polykin.transport.hmt

U_cylindrical_wall

U_cylindrical_wall(
    hi: float,
    ho: float,
    di: float,
    do: float,
    k: float,
    Rfi: float = 0.0,
    Rfo: float = 0.0,
) -> float

Calculate the overall heat transfer coefficient through a cylindrical wall with convection on both sides.

Under steady-state conditions, the overall heat transfer coefficient is given by the following expression:

\[ U_o = \left( \frac{d_o}{h_i d_i} + \frac{1}{h_o} + \frac{R_{f,i}d_o}{d_i} + R_{f,o} + \frac{d_o}{2k}\ln(d_o/d_i) \right)^{-1} \]

where \(h\) is the heat transfer coefficient, \(R_{f}\) is the fouling factor, \(d\) is the diameter, \(k\) is the wall thermal conductivity, and the subscripts \(i\) and \(o\) indicate inner and outer surfaces, respectively.

Tip

The overall heat transfer coefficient based on the inner surface, \(U_i\), can be computed using the indentity \(U_i d_i = U_o d_o\).

PARAMETER	DESCRIPTION
hi	Heat transfer coefficient at inner surface (W/(m²·K)). TYPE: float
ho	Heat transfer coefficient at outer surface (W/(m²·K)). TYPE: float
di	Inner diameter (m). TYPE: float
do	Outer diameter (m). TYPE: float
k	Wall thermal conductivity (W/(m·K)). TYPE: float
Rfi	Fouling factor at inner surface ((m²·K)/W). TYPE: float DEFAULT: 0.0
Rfo	Fouling factor at outer surface ((m²·K)/W). TYPE: float DEFAULT: 0.0

RETURNS	DESCRIPTION
float	Overall heat transfer coefficient based on outer surface (W/(m²·K)).

See also

• U_plane_wall: related method for a plane wall.

Examples:

Calculate the overall heat transfer coefficient for a carbon steel tube subjected to convection on both sides, with heat transfer coefficients of 2000 and 1000 W/(m²·K) for the inner and outer surfaces, respectively. The tube has inner and outer diameters of 40 mm and 50 mm. Neglect fouling effects.

>>> from polykin.transport import U_cylindrical_wall
>>> hi = 2e3   # W/(m²·K)
>>> ho = 1e3   # W/(m²·K)
>>> k = 6e2    # W/(m·K)
>>> di = 40e-3 # m
>>> do = 50e-3 # m
>>> Uo = U_cylindrical_wall(hi, ho, di, do, k)
>>> print(f"Uo={Uo:.1e} W/(m²·K)")
Uo=6.1e+02 W/(m²·K)

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

def U_cylindrical_wall(hi: float,
                       ho: float,
                       di: float,
                       do: float,
                       k: float,
                       Rfi: float = 0.0,
                       Rfo: float = 0.0) -> float:
    r"""Calculate the overall heat transfer coefficient through a cylindrical
    wall with convection on both sides.

    Under steady-state conditions, the overall heat transfer coefficient is
    given by the following expression:

    $$ U_o = \left( \frac{d_o}{h_i d_i} + \frac{1}{h_o} + \frac{R_{f,i}d_o}{d_i}
             + R_{f,o} + \frac{d_o}{2k}\ln(d_o/d_i) \right)^{-1} $$

    where $h$ is the heat transfer coefficient, $R_{f}$ is the fouling factor,
    $d$ is the diameter, $k$ is the wall thermal conductivity, and the subscripts
    $i$ and $o$ indicate inner and outer surfaces, respectively.

    !!! tip

        The overall heat transfer coefficient based on the inner surface, $U_i$,
        can be computed using the indentity $U_i d_i = U_o d_o$.

    Parameters
    ----------
    hi : float
        Heat transfer coefficient at inner surface (W/(m²·K)).
    ho : float
        Heat transfer coefficient at outer surface (W/(m²·K)).
    di : float
        Inner diameter (m).
    do : float
        Outer diameter (m).
    k : float
        Wall thermal conductivity (W/(m·K)).
    Rfi : float
        Fouling factor at inner surface ((m²·K)/W).
    Rfo : float
        Fouling factor at outer surface ((m²·K)/W).

    Returns
    -------
    float
        Overall heat transfer coefficient based on outer surface (W/(m²·K)).

    See also
    --------
    - [`U_plane_wall`](U_plane_wall.md): related method for a plane wall.

    Examples
    --------
    Calculate the overall heat transfer coefficient for a carbon steel tube 
    subjected to convection on both sides, with heat transfer coefficients 
    of 2000 and 1000 W/(m²·K) for the inner and outer surfaces, respectively. 
    The tube has inner and outer diameters of 40 mm and 50 mm. Neglect fouling
    effects.
    >>> from polykin.transport import U_cylindrical_wall
    >>> hi = 2e3   # W/(m²·K)
    >>> ho = 1e3   # W/(m²·K)
    >>> k = 6e2    # W/(m·K)
    >>> di = 40e-3 # m
    >>> do = 50e-3 # m
    >>> Uo = U_cylindrical_wall(hi, ho, di, do, k)
    >>> print(f"Uo={Uo:.1e} W/(m²·K)")
    Uo=6.1e+02 W/(m²·K)
    """
    return 1/(do/(hi*di) + 1/ho + Rfi*do/di + Rfo + do/(2*k)*np.log(do/di))