Prediction of Thermodynamic Properties of CO2 by Cubic and

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Prediction of Thermodynamic Properties of CO2 by Cubic and Multiparameter Equations of State for Fluid Dynamics Applications Denise S. Leal, Marcelo Embiruçu, Gloria M. N. Costa, and Karen V. Pontes*

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Industrial Engineering Graduate Program, Universidade Federal da Bahia, Rua Professor Aristides Novis, 02, EP-UFBA, Federaçaõ , Salvador, Bahia 40210-630, Brazil ABSTRACT: This paper makes a comprehensive analysis of equations of state (EOS) for the prediction of thermodynamic properties of CO2 which play a key role in fluid dynamics calculations: molar volume, throttling temperature, inversion curve, and velocity of sound. The cubic EOS Peng−Robinson and Soave−Redlich−Kwong are compared with the multiparameter Huang and Sterner and Pitzer EOS. We further propose a method to compute the throttling temperature, which relies on the residual property, unlike the usual practice which solely computes the Joule-Thomson coefficient or the inversion curve. The predicted properties are compared with experimental data from literature to verify the best EOS at different temperatures and pressure conditions for each investigated property. The Huang EOS is the best at predicting the molar volume, especially in supercritical conditions, and the inversion curve. The cubic EOS predict the throttling temperature with greater accuracy. The Sterner and Pitzer EOS outperforms the others when predicting the velocity of sound, especially in conditions near and above the critical point. These multiparameter EOS seem to balance the trade-off between complexity and accuracy. These findings might be useful for designing and monitoring processes that use CO2 over a wide range of temperature and pressure conditions.

I. INTRODUCTION There is increasing interest in supercritical carbon dioxide (sCO2) due to the several advantages it offers in many industrial applications, such as micronization for drug delivery,1,2 extrusion for food processing,3 energy power cycles,4 and enhanced oil recovery,5 to cite just a few.6 Its mild critical temperature (31.1 °C) enables near-critical operation at lower temperatures. The critical pressure, although relatively high (7.4 MPa), is common in large-scale processes working with supercritical CO2.7 Carbon dioxide is also nontoxic, nonflammable, and environmentally friendly, and is also known as a “green solvent”. These industrial applications require a knowledge of thermodynamic and transport properties over a wide range of operating conditions.1,8 The knowledge of the volume/density behavior as well as other thermodynamic properties (e.g., enthalpy) as a function of pressure and temperature plays an important role in the design and monitoring of these processes as well as in the product quality. The transport and thermodynamic properties have high relevance in different stages of the sCO2 technology, such as transport, expansion, processing, injecting, and storing processes. The knowledge of the specific volume/density as a function of pressure and temperature is very important to predict phenomena such as sudden (de)-pressurization of pipelines, liquid loading, and unloading of vessels, as well as for the design of equipment such as compressors, pumps, vessels, and pipelines.9 The velocity of sound can indicate the maximum velocity reached by the flow at the exit of a pipeline and the throat velocity in supersonic © XXXX American Chemical Society

nozzles, and it also can identify obstructions and leakage in gas pipelines and vibrations in compressors.10 In the rapid expansion of a supercritical solution (RESS) for the microencapsulation of pharmaceutical drugs, for example, the velocity of sound might be reached at the depressurization nozzle when the supercritical fluid is expanded to generate fine particles.1 The velocity of sound also plays an important role in the design and operation of pipelines used in several applications. In capture and geological sequestration and in the enhanced oil recovery (EOR) process, an abrupt pressure drop might also occur due to CO2 leakage after pipeline or equipment damage. As a result of the pressure drop, the velocity of gas might reach the velocity of sound. Another very important application of the velocity of sound is to establish fundamental equations of state for fluids because thermal and caloric properties might be derived from the velocity of sound measurements obtained experimentally with high precision.11,12 The Joule-Thomson inversion curve is important for monitoring the hazard assessment of pipeline depressurization13,14 as well as for the operation of separation processes for CO2 capture such as cryogenic distillation5 because it dictates whether the outflow stream will be cooled or heated due to the pressure drop. The inversion curve, though, solely indicates the regions where the fluid is cooled or heated after the expansion. The throttling temperature, on the other hand, can ascertain the temperature at the outlet of the Received: December 21, 2018 Accepted: March 4, 2019

A

DOI: 10.1021/acs.jced.8b01238 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

has dedicated attention to the modeling of the CO2 inversion curve28−32 or to the prediction of the velocity of sound of CO2;10,33 however, they do not compare the prediction of cubic and multiparameter EOS for all the properties investigated here. Furthermore, the usual approach is to compute the inversion curve instead of the throttling temperature. The frequently used cubic PR and SRK EOS provide poor predictions for the density around the critical point, which is key for sCO2 applications. Huang et al.34 propose an EOS for a wide range of temperatures (216−423 K) and pressures (up to 2010.3 MPa), which is particularly accurate in the critical region due the consideration of nonanalytical terms. Sterner and Pitzer35 introduce a new form of EOS that is valid over a broad range of temperatures (from 215 K to T > 2000 K) and pressures (from 0 to P > 10 GPa) to describe pure CO2. Span and Wagner22 develop an equation of state (SW) for the carbon dioxide covering the fluid region from the triple-point to pressures up to 800 MPa. This equation is based on Helmholz free energy and its parameters were fitted to experimental data of several properties, for example, P−ρ−T, velocity of sound, specific heat capacities, and the Joule-Thomson coefficient. These multiparameter EOS, on the one hand, require a large amount of data to fit the parameters, but on the other hand, they provide higher accuracy than the cubic EOS. Owing to the high accuracy of the SW EOS for predicting the thermodynamic properties for CO2 transport, it is used by several authors as a reference EOS.9,13,23 The SW EOS, though, has a complex structure and is CPU demanding, hindering its use in industrial applications. Numerous are the processes which operate with pure CO2 and then require the precise computation of some thermodynamic properties, such as specific volume, inversion curve, throttling temperature, and velocity of sound. However, the literature does not report a comprehensive analysis of these properties over a broad range of temperature and pressure conditions. The studies usually concentrate on just a few properties and, to the authors’ knowledge, do not compute the throttling temperature. The main goal of this paper, therefore, is to evaluate four EOS regarding the prediction of thermodynamic properties which are important for fluid dynamics and sCO2 applications, namely, the specific volume, the throttling temperature, the inversion curve, and the velocity of sound. We consider two cubic EOS, PR and SRK, as they are the cubic EOS most frequently used in the literature as the review above has demonstrated. Since the cubic EOS usually yield poor results for critical and supercritical regions, which are especially important for sCO2 applications, they are compared with two multiparameter EOS, the Huang34 and Sterner and Pitzer35 (SP) equations. Although these multiparameter EOS have been developed for CO2 and validated over a wide range of temperature and pressure conditions, including the critical point and supercritical conditions, they have rarely been used for sCO2 applications. Instead, the literature usually considers cubic EOS, reference equations such as SW or GERG EOS, or even more complex perturbation models such as the PC-SAFT equation. For some applications, though, complicated structures may reduce the applicability of the EOS due to the higher computational burden and less transparency for industrial implementation.36 There should be a balance between complexity and accuracy, which might be achieved by the multiparameter Huang and SP EOS. We further propose a method for ascertaining the throttling temperature based on residual properties, which has already presented promising results

expansion valve, which might be useful for monitoring phase separation, product quality, etc. A knowledge of pressure− volume−temperature (PVT) behavior, the Joule-Thomson effect, and the velocity of sound in the gas is therefore important to ensure the safe and efficient operation of several industrial sCO2 technologies. Despite the maturity of certain sCO2 industrial technologies, the transport and thermodynamic properties are usually monitored based on PVT, enthalpy, and/or entropy diagrams. The equations of state (EOS), however, correlate volume, pressure, and temperature, and can be used to compute key properties for the compressible flow of real gases, such as the specific volume, the throttling temperature and the inversion temperature as well as the velocity of sound. EOS therefore play a key role in the accurate modeling and simulation of CO2 flow in pipelines as well as in compressors, turbines, pumps, heaters, and separation processes. Despite this, pharmaceutical particle engineering applications, for example, only make use of EOS to study phase equilibrium and solubility.2 However, they can be of great value in predicting the temperature and pressure conditions during the expansion across the nozzle to control the quality of the particles formed. Power cycles could also be modeled to a great extent using EOS as the real work of compressors, turbines, and pumps can be ascertained through a real enthalpy and entropy difference between the inlet and outlet conditions.15 This approach would overcome some drawbacks of the usual approach based on the assumption of ideal behavior or on pressure−enthalpy/temperature−entropy diagrams to monitor the different stages of the process.4 The literature is usually concerned with the modeling of P−ρ−T behavior2,16 and with some specific thermodynamic and transport properties of pure CO2. Witkowski et al.14 proposed a model for the compression and transport of CO2. The model in the Aspen Plus software package uses Benedict-Web-Rubin with an extension by Starling (BWRS)17,18 and Soave−Redlich− Kwong (SRK)19 equations of state to predict the thermodynamic properties of the CO2 stream up to 50 bar and from 50 to 250 bar, respectively. No details, however, are given on the properties evaluated. Diamantonis et al.13 provide a literature review of thermodynamic and transport property models used for the capture, transport, and storage of CO2, mainly transport. They compute the inversion curve using the Perturbed ChainStatistical Associating Fluid Theory (PC-SAFT)20 and Peng− Robinson (PR),21 comparing the predictions against the reference equation of state Span and Wagner22 (SW). The PC-SAFT EOS presents better results, but no attempt is made to compute the throttling temperature. Raimondi23 compares the cubic PR and SRK equations with the SW and GERG model24 when evaluating the temperature change due to depressurization along a CO2 pipeline. They conclude that the cubic EOS produce large deviations around the region where liquid and vapor coexist, as a result a more complex model becomes a mandatory step in thermodynamic simulations for CO 2 transport. Zaho et al.25 investigate six EOS for predicting the density, specific heat, and velocity of sound for sCO2 Bryton power cycles. The authors conclude that the SW outperforms the cubic EOS as well as the Lee−Kesler−Plöcker26,27 and the BWRS, especially for the velocity of sound. Although Zaho et al.25 carry out a more comprehensive study, they do not model the inversion curve and the throttling temperature nor do they consider other multiparameter EOS. Consequently, the reference and more complex SW model outperforms the other equations. On the other hand, the thermodynamic community B



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Table 1. Equations of State Considered in This Study EOS

mathematical expression

acα(T )V V − Z= V−b RT[V (V + b) + b(V − b)] PR

Ωa ·R2Tc 2 Ω RT b = b c, Pc Pc ÉÑ2 ÅÄÅ Ñ ij ÅÅ T yzzzÑÑÑÑ α(T ) = ÅÅÅÅ1 + mjjjj1 − zzÑÑ ÅÅ j Tc z{ÑÑÑ ÅÇ k Ö

ac =

Z=

SRK

Ωa = 0.45724

Ωb = 0.07780

m = 0.375 + 1.54ω − 0.270ω2

acα(T ) V − V−b RT(V + b)

ΩaR2Tc 2 Ω RT b = b c , Ωa = 0.42748 Ωb = 0.08664 Pc Pc ÑÉÑ2 ÅÄÅ ij ÅÅ T yzzzÑÑÑÑ 2 α(T ) = ÅÅÅÅ1 + mjjjj1 − zzÑÑ m = 0.48 + 1.574ω − 0.17ω ÅÅ zÑÑ j T c ÅÇ {ÑÖ k

ac =

Z = 1 + b2ρ′ + b3ρ′ 2 + b4ρ′ 3 + b5ρ′ 4 + b6ρ′ 5 + b7ρ′ 2 exp(− c 21ρ′ 2 ) + b8ρ′ 4 exp(− c 21ρ′ 2 ) + c 22ρ′ exp(− c 27ΔT 2) Δρ Δρ exp(− c 25Δρ2 − c 27ΔT 2) + c 24 exp(− c 26Δρ2 − c 27ΔT 2) + c 23 ρ′ ρ′

c c c c c2 + 32 + 43 + 54 + 65 T′ T′ T′ T′ T′ c c b3 = c 7 + 8′′ + 92 T′ T c11 b4 = c10 + T′ c b5 = c12 + 13 T′ c14 b6 = T′ c c c b7 = 153 + 164 + 175 T′ T′ T′ c c c b8 = 183 + 194 + 205 T′ T′ T′ b2 = c1 +

HG

SP

yz a3 + 2a4ρ + 3a5ρ2 + 4a6ρ3 ji zz + a ρ exp(− a ρ) Z = 1 + a1ρ − ρjjjj 7 8 2 3 4 2z z a a a a a ( ) ρ ρ ρ ρ + + + + 2 3 4 5 6 { k + a 9ρ exp(− a10ρ)

ai = di ,1T −4 + di ,2T −2 + di ,3T −1 + di ,4 + di ,5T + di ,6T 2

conditions. This paper ends with the final conclusions (section V).

when applied to a polymerization system based on PC-SAFT EOS.37 This paper, therefore, aims to contribute to filling the gap between thermodynamics and fluid dynamics, providing a more comprehensive study of the prediction of important thermodynamic properties for the compressible flow of CO2 and for further sCO2 applications, covering liquid, superheated vapor and supercritical phases. The results should guide the choice regarding the best EOS for each property at given pressures and temperature conditions with acceptable accuracy and complexity. The paper is organized as follows. Section II presents the equations of state considered for predicting the thermodynamic properties, namely the specific volume, the throttling temperature, the inversion curve, and the velocity of sound. Each property is then detailed and modeled afterward in section III. The results section IV compares the prediction of each EOS with experimental data from the literature in order to indicate the accuracy of each model and the best EOS to describe each property investigated at given temperatures and pressure

II. EQUATIONS OF STATE Two cubic and two multiparameter Equations of State are considered for the prediction of the thermodynamic properties: Peng−Robinson,21 PR; Soave−Redlich−Kwong,19 SRK; Huang,34 HG; Sterner and Pitzer,35 SP. Table 1 presents the mathematical expression of each EOS, where Z is the compressibility factor, V is the molar volume, ρ is the density, Tc , Pc and ρc are the critical temperature, pressure, and density, respectively, ω is the acentric factor, T ′ = T /Tc ,, ρ′ = ρ /ρc , ΔT = 1 − T ′, Δρ = 1 − 1/ρ′, ci , i = 1, ..., 27 are constants,34 as di , j , i = 1, ..., 10, j = 1, ..., 6,35 and R is the ideal

gases constant (R = 82.06 cm3·atm·mol−1·K−1). C



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III. THERMODYNAMIC PROPERTIES III.1. Molar Volume. The molar volume is key for computing other state variables such as internal energy,

Figure 1. Representation of the pressure and temperature conditions of the experimental data used to validate the molar volume. The continuous and dashed lines connect at the critical point, CP (Tc = 304.127 K and Pc = 7.4 MPa).

enthalpy, and entropy, which are required for modeling properties such as throttling temperature, inversion curve, or velocity of sound. The knowledge of the density and molar volume at different pressure and temperature conditions is mandatory for the operation and design of equipment such as pipelines, compressor, vessels, and separation units. For a given temperature and pressure condition, the molar volume might be computed from the equations presented in Table 1 as the compressibility factor allows a better initial guess than the molar volume for solving the nonlinear equation. The cubic equations are solved using the method of Halley with analytical derivatives since it has faster convergence than the method of Newton.38 The multiparameter equations are solved using the function fzero from Matlab, which is a combination of bisection, secant, and inverse quadratic interpolation methods.39 All the computations are carried out in Matlab. To validate the EOS predictions, the results are compared with experimental data from the literature, which covered the liquid, superheated vapor, and supercritical condition. Figure 1 illustrates the pressure and temperature conditions of the experimental data in the CO2 phase diagram. The molar volume predicted by the EOS are further compared with data from NIST (National Institute of Standard and Technology), which provides the standards for thermophysical properties of pure fluids and fluid mixtures. In the case of CO2, most of the data in NIST is generated by the high accuracy Span and Wagner equation of state,40,41 reported as a reference EOS by several authors.13,23,25 The standard error (SE) and the average relative deviation (RD) are used to compare the prediction with the experimental data: N

SE =

∑ i=1

1 RD = N

(V iexp − V icalc)2 100 N

N

∑ i=1

|V iexp − V icalc| 100 V iexp

Figure 2. Algorithm proposed for interpolating T2 from tabulated experimental data from Price.44

(1)

Figure 3. Representation of the inversion curve and the isenthalpic curve in the PV diagram.

(2)

where V is the molar volume, N is the number of experimental data and the superscripts exp and cal indicate the experimental D



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It is possible to rewrite eq 3 in a more useful way so as to compute the Joule-Thomson coefficient:28 ÄÅ ÉÑ 1 ÅÅÅÅ RT 2 ij ∂Z yz ÑÑÑÑ jj zz Ñ μJT = Å Cp ÅÅÅÇ P k ∂T { P ÑÑÑÖ (4)

Table 2. Experimental Data for Validation of the Inversion Curve Predicted by the EoS author

temperature (K)

pressure (MPa)

Roebuck et al.47 Groot and Michels48 Price44 Perry43

246−281 298−423 373−1273 251−1290

2.0−24.3 35.8−77.7 18.3−89.3 5.1−81.1

where Cp is the real isobaric specific heat, given by42 ÄÅ É ÅÅ ∂P ÑÑÑ2 Å T ÅÅ ∂T ÑÑÑ ρ i ∂ 2P y dρ ÅÅ ρÑ ÑÖ jj zz o Cp = C p(T ) − T −R jj 2 zz 2 + Ç 2 ∂P 0 k ∂T { ρ ρ ∂ρ ρ

Table 3. Experimental Data for the Validation of the Velocity of Sound Predicted by the EOS author 49

Hodge Giordano et al.50 Herget51

temperature (K) 300 300 and 400 300−311

∫

( )

T

pressure (MPa)

C po is 43

0.1−6.1 14.8−2168.8 0.6−10.1

where R is the ideal gas constant and specific heat. For the CO2, C po is given by C po = 10.3 + 0.00574T −

and calculated values, respectively. As the molar volumes in the liquid phase, as well as the superheated vapor at T < Tc , are small, high relative deviations would result, leading to misinterpretation of the results. The standard error, therefore, is used instead of the relative deviation for the liquid phase and superheated vapor at T < Tc . The average relative deviation is computed for the critical point, supercritical phase, and superheated vapor at T > Tc . III.2. Throttling Temperature. Throttling devices are restrictions in a line which cause a significant pressure drop, such as a partially opened valve or a porous plug. When the fluid crosses a partially opened valve, there is a pressure drop but no work is done on or by the system. These restrictions, such as the partially opened valve, are usually small so the flow might be considered adiabatic. The kinetic and potential energy might also be neglected. The first law of thermodynamics, therefore, reduces to ΔH = 0, that is, the process is isenthalpic. If the fluid has ideal behavior, enthalpy depends solely on temperature, H = H(T ), meaning that the temperature remains constant after the throttling process. On the other hand, for real fluids, the enthalpy is a function of temperature and pressure, H = H(T , P), hence a temperature change upon pressure drop is to be observed after the throttling device. Some wellknown applications of the Joule-Thomson effect include refrigeration and liquefaction processes, in which the pressurized fluid passes through a valve to obtain the cooling effect.28 The change in temperature due to pressure change in an isenthalpic process is represented by the Joule-Thomson coefficient ( μJT ), defined by42 i ∂T y μJT = jjj zzz k ∂P {h

( )

(5)

the ideal isobaric

195500 T2

(6)

The usual practice in the literature is to calculate the JouleThomson coefficient to know if the fluid will be cooled or heated after the throttling process based on the sign of the coefficient.13,28,3−32 This information, however, might not suffice, as the knowledge of the outlet temperature itself is required to ascertain, for example, the amount of hydrates or subproducts formed.37 We therefore propose an approach to compute the throttling outlet temperature based on the concept of residual property. The change in enthalpy from a real state (T1 , P1) to a real state (T2 , P2) can be computed from the residual enthalpy (Δhr ) as Δh = h1id − h1 + h2id − h1id + h2 − h2id

(7)

Δh = −(Δhr)T1, P1 + Cpid (T2 − T1) + (Δhr)T2 , P2 = 0

(8)

or

ρ i ∂Z y

where the residual enthalpy is given by Δhr = −T RT

∫0

Δhr =T RT

V

∫∞

42

jj zz dρ + Z − 1 j z k ∂T { ρ ρ

ij ∂Z yz dV jj zz +Z−1 k ∂T {V V

(9)

(10)

for the multiparameter and cubic EOS, respectively. The derivatives in eqs 9 and 10 are obtained by each EOS explicit in Z and the integrals are obtained analytically to provide greater accuracy. For a given inlet condition and pressure drop, eq 8 is a function of the outlet temperature T2 alone, then it can be computed by solving the single-variable nonlinear equation in eq 8. It important to highlight that the method proposed here can compute the throttling temperature itself without requiring the complex expression of the real specific heat in (5). All the computations are carried out in Matlab, and the function fzero solves the single-variable nonlinear function in (8). To validate the proposed method for computing the throttling temperature and to ascertain the best EOS to predict it, experimental data from the literature are used. Price44 published Joule-Thomson coefficients computed from PV experimental measurements, according to the following equation, which is obtained from eq 4: ÅÄ ÑÉÑ ÑÑ 1 ÅÅÅÅ ij ∂PV yz ÑÑ j z PV T μJT = − Å j z ÑÑ PCp ÅÅÅÇ k ∂T { P ÑÖ (11)

(3)

The Joule-Thomson coefficient indicates whether the fluid is heated or cooled after the restriction: if it is positive, the fluid is cooled; if it is negative, the fluid is heated. The condensation caused by the expansion might form a liquid phase, which might be desired for cryogenic distillation or might not be desired as in the case of flowing fluids. Heating due to expansion, on the other hand, might favor side reactions and the formation of undesirable products.37 Drilling in gas reservoirs, which operate at high temperatures and pressures, in the range where the JouleThomson coefficient is negative, might lead to a temperature increase due to pressure relief, and even a small increase in temperature might significantly affect the operation of surface production facilities.28 E



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l o o (5m2 Ωa + 4Ωb ± 3m Ωa m2 Ωa + 8Ωb ) o 2 o o + Ω (1 m ) , o a o o 2(m2 Ωa − Ωb)2 o o o o o Ωb o o o if m ≥ o o Ωa o o Tr = o m o o o o (5m2 Ωa + 4Ωb − 3m Ωa m2 Ωa + 8Ωb ) o o o (1 + m)2 Ωa , o o o 2(m2 Ωa − Ωb)2 o o o o o Ωb o o o if m < o o Ωa o n

The tabulated data give the Joule-Thomson coefficient for several pressure and temperature conditions, instead of the outlet temperature (T2 ) as a function of inlet conditions (T1 , P1) and pressure drop (P2 ). To ascertain the experimental value of T2 for a given inlet condition and pressure drop, however, it is necessary to interpolate the data from Price44 using the algorithm proposed in Figure 2. An initial guess for T2 assumes that it is equal to T1, then the Joule-Thomson coefficient at the inlet (T1 , P1) and outlet conditions (T2 , P2 ) are read from the experimental data.44 The estimate of T2 is updated based on the mean value of the Joule-Thomson coefficient until the value of T2 does not change significantly between successive predictions. III.3. Inversion Curve. The points where the JouleThomson coefficient is equal to zero are the inversion points, which are connected to form the inversion curve in the temperature and pressure diagram, as depicted in Figure 3. It further illustrates an isenthalpic curve: starting at T1 , P1, if the outlet pressure P2 lies to the right-hand side of the inversion curve, the fluid will be heated (T2 > T1); if P2 lies to the left-hand side of the inversion curve, however, the fluid will be cooled (T2 < T1). The inversion curve delimits the region where the Joule-Thomson coefficient is positive or negative; hence, it is an important parameter for transport and separation processes, for example. The throttling process might be further used to obtain experimental properties of fluids, such as specific volume and specific heat using an equation of state. Therefore, a common approach in the literature is to use the Joule-Thomson coefficient and the inversion curve as a rigorous test to evaluate the prediction capability of an equation of state.31,45 On the basis of the principle that in the inversion curve the Joule-Thomson coefficient is equal to zero, the following equations can be derived from eq 4 for the cubic and multiparameter EOS, respectively: i ∂Z y i ∂Z y T jjj zzz + V jjj zzz = 0 ∂ T k {V k ∂V {T i ∂Z y i ∂Z y T jjj zzz + ρjjjj zzzz = 0 k ∂T { ρ k ∂ρ {T

(14)

Depending on the m value, two inversion temperatures can be predicted. Equation 14 is applied to the cubic equations of state with the α(Tr) function presented in Table 1 for the carbon dioxide (ω = 0.224 ) in order to compute the inversion temperatures at the zero-pressure intersection at the hightemperature gas range. Only the point D is evaluated here since it occurs at the temperature conditions which are usually of interest for the compressible flow analysis. The predictions of each EOS are validated with experimental data, according to Table 2. The data from Roebuck et al.47 and Groot and Michels48 cover only the lower branch of the inversion curve. The availability of experimental data of inversion curve is hindered by its complex acquisition. The fluid has to be submitted to temperatures and pressures which are much higher than the critical point and, because it is a differential property, there is a tendency to magnify the experimental errors as well.28 III.4. Velocity of Sound. The presence of valves and changes in the cross-sectional area, such as contraction, expansion, curves, elbows, and tees, for example, cause a local pressure drop and disturb the velocity profile. The velocity at which an infinitesimal disturbance (pressure wave) propagates through a compressible medium is known as the velocity of sound, which is mathematically defined as42

(12)

c=

ij ∂P yz jj zz j ∂ρ z k {S

(15)

where c is the velocity of sound, P is pressure, ρ is density and S is entropy. The assumptions made in developing equation 15 are10 the fluid is a continuum; changes in pressure are small so that changes in pressure are proportional to changes in density; viscous effects are negligible; the velocity of flow of the medium is small compared to the velocity of sound; heat conduction is neglected in all directions; the absorption coefficient in the medium is small. As already mentioned, monitoring the velocity of sound means that the maximum velocity reached by the flow at the exit of a pipeline can be computed, identifying obstructions and leakages in gas pipelines as well as vibrations in compressors. A more useful form for computing the velocity of sound, instead of the isentropic path of equation 15 is given by10

(13)

The derivatives in eqs 12 and 13 are obtained from the EOS explicit in Z , therefore they are functions of temperature and volume/density. For a given temperature, the nonlinear algebraic eq 12 or (13) is solved for V or ρ, respectively, using the function fzero in Matlab. The pressure of the inversion point is then computed from the EOS explicit in P , given the temperature and the volume or density. The inversion curve exhibits two zero-pressure intersections, points C and D in Figure 3 in the low- and high-temperature ranges, that is, the liquid and gas regions, respectively. As already mentioned, the prediction of the inversion curve is a wellrecognized severe test of an equation of state.31,45 Segura et al.46 use the zero-pressure intersections to demonstrate the limitation of Soave-type alpha functions, α(Tr), in cubic EOS for pure fluids. Depending on the α(Tr) function, multiple JouleThomson inversion curves can be predicted, indicating the existence of multiple stable critical points for pure fluids. According to Segura et al.,46 the inversion high-temperature at zero-pressure (point D) for SRK and PR, with the α(Tr) given in Table 1, is given by

c2 =

Cp ij ∂P yz jj zz Cv jk ∂ρ z{T

(16)

where Cp is the real isobaric specific heats, given by equation 5 and Cv is the real isochoric specific heat, given by Cv = Cvo(T ) − T F

∫0

ρ i ∂ 2P

yz dρ jj jj 2 zzz 2 k ∂T { ρ ρ

(17)



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Figure 4. Indication of the EOS with better accuracy for molar volume prediction in different pressure and temperature ranges of the PVT diagram.

where Cvo is the ideal isochoric specific heat, which correlates with C po , given by equation 6, according to43 C po − Cvo = R

P = P(ρ , T ) are known, the third function is calculated from eq 19. Therefore, eq 19 is useful to test the consistency of experimental data, thus the velocity of sound also plays a major role in testing equations of state.11 The derivatives in equations 16 and 19, as well as the derivatives to compute the real specific heats in (5) and (17), are obtained from the equations of state. If cubic equations of state are used, these equations might be written as a function of V instead of ρ using the definition V = 1/ρ. In order to compute the real specific heat, the integrals in eqs 5 and 17 have to obtained analytically or numerically. For greater precision, analytical expressions for the integrals are derived for both cubic (PR and SRK) and the HG EOS. The SP EOS, however, requires the handling of complex algebra to obtain analytical expressions, therefore the numerical approach for the integral is used. All the computations are carried out in Matlab and the numerical integral is computed by the function integral which uses the global adaptive quadrature.39 The prediction of the velocity of sound using different EOS is compared with the experimental data from the literature, according to Table 3. Hodge49 measures the velocity of sound in conditions below the critical point, whereas Giordano et al.50 present measurements in conditions above the critical point.

(18)

where R is the ideal gas constant, for which the value 82.06 cm3· atm·mol−1·K−1 is considered, in accordance with the parameters of the multiparameter EOS. To obtain the velocity of sound in m/s, therefore, a factor needs to be multiplied on the right-hand side of eq 16: FC = 101.325/MW , where MW is the molecular weight of the carbon dioxide. Kabelac11 uses an alternative equation to compute the velocity of sound from eq 15, which is thermodynamically equivalent to eq 16: i ∂P y T i ∂P y c 2 = jjjj zzzz + 2 jjj zzz ∂ ρ ρ Cv k ∂T { ρ k {T

(19)

where a factor FC = 101.325/MW also needs to be multiplied on the right-hand side to obtain the velocity of sound in m/s. These equations make it clear that the velocity of sound, on the one hand, is associated with the gradient of the state variable P = P(ρ , T ) and, on the other hand, it connects T , P and ρ with Cv . If c = c(ρ , T ) and P = P(ρ , T ) or Cv = Cv(ρ , T ) and G



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Figure 5. Relative deviation of the Huang EOS regarding molar volume in different pressure and temperature conditions.

Figure 6. Relative deviation of throttling temperature predicted by the four equations of state for different inlet pressures (P1, MPa) and pressure drop (ΔP , MPa): (a) Soave−Redlich−Kwong; (b) Peng−Robinson; (c) Huang; (d) Sterner-Pitzer.

Herget,51 on the other hand, measures a considerable amount of data, comprehending the critical point where a minimum of the velocity of sound is observed. The author observes that, in most of the region studied, the sensitivity of velocity with pressure is small, unlike in the immediate area of the critical point, where relatively small pressure changes may produce large or even enormous changes in the velocity of sound. Herget51 finds that

the velocity was measured over the entire range with an accuracy of 0.05%, except at the minimum of the velocity-pressure curve, where the error reached ±2.0%.

IV. RESULTS This section presents the predictions of each property using the four EOS studied, PR, SRK, HG, and SP. First the results for the H



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comparison. In an attempt to ascertain the most accurate EOS at given pressure and temperature conditions, the PVT diagram is discretized using temperature ranges of 50 and 100 K and pressure ranges of 2.5 and 10 MPa. Figure 4 illustrates the most accurate EOS at each pressure and temperature region where experimental data were available. In the liquid phase and superheated vapor at T < Tc , the standard error is computed. In the superheated vapor at T > Tc and supercritical phases, where the specific volumes are greater, the relative deviation is considered for validation. The figure only indicates the EOS with SE ≤ 1.00 for the liquid phase, SE ≤ 10.0 for the superheated vapor at T < Tc or RD ≤ 1.0% for the superheated vapor at T > Tc and supercritical phase, except those whose relative deviations are given in brackets. The EOS with the highest deviations/errors within these limits appear underlined in the figure, as indicated. The HG equation of state presents overall smaller deviations, even in the vicinity of the critical point, where the relative deviation is slightly higher (2.3%) but still much lower than the other equations of states, which could not predict the specific volume with reasonable deviations in this region. This finding confirms the claim of Huang et al.34 that their EOS is particularly accurate in the critical region due to the consideration of nonanalytical terms. The SP also performs reasonably well, even in the supercritical region where the cubic equations, mainly the SRK, cannot predict the specific volume as well as the other equations. The PR cubic equation of state is valid in most of the investigated supercritical space, except in the liquid and superheated vapor (T < Tc ) phases as well as around the critical point. The SRK equation performs better in the superheated vapor phase. The assumption of ideal gas is investigated, and the results in Figure 4 indicate that it is valid only in some specific regions outside the supercritical phase. Therefore, the multiparameter equations, specially the HG equation of state, present better predictions of the molar volume in the overall PVT diagram, as confirmed by the relative deviations presented in Figure 5 for different pressure and temperature conditions. Although the deviations of the Huang EOS around the critical point are higher, the other investigated EOS fail to predict the molar volume in this region. As already mentioned, several applications of CO2 operate in supercritical conditions, therefore the ideal assumption or the cubic equation of states could not represent the behavior of the fluid satisfactorily. The predictions from NIST present overall higher accuracy than the investigated EOS then, for the sake of simplicity, they are not represented in Figure 4. For the superheated vapor at T > Tc and supercritical regions, the maximum relative deviations are 0.1% and 0.4%, respectively. At the liquid phase and superheated vapor at T < Tc , the highest standard errors obtained by NIST are 6.00 × 10−5 and 2.16 × 10−3, respectively, whereas the Huang EOS gives 2.36 × 10−4 and 3.48 × 10−2 , respectively. As aforementioned, the NIST data are mostly computed by the Span and Wagner equation. Despite its highest accuracy to describe the thermodynamic properties, the SW model is much more complex than the HG and SP EOS; hence, convergence problems might arrive when the SW EOS has to be solved simultaneously with mass and energy balances,52−56 required to describe the compressible flow. As the modeling of the compressible flow is a very complex problem, a simpler but still accurate EOS, as the HG or SP, might ensure convergence and then a satisfactory description of the compressible flow.

Figure 7. Comparison of experimental data of the inversion curve with the prediction by the equations of state: (a) Soave−Redlich−Kwong (SRK) and Peng−Robinson (PR); (b) Huang and Sterner-Pitzer.

Table 4. Relative Deviation of the Inversion Curve Predicted by the EOS relative deviation lower branch upper branch inversion point total

SRK

PR

Huang

SP

11.30% 24.48% 6.61% 22.78%

18.67% 19.14% 9.55% 21.12%

13.02% 6.98% 4.45% 10.76%

16.49% 64.24% 7.55% 37.55%

molar volume are discussed, followed by the computation of the throttling temperature and the inversion curve. Finally, the results of the velocity of sound are presented. IV.1. Molar Volume. The analysis of the molar volume comprehends the following phases: liquid, superheated vapor, and supercritical, according to the PVT diagram in Figure 1. Besides the PR, SRK, HG, and SP EOS, the predictions using the ideal gas law and the NIST database are further considered for I



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Figure 8. Velocity of sound predicted by PR (a,b) and SRK (c,d) at different temperatures against experimental data from Hodge49 and Herget.51

IV.2. Throttling Temperature. This paper proposes a method to compute the throttling temperature to ascertain the outlet temperature after a throttling device. The Joule-Thomson coefficient on the other hand, only informs if the fluid is heated or cooled. The method is validated with experimental data from Price44 as discussed in the previous section. The relative deviations between the interpolated experimental data (Figure 2) and the prediction by the four studied EOS against inlet temperature are summarized in Figure 6 for several inlet pressures (P1) and pressure drop (ΔP ) conditions. The cubic EOS, namely SRK and PR, present overall better results. The multiparameter Huang and SP equations present higher deviations which decrease as the temperature increases. When ΔP = 10.1 MPa, the lower inlet pressure (30.4 MPa) yields higher deviations. Despite this, there seems to be no systematic correlation between the relative deviation and the pressure drop or the inlet pressure. The HG equation, despite outperforming the other equations for the molar volume prediction, cannot predict the throttling temperature as accurately as the cubic EOS do, especially for temperatures below 1000 K. The results indicate, however, that the method proposed here can efficiently compute the throttling temperature itself, providing further information for the monitoring of throttling processes. IV.3. Inversion Curve. Figure 7 illustrates the inversion curve predicted by the four equations of state (SRK, PR, HG, SP) against the experimental data from the literature (Table 2). The data from Price44 present a slope at high temperatures and, as observed by Chacin et al.,28 the SP EOS seems to follow this slope, unlike the other investigated EOS. Colina and OliveraFuentes29 investigate this behavior and conclude that the predictions of the SP at high temperatures are incorrect, arguing

that the ability of an EOS to successfully predict the high temperature branch of the inversion curve is directly related to its second and third virial coefficients. Diamantonis et al.13 compared inversion curves computed by PC-SAFT and PR against the reference EOS Span and Wagner and observed that the PR overshoots the maxima of the curves, as observed here. All the four EOS, cubic and multiparameter, can predict the trend of the lower branch of the inversion curve (P < 81.1 MPa and T < 600 K), as depicted in Figure 7. Table 4 summarizes the relative deviations for the lower and upper branch of the inversion curve, as well as the inversion point at the maximum temperature. The SRK presents lower deviations for the lower branch, similar to the Huang EOS, which outperforms at the upper branch (P < 81.1 MPa and T > 600 K) of the inversion curve. The Huang EOS, therefore, is the EOS indicated to predict the inversion curve but, if the lower branch alone is of interest, the cubic SRK can be used without any loss of accuracy. Applying eq 14 to the cubic SRK and PR with the α(Tr) function presented in Table 1 for the carbon dioxide (ω = 0.224 ), two inversion temperatures are predicted at the zero-pressure intersection at the high-temperature gas range: SRK predicts 1018 and 17737 K; PR predicts 1168 K and 24815 K. The lower temperatures 1018 K and 1168 K are in good agreement with the values obtained when extrapolating the SRK and PR curves in Figure 7, 1018 K and 1170 K, respectively, but are far from the experimental data near the zero-pressure intersection at high temperatures. This observation indicates that the α(Tr) in Table 1 used for the SRK and PR equations, although widely used in literature, might not be the best choice for the carbon dioxide. The higher temperatures, 17737 and 24815 K, are not usually applied for practical purposes. J



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Figure 9. Velocity of sound predicted by Huang (a, b) and SP (c, d) at different temperatures against experimental data from Hodge47 and Herget.51

IV.4. Velocity of Sound. The velocity of sound is predicted using the two approaches presented: eq 16 by Elizondo and Maddox10 and eq 19 by Kabelac.11 Both are equivalent, hence provide the same results. Figure 8 and Figure 9 compare the predicted velocity of sound with the experimental data from Hodge49 and Herget51 at different temperatures. The velocity of sound presents a minimum around the critical point, where the sensitivity regarding the pressure change and the experimental errors are greater, as observed by Herget.51 The phenomenology of the velocity of sound near and across the critical lines was investigated by Reis et al.,57 who formulate a thermodynamic link between the velocity of sound and isochoric heat capacity. All the equations of state present a minimum at the critical point

Table 5. Relative Deviation of the Velocity of Sound Predicted by the EOS Hodge49 and Herget51 EOS

1 ≤ P ≤ 10 MPa

P ≤ 6 MPa

Giordano et al.50

PR SRK Huang SP

15.77% 19.48% 10.61% 5.82%

1.10% 1.31% 0.71% 0.72%

29.75% 34.98% 30.97%a 5.20%

a

Up to 131 MPa and 300 K, as the EOS does not converge for higher temperatures and pressures.

Figure 10. Velocity of sound predicted by the EOS against experimental data from Giordano50 at 300 K (a) and 400 K (b). K



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can also be considered to predict the lower branch of the inversion curve without any loss of accuracy. The velocity of sound is better represented by the SP EOS, especially at pressures near and above the critical pressure, where the other EOS lose accuracy. When modeling the compressible flow, depending on the numerical method to solve the system of equations, the deviation of the property calculated by the EOS might propagate to other variables because the EOS is solved together with the mass, energy, and momentum balances. Therefore, an overall uncertainty analysis should be carried out for the particular problem. These findings contribute to sCO2 technology because the Huang and SP EOS have rarely been used for this, although these EOS were developed for CO2. On the one hand, the cubic EOS or the ideal gas assumption might not be valid in the usual operating conditions of these processes. On the other hand, the SW or the PC-SAFT EOS, for example, involve effort to predict properties such as the throttling temperature, inversion curve, and velocity of sound. This trade-off could be solved by the Huang and the SP EOS. The proposed method to compute the throttling temperature, validated with experimental data, provides additional information for process monitoring because the usual practice is to compute solely the Joule-Thomson coefficient. Our method does not require the computation of real specific heat and provides the temperature itself, which for some applications is more useful than just the information if the fluid is cooled or heated after the throttling process. This method might also be useful for other gases and applications. Finally, this paper presents a more comprehensive study of the key thermodynamic properties in fluid dynamics, addressing the gap between thermodynamics and fluid dynamics. The equations of state can accurately predict not only the specific volume but also the throttling temperature, the inversion curve, and the speed of sound. The findings might be of great value when designing and monitoring sCO2 processes because these properties play an important role in the process efficiency and product quality.

Table 6. Relative Deviation of the Velocity of Sound Predicted by SP and NIST Hodge49 and Herget51

1 ≤ P ≤ 10 MPa

P ≤ 6 MPa

T (K)

SP

NIST

SP

NIST

300.15 301.15 303.15 304.15 305.15 306.15 308.15 311.15 300.00 400.00a

0.32% 2.54% 9.06% 7.25% 8.33% 4.91% 5.10% 3.94%

0.26% 5.66% 5.92% 6.37% 4.56% 2.38% 1.72% 1.72%

0.13% 0.14% 0.20% 0.26% 0.23% 0.28% 0.37% 0.11%

0.11% 0.13% 0.19% 0.25% 0.23% 0.25% 0.38% 0.08%

Giordano et al.50 SP

NIST

5.46% 7.49%

6.49% 6.52%

a

Pressure up to 800 MPa.

but the predictions lose accuracy near and above the critical pressure. Up to 6 MPa, the relative deviations are lower than 1.4%, as Table 5 indicates. Above 6 MPa, though, only the multiparameter Sterner-Pitzer EOS can predict the velocity of sound with accuracy: the total deviation of the Sterner-Pitzer EOS reaches 5.82%, including the greatest errors around the critical point. This finding is confirmed when the predictions of the EOS are compared against the data of Giordano et al.,50 as Figure 10 illustrates. Since the data involve measurements at high pressures, the Sterner-Pitzer EOS is the only one which can predict the velocity of sound with accuracy. The Huang EOS, on the other hand, even converges in more severe temperature and pressure conditions. The predictions of the SP EOS are compared with the data generated by NIST since it computes the velocity of sound based on the benchmark Span and Wagner EOS. Table 6 summarizes the relative deviations of SP and NIST predictions against the experimental data from Hodge,49 Herget,51 and Giordano et al.50 at different temperatures. Within 1 ≤ P ≤ 10 MPa, which includes the critical point, the deviations are similar but NIST is generally more precise. For pressures up to 6 MPa, though, NIST is just slightly more precise. For the even higher pressure reported by Giordano et al.50 (see Figure 10), the SP is more precise at lower temperatures than the NIST data. Furthermore, the SW is valid until 800 MPa, whereas the SP can predict the velocity of sound for pressures up to 10 GPa.22,35,41 A much less complex equation of state, the Sterner Pitzer, then, can be used to predict the velocity of sound without loss of accuracy in many applications. This is a very useful finding, especially when modeling the compressible flow, where the EOS has to be solved simultaneously with the mass, energy, and momentum balance.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Gloria M. N. Costa: 0000-0002-1689-0090 Karen V. Pontes: 0000-0002-8208-0854 Funding

The authors thank the Fundaçaõ de Amparo à Pesquisa do Estado da Bahia (FAPESB), Coordenaçaõ de Aperfeiçoamento ́ Superior (CAPES), and the Conselho de Pessoal de Nivel ́ Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq), for their financial support.

V. CONCLUSIONS This paper presents the computation of thermodynamic properties, namely, molar volume/density, throttling temperature, inversion curve, and velocity of sound, using different equations of state. Regarding the molar volume/density, the multiparameter EOS outperform the cubic EOS, especially in the supercritical phase, where the sCO2 processes operate. Among the multiparameter EOS, the Huang EOS is generally more accurate than the SP EOS, especially in the vicinity of the critical point. Despite this, the cubic EOS yield more precise results for the throttling temperature. The inversion curve is more accurately predicted by the Huang EOS, but the SRK EOS

Notes

The authors declare no competing financial interest.

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REFERENCES

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Prediction of Thermodynamic Properties of CO2 by Cubic and

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