QUADRATURE ALGORITHMS FOR PHASE EQUILIBRIUM OF CONTINUOUS MIXTURES

Chicralla, Felipe C.; Lage, Paulo L. da C.; Secchi, Argimiro R.

doi:10.1590/0104-6632.20190363s20180522

Felipe C. Chicralla

Universidade Federal do Rio de Janeiro, Programa de Engenharia Química, Rio de Janeiro, RJ, Brasil. E-mail: paulo@peq.coppe.ufrj.br

http://orcid.org/0000-0003-1046-4741

Paulo L. da C. Lage

Universidade Federal do Rio de Janeiro, Programa de Engenharia Química, Rio de Janeiro, RJ, Brasil. E-mail: paulo@peq.coppe.ufrj.br

http://orcid.org/0000-0002-0396-5508

Argimiro R. Secchi

Universidade Federal do Rio de Janeiro, Programa de Engenharia Química, Rio de Janeiro, RJ, Brasil. E-mail: paulo@peq.coppe.ufrj.br

http://orcid.org/0000-0001-7297-3571

* Corresponding author: Paulo L. da C. Lage - E-mail: paulo@peq.coppe.ufrj.br

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Figure 1
Distribution functions 1 and 2 used to analyze the quadrature algorithms.

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Table 1
Parameters and truncation intervals for the two gamma distributions.

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Table 2
Adiabatic flash conditions.

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Table 3
Methods analyzed.

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Table 4
Parameters of the Jacobi polynomial families.

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Table 5
Computational cost for the generation of the Gauss-Christoffel quadrature for the distribution 1, in clocks.

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Table 6
Computational cost for the generation of the Gauss-Christoffel quadrature for the distribution 2, in clocks.

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Table 7
MSRE in the reconstructed moments for the distribution 1.

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Table 8
MSRE in the reconstructed moments for the distribution 2.

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Table 9
First 20 generalized and regular moments for the distribution 1 for several C values in the distribution variable definition.

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Table 10
First 20 generalized and regular moments for the distribution 2 for several C values in the distribution variable definition.

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Table 11
Maximum number of quadrature points obtained for the Gauss-Christoffel quadrature for the distribution 1 for several C values in the distribution variable definition and the corresponding MSRE values.

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Table 12
Maximum number of quadrature points obtained for the Gauss-Christoffel quadrature for the distribution 2 for several C values in the distribution variable definition and the corresponding MSRE values.

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Table 13
Generalized moments for the distribution 1 for some orthogonal polynomial families.

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Table 14
Generalized moments for the distribution 2 for some orthogonal polynomial families.

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Table 15
Maximum number of quadrature points obtained for the Gauss-Christoffel quadrature for the distribution 1 for some orthogonal polynomial families and the corresponding MSRE values.

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Table 16
Maximum number of quadrature points obtained for the Gauss-Christoffel quadrature for the distribution 2 for some orthogonal polynomial families and the corresponding MSRE values.

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Table 17
Stream properties for the adiabatic flash for distribution 1.

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Table 18
Stream properties for the adiabatic flash for distribution 2.

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Distribution 1	Distribution 2
A = 2.1	A = 4.0
B = 26.7	B = 35.0
M₀ = 100	M₀ = 100
M = [100, 300]	M = [100, 450]

	Distribution 1	Distribution 2
Feed temperature, T^F	500 K	625 K
Feed pressure, P^F	2 bar	3 bar
Flash pressure, P^flash	1 bar	1 bar

Method	Type of moments
LQMDA (Sack and Donovan, 1972)	Generalized
Chebyshev Algorithm (Chebyshev, 1858)	Generalized
LQMDA (Sack and Donovan, 1972)	Regular
Chebyshev Algorithm (Chebyshev, 1858)	Regular
GWA (Golub and Welsch, 1969)	Regular
PDA (Gordon, 1968) - PDA1	Regular
PDA (Lage, 2007) - PDA2	Regular

Polynomial Family	α	β
Chebyshev (1^st order)	-0.5	-0.5
Chebyshev (2^nd order)	0.5	0.5
Legendre	0	0

n	LQMDA	Cheb	LQMDA	Cheb	GWA	PDA2	PDA1
n	gen	gen	reg	reg	reg	reg	reg
3	135.3 ± 1.2	8.7 ± 0.5	6.0 ± 0.0	3.9 ± 0.4	4.5 ± 0.6	16.1 ± 0.6	6.7 ± 0.5
4	141.8 ± 6.1	10.7 ± 0.5	7.8 ± 0.5	5.9 ± 0.4	7.1 ± 0.6	22.7 ± 1.8	11.8 ± 1.1
5	143.9 ± 12.3	13.1 ± 2.0	10.0 ± 0.9	7.9 ± 0.8	9.4 ± 1.0	29.8 ± 3.0	11.3 ± 1.0
6	149.1 ± 0.9	16.4 ± 0.6	13.4 ± 0.6	11.5 ± 0.6	16.1 ± 0.4	37.1 ± 1.2	14.8 ± 0.5
7	151.7 ± 1.4	19.1 ± 0.4	15.9 ± 0.4	1.4 ± 0.6	17.5 ± 0.6	46.0 ± 1.2	NA
8	155.4 ± 2.9	23.6 ± 0.7	22.5 ± 0.8	17.9 ± 0.8	24.0 ± 4.0	59.3 ± 5.4	NA
9	157.9 ± 13.5	27.4 ± 3.4	27.9 ± 6.1	22.3 ± 2.3	30.2 ± 5.1	65.3 ± 8.3	NA
10	167.9 ± 4.9	34.8 ± 0.5	30.3 ± 3.9	32.1 ± 6.8	34.8 ± 0.5	86.1 ± 10.0	NA
11	174.0 ± 6.6	42.4 ± 5.6	35.9 ± 4.0	34.7 ± 1.0	42.9 ± 0.4	98.6 ± 5.6	NA
12	185.0 ± 6.9	51.4 ± 3.2	42.3 ± 0.5	39.6 ± 5.0	59.9 ± 6.5	112.8 ± 8.7	NA
13	188.0 ± 6.8	52.5 ± 2.1	NA	NA	NA	NA	NA
14	198.3 ± 4.2	60.1 ± 0.4	NA	NA	NA	NA	NA
15	211.0 ± 8.0	68.3 ± 0.7	NA	NA	NA	NA	NA
16	218.9 ± 7.3	79.9 ± 4.5	NA	NA	NA	NA	NA
17	228.3 ± 6.0	88.4 ± 2.8	NA	NA	NA	NA	NA
18	236.3 ± 6.8	91.6 ± 7.0	NA	NA	NA	NA	NA
19	249.0 ± 7.2	108.8 ± 1.3	NA	NA	NA	NA	NA
20	259.2 ± 4.7	119.5 ± 1.2	NA	NA	NA	NA	NA

n	LQMDA	Cheb	LQMDA	Cheb	GWA	PDA2	PDA1
n	gen	gen	reg	reg	reg	reg	reg
3	135.4 ± 1.1	8.4 ± 0.5	6.0 ± 0.0	3.9 ± 0.3	5.0 ± 0.0	16.5 ± 1.1	7.0 ± 0.0
4	142.4 ± 1.1	10.6 ± 0.5	8.0 ± 0.0	6.0 ± 0.0	7.3 ± 0.5	22.3 ± 1.3	11.5 ± 0.7
5	145.0 ± 1.3	12.9 ± 0.3	10.1 ± 0.3	8.4 ± 0.5	9.9 ± 0.3	30.9 ± 1.1	11.5 ± 0.5
6	148.3 ± 0.7	16.3 ± 0.5	13.5 ± 0.5	11.5 ± 0.5	16.3 ± 0.5	37.8 ± 1.1	14.9 ± 0.3
7	152.3 ± 1.5	20.0 ± 0.5	16.9 ± 0.3	17.2 ± 0.4	18.0 ± 0.5	46.2 ± 1.1	NA
8	156.7 ± 1.6	26.6 ± 10.0	22.3 ± 0.7	19.9 ± 5.3	22.6 ± 0.5	57.6 ± 0.7	NA
9	160.5 ± 1.7	27.9 ± 0.6	26.8 ± 0.6	23.4 ± 3.1	33.4 ± 5.5	69.7 ± 3.5	NA
10	167.0 ± 3.1	35.9 ± 0.3	29.8 ± 0.4	29.6 ± 0.5	35.1 ± 0.3	83.1 ± 0.7	NA
11	172.2 ± 4.1	42.5 ± 6.5	36.7 ± 7.5	35.9 ± 3.4	42.1 ± 0.3	99.7 ± 6.1	NA
12	179.2 ± 3.0	49.4 ± 5.9	NA	NA	NA	NA	NA
13	201.7 ± 6.4	57.2 ± 1.1	NA	NA	NA	NA	NA
14	200.4 ± 8.1	58.9 ± 3.9	NA	NA	NA	NA	NA
15	215.0 ± 5.4	68.0 ± 2.3	NA	NA	NA	NA	NA
16	224.8 ± 9.7	78.7 ± 0.7	NA	NA	NA	NA	NA
17	227.0 ± 5.6	83.8 ± 0.8	NA	NA	NA	NA	NA
18	231.9 ± 1.4	94.9 ± 0.6	NA	NA	NA	NA	NA
19	250.6 ± 7.8	111.0 ± 3.8	NA	NA	NA	NA	NA
20	252.4 ± 2.0	116.2 ± 1.1	NA	NA	NA	NA	NA

n	LQMDA	Cheb	LQMDA	Cheb	GWA	PDA2	PDA1
n	gen	gen	reg	reg	reg	reg	reg
3	7.02×10^-16	1.16×10^-15	9.01×10^-16	3.16×10^-15	3.25×10^-13	9.40×10^-16	9.40×10^-16
4	2.31×10^-15	4.73×10^-15	2.42×10^-15	1.87×10^-15	2.97×10^-4	7.78×10^-16	7.78×10^-16
5	9.74×10^-15	1.10×10^-14	6.60×10^-15	1.09×10^-15	2.42×10^-5	4.16×10^-15	4.16×10^-15
6	9.24×10^-15	1.28×10^-14	1.07×10^-14	7.35×10^-16	1.83×10^-6	3.40×10^-15	3.40×10^-15
7	2.61×10^-14	1.04×10^-14	1.07×10^-15	5.70×10^-15	1.33×10^-7	3.81×10^-15	NA
8	2.10×10^-14	1.01×10^-14	6.30×10^-15	6.58×10^-15	9.40×10^-9	1.28×10^-14	NA
9	5.70×10^-14	7.03×10^-14	4.21×10^-15	3.87×10^-15	6.50×10^-10	3.90×10^-16	NA
10	4.50×10^-14	4.31×10^-14	2.12×10^-14	7.98×10^-15	4.46×10^-11	2.83×10^-15	NA
11	1.80×10^-13	1.65×10^-13	2.57×10^-15	8.30×10^-15	2.76×10^-12	1.76×10^-14	NA
12	2.60×10^-13	8.14×10^-14	3.58×10^-15	6.71×10^-15	3.63×10^-12	6.61×10^-15	NA
13	5.77×10^-13	2.59×10^-13	NA	NA	NA	NA	NA
14	1.09×10^-12	4.95×10^-13	NA	NA	NA	NA	NA
15	2.38×10^-13	1.69×10^-13	NA	NA	NA	NA	NA
16	9.80×10^-13	2.46×10^-13	NA	NA	NA	NA	NA
17	4.78×10^-13	5.53×10^-13	NA	NA	NA	NA	NA
18	1.07×10^-12	8.09×10^-13	NA	NA	NA	NA	NA
19	1.77×10^-13	1.69×10^-13	NA	NA	NA	NA	NA
20	4.35×10^-13	2.51×10^-13	NA	NA	NA	NA	NA

n	LQMDA	Cheb	LQMDA	Cheb	GWA	PDA2	PDA1
n	gen	gen	reg	reg	reg	reg	reg
3	1.73×10^-14	3.73×10^-15	1.99×10^-15	4.17×10^-16	1.32×10^-13	2.61×10^-15	2.61×10^-15
4	4.64×10^-14	3.78×10^-14	2.56×10^-15	1.09×10^-15	1.12×10^-4	2.84×10^-15	2.84×10^-15
5	4.18×10^-14	4.21×10^-14	1.19×10^-15	1.19×10^-15	8.70×10^-6	4.23×10^-15	4.23×10^-15
6	2.20×10^-14	1.54×10^-14	5.35×10^-15	4.24×10^-15	6.37×10^-7	3.84×10^-15	3.84×10^-15
7	2.51×10^-14	3.57×10^-14	2.38×10^-15	3.80×10^-15	4.50×10^-8	3.44×10^-15	NA
8	4.90×10^-14	4.16×10^-14	3.84×10^-15	1.83×10^-15	3.10×10^-9	1.22×10^-15	NA
9	3.61×10^-14	5.33×10^-14	6.85×10^-15	2.20×10^-15	2.11×10^-10	6.61×10^-15	NA
10	4.83×10^-14	7.99×10^-14	2.82×10^-15	1.02×10^-14	1.38×10^-11	6.18×10^-15	NA
11	8.69×10^-14	1.47×10^-13	3.25×10^-15	3.06×10^-15	4.56×10^-13	4.43×10^-15	NA
12	7.25×10^-14	6.69×10^-14	NA	NA	NA	NA	NA
13	1.21×10^-13	7.38×10^-14	NA	NA	NA	NA	NA
14	1.66×10^-13	4.61×10^-14	NA	NA	NA	NA	NA
15	6.66×10^-14	4.69×10^-14	NA	NA	NA	NA	NA
16	6.87×10^-14	6.04×10^-14	NA	NA	NA	NA	NA
17	2.54×10^-13	2.58×10^-13	NA	NA	NA	NA	NA
18	2.79×10^-13	4.22×10^-13	NA	NA	NA	NA	NA
19	4.45×10^-13	4.07×10^-13	NA	NA	NA	NA	NA
20	3.61×10^-13	4.88×10^-14	NA	NA	NA	NA	NA

C	0.5		1.0		1.333		1.5		2.0
k	m_k^(P)	mk	m_k^(P)	mk	m_k^(P)	mk	m_k^(P)	mk	m_k^(P)	mk
0	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰
1	-1.35×10⁰	1.38×10^-1	-1.35×10⁰	2.75×10^-1	-1.35×10⁰	3.67×10^-1	-1.35×10⁰	4.13×10^-1	-1.35×10⁰	5.51×10^-1
2	1.34×10⁰	2.73×10^-2	1.34×10⁰	1.09×10^-1	1.34×10⁰	1.94×10^-1	1.34×10⁰	2.45×10^-1	1.34×10⁰	4.36×10^-1
3	-1.02×10⁰	6.85×10^-13	-1.02×10⁰	5.48×10^-2	-1.02×10⁰	1.30×10^-1	-1.02×10⁰	1.85×10^-1	-1.02×10⁰	4.38×10^-1
4	8.39×10^-1	2.02×10^-13	8.39×10^-1	3.24×10^-2	8.39×10^-1	1.02×10^-1	8.39×10^-1	1.64×10^-1	8.39×10^-1	5.18×10^-1
5	-5.85×10^-1	6.71×10^-4	-5.85×10^-1	2.15×10^-2	-5.85×10^-1	9.03×10^-2	-5.85×10^-1	1.63×10^-1	-5.85×10^-1	6.87×10^-1
6	5.12×10^-1	2.41×10^-4	5.12×10^-1	1.54×10^-2	5.12×10^-1	8.65×10^-2	5.12×10^-1	1.76×10^-1	5.12×10^-1	9.87×10^-1
7	-3.50×10^-1	9.19×10^-5	-3.50×10^-1	1.18×10^-2	-3.50×10^-1	8.80×10^-2	-3.50×10^-1	2.01×10^-1	-3.50×10^-1	1.51×10⁰
8	3.43×10^-1	3.66×10^-5	3.43×10^-1	9.38×10^-13	3.43×10^-1	9.35×10^-2	3.43×10^-1	2.40×10^-1	3.43×10^-1	2.40×10⁰
9	-2.25×10^-1	1.51×10^-5	-2.25×10^-1	7.73×10^-13	-2.25×10^-1	1.03×10^-1	-2.25×10^-1	2.97×10^-1	-2.25×10^-1	3.96×10⁰
10	2.50×10^-1	6.38×10^-6	2.50×10^-1	6.53×10^-13	2.50×10^-1	1.16×10^-1	2.50×10^-1	3.77×10^-1	2.50×10^-1	6.69×10⁰
11	-1.52×10^-1	2.75×10^-6	-1.52×10^-1	5.64×10^-13	-1.52×10^-1	1.33×10^-1	-1.52×10^-1	4.88×10^-1	-1.52×10^-1	1.16×10¹
12	1.93×10^-1	1.21×10^-6	1.93×10^-1	4.95×10^-13	1.93×10^-1	1.56×10^-1	1.93×10^-1	6.42×10^-1	1.93×10^-1	2.03×10¹
13	-1.05×10^-1	5.37×10^-7	-1.05×10^-1	4.40×10^-13	-1.05×10^-1	1.85×10^-1	-1.05×10^-1	8.57×10^-1	-1.05×10^-1	3.61×10¹
14	1.55×10^-1	2.42×10^-7	1.55×10^-1	3.96×10^-13	1.55×10^-1	2.21×10^-1	1.55×10^-1	1.16×10⁰	1.55×10^-1	6.49×10¹
15	-7.41×10^-2	1.10×10^-7	-7.41×10^-2	3.59×10^-13	-7.41×10^-2	2.68×10^-1	-7.41×10^-2	1.57×10⁰	-7.41×10^-2	1.18×10²
16	1.30×10^-1	5.02×10^-8	1.30×10^-1	3.29×10^-13	1.30×10^-1	3.27×10^-1	1.30×10^-1	2.16×10⁰	1.30×10^-1	2.15×10²
17	-5.20×10^-2	2.31×10^-8	-5.20×10^-2	3.03×10^-13	-5.20×10^-2	4.01×10^-1	-5.20×10^-2	2.98×10⁰	-5.20×10^-2	3.97×10²
18	1.11×10^-1	1.07×10^-8	1.11×10^-1	2.81×10^-13	1.11×10^-1	4.95×10^-1	1.11×10^-1	4.15×10⁰	1.11×10^-1	7.35×10²
19	-3.56×10^-2	4.98×10^-9	-3.56×10^-2	2.61×10^-13	-3.56×10^-2	6.15×10^-1	-3.56×10^-2	5.79×10⁰	-3.56×10^-2	1.37×10³

C	0.5		1.0		1.333		1.5		2.0
k	m_k^(P)	mk	m_k^(P)	mk	m_k^(P)	k	m_k^(P)	mk	m_k^(P)	mk
0	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰
1	-6.46×10^-1	1.96×10^-1	-6.46×10^-1	3.92×10^-1	-6.46×10^-1	5.23×10^-1	-6.46×10^-1	5.89×10^-1	-6.46×10^-1	7.85×10^-1
2	2.93×10^-1	4.71×10^-2	2.93×10^-1	1.89×10^-1	2.93×10^-1	3.35×10^-1	2.93×10^-1	4.24×10^-1	2.93×10^-1	7.54×10^-1
3	1.09×10^-1	1.32×10^-2	1.09×10^-1	1.05×10^-1	1.09×10^-1	2.50×10^-1	1.09×10^-1	3.56×10^-1	1.09×10^-1	8.44×10^-1
4	-2.17×10^-2	4.14×10^-13	-2.17×10^-2	6.62×10^-2	-2.17×10^-2	2.09×10^-1	-2.17×10^-2	3.35×10^-1	-2.17×10^-2	1.06×10⁰
5	1.25×10^-1	1.42×10^-13	1.25×10^-1	4.53×10^-2	1.25×10^-1	1.91×10^-1	1.25×10^-1	3.44×10^-1	1.25×10^-1	1.45×10⁰
6	4.10×10^-2	5.18×10^-4	4.10×10^-2	3.31×10^-2	4.10×10^-2	1.86×10^-1	4.10×10^-2	3.77×10^-1	4.10×10^-2	2.12×10⁰
7	7.69×10^-2	1.99×10^-4	7.69×10^-2	2.55×10^-2	7.69×10^-2	1.91×10^-1	7.69×10^-2	4.35×10^-1	7.69×10^-2	3.26×10⁰
8	6.49×10^-2	7.96×10^-5	6.49×10^-2	2.04×10^-2	6.49×10^-2	2.03×10^-1	6.49×10^-2	5.22×10^-1	6.49×10^-2	5.22×10⁰
9	6.89×10^-2	3.28×10^-5	6.89×10^-2	1.68×10^-2	6.89×10^-2	2.23×10^-1	6.89×10^-2	6.46×10^-1	6.89×10^-2	8.61×10⁰
10	6.83×10^-2	1.39×10^-5	6.83×10^-2	1.42×10^-2	6.83×10^-2	2.52×10^-1	6.83×10^-2	8.20×10^-1	6.83×10^-2	1.46×10¹
11	6.89×10^-2	5.98×10^-6	6.89×10^-2	1.23×10^-2	6.89×10^-2	2.89×10^-1	6.89×10^-2	1.06×10⁰	6.89×10^-2	2.51×10¹
12	6.91×10^-2	2.62×10^-6	6.91×10^-2	1.07×10^-2	6.91×10^-2	3.38×10^-1	6.91×10^-2	1.39×10⁰	6.91×10^-2	4.40×10¹
13	6.94×10^-2	1.16×10^-6	6.94×10^-2	9.53×10^-13	6.94×10^-2	4.00×10^-1	6.94×10^-2	1.85×10⁰	6.94×10^-2	7.81×10¹
14	6.97×10^-2	5.22×10^-7	6.97×10^-2	8.55×10^-13	6.97×10^-2	4.78×10^-1	6.97×10^-2	2.50×10⁰	6.97×10^-2	1.40×10²
15	7.00×10^-2	2.36×10^-7	7.00×10^-2	7.75×10^-13	7.00×10^-2	5.78×10^-1	7.00×10^-2	3.39×10⁰	7.00×10^-2	2.54×10²
16	7.02×10^-2	1.08×10^-7	7.02×10^-2	7.08×10^-13	7.02×10^-2	7.03×10^-1	7.02×10^-2	4.65×10⁰	7.02×10^-2	4.64×10²
17	7.05×10^-2	4.96×10^-8	7.05×10^-2	6.51×10^-13	7.05×10^-2	8.62×10^-1	7.05×10^-2	6.41×10⁰	7.05×10^-2	8.53×10²
18	7.07×10^-2	2.30×10^-8	7.07×10^-2	6.02×10^-13	7.07×10^-2	1.06×10⁰	7.07×10^-2	8.90×10⁰	7.07×10^-2	1.58×10³
19	7.09×10^-2	1.07×10^-8	7.09×10^-2	5.60×10^-13	7.09×10^-2	1.32×10⁰	7.09×10^-2	1.24×10¹	7.09×10^-2	2.93×10³

C	LQMDA-gen		Cheb-gen		LQMDA-reg		Cheb-reg		GWA		PDA2		PDA1
C	n_max	MSRE	n_max	MSRE	n_max	MSRE	n_max	MSRE	n_max	MSRE	n_max	MSRE	n_max	MSRE
0.500	85	5.57×10^-11	85	2.42×10^-11	12	3.58×10^-15	12	6.71×10^-15	12	3.63×10^-12	12	6.61×10^-15	6	3.40×10^-15
0.700	85	2.29×10^-11	85	3.14×10^-12	11	7.57×10^-15	11	9.90×10^-16	11	5.21×10^-15	11	4.42×10^-12	6	5.13×10^-15
0.900	85	3.43×10^-12	85	4.32×10^-11	11	1.07×10^-14	11	8.01×10^-15	11	4.92×10^-12	11	3.36×10^-15	6	6.64×10^-15
1.000	85	5.57×10^-11	85	2.42×10^-11	12	3.58×10^-15	12	6.71×10^-15	12	3.63×10^-12	12	6.61×10^-15	6	3.40×10^-15
1.100	85	3.35×10^-11	85	6.93×10^-12	11	1.91×10^-14	11	1.02×10^-14	11	4.45×10^-12	11	1.20×10^-15	6	2.17×10^-15
1.200	85	1.95×10^-11	85	2.90×10^-12	11	3.74×10^-15	11	9.52×10^-15	11	4.07×10^-12	11	1.49×10^-14	6	3.73×10^-15
1.300	85	2.89×10^-11	85	3.13×10^-11	12	4.87×10^-15	12	2.79×10^-15	12	6.61×10^-12	12	5.05×10^-15	6	3.74×10^-15
1.333	85	3.42×10^-12	85	1.83×10^-11	11	9.82×10^-15	11	1.19×10^-14	11	4.78×10^-12	11	3.85×10^-15	7	4.25×10^-12
1.500	85	2.60×10^-11	85	2.96×10^-12	11	7.79×10^-15	11	5.38×10^-15	11	4.53×10^-12	11	1.69×10^-14	7	8.43×10^-15
1.750	85	1.93×10^-11	85	4.96×10^-12	11	9.17×10^-15	11	9.79×10^-15	11	4.13×10^-12	11	1.78×10^-14	7	1.20×10^-14
2.000	85	5.57×10^-11	85	2.42×10^-11	12	3.58×10^-15	12	6.71×10^-15	12	3.63×10^-12	12	6.61×10^-15	7	3.81×10^-15
2.500	85	6.55×10^-12	85	9.60×10^-12	11	4.17×10^-15	11	8.29×10^-15	11	2.89×10^-12	11	2.07×10^-14	7	1.05×10^-14
3.000	85	2.60×10^-11	85	2.96×10^-12	11	7.79×10^-15	11	5.38×10^-15	11	4.53×10^-12	11	1.69×10^-14	8	4.66×10^-15

C	LQMDA-gen		Cheb-gen		LQMDA-reg		Cheb-reg		GWA		PDA2		PDA1
C	n_max	MSRE	n_max	MSRE	n_max	MSRE	n_max	MSRE	n_max	MSRE	n_max	MSRE	n_max	MSRE
0.500	84	4.90×10^-12	84	9.46×10^-12	11	3.25×10^-15	11	3.06×10^-15	11	4.56×10^-13	11	4.43×10^-15	6	3.84×10^-15
0.700	84	1.48×10^-13	84	2.73×10^-12	11	1.20×10^-14	11	1.25×10^-15	11	7.95×10^-14	11	1.25×10^-14	6	1.87×10^-15
0.900	84	8.41×10^-12	84	7.04×10^-12	11	3.42×10^-15	11	5.93×10^-15	11	9.65×10^-13	11	9.19×10^-16	6	1.09×10^-15
1.000	84	4.90×10^-12	84	9.46×10^-12	11	3.25×10^-15	11	3.06×10^-15	11	4.56×10^-13	11	4.43×10^-15	6	3.84×10^-15
1.100	84	4.76×10^-12	84	5.30×10^-12	11	2.74×10^-15	11	4.42×10^-15	11	2.20×10^-13	11	1.15×10^-14	6	6.78×10^-15
1.200	84	3.85×10^-12	84	4.70×10^-12	11	6.56×10^-15	11	1.49×10^-15	11	1.49×10^-12	11	7.28×10^-15	6	2.77×10^-15
1.300	84	2.84×10^-12	84	1.72×10^-11	11	3.01×10^-15	11	1.22×10^-15	11	2.33×10^-12	11	1.09×10^-14	7	7.56×10^-15
1.333	84	8.95×10^-12	84	7.35×10^-12	11	7.17×10^-15	11	1.37×10^-14	11	1.44×10^-12	11	3.50×10^-15	7	1.56×10^-15
1.500	84	1.01×10^-11	84	1.00×10^-11	11	7.12×10^-15	11	2.42×10^-15	11	3.98×10^-13	11	2.32×10^-15	7	4.82×10^-15
1.750	84	7.83×10^-12	84	2.91×10^-12	11	2.83×10^-15	11	2.75×10^-14	11	1.29×10^-12	11	5.51×10^-15	7	4.33×10^-15
2.000	84	4.90×10^-12	84	9.46×10^-12	11	3.25×10^-15	11	3.06×10^-15	11	4.56×10^-13	11	4.43×10^-15	7	3.44×10^-15
2.500	84	1.11×10^-11	84	5.93×10^-12	11	2.05×10^-15	11	1.53×10^-15	11	1.09×10^-12	11	6.01×10^-15	7	2.40×10^-15
3.000	84	1.01×10^-11	84	1.00×10^-11	11	7.12×10^-15	11	2.42×10^-15	11	3.98×10^-13	11	2.32×10^-15	8	1.64×10^-15

α	2	2	1	1	0	1	0.5	0	-0.5
β	2	1	2	1	1	0	0.5	0	-0.5
k	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)
0	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰
1	-1.35×10⁰	-6.23×10^-1	-1.62×10⁰	-8.98×10^-1	-1.17×10⁰	-1.74×10^-1	-6.74×10^-1	-4.49×10^-1	-2.25×10^-1
2	1.34×10⁰	3.33×10^-1	1.68×10⁰	5.05×10^-1	7.86×10^-1	-1.13×10^-1	2.11×10^-1	1.84×10^-13	-1.24×10^-1
3	-1.02×10⁰	-4.60×10^-2	-1.39×10⁰	-1.79×10^-1	-3.64×10^-1	1.41×10^-1	2.88×10^-2	1.29×10^-1	1.49×10^-1
4	8.39×10^-1	4.74×10^-2	1.07×10⁰	5.10×10^-2	1.20×10^-1	-5.86×10^-2	-7.17×10^-2	-9.51×10^-2	-6.62×10^-2
5	-5.85×10^-1	4.38×10^-2	-7.95×10^-1	5.11×10^-13	-2.25×10^-2	2.85×10^-2	5.58×10^-2	4.22×10^-2	8.73×10^-13
6	5.12×10^-1	2.76×10^-2	6.12×10^-1	-3.89×10^-13	-4.78×10^-13	3.33×10^-4	-2.99×10^-2	-1.30×10^-2	8.39×10^-13
7	-3.50×10^-1	4.21×10^-2	-4.70×10^-1	1.18×10^-2	8.56×10^-13	4.67×10^-13	1.73×10^-2	2.34×10^-13	-8.33×10^-13
8	3.43×10^-1	3.11×10^-2	3.81×10^-1	-1.91×10^-13	-6.94×10^-13	4.82×10^-13	-8.28×10^-13	3.54×10^-4	4.82×10^-13
9	-2.25×10^-1	3.87×10^-2	-3.05×10^-1	7.30×10^-13	4.97×10^-13	3.06×10^-13	5.97×10^-13	-6.71×10^-4	-2.51×10^-13
10	2.50×10^-1	3.31×10^-2	2.58×10^-1	2.15×10^-4	-3.54×10^-13	3.77×10^-13	-2.87×10^-13	5.15×10^-4	1.21×10^-13
11	-1.52×10^-1	3.72×10^-2	-2.13×10^-1	4.95×10^-13	2.57×10^-13	2.79×10^-13	2.81×10^-13	-3.49×10^-4	-6.91×10^-4
12	1.93×10^-1	3.41×10^-2	1.86×10^-1	1.02×10^-13	-1.92×10^-13	3.02×10^-13	-1.23×10^-13	2.35×10^-4	3.69×10^-4
13	-1.05×10^-1	3.66×10^-2	-1.56×10^-1	3.70×10^-13	1.47×10^-13	2.50×10^-13	1.61×10^-13	-1.63×10^-4	-2.59×10^-4
14	1.55×10^-1	3.47×10^-2	1.40×10^-1	1.31×10^-13	-1.15×10^-13	2.55×10^-13	-5.76×10^-4	1.15×10^-4	1.47×10^-4
15	-7.41×10^-2	3.62×10^-2	-1.19×10^-1	2.97×10^-13	9.19×10^-4	2.23×10^-13	1.04×10^-13	-8.21×10^-4	-1.21×10^-4
16	1.30×10^-1	3.50×10^-2	1.09×10^-1	1.38×10^-13	-7.51×10^-4	2.22×10^-13	-2.81×10^-4	5.89×10^-4	6.72×10^-4
17	-5.20×10^-2	3.61×10^-2	-9.36×10^-2	2.51×10^-13	6.32×10^-4	2.02×10^-13	7.36×10^-4	-3.99×10^-4	-6.26×10^-4
18	1.11×10^-1	3.52×10^-2	8.71×10^-2	1.34×10^-13	-5.47×10^-4	1.96×10^-13	-1.44×10^-4	2.63×10^-4	3.14×10^-4
19	-3.56×10^-2	3.61×10^-2	-7.51×10^-2	2.22×10^-13	4.96×10^-4	1.84×10^-13	5.71×10^-4	-1.20×10^-5	-3.27×10^-4

α	2	2	1	1	0	1	0.5	0	-0.5
β	2	1	2	1	1	0	0.5	0	-0.5
k	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)	m_k^(P)
0	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰	1.00×10⁰
1	-6.46×10^-1	-3.82×10^-2	-1.04×10⁰	-4.31×10^-1	-8.23×10^-1	1.77×10^-1	-3.23×10^-1	-2.15×10^-1	-1.08×10^-1
2	2.93×10^-1	-1.03×10^-1	5.43×10^-1	-5.73×10^-2	1.77×10^-1	-2.54×10^-1	-1.63×10^-1	-2.23×10^-1	-2.36×10^-1
3	1.09×10^-1	2.23×10^-1	-6.99×10^-2	1.95×10^-1	1.50×10^-1	9.30×10^-2	1.89×10^-1	1.62×10^-1	1.21×10^-1
4	-2.17×10^-2	4.03×10^-2	-6.92×10^-2	-7.58×10^-2	-1.43×10^-1	5.18×10^-2	-4.95×10^-2	-1.25×10^-2	2.20×10^-2
5	1.25×10^-1	6.92×10^-2	9.09×10^-2	2.40×10^-2	5.19×10^-2	-2.39×10^-2	-1.32×10^-2	-3.66×10^-2	-4.61×10^-2
6	4.10×10^-2	8.79×10^-2	-3.66×10^-2	2.04×10^-2	-3.21×10^-4	2.37×10^-2	2.41×10^-2	2.38×10^-2	1.78×10^-2
7	7.69×10^-2	6.75×10^-2	2.64×10^-2	-1.67×10^-13	-1.12×10^-2	9.28×10^-13	-9.00×10^-13	-6.10×10^-13	6.73×10^-4
8	6.49×10^-2	7.74×10^-2	5.01×10^-4	1.28×10^-2	7.96×10^-13	6.28×10^-13	4.42×10^-13	-1.04×10^-13	-4.48×10^-13
9	6.89×10^-2	7.32×10^-2	8.27×10^-13	4.97×10^-13	-3.68×10^-13	9.14×10^-13	1.26×10^-13	1.83×10^-13	2.46×10^-13
10	6.83×10^-2	7.43×10^-2	5.39×10^-13	7.00×10^-13	1.34×10^-13	6.30×10^-13	6.78×10^-4	-1.05×10^-13	-8.72×10^-4
11	6.89×10^-2	7.39×10^-2	5.58×10^-13	5.71×10^-13	-4.08×10^-4	6.59×10^-13	1.34×10^-13	4.26×10^-4	5.05×10^-5
12	6.91×10^-2	7.39×10^-2	5.07×10^-13	5.50×10^-13	1.09×10^-4	5.82×10^-13	7.11×10^-4	-1.39×10^-4	2.23×10^-5
13	6.94×10^-2	7.39×10^-2	4.79×10^-13	5.09×10^-13	-2.53×10^-5	5.48×10^-13	8.93×10^-4	3.92×10^-5	-9.16×10^-5
14	6.97×10^-2	7.39×10^-2	4.49×10^-13	4.78×10^-13	5.53×10^-6	5.09×10^-13	6.82×10^-4	-9.34×10^-6	4.95×10^-6
15	7.00×10^-2	7.40×10^-2	4.25×10^-13	4.50×10^-13	-2.12×10^-7	4.78×10^-13	6.84×10^-4	2.57×10^-6	-3.92×10^-5
16	7.02×10^-2	7.40×10^-2	4.02×10^-13	4.25×10^-13	6.74×10^-7	4.50×10^-13	5.93×10^-4	2.45×10^-7	-1.24×10^-5
17	7.05×10^-2	7.41×10^-2	3.83×10^-13	4.03×10^-13	1.39×10^-6	4.25×10^-13	5.65×10^-4	1.04×10^-6	-2.07×10^-5
18	7.07×10^-2	7.41×10^-2	3.65×10^-13	3.83×10^-13	1.44×10^-6	4.03×10^-13	5.12×10^-4	1.41×10^-6	-1.27×10^-5
19	7.09×10^-2	7.42×10^-2	3.49×10^-13	3.65×10^-13	2.40×10^-6	3.83×10^-13	4.83×10^-4	1.93×10^-6	-1.36×10^-5

Type		LQMDA		Cheb
Type		Gen		gen
α	β	n_max	MSRE	n_max	MSRE
2.0	2.0	85	5.57×10^-11	85	2.42×10^-11
2.0	1.0	85	4.31×10^-12	85	3.25×10^-12
1.0	2.0	85	2.86×10^-11	85	1.70×10^-11
1.0	1.0	85	9.07×10^-12	85	5.47×10^-11
0.0	1.0	85	4.13×10^-11	85	7.26×10^-12
1.0	0.0	85	5.46×10^-12	85	1.44×10^-11
0.5	0.5	85	2.99×10^-11	85	1.52×10^-11
0.0	0.0	85	2.36×10^-11	85	3.90×10^-11
-0.5	-0.5	85	2.16×10^-10	85	4.82×10^-10

Type		LQMDA		Cheb
Type		Gen		gen
α	β	n_max	MSRE	n_max	MSRE
2.0	2.0	84	4.90×10^-12	84	9.46×10^-12
2.0	1.0	84	2.32×10^-12	84	9.62×10^-13
1.0	2.0	84	1.02×10^-12	84	1.31×10^-11
1.0	1.0	84	3.61×10^-12	84	5.97×10^-12
0.0	1.0	84	1.51×10^-10	84	3.77×10^-10
1.0	0.0	83	3.60×10^-10	84	2.74×10^-10
0.5	0.5	84	1.56×10^-11	84	3.31×10^-12
0.0	0.0	84	6.85×10^-10	84	3.51×10^-10
-0.5	-0.5	81	1.72×10^-10	82	1.15×10^-10

n	T^bub	T^dew	T^flash	g^flash	$\bar{M}$ ^F	$\bar{M}$ ^V	$\bar{M}$ ^L	MSRE^F	MSRE^L	MSRE^V
	Gauss-Christoffel Quadrature
3	462.992	539.071	478.222	0.32949	155.087	131.721	166.568	1.04×10^-15	2.17×10^-15	4.05×10^-15
4	461.939	539.336	478.312	0.31792	155.087	132.229	165.741	1.86×10^-15	5.28×10^-15	7.28×10^-15
5	461.805	539.342	478.122	0.31756	155.087	132.281	165.699	7.51×10^-15	9.61×10^-15	4.99×10^-15
6	461.791	539.342	478.121	0.31780	155.087	132.247	165.726	3.03×10^-14	8.34×10^-14	1.84×10^-15
7	461.790	539.342	478.129	0.31779	155.087	132.248	165.725	1.50×10^-14	8.21×10^-14	2.49×10^-14
8	461.790	539.342	478.128	0.31779	155.087	132.250	165.724	1.80×10^-14	8.10×10^-14	4.30×10^-14
10	461.790	539.342	478.128	0.31779	155.087	132.250	165.724	5.29×10^-14	4.22×10^-13	1.41×10^-14
12	461.790	539.342	478.128	0.31779	155.087	132.250	165.724	2.29×10^-13	1.22×10^-13	1.08×10^-14
14	461.790	539.342	478.128	0.31779	155.087	132.250	165.724	6.07×10^-13	1.30×10^-13	1.20×10^-14
20	461.790	539.342	478.128	0.31779	155.087	132.250	165.724	2.03×10^-13	3.68×10^-13	4.85×10^-13
30	461.790	539.342	478.128	0.31779	155.087	132.250	165.724	1.85×10^-11	8.20×10^-13	1.48×10^-12
50	461.790	539.342	478.128	0.31779	155.087	132.250	165.724	7.96×10^-12	5.38×10^-13	1.82×10^-13
80	461.790	539.342	478.128	0.31779	155.087	132.250	165.724	2.02×10^-11	2.16×10^-12	1.65×10^-12
	Uniform Discretization
10	470.467	543.272	476.038	0.14931	157.854	133.576	162.115	NA	NA	NA
20	463.704	540.803	477.821	0.29001	155.739	132.517	165.225	NA	NA	NA
40	462.225	539.971	478.068	0.31161	155.268	132.304	165.662	NA	NA	NA
80	461.895	539.635	478.116	0.31624	155.146	132.260	165.731	NA	NA	NA
100	461.857	539.573	478.121	0.31677	155.130	132.255	165.735	NA	NA	NA
200	461.809	539.454	478.127	0.31748	155.104	132.250	165.735	NA	NA	NA
400	461.796	539.398	478.128	0.31767	155.094	132.250	165.731	NA	NA	NA
800	461.793	539.370	478.128	0.31773	155.090	132.250	165.728	NA	NA	NA
1000	461.793	539.364	478.128	0.31774	155.090	132.250	165.727	NA	NA	NA
2000	461.792	539.353	478.128	0.31776	155.088	132.250	165.725	NA	NA	NA
4000	461.792	539.347	478.128	0.31776	155.088	132.250	165.725	NA	NA	NA
8000	461.791	539.345	478.128	0.31777	155.087	132.250	165.724	NA	NA	NA
10000	461.791	539.344	478.128	0.31777	155.087	132.250	165.724	NA	NA	NA

n	T^bub	T^dew	T^flash	g^flash	$\bar{M}$ ^F	$\bar{M}$ ^V	$\bar{M}$ ^L	MSRE^F	MSRE^L	MSRE^V
	Gauss-Christoffel Quadrature
3	555.174	654.723	595.723	0.42818	237.324	196.014	268.257	1.73×10^-14	1.67×10^-15	3.95×10^-16
4	551.918	654.896	594.212	0.42635	237.324	197.314	267.060	4.64×10^-14	4.38×10^-15	4.35×10^-15
5	551.263	654.899	594.138	0.42662	237.324	196.756	267.509	4.18×10^-14	1.11×10^-14	4.21×10^-14
6	551.156	654.899	594.221	0.42623	237.324	196.843	267.396	2.20×10^-14	2.21×10^-14	2.43×10^-14
7	551.141	654.899	594.203	0.42644	237.324	196.852	267.415	2.51×10^-14	2.24×10^-14	1.21×10^-13
8	551.139	654.899	594.205	0.42641	237.324	196.844	267.417	4.90×10^-14	1.70×10^-14	5.45×10^-13
10	551.139	654.899	594.205	0.42640	237.324	196.846	267.415	4.83×10^-14	5.08×10^-14	5.64×10^-13
12	551.139	654.899	594.205	0.42640	237.324	196.846	267.415	7.25×10^-14	2.05×10^-13	1.20×10^-12
14	551.139	654.899	594.205	0.42640	237.324	196.846	267.415	1.66×10^-13	3.30×10^-13	1.96×10^-12
20	551.139	654.899	594.205	0.42640	237.324	196.846	267.415	3.61×10^-13	1.81×10^-13	1.79×10^-12
30	551.139	654.899	594.205	0.42640	237.324	196.846	267.415	4.68×10^-13	6.02×10^-14	2.82×10^-12
50	551.139	654.899	594.205	0.42640	237.324	196.846	267.415	1.76×10^-12	4.23×10^-13	1.34×10^-11
80	551.139	654.899	594.205	0.42640	237.324	196.846	267.415	5.04×10^-13	3.79×10^-12	9.91×10^-12
	Uniform Discretization
10	551.288	657.766	594.440	0.42681	237.954	196.675	268.691	NA	NA	NA
20	551.144	656.338	594.288	0.42502	237.718	196.798	267.966	NA	NA	NA
40	551.178	655.615	594.243	0.42559	237.527	196.826	267.683	NA	NA	NA
80	551.161	655.255	594.224	0.42599	237.426	196.836	267.548	NA	NA	NA
100	551.157	655.184	594.220	0.42607	237.405	196.838	267.522	NA	NA	NA
200	551.148	655.041	594.213	0.42624	237.365	196.842	267.468	NA	NA	NA
400	551.143	654.970	594.209	0.42632	237.344	196.844	267.442	NA	NA	NA
800	551.141	654.934	594.207	0.42636	237.334	196.845	267.428	NA	NA	NA
1000	551.140	654.927	594.207	0.42637	237.332	196.845	267.426	NA	NA	NA
2000	551.139	654.913	594.206	0.42639	237.328	196.845	267.420	NA	NA	NA
4000	551.139	654.906	594.206	0.42640	237.326	196.846	267.418	NA	NA	NA
8000	551.139	654.902	594.205	0.42640	237.325	196.846	267.416	NA	NA	NA
10000	551.139	654.901	594.205	0.42640	237.325	196.846	267.416	NA	NA	NA

\begin{array}{l} p_{- 1} (x) = 0 \\ p_{0} (x) = 1 \\ x p_{k} (x) = a_{k} p_{k + 1} (x) + b_{k} p_{k} (x) + c_{k} p_{k - 1} (x), k = 0,1,... \end{array}

\begin{array}{l} P_{- 1} (x) = 0 \\ P_{0} (x) = 1 \\ P_{k + 1} (x) = (x - θ_{k}) P_{k} (x) - η_{k}^{2} P_{k - 1} (x), k = 0,1,... \end{array}

{\tilde{A}}_{n} = (\begin{array}{l} θ_{0} & 1 & 0 & \dots & \dots & 0 \\ η_{1}^{2} & θ_{1} & 1 & ⋮ \\ 0 & ⋱ & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & 0 \\ ⋮ & η_{n - 2}^{2} & θ_{n - 2} & 1 \\ 0 & \dots & \dots & 0 & η_{n - 1}^{2} & θ_{n - 1} \end{array})

A_{n} = (\begin{array}{l} θ_{0} & η_{1} & 0 & \dots & \dots & 0 \\ η_{1} & θ_{1} & η_{2} & ⋮ \\ 0 & ⋱ & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & 0 \\ ⋮ & η_{n - 2} & θ_{n - 2} & η_{n - 1} \\ 0 & \dots & \dots & 0 & η_{n - 1} & θ_{n - 1} \end{array})

M S R E = \sqrt{\frac{1}{2 n} \sum_{k = 0}^{2 n - 1} {(\frac{μ_{k, d i s t} - μ_{k, r e c o n s t}}{μ_{k, d i s t}})}^{2}}

\begin{array}{l} C_{P}^{g i} = & (- 0.0919055 + 0.011308 T (K) - {6.3792010}^{- 6} T^{2} (K) + {1.4060510}^{- 9} T^{3} (K)) \\ (N_{c} + 0.284370) R \end{array}

[1] * Corresponding author: Paulo L. da C. Lage - E-mail: paulo@peq.coppe.ufrj.br

Brasil

Brasil

QUADRATURE ALGORITHMS FOR PHASE EQUILIBRIUM OF CONTINUOUS MIXTURES

Abstract