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polykin.properties.pvt¤

Tait ¤

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

This EoS implements the following explicit PVT dependence:

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

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 \(\hat{v}\) is the specific volume, \(T\) is the absolute temperature, \(P\) is the pressure, and \(A_i\) and \(B_i\) are constant parameters.

References

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

Parameter of equation [m³/kg].

TYPE: float

A1

Parameter of equation [m³/(kg·K)].

TYPE: float

A2

Parameter of equation [m³/(kg·K²)].

TYPE: float

B0

Parameter of equation [Pa].

TYPE: float

B1

Parameter of equation [K⁻¹].

TYPE: float

Tmin

Lower temperature bound [K].

TYPE: float DEFAULT: 0.0

Tmax

Upper temperature bound [K].

TYPE: float DEFAULT: inf

Pmin

Lower pressure bound [Pa].

TYPE: float DEFAULT: 0.0

Pmax

Upper pressure bound [Pa].

TYPE: float DEFAULT: inf

name

Name.

TYPE: str DEFAULT: ''

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

    This EoS implements the following explicit PVT dependence:

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

    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 $\hat{v}$ is the specific volume, $T$ is the absolute temperature, 
    $P$ is the pressure, and $A_i$ and $B_i$ are constant parameters.

    **References**

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

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

    Trange: tuple[float, float]
    Prange: tuple[float, float]
    symbol = r"$\hat{V}$"
    unit = "m³/kg"

    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:

        # 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')
        check_bounds(Tmin, 0, np.inf, 'Tmin')
        check_bounds(Tmax, 0, np.inf, 'Tmax')
        check_bounds(Tmax-Tmin, eps, np.inf, 'Tmax-Tmin')
        check_bounds(Pmin, 0, np.inf, 'Pmin')
        check_bounds(Pmax, 0, np.inf, 'Pmax')
        check_bounds(Pmax-Pmin, eps, np.inf, 'Pmax-Pmin')

        self.A0 = A0
        self.A1 = A1
        self.A2 = A2
        self.B0 = B0
        self.B1 = B1
        self.Trange = (Tmin, Tmax)
        self.Prange = (Pmin, Pmax)
        self.name = 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 [K⁻¹]     : {self.B1}"
        )

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

        Parameters
        ----------
        T : float | FloatArray
            Temperature [K].
        P : float | FloatArray
            Pressure [Pa].

        Returns
        -------
        float | FloatArray
            Specific volume [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*ln(1 + P/B))
        return v

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

        Parameters
        ----------
        T : float | FloatArray
            Temperature [K].

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

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

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

        Parameters
        ----------
        T : float | FloatArray
            Temperature [K].
        P : float | FloatArray
            Pressure [Pa].

        Returns
        -------
        float | FloatArray
            Thermal expansion coefficient, $\alpha$ [K⁻¹].
        """
        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: float | FloatArray,
             P: float | FloatArray
             ) -> float | FloatArray:
        r"""Calculate the isothermal compressibility coefficient, $\beta$.

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

        Parameters
        ----------
        T : float | FloatArray
            Temperature [K].
        P : float | FloatArray
            Pressure [Pa].

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

    def vs(self,
           T: float | FloatVectorLike,
           P: float | FloatVectorLike,
           Tunit: Literal['C', 'K'] = 'K',
           Punit: Literal['bar', 'MPa', 'Pa'] = 'Pa'
           ) -> 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 [`Tunit`].
        P : float | FloatArrayLike
            Pressure [`Punit`].
        Tunit : Literal['C', 'K']
            Temperature unit.
        Punit : Literal['bar', 'MPa', 'Pa']
            Pressure unit.

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

    @classmethod
    def from_database(cls,
                      name: str
                      ) -> Tait | None:
        r"""Construct `Tait` 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()}")

    @classmethod
    def get_database(cls) -> pd.DataFrame:
        r"""Get database with parameters for the Tait equation.

        Parameters from Table 3B-1 (p. 41) of Danner and High (2010).

        **References**

        *  Danner, Ronald P., and Martin S. High. Handbook of polymer
            solution thermodynamics. John Wiley & Sons, 2010, p. 41.
        """
        return load_PVT_parameters(method=cls.__name__)

alpha ¤

alpha(
    T: float | FloatArray, P: float | FloatArray
) -> float | FloatArray

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

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

Temperature [K].

TYPE: float | FloatArray

P

Pressure [Pa].

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

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

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

    Parameters
    ----------
    T : float | FloatArray
        Temperature [K].
    P : float | FloatArray
        Pressure [Pa].

    Returns
    -------
    float | FloatArray
        Thermal expansion coefficient, $\alpha$ [K⁻¹].
    """
    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: float | FloatArray, P: float | FloatArray
) -> float | FloatArray

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

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

Temperature [K].

TYPE: float | FloatArray

P

Pressure [Pa].

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

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

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

    Parameters
    ----------
    T : float | FloatArray
        Temperature [K].
    P : float | FloatArray
        Pressure [Pa].

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

eval ¤

eval(
    T: float | FloatArray, P: float | FloatArray
) -> float | FloatArray

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

PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float | FloatArray

P

Pressure [Pa].

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

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

    Parameters
    ----------
    T : float | FloatArray
        Temperature [K].
    P : float | FloatArray
        Pressure [Pa].

    Returns
    -------
    float | FloatArray
        Specific volume [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*ln(1 + P/B))
    return v

from_database classmethod ¤

from_database(name: str) -> Tait | None

Construct Tait with parameters from the database.

PARAMETER DESCRIPTION
name

Polymer code name.

TYPE: str

Source code in src/polykin/properties/pvt/tait.py 
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@classmethod
def from_database(cls,
                  name: str
                  ) -> Tait | None:
    r"""Construct `Tait` 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 Tait equation.

Parameters from Table 3B-1 (p. 41) of Danner and High (2010).

References

  • Danner, Ronald P., and Martin S. High. Handbook of polymer solution thermodynamics. John Wiley & Sons, 2010, p. 41.
Source code in src/polykin/properties/pvt/tait.py 
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@classmethod
def get_database(cls) -> pd.DataFrame:
    r"""Get database with parameters for the Tait equation.

    Parameters from Table 3B-1 (p. 41) of Danner and High (2010).

    **References**

    *  Danner, Ronald P., and Martin S. High. Handbook of polymer
        solution thermodynamics. John Wiley & Sons, 2010, p. 41.
    """
    return load_PVT_parameters(method=cls.__name__)

vs ¤

vs(
    T: float | FloatVectorLike,
    P: float | FloatVectorLike,
    Tunit: Literal["C", "K"] = "K",
    Punit: Literal["bar", "MPa", "Pa"] = "Pa",
) -> float | FloatArray

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

PARAMETER DESCRIPTION
T

Temperature [Tunit].

TYPE: float | FloatArrayLike

P

Pressure [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 [m³/kg].
Source code in src/polykin/properties/pvt/tait.py 
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def vs(self,
       T: float | FloatVectorLike,
       P: float | FloatVectorLike,
       Tunit: Literal['C', 'K'] = 'K',
       Punit: Literal['bar', 'MPa', 'Pa'] = 'Pa'
       ) -> 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 [`Tunit`].
    P : float | FloatArrayLike
        Pressure [`Punit`].
    Tunit : Literal['C', 'K']
        Temperature unit.
    Punit : Literal['bar', 'MPa', 'Pa']
        Pressure unit.

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

Parameter databank¤

Polymer A0 A1 A2 B0 B1 Tmin Tmax Pmin Pmax
BR 0.0010969 7.6789e-07 -2.2216e-10 1.7596e+08 0.0043355 277 328 100000 2.83e+08
HDPE 0.0011567 6.2888e-07 1.1268e-09 1.7867e+08 0.0047254 415 472 100000 2e+08
HMDS 0.0012727 1.6849e-06 4.3376e-09 5.891e+07 0.011203 298 343 0 9e+08
i-PB 0.0011561 6.1015e-07 8.3234e-10 1.8382e+08 0.0047833 407 514 0 1.96e+08
i-PMMA 0.0007977 5.5274e-07 -1.4503e-10 2.921e+08 0.004196 328 463 100000 2e+08
i-PP 0.0012033 4.8182e-07 7.7589e-10 1.4236e+08 0.0040184 447 571 0 1.96e+08
LDPE 0.0011004 1.4557e-06 -1.5749e-09 1.7598e+08 0.0046677 398 471 100000 2e+08
LLDPE 0.0011105 1.2489e-06 -4.0642e-10 1.7255e+08 0.0044256 420 473 100000 2e+08
PA 0.00078153 3.6134e-07 2.7519e-10 3.4019e+08 0.0038021 455 588 0 1.77e+08
PBMA 0.00093282 5.7856e-07 5.7343e-10 2.2569e+08 0.0053116 295 473 100000 2e+08
PC 0.00079165 4.4201e-07 2.8583e-10 3.1268e+08 0.0039728 430 610 0 1.77e+08
PCHMA 0.0008741 4.9035e-07 3.2707e-10 3.0545e+08 0.005503 383 472 100000 2e+08
PDMS 0.0010122 7.7266e-07 1.9944e-09 8.7746e+07 0.006256 298 343 0 1e+08
PDMS3 0.0010736 1.2837e-06 1.4565e-10 7.3947e+07 0.0080773 298 343 0 9e+08
PDMS10 0.0010536 1.1041e-06 4.6289e-10 8.1208e+07 0.0071257 298 343 0 9e+08
PDMS20 0.0010271 1.1054e-06 -2.7259e-10 8.5511e+07 0.0067944 298 343 0 9e+08
PDMS100 0.0010095 1.0662e-06 -3.6476e-10 8.8352e+07 0.0063228 298 343 0 9e+08
PDMS350 0.0010056 1.0003e-06 1.2039e-11 9.0488e+07 0.0064221 298 343 0 9e+08
PDMS1000 0.0010076 9.1603e-07 8.2159e-10 8.9137e+07 0.0063938 298 343 0 9e+08
PHENOXY 0.00083796 3.6449e-07 5.2933e-10 3.5434e+08 0.0043649 349 574 0 1.77e+08
PIB 0.001089 2.5554e-07 2.2682e-09 1.941e+08 0.0039995 326 383 0 1e+08
PMMA 0.00082396 3.049e-07 7.0201e-10 2.9803e+08 0.0043789 387 432 100000 2e+08
PMP 0.0012078 5.1461e-07 9.7366e-10 1.4978e+08 0.0046302 514 592 0 1.96e+08
POM 0.00083198 2.755e-07 2.2e-09 3.103e+08 0.0044652 462 492 0 1.96e+08
PoMS 0.00093905 5.1288e-07 5.9157e-11 2.469e+08 0.0036633 413 471 100000 1.8e+08
PS 0.00093805 3.3086e-07 6.691e-10 2.5001e+08 0.0041815 389 469 100000 2e+08
PTFE 0.00046867 1.1542e-07 1.1931e-09 4.091e+08 0.0092556 604 646 0 3.92e+08
PVAC 0.00082832 4.7205e-07 1.1364e-09 1.8825e+08 0.0038774 337 393 0 1e+08

Examples¤

Estimate the PVT properties of PMMA.

from polykin.properties.pvt import Tait

# Parameters from Handbook Polymer Solution Thermodynamics, p.39 
m = Tait(
    A0=8.2396e-4,
    A1=3.0490e-7,
    A2=7.0201e-10,
    B0=2.9803e8,
    B1=4.3789e-3,
    Tmin=387.15,
    Tmax=432.15,
    Pmin=0.1e6,
    Pmax=200e6,
    name="PMMA"
    )

print(m.vs(127., 1500, Tunit='C', Punit='bar'))
print(m.alpha(400., 1.5e8))
print(m.beta(400., 1.5e8))
0.0008247751539051464
0.00035883613484047545
2.95109155017507e-10

from polykin.properties.pvt import Tait

# Parameters retrieved from internal databank 
m = Tait.from_database("PMMA")

print(m.vs(127., 1500, Tunit='C', Punit='bar'))
print(m.alpha(400., 1.5e8))
print(m.beta(400., 1.5e8))
0.0008247751539051464
0.00035883613484047545
2.95109155017507e-10