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# -*- coding: utf-8 -*- 

'''Chemical Engineering Design Library (ChEDL). Utilities for process modeling. 

Copyright (C) 2016, Caleb Bell <Caleb.Andrew.Bell@gmail.com> 

 

Permission is hereby granted, free of charge, to any person obtaining a copy 

of this software and associated documentation files (the "Software"), to deal 

in the Software without restriction, including without limitation the rights 

to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 

copies of the Software, and to permit persons to whom the Software is 

furnished to do so, subject to the following conditions: 

 

The above copyright notice and this permission notice shall be included in all 

copies or substantial portions of the Software. 

 

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 

IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 

AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 

LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 

OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 

SOFTWARE.''' 

 

from __future__ import division 

from math import sin, exp, pi 

from scipy.constants import g, R 

 

__all__ = ['Reynolds', 'Prandtl', 'Grashof', 'Nusselt', 'Sherwood', 'Rayleigh', 

'Schmidt', 'Peclet_heat', 'Peclet_mass', 'Fourier_heat', 'Fourier_mass', 

'Graetz_heat', 'Lewis', 'Weber', 'Mach', 'Knudsen', 'Bond', 'Dean', 

'Froude', 'Strouhal', 'Biot', 'Stanton', 'Euler', 'Cavitation', 'Eckert', 

'Jakob', 'Power_number', 'Drag', 'Capillary', 'Bejan_L', 'Bejan_p', 'Boiling', 

'Confinement', 'Archimedes', 'Ohnesorge', 'Suratman', 'thermal_diffusivity', 'c_ideal_gas', 

'relative_roughness', 'nu_mu_converter', 'gravity', 

'K_from_f', 'K_from_L_equiv', 'L_equiv_from_K', 'dP_from_K', 'head_from_K', 'head_from_P', 

'P_from_head', 'Eotvos'] 

 

 

 

### Not quite dimensionless groups 

def thermal_diffusivity(k, rho, Cp): 

r'''Calculates thermal diffusivity or `alpha` for a fluid with the given 

parameters. 

 

.. math:: 

\alpha = \frac{k}{\rho Cp} 

 

Parameters 

---------- 

k : float 

Thermal conductivity, [W/m/K] 

Cp : float 

Heat capacity, [J/kg/K] 

rho : float 

Density, [kg/m^3] 

 

Returns 

------- 

alpha : float 

Thermal diffusivity, [m^2/s] 

 

Notes 

----- 

 

Examples 

-------- 

>>> thermal_diffusivity(0.02, 1., 1000.) 

2e-05 

 

References 

---------- 

.. [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: 

Van Nostrand Reinhold Co., 1984. 

''' 

return k/(rho*Cp) 

 

 

### Ideal gas fluid properties 

 

 

def c_ideal_gas(T, k, MW): 

r'''Calculates speed of sound `c` in an ideal gas at temperature T. 

 

.. math:: 

c = \sqrt{kR_{specific}T} 

 

Parameters 

---------- 

T : float 

Temperature of fluid, [K] 

k : float 

Isentropic exponent of fluid, [-] 

MW : float 

Molecular weight of fluid, [g/mol] 

 

Returns 

------- 

c : float 

Speed of sound in fluid, [m/s] 

 

Notes 

----- 

Used in compressible flow calculations. 

Note that the gas constant used is the specific gas constant: 

 

.. math:: 

R_{specific} = R\frac{1000}{MW} 

 

Examples 

-------- 

>>> c_ideal_gas(1.4, 303., 28.96) 

348.9820361755092 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

Rspecific = R*1000./MW 

return (k*Rspecific*T)**0.5 

 

 

### Dimensionless groups with documentation 

 

def Reynolds(V, D, 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. 

 

.. math:: 

Re = \frac{D \cdot V}{\nu} = \frac{\rho V D}{\mu} 

 

Inputs either of any of the following sets: 

 

* V, D, density `rho` and kinematic viscosity `mu` 

* V, D, and dynamic viscosity `nu` 

 

Parameters 

---------- 

D : float 

Diameter [m] 

V : float 

Velocity [m/s] 

rho : float, optional 

Density, [kg/m^3] 

mu : float, optional 

Dynamic viscosity, [Pa*s] 

nu : float, optional 

Kinematic viscosity, [m^2/s] 

 

Returns 

------- 

Re : float 

Reynolds number [] 

 

Notes 

----- 

.. math:: 

Re = \frac{\text{Momentum}}{\text{Viscosity}} 

 

An error is raised if none of the required input sets are provided. 

 

Examples 

-------- 

>>> Reynolds(2.5, 0.25, 1.1613, 1.9E-5) 

38200.65789473684 

>>> Reynolds(2.5, 0.25, nu=1.636e-05) 

38202.93398533008 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

if rho and mu: 

nu = mu/rho 

elif not nu: 

raise Exception('Either density and viscosity, or dynamic viscosity, \ 

is needed') 

return V*D/nu 

 

 

def Peclet_heat(V, L, rho=None, Cp=None, k=None, alpha=None): 

r'''Calculates heat transfer Peclet number or `Pe` for a specified velocity 

`V`, characteristic length `L`, and specified properties for the given 

fluid. 

 

.. 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/m^3] 

Cp : float, optional 

Heat capacity, [J/kg/K] 

k : float, optional 

Thermal conductivity, [W/m/K] 

alpha : float, optional 

Thermal diffusivity, [m^2/s] 

 

Returns 

------- 

Pe : float 

Peclet number (heat) [] 

 

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. 

 

Examples 

-------- 

>>> Peclet_heat(1.5, 2, 1000., 4000., 0.6) 

20000000.0 

>>> Peclet_heat(1.5, 2, alpha=1E-7) 

30000000.0 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

if rho and Cp and k: 

alpha = k/(rho*Cp) 

elif not alpha: 

raise Exception('Either heat capacity and thermal conductivity and\ 

density, or thermal diffusivity is needed') 

return V*L/alpha 

 

 

def Peclet_mass(V, L, D): 

r'''Calculates mass transfer Peclet number or `Pe` for a specified velocity 

`V`, characteristic length `L`, and diffusion coefficient `D`. 

 

.. math:: 

Pe = \frac{L V}{D} 

 

Parameters 

---------- 

V : float 

Velocity [m/s] 

L : float 

Characteristic length [m] 

D : float 

Diffusivity of a species, [m^2/s] 

 

Returns 

------- 

Pe : float 

Peclet number (mass) [] 

 

Notes 

----- 

.. math:: 

Pe = \frac{\text{Advective transport rate}}{\text{Diffusive transport rate}} 

 

Examples 

-------- 

>>> Peclet_mass(1.5, 2, 1E-9) 

3000000000.0 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

''' 

return V*L/D 

 

 

def Fourier_heat(t, L, rho=None, Cp=None, k=None, alpha=None): 

r'''Calculates heat transfer Fourier number or `Fo` for a specified time 

`t`, characteristic length `L`, and specified properties for the given 

fluid. 

 

.. math:: 

Fo = \frac{k t}{C_p \rho L^2} = \frac{\alpha t}{L^2} 

 

Inputs either of any of the following sets: 

 

* t, L, density `rho`, heat capcity `Cp`, and thermal conductivity `k` 

* t, L, and thermal diffusivity `alpha` 

 

Parameters 

---------- 

t : float 

time [s] 

L : float 

Characteristic length [m] 

rho : float, optional 

Density, [kg/m^3] 

Cp : float, optional 

Heat capacity, [J/kg/K] 

k : float, optional 

Thermal conductivity, [W/m/K] 

alpha : float, optional 

Thermal diffusivity, [m^2/s] 

 

Returns 

------- 

Fo : float 

Fourier number (heat) [] 

 

Notes 

----- 

.. math:: 

Fo = \frac{\text{Heat conduction rate}} 

{\text{Rate of thermal energy storage in a solid}} 

 

An error is raised if none of the required input sets are provided. 

 

Examples 

-------- 

>>> Fourier_heat(1.5, 2, 1000., 4000., 0.6) 

5.625e-08 

>>> Fourier_heat(1.5, 2, alpha=1E-7) 

3.75e-08 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

if rho and Cp and k: 

alpha = k/(rho*Cp) 

elif not alpha: 

raise Exception('Either heat capacity and thermal conductivity and\ 

density, or thermal diffusivity is needed') 

return t*alpha/L**2 

 

 

def Fourier_mass(t, L, D): 

r'''Calculates mass transfer Fourier number or `Fo` for a specified time 

`t`, characteristic length `L`, and diffusion coefficient `D`. 

 

.. math:: 

Fo = \frac{D t}{L^2} 

 

Parameters 

---------- 

t : float 

time [s] 

L : float 

Characteristic length [m] 

D : float 

Diffusivity of a species, [m^2/s] 

 

Returns 

------- 

Fo : float 

Fourier number (mass) [] 

 

Notes 

----- 

.. math:: 

Fo = \frac{\text{Diffusive transport rate}}{\text{Storage rate}} 

 

Examples 

-------- 

>>> Fourier_mass(1.5, 2, 1E-9) 

3.7500000000000005e-10 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

''' 

return t*D/L**2 

 

 

def Graetz_heat(V, D, x, rho=None, Cp=None, k=None, alpha=None): 

r'''Calculates Graetz number or `Gz` for a specified velocity 

`V`, diameter `D`, axial diatance `x`, and specified properties for the 

given fluid. 

 

.. math:: 

Gz = \frac{VD^2\cdot C_p \rho}{x\cdot k} = \frac{VD^2}{x \alpha} 

 

Inputs either of any of the following sets: 

 

* V, D, x, density `rho`, heat capcity `Cp`, and thermal conductivity `k` 

* V, D, x, and thermal diffusivity `alpha` 

 

Parameters 

---------- 

V : float 

Velocity, [m/s] 

D : float 

Diameter [m] 

x : float 

Axial distance [m] 

rho : float, optional 

Density, [kg/m^3] 

Cp : float, optional 

Heat capacity, [J/kg/K] 

k : float, optional 

Thermal conductivity, [W/m/K] 

alpha : float, optional 

Thermal diffusivity, [m^2/s] 

 

Returns 

------- 

Gz : float 

Graetz number [] 

 

Notes 

----- 

.. math:: 

Gz = \frac{\text{Time for radial heat diffusion in a fluid by conduction}} 

{\text{Time taken by fluid to reach distance x}} 

 

Gz = \frac{D}{x}RePr 

 

An error is raised if none of the required input sets are provided. 

 

Examples 

-------- 

>>> Graetz_heat(1.5, 0.25, 5, 800., 2200., 0.6) 

55000.0 

>>> Graetz_heat(1.5, 0.25, 5, alpha=1E-7) 

187500.0 

 

References 

---------- 

.. [1] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and 

David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: 

Wiley, 2011. 

''' 

if rho and Cp and k: 

alpha = k/(rho*Cp) 

elif not alpha: 

raise Exception('Either heat capacity and thermal conductivity and\ 

density, or thermal diffusivity is needed') 

return V*D**2/(x*alpha) 

 

 

def Schmidt(D, mu=None, nu=None, rho=None): 

r'''Calculates Schmidt number or `Sc` for a fluid with the given 

parameters. 

 

.. 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, [m^2/s] 

mu : float, optional 

Dynamic viscosity, [Pa*s] 

nu : float, optional 

Kinematic viscosity, [m^2/s] 

rho : float, optional 

Density, [kg/m^3] 

 

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. 

 

Examples 

-------- 

>>> Schmidt(D=2E-6, mu=4.61E-6, rho=800) 

0.00288125 

>>> Schmidt(D=1E-9, nu=6E-7) 

599.9999999999999 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

if rho and mu: 

return mu/(rho*D) 

elif nu: 

return nu/D 

else: 

raise Exception('Insufficient information provided for Schmidt number calculation') 

 

 

def Lewis(D=None, alpha=None, Cp=None, k=None, rho=None): 

r'''Calculates Lewis number or `Le` for a fluid with the given parameters. 

 

.. 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, [m^2/s] 

alpha : float, optional 

Thermal diffusivity, [m^2/s] 

Cp : float, optional 

Heat capacity, [J/kg/K] 

k : float, optional 

Thermal conductivity, [W/m/K] 

rho : float, optional 

Density, [kg/m^3] 

 

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. 

 

Examples 

-------- 

>>> Lewis(D=22.6E-6, alpha=19.1E-6) 

0.8451327433628318 

>>> Lewis(D=22.6E-6, rho=800., k=.2, Cp=2200) 

0.00502815768302494 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

.. [3] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. 

Berlin; New York:: Springer, 2010. 

''' 

if k and Cp and rho: 

alpha = k/(rho*Cp) 

elif alpha: 

pass 

else: 

raise Exception('Insufficient information provided for Le calculation') 

return alpha/D 

 

 

def Weber(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 

-------- 

>>> Weber(V=0.18, L=0.001, rho=900., sigma=0.01) 

2.916 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

.. [3] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. 

Berlin; New York:: Springer, 2010. 

''' 

return V**2*L*rho/sigma 

 

 

def Mach(V, c): 

r'''Calculates Mach number or `Ma` for a fluid of velocity `V` with speed 

of sound `c`. 

 

.. 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}} 

 

Examples 

-------- 

>>> Mach(33., 330) 

0.1 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return V/c 

 

 

def Confinement(D, rhol, rhog, sigma, g=g): 

r'''Calculates Confinement number or `Co` for a fluid in a channel of  

diameter `D` with liquid and gas densities `rhol` and `rhog` and surface 

tension `sigma`, under the influence of gravitational force `g`. 

 

.. math:: 

\text{Co}=\frac{\left[\frac{\sigma}{g(\rho_l-\rho_g)}\right]^{0.5}}{D} 

 

Parameters 

---------- 

D : float 

Diameter of channel, [m] 

rhol : float 

Density of liquid phase, [kg/m^3] 

rhog : float 

Density of gas phase, [kg/m^3] 

sigma : float 

Surface tension between liquid-gas phase, [N/m] 

g : float, optional 

Acceleration due to gravity, [m/s^2] 

 

Returns 

------- 

Co : float 

Confinement number [-] 

 

Notes 

----- 

Used in two-phase pressure drop and heat transfer correlations. First used 

in [1]_ according to [3]_. 

 

.. math:: 

\text{Co} = \frac{\frac{\text{surface tension force}} 

{\text{buoyancy force}}}{\text{Channel area}} 

 

Examples 

-------- 

>>> Confinement(0.001, 1077, 76.5, 4.27E-3) 

0.6596978265315191 

 

References 

---------- 

.. [1] Cornwell, Keith, and Peter A. Kew. "Boiling in Small Parallel  

Channels." In Energy Efficiency in Process Technology, edited by Dr P.  

A. Pilavachi, 624-638. Springer Netherlands, 1993.  

doi:10.1007/978-94-011-1454-7_56. 

.. [2] Kandlikar, Satish G. Heat Transfer and Fluid Flow in Minichannels  

and Microchannels. Elsevier, 2006. 

.. [3] Tran, T. N, M. -C Chyu, M. W Wambsganss, and D. M France. Two-Phase  

Pressure Drop of Refrigerants during Flow Boiling in Small Channels: An  

Experimental Investigation and Correlation Development." International  

Journal of Multiphase Flow 26, no. 11 (November 1, 2000): 1739-54.  

doi:10.1016/S0301-9322(99)00119-6. 

''' 

return (sigma/(g*(rhol-rhog)))**0.5/D 

 

 

def Knudsen(path, L): 

r'''Calculates Knudsen number or `Kn` for a fluid with mean free path 

`path` and for a characteristic length `L`. 

 

.. 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}} 

 

Examples 

-------- 

>>> Knudsen(1e-10, .001) 

1e-07 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return path/L 

 

 

def Prandtl(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. 

 

.. 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{momentum diffusivity}}{\text{thermal diffusivity}} 

 

An error is raised if none of the required input sets are provided. 

 

Examples 

-------- 

>>> Prandtl(Cp=1637., k=0.010, mu=4.61E-6) 

0.754657 

>>> Prandtl(Cp=1637., k=0.010, nu=6.4E-7, rho=7.1) 

0.7438528 

>>> Prandtl(nu=6.3E-7, alpha=9E-7) 

0.7000000000000001 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

.. [3] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. 

Berlin; New York:: Springer, 2010. 

''' 

if k and Cp and mu: 

return Cp*mu/k 

elif nu and rho and Cp and k: 

return nu*rho*Cp/k 

elif nu and alpha: 

return nu/alpha 

else: 

raise Exception('Insufficient information provided for Pr calculation') 

 

 

def Grashof(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 kinematic viscosity `mu` 

* L, beta, T1 and T2, and dynamic 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. 

Used in free convection problems only. 

 

Examples 

-------- 

Example 4 of [1]_, p. 1-21 (matches): 

 

>>> Grashof(L=0.9144, beta=0.000933, T1=178.2, rho=1.1613, mu=1.9E-5) 

4656936556.178915 

>>> Grashof(L=0.9144, beta=0.000933, T1=378.2, T2=200, nu=1.636e-05) 

4657491516.530312 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

if rho and mu: 

nu = mu/rho 

elif not nu: 

raise Exception('Either density and viscosity, or dynamic viscosity, \ 

is needed') 

return g*beta*abs(T2-T1)*L**3/nu**2 

 

 

def Bond(rhol, rhog, sigma, L): 

r'''Calculates Bond number, `Bo` also known as Eotvos number, 

for a fluid with the given liquid and gas densities, surface tension, 

and geometric parameter (usually length). 

 

.. math:: 

Bo = \frac{g(\rho_l-\rho_g)L^2}{\sigma} 

 

Parameters 

---------- 

rhol : float 

Density of liquid, [kg/m^3] 

rhog : float 

Density of gas, [kg/m^3] 

sigma : float 

Surface tension, [N/m] 

L : float 

Characteristic length, [m] 

 

Returns 

------- 

Bo : float 

Bond number [] 

 

Examples 

-------- 

>>> Bond(1000., 1.2, .0589, 2) 

665187.2339558573 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

''' 

return (g*(rhol-rhog)*L**2/sigma) 

 

Eotvos = Bond 

 

 

def Rayleigh(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. 

 

Examples 

-------- 

>>> Rayleigh(1.2, 4.6E9) 

5520000000.0 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return Pr*Gr 

 

 

def Froude(V, L, g=g, squared=False): 

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; Froude number is also often 

said to be defined as the square 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] 

squared : bool, optional 

Whether to return the squared form of Frounde number 

 

Returns 

------- 

Fr : float 

Froude number, [-] 

 

Notes 

----- 

Many alternate definitions including density ratios have been used. 

 

.. math:: 

Fr = \frac{\text{Inertial Force}}{\text{Gravity Force}} 

 

Examples 

-------- 

>>> Froude(1.83, L=2., g=1.63) 

1.0135432593877318 

>>> Froude(1.83, L=2., squared=True) 

0.17074638128208924 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

Fr = V/(L*g)**0.5 

if squared: 

Fr *= Fr 

return Fr 

 

 

def Strouhal(f, L, V): 

r'''Calculates Strouhal number `St` for a characteristic frequency `f`, 

characteristic length `L`, and velocity `V`. 

 

.. math:: 

St = \frac{fL}{V} 

 

Parameters 

---------- 

f : float 

Characteristic frequency, usually that of vortex shedding, [Hz] 

L : float 

Characteristic length, [m] 

V : float 

Velocity of the fluid, [m/s] 

 

Returns 

------- 

St : float 

Strouhal number, [-] 

 

Notes 

----- 

Sometimes abbreviated to S or Sr. 

 

.. math:: 

St = \frac{\text{Characteristif flow time}} 

{\text{Period of oscillation}} 

 

Examples 

-------- 

>>> Strouhal(8, 2., 4.) 

4.0 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return f*L/V 

 

 

def Nusselt(h, L, k): 

r'''Calculates Nusselt number `Nu` for a heat transfer coefficient `h`, 

characteristic length `L`, and thermal conductivity `k`. 

 

.. math:: 

Nu = \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 of fluid [W/m/K] 

 

Returns 

------- 

Nu : float 

Nusselt number, [-] 

 

Notes 

----- 

Do not confuse k, the thermal conductivity of the fluid, with that 

of within a solid object associated with! 

 

.. math:: 

Nu = \frac{\text{Convective heat transfer}} 

{\text{Conductive heat transfer}} 

 

Examples 

-------- 

>>> Nusselt(1000., 1.2, 300.) 

4.0 

>>> Nusselt(10000., .01, 4000.) 

0.025 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and 

David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: 

Wiley, 2011. 

''' 

return h*L/k 

 

 

def Sherwood(K, L, D): 

r'''Calculates Sherwood number `Sh` for a mass transfer coefficient `K`, 

characteristic length `L`, and diffusivity `D`. 

 

.. 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/s^2] 

 

Returns 

------- 

Sh : float 

Sherwood number, [-] 

 

Notes 

----- 

 

.. math:: 

Sh = \frac{\text{Mass transfer by convection}} 

{\text{Mass transfer by diffusion}} = \frac{K}{D/L} 

 

Examples 

-------- 

>>> Sherwood(1000., 1.2, 300.) 

4.0 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

''' 

return K*L/D 

 

 

def Biot(h, L, k): 

r'''Calculates Biot number `Br` 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 

----- 

Do not confuse k, the thermal conductivity within the object, with that 

of the medium h is calculated with! 

 

.. math:: 

Bi = \frac{\text{Surface thermal resistance}} 

{\text{Internal thermal resistance}} 

 

Examples 

-------- 

>>> Biot(1000., 1.2, 300.) 

4.0 

>>> Biot(10000., .01, 4000.) 

0.025 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return h*L/k 

 

 

def Stanton(h, V, rho, Cp): 

r'''Calculates Stanton number or `St` for a specified heat transfer 

coefficient `h`, velocity `V`, density `rho`, and heat capacity `Cp`. 

 

.. math:: 

St = \frac{h}{V\rho Cp} 

 

Parameters 

---------- 

h : float 

Heat transfer coefficient, [W/m^2/K] 

V : float 

Velocity, [m/s] 

rho : float 

Density, [kg/m^3] 

Cp : float 

Heat capacity, [J/kg/K] 

 

Returns 

------- 

St : float 

Stanton number [] 

 

Notes 

----- 

.. math:: 

St = \frac{\text{Heat transfer coefficient}}{\text{Thermal capacity}} 

 

Examples 

-------- 

>>> Stanton(5000, 5, 800, 2000.) 

0.000625 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [1] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and 

David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: 

Wiley, 2011. 

''' 

return h/(V*rho*Cp) 

 

 

def Euler(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/m^3] 

V : float 

Velocity of fluid, [m/s] 

 

Returns 

------- 

Eu : float 

Euler number [] 

 

Notes 

----- 

Used in pressure drop calculations. 

Rarely, this number is divided by two. 

Named after Leonhard Euler applied calculus to fluid dynamics. 

 

.. math:: 

Eu = \frac{\text{Pressure drop}}{2\cdot \text{velocity head}} 

 

Examples 

-------- 

>>> Euler(1E5, 1000., 4) 

6.25 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return dP/(rho*V**2) 

 

 

def Cavitation(P, Psat, rho, V): 

r'''Calculates Cavitation number or `Ca` for a fluid of velocity `V` with 

a pressure `P`, vapor pressure `Psat`, and density `rho`. 

 

.. math:: 

Ca = \sigma_c = \sigma = \frac{P-P_{sat}}{\frac{1}{2}\rho V^2} 

 

Parameters 

---------- 

P : float 

Internal pressure of the fluid, [Pa] 

Psat : float 

Vapor pressure of the fluid, [Pa] 

rho : float 

Density of the fluid, [kg/m^3] 

V : float 

Velocity of fluid, [m/s] 

 

Returns 

------- 

Ca : float 

Cavitation number [] 

 

Notes 

----- 

Used in determining if a flow through a restriction will cavitate. 

Sometimes, the multiplication by 2 will be omitted; 

 

.. math:: 

Ca = \frac{\text{Pressure - Vapor pressure}} 

{\text{Inertial pressure}} 

 

Examples 

-------- 

>>> Cavitation(2E5, 1E4, 1000, 10) 

3.8 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return (P-Psat)/(0.5*rho*V**2) 

 

 

def Eckert(V, Cp, dT): 

r'''Calculates Eckert number or `Ec` for a fluid of velocity `V` with 

a heat capacity `Cp`, between two temperature given as `dT`. 

 

.. math:: 

Ec = \frac{V^2}{C_p \Delta T} 

 

Parameters 

---------- 

V : float 

Velocity of fluid, [m/s] 

Cp : float 

Heat capacity of the fluid, [J/kg/K] 

dT : float 

Temperature difference, [K] 

 

Returns 

------- 

Ec : float 

Eckert number [] 

 

Notes 

----- 

Used in certain heat transfer calculations. Fairly rare. 

 

.. math:: 

Ec = \frac{\text{Kinetic energy} }{ \text{Enthalpy difference}} 

 

Examples 

-------- 

>>> Eckert(10, 2000., 25.) 

0.002 

 

References 

---------- 

.. [1] Goldstein, Richard J. ECKERT NUMBER. Thermopedia. Hemisphere, 2011. 

10.1615/AtoZ.e.eckert_number 

''' 

return V**2/(Cp*dT) 

 

 

def Jakob(Cp, Hvap, Te): 

r'''Calculates Jakob number or `Ja` for a boiling fluid with sensible heat 

capacity `Cp`, enthalpy of vaporization `Hvap`, and boiling at `Te` degrees 

above its saturation boiling point. 

 

.. math:: 

Ja = \frac{C_{P}\Delta T_e}{\Delta H_{vap}} 

 

Parameters 

---------- 

Cp : float 

Heat capacity of the fluid, [J/kg/K] 

Hvap : float 

Enthalpy of vaporization of the fluid at its saturation temperature [J/kg] 

Te : float 

Temperature difference above the fluid's saturation boiling temperature, [K] 

 

Returns 

------- 

Ja : float 

Jakob number [] 

 

Notes 

----- 

Used in boiling heat transfer analysis. Fairly rare. 

 

.. math:: 

Ja = \frac{\Delta \text{Sensible heat}}{\Delta \text{Latent heat}} 

 

Examples 

-------- 

>>> Jakob(4000., 2E6, 10.) 

0.02 

 

References 

---------- 

.. [1] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and 

David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: 

Wiley, 2011. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return Cp*Te/Hvap 

 

 

def Power_number(P, L, N, rho): 

r'''Calculates power number, `Po`, for an agitator applying a specified 

power `P` with a characteristic length `L`, rotationa speed `N`, to 

a fluid with a specified density `rho`. 

 

.. math:: 

Po = \frac{P}{\rho N^3 D^5} 

 

Parameters 

---------- 

P : float 

Power applied, [W] 

L : float 

Characteristic length, typically agitator diameter [m] 

N : float 

Speed [revolutions/second] 

rho : float 

Density of fluid, [kg/m^3] 

 

Returns 

------- 

Po : float 

Power number [] 

 

Notes 

----- 

Used in mixing calculations. 

 

.. math:: 

Po = \frac{\text{Power}}{\text{Rotational inertia}} 

 

Examples 

-------- 

>>> Power_number(P=180, L=0.01, N=2.5, rho=800.) 

144000000.0 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return P/(rho*N**3*L**5) 

 

 

def Drag(F, A, V, rho): 

r'''Calculates drag coefficient `Cd` for a given drag force `F`, 

projected area `A`, characteristic velocity `V`, and density `rho`. 

 

.. math:: 

C_D = \frac{F_d}{A\cdot\frac{1}{2}\rho V^2} 

 

Parameters 

---------- 

F : float 

Drag force, [N] 

A : float 

Projected area, [m^2] 

V : float 

Characteristic velocity, [m/s] 

rho : float 

Density, [kg/m^3] 

 

Returns 

------- 

Cd : float 

Drag coefficient, [-] 

 

Notes 

----- 

Used in flow around objects, or objects flowing within a fluid. 

 

.. math:: 

C_D = \frac{\text{Drag forces}}{\text{Projected area}\cdot 

\text{Velocity head}} 

 

Examples 

-------- 

>>> Drag(1000, 0.0001, 5, 2000) 

400.0 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return F/(A*rho*V**2/2.) 

 

 

def Capillary(V, mu, sigma): 

r'''Calculates Capillary number `Ca` for a characteristic velocity `V`, 

viscosity `mu`, and surface tension `sigma`. 

 

.. math:: 

Ca = \frac{V \mu}{\sigma} 

 

Parameters 

---------- 

V : float 

Characteristic velocity, [m/s] 

mu : float 

Dynamic viscosity, [Pa*s] 

sigma : float 

Surface tension, [N/m] 

 

Returns 

------- 

Ca : float 

Capillary number, [-] 

 

Notes 

----- 

Used in porous media calculations and film flow calculations. 

Surface tension may gas-liquid, or liquid-liquid. 

 

.. math:: 

Ca = \frac{\text{Viscous forces}} 

{\text{Surface forces}} 

 

Examples 

-------- 

>>> Capillary(1.2, 0.01, .1) 

0.12 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Kundu, Pijush K., Ira M. Cohen, and David R. Dowling. Fluid 

Mechanics. Academic Press, 2012. 

''' 

return V*mu/sigma 

 

 

def Archimedes(L, rhof, rhop, mu, g=g): 

r'''Calculates Archimedes number, `Ar`, for a fluid and particle with the 

given densities, characteristic length, viscosity, and gravity 

(usually diameter of particle). 

 

.. math:: 

Ar = \frac{L^3 \rho_f(\rho_p-\rho_f)g}{\mu^2} 

 

Parameters 

---------- 

L : float 

Characteristic length, typically particle diameter [m] 

rhof : float 

Density of fluid, [kg/m^3] 

rhop : float 

Density of particle, [kg/m^3] 

mu : float 

Viscosity of fluid, [N/m] 

g : float, optional 

Acceleration due to gravity, [m/s^2] 

 

Returns 

------- 

Ar : float 

Archimedes number [] 

 

Notes 

----- 

Used in fluid-particle interaction calculations. 

 

.. math:: 

Ar = \frac{\text{Gravitational force}}{\text{Viscous force}} 

 

Examples 

-------- 

>>> Archimedes(0.002, 2., 3000, 1E-3) 

470.4053872 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return L**3*rhof*(rhop-rhof)*g/mu**2 

 

 

def Ohnesorge(L, rho, mu, sigma): 

r'''Calculates Ohnesorge number, `Oh`, for a fluid with the given 

characteristic length, density, viscosity, and surface tension. 

 

.. math:: 

\text{Oh} = \frac{\mu}{\sqrt{\rho \sigma L }} 

 

Parameters 

---------- 

L : float 

Characteristic length [m] 

rho : float 

Density of fluid, [kg/m^3] 

mu : float 

Viscosity of fluid, [Pa*s] 

sigma : float 

Surface tension, [N/m] 

 

Returns 

------- 

Oh : float 

Ohnesorge number [] 

 

Notes 

----- 

Often used in spray calculations. Sometimes given the symbol Z. 

 

.. math:: 

Oh = \frac{\sqrt{\text{We}}}{\text{Re}}= \frac{\text{viscous forces}} 

{\sqrt{\text{Inertia}\cdot\text{Surface tension}} } 

 

Examples 

-------- 

>>> Ohnesorge(1E-4, 1000., 1E-3, 1E-1) 

0.01 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

''' 

return mu/(L*rho*sigma)**0.5 

 

 

def Suratman(L, rho, mu, sigma): 

r'''Calculates Suratman number, `Su`, for a fluid with the given 

characteristic length, density, viscosity, and surface tension. 

 

.. math:: 

\text{Su} = \frac{\rho\sigma L}{\mu^2} 

 

Parameters 

---------- 

L : float 

Characteristic length [m] 

rho : float 

Density of fluid, [kg/m^3] 

mu : float 

Viscosity of fluid, [Pa*s] 

sigma : float 

Surface tension, [N/m] 

 

Returns 

------- 

Su : float 

Suratman number [] 

 

Notes 

----- 

Also known as Laplace number. Used in two-phase flow, especially the  

bubbly-slug regime. No confusion regarding the definition of this group  

has been observed. 

 

.. math:: 

\text{Su} = \frac{\text{Re}^2}{\text{We}} =\frac{\text{Inertia}\cdot 

\text{Surface tension} }{\text{(viscous forces)}^2} 

 

The oldest reference to this group found by the author is in 1963, from  

[2]_. 

 

Examples 

-------- 

>>> Suratman(1E-4, 1000., 1E-3, 1E-1) 

10000.0 

 

References 

---------- 

.. [1] Sen, Nilava. "Suratman Number in Bubble-to-Slug Flow Pattern  

Transition under Microgravity." Acta Astronautica 65, no. 3-4 (August  

2009): 423-28. doi:10.1016/j.actaastro.2009.02.013. 

.. [2] Catchpole, John P., and George. Fulford. "DIMENSIONLESS GROUPS."  

Industrial & Engineering Chemistry 58, no. 3 (March 1, 1966): 46-60.  

doi:10.1021/ie50675a012. 

''' 

return rho*sigma*L/(mu*mu) 

 

 

def Bejan_L(dP, L, mu, alpha): 

r'''Calculates Bejan number of a length or `Be_L` for a fluid with the 

given parameters flowing over a characteristic length `L` and experiencing 

a pressure drop `dP`. 

 

.. math:: 

Be_L = \frac{\Delta P L^2}{\mu \alpha} 

 

Parameters 

---------- 

dP : float 

Pressure drop, [Pa] 

L : float 

Characteristic length, [m] 

mu : float, optional 

Dynamic viscosity, [Pa*s] 

alpha : float 

Thermal diffusivity, [m^2/s] 

 

Returns 

------- 

Be_L : float 

Bejan number with respect to length [] 

 

Notes 

----- 

Termed a dimensionless number by someone in 1988. 

 

Examples 

-------- 

>>> Bejan_L(1E4, 1, 1E-3, 1E-6) 

10000000000000.0 

 

References 

---------- 

.. [1] Awad, M. M. "The Science and the History of the Two Bejan Numbers." 

International Journal of Heat and Mass Transfer 94 (March 2016): 101-3. 

doi:10.1016/j.ijheatmasstransfer.2015.11.073. 

.. [2] Bejan, Adrian. Convection Heat Transfer. 4E. Hoboken, New Jersey: 

Wiley, 2013. 

''' 

return dP*L**2/(alpha*mu) 

 

 

def Bejan_p(dP, K, mu, alpha): 

r'''Calculates Bejan number of a permeability or `Be_p` for a fluid with 

the given parameters and a permeability `K` experiencing a pressure drop 

`dP`. 

 

.. math:: 

Be_p = \frac{\Delta P K}{\mu \alpha} 

 

Parameters 

---------- 

dP : float 

Pressure drop, [Pa] 

K : float 

Permeability, [m^2] 

mu : float, optional 

Dynamic viscosity, [Pa*s] 

alpha : float 

Thermal diffusivity, [m^2/s] 

 

Returns 

------- 

Be_p : float 

Bejan number with respect to pore characteristics [] 

 

Notes 

----- 

Termed a dimensionless number by someone in 1988. 

 

Examples 

-------- 

>>> Bejan_p(1E4, 1, 1E-3, 1E-6) 

10000000000000.0 

 

References 

---------- 

.. [1] Awad, M. M. "The Science and the History of the Two Bejan Numbers." 

International Journal of Heat and Mass Transfer 94 (March 2016): 101-3. 

doi:10.1016/j.ijheatmasstransfer.2015.11.073. 

.. [2] Bejan, Adrian. Convection Heat Transfer. 4E. Hoboken, New Jersey: 

Wiley, 2013. 

''' 

return dP*K/(alpha*mu) 

 

 

def Boiling(G, q, Hvap): 

r'''Calculates Boiling number or `Bg` using heat flux, two-phase mass flux, 

and heat of vaporization of the fluid flowing. Used in two-phase heat  

transfer calculations. 

 

.. math:: 

\text{Bg} = \frac{q}{G_{tp} \Delta H_{vap}} 

 

Parameters 

---------- 

G : float 

Two-phase mass flux in a channel (combined liquid and vapor) [kg/m^2/s] 

q : float 

Heat flux [W/m^2] 

Hvap : float 

Heat of vaporization of the fluid [J/kg] 

 

Returns 

------- 

Bg : float 

Boiling number [-] 

 

Notes 

----- 

Most often uses the symbol `Bo` instead of `Bg`, but this conflicts with 

Bond number. 

 

.. math:: 

\text{Bg} = \frac{\text{mass liquid evaporated / area heat transfer  

surface}}{\text{mass flow rate fluid / flow cross sectional area}} 

 

First defined in [4]_, though not named. 

 

Examples 

-------- 

>>> Boiling(300, 3000, 800000) 

1.25e-05 

 

References 

---------- 

.. [1] Winterton, Richard H.S. BOILING NUMBER. Thermopedia. Hemisphere,  

2011. 10.1615/AtoZ.b.boiling_number 

.. [2] Collier, John G., and John R. Thome. Convective Boiling and  

Condensation. 3rd edition. Clarendon Press, 1996. 

.. [3] Stephan, Karl. Heat Transfer in Condensation and Boiling. Translated 

by C. V. Green.. 1992 edition. Berlin; New York: Springer, 2013. 

.. [4] W. F. Davidson, P. H. Hardie, C. G. R. Humphreys, A. A. Markson,  

A. R. Mumford and T. Ravese "Studies of heat transmission through boiler 

tubing at pressures from 500 to 3300 pounds" Trans. ASME, Vol. 65, 9,  

February 1943, pp. 553-591.  

''' 

return q/(G*Hvap) 

 

 

def Dean(Re, Di, D): 

r'''Calculates Dean number, `De`, for a fluid with the Reynolds number `Re`, 

inner diameter `Di`, and a secondary diameter `D`. `D` may be the  

diameter of curvature, the diameter of a spiral, or some other dimension. 

 

.. math:: 

\text{De} = \sqrt{\frac{D_i}{D}} \text{Re} = \sqrt{\frac{D_i}{D}}  

\frac{\rho v D}{\mu} 

 

Parameters 

---------- 

Re : float 

Reynolds number [] 

Di : float 

Inner diameter [] 

D : float 

Diameter of curvature or outer spiral or other dimension [] 

 

Returns 

------- 

De : float 

Dean number [-] 

 

Notes 

----- 

Used in flow in curved geometry. 

 

.. math:: 

\text{De} = \frac{\sqrt{\text{centripetal forces}\cdot  

\text{inertial forces}}}{\text{viscous forces}} 

 

Examples 

-------- 

>>> Dean(10000, 0.1, 0.4) 

5000.0 

 

References 

---------- 

.. [1] Catchpole, John P., and George. Fulford. "DIMENSIONLESS GROUPS."  

Industrial & Engineering Chemistry 58, no. 3 (March 1, 1966): 46-60.  

doi:10.1021/ie50675a012. 

''' 

return (Di/D)**0.5*Re 

 

 

def relative_roughness(D, roughness=1.52e-06): 

r'''Calculates relative roughness `eD` using a diameter and the roughness 

of the material of the wall. Default roughness is that of steel. 

 

.. math:: 

eD=\frac{\epsilon}{D} 

 

Parameters 

---------- 

D : float 

Diameter of pipe, [m] 

roughness : float, optional 

Roughness of pipe wall [m] 

 

Returns 

------- 

eD : float 

Relative Roughness, [-] 

 

Examples 

-------- 

>>> relative_roughness(0.5, 1E-4) 

0.0002 

 

References 

---------- 

.. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook, 

Eighth Edition. McGraw-Hill Professional, 2007. 

.. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

return roughness/D 

 

 

### Misc utilities 

 

def nu_mu_converter(rho, mu=None, nu=None): 

r'''Calculates either kinematic or dynamic viscosity, depending on inputs. 

Used when one type of viscosity is known as well as density, to obtain 

the other type. Raises an error if both types of viscosity or neither type 

of viscosity is provided. 

 

.. math:: 

\nu = \frac{\mu}{\rho} 

 

\mu = \nu\rho 

 

Parameters 

---------- 

rho : float 

Density, [kg/m^3] 

mu : float, optional 

Dynamic viscosity, [Pa*s] 

nu : float, optional 

Kinematic viscosity, [m^2/s] 

 

Returns 

------- 

mu or nu : float 

Dynamic viscosity, [Pa*s] or Kinematic viscosity, [m^2/s] 

 

Examples 

-------- 

>>> nu_mu_converter(998., nu=1.0E-6) 

0.000998 

 

References 

---------- 

.. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and 

Applications. Boston: McGraw Hill Higher Education, 2006. 

''' 

if (nu and mu) or not rho or (not nu and not mu): 

raise Exception('Inputs must be rho and one of mu and nu.') 

if mu: 

return mu/rho 

elif nu: 

return nu*rho 

 

 

def gravity(latitude, H): 

r'''Calculates local acceleration due to gravity `g` according to [1]_. 

Uses latitude and height to calculate `g`. 

 

.. math:: 

g = 9.780356(1 + 0.0052885\sin^2\phi - 0.0000059^22\phi) 

- 3.086\times 10^{-6} H 

 

Parameters 

---------- 

latitude : float 

Degrees, [degrees] 

H : float 

Height above earth's surface [m] 

 

Returns 

------- 

g : float 

Acceleration due to gravity, [m/s^2] 

 

Notes 

----- 

Better models, such as EGM2008 exist. 

 

Examples 

-------- 

>>> gravity(55, 1E4) 

9.784151976863571 

 

References 

---------- 

.. [1] Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of 

Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014. 

''' 

lat = latitude*pi/180 

g = 9.780356*(1+0.0052885*sin(lat)**2 -0.0000059*sin(2*lat)**2)-3.086E-6*H 

return g 

 

### Friction loss conversion functions 

 

def K_from_f(fd, L, D): 

r'''Calculates loss coefficient, K, for a given section of pipe 

at a specified friction factor. 

 

.. math:: 

K = f_dL/D 

 

Parameters 

---------- 

fd : float 

friction factor of pipe, [] 

L : float 

Length of pipe, [m] 

D : float 

Inner diameter of pipe, [m] 

 

Returns 

------- 

K : float 

Loss coefficient, [] 

 

Notes 

----- 

For fittings with a specified L/D ratio, use D = 1 and set L to 

specified L/D ratio. 

 

Examples 

-------- 

>>> K_from_f(fd=0.018, L=100., D=.3) 

6.0 

''' 

return fd*L/D 

 

 

def K_from_L_equiv(L_D, fd=0.015): 

r'''Calculates loss coefficient, for a given equivalent length (L/D). 

 

.. math:: 

K = f_d \frac{L}{D} 

 

Parameters 

---------- 

L_D : float 

Length over diameter, [] 

fd : float, optional 

Darcy friction factor, [-] 

 

Returns 

------- 

K : float 

Loss coefficient, [] 

 

Notes 

----- 

Almost identical to `K_from_f`, but with a default friction factor for 

fully turbulent flow in steel pipes. 

 

Examples 

-------- 

>>> K_from_L_equiv(240.) 

3.5999999999999996 

''' 

return fd*L_D 

 

 

def L_equiv_from_K(K, fd=0.015): 

r'''Calculates equivalent length of pipe (L/D), for a given loss  

coefficient. 

 

.. math:: 

\frac{L}{D} = \frac{K}{f_d} 

 

Parameters 

---------- 

K : float 

Loss coefficient, [] 

fd : float, optional 

Darcy friction factor, [-] 

 

Returns 

------- 

L_D : float 

Length over diameter, [] 

 

Notes 

----- 

Assumes a default friction factor for fully turbulent flow in steel pipes. 

 

Examples 

-------- 

>>> L_equiv_from_K(3.6) 

240.00000000000003 

''' 

return K/fd 

 

 

def dP_from_K(K, rho, V): 

r'''Calculates pressure drop, for a given loss coefficient, 

at a specified density and velocity. 

 

.. math:: 

dP = 0.5K\rho V^2 

 

Parameters 

---------- 

K : float 

Loss coefficient, [] 

rho : float 

Density of fluid, [kg/m^3] 

V : float 

Velocity of fluid in pipe, [m/s] 

 

Returns 

------- 

dP : float 

Pressure drop, [Pa] 

 

Notes 

----- 

Loss ciefficient `K` is usually the sum of several factors, including 

the friction factor. 

 

Examples 

-------- 

>>> dP_from_K(K=10, rho=1000, V=3) 

45000.0 

''' 

return K*0.5*rho*V*V 

 

 

def head_from_K(K, V, g=g): 

r'''Calculates head loss, for a given loss coefficient, 

at a specified velocity. 

 

.. math:: 

\text{head} = \frac{K V^2}{2g} 

 

Parameters 

---------- 

K : float 

Loss coefficient, [] 

V : float 

Velocity of fluid in pipe, [m/s] 

g : float, optional 

Acceleration due to gravity, [m/s^2] 

 

Returns 

------- 

head : float 

Head loss, [m] 

 

Notes 

----- 

Loss ciefficient `K` is usually the sum of several factors, including 

the friction factor. 

 

Examples 

-------- 

>>> head_from_K(K=10, V=1.5) 

1.1471807396001694 

''' 

return K*0.5*V*V/g 

 

 

def head_from_P(P, rho, g=g): 

r'''Calculates head for a fluid of specified density at specified 

pressure. 

 

.. math:: 

\text{head} = {P\over{\rho g}} 

 

Parameters 

---------- 

P : float 

Pressure fluid in pipe, [Pa] 

rho : float 

Density of fluid, [kg/m^3] 

g : float, optional 

Acceleration due to gravity, [m/s^2] 

 

Returns 

------- 

head : float 

Head, [m] 

 

Notes 

----- 

By definition. Head varies with location, inversely propertional to the 

increase in gravitational constant. 

 

Examples 

-------- 

>>> head_from_P(P=98066.5, rho=1000) 

10.000000000000002 

''' 

return P/rho/g 

 

 

def P_from_head(head, rho, g=g): 

r'''Calculates head for a fluid of specified density at specified 

pressure. 

 

.. math:: 

P = \rho g \cdot \text{head} 

 

Parameters 

---------- 

head : float 

Head, [m] 

rho : float 

Density of fluid, [kg/m^3] 

g : float, optional 

Acceleration due to gravity, [m/s^2] 

 

Returns 

------- 

P : float 

Pressure fluid in pipe, [Pa] 

 

Notes 

----- 

 

Examples 

-------- 

>>> P_from_head(head=5., rho=800.) 

39226.6 

''' 

return head*rho*g 

 

 

 

### Synonyms 

alpha = thermal_diffusivity # synonym for thermal diffusivity 

Pr = Prandtl # Synonym