 2010.
PARAMETER DESCRIPTION
A0

Parameter of equation. Unit = m³/kg.

TYPE: float

A1

Parameter of equation. Unit = m³/(kg·K).

TYPE: float

A2

Parameter of equation. Unit = m³/(kg·K²).

TYPE: float

B0

Parameter of equation. Unit = Pa.

TYPE: float

B1

Parameter of equation. Unit = 1/K.

TYPE: float

Tmin

Lower temperature bound. Unit = K.

TYPE: float DEFAULT: 0.0

Tmax

Upper temperature bound. Unit = K.

TYPE: float DEFAULT: inf

Pmin

Lower pressure bound. Unit = Pa.

TYPE: float DEFAULT: 0.0

Pmax

Upper pressure bound. Unit = Pa.

TYPE: float DEFAULT: inf

name

Name.

TYPE: str DEFAULT: ''

Source code in src/polykin/properties/pvt_polymer/tait.py 
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class Tait(PolymerPVTEquation):
    r"""Tait equation of state for the specific volume of a liquid.

    This EoS implements the following explicit PVT dependence:

    $$\hat{V}(T,P)=\hat{V}(T,0)\left[1-C\ln\left(\frac{P}{B(T)}\right)\right]$$

    with:

    $$ \begin{gather*}
    \hat{V}(T,0) = A_0 + A_1(T - 273.15) + A_2(T - 273.15)^2 \\
    B(T) = B_0\exp\left [-B_1(T - 273.15)\right]
    \end{gather*} $$

    where $A_i$ and $B_i$ are constant parameters, $T$ is the absolute
    temperature, and $P$ is the pressure.

    **References**

    *   Danner, Ronald P., and Martin S. High. Handbook of polymer
        solution thermodynamics. John Wiley & Sons, 2010.

    Parameters
    ----------
    A0 : float
        Parameter of equation.
        Unit = m³/kg.
    A1 : float
        Parameter of equation.
        Unit = m³/(kg·K).
    A2 : float
        Parameter of equation.
        Unit = m³/(kg·K²).
    B0 : float
        Parameter of equation.
        Unit = Pa.
    B1 : float
        Parameter of equation.
        Unit = 1/K.
    Tmin : float
        Lower temperature bound.
        Unit = K.
    Tmax : float
        Upper temperature bound.
        Unit = K.
    Pmin : float
        Lower pressure bound.
        Unit = Pa.
    Pmax : float
        Upper pressure bound.
        Unit = Pa.
    name : str
        Name.
    """

    A0: float
    A1: float
    A2: float
    B0: float
    B1: float

    _C = 0.0894

    def __init__(self,
                 A0: float,
                 A1: float,
                 A2: float,
                 B0: float,
                 B1: float,
                 Tmin: float = 0.0,
                 Tmax: float = np.inf,
                 Pmin: float = 0.0,
                 Pmax: float = np.inf,
                 name: str = ''
                 ) -> None:
        """Construct `Tait` with the given parameters."""

        # Check bounds
        check_bounds(A0, 1e-4, 2e-3, 'A0')
        check_bounds(A1, 1e-7, 2e-6, 'A1')
        check_bounds(A2, -2e-9, 1e-8, 'A2')
        check_bounds(B0, 1e7, 1e9, 'B0')
        check_bounds(B1, 1e-3, 2e-2, 'B1')

        self.A0 = A0
        self.A1 = A1
        self.A2 = A2
        self.B0 = B0
        self.B1 = B1
        super().__init__(Tmin, Tmax, Pmin, Pmax, name)

    def __repr__(self) -> str:
        return (
            f"name:          {self.name}\n"
            f"symbol:        {self.symbol}\n"
            f"unit:          {self.unit}\n"
            f"Trange [K]:    {self.Trange}\n"
            f"Prange [Pa]:   {self.Prange}\n"
            f"A0 [m³/kg]:    {self.A0}\n"
            f"A1 [m³/kg.K]:  {self.A1}\n"
            f"A2 [m³/kg.K²]: {self.A2}\n"
            f"B0 [Pa]:       {self.B0}\n"
            f"B1 [1/K]:      {self.B1}"
        )

    def eval(self,
             T: Union[float, FloatArray],
             P: Union[float, FloatArray]
             ) -> Union[float, FloatArray]:
        r"""Evaluate specific volume, $\hat{V}$, at given SI conditions without
        unit conversions or checks.

        Parameters
        ----------
        T : float | FloatArray
            Temperature.
            Unit = K.
        P : float | FloatArray
            Pressure.
            Unit = Pa.

        Returns
        -------
        float | FloatArray
            Specific volume.
            Unit = m³/kg.
        """
        TC = T - 273.15
        V0 = self.A0 + self.A1*TC + self.A2*TC**2
        B = self._B(T)
        V = V0*(1 - self._C*log(1 + P/B))
        return V

    def _B(self,
           T: Union[float, FloatArray]
           ) -> Union[float, FloatArray]:
        r"""Parameter B(T).

        Parameters
        ----------
        T : float | FloatArray
            Temperature.
            Unit = K.

        Returns
        -------
        float | FloatArray
            B(T).
            Unit = Pa.
        """
        return self.B0*exp(-self.B1*(T - 273.15))

    def alpha(self,
              T: Union[float, FloatArray],
              P: Union[float, FloatArray]
              ) -> Union[float, FloatArray]:
        r"""Calculate thermal expansion coefficient, $\alpha$.

        $$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}$$

        Parameters
        ----------
        T : float | FloatArray
            Temperature.
            Unit = K.
        P : float | FloatArray
            Pressure.
            Unit = Pa.

        Returns
        -------
        float | FloatArray
            Thermal expansion coefficient, $\alpha$.
            Unit = 1/K.
        """
        A0 = self.A0
        A1 = self.A1
        A2 = self.A2
        TC = T - 273.15
        alpha0 = (A1 + 2*A2*TC)/(A0 + A1*TC + A2*TC**2)
        return alpha0 - P*self.B1*self.beta(T, P)

    def beta(self,
             T: Union[float, FloatArray],
             P: Union[float, FloatArray]
             ) -> Union[float, FloatArray]:
        r"""Calculate isothermal compressibility coefficient, $\beta$.

        $$\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

        Parameters
        ----------
        T : float | FloatArray
            Temperature.
            Unit = K.
        P : float | FloatArray
            Pressure.
            Unit = Pa.

        Returns
        -------
        float | FloatArray
            Isothermal compressibility coefficient, $\beta$.
            Unit = 1/Pa.
        """
        B = self._B(T)
        return (self._C/(P + B))/(1 - self._C*log(1 + P/B))

V ¤

V(
    T: Union[float, FloatArrayLike],
    P: Union[float, FloatArrayLike],
    Tunit: Literal["C", "K"] = "K",
    Punit: Literal["bar", "MPa", "Pa"] = "Pa",
) -> Union[float, FloatArray]

Evaluate the specific volume, \(\hat{V}\), at given temperature and pressure, including unit conversion and range check.

PARAMETER DESCRIPTION
T

Temperature. Unit = Tunit.

TYPE: float | FloatArrayLike

P

Pressure. Unit = Punit.

TYPE: float | FloatArrayLike

Tunit

Temperature unit.

TYPE: Literal['C', 'K'] DEFAULT: 'K'

Punit

Pressure unit.

TYPE: Literal['bar', 'MPa', 'Pa'] DEFAULT: 'Pa'

RETURNS DESCRIPTION
float | FloatArray

Specific volume. Unit = m³/kg.
Source code in src/polykin/properties/pvt_polymer/base.py 
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def V(self,
      T: Union[float, FloatArrayLike],
      P: Union[float, FloatArrayLike],
      Tunit: Literal['C', 'K'] = 'K',
      Punit: Literal['bar', 'MPa', 'Pa'] = 'Pa'
      ) -> Union[float, FloatArray]:
    r"""Evaluate the specific volume, $\hat{V}$, at given temperature and
    pressure, including unit conversion and range check.

    Parameters
    ----------
    T : float | FloatArrayLike
        Temperature.
        Unit = `Tunit`.
    P : float | FloatArrayLike
        Pressure.
        Unit = `Punit`.
    Tunit : Literal['C', 'K']
        Temperature unit.
    Punit : Literal['bar', 'MPa', 'Pa']
        Pressure unit.

    Returns
    -------
    float | FloatArray
        Specific volume.
        Unit = m³/kg.
    """
    TK = convert_check_temperature(T, Tunit, self.Trange)
    Pa = convert_check_pressure(P, Punit, self.Prange)
    return self.eval(TK, Pa)

__init__ ¤

__init__(
    A0: float,
    A1: float,
    A2: float,
    B0: float,
    B1: float,
    Tmin: float = 0.0,
    Tmax: float = np.inf,
    Pmin: float = 0.0,
    Pmax: float = np.inf,
    name: str = "",
) -> None

Construct Tait with the given parameters.

Source code in src/polykin/properties/pvt_polymer/tait.py 
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def __init__(self,
             A0: float,
             A1: float,
             A2: float,
             B0: float,
             B1: float,
             Tmin: float = 0.0,
             Tmax: float = np.inf,
             Pmin: float = 0.0,
             Pmax: float = np.inf,
             name: str = ''
             ) -> None:
    """Construct `Tait` with the given parameters."""

    # Check bounds
    check_bounds(A0, 1e-4, 2e-3, 'A0')
    check_bounds(A1, 1e-7, 2e-6, 'A1')
    check_bounds(A2, -2e-9, 1e-8, 'A2')
    check_bounds(B0, 1e7, 1e9, 'B0')
    check_bounds(B1, 1e-3, 2e-2, 'B1')

    self.A0 = A0
    self.A1 = A1
    self.A2 = A2
    self.B0 = B0
    self.B1 = B1
    super().__init__(Tmin, Tmax, Pmin, Pmax, name)

alpha ¤

alpha(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Calculate thermal expansion coefficient, \(\alpha\).

\[\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Thermal expansion coefficient, \(\alpha\). Unit = 1/K.
Source code in src/polykin/properties/pvt_polymer/tait.py 
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def alpha(self,
          T: Union[float, FloatArray],
          P: Union[float, FloatArray]
          ) -> Union[float, FloatArray]:
    r"""Calculate thermal expansion coefficient, $\alpha$.

    $$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}$$

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Thermal expansion coefficient, $\alpha$.
        Unit = 1/K.
    """
    A0 = self.A0
    A1 = self.A1
    A2 = self.A2
    TC = T - 273.15
    alpha0 = (A1 + 2*A2*TC)/(A0 + A1*TC + A2*TC**2)
    return alpha0 - P*self.B1*self.beta(T, P)

beta ¤

beta(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Calculate isothermal compressibility coefficient, \(\beta\).

\[\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Isothermal compressibility coefficient, \(\beta\). Unit = 1/Pa.
Source code in src/polykin/properties/pvt_polymer/tait.py 
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def beta(self,
         T: Union[float, FloatArray],
         P: Union[float, FloatArray]
         ) -> Union[float, FloatArray]:
    r"""Calculate isothermal compressibility coefficient, $\beta$.

    $$\beta=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}$$

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Isothermal compressibility coefficient, $\beta$.
        Unit = 1/Pa.
    """
    B = self._B(T)
    return (self._C/(P + B))/(1 - self._C*log(1 + P/B))

eval ¤

eval(
    T: Union[float, FloatArray], P: Union[float, FloatArray]
) -> Union[float, FloatArray]

Evaluate specific volume, \(\hat{V}\), at given SI conditions without unit conversions or checks.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float | FloatArray

P

Pressure. Unit = Pa.

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Specific volume. Unit = m³/kg.
Source code in src/polykin/properties/pvt_polymer/tait.py 
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def eval(self,
         T: Union[float, FloatArray],
         P: Union[float, FloatArray]
         ) -> Union[float, FloatArray]:
    r"""Evaluate specific volume, $\hat{V}$, at given SI conditions without
    unit conversions or checks.

    Parameters
    ----------
    T : float | FloatArray
        Temperature.
        Unit = K.
    P : float | FloatArray
        Pressure.
        Unit = Pa.

    Returns
    -------
    float | FloatArray
        Specific volume.
        Unit = m³/kg.
    """
    TC = T - 273.15
    V0 = self.A0 + self.A1*TC + self.A2*TC**2
    B = self._B(T)
    V = V0*(1 - self._C*log(1 + P/B))
    return V

from_database classmethod ¤

from_database(name: str) -> Optional[PolymerPVTEquation]

Construct PolymerPVTEquation with parameters from the database.

PARAMETER DESCRIPTION
name

Polymer code name.

TYPE: str

Source code in src/polykin/properties/pvt_polymer/base.py 
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@classmethod
def from_database(cls,
                  name: str
                  ) -> Optional[PolymerPVTEquation]:
    r"""Construct `PolymerPVTEquation` with parameters from the database.

    Parameters
    ----------
    name : str
        Polymer code name.
    """
    table = load_PVT_parameters(method=cls.__name__)
    try:
        mask = table.index == name
        parameters = table[mask].iloc[0, :].to_dict()
        return cls(**parameters, name=name)
    except IndexError:
        print(
            f"Error: '{name}' does not exist in polymer database.\n"
            f"Valid names are: {table.index.to_list()}")

get_database classmethod ¤

get_database() -> pd.DataFrame

Get database with parameters for the respective PVT equation.

Method Reference
Flory [2] Table 4.1.7 (p. 72-73)
Hartmann-Haque [2] Table 4.1.11 (p. 85-86)
Sanchez-Lacombe [2] Table 4.1.9 (p. 78-79)
Tait [1] Table 3B-1 (p. 41)

References

  1. Danner, Ronald P., and Martin S. High. Handbook of polymer solution thermodynamics. John Wiley & Sons, 2010.
  2. Caruthers et al. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions. AIChE, 1998.
Source code in src/polykin/properties/pvt_polymer/base.py 
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@classmethod
def get_database(cls) -> pd.DataFrame:
    r"""Get database with parameters for the respective PVT equation.

    | Method          | Reference                            |
    | :-----------    | ------------------------------------ |
    | Flory           | [2] Table 4.1.7  (p. 72-73)          |
    | Hartmann-Haque  | [2] Table 4.1.11 (p. 85-86)          |
    | Sanchez-Lacombe | [2] Table 4.1.9  (p. 78-79)          |
    | Tait            | [1] Table 3B-1 (p. 41)               |

    **References**

    1.  Danner, Ronald P., and Martin S. High. Handbook of polymer
        solution thermodynamics. John Wiley & Sons, 2010.
    2.  Caruthers et al. Handbook of Diffusion and Thermal Properties of
        Polymers and Polymer Solutions. AIChE, 1998.
    """
    return load_PVT_parameters(method=cls.__name__)