# -*- coding: utf-8 -*-
'''Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018 Caleb Bell <Caleb.Andrew.Bell@gmail.com>
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.'''
from __future__ import division
from math import (pi, sin, cos, tan, asin, acos, atan, acosh, log, radians,
degrees)
import numpy as np
from numpy.polynomial.chebyshev import chebval
from scipy.constants import inch
from scipy.interpolate import interp1d, InterpolatedUnivariateSpline
from scipy.integrate import quad
from scipy.optimize import newton, brenth
from scipy.special import ellipe
from fluids.optional.pychebfun import Chebfun
__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', 'V_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']
### 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 = (h*(2.*r - h))**0.5
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.'''
r = 0.5*D
a = (h*(2.*r - h))**0.5
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::
Af = 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'''
R = D/2.
Af = R*R*acos((R-h)/R) - (R-h)*(2*R*h - h*h)**0.5
M = abs((R-h)/R)
if h == R:
Vf = a*R*R/3.*pi
else:
K = acos(M) + M*M*M*acosh(1./M) - 2.*M*(1.-M*M)**0.5
if 0. <= h < R:
Vf = 2.*a*R*R/3*K
elif R < h <= 2*R:
Vf = 2.*a*R*R/3*(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::
Af = 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'''
R = 0.5*D
Af = R*R*acos((R-h)/R) - (R-h)*(2*R*h - h*h)**0.5
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::
Af = 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'''
R = 0.5*D
Af = R*R*acos((R-h)/R) - (R-h)*(2.*R*h - h*h)**0.5
Vf = 2.*a*R*R/3.*acos(1. - h/R) + 2.*a/9./R*(2*R*h - h**2)**0.5*(2*h - 3*R)*(h + R)
if headonly:
Vf = Vf/2.
else:
Vf += Af*L
return Vf
[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::
Af = 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'''
R = D/2.
r = (a**2 + R**2)/2./abs(a)
w = R - h
y = (2*R*h - h**2)**0.5
z = (r**2 - R**2)**0.5
Af = R**2*acos((R-h)/R) - (R-h)*(2*R*h - h**2)**0.5
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:
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:
def V_horiz_spherical_toint(x):
return (r**2 - x**2)*atan(((R**2 - x**2)/(r**2 - R**2))**0.5)
integrated = quad(V_horiz_spherical_toint, w, R)[0]
Vf = a/abs(a)*(2*integrated - Af*z)
if headonly:
Vf = Vf/2.
else:
Vf += Af*L
return Vf
[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
Dish-radius parameter; fD = dish radius [1/m]
k : float
knuckle-radius parameter ; kD = knuckle radius [1/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_torispherical(D=108., L=156., f=1., k=0.06, h=36)/231.
2028.626670842139
References
----------
.. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015.
http://www.webcalc.com.br/blog/Tank_Volume.PDF'''
R = D/2.
Af = R**2*acos((R-h)/R) - (R-h)*(2*R*h - h**2)**0.5
r = f*D
alpha = asin((1 - 2*k)/(2.*(f-k)))
a1 = r*(1-cos(alpha))
g = r*sin(alpha)
z = r*cos(alpha)
h1 = k*D*(1-sin(alpha))
h2 = D - h1
# Chebfun in Python failed on these functions
def V1_toint(x, w):
# No analytical integral available in MP
n = R - k*D + (k**2*D**2 - x**2)**0.5
ans = n**2*asin((n**2-w**2)**0.5/n) - w*(n**2 - w**2)**0.5
return ans
def V2_toint(x, w):
# No analytical integral available in MP
n = R - k*D + (k**2*D**2 - x**2)**0.5
ans = n**2*(acos(w/n) - acos(g/n)) - w*(n**2 - w**2)**0.5 + g*(n**2-g**2)**0.5
return ans
def V3_toint(x):
# 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
ans = (r**2-x**2)*atan((g**2-x**2)**0.5/z)
return ans
if 0 <= h <= h1:
w = R - h
Vf = 2*quad(V1_toint, 0, (2*k*D*h-h**2)**0.5, w)[0]
elif h1 < h < h2:
w = R - h
wmax1 = R - h1
V1max = quad(V1_toint, 0, (2*k*D*h1-h1**2)**0.5, wmax1)[0]
V2 = quad(V2_toint, 0, k*D*cos(alpha), w)[0]
V3 = quad(V3_toint, w, g)[0] - z/2.*(g**2*acos(w/g) -w*(2*g*(h-h1) - (h-h1)**2)**0.5)
Vf = 2*(V1max + V2 + V3)
else:
w = R - h
wmax1 = R - h1
wmax2 = R - h2
wwerird = R - (D - h)
V1max = quad(V1_toint, 0, (2*k*D*h1-h1**2)**0.5, wmax1)[0]
V1weird = quad(V1_toint, 0, (2*k*D*(D-h)-(D-h)**2)**0.5, wwerird)[0]
V2max = quad(V2_toint, 0, k*D*cos(alpha), wmax2)[0]
V3max = pi*a1/6.*(3*g**2 + a1**2)
Vf = 2*(2*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 < 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 < 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 < 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
Dish-radius parameter; fD = dish radius [1/m]
k : float
knuckle-radius parameter ; kD = knuckle radius [1/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_torispherical(D=132., f=1.0, k=0.06, h=24)/231.
904.0688283793511
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)
D1 = 2*f*D*sin(alpha)
s = (k*D*sin(alpha))**2
t = 2*a2
u = h - f*D*(1 - cos(alpha))
if 0 <= h <= a1:
Vf = pi*h**2/4*(2*a1 + D1**2/2/a1 - 4*h/3)
elif a1 < h <= a1 + a2:
Vf = (pi/4*(2*a1**3/3 + a1*D1**2/2.) + pi*u*((D/2. - k*D)**2 + s)
+ pi*t*u**2/2. - pi*u**3/3. + pi*D*(1 - 2*k)*((2*u-t)/4.*(s + t*u
- u**2)**0.5 + t*s**0.5/4. + k**2*D**2/2*(acos((t-2*u)/(2*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*s**0.5/4
+ 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 < 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 < 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 < 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
Dish-radius parameter; fD = dish radius [1/m]
k : float
knuckle-radius parameter ; kD = knuckle radius [1/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_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/
'''
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.*(s+t*u-u**2)**0.5 + t*s**0.5/4.
+ 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]_.
Formula below is for the full shape, the result of which is halved. The
formula also does not support `D` being larger than `a`; this is ensured
by simply swapping the variables if necessary, as geometrically the result
is the same. In the equations
.. math::
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}}
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
References
----------
.. [1] Weisstein, Eric W. "Spheroid." Text. Accessed March 14, 2016.
http://mathworld.wolfram.com/Spheroid.html.
'''
if D == a*2:
return pi*D**2/2 # necessary to avoid a division by zero when D == a
D = D/2.
D, a = min((D, a)), max((D, a))
e1 = (1 - D**2/a**2)**0.5
return (2*pi*a**2 + pi*D**2/e1*log((1+e1)/(1-e1)))/2.
[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 pi*D/2*(a**2 + (D/2)**2)**0.5
[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 pi*D/4*(a**2 + D**2)**0.5 + pi*D/2*a
[docs]def SA_torispheroidal(D, fd, fk):
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]
fd : float
Dish-radius parameter = f; fD = dish radius [1/m]
fk : float
knuckle-radius parameter = k; kD = knuckle radius [1/m]
Returns
-------
SA : float
Surface area [m^2]
Examples
--------
Example from [1]_.
>>> SA_torispheroidal(D=2.54, fd=1.039370079, fk=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.
'''
alpha_1 = fd*(1 - (1 - ((0.5-fk)/(fd-fk))**2)**0.5)
alpha_2 = fd - (fd**2 - 2*fd*fk + fk - 0.25)**0.5
alpha = alpha_1 # Up to top of dome
S1 = 2*pi*D**2*fd*alpha_1
alpha = alpha_2 # up to top of torus
S2_sub = asin((alpha-alpha_2)/fk) - asin((alpha_1-alpha_2)/fk)
S2 = 2*pi*D**2*fk*(alpha - alpha_1 + (0.5-fk)*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,
full_output=False):
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]
areas : tuple, only returned if full_output == True
(sideA_SA, sideB_SA, lateral_SA)
Other Parameters
----------------
full_output : bool, optional
Returns a tuple of (sideA_SA, sideB_SA, lateral_SA) if True
Examples
--------
Cylinder, Spheroid, Long Cones, and spheres. All checked.
>>> SA_tank(D=2, L=2)
18.84955592153876
>>> SA_tank(D=1., L=0, sideA='ellipsoidal', sideA_a=2, sideB='ellipsoidal',
... sideB_a=2)
28.480278854014387
>>> SA_tank(D=1., L=5, sideA='conical', sideA_a=2, sideB='conical',
... sideB_a=2)
22.18452243965656
>>> SA_tank(D=1., L=5, sideA='spherical', sideA_a=0.5, sideB='spherical',
... sideB_a=0.5)
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 = SA_partial_sphere(D=D, h=sideA_a)
elif sideA == 'torispherical':
sideA_SA = SA_torispheroidal(D=D, fd=sideA_f, fk=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 = SA_partial_sphere(D=D, h=sideB_a)
elif sideB == 'torispherical':
sideB_SA = SA_torispheroidal(D=D, fd=sideB_f, fk=sideB_k)
else:
sideB_SA = pi/4*D**2 # Circle
lateral_SA = pi*D*L
SA = sideA_SA + sideB_SA + lateral_SA
if full_output:
return SA, (sideA_SA, sideB_SA, lateral_SA)
else:
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
Dish-radius parameter; fD = dish radius [1/m]
k : float
knuckle-radius parameter ; kD = knuckle radius [1/m]
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
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
-------
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 not in [None, 'conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy']:
raise Exception('Unspoorted head type for side A')
if sideB not in [None, 'conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy']:
raise Exception('Unspoorted head type for side B')
R = D/2.
V = 0
if horizontal:
# Conical case
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':
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':
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':
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':
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 Exception('Input height is above top of tank')
Af = R**2*acos((R-h)/R) - (R-h)*(2*R*h - h**2)**0.5
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 -= V_vertical_torispherical(D, sideB_f, sideB_k, h=h2)
if h > L + sideA_a + sideB_a:
raise Exception('Input height is above top of tank')
return V
[docs]class TANK(object):
'''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'].
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]
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
----------
table : bool
Whether or not a table of heights-volumes has been generated
h_max : float
Height of the tank, [m]
V_total : float
Total volume of the tank as calculated [m^3]
heights : ndarray
Array of heights between 0 and h_max, [m]
volumes : ndarray
Array of volumes calculated from the heights, [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]
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 |
+----------------------+-----+-------+
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)
0.44209706414415384
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
(24.94775907657148, 5.118555935958284, 5.497246519930003, 14.331956620683192)
Solving for tank volumes, first horizontal, then vertical:
>>> TANK(D=10., horizontal=True, sideA='conical', sideB='conical', V=500).L
4.699531057009147
>>> TANK(L=4.69953105701, horizontal=True, sideA='conical', sideB='conical', V=500).D
9.999999999999407
>>> TANK(L_over_D=0.469953105701, horizontal=True, sideA='conical', sideB='conical', V=500).L
4.69953105700979
>>> TANK(D=10., horizontal=False, sideA='conical', sideB='conical', V=500).L
4.699531057009147
>>> TANK(L=4.69953105701, horizontal=False, sideA='conical', sideB='conical', V=500).D
9.999999999999407
>>> TANK(L_over_D=0.469953105701, horizontal=False, sideA='conical', sideB='conical', V=500).L
4.699531057009791
'''
table = False
chebyshev = False
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 + ' heads, sideA a=%f m, sideB a=%f m' % (self.sideA_a, self.sideB_a)
else:
if self.sideA:
A = '%s head on sideA with a=%f m' % (self.sideA, self.sideA_a)
else:
A = 'no head on sideA'
if self.sideB:
B = ' and %s head on sideB with a=%f m' % (self.sideB, self.sideB_a)
else:
B = ' and no head on sideB'
sides = A + B
return '<%s tank, V=%f m^3, D=%f m, L=%f m, %s.>' %(orient, self.V_total, self.D, self.L, sides)
def __init__(self, D=None, L=None, horizontal=True,
sideA=None, sideB=None, sideA_a=0, sideB_a=0,
sideA_f=1., sideA_k=0.06, sideB_f=1., sideB_k=0.06,
sideA_a_ratio=0.25, sideB_a_ratio=0.25, 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
self.sideA = sideA
self.sideA_a = sideA_a
self.sideA_f = sideA_f
self.sideA_k = sideA_k
self.sideA_a_ratio = sideA_a_ratio
self.sideB = sideB
self.sideB_a = sideB_a
self.sideB_f = sideB_f
self.sideB_k = sideB_k
self.sideB_a_ratio = sideB_a_ratio
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 and self.L:
# If L and D are known, get L_over_D
self.L_over_D = self.L/self.D
elif self.D and self.L_over_D:
# Otherwise, if L_over_D and D are provided, get L
self.L = self.D*self.L_over_D
elif self.L and self.L_over_D:
# Otherwise, if L_over_D and L are provided, get D
self.D = self.L/self.L_over_D
# Calculate diameter
self.R = self.D/2.
# If a_ratio is provided for either heads, use it.
if self.sideA and self.D:
if not self.sideA_a and self.sideA in ['conical', 'ellipsoidal', 'guppy', 'spherical']:
self.sideA_a = self.D*self.sideA_a_ratio
if self.sideB and self.D:
if not self.sideB_a and self.sideB in ['conical', 'ellipsoidal', 'guppy', 'spherical']:
self.sideB_a = self.D*self.sideB_a_ratio
# Calculate a for torispherical heads
if self.sideA == 'torispherical' and self.sideA_f and self.sideA_k:
self.sideA_a = a_torispherical(self.D, self.sideA_f, self.sideA_k)
if self.sideB == 'torispherical' and self.sideB_f and self.sideB_k:
self.sideB_a = a_torispherical(self.D, self.sideB_f, self.sideB_k)
# Calculate maximum tank height, h_max
if self.horizontal:
self.h_max = self.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)
# Set surface areas
self.A, (self.A_sideA, self.A_sideB, self.A_lateral) = SA_tank(
D=self.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,
full_output=True)
[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 Exception("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':
if not self.table:
self.set_table()
return float(self.interp_h_from_V(V))
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 Exception("Allowable methods are 'full' or 'chebyshev', "
"or 'brenth'.")
[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 = np.linspace(0, self.h_max, int(self.h_max/dx)+1)
else:
self.heights = np.linspace(0, self.h_max, n)
self.volumes = [self.V_from_h(h) for h in self.heights]
self.interp_h_from_V = InterpolatedUnivariateSpline(self.volumes, self.heights, ext=3)
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, [-]
'''
to_fit = lambda h: self.V_from_h(h, 'full')
self.c_forward = np.array(Chebfun.from_function(np.vectorize(to_fit),
[0.0, self.h_max], N=deg_forward).coefficients())
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()
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 = abs(Vtarget - a.V_total)
return error
[docs] 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 Exception('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 Exception('Only one of D, L, or L_over_D can be specified\
when solving for V')
if ((self.sideA and not self.sideA_a_ratio) or (self.sideB and not self.sideB_a_ratio)):
raise Exception('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(newton(solve_L, Lguess))
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 = (4*self.V/pi/self.L)**0.5
self.D = float(newton(solve_D, Dguess))
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(newton(solve_L_D, Lguess))
self.D = self.L/self.L_over_D
[docs]class HelicalCoil(object):
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, optional
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=%s m, total outer diameter=%s m, tube \
outer diameter=%s m, number of turns=%s, pitch=%s m' % (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:
H = H_total - Dt
if Do_total:
Do = Do_total - Dt
self.Do = Do
self.Dt = Dt
self.Do_total = self.Do+self.Dt
if N and pitch:
self.N = N
self.pitch = pitch
self.H = N*pitch
elif N and H:
self.N = N
self.H = H
self.pitch = self.H/N
if self.pitch < self.Dt:
raise Exception('Pitch is too small - tubes are colliding; maximum number of spirals is %f.'%(self.H/self.Dt))
elif H and pitch:
self.pitch = pitch
self.H = H
self.N = self.H/self.pitch
if self.pitch < self.Dt:
raise Exception('Pitch is too small - tubes are colliding; pitch must be larger than tube diameter.')
self.H_total = self.Dt + self.H
if self.Dt > self.Do:
raise Exception('Tube diameter is larger than helix outer diameter - not feasible.')
self.tube_circumference = pi*self.Do
self.tube_length = ((self.tube_circumference*self.N)**2 + self.H**2)**0.5
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
self.Di = Di
if 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]class PlateExchanger(object):
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 or tuple(2), 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. Many plate exchangers use two alternating patterns; use a tuple
of the two angles for that situation [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, [-]
plate_exchanger_identifier : str
Identifying string in format 'L' + wavelength + 'A' + amplitude + 'B'
+ chevron angle-chevron angle
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=%s degrees, area enhancement factor=%g' %(self.a, self.wavelength, '/'.join([str(i) for i in self.chevron_angles]), self.plate_enlargement_factor)
if self.width and self.length:
s += ', width=%g m, length=%g m' %(self.width, self.length)
if self.d_port:
s += ', port diameter=%g m' %(self.d_port)
if self.plates:
s += ', heat transfer area=%g m^2, %g plates>' %(self.A_heat_transfer, self.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.
'''
s = ('L' + str(round(self.wavelength*1000, 2))
+ 'A' + str(round(self.amplitude*1000, 2))
+ 'B' + '-'.join([str(i) for i in self.chevron_angles]))
return s
[docs] @staticmethod
def plate_enlargement_factor_analytical(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)
A = \frac{2\pi a}{\lambda}
Most plate heat exchangers approximate a sinusoidal geometry only.
Examples
--------
>>> PlateExchanger.plate_enlargement_factor_analytical(amplitude=5E-4, wavelength=3.7E-3)
1.1611862034509677
'''
b = 2.*amplitude
return 2.*ellipe(-b*b*pi*pi/(wavelength*wavelength))/pi
def __init__(self, amplitude, wavelength, chevron_angle=45, 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 isinstance(chevron_angle, tuple):
self.chevron_angles = chevron_angle
self.chevron_angle = self.beta = 0.5*(chevron_angle[0]+chevron_angle[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 = self.plate_enlargement_factor_analytical(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
self.width = width
self.length = length
self.thickness = thickness
self.d_port = d_port
self.plates = plates
if d_port and length:
self.length_port = self.length + self.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 and length:
self.A_plate_surface = self.length*self.width*self.plate_enlargement_factor # use this in Q = UAdT
if plates:
self.A_heat_transfer = (self.plates-2)*self.A_plate_surface # the two outermost sides aren't used
if width:
self.A_channel_flow = self.width*self.b # Use this to get G, kg/s/m^2
if plates:
self.channels = self.plates - 1
self.channels_per_fluid = 0.5*self.channels
[docs]class RectangularFinExchanger(object):
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.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(object):
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/(D_base**2 - self.D_throat**2)**0.5
D_inlet = 2*self.D_throat*((self.H_throat-H_inlet)**2 + b**2)**0.5/(2*b)
elif D_inlet is not None:
b = self.D_throat*(self.H_throat-H_inlet)/(D_inlet**2 - self.D_throat**2)**0.5
D_base = 2*self.D_throat*(self.H_throat**2 + b**2)**0.5/(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)/((self.D_outlet)**2 - self.D_throat**2)**0.5
# 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 = np.linspace(0, self.H_outlet, pts)
Rs = np.array([self.diameter(Z) for Z in Zs])/2
plt.plot(Zs, Rs)
plt.plot(Zs, -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*(H*H + b*b)**0.5/(2.0*b)
return R*2.0
[docs]class AirCooledExchanger(object):
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
--------
>>> 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 += '%s=%g, ' %(k, v)
except:
t += '%s=%s, ' %(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 Exception('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 = (pitch_parallel**2 + (0.5*pitch_normal)**2)**0.5
if fin_diameter is None and fin_height is None:
raise Exception('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 Exception('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/(pitch_normal**2 + pitch_parallel**2)**0.5))
pitch = (pitch_normal**2 + pitch_parallel**2)**0.5
else:
raise Exception('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