polykin.transport.heat

Nu_tank

Nu_tank(
    surface: Literal[
        "wall",
        "bottom-head",
        "helical-coil",
        "harp-coil-0",
        "harp-coil-45",
    ],
    impeller: Literal[
        "4BF",
        "4BP",
        "6BD",
        "HE3",
        "PROP",
        "anchor",
        "helical-ribbon",
    ],
    Re: float,
    Pr: float,
    mur: float,
    D_T: float = 1 / 3,
    H_T: float = 1.0,
    L_Ls: float = 1.0,
    d_T: float = 0.04,
    P_D: float = 1.0,
    nb: int = 2,
) -> float

Calculate the Nusselt number for a stirred tank.

This function calculates the Nusselt number based on impeller and surface type, and fluid dynamics parameters for a stirred tank, according to the correlations in chapter 14.4 of the Handbook of Industrial Mixing.

References

• Penney, W. R. and Atiemo-Obeng, V. A. "Heat Transfer" in "Handbook of Industrial Mixing: Science and Practice", Wiley, 2004.

PARAMETER	DESCRIPTION
surface	Heat transfer surface type. TYPE: Literal['wall', 'bottom-head', 'helical-coil', 'harp-coil-0', 'harp-coil-45']
impeller	Impeller type. TYPE: Literal['4BF', '4BP', '6BD', 'HE3', 'PROP', 'anchor', 'helical-ribbon']
Re	Impeller Reynolds number. TYPE: float
Pr	Prandtl number. TYPE: float
mur	Ratio of bulk viscosity to surface viscosity, \(\mu/\mu_s\). TYPE: float
D_T	Ratio of impeller diameter to tank diameter, \(D/T\). TYPE: float DEFAULT: 1 / 3
H_T	Ratio of liquid height to tank diameter, \(H/T\). TYPE: float DEFAULT: 1.0
L_Ls	Ratio of height of impeller blade to standard value, \(L/L_s\). TYPE: float DEFAULT: 1.0
d_T	Ratio of coil tube outer diameter to tank diameter, \(d/T\). TYPE: float DEFAULT: 0.04
P_D	Ratio of impeller blade pitch to impeller diameter. TYPE: float DEFAULT: 1.0
nb	Number of baffles or vertical tubes acting as baffles. TYPE: int DEFAULT: 2

RETURNS	DESCRIPTION
float	Nusselt number. Characteristic length depends on the surface type.

Examples:

Estimate the internal heat transfer coefficient for a 2-m diameter stirred tank equiped with a HE3 impeller operated at 120 rpm. Assume water properties and default geometry.

>>> from polykin.transport.heat import Nu_tank
>>> T = 2.     # m
>>> D = T/3    # m
>>> rho = 1e3  # kg/m³
>>> mu = 1e-3  # Pa.s
>>> k  = 0.6   # W/m.K
>>> cp = 4.2e3 # J/kg.K
>>> Re = (120/60) * D**2 * rho / mu
>>> Pr = mu*cp/k
>>> mur = 1.   # neglect temperature correction
>>> Nu = Nu_tank('wall', 'HE3', Re, Pr, mur)
>>> h = Nu*k/T
>>> print(f"h={h:.1e} W/m².K")
h=1.6e+03 W/m².K

Source code in src/polykin/transport/heat.py

def Nu_tank(
    surface: Literal['wall', 'bottom-head', 'helical-coil', 'harp-coil-0', 'harp-coil-45'],
    impeller: Literal['4BF', '4BP', '6BD', 'HE3', 'PROP', 'anchor', 'helical-ribbon'],
    Re: float,
    Pr: float,
    mur: float,
    D_T: float = 1/3,
    H_T: float = 1.,
    L_Ls: float = 1.,
    d_T: float = 0.04,
    P_D: float = 1.,
    nb: int = 2
) -> float:
    r"""Calculate the Nusselt number for a stirred tank.

    This function calculates the Nusselt number based on impeller and surface 
    type, and fluid dynamics parameters for a stirred tank, according to
    the correlations in chapter 14.4 of the Handbook of Industrial Mixing.

    **References**

    * Penney, W. R. and Atiemo-Obeng, V. A. "Heat Transfer" in "Handbook of
      Industrial Mixing: Science and Practice", Wiley, 2004.

    Parameters
    ----------
    surface : Literal['wall', 'bottom-head', 'helical-coil', 'harp-coil-0', 'harp-coil-45']
        Heat transfer surface type.
    impeller : Literal['4BF', '4BP', '6BD', 'HE3', 'PROP', 'anchor', 'helical-ribbon']
        Impeller type.
    Re : float
        Impeller Reynolds number.
    Pr : float
        Prandtl number.
    mur : float
        Ratio of bulk viscosity to surface viscosity, $\mu/\mu_s$.
    D_T : float
        Ratio of impeller diameter to tank diameter, $D/T$.
    H_T : float
        Ratio of liquid height to tank diameter, $H/T$.
    L_Ls : float
        Ratio of height of impeller blade to standard value, $L/L_s$.
    d_T : float, optional
        Ratio of coil tube outer diameter to tank diameter, $d/T$.
    P_D : float, optional
        Ratio of impeller blade pitch to impeller diameter.
    nb : int, optional
        Number of baffles or vertical tubes acting as baffles.

    Returns
    -------
    float
        Nusselt number. Characteristic length depends on the surface type.

    Examples
    --------
    Estimate the internal heat transfer coefficient for a 2-m diameter stirred
    tank equiped with a HE3 impeller operated at 120 rpm. Assume water properties
    and default geometry.
    >>> from polykin.transport.heat import Nu_tank
    >>> T = 2.     # m
    >>> D = T/3    # m
    >>> rho = 1e3  # kg/m³
    >>> mu = 1e-3  # Pa.s
    >>> k  = 0.6   # W/m.K
    >>> cp = 4.2e3 # J/kg.K
    >>> Re = (120/60) * D**2 * rho / mu
    >>> Pr = mu*cp/k
    >>> mur = 1.   # neglect temperature correction
    >>> Nu = Nu_tank('wall', 'HE3', Re, Pr, mur)
    >>> h = Nu*k/T
    >>> print(f"h={h:.1e} W/m².K")
    h=1.6e+03 W/m².K
    """

    # Default parameters
    K = 0.
    a = 2/3
    b = 1/3
    c = 0.14
    Gc = 1.

    impeller_error = False
    if surface == 'wall':
        if impeller == '6BD':
            K = 0.74
            Gc = H_T**-0.15 * L_Ls**0.2
        elif impeller == '4BF':
            K = 0.66
            Gc = H_T**-0.15 * L_Ls**0.2
        elif impeller == '4BP':
            K = 0.45
            Gc = H_T**-0.15 * L_Ls**0.2
        elif impeller == 'HE3':
            K = 0.31
            Gc = H_T**-0.15
        elif impeller == 'PROP':
            K = 0.50
            Gc = H_T**-0.15 * 1.29*P_D/(0.29 + P_D)
        elif impeller == 'anchor':
            if Re < 12:
                K = 0.
            elif Re >= 12 and Re < 100:
                K = 0.69
                a = 1/2
            elif Re >= 100:
                K = 0.32
        elif impeller == 'helical-ribbon':
            if Re < 13:
                K = 0.94
                a = 1/3
            elif Re >= 13 and Re < 210:
                K = 0.61
                a = 1/2
            else:
                K = 0.25
        else:
            impeller_error = True
    elif surface == 'bottom-head':
        if impeller == '6BD':
            K = 0.50
            Gc = H_T**-0.15 * L_Ls**0.2
        elif impeller == '4BF':
            K = 0.40
            Gc = H_T**-0.15 * L_Ls**0.2
        elif impeller == '4BP':
            K = 1.08
            Gc = H_T**-0.15 * L_Ls**0.2
        elif impeller == 'HE3':
            K = 0.90
            Gc = H_T**-0.15
        else:
            impeller_error = True
    elif surface == 'helical-coil':
        if impeller == 'PROP':
            K = 0.016
            a = 0.67
            b = 0.37
            Gc = (D_T/(1/3))**0.1 * (d_T/0.04)**0.5
        elif impeller == '6BD':
            K = 0.03
            Gc = H_T**-0.15 * L_Ls**0.2 * (D_T/(1/3))**0.1 * (d_T/0.04)**0.5
        else:
            impeller_error = True
    elif surface == 'harp-coil-0':
        if impeller == '4BF':
            K = 0.06  # the text mentions this value might be overestimated
            a = 0.65
            b = 0.3
            Gc = H_T**-0.15 * L_Ls**0.2 * (D_T/(1/3))**0.33 * (2/nb)**0.2
        else:
            impeller_error = True
    elif surface == 'harp-coil-45':
        if impeller == '6BD':
            # quite some doubts regarding this equation
            K = 0.021
            a = 0.67
            b = 0.4
            Gc = H_T**-0.15 * L_Ls**0.2 * (D_T/(1/3))**0.33 * (2/nb)**0.2
        else:
            impeller_error = True
    else:
        raise ValueError(f"Invalid heat transfer `surface`: {surface}.")

    if impeller_error:
        raise ValueError(
            f"Invalid combination of `surface`={surface} and `impeller`={impeller}.")

    return K * Re**a * Pr**b * mur**c * Gc