#!/usr/bin/python3
# -*- coding: utf-8 -*-
'''Pychemqt, Chemical Engineering Process simulator
Copyright (C) 2009-2025, Juan José Gómez Romera <jjgomera@gmail.com>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
This module implements physics adimensional groups
* :func:`Archimedes`: Archimedes number
* :func:`Bi`: Biot number
* :func:`Bo`: Bond number
* :func:`Eu`: Euler number
* :func:`Fo`: Fourier number
* :func:`Fr`: Froude number
* :func:`Ga`: Galilei number
* :func:`Gr`: Grashof number
* :func:`Gz`: Graetz number
* :func:`Kn`: Knudsen number
* :func:`Le`: Lewis number
* :func:`Ma`: Mach number
* :func:`Nu`: Nusselt number
* :func:`Pe`: Péclet number
* :func:`Pr`: Prandtl number
* :func:`Ra`: Rayleigh number
* :func:`Re`: Reynolds number
* :func:`Sh`: Sherwood number
* :func:`Sc`: Schmidt number
* :func:`St`: Stanton number
* :func:`We`: Weber number
* :func:`Dean`: Dean number
'''
from scipy.constants import g
from lib.unidades import Dimensionless
from lib.utilities import refDoc
__doi__ = {
1:
{"autor": "VDI-Gesellschaft",
"title": "VDI Heat Atlas 2nd Edition",
"ref": "Berlin, New York. Springer 2010.",
"doi": ""},
2:
{"autor": "",
"title": "Perry's Chemical Engineers' Handbook 9th Edition",
"ref": "McGraw-Hill (2019)",
"doi": ""}}
[docs]
@refDoc(__doi__, [1])
def Archimedes(L, rho_p, rho, mu=None, nu=None, g=g):
r"""Calculates Archimedes number `Ar` for two phases densities, a
geometric length `L` or any viscosity definition.
.. math::
Ar=\frac{gL^3\delta\rho}{\rho v^2}
Inputs either of any of the following sets:
* L, density of both phases (rho_p, rho) and kinematic viscosity `nu`
* L, density of both phases (rho_p, rho) and dynamic viscosity `mu`
Parameters
----------
L : float
Characteristic length [m]
rho_p : float
Density of particle phase [kg/m³]
rho : float
Density of bulk phase [kg/m³]
mu : float, optional
Dynamic viscosity, [Pa*s]
nu : float, optional
Kinematic viscosity, [m²/s]
g : float, optional
Acceleration due to gravity, [m/s²]
Returns
-------
Ar : float
Archimedes number, [-]
Notes
-----
It may be seen as a ratio of weight minus bouyancy and the inertial forces
It is often used in equations describing the motion of particles (solid
particles, drop, bubbles) in other fluid phase.
Examples
--------
>>> print("%0.0f" % Archimedes(5e-4, 2610, 0.6072, nu=48.09e-6))
2278
"""
if rho and mu:
nu = mu/rho
elif not nu:
raise Exception("undefined")
deltarho = abs(rho_p-rho)
return Dimensionless(g*L**3*deltarho/(rho*nu**2))
[docs]
@refDoc(__doi__, [1])
def Bi(h, L, k):
r"""Calculates Biot number `Bi` for heat transfer coefficient `h`,
geometric length `L`, and thermal conductivity `k`.
.. math::
Bi=\frac{hL}{k}
Parameters
----------
h : float
Heat transfer coefficient, [W/m^2/K]
L : float
Characteristic length, no typical definition [m]
k : float
Thermal conductivity, within the object [W/m/K]
Returns
-------
Bi : float
Biot number, [-]
Notes
-----
It may be seen as a ratio of two heat transfer resistences in series.
.. math::
Bi = \frac{\text{Surface thermal resistance}}
{\text{Internal thermal resistance}}
It is useful in calculations of transient heating or cooling processes
of solid bodies in liquid or gas flows.
Examples
--------
>>> Bi(60, 0.02, 0.15)
8.0
"""
return Dimensionless(h*L/k)
[docs]
@refDoc(__doi__, [2])
def Bo(rhol, rhog, sigma, L):
r"""Calculates Bond number, `Bo` also known as Eotvos number.
.. math::
Bo = \frac{g(\rho_l-\rho_g)L^2}{\sigma}
Parameters
----------
rhol : float
Density of liquid, [kg/m³]
rhog : float
Density of gas, [kg/m³]
sigma : float
Surface tension, [N/m]
L : float
Characteristic length, [m]
Returns
-------
Bo : float
Bond number, [-]
"""
return g*(rhol-rhog)*L**2/sigma
[docs]
@refDoc(__doi__, [2])
def Eu(dP, rho, V):
r"""Calculates Euler number or `Eu` for a fluid of velocity `V` and
density `rho` experiencing a pressure drop `dP`.
.. math::
Eu = \frac{\Delta P}{\rho V^2}
Parameters
----------
dP : float
Pressure drop experience by the fluid, [Pa]
rho : float
Density of the fluid, [kg/m³]
V : float
Velocity of fluid, [m/s]
Returns
-------
Eu : float
Euler number, [-]
Notes
-----
Used in pressure drop calculations.
.. math::
Eu = \frac{\text{Pressure drop}}{2\cdot \text{velocity head}}
"""
Eu = dP/(rho*V**2)
return Eu
[docs]
@refDoc(__doi__, [1])
def Fo(k, L, t):
r"""Calculates Fourier number `Fo`.
.. math::
Fo = \frac{kt}{l^2}
Parameters
----------
k : float
Thermal diffusivity [m²/s]
L : float
Characteristic length [m]
t : float
Time of cooling or heating [s]
Returns
-------
Fo : float
Fourier number, [-]
Notes
-----
Can be seen as a adimensional time.
It is commonly used in transient conduction problems.
Examples
--------
>>> print("%0.1f" % Fo(7e-6, 0.01, 60))
4.2
"""
return Dimensionless(k*t/L**2)
[docs]
@refDoc(__doi__, [1])
def Fr(V, L, g=g):
r"""Calculates Froude number `Fr` for velocity `V` and geometric length
`L`. If desired, gravity can be specified as well. Normally the function
returns the result of the equation below.
.. math::
Fr = \frac{V}{\sqrt{gL}}
Parameters
----------
V : float
Velocity of the particle or fluid, [m/s]
L : float
Characteristic length, no typical definition [m]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
Fr : float
Froude number, [-]
Notes
-----
Can be seen as a ratio of inertial force and gravity.
.. math::
Fr = \frac{\text{Inertial Force}}{\text{Gravity Force}}
Appears in problems of forced motion when gravity has same influence,
example in free liquid surfaces or multiphase flow.
Examples
--------
>>> print("%0.0f" % Fr(5, L=0.025))
102
"""
return Dimensionless(V**2/(L*g))
[docs]
@refDoc(__doi__, [1])
def Ga(L, rho=None, mu=None, nu=None, g=g):
r"""Calculates Galilei number `Ga`.
.. math::
Ar=\frac{gL^3}{v^2}
Inputs either of any of the following sets:
* L and kinematic viscosity `nu`
* L, density `rho` and dynamic viscosity `mu`
Parameters
----------
L : float
Characteristic length [m]
rho : float, optional
Density of bulk phase [kg/m³]
mu : float, optional
Dynamic viscosity, [Pa*s]
nu : float, optional
Kinematic viscosity, [m^2/s]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
Ga : float
Galilei number, [-]
Examples
--------
>>> print("%0.3f" % Ga(5e-4, nu=48.09e-6))
0.530
"""
if rho and mu:
nu = mu/rho
elif not nu:
raise Exception("undefined")
return Dimensionless(g*L**3/nu**2)
[docs]
@refDoc(__doi__, [1])
def Gr(L, beta, T1, T2=0, rho=None, mu=None, nu=None, g=g):
r"""Calculates Grashof number or `Gr` for a fluid with the given
properties, temperature difference, and characteristic length.
.. math::
Gr = = \frac{g\beta (T_s-T_\infty)L^3}{\nu^2}
= \frac{g\beta (T_s-T_\infty)L^3\rho^2}{\mu^2}
Inputs either of any of the following sets:
* L, beta, T1 and T2, and density `rho` and dynamic viscosity `mu`
* L, beta, T1 and T2, and kinematic viscosity `nu`
Parameters
----------
L : float
Characteristic length [m]
beta : float
Volumetric thermal expansion coefficient [1/K]
T1 : float
Temperature 1, usually a film temperature [K]
T2 : float, optional
Temperature 2, usually a bulk temperature (or 0 if only a difference
is provided to the function) [K]
rho : float, optional
Density, [kg/m^3]
mu : float, optional
Dynamic viscosity, [Pa*s]
nu : float, optional
Kinematic viscosity, [m^2/s]
g : float, optional
Acceleration due to gravity, [m/s^2]
Returns
-------
Gr : float
Grashof number, [-]
Notes
-----
.. math::
Gr = \frac{\text{Buoyancy forces}}{\text{Viscous forces}}
An error is raised if none of the required input sets are provided.
It is a relative difference of densities within one phase only (liquid
or gaseous), which occurs because of a spatial temperature diference.
Important to describe heat transfer in natural convection flow problems.
Examples
--------
>>> print("%0.2e" % Gr(L=0.6, beta=0.0031, T1=60, T2=20, nu=1.692e-5))
9.17e+08
"""
if rho and mu:
nu = mu/rho
elif not nu:
raise Exception("undefined")
Gr = g*beta*abs(T2-T1)*L**3/nu**2
return Dimensionless(Gr)
[docs]
@refDoc(__doi__, [1])
def Gz(k, D, t=None, L=None, V=None):
r"""Calculates Graetz number `Gz`.
.. math::
Gz = \frac{D^2}{kt}
Parameters
----------
k : float
Thermal diffusivity [m²/s]
D : float
Characteristic length [m]
t : float, optional
Time of cooling or heating [s]
L : float, optional
Second characteristic length, optional time definition [m]
V : float, optional
Mean flow velocity [m/s]
Returns
-------
Gz : float
Graetz number, [-]
Notes
-----
It is the reciprocal of Fourier number.
It is mainly used in calculations for steady flow, in which the time (the
residence time of the fluid in a heated or cooled portion of a channel)
can be expressed via the length L and the mean flow velocity V
Examples
--------
>>> print("%0.0f" % Gz(1.48e-7, 0.018, V=1.5, L=3))
1095
"""
if V and L:
t = L/V
elif not t:
raise Exception("undefined")
return Dimensionless(D**2/k/t)
[docs]
@refDoc(__doi__, [2])
def Kn(path, L):
r"""Calculates Knudsen number or `Kn`.
.. math::
Kn = \frac{\lambda}{L}
Parameters
----------
path : float
Mean free path between molecular collisions, [m]
L : float
Characteristic length, [m]
Returns
-------
Kn : float
Knudsen number, [-]
Notes
-----
Used in mass transfer calculations.
.. math::
Kn = \frac{\text{Mean free path length}}{\text{Characteristic length}}
"""
Kn = path/L
return Kn
[docs]
@refDoc(__doi__, [2])
def Le(D=None, alpha=None, Cp=None, k=None, rho=None):
r"""Calculates Lewis number or `Le`.
.. math::
Le = \frac{k}{\rho C_p D} = \frac{\alpha}{D}
Inputs can be either of the following sets:
* Diffusivity and Thermal diffusivity
* Diffusivity, heat capacity, thermal conductivity, and density
Parameters
----------
D : float
Diffusivity of a species, [m²/s]
alpha : float, optional
Thermal diffusivity, [m²/s]
Cp : float, optional
Heat capacity, [J/kg/K]
k : float, optional
Thermal conductivity, [W/m/K]
rho : float, optional
Density, [kg/m³]
Returns
-------
Le : float
Lewis number, [-]
Notes
-----
.. math::
Le=\frac{\text{Thermal diffusivity}}{\text{Mass diffusivity}} =
\frac{Sc}{Pr}
An error is raised if none of the required input sets are provided.
"""
if k and Cp and rho:
alpha = k/(rho*Cp)
elif alpha:
pass
else:
raise Exception("undefined")
Le = alpha/D
return Le
[docs]
@refDoc(__doi__, [2])
def Ma(V, c):
r"""Calculates Mach number or `Ma` for a fluid.
.. math::
Ma = \frac{V}{c}
Parameters
----------
V : float
Velocity of fluid, [m/s]
c : float
Speed of sound in fluid, [m/s]
Returns
-------
Ma : float
Mach number, [-]
Notes
-----
Used in compressible flow calculations.
.. math::
Ma = \frac{\text{fluid velocity}}{\text{sonic velocity}}
"""
return V/c
[docs]
@refDoc(__doi__, [1])
def Nu(h, L, k):
r"""Calculates Nusselt number or `Nu`.
.. math::
Nu = \frac{h L}{k}
Nusselt is the ratio of the convective heat transfer coefficient to the
heat transfer coefficient for conduction.
Parameters
----------
k : float
Thermal conductivity, [W/m/K]
L : float
Characteristic length, [m]
h : float
Heat transfer coefficient, [W/m²/K]
Returns
-------
Nu : float
Nusselt number, [-]
Notes
-----
Nusselt number is a dimensionless heat transfer coeffcient.
Be careful with the characteristic lenght definition, it deppend of system
configuration (internal diameter in a flow channel).
Examples
--------
>>> print("%0.1f" % Nu(102.3, 0.03927, 0.03181))
126.3
"""
return Dimensionless(h*L/k)
[docs]
@refDoc(__doi__, [2])
def Pe(V, L, rho=None, Cp=None, k=None, alpha=None):
r"""Calculates heat transfer Peclet number or `Pe`
.. math::
Pe = \frac{VL\rho C_p}{k} = \frac{LV}{\alpha}
Inputs either of any of the following sets:
* V, L, density `rho`, heat capcity `Cp`, and thermal conductivity `k`
* V, L, and thermal diffusivity `alpha`
Parameters
----------
V : float
Velocity [m/s]
L : float
Characteristic length [m]
rho : float, optional
Density, [kg/m³]
Cp : float, optional
Heat capacity, [J/kg/K]
k : float, optional
Thermal conductivity, [W/m/K]
alpha : float, optional
Thermal diffusivity, [m²/s]
Returns
-------
Pe : float
Peclet number, [-]
Notes
-----
.. math::
Pe = \frac{\text{Bulk heat transfer}}{\text{Conduction heat transfer}}
An error is raised if none of the required input sets are provided.
"""
if rho and Cp and k:
alpha = k/(rho*Cp)
elif not alpha:
raise Exception("undefined")
Pe = V*L/alpha
return Pe
[docs]
@refDoc(__doi__, [1])
def Pr(cp=None, k=None, mu=None, nu=None, rho=None, alpha=None):
r"""Calculates Prandtl number or `Pr` for a fluid with the given
parameters.
Prandtl number is the ratio of momentum diffusion to energy diffusion and
relates the velocity profile to the temperature profile.
.. math::
Pr = \frac{C_p \mu}{k} = \frac{\nu}{\alpha} = \frac{C_p \rho \nu}{k}
Inputs can be any of the following sets:
* Heat capacity, dynamic viscosity, and thermal conductivity
* Thermal diffusivity and kinematic viscosity
* Heat capacity, kinematic viscosity, thermal conductivity, and density
Parameters
----------
Cp : float
Heat capacity, [J/kg/K]
k : float
Thermal conductivity, [W/m/K]
mu : float, optional
Dynamic viscosity, [Pa*s]
nu : float, optional
Kinematic viscosity, [m^2/s]
rho : float
Density, [kg/m^3]
alpha : float
Thermal diffusivity, [m^2/s]
Returns
-------
Pr : float
Prandtl number, [-]
Notes
-----
.. math::
Pr=\frac{\text{kinematic viscosity}}{\text{thermal diffusivity}} = \
\frac{\text{momendum diffusivity}}{\text{thermal diffusivity}}
An error is raised if none of the required input sets are provided.
Examples
--------
>>> print("%0.2f" % Pr(cp=1821., k=0.134, mu=43.61e-5))
5.93
"""
if k and cp and mu:
Pr = cp*mu/k
elif nu and rho and cp and k:
Pr = nu*rho*cp/k
elif nu and alpha:
Pr = nu/alpha
else:
raise Exception("undefined")
return Dimensionless(Pr)
[docs]
@refDoc(__doi__, [1])
def Ra(Pr, Gr):
r"""Calculates Rayleigh number or `Ra` using Prandtl number `Pr` and
Grashof number `Gr` for a fluid with the given
properties, temperature difference, and characteristic length used
to calculate `Gr` and `Pr`.
.. math::
Ra = PrGr
Parameters
----------
Pr : float
Prandtl number, [-]
Gr : float
Grashof number, [-]
Returns
-------
Ra : float
Rayleigh number, [-]
Notes
-----
Used in free convection problems only.
"""
return Dimensionless(Pr*Gr)
[docs]
@refDoc(__doi__, [1])
def Re(D, V, rho=None, mu=None, nu=None):
r"""Calculates Reynolds number or `Re` for a fluid with the given
properties for the specified velocity and diameter.
Inputs either of any of the following sets:
* V, D, density `rho` and kinematic viscosity `mu`
* V, D, and dynamic viscosity `nu`
.. math::
Re = {D \cdot V}{\nu} = \frac{\rho V D}{\mu}
Reynolds is the ratio of inertial forces to viscous forces, used to
determine the type of single-phase flow.
* Re ≤ 2000: Laminar flow, fluid particles move parallel to the tube axis
* 2000 ≤ Re ≤ 10000: Transition from laminar to turbulent flow
* Re ≥ 10000: Turbulent flow, fluid particles move chaotically
Parameters
----------
D : float
Diameter [m]
V : float
Velocity [m/s]
rho : float, optional
Density, [kg/m^3]
mu : float, optional
Dynamic viscosity, [Pa*s]
Returns
-------
Re : float
Reynolds number, [-]
Notes
-----
.. math::
Re = \frac{\text{Momentum}}{\text{Viscosity}}
Can be seen as a ratio of inertial forces to frictional forces.
It's the crucial criterium to define the flow mode: laminar, turbulent.
Examples
--------
>>> Re(0.052, 1.05, rho=999.8, nu=1.3e-6)
42000.0
"""
if rho and mu:
nu = mu/rho
elif not nu:
raise Exception("undefined")
return Dimensionless(V*D/nu)
[docs]
@refDoc(__doi__, [2])
def Sh(K, L, D):
r"""Calculates Sherwood number or `Sh`.
.. math::
Sh = \frac{KL}{D}
Parameters
----------
K : float
Mass transfer coefficient, [m/s]
L : float
Characteristic length, no typical definition [m]
D : float
Diffusivity of a species [m/s²]
Returns
-------
Sh : float
Sherwood number, [-]
Notes
-----
.. math::
Sh = \frac{\text{Mass transfer by convection}}
{\text{Mass transfer by diffusion}} = \frac{K}{D/L}
"""
Sh = K*L/D
return Sh
[docs]
@refDoc(__doi__, [2])
def Sc(D, mu=None, nu=None, rho=None):
r"""Calculates Schmidt number or `Sc`.
.. math::
Sc = \frac{\mu}{D\rho} = \frac{\nu}{D}
Inputs can be any of the following sets:
* Diffusivity, dynamic viscosity, and density
* Diffusivity and kinematic viscosity
Parameters
----------
D : float
Diffusivity of a species, [m²/s]
mu : float, optional
Dynamic viscosity, [Pa·s]
nu : float, optional
Kinematic viscosity, [m²/s]
rho : float, optional
Density, [kg/m³]
Returns
-------
Sc : float
Schmidt number, [-]
Notes
-----
.. math::
Sc =\frac{\text{kinematic viscosity}}{\text{molecular diffusivity}}
= \frac{\text{viscous diffusivity}}{\text{species diffusivity}}
An error is raised if none of the required input sets are provided.
"""
if rho and mu:
Sc = mu/(rho*D)
elif nu:
Sc = nu/D
else:
raise Exception("undefined")
return Sc
[docs]
@refDoc(__doi__, [2])
def St(Nu=None, Pe=None, Re=None, Pr=None, alfa=None, rho=None, cp=None, V=None):
r"""Calculate Stanton number `St`
.. math::
St = \frac{Nu}{Pe} = \frac{Nu}{Re Pr} = \frac{\alpha}{\rho c_pV}
Inputs either of any of the following sets:
* Péclet and Nusselt number
* Nusselt, Reynolds and Prandt number
* V, density `rho`, heat specific `cp` and heat transfer coefficient `alfa`
Parameters
----------
Nu : float, optional
Nusselt number [-]
Pe : float, optional
Péclet number [-]
V : float, optional
Velocity of fluid [m/s]
alfa : float, optional
Heat transfer coefficient [W/m²/K]
rho : float, optional
Density [kg/m³]
cp : float, optional
Constant pressure specific heat [J/kg/K]
Returns
-------
St : float
Stanton number, [-]
"""
if Nu and Pe:
st = Nu/Pe
elif Nu and Re and Pr:
st = Nu/Re/Pr
elif alfa and rho and cp and V:
st = alfa/(rho*cp*V)
else:
raise Exception("undefined")
return Dimensionless(st)
[docs]
@refDoc(__doi__, [1])
def We(V, L, rho, sigma):
r"""Calculates Weber number, `We`, for a fluid with the given density,
surface tension, velocity, and geometric parameter (usually diameter
of bubble).
.. math::
We = \frac{V^2 L\rho}{\sigma}
Parameters
----------
V : float
Velocity of fluid, [m/s]
L : float
Characteristic length, typically bubble diameter [m]
rho : float
Density of fluid, [kg/m^3]
sigma : float
Surface tension, [N/m]
Returns
-------
We : float
Weber number, [-]
Notes
-----
Used in bubble calculations.
.. math::
We = \frac{\text{inertial force}}{\text{surface tension force}}
Examples
--------
>>> print("%0.2f" % We(V=11, L=0.005, rho=1.188, sigma=0.07278))
9.88
"""
return Dimensionless(V**2*L*rho/sigma)
[docs]
@refDoc(__doi__, [2])
def Dean(Re, di, Dc):
r"""Calculates Dean number, `De`, for a fluid with the Reynolds number `Re`,
tube diameter `di`, and helical coil diameter `Dc`.
.. math::
\text{De} = Re \sqrt{\frac{d_i}{D_c}}
Used in flow in curved geometry like helical coil.
Cited in [2]_, Eq 6-101
Parameters
----------
Re : float
Reynolds number, []
di : float
Inner tube diameter, [m]
Dc : float
Diameter of helical coil, [m]
Returns
-------
De : float
Dean number [-]
"""
return Re / (Dc/di)**0.5
if __name__ == "__main__":
import doctest
doctest.testmod()
