"""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.
This module contains correlations and functions for calculating pressure drop
from packings and demisters; separation efficiency of demisters; demister
pressure drop; and demister geometry.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/fluids/>`_
or contact the author at Caleb.Andrew.Bell@gmail.com.
.. contents:: :local:
Packing Pressure Drop
---------------------
.. autofunction:: fluids.packed_tower.Robbins
.. autofunction:: fluids.packed_tower.Stichlmair_dry
.. autofunction:: fluids.packed_tower.Stichlmair_wet
Packing Flooding
----------------
.. autofunction:: fluids.packed_tower.Stichlmair_flood
Demister Pressure Drop
----------------------
.. autofunction:: fluids.packed_tower.dP_demister_dry_Setekleiv_Svendsen
.. autofunction:: fluids.packed_tower.dP_demister_dry_Setekleiv_Svendsen_lit
.. autofunction:: fluids.packed_tower.dP_demister_wet_ElDessouky
Demister Separation Efficiency
------------------------------
.. autofunction:: fluids.packed_tower.separation_demister_ElDessouky
Demister Geometry
-----------------
.. autofunction:: fluids.packed_tower.voidage_experimental
.. autofunction:: fluids.packed_tower.specific_area_mesh
"""
from math import log, sqrt
from fluids.constants import g, pi
from fluids.numerics import newton_system, secant, solve_2_direct
__all__ = ['voidage_experimental', 'specific_area_mesh',
'Stichlmair_dry', 'Stichlmair_wet', 'Stichlmair_flood', 'Robbins',
'dP_demister_dry_Setekleiv_Svendsen_lit',
'dP_demister_dry_Setekleiv_Svendsen',
'dP_demister_wet_ElDessouky', 'separation_demister_ElDessouky']
### Demister
[docs]def dP_demister_dry_Setekleiv_Svendsen(S, voidage, vs, rho, mu, L=1.0):
r'''Calculates dry pressure drop across a demister, using the
correlation in [1]_. This model is for dry demisters with no holdup only.
.. math::
\frac{\Delta P \epsilon^2}{\rho_f v^2} = 10.29 - \frac{565}
{69.6SL - (SL)^2 - 779} - \frac{74.9}{160.9 - 4.85SL} + 45.33\left(
\frac{\mu_f \epsilon S^2 L}{\rho_f v}\right)^{0.75}
Parameters
----------
S : float
Specific area of the demister, normally ~250-1000 [m^2/m^3]
voidage : float
Voidage of bed of the demister material, normally ~0.98 []
vs : float
Superficial velocity of fluid, Q/A [m/s]
rho : float
Density of fluid [kg/m^3]
mu : float
Viscosity of fluid [Pa*s]
L : float, optional
Length of the demister [m]
Returns
-------
dP : float
Pressure drop across a dry demister [Pa]
Notes
-----
Useful at startup and in modeling. Dry pressure drop is normally negligible
compared to wet pressure drop. Coefficients obtained by evolutionary
programming and may not fit data outside of the limits of the variables.
Examples
--------
>>> dP_demister_dry_Setekleiv_Svendsen(S=250, voidage=.983, vs=1.2, rho=10, mu=3E-5, L=1)
320.3280788941329
References
----------
.. [1] Setekleiv, A. Eddie, and Hallvard F. Svendsen. "Dry Pressure Drop in
Spiral Wound Wire Mesh Pads at Low and Elevated Pressures." Chemical
Engineering Research and Design 109 (May 2016): 141-149.
doi:10.1016/j.cherd.2016.01.019.
'''
term = 10.29 - 565./(69.6*S*L - (S*L)**2 - 779) - 74.9/(160.9 - 4.85*S*L)
right = term + 45.33*(mu*voidage*S**2*L/rho/vs)**0.75
return right*rho*vs**2/voidage**2
[docs]def dP_demister_dry_Setekleiv_Svendsen_lit(S, voidage, vs, rho, mu, L=1.0):
r'''Calculates dry pressure drop across a demister, using the
correlation in [1]_. This model is for dry demisters with no holdup only.
Developed with literature data included as well as their own experimental
data.
.. math::
\frac{\Delta P \epsilon^2}{\rho_f v^2} = 7.3 - \frac{320}
{69.6SL - (SL)^2 - 779} - \frac{52.4}{161 - 4.85SL} + 27.2\left(
\frac{\mu_f \epsilon S^2 L}{\rho_f v}\right)^{0.75}
Parameters
----------
S : float
Specific area of the demister, normally ~250-1000 [m^2/m^3]
voidage : float
Voidage of bed of the demister material, normally ~0.98 []
vs : float
Superficial velocity of fluid, Q/A [m/s]
rho : float
Density of fluid [kg/m^3]
mu : float
Viscosity of fluid [Pa*s]
L : float, optional
Length of the demister [m]
Returns
-------
dP : float
Pressure drop across a dry demister [Pa]
Notes
-----
Useful at startup and in modeling. Dry pressure drop is normally negligible
compared to wet pressure drop. Coefficients obtained by evolutionary
programming and may not fit data outside of the limits of the variables.
Examples
--------
>>> dP_demister_dry_Setekleiv_Svendsen_lit(S=250, voidage=.983, vs=1.2, rho=10, mu=3E-5, L=1.0)
209.083848658307
References
----------
.. [1] Setekleiv, A. Eddie, and Hallvard F. Svendsen. "Dry Pressure Drop in
Spiral Wound Wire Mesh Pads at Low and Elevated Pressures." Chemical
Engineering Research and Design 109 (May 2016): 141-149.
doi:10.1016/j.cherd.2016.01.019.
'''
term = 7.3 - 320./(69.6*S*L - (S*L)**2 - 779) - 52.4/(161 - 4.85*S*L)
right = term + 27.2*(mu*voidage*S**2*L/rho/vs)**0.75
return right*rho*vs**2/voidage**2
[docs]def dP_demister_wet_ElDessouky(vs, voidage, d_wire, L=1.0):
r'''Calculates wet pressure drop across a demister, using the
correlation in [1]_. Uses only their own experimental data.
.. math::
\frac{\Delta P}{L} = 0.002357(1-\epsilon)^{0.375798}(V)^{0.81317}
(d_w)^{-1.56114147}
Parameters
----------
vs : float
Superficial velocity of fluid, Q/A [m/s]
voidage : float
Voidage of bed of the demister material, normally ~0.98 []
d_wire : float
Diameter of mesh wire,[m]
L : float, optional
Length of the demister [m]
Returns
-------
dP : float
Pressure drop across a dry demister [Pa]
Notes
-----
No dependency on the liquid properties is included here. Because of the
exponential nature of the correlation, the limiting pressure drop as V
is lowered is 0 Pa. A dry pressure drop correlation should be compared with
results from this at low velocities, and the larger of the
two pressure drops used.
The correlation in [1]_ was presented as follows, with wire diameter in
units of mm, density in kg/m^3, V in m/s, and dP in Pa/m.
.. math::
\Delta P = 3.88178(\rho_{mesh})^{0.375798}(V)^{0.81317}
(d_w)^{-1.56114147}
Here, the correlation is converted to base SI units and to use voidage;
not all demisters are stainless steel as in [1]_. A density of 7999 kg/m^3
was used in the conversion.
In [1]_, V ranged from 0.98-7.5 m/s, rho from 80.317-208.16 kg/m^3, depth
from 100 to 200 mm, wire diameter of 0.2mm to 0.32 mm, and particle
diameter from 1 to 5 mm.
Examples
--------
>>> dP_demister_wet_ElDessouky(6, 0.978, 0.00032)
688.9216420105029
References
----------
.. [1] El-Dessouky, Hisham T, Imad M Alatiqi, Hisham M Ettouney, and Noura
S Al-Deffeeri. "Performance of Wire Mesh Mist Eliminator." Chemical
Engineering and Processing: Process Intensification 39, no. 2 (March
2000): 129-39. doi:10.1016/S0255-2701(99)00033-1.
'''
return L*0.002356999643727531*(1-voidage)**0.375798*vs**0.81317*d_wire**-1.56114147
[docs]def separation_demister_ElDessouky(vs, voidage, d_wire, d_drop):
r'''Calculates droplet removal by a demister as a fraction from 0 to 1,
using the correlation in [1]_. Uses only their own experimental data.
.. math::
\eta = 0.85835(d_w)^{-0.28264}(1-\epsilon)^{0.099625}(V)^{0.106878}
(d_p)^{0.383197}
Parameters
----------
vs : float
Superficial velocity of fluid, Q/A [m/s]
voidage : float
Voidage of bed of the demister material, normally ~0.98 []
d_wire : float
Diameter of mesh wire,[m]
d_drop : float
Drop diameter, [m]
Returns
-------
eta : float
Fraction droplets removed by mass [-]
Notes
-----
No dependency on the liquid properties is included here. Because of the
exponential nature of the correlation, for smaller diameters separation
quickly lowers. This correlation can predict a separation larger than 1
for higher velocities, lower voidages, lower wire diameters, and large
droplet sizes. This function truncates these larger values to 1.
The correlation in [1]_ was presented as follows, with wire diameter in
units of mm, density in kg/m^3, V in m/s, separation in %, and particle
diameter in mm.
.. math::
\eta = 17.5047(d_w)^{-0.28264}(\rho_{mesh})^{0.099625}(V)^{0.106878}
(d_p)^{0.383197}
Here, the correlation is converted to base SI units and to use voidage;
not all demisters are stainless steel as in [1]_. A density of 7999 kg/m^3
was used in the conversion.
In [1]_, V ranged from 0.98-7.5 m/s, rho from 80.317-208.16 kg/m^3, depth
from 100 to 200 mm, wire diameter of 0.2 mm to 0.32 mm, and particle
diameter from 1 to 5 mm.
Examples
--------
>>> separation_demister_ElDessouky(1.35, 0.974, 0.0002, 0.005)
0.8982892997640582
References
----------
.. [1] El-Dessouky, Hisham T, Imad M Alatiqi, Hisham M Ettouney, and Noura
S Al-Deffeeri. "Performance of Wire Mesh Mist Eliminator." Chemical
Engineering and Processing: Process Intensification 39, no. 2 (March
2000): 129-39. doi:10.1016/S0255-2701(99)00033-1.
'''
eta = 0.858352355761947*d_wire**-0.28264*(1-voidage)**0.099625*vs**0.106878*d_drop**0.383197
return min(eta, 1.0)
[docs]def voidage_experimental(m, rho, D, H):
r'''Calculates voidage of a bed or mesh given an experimental weight and
fixed density, diameter, and height, as shown in [1]_. The formula is also
self-evident.
.. math::
\epsilon = 1 - \frac{\frac{m_{mesh}}{\frac{\pi}{4}d_{column}^2
L_{mesh}}}{\rho_{material}}
Parameters
----------
m : float
Mass of mesh or bed particles weighted, [kg]
rho : float
Density of solid particles or mesh [kg/m^3]
D : float
Diameter of the cylindrical bed [m]
H : float
Height of the demister or bed [m]
Returns
-------
voidage : float
Voidage of bed of the material []
Notes
-----
Should be trusted over manufacturer data.
Examples
--------
>>> voidage_experimental(m=126, rho=8000, D=1, H=1)
0.9799464771704212
References
----------
.. [1] Helsør, T., and H. Svendsen. "Experimental Characterization of
Pressure Drop in Dry Demisters at Low and Elevated Pressures." Chemical
Engineering Research and Design 85, no. 3 (2007): 377-85.
doi:10.1205/cherd06048.
'''
return 1 - m/(pi/4*D**2*H)/rho
[docs]def specific_area_mesh(voidage, d):
r'''Calculates the specific area of a wire mesh, as used in demisters or
filters. Shown in [1]_, and also self-evident and non-empirical.
Makes the ideal assumption that wires never touch.
.. math::
S = \frac{4(1-\epsilon)}{d_{wire}}
Parameters
----------
voidage : float
Voidage of the mesh []
d : float
Diameter of the wires making the mesh, [m]
Returns
-------
S : float
Specific area of the mesh [m^2/m^3]
Notes
-----
Should be preferred over manufacturer data. Can also be used to show that
manufacturer's data is inconsistent with their claimed voidage and wire
diameter.
Examples
--------
>>> specific_area_mesh(voidage=.934, d=3e-4)
879.9999999999994
References
----------
.. [1] Helsør, T., and H. Svendsen. "Experimental Characterization of
Pressure Drop in Dry Demisters at Low and Elevated Pressures." Chemical
Engineering Research and Design 85, no. 3 (2007): 377-85.
doi:10.1205/cherd06048.
'''
return 4*(1-voidage)/d
### Packing
[docs]def Stichlmair_dry(Vg, rhog, mug, voidage, specific_area, C1, C2, C3, H=1.):
r'''Calculates dry pressure drop across a packed column, using the
Stichlmair [1]_ correlation. Uses three regressed constants for each
type of packing, and voidage and specific area.
Pressure drop is given by:
.. math::
\Delta P_{dry} = \frac{3}{4} f_0 \frac{1-\epsilon}{\epsilon^{4.65}}
\rho_G \frac{H}{d_p}V_g^2
.. math::
f_0 = \frac{C_1}{Re_g} + \frac{C_2}{Re_g^{0.5}} + C_3
.. math::
d_p = \frac{6(1-\epsilon)}{a}
Parameters
----------
Vg : float
Superficial velocity of gas, Q/A [m/s]
rhog : float
Density of gas [kg/m^3]
mug : float
Viscosity of gas [Pa*s]
voidage : float
Voidage of bed of packing material []
specific_area : float
Specific area of the packing material [m^2/m^3]
C1 : float
Packing-specific constant []
C2 : float
Packing-specific constant []
C3 : float
Packing-specific constant []
H : float, optional
Height of packing [m]
Returns
-------
dP_dry : float
Pressure drop across dry packing [Pa]
Notes
-----
This model is used by most process simulation tools. If H is not provided,
it defaults to 1. If Z is not provided, it defaults to 1.
Examples
--------
>>> Stichlmair_dry(Vg=0.4, rhog=5., mug=5E-5, voidage=0.68,
... specific_area=260., C1=32., C2=7.0, C3=1.0)
236.80904286559885
References
----------
.. [1] Stichlmair, J., J. L. Bravo, and J. R. Fair. "General Model for
Prediction of Pressure Drop and Capacity of Countercurrent Gas/liquid
Packed Columns." Gas Separation & Purification 3, no. 1 (March 1989):
19-28. doi:10.1016/0950-4214(89)80016-7.
'''
dp = 6*(1-voidage)/specific_area
Re = Vg*rhog*dp/mug
f0 = C1/Re + C2/sqrt(Re) + C3
return 3/4.*f0*(1-voidage)/voidage**4.65*rhog*H/dp*Vg**2
def _Stichlmair_wet_err(dP_irr, h0, c1, dP_dry, H, voidage, c):
hT = h0*(1.0 + 20.0*dP_irr*dP_irr*c1)
err = dP_dry/H*((1-voidage+hT)/(1.0 - voidage))**((2.0 + c)/3.)*(voidage/(voidage-hT))**4.65 -dP_irr/H
return err
[docs]def Stichlmair_wet(Vg, Vl, rhog, rhol, mug, voidage, specific_area, C1, C2, C3, H=1.0):
r'''Calculates dry pressure drop across a packed column, using the
Stichlmair [1]_ correlation. Uses three regressed constants for each
type of packing, and voidage and specific area. This model is for irrigated
columns only.
Pressure drop is given by:
.. math::
\frac{\Delta P_{irr}}{H} = \frac{\Delta P_{dry}}{H}\left(\frac
{1-\epsilon + h_T}{1-\epsilon}\right)^{(2+c)/3}
\left(\frac{\epsilon}{\epsilon-h_T}\right)^{4.65}
.. math::
h_T = h_0\left[1 + 20\left(\frac{\Delta Pirr}{H\rho_L g}\right)^2\right]
.. math::
Fr_L = \frac{V_L^2 a}{g \epsilon^{4.65}}
.. math::
h_0 = 0.555 Fr_L^{1/3}
.. math::
c = \frac{-C_1/Re_g - C_2/(2Re_g^{0.5})}{f_0}
.. math::
\Delta P_{dry} = \frac{3}{4} f_0 \frac{1-\epsilon}{\epsilon^{4.65}}
\rho_G \frac{H}{d_p}V_g^2
.. math::
f_0 = \frac{C_1}{Re_g} + \frac{C_2}{Re_g^{0.5}} + C_3
.. math::
d_p = \frac{6(1-\epsilon)}{a}
Parameters
----------
Vg : float
Superficial velocity of gas, Q/A [m/s]
Vl : float
Superficial velocity of liquid, Q/A [m/s]
rhog : float
Density of gas [kg/m^3]
rhol : float
Density of liquid [kg/m^3]
mug : float
Viscosity of gas [Pa*s]
voidage : float
Voidage of bed of packing material []
specific_area : float
Specific area of the packing material [m^2/m^3]
C1 : float
Packing-specific constant []
C2 : float
Packing-specific constant []
C3 : float
Packing-specific constant []
H : float, optional
Height of packing [m]
Returns
-------
dP : float
Pressure drop across irrigated packing [Pa]
Notes
-----
This model is used by most process simulation tools. If H is not provided,
it defaults to 1. If Z is not provided, it defaults to 1.
A numerical solver is used and needed by this model. Its initial guess
is the dry pressure drop. Convergence problems may occur.
The model as described in [1]_ appears to have a typo, and could not match
the example. As described in [2]_, however, the model works.
Examples
--------
Example is from [1]_, matches.
>>> Stichlmair_wet(Vg=0.4, Vl = 5E-3, rhog=5., rhol=1200., mug=5E-5,
... voidage=0.68, specific_area=260., C1=32., C2=7., C3=1.)
539.876823725352
References
----------
.. [1] Stichlmair, J., J. L. Bravo, and J. R. Fair. "General Model for
Prediction of Pressure Drop and Capacity of Countercurrent Gas/liquid
Packed Columns." Gas Separation & Purification 3, no. 1 (March 1989):
19-28. doi:10.1016/0950-4214(89)80016-7.
.. [2] Piche, Simon R., Faical Larachi, and Bernard P. A. Grandjean.
"Improving the Prediction of Irrigated Pressure Drop in Packed
Absorption Towers." The Canadian Journal of Chemical Engineering 79,
no. 4 (August 1, 2001): 584-94. doi:10.1002/cjce.5450790417.
'''
dp = 6.0*(1.0 - voidage)/specific_area
Re = Vg*rhog*dp/mug
f0 = C1/Re + C2/sqrt(Re) + C3
dP_dry = 3/4.*f0*(1-voidage)/voidage**4.65*rhog*H/dp*Vg*Vg
c = (-C1/Re - C2/(2*sqrt(Re)))/f0
Frl = Vl**2*specific_area/(g*voidage**4.65)
h0 = 0.555*Frl**(1/3.)
c1 = 1.0/(H*rhol*g)
c1 *= c1
return secant(_Stichlmair_wet_err, dP_dry, args=(h0, c1, dP_dry, H, voidage, c))
def _Stichlmair_flood_f(inputs, Vl, rhog, rhol, mug, voidage, specific_area,
C1, C2, C3, H):
"""Internal function which calculates the errors of the two Stichlmair
objective functions, and their jacobian.
"""
Vg, dP_irr = float(inputs[0]), float(inputs[1])
dp = 6.0*(1.0 - voidage)/specific_area
Re = Vg*rhog*dp/mug
f0 = C1/Re + C2/sqrt(Re) + C3
dP_dry = 0.75*f0*(1.0 - voidage)/voidage**4.65*rhog*H/dp*Vg*Vg
c = (-C1/Re - 0.5*C2*1.0/sqrt(Re))/f0
Frl = Vl*Vl*specific_area/(g*voidage**4.65)
h0 = 0.555*Frl**(1/3.)
hT = h0*(1.0 + 20.0*(dP_irr/H/rhol/g)**2)
err1 = dP_dry/H*((1.0 - voidage + hT)/(1.0 - voidage))**((2.0 + c)/3.)*(voidage/(voidage-hT))**4.65 - dP_irr/H
term = (dP_irr/(rhol*g*H))**2
err2 = (1./term - 40.0*((2.0+c)/3.)*h0/(1.0 - voidage + h0*(1.0 + 20.0*term))
- 186.0*h0/(voidage - h0*(1.0 + 20.0*term)))
return err1, err2
def _Stichlmair_flood_f_and_jac(inputs, Vl, rhog, rhol, mug, voidage,
specific_area, C1, C2, C3, H):
"""Internal function which calculates the errors of the two Stichlmair
objective functions, and their jacobian.
Derived using SymPy on the main flooding function.
"""
Vg, dP_irr = inputs[0], inputs[1]
x0 = 1.0/H
x1 = Vg*Vg
x2 = voidage**(-4.65)
x3 = specific_area*x2
x4 = Vl*Vl*x3/g
x5 = x4**0.333333333333333
x6 = dP_irr*dP_irr
x7 = H*H
x8 = 1.0/x7
x9 = g*g
x10 = 1.0/x9
x11 = rhol*rhol
x12 = 1.0/x11
x13 = x5*(20.0*x10*x12*x6*x8 + 1.0)
x14 = 0.555*x13
x15 = (voidage/(voidage - x14))**4.65
x16 = 1.0/Vg
x17 = 1.0/rhog
x18 = voidage - 1.0
x19 = 1.0/x18
x20 = C1*mug*specific_area*x16*x17*x19
x21 = 2.44948974278318*C2
x22 = Vg*rhog/(mug*specific_area)
x23 = x21*1.0/sqrt(-x18*x22)
x24 = 6.0*C3 - x20 + x23
x25 = 1.0 - voidage
x26 = x14 + x25
x27 = -x19*x26
x28 = 2.0*C1*mug*specific_area*x16*x17/x25 + x21*1.0/sqrt(x22*x25)
x29 = 1.0/x24
x30 = x28*x29
x31 = x27**(-0.166666666666667*x30 + 0.666666666666667)
x32 = x11*x7*x9
x33 = 200.0*voidage
x34 = 111.0*x13
x35 = x33 - x34
x36 = 1.0/x35
x37 = -x33 + x34 + 200.0
x38 = 1.0/x37
x39 = 2.0*x20
x40 = -4.0*x20 + x23 + x29*(-x23 + x39)*(x23 - x39)
x41 = dP_irr*rhog*specific_area*x0*x1*x10*x12*x15*x2*x24*x31
x42 = dP_irr*x10*x12*x4**0.666666666666667*x8
F1, F2, dF1_dVg, dF2_dVg, dF1_dP_irr, dF2_dP_irr = (
-dP_irr*x0 + 0.0208333333333333*rhog*specific_area*x1*x15*x2*x24*x31,
x32/x6 - 20646.0*x36*x5 - x38*x5*(2960.0 - 740.0*x28*x29),
0.00173611111111111*Vg*rhog*x15*x3*x31*(144.0*C3 - 12.0*x20 + 18.0*x23 + x40*log(x27)),
x0*(430.125*x36*x41*x5 - 15.4166666666667*x38*x41*x5*(x30 - 4.0) - 1.0),
-1.85*x16*x29*x40*x5/x26,
3285600.0*x42*(-x30 + 4.0)*x38*x38- 91668240.0*x42*x36*x36 - 2.0*x32/(dP_irr*x6))
err = [0.0]*2
err[0] = F1
err[1] = F2
jac = [[dF1_dVg, dF2_dVg], [dF1_dP_irr, dF2_dP_irr]]# numba: delete
# jac = np.array([[dF1_dVg, dF2_dVg], [dF1_dP_irr, dF2_dP_irr]]) # numba: uncomment
return err, jac
[docs]def Stichlmair_flood(Vl, rhog, rhol, mug, voidage, specific_area, C1, C2, C3,
H=1.0):
r'''Calculates gas rate for flooding of a packed column, using the
Stichlmair [1]_ correlation. Uses three regressed constants for each
type of packing, and voidage and specific area.
Pressure drop is given by:
.. math::
\frac{\Delta P_{irr}}{H} = \frac{\Delta P_{dry}}{H}\left(\frac
{1-\epsilon + h_T}{1-\epsilon}\right)^{(2+c)/3}
\left(\frac{\epsilon}{\epsilon-h_T}\right)^{4.65}
.. math::
h_T = h_0\left[1 + 20\left(\frac{\Delta Pirr}{H\rho_L g}\right)^2\right]
.. math::
Fr_L = \frac{V_L^2 a}{g \epsilon^{4.65}}
.. math::
h_0 = 0.555 Fr_L^{1/3}
.. math::
c = \frac{-C_1/Re_g - C_2/(2Re_g^{0.5})}{f_0}
.. math::
\Delta P_{dry} = \frac{3}{4} f_0 \frac{1-\epsilon}{\epsilon^{4.65}}
\rho_G \frac{H}{d_p}V_g^2
.. math::
f_0 = \frac{C_1}{Re_g} + \frac{C_2}{Re_g^{0.5}} + C_3
.. math::
d_p = \frac{6(1-\epsilon)}{a}
Parameters
----------
Vl : float
Superficial velocity of liquid, Q/A [m/s]
rhog : float
Density of gas [kg/m^3]
rhol : float
Density of liquid [kg/m^3]
mug : float
Viscosity of gas [Pa*s]
voidage : float
Voidage of bed of packing material []
specific_area : float
Specific area of the packing material [m^2/m^3]
C1 : float
Packing-specific constant []
C2 : float
Packing-specific constant []
C3 : float
Packing-specific constant []
H : float, optional
Height of packing [m]
Returns
-------
Vg : float
Superficial velocity of gas, Q/A [m/s]
Notes
-----
A numerical solver is used to solve this model.
Examples
--------
Example is from [1]_, matches.
>>> Stichlmair_flood(Vl = 5E-3, rhog=5., rhol=1200., mug=5E-5,
... voidage=0.68, specific_area=260., C1=32., C2=7., C3=1.)
0.6394323542746928
References
----------
.. [1] Stichlmair, J., J. L. Bravo, and J. R. Fair. "General Model for
Prediction of Pressure Drop and Capacity of Countercurrent Gas/liquid
Packed Columns." Gas Separation & Purification 3, no. 1 (March 1989):
19-28. doi:10.1016/0950-4214(89)80016-7.
'''
guess = [0.0]*2
guess[0] = Vl*100.0
guess[1] = 1000.0
return newton_system(_Stichlmair_flood_f_and_jac, x0=guess, jac=True,
args=(Vl, rhog, rhol, mug, voidage, specific_area, C1,
C2, C3, H), ytol=1e-11, solve_func=solve_2_direct)[0][0]
[docs]def Robbins(L, G, rhol, rhog, mul, H=1.0, Fpd=24.0):
r'''Calculates pressure drop across a packed column, using the Robbins
equation.
Pressure drop is given by:
.. math::
\Delta P = C_3 G_f^2 10^{C_4L_f}+0.4[L_f/20000]^{0.1}[C_3G_f^210^{C_4L_f}]^4
.. math::
G_f=G[0.075/\rho_g]^{0.5}[F_{pd}/20]^{0.5}=986F_s[F_{pd}/20]^{0.5}
.. math::
L_f=L[62.4/\rho_L][F_{pd}/20]^{0.5}\mu^{0.1}
.. math::
F_s=V_s\rho_g^{0.5}
Parameters
----------
L : float
Specific liquid mass flow rate [kg/s/m^2]
G : float
Specific gas mass flow rate [kg/s/m^2]
rhol : float
Density of liquid [kg/m^3]
rhog : float
Density of gas [kg/m^3]
mul : float
Viscosity of liquid [Pa*s]
H : float
Height of packing [m]
Fpd : float
Robbins packing factor (tabulated for packings) [1/ft]
Returns
-------
dP : float
Pressure drop across packing [Pa]
Notes
-----
Perry's displayed equation has a typo in a superscript.
This model is based on the example in Perry's.
Examples
--------
>>> Robbins(L=12.2, G=2.03, rhol=1000., rhog=1.1853, mul=0.001, H=2.0, Fpd=24.0)
619.6624593438102
References
----------
.. [1] Robbins [Chem. Eng. Progr., p. 87 (May 1991) Improved Pressure Drop
Prediction with a New Correlation.
'''
# Convert SI units to imperial for use in correlation
L = L*737.33812 # kg/s/m^2 to lb/hr/ft^2
G = G*737.33812 # kg/s/m^2 to lb/hr/ft^2
rhol = rhol*0.062427961 # kg/m^3 to lb/ft^3
rhog = rhog*0.062427961 # kg/m^3 to lb/ft^3
mul = mul*1000.0 # Pa*s to cP
C3 = 7.4E-8
C4 = 2.7E-5
Fpd_root_term = sqrt(.05*Fpd)
Lf = L*(62.4/rhol)*Fpd_root_term*mul**0.1
Gf = G*sqrt(0.075/rhog)*Fpd_root_term
Gf2 = Gf*Gf
C4LF_10_GF2_C3 = C3*Gf2*10.0**(C4*Lf)
C4LF_10_GF2_C3_2 = C4LF_10_GF2_C3*C4LF_10_GF2_C3
dP = C4LF_10_GF2_C3 + 0.4*(5e-5*Lf)**0.1*(C4LF_10_GF2_C3_2*C4LF_10_GF2_C3_2)
return dP*817.22083*H # in. H2O to Pa/m