User:Maistral01/sandbox

• Comment: I cannot judge whether this is notable because there are insufficient references to verify that it is notable. One paper does not make it notable, the more so since it appears to be cited by no-one 🇺🇦 Fiddle^Timtrent Faddle^Talk to me 🇺🇦 16:56, 1 October 2024 (UTC)

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The Almajose-Dalida equation of state (ADEOS) is a three-parameter cubic equation of state designed to predict the thermodynamic properties of pure substances and mixtures more accurately. The EOS optimizes the attraction term in the equation, balancing computational simplicity with precision in key thermodynamic properties such as vapor pressure, liquid density, and enthalpy of vaporization. It is particularly useful in chemical engineering applications and process simulations.

Mathematical formulation

The Almajose-Dalida equation follows the form:

$P={\dfrac {RT}{V-b}}-{\dfrac {a(T)}{V^{2}+4cV-2c^{2}}}$

Where:

• $P$ = pressure,
• $T$ = temperature,
• $V$ = molar volume,
• $R$ = universal gas constant,
• $a(T)$ = attraction parameter, which depends on temperature,
• $b$ = co-volume parameter,
• $c$ = volume-dependent attraction parameter.

Equation of state parameters

The equation of state parameters are formulated as:

$a(T)=a_{c}\cdot \alpha (T)$

$a_{c}=\Omega _{A}\cdot {\frac {R^{2}\cdot T_{c}^{2}}{P_{c}}}$

$b=\Omega _{B}\cdot {\frac {R\cdot T_{c}}{P_{c}}}$

$c=\Omega _{C}\cdot {\frac {R\cdot T_{c}}{P_{c}}}$

The alpha function, $\alpha (T)$, follows the form prescribed by Soave:

$\alpha (T)=\left(1+\tau \cdot \left(1-{\sqrt {\frac {T}{T_{c}}}}\right)\right)^{2}$

An inherent difference of three-parameter equations of state to that of two-parameter equations such as Soave-Redlich-Kwong and Peng-Robinson is that the equation of state parameters $\Omega _{A}$, $\Omega _{B}$, and $Z_{c}$ do not take fixed values. This is due to the fact that three-parameter equations of state are underdetermined; that is, the application of the critical conditions for a three-parameter equation results to three equations with four unknowns, $\Omega _{A}$, $\Omega _{B}$, $\Omega _{C}$, and $Z_{c}$. As such, equation of state values for the Almajose-Dalida are tabulated.

Almajose-Dalida equation of state constants for select fluids

i	Compound Name	$\Omega _{A}$	$\Omega _{B}$	$\Omega _{C}$	$\tau$
1	Methane	0.43305	0.08480	0.02502	0.46606
2	Ethane	0.43477	0.08254	0.02690	0.60228
3	Propane	0.43665	0.08035	0.02881	0.68545
4	Benzene	0.43935	0.07759	0.03137	0.76135
5	Toluene	0.44180	0.07538	0.03353	0.83182
6	Water	0.47090	0.06071	0.05314	0.80280
7	Carbon Dioxide	0.44186	0.07534	0.03358	0.75373
8	Nitrogen	0.43242	0.08568	0.02431	0.51544
9	Hydrogen	0.42616	0.09788	0.01546	0.08676
10	Oxygen	0.43201	0.08629	0.02383	0.48734
11	Ammonia	0.45703	0.06585	0.04478	0.73905
12	Methanol	0.46690	0.06196	0.05086	1.06588
13	R134a (Tetrafluoroethane)	0.44348	0.07402	0.03493	0.91087
14	R1233zd(E)	0.43961	0.07734	0.03160	0.89251
15	R1234yf	0.44889	0.07024	0.03914	0.80875

Generalized Almajose-Dalida

The generalized ADEOS equation uses correlating equations to predict the equation of state parameters, usually in terms of the acentric factor or the critical compressibility factor. The correlating equations are:

$\Omega _{A}=0.917404-3.114268\cdot Z_{C}+4.972737\cdot Z_{C}^{2}$

$\Omega _{B}=0.030404-0.044936\cdot Z_{C}+0.815415\cdot Z_{C}^{2}$

$\Omega _{C}=0.252264-1.254946\cdot Z_{C}+1.615419\cdot Z_{C}^{2}$

The Soave alpha function parameter has also been correlated as:

$\tau =0.447972+1.562857\cdot \omega -0.297088\cdot \omega ^{2}$

Departure property derivations

The departure quantities are calculated as follows:

Fugacity

$\ln(\phi )=Z-1-\ln(Z-B)+{\frac {A}{2{\sqrt {6}}\cdot C}}\cdot \ln \left({\frac {Z+(2-{\sqrt {6}})\cdot C}{Z+(2+{\sqrt {6}})\cdot C}}\right)$

Departure enthalpy

${\frac {H^{R}}{RT}}=Z-1-\left(A-{\frac {P}{R^{2}T^{2}}}{\frac {\partial a}{\partial T}}\right)\cdot {\frac {1}{2{\sqrt {6}}\cdot C}}\cdot \ln \left({\frac {Z+(2+{\sqrt {6}})\cdot C}{Z+(2-{\sqrt {6}})\cdot C}}\right)$

Departure entropy

${\frac {S^{R}}{R}}=\ln(Z-B)-{\frac {P}{R^{2}T}}{\frac {\partial a}{\partial T}}\cdot {\frac {1}{2{\sqrt {6}}\cdot C}}\cdot \ln \left({\frac {Z+(2+{\sqrt {6}})\cdot C}{Z+(2-{\sqrt {6}})\cdot C}}\right)$

Development of the optimized equation

The Almajose-Dalida was developed via the general three-parameter equation of state:

$P={\frac {RT}{V-b}}-{\frac {a}{V^{2}+k_{1}bV+k_{2}cV+k_{3}b^{2}+k_{4}c^{2}+k_{5}bc}}$

The coefficients of the quadratic denominator polynomial of the general three-parameter equation of state are then optimized to determine the best form of the attractive term via integer optimization against 65 fluids commonly used in process simulations. From their study, two equations have been determined:

$P={\frac {RT}{V-b}}-{\frac {a}{V^{2}+4bV+3cV-3c^{2}-bc}}$

$P={\dfrac {RT}{V-b}}-{\dfrac {a(T)}{V^{2}+4cV-2c^{2}}}$

While their study has shown that the first equation provides the least error, the second equation carries over the mathematical elegance of a two-parameter equation of state, which is taken of greater importance considering the very small error advantage of the first equation