"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling. Copyright (C) 2016, 2017, 2018, 2019, 2020 Caleb Bell. Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. This module contains functionality for calculating parameters about different geometrical forms that come up in engineering practice. Implemented geometry objects are tanks, helical coils, cooling towers, air coolers, compact heat exchangers, and plate and frame heat exchangers. Additional functionality for typical catalyst/adsorbent pellet shapes is also included. For reporting bugs, adding feature requests, or submitting pull requests, please use the `GitHub issue tracker `_ or contact the author at Caleb.Andrew.Bell@gmail.com. .. contents:: :local: Main Interfaces --------------- .. autoclass :: TANK :members: :undoc-members: :show-inheritance: .. autofunction:: V_tank .. autofunction:: V_from_h .. autofunction:: SA_tank .. autofunction:: SA_from_h .. autoclass :: HelicalCoil :members: :undoc-members: :show-inheritance: .. autoclass :: PlateExchanger :members: :undoc-members: :show-inheritance: .. autoclass :: AirCooledExchanger :members: :undoc-members: :show-inheritance: .. autoclass :: HyperbolicCoolingTower :members: :undoc-members: :show-inheritance: .. autoclass :: RectangularFinExchanger :members: :undoc-members: :show-inheritance: .. autoclass :: RectangularOffsetStripFinExchanger :members: :undoc-members: :show-inheritance: Tank Volume Functions --------------------- .. autofunction:: V_partial_sphere .. autofunction:: V_horiz_conical .. autofunction:: V_horiz_ellipsoidal .. autofunction:: V_horiz_guppy .. autofunction:: V_horiz_spherical .. autofunction:: V_horiz_torispherical .. autofunction:: V_vertical_conical .. autofunction:: V_vertical_ellipsoidal .. autofunction:: V_vertical_spherical .. autofunction:: V_vertical_torispherical .. autofunction:: V_vertical_conical_concave .. autofunction:: V_vertical_ellipsoidal_concave .. autofunction:: V_vertical_spherical_concave .. autofunction:: V_vertical_torispherical_concave Tank Surface Area Functions --------------------------- .. autofunction:: SA_partial_sphere .. autofunction:: SA_ellipsoidal_head .. autofunction:: SA_conical_head .. autofunction:: SA_guppy_head .. autofunction:: SA_torispheroidal .. autofunction:: SA_partial_cylindrical_body .. autofunction:: SA_partial_horiz_conical_head .. autofunction:: SA_partial_horiz_spherical_head .. autofunction:: SA_partial_horiz_ellipsoidal_head .. autofunction:: SA_partial_horiz_guppy_head .. autofunction:: SA_partial_horiz_torispherical_head .. autofunction:: SA_partial_vertical_conical_head .. autofunction:: SA_partial_vertical_ellipsoidal_head .. autofunction:: SA_partial_vertical_spherical_head .. autofunction:: SA_partial_vertical_torispherical_head Miscellaneous Geometry Functions -------------------------------- .. autofunction:: pitch_angle_solver .. autofunction:: plate_enlargement_factor .. autofunction:: a_torispherical .. autofunction:: A_partial_circle .. autofunction:: circle_segment_h_from_A Pellet Properties ----------------- .. autofunction:: sphericity .. autofunction:: aspect_ratio .. autofunction:: circularity .. autofunction:: A_cylinder .. autofunction:: V_cylinder .. autofunction:: A_hollow_cylinder .. autofunction:: V_hollow_cylinder .. autofunction:: A_multiple_hole_cylinder .. autofunction:: V_multiple_hole_cylinder """ from cmath import sqrt as csqrt from math import acos, acosh, asin, atan, cos, degrees, isclose, log, log1p, pi, radians, sin, sqrt, tan from fluids.numerics import cacos, catan, chebval, derivative, ellipe, ellipeinc, ellipkinc, linspace, newton, quad, secant, translate_bound_func __all__ = ['TANK', 'HelicalCoil', 'PlateExchanger', 'RectangularFinExchanger', 'RectangularOffsetStripFinExchanger', 'HyperbolicCoolingTower', 'AirCooledExchanger', 'SA_partial_sphere', 'V_partial_sphere', 'V_horiz_conical', 'V_horiz_ellipsoidal', 'V_horiz_guppy', 'V_horiz_spherical', 'V_horiz_torispherical', 'V_vertical_conical', 'V_vertical_ellipsoidal', 'V_vertical_spherical', 'V_vertical_torispherical', 'V_vertical_conical_concave', 'V_vertical_ellipsoidal_concave', 'V_vertical_spherical_concave', 'V_vertical_torispherical_concave', 'a_torispherical', 'SA_ellipsoidal_head', 'SA_conical_head', 'SA_guppy_head', 'SA_torispheroidal', 'SA_partial_cylindrical_body', 'A_partial_circle', 'SA_partial_horiz_conical_head', 'SA_partial_horiz_spherical_head', 'SA_partial_horiz_guppy_head', 'SA_partial_horiz_ellipsoidal_head', 'SA_partial_horiz_torispherical_head', 'SA_partial_vertical_conical_head', 'SA_partial_vertical_spherical_head', 'SA_partial_vertical_torispherical_head', 'SA_partial_vertical_ellipsoidal_head', 'V_from_h', 'V_tank', 'SA_from_h', 'SA_tank', 'sphericity', 'aspect_ratio', 'circularity', 'A_cylinder', 'V_cylinder', 'A_hollow_cylinder', 'V_hollow_cylinder', 'A_multiple_hole_cylinder', 'V_multiple_hole_cylinder', 'pitch_angle_solver', 'plate_enlargement_factor', 'circle_segment_h_from_A'] ### Spherical Vessels, partially filled 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. ''' if h > D: h = D elif h < 0.0: h = 0.0 r = D*0.5 a = sqrt(h*(2.*r - h)) return pi*(a*a + h*h) def V_partial_sphere(D, h): r'''Calculates volume of a partial sphere according to [1]_. If h is half of D, the shape is half a sphere. No bottom is considered in this function. Valid inputs are positive values of D and h, with h always smaller or equal to D. .. math:: a = \sqrt{h(2r - h)} .. math:: V = 1/6 \pi h(3a^2 + h^2) Parameters ---------- D : float Diameter of the sphere, [m] h : float Height, as measured up to where the sphere is cut off, [m] Returns ------- V : float Volume [m^3] Examples -------- >>> V_partial_sphere(1., 0.7) 0.4105014400690663 References ---------- .. [1] Weisstein, Eric W. "Spherical Cap." Text. Accessed December 22, 2015. http://mathworld.wolfram.com/SphericalCap.html. ''' if h <= 0.0: return 0.0 if h > D: h = D r = 0.5*D a = sqrt(h*(2.*r - h)) return (1/6.)*pi*h*(3.*a*a + h*h) ### Functions as developed by Dan Jones def V_horiz_conical(D, L, a, h, headonly=False): r'''Calculates volume of a tank with conical ends, according to [1]_. .. math:: V_f = A_fL + \frac{2aR^2}{3}K, \;\;0 \le h < R\\ .. math:: V_f = A_fL + \frac{2aR^2}{3}\pi/2,\;\; h = R\\ .. math:: V_f = A_fL + \frac{2aR^2}{3}(\pi-K), \;\; R< h \le 2R .. math:: K = \cos^{-1} M + M^3\cosh^{-1} \frac{1}{M} - 2M\sqrt{1 - M^2} .. math:: M = \left|\frac{R-h}{R}\right| .. math:: A_f = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2} Parameters ---------- D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] a : float Distance the cone head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_horiz_conical(D=108., L=156., a=42., h=36)/231 2041.1923581273443 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' if h <= 0.0: return 0.0 elif h > D: h = D R = 0.5*D R_third = R*(1.0/3.0) t0 = (R-h)/R Af = R*R*acos(t0) - (R-h)*sqrt(h*(R + R - h)) M = abs(t0) if h == R: Vf = a*R*R_third*pi else: K = acos(M) + M*M*M*acosh(1./M) - 2.*M*sqrt(1.-M*M) if 0. <= h < R: Vf = 2.*a*R*R_third*K else: # elif R < h <= 2.0*R: Vf = 2.*a*R*R_third*(pi - K) if headonly: Vf = 0.5*Vf else: Vf += Af*L return Vf def V_horiz_ellipsoidal(D, L, a, h, headonly=False): r'''Calculates volume of a tank with ellipsoidal ends, according to [1]_. .. math:: V_f = A_fL + \pi a h^2\left(1 - \frac{h}{3R}\right) .. math:: A_f = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2} Parameters ---------- D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] a : float Distance the ellipsoidal head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_horiz_ellipsoidal(D=108, L=156, a=42, h=36)/231. 2380.9565415578145 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' if h <= 0.0: return 0.0 elif h > D: h = D R = 0.5*D Af = R*R*acos((R-h)/R) - (R-h)*sqrt(2*R*h - h*h) Vf = pi*a*h*h*(1 - h/(3.*R)) if headonly: Vf = 0.5*Vf else: Vf += Af*L return Vf def V_horiz_guppy(D, L, a, h, headonly=False): r'''Calculates volume of a tank with guppy heads, according to [1]_. .. math:: V_f = A_fL + \frac{2aR^2}{3}\cos^{-1}\left(1 - \frac{h}{R}\right) +\frac{2a}{9R}\sqrt{2Rh - h^2}(2h-3R)(h+R) .. math:: A_f = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2} Parameters ---------- D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] a : float Distance the guppy head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_horiz_guppy(D=108., L=156., a=42., h=36)/231. 1931.7208029476762 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' if h <= 0.0: return 0.0 elif h > D: h = D R = 0.5*D x0 = sqrt(2.*R*h - h*h) Af = R*R*acos((R-h)/R) - (R-h)*x0 Vf = (2./3.0)*a*R*R*acos(1. - h/R) + (2./9.)*a/R*x0*(2.0*h - 3.0*R)*(h + R) if headonly: Vf = Vf*0.5 else: Vf += Af*L return Vf def _V_horiz_spherical_toint(x, r2, R2, den_inv): x2 = x*x return (r2 - x2)*atan(sqrt((R2 - x2)*den_inv)) def V_horiz_spherical(D, L, a, h, headonly=False): r'''Calculates volume of a tank with spherical heads, according to [1]_. .. math:: V_f = A_fL + \frac{\pi a}{6}(3R^2 + a^2),\;\; h = R, |a|\le R .. math:: V_f = A_fL + \frac{\pi a}{3}(3R^2 + a^2),\;\; h = D, |a|\le R .. math:: V_f = A_fL + \pi a h^2\left(1 - \frac{h}{3R}\right),\;\; h = 0, \text{ or } |a| = 0, R, -R .. math:: V_f = A_fL + \frac{a}{|a|}\left\{\frac{2r^3}{3}\left[\cos^{-1} \frac{R^2 - rw}{R(w-r)} + \cos^{-1}\frac{R^2 + rw}{R(w+r)} - \frac{z}{r}\left(2 + \left(\frac{R}{r}\right)^2\right) \cos^{-1}\frac{w}{R}\right] - 2\left(wr^2 - \frac{w^3}{3}\right) \tan^{-1}\frac{y}{z} + \frac{4wyz}{3}\right\} ,\;\; h \ne R, D; a \ne 0, R, -R, |a| \ge 0.01D .. math:: V_f = A_fL + \frac{a}{|a|}\left[2\int_w^R(r^2 - x^2)\tan^{-1} \sqrt{\frac{R^2-x^2}{r^2-R^2}}dx - A_f z\right] ,\;\; h \ne R, D; a \ne 0, R, -R, |a| < 0.01D .. math:: A_f = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2} .. math:: r = \frac{a^2 + R^2}{2|a|} .. math:: w = R - h .. math:: y = \sqrt{2Rh-h^2} .. math:: z = \sqrt{r^2 - R^2} Parameters ---------- D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] a : float Distance the spherical head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_horiz_spherical(D=108., L=156., a=42., h=36)/231. 2303.96151169861 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' if h <= 0.0: return 0.0 elif h > D: h = D R = 0.5*D r = 0.5*(a*a + R*R)/abs(a) w = R - h y = sqrt(2.0*R*h - h*h) if isclose(r, R, rel_tol=2e-15): # Handle the case of small issues in calculation of `r` blowing up the calculation z = 0.0 else: z = sqrt(r*r - R*R) Af = R*R*acos((R-h)/R) - (R-h)*sqrt(2.0*R*h - h*h) if h == R and abs(a) <= R: Vf = (pi/6.0)*a*(3.0*R*R + a*a) elif h == D and abs(a) <= R: Vf = (pi/3.0)*a*(3.0*R*R + a*a) elif h == 0 or a in (0.0, R, -R) or z == 0.0: Vf = pi*a*h*h*(1.0 - h/R*(1.0/3.0)) elif abs(a) >= 0.01*D: R_r = R/r Vf = a/abs(a)*( (2.0/3.0)*r*r*r*(acos((R*R - r*w)/(R*(w-r))) + acos((R*R+r*w)/(R*(w+r))) - z/r*(2.0+R_r*R_r)*acos(w/R)) - 2.0*(w*r*r - w*w*w*(1.0/3.0))*atan(y/z) + (4.0/3.0)*w*y*z) else: r2 = r*r R2 = R*R den_inv = 1.0/(r2 - R2) integrated = quad(_V_horiz_spherical_toint, w, R, args=(r2, R2, den_inv))[0] # , epsrel=1.49e-13, Vf = a/abs(a)*(2.0*integrated - Af*z) if headonly: Vf = 0.5*Vf else: Vf += Af*L return Vf def V_horiz_torispherical_toint_1(x, w, c10, c11): # No analytical integral available in MP n = c11 + sqrt(c10 - x*x) n2 = n*n t = sqrt(n2 - w*w) return n2*asin(t/n) - w*t def V_horiz_torispherical_toint_2(x, w, c10, c11, g, g2): # No analytical integral available in MP n = c11 + sqrt(c10 - x*x) n2 = n*n n_inv = 1.0/n ans = n2*(acos(w*n_inv) - acos(g*n_inv)) - w*sqrt(n2 - w*w) + g*sqrt(n2 - g2) return ans def V_horiz_torispherical_toint_3(x, r2, g2, z_inv): # There is an analytical integral in MP, but for all cases we seem to # get ZeroDivisionError: 0.0 cannot be raised to a negative power x2 = x*x return (r2 - x2)*atan(sqrt(g2 - x2)*z_inv) def V_horiz_torispherical(D, L, f, k, h, headonly=False): r'''Calculates volume of a tank with torispherical heads, according to [1]_. .. math:: V_f = A_fL + 2V_1, \;\; 0 \le h \le h_1\\ V_f = A_fL + 2(V_{1,max} + V_2 + V_3), \;\; h_1 < h < h_2\\ V_f = A_fL + 2[2V_{1,max} - V_1(h=D-h) + V_{2,max} + V_{3,max}] , \;\; h_2 \le h \le D .. math:: V_1 = \int_0^{\sqrt{2kDh - h^2}} \left[n^2\sin^{-1}\frac{\sqrt {n^2-w^2}}{n} - w\sqrt{n^2-w^2}\right]dx .. math:: V_2 = \int_0^{kD\cos\alpha}\left[n^2\left(\cos^{-1}\frac{w}{n} - \cos^{-1}\frac{g}{n}\right) - w\sqrt{n^2 - w^2} + g\sqrt{n^2 - g^2}\right]dx .. math:: V_3 = \int_w^g(r^2 - x^2)\tan^{-1}\frac{\sqrt{g^2 - x^2}}{z}dx - \frac{z}{2}\left(g^2\cos^{-1}\frac{w}{g} - w\sqrt{2g(h-h_1) - (h-h_1)^2}\right) .. math:: V_{1,max} = v_1(h=h_1) .. math:: v_{2,max} = v_2(h=h_2) .. math:: v_{3,max} = \frac{\pi a_1}{6}(3g^2 + a_1^2) .. math:: a_1 = fD(1-\cos\alpha) .. math:: \alpha = \sin^{-1}\frac{1-2k}{2(f-k)} .. math:: n = R - kD + \sqrt{k^2D^2-x^2} .. math:: g = r\sin\alpha .. math:: r = fD .. math:: h_2 = D - h_1 .. math:: w = R - h .. math:: z = \sqrt{r^2- g^2} Parameters ---------- D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] f : float Dimensionless dish-radius parameter; also commonly given as the product of `f` and `D` (`fD`), which is called both dish radius and also crown radius and has units of length, [-] k : float Dimensionless knuckle-radius parameter; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_horiz_torispherical(D=108., L=156., f=1., k=0.06, h=36)/231. 2028.62667 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' # print((D, L, f, k, h, headonly)) if h <= 0.0: return 0.0 if h > D or isclose(h, D, rel_tol=2e-15): h = D if f is None or k is None: raise ValueError("Missing f or k") R = 0.5*D R2 = R*R hh = h*h Af = R2*acos((R-h)/R) - (R-h)*sqrt(2.0*R*h - hh) r = f*D alpha = asin((1.0 - 2.0*k)/(2.*(f - k))) cos_alpha = cos(alpha) sin_alpha = sin(alpha) a1 = r*(1.0 - cos_alpha) g = r*sin_alpha z = r*cos_alpha h1 = k*D*(1.0 - sin_alpha) h2 = D - h1 # Chebfun in Python failed on these functions c10 = k*k*D*D c11 = R - k*D g2 = g*g r2 = r*r if 0.0 <= h <= h1: w = R - h Vf = 2.0*quad(V_horiz_torispherical_toint_1, 0.0, sqrt(2.0*k*D*h - hh), (w, c10, c11))[0] elif h1 < h < h2: w = R - h wmax1 = R - h1 V1max = quad(V_horiz_torispherical_toint_1, 0.0, sqrt(2.0*k*D*h1 - h1*h1), (wmax1,c10, c11))[0] V2 = quad(V_horiz_torispherical_toint_2, 0.0, k*D*cos_alpha, (w, c10, c11, g, g2))[0] V3 = quad(V_horiz_torispherical_toint_3, w, g , (r2, g2, 1.0/z))[0] - 0.5*z*(g*g*acos(w/g) -w*sqrt(2.0*g*(h-h1) - (h-h1)*(h-h1))) Vf = 2.0*(V1max + V2 + V3) else: w = R - h wmax1 = R - h1 wmax2 = R - h2 wwerird = R - (D - h) upper_1 = sqrt(2.0*k*D*h1-h1*h1) upper_2 = sqrt(2.0*k*D*(D-h)-(D-h)*(D-h)) upper_3 = k*D*cos_alpha V1max = quad(V_horiz_torispherical_toint_1, 0.0, upper_1, (wmax1,c10, c11))[0] if upper_2 != 0.0: V1weird = quad(V_horiz_torispherical_toint_1, 0.0, upper_2, (wwerird,c10, c11))[0] else: V1weird = 0.0 V2max = quad(V_horiz_torispherical_toint_2, 0.0, upper_3, (wmax2, c10, c11, g, g2))[0] V3max = (pi/6.0)*a1*(3.0*g*g + a1*a1) Vf = 2.0*(2.0*V1max - V1weird + V2max + V3max) if headonly: Vf = 0.5*Vf else: Vf += Af*L return Vf ### Begin vertical tanks def V_vertical_conical(D, a, h): r'''Calculates volume of a vertical tank with a convex conical bottom, according to [1]_. No provision for the top of the tank is made here. .. math:: V_f = \frac{\pi}{4}\left(\frac{Dh}{a}\right)^2\left(\frac{h}{3}\right),\; h < a .. math:: V_f = \frac{\pi D^2}{4}\left(h - \frac{2a}{3}\right),\; h\ge a Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the cone head extends under the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_vertical_conical(132., 33., 24)/231. 250.67461381371024 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' if h <= 0.0: return 0.0 # vertical tanks can have `h` arbitrarily high unlike horizontal tanks if h < a: ratio = D*h/a Vf = 0.25*pi*ratio*ratio*(1.0/3.0)*h else: Vf = 0.25*pi*D*D*(h - a*(2.0/3.)) return Vf def V_vertical_ellipsoidal(D, a, h): r'''Calculates volume of a vertical tank with a convex ellipsoidal bottom, according to [1]_. No provision for the top of the tank is made here. .. math:: V_f = \frac{\pi}{4}\left(\frac{Dh}{a}\right)^2 \left(a - \frac{h}{3}\right),\; h < a .. math:: V_f = \frac{\pi D^2}{4}\left(h - \frac{a}{3}\right),\; h \ge a Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the ellipsoid head extends under the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_vertical_ellipsoidal(132., 33., 24)/231. 783.3581681678445 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' if h <= 0.0: return 0.0 if h < a: ratio = D*h/a Vf = 0.25*pi*ratio*ratio*(a - h*(1.0/3.)) else: Vf = 0.25*pi*D*D*(h - a*(1.0/3.)) return Vf def V_vertical_spherical(D, a, h): r'''Calculates volume of a vertical tank with a convex spherical bottom, according to [1]_. No provision for the top of the tank is made here. .. math:: V_f = \frac{\pi h^2}{4}\left(2a + \frac{D^2}{2a} - \frac{4h}{3}\right),\; h < a .. math:: V_f = \frac{\pi}{4}\left(\frac{2a^3}{3} - \frac{aD^2}{2} + hD^2\right),\; h\ge a Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the spherical head extends under the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_vertical_spherical(132., 33., 24)/231. 583.6018352850442 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' if h <= 0.0: return 0.0 if h < a: Vf = 0.25*pi*h*h*(2.0*a + 0.5*D*D/a - (4/3.0)*h) else: Vf = 0.25*pi*((2.0/3.0)*a*a*a - 0.5*a*D*D+ h*D*D) return Vf def V_vertical_torispherical(D, f, k, h): r'''Calculates volume of a vertical tank with a convex torispherical bottom, according to [1]_. No provision for the top of the tank is made here. .. math:: V_f = \frac{\pi h^2}{4}\left(2a_1 + \frac{D_1^2}{2a_1} - \frac{4h}{3}\right),\; 0 \le h \le a_1 .. math:: V_f = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right) +\pi u\left[\left(\frac{D}{2}-kD\right)^2 +s\right] + \frac{\pi tu^2}{2} - \frac{\pi u^3}{3} + \pi D(1-2k)\left[ \frac{2u-t}{4}\sqrt{s+tu-u^2} + \frac{t\sqrt{s}}{4} + \frac{k^2D^2}{2}\left(\cos^{-1}\frac{t-2u}{2kD}-\alpha\right)\right] ,\; a_1 < h \le a_1 + a_2 .. math:: V_f = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right) +\frac{\pi t}{2}\left[\left(\frac{D}{2}-kD\right)^2 +s\right] +\frac{\pi t^3}{12} + \pi D(1-2k)\left[\frac{t\sqrt{s}}{4} + \frac{k^2D^2}{2}\sin^{-1}(\cos\alpha)\right] + \frac{\pi D^2}{4}[h-(a_1+a_2)] ,\; a_1 + a_2 < h .. math:: \alpha = \sin^{-1}\frac{1-2k}{2(f-k)} .. math:: a_1 = fD(1-\cos\alpha) .. math:: a_2 = kD\cos\alpha .. math:: D_1 = 2fD\sin\alpha .. math:: s = (kD\sin\alpha)^2 .. math:: t = 2a_2 .. math:: u = h - fD(1-\cos\alpha) Parameters ---------- D : float Diameter of the main cylindrical section, [m] f : float Dimensionless dish-radius parameter; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] k : float Dimensionless knuckle-radius parameter; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] h : float Height, as measured up to where the fluid ends, [m] Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_vertical_torispherical(D=132., f=1.0, k=0.06, h=24)/231. 904.0688283793 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' if h <= 0.0: return 0.0 if f is None or k is None: raise ValueError("f and k are required") alpha = asin((1.0 - 2.0*k)/(2.0*(f-k))) sin_alpha = sin(alpha) cos_alpha = cos(alpha) a1 = f*D*(1.0 - cos_alpha) a2 = k*D*cos_alpha D1 = 2.0*f*D*sin_alpha x1 = k*D*sin_alpha s = x1*x1 t = a2 + a2 u = h - f*D*(1.0 - cos_alpha) h2 = h*h if 0.0 <= h <= a1: Vf = 0.25*pi*h2*(a1 + a1 + 0.5*D1*D1/a1 - (4.0/3.0)*h) elif a1 < h <= a1 + a2: x2 = (0.5*D - k*D) u2 = u*u Vf = (0.25*pi*a1*((2.0/3.0)*a1*a1 + 0.5*D1*D1) + pi*u*(x2*x2 + s) + pi*u2*(0.5*t - u*(1.0/3.)) + pi*D*(1.0 - 2.0*k)*(0.25*(2.0*u - t)*sqrt(s + t*u - u2) + 0.25*t*sqrt(s) + 0.5*k*k*D*D*(acos((t - 2.0*u)/(2.0*k*D)) - alpha))) else: ratio = (0.5*D - k*D) Vf = 0.25*pi*((2.0/3.0)*a1*a1*a1 + 0.5*a1*D1*D1) + 0.5*pi*t*(ratio*ratio + s) + pi*t*t*t*(1.0/12.) + pi*D*(1.0 - 2.0*k)*(0.25*t*sqrt(s) + k*k*D*D*(1.0/2.0)*asin(cos(alpha))) + 0.25*pi*D*D*(h - (a1 + a2)) return Vf ### Begin vertical tanks with concave heads 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)/231 251.158255657951 References ---------- .. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/ ''' if h <= 0.0: return 0.0 if h < abs(a): a_plus_h = a + h Vf = pi*D*D*(1.0/12.)*(3.0*h + a - a_plus_h*a_plus_h*a_plus_h/(a*a)) else: Vf = pi*D*D*(1.0/12.)*(3.0*h + a) return Vf def V_vertical_ellipsoidal_concave(D, a, h): r'''Calculates volume of a vertical tank with a concave ellipsoidal bottom, according to [1]_. No provision for the top of the tank is made here. .. math:: V = \frac{\pi D^2}{12} \left(3h + 2a - \frac{(a+h)^2(2a-h)}{a^2}\right) ,\;\; 0 \le h < |a| .. math:: V = \frac{\pi D^2}{12} (3h + 2a ),\;\; h \ge |a| Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Negative distance the eppilsoid head extends inside the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_vertical_ellipsoidal_concave(D=113., a=-33, h=15)/231 44.84968851034856 References ---------- .. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/ ''' if h <= 0.0: return 0.0 if h < abs(a): a_plus_h = a + h Vf = pi*D*D*(1.0/12.)*(3.0*h + 2.0*a - a_plus_h*a_plus_h*(2.0*a-h)/(a*a)) else: Vf = pi*D*D*(1.0/12.)*(3.0*h + 2.0*a) return Vf def V_vertical_spherical_concave(D, a, h): r'''Calculates volume of a vertical tank with a concave spherical bottom, according to [1]_. No provision for the top of the tank is made here. .. math:: V = \frac{\pi}{12}\left[3D^2h + \frac{a}{2}(3D^2 + 4a^2) + (a+h)^3 \left(4 - \frac{3D^2 + 12a^2}{2a(a+h)}\right)\right],\;\; 0 \le h < |a| .. math:: V = \frac{\pi}{12}\left[3D^2h + \frac{a}{2}(3D^2 + 4a^2) \right] ,\;\; h \ge |a| Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Negative distance the spherical head extends inside the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_vertical_spherical_concave(D=113., a=-33, h=15)/231 112.81405437348528 References ---------- .. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/ ''' if h <= 0.0: return 0.0 D2 = D*D a2 = a*a if h < abs(a): a_plus_h = a + h Vf = pi*(1.0/12)*(3.0*D2*h + a*(1.0/2.)*(3.0*D2 + 4.0*a2) + a_plus_h*a_plus_h*a_plus_h*(4.0 - (3.0*D2+12.0*a2)/(2.*a*a_plus_h))) else: Vf = pi*(1.0/12)*(3.0*D2*h + a*(1.0/2.)*(3.0*D2 + 4.0*a2)) return Vf def V_vertical_torispherical_concave(D, f, k, h): r'''Calculates volume of a vertical tank with a concave torispherical bottom, according to [1]_. No provision for the top of the tank is made here. .. math:: V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + v_1(h=a_1 + a_2 -h),\; 0 \le h < a_2 .. math:: V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + v_2(h=a_1 + a_2 -h),\; a_2 \le h < a_1 + a_2 .. math:: V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + 0,\; h \ge a_1 + a_2 .. math:: v_1 = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right) +\pi u\left[\left(\frac{D}{2}-kD\right)^2 +s\right] + \frac{\pi tu^2}{2} - \frac{\pi u^3}{3} + \pi D(1-2k)\left[ \frac{2u-t}{4}\sqrt{s+tu-u^2} + \frac{t\sqrt{s}}{4} + \frac{k^2D^2}{2}\left(\cos^{-1}\frac{t-2u}{2kD}-\alpha\right)\right] .. math:: v_2 = \frac{\pi h^2}{4}\left(2a_1 + \frac{D_1^2}{2a_1} - \frac{4h}{3}\right) .. math:: \alpha = \sin^{-1}\frac{1-2k}{2(f-k)} .. math:: a_1 = fD(1-\cos\alpha) .. math:: a_2 = kD\cos\alpha .. math:: D_1 = 2fD\sin\alpha .. math:: s = (kD\sin\alpha)^2 .. math:: t = 2a_2 .. math:: u = h - fD(1-\cos\alpha) Parameters ---------- D : float Diameter of the main cylindrical section, [m] f : float Dimensionless dish-radius parameter; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] k : float Dimensionless knuckle-radius parameter; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] h : float Height, as measured up to where the fluid ends, [m] Returns ------- V : float Volume [m^3] Examples -------- Matching example from [1]_, with inputs in inches and volume in gallons. >>> V_vertical_torispherical_concave(D=113., f=0.71, k=0.081, h=15)/231 103.88569287163769 References ---------- .. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/ ''' if h <= 0.0: return 0.0 alpha = asin((1.0-2.0*k)/(2.*(f-k))) cos_alpha = cos(alpha) sin_alpha = sin(alpha) a1 = f*D*(1-cos_alpha) a2 = k*D*cos_alpha D1 = 2.0*f*D*sin_alpha s = (k*D*sin_alpha) s *= s t = 2.0*a2 def V1(h): u = h-f*D*(1.0-cos_alpha) ratio = (0.5*D-k*D) v1 = 0.25*pi*((2.0/3.0)*a1*a1*a1 + 0.5*a1*D1*D1) + pi*u*(ratio*ratio +s) v1 += u*u*(0.5*pi*t - pi*(1.0/3.)*u) v1 += pi*D*(1.0-2.0*k)*((2.0*u-t)*0.25*sqrt(s+t*u-u*u) + 0.25*t*sqrt(s) + k*k*D*D*0.5*(acos((t-2.0*u)/(2.0*k*D)) -alpha)) return v1 def V2(h): v2 = 0.25*pi*h*h*(2.0*a1 + D1*D1/(2.*a1) - 4/3.0*h) return v2 if 0 <= h < a2: Vf = 0.25*pi*D*D*h - V1(a1+a2) + V1(a1+a2-h) elif a2 <= h < a1 + a2: Vf = 0.25*pi*D*D*h - V1(a1+a2) + V2(a1+a2-h) else: Vf = 0.25*pi*D*D*h - V1(a1+a2) return Vf ### Total surface area of heads, orientation-independent def SA_ellipsoidal_head(D, a): r'''Calculates the surface area of an ellipsoidal head according to [1]_ and [2]_. The formula below is for the full shape, the result of which is halved. The formula is for :math:`a < R`. In the equations, `a` is the same and `c` is `D`. .. math:: \text{SA} = 2\pi a^2 + \frac{\pi c^2}{e_1}\ln\left(\frac{1+e_1}{1-e_1} \right) .. math:: e_1 = \sqrt{1 - \frac{c^2}{a^2}} For the case of :math:`a \ge R` from [2]_, which is needed to make the tank head volume grow linearly with length: .. math:: \text{SA} = 2\pi R^2 + \frac{2\pi a^2 R}{\sqrt{a^2 - R^2}}\cos^{-1}\frac{R}{|a|} Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the ellipsoidal head extends, [m] Returns ------- SA : float Surface area [m^2] Examples -------- Spherical case >>> SA_ellipsoidal_head(2, 1) 6.283185307179586 >>> SA_ellipsoidal_head(2, 1.5) 8.459109081729984 References ---------- .. [1] Weisstein, Eric W. "Spheroid." Text. Accessed March 14, 2016. http://mathworld.wolfram.com/Spheroid.html. .. [2] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' if D == a*2.0: return 0.5*pi*D*D # necessary to avoid a division by zero when D == a R = 0.5*D if a < R: R, a = min((R, a)), max((R, a)) e1 = sqrt(1.0 - R*R/(a*a)) if e1 != 1.0: log_term = log1p(e1) - log1p(-e1) else: # Limit as a goes to zero relative to D; may only be ~6 orders of # magnitude smaller than D and will still occur log_term = 0.0 return (2.0*pi*a*a + pi*R*R/e1*log_term)*0.5 else: return pi*R*R + pi*a*a*R*1.0/sqrt(a*a - R*R)*acos(R/abs(a)) def SA_conical_head(D, a): r'''Calculates the surface area of a conical head according to [1]_. .. math:: SA = \frac{\pi D}{2} \sqrt{a^2 + \left(\frac{D}{2}\right)^2} Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the conical head extends, [m] Returns ------- SA : float Surface area [m^2] Examples -------- >>> SA_conical_head(2, 1) 4.442882938158366 References ---------- .. [1] Weisstein, Eric W. "Cone." Text. Accessed March 14, 2016. http://mathworld.wolfram.com/Cone.html. ''' return 0.5*pi*D*sqrt(a*a + 0.25*D*D) def SA_guppy_head(D, a): r'''Calculates the surface area of a guppy head according to [1]_. Some work was involved in combining formulas for the ellipse of the head, and the conic section on the sides. .. math:: SA = \frac{\pi D}{4}\sqrt{D^2 + a^2} + \frac{\pi D}{2}a Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the conical head extends, [m] Returns ------- SA : float Surface area [m^2] Examples -------- >>> SA_guppy_head(2, 1) 6.654000019110157 References ---------- .. [1] Weisstein, Eric W. "Cone." Text. Accessed March 14, 2016. http://mathworld.wolfram.com/Cone.html. ''' return 0.25*pi*D*sqrt(a*a + D*D) + 0.5*pi*D*a def SA_torispheroidal(D, f, k): r'''Calculates surface area of a torispherical head according to [1]_. Somewhat involved. Equations are adapted to be used for a full head. .. math:: SA = S_1 + S_2 .. math:: S_1 = 2\pi D^2 f_d \alpha .. math:: S_2 = 2\pi D^2 f_k\left(\alpha - \alpha_1 + (0.5 - f_k)\left(\sin^{-1} \left(\frac{\alpha-\alpha_2}{f_k}\right) - \sin^{-1}\left(\frac{ \alpha_1-\alpha_2}{f_k}\right)\right)\right) .. math:: \alpha_1 = f_d\left(1 - \sqrt{1 - \left(\frac{0.5 - f_k}{f_d-f_k} \right)^2}\right) .. math:: \alpha_2 = f_d - \sqrt{f_d^2 - 2f_d f_k + f_k - 0.25} .. math:: \alpha = \frac{a}{D_i} Parameters ---------- D : float Diameter of the main cylindrical section, [m] f : float Dimensionless dish-radius parameter; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] k : float Dimensionless knuckle-radius parameter; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] Returns ------- SA : float Surface area [m^2] Examples -------- Example from [1]_. >>> SA_torispheroidal(D=2.54, f=1.039370079, k=0.062362205) 6.00394283477063 References ---------- .. [1] Honeywell. "Calculate Surface Areas and Cross-sectional Areas in Vessels with Dished Heads". https://www.honeywellprocess.com/library/marketing/whitepapers/WP-VesselsWithDishedHeads-UniSimDesign.pdf Whitepaper. 2014. ''' D2 = D*D x1 = 2.0*pi*D2 k_inv = 1.0/k x2 = ((0.5 - k)/(f-k)) alpha_1 = f*(1.0 - sqrt(1.0 - x2*x2)) alpha_2 = f - sqrt(f*f - 2.0*f*k + k - 0.25) alpha = alpha_1 # Up to top of dome S1 = x1*f*alpha_1 alpha = alpha_2 # up to top of torus S2_sub = asin((alpha-alpha_2)*k_inv) - asin((alpha_1-alpha_2)*k_inv) S2 = x1*k*(alpha - alpha_1 + (0.5 - k) *S2_sub) return S1 + S2 def SA_tank(D, L, sideA=None, sideB=None, sideA_a=0, sideB_a=0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None): r'''Calculates the surface are of a cylindrical tank with optional heads. In the degenerate case of being provided with only `D` and `L`, provides the surface area of a cylinder. Parameters ---------- D : float Diameter of the cylindrical section of the tank, [m] L : float Length of the main cylindrical section of the tank, [m] sideA : string, optional The left (or bottom for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical']. sideB : string, optional The right (or top for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical']. sideA_a : float, optional The distance the head as specified by sideA extends down or to the left from the main cylindrical section, [m] sideB_a : float, optional The distance the head as specified by sideB extends up or to the right from the main cylindrical section, [m] sideA_f : float, optional Dish-radius parameter for side A; fD = dish radius [1/m] sideA_k : float, optional knuckle-radius parameter for side A; kD = knuckle radius [1/m] sideB_f : float, optional Dish-radius parameter for side B; fD = dish radius [1/m] sideB_k : float, optional knuckle-radius parameter for side B; kD = knuckle radius [1/m] Returns ------- SA : float Surface area of the tank [m^2] sideA_SA : float Surface area of only `sideA` [m^2] sideB_SA : float Surface area of only `sideB` [m^2] lateral_SA : float Surface area of cylindrical section of tank [m^2] Examples -------- Cylinder, Spheroid, Long Cones, and spheres. All checked. >>> SA_tank(D=2, L=2)[0] 18.84955592153876 >>> SA_tank(D=1., L=0, sideA='ellipsoidal', sideA_a=2, sideB='ellipsoidal', ... sideB_a=2)[0] 10.124375616183 >>> SA_tank(D=1., L=5, sideA='conical', sideA_a=2, sideB='conical', ... sideB_a=2)[0] 22.18452243965 >>> SA_tank(D=1., L=5, sideA='spherical', sideA_a=0.5, sideB='spherical', ... sideB_a=0.5)[0] 18.8495559215 ''' # Side A if sideA == 'conical': sideA_SA = SA_conical_head(D=D, a=sideA_a) elif sideA == 'ellipsoidal': sideA_SA = SA_ellipsoidal_head(D=D, a=sideA_a) elif sideA == 'guppy': sideA_SA = SA_guppy_head(D=D, a=sideA_a) elif sideA == 'spherical': sideA_SA = pi * (sideA_a * sideA_a + 0.25 * D * D) elif sideA == 'torispherical': if sideA_f is None or sideA_k is None: raise ValueError("Missing torispherical 'f' or 'k' parameter for sideA") sideA_SA = SA_torispheroidal(D=D, f=sideA_f, k=sideA_k) else: sideA_SA = 0.25*pi*D*D # Circle # Side B # Calculate side B (reuse side A calculation if parameters are identical) if (sideA == sideB and sideA_a == sideB_a and sideA_f == sideB_f and sideA_k == sideB_k): sideB_SA = sideA_SA else: if sideB == 'conical': sideB_SA = SA_conical_head(D=D, a=sideB_a) elif sideB == 'ellipsoidal': sideB_SA = SA_ellipsoidal_head(D=D, a=sideB_a) elif sideB == 'guppy': sideB_SA = SA_guppy_head(D=D, a=sideB_a) elif sideB == 'spherical': sideB_SA = pi*(sideB_a*sideB_a + 0.25*D*D)#SA_partial_sphere(D=D, h=sideB_a) elif sideB == 'torispherical': if sideB_f is None or sideB_k is None: raise ValueError("Missing torispherical 'f' or 'k' parameter for sideB") sideB_SA = SA_torispheroidal(D=D, f=sideB_f, k=sideB_k) else: sideB_SA = 0.25*pi*D*D # Circle lateral_SA = pi*D*L SA = sideA_SA + sideB_SA + lateral_SA return SA, sideA_SA, sideB_SA, lateral_SA def V_tank(D, L, horizontal=True, sideA=None, sideB=None, sideA_a=0.0, sideB_a=0.0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None): r'''Calculates the total volume of a vertical or horizontal tank with different head types. Parameters ---------- D : float Diameter of the cylindrical section of the tank, [m] L : float Length of the main cylindrical section of the tank, [m] horizontal : bool, optional Whether or not the tank is a horizontal or vertical tank sideA : string, optional The left (or bottom for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical']. sideB : string, optional The right (or top for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical']. sideA_a : float, optional The distance the head as specified by sideA extends down or to the left from the main cylindrical section, [m] sideB_a : float, optional The distance the head as specified by sideB extends up or to the right from the main cylindrical section, [m] sideA_f : float, optional Dimensionless dish-radius parameter for side A; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideA_k : float, optional Dimensionless knuckle-radius parameter for side A; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] sideB_f : float, optional Dimensionless dish-radius parameter for side B; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideB_k : float, optional Dimensionless knuckle-radius parameter for side B; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] Returns ------- V : float Total volume [m^3] sideA_V : float Volume of only `sideA` [m^3] sideB_V : float Volume of only `sideB` [m^3] lateral_V : float Volume of cylindrical section of tank [m^3] Examples -------- >>> V_tank(D=1.5, L=5., horizontal=False, sideA='conical', ... sideB='conical', sideA_a=2., sideB_a=1.) (10.602875205865551, 1.1780972450961726, 0.5890486225480863, 8.835729338221293) ''' if sideA is not None and sideA not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'): raise ValueError('Unspoorted head type for side A') if sideB is not None and sideB not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'): raise ValueError('Unspoorted head type for side B') R = 0.5*D sideA_V = sideB_V = lateral_V = 0.0 if horizontal: # Conical case if sideA == 'conical' and sideB == 'conical' and sideA_a == sideB_a: sideB_V = sideA_V = V_horiz_conical(D, L, sideA_a, D, headonly=True) else: if sideA == 'conical': sideA_V = V_horiz_conical(D, L, sideA_a, D, headonly=True) if sideB == 'conical': sideB_V = V_horiz_conical(D, L, sideB_a, D, headonly=True) # Elliosoidal case if sideA == 'ellipsoidal' and sideB == 'ellipsoidal' and sideA_a == sideB_a: sideB_V = sideA_V = V_horiz_ellipsoidal(D, L, sideA_a, D, headonly=True) else: if sideA == 'ellipsoidal': sideA_V = V_horiz_ellipsoidal(D, L, sideA_a, D, headonly=True) if sideB == 'ellipsoidal': sideB_V = V_horiz_ellipsoidal(D, L, sideB_a, D, headonly=True) # Guppy case if sideA == 'guppy' and sideB == 'guppy' and sideA_a == sideB_a: sideB_V = sideA_V = V_horiz_guppy(D, L, sideA_a, D, headonly=True) else: if sideA == 'guppy': sideA_V = V_horiz_guppy(D, L, sideA_a, D, headonly=True) if sideB == 'guppy': sideB_V = V_horiz_guppy(D, L, sideB_a, D, headonly=True) # Spherical case if sideA == 'spherical' and sideB == 'spherical' and sideA_a == sideB_a: sideB_V = sideA_V = V_horiz_spherical(D, L, sideA_a, D, headonly=True) else: if sideA == 'spherical': sideA_V = V_horiz_spherical(D, L, sideA_a, D, headonly=True) if sideB == 'spherical': sideB_V = V_horiz_spherical(D, L, sideB_a, D, headonly=True) # Torispherical case if (sideA == 'torispherical' and sideB == 'torispherical' and (sideA_f == sideB_f) and (sideA_k == sideB_k)): sideB_V = sideA_V = V_horiz_torispherical(D, L, sideA_f, sideA_k, D, headonly=True) else: if sideA == 'torispherical': sideA_V = V_horiz_torispherical(D, L, sideA_f, sideA_k, D, headonly=True) if sideB == 'torispherical': sideB_V = V_horiz_torispherical(D, L, sideB_f, sideB_k, D, headonly=True) Af = R*R*acos((R-D)/R) - (R-D)*sqrt(2.0*R*D - D*D) lateral_V = L*Af else: # Bottom head if sideA == 'conical' and sideB == 'conical' and sideA_a == sideB_a: sideB_V = sideA_V = V_vertical_conical(D, sideA_a, h=sideA_a) else: if sideA == 'conical': sideA_V = V_vertical_conical(D, sideA_a, h=sideA_a) if sideB == 'conical': sideB_V = V_vertical_conical(D, sideB_a, h=sideB_a) if sideA == 'ellipsoidal' and sideB == 'ellipsoidal' and sideA_a == sideB_a: sideB_V = sideA_V = V_vertical_ellipsoidal(D, sideA_a, h=sideA_a) else: if sideA == 'ellipsoidal': sideA_V = V_vertical_ellipsoidal(D, sideA_a, h=sideA_a) if sideB == 'ellipsoidal': sideB_V = V_vertical_ellipsoidal(D, sideB_a, h=sideB_a) if sideA == 'spherical' and sideB == 'spherical' and sideA_a == sideB_a: sideB_V = sideA_V = V_vertical_spherical(D, sideA_a, h=sideA_a) else: if sideA == 'spherical': sideA_V = V_vertical_spherical(D, sideA_a, h=sideA_a) if sideB == 'spherical': sideB_V = V_vertical_spherical(D, sideB_a, h=sideB_a) if sideA == 'torispherical' and sideB == 'torispherical' and sideA_f == sideB_f and sideA_k == sideB_k: sideB_V = sideA_V = V_vertical_torispherical(D, sideA_f, sideA_k, h=sideA_a) else: if sideA == 'torispherical': sideA_V = V_vertical_torispherical(D, sideA_f, sideA_k, h=sideA_a) if sideB == 'torispherical': sideB_V = V_vertical_torispherical(D, sideB_f, sideB_k, h=sideB_a) # Cylindrical section lateral_V = 0.25 * pi * D * D * L return lateral_V + sideA_V + sideB_V, sideA_V, sideB_V, lateral_V def SA_partial_cylindrical_body(L, D, h): r'''Calculates the partial area of a cylinder's body in the context of a horizontal cylindrical vessel and liquid partially filling it. This computes the wetted surface area of the bottom of the cylinder. .. math:: \text{SA} = L D \cos^{-1}\left(\frac{D - 2h}{D}\right) Parameters ---------- L : float Length of the cylinder, [m] D : float Diameter of the cylinder, [m] h : float Height measured from bottom of cylinder to liquid level, [m] Returns ------- SA_partial : float Partial (wetted) surface area, [m^2] Notes ----- This method is undefined for :math:`h > D`. and :math:`h < 0`, but those cases are handled by returning the full surface area and the zero respectively. Examples -------- >>> SA_partial_cylindrical_body(L=200.0, D=96., h=22.0) 19168.852890279868 References ---------- .. [1] Weisstein, Eric W. "Circular Segment." Text. Wolfram Research, Inc. Accessed May 10, 2020. https://mathworld.wolfram.com/CircularSegment.html. ''' if h < 0.0: return 0.0 elif h > D: h = D C = D*acos((D - h - h)/D) return C*L def A_partial_circle(D, h): r'''Calculates the partial area of a circle, in the context of the circle being an end cap to a horizontal cylindrical vessel and liquid partially filling it. This computes the wetted surface area of one of the end caps. Multiply this by two to obtain the wetted area of two end caps. .. math:: \text{SA} = R^2\cos^{-1}\frac{(R - h)}{R} - (R - h)\sqrt{(2Rh - h^2)} Parameters ---------- D : float Diameter of the circle, [m] h : float Height measured from bottom of circle to liquid level, [m] Returns ------- SA_partial : float Partial (wetted) surface area, [m^2] Notes ----- This method is undefined for :math:`h > D` and :math:`h < 0`, but those cases are handled by returning the full surface area and the zero respectively. Examples -------- >>> A_partial_circle(D=96., h=22.0) 1251.2018147383194 References ---------- .. [1] Weisstein, Eric W. "Circular Segment." Text. Wolfram Research, Inc. Accessed May 10, 2020. https://mathworld.wolfram.com/CircularSegment.html. ''' if h > D: h = D # Catch the case of a computed `h` being trivially larger than `D` due to floating point elif h < 0.0: return 0.0 R = 0.5*D SA = R*R*acos((R - h)/R) - (R - h)*sqrt(2.0*R*h - h*h) if SA < 0.0: SA = 0.0 # Catch trig errors return SA def circle_segment_area_inner(h, R, A_expect): # 2 sqrt, 1 acos, 4 division x0 = R*R x1 = -h x2 = R + x1 x3 = sqrt(h*(2.0*R + x1)) x4 = x2*x2 A_err = x0*acos(x2/R) - x2*x3 - A_expect der = R/sqrt(1.0 - x4/x0) + x3 - x4/x3 return A_err, der def circle_segment_h_from_A(A, D): r'''Calculates the height of a chord of a circle given the area of that circle segment. This is a numerical problem, solving the following equation for `h`. .. math:: \text{A} = R^2\cos^{-1}\frac{(R - h)}{R} - (R - h)\sqrt{(2Rh - h^2)} Parameters ---------- A : float Circle section area, [m^2] D : float Diameter of the circle, [m] Returns ------- h : float Height measured from bottom of circle to the end of the circle section, [m] Notes ----- Examples -------- >>> circle_segment_h_from_A(A=1251.2018147383194, D=96.) 22.0 References ---------- .. [1] Weisstein, Eric W. "Circular Segment." Text. Wolfram Research, Inc. Accessed May 10, 2020. https://mathworld.wolfram.com/CircularSegment.html. ''' if A == 0.0: return 0.0 R = 0.5*D return newton(circle_segment_area_inner, x0=0.25*R, fprime=True, high=R, low=0.0, args=(R, A), xtol=1e-12, bisection=True) def SA_partial_horiz_conical_head(D, a, h): r'''Calculates the partial area of a conical tank head in the context of a horizontal vessel and liquid partially filling it. This computes the wetted surface area of one of the conical heads only. .. math:: \text{SA} = \frac{\sqrt{(a^2 + R^2)}}{R}\left[R^2\cos^{-1}\left(\frac{ (R-h)}{R}\right) - (R-h)\sqrt{(2Rh - h^2)}\right] Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the cone head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one conical tank head, [m^2] Notes ----- This method is undefined for :math:`h > D` and :math:`h < 0`, but those cases are handled by returning the full surface area and the zero respectively. Examples -------- >>> SA_partial_horiz_conical_head(D=72., a=48.0, h=24.0) 1980.0498315169873 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' if h > D: h = D elif h < 0: return 0.0 R = 0.5*D R_inv = 1.0/R return sqrt(a*a + R*R)*R_inv*(R*R*acos((R-h)*R_inv) - (R-h)*sqrt(2.0*R*h - h*h)) def _SA_partial_horiz_spherical_head_to_int(x, R2, a4, c1, c2): x2 = x*x to_pow = (R2 - x2)/(c2 - a4*x2) if to_pow < 0.0: to_pow = 0.0 num = c1*sqrt(to_pow) if num > 1.0: # Sometimes, the numerical error will result in trying to asin a number just slightly higher than 1 unless we catch it return 0.5*pi return asin(num) def SA_partial_horiz_spherical_head(D, a, h): r'''Calculates the partial area of a spherical tank head in the context of a horizontal vessel and liquid partially filling it. This computes the wetted surface area of one of the spherical heads only. .. math:: \text{SA} = \frac{a^2 + R^2}{|a|}\int_{R-h}^R \sin^{-1} \frac{2|a|\sqrt{R^2-x^2}} {\sqrt{(a^2+R^2)^2 - (2ax)^2}} dx For the special case of :math:`|a| = R` : .. math:: \text{SA} = \pi R h Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the spherical head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one spherical tank head, [m^2] Notes ----- This method is undefined for :math:`h > D` and :math:`h < 0`, but those cases are handled by returning the full surface area and the zero respectively. A symbolic attempt did not suggest any analytical integrals were available. Examples -------- >>> SA_partial_horiz_spherical_head(D=72., a=48.0, h=24.0) 2027.2672 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' R = 0.5*D if a == R: return pi*R*h elif h < 0.0: return 0.0 elif h > D: h = D fact = (a*a + R*R)/abs(a) R2 = R*R a2 = a + a a4 = a2*a2 c1 = 2.0*abs(a) c2 = (a*a + R2)*(a*a + R2) SA = fact*quad(_SA_partial_horiz_spherical_head_to_int, R-h, R, args=(R2, a4, c1, c2))[0] return SA def _SA_partial_horiz_ellipsoidal_head_to_int_dbl(x, y, c1, R2, R4, h): y2 = y*y x2 = x*x num = c1*(x2 + y2) - R4 den = x2 + (y2 - R2) # Brackets help numerical truncation; zero div without it to_sqrt = num/den if to_sqrt < 0.0: # Equation is undefined for y == R when x is zero; avoid it return _SA_partial_horiz_ellipsoidal_head_to_int_dbl(x, y*(1.0 - 1e-12), c1, R2, R4, h) return sqrt(to_sqrt) def _SA_partial_horiz_ellipsoidal_head_limits(x, c1, R2, R4, h): return [0.0, sqrt(R2 - x*x)] def _SA_partial_horiz_ellipsoidal_head_limits2(c1, R2, R4, h): R = sqrt(R2) return [R-h, R] def _SA_partial_horiz_ellipsoidal_head_to_int(y, c1, R2, R4): y2 = y*y t0 = c1*y2 x6 = c1*(y2 - R2)/(t0 - R4) ans = sqrt(R4 - t0)*float(ellipe(x6)) return ans def SA_partial_horiz_ellipsoidal_head(D, a, h): r'''Calculates the partial area of a ellipsoidal tank head in the context of a horizontal vessel and liquid partially filling it. This computes the wetted surface area of one of the ellipsoidal heads only. .. math:: \text{SA} = \frac{2}{R} \int_{R-h}^R \int_0^{\sqrt{R^2 - x^2}} \sqrt{ \frac{(R^2 - a^2)x^2 + (R^2 - a^2)y^2 - R^4} {x^2 + y^2 - R^2}} dy dx After extensive manipulation, the first integral was solved analytically, extending the result of [1]_ with greater performance. .. math:: \text{SA} = \frac{2}{R} \int_{R-h}^R \frac{\left(\frac{R^{4} - R^{2} \left(R^{2} - a^{2}\right)}{R^{2} - y^{2}}\right)^{0.5} \left(R^{2} - y^{2}\right)^{0.5} E{\left(\frac{\left(- R^{2} + y^{2}\right) \left(R^{2} - a^{2}\right)}{- R^{4} + y^{2} \left(R^{2} - a^{2}\right)} \right)}} {\left(\frac{R^{4} - R^{2} \left(R^{2} - a^{2}\right)}{R^{4} - y^{2} \left(R^{2} - a^{2}\right)}\right)^{0.5}} Where :math:`E(x)` is the complete elliptic integral of the second kind, calculated with SciPy's link to the cephes library. Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the ellipsoidal head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one ellipsoidal tank head, [m^2] Notes ----- This method is undefined for :math:`h > D` and :math:`h < 0`, but those cases are handled by returning the full surface area and the zero respectively. The original numerical double integral is extremely nasty - there are places where f(x) -> infinity but that have a bounded area. quadpack's numerical integration handles this well, but adaptive inetgration which is not aware of singularities does not. Examples -------- >>> SA_partial_horiz_ellipsoidal_head(D=72., a=48.0, h=24.0) 3401.233622547 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' R = 0.5*D if h < 0.0: return 0.0 elif h > D: h = D R2 = R*R R4 = R2*R2 a2 = a*a c1 = R2 - a2 # from fluids.numerics import dblquad # from scipy.integrate import dblquad, nquad ## quad_val = nquad(_SA_partial_horiz_ellipsoidal_head_to_int, ranges=[_SA_partial_horiz_ellipsoidal_head_limits, _SA_partial_horiz_ellipsoidal_head_limits2], ## args=(c1, R2, R4, h))[0] # quad_val = dblquad(_SA_partial_horiz_ellipsoidal_head_to_int, R-h, R, lambda x: 0.0, lambda x: (R2 - x*x)**0.5, # args=(c1, R2, R4, h))[0] quad_val = quad(_SA_partial_horiz_ellipsoidal_head_to_int, R-h, R, args=(c1, R2, R4))[0] SA = 2.0/R*quad_val return SA def _SA_partial_horiz_guppy_head_to_int(x, a, R): x0 = a*a x1 = R - x x2 = x1*x1 x3 = 1.0/x2 x4 = x0*x3 + 1.0 x5 = R*R x6 = x*x x7 = x5 - x6 x8 = sqrt(x7) x9 = x4*x8 x10 = x0 + 4.0*x5 x17 = sqrt(sqrt(x10)) x11 = x17*x17 x12 = 1.0/x11 x13 = a*x7 x14 = x12*x13*x3 + 1.0 x15 = a*x12 x16 = sqrt(a) x18 = 2*atan(x16*x8/(x1*x17)) x19 = 0.5 - 0.5*x15 x100 = (-2.0*R*x*x11 + x11*x5 + x11*x6 + x13) x20 = x1*x14*sqrt(x2*x5*(x0 + x2)/(x100*x100))/x5 return 0.08333333333333333*( (-4.0*x10**0.75*x16*x20*ellipeinc(x18, x19) + 4.0*x9 + 2.0*x17*x20*(a*x11 + x10)*ellipkinc(x18, x19)/x16 + 8.0*x15*x9/x14)*1.0/sqrt(x4)) def SA_partial_horiz_guppy_head(D, a, h): r'''Calculates the partial area of a guppy tank head in the context of a horizontal vessel and liquid partially filling it. This computes the wetted surface area of one of the guppy heads only. .. math:: \text{SA} = 2\int_{-R}^{h-R}\int_{0}^{\sqrt{R^2 - x^2}} \sqrt{1 + \left(\frac{a}{2R}\left(1 - \frac{y^2}{(R-x)^2} \right) \right)^2 + \left(\frac{ay}{R(R-x)} \right)^2 } dy dx After extensive manipulation, the first integral was solved analytically, extending the result of [1]_. Even with the special functions, this form has somewhat greater performance (and improved precision). .. math:: \text{SA} = 2 \int_{-R}^{h-R} \frac{\frac{2 a \left(4 + \frac{a^{2} \left(2 R^{2} - 2 R y\right)^{2}}{R^{2} \left(R - y\right)^{4}}\right) \sqrt{R^{2} - y^{2}}}{\sqrt{4 R^{2} + a^{2}} \left(\frac{a \left(R^{2} - y^{2}\right)}{\left(R - y\right)^{2} \sqrt{4 R^{2} + a^{2}}} + 1\right)} + \left(4 + \frac{a^{2} \left(2 R^{2} - 2 R y\right) ^{2}}{R^{2} \left(R - y\right)^{4}}\right) \sqrt{R^{2} - y^{2}} - \frac{2 \sqrt{a} \sqrt{\frac{4 R^{2} \left(R - y\right)^{4} + a^{2} \left(2 R^{2} - 2 R y\right)^{2}}{\left(R^{2} \sqrt{4 R^{2} + a^{2}} - 2 R y \sqrt{4 R^{2} + a^{2}} + a \left(R^{2} - y^{2}\right) + y^{2} \sqrt{4 R^{2} + a^{2}}\right)^{2}}} \left(R - y\right) \left(4 R^{2} + a^{2}\right)^{0.75} \left(\frac{a \left(R^{2} - y^{2}\right)}{\left(R - y\right)^{2} \sqrt{4 R^{2} + a^{2}}} + 1\right) \operatorname{ellipeinc}{\left(2 \operatorname{atan}{\left(\frac{\sqrt{a} \sqrt{R^{2} - y^{2}}} {\left(R - y\right) \left(4 R^{2} + a^{2}\right)^{0.25}} \right)}, - \frac{a}{2 \sqrt{4 R^{2} + a^{2}}} + 0.5 \right)}}{R^{2}} + \frac{1.0 \sqrt{\frac{4 R^{2} \left(R - y\right)^{4} + a^{2} \left(2 R^{2} - 2 R y\right)^{2}}{\left(R^{2} \sqrt{4 R^{2} + a^{2}} - 2 R y \sqrt{4 R^{2} + a^{2}} + a \left(R^{2} - y^{2}\right) + y^{2} \sqrt{4 R^{2} + a^{2}}\right)^{2}}} \left(R - y\right) \left(4 R^{2} + a^{2}\right)^{0.25} \left(\frac{a \left(R^{2} - y^{2}\right)}{\left(R - y\right)^{2} \sqrt{4 R^{2} + a^{2}}} + 1\right) \left(4 R^{2} + a^{2} + a \sqrt{4 R^{2} + a^{2}}\right) \operatorname{ellipkinc}{\left(2 \operatorname{atan}{\left(\frac{\sqrt{a} \sqrt{R^{2} - y^{2}}}{\left(R - y\right) \left(4 R^{2} + a^{2}\right)^{0.25}} \right)},- \frac{a}{2 \sqrt{4 R^{2} + a^{2}}} + 0.5 \right)}}{R^{2} \sqrt{a}}}{6 \sqrt{4 + \frac{a^{2} \left(2 R^{2} - 2 R y\right)^{2}}{R^{2} \left(R - y\right)^{4}}}} Where ellipeinc is the incomplete elliptic integral of the second kind, and ellipkinc is the incomplete elliptic integral of the first kind, both calculated with SciPy's link to the cephes library. Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the guppy head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one guppy tank head, [m^2] Notes ----- This method is undefined for :math:`h > D` and :math:`h < 0`, but those cases are handled by returning the full surface area and the zero respectively. The analytical integral was derived with Rubi. Examples -------- >>> SA_partial_horiz_guppy_head(D=72., a=48.0, h=24.0) 1467.8949 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' R = 0.5*D if a == R: return pi*R*h elif h < 0.0: return 0.0 elif h > D: h = D if -R == h-R: return 0.0 # c1 = a/(2.0*R) # c2 = c1*c1 # a_R_ratio = a/R # a_R_ratio2 = a_R_ratio*a_R_ratio # from scipy.integrate import dblquad # def to_quad(y, x): # t1 = y/(R-x) # t1 *= t1 # t2 = (1.0 - t1) # return (1.0 + c2*t2*t2 + a_R_ratio2*t1)**0.5 # quad_val = dblquad(to_quad, -R, h-R, lambda x: 0.0, lambda x: (R*R - x*x)**0.5)[0] quad_val = float(quad(_SA_partial_horiz_guppy_head_to_int, -R, h-R, args=(a, R))[0]) SA = 2.0*quad_val return SA def _SA_partial_horiz_torispherical_head_int_1(x, b, c): x = float(x) # double check python float here to avoid numpy not erroring on sqrt # May be best to always use complex numbers here # print('_SA_partial_horiz_torispherical_head_int_1', x, b, c) x0 = x*x x1 = b - x0 x2 = sqrt(x1) x3 = -b + x0 x4 = c*c try: x5 = 1.0/sqrt(x1 - x4) except: x5 = 1.0/csqrt(x1 - x4) x6 = x3 + x4 x7 = sqrt(b) try: x3_pow = x3**(-1.5) except: x3_pow = (x3+0j)**(-1.5) ans = (x*cacos(c/x2) + x3_pow*x5*(-c*x1*csqrt(-x6*x6)*catan(x*x2/(csqrt(x3)*csqrt(x6))) + x6*x7*csqrt(-x1*x1)*catan(c*x*x5/x7))/csqrt(-x6/x1)) # print('_SA_partial_horiz_torispherical_head_int_1', abs(ans.real)) return abs(ans.real) def _SA_partial_horiz_torispherical_head_int_2(y, t2, s, c1): # May be best to always use complex numbers here # from mpmath import mp, mpf, atanh as catanh # mp.dps=30 # y, t2, s, c1 = mpf(y), mpf(t2), mpf(s), mpf(c1) y = float(y) # double check python float here to avoid numpy not erroring on sqrt y2 = y*y try: x10 = sqrt(t2 - y2) try: # Some tiny heights make the square root slightly under 0 x = (sqrt(c1 - y2 + (s+s)*x10)).real except: # Python 2 compat - don't take the square root of a negative number with no complex part x = (csqrt(c1 - y2 + (s+s)*x10 + 0.0j)).real except: x10 = csqrt(t2 - y2+0.0j) x = (csqrt(c1 - y2 + (s+s)*x10 + 0.0j)).real try: x0 = t2 - y2 x1 = s*x10.real t10 = x1 + x1 + s*s + x0 # x3, x4 present a very nasty numerical problem. # issue occurs when h == R, x3 is really equal to R**2 - 2*R*h + h**2 x3 = t10 - x*x x4 = sqrt(x3) # One solution is to use higher precision everywhere ans = x4*sqrt(t2*t10/(x0*x3))*catan(x/x4).real except: ans = 0.0 # ans = sqrt((t2* (s**2+t2-x**2+2.0*s* sqrt(t2-x**2)))/((t2-x**2)* (s**2+t2-x**2+2 *s* sqrt(t2-x**2)-y**2)))* sqrt(s**2+t2-x**2+2 *s* sqrt(t2-x**2)-y**2) *atan(y/sqrt(s**2+t2-x**2+2 *s* sqrt(t2-x**2)-y**2)) # print(float(y), float(t2), float(s), float(c1), float(ans.real)) # return float(ans.real) return ans.real def _SA_partial_horiz_torispherical_head_int_3(y, x, s, t2): x2 = x*x y2 = y*y x10 = sqrt(t2 - x2) num = (s + x10)*(s + x10)*x2 + (t2 - x2)*y2 den = (t2 - x2)*(s*s + t2 - x2 - y2 + 2.0*s*x10) f = sqrt(1.0 + num/den) return f def SA_partial_horiz_torispherical_head(D, f, k, h): r'''Calculates the partial area of a torispherical tank head in the context of a horizontal vessel and liquid partially filling it. This computes the wetted surface area of one of the torispherical heads only. The expressions used are quite complicated; see [1]_ for more details. Parameters ---------- D : float Diameter of the main cylindrical section, [m] f : float Dimensionless dish-radius parameter; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] k : float Dimensionless knuckle-radius parameter; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] h : float Height, as measured up to where the fluid ends, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one torispherical tank head, [m^2] Notes ----- This method is undefined for :math:`h > D` and :math:`h < 0`, but those cases are handled by returning the full surface area and the zero respectively. One integral: .. math:: \int_{R-h}^{fD\sin \alpha} \cos^{-1} \frac{fD\cos \alpha}{\sqrt{f^2 D^2 - x^2}} dx Can be computed as follows, using WolframAlpha. .. math:: x \operatorname{acos}{\left(\frac{c}{\sqrt{b - x^{2}}} \right)} + \frac{\sqrt{b} \sqrt{- \left(b - x^{2}\right)^{2}} \left(- b + c^{2} + x^{2}\right) \operatorname{atan}{\left(\frac{c x}{\sqrt{b} \sqrt{b - c^{2} - x^{2}}} \right)} + c \sqrt{- \left(- b + c^{2} + x^{2}\right)^{2}} \left(- b + x^{2}\right) \operatorname{atan}{\left(\frac{x \sqrt{b - x^{2}}} {\sqrt{- b + x^{2}} \sqrt{- b + c^{2} + x^{2}}} \right)}}{\sqrt{ \frac{- b + c^{2} + x^{2}}{- b + x^{2}}} \left(- b + x^{2}\right)^{1.5} \sqrt{b - c^{2} - x^{2}}} With the following constants: .. math:: c = fD\cos \alpha .. math:: b = f^2 D^2 The other integral is a double integral. There is an analytical integral available for the first integral, which takes the form: .. math:: 2 \sqrt{\frac{R^{2} k^{2} \left(4 R^{2} k^{2} - y^{2} + \left(- 2 R k + R\right)^{2} + 2 \left(- 2 R k + R\right) \sqrt{4 R^{2} k^{2} - y^{2}} \right)}{\left(4 R^{2} k^{2} - y^{2}\right) \left(\left(R - h\right)^{2} - \left(- 4 R k + 2 R\right) \sqrt{4 R^{2} k^{2} - y^{2}} + 2 \left(- 2 R k + R\right) \sqrt{4 R^{2} k^{2} - y^{2}} \right)}} \sqrt{\left(R - h\right)^{2} - \left(- 4 R k + 2 R\right) \sqrt{4 R^{2} k^{2} - y^{2}} + 2 \left(- 2 R k + R\right) \sqrt{4 R^{2} k^{2} - y^{2}}} \operatorname{atan}{\left(\frac{ \sqrt{4 R^{2} k^{2} - y^{2} - \left(R - h\right)^{2} + \left(- 4 R k + 2 R\right) \sqrt{4 R^{2} k^{2} - y^{2}} + \left(- 2 R k + R\right) ^{2}}}{\sqrt{\left(R - h\right)^{2} - \left(- 4 R k + 2 R\right) \sqrt{4 R^{2} k^{2} - y^{2}} + 2 \left(- 2 R k + R\right) \sqrt{4 R^{2} k^{2} - y^{2}}}} \right)} Examples -------- >>> SA_partial_horiz_torispherical_head(D=72., f=1, k=.06, h=24.0) 1471.201832459 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' if h <= 0.0: return 0.0 elif h > D: h = D R = D/2. r = f*D alpha = asin((1.0 - 2.0*k)/(2.*(f-k))) cos_alpha = cos(alpha) sin_alpha = sin(alpha) s = R - k*D t = k*D s2 = s*s t2 = t*t a1 = r*(1.0 - cos_alpha) a2 = k*D*cos_alpha c = f*D*cos_alpha b = f*f*D*D c1 = s2 + t2 - (R - h)**2 def G_lim(x): # numba: delete x2 = x*x # numba: delete try: # numba: delete G = sqrt(c1 - x2 + (s+s)*sqrt(t2 - x2)) # numba: delete except: # numba: delete # Python 2 compat - don't take the square root of a negative number with no complex part # numba: delete G = sqrt(c1 - x2 + (s+s)*sqrt(t2 - x2+0.0j) + 0.0j) # numba: delete return G.real # Some tiny heights make the square root slightly under 0 # numba: delete limit_1 = k*D*(1.0 - sin_alpha) if h < limit_1: SA = quad(_SA_partial_horiz_torispherical_head_int_2, 0.0, sqrt(2*k*D*h - h*h), args=(t2, s, c1))[0] return 2.0*SA elif limit_1 < h <= R: if (D*.499 < h < D*.501): # numba: delete from scipy.integrate import dblquad # numba: delete SA = 2.0*dblquad(_SA_partial_horiz_torispherical_head_int_3, 0.0, a2, lambda x: 0, G_lim, args=(s, t2))[0] # numba: delete # print(SA/2.) else: # numba: delete # Numerical issues SA = 2.0*quad(_SA_partial_horiz_torispherical_head_int_2, 0.0, a2, args=(t2, s, c1))[0] # numba: delete # print('SA', SA) # SA = 2.0*quad(_SA_partial_horiz_torispherical_head_int_2, 0.0, a2, args=(t2, s, c1))[0] # numba: uncomment try: high = _SA_partial_horiz_torispherical_head_int_1(f*D*sin_alpha, b, c) except: # Expression with the substitution is equally complicated high = _SA_partial_horiz_torispherical_head_int_1(f*D*sin_alpha*(1.0 + 1e-14), b, c) int_1_term1 = high - _SA_partial_horiz_torispherical_head_int_1(R-h, b, c) SA += 2.0*f*D*int_1_term1 else: SA = 2.0*pi*f*D*a1 + 2.0*pi*k*D*(a2 + (R - k*D)*asin(a2/(k*D))) SA -= SA_partial_horiz_torispherical_head(D, f, k, h=D-h) return SA def SA_partial_vertical_conical_head(D, a, h): r'''Calculates the partial area of a conical tank head in the context of a vertical vessel and liquid partially filling it. This computes the wetted surface area of one of the conical heads only, and is valid for `h` up to `a` only. .. math:: \text{SA} = \frac{\pi R h^2 \sqrt{a^2 + R^2}}{a^2} Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the cone head extends beneath the vertical tank, [m] h : float Height, as measured up to where the fluid ends or the top of the conical head, whichever is less, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one conical tank head extending beneath the vessel, [m^2] Notes ----- This method is undefined for :math:`h < 0`, but this is handled by returning zero. Examples -------- >>> SA_partial_vertical_conical_head(D=72., a=48.0, h=24.0) 1696.4600329384882 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' if a == 0.0: return 0.25*pi*D*D elif h <= 0.0: return 0.0 elif h > a: h = a R = 0.5*D SA = pi*R*h*h*sqrt(a*a + R*R)/(a*a) return SA def SA_partial_vertical_ellipsoidal_head(D, a, h): r'''Calculates the partial area of a ellipsoidal tank head in the context of a vertical vessel and liquid partially filling it. This computes the wetted surface area of one of the ellipsoidal heads only, and is valid for `h` up to `a` only. If :math:`a > R`: .. math:: \text{SA} = \pi R^2 - \frac{\pi (a - h)R\sqrt{a^4 - (a-h)^2(a^2-R^2)}}{a^2} + \frac{\pi a^2 R}{\sqrt{a^2 - R^2}}\left( \cos^{-1} \frac{R}{a} - \sin^{-1} \frac{(a-h)\sqrt{a^2-R^2}}{a^2} \right) Otherwise for :math:`0 < a < R`: .. math:: \text{SA} = \pi R^2 - \frac{\pi (a - h)R\sqrt{a^4 - (a-h)^2(a^2-R^2)}}{a^2} + \frac{\pi a^2 R}{\sqrt{a^2 - R^2}}\ln \left(\frac{a(\sqrt{R^2 - a^2} + R)} {(a-h)\sqrt{R^2 - a^2} + \sqrt{a^4 + (a-h)^2(R^2 - a^2)}} \right) Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the ellipsoidal head extends beneath the vertical tank, [m] h : float Height, as measured up to where the fluid ends or the top of the ellipsoidal head, whichever is less, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one ellipsoidal tank head extending beneath the vessel, [m^2] Notes ----- This method is undefined for :math:`h < 0`, but this is handled by returning zero. Examples -------- >>> SA_partial_vertical_ellipsoidal_head(D=72., a=48.0, h=24.0) 4675.23789137632 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' if a == 0.0: return 0.25*pi*D*D elif h <= 0.0: return 0.0 elif h > a: h = a # h should be less than a R = 0.5*D SA = pi*R*R a2 = a*a a_inv = 1.0/a R2 = R*R SA -= pi*(a - h)*R*sqrt(a2*a2 - (a-h)*(a-h)*(a2 - R2))*a_inv*a_inv if a > R: # This one has issues around a == R SA += pi*a2*R/sqrt(a2 - R2)*(acos(R*a_inv) - asin((a-h)*sqrt(a2 - R2)*a_inv*a_inv)) elif a == R: # Special case avoids zero division return pi*D*h else: x1 = sqrt(R2 - a2) num = a*(x1 + R) den = (a-h)*x1 + sqrt(a2*a2 + (a-h)*(a-h)*(R2 - a2)) SA += pi*a2*R/x1*log(num/den) return SA def SA_partial_vertical_spherical_head(D, a, h): r'''Calculates the partial area of a spherical tank head in the context of a vertical vessel and liquid partially filling it. This computes the wetted surface area of one of the conical heads only, and is valid for `h` up to `a` only. .. math:: \text{SA} = \pi h \left(\frac{a^2 + R^2}{a}\right) Parameters ---------- D : float Diameter of the main cylindrical section, [m] a : float Distance the spherical head extends beneath the vertical tank, [m] h : float Height, as measured up to where the fluid ends or the top of the spherical head, whichever is less, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one spherical tank head extending beneath the vessel, [m^2] Notes ----- This method is undefined for :math:`h < 0`, but this is handled by returning zero. Examples -------- >>> SA_partial_vertical_spherical_head(72, a=24, h=12) 2940.5307237600464 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' if a == 0.0: return 0.25*pi*D*D elif h <= 0.0: return 0.0 elif h > a: h = a R = 0.5*D SA = pi*h*((a*a + R*R)/a) return SA def SA_partial_vertical_torispherical_head(D, f, k, h): r'''Calculates the partial area of a torispherical tank head in the context of a vertical vessel and liquid partially filling it. This computes the wetted surface area of one of the torispherical heads only. if :math:`a_1 <= h`: .. math:: \text{SA} = 2\pi f D h if :math:`a_1 \le h \le a`: .. math:: \text{SA} = 2\pi f D a_1 + 2\pi k D\left( h - a_1 + (R - kD) \left( \sin^{-1} \frac{a_2}{kD} - \sin^{-1} \frac{a-h}{kD} \right) \right) .. math:: \alpha = \sin^{-1}\frac{1-2k}{2(f-k)} .. math:: a_1 = fD(1-\cos\alpha) .. math:: a_2 = kD\cos\alpha Parameters ---------- D : float Diameter of the main cylindrical section, [m] f : float Dimensionless dish-radius parameter; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] k : float Dimensionless knuckle-radius parameter; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] h : float Height, as measured up to where the fluid ends or the top of the torispherical head, whichever is less, [m] Returns ------- SA_partial : float Partial (wetted) surface area of one torispherical tank head, [m^2] Notes ----- This method is undefined for :math:`h > D` and :math:`h < 0`, but those cases are handled by returning the full surface area and the zero respectively. Examples -------- This method is undefined for :math:`h < 0`, but this is handled by returning zero. References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf ''' if h <= 0.0: return 0.0 R = 0.5*D alpha = asin((1.0 - 2.0*k)/(2.0*(f-k))) cos_alpha = cos(alpha) a1 = f*D*(1.0 - cos_alpha) a2 = k*D*cos_alpha a = a1 + a2 if h > a: h = a if h < a1: SA = 2.0*pi*f*D*h elif a1 <= h <= a: SA = 2.0*pi*f*D*a1 kD_inv = 1.0/(k*D) SA += 2.0*pi*k*D*(h - a1 + (R - k*D)*(asin(a2*kD_inv) - asin((a-h)*kD_inv))) else: # This case should not occur due to the earlier checks return 0.0 return SA def a_torispherical(D, f, k): r'''Calculates depth of a torispherical head according to [1]_. .. math:: a = a_1 + a_2 .. math:: \alpha = \sin^{-1}\frac{1-2k}{2(f-k)} .. math:: a_1 = fD(1-\cos\alpha) .. math:: a_2 = kD\cos\alpha Parameters ---------- D : float Diameter of the main cylindrical section, [m] f : float Dimensionless dish-radius parameter; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] k : float Dimensionless knuckle-radius parameter; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] Returns ------- a : float Depth of head [m] Examples -------- Example from [1]_. >>> a_torispherical(D=96., f=0.9, k=0.2) 25.684268924767125 References ---------- .. [1] Jones, D. "Calculating Tank Volume." Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF ''' alpha = asin((1.0-2.0*k)/(2.0*(f-k))) cos_alpha = cos(alpha) a1 = f*D*(1 - cos_alpha) a2 = k*D*cos_alpha return a1 + a2 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]_. If the height specified is above the height of the tank, it is truncated to the top of the tank. Parameters ---------- h : float Height of the liquid in the tank, [m] D : float Diameter of the cylindrical section of the tank, [m] L : float Length of the main cylindrical section of the tank, [m] horizontal : bool, optional Whether or not the tank is a horizontal or vertical tank sideA : string, optional The left (or bottom for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical']. sideB : string, optional The right (or top for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical']. sideA_a : float, optional The distance the head as specified by sideA extends down or to the left from the main cylindrical section, [m] sideB_a : float, optional The distance the head as specified by sideB extends up or to the right from the main cylindrical section, [m] sideA_f : float, optional Dimensionless dish-radius parameter for side A; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideA_k : float, optional Dimensionless knuckle-radius parameter for side A; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] sideB_f : float, optional Dimensionless dish-radius parameter for side B; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideB_k : float, optional Dimensionless knuckle-radius parameter for side B; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] Returns ------- V : float Volume up to h [m^3] Examples -------- >>> V_from_h(h=7, D=1.5, L=5., horizontal=False, sideA='conical', ... sideB='conical', sideA_a=2., sideB_a=1.) 10.013826583317465 References ---------- .. [1] Jones, D. "Compute Fluid Volumes in Vertical Tanks." Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/ ''' if sideA is not None and sideA not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'): raise ValueError('Unspoorted head type for side A') if sideB is not None and sideB not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'): raise ValueError('Unspoorted head type for side B') R = 0.5*D V = 0.0 if horizontal: if h > D: # Must be before Af, which will raise a domain error h = D # Conical case if sideA == 'conical' and sideB == 'conical' and sideA_a == sideB_a: V += 2.0*V_horiz_conical(D, L, sideA_a, h, headonly=True) else: if sideA == 'conical': V += V_horiz_conical(D, L, sideA_a, h, headonly=True) if sideB == 'conical': V += V_horiz_conical(D, L, sideB_a, h, headonly=True) # Elliosoidal case if sideA == 'ellipsoidal' and sideB == 'ellipsoidal' and sideA_a == sideB_a: V += 2.0*V_horiz_ellipsoidal(D, L, sideA_a, h, headonly=True) else: if sideA == 'ellipsoidal': V += V_horiz_ellipsoidal(D, L, sideA_a, h, headonly=True) if sideB == 'ellipsoidal': V += V_horiz_ellipsoidal(D, L, sideB_a, h, headonly=True) # Guppy case if sideA == 'guppy' and sideB == 'guppy' and sideA_a == sideB_a: V += 2.0*V_horiz_guppy(D, L, sideA_a, h, headonly=True) else: if sideA == 'guppy': V += V_horiz_guppy(D, L, sideA_a, h, headonly=True) if sideB == 'guppy': V += V_horiz_guppy(D, L, sideB_a, h, headonly=True) # Spherical case if sideA == 'spherical' and sideB == 'spherical' and sideA_a == sideB_a: V += 2.0*V_horiz_spherical(D, L, sideA_a, h, headonly=True) else: if sideA == 'spherical': V += V_horiz_spherical(D, L, sideA_a, h, headonly=True) if sideB == 'spherical': V += V_horiz_spherical(D, L, sideB_a, h, headonly=True) # Torispherical case if (sideA == 'torispherical' and sideB == 'torispherical' and (sideA_f == sideB_f) and (sideA_k == sideB_k)): V += 2.0*V_horiz_torispherical(D, L, sideA_f, sideA_k, h, headonly=True) else: if sideA == 'torispherical': V += V_horiz_torispherical(D, L, sideA_f, sideA_k, h, headonly=True) if sideB == 'torispherical': V += V_horiz_torispherical(D, L, sideB_f, sideB_k, h, headonly=True) Af = R*R*acos((R-h)/R) - (R-h)*sqrt(2.0*R*h - h*h) V += L*Af else: max_h = L + sideA_a + sideB_a if h > max_h: h = max_h # 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 += 0.25*pi*D*D*L # All middle elif h > sideA_a: V += 0.25*pi*D*D*(h - sideA_a) # Partial middle # Top head if h > sideA_a + L: h2 = sideB_a - (h - sideA_a - L) if sideB == 'conical': V += V_vertical_conical(D, sideB_a, h=sideB_a) V -= V_vertical_conical(D, sideB_a, h=h2) if sideB == 'ellipsoidal': V += V_vertical_ellipsoidal(D, sideB_a, h=sideB_a) V -= V_vertical_ellipsoidal(D, sideB_a, h=h2) if sideB == 'spherical': V += V_vertical_spherical(D, sideB_a, h=sideB_a) V -= V_vertical_spherical(D, sideB_a, h=h2) if sideB == 'torispherical': V += V_vertical_torispherical(D, sideB_f, sideB_k, h=sideB_a) V -= max(0.0, V_vertical_torispherical(D, sideB_f, sideB_k, h=h2)) return V def SA_from_h(h, D, L, horizontal=True, sideA=None, sideB=None, sideA_a=0.0, sideB_a=0.0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None): r'''Calculates partially full wetted surface area of a vertical or horizontal tank with different head types according to [1]_. Parameters ---------- h : float Height of the liquid in the tank, [m] D : float Diameter of the cylindrical section of the tank, [m] L : float Length of the main cylindrical section of the tank, [m] horizontal : bool, optional Whether or not the tank is a horizontal or vertical tank sideA : string, optional The left (or bottom for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical']. sideB : string, optional The right (or top for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical']. sideA_a : float, optional The distance the head as specified by sideA extends down or to the left from the main cylindrical section, [m] sideB_a : float, optional The distance the head as specified by sideB extends up or to the right from the main cylindrical section, [m] sideA_f : float, optional Dimensionless dish-radius parameter for side A; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideA_k : float, optional Dimensionless knuckle-radius parameter for side A; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] sideB_f : float, optional Dimensionless dish-radius parameter for side B; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideB_k : float, optional Dimensionless knuckle-radius parameter for side B; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] Returns ------- SA : float Wetted wall surface area up to h [m^3] Examples -------- >>> SA_from_h(h=7, D=1.5, L=5., horizontal=False, sideA='conical', ... sideB='conical', sideA_a=2., sideB_a=1.) 28.59477853914843 References ---------- .. [1] Jones, D. "Calculating Tank Wetted Area." Text. Chemical Processing. April 2017. https://www.chemicalprocessing.com/assets/Uploads/calculating-tank-wetted-area.pdf http://www.chemicalprocessing.com/articles/2003/193/ ''' if sideA is not None and sideA not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'): raise ValueError('Unspoorted head type for side A') if sideB is not None and sideB not in ('conical', 'ellipsoidal', 'torispherical', 'spherical', 'guppy'): raise ValueError('Unspoorted head type for side B') R = 0.5*D SA = 0.0 if horizontal: if h > D: h = D # Conical case if sideA == 'conical': SA += SA_partial_horiz_conical_head(D, sideA_a, h) if sideB == 'conical': SA += SA_partial_horiz_conical_head(D, sideB_a, h) # Elliosoidal case if sideA == 'ellipsoidal': SA += SA_partial_horiz_ellipsoidal_head(D, sideA_a, h) if sideB == 'ellipsoidal': SA += SA_partial_horiz_ellipsoidal_head(D, sideB_a, h) # Guppy case if sideA == 'guppy': SA += SA_partial_horiz_guppy_head(D, sideA_a, h) if sideB == 'guppy': SA += SA_partial_horiz_guppy_head(D, sideB_a, h) # Spherical case if sideA == 'spherical': SA += SA_partial_horiz_spherical_head(D, sideA_a, h) if sideB == 'spherical': SA += SA_partial_horiz_spherical_head(D, sideB_a, h) # Torispherical case if sideA == 'torispherical': if sideA_f is not None and sideA_k is not None: SA += SA_partial_horiz_torispherical_head(D, sideA_f, sideA_k, h) else: raise ValueError("Torispherical sideA but no `f` and `k` provided") if sideB == 'torispherical': if sideB_f is not None and sideB_k is not None: SA += SA_partial_horiz_torispherical_head(D, sideB_f, sideB_k, h) else: raise ValueError("Torispherical sideB but no `f` and `k` provided") # Flat case if sideA is None: SA += A_partial_circle(D, h) if sideB is None: SA += A_partial_circle(D, h) SA += L*D*acos((D - h - h)/D) else: max_h = L + sideA_a + sideB_a if h > max_h: h = max_h # Bottom head if sideA in ('conical', 'ellipsoidal', 'torispherical', 'spherical'): if sideA == 'conical': SA += SA_partial_vertical_conical_head(D, sideA_a, h=min(sideA_a, h)) elif sideA == 'ellipsoidal': SA += SA_partial_vertical_ellipsoidal_head(D, sideA_a, h=min(sideA_a, h)) elif sideA == 'spherical': SA += SA_partial_vertical_spherical_head(D, sideA_a, h=min(sideA_a, h)) elif sideA == 'torispherical': if sideA_f is not None and sideA_k is not None: SA += SA_partial_vertical_torispherical_head(D, sideA_f, sideA_k, h=min(sideA_a, h)) else: raise ValueError("Torispherical sideA but no `f` and `k` provided") elif sideA is None: SA += 0.25*pi*D*D # Cylindrical section if h >= sideA_a + L: SA += pi*D*L # All middle elif h > sideA_a: SA += pi*D*(h - sideA_a) # Partial middle # Top head if h >= sideA_a + L: # greater or equals is needed! Flat head on top adds lots of area. h2 = sideB_a - (h - sideA_a - L) if sideB == 'conical': if sideB_a == 0.0: SA += 0.25*pi*D*D else: SA += SA_partial_vertical_conical_head(D, sideB_a, h=sideB_a) SA -= SA_partial_vertical_conical_head(D, sideB_a, h=h2) elif sideB == 'ellipsoidal': if sideB_a == 0.0: SA += 0.25*pi*D*D else: SA += SA_partial_vertical_ellipsoidal_head(D, sideB_a, h=sideB_a) SA -= SA_partial_vertical_ellipsoidal_head(D, sideB_a, h=h2) elif sideB == 'spherical': if sideB_a == 0.0: SA += 0.25*pi*D*D else: SA += SA_partial_vertical_spherical_head(D, sideB_a, h=sideB_a) SA -= SA_partial_vertical_spherical_head(D, sideB_a, h=h2) elif sideB == 'torispherical': if sideB_a == 0.0: SA += 0.25*pi*D*D else: if sideB_f is not None and sideB_k is not None: SA += SA_partial_vertical_torispherical_head(D, sideB_f, sideB_k, h=sideB_a) SA -= max(0.0, SA_partial_vertical_torispherical_head(D, sideB_f, sideB_k, h=h2)) else: raise ValueError("Torispherical sideB but no `f` and `k` provided") elif sideB is None and h == sideA_a + L: # End cap if flat SA += 0.25*pi*D*D return SA def tank_from_two_specs_err(guess, spec0, spec1, spec0_name, spec1_name, h, horizontal, sideA, sideB, sideA_a, sideB_a, sideA_f, sideA_k, sideB_f, sideB_k, sideA_a_ratio, sideB_a_ratio): D, L_over_D = float(guess[0]), float(guess[1]) obj = TANK(D=D, L_over_D=L_over_D, horizontal=horizontal, sideA=sideA, sideB=sideB, sideA_a=sideA_a, sideB_a=sideB_a, sideA_f=sideA_f, sideA_k=sideA_k, sideB_f=sideB_f, sideB_k=sideB_k, sideA_a_ratio=sideA_a_ratio, sideB_a_ratio=sideB_a_ratio) # ensure h is always under the top h = min(h, obj.h_max) if spec0_name == 'V': err0 = obj.V_total - spec0 elif spec0_name == 'SA': err0 = obj.A - spec0 elif spec0_name == 'V_partial': err0 = obj.V_from_h(h) - spec0 elif spec0_name == 'SA_partial': err0 = obj.SA_from_h(h) - spec0 elif spec0_name == 'A_cross': err0 = obj.A_cross_sectional(h) - spec0 if spec1_name == 'V': err1 = obj.V_total - spec1 elif spec1_name == 'SA': err1 = obj.A - spec1 elif spec1_name == 'V_partial': err1 = obj.V_from_h(h) - spec1 elif spec1_name == 'SA_partial': err1 = obj.SA_from_h(h) - spec1 elif spec1_name == 'A_cross': err1 = obj.A_cross_sectional(h) - spec1 # print(err0, err1, D, L_over_D, h) return [err0, err1] class TANK: """Class representing tank volumes and levels. All parameters are also attributes. Parameters ---------- D : float Diameter of the cylindrical section of the tank, [m] L : float Length of the main cylindrical section of the tank, [m] horizontal : bool, optional Whether or not the tank is a horizontal or vertical tank sideA : string, optional The left (or bottom for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical', 'same']. sideB : string, optional The right (or top for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical', 'same']. sideA_a : float, optional The distance the head as specified by sideA extends down or to the left from the main cylindrical section, [m] sideB_a : float, optional The distance the head as specified by sideB extends up or to the right from the main cylindrical section, [m] sideA_f : float, optional Dimensionless dish-radius parameter for side A; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideA_k : float, optional Dimensionless knuckle-radius parameter for side A; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] sideB_f : float, optional Dimensionless dish-radius parameter for side B; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideB_k : float, optional Dimensionless knuckle-radius parameter for side B; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] sideA_a_ratio : float, optional Ratio for `a` parameter; can be used instead of specifying an absolute value, [-] sideB_a_ratio : float, optional Ratio for `a` parameter; can be used instead of specifying an absolute value, [-] L_over_D : float, optional Ratio of length over diameter, used only when D and L are both unspecified but V is, [-] V : float, optional Volume of the tank; solved for if specified, using sideA_a_ratio/sideB_a_ratio, sideA, sideB, horizontal, and one of L_over_D, L, or D, [m^3] Attributes ---------- h_max : float Height of the tank, [m] V_total : float Total volume of the tank as calculated [m^3] sideA_V : float Volume of only `sideA` [m^3] sideB_V : float Volume of only `sideB` [m^3] lateral_V : float Volume of cylindrical section of tank [m^3] A : float Total surface area of the tank, [m^2] A_sideA : float Surface area of sideA, [m^2] A_sideB : float Surface area of sideB, [m^2] A_lateral : float Surface area of the lateral side, [m^2] A_sideA_extra : float Additional surface area of sideA beyond that of a flat disk, [m^2] A_sideB_extra : float Additional surface area of sideB beyond that of a flat disk, [m^2] table : bool Whether or not a table of heights-volumes has been generated heights : ndarray Array of heights between 0 and h_max, [m] volumes : ndarray Array of volumes calculated from the heights, [m^3] c_forward : ndarray Coefficients for the Chebyshev approximations in calculating V from h, [-] c_backward : ndarray Coefficients for the Chebyshev approximations in calculating h from V, [-] Notes ----- For torpsherical tank heads, the following `f` and `k` parameters are used in standards. The default is ASME F&D. +----------------------+-----+-------+ | | f | k | +======================+=====+=======+ | 2:1 semi-elliptical | 0.9 | 0.17 | +----------------------+-----+-------+ | ASME F&D | 1 | 0.06 | +----------------------+-----+-------+ | ASME 80/6 | 0.8 | 0.06 | +----------------------+-----+-------+ | ASME 80/10 F&D | 0.8 | 0.1 | +----------------------+-----+-------+ | DIN 28011 | 1 | 0.1 | +----------------------+-----+-------+ | DIN 28013 | 0.8 | 0.154 | +----------------------+-----+-------+ For the following cases, numerical integrals are used. V_horiz_spherical V_horiz_torispherical SA_partial_horiz_spherical_head SA_partial_horiz_ellipsoidal_head SA_partial_horiz_guppy_head SA_partial_horiz_torispherical_head Examples -------- Total volume of a tank: >>> TANK(D=1.2, L=4, horizontal=False).V_total 4.523893421169302 Volume of a tank at a given height: >>> TANK(D=1.2, L=4, horizontal=False).V_from_h(.5) 0.5654866776461628 Height of liquid for a given volume: >>> TANK(D=1.2, L=4, horizontal=False).h_from_V(.5) 0.442097064 Surface area of a tank with a conical head: >>> T1 = TANK(V=10, L_over_D=0.7, sideB='conical', sideB_a=0.5) >>> T1.A, T1.A_sideA, T1.A_sideB, T1.A_lateral (24.94775907, 5.118555, 5.497246, 14.331956) Solving for tank volumes, first horizontal, then vertical: >>> TANK(D=10., horizontal=True, sideA='conical', sideB='conical', V=500).L 4.699531 >>> TANK(L=4.69953105701, horizontal=True, sideA='conical', sideB='conical', V=500).D 9.9999999 >>> TANK(L_over_D=0.469953105701, horizontal=True, sideA='conical', sideB='conical', V=500).L 4.6995310 >>> TANK(D=10., horizontal=False, sideA='conical', sideB='conical', V=500).L 4.699531 >>> TANK(L=4.69953105701, horizontal=False, sideA='conical', sideB='conical', V=500).D 9.99999999 >>> TANK(L_over_D=0.469953105701, horizontal=False, sideA='conical', sideB='conical', V=500).L 4.699531057 """ table = False chebyshev = False __full_path__ = __module__ + '.TANK' def __str__(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 + (f' heads, a={self.sideA_a:f} m') else: sides = self.sideA + f' heads, sideA a={self.sideA_a:f} m, sideB a={self.sideB_a:f} m' else: if self.sideA: A = f'{self.sideA} head on sideA with a={self.sideA_a:f} m' else: A = 'no head on sideA' if self.sideB: B = f' and {self.sideB} head on sideB with a={self.sideB_a:f} m' else: B = ' and no head on sideB' sides = A + B return f'<{orient} tank, V={self.V_total:f} m^3, D={self.D:f} m, L={self.L:f} m, {sides}.>' def __repr__(self): attributes = ', '.join(f"{slot}={getattr(self, slot)!r}" for slot in self.model_inputs if getattr(self, slot) is not None) return f"{self.__class__.__name__}({attributes})" def __hash__(self): return hash(tuple(getattr(self, slot) for slot in self.model_inputs)) model_inputs = ('D', 'L', 'horizontal', 'sideA', 'sideB', 'sideA_a', 'sideB_a', 'sideA_f', 'sideA_k', 'sideB_f', 'sideB_k', 'sideA_a_ratio', 'sideB_a_ratio', 'L_over_D', 'V') def __init__(self, D=None, L=None, horizontal=True, sideA=None, sideB=None, sideA_a=None, sideB_a=None, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None, sideA_a_ratio=None, sideB_a_ratio=None, L_over_D=None, V=None): self.D = D self.L = L self.L_over_D = L_over_D self.V = V self.horizontal = horizontal sideA_same, sideB_same = sideA == 'same', sideB == 'same' if sideA_same and not sideB_same: sideA, sideA_a, sideA_a_ratio, sideA_f, sideA_k = sideB, sideB_a, sideB_a_ratio, sideB_f, sideB_k elif sideB_same and not sideA_same: sideB, sideB_a, sideB_a_ratio, sideB_f, sideB_k = sideA, sideA_a, sideA_a_ratio, sideA_f, sideA_k elif sideA_same and sideB_same: raise ValueError("Cannot specify both sides as same") self.sideA = sideA if sideA is None and sideA_a is None: sideA_a = 0.0 self.sideA_a = sideA_a if sideA_a is None and sideA_a_ratio is None and (sideA is not None and sideA != 'torispherical'): sideA_a_ratio = 0.25 self.sideA_a_ratio = sideA_a_ratio if sideA_a is None and sideA == 'torispherical': if sideA_f is None: sideA_f = 1.0 if sideA_k is None: sideA_k = 0.06 self.sideA_f = sideA_f self.sideA_k = sideA_k self.sideB = sideB if sideB is None and sideB_a is None: sideB_a = 0.0 self.sideB_a = sideB_a if sideB_a is None and sideB_a_ratio is None and (sideB is not None and sideB != 'torispherical'): sideB_a_ratio = 0.25 self.sideB_a_ratio = sideB_a_ratio if sideB_a is None and sideB == 'torispherical': if sideB_f is None: sideB_f = 1.0 if sideB_k is None: sideB_k = 0.06 self.sideB_f = sideB_f self.sideB_k = sideB_k if self.horizontal: self.vertical = False self.orientation = 'horizontal' self.angle = 0 else: self.vertical = True self.orientation = 'vertical' self.angle = 90 # If V is specified and either L or D are known, solve for L, D, L_over_D if self.V: self._solve_tank_for_V() self.set_misc() def set_misc(self): """Set more parameters, after the tank is better defined than in the __init__ function. Notes ----- Two of D, L, and L_over_D must be known when this function runs. The other one is set from the other two first thing in this function. a_ratio parameters are used to calculate a values for the heads here, if applicable. Radius is calculated here. Maximum tank height is calculated here. V_total is calculated here. """ if self.D is not None and self.L is not None: # If L and D are known, get L_over_D self.L_over_D = self.L/self.D elif self.D is not None and self.L_over_D is not None: # Otherwise, if L_over_D and D are provided, get L self.L = self.D*self.L_over_D elif self.L is not None and self.L_over_D is not None: # Otherwise, if L_over_D and L are provided, get D self.D = self.L/self.L_over_D D = self.D # Calculate diameter self.R = self.D/2. # If a_ratio is provided for either heads, use it. if self.sideA is not None and D is not None and self.sideA_a is None and self.sideA in ('conical', 'ellipsoidal', 'guppy', 'spherical'): self.sideA_a = D*self.sideA_a_ratio if self.sideB is not None and D is not None and self.sideB_a is None and self.sideB in ('conical', 'ellipsoidal', 'guppy', 'spherical'): self.sideB_a = D*self.sideB_a_ratio # Calculate a for torispherical heads if self.sideA == 'torispherical' and self.sideA_f is not None and self.sideA_k is not None: self.sideA_a = a_torispherical(D, self.sideA_f, self.sideA_k) if self.sideB == 'torispherical' and self.sideB_f is not None and self.sideB_k is not None: self.sideB_a = a_torispherical(D, self.sideB_f, self.sideB_k) # Ensure the correct a_ratios are set, whether there is a default being used or not if self.sideA_a_ratio is None and self.sideA_a is not None: self.sideA_a_ratio = self.sideA_a/D elif self.sideA_a_ratio is not None and self.sideA_a is not None and self.sideA_a != D*self.sideA_a_ratio: self.sideA_a_ratio = self.sideA_a/D if self.sideB_a_ratio is None and self.sideB_a is not None: self.sideB_a_ratio = self.sideB_a/D elif self.sideB_a_ratio is not None and self.sideB_a is not None and self.sideB_a != D*self.sideB_a_ratio: self.sideB_a_ratio = self.sideB_a/D # Calculate maximum tank height, h_max if self.horizontal: self.h_max = D else: self.h_max = self.L if self.sideA_a: self.h_max += self.sideA_a if self.sideB_a: self.h_max += self.sideB_a # Set maximum height # self.V_total = self.V_from_h(self.h_max) self.V_total, self.V_sideA, self.V_sideB, self.V_lateral = V_tank( D=D, L=self.L, sideA=self.sideA, sideB=self.sideB, sideA_a=self.sideA_a, sideB_a=self.sideB_a, sideA_f=self.sideA_f, sideA_k=self.sideA_k, sideB_f=self.sideB_f, sideB_k=self.sideB_k, horizontal=self.horizontal) # Set surface areas self.A, self.A_sideA, self.A_sideB, self.A_lateral = SA_tank( D=D, L=self.L, sideA=self.sideA, sideB=self.sideB, sideA_a=self.sideA_a, sideB_a=self.sideB_a, sideA_f=self.sideA_f, sideA_k=self.sideA_k, sideB_f=self.sideB_f, sideB_k=self.sideB_k) A_circular_plate = 0.25*pi*D*D self.A_sideA_extra = self.A_sideA - A_circular_plate self.A_sideB_extra = self.A_sideB - A_circular_plate @staticmethod def from_two_specs(spec0, spec1, spec0_name='V', spec1_name='A_cross', h=None, horizontal=True, sideA=None, sideB=None, sideA_a=None, sideB_a=None, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None, sideA_a_ratio=None, sideB_a_ratio=None): r'''Method to create a new tank instance according to two specifications which are not direct geometry parameters. The allowable options are 'V', 'SA', 'V_partial', 'SA_partial', and 'A_cross', the later three of which require `h` to be specified. Parameters ---------- spec0 : float Goal for `spec0_name`, [-] spec1 : float Goal for `spec1_name`, [-] spec0_name : str One of 'V', 'SA', 'V_partial', 'SA_partial', and 'A_cross' [-] spec1_name : str One of 'V', 'SA', 'V_partial', 'SA_partial', and 'A_cross' [-] h : float Height at which to calculate the specs, [m] horizontal : bool, optional Whether or not the tank is a horizontal or vertical tank sideA : string, optional The left (or bottom for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical', 'same']. sideB : string, optional The right (or top for vertical) head of the tank's type; one of [None, 'conical', 'ellipsoidal', 'torispherical', 'guppy', 'spherical', 'same']. sideA_a : float, optional The distance the head as specified by sideA extends down or to the left from the main cylindrical section, [m] sideB_a : float, optional The distance the head as specified by sideB extends up or to the right from the main cylindrical section, [m] sideA_f : float, optional Dimensionless dish-radius parameter for side A; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideA_k : float, optional Dimensionless knuckle-radius parameter for side A; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] sideB_f : float, optional Dimensionless dish-radius parameter for side B; also commonly given as the product of `f` and `D` (`fD`), which is called dish radius and has units of length, [-] sideB_k : float, optional Dimensionless knuckle-radius parameter for side B; also commonly given as the product of `k` and `D` (`kD`), which is called the knuckle radius and has units of length, [-] Returns ------- TANK : TANK Tank object at solved specifications, [-] Notes ----- Limited testing has been done on this method. The bounds are D between 0.1 mm and 10 km, with L_D ratios of 1e-4 to 1e4. ''' args = (spec0, spec1, spec0_name, spec1_name, h, horizontal, sideA, sideB, sideA_a, sideB_a, sideA_f, sideA_k, sideB_f, sideB_k, sideA_a_ratio, sideB_a_ratio) new_f, translate_into, translate_outof = translate_bound_func(tank_from_two_specs_err, bounds=[(1e-4, 1e4), (1e-4, 1e4)]) # Diameter and length/diameter as iteration variables guess = translate_into([1.0, 3.0]) from scipy.optimize import fsolve ans = fsolve(new_f, guess, args=args, xtol=1e-10, factor=.1) val0, val1 = translate_outof(ans) return TANK(D=float(val0), L_over_D=float(val1), horizontal=horizontal, sideA=sideA, sideB=sideB, sideA_a=sideA_a, sideB_a=sideB_a, sideA_f=sideA_f, sideA_k=sideA_k, sideB_f=sideB_f, sideB_k=sideB_k, sideA_a_ratio=sideA_a_ratio, sideB_a_ratio=sideB_a_ratio,) def add_thickness(self, thickness, sideA_thickness=None, sideB_thickness=None): r'''Method to create a new tank instance with the same parameters as itself, except with an added thickness to it. This is useful to obtain ex. the inside of a tank and the outside; their different in volumes is the volume of the shell, and could be used to determine weight. Parameters ---------- thickness : float Thickness to add to the tank diameter, [m] sideA_thickness : float, optional The thickness to add to the sideA head; if not specified, it will be `thickness`, [m] sideB_thickness : float, optional The thickness to add to the sideB head; if not specified, it will be `thickness`, [m] Returns ------- TANK : TANK Tank object, [-] Notes ----- Be careful not to specify a negative thickness larger than the heads' lengths, or the head will become concave! The same applies to adding a thickness to convex heads - they can become convex. ''' kwargs = dict(D=self.D, L=self.L, horizontal=self.horizontal, sideA=self.sideA, sideB=self.sideB, sideA_a=self.sideA_a, sideB_a=self.sideB_a, sideA_f=self.sideA_f, sideA_k=self.sideA_k, sideB_f=self.sideB_f, sideB_k=self.sideB_k) if sideA_thickness is None: sideA_thickness = thickness if sideB_thickness is None: sideB_thickness = thickness # Do not transfer a_ratios or volume or L_over_D kwargs['D'] += 2.0*thickness kwargs['L'] += sideA_thickness + sideB_thickness # For torispherical vessels, the heads are defined from the `f` and `k` # parameters which are already functions of diameter, and so will be # fixed automatically; if the `a` parameters are specified they would # not be corrected if self.sideA != 'torispherical': kwargs['sideA_a'] += sideA_thickness else: del kwargs['sideA_a'] if self.sideB != 'torispherical': kwargs['sideB_a'] += sideB_thickness else: del kwargs['sideB_a'] return TANK(**kwargs) def SA_from_h(self, h, method='full'): r'''Method to calculate the volume of liquid in a fully defined tank given a specified height `h`. `h` must be under the maximum height. Parameters ---------- h : float Height specified, [m] method : str, optional 'full' (calculated rigorously) ; nothing else is implemented Returns ------- SA : float Surface area of liquid in the tank up to the specified height, [m^2] Notes ----- ''' if method == 'full': return SA_from_h(h, self.D, self.L, self.horizontal, self.sideA, self.sideB, self.sideA_a, self.sideB_a, self.sideA_f, self.sideA_k, self.sideB_f, self.sideB_k) else: raise ValueError("Allowable methods are 'full' .") def V_from_h(self, h, method='full'): r'''Method to calculate the volume of liquid in a fully defined tank given a specified height `h`. `h` must be under the maximum height. If the method is 'chebyshev', and the coefficients have not yet been calculated, they are created by calling `set_chebyshev_approximators`. Parameters ---------- h : float Height specified, [m] method : str One of 'full' (calculated rigorously) or 'chebyshev' Returns ------- V : float Volume of liquid in the tank up to the specified height, [m^3] Notes ----- ''' if method == 'full': return V_from_h(h, self.D, self.L, self.horizontal, self.sideA, self.sideB, self.sideA_a, self.sideB_a, self.sideA_f, self.sideA_k, self.sideB_f, self.sideB_k) elif method == 'chebyshev': if not self.chebyshev: self.set_chebyshev_approximators() return self.V_from_h_cheb(h) else: raise ValueError("Allowable methods are 'full' or 'chebyshev'.") def h_from_V(self, V, method='spline'): r'''Method to calculate the height of liquid in a fully defined tank given a specified volume of liquid in it `V`. `V` must be under the maximum volume. If the method is 'spline', and the interpolation table is not yet defined, creates it by calling the method set_table. If the method is 'chebyshev', and the coefficients have not yet been calculated, they are created by calling `set_chebyshev_approximators`. Parameters ---------- V : float Volume of liquid in the tank up to the desired height, [m^3] method : str One of 'spline', 'chebyshev', or 'brenth' Returns ------- h : float Height of liquid at which the volume is as desired, [m] ''' if method == 'spline': try: if not self.table: self.set_table() return float(self.interp_h_from_V(V)) except: # Missing scipy return self.h_from_V(V, 'brenth') elif method == 'chebyshev': if not self.chebyshev: self.set_chebyshev_approximators() return self.h_from_V_cheb(V) elif method == 'brenth': to_solve = lambda h : self.V_from_h(h, method='full') - V return secant(to_solve, x0=0.5*self.h_max, low=0, high=self.h_max, bisection=True) else: raise ValueError("Allowable methods are 'full' or 'chebyshev', " "or 'brenth'.") def A_cross_sectional(self, h, method='full'): r'''Method to calculate the cross-sectional liquid surface area from which gas can evolve in a fully defined tank given a specified height `h`. `h` must be under the maximum height. This is calculated by numeric differentiation for most cases. Parameters ---------- h : float Height specified, [m] method : str, optional 'full' (calculated rigorously) or 'chebyshev', [-] Returns ------- A_cross : float Surface area of liquid in the tank up to the specified height, [m^2] Notes ----- ''' # The derivative will give bad values in some cases, when right up against boundaries # Analytical formulations can be done, but will be lots of code if h in (self.h_max, 0.0): return 0.0 return derivative(lambda h: self.V_from_h(h), h, dx=1e-7*h, order=3, n=1) def set_table(self, n=100, dx=None): r'''Method to set an interpolation table of liquids levels versus volumes in the tank, for a fully defined tank. Normally run by the h_from_V method, this may be run prior to its use with a custom specification. Either the number of points on the table, or the vertical distance between steps may be specified. Parameters ---------- n : float, optional Number of points in the interpolation table, [-] dx : float, optional Vertical distance between steps in the interpolation table, [m] ''' if dx: self.heights = linspace(0.0, self.h_max, int(self.h_max/dx)+1) else: self.heights = linspace(0.0, self.h_max, n) self.volumes = [self.V_from_h(h) for h in self.heights] from scipy.interpolate import UnivariateSpline # TODO replace with splrep/splev to avoid the object self.interp_h_from_V = UnivariateSpline(self.volumes, self.heights, ext=3, s=0.0) self.table = True def set_chebyshev_approximators(self, deg_forward=50, deg_backwards=200): r'''Method to derive and set coefficients for chebyshev polynomial function approximation of the height-volume and volume-height relationship. A single set of chebyshev coefficients is used for the entire height- volume and volume-height relationships respectively. The forward relationship, `V_from_h`, requires far fewer coefficients in its fit than the reverse to obtain the same relative accuracy. Optionally, deg_forward or deg_backwards can be set to None to try to automatically fit the series to machine precision. Parameters ---------- deg_forward : int, optional The degree of the chebyshev polynomial to be created for the `V_from_h` curve, [-] deg_backwards : int, optional The degree of the chebyshev polynomial to be created for the `h_from_V` curve, [-] ''' import numpy as np from fluids.optional.pychebfun import Chebfun to_fit = lambda h: self.V_from_h(h, 'full') # These high-degree polynomials cannot safety be evaluated using Horner's methods # chebval is 2.5x as slow but 100% required; depending on the geometry, but # experience shows typically at around 40 coefficients the results are become junk self.c_forward = Chebfun.from_function(np.vectorize(to_fit), [0.0, self.h_max], N=deg_forward).coefficients().tolist() self.V_from_h_cheb = lambda x : chebval((2.0*x-self.h_max)/(self.h_max), self.c_forward) to_fit = lambda h: self.h_from_V(h, 'brenth') self.c_backward = Chebfun.from_function(np.vectorize(to_fit), [0.0, self.V_total], N=deg_backwards).coefficients().tolist() # TODO do not use lambda 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. """ # print('Vtarget, D, L, L_D', Vtarget, D, L, L/D) a = TANK(D=float(D), L=float(L), horizontal=horizontal, sideA=sideA, sideB=sideB, sideA_a=sideA_a, sideB_a=sideB_a, sideA_f=sideA_f, sideA_k=sideA_k, sideB_f=sideB_f, sideB_k=sideB_k, sideA_a_ratio=sideA_a_ratio, sideB_a_ratio=sideB_a_ratio) error = (Vtarget - a.V_total) # print(error, 'error') return error def _solve_tank_for_V(self): """Method which is called to solve for tank geometry when a certain volume is specified. Will be called by the __init__ method if V is set. Notes ----- Raises an error if L and either of sideA_a or sideB_a are specified; these can only be set once D is known. Raises an error if more than one of D, L, or L_over_D are specified. Raises an error if the head ratios are not provided. Calculates initial guesses assuming no heads are present, and then uses fsolve to determine the correct dimensions for the tank. Tested, but bugs and limitations are expected here. """ if self.L and (self.sideA_a or self.sideB_a): raise ValueError('Cannot specify head sizes when solving for V') if (self.D and self.L) or (self.D and self.L_over_D) or (self.L and self.L_over_D): raise ValueError('Only one of D, L, or L_over_D can be specified\ when solving for V') if ((self.sideA is not None and (self.sideA_a_ratio is None and self.sideA_a is None) and self.sideA != 'torispherical') or (self.sideB is not None and (self.sideB_a_ratio is None and self.sideB_a is None) and self.sideB != 'torispherical')): raise ValueError('When heads are specified, head parameter ratios are required') if self.D: # Iterate until L is appropriate solve_L = lambda L: self._V_solver_error(self.V, self.D, L, self.horizontal, self.sideA, self.sideB, self.sideA_a, self.sideB_a, self.sideA_f, self.sideA_k, self.sideB_f, self.sideB_k, self.sideA_a_ratio, self.sideB_a_ratio) Lguess = self.V/(pi/4*self.D**2) self.L = float(secant(solve_L, Lguess, xtol=1e-13)) elif self.L: # Iterate until D is appropriate solve_D = lambda D: self._V_solver_error(self.V, D, self.L, self.horizontal, self.sideA, self.sideB, self.sideA_a, self.sideB_a, self.sideA_f, self.sideA_k, self.sideB_f, self.sideB_k, self.sideA_a_ratio, self.sideB_a_ratio) Dguess = sqrt(4*self.V/pi/self.L) self.D = float(secant(solve_D, Dguess, xtol=1e-13)) else: # Use L_over_D until L and D are appropriate Lguess = (4*self.V*self.L_over_D**2/pi)**(1/3.) solve_L_D = lambda L: self._V_solver_error(self.V, L/self.L_over_D, L, self.horizontal, self.sideA, self.sideB, self.sideA_a, self.sideB_a, self.sideA_f, self.sideA_k, self.sideB_f, self.sideB_k, self.sideA_a_ratio, self.sideB_a_ratio) self.L = float(secant(solve_L_D, Lguess, xtol=1e-13)) self.D = self.L/self.L_over_D class HelicalCoil: r'''Class representing a helical coiled tube, as are found in many heated tanks and some small nuclear reactors. All parameters are also attributes. One set of the following parameters is required; inner tube diameter is optional. * Tube outer diameter, coil outer diameter, pitch, number of coil turns * Tube outer diameter, coil outer diameter, pitch, height * Tube outer diameter, coil outer diameter, number of coil turns, height Parameters ---------- Dt : float Outer diameter of the tube wound to make up the helical spiral, [m] Do : float Diameter of the spiral as measured from the center of the coil on one side to the center of the coil on the other side, [m] Do_total : float, optional Diameter of the spiral as measured from one edge of the tube to the other edge; equal to Do + Dt; either `Do` or `Do_total` may be specified and the other will be calculated [m] pitch : float, optional Height change from one coil to the next as measured from the middles of the tube, [m] H : float, optional Height of the spiral, as measured from the middle of the bottom of the tube to the middle of the top of the tube, [m] H_total : float, optional Height of the spiral as measured from one edge of the tube to the other edge; equal to `H_total` + `Dt`; either may be specified and the other will be calculated [m] N : float, optional Number of coil turns; may be specified along with `pitch` instead of specifying `H` or `H_total`, [-] Di : float, optional Inner diameter of the tube; if specified, inside and annulus properties will be calculated, [m] Attributes ---------- tube_circumference : float Circumference of the tube as measured though its center, not inner or outer edges; :math:`C = \pi D_o`, [m] tube_length : float Length of tube used to make the helical coil; :math:`L = \sqrt{(\pi D_o\cdot N)^2 + H^2}`, [m] surface_area : float Surface area of the outer surface of the helical coil; :math:`A_t = \pi D_t L`, [m^2] inner_surface_area : float Surface area of the inner surface of the helical coil; calculated if `Di` is supplied; :math:`A_{inside} = \pi D_i L`, [m^2] inlet_area : float Area of the inlet to the helical coil; calculated if `Di` is supplied; :math:`A_{inlet} = \frac{\pi}{4} D_i^2`, [m^2] inner_volume : float Volume of the tube as would be filled by a fluid, useful for weight calculations; calculated if `Di` is supplied; :math:`V_{inside} = A_i L`, [m^3] annulus_area : float Area of the annulus (wall of the pipe); calculated if `Di` is supplied; :math:`A_a = \frac{\pi}{4} (D_t^2 - D_i^2)`, [m^2] annulus_volume : float Volume of the annulus (wall of the pipe); calculated if `Di` is supplied, useful for weight calculations; :math:`V_a = A_a L`, [m^3] total_volume : float Total volume occupied by the pipe and the fluid inside it; :math:`V = D_t L`, [m^3] helix_angle : float Angle between the pitch and coil diameter; used in some calculations; :math:`\alpha = \arctan \left(\frac{p_t}{\pi D_o}\right)`, [radians] curvature : float Coil curvature, useful in some calculations; :math:`\delta = \frac{D_t}{D_o[1 + 4\pi^2 \tan^2(\alpha)]}`, [-] Notes ----- `Do` must be larger than `Dt`. Examples -------- >>> C1 = HelicalCoil(Do=30, H=20, pitch=5, Dt=2) >>> C1.N, C1.tube_length, C1.surface_area (4.0, 377.5212621504738, 2372.0360474917497) Same coil, with the inputs one would physically measure from the coil, and a specified inlet diameter: >>> C1 = HelicalCoil(Do_total=32, H_total=22, pitch=5, Dt=2, Di=1.8) >>> C1.N, C1.tube_length, C1.surface_area (4.0, 377.5212621504738, 2372.0360474917497) >>> C1.inner_surface_area, C1.inlet_area, C1.inner_volume, C1.total_volume, C1.annulus_volume (2134.832442742575, 2.5446900494077327, 960.6745992341587, 1186.0180237458749, 225.3434245117162) References ---------- .. [1] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' def __repr__(self): # pragma : no cover s = f' self.Do: raise ValueError('Tube diameter is larger than helix outer diameter - not feasible.') self.tube_circumference = pi*self.Do self.tube_length = sqrt((self.tube_circumference*self.N)**2 + self.H**2) self.surface_area = self.tube_length*pi*self.Dt #print(pi*self.tube_length*self.Dt) == surface_area self.helix_angle = atan(self.pitch/(pi*self.Do)) self.curvature = self.Dt/self.Do/(1. + 4*pi**2*tan(self.helix_angle)**2) #print(self.N*pi*self.Do/cos(self.helix_angle)) # Confirms the length with another formula self.total_inlet_area = pi/4.*self.Dt**2 self.total_volume = self.total_inlet_area*self.tube_length if Di is not None: self.Di = Di self.inner_surface_area = self.tube_length*pi*self.Di self.inlet_area = pi/4.*self.Di**2 self.inner_volume = self.inlet_area*self.tube_length self.annulus_area = self.total_inlet_area - self.inlet_area self.annulus_volume = self.total_volume - self.inner_volume def plate_enlargement_factor(amplitude, wavelength): r'''Calculates the enhancement factor of the sinusoidal waves of the plate heat exchanger. This is the multiplier for the flat plate area to obtain the actual area available for heat transfer. Obtained from the following integral: .. math:: \phi = \frac{\text{Effective area}}{\text{Projected area}} = \frac{\int_0^\lambda\sqrt{1 + \left(\frac{\gamma\pi}{2}\right)^2 \cos^2\left(\frac{2\pi}{\lambda}x\right)}dx}{\lambda} .. math:: \gamma = \frac{4a}{\lambda} The solution to the integral is: .. math:: \phi = \frac{2E\left(\frac{-4a^2\pi^2}{\lambda^2}\right)}{\pi} where E is the complete elliptic integral of the second kind, calculated with SciPy. Parameters ---------- amplitude : float Half the height of the wave of the ridges, [m] wavelength : float Distance between the bottoms of two of the ridges (sometimes called pitch), [m] Returns ------- plate_enlargement_factor : float The extra surface area multiplier as compared to a flat plate caused the corrugations, [-] Notes ----- This is the exact analytical integral, obtained via Mathematica, Maple, and quite a bit of trial and error. It is confirmed via numerical integration. The expression normally given is an approximation as follows: .. math:: \phi = \frac{1}{6}\left(1+\sqrt{1+A^2} + 4\sqrt{1+A^2/2}\right) .. math:: A = \frac{2\pi a}{\lambda} Most plate heat exchangers approximate a sinusoidal geometry only. Examples -------- >>> plate_enlargement_factor(amplitude=5E-4, wavelength=3.7E-3) 1.1611862034509677 ''' b = 2.*amplitude return 2.*float(ellipe(-b*b*pi*pi/(wavelength*wavelength)))/pi class PlateExchanger: r'''Class representing a plate heat exchanger with sinusoidal ridges. All parameters are also attributes. Parameters ---------- amplitude : float Half the height of the wave of the ridges, [m] wavelength : float Distance between the bottoms of two of the ridges (sometimes called pitch), [m] chevron_angle : float, optional Angle of the plate corrugations with respect to the vertical axis (the direction of flow if the plates were straight), between 0 and 90, [degrees] chevron_angles : tuple(2), optional Many plate exchangers use two alternating patterns; for those cases provide tuple of the two angles for that situation and the argument `chevron_angle` is ignored, [degrees] width : float, optional Width of the plates in the heat exchanger, between the gaskets, [m] length : float, optional Length of the heat exchanger as measured from one port to the other, excluding the diameter of the ports themselves (little useful heat transfer happens there), [m] thickness : float, optional Thickness of the metal making up the plates, [m] d_port : float, optional The diameter of the ports in the plates, [m] plates : int, optional The number of plates in the heat exchanger, including the two not used for heat transfer at the beginning and end [-] Attributes ---------- chevron_angles : tuple(2) The two specified angles (repeated value if only one specified), [degrees] chevron_angle : float The averaged angle of the chevrons, [degrees] inclination_angle : float 90 - `chevron_angle`, used in many publications instead of `chevron_angle`, [degrees] plate_corrugation_aspect_ratio : float The aspect ratio of the corrugations :math:`\gamma = \frac{4a}{\lambda}`, [-] plate_enlargement_factor : float The extra surface area multiplier as compared to a flat plate caused the corrugations, [-] D_eq : float Equivalent diameter of the channels, :math:`D_{eq} = 4a` [m] D_hydraulic : float Hydraulic diameter of the channels, :math:`D_{hyd} = \frac{4a}{\phi}` [m] length_port : float Port center to port center along the direction of flow, [m] A_plate_surface : float The surface area of one plate in the heat exchanger, including the extra due to corrugations (excluding the bit between the ports), :math:`A_p = L\cdot W\cdot \phi` [m^2] A_heat_transfer : float The total surface area available for heat transfer in the exchanger, the multiple of `A_plate_surface` by the number of plates after removing the two on the edges, [m^2] A_channel_flow : float The area for the fluid to flow in one channel, :math:`W\cdot b` [m^2] channels : int The number of plates minus one, [-] channels_per_fluid : int Half the number of total channels, [-] Notes ----- Only wavelength and amplitude are required as inputs to this function. Examples -------- >>> PlateExchanger(amplitude=5E-4, wavelength=3.7E-3, length=1.2, width=.3, ... d_port=.05, plates=51) 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 = '' else: s += '>' return s @property def plate_exchanger_identifier(self): """Method to create an identifying string in format 'L' + wavelength + 'A' + amplitude + 'B' + chevron angle-chevron angle. Wavelength and amplitude are specified in units of mm and rounded to two decimal places. """ wave_rounded = round(self.wavelength*1000, 2) amplitude_rounded = round(self.amplitude*1000, 2) a1 = self.chevron_angles[0] a2 = self.chevron_angles[1] s = (f'L{wave_rounded}A{amplitude_rounded}B{a1}-{a2}') return s def __init__(self, amplitude, wavelength, chevron_angle=45, chevron_angles=None, width=None, length=None, thickness=None, d_port=None, plates=None): self.amplitude = self.a = amplitude # half a sine wave's height self.b = 2*self.amplitude # Used in some models. From a flat plate, a press goes down this far into the plate. Also called the hot and cold gap self.wavelength = self.pitch = wavelength # self.lambda if chevron_angles is not None: self.chevron_angles = chevron_angles self.chevron_angle = self.beta = 0.5*(chevron_angles[0] + chevron_angles[1]) else: self.chevron_angle = self.beta = chevron_angle # between 0 and 90 self.chevron_angles = (chevron_angle, chevron_angle) self.inclination_angle = 90 - self.chevron_angle # Used in some definitions instead self.plate_corrugation_aspect_ratio = self.gamma = 4*self.a/self.wavelength self.plate_enlargement_factor = plate_enlargement_factor(self.amplitude, self.wavelength) self.D_eq = 4*self.amplitude # Equivalent diameter for inter-plate spacing self.D_hydraulic = 4*self.amplitude/self.plate_enlargement_factor # Get better results when correlations use this if width is not None: self.width = width if length is not None: self.length = length if thickness is not None: self.thickness = thickness if d_port is not None: self.d_port = d_port if plates is not None: self.plates = plates if d_port is not None and length is not None: self.length_port = length + d_port # port center to port center along the direction of flow # There is another larger length as well, including both port diameters if width is not None and length is not None: self.A_plate_surface = length*width*self.plate_enlargement_factor # use this in Q = UAdT if plates is not None: self.A_heat_transfer = (plates-2)*self.A_plate_surface # the two outermost sides aren't used if width is not None: self.A_channel_flow = self.width*self.b # Use this to get G, kg/s/m^2 if plates is not None: self.channels = self.plates - 1 self.channels_per_fluid = 0.5*self.channels class RectangularFinExchanger: r'''Class representing a plate-fin heat exchanger with straight rectangular fins. All parameters are also attributes. Parameters ---------- fin_height : float The total distance between the two metal plates sandwiching the fins and holding them together (abbreviated `h`), [m] fin_thickness : float The thickness of the material the fins were formed from (abbreviated `t`), [m] fin_spacing : float The unit cell spacing from one fin to the next; the space between the sides of two fins plus one thickness (abbreviated `s`), [m] length : float, optional The total length of the flow passage of the plate-fin exchanger (abbreviated `L`), [m] width : float, optional The total width of the space the fins are in; this is also :math:`N_{fins}\times s` (abbreviated `W`), [m] layers : int, optional The number of layers in the plate-fin exchanger; note these HX almost always single-pass only, [-] plate_thickness : float, optional The thickness of the metal separator between layers, [m] flow : str, optional One of 'counterflow', 'crossflow', or 'parallelflow' Attributes ---------- channel_height : float The height of the channel the fluid flows in :math:`\text{channel height } = \text{fin height} - \text{fin thickness}`, [m] channel_width : float The width of the channel the fluid flows in :math:`\text{channel width } = \text{fin spacing} - \text{fin thickness}`, [m] fin_count : int The number of fins per unit length of the layer, :math:`\text{fin count} = \frac{1}{\text{fin spacing}}`, [1/m] blockage_ratio : float The fraction of the layer which is blocked to flow by the fins, :math:`\text{blockage ratio} = \frac{s\cdot h - s\cdot t - t(h-t)}{s\cdot h}`, [m] A_channel : float Flow area of a single channel in a single layer, :math:`\text{channel area} = (s-t)(h-t)`, [m] P_channel : float Wetted perimeter of a single channel in a single layer, :math:`\text{channel perimeter} = 2(s-t) + 2(h-t)`, [m] Dh : float Hydraulic diameter of a single channel in a single layer, :math:`D_{hydraulic} = \frac{4 A_{channel}}{P_{channel}}`, [m] layer_thickness : float The thickness of a single layer - the sum of a fin height and a plate thickness, [m] layer_fin_count : int The number of fins in a layer; rounded to the nearest whole fin, [-] A_HX_layer : float The surface area including fins for heat transfer in one layer of the HX, [m^2] A_HX : float The total surface area of the heat exchanger with all layers combined, [m^2] height : float The height of all the layers of the heat exchanger combined, plus one extra plate thickness, [m] volume : float The product of the height, width, and length of the HX, [m^3] A_specific_HX : float The specific surface area of the heat exchanger - square meters per meter cubed, [m^3] Notes ----- The only required parameters are the fin geometry itself; `fin_height`, `fin_thickness`, and `fin_spacing`. Examples -------- >>> PFE = RectangularFinExchanger(0.03, 0.001, 0.012) >>> PFE.Dh 0.01595 References ---------- .. [1] Yang, Yujie, and Yanzhong Li. "General Prediction of the Thermal Hydraulic Performance for Plate-Fin Heat Exchanger with Offset Strip Fins." International Journal of Heat and Mass Transfer 78 (November 1, 2014): 860-70. doi:10.1016/j.ijheatmasstransfer.2014.07.060. .. [2] Sheik Ismail, L., R. Velraj, and C. Ranganayakulu. "Studies on Pumping Power in Terms of Pressure Drop and Heat Transfer Characteristics of Compact Plate-Fin Heat Exchangers-A Review." Renewable and Sustainable Energy Reviews 14, no. 1 (January 2010): 478-85. doi:10.1016/j.rser.2009.06.033. ''' def __init__(self, fin_height, fin_thickness, fin_spacing, length=None, width=None, layers=None, plate_thickness=None, flow='crossflow'): self.h = self.fin_height = fin_height # including 2x thickness self.t = self.fin_thickness = fin_thickness self.s = self.fin_spacing = fin_spacing self.L = self.length = length self.W = self.width = width self.layers = layers self.flow = flow self.plate_thickness = plate_thickness self.channel_height = self.fin_height - self.fin_thickness self.channel_width = self.fin_spacing - self.fin_thickness self.fin_count = 1./self.fin_spacing self.blockage_ratio = (self.s*self.h - self.s*self.t - (self.h-self.t)*self.t)/(self.s*self.h) self.A_channel = (self.s-self.t)*(self.h-self.t) self.P_channel = 2*(self.s-self.t) + 2*(self.h-self.t) self.Dh = 4*self.A_channel/self.P_channel self.set_overall_geometry() 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 class RectangularOffsetStripFinExchanger(RectangularFinExchanger): def __init__(self, fin_length, fin_height, fin_thickness, fin_spacing, length=None, width=None, layers=None, plate_thickness=None, flow='crossflow'): self.l = self.fin_length = fin_length self.h = self.fin_height = fin_height self.t = self.fin_thickness = fin_thickness self.s = self.fin_spacing = fin_spacing self.blockage_ratio = self.omega = 2*self.t/self.s*(1. - self.t/self.h) + self.t/self.h*(1 - 2*self.t/self.s) # Kim blockage ratio beta self.blockage_ratio_Kim = self.t/self.h + self.t/self.s - self.t**2/(self.h*self.s) # Definitions as in the paper with the most common correlation self.alpha = self.s/self.h # "General prediction" uses t/h here self.delta = self.t/self.l self.gamma = self.t/self.s # free flow area self.A_channel = (self.h - self.t)*(self.s - self.t) self.A = 2.*(self.l*(self.h-self.t) + self.l*(self.s-self.t) + self.t*(self.h-self.t)) + self.t*(self.s-2*self.t) self.Dh = 4.*self.l*self.A_channel/self.A # not the standard definition self.P_channel = 2*(self.s-self.t) + 2*(self.h-self.t) self.Dh_Kays_London = 4*self.A_channel/(2*(self.h -self.t)+ 2*(self.s -self.t)) # Does not consider the fronts of backs of the fins, only the 2d shape self.Dh_Joshi_Webb = 2*self.l*(self.h - self.t)*(self.s - 2*self.t)/(self.l*(self.h-self.t) + self.l*(self.s - self.t) + self.t*(self.h - self.t)) self.L = self.length = length self.W = self.width = width self.layers = layers self.flow = flow self.plate_thickness = plate_thickness self.fin_count = 1./self.fin_spacing self.set_overall_geometry() class HyperbolicCoolingTower: r'''Class representing the geometry of a hyperbolic cooling tower, as used in many industries especially the poewr industry. All parameters are also attributes. `H_inlet`, `D_outlet`, and `H_outlet` are always required. Additionally, one set of the following parameters is required; `H_support`, `D_support`, `n_support`, and `inlet_rounding` are all optional as well. * Inlet diameter * Inlet diameter and throat diameter * Inlet diameter and throat height * Inlet diameter, throat diameter, and throat height * Base diameter, throat diameter, and throat height If the inlet diameter is provided but the throat diameter and/or the throat height are missing, two heuristics are used to estimate them (to avoid these heuristics simply specify the values): * Assume the throat elevation is 2/3 the elevation of the tower. * Assume the throat diameter is 63% the diameter of the inlet. Parameters ---------- H_inlet : float Height of the inlet zone of the cooling tower (also called rain zone), [m] D_outlet : float The inside diameter of the cooling tower outlet (top of the tower; the elevation the concrete section ends), [m] H_outlet : float The height of the cooling tower outlet (top of the tower;the elevation the concrete section ends), [m] D_inlet : float, optional The inside diameter of the cooling tower inlet at the elevation the concrete section begins, [m] D_base : float, optional The diameter of the cooling tower at the very base of the tower (the bottom of the inlet zone, at the elevation of the ground), [m] D_throat : float, optional The diameter of the cooling tower at its minimum section, called its throat; where the two hyperbolas meet, [m] h_throat : float, optional The elevation of the cooling tower's throat (its minimum section; where the two hyperbolas meet), [m] inlet_rounding : float, optional Radius of an optional rounded protrusion from the lip of the cooling tower shell base, which curves upwards from the lip (used to reduce the dead zone area rather than having a flat lip), [m] H_support : float, optional The height of each support column, [m] D_support : float, optional The diameter of each support column, [m] n_support : int, optional The number of support columns of the cooling tower, [m] Attributes ---------- b_lower : float The `b` parameter in the hyperbolic equation for the lower section of the cooling tower, [m] b_upper : float The `b` parameter in the hyperbolic equation for the upper section of the cooling tower, [m] Notes ----- Note there are two hyperbolas in a hyperbolic cooling tower - one under the throat and one above it; they are not necessarily the same. A hyperbolic cooling tower is not the absolute optimal design, but is is close. The optimality is determined by the amount of material required to build it while maintaining its rigidity. For thermal design purposes, a hyperbolic model covers any minor variation quite well. Examples -------- >>> ct = HyperbolicCoolingTower(D_outlet=89.0, H_outlet=200, D_inlet=136.18, H_inlet=14.5) >>> ct >>> 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 = """""" s = s%(self.D_inlet, self.D_outlet, self.H_inlet, self.H_outlet, self.D_throat, self.H_throat, self.D_base) return s def __init__(self, H_inlet, D_outlet, H_outlet, D_inlet=None, D_base=None, D_throat=None, H_throat=None, H_support=None, D_support=None, n_support=None, inlet_rounding=None): self.D_outlet = D_outlet self.H_inlet = H_inlet self.H_outlet = H_outlet if H_throat is None: H_throat = 2/3.0*H_outlet self.H_throat = H_throat if D_throat is None: if D_inlet is not None: D_throat = 0.63*D_inlet else: raise ValueError('Provide either `D_throat`, or `D_inlet` so it may be estimated.') self.D_throat = D_throat if D_inlet is None and D_base is None: raise ValueError('Need `D_inlet` or `D_base`') if D_base is not None: b = self.D_throat*self.H_throat/sqrt(D_base**2 - self.D_throat**2) D_inlet = 2*self.D_throat*sqrt((self.H_throat-H_inlet)**2 + b**2)/(2*b) elif D_inlet is not None: b = self.D_throat*(self.H_throat-H_inlet)/sqrt(D_inlet**2 - self.D_throat**2) D_base = 2*self.D_throat*sqrt(self.H_throat**2 + b**2)/(2*b) self.D_inlet = D_inlet self.D_base = D_base self.b_lower = b # Upper b parameter self.b_upper = self.D_throat*(self.H_outlet - self.H_throat)/sqrt((self.D_outlet)**2 - self.D_throat**2) # May or may not be specified self.H_support = H_support self.D_support = D_support self.n_support = n_support self.inlet_rounding = inlet_rounding def plot(self, pts=100): # pragma: no cover import matplotlib.pyplot as plt Zs = linspace(0, self.H_outlet, pts) Rs = [self.diameter(Z)*0.5 for Z in Zs] plt.plot(Zs, Rs) plt.plot(Zs, [-v for v in Rs]) plt.show() def diameter(self, H): r'''Calculates cooling tower diameter at a specified height, using the formulas for either hyperbola, depending on the height specified. .. math:: D = D_{throat}\frac{\sqrt{H^2 + b^2}}{b} The value of `H` and `b` used in the above equation is as follows: * `H_throat` - H and `b_lower` if under the throat * `H` - `H_throat` and `b_upper`, if above the throat Parameters ---------- H : float Height at which to calculate the cooling tower diameter, [m] Returns ------- D : float Diameter of the cooling tower at the specified height, [m] ''' # Compute the diameter at H if H <= self.H_throat: # Height relative to throat height H = self.H_throat - H b = self.b_lower else: H = H - self.H_throat b = self.b_upper R = self.D_throat*sqrt(H*H + b*b)/(2.0*b) return R*2.0 class AirCooledExchanger: r'''Class representing the geometry of an air cooled heat exchanger with one or more tube bays, fans, or bundles. All parameters are also attributes. The minimum information required to describe an air cooler is as follows: * `tube_rows` * `tube_passes` * `tubes_per_row` * `tube_length` * `tube_diameter` * `fin_thickness` Two of `angle`, `pitch`, `pitch_parallel`, and `pitch_normal` (`pitch_ratio` may take the place of `pitch`) Either `fin_diameter` or `fin_height`. Either `fin_density` or `fin_interval`. Parameters ---------- tube_rows : int Number of tube rows per bundle, [-] tube_passes : int Number of tube passes (times the fluid travels across one tube length), [-] tubes_per_row : float Number of tubes per row per bundle, [-] tube_length : float Total length of the tube bundle tubes, [m] tube_diameter : float Diameter of the bare tube, [m] fin_thickness : float Thickness of the fins, [m] angle : float, optional Angle of the tube layout, [degrees] pitch : float, optional Shortest distance between tube centers; defined in relation to the flow direction only, [m] pitch_parallel : float, optional Distance between tube center along a line parallel to the flow; has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m] pitch_normal : float, optional Distance between tube centers in a line 90° to the line of flow; has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m] pitch_ratio : float, optional Ratio of the pitch to bare tube diameter, [-] fin_diameter : float, optional Outer diameter of each tube after including the fin on both sides, [m] fin_height : float, optional Height above bare tube of the tube fins, [m] fin_density : float, optional Number of fins per meter of tube, [1/m] fin_interval : float, optional Space between each fin, including the thickness of one fin at its base, [m] parallel_bays : int, optional Number of bays in the unit, [-] bundles_per_bay : int, optional Number of tube bundles per bay, [-] fans_per_bay : int, optional Number of fans per bay, [-] corbels : bool, optional Whether or not the air cooler has corbels, which increase the air velocity by adding half a tube to the sides for the case of non-rectangular tube layouts, [-] tube_thickness : float, optional Thickness of the bare metal tubes, [m] fan_diameter : float, optional Diameter of air cooler fan, [m] Attributes ---------- bare_length : float Length of bare tube between two fins :math:`\text{bare length} = \text{fin interval} - t_{fin}`, [m] tubes_per_bundle : float Total number of tubes per bundle :math:`N_{tubes/bundle} = N_{tubes/row} \cdot N_{rows}`, [-] tubes_per_bay : float Total number of tubes per bay :math:`N_{tubes/bay} = N_{tubes/bundle} \cdot N_{bundles/bay}`, [-] tubes : float Total number of tubes in all bundles in all bays combined :math:`N_{tubes} = N_{tubes/bay} \cdot N_{bays}`, [-] pitch_diagonal : float Distance between tube centers in a diagonal line between one normal tube and one parallel tube; :math:`s_D = \left[s_L^2 + \left(\frac{s_T}{2}\right)^2\right]^{0.5}`, [m] A_bare_tube_per_tube : float Area of the bare tube including the portion hidden by the fin per tube :math:`A_{bare,total/tube} = \pi D_{tube} L_{tube}`, [m^2] A_bare_tube_per_row : float Area of the bare tube including the portion hidden by the fin per tube row :math:`A_{bare,total/row} = \pi D_{tube} L_{tube} N_{tubes/row}`, [m^2] A_bare_tube_per_bundle : float Area of the bare tube including the portion hidden by the fin per bundle :math:`A_{bare,total/bundle} = \pi D_{tube} L_{tube} N_{tubes/bundle}`, [m^2] A_bare_tube_per_bay : float Area of the bare tube including the portion hidden by the fin per bay :math:`A_{bare,total/bay} = \pi D_{tube} L_{tube} N_{tubes/bay}`, [m^2] A_bare_tube : float Area of the bare tube including the portion hidden by the fin per in all bundles and bays combined :math:`A_{bare,total} = \pi D_{tube} L_{tube} N_{tubes}`, [m^2] A_tube_showing_per_tube : float Area of the bare tube which is exposed per tube :math:`A_{bare, showing/tube} = \pi D_{tube} L_{tube} \left(1 - \frac{t_{fin}} {\text{fin interval}} \right)`, [m^2] A_tube_showing_per_row : float Area of the bare tube which is exposed per tube row, [m^2] A_tube_showing_per_bundle : float Area of the bare tube which is exposed per bundle, [m^2] A_tube_showing_per_bay : float Area of the bare tube which is exposed per bay, [m^2] A_tube_showing : float Area of the bare tube which is exposed in all bundles and bays combined, [m^2] A_per_fin : float Surface area per fin :math:`A_{fin} = 2 \frac{\pi}{4} (D_{fin}^2 - D_{tube}^2) + \pi D_{fin} t_{fin}`, [m^2] A_fin_per_tube : float Surface area of all fins per tube :math:`A_{fin/tube} = N_{fins/m} L_{tube} A_{fin}`, [m^2] A_fin_per_row : float Surface area of all fins per row, [m^2] A_fin_per_bundle : float Surface area of all fins per bundle, [m^2] A_fin_per_bay : float Surface area of all fins per bay, [m^2] A_fin : float Surface area of all fins in all bundles and bays combined, [m^2] A_per_tube : float Surface area of combined finned and non-fined area exposed for heat transfer per tube :math:`A_{tube} = A_{bare, showing/tube} + A_{fin/tube}`, [m^2] A_per_row : float Surface area of combined finned and non-finned area exposed for heat transfer per tube row, [m^2] A_per_bundle : float Surface area of combined finned and non-finned area exposed for heat transfer per tube bundle, [m^2] A_per_bay : float Surface area of combined finned and non-finned area exposed for heat transfer per bay, [m^2] A : float Surface area of combined finned and non-finned area exposed for heat transfer in all bundles and bays combined, [m^2] A_increase : float Ratio of actual surface area to bare tube surface area :math:`A_{increase} = \frac{A_{tube}}{A_{bare, total/tube}}`, [-] A_tube_flow : float The area for the fluid to flow in one tube, :math:`\pi/4\cdot D_i^2`, [m^2] A_tube_flow_per_row : float The area for the fluid to flow in one row, :math:`\pi/4\cdot D_i^2 N_{tubes/row}`, [m^2] A_tube_flow_per_bundle : float The area for the fluid to flow in one bundle, :math:`\pi/4\cdot D_i^2 N_{tubes/bundle}`, [m^2] A_tube_flow_per_bay : float The area for the fluid to flow in one bay, :math:`\pi/4\cdot D_i^2 N_{tubes/bay}`, [m^2] A_tube_flow_total : float The area for the fluid to flow in all tubes combined, as if there were a single pass [m^2] channels : int The number of tubes the fluid flows through at the inlet header, [-] tube_volume_per_tube : float Fluid volume per tube inside :math:`V_{tube, flow} = \frac{\pi}{4} D_{i}^2 L_{tube}`, [m^3] tube_volume_per_row : float Fluid volume of tubes per row, [m^3] tube_volume_per_bundle : float Fluid volume of tubes per bundle, [m^3] tube_volume_per_bay : float Fluid volume of tubes per bay, [m^3] tube_volume : float Fluid volume of tubes in all bundles and bays combined, [m^3] A_diagonal_per_bundle : float Air flow area along the diagonal plane per bundle :math:`A_d = 2 N_{tubes/row} L_{tube} (P_d - D_{tube} - 2 N_{fins/m} h_{fin} t_{fin}) + A_\text{extra,side}`, [m^2] A_normal_per_bundle : float Air flow area along the normal (transverse) plane; this is normally the minimum flow area, except for some staggered configurations :math:`A_t = N_{tubes/row} L_{tube} (P_t - D_{tube} - 2 N_{fins/m} h_{fin} t_{fin}) + A_\text{extra,side}`, [m^2] A_min_per_bundle : float Minimum air flow area per bundle; this is the characteristic area for velocity calculation in most finned tube convection correlations :math:`A_{min} = min(A_d, A_t)`, [m^2] A_min_per_bay : float Minimum air flow area per bay, [m^2] A_min : float Minimum air flow area, [m^2] A_face_per_bundle : float Face area per bundle :math:`A_{face} = P_{T} (1+N_{tubes/row}) L_{tube}`; if corbels are used, add 0.5 to tubes/row instead of 1, [m^2] A_face_per_bay : float Face area per bay, [m^2] A_face : float Total face area, [m^2] flow_area_contraction_ratio : float Ratio of `A_min` to `A_face`, [-] Notes ----- Examples -------- >>> from scipy.constants import inch >>> AC = AirCooledExchanger(tube_rows=4, tube_passes=4, tubes_per_row=56, tube_length=10.9728, ... tube_diameter=1*inch, fin_thickness=0.013*inch, fin_density=10/inch, ... angle=30, pitch=2.5*inch, fin_height=0.625*inch, tube_thickness=0.00338, ... bundles_per_bay=2, parallel_bays=3, corbels=True) References ---------- .. [1] Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983. ''' def __repr__(self): attributes = ', '.join(f"{slot}={getattr(self, slot)!r}" for slot in self.model_inputs) return f"{self.__class__.__name__}({attributes})" def __hash__(self): return hash(tuple(getattr(self, slot) for slot in self.model_inputs)) # pitch_ratio is skipped in favor of pitch; model_inputs = ('tube_rows', 'tube_passes', 'tubes_per_row', 'tube_length', 'tube_diameter', 'fin_thickness', 'angle', 'pitch', 'pitch_parallel', 'pitch_normal', 'fin_diameter', 'fin_height', 'fin_density', 'fin_interval', 'parallel_bays', 'bundles_per_bay', 'fans_per_bay', 'corbels', 'tube_thickness', 'fan_diameter') def __init__(self, tube_rows, tube_passes, tubes_per_row, tube_length, tube_diameter, fin_thickness, angle=None, pitch=None, pitch_parallel=None, pitch_normal=None, pitch_ratio=None, fin_diameter=None, fin_height=None, fin_density=None, fin_interval=None, parallel_bays=1, bundles_per_bay=1, fans_per_bay=1, corbels=False, tube_thickness=None, fan_diameter=None): # TODO: fin types self.tube_rows = tube_rows self.tube_passes = tube_passes self.tubes_per_row = tubes_per_row self.tube_length = tube_length self.tube_diameter = tube_diameter self.fin_thickness = fin_thickness self.fan_diameter = fan_diameter if pitch_ratio is not None: if pitch is not None: pitch = self.tube_diameter*pitch_ratio else: raise ValueError('Specify only one of `pitch_ratio` or `pitch`') angle, pitch, pitch_parallel, pitch_normal = pitch_angle_solver( angle=angle, pitch=pitch, pitch_parallel=pitch_parallel, pitch_normal=pitch_normal) self.angle = angle self.pitch = pitch self.pitch_ratio = pitch/self.tube_diameter self.pitch_parallel = pitch_parallel self.pitch_normal = pitch_normal self.pitch_diagonal = sqrt(pitch_parallel**2 + (0.5*pitch_normal)**2) if fin_diameter is None and fin_height is None: raise ValueError('Specify only one of `fin_diameter` or `fin_height`') elif fin_diameter is not None: fin_height = 0.5*(fin_diameter - tube_diameter) elif fin_height is not None: fin_diameter = tube_diameter + 2.0*fin_height self.fin_height = fin_height self.fin_diameter = fin_diameter if fin_density is None and fin_interval is None: raise ValueError('Specify only one of `fin_density` or `fin_interval`') elif fin_density is not None: fin_interval = 1.0/fin_density elif fin_interval is not None: fin_density = 1.0/fin_interval self.fin_interval = fin_interval self.fin_density = fin_density self.parallel_bays = parallel_bays self.bundles_per_bay = bundles_per_bay self.fans_per_bay = fans_per_bay self.corbels = corbels self.tube_thickness = tube_thickness if self.fin_interval: self.bare_length = self.fin_interval - self.fin_thickness else: self.bare_length = None self.tubes_per_bundle = self.tubes_per_row*self.tube_rows self.tubes_per_bay = self.tubes_per_bundle*self.bundles_per_bay self.tubes = self.tubes_per_bay*self.parallel_bays self.A_bare_tube_per_tube = pi*self.tube_diameter*self.tube_length self.A_bare_tube_per_row = self.A_bare_tube_per_tube*self.tubes_per_row self.A_bare_tube_per_bundle = self.A_bare_tube_per_tube*self.tubes_per_bundle self.A_bare_tube_per_bay = self.A_bare_tube_per_tube*self.tubes_per_bay self.A_bare_tube = self.A_bare_tube_per_tube*self.tubes self.A_tube_showing_per_tube = pi*self.tube_diameter*self.tube_length*(1.0 - self.fin_thickness/self.fin_interval) self.A_tube_showing_per_row = self.A_tube_showing_per_tube*self.tubes_per_row self.A_tube_showing_per_bundle = self.A_tube_showing_per_tube*self.tubes_per_bundle self.A_tube_showing_per_bay = self.A_tube_showing_per_tube*self.tubes_per_bay self.A_tube_showing = self.A_tube_showing_per_tube*self.tubes self.A_per_fin = (2.0*pi/4.0*(self.fin_diameter**2 - self.tube_diameter**2) + pi*self.fin_diameter*self.fin_thickness) # pi*D*L(fin) self.A_fin_per_tube = self.fin_density*self.tube_length*self.A_per_fin self.A_fin_per_row = self.A_fin_per_tube*self.tubes_per_row self.A_fin_per_bundle = self.A_fin_per_tube*self.tubes_per_bundle self.A_fin_per_bay = self.A_fin_per_tube*self.tubes_per_bay self.A_fin = self.A_fin_per_tube*self.tubes self.A_per_tube = self.A_tube_showing_per_tube + self.A_fin_per_tube self.A_per_row = self.A_tube_showing_per_row + self.A_fin_per_row self.A_per_bundle = self.A_tube_showing_per_bundle + self.A_fin_per_bundle self.A_per_bay = self.A_tube_showing_per_bay + self.A_fin_per_bay self.A = self.A_tube_showing + self.A_fin self.A_increase = self.A/self.A_bare_tube # TODO A_extra could be calculated based on a fixed width and height of the bay A_extra = 0.0 self.A_diagonal_per_bundle = 2.0*self.tubes_per_row*self.tube_length*(self.pitch_diagonal - self.tube_diameter - 2.0*fin_density*self.fin_height*self.fin_thickness) + A_extra self.A_normal_per_bundle = self.tubes_per_row*self.tube_length*(self.pitch_normal - self.tube_diameter - 2.0*fin_density*self.fin_height*self.fin_thickness) + A_extra self.A_min_per_bundle = min(self.A_diagonal_per_bundle, self.A_normal_per_bundle) self.A_min_per_bay = self.A_min_per_bundle*self.bundles_per_bay self.A_min = self.A_min_per_bay*self.parallel_bays i = 0.5 if self.corbels else 1.0 self.A_face_per_bundle = self.pitch_normal*self.tube_length*(self.tubes_per_row + i) self.A_face_per_bay = self.A_face_per_bundle*self.bundles_per_bay self.A_face = self.A_face_per_bay*self.parallel_bays self.flow_area_contraction_ratio = self.A_min/self.A_face if self.tube_thickness is not None: self.Di = self.tube_diameter - self.tube_thickness*2.0 self.A_tube_flow = pi/4.0*self.Di*self.Di self.A_tube_flow_per_row = self.A_tube_flow*self.tubes_per_row self.A_tube_flow_per_bundle = self.A_tube_flow*self.tubes_per_bundle self.A_tube_flow_per_bay = self.A_tube_flow*self.tubes_per_bay self.A_tube_flow_total = self.A_tube_flow*self.tubes 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' def pitch_angle_solver(angle=None, pitch=None, pitch_parallel=None, pitch_normal=None): r'''Utility to take any two of `angle`, `pitch`, `pitch_parallel`, and `pitch_normal` and calculate the other two. This is useful for applications with tube banks, as in shell and tube heat exchangers or air coolers and allows for a wider range of user input. .. math:: \text{pitch normal} = \text{pitch} \cdot \sin(\text{angle}) .. math:: \text{pitch parallel} = \text{pitch} \cdot \cos(\text{angle}) Parameters ---------- angle : float, optional The angle of the tube layout, [degrees] pitch : float, optional The shortest distance between tube centers; defined in relation to the flow direction only, [m] pitch_parallel : float, optional The distance between tube center along a line parallel to the flow; has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m] pitch_normal : float, optional The distance between tube centers in a line 90° to the line of flow; has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m] Returns ------- angle : float The angle of the tube layout, [degrees] pitch : float The shortest distance between tube centers; defined in relation to the flow direction only, [m] pitch_parallel : float The distance between tube center along a line parallel to the flow; has been called `longitudinal` pitch, `pp`, `s2`, `SL`, and `p2`, [m] pitch_normal : float The distance between tube centers in a line 90° to the line of flow; has been called the `transverse` pitch, `pn`, `s1`, `ST`, and `p1`, [m] Notes ----- For the 90 and 0 degree case, the normal or parallel pitches can be zero; given the angle and the zero value, obviously is it not possible to calculate the pitch and a math error will be raised. No exception will be raised if three or four inputs are provided; the other two will simply be calculated according to the list of if statements used. An exception will be raised if only one input is provided. Examples -------- >>> pitch_angle_solver(pitch=1, angle=30) (30, 1, 0.8660254037844387, 0.49999999999999994) References ---------- .. [1] Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983. ''' if angle is not None and pitch is not None: pitch_normal = pitch*sin(radians(angle)) pitch_parallel = pitch*cos(radians(angle)) elif angle is not None and pitch_normal is not None: pitch = pitch_normal/sin(radians(angle)) pitch_parallel = pitch*cos(radians(angle)) elif angle is not None and pitch_parallel is not None: pitch = pitch_parallel/cos(radians(angle)) pitch_normal = pitch*sin(radians(angle)) elif pitch_normal is not None and pitch is not None: angle = degrees(asin(pitch_normal/pitch)) pitch_parallel = pitch*cos(radians(angle)) elif pitch_parallel is not None and pitch is not None: angle = degrees(acos(pitch_parallel/pitch)) pitch_normal = pitch*sin(radians(angle)) elif pitch_parallel is not None and pitch_normal is not None: angle = degrees(asin(pitch_normal/sqrt(pitch_normal**2 + pitch_parallel**2))) pitch = sqrt(pitch_normal**2 + pitch_parallel**2) else: raise ValueError('Two of the arguments are required') return angle, pitch, pitch_parallel, pitch_normal 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 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 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 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 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 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 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 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 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