"""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 equations for modeling flow where density changes significantly during the process - compressible flow. Also included are equations for choked flow - the phenomenon where the velocity of a fluid reaches its speed of sound. 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: Compression Processes --------------------- .. autofunction:: isothermal_work_compression .. autofunction:: isentropic_work_compression .. autofunction:: isentropic_T_rise_compression .. autofunction:: isentropic_efficiency .. autofunction:: polytropic_exponent Compressible Flow ----------------- .. autofunction:: isothermal_gas Empirical Compressible Flow --------------------------- .. autofunction:: Panhandle_A .. autofunction:: Panhandle_B .. autofunction:: Weymouth .. autofunction:: Spitzglass_high .. autofunction:: Spitzglass_low .. autofunction:: Oliphant .. autofunction:: Fritzsche .. autofunction:: Muller .. autofunction:: IGT Critical Flow ------------- .. autofunction:: T_critical_flow .. autofunction:: P_critical_flow .. autofunction:: is_critical_flow .. autofunction:: P_isothermal_critical_flow .. autofunction:: P_upstream_isothermal_critical_flow Stagnation Point ---------------- .. autofunction:: stagnation_energy .. autofunction:: P_stagnation .. autofunction:: T_stagnation .. autofunction:: T_stagnation_ideal """ from math import exp, isinf, log, pi, sqrt from fluids.constants import R from fluids.numerics import brenth, lambertw, secant __all__ = ['Panhandle_A', 'Panhandle_B', 'Weymouth', 'Spitzglass_high', 'Spitzglass_low', 'Oliphant', 'Fritzsche', 'Muller', 'IGT', 'isothermal_gas', 'isothermal_work_compression', 'polytropic_exponent', 'isentropic_work_compression', 'isentropic_efficiency', 'isentropic_T_rise_compression', 'T_critical_flow', 'P_critical_flow', 'P_isothermal_critical_flow', 'is_critical_flow', 'stagnation_energy', 'P_stagnation', 'T_stagnation', 'T_stagnation_ideal'] def isothermal_work_compression(P1, P2, T, Z=1.0): r'''Calculates the work of compression or expansion of a gas going through an isothermal process. .. math:: W = zRT\ln\left(\frac{P_2}{P_1}\right) Parameters ---------- P1 : float Inlet pressure, [Pa] P2 : float Outlet pressure, [Pa] T : float Temperature of the gas going through an isothermal process, [K] Z : float Constant compressibility factor of the gas, [-] Returns ------- W : float Work performed per mole of gas compressed/expanded [J/mol] Notes ----- The full derivation with all forms is as follows: .. math:: W = \int_{P_1}^{P_2} V dP = zRT\int_{P_1}^{P_2} \frac{1}{P} dP .. math:: W = zRT\ln\left(\frac{P_2}{P_1}\right) = P_1 V_1 \ln\left(\frac{P_2} {P_1}\right) = P_2 V_2 \ln\left(\frac{P_2}{P_1}\right) The substitutions are according to the ideal gas law with compressibility: .. math: PV = ZRT The work of compression/expansion is the change in enthalpy of the gas. Returns negative values for expansion and positive values for compression. An average compressibility factor can be used where Z changes. For further accuracy, this expression can be used repeatedly with small changes in pressure and the work from each step summed. This is the best possible case for compression; all actual compresssors require more work to do the compression. By making the compression take a large number of stages and cooling the gas between stages, this can be approached reasonable closely. Integrally geared compressors are often used for this purpose. Examples -------- >>> isothermal_work_compression(1E5, 1E6, 300) 5743.427304244769 References ---------- .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. ''' return Z*R*T*log(P2/P1) def isentropic_work_compression(T1, k, Z=1.0, P1=None, P2=None, W=None, eta=None): r'''Calculation function for dealing with compressing or expanding a gas going through an isentropic, adiabatic process assuming constant Cp and Cv. The polytropic model is the same equation; just provide `n` instead of `k` and use a polytropic efficiency for `eta` instead of a isentropic efficiency. Can calculate any of the following, given all the other inputs: * W, Work of compression * P2, Pressure after compression * P1, Pressure before compression * eta, isentropic efficiency of compression .. math:: W = \left(\frac{k}{k-1}\right)ZRT_1\left[\left(\frac{P_2}{P_1} \right)^{(k-1)/k}-1\right]/\eta_{isentropic} Parameters ---------- T1 : float Initial temperature of the gas, [K] k : float Isentropic exponent of the gas (Cp/Cv) or polytropic exponent `n` to use this as a polytropic model instead [-] Z : float, optional Constant compressibility factor of the gas, [-] P1 : float, optional Inlet pressure, [Pa] P2 : float, optional Outlet pressure, [Pa] W : float, optional Work performed per mole of gas compressed/expanded [J/mol] eta : float, optional Isentropic efficiency of the process or polytropic efficiency of the process to use this as a polytropic model instead [-] Returns ------- W, P1, P2, or eta : float The missing input which was solved for [base SI] Notes ----- For the same compression ratio, this is always of larger magnitude than the isothermal case. The full derivation is as follows: For constant-heat capacity "isentropic" fluid, .. math:: V = \frac{P_1^{1/k}V_1}{P^{1/k}} .. math:: W = \int_{P_1}^{P_2} V dP = \int_{P_1}^{P_2}\frac{P_1^{1/k}V_1} {P^{1/k}}dP .. math:: W = \frac{P_1^{1/k} V_1}{1 - \frac{1}{k}}\left[P_2^{1-1/k} - P_1^{1-1/k}\right] After performing the integration and substantial mathematical manipulation we can obtain: .. math:: W = \left(\frac{k}{k-1}\right) P_1 V_1 \left[\left(\frac{P_2}{P_1} \right)^{(k-1)/k}-1\right] Using PV = ZRT: .. math:: W = \left(\frac{k}{k-1}\right)ZRT_1\left[\left(\frac{P_2}{P_1} \right)^{(k-1)/k}-1\right] The work of compression/expansion is the change in enthalpy of the gas. Returns negative values for expansion and positive values for compression. An average compressibility factor should be used as Z changes. For further accuracy, this expression can be used repeatedly with small changes in pressure and new values of isentropic exponent, and the work from each step summed. For the polytropic case this is not necessary, as `eta` corrects for the simplification. Examples -------- >>> isentropic_work_compression(P1=1E5, P2=1E6, T1=300, k=1.4, eta=0.78) 10416.876986384483 References ---------- .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. ''' if W is None and eta is not None and P1 is not None and P2 is not None: return k/(k - 1.0)*Z*R*T1*((P2/P1)**((k-1.)/k) - 1.0)/eta elif P1 is None and eta is not None and W is not None and P2 is not None: return P2*(1.0 + W*eta/(R*T1*Z) - W*eta/(R*T1*Z*k))**(-k/(k - 1.0)) elif P2 is None and eta is not None and W is not None and P1 is not None: return P1*(1.0 + W*eta/(R*T1*Z) - W*eta/(R*T1*Z*k))**(k/(k - 1.0)) elif eta is None and P1 is not None and P2 is not None and W is not None: return R*T1*Z*k*((P2/P1)**((k - 1.0)/k) - 1.0)/(W*(k - 1.0)) else: raise ValueError('Three of W, P1, P2, and eta must be specified.') def isentropic_T_rise_compression(T1, P1, P2, k, eta=1): r'''Calculates the increase in temperature of a fluid which is compressed or expanded under isentropic, adiabatic conditions assuming constant Cp and Cv. The polytropic model is the same equation; just provide `n` instead of `k` and use a polytropic efficienty for `eta` instead of a isentropic efficiency. .. math:: T_2 = T_1 + \frac{\Delta T_s}{\eta_s} = T_1 \left\{1 + \frac{1} {\eta_s}\left[\left(\frac{P_2}{P_1}\right)^{(k-1)/k}-1\right]\right\} Parameters ---------- T1 : float Initial temperature of gas [K] P1 : float Initial pressure of gas [Pa] P2 : float Final pressure of gas [Pa] k : float Isentropic exponent of the gas (Cp/Cv) or polytropic exponent `n` to use this as a polytropic model instead [-] eta : float Isentropic efficiency of the process or polytropic efficiency of the process to use this as a polytropic model instead [-] Returns ------- T2 : float Final temperature of gas [K] Notes ----- For the ideal case of `eta` = 1, the model simplifies to: .. math:: \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(k-1)/k} Examples -------- >>> isentropic_T_rise_compression(286.8, 54050, 432400, 1.4) 519.5230938217768 References ---------- .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. .. [2] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. ''' dT = T1*((P2/P1)**((k - 1.0)/k) - 1.0)/eta return T1 + dT def isentropic_efficiency(P1, P2, k, eta_s=None, eta_p=None): r'''Calculates either isentropic or polytropic efficiency from the other type of efficiency. .. math:: \eta_s = \frac{(P_2/P_1)^{(k-1)/k}-1} {(P_2/P_1)^{\frac{k-1}{k\eta_p}}-1} .. math:: \eta_p = \frac{\left(k - 1\right) \ln{\left (\frac{P_{2}}{P_{1}} \right )}}{k \ln{\left (\frac{1}{\eta_{s}} \left(\eta_{s} + \left(\frac{P_{2}}{P_{1}}\right)^{\frac{1}{k} \left(k - 1\right)} - 1\right) \right )}} Parameters ---------- P1 : float Initial pressure of gas [Pa] P2 : float Final pressure of gas [Pa] k : float Isentropic exponent of the gas (Cp/Cv) [-] eta_s : float, optional Isentropic (adiabatic) efficiency of the process, [-] eta_p : float, optional Polytropic efficiency of the process, [-] Returns ------- eta_s or eta_p : float Isentropic or polytropic efficiency, depending on input, [-] Notes ----- The form for obtained `eta_p` from `eta_s` was derived with SymPy. Examples -------- >>> isentropic_efficiency(1E5, 1E6, 1.4, eta_p=0.78) 0.7027614191263858 References ---------- .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. ''' if eta_s is None and eta_p is not None: return ((P2/P1)**((k-1.0)/k)-1.0)/((P2/P1)**((k-1.0)/(k*eta_p))-1.0) elif eta_p is None and eta_s is not None: return (k - 1.0)*log(P2/P1)/(k*log( (eta_s + (P2/P1)**((k - 1.0)/k) - 1.0)/eta_s)) else: raise ValueError('Either eta_s or eta_p is required') def polytropic_exponent(k, n=None, eta_p=None): r'''Calculates one of: * Polytropic exponent from polytropic efficiency * Polytropic efficiency from the polytropic exponent .. math:: n = \frac{k\eta_p}{1 - k(1-\eta_p)} .. math:: \eta_p = \frac{\left(\frac{n}{n-1}\right)}{\left(\frac{k}{k-1} \right)} = \frac{n(k-1)}{k(n-1)} Parameters ---------- k : float Isentropic exponent of the gas (Cp/Cv) [-] n : float, optional Polytropic exponent of the process [-] eta_p : float, optional Polytropic efficiency of the process, [-] Returns ------- n or eta_p : float Polytropic exponent or polytropic efficiency, depending on input, [-] Notes ----- Examples -------- >>> polytropic_exponent(1.4, eta_p=0.78) 1.5780346820809246 References ---------- .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf Professional Publishing, 2009. ''' if n is None and eta_p is not None: return k*eta_p/(1.0 - k*(1.0 - eta_p)) elif eta_p is None and n is not None: return n*(k - 1.0)/(k*(n - 1.0)) else: raise ValueError('Either n or eta_p is required') def T_critical_flow(T, k): r'''Calculates critical flow temperature `Tcf` for a fluid with the given isentropic coefficient. `Tcf` is in a flow (with Ma=1) whose stagnation conditions are known. Normally used with converging/diverging nozzles. .. math:: \frac{T^*}{T_0} = \frac{2}{k+1} Parameters ---------- T : float Stagnation temperature of a fluid with Ma=1 [K] k : float Isentropic coefficient [] Returns ------- Tcf : float Critical flow temperature at Ma=1 [K] Notes ----- Assumes isentropic flow. Examples -------- Example 12.4 in [1]_: >>> T_critical_flow(473, 1.289) 413.2809086937528 References ---------- .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. ''' return T*2.0/(k + 1.0) def P_critical_flow(P, k): r'''Calculates critical flow pressure `Pcf` for a fluid with the given isentropic coefficient. `Pcf` is in a flow (with Ma=1) whose stagnation conditions are known. Normally used with converging/diverging nozzles. .. math:: \frac{P^*}{P_0} = \left(\frac{2}{k+1}\right)^{k/(k-1)} Parameters ---------- P : float Stagnation pressure of a fluid with Ma=1 [Pa] k : float Isentropic coefficient [] Returns ------- Pcf : float Critical flow pressure at Ma=1 [Pa] Notes ----- Assumes isentropic flow. Examples -------- Example 12.4 in [1]_: >>> P_critical_flow(1400000, 1.289) 766812.9022792266 References ---------- .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. ''' return P*(2.0/(k + 1.))**(k/(k - 1.0)) def P_isothermal_critical_flow(P, fd, D, L): r'''Calculates critical flow pressure `Pcf` for a fluid flowing isothermally and suffering pressure drop caused by a pipe's friction factor. .. math:: P_2 = P_{1} e^{\frac{1}{2 D} \left(D \left(\operatorname{LambertW} {\left (- e^{\frac{1}{D} \left(- D - L f_d\right)} \right )} + 1\right) + L f_d\right)} Parameters ---------- P : float Inlet pressure [Pa] fd : float Darcy friction factor for flow in pipe [-] D : float Diameter of pipe, [m] L : float Length of pipe, [m] Returns ------- Pcf : float Critical flow pressure of a compressible gas flowing from `P1` to `Pcf` in a tube of length L and friction factor `fd` [Pa] Notes ----- Assumes isothermal flow. Developed based on the `isothermal_gas` model, using SymPy. The isothermal gas model is solved for maximum mass flow rate; any pressure drop under it is impossible due to the formation of a shock wave. Examples -------- >>> P_isothermal_critical_flow(P=1E6, fd=0.00185, L=1000., D=0.5) 389699.73176 References ---------- .. [1] Wilkes, James O. Fluid Mechanics for Chemical Engineers with Microfluidics and CFD. 2 edition. Upper Saddle River, NJ: Prentice Hall, 2005. ''' # Correct branch of lambertw found by trial and error lambert_term = float((lambertw(-exp((-D - L*fd)/D), -1)).real) return P*exp((D*(lambert_term + 1.0) + L*fd)/(2.0*D)) def P_upstream_isothermal_critical_flow(P, fd, D, L): """Not part of the public API. Reverses `P_isothermal_critical_flow`. Examples -------- >>> P_upstream_isothermal_critical_flow(P=389699.7317645518, fd=0.00185, ... L=1000., D=0.5) 1000000.00000 """ lambertw_term = float(lambertw(-exp(-(fd*L+D)/D), -1).real) return exp(-0.5*(D*lambertw_term+fd*L+D)/D)*P def is_critical_flow(P1, P2, k): r'''Determines if a flow of a fluid driven by pressure gradient P1 - P2 is critical, for a fluid with the given isentropic coefficient. This function calculates critical flow pressure, and checks if this is larger than P2. If so, the flow is critical and choked. Parameters ---------- P1 : float Higher, source pressure [Pa] P2 : float Lower, downstream pressure [Pa] k : float Isentropic coefficient [] Returns ------- flowtype : bool True if the flow is choked; otherwise False Notes ----- Assumes isentropic flow. Uses P_critical_flow function. Examples -------- Examples 1-2 from API 520. >>> is_critical_flow(670E3, 532E3, 1.11) False >>> is_critical_flow(670E3, 101E3, 1.11) True References ---------- .. [1] API. 2014. API 520 - Part 1 Sizing, Selection, and Installation of Pressure-relieving Devices, Part I - Sizing and Selection, 9E. ''' Pcf = P_critical_flow(P1, k) return Pcf > P2 def stagnation_energy(V): r'''Calculates the increase in enthalpy `dH` which is provided by a fluid's velocity `V`. .. math:: \Delta H = \frac{V^2}{2} Parameters ---------- V : float Velocity [m/s] Returns ------- dH : float Incease in enthalpy [J/kg] Notes ----- The units work out. This term is pretty small, but not trivial. Examples -------- >>> stagnation_energy(125) 7812.5 References ---------- .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. ''' return 0.5*V*V def P_stagnation(P, T, Tst, k): r'''Calculates stagnation flow pressure `Pst` for a fluid with the given isentropic coefficient and specified stagnation temperature and normal temperature. Normally used with converging/diverging nozzles. .. math:: \frac{P_0}{P}=\left(\frac{T_0}{T}\right)^{\frac{k}{k-1}} Parameters ---------- P : float Normal pressure of a fluid [Pa] T : float Normal temperature of a fluid [K] Tst : float Stagnation temperature of a fluid moving at a certain velocity [K] k : float Isentropic coefficient [] Returns ------- Pst : float Stagnation pressure of a fluid moving at a certain velocity [Pa] Notes ----- Assumes isentropic flow. Examples -------- Example 12-1 in [1]_. >>> P_stagnation(54050., 255.7, 286.8, 1.4) 80772.80495900588 References ---------- .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. ''' return P*(Tst/T)**(k/(k - 1.0)) def T_stagnation(T, P, Pst, k): r'''Calculates stagnation flow temperature `Tst` for a fluid with the given isentropic coefficient and specified stagnation pressure and normal pressure. Normally used with converging/diverging nozzles. .. math:: T=T_0\left(\frac{P}{P_0}\right)^{\frac{k-1}{k}} Parameters ---------- T : float Normal temperature of a fluid [K] P : float Normal pressure of a fluid [Pa] Pst : float Stagnation pressure of a fluid moving at a certain velocity [Pa] k : float Isentropic coefficient [] Returns ------- Tst : float Stagnation temperature of a fluid moving at a certain velocity [K] Notes ----- Assumes isentropic flow. Examples -------- Example 12-1 in [1]_. >>> T_stagnation(286.8, 54050, 54050*8, 1.4) 519.5230938217768 References ---------- .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. ''' return T*(Pst/P)**((k - 1.0)/k) def T_stagnation_ideal(T, V, Cp): r'''Calculates the ideal stagnation temperature `Tst` calculated assuming the fluid has a constant heat capacity `Cp` and with a specified velocity `V` and temperature `T`. .. math:: T^* = T + \frac{V^2}{2C_p} Parameters ---------- T : float Tempearture [K] V : float Velocity [m/s] Cp : float Ideal heat capacity [J/kg/K] Returns ------- Tst : float Stagnation temperature [J/kg] Examples -------- Example 12-1 in [1]_. >>> T_stagnation_ideal(255.7, 250, 1005.) 286.79452736318405 References ---------- .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. ''' return T + 0.5*V*V/Cp def isothermal_gas_err_P1(P1, fd, rho, P2, L, D, m): return m - isothermal_gas(rho, fd, P1=P1, P2=P2, L=L, D=D) def isothermal_gas_err_P2(P2, rho, fd, P1, L, D, m): return m - isothermal_gas(rho, fd, P1=P1, P2=P2, L=L, D=D) def isothermal_gas_err_P2_basis(P1, P2, rho, fd, m, L, D): return abs(P2 - isothermal_gas(rho, fd, m=m, P1=P1, P2=None, L=L, D=D)) def isothermal_gas_err_D(D, m, rho, fd, P1, P2, L): return m - isothermal_gas(rho, fd, P1=P1, P2=P2, L=L, D=D) def isothermal_gas(rho, fd, P1=None, P2=None, L=None, D=None, m=None): r'''Calculation function for dealing with flow of a compressible gas in a pipeline for the complete isothermal flow equation. Can calculate any of the following, given all other inputs: * Mass flow rate * Upstream pressure (numerical) * Downstream pressure (analytical or numerical if an overflow occurs) * Diameter of pipe (numerical) * Length of pipe A variety of forms of this equation have been presented, differing in their use of the ideal gas law and choice of gas constant. The form here uses density explicitly, allowing for non-ideal values to be used. .. math:: \dot m^2 = \frac{\left(\frac{\pi D^2}{4}\right)^2 \rho_{avg} \left(P_1^2-P_2^2\right)}{P_1\left(f_d\frac{L}{D} + 2\ln\frac{P_1}{P_2} \right)} Parameters ---------- rho : float Average density of gas in pipe, [kg/m^3] fd : float Darcy friction factor for flow in pipe [-] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] m : float, optional Mass flow rate of gas through pipe, [kg/s] Returns ------- m, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- The solution for P2 has the following closed form, derived using Maple: .. math:: P_2={P_1 \left( {{ e}^{0.5\cdot{\frac {1}{{m}^{2}} \left( -C{m}^{2} +\text{ lambertW} \left(-{\frac {BP_1}{{m}^{2}}{{ e}^{-{\frac {-C{m}^{ 2}+BP_1}{{m}^{2}}}}}}\right){}{m}^{2}+BP_1 \right) }}} \right) ^{-1}} .. math:: B = \frac{\pi^2 D^4}{4^2} \rho_{avg} .. math:: C = f_d \frac{L}{D} A wide range of conditions are impossible due to choked flow. See `P_isothermal_critical_flow` for details. An exception is raised when they occur. The 2 multiplied by the logarithm is often shown as a power of the pressure ratio; this is only the case when the pressure ratio is raised to the power of 2 before its logarithm is taken. A number of limitations exist for this model: * Density dependence is that of an ideal gas. * If calculating the pressure drop, the average gas density cannot be known immediately; iteration must be used to correct this. * The friction factor depends on both the gas density and velocity, so it should be solved for iteratively as well. It changes throughout the pipe as the gas expands and velocity increases. * The model is not easily adapted to include elevation effects due to the acceleration term included in it. * As the gas expands, it will change temperature slightly, further altering the density and friction factor. There are many commercial packages which perform the actual direct integration of the flow, such as OLGA Dynamic Multiphase Flow Simulator, or ASPEN Hydraulics. This expression has also been presented with the ideal gas assumption directly incorporated into it [4]_ (note R is the specific gas constant, in units of J/kg/K): .. math:: \dot m^2 = \frac{\left(\frac{\pi D^2}{4}\right)^2 \left(P_1^2-P_2^2\right)}{RT\left(f_d\frac{L}{D} + 2\ln\frac{P_1}{P_2} \right)} Examples -------- >>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, P2=9E5, L=1000, D=0.5) 145.4847572636031 References ---------- .. [1] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. .. [2] Kim, J. and Singh, N. "A Novel Equation for Isothermal Pipe Flow.". Chemical Engineering, June 2012, http://www.chemengonline.com/a-novel-equation-for-isothermal-pipe-flow/?printmode=1 .. [3] Wilkes, James O. Fluid Mechanics for Chemical Engineers with Microfluidics and CFD. 2 edition. Upper Saddle River, NJ: Prentice Hall, 2005. .. [4] Rennels, Donald C., and Hobart M. Hudson. Pipe Flow: A Practical and Comprehensive Guide. 1st edition. Hoboken, N.J: Wiley, 2012. ''' if m is None and P1 is not None and P2 is not None and L is not None and D is not None: Pcf = P_isothermal_critical_flow(P=P1, fd=fd, D=D, L=L) if P2 < Pcf: raise ValueError('Given outlet pressure is not physically possible ' # numba: delete f'due to the formation of choked flow at P2={Pcf:f}, specified outlet pressure was {P2:f}') # numba: delete # raise ValueError("Not possible") # numba: uncomment if P2 > P1: raise ValueError('Specified outlet pressure is larger than the ' 'inlet pressure; fluid will flow backwards.') return sqrt(0.0625*pi*pi*D**4*rho/(P1*(fd*L/D + 2.0*log(P1/P2)))*(P1*P1 - P2*P2)) elif L is None and P1 is not None and P2 is not None and D is not None and m is not None: return D*(pi*pi*D**4*rho*(P1*P1 - P2*P2) - 32.0*P1*m*m*log(P1/P2))/(16.0*P1*fd*m*m) elif P1 is None and L is not None and P2 is not None and D is not None and m is not None: Pcf = P_upstream_isothermal_critical_flow(P=P2, fd=fd, D=D, L=L) try: # Use the explicit solution for P2 with different P1 guesses; # newton doesn't like solving for m. P1 = secant(isothermal_gas_err_P2_basis, (P2+Pcf)/2., args=(P2, rho, fd, m, L, D)) if not (P2 <= P1): raise ValueError("Failed") return P1 except: try: return brenth(isothermal_gas_err_P1, P2, Pcf, args=(fd, rho, P2, L, D, m)) except: m_max = isothermal_gas(rho, fd, P1=Pcf, P2=P2, L=L, D=D) # numba: delete raise ValueError(f'The desired mass flow rate of {m:f} kg/s cannot ' # numba: delete 'be achieved with the specified downstream pressure; the maximum flowrate is ' # numba: delete f'{m_max:f} kg/s at an upstream pressure of {Pcf:f} Pa') # numba: delete # raise ValueError("Failed") # numba: uncomment elif P2 is None and L is not None and P1 is not None and D is not None and m is not None: try: Pcf = P_isothermal_critical_flow(P=P1, fd=fd, D=D, L=L) m_max = isothermal_gas(rho, fd, P1=P1, P2=Pcf, L=L, D=D) if not (m <= m_max): raise ValueError("Failed") C = fd*L/D B = (pi/4*D**2)**2*rho arg = -B/m**2*P1*exp(-(-C*m**2+B*P1)/m**2) # Consider the two real branches of the lambertw function. # The k=-1 branch produces the higher P2 values; the k=0 branch is # physically impossible. lambert_ans = float(lambertw(arg, k=-1).real) # Large overflow problem here; also divide by zero problems! # Fail and try a numerical solution if it doesn't work. if isinf(lambert_ans): raise ValueError("Should not be infinity") P2 = P1/exp((-C*m**2+lambert_ans*m**2+B*P1)/m**2/2.) if not (P2 < P1): raise ValueError("Should not be the case") return P2 except: Pcf = P_isothermal_critical_flow(P=P1, fd=fd, D=D, L=L) try: return brenth(isothermal_gas_err_P2, Pcf, P1, args=(rho, fd, P1, L, D, m)) except: m_max = isothermal_gas(rho, fd, P1=P1, P2=Pcf, L=L, D=D) raise ValueError('The desired mass flow rate cannot be achieved ' # numba: delete f'with the specified upstream pressure of {P1:f} Pa; the maximum flowrate is {m_max:f} ' # numba: delete f'kg/s at a downstream pressure of {Pcf:f}') # numba: delete # raise ValueError("Failed") # numba: uncomment # A solver which respects its boundaries is required here. # brenth cuts the time down from 2 ms to 200 mircoseconds. # Is is believed Pcf and P1 will always bracked the root, however # leave the commented code for testing elif D is None and P2 is not None and P1 is not None and L is not None and m is not None: return secant(isothermal_gas_err_D, 0.1, args=(m, rho, fd, P1, P2, L)) else: raise ValueError('This function solves for either mass flow, upstream \ pressure, downstream pressure, diameter, or length; all other inputs \ must be provided.') def Panhandle_A(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=0.92): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the Panhandle A formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe * Length of pipe A variety of different constants and expressions have been presented for the Panhandle A equation. Here, a new form is developed with all units in base SI, based on the work of [1]_. .. math:: Q = 158.02053 E \left(\frac{T_s}{P_s}\right)^{1.0788}\left[\frac{P_1^2 -P_2^2}{L \cdot {SG}^{0.8539} T_{avg}Z_{avg}}\right]^{0.5394}D^{2.6182} Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- [1]_'s original constant was 4.5965E-3, and it has units of km (length), kPa, mm (diameter), and flowrate in m^3/day. The form in [2]_ has the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is 1.9152E-4. The GPSA [3]_ has a leading constant of 0.191, a bracketed power of 0.5392, a specific gravity power of 0.853, and otherwise the same constants. It is in units of mm (diameter) and kPa and m^3/day; length is stated to be in km, but according to the errata is in m. [4]_ has a leading constant of 1.198E7, a specific gravity of power of 0.8541, and a power of diameter which is under the root of 4.854 and is otherwise the same. It has units of kPa and m^3/day, but is otherwise in base SI units. [5]_ has a leading constant of 99.5211, but its reference correction has no exponent; other exponents are the same as here. It is entirely in base SI units. [6]_ has pressures in psi, diameter in inches, length in miles, Q in ft^3/day, T in degrees Rankine, and a constant of 435.87. Its reference condition power is 1.07881, and it has a specific gravity correction outside any other term with a power of 0.4604. Examples -------- >>> Panhandle_A(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 42.56082051195928 References ---------- .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. .. [2] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. .. [3] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. .. [4] Campbell, John M. Gas Conditioning and Processing, Vol. 2: The Equipment Modules. 7th edition. Campbell Petroleum Series, 1992. .. [5] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations for Steady State Flow in Natural Gas Pipelines." Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 262-73. doi:10.1590/S1678-58782007000300005. .. [6] Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla: Krieger Pub Co, 1991. ''' c1 = 1.0788 c2 = 0.8539 c3 = 0.5394 c4 = 2.6182 c5 = 158.0205328706957220332831680508433862787 # 45965*10**(591/1250)/864 if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return c5*E*(Ts/Ps)**c1*((P1**2 - P2**2)/(L*SG**c2*Tavg*Zavg))**c3*D**c4 elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return (Q*(Ts/Ps)**(-c1)*(SG**(-c2)*(P1**2 - P2**2)/(L*Tavg*Zavg))**(-c3)/(E*c5))**(1./c4) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return sqrt(L*SG**c2*Tavg*Zavg*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(1./c3) + P2**2) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return sqrt(-L*SG**c2*Tavg*Zavg*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(1./c3) + P1**2) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return SG**(-c2)*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(-1./c3)*(P1**2 - P2**2)/(Tavg*Zavg) else: raise ValueError('This function solves for either flow, upstream \ pressure, downstream pressure, diameter, or length; all other inputs \ must be provided.') def Panhandle_B(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=0.92): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the Panhandle B formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe * Length of pipe A variety of different constants and expressions have been presented for the Panhandle B equation. Here, a new form is developed with all units in base SI, based on the work of [1]_. .. math:: Q = 152.88116 E \left(\frac{T_s}{P_s}\right)^{1.02}\left[\frac{P_1^2 -P_2^2}{L \cdot {SG}^{0.961} T_{avg}Z_{avg}}\right]^{0.51}D^{2.53} Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- [1]_'s original constant was 1.002E-2, and it has units of km (length), kPa, mm (diameter), and flowrate in m^3/day. The form in [2]_ has the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is 4.1749E-4. The GPSA [3]_ has a leading constant of 0.339, and otherwise the same constants. It is in units of mm (diameter) and kPa and m^3/day; length is stated to be in km, but according to the errata is in m. [4]_ has a leading constant of 1.264E7, a diameter power of 4.961 which is also under the 0.51 power, and is otherwise the same. It has units of kPa and m^3/day, but is otherwise in base SI units. [5]_ has a leading constant of 135.8699, but its reference correction has no exponent and its specific gravity has a power of 0.9608; the other exponents are the same as here. It is entirely in base SI units. [6]_ has pressures in psi, diameter in inches, length in miles, Q in ft^3/day, T in degrees Rankine, and a constant of 737 with the exponents the same as here. Examples -------- >>> Panhandle_B(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 42.35366178004172 References ---------- .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. .. [2] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. .. [3] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. .. [4] Campbell, John M. Gas Conditioning and Processing, Vol. 2: The Equipment Modules. 7th edition. Campbell Petroleum Series, 1992. .. [5] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations for Steady State Flow in Natural Gas Pipelines." Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 262-73. doi:10.1590/S1678-58782007000300005. .. [6] Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla: Krieger Pub Co, 1991. ''' c1 = 1.02 # reference condition power c2 = 0.961 # sg power c3 = 0.51 # main power c4 = 2.53 # diameter power c5 = 152.8811634298055458624385985866624419060 # 4175*10**(3/25)/36 if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return c5*E*(Ts/Ps)**c1*((P1**2 - P2**2)/(L*SG**c2*Tavg*Zavg))**c3*D**c4 elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return (Q*(Ts/Ps)**(-c1)*(SG**(-c2)*(P1**2 - P2**2)/(L*Tavg*Zavg))**(-c3)/(E*c5))**(1./c4) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return sqrt(L*SG**c2*Tavg*Zavg*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(1./c3) + P2**2) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return sqrt(-L*SG**c2*Tavg*Zavg*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(1./c3) + P1**2) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return SG**(-c2)*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(-1./c3)*(P1**2 - P2**2)/(Tavg*Zavg) else: raise ValueError('This function solves for either flow, upstream \ pressure, downstream pressure, diameter, or length; all other inputs \ must be provided.') def Weymouth(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=0.92): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the Weymouth formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe * Length of pipe A variety of different constants and expressions have been presented for the Weymouth equation. Here, a new form is developed with all units in base SI, based on the work of [1]_. .. math:: Q = 137.32958 E \frac{T_s}{P_s}\left[\frac{P_1^2 -P_2^2}{L \cdot {SG} \cdot T_{avg}Z_{avg}}\right]^{0.5}D^{2.667} Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- [1]_'s original constant was 3.7435E-3, and it has units of km (length), kPa, mm (diameter), and flowrate in m^3/day. The form in [2]_ has the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is 1.5598E-4. The GPSA [3]_ has a leading constant of 0.1182, and otherwise the same constants. It is in units of mm (diameter) and kPa and m^3/day; length is stated to be in km, but according to the errata is in m. [4]_ has a leading constant of 1.162E7, a diameter power of 5.333 which is also under the 0.50 power, and is otherwise the same. It has units of kPa and m^3/day, but is otherwise in base SI units. [5]_ has a leading constant of 137.2364; the other exponents are the same as here. It is entirely in base SI units. [6]_ has pressures in psi, diameter in inches, length in miles, Q in ft^3/hour, T in degrees Rankine, and a constant of 18.062 with the exponents the same as here. Examples -------- >>> Weymouth(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 32.07729055913029 References ---------- .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. .. [2] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. .. [3] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. .. [4] Campbell, John M. Gas Conditioning and Processing, Vol. 2: The Equipment Modules. 7th edition. Campbell Petroleum Series, 1992. .. [5] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations for Steady State Flow in Natural Gas Pipelines." Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 262-73. doi:10.1590/S1678-58782007000300005. .. [6] Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla: Krieger Pub Co, 1991. ''' c3 = 0.5 # main power c4 = 2.667 # diameter power c5 = 137.3295809942512546732179684618143090992 # 37435*10**(501/1000)/864 if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return c5*E*(Ts/Ps)*((P1**2 - P2**2)/(L*SG*Tavg*Zavg))**c3*D**c4 elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return (Ps*Q*((P1**2 - P2**2)/(L*SG*Tavg*Zavg))**(-c3)/(E*Ts*c5))**(1./c4) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return sqrt(L*SG*Tavg*Zavg*(D**(-c4)*Ps*Q/(E*Ts*c5))**(1./c3) + P2**2) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return sqrt(-L*SG*Tavg*Zavg*(D**(-c4)*Ps*Q/(E*Ts*c5))**(1./c3) + P1**2) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return (D**(-c4)*Ps*Q/(E*Ts*c5))**(-1./c3)*(P1**2 - P2**2)/(SG*Tavg*Zavg) else: raise ValueError('This function solves for either flow, upstream \ pressure, downstream pressure, diameter, or length; all other inputs \ must be provided.') def _to_solve_Spitzglass_high(D, Q, SG, Tavg, L, P1, P2, Ts, Ps, Zavg, E): return Q - Spitzglass_high(SG=SG, Tavg=Tavg, L=L, D=D, P1=P1, P2=P2, Ts=Ts, Ps=Ps,Zavg=Zavg, E=E) def Spitzglass_high(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=1.): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the Spitzglass (high pressure drop) formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe (numerical solution) * Length of pipe A variety of different constants and expressions have been presented for the Spitzglass (high pressure drop) formula. Here, the form as in [1]_ is used but with a more precise metric conversion from inches to m. .. math:: Q = 125.1060 E \left(\frac{T_s}{P_s}\right)\left[\frac{P_1^2 -P_2^2}{L \cdot {SG} T_{avg}Z_{avg} (1 + 0.09144/D + \frac{150}{127}D)} \right]^{0.5}D^{2.5} Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- This equation is often presented without any correction for reference conditions for specific gravity. This model is also presented in [2]_ with a leading constant of 1.0815E-2, the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour. Examples -------- >>> Spitzglass_high(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 29.42670246281681 References ---------- .. [1] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations for Steady State Flow in Natural Gas Pipelines." Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 262-73. doi:10.1590/S1678-58782007000300005. .. [2] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. ''' c3 = 1.181102362204724409448818897637795275591 # 0.03/inch or 150/127 c4 = 0.09144 c5 = 125.1060 if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return (c5*E*Ts/Ps*D**2.5*sqrt((P1**2-P2**2) /(L*SG*Zavg*Tavg*(1 + c4/D + c3*D)))) elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return secant(_to_solve_Spitzglass_high, 0.5, args=(Q, SG, Tavg, L, P1, P2, Ts, Ps, Zavg, E)) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return sqrt((D**6*E**2*P2**2*Ts**2*c5**2 + D**2*L*Ps**2*Q**2*SG*Tavg*Zavg*c3 + D*L*Ps**2*Q**2*SG*Tavg*Zavg + L*Ps**2*Q**2*SG*Tavg*Zavg*c4)/(D**6*E**2*Ts**2*c5**2)) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return sqrt((D**6*E**2*P1**2*Ts**2*c5**2 - D**2*L*Ps**2*Q**2*SG*Tavg*Zavg*c3 - D*L*Ps**2*Q**2*SG*Tavg*Zavg - L*Ps**2*Q**2*SG*Tavg*Zavg*c4)/(D**6*E**2*Ts**2*c5**2)) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return (D**6*E**2*Ts**2*c5**2*(P1**2 - P2**2) /(Ps**2*Q**2*SG*Tavg*Zavg*(D**2*c3 + D + c4))) else: raise ValueError('This function solves for either flow, upstream \ pressure, downstream pressure, diameter, or length; all other inputs \ must be provided.') def _to_solve_Spitzglass_low(D, Q, SG, Tavg, L, P1, P2, Ts, Ps, Zavg, E): return Q - Spitzglass_low(SG=SG, Tavg=Tavg, L=L, D=D, P1=P1, P2=P2, Ts=Ts, Ps=Ps, Zavg=Zavg, E=E) def Spitzglass_low(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=1.): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the Spitzglass (low pressure drop) formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe (numerical solution) * Length of pipe A variety of different constants and expressions have been presented for the Spitzglass (low pressure drop) formula. Here, the form as in [1]_ is used but with a more precise metric conversion from inches to m. .. math:: Q = 125.1060 E \left(\frac{T_s}{P_s}\right)\left[\frac{2(P_1 -P_2)(P_s+1210)}{L \cdot {SG} \cdot T_{avg}Z_{avg} (1 + 0.09144/D + \frac{150}{127}D)}\right]^{0.5}D^{2.5} Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- This equation is often presented without any correction for reference conditions for specific gravity. This model is also presented in [2]_ with a leading constant of 5.69E-2, the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour. However, it is believed to contain a typo, and gives results <1/3 of the correct values. It is also present in [2]_ in imperial form; this is believed correct, but makes a slight assumption not done in [1]_. This model is present in [3]_ without reference corrections. The 1210 constant in [1]_ is an approximation necessary for the reference correction to function without a square of the pressure difference. The GPSA version is as follows, and matches this formulation very closely: .. math:: Q = 0.821 \left[\frac{(P_1-P_2)D^5}{L \cdot {SG} (1 + 91.44/D + 0.0018D)}\right]^{0.5} The model is also shown in [4]_, with diameter in inches, length in feet, flow in MMSCFD, pressure drop in inH2O, and a rounded leading constant of 0.09; this makes its predictions several percent higher than the model here. Examples -------- >>> Spitzglass_low(D=0.154051, P1=6720.3199, P2=0, L=54.864, SG=0.6, Tavg=288.7) 0.9488775242530617 References ---------- .. [1] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations for Steady State Flow in Natural Gas Pipelines." Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 262-73. doi:10.1590/S1678-58782007000300005. .. [2] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. .. [3] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. .. [4] PetroWiki. "Pressure Drop Evaluation along Pipelines" Accessed September 11, 2016. http://petrowiki.org/Pressure_drop_evaluation_along_pipelines#Spitzglass_equation_2. ''' c3 = 1.181102362204724409448818897637795275591 # 0.03/inch or 150/127 c4 = 0.09144 c5 = 125.1060 if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return c5*Ts/Ps*D**2.5*E*sqrt(((P1-P2)*2*(Ps+1210.))/(L*SG*Tavg*Zavg*(1 + c4/D + c3*D))) elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return secant(_to_solve_Spitzglass_low, 0.5, args=(Q, SG, Tavg, L, P1, P2, Ts, Ps, Zavg, E)) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return 0.5*(2.0*D**6*E**2*P2*Ts**2*c5**2*(Ps + 1210.0) + D**2*L*Ps**2*Q**2*SG*Tavg*Zavg*c3 + D*L*Ps**2*Q**2*SG*Tavg*Zavg + L*Ps**2*Q**2*SG*Tavg*Zavg*c4)/(D**6*E**2*Ts**2*c5**2*(Ps + 1210.0)) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return 0.5*(2.0*D**6*E**2*P1*Ts**2*c5**2*(Ps + 1210.0) - D**2*L*Ps**2*Q**2*SG*Tavg*Zavg*c3 - D*L*Ps**2*Q**2*SG*Tavg*Zavg - L*Ps**2*Q**2*SG*Tavg*Zavg*c4)/(D**6*E**2*Ts**2*c5**2*(Ps + 1210.0)) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return 2.0*D**6*E**2*Ts**2*c5**2*(P1*Ps + 1210.0*P1 - P2*Ps - 1210.0*P2)/(Ps**2*Q**2*SG*Tavg*Zavg*(D**2*c3 + D + c4)) else: raise ValueError('This function solves for either flow, upstream \ pressure, downstream pressure, diameter, or length; all other inputs \ must be provided.') def _to_solve_Oliphant(D, Q, SG, Tavg, L, P1, P2, Ts, Ps, Zavg, E): return Q - Oliphant(SG=SG, Tavg=Tavg, L=L, D=D, P1=P1, P2=P2, Ts=Ts, Ps=Ps, Zavg=Zavg, E=E) def Oliphant(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=0.92): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the Oliphant formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe (numerical solution) * Length of pipe This model is a more complete conversion to metric of the Imperial version presented in [1]_. .. math:: Q = 84.5872\left(D^{2.5} + 0.20915D^3\right)\frac{T_s}{P_s}\left(\frac {P_1^2 - P_2^2}{L\cdot {SG} \cdot T_{avg}}\right)^{0.5} Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- Recommended in [1]_ for use between vacuum and 100 psi. The model is simplified by grouping constants here; however, it is presented in the imperial unit set inches (diameter), miles (length), psi, Rankine, and MMSCFD in [1]_: .. math:: Q = 42(24)\left(D^{2.5} + \frac{D^3}{30}\right)\left(\frac{14.4}{P_s} \right)\left(\frac{T_s}{520}\right)\left[\left(\frac{0.6}{SG}\right) \left(\frac{520}{T_{avg}}\right)\left(\frac{P_1^2 - P_2^2}{L}\right) \right]^{0.5} Examples -------- >>> Oliphant(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 28.851535408143057 References ---------- .. [1] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors Suppliers Association, Tulsa, OK, 2012. .. [2] F. N. Oliphant, "Production of Natural Gas," Report. USGS, 1902. ''' # c1 = 42*24*Q*foot**3/day*(mile)**0.5*9/5.*(5/9.)**0.5*psi*(1/psi)*14.4/520.*0.6**0.5*520**0.5/inch**2.5 c1 = 84.587176139918568651410168968141078948974609375000 c2 = 0.2091519350460528670065940559652517549694 # 1/(30.*0.0254**0.5) if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return c1*(D**2.5 + c2*D**3)*Ts/Ps*sqrt((P1**2-P2**2)/(L*SG*Tavg)) elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return secant(_to_solve_Oliphant, 0.5, args=(Q, SG, Tavg, L, P1, P2, Ts, Ps, Zavg, E)) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return sqrt(L*Ps**2*Q**2*SG*Tavg/(Ts**2*c1**2*(D**3*c2 + D**2.5)**2) + P2**2) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return sqrt(-L*Ps**2*Q**2*SG*Tavg/(Ts**2*c1**2*(D**3*c2 + D**2.5)**2) + P1**2) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return Ts**2*c1**2*(P1**2 - P2**2)*(D**3*c2 + D**2.5)**2/(Ps**2*Q**2*SG*Tavg) else: raise ValueError('This function solves for either flow, upstream \ pressure, downstream pressure, diameter, or length; all other inputs \ must be provided.') def Fritzsche(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=1.0): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the Fritzsche formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe * Length of pipe A variety of different constants and expressions have been presented for the Fritzsche formula. Here, the form as in [1]_ is used but with all inputs in base SI units. .. math:: Q = 93.500 \frac{T_s}{P_s}\left(\frac{P_1^2 - P_2^2} {L\cdot {SG}^{0.8587} \cdot T_{avg}}\right)^{0.538}D^{2.69} Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- This model is also presented in [1]_ with a leading constant of 2.827, the same exponents as used here, units of mm (diameter), kPa, km (length), and flow in m^3/hour. This model is shown in base SI units in [2]_, and with a leading constant of 94.2565, a diameter power of 2.6911, main group power of 0.5382 and a specific gravity power of 0.858. The difference is very small. Examples -------- >>> Fritzsche(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15) 39.421535157535565 References ---------- .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. .. [2] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations for Steady State Flow in Natural Gas Pipelines." Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 262-73. doi:10.1590/S1678-58782007000300005. ''' # Rational('2.827E-3')/(3600*24)*(1000)**Rational('2.69')*(1000)**Rational('0.538')*1000/(1000**2)**Rational('0.538') c5 = 93.50009798751128188757518688244137811221 # 14135*10**(57/125)/432 c2 = 0.8587 c3 = 0.538 c4 = 2.69 if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return c5*E*(Ts/Ps)*((P1**2 - P2**2)/(SG**c2*Tavg*L*Zavg))**c3*D**c4 elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return (Ps*Q*(SG**(-c2)*(P1**2 - P2**2)/(L*Tavg*Zavg))**(-c3)/(E*Ts*c5))**(1./c4) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return sqrt(L*SG**c2*Tavg*Zavg*(D**(-c4)*Ps*Q/(E*Ts*c5))**(1./c3) + P2**2) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return sqrt(-L*SG**c2*Tavg*Zavg*(D**(-c4)*Ps*Q/(E*Ts*c5))**(1./c3) + P1**2) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return SG**(-c2)*(D**(-c4)*Ps*Q/(E*Ts*c5))**(-1./c3)*(P1**2 - P2**2)/(Tavg*Zavg) else: raise ValueError('This function solves for either flow, upstream pressure, downstream pressure, diameter, or length; all other inputs must be provided.') def Muller(SG, Tavg, mu, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=1.0): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the Muller formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe * Length of pipe A variety of different constants and expressions have been presented for the Muller formula. Here, the form as in [1]_ is used but with all inputs in base SI units. .. math:: Q = 15.7743\frac{T_s}{P_s}E\left(\frac{P_1^2 - P_2^2}{L \cdot Z_{avg} \cdot T_{avg}}\right)^{0.575} \left(\frac{D^{2.725}}{\mu^{0.15} SG^{0.425}}\right) Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] mu : float Average viscosity of the fluid in the pipeline, [Pa*s] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- This model is presented in [1]_ with a leading constant of 0.4937, the same exponents as used here, units of inches (diameter), psi, feet (length), Rankine, pound/(foot*second) for viscosity, and 1000 ft^3/hour. This model is also presented in [2]_ in both SI and imperial form. The SI form was incorrectly converted and yields much higher flow rates. The imperial version has a leading constant of 85.7368, the same powers as used here except with rounded values of powers of viscosity (0.2609) and specific gravity (0.7391) rearranged to be inside the bracketed group; its units are inches (diameter), psi, miles (length), Rankine, pound/(foot*second) for viscosity, and ft^3/day. This model is shown in base SI units in [3]_, and with a leading constant of 15.7650, a diameter power of 2.724, main group power of 0.5747, a specific gravity power of 0.74, and a viscosity power of 0.1494. Examples -------- >>> Muller(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, mu=1E-5, ... Tavg=277.15) 60.45796698148659 References ---------- .. [1] Mohitpour, Mo, Golshan, and Allan Murray. Pipeline Design and Construction: A Practical Approach. 3rd edition. New York: Amer Soc Mechanical Engineers, 2006. .. [2] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. .. [3] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations for Steady State Flow in Natural Gas Pipelines." Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 262-73. doi:10.1590/S1678-58782007000300005. ''' # 1000*foot**3/hour*0.4937/inch**2.725*foot**0.575*(5/9.)**0.575*9/5.*(pound/foot)**0.15*psi*(1/psi**2)**0.575 c5 = 15.77439908642077352939746374951659525108 # 5642991*196133**(17/20)*2**(3/5)*3**(11/40)*5**(7/40)/30645781250 c2 = 0.575 # main power c3 = 2.725 # D power c4 = 0.425 # SG power c1 = 0.15 # mu power if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return c5*Ts/Ps*E*((P1**2-P2**2)/Tavg/L/Zavg)**c2*D**c3/SG**c4/mu**c1 elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return (Ps*Q*SG**c4*mu**c1*((P1**2 - P2**2)/(L*Tavg*Zavg))**(-c2)/(E*Ts*c5))**(1./c3) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return sqrt(L*Tavg*Zavg*(D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(1/c2) + P2**2) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return sqrt(-L*Tavg*Zavg*(D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(1/c2) + P1**2) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return (D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(-1/c2)*(P1**2 - P2**2)/(Tavg*Zavg) else: raise ValueError('This function solves for either flow, upstream pressure, downstream pressure, diameter, or length; all other inputs must be provided.') def IGT(SG, Tavg, mu, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7, Ps=101325., Zavg=1.0, E=1.0): r'''Calculation function for dealing with flow of a compressible gas in a pipeline with the IGT formula. Can calculate any of the following, given all other inputs: * Flow rate * Upstream pressure * Downstream pressure * Diameter of pipe * Length of pipe A variety of different constants and expressions have been presented for the IGT formula. Here, the form as in [1]_ is used but with all inputs in base SI units. .. math:: Q = 24.6241\frac{T_s}{P_s}E\left(\frac{P_1^2 - P_2^2}{L \cdot Z_{avg} \cdot T_{avg}}\right)^{5/9} \left(\frac{D^{8/3}}{\mu^{1/9} SG^{4/9}}\right) Parameters ---------- SG : float Specific gravity of fluid with respect to air at the reference temperature and pressure `Ts` and `Ps`, [-] Tavg : float Average temperature of the fluid in the pipeline, [K] mu : float Average viscosity of the fluid in the pipeline, [Pa*s] L : float, optional Length of pipe, [m] D : float, optional Diameter of pipe, [m] P1 : float, optional Inlet pressure to pipe, [Pa] P2 : float, optional Outlet pressure from pipe, [Pa] Q : float, optional Flow rate of gas through pipe at `Ts` and `Ps`, [m^3/s] Ts : float, optional Reference temperature for the specific gravity of the gas, [K] Ps : float, optional Reference pressure for the specific gravity of the gas, [Pa] Zavg : float, optional Average compressibility factor for gas, [-] E : float, optional Pipeline efficiency, a correction factor between 0 and 1 Returns ------- Q, P1, P2, D, or L : float The missing input which was solved for [base SI] Notes ----- This model is presented in [1]_ with a leading constant of 0.6643, the same exponents as used here, units of inches (diameter), psi, feet (length), Rankine, pound/(foot*second) for viscosity, and 1000 ft^3/hour. This model is also presented in [2]_ in both SI and imperial form. Both forms are correct. The imperial version has a leading constant of 136.9, the same powers as used here except with rounded values of powers of viscosity (0.2) and specific gravity (0.8) rearranged to be inside the bracketed group; its units are inches (diameter), psi, miles (length), Rankine, pound/(foot*second) for viscosity, and ft^3/day. This model is shown in base SI units in [3]_, and with a leading constant of 24.6145, and the same powers as used here. Examples -------- >>> IGT(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, mu=1E-5, Tavg=277.15) 48.92351786788815 References ---------- .. [1] Mohitpour, Mo, Golshan, and Allan Murray. Pipeline Design and Construction: A Practical Approach. 3rd edition. New York: Amer Soc Mechanical Engineers, 2006. .. [2] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. .. [3] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations for Steady State Flow in Natural Gas Pipelines." Journal of the Brazilian Society of Mechanical Sciences and Engineering 29, no. 3 (September 2007): 262-73. doi:10.1590/S1678-58782007000300005. ''' # 1000*foot**3/hour*0.6643/inch**(8/3.)*foot**(5/9.)*(5/9.)**(5/9.)*9/5.*(pound/foot)**(1/9.)*psi*(1/psi**2)**(5/9.) c5 = 24.62412451461407054875301709443930350550 # 1084707*196133**(8/9)*2**(1/9)*6**(1/3)/4377968750 c2 = 5/9. # main power c3 = 8/3. # D power c4 = 4/9. # SG power c1 = 1/9. # mu power if Q is None and L is not None and D is not None and P1 is not None and P2 is not None: return c5*Ts/Ps*E*((P1**2-P2**2)/Tavg/L/Zavg)**c2*D**c3/SG**c4/mu**c1 elif D is None and L is not None and Q is not None and P1 is not None and P2 is not None: return (Ps*Q*SG**c4*mu**c1*((P1**2 - P2**2)/(L*Tavg*Zavg))**(-c2)/(E*Ts*c5))**(1./c3) elif P1 is None and L is not None and Q is not None and D is not None and P2 is not None: return sqrt(L*Tavg*Zavg*(D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(1/c2) + P2**2) elif P2 is None and L is not None and Q is not None and D is not None and P1 is not None: return sqrt(-L*Tavg*Zavg*(D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(1/c2) + P1**2) elif L is None and P2 is not None and Q is not None and D is not None and P1 is not None: return (D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(-1/c2)*(P1**2 - P2**2)/(Tavg*Zavg) else: raise ValueError('This function solves for either flow, upstream pressure, downstream pressure, diameter, or length; all other inputs must be provided.')