def Q_weir_V_Shen(h1, angle=90):
r'''Calculates the flow rate across a V-notch (triangular) weir from
the height of the liquid above the tip of the notch, and with the angle
of the notch. Most of these type of weir are 90 degrees. Model from [1]_
as reproduced in [2]_.
Flow rate is given by:
.. math::
Q = C \tan\left(\frac{\theta}{2}\right)\sqrt{g}(h_1 + k)^{2.5}
Parameters
----------
h1 : float
Height of the fluid above the notch [m]
angle : float, optional
Angle of the notch [degrees]
Returns
-------
Q : float
Volumetric flow rate across the weir [m^3/s]
Notes
-----
angles = [20, 40, 60, 80, 100]
Cs = [0.59, 0.58, 0.575, 0.575, 0.58]
k = [0.0028, 0.0017, 0.0012, 0.001, 0.001]
The following limits apply to the use of this equation:
h1 >= 0.05 m
h2 > 0.45 m
h1/h2 <= 0.4 m
b > 0.9 m
.. math::
\frac{h_1}{b}\tan\left(\frac{\theta}{2}\right) < 2
Flows are lower than obtained by the curves at
http://www.lmnoeng.com/Weirs/vweir.php.
Examples
--------
>>> Q_weir_V_Shen(0.6, angle=45)
0.21071725775478228
References
----------
.. [1] Shen, John. "Discharge Characteristics of Triangular-Notch
Thin-Plate Weirs : Studies of Flow to Water over Weirs and Dams."
USGS Numbered Series. Water Supply Paper. U.S. Geological Survey :
U.S. G.P.O., 1981
.. [2] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
Van Nostrand Reinhold Co., 1984.
'''
C = interp(angle, angles_Shen, Cs_Shen)
k = interp(angle, angles_Shen, k_Shen)
return C*tan(radians(angle)/2)*g**0.5*(h1 + k)**2.5
def Q_weir_V_Shen(h1, angle=90):
r'''Calculates the flow rate across a V-notch (triangular) weir from
the height of the liquid above the tip of the notch, and with the angle
of the notch. Most of these type of weir are 90 degrees. Model from [1]_
as reproduced in [2]_.
Flow rate is given by:
.. math::
Q = C \tan\left(\frac{\theta}{2}\right)\sqrt{g}(h_1 + k)^{2.5}
Parameters
----------
h1 : float
Height of the fluid above the notch [m]
angle : float, optional
Angle of the notch [degrees]
Returns
-------
Q : float
Volumetric flow rate across the weir [m^3/s]
Notes
-----
angles = [20, 40, 60, 80, 100]
Cs = [0.59, 0.58, 0.575, 0.575, 0.58]
k = [0.0028, 0.0017, 0.0012, 0.001, 0.001]
The following limits apply to the use of this equation:
h1 >= 0.05 m
h2 > 0.45 m
h1/h2 <= 0.4 m
b > 0.9 m
.. math::
\frac{h_1}{b}\tan\left(\frac{\theta}{2}\right) < 2
Flows are lower than obtained by the curves at
http://www.lmnoeng.com/Weirs/vweir.php.
Examples
--------
>>> Q_weir_V_Shen(0.6, angle=45)
0.21071725775478228
References
----------
.. [1] Shen, John. "Discharge Characteristics of Triangular-Notch
Thin-Plate Weirs : Studies of Flow to Water over Weirs and Dams."
USGS Numbered Series. Water Supply Paper. U.S. Geological Survey :
U.S. G.P.O., 1981
.. [2] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
Van Nostrand Reinhold Co., 1984.
'''
C = interp(angle, angles_Shen, Cs_Shen)
k = interp(angle, angles_Shen, k_Shen)
return C*tan(radians(angle)/2)*g**0.5*(h1 + k)**2.5
def API520_B(Pset, Pback, overpressure=0.1):
r'''Calculates capacity correction due to backpressure on balanced
spring-loaded PRVs in vapor service. For pilot operated valves,
this is always 1. Applicable up to 50% of the percent gauge backpressure,
For use in API 520 relief valve sizing. 1D interpolation among a table with
53 backpressures is performed.
Parameters
----------
Pset : float
Set pressure for relief [Pa]
Pback : float
Backpressure, [Pa]
overpressure : float, optional
The maximum fraction overpressure; one of 0.1, 0.16, or 0.21, [-]
Returns
-------
Kb : float
Correction due to vapor backpressure [-]
Notes
-----
If the calculated gauge backpressure is less than 30%, 38%, or 50% for
overpressures of 0.1, 0.16, or 0.21, a value of 1 is returned.
Percent gauge backpressure must be under 50%.
Examples
--------
Custom examples from figure 30:
>>> API520_B(1E6, 5E5)
0.7929945420944432
References
----------
.. [1] API Standard 520, Part 1 - Sizing and Selection.
'''
gauge_backpressure = (Pback - atm) / (Pset - atm) * 100.0 # in percent
if overpressure not in (0.1, 0.16, 0.21):
raise Exception('Only overpressure of 10%, 16%, or 21% are permitted')
if (overpressure == 0.1 and gauge_backpressure < 30.0) or (
overpressure == 0.16
and gauge_backpressure < 38.0) or (overpressure == 0.21
and gauge_backpressure < 50.0):
return 1.0
elif gauge_backpressure > 50.0:
raise Exception('Gauge pressure must be < 50%')
if overpressure == 0.16:
Kb = interp(gauge_backpressure, Kb_16_over_x, Kb_16_over_y)
elif overpressure == 0.1:
Kb = interp(gauge_backpressure, Kb_10_over_x, Kb_10_over_y)
return Kb
def API520_B(Pset, Pback, overpressure=0.1):
r'''Calculates capacity correction due to backpressure on balanced
spring-loaded PRVs in vapor service. For pilot operated valves,
this is always 1. Applicable up to 50% of the percent gauge backpressure,
For use in API 520 relief valve sizing. 1D interpolation among a table with
53 backpressures is performed.
Parameters
----------
Pset : float
Set pressure for relief [Pa]
Pback : float
Backpressure, [Pa]
overpressure : float, optional
The maximum fraction overpressure; one of 0.1, 0.16, or 0.21, []
Returns
-------
Kb : float
Correction due to vapor backpressure [-]
Notes
-----
If the calculated gauge backpressure is less than 30%, 38%, or 50% for
overpressures of 0.1, 0.16, or 0.21, a value of 1 is returned.
Percent gauge backpressure must be under 50%.
Examples
--------
Custom examples from figure 30:
>>> API520_B(1E6, 5E5)
0.7929945420944432
References
----------
.. [1] API Standard 520, Part 1 - Sizing and Selection.
'''
gauge_backpressure = (Pback-atm)/(Pset-atm)*100 # in percent
if overpressure not in [0.1, 0.16, 0.21]:
raise Exception('Only overpressure of 10%, 16%, or 21% are permitted')
if (overpressure == 0.1 and gauge_backpressure < 30) or (
overpressure == 0.16 and gauge_backpressure < 38) or (
overpressure == 0.21 and gauge_backpressure < 50):
return 1
elif gauge_backpressure > 50:
raise Exception('Gauge pressure must be < 50%')
if overpressure == 0.16:
Kb = interp(gauge_backpressure, Kb_16_over_x, Kb_16_over_y)
elif overpressure == 0.1:
Kb = interp(gauge_backpressure, Kb_10_over_x, Kb_10_over_y)
return Kb
def round_edge_screen(alpha, Re, angle=0.0):
r'''Returns the loss coefficient for a round edged wire screen or bar
screen, as shown in [1]_. Angle of inclination may be specified as well.
Parameters
----------
alpha : float
Fraction of screen open to flow [-]
Re : float
Reynolds number of flow through screen with D = space between rods, []
angle : float, optional
Angle of inclination, with 0 being straight and 90 being parallel to
flow [degrees]
Returns
-------
K : float
Loss coefficient [-]
Notes
-----
Linear interpolation between a table of values. Re table extends
from 20 to 400, with constant values outside of the table. This behavior
should be adequate.
alpha should be between 0.05 and 0.8.
If angle is over 85 degrees, the value at 85 degrees is used.
The velocity the loss coefficient relates to is the approach velocity
before the screen.
Examples
--------
>>> round_edge_screen(0.5, 100)
2.0999999999999996
>>> round_edge_screen(0.5, 100, 45)
1.05
References
----------
.. [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
Van Nostrand Reinhold Co., 1984.
'''
beta = interp(Re, round_Res, round_betas)
alpha2 = alpha * alpha
K = beta * (1.0 - alpha2) / alpha2
if angle is not None:
if angle <= 45.0:
v = cos(radians(angle))
K *= v * v
else:
K *= interp(angle, round_thetas, round_gammas)
return K
def round_edge_screen(alpha, Re, angle=0):
r'''Returns the loss coefficient for a round edged wire screen or bar
screen, as shown in [1]_. Angle of inclination may be specified as well.
Parameters
----------
alpha : float
Fraction of screen open to flow [-]
Re : float
Reynolds number of flow through screen with D = space between rods, []
angle : float, optional
Angle of inclination, with 0 being straight and 90 being parallel to
flow [degrees]
Returns
-------
K : float
Loss coefficient [-]
Notes
-----
Linear interpolation between a table of values. Re table extends
from 20 to 400, with constant values outside of the table. This behavior
should be adequate.
alpha should be between 0.05 and 0.8.
If angle is over 85 degrees, the value at 85 degrees is used.
Examples
--------
>>> round_edge_screen(0.5, 100)
2.0999999999999996
>>> round_edge_screen(0.5, 100, 45)
1.05
References
----------
.. [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
Van Nostrand Reinhold Co., 1984.
'''
beta = interp(Re, round_Res, round_betas)
alpha2 = alpha*alpha
K = beta*(1.0 - alpha2)/alpha2
if angle:
if angle <= 45:
K *= cos(radians(angle))**2
else:
K *= interp(angle, round_thetas, round_gammas)
return K
def square_edge_screen(alpha):
r'''Returns the loss coefficient for a square wire screen or square bar
screen or perforated plate with squared edges, as shown in [1]_.
Parameters
----------
alpha : float
Fraction of screen open to flow [-]
Returns
-------
K : float
Loss coefficient [-]
Notes
-----
Linear interpolation between a table of values.
The velocity the loss coefficient relates to is the approach velocity
before the screen.
Examples
--------
>>> square_edge_screen(0.99)
0.008000000000000007
References
----------
.. [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
Van Nostrand Reinhold Co., 1984.
'''
return interp(alpha, square_alphas, square_Ks)
def square_edge_screen(alpha):
r'''Returns the loss coefficient for a square wire screen or square bar
screen or perforated plate with squared edges, as shown in [1]_.
Parameters
----------
alpha : float
Fraction of screen open to flow [-]
Returns
-------
K : float
Loss coefficient [-]
Notes
-----
Linear interpolation between a table of values.
Examples
--------
>>> square_edge_screen(0.99)
0.008000000000000007
References
----------
.. [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
Van Nostrand Reinhold Co., 1984.
'''
return interp(alpha, square_alphas, square_Ks)
def test_interp():
from fluids.numerics import interp
# Real world test data
a = [
0.29916, 0.29947, 0.31239, 0.31901, 0.32658, 0.33729, 0.34202, 0.34706,
0.35903, 0.36596, 0.37258, 0.38487, 0.38581, 0.40125, 0.40535, 0.41574,
0.42425, 0.43401, 0.44788, 0.45259, 0.47181, 0.47309, 0.49354, 0.49924,
0.51653, 0.5238, 0.53763, 0.54806, 0.55684, 0.57389, 0.58235, 0.59782,
0.60156, 0.62265, 0.62649, 0.64948, 0.65099, 0.6687, 0.67587, 0.68855,
0.69318, 0.70618, 0.71333, 0.72351, 0.74954, 0.74965
]
b = [
0.164534, 0.164504, 0.163591, 0.163508, 0.163439, 0.162652, 0.162224,
0.161866, 0.161238, 0.160786, 0.160295, 0.15928, 0.159193, 0.157776,
0.157467, 0.156517, 0.155323, 0.153835, 0.151862, 0.151154, 0.14784,
0.147613, 0.144052, 0.14305, 0.140107, 0.138981, 0.136794, 0.134737,
0.132847, 0.129303, 0.127637, 0.124758, 0.124006, 0.119269, 0.118449,
0.113605, 0.113269, 0.108995, 0.107109, 0.103688, 0.102529, 0.099567,
0.097791, 0.095055, 0.087681, 0.087648
]
xs = np.linspace(0.29, 0.76, 100)
ys = [interp(xi, a, b) for xi in xs.tolist()]
ys_numpy = np.interp(xs, a, b)
assert_allclose(ys, ys_numpy, atol=1e-12, rtol=1e-11)
def RI_to_brix(RI):
"""Convert a standard refractive index measurement to the `brix` scale.
Parameters
----------
RI : float
Refractive index, [-]
Returns
-------
brix : float
Degrees brix to be converted, [°Bx]
Notes
-----
The scale is officially defined from 0 to 85; but the data source contains
values up to 95.
Linear extrapolation to values under 0 or above 95 is performed.
The ICUMSA (International Committee of Uniform Method of Sugar Analysis)
published a document setting out the reference values in 1974; but an
original data source has not been found and reviewed.
Examples
--------
>>> RI_to_brix(1.341452)
5.800000000000059
>>> RI_to_brix(1.33299)
0.0
>>> RI_to_brix(1.532)
95.0
References
----------
.. [1] "Refractometer Data Book-Refractive Index and Brix | ATAGO CO.,
LTD." Accessed June 13, 2020.
https://www.atago.net/en/databook-refractometer_relationship.php.
"""
return interp(RI, ICUMSA_1974_RIs, ICUMSA_1974_brix, extrapolate=True)
def refractory_VDI_Cp(ID, T=None):
r'''Returns heat capacity of a refractory material from a table in
[1]_. Here, heat capacity is a function of temperature between
673.15 K and 1473.15 K according to linear interpolation among 5
equally-spaced points. Here, heat capacity is not a function of
porosity, affects it. If T is outside the acceptable range, it is
rounded to the nearest limit. If T is not provided, the lowest temperature's
value is provided.
Parameters
----------
ID : str
ID corresponding to a material in the dictionary `refractories`
T : float, optional
Temperature of the refractory material, [K]
Returns
-------
Cp : float
Heat capacity of the refractory material, [W/m/K]
Examples
--------
>>> [refractory_VDI_Cp('Fused silica', i) for i in [None, 200, 1000, 1500]]
[917.0, 917.0, 956.78225, 982.0]
References
----------
.. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
Berlin; New York:: Springer, 2010.
'''
if T is None:
return float(refractories[ID][2][0])
else:
Cps = refractories[ID][2]
if T < _refractory_Ts[0]:
T = _refractory_Ts[0]
elif T > _refractory_Ts[-1]:
T = _refractory_Ts[-1]
return float(interp(T, _refractory_Ts, Cps))
def brix_to_RI(brix):
"""Convert a refractive index measurement on the `brix` scale to a standard
refractive index.
Parameters
----------
brix : float
Degrees brix to be converted, [°Bx]
Returns
-------
RI : float
Refractive index, [-]
Notes
-----
The scale is officially defined from 0 to 85; but the data source contains
values up to 95. Linear extrapolation outside of the bounds is performed;
and a table of 96 values are linearly interpolated.
The ICUMSA (International Committee of Uniform Method of Sugar Analysis)
published a document setting out the reference values in 1974; but an
original data source has not been found and reviewed.
Examples
--------
>>> brix_to_RI(5.8)
1.341452
>>> brix_to_RI(0.0)
1.33299
>>> brix_to_RI(95.0)
1.532
References
----------
.. [1] "Refractometer Data Book-Refractive Index and Brix | ATAGO CO.,
LTD." Accessed June 13, 2020.
https://www.atago.net/en/databook-refractometer_relationship.php.
"""
return interp(brix, ICUMSA_1974_brix, ICUMSA_1974_RIs, extrapolate=True)
def API520_W(Pset, Pback):
r'''Calculates capacity correction due to backpressure on balanced
spring-loaded PRVs in liquid service. For pilot operated valves,
this is always 1. Applicable up to 50% of the percent gauge backpressure,
For use in API 520 relief valve sizing. 1D interpolation among a table with
53 backpressures is performed.
Parameters
----------
Pset : float
Set pressure for relief [Pa]
Pback : float
Backpressure, [Pa]
Returns
-------
KW : float
Correction due to liquid backpressure [-]
Notes
-----
If the calculated gauge backpressure is less than 15%, a value of 1 is
returned.
Examples
--------
Custom example from figure 31:
>>> API520_W(1E6, 3E5) # 22% overpressure
0.9511471848008564
References
----------
.. [1] API Standard 520, Part 1 - Sizing and Selection.
'''
gauge_backpressure = (Pback - atm) / (Pset - atm) * 100.0 # in percent
if gauge_backpressure < 15.0:
return 1.0
return interp(gauge_backpressure, Kw_x, Kw_y)
def API520_W(Pset, Pback):
r'''Calculates capacity correction due to backpressure on balanced
spring-loaded PRVs in liquid service. For pilot operated valves,
this is always 1. Applicable up to 50% of the percent gauge backpressure,
For use in API 520 relief valve sizing. 1D interpolation among a table with
53 backpressures is performed.
Parameters
----------
Pset : float
Set pressure for relief [Pa]
Pback : float
Backpressure, [Pa]
Returns
-------
KW : float
Correction due to liquid backpressure [-]
Notes
-----
If the calculated gauge backpressure is less than 15%, a value of 1 is
returned.
Examples
--------
Custom example from figure 31:
>>> API520_W(1E6, 3E5) # 22% overpressure
0.9511471848008564
References
----------
.. [1] API Standard 520, Part 1 - Sizing and Selection.
'''
gauge_backpressure = (Pback-atm)/(Pset-atm)*100.0 # in percent
if gauge_backpressure < 15.0:
return 1.0
return interp(gauge_backpressure, Kw_x, Kw_y)
def round_edge_open_mesh(alpha, subtype='diamond pattern wire', angle=0.0):
r'''Returns the loss coefficient for a round edged open net/screen
made of one of the following patterns, according to [1]_:
'round bar screen':
.. math::
K = 0.95(1-\alpha) + 0.2(1-\alpha)^2
'diamond pattern wire':
.. math::
K = 0.67(1-\alpha) + 1.3(1-\alpha)^2
'knotted net':
.. math::
K = 0.70(1-\alpha) + 4.9(1-\alpha)^2
'knotless net':
.. math::
K = 0.72(1-\alpha) + 2.1(1-\alpha)^2
Parameters
----------
alpha : float
Fraction of net/screen open to flow [-]
subtype : str
One of 'round bar screen', 'diamond pattern wire', 'knotted net' or
'knotless net'.
angle : float, optional
Angle of inclination, with 0 being straight and 90 being parallel to
flow [degrees]
Returns
-------
K : float
Loss coefficient [-]
Notes
-----
`alpha` should be between 0.85 and 1 for these correlations.
Flow should be turbulent, with Re > 500.
The velocity the loss coefficient relates to is the approach velocity
before the mesh.
Examples
--------
>>> round_edge_open_mesh(0.96, angle=33.)
0.02031327712601458
References
----------
.. [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
Van Nostrand Reinhold Co., 1984.
'''
one_m_alpha = (1.0 - alpha)
if subtype == 'round bar screen':
K = 0.95 + 0.2 * one_m_alpha
elif subtype == 'diamond pattern wire':
K = 0.67 + 1.3 * one_m_alpha
elif subtype == 'knotted net':
K = 0.70 + 4.9 * one_m_alpha
elif subtype == 'knotless net':
K = 0.72 + 2.1 * one_m_alpha
else:
raise ValueError('Subtype not recognized')
K *= one_m_alpha
if angle is not None:
if angle < 45.0:
K *= cos(radians(angle))**2.0
else:
K *= interp(angle, round_thetas, round_gammas)
return K
def round_edge_open_mesh(alpha, subtype='diamond pattern wire', angle=0):
r'''Returns the loss coefficient for a round edged open net/screen
made of one of the following patterns, according to [1]_:
'round bar screen':
.. math::
K = 0.95(1-\alpha) + 0.2(1-\alpha)^2
'diamond pattern wire':
.. math::
K = 0.67(1-\alpha) + 1.3(1-\alpha)^2
'knotted net':
.. math::
K = 0.70(1-\alpha) + 4.9(1-\alpha)^2
'knotless net':
.. math::
K = 0.72(1-\alpha) + 2.1(1-\alpha)^2
Parameters
----------
alpha : float
Fraction of net/screen open to flow [-]
subtype : str
One of 'round bar screen', 'diamond pattern wire', 'knotted net' or
'knotless net'.
angle : float, optional
Angle of inclination, with 0 being straight and 90 being parallel to
flow [degrees]
Returns
-------
K : float
Loss coefficient [-]
Notes
-----
`alpha` should be between 0.85 and 1 for these correlations.
Flow should be turbulent, with Re > 500.
Examples
--------
>>> round_edge_open_mesh(0.96, angle=33.)
0.02031327712601458
References
----------
.. [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
Van Nostrand Reinhold Co., 1984.
'''
if subtype == 'round bar screen':
K = 0.95*(1-alpha) + 0.2*(1-alpha)**2
elif subtype == 'diamond pattern wire':
K = 0.67*(1-alpha) + 1.3*(1-alpha)**2
elif subtype == 'knotted net':
K = 0.70*(1-alpha) + 4.9*(1-alpha)**2
elif subtype == 'knotless net':
K = 0.72*(1-alpha) + 2.1*(1-alpha)**2
else:
raise Exception('Subtype not recognized')
if angle:
if angle < 45:
K *= cos(radians(angle))**2
else:
K *= interp(angle, round_thetas, round_gammas)
return K
