"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2019, 2020 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 functionality for calculating parameters about different
geometrical forms that come up in engineering practice.
Implemented geometry objects are tanks, helical coils, cooling towers,
air coolers, compact heat exchangers, and plate and frame heat exchangers.
Additional functionality for typical catalyst/adsorbent pellet shapes is also
included.
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:
Main Interfaces
---------------
.. autoclass :: TANK
:members:
:undoc-members:
:show-inheritance:
.. autofunction:: V_tank
.. autofunction:: V_from_h
.. autofunction:: SA_tank
.. autofunction:: SA_from_h
.. autoclass :: HelicalCoil
:members:
:undoc-members:
:show-inheritance:
.. autoclass :: PlateExchanger
:members:
:undoc-members:
:show-inheritance:
.. autoclass :: AirCooledExchanger
:members:
:undoc-members:
:show-inheritance:
.. autoclass :: HyperbolicCoolingTower
:members:
:undoc-members:
:show-inheritance:
.. autoclass :: RectangularFinExchanger
:members:
:undoc-members:
:show-inheritance:
.. autoclass :: RectangularOffsetStripFinExchanger
:members:
:undoc-members:
:show-inheritance:
Tank Volume Functions
---------------------
.. autofunction:: V_partial_sphere
.. autofunction:: V_horiz_conical
.. autofunction:: V_horiz_ellipsoidal
.. autofunction:: V_horiz_guppy
.. autofunction:: V_horiz_spherical
.. autofunction:: V_horiz_torispherical
.. autofunction:: V_vertical_conical
.. autofunction:: V_vertical_ellipsoidal
.. autofunction:: V_vertical_spherical
.. autofunction:: V_vertical_torispherical
.. autofunction:: V_vertical_conical_concave
.. autofunction:: V_vertical_ellipsoidal_concave
.. autofunction:: V_vertical_spherical_concave
.. autofunction:: V_vertical_torispherical_concave
Tank Surface Area Functions
---------------------------
.. autofunction:: SA_partial_sphere
.. autofunction:: SA_ellipsoidal_head
.. autofunction:: SA_conical_head
.. autofunction:: SA_guppy_head
.. autofunction:: SA_torispheroidal
.. autofunction:: SA_partial_cylindrical_body
.. autofunction:: SA_partial_horiz_conical_head
.. autofunction:: SA_partial_horiz_spherical_head
.. autofunction:: SA_partial_horiz_ellipsoidal_head
.. autofunction:: SA_partial_horiz_guppy_head
.. autofunction:: SA_partial_horiz_torispherical_head
.. autofunction:: SA_partial_vertical_conical_head
.. autofunction:: SA_partial_vertical_ellipsoidal_head
.. autofunction:: SA_partial_vertical_spherical_head
.. autofunction:: SA_partial_vertical_torispherical_head
Miscellaneous Geometry Functions
--------------------------------
.. autofunction:: pitch_angle_solver
.. autofunction:: plate_enlargement_factor
.. autofunction:: a_torispherical
.. autofunction:: A_partial_circle
.. autofunction:: circle_segment_h_from_A
Pellet Properties
-----------------
.. autofunction:: sphericity
.. autofunction:: aspect_ratio
.. autofunction:: circularity
.. autofunction:: A_cylinder
.. autofunction:: V_cylinder
.. autofunction:: A_hollow_cylinder
.. autofunction:: V_hollow_cylinder
.. autofunction:: A_multiple_hole_cylinder
.. autofunction:: V_multiple_hole_cylinder
"""
from cmath import sqrt as csqrt
from math import acos, acosh, asin, atan, cos, degrees, log, pi, radians, sin, sqrt, tan
from fluids.numerics import brenth, cacos, catan, chebval, derivative, ellipe, ellipeinc, ellipkinc, linspace, newton, quad, secant, translate_bound_func
__all__ = ['TANK', 'HelicalCoil', 'PlateExchanger', 'RectangularFinExchanger',
'RectangularOffsetStripFinExchanger', 'HyperbolicCoolingTower',
'AirCooledExchanger',
'SA_partial_sphere',
'V_partial_sphere', 'V_horiz_conical',
'V_horiz_ellipsoidal', 'V_horiz_guppy', 'V_horiz_spherical',
'V_horiz_torispherical', 'V_vertical_conical',
'V_vertical_ellipsoidal', 'V_vertical_spherical',
'V_vertical_torispherical', 'V_vertical_conical_concave',
'V_vertical_ellipsoidal_concave', 'V_vertical_spherical_concave',
'V_vertical_torispherical_concave', 'a_torispherical',
'SA_ellipsoidal_head', 'SA_conical_head', 'SA_guppy_head',
'SA_torispheroidal', 'SA_partial_cylindrical_body',
'A_partial_circle', 'SA_partial_horiz_conical_head',
'SA_partial_horiz_spherical_head', 'SA_partial_horiz_guppy_head',
'SA_partial_horiz_ellipsoidal_head', 'SA_partial_horiz_torispherical_head',
'SA_partial_vertical_conical_head', 'SA_partial_vertical_spherical_head',
'SA_partial_vertical_torispherical_head', 'SA_partial_vertical_ellipsoidal_head',
'V_from_h', 'V_tank', 'SA_from_h', 'SA_tank', 'sphericity',
'aspect_ratio', 'circularity', 'A_cylinder', 'V_cylinder',
'A_hollow_cylinder', 'V_hollow_cylinder',
'A_multiple_hole_cylinder', 'V_multiple_hole_cylinder',
'pitch_angle_solver', 'plate_enlargement_factor', 'circle_segment_h_from_A']
### Spherical Vessels, partially filled
[docs]def SA_partial_sphere(D, h):
r'''Calculates surface area of a partial sphere according to [1]_.
If h is half of D, the shape is half a sphere. No bottom is considered in
this function. Valid inputs are positive values of D and h, with h always
smaller or equal to D.
.. math::
a = \sqrt{h(2r - h)}
.. math::
A = \pi(a^2 + h^2)
Parameters
----------
D : float
Diameter of the sphere, [m]
h : float
Height, as measured from the cap to where the sphere is cut off [m]
Returns
-------
SA : float
Surface area [m^2]
Examples
--------
>>> SA_partial_sphere(1., 0.7)
2.199114857512855
References
----------
.. [1] Weisstein, Eric W. "Spherical Cap." Text. Accessed December 22, 2015.
http://mathworld.wolfram.com/SphericalCap.html.
'''
r = D*0.5
a = sqrt(h*(2.*r - h))
return pi*(a*a + h*h)
[docs]def V_partial_sphere(D, h):
r'''Calculates volume of a partial sphere according to [1]_.
If h is half of D, the shape is half a sphere. No bottom is considered in
this function. Valid inputs are positive values of D and h, with h always
smaller or equal to D.
.. math::
a = \sqrt{h(2r - h)}
.. math::
V = 1/6 \pi h(3a^2 + h^2)
Parameters
----------
D : float
Diameter of the sphere, [m]
h : float
Height, as measured up to where the sphere is cut off, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
>>> V_partial_sphere(1., 0.7)
0.4105014400690663
References
----------
.. [1] Weisstein, Eric W. "Spherical Cap." Text. Accessed December 22, 2015.
http://mathworld.wolfram.com/SphericalCap.html.
'''
if h <= 0.0:
return 0.0
r = 0.5*D
a = sqrt(h*(2.*r - h))
return 1/6.*pi*h*(3.*a*a + h*h)
#def V_horizontal_bullet(D, L, H, b=None):
# # As in GPSA
# if not b:
# b = 0.25*D # elliptical 2:1 heads
# Ze = H/D
# Zc = H/D
# K1 = 2*b/D
# alpha = 2*atan(H/sqrt(2*H*D/2 - H**2))
# fZc = (alpha - sin(alpha)*cos(alpha))/pi
# fZe = -H**2/D**2*(-3 + 2*H/D)
# V = 1/6.*pi*K1*D**3*fZe + 1/4.*pi*D**2*L*fZc
# return V
#print(V_horizontal_bullet(1., 5., .4999999999999, 0.000000000000000001))
#def V_vertical_bullet(D, L, H, b=None):
# K1 = 2*b/D
# Ze = (H1 + H2)/K1*D # is divided by D?
# fZe = -((H1 + H2))
#
# V = 1/6.*pi*K1*D**3*fZe + 1/4.*pi*D**2*L*fZc
# return V
### Functions as developed by Dan Jones
[docs]def V_horiz_conical(D, L, a, h, headonly=False):
r'''Calculates volume of a tank with conical ends, according to [1]_.
.. math::
V_f = A_fL + \frac{2aR^2}{3}K, \;\;0 \le h < R\\
.. math::
V_f = A_fL + \frac{2aR^2}{3}\pi/2,\;\; h = R\\
.. math::
V_f = A_fL + \frac{2aR^2}{3}(\pi-K), \;\; R< h \le 2R
.. math::
K = \cos^{-1} M + M^3\cosh^{-1} \frac{1}{M} - 2M\sqrt{1 - M^2}
.. math::
M = \left|\frac{R-h}{R}\right|
.. math::
A_f = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
L : float
Length of the main cylindrical section, [m]
a : float
Distance the cone head extends on one side, [m]
h : float
Height, as measured up to where the fluid ends, [m]
headonly : bool, optional
Function returns only the volume of a single head side if True
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_horiz_conical(D=108., L=156., a=42., h=36)/231
2041.1923581273443
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
R = 0.5*D
R_third = R/3.0
t0 = (R-h)/R
if t0 < -1.0 or t0 > 1.0:
raise ValueError("Unphysical height")
Af = R*R*acos(t0) - (R-h)*sqrt(h*(R + R - h))
M = abs(t0)
if h == R:
Vf = a*R*R_third*pi
else:
K = acos(M) + M*M*M*acosh(1./M) - 2.*M*sqrt(1.-M*M)
if 0. <= h < R:
Vf = 2.*a*R*R_third*K
else:
# elif R < h <= 2.0*R:
Vf = 2.*a*R*R_third*(pi - K)
if headonly:
Vf = 0.5*Vf
else:
Vf += Af*L
return Vf
[docs]def V_horiz_ellipsoidal(D, L, a, h, headonly=False):
r'''Calculates volume of a tank with ellipsoidal ends, according to [1]_.
.. math::
V_f = A_fL + \pi a h^2\left(1 - \frac{h}{3R}\right)
.. math::
A_f = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
L : float
Length of the main cylindrical section, [m]
a : float
Distance the ellipsoidal head extends on one side, [m]
h : float
Height, as measured up to where the fluid ends, [m]
headonly : bool, optional
Function returns only the volume of a single head side if True
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_horiz_ellipsoidal(D=108, L=156, a=42, h=36)/231.
2380.9565415578145
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
R = 0.5*D
Af = R*R*acos((R-h)/R) - (R-h)*sqrt(2*R*h - h*h)
Vf = pi*a*h*h*(1 - h/(3.*R))
if headonly:
Vf = 0.5*Vf
else:
Vf += Af*L
return Vf
[docs]def V_horiz_guppy(D, L, a, h, headonly=False):
r'''Calculates volume of a tank with guppy heads, according to [1]_.
.. math::
V_f = A_fL + \frac{2aR^2}{3}\cos^{-1}\left(1 - \frac{h}{R}\right)
+\frac{2a}{9R}\sqrt{2Rh - h^2}(2h-3R)(h+R)
.. math::
A_f = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
L : float
Length of the main cylindrical section, [m]
a : float
Distance the guppy head extends on one side, [m]
h : float
Height, as measured up to where the fluid ends, [m]
headonly : bool, optional
Function returns only the volume of a single head side if True
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_horiz_guppy(D=108., L=156., a=42., h=36)/231.
1931.7208029476762
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
R = 0.5*D
x0 = sqrt(2.*R*h - h*h)
Af = R*R*acos((R-h)/R) - (R-h)*x0
Vf = 2.*a*R*R/3.*acos(1. - h/R) + 2.*a/9./R*x0*(2.0*h - 3.0*R)*(h + R)
if headonly:
Vf = Vf*0.5
else:
Vf += Af*L
return Vf
def _V_horiz_spherical_toint(x, r2, R2, den_inv):
x2 = x*x
return (r2 - x2)*atan(sqrt((R2 - x2)*den_inv))
[docs]def V_horiz_spherical(D, L, a, h, headonly=False):
r'''Calculates volume of a tank with spherical heads, according to [1]_.
.. math::
V_f = A_fL + \frac{\pi a}{6}(3R^2 + a^2),\;\; h = R, |a|\le R
.. math::
V_f = A_fL + \frac{\pi a}{3}(3R^2 + a^2),\;\; h = D, |a|\le R
.. math::
V_f = A_fL + \pi a h^2\left(1 - \frac{h}{3R}\right),\;\; h = 0,
\text{ or } |a| = 0, R, -R
.. math::
V_f = A_fL + \frac{a}{|a|}\left\{\frac{2r^3}{3}\left[\cos^{-1}
\frac{R^2 - rw}{R(w-r)} + \cos^{-1}\frac{R^2 + rw}{R(w+r)}
- \frac{z}{r}\left(2 + \left(\frac{R}{r}\right)^2\right)
\cos^{-1}\frac{w}{R}\right] - 2\left(wr^2 - \frac{w^3}{3}\right)
\tan^{-1}\frac{y}{z} + \frac{4wyz}{3}\right\}
,\;\; h \ne R, D; a \ne 0, R, -R, |a| \ge 0.01D
.. math::
V_f = A_fL + \frac{a}{|a|}\left[2\int_w^R(r^2 - x^2)\tan^{-1}
\sqrt{\frac{R^2-x^2}{r^2-R^2}}dx - A_f z\right]
,\;\; h \ne R, D; a \ne 0, R, -R, |a| < 0.01D
.. math::
A_f = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2}
.. math::
r = \frac{a^2 + R^2}{2|a|}
.. math::
w = R - h
.. math::
y = \sqrt{2Rh-h^2}
.. math::
z = \sqrt{r^2 - R^2}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
L : float
Length of the main cylindrical section, [m]
a : float
Distance the spherical head extends on one side, [m]
h : float
Height, as measured up to where the fluid ends, [m]
headonly : bool, optional
Function returns only the volume of a single head side if True
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_horiz_spherical(D=108., L=156., a=42., h=36)/231.
2303.9615116986183
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
R = D/2.
r = (a*a + R*R)/2./abs(a)
w = R - h
y = sqrt(2*R*h - h**2)
z = sqrt(r**2 - R**2)
Af = R**2*acos((R-h)/R) - (R-h)*sqrt(2*R*h - h**2)
if h == R and abs(a) <= R:
Vf = pi*a/6*(3*R**2 + a**2)
elif h == D and abs(a) <= R:
Vf = pi*a/3*(3*R**2 + a**2)
elif h == 0 or a == 0 or a == R or a == -R or z == 0.0:
Vf = pi*a*h**2*(1 - h/3./R)
elif abs(a) >= 0.01*D:
Vf = a/abs(a)*(
2*r**3/3.*(acos((R**2 - r*w)/(R*(w-r))) + acos((R**2+r*w)/(R*(w+r)))
- z/r*(2+(R/r)**2)*acos(w/R))
- 2*(w*r**2 - w**3/3)*atan(y/z) + 4*w*y*z/3)
else:
r2 = r*r
R2 = R*R
den_inv = 1.0/(r2 - R2)
integrated = quad(_V_horiz_spherical_toint, w, R, args=(r2, R2, den_inv))[0] # , epsrel=1.49e-13,
Vf = a/abs(a)*(2*integrated - Af*z)
if headonly:
Vf = Vf/2.
else:
Vf += Af*L
return Vf
def V_horiz_torispherical_toint_1(x, w, c10, c11):
# No analytical integral available in MP
n = c11 + sqrt(c10 - x*x)
n2 = n*n
t = sqrt(n2 - w*w)
return n2*asin(t/n) - w*t
def V_horiz_torispherical_toint_2(x, w, c10, c11, g, g2):
# No analytical integral available in MP
n = c11 + sqrt(c10 - x*x)
n2 = n*n
n_inv = 1.0/n
ans = n2*(acos(w*n_inv) - acos(g*n_inv)) - w*sqrt(n2 - w*w) + g*sqrt(n2 - g2)
return ans
def V_horiz_torispherical_toint_3(x, r2, g2, z_inv):
# There is an analytical integral in MP, but for all cases we seem to
# get ZeroDivisionError: 0.0 cannot be raised to a negative power
x2 = x*x
return (r2 - x2)*atan(sqrt(g2 - x2)*z_inv)
[docs]def V_horiz_torispherical(D, L, f, k, h, headonly=False):
r'''Calculates volume of a tank with torispherical heads, according to [1]_.
.. math::
V_f = A_fL + 2V_1, \;\; 0 \le h \le h_1\\
V_f = A_fL + 2(V_{1,max} + V_2 + V_3), \;\; h_1 < h < h_2\\
V_f = A_fL + 2[2V_{1,max} - V_1(h=D-h) + V_{2,max} + V_{3,max}]
, \;\; h_2 \le h \le D
.. math::
V_1 = \int_0^{\sqrt{2kDh - h^2}} \left[n^2\sin^{-1}\frac{\sqrt
{n^2-w^2}}{n} - w\sqrt{n^2-w^2}\right]dx
.. math::
V_2 = \int_0^{kD\cos\alpha}\left[n^2\left(\cos^{-1}\frac{w}{n}
- \cos^{-1}\frac{g}{n}\right) - w\sqrt{n^2 - w^2} + g\sqrt{n^2
- g^2}\right]dx
.. math::
V_3 = \int_w^g(r^2 - x^2)\tan^{-1}\frac{\sqrt{g^2 - x^2}}{z}dx
- \frac{z}{2}\left(g^2\cos^{-1}\frac{w}{g} - w\sqrt{2g(h-h_1)
- (h-h_1)^2}\right)
.. math::
V_{1,max} = v_1(h=h_1)
.. math::
v_{2,max} = v_2(h=h_2)
.. math::
v_{3,max} = \frac{\pi a_1}{6}(3g^2 + a_1^2)
.. math::
a_1 = fD(1-\cos\alpha)
.. math::
\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}
.. math::
n = R - kD + \sqrt{k^2D^2-x^2}
.. math::
g = r\sin\alpha
.. math::
r = fD
.. math::
h_2 = D - h_1
.. math::
w = R - h
.. math::
z = \sqrt{r^2- g^2}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
L : float
Length of the main cylindrical section, [m]
f : float
Dimensionless dish-radius parameter; also commonly given as the
product of `f` and `D` (`fD`), which is called both dish radius and
also crown radius and has units of length, [-]
k : float
Dimensionless knuckle-radius parameter; also commonly given as the
product of `k` and `D` (`kD`), which is called the knuckle radius
and has units of length, [-]
h : float
Height, as measured up to where the fluid ends, [m]
headonly : bool, optional
Function returns only the volume of a single head side if True
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_horiz_torispherical(D=108., L=156., f=1., k=0.06, h=36)/231.
2028.62667
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
if f is None or k is None:
raise ValueError("Missing f or k")
R = 0.5*D
R2 = R*R
hh = h*h
Af = R2*acos((R-h)/R) - (R-h)*sqrt(2.0*R*h - hh)
r = f*D
alpha = asin((1.0 - 2.0*k)/(2.*(f - k)))
cos_alpha = cos(alpha)
sin_alpha = sin(alpha)
a1 = r*(1.0 - cos_alpha)
g = r*sin_alpha
z = r*cos_alpha
h1 = k*D*(1.0 - sin_alpha)
h2 = D - h1
# Chebfun in Python failed on these functions
c10 = k*k*D*D
c11 = R - k*D
g2 = g*g
r2 = r*r
if 0.0 <= h <= h1:
w = R - h
Vf = 2.0*quad(V_horiz_torispherical_toint_1, 0.0, sqrt(2.0*k*D*h - hh), (w, c10, c11))[0]
elif h1 < h < h2:
w = R - h
wmax1 = R - h1
V1max = quad(V_horiz_torispherical_toint_1, 0.0, sqrt(2.0*k*D*h1 - h1*h1), (wmax1,c10, c11))[0]
V2 = quad(V_horiz_torispherical_toint_2, 0.0, k*D*cos_alpha, (w, c10, c11, g, g2))[0]
V3 = quad(V_horiz_torispherical_toint_3, w, g , (r2, g2, 1.0/z))[0] - 0.5*z*(g*g*acos(w/g) -w*sqrt(2.0*g*(h-h1) - (h-h1)*(h-h1)))
Vf = 2.0*(V1max + V2 + V3)
else:
w = R - h
wmax1 = R - h1
wmax2 = R - h2
wwerird = R - (D - h)
V1max = quad(V_horiz_torispherical_toint_1, 0.0, sqrt(2.0*k*D*h1-h1*h1), (wmax1,c10, c11))[0]
V1weird = quad(V_horiz_torispherical_toint_1, 0.0, sqrt(2.0*k*D*(D-h)-(D-h)*(D-h)), (wwerird,c10, c11))[0]
V2max = quad(V_horiz_torispherical_toint_2, 0.0, k*D*cos_alpha, (wmax2, c10, c11, g, g2))[0]
V3max = pi*a1/6.*(3.0*g*g + a1*a1)
Vf = 2.0*(2.0*V1max - V1weird + V2max + V3max)
if headonly:
Vf = Vf/2.
else:
Vf += Af*L
return Vf
### Begin vertical tanks
[docs]def V_vertical_conical(D, a, h):
r'''Calculates volume of a vertical tank with a convex conical bottom,
according to [1]_. No provision for the top of the tank is made here.
.. math::
V_f = \frac{\pi}{4}\left(\frac{Dh}{a}\right)^2\left(\frac{h}{3}\right),\; h < a
.. math::
V_f = \frac{\pi D^2}{4}\left(h - \frac{2a}{3}\right),\; h\ge a
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the cone head extends under the main cylinder, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_vertical_conical(132., 33., 24)/231.
250.67461381371024
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
if h < a:
Vf = pi/4*(D*h/a)**2*(h/3.)
else:
Vf = pi*D**2/4*(h - 2*a/3.)
return Vf
[docs]def V_vertical_ellipsoidal(D, a, h):
r'''Calculates volume of a vertical tank with a convex ellipsoidal bottom,
according to [1]_. No provision for the top of the tank is made here.
.. math::
V_f = \frac{\pi}{4}\left(\frac{Dh}{a}\right)^2 \left(a - \frac{h}{3}\right),\; h < a
.. math::
V_f = \frac{\pi D^2}{4}\left(h - \frac{a}{3}\right),\; h \ge a
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the ellipsoid head extends under the main cylinder, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_vertical_ellipsoidal(132., 33., 24)/231.
783.3581681678445
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
if h < a:
Vf = pi/4*(D*h/a)**2*(a - h/3.)
else:
Vf = pi*D**2/4*(h - a/3.)
return Vf
[docs]def V_vertical_spherical(D, a, h):
r'''Calculates volume of a vertical tank with a convex spherical bottom,
according to [1]_. No provision for the top of the tank is made here.
.. math::
V_f = \frac{\pi h^2}{4}\left(2a + \frac{D^2}{2a} - \frac{4h}{3}\right),\; h < a
.. math::
V_f = \frac{\pi}{4}\left(\frac{2a^3}{3} - \frac{aD^2}{2} + hD^2\right),\; h\ge a
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the spherical head extends under the main cylinder, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_vertical_spherical(132., 33., 24)/231.
583.6018352850442
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
if h < a:
Vf = pi*h**2/4*(2*a + D**2/2/a - 4*h/3)
else:
Vf = pi/4*(2*a**3/3 - a*D**2/2 + h*D**2)
return Vf
[docs]def V_vertical_torispherical(D, f, k, h):
r'''Calculates volume of a vertical tank with a convex torispherical bottom,
according to [1]_. No provision for the top of the tank is made here.
.. math::
V_f = \frac{\pi h^2}{4}\left(2a_1 + \frac{D_1^2}{2a_1}
- \frac{4h}{3}\right),\; 0 \le h \le a_1
.. math::
V_f = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right)
+\pi u\left[\left(\frac{D}{2}-kD\right)^2 +s\right]
+ \frac{\pi tu^2}{2} - \frac{\pi u^3}{3} + \pi D(1-2k)\left[
\frac{2u-t}{4}\sqrt{s+tu-u^2} + \frac{t\sqrt{s}}{4}
+ \frac{k^2D^2}{2}\left(\cos^{-1}\frac{t-2u}{2kD}-\alpha\right)\right]
,\; a_1 < h \le a_1 + a_2
.. math::
V_f = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right)
+\frac{\pi t}{2}\left[\left(\frac{D}{2}-kD\right)^2 +s\right]
+\frac{\pi t^3}{12} + \pi D(1-2k)\left[\frac{t\sqrt{s}}{4}
+ \frac{k^2D^2}{2}\sin^{-1}(\cos\alpha)\right]
+ \frac{\pi D^2}{4}[h-(a_1+a_2)] ,\; a_1 + a_2 < h
.. math::
\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}
.. math::
a_1 = fD(1-\cos\alpha)
.. math::
a_2 = kD\cos\alpha
.. math::
D_1 = 2fD\sin\alpha
.. math::
s = (kD\sin\alpha)^2
.. math::
t = 2a_2
.. math::
u = h - fD(1-\cos\alpha)
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
f : float
Dimensionless dish-radius parameter; also commonly given as the
product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
k : float
Dimensionless knuckle-radius parameter; also commonly given as the
product of `k` and `D` (`kD`), which is called the knuckle radius
and has units of length, [-]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_vertical_torispherical(D=132., f=1.0, k=0.06, h=24)/231.
904.0688283793
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
if h <= 0.0:
return 0.0
if f is None or k is None:
raise ValueError("f and k are required")
alpha = asin((1.0 - 2.0*k)/(2.0*(f-k)))
sin_alpha = sin(alpha)
cos_alpha = cos(alpha)
a1 = f*D*(1.0 - cos_alpha)
a2 = k*D*cos_alpha
D1 = 2.0*f*D*sin_alpha
x1 = k*D*sin_alpha
s = x1*x1
t = a2 + a2
u = h - f*D*(1.0 - cos_alpha)
h2 = h*h
if 0.0 <= h <= a1:
Vf = 0.25*pi*h2*(a1 + a1 + 0.5*D1*D1/a1 - (4.0/3.0)*h)
elif a1 < h <= a1 + a2:
x2 = (0.5*D - k*D)
u2 = u*u
Vf = (0.25*pi*a1*((2.0/3.0)*a1*a1 + 0.5*D1*D1) + pi*u*(x2*x2 + s)
+ pi*u2*(0.5*t - u/3.) + pi*D*(1.0 - 2.0*k)*(0.25*(2.0*u - t)*sqrt(s + t*u
- u2) + 0.25*t*sqrt(s) + 0.5*k*k*D*D*(acos((t - 2.0*u)/(2.0*k*D)) - alpha)))
else:
Vf = pi/4*(2*a1**3/3. + a1*D1**2/2.) + pi*t/2.*((D/2 - k*D)**2
+ s) + pi*t**3/12. + pi*D*(1 - 2*k)*(t*sqrt(s)/4
+ k**2*D**2/2*asin(cos(alpha))) + pi*D**2/4*(h - (a1 + a2))
return Vf
### Begin vertical tanks with concave heads
[docs]def V_vertical_conical_concave(D, a, h):
r'''Calculates volume of a vertical tank with a concave conical bottom,
according to [1]_. No provision for the top of the tank is made here.
.. math::
V = \frac{\pi D^2}{12} \left(3h + a - \frac{(a+h)^3}{a^2}\right)
,\;\; 0 \le h < |a|
.. math::
V = \frac{\pi D^2}{12} (3h + a ),\;\; h \ge |a|
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Negative distance the cone head extends inside the main cylinder, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_vertical_conical_concave(D=113., a=-33, h=15)/231
251.15825565795188
References
----------
.. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical
Processing. December 18, 2003.
http://www.chemicalprocessing.com/articles/2003/193/
'''
if h <= 0.0:
return 0.0
if h < abs(a):
Vf = pi*D**2/12.*(3*h + a - (a+h)**3/a**2)
else:
Vf = pi*D**2/12.*(3*h + a)
return Vf
[docs]def V_vertical_ellipsoidal_concave(D, a, h):
r'''Calculates volume of a vertical tank with a concave ellipsoidal bottom,
according to [1]_. No provision for the top of the tank is made here.
.. math::
V = \frac{\pi D^2}{12} \left(3h + 2a - \frac{(a+h)^2(2a-h)}{a^2}\right)
,\;\; 0 \le h < |a|
.. math::
V = \frac{\pi D^2}{12} (3h + 2a ),\;\; h \ge |a|
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Negative distance the eppilsoid head extends inside the main cylinder, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_vertical_ellipsoidal_concave(D=113., a=-33, h=15)/231
44.84968851034856
References
----------
.. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical
Processing. December 18, 2003.
http://www.chemicalprocessing.com/articles/2003/193/
'''
if h <= 0.0:
return 0.0
if h < abs(a):
Vf = pi*D**2/12.*(3*h + 2*a - (a+h)**2*(2*a-h)/a**2)
else:
Vf = pi*D**2/12.*(3*h + 2*a)
return Vf
[docs]def V_vertical_spherical_concave(D, a, h):
r'''Calculates volume of a vertical tank with a concave spherical bottom,
according to [1]_. No provision for the top of the tank is made here.
.. math::
V = \frac{\pi}{12}\left[3D^2h + \frac{a}{2}(3D^2 + 4a^2) + (a+h)^3
\left(4 - \frac{3D^2 + 12a^2}{2a(a+h)}\right)\right],\;\; 0 \le h < |a|
.. math::
V = \frac{\pi}{12}\left[3D^2h + \frac{a}{2}(3D^2 + 4a^2) \right]
,\;\; h \ge |a|
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Negative distance the spherical head extends inside the main cylinder, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_vertical_spherical_concave(D=113., a=-33, h=15)/231
112.81405437348528
References
----------
.. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical
Processing. December 18, 2003.
http://www.chemicalprocessing.com/articles/2003/193/
'''
if h <= 0.0:
return 0.0
if h < abs(a):
Vf = pi/12*(3*D**2*h + a/2.*(3*D**2 + 4*a**2) + (a+h)**3*(4 - (3*D**2+12*a**2)/(2.*a*(a+h))))
else:
Vf = pi/12*(3*D**2*h + a/2.*(3*D**2 + 4*a**2))
return Vf
[docs]def V_vertical_torispherical_concave(D, f, k, h):
r'''Calculates volume of a vertical tank with a concave torispherical bottom,
according to [1]_. No provision for the top of the tank is made here.
.. math::
V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + v_1(h=a_1 + a_2 -h),\; 0 \le h < a_2
.. math::
V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + v_2(h=a_1 + a_2 -h),\; a_2 \le h < a_1 + a_2
.. math::
V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + 0,\; h \ge a_1 + a_2
.. math::
v_1 = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right)
+\pi u\left[\left(\frac{D}{2}-kD\right)^2 +s\right]
+ \frac{\pi tu^2}{2} - \frac{\pi u^3}{3} + \pi D(1-2k)\left[
\frac{2u-t}{4}\sqrt{s+tu-u^2} + \frac{t\sqrt{s}}{4}
+ \frac{k^2D^2}{2}\left(\cos^{-1}\frac{t-2u}{2kD}-\alpha\right)\right]
.. math::
v_2 = \frac{\pi h^2}{4}\left(2a_1 + \frac{D_1^2}{2a_1} - \frac{4h}{3}\right)
.. math::
\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}
.. math::
a_1 = fD(1-\cos\alpha)
.. math::
a_2 = kD\cos\alpha
.. math::
D_1 = 2fD\sin\alpha
.. math::
s = (kD\sin\alpha)^2
.. math::
t = 2a_2
.. math::
u = h - fD(1-\cos\alpha)
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
f : float
Dimensionless dish-radius parameter; also commonly given as the
product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
k : float
Dimensionless knuckle-radius parameter; also commonly given as the
product of `k` and `D` (`kD`), which is called the knuckle radius
and has units of length, [-]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
Matching example from [1]_, with inputs in inches and volume in gallons.
>>> V_vertical_torispherical_concave(D=113., f=0.71, k=0.081, h=15)/231
103.88569287163769
References
----------
.. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical
Processing. December 18, 2003.
http://www.chemicalprocessing.com/articles/2003/193/
'''
if h <= 0.0:
return 0.0
alpha = asin((1-2*k)/(2.*(f-k)))
a1 = f*D*(1-cos(alpha))
a2 = k*D*cos(alpha)
D1 = 2*f*D*sin(alpha)
s = (k*D*sin(alpha))**2
t = 2*a2
def V1(h):
u = h-f*D*(1-cos(alpha))
v1 = pi/4*(2*a1**3/3. + a1*D1**2/2.) + pi*u*((D/2.-k*D)**2 +s)
v1 += pi*t*u**2/2. - pi*u**3/3.
v1 += pi*D*(1-2*k)*((2*u-t)/4.*sqrt(s+t*u-u**2) + t*sqrt(s)/4.
+ k**2*D**2/2.*(acos((t-2*u)/(2*k*D)) -alpha))
return v1
def V2(h):
v2 = pi*h**2/4.*(2*a1 + D1**2/(2.*a1) - 4*h/3.)
return v2
if 0 <= h < a2:
Vf = pi*D**2*h/4 - V1(a1+a2) + V1(a1+a2-h)
elif a2 <= h < a1 + a2:
Vf = pi*D**2*h/4 - V1(a1+a2) + V2(a1+a2-h)
else:
Vf = pi*D**2*h/4 - V1(a1+a2)
return Vf
### Total surface area of heads, orientation-independent
[docs]def SA_ellipsoidal_head(D, a):
r'''Calculates the surface area of an ellipsoidal head according to [1]_ and [2]_.
The formula below is for the full shape, the result of which is halved. The
formula is for :math:`a < R`. In the equations, `a` is the same and `c` is `D`.
.. math::
\text{SA} = 2\pi a^2 + \frac{\pi c^2}{e_1}\ln\left(\frac{1+e_1}{1-e_1}
\right)
.. math::
e_1 = \sqrt{1 - \frac{c^2}{a^2}}
For the case of :math:`a \ge R` from [2]_, which is needed to make the tank
head volume grow linearly with length:
.. math::
\text{SA} = 2\pi R^2 + \frac{2\pi a^2 R}{\sqrt{a^2 - R^2}}\cos^{-1}\frac{R}{|a|}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the ellipsoidal head extends, [m]
Returns
-------
SA : float
Surface area [m^2]
Examples
--------
Spherical case
>>> SA_ellipsoidal_head(2, 1)
6.283185307179586
>>> SA_ellipsoidal_head(2, 1.5)
8.459109081729984
References
----------
.. [1] Weisstein, Eric W. "Spheroid." Text. Accessed March 14, 2016.
http://mathworld.wolfram.com/Spheroid.html.
.. [2] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
if D == a*2.0:
return 0.5*pi*D*D # necessary to avoid a division by zero when D == a
R = 0.5*D
if a < R:
R, a = min((R, a)), max((R, a))
e1 = sqrt(1.0 - R*R/(a*a))
if e1 != 1.0:
# try:
log_term = log((1.0 + e1)/(1.0 - e1))
# except ZeroDivisionError:
else:
# Limit as a goes to zero relative to D; may only be ~6 orders of
# magnitude smaller than D and will still occur
log_term = 0.0
return (2.0*pi*a*a + pi*R*R/e1*log_term)*0.5
else:
return pi*R*R + pi*a*a*R*1.0/sqrt(a*a - R*R)*acos(R/abs(a))
[docs]def SA_conical_head(D, a):
r'''Calculates the surface area of a conical head according to [1]_.
.. math::
SA = \frac{\pi D}{2} \sqrt{a^2 + \left(\frac{D}{2}\right)^2}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the conical head extends, [m]
Returns
-------
SA : float
Surface area [m^2]
Examples
--------
>>> SA_conical_head(2, 1)
4.442882938158366
References
----------
.. [1] Weisstein, Eric W. "Cone." Text. Accessed March 14, 2016.
http://mathworld.wolfram.com/Cone.html.
'''
return 0.5*pi*D*sqrt(a*a + 0.25*D*D)
[docs]def SA_guppy_head(D, a):
r'''Calculates the surface area of a guppy head according to [1]_.
Some work was involved in combining formulas for the ellipse of the head,
and the conic section on the sides.
.. math::
SA = \frac{\pi D}{4}\sqrt{D^2 + a^2} + \frac{\pi D}{2}a
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the conical head extends, [m]
Returns
-------
SA : float
Surface area [m^2]
Examples
--------
>>> SA_guppy_head(2, 1)
6.654000019110157
References
----------
.. [1] Weisstein, Eric W. "Cone." Text. Accessed March 14, 2016.
http://mathworld.wolfram.com/Cone.html.
'''
return 0.25*pi*D*sqrt(a*a + D*D) + 0.5*pi*D*a
[docs]def SA_torispheroidal(D, f, k):
r'''Calculates surface area of a torispherical head according to [1]_.
Somewhat involved. Equations are adapted to be used for a full head.
.. math::
SA = S_1 + S_2
.. math::
S_1 = 2\pi D^2 f_d \alpha
.. math::
S_2 = 2\pi D^2 f_k\left(\alpha - \alpha_1 + (0.5 - f_k)\left(\sin^{-1}
\left(\frac{\alpha-\alpha_2}{f_k}\right) - \sin^{-1}\left(\frac{
\alpha_1-\alpha_2}{f_k}\right)\right)\right)
.. math::
\alpha_1 = f_d\left(1 - \sqrt{1 - \left(\frac{0.5 - f_k}{f_d-f_k}
\right)^2}\right)
.. math::
\alpha_2 = f_d - \sqrt{f_d^2 - 2f_d f_k + f_k - 0.25}
.. math::
\alpha = \frac{a}{D_i}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
f : float
Dimensionless dish-radius parameter; also commonly given as the
product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
k : float
Dimensionless knuckle-radius parameter; also commonly given as the
product of `k` and `D` (`kD`), which is called the knuckle radius
and has units of length, [-]
Returns
-------
SA : float
Surface area [m^2]
Examples
--------
Example from [1]_.
>>> SA_torispheroidal(D=2.54, f=1.039370079, k=0.062362205)
6.00394283477063
References
----------
.. [1] Honeywell. "Calculate Surface Areas and Cross-sectional Areas in
Vessels with Dished Heads". https://www.honeywellprocess.com/library/marketing/whitepapers/WP-VesselsWithDishedHeads-UniSimDesign.pdf
Whitepaper. 2014.
'''
D2 = D*D
x1 = 2.0*pi*D2
k_inv = 1.0/k
x2 = ((0.5 - k)/(f-k))
alpha_1 = f*(1.0 - sqrt(1.0 - x2*x2))
alpha_2 = f - sqrt(f*f - 2.0*f*k + k - 0.25)
alpha = alpha_1 # Up to top of dome
S1 = x1*f*alpha_1
alpha = alpha_2 # up to top of torus
S2_sub = asin((alpha-alpha_2)*k_inv) - asin((alpha_1-alpha_2)*k_inv)
S2 = x1*k*(alpha - alpha_1 + (0.5 - k) *S2_sub)
return S1 + S2
[docs]def SA_tank(D, L, sideA=None, sideB=None, sideA_a=0,
sideB_a=0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None):
r'''Calculates the surface are of a cylindrical tank with optional heads.
In the degenerate case of being provided with only `D` and `L`, provides
the surface area of a cylinder.
Parameters
----------
D : float
Diameter of the cylindrical section of the tank, [m]
L : float
Length of the main cylindrical section of the tank, [m]
sideA : string, optional
The left (or bottom for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical'].
sideB : string, optional
The right (or top for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical'].
sideA_a : float, optional
The distance the head as specified by sideA extends down or to the left
from the main cylindrical section, [m]
sideB_a : float, optional
The distance the head as specified by sideB extends up or to the right
from the main cylindrical section, [m]
sideA_f : float, optional
Dish-radius parameter for side A; fD = dish radius [1/m]
sideA_k : float, optional
knuckle-radius parameter for side A; kD = knuckle radius [1/m]
sideB_f : float, optional
Dish-radius parameter for side B; fD = dish radius [1/m]
sideB_k : float, optional
knuckle-radius parameter for side B; kD = knuckle radius [1/m]
Returns
-------
SA : float
Surface area of the tank [m^2]
sideA_SA : float
Surface area of only `sideA` [m^2]
sideB_SA : float
Surface area of only `sideB` [m^2]
lateral_SA : float
Surface area of cylindrical section of tank [m^2]
Examples
--------
Cylinder, Spheroid, Long Cones, and spheres. All checked.
>>> SA_tank(D=2, L=2)[0]
18.84955592153876
>>> SA_tank(D=1., L=0, sideA='ellipsoidal', sideA_a=2, sideB='ellipsoidal',
... sideB_a=2)[0]
10.124375616183062
>>> SA_tank(D=1., L=5, sideA='conical', sideA_a=2, sideB='conical',
... sideB_a=2)[0]
22.18452243965656
>>> SA_tank(D=1., L=5, sideA='spherical', sideA_a=0.5, sideB='spherical',
... sideB_a=0.5)[0]
18.84955592153876
'''
# Side A
if sideA == 'conical':
sideA_SA = SA_conical_head(D=D, a=sideA_a)
elif sideA == 'ellipsoidal':
sideA_SA = SA_ellipsoidal_head(D=D, a=sideA_a)
elif sideA == 'guppy':
sideA_SA = SA_guppy_head(D=D, a=sideA_a)
elif sideA == 'spherical':
sideA_SA = pi*(sideA_a*sideA_a + 0.25*D*D) # (SA_partial_sphere(D=D, h=sideA_a)
elif sideA == 'torispherical':
if sideA_f is None:
raise ValueError("Missing torispherical `f` parameter for sideA")
if sideA_k is None:
raise ValueError("Missing torispherical `k` parameter for sideA")
sideA_SA = SA_torispheroidal(D=D, f=sideA_f, k=sideA_k)
else:
sideA_SA = pi/4*D**2 # Circle
# Side B
if sideB == 'conical':
sideB_SA = SA_conical_head(D=D, a=sideB_a)
elif sideB == 'ellipsoidal':
sideB_SA = SA_ellipsoidal_head(D=D, a=sideB_a)
elif sideB == 'guppy':
sideB_SA = SA_guppy_head(D=D, a=sideB_a)
elif sideB == 'spherical':
sideB_SA = pi*(sideB_a*sideB_a + 0.25*D*D)#SA_partial_sphere(D=D, h=sideB_a)
elif sideB == 'torispherical':
if sideB_f is None:
raise ValueError("Missing torispherical `f` parameter for sideB")
if sideB_k is None:
raise ValueError("Missing torispherical `k` parameter for sideB")
sideB_SA = SA_torispheroidal(D=D, f=sideB_f, k=sideB_k)
else:
sideB_SA = pi/4*D**2 # Circle
lateral_SA = pi*D*L
SA = sideA_SA + sideB_SA + lateral_SA
return SA, sideA_SA, sideB_SA, lateral_SA
[docs]def V_tank(D, L, horizontal=True, sideA=None, sideB=None, sideA_a=0.0,
sideB_a=0.0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None):
r'''Calculates the total volume of a vertical or horizontal tank with
different head types.
Parameters
----------
D : float
Diameter of the cylindrical section of the tank, [m]
L : float
Length of the main cylindrical section of the tank, [m]
horizontal : bool, optional
Whether or not the tank is a horizontal or vertical tank
sideA : string, optional
The left (or bottom for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical'].
sideB : string, optional
The right (or top for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical'].
sideA_a : float, optional
The distance the head as specified by sideA extends down or to the left
from the main cylindrical section, [m]
sideB_a : float, optional
The distance the head as specified by sideB extends up or to the right
from the main cylindrical section, [m]
sideA_f : float, optional
Dimensionless dish-radius parameter for side A; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideA_k : float, optional
Dimensionless knuckle-radius parameter for side A; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
sideB_f : float, optional
Dimensionless dish-radius parameter for side B; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideB_k : float, optional
Dimensionless knuckle-radius parameter for side B; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
Returns
-------
V : float
Total volume [m^3]
sideA_V : float
Volume of only `sideA` [m^3]
sideB_V : float
Volume of only `sideB` [m^3]
lateral_V : float
Volume of cylindrical section of tank [m^3]
Examples
--------
>>> V_tank(D=1.5, L=5., horizontal=False, sideA='conical',
... sideB='conical', sideA_a=2., sideB_a=1.)
(10.602875205865551, 1.1780972450961726, 0.5890486225480863, 8.835729338221293)
'''
if sideA is not None and sideA not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'):
raise ValueError('Unspoorted head type for side A')
if sideB is not None and sideB not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'):
raise ValueError('Unspoorted head type for side B')
R = 0.5*D
sideA_V = sideB_V = lateral_V = 0.0
if horizontal:
# Conical case
if sideA == 'conical' and sideB == 'conical' and sideA_a == sideB_a:
sideB_V = sideA_V = V_horiz_conical(D, L, sideA_a, D, headonly=True)
else:
if sideA == 'conical':
sideA_V = V_horiz_conical(D, L, sideA_a, D, headonly=True)
if sideB == 'conical':
sideB_V = V_horiz_conical(D, L, sideB_a, D, headonly=True)
# Elliosoidal case
if sideA == 'ellipsoidal' and sideB == 'ellipsoidal' and sideA_a == sideB_a:
sideB_V = sideA_V = V_horiz_ellipsoidal(D, L, sideA_a, D, headonly=True)
else:
if sideA == 'ellipsoidal':
sideA_V = V_horiz_ellipsoidal(D, L, sideA_a, D, headonly=True)
if sideB == 'ellipsoidal':
sideB_V = V_horiz_ellipsoidal(D, L, sideB_a, D, headonly=True)
# Guppy case
if sideA == 'guppy' and sideB == 'guppy' and sideA_a == sideB_a:
sideB_V = sideA_V = V_horiz_guppy(D, L, sideA_a, D, headonly=True)
else:
if sideA == 'guppy':
sideA_V = V_horiz_guppy(D, L, sideA_a, D, headonly=True)
if sideB == 'guppy':
sideB_V = V_horiz_guppy(D, L, sideB_a, D, headonly=True)
# Spherical case
if sideA == 'spherical' and sideB == 'spherical' and sideA_a == sideB_a:
sideB_V = sideA_V = V_horiz_spherical(D, L, sideA_a, D, headonly=True)
else:
if sideA == 'spherical':
sideA_V = V_horiz_spherical(D, L, sideA_a, D, headonly=True)
if sideB == 'spherical':
sideB_V = V_horiz_spherical(D, L, sideB_a, D, headonly=True)
# Torispherical case
if (sideA == 'torispherical' and sideB == 'torispherical'
and (sideA_f == sideB_f) and (sideA_k == sideB_k)):
sideB_V = sideA_V = V_horiz_torispherical(D, L, sideA_f, sideA_k, D, headonly=True)
else:
if sideA == 'torispherical':
sideA_V = V_horiz_torispherical(D, L, sideA_f, sideA_k, D, headonly=True)
if sideB == 'torispherical':
sideB_V = V_horiz_torispherical(D, L, sideB_f, sideB_k, D, headonly=True)
Af = R*R*acos((R-D)/R) - (R-D)*sqrt(2.0*R*D - D*D)
lateral_V = L*Af
else:
# Bottom head
if sideA == 'conical':
sideA_V = V_vertical_conical(D, sideA_a, h=sideA_a)
if sideA == 'ellipsoidal':
sideA_V = V_vertical_ellipsoidal(D, sideA_a, h=sideA_a)
if sideA == 'spherical':
sideA_V = V_vertical_spherical(D, sideA_a, h=sideA_a)
if sideA == 'torispherical':
sideA_V = V_vertical_torispherical(D, sideA_f, sideA_k, h=sideA_a)
# Cylindrical section
lateral_V = pi/4*D**2*L # All middle
if sideB == 'conical':
sideB_V = V_vertical_conical(D, sideB_a, h=sideB_a)
if sideB == 'ellipsoidal':
sideB_V = V_vertical_ellipsoidal(D, sideB_a, h=sideB_a)
if sideB == 'spherical':
sideB_V = V_vertical_spherical(D, sideB_a, h=sideB_a)
if sideB == 'torispherical':
sideB_V = V_vertical_torispherical(D, sideB_f, sideB_k, h=sideB_a)
return lateral_V + sideA_V + sideB_V, sideA_V, sideB_V, lateral_V
[docs]def SA_partial_cylindrical_body(L, D, h):
r'''Calculates the partial area of a cylinder's body in the context of
a horizontal cylindrical vessel and liquid partially
filling it. This computes the wetted surface area of the bottom of the
cylinder.
.. math::
\text{SA} = L D \cos^{-1}\left(\frac{D - 2h}{D}\right)
Parameters
----------
L : float
Length of the cylinder, [m]
D : float
Diameter of the cylinder, [m]
h : float
Height measured from bottom of cylinder to liquid level, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area, [m^2]
Notes
-----
This method is undefined for :math:`h > D`. and :math:`h < 0`, but those
cases are handled by returning the full surface area and the zero
respectively.
Examples
--------
>>> SA_partial_cylindrical_body(L=200.0, D=96., h=22.0)
19168.852890279868
References
----------
.. [1] Weisstein, Eric W. "Circular Segment." Text. Wolfram Research, Inc.
Accessed May 10, 2020. https://mathworld.wolfram.com/CircularSegment.html.
'''
if h < 0.0:
return 0.0
elif h > D:
h = D
C = D*acos((D - h - h)/D)
return C*L
[docs]def A_partial_circle(D, h):
r'''Calculates the partial area of a circle, in the context of the circle
being an end cap to a horizontal cylindrical vessel and liquid partially
filling it. This computes the wetted surface area of one of the end caps.
Multiply this by two to obtain the wetted area of two end caps.
.. math::
\text{SA} = R^2\cos^{-1}\frac{(R - h)}{R} - (R - h)\sqrt{(2Rh - h^2)}
Parameters
----------
D : float
Diameter of the circle, [m]
h : float
Height measured from bottom of circle to liquid level, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area, [m^2]
Notes
-----
This method is undefined for :math:`h > D` and :math:`h < 0`, but those
cases are handled by returning the full surface area and the zero
respectively.
Examples
--------
>>> A_partial_circle(D=96., h=22.0)
1251.2018147383194
References
----------
.. [1] Weisstein, Eric W. "Circular Segment." Text. Wolfram Research, Inc.
Accessed May 10, 2020. https://mathworld.wolfram.com/CircularSegment.html.
'''
if h > D:
h = D # Catch the case of a computed `h` being trivially larger than `D` due to floating point
elif h < 0.0:
return 0.0
R = 0.5*D
SA = R*R*acos((R - h)/R) - (R - h)*sqrt(2.0*R*h - h*h)
if SA < 0.0:
SA = 0.0 # Catch trig errors
return SA
def circle_segment_area_inner(h, R, A_expect):
# 2 sqrt, 1 acos, 4 division
x0 = R*R
x1 = -h
x2 = R + x1
x3 = sqrt(h*(2.0*R + x1))
x4 = x2*x2
A_err = x0*acos(x2/R) - x2*x3 - A_expect
der = R/sqrt(1.0 - x4/x0) + x3 - x4/x3
return A_err, der
[docs]def circle_segment_h_from_A(A, D):
r'''Calculates the height of a chord of a circle given the area of that
circle segment. This is a numerical problem, solving the
following equation for `h`.
.. math::
\text{A} = R^2\cos^{-1}\frac{(R - h)}{R} - (R - h)\sqrt{(2Rh - h^2)}
Parameters
----------
A : float
Circle section area, [m^2]
D : float
Diameter of the circle, [m]
Returns
-------
h : float
Height measured from bottom of circle to the end of the circle section,
[m]
Notes
-----
Examples
--------
>>> circle_segment_h_from_A(A=1251.2018147383194, D=96.)
22.0
References
----------
.. [1] Weisstein, Eric W. "Circular Segment." Text. Wolfram Research, Inc.
Accessed May 10, 2020. https://mathworld.wolfram.com/CircularSegment.html.
'''
if A == 0.0:
return 0.0
R = 0.5*D
return newton(circle_segment_area_inner, x0=0.25*R, fprime=True, high=R, low=0.0,
args=(R, A), xtol=1e-12, bisection=True)
[docs]def SA_partial_horiz_conical_head(D, a, h):
r'''Calculates the partial area of a conical tank head in the context of
a horizontal vessel and liquid partially
filling it. This computes the wetted surface area of one of the conical
heads only.
.. math::
\text{SA} = \frac{\sqrt{(a^2 + R^2)}}{R}\left[R^2\cos^{-1}\left(\frac{
(R-h)}{R}\right) - (R-h)\sqrt{(2Rh - h^2)}\right]
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the cone head extends on one side, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one conical tank head, [m^2]
Notes
-----
This method is undefined for :math:`h > D` and :math:`h < 0`, but those
cases are handled by returning the full surface area and the zero
respectively.
Examples
--------
>>> SA_partial_horiz_conical_head(D=72., a=48.0, h=24.0)
1980.0498315169873
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
if h > D:
h = D
elif h < 0:
return 0.0
R = 0.5*D
R_inv = 1.0/R
return sqrt(a*a + R*R)*R_inv*(R*R*acos((R-h)*R_inv) - (R-h)*sqrt(2.0*R*h - h*h))
def _SA_partial_horiz_spherical_head_to_int(x, R2, a4, c1, c2):
x2 = x*x
to_pow = (R2 - x2)/(c2 - a4*x2)
if to_pow < 0.0:
to_pow = 0.0
num = c1*sqrt(to_pow)
try:
return asin(num)
except:
# Tried to asin a number just slightly higher than 1
return 0.5*pi
[docs]def SA_partial_horiz_spherical_head(D, a, h):
r'''Calculates the partial area of a spherical tank head in the context of
a horizontal vessel and liquid partially
filling it. This computes the wetted surface area of one of the spherical
heads only.
.. math::
\text{SA} = \frac{a^2 + R^2}{|a|}\int_{R-h}^R
\sin^{-1} \frac{2|a|\sqrt{R^2-x^2}} {\sqrt{(a^2+R^2)^2 - (2ax)^2}} dx
For the special case of :math:`|a| = R` :
.. math::
\text{SA} = \pi R h
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the spherical head extends on one side, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one spherical tank head, [m^2]
Notes
-----
This method is undefined for :math:`h > D` and :math:`h < 0`, but those
cases are handled by returning the full surface area and the zero
respectively.
A symbolic attempt did not suggest any analytical integrals were available.
Examples
--------
>>> SA_partial_horiz_spherical_head(D=72., a=48.0, h=24.0)
2027.2672
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
R = 0.5*D
if a == R:
return pi*R*h
elif h < 0.0:
return 0.0
elif h > D:
h = D
fact = (a*a + R*R)/abs(a)
R2 = R*R
a2 = a + a
a4 = a2*a2
c1 = 2.0*abs(a)
c2 = (a*a + R2)*(a*a + R2)
SA = fact*quad(_SA_partial_horiz_spherical_head_to_int, R-h, R, args=(R2, a4, c1, c2))[0]
return SA
def _SA_partial_horiz_ellipsoidal_head_to_int_dbl(x, y, c1, R2, R4, h):
y2 = y*y
x2 = x*x
num = c1*(x2 + y2) - R4
den = x2 + (y2 - R2) # Brackets help numerical truncation; zero div without it
try:
return sqrt(num/den)
except:
# Equation is undefined for y == R when x is zero; avoid it
return _SA_partial_horiz_ellipsoidal_head_to_int_dbl(x, y*(1.0 - 1e-12), c1, R2, R4, h)
def _SA_partial_horiz_ellipsoidal_head_limits(x, c1, R2, R4, h):
return [0.0, sqrt(R2 - x*x)]
def _SA_partial_horiz_ellipsoidal_head_limits2(c1, R2, R4, h):
R = sqrt(R2)
return [R-h, R]
def _SA_partial_horiz_ellipsoidal_head_to_int(y, c1, R2, R4):
y2 = y*y
t0 = c1*y2
x6 = c1*(y2 - R2)/(t0 - R4)
ans = sqrt(R4 - t0)*float(ellipe(x6))
return ans
[docs]def SA_partial_horiz_ellipsoidal_head(D, a, h):
r'''Calculates the partial area of a ellipsoidal tank head in the context of
a horizontal vessel and liquid partially
filling it. This computes the wetted surface area of one of the ellipsoidal
heads only.
.. math::
\text{SA} = \frac{2}{R} \int_{R-h}^R \int_0^{\sqrt{R^2 - x^2}}
\sqrt{ \frac{(R^2 - a^2)x^2
+ (R^2 - a^2)y^2 - R^4} {x^2 + y^2 - R^2}} dy dx
After extensive manipulation, the first integral was solved analytically,
extending the result of [1]_ with greater performance.
.. math::
\text{SA} = \frac{2}{R} \int_{R-h}^R \frac{\left(\frac{R^{4} - R^{2}
\left(R^{2} - a^{2}\right)}{R^{2} - y^{2}}\right)^{0.5} \left(R^{2}
- y^{2}\right)^{0.5} E{\left(\frac{\left(- R^{2} + y^{2}\right) \left(R^{2}
- a^{2}\right)}{- R^{4} + y^{2} \left(R^{2} - a^{2}\right)} \right)}}
{\left(\frac{R^{4} - R^{2} \left(R^{2} - a^{2}\right)}{R^{4} - y^{2}
\left(R^{2} - a^{2}\right)}\right)^{0.5}}
Where :math:`E(x)` is the complete elliptic integral of the second kind,
calculated with SciPy's link to the cephes library.
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the ellipsoidal head extends on one side, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one ellipsoidal tank head, [m^2]
Notes
-----
This method is undefined for :math:`h > D` and :math:`h < 0`, but those
cases are handled by returning the full surface area and the zero
respectively.
The original numerical double integral is extremely nasty - there are places
where f(x) -> infinity but that have a bounded area. quadpack's numerical
integration handles this well, but adaptive inetgration which is not
aware of singularities does not.
Examples
--------
>>> SA_partial_horiz_ellipsoidal_head(D=72., a=48.0, h=24.0)
3401.233622547
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
R = 0.5*D
if h < 0.0:
return 0.0
elif h > D:
h = D
R2 = R*R
R4 = R2*R2
a2 = a*a
c1 = R2 - a2
# from fluids.numerics import dblquad
# from scipy.integrate import dblquad, nquad
## quad_val = nquad(_SA_partial_horiz_ellipsoidal_head_to_int, ranges=[_SA_partial_horiz_ellipsoidal_head_limits, _SA_partial_horiz_ellipsoidal_head_limits2],
## args=(c1, R2, R4, h))[0]
# quad_val = dblquad(_SA_partial_horiz_ellipsoidal_head_to_int, R-h, R, lambda x: 0.0, lambda x: (R2 - x*x)**0.5,
# args=(c1, R2, R4, h))[0]
quad_val = quad(_SA_partial_horiz_ellipsoidal_head_to_int, R-h, R, args=(c1, R2, R4))[0]
SA = 2.0/R*quad_val
return SA
def _SA_partial_horiz_guppy_head_to_int(x, a, R):
x0 = a*a
x1 = R - x
x2 = x1*x1
x3 = 1.0/x2
x4 = x0*x3 + 1.0
x5 = R*R
x6 = x*x
x7 = x5 - x6
x8 = sqrt(x7)
x9 = x4*x8
x10 = x0 + 4.0*x5
x17 = sqrt(sqrt(x10))
x11 = x17*x17
x12 = 1.0/x11
x13 = a*x7
x14 = x12*x13*x3 + 1.0
x15 = a*x12
x16 = sqrt(a)
x18 = 2*atan(x16*x8/(x1*x17))
x19 = 0.5 - 0.5*x15
x100 = (-2.0*R*x*x11 + x11*x5 + x11*x6 + x13)
x20 = x1*x14*sqrt(x2*x5*(x0 + x2)/(x100*x100))/x5
return 0.08333333333333333*(
(-4.0*x10**0.75*x16*x20*ellipeinc(x18, x19) + 4.0*x9
+ 2.0*x17*x20*(a*x11 + x10)*ellipkinc(x18, x19)/x16
+ 8.0*x15*x9/x14)*1.0/sqrt(x4))
[docs]def SA_partial_horiz_guppy_head(D, a, h):
r'''Calculates the partial area of a guppy tank head in the context of
a horizontal vessel and liquid partially
filling it. This computes the wetted surface area of one of the guppy
heads only.
.. math::
\text{SA} = 2\int_{-R}^{h-R}\int_{0}^{\sqrt{R^2 - x^2}}
\sqrt{1 + \left(\frac{a}{2R}\left(1 - \frac{y^2}{(R-x)^2} \right)
\right)^2
+ \left(\frac{ay}{R(R-x)} \right)^2 } dy dx
After extensive manipulation, the first integral was solved analytically,
extending the result of [1]_. Even with the special functions, this
form has somewhat greater performance (and improved precision).
.. math::
\text{SA} = 2 \int_{-R}^{h-R} \frac{\frac{2 a \left(4 + \frac{a^{2}
\left(2 R^{2} - 2 R y\right)^{2}}{R^{2} \left(R - y\right)^{4}}\right)
\sqrt{R^{2} - y^{2}}}{\sqrt{4 R^{2} + a^{2}} \left(\frac{a \left(R^{2}
- y^{2}\right)}{\left(R - y\right)^{2} \sqrt{4 R^{2} + a^{2}}}
+ 1\right)} + \left(4 + \frac{a^{2} \left(2 R^{2} - 2 R y\right)
^{2}}{R^{2} \left(R - y\right)^{4}}\right) \sqrt{R^{2} - y^{2}}
- \frac{2 \sqrt{a} \sqrt{\frac{4 R^{2} \left(R - y\right)^{4}
+ a^{2} \left(2 R^{2} - 2 R y\right)^{2}}{\left(R^{2} \sqrt{4 R^{2}
+ a^{2}} - 2 R y \sqrt{4 R^{2} + a^{2}} + a \left(R^{2}
- y^{2}\right) + y^{2} \sqrt{4 R^{2} + a^{2}}\right)^{2}}}
\left(R - y\right) \left(4 R^{2} + a^{2}\right)^{0.75}
\left(\frac{a \left(R^{2} - y^{2}\right)}{\left(R - y\right)^{2}
\sqrt{4 R^{2} + a^{2}}} + 1\right) \operatorname{ellipeinc}{\left(2
\operatorname{atan}{\left(\frac{\sqrt{a} \sqrt{R^{2} - y^{2}}}
{\left(R - y\right) \left(4 R^{2} + a^{2}\right)^{0.25}} \right)},
- \frac{a}{2 \sqrt{4 R^{2} + a^{2}}} + 0.5 \right)}}{R^{2}}
+ \frac{1.0 \sqrt{\frac{4 R^{2} \left(R - y\right)^{4} + a^{2}
\left(2 R^{2} - 2 R y\right)^{2}}{\left(R^{2} \sqrt{4 R^{2}
+ a^{2}} - 2 R y \sqrt{4 R^{2} + a^{2}} + a \left(R^{2} - y^{2}\right)
+ y^{2} \sqrt{4 R^{2} + a^{2}}\right)^{2}}} \left(R - y\right)
\left(4 R^{2} + a^{2}\right)^{0.25} \left(\frac{a \left(R^{2}
- y^{2}\right)}{\left(R - y\right)^{2} \sqrt{4 R^{2} + a^{2}}}
+ 1\right) \left(4 R^{2} + a^{2} + a \sqrt{4 R^{2} + a^{2}}\right)
\operatorname{ellipkinc}{\left(2 \operatorname{atan}{\left(\frac{\sqrt{a}
\sqrt{R^{2} - y^{2}}}{\left(R - y\right) \left(4 R^{2}
+ a^{2}\right)^{0.25}} \right)},- \frac{a}{2 \sqrt{4 R^{2} + a^{2}}}
+ 0.5 \right)}}{R^{2} \sqrt{a}}}{6 \sqrt{4 + \frac{a^{2} \left(2 R^{2}
- 2 R y\right)^{2}}{R^{2} \left(R - y\right)^{4}}}}
Where ellipeinc is the incomplete elliptic integral of the second kind,
and ellipkinc is the incomplete elliptic integral of the first kind,
both calculated with SciPy's link to the cephes library.
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the guppy head extends on one side, [m]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one guppy tank head, [m^2]
Notes
-----
This method is undefined for :math:`h > D` and :math:`h < 0`, but those
cases are handled by returning the full surface area and the zero
respectively.
The analytical integral was derived with Rubi.
Examples
--------
>>> SA_partial_horiz_guppy_head(D=72., a=48.0, h=24.0)
1467.8949
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
R = 0.5*D
if a == R:
return pi*R*h
elif h < 0.0:
return 0.0
elif h > D:
h = D
if -R == h-R:
return 0.0
# c1 = a/(2.0*R)
# c2 = c1*c1
# a_R_ratio = a/R
# a_R_ratio2 = a_R_ratio*a_R_ratio
# from scipy.integrate import dblquad
# def to_quad(y, x):
# t1 = y/(R-x)
# t1 *= t1
# t2 = (1.0 - t1)
# return (1.0 + c2*t2*t2 + a_R_ratio2*t1)**0.5
# quad_val = dblquad(to_quad, -R, h-R, lambda x: 0.0, lambda x: (R*R - x*x)**0.5)[0]
quad_val = quad(_SA_partial_horiz_guppy_head_to_int, -R, h-R, args=(a, R))[0]
SA = 2.0*quad_val
return SA
def _SA_partial_horiz_torispherical_head_int_1(x, b, c):
x0 = x*x
x1 = b - x0
x2 = sqrt(x1)
x3 = -b + x0
x4 = c*c
try:
x5 = 1.0/sqrt(x1 - x4)
except:
x5 = 1.0/csqrt(x1 - x4)
x6 = x3 + x4
x7 = sqrt(b)
try:
x3_pow = x3**(-1.5)
except:
x3_pow = (x3+0j)**(-1.5)
ans = (x*cacos(c/x2) + x3_pow*x5*(-c*x1*csqrt(-x6*x6)*catan(x*x2/(csqrt(x3)*csqrt(x6)))
+ x6*x7*csqrt(-x1*x1)*catan(c*x*x5/x7))/csqrt(-x6/x1))
return abs(ans.real)
def _SA_partial_horiz_torispherical_head_int_2(y, t2, s, c1):
# from mpmath import mp, mpf, atanh as catanh
# mp.dps=30
# y, t2, s, c1 = mpf(y), mpf(t2), mpf(s), mpf(c1)
y2 = y*y
try:
x10 = sqrt(t2 - y2)
try:
# Some tiny heights make the square root slightly under 0
x = (sqrt(c1 - y2 + (s+s)*x10)).real
except:
# Python 2 compat - don't take the square root of a negative number with no complex part
x = (csqrt(c1 - y2 + (s+s)*x10 + 0.0j)).real
except:
x10 = csqrt(t2 - y2+0.0j)
x = (csqrt(c1 - y2 + (s+s)*x10 + 0.0j)).real
try:
x0 = t2 - y2
x1 = s*x10.real
t10 = x1 + x1 + s*s + x0
# x3, x4 present a very nasty numerical problem.
# issue occurs when h == R, x3 is really equal to R**2 - 2*R*h + h**2
x3 = t10 - x*x
x4 = sqrt(x3)
# One solution is to use higher precision everywhere
ans = x4*sqrt(t2*t10/(x0*x3))*catan(x/x4).real
except:
ans = 0.0
# ans = sqrt((t2* (s**2+t2-x**2+2.0*s* sqrt(t2-x**2)))/((t2-x**2)* (s**2+t2-x**2+2 *s* sqrt(t2-x**2)-y**2)))* sqrt(s**2+t2-x**2+2 *s* sqrt(t2-x**2)-y**2) *atan(y/sqrt(s**2+t2-x**2+2 *s* sqrt(t2-x**2)-y**2))
# print(float(y), float(t2), float(s), float(c1), float(ans.real))
# return float(ans.real)
return ans.real
def _SA_partial_horiz_torispherical_head_int_3(y, x, s, t2):
x2 = x*x
y2 = y*y
x10 = sqrt(t2 - x2)
num = (s + x10)*(s + x10)*x2 + (t2 - x2)*y2
den = (t2 - x2)*(s*s + t2 - x2 - y2 + 2.0*s*x10)
f = sqrt(1.0 + num/den)
return f
[docs]def SA_partial_horiz_torispherical_head(D, f, k, h):
r'''Calculates the partial area of a torispherical tank head in the context of
a horizontal vessel and liquid partially
filling it. This computes the wetted surface area of one of the torispherical
heads only.
The expressions used are quite complicated; see [1]_ for more details.
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
f : float
Dimensionless dish-radius parameter; also commonly given as the
product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
k : float
Dimensionless knuckle-radius parameter; also commonly given as the
product of `k` and `D` (`kD`), which is called the knuckle radius
and has units of length, [-]
h : float
Height, as measured up to where the fluid ends, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one torispherical tank head, [m^2]
Notes
-----
This method is undefined for :math:`h > D` and :math:`h < 0`, but those
cases are handled by returning the full surface area and the zero
respectively.
One integral:
.. math::
\int_{R-h}^{fD\sin \alpha} \cos^{-1} \frac{fD\cos \alpha}{\sqrt{f^2 D^2
- x^2}} dx
Can be computed as follows, using WolframAlpha.
.. math::
x \operatorname{acos}{\left(\frac{c}{\sqrt{b - x^{2}}} \right)} + \frac{\sqrt{b}
\sqrt{- \left(b - x^{2}\right)^{2}} \left(- b + c^{2} + x^{2}\right)
\operatorname{atan}{\left(\frac{c x}{\sqrt{b} \sqrt{b - c^{2} - x^{2}}}
\right)} + c \sqrt{- \left(- b + c^{2} + x^{2}\right)^{2}} \left(- b
+ x^{2}\right) \operatorname{atan}{\left(\frac{x \sqrt{b - x^{2}}}
{\sqrt{- b + x^{2}} \sqrt{- b + c^{2} + x^{2}}} \right)}}{\sqrt{
\frac{- b + c^{2} + x^{2}}{- b + x^{2}}} \left(- b + x^{2}\right)^{1.5}
\sqrt{b - c^{2} - x^{2}}}
With the following constants:
.. math::
c = fD\cos \alpha
.. math::
b = f^2 D^2
The other integral is a double integral. There is an analytical integral
available for the first integral, which takes the form:
.. math::
2 \sqrt{\frac{R^{2} k^{2} \left(4 R^{2} k^{2} - y^{2} + \left(- 2 R k
+ R\right)^{2} + 2 \left(- 2 R k + R\right) \sqrt{4 R^{2} k^{2} - y^{2}}
\right)}{\left(4 R^{2} k^{2} - y^{2}\right) \left(\left(R
- h\right)^{2} - \left(- 4 R k + 2 R\right) \sqrt{4 R^{2} k^{2}
- y^{2}} + 2 \left(- 2 R k + R\right) \sqrt{4 R^{2} k^{2} - y^{2}}
\right)}} \sqrt{\left(R - h\right)^{2} - \left(- 4 R k + 2 R\right)
\sqrt{4 R^{2} k^{2} - y^{2}} + 2 \left(- 2 R k + R\right)
\sqrt{4 R^{2} k^{2} - y^{2}}} \operatorname{atan}{\left(\frac{
\sqrt{4 R^{2} k^{2} - y^{2} - \left(R - h\right)^{2} + \left(- 4 R k
+ 2 R\right) \sqrt{4 R^{2} k^{2} - y^{2}} + \left(- 2 R k + R\right)
^{2}}}{\sqrt{\left(R - h\right)^{2} - \left(- 4 R k + 2 R\right)
\sqrt{4 R^{2} k^{2} - y^{2}} + 2 \left(- 2 R k + R\right)
\sqrt{4 R^{2} k^{2} - y^{2}}}} \right)}
Examples
--------
>>> SA_partial_horiz_torispherical_head(D=72., f=1, k=.06, h=24.0)
1471.201832459
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
if h <= 0.0:
return 0.0
elif h > D:
h = D
R = D/2.
r = f*D
alpha = asin((1.0 - 2.0*k)/(2.*(f-k)))
cos_alpha = cos(alpha)
sin_alpha = sin(alpha)
s = R - k*D
t = k*D
s2 = s*s
t2 = t*t
a1 = r*(1.0 - cos_alpha)
a2 = k*D*cos_alpha
c = f*D*cos_alpha
b = f*f*D*D
c1 = s2 + t2 - (R - h)**2
def G_lim(x): # numba: delete
x2 = x*x # numba: delete
try: # numba: delete
G = sqrt(c1 - x2 + (s+s)*sqrt(t2 - x2)) # numba: delete
except: # numba: delete
# Python 2 compat - don't take the square root of a negative number with no complex part # numba: delete
G = sqrt(c1 - x2 + (s+s)*sqrt(t2 - x2+0.0j) + 0.0j) # numba: delete
return G.real # Some tiny heights make the square root slightly under 0 # numba: delete
limit_1 = k*D*(1.0 - sin_alpha)
if h < limit_1:
SA = quad(_SA_partial_horiz_torispherical_head_int_2, 0.0, sqrt(2*k*D*h - h*h), args=(t2, s, c1))[0]
return 2.0*SA
elif limit_1 < h <= R:
if (D*.499 < h < D*.501): # numba: delete
from scipy.integrate import dblquad # numba: delete
SA = 2.0*dblquad(_SA_partial_horiz_torispherical_head_int_3, 0.0, a2, lambda x: 0, G_lim, args=(s, t2))[0] # numba: delete
else: # numba: delete
# Numerical issues
SA = 2.0*quad(_SA_partial_horiz_torispherical_head_int_2, 0.0, a2, args=(t2, s, c1))[0] # numba: delete
# SA = 2.0*quad(_SA_partial_horiz_torispherical_head_int_2, 0.0, a2, args=(t2, s, c1))[0] # numba: uncomment
try:
high = _SA_partial_horiz_torispherical_head_int_1(f*D*sin_alpha, b, c)
except:
# Expression with the substitution is equally complicated
high = _SA_partial_horiz_torispherical_head_int_1(f*D*sin_alpha*(1.0 + 1e-14), b, c)
int_1_term1 = high - _SA_partial_horiz_torispherical_head_int_1(R-h, b, c)
SA += 2.0*f*D*int_1_term1
else:
SA = 2.0*pi*f*D*a1 + 2*pi*k*D*(a2 + (R - k*D)*asin(a2/(k*D)))
SA -= SA_partial_horiz_torispherical_head(D, f, k, h=D-h)
return SA
[docs]def SA_partial_vertical_conical_head(D, a, h):
r'''Calculates the partial area of a conical tank head in the context of
a vertical vessel and liquid partially
filling it. This computes the wetted surface area of one of the conical
heads only, and is valid for `h` up to `a` only.
.. math::
\text{SA} = \frac{\pi R h^2 \sqrt{a^2 + R^2}}{a^2}
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the cone head extends beneath the vertical tank, [m]
h : float
Height, as measured up to where the fluid ends or the top of the
conical head, whichever is less, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one conical tank head extending
beneath the vessel, [m^2]
Notes
-----
This method is undefined for :math:`h < 0`, but this is handled
by returning zero.
Examples
--------
>>> SA_partial_vertical_conical_head(D=72., a=48.0, h=24.0)
1696.4600329384882
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
if a == 0.0:
return 0.25*pi*D*D
elif h <= 0.0:
return 0.0
R = 0.5*D
SA = pi*R*h*h*sqrt(a*a + R*R)/(a*a)
return SA
[docs]def SA_partial_vertical_ellipsoidal_head(D, a, h):
r'''Calculates the partial area of a ellipsoidal tank head in the context of
a vertical vessel and liquid partially
filling it. This computes the wetted surface area of one of the ellipsoidal
heads only, and is valid for `h` up to `a` only.
If :math:`a > R`:
.. math::
\text{SA} = \pi R^2 - \frac{\pi (a - h)R\sqrt{a^4 - (a-h)^2(a^2-R^2)}}{a^2}
+ \frac{\pi a^2 R}{\sqrt{a^2 - R^2}}\left(
\cos^{-1} \frac{R}{a} - \sin^{-1} \frac{(a-h)\sqrt{a^2-R^2}}{a^2}
\right)
Otherwise for :math:`0 < a < R`:
.. math::
\text{SA} = \pi R^2 - \frac{\pi (a - h)R\sqrt{a^4 - (a-h)^2(a^2-R^2)}}{a^2}
+ \frac{\pi a^2 R}{\sqrt{a^2 - R^2}}\ln \left(\frac{a(\sqrt{R^2 - a^2} + R)}
{(a-h)\sqrt{R^2 - a^2} + \sqrt{a^4 + (a-h)^2(R^2 - a^2)}}
\right)
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the ellipsoidal head extends beneath the vertical tank, [m]
h : float
Height, as measured up to where the fluid ends or the top of the
ellipsoidal head, whichever is less, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one ellipsoidal tank head extending
beneath the vessel, [m^2]
Notes
-----
This method is undefined for :math:`h < 0`, but this is handled
by returning zero.
Examples
--------
>>> SA_partial_vertical_ellipsoidal_head(D=72., a=48.0, h=24.0)
4675.23789137632
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
if a == 0.0:
return 0.25*pi*D*D
elif h <= 0.0:
return 0.0
# h should be less than a
R = 0.5*D
SA = pi*R*R
a2 = a*a
a_inv = 1.0/a
R2 = R*R
SA -= pi*(a - h)*R*sqrt(a2*a2 - (a-h)*(a-h)*(a2 - R2))*a_inv*a_inv
if a > R:
# This one has issues around a == R
SA += pi*a2*R/sqrt(a2 - R2)*(acos(R*a_inv) - asin((a-h)*sqrt(a2 - R2)*a_inv*a_inv))
elif a == R:
# Special case avoids zero division
return pi*D*h
else:
x1 = sqrt(R2 - a2)
num = a*(x1 + R)
den = (a-h)*x1 + sqrt(a2*a2 + (a-h)*(a-h)*(R2 - a2))
SA += pi*a2*R/x1*log(num/den)
return SA
[docs]def SA_partial_vertical_spherical_head(D, a, h):
r'''Calculates the partial area of a spherical tank head in the context of
a vertical vessel and liquid partially
filling it. This computes the wetted surface area of one of the conical
heads only, and is valid for `h` up to `a` only.
.. math::
\text{SA} = \pi h \left(\frac{a^2 + R^2}{a}\right)
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
a : float
Distance the spherical head extends beneath the vertical tank, [m]
h : float
Height, as measured up to where the fluid ends or the top of the
spherical head, whichever is less, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one spherical tank head extending
beneath the vessel, [m^2]
Notes
-----
This method is undefined for :math:`h < 0`, but this is handled
by returning zero.
Examples
--------
>>> SA_partial_vertical_spherical_head(72, a=24, h=12)
2940.5307237600464
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
if a == 0.0:
return 0.25*pi*D*D
elif h <= 0.0:
return 0.0
R = 0.5*D
SA = pi*h*((a*a + R*R)/a)
return SA
[docs]def SA_partial_vertical_torispherical_head(D, f, k, h):
r'''Calculates the partial area of a torispherical tank head in the context of
a vertical vessel and liquid partially
filling it. This computes the wetted surface area of one of the torispherical
heads only.
if :math:`a_1 <= h`:
.. math::
\text{SA} = 2\pi f D h
if :math:`a_1 \le h \le a`:
.. math::
\text{SA} = 2\pi f D a_1 + 2\pi k D\left(
h - a_1 + (R - kD) \left(
\sin^{-1} \frac{a_2}{kD} - \sin^{-1} \frac{a-h}{kD} \right) \right)
.. math::
\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}
.. math::
a_1 = fD(1-\cos\alpha)
.. math::
a_2 = kD\cos\alpha
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
f : float
Dimensionless dish-radius parameter; also commonly given as the
product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
k : float
Dimensionless knuckle-radius parameter; also commonly given as the
product of `k` and `D` (`kD`), which is called the knuckle radius
and has units of length, [-]
h : float
Height, as measured up to where the fluid ends or the top of the
torispherical head, whichever is less, [m]
Returns
-------
SA_partial : float
Partial (wetted) surface area of one torispherical tank head, [m^2]
Notes
-----
This method is undefined for :math:`h > D` and :math:`h < 0`, but those
cases are handled by returning the full surface area and the zero
respectively.
Examples
--------
This method is undefined for :math:`h < 0`, but this is handled
by returning zero.
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
'''
if h <= 0.0:
return 0.0
R = 0.5*D
alpha = asin((1.0 - 2.0*k)/(2.0*(f-k)))
cos_alpha = cos(alpha)
a1 = f*D*(1.0 - cos_alpha)
a2 = k*D*cos_alpha
a = a1 + a2
if h < a1:
SA = 2.0*pi*f*D*h
elif a1 <= h <= a:
SA = 2.0*pi*f*D*a1
kD_inv = 1.0/(k*D)
SA += 2.0*pi*k*D*(h - a1 + (R - k*D)*(asin(a2*kD_inv) - asin((a-h)*kD_inv)))
return SA
[docs]def a_torispherical(D, f, k):
r'''Calculates depth of a torispherical head according to [1]_.
.. math::
a = a_1 + a_2
.. math::
\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}
.. math::
a_1 = fD(1-\cos\alpha)
.. math::
a_2 = kD\cos\alpha
Parameters
----------
D : float
Diameter of the main cylindrical section, [m]
f : float
Dimensionless dish-radius parameter; also commonly given as the
product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
k : float
Dimensionless knuckle-radius parameter; also commonly given as the
product of `k` and `D` (`kD`), which is called the knuckle radius
and has units of length, [-]
Returns
-------
a : float
Depth of head [m]
Examples
--------
Example from [1]_.
>>> a_torispherical(D=96., f=0.9, k=0.2)
25.684268924767125
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF
'''
alpha = asin((1-2*k)/(2*(f-k)))
a1 = f*D*(1 - cos(alpha))
a2 = k*D*cos(alpha)
return a1 + a2
[docs]def V_from_h(h, D, L, horizontal=True, sideA=None, sideB=None, sideA_a=0,
sideB_a=0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None):
r'''Calculates partially full volume of a vertical or horizontal tank with
different head types according to [1]_.
Parameters
----------
h : float
Height of the liquid in the tank, [m]
D : float
Diameter of the cylindrical section of the tank, [m]
L : float
Length of the main cylindrical section of the tank, [m]
horizontal : bool, optional
Whether or not the tank is a horizontal or vertical tank
sideA : string, optional
The left (or bottom for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical'].
sideB : string, optional
The right (or top for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical'].
sideA_a : float, optional
The distance the head as specified by sideA extends down or to the left
from the main cylindrical section, [m]
sideB_a : float, optional
The distance the head as specified by sideB extends up or to the right
from the main cylindrical section, [m]
sideA_f : float, optional
Dimensionless dish-radius parameter for side A; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideA_k : float, optional
Dimensionless knuckle-radius parameter for side A; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
sideB_f : float, optional
Dimensionless dish-radius parameter for side B; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideB_k : float, optional
Dimensionless knuckle-radius parameter for side B; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
Returns
-------
V : float
Volume up to h [m^3]
Examples
--------
>>> V_from_h(h=7, D=1.5, L=5., horizontal=False, sideA='conical',
... sideB='conical', sideA_a=2., sideB_a=1.)
10.013826583317465
References
----------
.. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical
Processing. December 18, 2003.
http://www.chemicalprocessing.com/articles/2003/193/
'''
if sideA is not None and sideA not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'):
raise ValueError('Unspoorted head type for side A')
if sideB is not None and sideB not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'):
raise ValueError('Unspoorted head type for side B')
R = 0.5*D
V = 0.0
if horizontal:
# Conical case
if sideA == 'conical' and sideB == 'conical' and sideA_a == sideB_a:
V += 2.0*V_horiz_conical(D, L, sideA_a, h, headonly=True)
else:
if sideA == 'conical':
V += V_horiz_conical(D, L, sideA_a, h, headonly=True)
if sideB == 'conical':
V += V_horiz_conical(D, L, sideB_a, h, headonly=True)
# Elliosoidal case
if sideA == 'ellipsoidal' and sideB == 'ellipsoidal' and sideA_a == sideB_a:
V += 2.0*V_horiz_ellipsoidal(D, L, sideA_a, h, headonly=True)
else:
if sideA == 'ellipsoidal':
V += V_horiz_ellipsoidal(D, L, sideA_a, h, headonly=True)
if sideB == 'ellipsoidal':
V += V_horiz_ellipsoidal(D, L, sideB_a, h, headonly=True)
# Guppy case
if sideA == 'guppy' and sideB == 'guppy' and sideA_a == sideB_a:
V += 2.0*V_horiz_guppy(D, L, sideA_a, h, headonly=True)
else:
if sideA == 'guppy':
V += V_horiz_guppy(D, L, sideA_a, h, headonly=True)
if sideB == 'guppy':
V += V_horiz_guppy(D, L, sideB_a, h, headonly=True)
# Spherical case
if sideA == 'spherical' and sideB == 'spherical' and sideA_a == sideB_a:
V += 2.0*V_horiz_spherical(D, L, sideA_a, h, headonly=True)
else:
if sideA == 'spherical':
V += V_horiz_spherical(D, L, sideA_a, h, headonly=True)
if sideB == 'spherical':
V += V_horiz_spherical(D, L, sideB_a, h, headonly=True)
# Torispherical case
if (sideA == 'torispherical' and sideB == 'torispherical'
and (sideA_f == sideB_f) and (sideA_k == sideB_k)):
V += 2.0*V_horiz_torispherical(D, L, sideA_f, sideA_k, h, headonly=True)
else:
if sideA == 'torispherical':
V += V_horiz_torispherical(D, L, sideA_f, sideA_k, h, headonly=True)
if sideB == 'torispherical':
V += V_horiz_torispherical(D, L, sideB_f, sideB_k, h, headonly=True)
if h > D: # Must be before Af, which will raise a domain error
raise ValueError('Input height is above top of tank')
Af = R*R*acos((R-h)/R) - (R-h)*sqrt(2.0*R*h - h*h)
V += L*Af
else:
# Bottom head
if sideA in ('conical', 'ellipsoidal', 'torispherical', 'spherical'):
if sideA == 'conical':
V += V_vertical_conical(D, sideA_a, h=min(sideA_a, h))
if sideA == 'ellipsoidal':
V += V_vertical_ellipsoidal(D, sideA_a, h=min(sideA_a, h))
if sideA == 'spherical':
V += V_vertical_spherical(D, sideA_a, h=min(sideA_a, h))
if sideA == 'torispherical':
V += V_vertical_torispherical(D, sideA_f, sideA_k, h=min(sideA_a, h))
# Cylindrical section
if h >= sideA_a + L:
V += pi/4*D**2*L # All middle
elif h > sideA_a:
V += pi/4*D**2*(h - sideA_a) # Partial middle
# Top head
if h > sideA_a + L:
h2 = sideB_a - (h - sideA_a - L)
if sideB == 'conical':
V += V_vertical_conical(D, sideB_a, h=sideB_a)
V -= V_vertical_conical(D, sideB_a, h=h2)
if sideB == 'ellipsoidal':
V += V_vertical_ellipsoidal(D, sideB_a, h=sideB_a)
V -= V_vertical_ellipsoidal(D, sideB_a, h=h2)
if sideB == 'spherical':
V += V_vertical_spherical(D, sideB_a, h=sideB_a)
V -= V_vertical_spherical(D, sideB_a, h=h2)
if sideB == 'torispherical':
V += V_vertical_torispherical(D, sideB_f, sideB_k, h=sideB_a)
V -= max(0.0, V_vertical_torispherical(D, sideB_f, sideB_k, h=h2))
if h > L + sideA_a + sideB_a:
raise ValueError('Input height is above top of tank')
return V
[docs]def SA_from_h(h, D, L, horizontal=True, sideA=None, sideB=None, sideA_a=0.0,
sideB_a=0.0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None):
r'''Calculates partially full wetted surface area of a vertical or horizontal tank with
different head types according to [1]_.
Parameters
----------
h : float
Height of the liquid in the tank, [m]
D : float
Diameter of the cylindrical section of the tank, [m]
L : float
Length of the main cylindrical section of the tank, [m]
horizontal : bool, optional
Whether or not the tank is a horizontal or vertical tank
sideA : string, optional
The left (or bottom for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical'].
sideB : string, optional
The right (or top for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical'].
sideA_a : float, optional
The distance the head as specified by sideA extends down or to the left
from the main cylindrical section, [m]
sideB_a : float, optional
The distance the head as specified by sideB extends up or to the right
from the main cylindrical section, [m]
sideA_f : float, optional
Dimensionless dish-radius parameter for side A; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideA_k : float, optional
Dimensionless knuckle-radius parameter for side A; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
sideB_f : float, optional
Dimensionless dish-radius parameter for side B; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideB_k : float, optional
Dimensionless knuckle-radius parameter for side B; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
Returns
-------
SA : float
Wetted wall surface area up to h [m^3]
Examples
--------
>>> SA_from_h(h=7, D=1.5, L=5., horizontal=False, sideA='conical',
... sideB='conical', sideA_a=2., sideB_a=1.)
28.59477853914843
References
----------
.. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing.
April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf
http://www.chemicalprocessing.com/articles/2003/193/
'''
if sideA is not None and sideA not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'):
raise ValueError('Unspoorted head type for side A')
if sideB is not None and sideB not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'):
raise ValueError('Unspoorted head type for side B')
R = 0.5*D
SA = 0.0
if horizontal:
# Conical case
if sideA == 'conical':
SA += SA_partial_horiz_conical_head(D, sideA_a, h)
if sideB == 'conical':
SA += SA_partial_horiz_conical_head(D, sideB_a, h)
# Elliosoidal case
if sideA == 'ellipsoidal':
SA += SA_partial_horiz_ellipsoidal_head(D, sideA_a, h)
if sideB == 'ellipsoidal':
SA += SA_partial_horiz_ellipsoidal_head(D, sideB_a, h)
# Guppy case
if sideA == 'guppy':
SA += SA_partial_horiz_guppy_head(D, sideA_a, h)
if sideB == 'guppy':
SA += SA_partial_horiz_guppy_head(D, sideB_a, h)
# Spherical case
if sideA == 'spherical':
SA += SA_partial_horiz_spherical_head(D, sideA_a, h)
if sideB == 'spherical':
SA += SA_partial_horiz_spherical_head(D, sideB_a, h)
# Torispherical case
if sideA == 'torispherical':
if sideA_f is not None and sideA_k is not None:
SA += SA_partial_horiz_torispherical_head(D, sideA_f, sideA_k, h)
else:
raise ValueError("Torispherical sideA but no `f` and `k` provided")
if sideB == 'torispherical':
if sideB_f is not None and sideB_k is not None:
SA += SA_partial_horiz_torispherical_head(D, sideB_f, sideB_k, h)
else:
raise ValueError("Torispherical sideB but no `f` and `k` provided")
# Flat case
if sideA is None:
SA += A_partial_circle(D, h)
if sideB is None:
SA += A_partial_circle(D, h)
if h > D: # Must be before Af, which will raise a domain error
raise ValueError('Input height is above top of tank')
SA += L*D*acos((D - h - h)/D)
else:
# Bottom head
if sideA in ('conical', 'ellipsoidal', 'torispherical', 'spherical'):
if sideA == 'conical':
SA += SA_partial_vertical_conical_head(D, sideA_a, h=min(sideA_a, h))
elif sideA == 'ellipsoidal':
SA += SA_partial_vertical_ellipsoidal_head(D, sideA_a, h=min(sideA_a, h))
elif sideA == 'spherical':
SA += SA_partial_vertical_spherical_head(D, sideA_a, h=min(sideA_a, h))
elif sideA == 'torispherical':
if sideA_f is not None and sideA_k is not None:
SA += SA_partial_vertical_torispherical_head(D, sideA_f, sideA_k, h=min(sideA_a, h))
else:
raise ValueError("Torispherical sideA but no `f` and `k` provided")
elif sideA is None:
SA += 0.25*pi*D*D
# Cylindrical section
if h >= sideA_a + L:
SA += pi*D*L # All middle
elif h > sideA_a:
SA += pi*D*(h - sideA_a) # Partial middle
# Top head
if h >= sideA_a + L: # greater or equals is needed! Flat head on top adds lots of area.
h2 = sideB_a - (h - sideA_a - L)
if sideB == 'conical':
if sideB_a == 0.0:
SA += 0.25*pi*D*D
else:
SA += SA_partial_vertical_conical_head(D, sideB_a, h=sideB_a)
SA -= SA_partial_vertical_conical_head(D, sideB_a, h=h2)
elif sideB == 'ellipsoidal':
if sideB_a == 0.0:
SA += 0.25*pi*D*D
else:
SA += SA_partial_vertical_ellipsoidal_head(D, sideB_a, h=sideB_a)
SA -= SA_partial_vertical_ellipsoidal_head(D, sideB_a, h=h2)
elif sideB == 'spherical':
if sideB_a == 0.0:
SA += 0.25*pi*D*D
else:
SA += SA_partial_vertical_spherical_head(D, sideB_a, h=sideB_a)
SA -= SA_partial_vertical_spherical_head(D, sideB_a, h=h2)
elif sideB == 'torispherical':
if sideB_a == 0.0:
SA += 0.25*pi*D*D
else:
if sideB_f is not None and sideB_k is not None:
SA += SA_partial_vertical_torispherical_head(D, sideB_f, sideB_k, h=sideB_a)
SA -= max(0.0, SA_partial_vertical_torispherical_head(D, sideB_f, sideB_k, h=h2))
else:
raise ValueError("Torispherical sideB but no `f` and `k` provided")
elif sideB is None and h == sideA_a + L:
# End cap if flat
SA += 0.25*pi*D*D
if h > L + sideA_a + sideB_a:
raise ValueError('Input height is above top of tank')
return SA
def tank_from_two_specs_err(guess, spec0, spec1, spec0_name, spec1_name,
h, horizontal, sideA, sideB, sideA_a, sideB_a,
sideA_f, sideA_k, sideB_f, sideB_k,
sideA_a_ratio, sideB_a_ratio):
D, L_over_D = float(guess[0]), float(guess[1])
obj = TANK(D=D, L_over_D=L_over_D, horizontal=horizontal,
sideA=sideA, sideB=sideB, sideA_a=sideA_a, sideB_a=sideB_a,
sideA_f=sideA_f, sideA_k=sideA_k, sideB_f=sideB_f, sideB_k=sideB_k,
sideA_a_ratio=sideA_a_ratio, sideB_a_ratio=sideB_a_ratio)
# ensure h is always under the top
h = min(h, obj.h_max)
if spec0_name == 'V':
err0 = obj.V_total - spec0
elif spec0_name == 'SA':
err0 = obj.A - spec0
elif spec0_name == 'V_partial':
err0 = obj.V_from_h(h) - spec0
elif spec0_name == 'SA_partial':
err0 = obj.SA_from_h(h) - spec0
elif spec0_name == 'A_cross':
err0 = obj.A_cross_sectional(h) - spec0
if spec1_name == 'V':
err1 = obj.V_total - spec1
elif spec1_name == 'SA':
err1 = obj.A - spec1
elif spec1_name == 'V_partial':
err1 = obj.V_from_h(h) - spec1
elif spec1_name == 'SA_partial':
err1 = obj.SA_from_h(h) - spec1
elif spec1_name == 'A_cross':
err1 = obj.A_cross_sectional(h) - spec1
# print(err0, err1, D, L_over_D, h)
return [err0, err1]
[docs]class TANK:
"""Class representing tank volumes and levels. All parameters are also
attributes.
Parameters
----------
D : float
Diameter of the cylindrical section of the tank, [m]
L : float
Length of the main cylindrical section of the tank, [m]
horizontal : bool, optional
Whether or not the tank is a horizontal or vertical tank
sideA : string, optional
The left (or bottom for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical',
'same'].
sideB : string, optional
The right (or top for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical',
'same'].
sideA_a : float, optional
The distance the head as specified by sideA extends down or to the left
from the main cylindrical section, [m]
sideB_a : float, optional
The distance the head as specified by sideB extends up or to the right
from the main cylindrical section, [m]
sideA_f : float, optional
Dimensionless dish-radius parameter for side A; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideA_k : float, optional
Dimensionless knuckle-radius parameter for side A; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
sideB_f : float, optional
Dimensionless dish-radius parameter for side B; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideB_k : float, optional
Dimensionless knuckle-radius parameter for side B; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
sideA_a_ratio : float, optional
Ratio for `a` parameter; can be used instead of specifying an absolute
value, [-]
sideB_a_ratio : float, optional
Ratio for `a` parameter; can be used instead of specifying an absolute
value, [-]
L_over_D : float, optional
Ratio of length over diameter, used only when D and L are both
unspecified but V is, [-]
V : float, optional
Volume of the tank; solved for if specified, using
sideA_a_ratio/sideB_a_ratio, sideA, sideB, horizontal, and one
of L_over_D, L, or D, [m^3]
Attributes
----------
h_max : float
Height of the tank, [m]
V_total : float
Total volume of the tank as calculated [m^3]
sideA_V : float
Volume of only `sideA` [m^3]
sideB_V : float
Volume of only `sideB` [m^3]
lateral_V : float
Volume of cylindrical section of tank [m^3]
A : float
Total surface area of the tank, [m^2]
A_sideA : float
Surface area of sideA, [m^2]
A_sideB : float
Surface area of sideB, [m^2]
A_lateral : float
Surface area of the lateral side, [m^2]
A_sideA_extra : float
Additional surface area of sideA beyond that of a flat disk, [m^2]
A_sideB_extra : float
Additional surface area of sideB beyond that of a flat disk, [m^2]
table : bool
Whether or not a table of heights-volumes has been generated
heights : ndarray
Array of heights between 0 and h_max, [m]
volumes : ndarray
Array of volumes calculated from the heights, [m^3]
c_forward : ndarray
Coefficients for the Chebyshev approximations in calculating V from h,
[-]
c_backward : ndarray
Coefficients for the Chebyshev approximations in calculating h from V,
[-]
Notes
-----
For torpsherical tank heads, the following `f` and `k` parameters are used
in standards. The default is ASME F&D.
+----------------------+-----+-------+
| | f | k |
+======================+=====+=======+
| 2:1 semi-elliptical | 0.9 | 0.17 |
+----------------------+-----+-------+
| ASME F&D | 1 | 0.06 |
+----------------------+-----+-------+
| ASME 80/6 | 0.8 | 0.06 |
+----------------------+-----+-------+
| ASME 80/10 F&D | 0.8 | 0.1 |
+----------------------+-----+-------+
| DIN 28011 | 1 | 0.1 |
+----------------------+-----+-------+
| DIN 28013 | 0.8 | 0.154 |
+----------------------+-----+-------+
For the following cases, numerical integrals are used.
V_horiz_spherical
V_horiz_torispherical
SA_partial_horiz_spherical_head
SA_partial_horiz_ellipsoidal_head
SA_partial_horiz_guppy_head
SA_partial_horiz_torispherical_head
Examples
--------
Total volume of a tank:
>>> TANK(D=1.2, L=4, horizontal=False).V_total
4.523893421169302
Volume of a tank at a given height:
>>> TANK(D=1.2, L=4, horizontal=False).V_from_h(.5)
0.5654866776461628
Height of liquid for a given volume:
>>> TANK(D=1.2, L=4, horizontal=False).h_from_V(.5)
0.442097064
Surface area of a tank with a conical head:
>>> T1 = TANK(V=10, L_over_D=0.7, sideB='conical', sideB_a=0.5)
>>> T1.A, T1.A_sideA, T1.A_sideB, T1.A_lateral
(24.94775907, 5.118555, 5.497246, 14.331956)
Solving for tank volumes, first horizontal, then vertical:
>>> TANK(D=10., horizontal=True, sideA='conical', sideB='conical', V=500).L
4.699531
>>> TANK(L=4.69953105701, horizontal=True, sideA='conical', sideB='conical', V=500).D
9.9999999
>>> TANK(L_over_D=0.469953105701, horizontal=True, sideA='conical', sideB='conical', V=500).L
4.6995310
>>> TANK(D=10., horizontal=False, sideA='conical', sideB='conical', V=500).L
4.699531
>>> TANK(L=4.69953105701, horizontal=False, sideA='conical', sideB='conical', V=500).D
9.99999999
>>> TANK(L_over_D=0.469953105701, horizontal=False, sideA='conical', sideB='conical', V=500).L
4.699531057
"""
table = False
chebyshev = False
__full_path__ = __module__ + '.TANK'
def __repr__(self): # pragma: no cover
orient = 'Horizontal' if self.horizontal else 'Vertical'
if self.sideA is None and self.sideB is None:
sides = 'no heads'
elif self.sideA == self.sideB:
if self.sideA_a == self.sideB_a:
sides = self.sideA + (' heads, a=%f m' %(self.sideA_a))
else:
sides = self.sideA + f' heads, sideA a={self.sideA_a:f} m, sideB a={self.sideB_a:f} m'
else:
if self.sideA:
A = f'{self.sideA} head on sideA with a={self.sideA_a:f} m'
else:
A = 'no head on sideA'
if self.sideB:
B = f' and {self.sideB} head on sideB with a={self.sideB_a:f} m'
else:
B = ' and no head on sideB'
sides = A + B
return f'<{orient} tank, V={self.V_total:f} m^3, D={self.D:f} m, L={self.L:f} m, {sides}.>'
def __init__(self, D=None, L=None, horizontal=True,
sideA=None, sideB=None, sideA_a=None, sideB_a=None,
sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None,
sideA_a_ratio=None, sideB_a_ratio=None, L_over_D=None, V=None):
self.D = D
self.L = L
self.L_over_D = L_over_D
self.V = V
self.horizontal = horizontal
sideA_same, sideB_same = sideA == 'same', sideB == 'same'
if sideA_same and not sideB_same:
sideA, sideA_a, sideA_a_ratio, sideA_f, sideA_k = sideB, sideB_a, sideB_a_ratio, sideB_f, sideB_k
elif sideB_same and not sideA_same:
sideB, sideB_a, sideB_a_ratio, sideB_f, sideB_k = sideA, sideA_a, sideA_a_ratio, sideA_f, sideA_k
elif sideA_same and sideB_same:
raise ValueError("Cannot specify both sides as same")
self.sideA = sideA
if sideA is None and sideA_a is None:
sideA_a = 0.0
self.sideA_a = sideA_a
if sideA_a is None and sideA_a_ratio is None and (sideA is not None and sideA != 'torispherical'):
sideA_a_ratio = 0.25
self.sideA_a_ratio = sideA_a_ratio
if sideA_a is None and sideA == 'torispherical':
if sideA_f is None:
sideA_f = 1.0
if sideA_k is None:
sideA_k = 0.06
self.sideA_f = sideA_f
self.sideA_k = sideA_k
self.sideB = sideB
if sideB is None and sideB_a is None:
sideB_a = 0.0
self.sideB_a = sideB_a
if sideB_a is None and sideB_a_ratio is None and (sideB is not None and sideB != 'torispherical'):
sideB_a_ratio = 0.25
self.sideB_a_ratio = sideB_a_ratio
if sideB_a is None and sideB == 'torispherical':
if sideB_f is None:
sideB_f = 1.0
if sideB_k is None:
sideB_k = 0.06
self.sideB_f = sideB_f
self.sideB_k = sideB_k
if self.horizontal:
self.vertical = False
self.orientation = 'horizontal'
self.angle = 0
else:
self.vertical = True
self.orientation = 'vertical'
self.angle = 90
# If V is specified and either L or D are known, solve for L, D, L_over_D
if self.V:
self._solve_tank_for_V()
self.set_misc()
[docs] def set_misc(self):
"""Set more parameters, after the tank is better defined than in the
__init__ function.
Notes
-----
Two of D, L, and L_over_D must be known when this function runs.
The other one is set from the other two first thing in this function.
a_ratio parameters are used to calculate a values for the heads here,
if applicable.
Radius is calculated here.
Maximum tank height is calculated here.
V_total is calculated here.
"""
if self.D is not None and self.L is not None:
# If L and D are known, get L_over_D
self.L_over_D = self.L/self.D
elif self.D is not None and self.L_over_D is not None:
# Otherwise, if L_over_D and D are provided, get L
self.L = self.D*self.L_over_D
elif self.L is not None and self.L_over_D is not None:
# Otherwise, if L_over_D and L are provided, get D
self.D = self.L/self.L_over_D
D = self.D
# Calculate diameter
self.R = self.D/2.
# If a_ratio is provided for either heads, use it.
if self.sideA is not None and D is not None and self.sideA_a is None and self.sideA in ('conical', 'ellipsoidal', 'guppy', 'spherical'):
self.sideA_a = D*self.sideA_a_ratio
if self.sideB is not None and D is not None and self.sideB_a is None and self.sideB in ('conical', 'ellipsoidal', 'guppy', 'spherical'):
self.sideB_a = D*self.sideB_a_ratio
# Calculate a for torispherical heads
if self.sideA == 'torispherical' and self.sideA_f is not None and self.sideA_k is not None:
self.sideA_a = a_torispherical(D, self.sideA_f, self.sideA_k)
if self.sideB == 'torispherical' and self.sideB_f is not None and self.sideB_k is not None:
self.sideB_a = a_torispherical(D, self.sideB_f, self.sideB_k)
# Ensure the correct a_ratios are set, whether there is a default being used or not
if self.sideA_a_ratio is None and self.sideA_a is not None:
self.sideA_a_ratio = self.sideA_a/D
elif self.sideA_a_ratio is not None and self.sideA_a is not None and self.sideA_a != D*self.sideA_a_ratio:
self.sideA_a_ratio = self.sideA_a/D
if self.sideB_a_ratio is None and self.sideB_a is not None:
self.sideB_a_ratio = self.sideB_a/D
elif self.sideB_a_ratio is not None and self.sideB_a is not None and self.sideB_a != D*self.sideB_a_ratio:
self.sideB_a_ratio = self.sideB_a/D
# Calculate maximum tank height, h_max
if self.horizontal:
self.h_max = D
else:
self.h_max = self.L
if self.sideA_a:
self.h_max += self.sideA_a
if self.sideB_a:
self.h_max += self.sideB_a
# Set maximum height
# self.V_total = self.V_from_h(self.h_max)
self.V_total, self.V_sideA, self.V_sideB, self.V_lateral = V_tank(
D=D, L=self.L, sideA=self.sideA, sideB=self.sideB, sideA_a=self.sideA_a,
sideB_a=self.sideB_a, sideA_f=self.sideA_f, sideA_k=self.sideA_k,
sideB_f=self.sideB_f, sideB_k=self.sideB_k, horizontal=self.horizontal)
# Set surface areas
self.A, self.A_sideA, self.A_sideB, self.A_lateral = SA_tank(
D=D, L=self.L, sideA=self.sideA, sideB=self.sideB, sideA_a=self.sideA_a,
sideB_a=self.sideB_a, sideA_f=self.sideA_f, sideA_k=self.sideA_k,
sideB_f=self.sideB_f, sideB_k=self.sideB_k)
A_circular_plate = 0.25*pi*D*D
self.A_sideA_extra = self.A_sideA - A_circular_plate
self.A_sideB_extra = self.A_sideB - A_circular_plate
[docs] @staticmethod
def from_two_specs(spec0, spec1, spec0_name='V', spec1_name='A_cross',
h=None, horizontal=True,
sideA=None, sideB=None, sideA_a=None, sideB_a=None,
sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None,
sideA_a_ratio=None, sideB_a_ratio=None):
r'''Method to create a new tank instance according to two
specifications which are not direct geometry parameters.
The allowable options are 'V', 'SA', 'V_partial', 'SA_partial',
and 'A_cross', the later three of which require `h` to be specified.
Parameters
----------
spec0 : float
Goal for `spec0_name`, [-]
spec1 : float
Goal for `spec1_name`, [-]
spec0_name : str
One of 'V', 'SA', 'V_partial', 'SA_partial', and 'A_cross' [-]
spec1_name : str
One of 'V', 'SA', 'V_partial', 'SA_partial', and 'A_cross' [-]
h : float
Height at which to calculate the specs, [m]
horizontal : bool, optional
Whether or not the tank is a horizontal or vertical tank
sideA : string, optional
The left (or bottom for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical',
'same'].
sideB : string, optional
The right (or top for vertical) head of the tank's type; one of
[None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical',
'same'].
sideA_a : float, optional
The distance the head as specified by sideA extends down or to the left
from the main cylindrical section, [m]
sideB_a : float, optional
The distance the head as specified by sideB extends up or to the right
from the main cylindrical section, [m]
sideA_f : float, optional
Dimensionless dish-radius parameter for side A; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideA_k : float, optional
Dimensionless knuckle-radius parameter for side A; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
sideB_f : float, optional
Dimensionless dish-radius parameter for side B; also commonly given as
the product of `f` and `D` (`fD`), which is called dish radius and
has units of length, [-]
sideB_k : float, optional
Dimensionless knuckle-radius parameter for side B; also commonly given
as the product of `k` and `D` (`kD`), which is called the knuckle
radius and has units of length, [-]
Returns
-------
TANK : TANK
Tank object at solved specifications, [-]
Notes
-----
Limited testing has been done on this method. The bounds are D between
0.1 mm and 10 km, with L_D ratios of 1e-4 to 1e4.
'''
args = (spec0, spec1, spec0_name, spec1_name,
h, horizontal, sideA, sideB, sideA_a, sideB_a,
sideA_f, sideA_k, sideB_f, sideB_k,
sideA_a_ratio, sideB_a_ratio)
new_f, translate_into, translate_outof = translate_bound_func(tank_from_two_specs_err,
bounds=[(1e-4, 1e4), (1e-4, 1e4)])
# Diameter and length/diameter as iteration variables
guess = translate_into([1.0, 3.0])
from scipy.optimize import fsolve
ans = fsolve(new_f, guess, args=args, xtol=1e-10, factor=.1)
val0, val1 = translate_outof(ans)
return TANK(D=float(val0), L_over_D=float(val1), horizontal=horizontal,
sideA=sideA, sideB=sideB, sideA_a=sideA_a, sideB_a=sideB_a,
sideA_f=sideA_f, sideA_k=sideA_k, sideB_f=sideB_f, sideB_k=sideB_k,
sideA_a_ratio=sideA_a_ratio, sideB_a_ratio=sideB_a_ratio,)
[docs] def add_thickness(self, thickness, sideA_thickness=None,
sideB_thickness=None):
r'''Method to create a new tank instance with the same parameters as
itself, except with an added thickness to it. This is useful to obtain
ex. the inside of a tank and the outside; their different in volumes is
the volume of the shell, and could be used to determine weight.
Parameters
----------
thickness : float
Thickness to add to the tank diameter, [m]
sideA_thickness : float, optional
The thickness to add to the sideA head; if not specified,
it will be `thickness`, [m]
sideB_thickness : float, optional
The thickness to add to the sideB head; if not specified,
it will be `thickness`, [m]
Returns
-------
TANK : TANK
Tank object, [-]
Notes
-----
Be careful not to specify a negative thickness larger than the heads'
lengths, or the head will become concave! The same applies to adding
a thickness to convex heads - they can become convex.
'''
kwargs = dict(D=self.D, L=self.L, horizontal=self.horizontal,
sideA=self.sideA, sideB=self.sideB, sideA_a=self.sideA_a,
sideB_a=self.sideB_a, sideA_f=self.sideA_f,
sideA_k=self.sideA_k, sideB_f=self.sideB_f, sideB_k=self.sideB_k)
if sideA_thickness is None:
sideA_thickness = thickness
if sideB_thickness is None:
sideB_thickness = thickness
# Do not transfer a_ratios or volume or L_over_D
kwargs['D'] += 2.0*thickness
kwargs['L'] += sideA_thickness + sideB_thickness
# For torispherical vessels, the heads are defined from the `f` and `k`
# parameters which are already functions of diameter, and so will be
# fixed automatically; if the `a` parameters are specified they would
# not be corrected
if self.sideA != 'torispherical':
kwargs['sideA_a'] += sideA_thickness
else:
del kwargs['sideA_a']
if self.sideB != 'torispherical':
kwargs['sideB_a'] += sideB_thickness
else:
del kwargs['sideB_a']
return TANK(**kwargs)
[docs] def SA_from_h(self, h, method='full'):
r'''Method to calculate the volume of liquid in a fully defined tank
given a specified height `h`. `h` must be under the maximum height.
Parameters
----------
h : float
Height specified, [m]
method : str, optional
'full' (calculated rigorously) ; nothing else is implemented
Returns
-------
SA : float
Surface area of liquid in the tank up to the specified height, [m^2]
Notes
-----
'''
if method == 'full':
return SA_from_h(h, self.D, self.L, self.horizontal, self.sideA,
self.sideB, self.sideA_a, self.sideB_a,
self.sideA_f, self.sideA_k, self.sideB_f,
self.sideB_k)
else:
raise ValueError("Allowable methods are 'full' .")
[docs] def V_from_h(self, h, method='full'):
r'''Method to calculate the volume of liquid in a fully defined tank
given a specified height `h`. `h` must be under the maximum height.
If the method is 'chebyshev', and the coefficients have not yet been
calculated, they are created by calling `set_chebyshev_approximators`.
Parameters
----------
h : float
Height specified, [m]
method : str
One of 'full' (calculated rigorously) or 'chebyshev'
Returns
-------
V : float
Volume of liquid in the tank up to the specified height, [m^3]
Notes
-----
'''
if method == 'full':
return V_from_h(h, self.D, self.L, self.horizontal, self.sideA,
self.sideB, self.sideA_a, self.sideB_a,
self.sideA_f, self.sideA_k, self.sideB_f,
self.sideB_k)
elif method == 'chebyshev':
if not self.chebyshev:
self.set_chebyshev_approximators()
return self.V_from_h_cheb(h)
else:
raise ValueError("Allowable methods are 'full' or 'chebyshev'.")
[docs] def h_from_V(self, V, method='spline'):
r'''Method to calculate the height of liquid in a fully defined tank
given a specified volume of liquid in it `V`. `V` must be under the
maximum volume. If the method is 'spline', and the interpolation table
is not yet defined, creates it by calling the method set_table. If the
method is 'chebyshev', and the coefficients have not yet been
calculated, they are created by calling `set_chebyshev_approximators`.
Parameters
----------
V : float
Volume of liquid in the tank up to the desired height, [m^3]
method : str
One of 'spline', 'chebyshev', or 'brenth'
Returns
-------
h : float
Height of liquid at which the volume is as desired, [m]
'''
if method == 'spline':
try:
if not self.table:
self.set_table()
return float(self.interp_h_from_V(V))
except:
# Missing scipy
return self.h_from_V(V, 'brenth')
elif method == 'chebyshev':
if not self.chebyshev:
self.set_chebyshev_approximators()
return self.h_from_V_cheb(V)
elif method == 'brenth':
to_solve = lambda h : self.V_from_h(h, method='full') - V
return brenth(to_solve, self.h_max, 0)
else:
raise ValueError("Allowable methods are 'full' or 'chebyshev', "
"or 'brenth'.")
[docs] def A_cross_sectional(self, h, method='full'):
r'''Method to calculate the cross-sectional liquid surface area
from which gas can evolve in a fully defined tank
given a specified height `h`. `h` must be under the maximum height.
This is calculated by numeric differentiation for most cases.
Parameters
----------
h : float
Height specified, [m]
method : str, optional
'full' (calculated rigorously) or 'chebyshev', [-]
Returns
-------
A_cross : float
Surface area of liquid in the tank up to the specified height, [m^2]
Notes
-----
'''
# The derivative will give bad values in some cases, when right up against boundaries
# Analytical formulations can be done, but will be lots of code
return derivative(lambda h: self.V_from_h(h), h, dx=1e-7*h, order=3, n=1)
[docs] def set_table(self, n=100, dx=None):
r'''Method to set an interpolation table of liquids levels versus
volumes in the tank, for a fully defined tank. Normally run by the
h_from_V method, this may be run prior to its use with a custom
specification. Either the number of points on the table, or the
vertical distance between steps may be specified.
Parameters
----------
n : float, optional
Number of points in the interpolation table, [-]
dx : float, optional
Vertical distance between steps in the interpolation table, [m]
'''
if dx:
self.heights = linspace(0.0, self.h_max, int(self.h_max/dx)+1)
else:
self.heights = linspace(0.0, self.h_max, n)
self.volumes = [self.V_from_h(h) for h in self.heights]
from scipy.interpolate import UnivariateSpline
self.interp_h_from_V = UnivariateSpline(self.volumes, self.heights, ext=3, s=0.0)
self.table = True
[docs] def set_chebyshev_approximators(self, deg_forward=50, deg_backwards=200):
r'''Method to derive and set coefficients for chebyshev polynomial
function approximation of the height-volume and volume-height
relationship.
A single set of chebyshev coefficients is used for the entire height-
volume and volume-height relationships respectively.
The forward relationship, `V_from_h`, requires
far fewer coefficients in its fit than the reverse to obtain the same
relative accuracy.
Optionally, deg_forward or deg_backwards can be set to None to try to
automatically fit the series to machine precision.
Parameters
----------
deg_forward : int, optional
The degree of the chebyshev polynomial to be created for the
`V_from_h` curve, [-]
deg_backwards : int, optional
The degree of the chebyshev polynomial to be created for the
`h_from_V` curve, [-]
'''
import numpy as np
from fluids.optional.pychebfun import Chebfun
to_fit = lambda h: self.V_from_h(h, 'full')
# These high-degree polynomials cannot safety be evaluated using Horner's methods
# chebval is 2.5x as slow but 100% required; around 40 coefficients results are junk
self.c_forward = Chebfun.from_function(np.vectorize(to_fit),
[0.0, self.h_max], N=deg_forward).coefficients().tolist()
self.V_from_h_cheb = lambda x : chebval((2.0*x-self.h_max)/(self.h_max), self.c_forward)
to_fit = lambda h: self.h_from_V(h, 'brenth')
self.c_backward = Chebfun.from_function(np.vectorize(to_fit), [0.0, self.V_total], N=deg_backwards).coefficients().tolist()
self.h_from_V_cheb = lambda x : chebval((2.0*x-self.V_total)/(self.V_total), self.c_backward)
self.chebyshev = True
def _V_solver_error(self, Vtarget, D, L, horizontal, sideA, sideB, sideA_a,
sideB_a, sideA_f, sideA_k, sideB_f, sideB_k,
sideA_a_ratio, sideB_a_ratio):
"""Function which uses only the variables given, and the TANK class
itself, to determine how far from the desired volume, Vtarget, the
volume produced by the specified parameters in a new TANK instance is.
Should only be used by _solve_tank_for_V method.
"""
a = TANK(D=float(D), L=float(L), horizontal=horizontal, sideA=sideA, sideB=sideB,
sideA_a=sideA_a, sideB_a=sideB_a, sideA_f=sideA_f,
sideA_k=sideA_k, sideB_f=sideB_f, sideB_k=sideB_k,
sideA_a_ratio=sideA_a_ratio, sideB_a_ratio=sideB_a_ratio)
error = (Vtarget - a.V_total)
return error
def _solve_tank_for_V(self):
"""Method which is called to solve for tank geometry when a certain
volume is specified. Will be called by the __init__ method if V is set.
Notes
-----
Raises an error if L and either of sideA_a or sideB_a are specified;
these can only be set once D is known.
Raises an error if more than one of D, L, or L_over_D are specified.
Raises an error if the head ratios are not provided.
Calculates initial guesses assuming no heads are present, and then uses
fsolve to determine the correct dimensions for the tank.
Tested, but bugs and limitations are expected here.
"""
if self.L and (self.sideA_a or self.sideB_a):
raise ValueError('Cannot specify head sizes when solving for V')
if (self.D and self.L) or (self.D and self.L_over_D) or (self.L and self.L_over_D):
raise ValueError('Only one of D, L, or L_over_D can be specified\
when solving for V')
if ((self.sideA is not None and (self.sideA_a_ratio is None and self.sideA_a is None) and self.sideA != 'torispherical')
or (self.sideB is not None and (self.sideB_a_ratio is None and self.sideB_a is None) and self.sideB != 'torispherical')):
raise ValueError('When heads are specified, head parameter ratios are required')
if self.D:
# Iterate until L is appropriate
solve_L = lambda L: self._V_solver_error(self.V, self.D, L, self.horizontal, self.sideA, self.sideB, self.sideA_a, self.sideB_a, self.sideA_f, self.sideA_k, self.sideB_f, self.sideB_k, self.sideA_a_ratio, self.sideB_a_ratio)
Lguess = self.V/(pi/4*self.D**2)
self.L = float(secant(solve_L, Lguess, xtol=1e-13))
elif self.L:
# Iterate until D is appropriate
solve_D = lambda D: self._V_solver_error(self.V, D, self.L, self.horizontal, self.sideA, self.sideB, self.sideA_a, self.sideB_a, self.sideA_f, self.sideA_k, self.sideB_f, self.sideB_k, self.sideA_a_ratio, self.sideB_a_ratio)
Dguess = sqrt(4*self.V/pi/self.L)
self.D = float(secant(solve_D, Dguess, xtol=1e-13))
else:
# Use L_over_D until L and D are appropriate
Lguess = (4*self.V*self.L_over_D**2/pi)**(1/3.)
solve_L_D = lambda L: self._V_solver_error(self.V, L/self.L_over_D, L, self.horizontal, self.sideA, self.sideB, self.sideA_a, self.sideB_a, self.sideA_f, self.sideA_k, self.sideB_f, self.sideB_k, self.sideA_a_ratio, self.sideB_a_ratio)
self.L = float(secant(solve_L_D, Lguess, xtol=1e-13))
self.D = self.L/self.L_over_D
[docs]class HelicalCoil:
r'''Class representing a helical coiled tube, as are found in many heated
tanks and some small nuclear reactors. All parameters are also attributes.
One set of the following parameters is required; inner tube diameter is
optional.
* Tube outer diameter, coil outer diameter, pitch, number of coil turns
* Tube outer diameter, coil outer diameter, pitch, height
* Tube outer diameter, coil outer diameter, number of coil turns, height
Parameters
----------
Dt : float
Outer diameter of the tube wound to make up the helical spiral, [m]
Do : float
Diameter of the spiral as measured from the center of the coil on one
side to the center of the coil on the other side, [m]
Do_total : float, optional
Diameter of the spiral as measured from one edge of the tube to the
other edge; equal to Do + Dt; either `Do` or `Do_total` may be
specified and the other will be calculated [m]
pitch : float, optional
Height change from one coil to the next as measured from the middles
of the tube, [m]
H : float, optional
Height of the spiral, as measured from the middle of the bottom of the
tube to the middle of the top of the tube, [m]
H_total : float, optional
Height of the spiral as measured from one edge of the tube to the other
edge; equal to `H_total` + `Dt`; either may be specified and the other
will be calculated [m]
N : float, optional
Number of coil turns; may be specified along with `pitch` instead of
specifying `H` or `H_total`, [-]
Di : float, optional
Inner diameter of the tube; if specified, inside and annulus properties
will be calculated, [m]
Attributes
----------
tube_circumference : float
Circumference of the tube as measured though its center, not inner or
outer edges; :math:`C = \pi D_o`, [m]
tube_length : float
Length of tube used to make the helical coil;
:math:`L = \sqrt{(\pi D_o\cdot N)^2 + H^2}`, [m]
surface_area : float
Surface area of the outer surface of the helical coil;
:math:`A_t = \pi D_t L`, [m^2]
inner_surface_area : float
Surface area of the inner surface of the helical coil; calculated if
`Di` is supplied; :math:`A_{inside} = \pi D_i L`, [m^2]
inlet_area : float
Area of the inlet to the helical coil; calculated if
`Di` is supplied; :math:`A_{inlet} = \frac{\pi}{4} D_i^2`, [m^2]
inner_volume : float
Volume of the tube as would be filled by a fluid, useful for weight
calculations; calculated if `Di` is supplied;
:math:`V_{inside} = A_i L`, [m^3]
annulus_area : float
Area of the annulus (wall of the pipe); calculated if `Di` is supplied;
:math:`A_a = \frac{\pi}{4} (D_t^2 - D_i^2)`, [m^2]
annulus_volume : float
Volume of the annulus (wall of the pipe); calculated if `Di`
is supplied, useful for weight calculations; :math:`V_a = A_a L`, [m^3]
total_volume : float
Total volume occupied by the pipe and the fluid inside it;
:math:`V = D_t L`, [m^3]
helix_angle : float
Angle between the pitch and coil diameter; used in some calculations;
:math:`\alpha = \arctan \left(\frac{p_t}{\pi D_o}\right)`, [radians]
curvature : float
Coil curvature, useful in some calculations;
:math:`\delta = \frac{D_t}{D_o[1 + 4\pi^2 \tan^2(\alpha)]}`, [-]
Notes
-----
`Do` must be larger than `Dt`.
Examples
--------
>>> C1 = HelicalCoil(Do=30, H=20, pitch=5, Dt=2)
>>> C1.N, C1.tube_length, C1.surface_area
(4.0, 377.5212621504738, 2372.0360474917497)
Same coil, with the inputs one would physically measure from the coil,
and a specified inlet diameter:
>>> C1 = HelicalCoil(Do_total=32, H_total=22, pitch=5, Dt=2, Di=1.8)
>>> C1.N, C1.tube_length, C1.surface_area
(4.0, 377.5212621504738, 2372.0360474917497)
>>> C1.inner_surface_area, C1.inlet_area, C1.inner_volume, C1.total_volume, C1.annulus_volume
(2134.832442742575, 2.5446900494077327, 960.6745992341587, 1186.0180237458749, 225.3434245117162)
References
----------
.. [1] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and
Correlations for Convection Heat Transfer and Pressure Losses in
Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0
(June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.
'''
def __repr__(self): # pragma : no cover
s = '<Helical coil, total height={} m, total outer diameter={} m, tube \
outer diameter={} m, number of turns={}, pitch={} m'.format(self.H_total, self.Do_total, self.Dt, self.N, self.pitch)
if self.Di:
s += ', inside diameter %s m' %(self.Di)
s += '>'
return s
def __init__(self, Dt, Do=None, pitch=None, H=None, N=None, H_total=None,
Do_total=None, Di=None):
# H goes from center of tube in bottom of coil to center of tube in top of coil
# Do goes from the center of the spiral to the center of the outer tube
if H_total is not None:
H = H_total - Dt
if Do_total is not None:
Do = Do_total - Dt
self.Do = Do
self.Dt = Dt
self.Do_total = self.Do+self.Dt
if N is not None and pitch is not None:
self.N = N
self.pitch = pitch
self.H = N*pitch
elif N is not None and H is not None:
self.N = N
self.H = H
self.pitch = self.H/N
if self.pitch < self.Dt:
raise ValueError('Pitch is too small - tubes are colliding')#; maximum number of spirals is %f.'%(self.H/self.Dt))
elif H is not None and pitch is not None:
self.pitch = pitch
self.H = H
self.N = self.H/self.pitch
if self.pitch < self.Dt:
raise ValueError('Pitch is too small - tubes are colliding; pitch must be larger than tube diameter.')
if self.H is not None: # numba
self.H_total = self.Dt + self.H
if self.Dt > self.Do:
raise ValueError('Tube diameter is larger than helix outer diameter - not feasible.')
self.tube_circumference = pi*self.Do
self.tube_length = sqrt((self.tube_circumference*self.N)**2 + self.H**2)
self.surface_area = self.tube_length*pi*self.Dt
#print(pi*self.tube_length*self.Dt) == surface_area
self.helix_angle = atan(self.pitch/(pi*self.Do))
self.curvature = self.Dt/self.Do/(1. + 4*pi**2*tan(self.helix_angle)**2)
#print(self.N*pi*self.Do/cos(self.helix_angle)) # Confirms the length with another formula
self.total_inlet_area = pi/4.*self.Dt**2
self.total_volume = self.total_inlet_area*self.tube_length
if Di is not None:
self.Di = Di
self.inner_surface_area = self.tube_length*pi*self.Di
self.inlet_area = pi/4.*self.Di**2
self.inner_volume = self.inlet_area*self.tube_length
self.annulus_area = self.total_inlet_area - self.inlet_area
self.annulus_volume = self.total_volume - self.inner_volume
[docs]def plate_enlargement_factor(amplitude, wavelength):
r'''Calculates the enhancement factor of the sinusoidal waves of the
plate heat exchanger. This is the multiplier for the flat plate area
to obtain the actual area available for heat transfer. Obtained from
the following integral:
.. math::
\phi = \frac{\text{Effective area}}{\text{Projected area}}
= \frac{\int_0^\lambda\sqrt{1 + \left(\frac{\gamma\pi}{2}\right)^2
\cos^2\left(\frac{2\pi}{\lambda}x\right)}dx}{\lambda}
.. math::
\gamma = \frac{4a}{\lambda}
The solution to the integral is:
.. math::
\phi = \frac{2E\left(\frac{-4a^2\pi^2}{\lambda^2}\right)}{\pi}
where E is the complete elliptic integral of the second kind,
calculated with SciPy.
Parameters
----------
amplitude : float
Half the height of the wave of the ridges, [m]
wavelength : float
Distance between the bottoms of two of the ridges (sometimes called
pitch), [m]
Returns
-------
plate_enlargement_factor : float
The extra surface area multiplier as compared to a flat plate
caused the corrugations, [-]
Notes
-----
This is the exact analytical integral, obtained via Mathematica, Maple,
and quite a bit of trial and error. It is confirmed via numerical
integration. The expression normally given is an
approximation as follows:
.. math::
\phi = \frac{1}{6}\left(1+\sqrt{1+A^2} + 4\sqrt{1+A^2/2}\right)
.. math::
A = \frac{2\pi a}{\lambda}
Most plate heat exchangers approximate a sinusoidal geometry only.
Examples
--------
>>> plate_enlargement_factor(amplitude=5E-4, wavelength=3.7E-3)
1.1611862034509677
'''
b = 2.*amplitude
return 2.*float(ellipe(-b*b*pi*pi/(wavelength*wavelength)))/pi
[docs]class PlateExchanger:
r'''Class representing a plate heat exchanger with sinusoidal ridges.
All parameters are also attributes.
Parameters
----------
amplitude : float
Half the height of the wave of the ridges, [m]
wavelength : float
Distance between the bottoms of two of the ridges (sometimes called
pitch), [m]
chevron_angle : float, optional
Angle of the plate corrugations with respect to the vertical axis
(the direction of flow if the plates were straight), between 0 and
90, [degrees]
chevron_angles : tuple(2), optional
Many plate exchangers use two alternating patterns; for those cases
provide tuple of the two angles for that situation and the argument
`chevron_angle` is ignored, [degrees]
width : float, optional
Width of the plates in the heat exchanger, between the gaskets, [m]
length : float, optional
Length of the heat exchanger as measured from one port to the other,
excluding the diameter of the ports themselves (little useful heat
transfer happens there), [m]
thickness : float, optional
Thickness of the metal making up the plates, [m]
d_port : float, optional
The diameter of the ports in the plates, [m]
plates : int, optional
The number of plates in the heat exchanger, including the two not
used for heat transfer at the beginning and end [-]
Attributes
----------
chevron_angles : tuple(2)
The two specified angles (repeated value if only one specified), [degrees]
chevron_angle : float
The averaged angle of the chevrons, [degrees]
inclination_angle : float
90 - `chevron_angle`, used in many publications instead of `chevron_angle`,
[degrees]
plate_corrugation_aspect_ratio : float
The aspect ratio of the corrugations
:math:`\gamma = \frac{4a}{\lambda}`, [-]
plate_enlargement_factor : float
The extra surface area multiplier as compared to a flat plate
caused the corrugations, [-]
D_eq : float
Equivalent diameter of the channels, :math:`D_{eq} = 4a` [m]
D_hydraulic : float
Hydraulic diameter of the channels, :math:`D_{hyd} = \frac{4a}{\phi}` [m]
length_port : float
Port center to port center along the direction of flow, [m]
A_plate_surface : float
The surface area of one plate in the heat exchanger, including the
extra due to corrugations (excluding the bit between the ports),
:math:`A_p = L\cdot W\cdot \phi` [m^2]
A_heat_transfer : float
The total surface area available for heat transfer in the exchanger,
the multiple of `A_plate_surface` by the number of plates after
removing the two on the edges, [m^2]
A_channel_flow : float
The area for the fluid to flow in one channel, :math:`W\cdot b` [m^2]
channels : int
The number of plates minus one, [-]
channels_per_fluid : int
Half the number of total channels, [-]
Notes
-----
Only wavelength and amplitude are required as inputs to this function.
Examples
--------
>>> PlateExchanger(amplitude=5E-4, wavelength=3.7E-3, length=1.2, width=.3,
... d_port=.05, plates=51)
<Plate heat exchanger, amplitude=0.0005 m, wavelength=0.0037 m, chevron_angles=45/45 degrees, area enhancement factor=1.16119, width=0.3 m, length=1.2 m, port diameter=0.05 m, heat transfer area=20.4833 m^2, 51 plates>
References
----------
.. [1] Amalfi, Raffaele L., Farzad Vakili-Farahani, and John R. Thome.
"Flow Boiling and Frictional Pressure Gradients in Plate Heat Exchangers.
Part 1: Review and Experimental Database." International Journal of
Refrigeration 61 (January 2016): 166-84. doi:10.1016/j.ijrefrig.2015.07.010.
'''
def __repr__(self): # pragma : no cover
s = '<Plate heat exchanger, amplitude={:g} m, wavelength={:g} m, \
chevron_angles={} degrees, area enhancement factor={:g}'.format(self.a, self.wavelength, '/'.join([str(i) for i in self.chevron_angles]), self.plate_enlargement_factor)
if self.width and self.length:
s += f', width={self.width:g} m, length={self.length:g} m'
if self.d_port:
s += ', port diameter=%g m' %(self.d_port)
if self.plates:
s += f', heat transfer area={self.A_heat_transfer:g} m^2, {self.plates:g} plates>'
else:
s += '>'
return s
@property
def plate_exchanger_identifier(self):
"""Method to create an identifying string in format 'L' + wavelength +
'A' + amplitude + 'B' + chevron angle-chevron angle.
Wavelength and amplitude are specified in units of mm and rounded to two
decimal places.
"""
wave_rounded = round(self.wavelength*1000, 2)
amplitude_rounded = round(self.amplitude*1000, 2)
a1 = self.chevron_angles[0]
a2 = self.chevron_angles[1]
s = (f'L{wave_rounded}A{amplitude_rounded}B{a1}-{a2}')
return s
def __init__(self, amplitude, wavelength, chevron_angle=45,
chevron_angles=None, width=None, length=None, thickness=None,
d_port=None, plates=None):
self.amplitude = self.a = amplitude # half a sine wave's height
self.b = 2*self.amplitude # Used in some models. From a flat plate, a press goes down this far into the plate. Also called the hot and cold gap
self.wavelength = self.pitch = wavelength # self.lambda
if chevron_angles is not None:
self.chevron_angles = chevron_angles
self.chevron_angle = self.beta = 0.5*(chevron_angles[0] + chevron_angles[1])
else:
self.chevron_angle = self.beta = chevron_angle # between 0 and 90
self.chevron_angles = (chevron_angle, chevron_angle)
self.inclination_angle = 90 - self.chevron_angle # Used in some definitions instead
self.plate_corrugation_aspect_ratio = self.gamma = 4*self.a/self.wavelength
self.plate_enlargement_factor = plate_enlargement_factor(self.amplitude, self.wavelength)
self.D_eq = 4*self.amplitude # Equivalent diameter for inter-plate spacing
self.D_hydraulic = 4*self.amplitude/self.plate_enlargement_factor # Get better results when correlations use this
if width is not None:
self.width = width
if length is not None:
self.length = length
if thickness is not None:
self.thickness = thickness
if d_port is not None:
self.d_port = d_port
if plates is not None:
self.plates = plates
if d_port is not None and length is not None:
self.length_port = length + d_port # port center to port center along the direction of flow
# There is another larger length as well, including both port diameters
if width is not None and length is not None:
self.A_plate_surface = length*width*self.plate_enlargement_factor # use this in Q = UAdT
if plates is not None:
self.A_heat_transfer = (plates-2)*self.A_plate_surface # the two outermost sides aren't used
if width is not None:
self.A_channel_flow = self.width*self.b # Use this to get G, kg/s/m^2
if plates is not None:
self.channels = self.plates - 1
self.channels_per_fluid = 0.5*self.channels
[docs]class RectangularFinExchanger:
r'''Class representing a plate-fin heat exchanger with straight rectangular
fins. All parameters are also attributes.
Parameters
----------
fin_height : float
The total distance between the two metal plates sandwiching the fins
and holding them together (abbreviated `h`), [m]
fin_thickness : float
The thickness of the material the fins were formed from
(abbreviated `t`), [m]
fin_spacing : float
The unit cell spacing from one fin to the next; the space between the
sides of two fins plus one thickness (abbreviated `s`), [m]
length : float, optional
The total length of the flow passage of the plate-fin exchanger
(abbreviated `L`), [m]
width : float, optional
The total width of the space the fins are in; this is also
:math:`N_{fins}\times s` (abbreviated `W`), [m]
layers : int, optional
The number of layers in the plate-fin exchanger; note these HX almost
always single-pass only, [-]
plate_thickness : float, optional
The thickness of the metal separator between layers, [m]
flow : str, optional
One of 'counterflow', 'crossflow', or 'parallelflow'
Attributes
----------
channel_height : float
The height of the channel the fluid flows in
:math:`\text{channel height } = \text{fin height} - \text{fin thickness}`, [m]
channel_width : float
The width of the channel the fluid flows in
:math:`\text{channel width } = \text{fin spacing} - \text{fin thickness}`, [m]
fin_count : int
The number of fins per unit length of the layer,
:math:`\text{fin count} = \frac{1}{\text{fin spacing}}`, [1/m]
blockage_ratio : float
The fraction of the layer which is blocked to flow by the fins,
:math:`\text{blockage ratio} = \frac{s\cdot h - s\cdot t - t(h-t)}{s\cdot h}`,
[m]
A_channel : float
Flow area of a single channel in a single layer,
:math:`\text{channel area} = (s-t)(h-t)`, [m]
P_channel : float
Wetted perimeter of a single channel in a single layer,
:math:`\text{channel perimeter} = 2(s-t) + 2(h-t)`, [m]
Dh : float
Hydraulic diameter of a single channel in a single layer,
:math:`D_{hydraulic} = \frac{4 A_{channel}}{P_{channel}}`, [m]
layer_thickness : float
The thickness of a single layer - the sum of a fin height and
a plate thickness, [m]
layer_fin_count : int
The number of fins in a layer; rounded to the nearest whole fin, [-]
A_HX_layer : float
The surface area including fins for heat transfer in one layer of the
HX, [m^2]
A_HX : float
The total surface area of the heat exchanger with all layers combined,
[m^2]
height : float
The height of all the layers of the heat exchanger combined, plus one
extra plate thickness, [m]
volume : float
The product of the height, width, and length of the HX, [m^3]
A_specific_HX : float
The specific surface area of the heat exchanger - square meters per
meter cubed, [m^3]
Notes
-----
The only required parameters are the fin geometry itself; `fin_height`,
`fin_thickness`, and `fin_spacing`.
Examples
--------
>>> PFE = RectangularFinExchanger(0.03, 0.001, 0.012)
>>> PFE.Dh
0.01595
References
----------
.. [1] Yang, Yujie, and Yanzhong Li. "General Prediction of the Thermal
Hydraulic Performance for Plate-Fin Heat Exchanger with Offset Strip
Fins." International Journal of Heat and Mass Transfer 78 (November 1,
2014): 860-70. doi:10.1016/j.ijheatmasstransfer.2014.07.060.
.. [2] Sheik Ismail, L., R. Velraj, and C. Ranganayakulu. "Studies on
Pumping Power in Terms of Pressure Drop and Heat Transfer
Characteristics of Compact Plate-Fin Heat Exchangers-A Review."
Renewable and Sustainable Energy Reviews 14, no. 1 (January 2010):
478-85. doi:10.1016/j.rser.2009.06.033.
'''
def __init__(self, fin_height, fin_thickness, fin_spacing, length=None, width=None, layers=None, plate_thickness=None, flow='crossflow'):
self.h = self.fin_height = fin_height # including 2x thickness
self.t = self.fin_thickness = fin_thickness
self.s = self.fin_spacing = fin_spacing
self.L = self.length = length
self.W = self.width = width
self.layers = layers
self.flow = flow
self.plate_thickness = plate_thickness
self.channel_height = self.fin_height - self.fin_thickness
self.channel_width = self.fin_spacing - self.fin_thickness
self.fin_count = 1./self.fin_spacing
self.blockage_ratio = (self.s*self.h - self.s*self.t - (self.h-self.t)*self.t)/(self.s*self.h)
self.A_channel = (self.s-self.t)*(self.h-self.t)
self.P_channel = 2*(self.s-self.t) + 2*(self.h-self.t)
self.Dh = 4*self.A_channel/self.P_channel
self.set_overall_geometry()
[docs] def set_overall_geometry(self):
if self.plate_thickness:
self.layer_thickness = self.plate_thickness + self.fin_height
if self.length and self.width:
self.layer_fin_count = round(self.fin_count*self.width, 0)
if hasattr(self, 'SA_fin'):
self.A_HX_layer = self.layer_fin_count*self.SA_fin*self.length
else:
self.A_HX_layer = self.P_channel*self.length*self.layer_fin_count
if self.layers:
self.A_HX = self.layers*self.A_HX_layer
if self.plate_thickness:
self.height = self.layer_thickness*self.layers + self.plate_thickness
self.volume = (self.length*self.width*self.height)
self.A_specific_HX = self.A_HX/self.volume
[docs]class RectangularOffsetStripFinExchanger(RectangularFinExchanger):
def __init__(self, fin_length, fin_height, fin_thickness, fin_spacing, length=None, width=None, layers=None, plate_thickness=None, flow='crossflow'):
self.l = self.fin_length = fin_length
self.h = self.fin_height = fin_height
self.t = self.fin_thickness = fin_thickness
self.s = self.fin_spacing = fin_spacing
self.blockage_ratio = self.omega = 2*self.t/self.s*(1. - self.t/self.h) + self.t/self.h*(1 - 2*self.t/self.s)
# Kim blockage ratio beta
self.blockage_ratio_Kim = self.t/self.h + self.t/self.s - self.t**2/(self.h*self.s)
# Definitions as in the paper with the most common correlation
self.alpha = self.s/self.h # "General prediction" uses t/h here
self.delta = self.t/self.l
self.gamma = self.t/self.s
# free flow area
self.A_channel = (self.h - self.t)*(self.s - self.t)
self.A = 2.*(self.l*(self.h-self.t) + self.l*(self.s-self.t) + self.t*(self.h-self.t)) + self.t*(self.s-2*self.t)
self.Dh = 4.*self.l*self.A_channel/self.A # not the standard definition
self.P_channel = 2*(self.s-self.t) + 2*(self.h-self.t)
self.Dh_Kays_London = 4*self.A_channel/(2*(self.h -self.t)+ 2*(self.s -self.t))
# Does not consider the fronts of backs of the fins, only the 2d shape
self.Dh_Joshi_Webb = 2*self.l*(self.h - self.t)*(self.s - 2*self.t)/(self.l*(self.h-self.t) + self.l*(self.s - self.t) + self.t*(self.h - self.t))
self.L = self.length = length
self.W = self.width = width
self.layers = layers
self.flow = flow
self.plate_thickness = plate_thickness
self.fin_count = 1./self.fin_spacing
self.set_overall_geometry()
[docs]class HyperbolicCoolingTower:
r'''Class representing the geometry of a hyperbolic cooling tower, as used
in many industries especially the poewr industry. All parameters are also
attributes.
`H_inlet`, `D_outlet`, and `H_outlet` are always required. Additionally,
one set of the following parameters is required; `H_support`, `D_support`,
`n_support`, and `inlet_rounding` are all optional as well.
* Inlet diameter
* Inlet diameter and throat diameter
* Inlet diameter and throat height
* Inlet diameter, throat diameter, and throat height
* Base diameter, throat diameter, and throat height
If the inlet diameter is provided but the throat diameter and/or the throat
height are missing, two heuristics are used to estimate them (to avoid
these heuristics simply specify the values):
* Assume the throat elevation is 2/3 the elevation of the tower.
* Assume the throat diameter is 63% the diameter of the inlet.
Parameters
----------
H_inlet : float
Height of the inlet zone of the cooling tower (also called rain zone),
[m]
D_outlet : float
The inside diameter of the cooling tower outlet (top of the tower; the
elevation the concrete section ends), [m]
H_outlet : float
The height of the cooling tower outlet (top of the tower;the
elevation the concrete section ends), [m]
D_inlet : float, optional
The inside diameter of the cooling tower inlet at the elevation the
concrete section begins, [m]
D_base : float, optional
The diameter of the cooling tower at the very base of the tower (the
bottom of the inlet zone, at the elevation of the ground), [m]
D_throat : float, optional
The diameter of the cooling tower at its minimum section, called its
throat; where the two hyperbolas meet, [m]
h_throat : float, optional
The elevation of the cooling tower's throat (its minimum section; where
the two hyperbolas meet), [m]
inlet_rounding : float, optional
Radius of an optional rounded protrusion from the lip of the cooling
tower shell base, which curves upwards from the lip (used to reduce
the dead zone area rather than having a flat lip), [m]
H_support : float, optional
The height of each support column, [m]
D_support : float, optional
The diameter of each support column, [m]
n_support : int, optional
The number of support columns of the cooling tower, [m]
Attributes
----------
b_lower : float
The `b` parameter in the hyperbolic equation for the lower section of
the cooling tower, [m]
b_upper : float
The `b` parameter in the hyperbolic equation for the upper section of
the cooling tower, [m]
Notes
-----
Note there are two hyperbolas in a hyperbolic cooling tower - one under the
throat and one above it; they are not necessarily the same.
A hyperbolic cooling tower is not the absolute optimal design, but is is
close. The optimality is determined by the amount of material required to
build it while maintaining its rigidity. For thermal design purposes,
a hyperbolic model covers any minor variation quite well.
Examples
--------
>>> ct = HyperbolicCoolingTower(D_outlet=89.0, H_outlet=200, D_inlet=136.18, H_inlet=14.5)
>>> ct
<Hyperbolic cooling tower, inlet diameter=136.18 m, outlet diameter=89 m, inlet height=14.5 m, outlet height=200 m, throat diameter=85.7934 m, throat height=133.333 m, base diameter=146.427 m>
>>> ct.diameter(5)
142.84514486126062
References
----------
.. [1] Chen, W. F., and E. M. Lui, eds. Handbook of Structural Engineering,
Second Edition. Boca Raton, Fla: CRC Press, 2005.
.. [2] Ansary, A. M. El, A. A. El Damatty, and A. O. Nassef. Optimum Shape
and Design of Cooling Towers, 2011.
'''
def __repr__(self): # pragma : no cover
s = """<Hyperbolic cooling tower, inlet diameter=%g m, outlet diameter=%g m, inlet height=%g m, \
outlet height=%g m, throat diameter=%g m, throat height=%g m, base diameter=%g m>"""
s = s%(self.D_inlet, self.D_outlet, self.H_inlet, self.H_outlet, self.D_throat, self.H_throat, self.D_base)
return s
def __init__(self, H_inlet, D_outlet, H_outlet, D_inlet=None, D_base=None,
D_throat=None, H_throat=None,
H_support=None, D_support=None, n_support=None,
inlet_rounding=None):
self.D_outlet = D_outlet
self.H_inlet = H_inlet
self.H_outlet = H_outlet
if H_throat is None:
H_throat = 2/3.0*H_outlet
self.H_throat = H_throat
if D_throat is None:
if D_inlet is not None:
D_throat = 0.63*D_inlet
else:
raise ValueError('Provide either `D_throat`, or `D_inlet` so it may be estimated.')
self.D_throat = D_throat
if D_inlet is None and D_base is None:
raise ValueError('Need `D_inlet` or `D_base`')
if D_base is not None:
b = self.D_throat*self.H_throat/sqrt(D_base**2 - self.D_throat**2)
D_inlet = 2*self.D_throat*sqrt((self.H_throat-H_inlet)**2 + b**2)/(2*b)
elif D_inlet is not None:
b = self.D_throat*(self.H_throat-H_inlet)/sqrt(D_inlet**2 - self.D_throat**2)
D_base = 2*self.D_throat*sqrt(self.H_throat**2 + b**2)/(2*b)
self.D_inlet = D_inlet
self.D_base = D_base
self.b_lower = b
# Upper b parameter
self.b_upper = self.D_throat*(self.H_outlet - self.H_throat)/sqrt((self.D_outlet)**2 - self.D_throat**2)
# May or may not be specified
self.H_support = H_support
self.D_support = D_support
self.n_support = n_support
self.inlet_rounding = inlet_rounding
[docs] def plot(self, pts=100): # pragma: no cover
import matplotlib.pyplot as plt
Zs = linspace(0, self.H_outlet, pts)
Rs = [self.diameter(Z)*0.5 for Z in Zs]
plt.plot(Zs, Rs)
plt.plot(Zs, [-v for v in Rs])
plt.show()
[docs] def diameter(self, H):
r'''Calculates cooling tower diameter at a specified height, using
the formulas for either hyperbola, depending on the height specified.
.. math::
D = D_{throat}\frac{\sqrt{H^2 + b^2}}{b}
The value of `H` and `b` used in the above equation is as follows:
* `H_throat` - H and `b_lower` if under the throat
* `H` - `H_throat` and `b_upper`, if above the throat
Parameters
----------
H : float
Height at which to calculate the cooling tower diameter, [m]
Returns
-------
D : float
Diameter of the cooling tower at the specified height, [m]
'''
# Compute the diameter at H
if H <= self.H_throat:
# Height relative to throat height
H = self.H_throat - H
b = self.b_lower
else:
H = H - self.H_throat
b = self.b_upper
R = self.D_throat*sqrt(H*H + b*b)/(2.0*b)
return R*2.0
[docs]class AirCooledExchanger:
r'''Class representing the geometry of an air cooled heat exchanger with
one or more tube bays, fans, or bundles.
All parameters are also attributes.
The minimum information required to describe an air cooler is as follows:
* `tube_rows`
* `tube_passes`
* `tubes_per_row`
* `tube_length`
* `tube_diameter`
* `fin_thickness`
Two of `angle`, `pitch`, `pitch_parallel`, and `pitch_normal`
(`pitch_ratio` may take the place of `pitch`)
Either `fin_diameter` or `fin_height`.
Either `fin_density` or `fin_interval`.
Parameters
----------
tube_rows : int
Number of tube rows per bundle, [-]
tube_passes : int
Number of tube passes (times the fluid travels across one tube length),
[-]
tubes_per_row : float
Number of tubes per row per bundle, [-]
tube_length : float
Total length of the tube bundle tubes, [m]
tube_diameter : float
Diameter of the bare tube, [m]
fin_thickness : float
Thickness of the fins, [m]
angle : float, optional
Angle of the tube layout, [degrees]
pitch : float, optional
Shortest distance between tube centers; defined in relation to the
flow direction only, [m]
pitch_parallel : float, optional
Distance between tube center along a line parallel to the flow;
has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m]
pitch_normal : float, optional
Distance between tube centers in a line 90° to the line of flow;
has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m]
pitch_ratio : float, optional
Ratio of the pitch to bare tube diameter, [-]
fin_diameter : float, optional
Outer diameter of each tube after including the fin on both sides,
[m]
fin_height : float, optional
Height above bare tube of the tube fins, [m]
fin_density : float, optional
Number of fins per meter of tube, [1/m]
fin_interval : float, optional
Space between each fin, including the thickness of one fin at its
base, [m]
parallel_bays : int, optional
Number of bays in the unit, [-]
bundles_per_bay : int, optional
Number of tube bundles per bay, [-]
fans_per_bay : int, optional
Number of fans per bay, [-]
corbels : bool, optional
Whether or not the air cooler has corbels, which increase the air
velocity by adding half a tube to the sides for the case of
non-rectangular tube layouts, [-]
tube_thickness : float, optional
Thickness of the bare metal tubes, [m]
fan_diameter : float, optional
Diameter of air cooler fan, [m]
Attributes
----------
bare_length : float
Length of bare tube between two fins
:math:`\text{bare length} = \text{fin interval} - t_{fin}`, [m]
tubes_per_bundle : float
Total number of tubes per bundle
:math:`N_{tubes/bundle} = N_{tubes/row} \cdot N_{rows}`, [-]
tubes_per_bay : float
Total number of tubes per bay
:math:`N_{tubes/bay} = N_{tubes/bundle} \cdot N_{bundles/bay}`, [-]
tubes : float
Total number of tubes in all bundles in all bays combined
:math:`N_{tubes} = N_{tubes/bay} \cdot N_{bays}`, [-]
pitch_diagonal : float
Distance between tube centers in a diagonal line between one normal
tube and one parallel tube;
:math:`s_D = \left[s_L^2 + \left(\frac{s_T}{2}\right)^2\right]^{0.5}`,
[m]
A_bare_tube_per_tube : float
Area of the bare tube including the portion hidden by the fin per
tube :math:`A_{bare,total/tube} = \pi D_{tube} L_{tube}`, [m^2]
A_bare_tube_per_row : float
Area of the bare tube including the portion hidden by the fin per
tube row
:math:`A_{bare,total/row} = \pi D_{tube} L_{tube} N_{tubes/row}`, [m^2]
A_bare_tube_per_bundle : float
Area of the bare tube including the portion hidden by the fin per
bundle :math:`A_{bare,total/bundle} = \pi D_{tube} L_{tube}
N_{tubes/bundle}`, [m^2]
A_bare_tube_per_bay : float
Area of the bare tube including the portion hidden by the fin per
bay :math:`A_{bare,total/bay} = \pi D_{tube} L_{tube} N_{tubes/bay}`,
[m^2]
A_bare_tube : float
Area of the bare tube including the portion hidden by the fin per
in all bundles and bays combined :math:`A_{bare,total} = \pi D_{tube}
L_{tube} N_{tubes}`, [m^2]
A_tube_showing_per_tube : float
Area of the bare tube which is exposed per tube :math:`A_{bare,
showing/tube} = \pi D_{tube} L_{tube} \left(1 - \frac{t_{fin}}
{\text{fin interval}} \right)`, [m^2]
A_tube_showing_per_row : float
Area of the bare tube which is exposed per tube row, [m^2]
A_tube_showing_per_bundle : float
Area of the bare tube which is exposed per bundle, [m^2]
A_tube_showing_per_bay : float
Area of the bare tube which is exposed per bay, [m^2]
A_tube_showing : float
Area of the bare tube which is exposed in all bundles and bays
combined, [m^2]
A_per_fin : float
Surface area per fin :math:`A_{fin} = 2 \frac{\pi}{4} (D_{fin}^2 -
D_{tube}^2) + \pi D_{fin} t_{fin}`, [m^2]
A_fin_per_tube : float
Surface area of all fins per tube
:math:`A_{fin/tube} = N_{fins/m} L_{tube} A_{fin}`, [m^2]
A_fin_per_row : float
Surface area of all fins per row, [m^2]
A_fin_per_bundle : float
Surface area of all fins per bundle, [m^2]
A_fin_per_bay : float
Surface area of all fins per bay, [m^2]
A_fin : float
Surface area of all fins in all bundles and bays combined, [m^2]
A_per_tube : float
Surface area of combined finned and non-fined area exposed for heat
transfer per tube :math:`A_{tube} = A_{bare, showing/tube}
+ A_{fin/tube}`, [m^2]
A_per_row : float
Surface area of combined finned and non-finned area exposed for heat
transfer per tube row, [m^2]
A_per_bundle : float
Surface area of combined finned and non-finned area exposed for heat
transfer per tube bundle, [m^2]
A_per_bay : float
Surface area of combined finned and non-finned area exposed for heat
transfer per bay, [m^2]
A : float
Surface area of combined finned and non-finned area exposed for heat
transfer in all bundles and bays combined, [m^2]
A_increase : float
Ratio of actual surface area to bare tube surface area
:math:`A_{increase} = \frac{A_{tube}}{A_{bare, total/tube}}`, [-]
A_tube_flow : float
The area for the fluid to flow in one tube, :math:`\pi/4\cdot D_i^2`,
[m^2]
channels : int
The number of tubes the fluid flows through at the inlet header, [-]
tube_volume_per_tube : float
Fluid volume per tube inside :math:`V_{tube, flow} = \frac{\pi}{4}
D_{i}^2 L_{tube}`, [m^3]
tube_volume_per_row : float
Fluid volume of tubes per row, [m^3]
tube_volume_per_bundle : float
Fluid volume of tubes per bundle, [m^3]
tube_volume_per_bay : float
Fluid volume of tubes per bay, [m^3]
tube_volume : float
Fluid volume of tubes in all bundles and bays combined, [m^3]
A_diagonal_per_bundle : float
Air flow area along the diagonal plane per bundle
:math:`A_d = 2 N_{tubes/row} L_{tube} (P_d - D_{tube} - 2 N_{fins/m} h_{fin} t_{fin}) + A_\text{extra,side}`, [m^2]
A_normal_per_bundle : float
Air flow area along the normal (transverse) plane; this is normally
the minimum flow area, except for some staggered configurations
:math:`A_t = N_{tubes/row} L_{tube} (P_t - D_{tube} - 2 N_{fins/m} h_{fin} t_{fin}) + A_\text{extra,side}`, [m^2]
A_min_per_bundle : float
Minimum air flow area per bundle; this is the characteristic area for
velocity calculation in most finned tube convection correlations
:math:`A_{min} = min(A_d, A_t)`, [m^2]
A_min_per_bay : float
Minimum air flow area per bay, [m^2]
A_min : float
Minimum air flow area, [m^2]
A_face_per_bundle : float
Face area per bundle :math:`A_{face} = P_{T} (1+N_{tubes/row})
L_{tube}`; if corbels are used, add 0.5 to tubes/row instead of 1,
[m^2]
A_face_per_bay : float
Face area per bay, [m^2]
A_face : float
Total face area, [m^2]
flow_area_contraction_ratio : float
Ratio of `A_min` to `A_face`, [-]
Notes
-----
Examples
--------
>>> from scipy.constants import inch
>>> AC = AirCooledExchanger(tube_rows=4, tube_passes=4, tubes_per_row=56, tube_length=10.9728,
... tube_diameter=1*inch, fin_thickness=0.013*inch, fin_density=10/inch,
... angle=30, pitch=2.5*inch, fin_height=0.625*inch, tube_thickness=0.00338,
... bundles_per_bay=2, parallel_bays=3, corbels=True)
References
----------
.. [1] Schlunder, Ernst U, and International Center for Heat and Mass
Transfer. Heat Exchanger Design Handbook. Washington:
Hemisphere Pub. Corp., 1983.
'''
def __repr__(self):
s = '<Air Cooler Geometry, %s>'
t = ''
for k, v in self.__dict__.items():
try:
t += f'{k}={v:g}, '
except:
t += f'{k}={v}, '
t = t[0:-2]
return s%t
def __init__(self, tube_rows, tube_passes, tubes_per_row, tube_length,
tube_diameter, fin_thickness,
angle=None, pitch=None, pitch_parallel=None, pitch_normal=None,
pitch_ratio=None,
fin_diameter=None, fin_height=None,
fin_density=None, fin_interval=None,
parallel_bays=1, bundles_per_bay=1, fans_per_bay=1,
corbels=False, tube_thickness=None, fan_diameter=None):
# TODO: fin types
self.tube_rows = tube_rows
self.tube_passes = tube_passes
self.tubes_per_row = tubes_per_row
self.tube_length = tube_length
self.tube_diameter = tube_diameter
self.fin_thickness = fin_thickness
self.fan_diameter = fan_diameter
if pitch_ratio is not None:
if pitch is not None:
pitch = self.tube_diameter*pitch_ratio
else:
raise ValueError('Specify only one of `pitch_ratio` or `pitch`')
angle, pitch, pitch_parallel, pitch_normal = pitch_angle_solver(
angle=angle, pitch=pitch, pitch_parallel=pitch_parallel,
pitch_normal=pitch_normal)
self.angle = angle
self.pitch = pitch
self.pitch_parallel = pitch_parallel
self.pitch_normal = pitch_normal
self.pitch_diagonal = sqrt(pitch_parallel**2 + (0.5*pitch_normal)**2)
if fin_diameter is None and fin_height is None:
raise ValueError('Specify only one of `fin_diameter` or `fin_height`')
elif fin_diameter is not None:
fin_height = 0.5*(fin_diameter - tube_diameter)
elif fin_height is not None:
fin_diameter = tube_diameter + 2.0*fin_height
self.fin_height = fin_height
self.fin_diameter = fin_diameter
if fin_density is None and fin_interval is None:
raise ValueError('Specify only one of `fin_density` or `fin_interval`')
elif fin_density is not None:
fin_interval = 1.0/fin_density
elif fin_interval is not None:
fin_density = 1.0/fin_interval
self.fin_interval = fin_interval
self.fin_density = fin_density
self.parallel_bays = parallel_bays
self.bundles_per_bay = bundles_per_bay
self.fans_per_bay = fans_per_bay
self.corbels = corbels
self.tube_thickness = tube_thickness
if self.fin_interval:
self.bare_length = self.fin_interval - self.fin_thickness
else:
self.bare_length = None
self.tubes_per_bundle = self.tubes_per_row*self.tube_rows
self.tubes_per_bay = self.tubes_per_bundle*self.bundles_per_bay
self.tubes = self.tubes_per_bay*self.parallel_bays
self.A_bare_tube_per_tube = pi*self.tube_diameter*self.tube_length
self.A_bare_tube_per_row = self.A_bare_tube_per_tube*self.tubes_per_row
self.A_bare_tube_per_bundle = self.A_bare_tube_per_tube*self.tubes_per_bundle
self.A_bare_tube_per_bay = self.A_bare_tube_per_tube*self.tubes_per_bay
self.A_bare_tube = self.A_bare_tube_per_tube*self.tubes
self.A_tube_showing_per_tube = pi*self.tube_diameter*self.tube_length*(1.0 - self.fin_thickness/self.fin_interval)
self.A_tube_showing_per_row = self.A_tube_showing_per_tube*self.tubes_per_row
self.A_tube_showing_per_bundle = self.A_tube_showing_per_tube*self.tubes_per_bundle
self.A_tube_showing_per_bay = self.A_tube_showing_per_tube*self.tubes_per_bay
self.A_tube_showing = self.A_tube_showing_per_tube*self.tubes
self.A_per_fin = (2.0*pi/4.0*(self.fin_diameter**2 - self.tube_diameter**2)
+ pi*self.fin_diameter*self.fin_thickness) # pi*D*L(fin)
self.A_fin_per_tube = self.fin_density*self.tube_length*self.A_per_fin
self.A_fin_per_row = self.A_fin_per_tube*self.tubes_per_row
self.A_fin_per_bundle = self.A_fin_per_tube*self.tubes_per_bundle
self.A_fin_per_bay = self.A_fin_per_tube*self.tubes_per_bay
self.A_fin = self.A_fin_per_tube*self.tubes
self.A_per_tube = self.A_tube_showing_per_tube + self.A_fin_per_tube
self.A_per_row = self.A_tube_showing_per_row + self.A_fin_per_row
self.A_per_bundle = self.A_tube_showing_per_bundle + self.A_fin_per_bundle
self.A_per_bay = self.A_tube_showing_per_bay + self.A_fin_per_bay
self.A = self.A_tube_showing + self.A_fin
self.A_increase = self.A/self.A_bare_tube
# TODO A_extra could be calculated based on a fixed width and height of the bay
A_extra = 0.0
self.A_diagonal_per_bundle = 2.0*self.tubes_per_row*self.tube_length*(self.pitch_diagonal - self.tube_diameter - 2.0*fin_density*self.fin_height*self.fin_thickness) + A_extra
self.A_normal_per_bundle = self.tubes_per_row*self.tube_length*(self.pitch_normal - self.tube_diameter - 2.0*fin_density*self.fin_height*self.fin_thickness) + A_extra
self.A_min_per_bundle = min(self.A_diagonal_per_bundle, self.A_normal_per_bundle)
self.A_min_per_bay = self.A_min_per_bundle*self.bundles_per_bay
self.A_min = self.A_min_per_bay*self.parallel_bays
i = 0.5 if self.corbels else 1.0
self.A_face_per_bundle = self.pitch_normal*self.tube_length*(self.tubes_per_row + i)
self.A_face_per_bay = self.A_face_per_bundle*self.bundles_per_bay
self.A_face = self.A_face_per_bay*self.parallel_bays
self.flow_area_contraction_ratio = self.A_min/self.A_face
if self.tube_thickness is not None:
self.Di = self.tube_diameter - self.tube_thickness*2.0
self.A_tube_flow = pi/4.0*self.Di*self.Di
self.tube_volume_per_tube = self.A_tube_flow*self.tube_length
self.tube_volume_per_row = self.tube_volume_per_tube*self.tubes_per_row
self.tube_volume_per_bundle = self.tube_volume_per_tube*self.tubes_per_bundle
self.tube_volume_per_bay = self.tube_volume_per_tube*self.tubes_per_bay
self.tube_volume = self.tube_volume_per_tube*self.tubes
else:
self.Di = None
self.A_tube_flow = None
self.tube_volume_per_tube = None
self.tube_volume_per_row = None
self.tube_volume_per_bundle = None
self.tube_volume_per_bay = None
self.tube_volume = None
# TODO: Support different numbers of tube rows per pass - maybe pass
# a list of rows per pass to tube_passes?
if self.tube_rows % self.tube_passes == 0:
self.channels = self.tubes_per_bundle/self.tube_passes
else:
self.channels = self.tubes_per_row
if self.angle == 30:
self.pitch_str = 'triangular'
self.pitch_class = 'staggered'
elif self.angle == 60:
self.pitch_str = 'rotated triangular'
self.pitch_class = 'staggered'
elif self.angle == 45:
self.pitch_str = 'rotated square'
self.pitch_class = 'in-line'
elif self.angle == 90:
self.pitch_str = 'square'
self.pitch_class = 'in-line'
else:
self.pitch_str = 'custom'
self.pitch_class = 'custom'
[docs]def pitch_angle_solver(angle=None, pitch=None, pitch_parallel=None,
pitch_normal=None):
r'''Utility to take any two of `angle`, `pitch`, `pitch_parallel`, and
`pitch_normal` and calculate the other two. This is useful for applications
with tube banks, as in shell and tube heat exchangers or air coolers and
allows for a wider range of user input.
.. math::
\text{pitch normal} = \text{pitch} \cdot \sin(\text{angle})
.. math::
\text{pitch parallel} = \text{pitch} \cdot \cos(\text{angle})
Parameters
----------
angle : float, optional
The angle of the tube layout, [degrees]
pitch : float, optional
The shortest distance between tube centers; defined in relation to the
flow direction only, [m]
pitch_parallel : float, optional
The distance between tube center along a line parallel to the flow;
has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m]
pitch_normal : float, optional
The distance between tube centers in a line 90° to the line of flow;
has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m]
Returns
-------
angle : float
The angle of the tube layout, [degrees]
pitch : float
The shortest distance between tube centers; defined in relation to the
flow direction only, [m]
pitch_parallel : float
The distance between tube center along a line parallel to the flow;
has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m]
pitch_normal : float
The distance between tube centers in a line 90° to the line of flow;
has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m]
Notes
-----
For the 90 and 0 degree case, the normal or parallel pitches can be zero;
given the angle and the zero value, obviously is it not possible to
calculate the pitch and a math error will be raised.
No exception will be raised if three or four inputs are provided; the other
two will simply be calculated according to the list of if statements used.
An exception will be raised if only one input is provided.
Examples
--------
>>> pitch_angle_solver(pitch=1, angle=30)
(30, 1, 0.8660254037844387, 0.49999999999999994)
References
----------
.. [1] Schlunder, Ernst U, and International Center for Heat and Mass
Transfer. Heat Exchanger Design Handbook. Washington:
Hemisphere Pub. Corp., 1983.
'''
if angle is not None and pitch is not None:
pitch_normal = pitch*sin(radians(angle))
pitch_parallel = pitch*cos(radians(angle))
elif angle is not None and pitch_normal is not None:
pitch = pitch_normal/sin(radians(angle))
pitch_parallel = pitch*cos(radians(angle))
elif angle is not None and pitch_parallel is not None:
pitch = pitch_parallel/cos(radians(angle))
pitch_normal = pitch*sin(radians(angle))
elif pitch_normal is not None and pitch is not None:
angle = degrees(asin(pitch_normal/pitch))
pitch_parallel = pitch*cos(radians(angle))
elif pitch_parallel is not None and pitch is not None:
angle = degrees(acos(pitch_parallel/pitch))
pitch_normal = pitch*sin(radians(angle))
elif pitch_parallel is not None and pitch_normal is not None:
angle = degrees(asin(pitch_normal/sqrt(pitch_normal**2 + pitch_parallel**2)))
pitch = sqrt(pitch_normal**2 + pitch_parallel**2)
else:
raise ValueError('Two of the arguments are required')
return angle, pitch, pitch_parallel, pitch_normal
[docs]def sphericity(A, V):
r'''Returns the sphericity of a particle of surface area `A` and volume
`V`. Sphericity is the ratio of the surface area of a sphere with the same
volume as the particle (equivalent diameter) to the actual surface area of
the particle.
.. math::
\Psi = \frac{\text{A of sphere with } V_p } {{A}_p}
= \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}
Parameters
----------
A : float
Surface area of particle, [m^2]
V : float
Volume of particle, [m^3]
Returns
-------
Psi : float
Sphericity [-]
Notes
-----
All non-spherical particles have spericities less than 1 but greater than 0.
Many common geometrical shapes have their results calculated exactly in [2]_.
Examples
--------
>>> sphericity(10., 2.)
0.767663317071005
For a cube of side length a=3, the surface area is 6*a^2=54 and volume a^3=27.
Its sphericity is then:
>>> sphericity(A=54, V=27)
0.8059959770082346
References
----------
.. [1] Rhodes, Martin J., ed. Introduction to Particle Technology. 2E.
Chichester, England ; Hoboken, NJ: Wiley, 2008.
.. [2] "Sphericity." Wikipedia, March 8, 2017.
https://en.wikipedia.org/w/index.php?title=Sphericity&oldid=769183043
'''
return pi**(1/3.)*(6*V)**(2/3.)/A
[docs]def aspect_ratio(Dmin, Dmax):
r'''Returns the aspect ratio of a shape with minimum and maximum dimension,
`Dmin` and `Dmax`.
.. math::
A_R = \frac{D_{min}}{D_{max}}
Parameters
----------
Dmin : float
Minimum dimension, [m]
Dmax : float
Maximum dimension, [m]
Returns
-------
a_r : float
Aspect ratio [-]
Examples
--------
>>> aspect_ratio(.2, 2)
0.1
'''
return Dmin/Dmax
[docs]def circularity(A, P):
r'''Returns the circularity of a shape with area `A` and perimeter `P`.
.. math::
f_{circ} = \frac {4 \pi A} {P^2}
Defined to be 1 for a circle. Used to characterize particles. Any
non-circular shape must have a circularity less than one.
Parameters
----------
A : float
Area of the shape, [m^2]
P : float
Perimeter of the shape, [m]
Returns
-------
f_circ : float
Circularity of the shape [-]
Examples
--------
Square, side length = 2 (all squares are the same):
>>> circularity(A=(2*2), P=4*2)
0.7853981633974483
Rectangle, one side length = 1, second side length = 100
>>> D1 = 1
>>> D2 = 100
>>> A = D1*D2
>>> P = 2*D1 + 2*D2
>>> circularity(A, P)
0.030796908671598795
'''
return 4*pi*A/P**2
[docs]def A_cylinder(D, L):
r'''Returns the surface area of a cylinder.
.. math::
A = \pi D L + 2\cdot \frac{\pi D^2}{4}
Parameters
----------
D : float
Diameter of the cylinder, [m]
L : float
Length of the cylinder, [m]
Returns
-------
A : float
Surface area [m^2]
Examples
--------
>>> A_cylinder(0.01, .1)
0.0032986722862692833
'''
cap = pi*D**2/4*2
side = pi*D*L
return cap + side
[docs]def V_cylinder(D, L):
r'''Returns the volume of a cylinder.
.. math::
V = \frac{\pi D^2}{4}L
Parameters
----------
D : float
Diameter of the cylinder, [m]
L : float
Length of the cylinder, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
>>> V_cylinder(0.01, .1)
7.853981633974484e-06
'''
return pi*D**2/4*L
[docs]def A_hollow_cylinder(Di, Do, L):
r'''Returns the surface area of a hollow cylinder.
.. math::
A = \pi D_o L + \pi D_i L + 2\cdot \frac{\pi D_o^2}{4}
- 2\cdot \frac{\pi D_i^2}{4}
Parameters
----------
Di : float
Diameter of the hollow in the cylinder, [m]
Do : float
Diameter of the exterior of the cylinder, [m]
L : float
Length of the cylinder, [m]
Returns
-------
A : float
Surface area [m^2]
Examples
--------
>>> A_hollow_cylinder(0.005, 0.01, 0.1)
0.004830198704894308
'''
side_o = pi*Do*L
side_i = pi*Di*L
cap_circle = pi*Do**2/4*2
cap_removed = pi*Di**2/4*2
return side_o + side_i + cap_circle - cap_removed
[docs]def V_hollow_cylinder(Di, Do, L):
r'''Returns the volume of a hollow cylinder.
.. math::
V = \frac{\pi D_o^2}{4}L - L\frac{\pi D_i^2}{4}
Parameters
----------
Di : float
Diameter of the hollow in the cylinder, [m]
Do : float
Diameter of the exterior of the cylinder, [m]
L : float
Length of the cylinder, [m]
Returns
-------
V : float
Volume [m^3]
Examples
--------
>>> V_hollow_cylinder(0.005, 0.01, 0.1)
5.890486225480862e-06
'''
return pi*Do**2/4*L - pi*Di**2/4*L
[docs]def A_multiple_hole_cylinder(Do, L, holes):
r'''Returns the surface area of a cylinder with multiple holes.
Calculation will naively return a negative value or other impossible
result if the number of cylinders added is physically impossible.
Holes may be of different shapes, but must be perpendicular to the
axis of the cylinder.
.. math::
A = \pi D_o L + 2\cdot \frac{\pi D_o^2}{4} +
\sum_{i}^n \left( \pi D_i L - 2\cdot \frac{\pi D_i^2}{4}\right)
Parameters
----------
Do : float
Diameter of the exterior of the cylinder, [m]
L : float
Length of the cylinder, [m]
holes : list
List of tuples containing (diameter, count) pairs of descriptions for
each of the holes sizes.
Returns
-------
A : float
Surface area [m^2]
Examples
--------
>>> A_multiple_hole_cylinder(0.01, 0.1, [(0.005, 1)])
0.004830198704894308
'''
side_o = pi*Do*L
cap_circle = pi*Do**2/4*2
A = cap_circle + side_o
for Di, n in holes:
side_i = pi*Di*L
cap_removed = pi*Di**2/4*2
A = A + side_i*n - cap_removed*n
return A
[docs]def V_multiple_hole_cylinder(Do, L, holes):
r'''Returns the solid volume of a cylinder with multiple cylindrical holes.
Calculation will naively return a negative value or other impossible
result if the number of cylinders added is physically impossible.
.. math::
V = \frac{\pi D_o^2}{4}L - L\frac{\pi D_i^2}{4}
Parameters
----------
Do : float
Diameter of the exterior of the cylinder, [m]
L : float
Length of the cylinder, [m]
holes : list
List of tuples containing (diameter, count) pairs of descriptions for
each of the holes sizes.
Returns
-------
V : float
Volume [m^3]
Examples
--------
>>> V_multiple_hole_cylinder(0.01, 0.1, [(0.005, 1)])
5.890486225480862e-06
'''
V = pi*Do**2/4*L
for Di, n in holes:
V -= pi*Di*Di/4*L*n
return V