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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 exp, pi 

 

__all__ = ['dP_packed_bed', 'Ergun', 'Kuo_Nydegger', 'Jones_Krier', 'Carman', 'Hicks', 

'Brauer', 'KTA', 'Erdim_Akgiray_Demir', 'Fahien_Schriver', 'Idelchik', 

'Harrison_Brunner_Hecker', 'Montillet_Akkari_Comiti', 

'voidage_Benyahia_Oneil', 

'voidage_Benyahia_Oneil_spherical', 'voidage_Benyahia_Oneil_cylindrical'] 

 

 

 

def Ergun(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, as shown in [2]_ and [3]_. Eighteenth most 

accurate correlation overall in the review of [2]_. 

 

Most often presented in the following form: 

 

.. math:: 

\Delta P = \frac{150\mu (1-\epsilon)^2 v_s L}{\epsilon^3 d_p^2} 

+ \frac{1.75 (1-\epsilon) \rho v_s^2 L}{\epsilon^3 d_p} 

 

It is also often presented with a term for sphericity, which is multiplied 

by particle diameter everywhere in the equation. However, this is highly 

emperical and better correlations for beds of differently-shaped particles 

exist. To use sphericity in this model, multiple the input particle 

diameter by the spericity separately. 

 

In the review of [2]_, it is expressed in terms of a parameter `fp`, shown 

below. This is a convenient means of expressing all forms of pressure drop 

in packed beds correlations in a way that allows for easy comparison. 

 

.. math:: 

f_p = \left(150 + 1.75\left(\frac{Re}{1-\epsilon}\right)\right) 

\frac{(1-\epsilon)^2}{\epsilon^3 Re} 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re = \frac{\rho v_s d_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

The first term in this equation represents laminar loses, and the second, 

turbulent loses. Developed with data from spheres, sand, and pulverized 

coke. Fluids tested were carbon dioxide, nitrogen, methane, and hydrogen. 

 

Validity range shown in [3]_ is :math:`1 < Re_{Erg} < 2300`. 

Overpredicts pressure drop for :math:`Re_{Erg} > 700`. 

 

Examples 

-------- 

>>> Ergun(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1338.8671874999995 

 

References 

---------- 

.. [1] Ergun, S. (1952) "Fluid flow through packed columns", 

Chem. Eng. Prog., 48, 89-94. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

.. [3] Jones, D. P., and H. Krier. "Gas Flow Resistance Measurements 

Through Packed Beds at High Reynolds Numbers." Journal of Fluids 

Engineering 105, no. 2 (June 1, 1983): 168-172. doi:10.1115/1.3240959. 

''' 

Re = dp*rho*vs/mu 

fp = (150 + 1.75*(Re/(1-voidage)))*(1-voidage)**2/(voidage**3*Re) 

return fp*rho*vs**2*L/dp 

 

 

def Kuo_Nydegger(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, as shown in [2]_ and [3]. Thirty-eighth most 

accurate correlation overall in the review of [2]_. 

 

.. math:: 

f_p = \left(276.23 + 5.05\left(\frac{Re}{1-\epsilon}\right)^{0.87} 

\right)\frac{(1-\epsilon)^2}{\epsilon^3 Re} 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re = \frac{\rho v_s d_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

Validity range shown in [2]_ as for a range of 

:math:`460 < Re < 14600`. 

:math:`0.3760 < \epsilon < 0.3901`. 

Developed with data from rough granular ball propellants beds, with air. 

 

Examples 

-------- 

>>> Kuo_Nydegger(dp=8E-1, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

0.025651460973648624 

 

References 

---------- 

.. [1] Kuo, K. K. and Nydegger, C., "Flow Resistance Measurement and 

Correlation in Packed Beds of WC 870 Ball Propellants," Journal of 

Ballistics , Vol. 2, No. 1, pp. 1-26, 1978. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

.. [3] Jones, D. P., and H. Krier. "Gas Flow Resistance Measurements 

Through Packed Beds at High Reynolds Numbers." Journal of Fluids 

Engineering 105, no. 2 (June 1, 1983): 168-172. doi:10.1115/1.3240959. 

''' 

Re = dp*rho*vs/mu 

fp = (276.23 + 5.05*(Re/(1-voidage))**0.87)*(1-voidage)**2/(voidage**3*Re) 

return fp*rho*vs**2*L/dp 

 

 

def Jones_Krier(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, also shown in [2]_. Tenth most accurate 

correlation overall in the review of [2]_. 

 

.. math:: 

f_p = \left(150 + 3.89\left(\frac{Re}{1-\epsilon}\right)^{0.87}\right) 

\frac{(1-\epsilon)^2}{\epsilon^3 Re} 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re = \frac{\rho v_s d_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

Validity range shown in [1]_ as for a range of 

:math:`733 < Re < 126,670`. 

:math:`0.3804 < \epsilon < 0.4304`. 

Developed from data of spherical glass beads. 

 

Examples 

-------- 

>>> Jones_Krier(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1362.2719449873746 

 

References 

---------- 

.. [1] Jones, D. P., and H. Krier. "Gas Flow Resistance Measurements 

Through Packed Beds at High Reynolds Numbers." Journal of Fluids 

Engineering 105, no. 2 (June 1, 1983): 168-172. doi:10.1115/1.3240959. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

''' 

Re = dp*rho*vs/mu 

fp = (150 + 3.89*(Re/(1-voidage))**0.87)*(1-voidage)**2/(voidage**3*Re) 

return fp*rho*vs**2*L/dp 

 

 

def Carman(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, as shown in [2]_. Fifth most accurate 

correlation overall in the review of [2]_. Also shown in [3]_. 

 

.. math:: 

f_p = \left(180 + 2.871\left(\frac{Re}{1-\epsilon}\right)^{0.9}\right) 

\frac{(1-\epsilon)^2}{\epsilon^3 Re} 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re = \frac{\rho v_s d_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

Valid in [1]_, [2]_, and [3]_ for a range of 

:math:`300 < Re_{Erg} < 60,000`. 

 

Examples 

-------- 

>>> Carman(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1614.721678121775 

 

References 

---------- 

.. [1] P.C. Carman, Fluid flow through granular beds, Transactions of the 

London Institute of Chemical Engineers 15 (1937) 150-166. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

.. [2] Allen, K. G., T. W. von Backstrom, and D. G. Kroger. "Packed Bed 

Pressure Drop Dependence on Particle Shape, Size Distribution, Packing 

Arrangement and Roughness." Powder Technology 246 (September 2013): 

590-600. doi:10.1016/j.powtec.2013.06.022. 

''' 

Re = dp*rho*vs/mu 

fp = (180 + 2.871*(Re/(1-voidage))**0.9)*(1-voidage)**2/(voidage**3*Re) 

return fp*rho*vs**2*L/dp 

 

 

def Hicks(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, as shown in [2]_. Twenty-third most accurate 

correlation overall in the review of [2]_. Also shown in [3]_. 

 

.. math:: 

f_p = 6.8 \frac{(1-\epsilon)^{1.2}}{Re^{0.2}\epsilon^3} 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re = \frac{\rho v_s d_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

Valid in [1]_, [2]_, and [3]_ for a range of 

:math:`300 < Re_{Erg} < 60,000`. 

 

Examples 

-------- 

>>> Hicks(dp=0.01, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

3.631703956680737 

 

References 

---------- 

.. [1] Hicks, R. E. "Pressure Drop in Packed Beds of Spheres." Industrial 

Engineering Chemistry Fundamentals 9, no. 3 (August 1, 1970): 500-502. 

doi:10.1021/i160035a032. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

.. [2] Allen, K. G., T. W. von Backstrom, and D. G. Kroger. "Packed Bed 

Pressure Drop Dependence on Particle Shape, Size Distribution, Packing 

Arrangement and Roughness." Powder Technology 246 (September 2013): 

590-600. doi:10.1016/j.powtec.2013.06.022. 

''' 

Re = dp*rho*vs/mu 

fp = 6.8*(1-voidage)**1.2/Re**0.2/voidage**3 

return fp*rho*vs**2*L/dp 

 

 

def Brauer(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, as shown in [2]_. Seventh most accurate 

correlation overall in the review of [2]_. Also shown in [3]_. 

 

.. math:: 

f_p = \left(160 + 3\left(\frac{Re}{1-\epsilon}\right)^{0.9}\right) 

\frac{(1-\epsilon)^2}{\epsilon^3 Re} 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re = \frac{\rho v_s d_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

Original has not been reviewed. 

In [2]_, is stated as for a range of :math:`2 < Re_{Erg} < 20,000`. 

In [3]_, is stated as for a range of :math:`0.01 < Re_{Erg} < 40,000`. 

 

Examples 

-------- 

>>> Brauer(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1441.5479196020563 

 

References 

---------- 

.. [1] H. Brauer, Grundlagen der Einphasen -und Mehrphasenstromungen, 

Sauerlander AG, Aarau, 1971. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

.. [2] Allen, K. G., T. W. von Backstrom, and D. G. Kroger. "Packed Bed 

Pressure Drop Dependence on Particle Shape, Size Distribution, Packing 

Arrangement and Roughness." Powder Technology 246 (September 2013): 

590-600. doi:10.1016/j.powtec.2013.06.022. 

''' 

Re = dp*rho*vs/mu 

fp = (160 + 3.1*(Re/(1-voidage))**0.9)*(1-voidage)**2/(voidage**3*Re) 

return fp*rho*vs**2*L/dp 

 

 

def KTA(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, as shown in [2]_. Third most accurate 

correlation overall in the review of [2]_. 

 

.. math:: 

f_p = \left(160 + 3\left(\frac{Re}{1-\epsilon}\right)^{0.9}\right) 

\frac{(1-\epsilon)^2}{\epsilon^3 Re} 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re= \frac{\rho v_s d_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

Developed for gas flow through pebbles. 

In [2]_, stated as for a range of :math:`1 < RE_{Erg} <100,000`. 

In [1]_, a limit on porosity is stated as :math:`0.36 < \epsilon < 0.42`. 

 

 

Examples 

-------- 

>>> KTA(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1440.409277034248 

 

References 

---------- 

.. [1] KTA. KTA 3102.3 Reactor Core Design of High-Temperature Gas-Cooled 

Reactors Part 3: Loss of Pressure through Friction in Pebble Bed Cores. 

Germany, 1981. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

''' 

Re = dp*rho*vs/mu 

fp = (160 + 3*(Re/(1-voidage))**0.9)*(1-voidage)**2/(voidage**3*Re) 

return fp*rho*vs**2*L/dp 

 

 

def Erdim_Akgiray_Demir(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, claiming to be the best model to date. 

 

.. math:: 

f_v = 160 + 2.81Re_{Erg}^{0.904} 

 

f_v = \frac{\Delta P d_p^2}{\mu v_s L}\frac{\epsilon^3}{(1-\epsilon)^2} 

 

Re_{Erg} = \frac{\rho v_s d_p}{\mu(1-\epsilon)} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

Developed with data in the range of: 

 

.. math:: 

2 < Re_{Erg} <3582\\ 

4 < d_t/d_p < 34.1\\ 

0.377 < \epsilon <0.470 

 

Examples 

-------- 

>>> Erdim_Akgiray_Demir(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1438.2826958844414 

 

References 

---------- 

.. [1] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

''' 

Rem = dp*rho*vs/mu/(1-voidage) 

fv = 160 + 2.81*Rem**0.904 

return fv*(mu*vs*L/dp**2)*(1-voidage)**2/voidage**3 

 

 

def Fahien_Schriver(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, as shown in [2]_. Second most accurate 

correlation overall in the review of [2]_. 

 

.. math:: 

f_p = \left(q\frac{f_{1L}}{Re_{Erg}} + (1-q)\left(f_2 + \frac{f_{1T}} 

{Re_{Erg}}\right)\right)\frac{1-\epsilon}{\epsilon^3} 

 

q = \exp\left(-\frac{\epsilon^2(1-\epsilon)}{12.6}Re_{Erg}\right) 

 

f_{1L}=\frac{136}{(1-\epsilon)^{0.38}} 

 

f_{1T} = \frac{29}{(1-\epsilon)^{1.45}\epsilon^2} 

 

f_2 = \frac{1.87\epsilon^{0.75}}{(1-\epsilon)^{0.26}} 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re_{Erg} = \frac{\rho v_s d_p}{\mu(1-\epsilon)} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

No range of validity available. 

 

Examples 

-------- 

>>> Fahien_Schriver(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1470.6175541844711 

 

References 

---------- 

.. [1] R.W. Fahien, C.B. Schriver, Paper presented at the 1961 Denver 

meeting of AIChE, in: R.W. Fahien, Fundamentals of Transport Phenomena, 

McGraw-Hill, New York, 1983. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

''' 

Rem = dp*rho*vs/mu/(1-voidage) 

q = exp(-voidage**2*(1-voidage)/12.6*Rem) 

f1L = 136/(1-voidage)**0.38 

f1T = 29/((1-voidage)**1.45*voidage**2) 

f2 = 1.87*voidage**0.75/(1-voidage)**0.26 

fp = (q*f1L/Rem + (1-q)*(f2 + f1T/Rem))*(1-voidage)/voidage**3 

return fp*rho*vs**2*L/dp 

 

 

def Idelchik(dp, voidage, vs, rho, mu, L=1): 

r'''Calculates pressure drop across a packed bed of spheres as in [2]_, 

originally in [1]_. 

 

.. math:: 

\frac{\Delta P}{L\rho v_s^2} d_p = \frac{0.765}{\epsilon^{4.2}} 

\left(\frac{30}{Re_l} + \frac{3}{Re_l^{0.7}} + 0.3\right) 

 

Re_l = (0.45/\epsilon^{0.5})Re_{Erg} 

 

Re_{Erg} = \frac{\rho v_s D_p}{\mu(1-\epsilon)} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

:math:`0.001 < Re_{Erg} <1000` 

This equation is valid for void fractions between 0.3 and 0.8. Cited as 

by Bernshtein. 

 

Examples 

-------- 

>>> Idelchik(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1571.909125999067 

 

References 

---------- 

.. [1] Idelchik, I. E. Flow Resistance: A Design Guide for Engineers. 

Hemisphere Publishing Corporation, New York, 1989. 

.. [2] Allen, K. G., T. W. von Backstrom, and D. G. Kroger. "Packed Bed 

Pressure Drop Dependence on Particle Shape, Size Distribution, Packing 

Arrangement and Roughness." Powder Technology 246 (September 2013): 

590-600. doi:10.1016/j.powtec.2013.06.022. 

''' 

Re = rho*vs*dp/mu/(1-voidage) 

Re = (0.45/voidage**0.5)*Re 

right = 0.765/voidage**4.2*(30./Re + 3./Re**0.7 + 0.3) 

left = dp/L/rho/vs**2 

return right/left 

 

 

def Harrison_Brunner_Hecker(dp, voidage, vs, rho, mu, L=1, Dt=None): 

r'''Calculates pressure drop across a packed bed of spheres using a 

correlation developed in [1]_, also shown in [2]_. Fourth most accurate 

correlation overall in the review of [2]_. 

Applies a wall correction if diameter of tube is provided. 

 

.. math:: 

f_p = \left(119.8A + 4.63B\left(\frac{Re}{1-\epsilon}\right)^{5/6} 

\right)\frac{(1-\epsilon)^2}{\epsilon^3 Re} 

 

A = \left(1 + \pi \frac{d_p}{6(1-\epsilon)D_t}\right)^2 

 

B = 1 - \frac{\pi^2 d_p}{24D_t}\left(1 - \frac{0.5d_p}{D_t}\right) 

 

f_p = \frac{\Delta P d_p}{\rho v_s^2 L} 

 

Re = \frac{\rho v_s d_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

Dt : float, optional 

Diameter of the tube, [m] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

 

Notes 

----- 

Uses data from other sources only. Correlation will underestimate pressure 

drop if tube diameter is not provided. Limits are specified in [1]_ as: 

.. math:: 

0.72 < Re < 7700 \\ 

8.3 < d_t/d_p < 50 \\ 

0.33 < \epsilon < 0.88 

 

Examples 

-------- 

>>> Harrison_Brunner_Hecker(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3, Dt=1E-2) 

1255.1625662548427 

 

References 

---------- 

.. [1] KTA. KTA 3102.3 Reactor Core Design of High-Temperature Gas-Cooled 

Reactors Part 3: Loss of Pressure through Friction in Pebble Bed Cores. 

Germany, 1981. 

.. [2] Erdim, Esra, Ömer Akgiray, and İbrahim Demir. "A Revisit of Pressure 

Drop-Flow Rate Correlations for Packed Beds of Spheres." Powder 

Technology 283 (October 2015): 488-504. doi:10.1016/j.powtec.2015.06.017. 

''' 

Re = dp*rho*vs/mu 

if not Dt: 

A, B = 1, 1 

else: 

A = (1 + pi*dp/(6*(1-voidage)*Dt))**2 

B = 1 - pi**2*dp/24/Dt*(1 - dp/(2*Dt)) 

fp = (119.8*A + 4.63*B*(Re/(1-voidage))**(5/6.))*(1-voidage)**2/(voidage**3*Re) 

return fp*rho*vs**2*L/dp 

 

 

def Montillet_Akkari_Comiti(dp, voidage, vs, rho, mu, L=1, Dt=None): 

r'''Calculates pressure drop across a packed bed of spheres as in [2]_, 

originally in [1]_. Wall effect adjustment is used if `Dt` is provided. 

 

.. math:: 

\frac{\Delta P}{L\rho V_s^2} D_p \frac{\epsilon^3}{(1-\epsilon)} 

= a\left(\frac{D_c}{D_p}\right)^{0.20} 

\left(\frac{1000}{Re_{p}} + \frac{60}{Re_{p}^{0.5}} + 12 \right) 

 

Re_{p} = \frac{\rho v_s D_p}{\mu} 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

Dt : float, optional 

Diameter of the tube, [m] 

 

Returns 

------- 

dP : float 

Pressure drop across bed [Pa] 

 

Notes 

----- 

:math:`10 < REp <2500` 

if Dc/D > 50, set to 2.2. 

a = 0.061 for epsilon < 0.4, 0.050 for > 0.4. 

 

Examples 

-------- 

Custom example: 

 

>>> Montillet_Akkari_Comiti(dp=0.0008, voidage=0.4, L=0.5, vs=0.00132629120, rho=1000., mu=1.00E-003) 

1148.1905244077548 

 

References 

---------- 

.. [1] Montillet, A., E. Akkari, and J. Comiti. "About a Correlating 

Equation for Predicting Pressure Drops through Packed Beds of Spheres 

in a Large Range of Reynolds Numbers." Chemical Engineering and 

Processing: Process Intensification 46, no. 4 (April 2007): 329-33. 

doi:10.1016/j.cep.2006.07.002. 

.. [2] Allen, K. G., T. W. von Backstrom, and D. G. Kroger. "Packed Bed 

Pressure Drop Dependence on Particle Shape, Size Distribution, Packing 

Arrangement and Roughness." Powder Technology 246 (September 2013): 

590-600. doi:10.1016/j.powtec.2013.06.022. 

''' 

Re = rho*vs*dp/mu 

if voidage < 0.4: 

a = 0.061 

else: 

a = 0.05 

if not Dt or Dt/dp > 50: 

Dterm = 2.2 

else: 

Dterm = (Dt/dp)**0.2 

right = a*Dterm*(1000./Re + 60/Re**0.5 + 12) 

left = dp/L/rho/vs**2*voidage**3/(1-voidage) 

return right/left 

 

 

 

 

# Format: Nice nane : (formula, uses_dt) 

packed_beds_correlations = { 

'Ergun': (Ergun, False), 

'Kuo & Nydegger': (Kuo_Nydegger, False), 

'Jones & Krier': (Jones_Krier, False), 

'Carman': (Carman, False), 

'Hicks': (Hicks, False), 

'Brauer': (Brauer, False), 

'KTA': (KTA, False), 

'Erdim, Akgiray & Demir': (Erdim_Akgiray_Demir, False), 

'Fahien & Schriver': (Fahien_Schriver, False), 

'Idelchik': (Idelchik, False), 

'Harrison, Brunner & Hecker': (Harrison_Brunner_Hecker, True), 

'Montillet, Akkari & Comiti': (Montillet_Akkari_Comiti, True) 

} 

 

def dP_packed_bed(dp, voidage, vs, rho, mu, L=1, Dt=None, sphericity=None, 

AvailableMethods=False, Method=None): 

r'''This function handles choosing which pressure drop in a packed bed 

correlation is used. Automatically select which correlation 

to use if none is provided. Returns None if insufficient information is 

provided. 

 

Prefered correlations are 'Erdim, Akgiray & Demir' when tube 

diameter is not provided, and 'Harrison, Brunner & Hecker' when tube 

diameter is provided. 

 

Examples 

-------- 

>>> dP_packed_bed(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3) 

1438.2826958844414 

>>> dP_packed_bed(dp=8E-4, voidage=0.4, vs=1E-3, rho=1E3, mu=1E-3, Dt=0.01) 

1255.1625662548427 

 

Parameters 

---------- 

dp : float 

Particle diameter of spheres [m] 

voidage : float 

Void fraction of bed packing [-] 

vs : float 

Superficial velocity of the fluid [m/s] 

rho : float 

Density of the fluid [kg/m^3] 

mu : float 

Viscosity of the fluid, [Pa*S] 

L : float, optional 

Length the fluid flows in the packed bed [m] 

Dt : float, optional 

Diameter of the tube, [m] 

sphericity : float, optional 

Sphericity of the particles [-] 

 

Returns 

------- 

dP : float 

Pressure drop across the bed [Pa] 

methods : list, only returned if AvailableMethods == True 

List of methods which can be used to calculate `dP` with the given inputs 

 

Other Parameters 

---------------- 

Method : string, optional 

A string of the function name to use, as in the dictionary 

packed_beds_correlations 

AvailableMethods : bool, optional 

If True, function will consider which methods which can be used to 

calculate `dP` with the given inputs and return them as a list 

''' 

def list_methods(): 

methods = [] 

if all((dp, voidage, vs, rho, mu, L)): 

for key, values in packed_beds_correlations.items(): 

if Dt or not values[1]: 

methods.append(key) 

if 'Harrison, Brunner & Hecker' in methods: 

methods.remove('Harrison, Brunner & Hecker') 

methods.insert(0, 'Harrison, Brunner & Hecker') 

elif 'Erdim, Akgiray & Demir' in methods: 

methods.remove('Erdim, Akgiray & Demir') 

methods.insert(0, 'Erdim, Akgiray & Demir') 

return methods 

 

if AvailableMethods: 

return list_methods() 

if not Method: 

Method = list_methods()[0] 

 

if dp and sphericity: 

dp = dp*sphericity 

if Method in packed_beds_correlations: 

if packed_beds_correlations[Method][1]: 

return packed_beds_correlations[Method][0](dp=dp, voidage=voidage, vs=vs, rho=rho, mu=mu, L=L, Dt=Dt) 

else: 

return packed_beds_correlations[Method][0](dp=dp, voidage=voidage, vs=vs, rho=rho, mu=mu, L=L) 

else: 

raise Exception('Failure in in function') 

 

 

#import matplotlib.pyplot as plt 

#import numpy as np 

# 

#voidage = 0.4 

#rho = 1000. 

#mu = 1E-3 

#vs = 0.1 

#dp = 0.0001 

#methods = dP_packed_bed(dp, voidage, vs, rho, mu, L=1, AvailableMethods=True) 

#dps = np.logspace(-4, -1, 100) 

# 

#for method in methods: 

# dPs = [dP_packed_bed(dp, voidage, vs, rho, mu, Method=method) for dp in dps] 

# plt.semilogx(dps, dPs, label=method) 

#plt.legend() 

#plt.show() 

 

 

 

### Voidage correlations 

 

def voidage_Benyahia_Oneil(Dpe, Dt, sphericity): 

r'''Calculates voidage of a bed of arbitraryily shaped uniform particles 

packed into a bed or tube of diameter `Dt`, with equivalent sphere diameter 

`Dp`. Shown in [1]_, and cited by various authors. Correlations exist 

also for spheres, solid cylinders, hollow cylinders, and 4-hole cylinders. 

Based on a series of physical measurements. 

 

.. math:: 

\epsilon = 0.1504 + \frac{0.2024}{\phi} + \frac{1.0814} 

{\left(\frac{d_{t}}{d_{pe}}+0.1226\right)^2} 

 

Parameters 

---------- 

Dpe : float 

Equivalent spherical particle diameter, [m] 

Dt : float 

Diameter of the tube, [m] 

sphericity : float 

Sphericity of particles in bed [] 

 

Returns 

------- 

voidage : float 

Void fraction of bed packing [] 

 

Notes 

----- 

Average error of 5.2%; valid 1.5 < dtube/dp < 50 and 0.42 < sphericity < 1 

 

Examples 

-------- 

>>> voidage_Benyahia_Oneil(1E-3, 1E-2, .8) 

0.41395363849210065 

 

References 

---------- 

.. [1] Benyahia, F., and K. E. O’Neill. "Enhanced Voidage Correlations for 

Packed Beds of Various Particle Shapes and Sizes." Particulate Science 

and Technology 23, no. 2 (April 1, 2005): 169-77. 

doi:10.1080/02726350590922242. 

''' 

return 0.1504 + 0.2024/sphericity + 1.0814/(Dt/Dpe + 0.1226)**2 

 

 

def voidage_Benyahia_Oneil_spherical(Dp, Dt): 

r'''Calculates voidage of a bed of spheres 

packed into a bed or tube of diameter `Dt`, with sphere diameters 

`Dp`. Shown in [1]_, and cited by various authors. Correlations exist 

also for solid cylinders, hollow cylinders, and 4-hole cylinders. 

Based on a series of physical measurements. 

 

.. math:: 

\epsilon = 0.390+\frac{1.740}{\left(\frac{d_{cyl}}{d_p}+1.140\right)^2} 

 

Parameters 

---------- 

Dp : float 

Spherical particle diameter, [m] 

Dt : float 

Diameter of the tube, [m] 

 

Returns 

------- 

voidage : float 

Void fraction of bed packing [] 

 

Notes 

----- 

Average error 1.5%, 1.5 < ratio < 50. 

 

Examples 

-------- 

>>> voidage_Benyahia_Oneil_spherical(.001, .05) 

0.3906653157443224 

 

References 

---------- 

.. [1] Benyahia, F., and K. E. O’Neill. "Enhanced Voidage Correlations for 

Packed Beds of Various Particle Shapes and Sizes." Particulate Science 

and Technology 23, no. 2 (April 1, 2005): 169-77. 

doi:10.1080/02726350590922242. 

''' 

return 0.390 + 1.740/(Dt/Dp + 1.140)**2 

 

 

def voidage_Benyahia_Oneil_cylindrical(Dpe, Dt, sphericity): 

r'''Calculates voidage of a bed of cylindrical uniform particles 

packed into a bed or tube of diameter `Dt`, with equivalent sphere diameter 

`Dpe`. Shown in [1]_, and cited by various authors. Correlations exist 

also for spheres, solid cylinders, hollow cylinders, and 4-hole cylinders. 

Based on a series of physical measurements. 

 

.. math:: 

\epsilon = 0.373+\frac{1.703}{\left(\frac{d_{cyl}}{d_p}+0.611\right)^2} 

 

Parameters 

---------- 

Dpe : float 

Equivalent spherical particle diameter, [m] 

Dt : float 

Diameter of the tube, [m] 

sphericity : float 

Sphericity of particles in bed [] 

 

Returns 

------- 

voidage : float 

Void fraction of bed packing [] 

 

Notes 

----- 

Average error 0.016%; 1.7 < ratio < 26.3. 

 

Examples 

-------- 

>>> voidage_Benyahia_Oneil_cylindrical(.01, .1, .6) 

0.38812523109607894 

 

References 

---------- 

.. [1] Benyahia, F., and K. E. O’Neill. "Enhanced Voidage Correlations for 

Packed Beds of Various Particle Shapes and Sizes." Particulate Science 

and Technology 23, no. 2 (April 1, 2005): 169-77. 

doi:10.1080/02726350590922242. 

''' 

return 0.373 + 1.703/(Dt/Dpe + 0.611)**2