A Possible Explicit Equation Fitting Method for the Gaseous Heat Capacity Near the Critical Point Based on Density and Temperature

Abstract

CO₂ is a potential fluid for absorbing and accumulating thermal energy; an accurate and fast calculation method for the heat capacity is essential for the study of the flow state near the critical point. However, the calculation of the heat capacity near the critical point by the equations suggested by NIST can easily be divergent, such as for CO₂, nitrogen, methane, etc. Therefore, an explicit fitting equation was studied. The fitting equation, which used density and temperature as variables and contained three constants, was derived from the nature of heat capacity change (molecular kinetic energy and potential energy). Based on the heat capacity data of the NIST WebBook, the heat capacity of CO₂ is taken as the example for the equation deduction and parameter fitting. The three constants were defined in order by Origin fitting software. By this new approach, it is found that the heat capacity at the critical point is below 1% deviant from that of the NIST WebBook. Moreover, the heat capacities that are difficult to be calculated in the NIST WebBook are well calculated. The study shows that the fitting equation is efficient for the prediction of heat capacity of gases near the critical point.

1. Introduction

Supercritical carbon dioxide (SC-CO₂ for short) can produce large temperature differences with pressure changes near the critical point. Therefore, it has become an effective medium for jet rock breaking, fracturing, thermal cycling, and other deep earth resource exploitation technologies. SC-CO₂, as a jet medium, can improve the mechanical penetration rate, increase oil and gas production, save water resources and energy, and reduce emissions, and it has become an anhydrous drilling fluid and fracturing fluid with great application potential [1,2,3]. The threshold pressure of the SC-CO₂ jet for breaking granite and shale is two-thirds and one-half that of a water jet, respectively [4]. SC-CO₂ coiled-tubing slim-hole jet-assisted rock breaking drilling can greatly reduce the drilling cost due to the low threshold pressure of jet rock breaking, low circulating pressure loss, the improved service life of drilling tools, and accessible underbalanced drilling conditions [4,5]. Compared with water jet fracturing, the initiation pressure of SC-CO₂ jet fracturing is less than half of that of water jet fracturing [6]. With SC-CO₂ as the fracturing fluid, clay hydration and expansion as well as water shortage can be solved. Moreover, shale has stronger adsorption of CO₂, which can replace methane and realize CO₂ geological storage [7,8,9,10]. The common basic problem in SC-CO₂ jet drilling and SC-CO₂ jet fracturing is the mechanism of the SC-CO₂ jet rock breaking. At present, a lot of experiments on SC-CO₂ jet rock breaking have been conducted, proving that SC-CO₂ jets have greater advantages than water jets in rock breaking [4,5,11,12,13,14,15,16]. What is interesting about the mechanism is that a large temperature gradient and high-temperature stress can be generated under a SC-CO₂ jet, and the higher temperature stress is the main reason why the SC-CO₂ jet is superior to the water jet in rock breaking [17,18,19,20]. Nevertheless, there are few studies on the SC-CO₂ jet flow field and stress field near the critical point due to the drastic changes in density, heat capacity, etc.

Geothermal and solar electricity generation is a renewable generating technology that can enable significant reduction in fossil fuel consumption. In environmental, cost, and safety terms, CO₂ is regarded as an ideal natural heat transfer medium [21,22,23,24,25,26,27,28]. The SC-CO₂-based Brayton cycle has been seen as a good alternative to conventional power cycles because of high cycle efficiency, compact turbo machinery, and compact heat exchangers [29,30,31,32,33]. The big change in SC-CO₂ heat capacity near the critical point makes it efficient to store and release energy. If we control the CO₂ phase of flow under the supercritical state, there will be no phase change such as water gasification and the cost of the equipment controlling phase change can be saved [29]. Moreover, the heat transfer rate is high near the critical point [34,35,36]. Based on the theory above, Ni et al. have successfully applied invention patents using SC-CO₂ near critical point as a transmission media to more efficiently generate electricity [37] and make use of geothermal resources to supply heating and exploit oil [38]. As the patents described, the cycle efficiency of SC-CO₂ near the critical point is more than two times that of water in the conditions where the energy of water vaporization can be transferred to electricity 100%. The main problem is how to predict and control the flow state near the critical point. Simulation is a good method for flow prediction and design of the experimental setup.

Recently, drug processing using SCFs has been proposed as a sufficient alternative for conventional techniques. SC-CO₂ is non-toxic, low-cost, non-flammable, and easyily accessible with moderate critical pressure and near ambient critical temperature. Amongst SCFs, SC-CO₂ and the mixtures containing high amounts of SC-CO₂ are the most used SCFs. Modeling is an efficient method for the study of drug solubilities and drug particle formation and delivery [39,40]. To predict the temperature, pressure, density, and velocity of fluids in the wide range of operating conditions, equations of density and heat capacity have an important effect on the accurate correlative capability [41].

In a word, the application of SC-CO₂ in the above technical fields is closely related to the fluid properties and flow field of carbon dioxide. Mature commercial software such as FLUENT and CFX are efficient and economical methods to study the flow field, and the heat capacity is an essential input parameter. For the accurate calculation of the heat capacity of carbon dioxide and its mixture, NIST real gas equations, such as the Span–Wagner (S-W) equation of state [42] and GERG-2008 wide-range equation [43], can satisfy both the calculation accuracy and application range in engineering. In addition, NIST has provided an effective software named REFPROP for the fluid properties calculation of density, heat capacity, and so on. However, although the FLUENT and NIST WebBook [44] used the same real gas equations for the calculation of isobaric heat capacity (C_p for short), such as those for carbon dioxide, methane, nitrogen, and water, the heat capacity near the critical point can easily diverge in the fluent simulation and NIST WebBook calculation (as shown in

Table 1. Points where critical heat capacities diverge easily in the NIST WebBook [44].

Fluid	Temperature (K)	Pressure (MPa)	ρ (kg/m3)	Cp (J/g·K)	Phase
CO2	303.7	7.3725	608.81298	undefined	liquid
CO2	304.2	7.3675	362.24288	undefined	vapor
CO2	304.5	7.3835	342.33699	undefined	supercritical
CH4	190.1	4.5955	214.82531	undefined	liquid
CH4	190.6	4.5960	131.59005	undefined	vapor
CH4	190.8	4.6035	122.47476	undefined	supercritical
N2	125.7	3.396	428.58276	undefined	liquid
N2	126.2	3.392	255.06112	undefined	vapor
N2	126.4	3.4	230.87678	undefined	supercritical
H2O	646.6	22.064	423.77596	undefined	liquid
H2O	647.2	22.06	250.46303	undefined	vapor
H2O	647.5	22.066	230.395	undefined	supercritical

). By exploring the deep reason, one possible reason is that because the heat capacity calculation equation is implicit and involves multiple intermediate coefficients, if the significant digit of the parameter is reduced for the sake of fast calculation, numerical truncation and cumulative error will be generated, resulting in the situation that the heat capacity cannot be calculated.

In terms of the gas heat capacity calculation, researchers have conducted a lot of valuable research. For example, C N Yang and C P Yang studied the critical point in the liquid–gas transition effect on the heat capacity of oxygen and noble gases [45]. W Botch and M Fixman founded a theory for dynamic heat capacity associated with long-wavelength density fluctuations in the critical region [46]. G Orkoulas et al. established that the divergence is shared almost equally between the second temperature derivatives of the pressure and the chemical potential [47]. Previous studies have shown that the heat capacity increases substantially near the critical point and density fluctuation and temperature-related parameters are the key factors affecting the heat capacity of gases. Therefore, this study derives the fitting equation method to directly solve the isobaric heat capacity calculation equation by density and temperature based on the authentic NIST WebBook data [44] and the nature of the heat capacity change (molecular kinetic energy and potential energy). The heat capacity of CO₂ is taken as the example for the equation deduction and parameter fitting. Compared with the existing implicit function equation for solving the heat capacity, this new explicit method is simpler and faster because the intermediate coefficients calculation is omitted and lots of iterative calculations are avoided. The heat capacity near the critical point is hard to diverge because of the smaller truncated cumulative error. Additionally, it gives a simple method for deducing the critical heat capacity by fitting the experimental data near the critical region. In particular, the current equations of states, namely the Peng–Robinson (PR), Soave–Redlich–Kwong (SRK), etc., can calculate the density, temperature, and pressure of gases, but there are no effective corresponding heat capacity equations. Therefore, the heat capacity calculation method established in this paper can also be an important supplement to the above gases’ equations of state.

2. Identification of the Heat Capacity Fitting Equation

The internal energy of gas molecules can be divided into two parts, molecular kinetic energy and molecular potential energy, among which molecular kinetic energy is mainly related to the temperature. The isobaric heat capacity of fluids with constant density can be calculated according to Equation (1):

$$C_p=\frac{\Delta E_p+\Delta E_k}{\Delta T}$$

(1)

where $C_p$ is the isobaric heat capacity, $E_p$ is the molecular potential energy, $E_k$ is the molecular kinetic energy, and $T$ is the temperature.

The identification of the potential energy function is the key to determine the fitting of Equation (1), and the potential energy function can be divided into the model fluid potential energy function, the semi-empirical potential energy function, and the ab initio potential energy function [48]. In these three types of functions, the semi-empirical potential energy function is more welcome and popular due to its good calculation precision and simple form. The commonly used empirical potential energy model can be further divided into the two-center Lennard-Jones plus quadrupole model (2CLJQ) [49,50,51,52] and the three-center Lennard-Jones plus quadrupole model (3CLJQ) [53]. Among these two models, 2CLJQ was introduced for accurate simulation and calculation of the properties of supercritical carbon dioxide [54]. In addition, Meng Long et al. proposed a 2CLJDQP model considering the long-range electrostatic and induction effects [55], and Wang Sheng et al. improved the 2CLJDQP model by introducing potential energy parameters related to temperature [56].

Molecular potential energy can be mainly divided into intermolecular potential energy and intramolecular potential energy. It is found that intermolecular attraction plays a more important role when the pressure is low, and then the fluid density fluctuates obviously, whereas the intermolecular repulsive force leads to weak density fluctuation when the pressure is high [57]. As the heat capacity is mainly influenced by density, intramolecular potential energy is not considered in this study, which can also simplify the fitting process. Although the later developed models such as 2CLJDQP have higher accuracy, they will make the fitted equations and processes more complicated. The intermolecular potential energy of 2CLJQ can be expressed by Equation (2) [58,59]; it has been proved to be useful for the isobaric heat capacity of working fluid in a supercritical CO₂ Brayton cycle. Since it is a function of density and temperature, the molecular potential energy is exclusively related to the temperature when the density is constant. The main purpose of considering potential energy is to obtain a semi-empirical formula based on the principles of a simple form and few parameters for fitting; the relationship of the energy and size terms of the 2CLJQ model to critical points [60] is also ignored.

$$E_{p\left(r,T\right)}=\frac{{\epsilon }^{\left(eff\right)}}{n-6}\left\{6{\left[\frac{r_m{}^{\left(eff\right)}}{r}\right]}^n-n{\left[\frac{r_m{}^{\left(eff\right)}}{r}\right]}^6\right\}$$

(2)

where ${\epsilon }^{(eff)}$ is the potential well depth determined by the temperature, $r_m{}^{(eff)}$ is the molecular equilibrium spacing determined by the temperature, $r$ is the intermolecular spacing, and n is a constant.

${\epsilon }^{(eff)}$ and $r_m{}^{(eff)}$ can be approximately expressed as Equations (3) and (4).

$${\epsilon }^{(eff)}=\epsilon {\left(\frac{r_m}{r_m{}^{(eff)}}\right)}^6$$

(3)

$$r_m{}^{(eff)}=r_m+dT$$

(4)

where $\epsilon$ and $r_m$ are the potential well depth and the molecular equilibrium spacing, respectively, whereas d is a constant.

If we substitute Equations (3) and (4) into Equation (2), we can obtain Equation (5), which expresses the relationship between molecular potential energy and temperature. When the intermolecular distance is constant, Equation (5) is derivative, which can bring us Equations (6) and (7) that describe the variation in molecular potential energy and density potential energy with temperature, respectively.

$$E_{p\left(r,T\right)}=\frac{\epsilon r_m{}^6}{n-6}\left(6\frac{{\left(r_m+dT\right)}^{n-6}}{r^n}-\frac{n}{r^6}\right)$$

(5)

$$\frac{\Delta E_{p\left(r,T\right)}}{\Delta T}=\frac{\partial E_{p\left(r,T\right)}}{\partial T}=\frac{6\epsilon r_m{}^6{d}^{n-6}{\left(r_m/d+T\right)}^{n-7}}{r^n}$$

(6)

$$\frac{\Delta E_{p\left(\rho ,T\right)}}{\Delta T}=\frac{6\epsilon {\rho }^{n/3}{N}^{n/3-2}{d}^{n-6}{\left[{\left(M/{\rho }_m\right)}^{1/3}/d+T\right]}^{n-7}}{{\rho }_m{}^2{M}_{}{}^n}$$

(7)

In Equation (7), $\rho$ is the fluid density, $N$ is the molar constant, $M$ is the molecular weight, and ${\rho }_m$ is the fluid density when molecules are in the ground state.

When the density is constant, all the parameters except the temperature in Equation (7) are constant. Therefore, Equation (7) can be simplified as a relationship involving three constants, a, T₀, and c, as shown in Equation (8).

$$\frac{\Delta E_p^{}{}_{\left(\rho ,T\right)}}{\Delta T}=\frac{a}{{(T-T_0)}^c}$$

(8)

$$a=\frac{6\epsilon {\rho }^{n/3}{N}^{n/3-2}{d}^{n-6}}{{\rho }_m{}^2{M}_{}{}^n}$$

(9)

$$T_0=-{\left(M/{\rho }_m\right)}^{1/3}/d$$

(10)

$$c=7-n$$

(11)

The influence of the molecular kinetic energy on heat capacity can be neglected at the critical point because heat capacity significantly varies there and molecular potential energy dominants. In order to verify the correctness of Equation (8), the heat capacities of methane, water, nitrogen, and carbon dioxide under the critical density are fitted (corresponding to Figure 1, Figure 2, Figure 3 and Figure 4, respectively).

Table 2. Fitting results of critical heat capacity with different fluids.

Fluid	Critical Density kg/m3	a	c	T0	R-Square	MARE (%)
CH4	162.66	247.6514	1.18539	190.5413	0.99993	90.2242
N2	313.3	81.74206	1.10026	126.1819	0.99993	41.4713
H2O	322	1228.679	1.14402	647.0782	0.99995	27.5464
CO2	467.6	156.7309	304.1184	1.12325	0.99997	65.2738

displays the fitting values, determination coefficients (R-square), and mean absolute relative error (MARE) of the parameters in Equation (8). It is found that though the molecular kinetic energy is not considered, Equation (8) well applies the heat capacity near the critical temperature, demonstrating the effectiveness of the fitting near the critical point. However, as the temperature rises, the fitting error becomes bigger, finally resulting in a big absolute relative error.

The molecular kinetic energy cannot be neglected anymore when the temperature is away from the critical point, as it contributes more to the heat capacity when the effect of the molecular potential energy decreases. Figure 5 and Figure 6 show the fitting results of the CO₂ heat capacity with densities of 10 kg/m³ and 50 kg/m³, from which we can further find that Equation (8) is not applicable for the fitting far away from the fluid critical point.

It has been demonstrated that the molecular kinetic energy can be approximately calculated by the regression of a one-variable polynomial equation [61,62]. By the law that explains the variation in the molecular potential energy with the distance to the molecules [63], we can know that the molecular potential energy rarely varies with the temperature when the intermolecular distance is very large; therefore, the variation in heat capacity can be regarded as the variation in molecular kinetic energy with temperature.

We use the CO₂ heat capacity corresponding to density of 0.0001 kg/m³ to approximate the change in CO₂ kinetic energy per unit mass. Then, based on the calculation data of the NIST WebBook [44], we employed OriginPro to fit the relationship between CO₂ kinetic energy and temperature (as shown in Equation (12)) with a determination coefficient of 0.99999. Next, we substitute Equations (12) and (8) into Equation (1) and can obtain the CO₂ heat capacity fitting Equation (13).

$$\frac{\Delta E_k}{\Delta T}=C{p}_{\rho =0.0001}=0.58496-\frac{2.10229T}{{10}^4}+\frac{8.49535T^2}{{10}^6}-\frac{2.36752T^3}{{10}^8}+\frac{2.84159T^4}{{10}^{11}}-\frac{1.29972T^5}{{10}^{14}}$$

(12)

$$C{p}_{{\mathrm{CO}}_2}=\frac{a}{{(T-T_0)}^c}+0.58496-\frac{2.10229T}{{10}^4}+\frac{8.49535T^2}{{10}^6}-\frac{2.36752T^3}{{10}^8}+\frac{2.84159T^4}{{10}^{11}}-\frac{1.29972T^5}{{10}^{14}}$$

(13)

Equation (13) is then used to again test the CO₂ heat capacity fitting effect. Figure 7 and Figure 8 are the CO₂ heat capacity fitting curves considering molecular kinetic energy with respective densities of 10 kg/m³ and 50 kg/m³. It is obvious that Equation (13), which considers molecular kinetic energy, can well fit the CO₂ heat capacity with low density.

We used the software OriginPro to further verify the fitting precision of CO₂ heat capacity with densities of 1 kg/m³, 467.6 kg/m³ (critical density), and 1178 kg/m³ (maximum density), as shown in Figure 9, Figure 10 and Figure 11 and

Table 3. Fitting results for CO₂ heat capacity with different densities.

ρ kg/m3	a	c	T0	R-Square	MARE (%)
1	6.765	1.45845	128.1503	1	0.0001
10	71.9377	1.47141	134.17161	1	0.0004
50	76.80196	1.23533	190.82847	0.99947	0.0012
467.6	155.88611	1.12793	304.09746	0.99996	0.7123
1178	44.8895	0.79192	119.4896	0.99996	0.0002

. We can see that the mean absolute relative error (MARE) of the parameters in Equation (13) is less than 1/50 that of Equation (8) and that both the determination coefficients (R-square) and mean absolute relative error (MARE) reach excellent values. Equation (13) is demonstrated to well fit the variation in heat capacity with temperature. Therefore, we finally identify Equation (13) as the fitting Equation of CO₂ heat capacity as it fully considers the variation in molecular kinetic energy and potential energy. Figure 12 shows that the contribution ratio of the molecular kinetic energy to the total fitting CO₂ heat capacity (Ratio-C_p of E_k) increases with increases in temperature, and the maximum value reaches about 0.35, which proves the importance of considering the molecular kinetic energy.

3. Fitting Equation Parameters Determination

During the first fitting, we find that the three parameters influence each other. Therefore, it is necessary to implement three rounds of fitting to identify c, T₀, and a in order. The first fitting determined the relationship between constant c and density; the second fitting determined the relationship between constant T₀ and density; the third fitting determined the relationship between constant a and density. The first and final fitting results of a, c, and T₀ are shown in Figure 13, Figure 14 and Figure 15, which have an average determination coefficient of 0.99993, which indicates very high fitting precision.

It can be seen from the first fitting dot of Figure 14 that the variation of T₀ is more regular, which is totally different from that of a and c, especially when the density is low or high. The reason could be that more deviation from the critical density means a smaller change in molecular potential energy, which makes it difficult for the fitting parameters to reflect the influence of molecular potential energy on heat capacity. If we consider c as an impact factor for the influence of molecular potential energy on heat capacity, we might obtain a larger c with larger molecular potential energy at the critical density point, whereas c tends to be a constant when the density is low or high. Therefore, we should simplify the fitting Equation of c. Existing multiple library functions in the OriginPro were compared and studied to finally identify the fitting Equation of c and the related parameters, as expressed as Equation (14) and shown in

Table 4. Fitting parameters of c (R-square > 0.9999, MARE < 0.0001).

ρ kg/m3	xc	A	A1	A2	A3	A4	A5
419–445	445	1.14733	−0.135	58.6	−560	−2143	45,108
445–467.6	467.6	1.12795	−0.12779	−22.8665	1322.2	−40188	430,658
467.6–495	467.6	1.12795	−0.62095	-19.3	840.2	−28050	318,116
495–518	495	1.18665	0.42544	74	−680	2790	0
518–570	570	0.8	−0.55794	−17.6784	−107.511	1356.378	0

. The final fitting dot of Figure 13 shows the relationship between fitting c and density.

$$c=\left\{\begin{array}{ll}1& \left(\rho \le 418\right)\\ A/\left[1+{\displaystyle \sum _{\mathrm{i}=1}^5{A}_i{(\left|\rho -x_c\right|/500)}^{\mathrm{i}}}\right]& (419\le \rho \le 518)\\ A/\left[1+{\displaystyle \sum _{\mathrm{i}=1}^5{A}_i{(\left|\rho -x_c\right|/500)}^{\mathrm{i}}}\right]& (518\le \rho \le 570)\\ 0.8& \left(\rho \ge 570\right)\end{array}\right\}$$

(14)

When $\rho$ ≤ 418 kg/m³ or $\rho$ ≥ 519 kg/m³, the maximum heat capacity can be calculated based on the NIST WebBook. However, in the transition zone where 418 kg/m³ ≤ $\rho$ ≤ 519 kg/m³, the maximum heat capacity cannot be calculated. Therefore, there should be a large mutation within or near the density range of 418–419 kg/m³ and 518–519 kg/m³. In this context, we adopt the linear interpolation method to calculate the heat capacity of the transition zone, expressed as Equation (15).

$$Cp=\left\{\begin{array}{ll}C{p}_{\rho =418}+(C{p}_{\rho =419}-C{p}_{\rho =418}\left)\right(\rho -418)& (418\le \rho \le 419)\\ C{p}_{\rho =518}+(C{p}_{\rho =519}-C{p}_{\rho =518}\left)\right(\rho -518)& (518\le \rho \le 519)\end{array}\right\}$$

(15)

In the fitting process, it is found that the heat capacity fitting precision is more sensitive to changes in T₀, whereas T₀ changes very regularly. Therefore, we then fit the T₀ through the piecewise Equation (16).

Table 5. Fitting parameters of T₀ (R-square > 0.9999, MARE < 0.0001).

ρ kg/m3	xc	A	A1	A2	A3	A4	A5
0.01–250	250	290.17571	0.33243	0.2834	7.34123	−26.56616	40.9466
250–330	330	300.52514	0.11042	1.85179	−26.71893	209.13131	−543.8088
330–375	375	303.00611	0.06046	0.13587	6.2638	−67.77213	269.90062
375–418	418	303.94059	0.0181	0.28438	−5.80458	100.62786	−507.6728
419–445	445	304.09399	0.00631	0.10056	−0.81355	20.9428	−126.4
445–467.6	467.6	304.11792	−0.00025	−0.02075	0.64954	37.95686	−462
467.6–495	467.6	304.11792	−0.00019	−0.01384	−0.87259	70.33025	−644.07
495–518	495	304.08123	0.00786	0.09597	−0.2174	2.16398	0
519–570	519	303.92747	0.0188	0.01178	1.83036	−10.15248	47.55789
570–620	570	302.89499	0.0643	0.22867	1.40737	−1.70157
620–700	620	299.90971	0.135	0.57585	−2.28116	18.71844	−46.52369
700–850	700	289.93954	0.25501	1.83235	−14.84521	67.05886	−97.84919
850–1178	850	253.00199	0.6462	0.74432	0.24847	1.81233	0

shows the fitting parameters. The final fitting dot of Figure 14 shows the relationship between fitting T₀ and density.

In the fitting process, it is found that the heat capacity fitting precision is more sensitive to changes in T₀, whereas T₀ changes very regularly. Therefore, we then fit the T₀ through the piecewise Equation (16). Table 5 shows the fitting parameters. The final fitting dot of Figure 14 shows the relationship between fitting T₀ and density.

$$T_0=A/\left[1+{\displaystyle \sum _{\mathrm{i}=1}^5{A}_i{(\left|\rho -x_c\right|/500)}^{\mathrm{i}}}\right](0.01\le \rho \le 1178)$$

(16)

Based on the calculated c and T₀ by Equations (14) and (16), the equation is fitted for the third time, which brings in variations of a. This has a high determination coefficient of 0.99925, still indicating a very high fitting precision. Then a is fitted by the piecewise Equation (17).

Table 6. Fitting parameters of a (R-square > 0.9999, MARE < 0.0001).

ρ kg/m3	ρ0	A	A1	A2	A3	A4	A5
250–330	330	82.24941	2.17897	−77.69272	1477.057	−10,862.85	27,604.2
330–375	375	95.04348	1.86662	22.971	−945.3032	11,954.6	−49,742.9
375–418	375	95.04348	−3.58088	129.32	−4664.13	60,103.1	−264,235.5
419–445	445	161.57195	0.60004	38.45717	1152.5	−23,625.2	139,048
445–467.6	467.6	155.88555	−1.91731	−9.8931	2295	−52,762.5	423,919
467.6–495	467.6	155.88555	0.80828	−24.82	2681.87	−58,726.45	435,333
495–518	495	142.14881	2.91935	58.266	1210.6	−20,377.8	125,216.9
519–570	570	49.33043	−4.12187	−38.1146	306.7	−316.4	116.7
570–620	570	49.33043	4.29655	−91.66	2592.7	−31861	132,164
620–700	620	39.7411	1.49342	−10.76	92.021	−479.35	920.58
700–850	700	35.39488	0.40317	−4.7353	39.246	−165.635	238.462
850–1178	850	35.67187	−0.23465	0.04875	−2.13036	4.66409	−3.08148

shows the fitting parameters. The final fitting dot of Figure 14 shows the relationship between fitting a and density.

$$a=\left\{\begin{array}{ll}(67.24968+167.4\rho )/(\rho +354)& \left(0\le \rho \le 250\right)\\ A/\left[1+{\displaystyle \sum _{\mathrm{i}=1}^5{A}_i{(\left|\rho -{\rho }_0\right|/500)}^{\mathrm{i}}}\right]& (250\le \rho \le 1178)\end{array}\right\}$$

(17)

4. Validation of Fitting Precision

Based on the fitting equations of a, c and T₀, we compile the C++ program to calculate the heat capacity and validate the results near the critical temperature point with temperatures of 304.1 K and 304.3 K. Figure 16 and Figure 17 show the comparison result at the temperatures of 304.1 K and 304.3 K.

Table 7. Points where critical CO₂ heat capacities are predicted successfully.

ρ (kg/m3)	NIST WebBook Cp [44]	NIST REFPROP Cp	Fitting Cp	MARE (%)
385	128.03301	128.03301	129.29207	0.98339
390	Undefined	158.72872	160.41812	1.06433
430	Undefined	2614.91213	2465.61014	5.70964
435	Undefined	Undefined	4092.02132	Undefined
445	Undefined	Undefined	57112.40951	Undefined
450	Undefined	Undefined	Undefined	Undefined
490	Undefined	Undefined	Undefined	Undefined
495	Undefined	Undefined	15914.15082	Undefined
505	Undefined	Undefined	2453.76021	Undefined
510	Undefined	1457.67440	1405.32011	3.59163
560	Undefined	60.68897	60.98897	0.49432
565	50.74832	50.74832	51.04832	0.59115

shows the points where the critical CO₂ heat capacities in the NIST WebBook and NIST REFPROP are predicted successfully using the fitting method. Through comparison with the NIST WebBook and NIST REFPROP data, the mean absolute relative errors (MAREs) were below 0.674% and 0.337%, respectively. The biggest MARE happens around the critical density, where the heat capacity and its change rate with density is high. There are various reasons for the MARE, and the main reason should be the measures taken for the fitting simplification, such as the assumption of the molecular kinetic energy density independence, the disregard of the influence of the intramolecular potential energy, and the mandatory assignment of c away from the critical density. Nevertheless, compared with the NIST REFPROP heat capacity, the fitting equation still has acceptably high accuracy. In addition, several traditionally unavailable heat capacities in the NIST WebBook and NIST REFPROP are well predicted, such as the point with a temperature of 304.1 K and densities of 435–445 kg/m³ and 510–560 kg/m³. The fitting result shows that the fitting method established in the article has better convergence in the heat capacity calculation. The detailed C++ program and fitting data is in the Supplementary Materials.

5. Conclusions

Currently, existing accurate equations for the calculation of the heat capacities of carbon dioxide, nitrogen, methane, and water, are easy to diverge for heat capacity calculations near the critical temperature and pressure. To solve this problem, a fitting equation for heat capacity with constant density was derived through analyses of the relationships among molecular potential energy, molecular kinetic energy, density, and temperature, which was further validated afterwards. The variations of the fitting parameters with density were fitted in order, which brings the carbon dioxide heat capacity variation with density and temperature. Based on this, a corresponding C++ program is compiled. Moreover, the relative error compared with the NIST WebBook data is identified to be within 1% with temperatures of 304.1 K and 304.3 K. In addition, for the areas where the NIST WebBook and NIST REFPROP data diverge easily, such as densities in the ranges 435–445 kg/m³ and 510–560 kg/m³ at 304.1 K, the fitting equation also made a good calculation. Moreover, due to the avoidance of complex intermediate calculation processes, the heat capacity calculation speed is greatly improved, which is more suitable for the calculation and simulation of large-scale engineering. Therefore, this fitting formula can be used as an effective supplement to the heat capacity calculation equation and as a prediction method for fluid heat capacity experimental data.

This paper has proved that the heat capacity variation near the critical point can be effectively fitted by the semi-empirical formula determined by molecular potential energy and kinetic energy. However, in order to reduce the complexity of the fitting formula and the difficulty of the fitting process, a simplified and relatively simple molecular potential energy equation was used that resulted in a decline in the fitting accuracy; this deficiency should be further studied in the future.