feos-ad 0.2.3

FeOs-AD - Implicit automatic differentiation of equations of state and phase equilibria.
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242

use super::{A0, A1, A2, AD, B0, B1, B2, BD, CD, MAX_ETA};
use crate::{NamedParameters, ParametersAD, ResidualHelmholtzEnergy};
use nalgebra::SVector;
use num_dual::DualNum;
use std::f64::consts::{FRAC_PI_6, PI};

const PI_SQ_43: f64 = 4.0 / 3.0 * PI * PI;

/// Optimized implementation of PC-SAFT for a single component.
#[derive(Clone, Copy)]
pub struct PcSaftPure<const N: usize>(pub [f64; N]);

fn helmholtz_energy_density_non_assoc<D: DualNum<f64> + Copy>(
    m: D,
    sigma: D,
    epsilon_k: D,
    mu: D,
    temperature: D,
    density: D,
) -> (D, [D; 2]) {
    // temperature dependent segment diameter
    let diameter = sigma * (-(epsilon_k * (-3.) / temperature).exp() * 0.12 + 1.0);

    let eta = m * density * diameter.powi(3) * FRAC_PI_6;
    let eta2 = eta * eta;
    let eta3 = eta2 * eta;
    let eta_m1 = (-eta + 1.0).recip();
    let eta_m2 = eta_m1 * eta_m1;
    let etas = [
        D::one(),
        eta,
        eta2,
        eta3,
        eta2 * eta2,
        eta2 * eta3,
        eta3 * eta3,
    ];

    // hard sphere
    let hs = m * density * (eta * 4.0 - eta2 * 3.0) * eta_m2;

    // hard chain
    let g = (-eta * 0.5 + 1.0) * eta_m1 * eta_m2;
    let hc = -density * (m - 1.0) * g.ln();

    // dispersion
    let e = epsilon_k / temperature;
    let s3 = sigma.powi(3);
    let mut i1 = D::zero();
    let mut i2 = D::zero();
    let m1 = (m - 1.0) / m;
    let m2 = (m - 2.0) / m;
    for i in 0..7 {
        i1 += (m1 * (m2 * A2[i] + A1[i]) + A0[i]) * etas[i];
        i2 += (m1 * (m2 * B2[i] + B1[i]) + B0[i]) * etas[i];
    }
    let c1 = (m * (eta * 8.0 - eta2 * 2.0) * eta_m2 * eta_m2 + 1.0
        - (m - 1.0) * (eta * 20.0 - eta2 * 27.0 + eta2 * eta * 12.0 - eta2 * eta2 * 2.0)
            / ((eta - 1.0) * (eta - 2.0)).powi(2))
    .recip();
    let i = i1 * 2.0 + c1 * i2 * m * e;
    let disp = -density * density * m.powi(2) * e * s3 * i * PI;

    // dipoles
    let mu2 = mu.powi(2) / (m * temperature * 1.380649e-4);
    let m_dipole = if m.re() > 2.0 { D::from(2.0) } else { m };
    let m1 = (m_dipole - 1.0) / m_dipole;
    let m2 = m1 * (m_dipole - 2.0) / m_dipole;
    let mut j1 = D::zero();
    let mut j2 = D::zero();
    for i in 0..5 {
        let a = m2 * AD[i][2] + m1 * AD[i][1] + AD[i][0];
        let b = m2 * BD[i][2] + m1 * BD[i][1] + BD[i][0];
        j1 += (a + b * e) * etas[i];
        if i < 4 {
            j2 += (m2 * CD[i][2] + m1 * CD[i][1] + CD[i][0]) * etas[i];
        }
    }

    // mu is factored out of these expressions to deal with the case where mu=0
    let phi2 = -density * density * j1 / s3 * PI;
    let phi3 = -density * density * density * j2 / s3 * PI_SQ_43;
    let dipole = phi2 * phi2 * mu2 * mu2 / (phi2 - phi3 * mu2);

    ((hs + hc + disp + dipole) * temperature, [eta, eta_m1])
}

fn helmholtz_energy_density<D: DualNum<f64> + Copy>(
    parameters: &[D; 8],
    temperature: D,
    density: D,
) -> D {
    let [m, sigma, epsilon_k, mu, kappa_ab, epsilon_k_ab, na, nb] = *parameters;
    let (non_assoc, [eta, eta_m1]) =
        helmholtz_energy_density_non_assoc(m, sigma, epsilon_k, mu, temperature, density);

    // association
    let delta_assoc = ((epsilon_k_ab / temperature).exp() - 1.0) * sigma.powi(3) * kappa_ab;
    let k = eta * eta_m1;
    let delta = (k * (k * 0.5 + 1.5) + 1.0) * eta_m1 * delta_assoc;
    let rhoa = na * density;
    let rhob = nb * density;
    let aux = (rhoa - rhob) * delta + 1.0;
    let sqrt = (aux * aux + rhob * delta * 4.0).sqrt();
    let xa = (sqrt + 1.0 + (rhob - rhoa) * delta).recip() * 2.0;
    let xb = (sqrt + 1.0 - (rhob - rhoa) * delta).recip() * 2.0;
    let assoc = rhoa * (xa.ln() - xa * 0.5 + 0.5) + rhob * (xb.ln() - xb * 0.5 + 0.5);

    non_assoc + assoc * temperature
}

impl<const N: usize> ParametersAD for PcSaftPure<N> {
    type Parameters<D: DualNum<f64> + Copy> = [D; N];

    fn params<D: DualNum<f64> + Copy>(&self) -> Self::Parameters<D> {
        self.0.map(D::from)
    }

    fn params_from_inner<D: DualNum<f64> + Copy, D2: DualNum<f64, Inner = D> + Copy>(
        parameters: &Self::Parameters<D>,
    ) -> Self::Parameters<D2> {
        parameters.map(D2::from_inner)
    }
}

impl ResidualHelmholtzEnergy<1> for PcSaftPure<8> {
    const RESIDUAL: &str = "PC-SAFT (pure)";

    fn compute_max_density(&self, _: &SVector<f64, 1>) -> f64 {
        let m = self.0[0];
        let sigma = self.0[1];
        MAX_ETA / (FRAC_PI_6 * m * sigma.powi(3))
    }

    fn residual_helmholtz_energy_density<D: DualNum<f64> + Copy>(
        parameters: &Self::Parameters<D>,
        temperature: D,
        partial_density: &SVector<D, 1>,
    ) -> D {
        let density = partial_density.data.0[0][0];
        helmholtz_energy_density(parameters, temperature, density)
    }
}

impl ResidualHelmholtzEnergy<1> for PcSaftPure<4> {
    const RESIDUAL: &str = "PC-SAFT (pure)";

    fn compute_max_density(&self, _: &SVector<f64, 1>) -> f64 {
        let m = self.0[0];
        let sigma = self.0[1];
        MAX_ETA / (FRAC_PI_6 * m * sigma.powi(3))
    }

    fn residual_helmholtz_energy_density<D: DualNum<f64> + Copy>(
        parameters: &Self::Parameters<D>,
        temperature: D,
        partial_density: &SVector<D, 1>,
    ) -> D {
        let density = partial_density.data.0[0][0];
        let [m, sigma, epsilon_k, mu] = *parameters;
        helmholtz_energy_density_non_assoc(m, sigma, epsilon_k, mu, temperature, density).0
    }
}

impl<const N: usize> NamedParameters for PcSaftPure<N> {
    fn index_parameters_mut<'a, D: DualNum<f64> + Copy>(
        parameters: &'a mut [D; N],
        index: &str,
    ) -> &'a mut D {
        match index {
            "m" => &mut parameters[0],
            "sigma" => &mut parameters[1],
            "epsilon_k" => &mut parameters[2],
            "mu" => &mut parameters[3],
            "kappa_ab" => &mut parameters[4],
            "epsilon_k_ab" => &mut parameters[5],
            "na" => &mut parameters[6],
            "nb" => &mut parameters[7],
            _ => panic!("{index} is not a valid PC-SAFT parameter!"),
        }
    }
}

#[cfg(test)]
pub mod test {
    use super::{PcSaftPure, ResidualHelmholtzEnergy};
    use crate::eos::pcsaft::test::pcsaft;
    use approx::assert_relative_eq;
    use feos_core::{Contributions::Total, EosResult, ReferenceSystem, State};
    use nalgebra::SVector;
    use ndarray::arr1;
    use quantity::{KELVIN, KILO, METER, MOL};

    #[test]
    fn test_pcsaft_pure() -> EosResult<()> {
        let (pcsaft, eos) = pcsaft()?;
        let pcsaft = pcsaft.0;

        let temperature = 300.0 * KELVIN;
        let volume = 2.3 * METER * METER * METER;
        let moles = arr1(&[1.3]) * KILO * MOL;

        let state = State::new_nvt(&eos, temperature, volume, &moles)?;
        let a_feos = state.residual_molar_helmholtz_energy();
        let mu_feos = state.residual_chemical_potential();
        let p_feos = state.pressure(Total);
        let s_feos = state.residual_molar_entropy();
        let h_feos = state.residual_molar_enthalpy();

        let total_moles = moles.sum();
        let t = temperature.to_reduced();
        let v = (volume / total_moles).to_reduced();
        let x = SVector::from_fn(|i, _| moles.get(i).convert_into(total_moles));
        let a_ad = PcSaftPure::residual_molar_helmholtz_energy(&pcsaft, t, v, &x);
        let mu_ad = PcSaftPure::residual_chemical_potential(&pcsaft, t, v, &x);
        let p_ad = PcSaftPure::pressure(&pcsaft, t, v, &x);
        let s_ad = PcSaftPure::residual_molar_entropy(&pcsaft, t, v, &x);
        let h_ad = PcSaftPure::residual_molar_enthalpy(&pcsaft, t, v, &x);

        println!("\nMolar Helmholtz energy:\n{}", a_feos.to_reduced(),);
        println!("{a_ad}");
        assert_relative_eq!(a_feos.to_reduced(), a_ad, max_relative = 1e-14);

        println!("\nChemical potential:\n{}", mu_feos.get(0).to_reduced());
        println!("{}", mu_ad[0]);
        assert_relative_eq!(mu_feos.get(0).to_reduced(), mu_ad[0], max_relative = 1e-14);

        println!("\nPressure:\n{}", p_feos.to_reduced());
        println!("{p_ad}");
        assert_relative_eq!(p_feos.to_reduced(), p_ad, max_relative = 1e-14);

        println!("\nMolar entropy:\n{}", s_feos.to_reduced());
        println!("{s_ad}");
        assert_relative_eq!(s_feos.to_reduced(), s_ad, max_relative = 1e-14);

        println!("\nMolar enthalpy:\n{}", h_feos.to_reduced());
        println!("{h_ad}");
        assert_relative_eq!(h_feos.to_reduced(), h_ad, max_relative = 1e-14);

        Ok(())
    }
}