"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2020 Caleb Bell <Caleb.Andrew.Bell@gmail.com>
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SOFTWARE.
This module contains some basic functions for fluid mechanics mixing
calculations.
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:
Misc Functions
--------------
.. autofunction:: size_tee
.. autofunction:: COV_motionless_mixer
.. autofunction:: K_motionless_mixer
.. autofunction:: agitator_time_homogeneous
.. autofunction:: Kp_helical_ribbon_Rieger
.. autofunction:: time_helical_ribbon_Grenville
"""
from math import log, pi, sqrt
from fluids.constants import g
__all__ = ['agitator_time_homogeneous',
'Kp_helical_ribbon_Rieger', 'time_helical_ribbon_Grenville', 'size_tee',
'COV_motionless_mixer', 'K_motionless_mixer']
max_Fo_for_turbulent = 1/1225.
min_regime_constant_for_turbulent = 6370.
def adjust_homogeneity(fraction):
'''Base: 95% homogeneity'''
multiplier = log(1-fraction)/log(0.05)
return multiplier
[docs]def agitator_time_homogeneous(N, P, T, H, mu, rho, D=None, homogeneity=.95):
r'''Calculates time for a fluid mizing in a tank with an impeller to
reach a specified level of homogeneity, according to [1]_.
.. math::
N_p = \frac{Pg}{\rho N^3 D^5}
.. math::
Re_{imp} = \frac{\rho D^2 N}{\mu}
.. math::
\text{constant} = N_p^{1/3} Re_{imp}
.. math::
Fo = 5.2/\text{constant} \text{for turbulent regime}
.. math::
Fo = (183/\text{constant})^2 \text{for transition regime}
Parameters
----------
N : float:
Speed of impeller, [revolutions/s]
P : float
Actual power required to mix, ignoring mechanical inefficiencies [W]
T : float
Tank diameter, [m]
H : float
Tank height, [m]
mu : float
Mixture viscosity, [Pa*s]
rho : float
Mixture density, [kg/m^3]
D : float, optional
Impeller diameter [m]
homogeneity : float, optional
Fraction completion of mixing, []
Returns
-------
t : float
Time for specified degree of homogeneity [s]
Notes
-----
If impeller diameter is not specified, assumed to be 0.5 tank diameters.
The first example is solved forward rather than backwards here. A rather
different result is obtained, but is accurate.
No check to see if the mixture if laminar is currently implemented.
This would under predict the required time.
Examples
--------
>>> agitator_time_homogeneous(D=36*.0254, N=56/60., P=957., T=1.83, H=1.83, mu=0.018, rho=1020, homogeneity=.995)
15.143198226374668
>>> agitator_time_homogeneous(D=1, N=125/60., P=298., T=3, H=2.5, mu=.5, rho=980, homogeneity=.95)
67.7575069865228
References
----------
.. [1] Paul, Edward L, Victor A Atiemo-Obeng, and Suzanne M Kresta.
Handbook of Industrial Mixing: Science and Practice.
Hoboken, N.J.: Wiley-Interscience, 2004.
'''
if not D:
D = T*0.5
Np = P*g/rho/N**3/D**5
Re_imp = rho/mu*D**2*N
regime_constant = Np**(1/3.)*Re_imp
if regime_constant >= min_regime_constant_for_turbulent:
Fo = (5.2/regime_constant)
else:
Fo = (183./regime_constant)**2
time = rho*T**1.5*sqrt(H)/mu*Fo
multiplier = adjust_homogeneity(homogeneity)
return time*multiplier
[docs]def Kp_helical_ribbon_Rieger(D, h, nb, pitch, width, T):
r'''Calculates product of power number and Reynolds number for a
specified geometry for a heilical ribbon mixer in the laminar regime.
One of several correlations listed in [1]_, it used more data than other
listed correlations and was recommended.
.. math::
K_p = 82.8\frac{h}{D}\left(\frac{c}{D}\right)^{-0.38} \left(\frac{p}{D}\right)^{-0.35}
\left(\frac{w}{D}\right)^{0.20} n_b^{0.78}
Parameters
----------
D : float
Impeller diameter [m]
h : float
Ribbon mixer height, [m]
nb : float:
Number of blades, [-]
pitch : float
Height of one turn around a helix [m]
width : float
Width of one blade [m]
T : float
Tank diameter, [m]
Returns
-------
Kp : float
Product of Power number and Reynolds number for laminar regime []
Notes
-----
Example is from example 9-6 in [1]_. Confirmed.
Examples
--------
>>> Kp_helical_ribbon_Rieger(D=1.9, h=1.9, nb=2, pitch=1.9, width=.19, T=2)
357.39749163259256
References
----------
.. [1] Paul, Edward L, Victor A Atiemo-Obeng, and Suzanne M Kresta.
Handbook of Industrial Mixing: Science and Practice.
Hoboken, N.J.: Wiley-Interscience, 2004.
.. [2] Rieger, F., V. Novak, and D. Havelkov (1988). The influence of the
geometrical shape on the power requirements of ribbon impellers,
Int. Chem. Eng., 28, 376-383.
'''
c = 0.5*(T - D)
return 82.8*h/D*(c/D)**-.38*(pitch/D)**-0.35*(width/D)**0.2*nb**0.78
[docs]def time_helical_ribbon_Grenville(Kp, N):
r'''Calculates product of time required for mixing in a helical ribbon
coil in the laminar regime according to the Grenville [2]_ method
recommended in [1]_.
.. math::
t = 896\times10^3K_p^{-1.69}/N
Parameters
----------
Kp : float
Product of power number and Reynolds number for laminar regime []
N : float
Speed of impeller, [revolutions/s]
Returns
-------
t : float
Time for homogeneity [s]
Notes
-----
Degree of homogeneity is not specified.
Example is from example 9-6 in [1]_. Confirmed.
Examples
--------
>>> time_helical_ribbon_Grenville(357.4, 4/60.)
650.980654028894
References
----------
.. [1] Paul, Edward L, Victor A Atiemo-Obeng, and Suzanne M Kresta.
Handbook of Industrial Mixing: Science and Practice.
Hoboken, N.J.: Wiley-Interscience, 2004.
.. [2] Grenville, R. K., T. M. Hutchinson, and R. W. Higbee (2001).
Optimisation of helical ribbon geometry for blending in the laminar
regime, presented at MIXING XVIII, NAMF.
'''
return 896E3*Kp**-1.69/N
### Tee mixer
[docs]def size_tee(Q1, Q2, D, D2, n=1, pipe_diameters=5):
r'''Calculates CoV of an optimal or specified tee for mixing at a tee
according to [1]_. Assumes turbulent flow.
The smaller stream in injected into the main pipe, which continues
straight.
COV calculation is according to [2]_.
Parameters
----------
Q1 : float
Volumetric flow rate of larger stream [m^3/s]
Q2 : float
Volumetric flow rate of smaller stream [m^3/s]
D : float
Diameter of pipe after tee [m]
D2 : float
Diameter of mixing inlet, optional (optimally calculated if not
specified) [m]
n : float
Number of jets, 1 to 4 []
pipe_diameters : float
Number of diameters along tail pipe for CoV calculation, 0 to 5 []
Returns
-------
CoV : float
Standard deviation of dimensionless concentration [-]
Notes
-----
Not specified if this works for liquid also, though probably not.
Example is from example Example 9-6 in [1]_. Low precision used in example.
Examples
--------
>>> size_tee(Q1=11.7, Q2=2.74, D=0.762, D2=None, n=1, pipe_diameters=5)
0.2940930233038544
References
----------
.. [1] Paul, Edward L, Victor A Atiemo-Obeng, and Suzanne M Kresta.
Handbook of Industrial Mixing: Science and Practice.
Hoboken, N.J.: Wiley-Interscience, 2004.
.. [2] Giorges, Aklilu T. G., Larry J. Forney, and Xiaodong Wang.
"Numerical Study of Multi-Jet Mixing." Chemical Engineering Research and
Design, Fluid Flow, 79, no. 5 (July 2001): 515-22.
doi:10.1205/02638760152424280.
'''
V1 = Q1/(pi/4*D**2)
# Cv = Q2/(Q1 + Q2)
# COV0 = sqrt((1-Cv)/Cv)
if D2 is None:
D2 = (Q2/Q1)**(2/3.)*D
V2 = Q2/(pi/4*D2**2)
B = n**2*(D2/D)**2*(V2/V1)**2
if not n == 1 and not n == 2 and not n == 3 and not n ==4:
raise ValueError('Only 1 or 4 side streams investigated')
if n == 1:
if B < 0.7:
E = 1.33
else:
E = 1/33. + 0.95*log(B/0.7)
elif n == 2:
if B < 0.8:
E = 1.44
else:
E = 1.44 + 0.95*log(B/0.8)**1.5
elif n == 3:
if B < 0.8:
E = 1.75
else:
E = 1.75 + 0.95*log(B/0.8)**1.8
else:
if B < 2:
E = 1.97
else:
E = 1.97 + 0.95*log(B/2.)**2
COV = sqrt(0.32/B**0.86*(pipe_diameters)**-E)
return COV
### Commercial motionless mixers
"""Data from:
Paul, Edward L, Victor A Atiemo-Obeng, and Suzanne M Kresta.
Handbook of Industrial Mixing: Science and Practice.
Hoboken, N.J.: Wiley-Interscience, 2004."""
StatixMixers = {}
StatixMixers['KMS'] = {'Name': 'KMS', 'Vendor': 'Chemineer', 'Description': 'Twisted ribbon. Alternating left and right twists.', 'KL': 6.9, 'KiL': 0.87, 'KT': 150, 'KiT': 0.5}
StatixMixers['SMX'] = {'Name': 'SMX', 'Vendor': 'Koch-Glitsch', 'Description': 'Guide vanes 45 degrees to pipe axis. Adjacent elements rotated 90 degrees.', 'KL': 37.5, 'KiL': 0.63, 'KT': 500, 'KiT': 0.46}
StatixMixers['SMXL'] = {'Name': 'SMXL', 'Vendor': 'Koch-Glitsch', 'Description': 'Similar to SMX, but intersection bars at 30 degrees to pipe axis.', 'KL': 7.8, 'KiL': 0.85, 'KT': 100, 'KiT': 0.87}
StatixMixers['SMF'] = {'Name': 'SMF', 'Vendor': 'Koch-Glitsch', 'Description': 'Three guide vanes projecting from the tube wall in a way as to not contact. Designed for applications subject to plugging.', 'KL': 5.6, 'KiL': 0.83, 'KT': 130, 'KiT': 0.4}
[docs]def COV_motionless_mixer(Ki, Q1, Q2, pipe_diameters):
r'''Calculates CoV of a motionless mixer with a regression parameter in
[1]_ and originally in [2]_.
.. math::
\frac{CoV}{CoV_0} = K_i^{L/D}
Parameters
----------
Ki : float
Correlation parameter specific to a mixer's design, [-]
Q1 : float
Volumetric flow rate of larger stream [m^3/s]
Q2 : float
Volumetric flow rate of smaller stream [m^3/s]
pipe_diameters : float
Number of diameters along tail pipe for CoV calculation, 0 to 5 []
Returns
-------
CoV : float
Standard deviation of dimensionless concentration [-]
Notes
-----
Example 7-8.3.2 in [1]_, solved backwards.
Examples
--------
>>> COV_motionless_mixer(Ki=.33, Q1=11.7, Q2=2.74, pipe_diameters=4.74/.762)
0.0020900028665727685
References
----------
.. [1] Paul, Edward L, Victor A Atiemo-Obeng, and Suzanne M Kresta.
Handbook of Industrial Mixing: Science and Practice.
Hoboken, N.J.: Wiley-Interscience, 2004.
.. [2] Streiff, F. A., S. Jaffer, and G. Schneider (1999). Design and
application of motionless mixer technology, Proc. ISMIP3, Osaka,
pp. 107-114.
'''
Cv = Q2/(Q1 + Q2)
COV0 = sqrt((1-Cv)/Cv)
COVr = Ki**(pipe_diameters)
COV = COV0*COVr
return COV
[docs]def K_motionless_mixer(K, L, D, fd):
r'''Calculates loss coefficient of a motionless mixer with a regression
parameter in [1]_ and originally in [2]_.
.. math::
K = K_{L/T}f\frac{L}{D}
Parameters
----------
K : float
Correlation parameter specific to a mixer's design, [-]
Also specific to laminar or turbulent regime.
L : float
Length of the motionless mixer [m]
D : float
Diameter of pipe [m]
fd : float
Darcy friction factor [-]
Returns
-------
K : float
Loss coefficient of mixer [-]
Notes
-----
Related to example 7-8.3.2 in [1]_.
Examples
--------
>>> K_motionless_mixer(K=150, L=.762*5, D=.762, fd=.01)
7.5
References
----------
.. [1] Paul, Edward L, Victor A Atiemo-Obeng, and Suzanne M Kresta.
Handbook of Industrial Mixing: Science and Practice.
Hoboken, N.J.: Wiley-Interscience, 2004.
.. [2] Streiff, F. A., S. Jaffer, and G. Schneider (1999). Design and
application of motionless mixer technology, Proc. ISMIP3, Osaka,
pp. 107-114.
'''
return L/D*fd*K