Design Optimization

Using data from an ebulliometer, determine parameters for a Vapor Liquid Equilibrium chart using the Wilson activity coefficient model with measured data for an ethanol-cyclohexane mixture at ambient pressure. Use the results to determine whether there is:

from gekko import GEKKO import numpy as np import pandas as pd import matplotlib.pyplot as plt from scipy.optimize import fsolve from scipy.optimize import curve_fit from scipy.interpolate import interp1d

1. from numpy import *

R = 1.9858775 # cal/mol K

def vaporP(T: float, component: list):

    '''
    :param T: temp [K]
    :param component: [vapor pressure parameters]
    :return: Vapor Pressure [Pa]
    '''
    A = component[0]
    B = component[1]
    C = component[2]
    D = component[3]
    E = component[4]
    return np.exp(A + B/T + C*np.log(T) + D*T**E)

def liquidMolarVolume(T: float, component: list):

    '''
    :param T: temp in [K}
    :param component: component liquid molar volume
    : constants in list or array
    :return: liquid molar volume []
    '''
    A = component[0]
    B = component[1]
    C = component[2]
    D = component[3]
    return 1 / A * B ** (1 + (1 - T / C) ** D)

def Lij(aij: float, T: float, Vlvali: float, Vlvalj: float):

    # Vl1 = liquidMolarVolume(T, Vlvali)
    # Vl2 = liquidMolarVolume(T, Vlvalj)
    Vl1 = Vlvali
    Vl2 = Vlvalj

    return Vl2 / Vl1 * np.exp(-aij / (R * T))

def Gamma(a12: float, a21: float, x1: float, Temp: float, Component: int):

    '''
    :param Component:
    :param a12:
    :param a21:
    :param x1:
    :param Temp:
    :return:
    '''
    # Liquid molar volume
    Vethanol = 5.8492e-2
    VCyclochexane = 0.10882

    x2 = 1.0 - x1
    L12 = Lij(a12, Temp, Vethanol, VCyclochexane)
    L21 = Lij(a21, Temp,  VCyclochexane, Vethanol)
    L11 = 1.0
    L22 = 1.0

    if Component == 1:
        A = x1 + x2 * L12
        B = x1 * L11 / (x1 * L11 + x2 * L12)            +x2 * L21 / (x1 * L21 + x2 * L22)

    elif Component == 2:

        A = x2  + x1 * L21
        B = x2 * L22 / (x2 * L22 + x1 * L21)            +x1 * L12 / (x2 * L12 + x1 * L11)

    return np.exp(1.0 - np.log(A) - B)

1. yi * P = xi * γi * Pisat
2. Component 1: Ethanol
3. Component 2: Cyclohexane

def massTOMoleF(x: list):

    #Converts RI to mole fraction of Ethanol
    MW_Ethanol = 46.06844
    MW_cyclohexane = 84.15948
    xeth = 49.261*x**2 - 152.51*x + 117.32
    xcyc = 1-xeth
    moleth = xeth / MW_Ethanol
    molcyc = xcyc/ MW_cyclohexane
    return moleth/ (moleth + molcyc)

1. Read in actual data
2. filename = 'RealData.csv'

filename = 'http://apmonitor.com/me575/uploads/Main/vle_data.txt' data = pd.read_csv(filename)

temp_data = data['Temp_D'].values + 273.15 # K IndexMeasuredY = data['Distills_RI'] IndexMeasuredX = data['Bottoms_RI'] x1_data = massTOMoleF(IndexMeasuredX) x1_data[0] = 0 x1_data[1] = 1 y1_data = massTOMoleF(IndexMeasuredY) y1_data[0] = 0 y1_data[1] = 1

1. Ethanol & Cyclohexane Properties ---------
2. Psat paramater values

PsatC2H5OH = [73.304, -7122.3, -7.1424, 2.8853e-6, 2.0] PsatC6H12 = [51.087, -5226.4, -4.2278, 9.7554e-18, 6.0]

1. thermo Liquid Molar Volume paramaters

Vl_C2H5OH = [1.6288, 0.27469, 514.0, 0.23178] Vl_C6H12 = [0.88998, 0.27376, 553.8, 0.28571]

1. Build Model ------------------------------

m = GEKKO()

1. Define Variables & Parameters

a12_ = m.FV(value = 2026.3) a12_.STATUS = 1 a21_ = m.FV(value = 390.4) a21_.STATUS = 1 Temp = m.Var(333) x1 = m.Param(value = np.array(x1_data)) y1 = m.CV(value = np.array(y1_data)) y1.FSTATUS = 1 # min fstatus * (meas-pred)^2 y2 = m.Intermediate(1.0 - y1) P = 85900 # Pa in Provo, UT

1. Calc Vapor pressures of ethanol & cyclohexane at
2. different temps using thermo parameters
3. Psat = exp(A + B / T + C * log(T) + D * T ** E)

PsatEthanol = m.Intermediate(

        m.exp(
                PsatC2H5OH[0] +
                PsatC2H5OH[1] / Temp +
                PsatC2H5OH[2] * m.log(Temp) +
                PsatC2H5OH[3] * Temp ** PsatC2H5OH[4]
        )

)

PsatCyclohexane = m.Intermediate(

        m.exp(
                PsatC6H12[0] +
                PsatC6H12[1] / Temp +
                PsatC6H12[2] * m.log(Temp) +
                PsatC6H12[3] * Temp ** PsatC6H12[4]
        )

)

1. Calculate Liquid Molar volumes of components 1 and 2:
2. ethanol & cyclohexane, respectively

LMV_C2H5OH = 5.8492e-2 LMV_C6H12 = 0.10882

1. Construct parameters for Wilson Model

L12 = m.Intermediate(LMV_C6H12 / LMV_C2H5OH * m.exp(-a12_ / (R * Temp))) L21 = m.Intermediate(LMV_C2H5OH / LMV_C6H12 * m.exp(-a21_ / (R * Temp))) L11 = 1.0 L22 = 1.0

x2 = m.Intermediate(1.0 - x1)

A1 = m.Intermediate(x1 + x2 * L12) B1 = m.Intermediate(x1 * L11 / (x1 * L11 + x2 * L12) + x2 * L21 / (x1 * L21 + x2 * L22))

A2 = m.Intermediate(x2 + x1 * L21) B2 = m.Intermediate(x2 * L22 / (x2 * L22 + x1 * L21) + x1 * L12 / (x2 * L12 + x1 * L11))

1. Find Gammas for each component

G_C2H5OH = m.Intermediate(m.exp(1.0 - m.log(A1) - B1)) G_C6H12 = m.Intermediate(m.exp(1.0 - m.log(A2) - B2))

m.Equation(y2 == (G_C6H12 * PsatCyclohexane / P) * x2) m.Equation(y1 == (G_C2H5OH * PsatEthanol / P) * x1)

1. Options

m.options.IMODE = 2 m.options.EV_TYPE = 2 # Objective type, SSE m.options.NODES = 3 # Collocation nodes m.options.SOLVER = 3 # IPOPT

m.solve()

regressed_a12 = a12_.value[-1] regressed_a21 = a21_.value[-1]

a12_Hysys = 2026.3 a21_Hysys = 390.4

x1_linspace = np.linspace(0, 1.0, 100) y1_predicted = [] T_predicted = []

print("Activity Coefficient Parameters") print('Model a12 a21') print('Gekko: ' + str(round(regressed_a12,2)) + str(round(regressed_a21,2))) print('Hysys: ' + str(round(a12_Hysys,2)) + str(round(a21_Hysys,2)))

def getValues(y1_and_Temp: list, a12: float, a21: float, P: float, x1: float):

    y1 = y1_and_Temp[0]
    T = y1_and_Temp[1]

    y2 = 1 - y1
    x2 = 1 - x1

    A = y1*P - x1*Gamma(a12,a21,x1,T,1)         * vaporP(T,PsatC2H5OH)
    B = y2*P - x2*Gamma(a12,a21,x1,T,2)         * vaporP(T,PsatC6H12)

    return [A,B]

TempInterp = interp1d(x1_data, temp_data) for i in range(len(x1_linspace)):

    answers = fsolve(getValues, [x1_linspace[i],                     TempInterp(x1_linspace[i])],                     args = (regressed_a12,                              regressed_a21,                              P, x1_linspace[i]))
    y1_predicted.append(answers[0])
    T_predicted.append(answers[1])

GammaPred = Gamma(regressed_a12, regressed_a21, x1_data, temp_data, 1) GammaPredHysys = Gamma(a12_Hysys, a21_Hysys, x1_data, temp_data, 2)

VP_Ethanol = vaporP(temp_data, PsatC2H5OH)

y1Predicted = (GammaPred * VP_Ethanol / P) * x1_data y1PredictedHysys = (GammaPredHysys * VP_Ethanol / P) * x1_data

plt.figure(1) plt.plot(x1_data,y1_data,"bx",label = "Measured Data") plt.plot(x1_linspace,x1_linspace,"-", color="#8bd8bd", label="Reference",markersize=7) plt.plot(x1_linspace,y1_predicted,"--", color="#243665",label = "Model",markersize=7) plt.plot(x1_data, y1Predicted, "o",

         color="#ec8b5e",label="Predicted",markersize=5)

plt.legend(loc = "best") plt.xlabel("x") plt.ylabel("y") plt.grid()

plt.figure(2) plt.plot(x1_data, temp_data, "go") plt.plot(y1_data, temp_data, "bo") plt.plot(x1_linspace, T_predicted, "r--",) plt.plot(y1_predicted, T_predicted, "k--",) plt.legend(["$x_1$ Measured", "$y_1$ Measured", "$x_1$ Predicted", "$y_1$ Predicted"]) plt.xlabel("$x, y$") plt.ylabel("$Temp$") plt.grid()

plt.show()

APM Python

Python GEKKO

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• Nonlinear Confidence Interval in Python (zip)

• Nonlinear Confidence Interval in Excel (zip)

Nonlinear Confidence Interval

• Nonlinear Confidence Interval in Python (zip)

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Nonlinear confidence intervals can be visualized as a function of 2 parameters. In this case, both parameters are simultaneously varied to find the confidence region. The confidence interval is determined with an F-test that specifies an upper limit to the deviation from the optimal solution

with p=2 (number of parameters), n=number of measurements, theta=[parameter 1, parameter 2] (parameters), theta^* as the optimal parameters, SSE as the sum of squared errors, and the F statistic that has 3 arguments (alpha=confidence level, degrees of freedom 1, and degrees of freedom 2). For many problems, this creates a multi-dimensional nonlinear confidence region. In the case of 2 parameters, the nonlinear confidence region is a 2-dimensional space. Below is an example that shows the confidence region for the dye fading experiment confidence region for forward and reverse activation energies.

The optimal parameter values are in the 95% confidence region. This plot demonstrates that the 2D confidence region is not necessarily symmetric.

Thermodynamic Parameter Estimation

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Using data from an ebulliometer, determine parameters for the Wilson activity coefficient model using the measured data for an ethanol-cyclohexane mixture at ambient pressure. Use the results to determine whether there is:

where y₁is the vapor mole fraction, P is the pressure, x₁ is the liquid mole fraction, gamma₁ is the activity coefficient that is different than 1.0 for non-ideal mixtures, and P₁^sat is the pure component vapor pressure. The same equation also applies to component 2 in the mixture with the corresponding equation with subscript 2.

There are correlations for P^sat₁ and density (rho) for many common pure components from BYU's DIPPR database. In this case P₁^sat is a function of temperature according to

and density rho or molar volume v is also a function of temperature according to

(:title Estimate Thermodynamic Parameters from Data:) (:keywords VLE, wilson equation, nonlinear, optimization, engineering optimization, dynamic estimation, interior point, active set, differential, algebraic, modeling language, university course:) (:description Case study on data reconciliation for thermodynamic properties using optimization techniques in engineering:)

Case Study on Thermodynamic Parameter Estimation

Using data from an ebulliometer, determine parameters for the Wilson activity coefficient model using the measured data for an ethanol-cyclohexane mixture at ambient pressure. Use the results to determine whether there is:

• An azeotrope in the system and, if so, at what composition
• The values of the activity coefficients at the infinitely dilute compositions
  • gamma₁ at x₁=0
  • gamma₂ at x₁=1

The liquid and vapor compositions of this binary mixture are related by the following thermodynamic relationships

where y₁is the vapor mole fraction, P is the pressure, x₁ is the liquid mole fraction, gamma₁ is the activity coefficient that is different than 1.0 for non-ideal mixtures, and P^sat₁ is the pure component vapor pressure. The same equation also applies to component 2 in the mixture with the corresponding equation with subscript 2.

The Wilson equation is used to predict the activity coefficients gamma₂ and gamma₂ over the range of liquid compositions.

There are correlations for P^sat₁ for many common pure components from BYU's DIPPR database. In this case P^sat₁ is a function of temperature according to

The number of degrees of freedom in a multi-component and multi-phase system is given by DOF = 2 + #Components - #Phases. In this case, there are two phases (liquid and vapor) and two components (ethanol and cyclohexane). This leads to two degrees of freedom that must be specified. In this case, we can chose to fix two of the four measured values for this system with either x₁, y₁, P, or T. It is recommended to fix the values of x₁ and P as shown in the tutorial below.

Background on Parameter Estimation

A common application of optimization is to estimate parameters from experimental data. One of the most common forms of parameter estimation is the least squares objective with (model-measurement)^2 summed over all of the data points. The optimization problem is subject to the model equations that relate the model parameters or exogenous inputs to the predicted measurements. The model predictions are connected by common parameters that are adjusted to minimize the sum of squared errors.

Tutorial on Parameter Estimation

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