Analysis of a New Liquefaction Combined with Desublimation System for CO₂ Separation Based on N₂/CO₂ Phase Equilibrium

Abstract

Cryogenic CO₂ capture is considered as a promising CO₂ capture method due to its energy saving and environmental friendliness. The phase equilibrium analysis of CO₂-mixtures at low temperature is crucial for the design and operation of a cryogenic system because it plays an important role in analysis of recovery and purity of the captured CO₂. After removal of water and toxic gas, the main components in typical boiler gases are N₂/CO₂. Therefore, this paper evaluates the reliabilities of different cubic equations of state (EOS) and mixing rules for N₂/CO₂. The results show that Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) fit the experimental data well, PR combined with the van der Waals (vdW) mixing rule is more accurate than the other models. With temperature decrease, the accuracy of the model improves and the deviation of the N₂ vapor fraction is 0.43% at 220 K. Based on the selected calculation model, the thermodynamic properties of N₂/CO₂ at low temperature are analyzed. According to the results, a new liquefaction combined with a desublimation system is proposed. The total recovery and purity of CO₂ production of the new system are satisfactory enough for engineering applications. Additionally, the total energy required by the new system to capture the CO₂ is about 3.108 MJ·kg⁻¹ CO₂, which appears to be at least 9% lower than desublimation separation when the initial concentration of CO₂ is 40%.

1. Introduction

Carbon dioxide capture and storage (CCS), which involves capture, transport and storage, is proposed as an important strategy to reduce greenhouse gas emissions significantly [1]. Of these three steps, CO₂ capture is recognized as a promising and relatively quick solution to reduce global CO₂ emissions. Many different types of CO₂ capture methods have been studied by researchers all over the world. The main CO₂ capture methods contain chemical absorption, physical adsorption, membrane separation, and cryogenic distillation. Among these technologies, cryogenic CO₂ capture is attracting increased attention due to its energy saving and environmental friendliness [2]. Kelley et al. [3] proposed the Controlled Freeze Zone™ technology which is capable of removing CO₂ and H₂S from natural gas. In the process, CO₂ is frozen out and remelted in a distillation tower. Song et al. [2] developed a novel CO₂ capture process based on Stirling coolers (SC); the CO₂ in the gas stream can be captured in solid form under the cryogenic condition, and frosted on the cold head of an SC. Theunissen et al. [4] reported condensed rotational separations for CO₂ removal from contaminated natural gas. Zanganeh et al. [5] designed a CO₂ cryogenic separate system where flue gas is compressed, cooled and dried, then the CO₂ is condensed into a liquid. The above strategies capture CO₂ by liquefaction or desublimation separately, however, at a nominal 14% CO₂ in typical flue gas, no liquid forms at any temperature or pressure occur, and desublimation separation is too energy-intensive to apply alone. For these reasons, a new liquefaction combined with desublimation system is proposed. The phase equilibrium of CO₂-mixtures is of great importance to the design and operation of a cryogenic system. Depending on the sources of CO₂-mixtures, the purity of CO₂ stream is varied. Generally, the common impurities in flue gas include N₂, O₂, Ar, CH₄, H₂S, SO₂ and H₂O [6]. After removal of water and toxic gas, the main impurity in typical boiler gases is N₂ [7], the permanent gas N₂ is the main obstacle in the separation of CO₂. Therefore, N₂/CO₂ phase equilibrium analysis at a low temperature is crucial for the investigation and design of a cryogenic CO₂ capture system. The experimental data of N₂/CO₂ covers pressures from 0.6 MPa to 13.95 MPa and temperatures from 218.15 K to 403.15 K, with CO₂ liquid mole fractions ranging from 0.43 to 1, and CO₂ gas mole fractions ranging from 0.153 to 1 [8]. However, the experimental data are not continuous, there are some gaps between the available experimental data and the requirements of engineering applications. Therefore, semi-empirical equations of state (EOS) are usually used to satisfy the requirements with respect to the design and operation of cryogenic CO₂ capture system. Cubic EOS offer a compromise between the generality and simplicity that are often used for many technical applications [9].

For N₂/CO₂ phase equilibrium calculations, different cubic EOSs and their reliability have been studied by some investigators. Dorau et al. [10] conducted N₂/CO₂ equilibrium experiments and the results showed the Peng-Robinson (PR) EOS correlates the experimental data well. However, the concrete accuracy was not given and experiments were conducted only at 223.15 K and 273.15 K. Thiery et al. [11] investigated Soave-Redlich-Kwong (SRK) EOS for the vapor liquid equilibrium (VLE) calculations of N₂/CO₂. The results indicated that the average deviation for the saturated pressures was around 4% in the temperature range of 218.15–273.15 K. Duan and Hu [12] developed a new cubic EOS, modeling the VLE properties of natural fluids based on SRK EOS. The calculations of the VLE were in good agreement with experiments for the N₂/CO₂ mixture. Li and Yan [13] evaluated the reliabilities of seven cubic EOS for predicting volumes of binary CO₂ mixtures. Comparatively, the calculated results of PR and Patel-Teja (PT) are in good agreement with the experimental data especially in the liquid phase for the N₂/CO₂ mixture, the PR is superior in the calculation of liquid volumes, where the absolute average deviations is 1.74%, but the analysis was only carried out at 270 K. Ahmad et al. [14] stated that SRK provides satisfactory predictions that match well with the experimental data, especially with high concentration of CO₂ (97.5%) in the case of N₂/CO₂ mixtures. For O₂/CO₂ mixtures, the SRK can correctly predict the dew-point line and the bubble point line with high concentrations of CO₂.

Due to the semi-empirical EOS that were developed by using pure component data, the application of EOSs for multi-component is based on different mixing rules. Some EOS are not of sufficient accuracy, mainly because of inadequate use of the mixing rules. In general, the same equation of state, combining different mixing rules, may lead to different results. Raabe and Köhler [15] investigated the performances of SRK and PR cubic EOS with different types of mixing rules (including van der Waals (vdW) mixing rule and some Gibbs energy (G^E) mixing rules that incorporate an activity coefficient model into the EOS model by equating the excess G^E) in the N₂/C₂H₆ system. The results showed that the vdW2 mixing rule, which introduced two variable parameters, was able to achieve high accuracy. Faúndez and Valderrama [16] used PR EOS combined with the Wong-Sandler (WS) mixing rule to study the hydrocarbon + alcohol mixtures at low and moderate pressure. It was concluded that the model could correlate equilibrium data in complex mixtures with accuracy similar to that of other models. However, it is difficult to evaluate precisely and systematically the relative merits of different mixing rules from existing papers for N₂/CO₂ mixture, since there is no widely used basis of comparison. In general, all of the studied EOS and mixing rules have various performances for various mixtures.

In this paper, the reliability of different cubic EOS combined with various mixing rules for the VLE properties of N₂/CO₂ mixture are evaluated. Based on the comparison with the collected experimental data, the most accurate model to conduct N₂/CO₂ phase equilibrium analysis at low temperature is identified. Additionally, the thermodynamic properties of N₂/CO₂ at low temperature are analyzed based on the selected calculation model. According to the results, a new liquefaction combined with desublimation system is proposed. Finally, the recovery and purity of CO₂ and the energy consumption of the new system are analyzed.

2. Thermodynamic Models

2.1. Cubic Equations of State

The structure of cubic equations is simple, which makes them popular in engineering applications. The cubic equations used for N₂/CO₂ phase equilibrium calculations in this study are the Redlich-Kwong (RK) [17], the SRK [18] and the PR [19]. All studied EOS are summarized in Table 1.

RK EOS is the earliest modification of van der Waals EOS. Its precision of vapor phase property predictions is improved, but the temperature function is too simple, thus, the application scope is limited. SRK EOS is a modification of RK EOS by introducing a temperature-dependent function. It improves the accuracy of liquid phase properties prediction and its application range becomes wider, it is the first widely accepted cubic equation in the engineering field. PR EOS is modified on the basis of SRK EOS, which turns the hard sphere model from a cubic equation to a general form and analyzes the relationship between gravity and molecular density elaborately.

Table 1. Summary of studied equations of state (EOS) for N₂/CO₂. RK: Redlich-Kwong; SRK: Soave-Redlich-Kwong; and PR: Peng-Robinson.

EOS	Function form	Coefficient
RK	$P=\frac{RT}{v-b}-\frac{a}{T^{0.5}v(v+b)}$	$a=0.42748R^2{T}_{\text{C}}^{2.5}/P_{\text{C}}$
		$b=0.08664R{T}_{\text{C}}/P_{\text{C}}$
SRK	$P=\frac{RT}{v-b}-\frac{a}{v(\text{v}+b)}$	$a=0.42748R^2{T}_{\text{C}}^2\text{α}(T)/P_{\text{C}}$
		$b=0.08664R{T}_{\text{C}}/P_{\text{C}}$
		$\text{α}(T)={\left[1+m(1-T_{\text{R}}^{0.5})\right]}^2, T_{\text{R}}=T/T_{\text{C}}$
		$m=0.48+1.574\text{ω}-0.176{\text{ω}}^2$
PR	$P=\frac{RT}{v-b}-\frac{a}{v(v+b)+b(v-b)}$	$a=0.45724R^2{T}_{\text{C}}^2\text{α}(T)/P_{\text{C}}$
		$b=0.07780R{T}_{\text{C}}/P_{\text{C}}$
		$\text{α}(T)={\left[1+m(1-T_{\text{R}}^{0.5})\right]}^2, T_{\text{R}}=T/T_{\text{C}}$
		$m=0.3746+1.5423\text{ω}-0.2699{\text{ω}}^2$

Table 1. Summary of studied equations of state (EOS) for N₂/CO₂. RK: Redlich-Kwong; SRK: Soave-Redlich-Kwong; and PR: Peng-Robinson.

EOS	Function form	Coefficient
RK	$P=\frac{RT}{v-b}-\frac{a}{T^{0.5}v(v+b)}$	$a=0.42748R^2{T}_{\text{C}}^{2.5}/P_{\text{C}}$
		$b=0.08664R{T}_{\text{C}}/P_{\text{C}}$
SRK	$P=\frac{RT}{v-b}-\frac{a}{v(\text{v}+b)}$	$a=0.42748R^2{T}_{\text{C}}^2\text{α}(T)/P_{\text{C}}$
		$b=0.08664R{T}_{\text{C}}/P_{\text{C}}$
		$\text{α}(T)={\left[1+m(1-T_{\text{R}}^{0.5})\right]}^2, T_{\text{R}}=T/T_{\text{C}}$
		$m=0.48+1.574\text{ω}-0.176{\text{ω}}^2$
PR	$P=\frac{RT}{v-b}-\frac{a}{v(v+b)+b(v-b)}$	$a=0.45724R^2{T}_{\text{C}}^2\text{α}(T)/P_{\text{C}}$
		$b=0.07780R{T}_{\text{C}}/P_{\text{C}}$
		$\text{α}(T)={\left[1+m(1-T_{\text{R}}^{0.5})\right]}^2, T_{\text{R}}=T/T_{\text{C}}$
		$m=0.3746+1.5423\text{ω}-0.2699{\text{ω}}^2$

2.2. Mixing Rules

Three mixing rules widely used in industry are included in this study: vdW, modified Huron-Vidal second-order model (MHV2) [20], and WS [21] mixing rules. These three mixing rules are summarized in Table 2.

Table 2. Summary of studied mixing rules. vdW: van der Waals; WS: Wong-Sandler; and MHV2: modified Huron-Vidal second-order model.

Mixing rules	a	b
vdW	$a={\displaystyle \sum _i{\displaystyle \sum _j{x}_i}}{x}_j{a}_{ij}, a_{ij}=\left(1-k_{ij}\right)\sqrt{a_i{a}_j}$	$b={\displaystyle \sum _i{x}_i}{b}_i$
WS	$a=RT\frac{\left({{\displaystyle {\Sigma }_i{\displaystyle {\Sigma }_j{x}_i{x}_j\left(b-\left(a/RT\right)\right)}}}_{ij}\right)\left({\displaystyle {\Sigma }_i{x}_i}\left(a_i/b_{i}RT\right)+\left(G^{\text{ex}}/CRT\right)\right)}{1-\left({\displaystyle {\Sigma }_i{x}_i}\left(a_i/b_{i}RT\right)+\left(G^{\text{ex}}/CRT\right)\right)}$	$b=\frac{{{\displaystyle {\Sigma }_i{\displaystyle {\Sigma }_j{x}_i{x}_j\left(b-\left(a/RT\right)\right)}}}_{ij}}{1-\left({\displaystyle {\Sigma }_i{x}_i}\left(a_i/b_{i}RT\right)+\left(G^{\text{ex}}/CRT\right)\right)}$
MHV2	$\frac{G_{\text{m}}^{\mathrm{E}}\left(p=0\right)}{RT}+{\displaystyle \sum _i{x}_i}\ln \left(\frac{b_i}{b}\right)=q\left(a\right)-{\displaystyle \sum _i{x}_{i}q\left(a_i\right)}$

Table 2. Summary of studied mixing rules. vdW: van der Waals; WS: Wong-Sandler; and MHV2: modified Huron-Vidal second-order model.

Mixing rules	a	b
vdW	$a={\displaystyle \sum _i{\displaystyle \sum _j{x}_i}}{x}_j{a}_{ij}, a_{ij}=\left(1-k_{ij}\right)\sqrt{a_i{a}_j}$	$b={\displaystyle \sum _i{x}_i}{b}_i$
WS	$a=RT\frac{\left({{\displaystyle {\Sigma }_i{\displaystyle {\Sigma }_j{x}_i{x}_j\left(b-\left(a/RT\right)\right)}}}_{ij}\right)\left({\displaystyle {\Sigma }_i{x}_i}\left(a_i/b_{i}RT\right)+\left(G^{\text{ex}}/CRT\right)\right)}{1-\left({\displaystyle {\Sigma }_i{x}_i}\left(a_i/b_{i}RT\right)+\left(G^{\text{ex}}/CRT\right)\right)}$	$b=\frac{{{\displaystyle {\Sigma }_i{\displaystyle {\Sigma }_j{x}_i{x}_j\left(b-\left(a/RT\right)\right)}}}_{ij}}{1-\left({\displaystyle {\Sigma }_i{x}_i}\left(a_i/b_{i}RT\right)+\left(G^{\text{ex}}/CRT\right)\right)}$
MHV2	$\frac{G_{\text{m}}^{\mathrm{E}}\left(p=0\right)}{RT}+{\displaystyle \sum _i{x}_i}\ln \left(\frac{b_i}{b}\right)=q\left(a\right)-{\displaystyle \sum _i{x}_{i}q\left(a_i\right)}$

The vdW mixing rule is the simplest mixing rule, it can be applied for non-polar and weak polar substances, but it is not suitable for highly non-ideal systems. The MHV2 mixing rule could directly associate the parameters obtained from low-pressure phase equilibrium by the G^E model; however, with the extrapolation model, the error will be larger. The WS mixing rule can be used for phase equilibrium calculation in high and low pressures, it has good extrapolation performance and is suitable for polar and non-polar system.

3. The New Liquefaction Combined with Desublimation System

The schematic of the liquefaction combined with desublimation CO₂ capture process is shown in Figure 1. The whole capture process can be described as follows: firstly, the flue gas, mainly including N₂/CO₂, is compressed to a certain pressure, about 2–5 MPa, by the compressors; then the gas is cooled to a temperature slightly above the dew point of CO₂ by cooling with water, low temperature effluent streams, and the liquid CO₂ sequentially. After the pre-chilling step, the gas stream is introduced into the gas-liquid separator, the temperature of separator, which depends on various components, is in the range of 210–283 K. In this section, the CO₂ in the flue gas is condensed into a liquid and then the residual gas (mainly as N₂) is fed into the desublimation separation tower. The pressure of this unit is equal to gas-liquid separator, but the temperature is lower. In the desublimation separation tower, the CO₂ in the mixture is frosted into dry ice, which is melted into a liquid by cooling the incoming gases. Additionally, the dry ice can provide cooling capacity for the low-temperature condensation of the cascade refrigerating machine and reduce the energy consumption. In this way, the CO₂ is recovered from flue gas and finally changes into a liquid phase and a gaseous nitrogen stream. The liquid CO₂, which is formed by dry ice melting and liquefaction, is compressed to about 10 MPa by a pump and is stored in the storage tank at a normal temperature (about 293 K).

Figure 1. Schematic diagram of liquefaction combined with desublimation CO₂ capture process.

Figure 1. Schematic diagram of liquefaction combined with desublimation CO₂ capture process.

4. Method of Analysis

4.1. Thermo Model

From thermodynamics, a system of multiple components at a given temperature and pressure is in equilibrium when the fugacities of each component, in all phases, are equal. According to this principle, gas-liquid and gas-solid phase equilibrium calculations can be conducted [22].

In the phase equilibrium calculations, the EOS are usually rewritten into the form of the compressibility factor z. The i component’s fugacity coefficient derived by the RK and SRK EOS is as follows:

$${\text{lnφ}}_i=\frac{b_i}{b}(z-1)-\ln (z-B)-\frac{A}{B}\left(\frac{2{\displaystyle \sum _j{x}_j{a}_{ij}}}{a}-\frac{b_i}{b}\right)\ln (1+\frac{B}{z})$$

(1)

The i component’s fugacity coefficient derived by the PR equation of state is as follows:

$${\text{φ}}_i(T,p,x_i)=\exp [\frac{b_i}{b}(z-1)-\ln (z-B)-\frac{A}{2.828B}\times \left(\frac{2{\displaystyle \sum _j{x}_j{(a_i{a}_j)}^{0.5}(1-k_{ij})}}{a}-\frac{b_i}{b}\right)\times \ln (\frac{z+2.414B}{z-0.414B})]$$

(2)

where z is the compressibility factor, z = PV/RT, z may have three real roots, assumption: z₁ > z₂ > z₃; for gas, gas compressibility factor z^V = z₁; for liquid, liquid compressibility factor z^L = z₃, when there is only one real root, it is the required compression factor; A = ap/(R²T²); B = bp/RT; p represents the system pressure; and T represents the system temperature.

4.2. The Separation Performance

The separation performance of the cryogenic CO₂ capture system can be expressed in two parameters: the CO₂ purity and the CO₂ recovery. The CO₂ purity is the concentration of CO₂ in the liquid production. The recovery is defined as the fraction of the total amount of CO₂ in the feed that ends up in the liquid product. The recovery can be calculated by the vapor-liquid equilibrium and material balance equation at various initial conditions. An expression for the CO₂ recovery of a binary gas mixture is given as:

$$\text{ξ}=\frac{(z_{{\text{CO}}_2}-y_{{\text{CO}}_2})\times x_{{\text{CO}}_2}}{(x_{{\text{CO}}_2}-y_{{\text{CO}}_2})\times z_{{\text{CO}}_2}}$$

(3)

where ξ denotes the CO₂ recovery; $x_{{\text{CO}}_2}$, $y_{{\text{CO}}_2}$, and $z_{{\text{CO}}_2}$ represent the mole fraction of CO₂ in the liquid product, residual gas, and feed stream, respectively.

4.3. Energy Consumption

The energy consumption of the cryogenic CO₂ capture system mainly includes compression and refrigeration. The whole energy consumption can be calculated by the following formula:

$$W=W_1+W_2-W_3-W_4+W_5$$

(4)

where W₁ is the compression work; W₂ is external work of refrigeration; W₃ is expansion recovery work; W₄ is refrigeration work, which is saved by recycling cooling capacity; and W₅ is pump power.

The cooling capacity and heat transfer process can be analyzed by the way of the enthalpy difference. The calculation formula of enthalpy can be obtained by the PR EOS:

$$h=h_0-h_{\text{r}}$$

(5)

$$h_0=h_0^0+{\displaystyle {\int }_{T_0}^T{C}_{\text{p}0}}\text{d}T$$

(6)

$$h_{\text{r}}=a_{\text{r}}+T{s}_{\text{r}}+RT-pv$$

(7)

where excess free energy, and excess entropy equations are as Equations (8) and (9) show:

$$a_r=-R\ln \frac{v-b}{v}-\frac{a}{2\sqrt{2}b}\ln \frac{v-0.414b}{v+2.414b}+R\ln \frac{v}{v_0}$$

(8)

$$s_r=-R\ln \frac{v-b}{v}+\frac{\text{β}}{2\sqrt{2}b}\ln \frac{v-0.414b}{v+2.414b}-R\ln \frac{v}{v_0}s$$

(9)

where v₀ is the specific volume of ideal gas; h₀ and s₀ are the specific enthalpy and entropy of ideal gas; p₀ and T₀ are the pressure and temperature of reference state; h₀⁰ is the specific enthalpy of reference state; and C_p0 is the specific heat of ideal gas.

The work consumption of unit CO₂ separation and liquefaction can be calculated by the following formula:

$$W_{\text{0}}=W/m$$

(10)

where W₀ is work consumption of unit CO₂ separation and liquefaction; and m is the whole mass of the obtained CO₂.

Assuming the primary energy conversion rate is 30%, the primary energy consumption can be calculated by the Equation (11):

$$Q_0=W_0/\text{η}$$

(11)

where Q₀ is energy consumption of unit CO₂ separation and liquefaction; and η is the primary energy conversion rate, its value is 30%.

5. Result and Discussion

5.1. Selection and Calculation of the Equations of State

This section mainly discusses the accuracy of different EOS. Due to the fact that the classical vdW mixing rule is widely used to model gas processing systems, it is selected to eliminate the differences caused by the various mixing rules. The liquid N₂ concentration is kept constant in the phase equilibrium calculations.

In Figure 2, the calculated results are compared with experimental data [23,24,25] at 273.2 K and 220 K. Along with the increment of pressure, the performance of all equations decreases. Comparatively, RK has the worst accuracy. It is clear that the calculated results of PR and SRK fit the experimental data well, and, comparatively, PR has a better accuracy. Therefore, further analysis should be carried out using PR or SRK EOS.

Figure 2. Comparison of p-x curves diagram of N₂-CO₂ system at different temperatures: (a) 273.2 K; and (b) 220 K.

Figure 2. Comparison of p-x curves diagram of N₂-CO₂ system at different temperatures: (a) 273.2 K; and (b) 220 K.

5.2. Selection and Calculation of the Mixing Rules

Due to the fact that PR and SRK EOS have better accuracies, this section combines PR and SRK EOS with three kinds of mixing rules to predict VLE properties. The comparisons between calculated results and experimental data [23,24,25,26] at 293.2 K, 273.2 K, 240 K and 220 K are shown in Figure 3.

Figure 3. Comparison of varied mixing rules at different temperatures: (a) 293.2 K; (b) 273.2 K; (c) 240 K; and (d) 220 K.

Figure 3. Comparison of varied mixing rules at different temperatures: (a) 293.2 K; (b) 273.2 K; (c) 240 K; and (d) 220 K.

Table 3 summarizes the average relative deviation between calculated results and experimental data on P_s and the N₂ vapor fraction of N₂/CO₂.

Table 3. Average relative deviation between calculated results and experimental data (%).

Models	293.2 K		273.2 K		240 K		220 K
	Ps	Vapor fraction	Ps	Vapor fraction	Ps	Vapor fraction	Ps	Vapor fraction
SRK-vdW	0.92	5.54	3.83	6.95	8.88	2.38	8.45	0.84
SRK-MHV2	9.93	9.09	17.60	15.22	39.10	15.55	55.78	13.17
SRK-WS	9.42	9.07	15.09	13.30	33.64	11.22	47.41	9.52
PR-vdW	1.37	3.46	2.69	5.15	7.37	1.43	6.31	0.43
PR-MHV2	7.11	6.49	12.74	11.33	32.99	12.76	50.74	11.51
PR-WS	8.76	10.28	14.43	13.88	33.60	11.31	48.11	9.76

Table 3. Average relative deviation between calculated results and experimental data (%).

Models	293.2 K		273.2 K		240 K		220 K
	Ps	Vapor fraction	Ps	Vapor fraction	Ps	Vapor fraction	Ps	Vapor fraction
SRK-vdW	0.92	5.54	3.83	6.95	8.88	2.38	8.45	0.84
SRK-MHV2	9.93	9.09	17.60	15.22	39.10	15.55	55.78	13.17
SRK-WS	9.42	9.07	15.09	13.30	33.64	11.22	47.41	9.52
PR-vdW	1.37	3.46	2.69	5.15	7.37	1.43	6.31	0.43
PR-MHV2	7.11	6.49	12.74	11.33	32.99	12.76	50.74	11.51
PR-WS	8.76	10.28	14.43	13.88	33.60	11.31	48.11	9.76

It can be seen that the same equation of state, combining different mixing rules, could lead to different results. The performances of models are normally getting worse as the conditions are approaching the critical points of the mixtures.

Comparatively, PR and SRK EOS, combined with the vdW mixing rule, have better performances. The average deviations on P_s or the N₂ vapor fraction are within 9%. In general, PR EOS combined with the vdW mixing rule is more accurate than the other models, where the maximum deviation is 7.37% in the calculations on P_s. With a temperature decrease, the accuracy of the model (PR EOS combined with the vdW mixing rule) improving, the deviation of N₂ vapor fraction is 0.43% at 220 K.

Therefore, in this paper, we select PR EOS, combined with the vdW mixing rule, to analyze the new cryogenic CO₂ capture system. In addition, we assume the dry ice as a pure CO₂ solid phase. The gas-solid phase equilibrium calculation focuses on the gas phase, thus, we also use the selected model to calculate gas-solid phase equilibrium [22].

5.3. Thermodynamic Properties of N₂/CO₂ at Low Temperature

Recent work indicates that a combination of oxy-fuel and post-combustion strategies can produce a CO₂-rich flue gas stream that can be passed through a cryogenic separation process to capture CO₂ [27,28]. Zanganeh et al. [29] proposed an approach that allows air to be partially used in oxy-fired coal power plants, the concentration of CO₂ in flue gas is about 30%–90%. In this paper, 40% CO₂ concentration is selected as an example to analyze the new liquefaction, combined with desublimation system. After the CO₂ is separated from flue gas, the product is compressed to pipeline pressure (typically 10 MPa). Generally, increasing the condensing pressure will allow the CO₂ to condense at higher temperatures. However, the higher pressures would quite severely constrain suitable materials of construction, thus, 2–5 MPa is preferable to separate CO₂.

The bubble point, dew point, and desublimation temperature of the N₂/CO₂ binary system are calculated by PR EOS combined with the vdW mixing rule. In the case of 2 MPa, the calculated results are presented in Figure 4.

When the temperature is dropped to the dew point, CO₂ is firstly condensed to a liquid phase. However, when the concentration of CO₂ is reduced to 21.7%, the dew point temperature is equal to the desublimation temperature. Along with a decreasing of CO₂ mole fraction, the desublimation temperature is higher than the dew point temperature, thus, the CO₂ will not form a liquid phase at any temperature at 2 MPa. Rather, the CO₂ desublimation forms an essentially pure solid phase rather than a liquid solution.

Figure 4. T-x curves of N₂/CO₂ binary system at a pressure of 2 MPa.

Figure 4. T-x curves of N₂/CO₂ binary system at a pressure of 2 MPa.

Figure 5 indicates the influences of temperature and pressure on the recovery and purity of CO₂ liquefied in the case of a 40% CO₂ initial concentration. From the results, it can be concluded that the recovery of liquid CO₂ decreases and purity increases with the increase of temperature. With an increase of pressure, CO₂ liquefied recovery increases while the purity decreases. Thus, it is difficult to achieve a high recovery and purity simultaneously by liquefaction. However, desublimation separation is too energy-intensive to apply alone. In this paper, the liquefaction and desublimation process are combined together to avoid the energy penalty and to achieve high recovery and purity.

Figure 5. (a) Recovery of liquid CO₂ at different conditions; and (b) purity of liquid CO₂ at different conditions.

Figure 5. (a) Recovery of liquid CO₂ at different conditions; and (b) purity of liquid CO₂ at different conditions.

In the studied case, the total flow rate for the flue gas is 143.33 kg/s and initial CO₂ concentration is 40%. A total of 58.89% of CO₂ can be separated by liquefaction at 2 MPa when the temperature is 208.37 K. The residual CO₂ can be captured by the desublimation separation tower. The total CO₂ recovery of the studied system can reach 90% when the desublimation capture temperature is 192.02 K.

According to the results above, a new liquefaction combined with a desublimation system is proposed, as shown in Figure 1. Table 4 shows the parameters of key state points in the studied system. In the process, a three-stage compression with intercooling is used to avoid high exhaust temperatures and decrease the compression work. In order to improve the efficiency of the expansion and avoid dry ice formed through the expansion process, two stages of expansion are conducted in the process.

Table 4. The parameters of key state point.

No.	CO2 mole fraction (%)	T (K)	P (MPa)
1	40	298.15	0.1
2	40	298.15	2
3	40	223.85	2
4	21.69	208.37	2
5	6.31	192.02	2
6	6.31	170.17	0.1
7	97.41	208.37	2
8	98.29	211.95	2
9	98.29	293.15	10

Table 4. The parameters of key state point.

No.	CO2 mole fraction (%)	T (K)	P (MPa)
1	40	298.15	0.1
2	40	298.15	2
3	40	223.85	2
4	21.69	208.37	2
5	6.31	192.02	2
6	6.31	170.17	0.1
7	97.41	208.37	2
8	98.29	211.95	2
9	98.29	293.15	10

5.4. The Results of Energy Calculations

The energy consumptions of each part are listed in Table 5.

Table 5. Energy consumptions for CO₂ separation system.

Parameters	Value	Dimension
Compression work	44,036.9	kW
Expansion work	10,715.9	kW
Cold consumption by liquefaction	16,139.01	kW
Power consumption by liquefaction	15,979.22	kW
Cold consumption by desublimation	13,841.99	kW
Power consumption by desublimation	17,814.66	kW
Save power from cold energy recovery	6171.31	kW
Pump work	604.92	kW
Total work	61,548.5	kW
Unit power consumption	0.9326	MJ·kg−1
Unit energy consumption	3.108	MJ·kg−1

Table 5. Energy consumptions for CO₂ separation system.

Parameters	Value	Dimension
Compression work	44,036.9	kW
Expansion work	10,715.9	kW
Cold consumption by liquefaction	16,139.01	kW
Power consumption by liquefaction	15,979.22	kW
Cold consumption by desublimation	13,841.99	kW
Power consumption by desublimation	17,814.66	kW
Save power from cold energy recovery	6171.31	kW
Pump work	604.92	kW
Total work	61,548.5	kW
Unit power consumption	0.9326	MJ·kg−1
Unit energy consumption	3.108	MJ·kg−1

The calculated results show that the total energy required by the system to capture the CO₂ is about 3.108 MJ/kgCO₂. In order to compare the energy consumptions of various capture processes, three separation methods are analyzed in the case of an initial 40% CO₂ concentration, which include desublimation, liquefaction combined with desublimation, and liquefaction. The results are listed in Table 6.

When the liquefaction separation is applied alone, the biggest liquefied recovery of CO₂ is 85.44% with a purity 90.44% at 6 MPa. The energy consumption is lower than other methods, but the total recovery and purity of CO₂ production are not satisfactory enough for engineering applications. When the desublimation method is used alone, the dry ice can be regarded as a pure CO₂ solid, thus, pure CO₂ liquid will be produced after melting. The CO₂ could freeze at a lower pressure while the total recovery and purity of CO₂ are high enough. However, the process is energy-intensive. As for the new method, which combines liquefaction with desublimation, the calculated results indicate clear advantages for this system. The total recovery and purity of CO₂ are highly favorable. In addition, the new process appears to be at least 9% lower than desublimation separation.

Table 6. Energy consumption comparisons of three separation methods.

Separation methods	Pressure (MPa)	Liquefied recovery (%)	Liquid CO2 purity (%)	Solid CO2 recovery (%)	Total recovery (%)	Energy consumption (MJ·kg−1)
Desublimation	0.55	0	100	90	90	3.416
Liquefaction combined desublimation	2	58.89	97.41	31.11	90	3.108
Liquefaction combined desublimation	3	73.04	95.82	16.96	90	3.162
Liquefaction combined desublimation	4	79.27	94.23	10.73	90	3.193
Liquefaction separation	6	85.44	90.44	0	85.44	-

Table 6. Energy consumption comparisons of three separation methods.

Separation methods	Pressure (MPa)	Liquefied recovery (%)	Liquid CO2 purity (%)	Solid CO2 recovery (%)	Total recovery (%)	Energy consumption (MJ·kg−1)
Desublimation	0.55	0	100	90	90	3.416
Liquefaction combined desublimation	2	58.89	97.41	31.11	90	3.108
Liquefaction combined desublimation	3	73.04	95.82	16.96	90	3.162
Liquefaction combined desublimation	4	79.27	94.23	10.73	90	3.193
Liquefaction separation	6	85.44	90.44	0	85.44	-

6. Conclusions

(1) For N₂/CO₂ phase equilibrium calculations, the calculated results of PR and SRK fit the experimental data well, PR EOS combined with the vdW mixing rule is more accurate than the other models. With temperature decrease, the accuracy of the selected model improves and the deviation of the N₂ vapor fraction is 0.43% at 220 K.

(2) It is difficult to achieve a high recovery and purity simultaneously by liquefaction.

(3) The total recovery and purity of CO₂ production of the new method, which is combined with liquefaction and desublimation, are satisfactory. Additionally, the total energy required by the new system to capture the CO₂ is about 3.108 MJ·kg⁻¹ CO₂, which appears to be at least 9% lower than desublimation separation when the initial concentration of CO₂ is 40%.