# -*- coding: utf-8 -*- """Chemical Engineering Design Library (ChEDL). Utilities for process modeling. Copyright (C) 2016, 2017, 2018, 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 rating and designing vapor-liquid separators. 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: Functions --------- .. autofunction :: v_Sounders_Brown .. autofunction :: K_separator_Watkins .. autofunction :: K_separator_demister_York .. autofunction :: K_Sounders_Brown_theoretical """ from __future__ import division from math import log, exp, sqrt from fluids.constants import g, foot, psi from fluids.numerics import splev, implementation_optimize_tck __all__ = ['v_Sounders_Brown', 'K_separator_Watkins', 'K_separator_demister_York', 'K_Sounders_Brown_theoretical'] # 92 points taken from a 2172x3212 page scan, after dewarping the scan, # digitization with Engauge Digitizer, and extensive checking; every 5th point # it produced was selected plus the last point. The initial value is adjusted # to be the lower limit of the graph. tck_Watkins = implementation_optimize_tck([[-5.115995809754082, -5.115995809754082, -5.115995809754082, -5.115995809754082, -4.160106231099973, -3.209113630523477, -1.2175106961204154, 0.4587657198189318, 1.1197669427405068, 1.6925908552310418, 1.6925908552310418, 1.6925908552310418, 1.6925908552310418], [-1.4404286048266364, -1.2375168139385286, -0.9072614905522024, -0.7662335745829165, -0.944537665617708, -1.957339717378027, -3.002614318094637, -3.5936804378352956, -3.8779153181940553, 0.0, 0.0, 0.0, 0.0], 3]) def K_separator_Watkins(x, rhol, rhog, horizontal=False, method='spline'): r'''Calculates the Sounders-Brown `K` factor as used in determining maximum allowable gas velocity in a two-phase separator in either a horizontal or vertical orientation. This function approximates a graph published in [1]_ to determine `K` as used in the following equation: .. math:: v_{max} = K_{SB}\sqrt{\frac{\rho_l-\rho_g}{\rho_g}} The graph has `K_{SB}` on its y-axis, and the following as its x-axis: .. math:: \frac{m_l}{m_g}\sqrt{\rho_g/\rho_l} = \frac{(1-x)}{x}\sqrt{\rho_g/\rho_l} Cubic spline interpolation is the default method of retrieving a value from the graph, which was digitized with Engauge-Digitizer. Also supported are two published curve fits to the graph. The first is that of Blackwell (1984) [2]_, as follows: .. math:: K_{SB} = \exp(-1.942936 -0.814894X -0.179390 X^2 -0.0123790 X^3 + 0.000386235 X^4 + 0.000259550 X^5) X = \ln\left[\frac{(1-x)}{x}\sqrt{\rho_g/\rho_l}\right] The second is that of Branan (1999), as follows: .. math:: K_{SB} = \exp(-1.877478097 -0.81145804597X -0.1870744085 X^2 -0.0145228667 X^3 -0.00101148518 X^4) X = \ln\left[\frac{(1-x)}{x}\sqrt{\rho_g/\rho_l}\right] Parameters ---------- x : float Quality of fluid entering separator, [-] rhol : float Density of liquid phase [kg/m^3] rhog : float Density of gas phase [kg/m^3] horizontal : bool, optional Whether to use the vertical or horizontal value; horizontal is 1.25 higher method : str One of 'spline, 'blackwell', or 'branan' Returns ------- K : float Sounders Brown horizontal or vertical `K` factor for two-phase separator design only, [m/s] Notes ----- Both the 'branan' and 'blackwell' models are used frequently. However, the spline is much more accurate. No limits checking is enforced. However, the x-axis spans only 0.006 to 5.4, and the function should not be used outside those limits. Examples -------- >>> K_separator_Watkins(0.88, 985.4, 1.3, horizontal=True) 0.07951613600476297 References ---------- .. [1] Watkins (1967). Sizing Separators and Accumulators, Hydrocarbon Processing, November 1967. .. [2] Blackwell, W. Wayne. Chemical Process Design on a Programmable Calculator. New York: Mcgraw-Hill, 1984. .. [3] Branan, Carl R. Pocket Guide to Chemical Engineering. 1st edition. Houston, Tex: Gulf Professional Publishing, 1999. ''' factor = (1. - x)/x*sqrt(rhog/rhol) if method == 'spline': K = exp(float(splev(log(factor), tck_Watkins))) elif method == 'blackwell': X = log(factor) A = -1.877478097 B = -0.81145804597 C = -0.1870744085 D = -0.0145228667 E = -0.00101148518 K = exp(A + X*(B + X*(C + X*(D + E*X)))) elif method == 'branan': X = log(factor) A = -1.942936 B = -0.814894 C = -0.179390 D = -0.0123790 E = 0.000386235 F = 0.000259550 K = exp(A + X*(B + X*(C + X*(D + X*(E + F*X))))) else: raise ValueError("Only methods 'spline', 'branan', and 'blackwell' are supported.") K *= foot # Converts units of ft/s to m/s; the graph and all fits are in ft/s if horizontal: K *= 1.25 # Watkins recommends a factor of 1.25 for horizontal separators over vertical separators return K def K_separator_demister_York(P, horizontal=False): r'''Calculates the Sounders Brown `K` factor as used in determining maximum permissible gas velocity in a two-phase separator in either a horizontal or vertical orientation, *with a demister*. This function is a curve fit to [1]_ published in [2]_ and is widely used. For 1 < P < 15 psia: .. math:: K = 0.1821 + 0.0029P + 0.0460\ln P For 15 <= P <= 40 psia: .. math:: K = 0.35 For P < 5500 psia: .. math:: K = 0.430 - 0.023\ln P In the above equations, P is in units of psia. Parameters ---------- P : float Pressure of separator, [Pa] horizontal : bool, optional Whether to use the vertical or horizontal value; horizontal is 1.25 times higher, [-] Returns ------- K : float Sounders Brown Horizontal or vertical `K` factor for two-phase separator design with a demister, [m/s] Notes ----- If the input pressure is under 1 psia, 1 psia is used. If the input pressure is over 5500 psia, 5500 psia is used. Examples -------- >>> K_separator_demister_York(975*psi) 0.08281536035331669 References ---------- .. [2] Otto H. York Company, "Mist Elimination in Gas Treatment Plants and Refineries," Engineering, Parsippany, NJ. .. [1] Svrcek, W. Y., and W. D. Monnery. "Design Two-Phase Separators within the Right Limits" Chemical Engineering Progress, (October 1, 1993): 53-60. ''' P = P/psi # Correlation in terms of psia if P < 15: if P < 1: P = 1 # Prevent negative K values, but as a consequence be # optimistic for K values; limit is 0.185 ft/s but real values # should probably be lower K = 0.1821 + 0.0029*P + 0.0460*log(P) elif P < 40: K = 0.35 else: if P > 5500: P = 5500 # Do not allow for lower K values above 5500 psia, as # the limit is stated to be 5500 K = 0.430 - 0.023*log(P) K *= foot # Converts units of ft/s to m/s; the graph and all fits are in ft/s if horizontal: # Watkins recommends a factor of 1.25 for horizontal separators over # vertical separators as well K *= 1.25 return K def v_Sounders_Brown(K, rhol, rhog): r'''Calculates the maximum allowable vapor velocity in a two-phase separator to permit separation between entrained droplets and the gas using an empirical `K` factor, named after Sounders and Brown [1]_. This is a simplifying expression for terminal velocity and drag on particles. .. math:: v_{max} = K_{SB} \sqrt{\frac{\rho_l-\rho_g}{\rho_g}} Parameters ---------- K : float Sounders Brown `K` factor for two-phase separator design, [m/s] rhol : float Density of liquid phase [kg/m^3] rhog : float Density of gas phase [kg/m^3] Returns ------- v_max : float Maximum allowable vapor velocity in a two-phase separator to permit separation between entrained droplets and the gas, [m/s] Notes ----- The Sounders Brown K factor is related to the terminal velocity as shown in the following expression. .. math:: v_{term} = v_{max} = \sqrt{\frac{4 g d_p (\rho_p-\rho_f)}{3 C_D \rho_f }} v_{term} = \sqrt{\frac{(\rho_p-\rho_f)}{\rho_f}} \sqrt{\frac{4 g d_p}{3 C_D}} v_{term} = K_{SB} \sqrt{\frac{4 g d_p}{3 C_D}} Note this form corresponds to the Newton's law range (Re > 500), but in reality droplets are normally in the intermediate or Stoke's law region [2]_. For this reason using the drag coefficient expression directly is cleaner, but identical results can be found with the Sounders Brown equation. Examples -------- >>> v_Sounders_Brown(K=0.08, rhol=985.4, rhog=1.3) 2.2010906387516167 References ---------- .. [1] Souders, Mott., and George Granger. Brown. "Design of Fractionating Columns I. Entrainment and Capacity." Industrial & Engineering Chemistry 26, no. 1 (January 1, 1934): 98-103. https://doi.org/10.1021/ie50289a025. .. [2] Vasude, Gael D. Ulrich and Palligarnai T. Chemical Engineering Process Design and Economics : A Practical Guide. 2nd edition. Durham, N.H: Process Publishing, 2004. ''' return K*sqrt((rhol - rhog)/rhog) def K_Sounders_Brown_theoretical(D, Cd, g=g): r'''Converts a known drag coefficient into a Sounders-Brown `K` factor for two-phase separator design. This factor is the traditional way for separator diameters to be obtained although it is unnecessary and the theoretical drag coefficient method can be used instead. .. math:: K_{SB} = \sqrt{\frac{(\rho_p-\rho_f)}{\rho_f}} = \sqrt{\frac{4 g d_p}{3 C_D}} Parameters ---------- D : float Design diameter of the droplets, [m] Cd : float Drag coefficient [-] g : float, optional Acceleration due to gravity, [m/s^2] Returns ------- K : float Sounders Brown `K` factor for two-phase separator design, [m/s] Notes ----- Drag coefficient is a function of velocity; so iteration is needed to obtain the most correct answer. The following example shows the use of iteration to obtain the final velocity: >>> from fluids import * >>> V = 2.0 >>> D = 150E-6 >>> rho = 1.3 >>> rhol = 700. >>> mu = 1E-5 >>> for i in range(10): ... Re = Reynolds(V=V, rho=rho, mu=mu, D=D) ... Cd = drag_sphere(Re) ... K = K_Sounders_Brown_theoretical(D=D, Cd=Cd) ... V = v_Sounders_Brown(K, rhol=rhol, rhog=rho) ... print('%.14f' %V) 0.76093307417658 0.56242939340131 0.50732895050696 0.48957142095508 0.48356021946899 0.48149076033622 0.48077414934614 0.48052549959141 0.48043916249756 0.48040917690193 The use of Sounders-Brown constants can be replaced as follows (the v_terminal method includes its own solver for terminal velocity): >>> from fluids.drag import v_terminal >>> v_terminal(D=D, rhop=rhol, rho=rho, mu=mu) 0.4803932186998 Examples -------- >>> K_Sounders_Brown_theoretical(D=150E-6, Cd=0.5) 0.06263114241333939 References ---------- .. [1] Svrcek, W. Y., and W. D. Monnery. "Design Two-Phase Separators within the Right Limits" Chemical Engineering Progress, (October 1, 1993): 53-60. ''' return sqrt((4.0/3.0)*g*D/(Cd))