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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__)

Parameter databank¤

Polymer P0 V0 T0 Tmin Tmax Pmin Pmax
HDPE 5.376e+08 0.0009818 6548 415 473 0 2e+08
HDPE 3.767e+08 0.001013 7002 413 476 0 1.96e+08
HMLPE 6.043e+08 0.0009855 6458 410 473 0 2e+08
BPE 4.531e+08 0.0009992 6710 398 471 0 2e+08
LDPE 5.292e+08 0.0009794 6356 394 448 0 1.96e+08
LDPE-A 4.695e+08 0.0009963 6774 385 498 0 1.96e+08
LDPE-B 4.564e+08 0.001003 6896 385 498 0 1.96e+08
LDPE-C 4.71e+08 0.0009974 6809 385 498 0 1.96e+08
PIB 3.96e+08 0.0009455 7396 326 383 0 1e+08
i-PP 3.974e+08 0.001007 7011 443 570 0 1.96e+08
a-PP 4.059e+08 0.0009755 6351 353 393 0 1e+08
i-PP 4.039e+08 0.0009897 6838 406 519 0 1.96e+08
PMP 3.95e+08 0.00102 7079 514 592 0 1.96e+08
PMMA 5.688e+08 0.0007204 7717 387 432 0 2e+08
PCHMA 4.614e+08 0.0007772 7700 321 471 0 2e+08
PNBMA 5.096e+08 0.0008087 6794 307 473 0 2e+08
PS 4.052e+08 0.0008277 8118 388 469 0 2e+08
POMS 4.415e+08 0.0008457 8463 412 471 0 1.8e+08
PVAC 5.997e+08 0.000709 6449 337 393 0 1e+08
PDMS 3.269e+08 0.0008264 5184 298 343 0 1e+08
PTFE 3.078e+08 0.000878 5070 298 343 0 9e+08
PSF 3.133e+08 0.0008694 5288 298 343 0 9e+08
PBD 3.156e+08 0.0008531 5395 298 343 0 9e+08
PEO 3.23e+08 0.0008412 5470 298 343 0 9e+08
PTHF 3.115e+08 0.0008403 5554 298 343 0 9e+08
PET 3.115e+08 0.0008403 5554 298 343 0 9e+08
PPO 4.049e+08 0.0004215 7088 603 645 0 3.9e+08
PC 7.382e+08 0.0006847 8664 475 644 0 1.96e+08
PAR 4.544e+08 0.0009173 5522 277 328 0 2.83e+08
PH 6.016e+08 0.0007719 7147 361 497 0 6.8e+07
PEEK 4.598e+08 0.0008774 7006 335 439 0 7.8e+07
PVC 8.51e+08 0.0006452 8215 547 615 0 1.96e+08
PA6 6.509e+08 0.0007472 7360 476 593 0 1.76e+08
PA66 6.71e+08 0.000707 8039 424 613 0 1.76e+08
PVME 6.512e+08 0.0006991 8470 450 583 0 1.76e+08
PMA 7.132e+08 0.0007389 7869 341 573 0 1.76e+08
PEA 8.329e+08 0.0006642 8667 619 671 0 2e+08
PEMA 5.018e+08 0.0006201 7752 373 423 0 2e+08
TMPC 4.11e+08 0.0006896 9182 509 569 0 1.96e+08
HFPC 5.583e+08 0.0006885 7865 519 571 0 1.96e+08
BCPC 5.128e+08 0.0008266 6607 303 471 0 2e+08
PECH 5.992e+08 0.0007277 6894 310 493 0 1.96e+08
PCL 5.115e+08 0.0007563 6599 310 490 0 1.96e+08

Examples¤

Estimate the PVT properties of PMMA.

from polykin.properties.pvt_polymer import Flory

# Parameters from Handbook of Diffusion and Thermal Properties of Polymers
# and Polymer Solutions, p.72. 
m = Flory(
    V0=0.7204e-3,
    T0=7717.,
    P0=568.8e6,
    Tmin=387.15,
    Tmax=432.15,
    Pmin=0.1e6,
    Pmax=200e6,
    name="PMMA"
    )

print(m.V(127., 1500, Tunit='C', Punit='bar'))
print(m.alpha(400., 1.5e8))
print(m.beta(400., 1.5e8))
0.0008248219623602766
0.0003709734601636698
2.5411526814543353e-10

from polykin.properties.pvt_polymer import Flory

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

print(m.V(127., 1500, Tunit='C', Punit='bar'))
print(m.alpha(400., 1.5e8))
print(m.beta(400., 1.5e8))
0.0008248219623602766
0.0003709734601636698
2.5411526814543353e-10