Theoretical Methodology of a High-Flux Coal-Direct Chemical Looping Combustion System

Abstract

This study, as an extension of our previous experimental tests, presented a mechanism analysis of air reactor (AR) coupling in a high-flux coal-direct chemical looping combustion (CDCLC) system and provided a theoretical methodology to the system optimal design with favorable operation stability and low gas leakages. Firstly, it exhibited the dipleg flow diagrams of the CDCLC system and concluded the feasible gas–solid flow states for solid circulation and gas leakage control. On this basis, the semi-theoretical formulas of gas leakages were proposed to predict the optimal regions of the pressure gradients of the AR. Meanwhile, an empirical formula of critical sealing was also developed to identify the advent of circulation collapse so as to ensure the operation stability of the whole system. Furthermore, the theoretical methodology was applied in the condition design of the cold system. The favorable gas–solid flow behaviors together with the good control of gas leakages demonstrated the feasibility of the theoretical methodology. Finally, the theoretical methodology was adopted to carry out a capability assessment of the high-flux CDCLC system under a hot state in terms of the restraint of gas leakages and the stability of solid circulation.

1. Introduction

Coal-direct chemical looping combustion (CDCLC) has been demonstrated as an attractive combustion technology of coal with the inherent feature for CO₂ capture [1,2]. The CDCLC concept is typically implemented in two interconnected reactors, the so-called fuel reactor (FR) and the air reactor (AR), with an oxygen carrier (OC) circulating in between to transfer oxygen and heat. Specifically, in the FR, the fuel is first devolatilized and gasified by the gasification agent steam, and then the gasification products (mainly CO, H₂, and CH₄) are further oxidized to CO₂ and H₂O by the OC. In the AR, the reduced OC from the FR is oxidized by the air for regeneration, and then will be recirculated back to the FR. By means of the OC particles that deliver oxygen from the AR to FR, the direct mixing of the fuel and air can be avoided, and further highly purified CO₂, without the dilution of N₂, can be acquired at the outlet of the FR via the condensation of steam [3,4,5,6,7,8,9,10,11,12].

Alternatively, a characterized CDCLC system consisting of a high-flux circulating fluidized bed (HFCFB) riser as the FR and a counter-flow moving bed (CFMB) as the AR was proposed in our previous studies [13,14,15], as shown in Figure 1. The main advantages of this design are that the HFCFB FR can provide high solids concentration over the whole reactor height for favorable gas–solid contact efficiency and reaction performance, and that the CFMB AR possesses steady solids flow and low-pressure drop. Besides, an inertial separator, connecting the two reactors, was specially designed as the carbon stripper to separate the coarse OC particles off the FR into the AR for regeneration, and also recirculate the fine particles of the unconverted coal char back to the FR for further conversion.

Up to now, we have successively developed cold [14] and pilot-scale hot [15] experimental systems of this high-flux CDCLC concept, preliminarily realizing the whole-system stable operation with acceptable gas–solid reaction performance under certain conditions. Similar feasibility studies have been experimentally conducted in different pilot-scale CDCLC units, e.g., the 10 kW_th [3] and 100 kW_th [9] units at Chalmers University of Technology (Sweden), the 10 kW_th [4] and 50 kW_th [5] units at Southeast University (China), the 1 MW_th unit from Technische Universität Darmstadt (Germany) [10], the 25 kW_th unit at Hamburg University of Technology (Germany) [11], the 25 kW_th unit from Ohio State University (America) [12], the 50 kW_th unit at Instituto de Carboquimica (ICB-CSIC) (Spain) [16], and the 5 kW_th CDCLC reactor at Huazhong University of Science and Technology (China) [17]. However, despite promising experimental results obtained in pilot-scale units, the CDCLC technology for CO₂ capture has to be further developed towards large-scale commercial applications. In this aspect, it is essential to develop theoretical methodologies, beside experimental studies, for a better understanding of hydrodynamic and reaction mechanisms in CDCLC processes, which can provide vital references to the design, operation, and process optimization of the future large-scale CDCLC power plants. By far, compared to the extensive experimental studies, few studies are available in the literature on the development of theoretical methodologies in terms of hydrodynamics and/or reaction mechanisms for CDCLC processes. Su et al. [18], based on the hydrodynamic equations for fluidized beds and the reaction kinetics, simulated the CDCLC process in a dual circulating fluidized bed (DCFB) system. Ohlemüller et al. [19] developed a process simulation model to predict the flow and reaction performances of a 1 MW_th unit at Technische Universität Darmstadt.

In our previous experimental tests, we have found that the coupling of the CFMB AR into the downcomer of the HFCFB FR makes the hydrodynamic mechanism of the whole system much more complicated, and hence leads to crucial effects on the operation independence of the two reactors (i.e., FR and AR) in terms of solid circulation stability and gas leakages [20]. In this context, it is necessary to carry out an in-depth mechanism investigation of this high-flux CDCLC system coupled by a CFMB AR, which is significant to the design and operation processes of the future CDCLC applications. Therefore, the objective of this study is to develop a theoretical methodology to illustrate the fundamental hydrodynamics of our high-flux CDCLC system, extending from the previous experimental studies. The main contributions of this work are listed as follows: (1) the screening of the feasible gas–solid flow states for solid circulation and gas leakage control on the strength of the dipleg flow diagrams of the CDCLC system; (2) the development of the semi-theoretical formulas of gas leakages to predict the optimal regions of the pressure gradients of the AR; (3) the development of the empirical formula of critical sealing to identify the advent of circulation collapse so as to ensure the whole-system operation stability; (4) the feasibility validation of the theoretical methodology through its application in the cold-state condition design; (5) the successful application of the theoretical methodology into the capability assessment of the high-flux CDCLC system under a hot state, in terms of the restraint of gas leakages and the stability of solid circulation.

2. Materials and Methods

2.1. Visualization Experimental Device

The visualization experimental device of the high-flux CDCLC system consists primarily of a FR, an inertial separator, a downcomer, an AR, and a J-valve. During the operation process, the FR, with an inner diameter of 60 mm and a height of 5.8 m, was operated in dense suspension upflow (DSU) regime with high solid circulation flux and solids holdup. An inertial separator was installed following the FR, and was used as the carbon stripper to separate the gas stream and elutriated particles off the FR. The particle outlet of the inertial separator was connected with the downcomer, and further the AR which was operated in moving bed regime with an inner diameter of 418 mm and a height of 0.7 m. After leaving the AR, the OC particles were transported back to the FR with the help of the J-valve. The drawing presented in Figure 2 schematically shows how the different sections of the visualization experimental setup are interconnected. The more detailed description can be found in our previous experimental studies with this system [14,20].

The tracer gas (99.99% CO) concentrations were continuously measured with a gas analyzer (MRU, Neckarsulm, Germany) at the outlets of the two reactors. The pressures were measured with pressure gages and a multi-channel differential pressure transducer. The gas flow rates were adjusted and measured by calibrated rotameters (Changzhou shuanghuan thermal instrument co., LTD, Changzhou, China) and then normalized to the standard state (labeled with a subscript sta). Specifically, Q_1,sta, Q_2,sta, Q_3,sta, and Q_4,sta represent the inlet air flow rate of the FR, the fluidizing air flow rate of the J-valve, the aeration air flow rate of the J-valve, and the inlet air flow rate of the AR, respectively. Q_a,sta and Q_b,sta represent the outlet air flow rates of the FR and the AR, respectively.

2.2. Materials

The OC material used in this study was a natural iron ore from Harbin, China with an average particle diameter of 0.43 mm and bulk density of 1577 kg/m³. The minimum fluidization gas velocity under the cold condition was 0.187 m/s [20].

2.3. Performance Indicators

The upper pressure gradient (ΔP₁/H₁) represents the pressure gradient between the AR and the carbon stripper, which was expressed as Equation (1). The lower pressure gradient (ΔP₂/H₂) represents the pressure gradient between the J-valve and the AR, which was expressed as Equation (2) [20].

$$\Delta P_1/H_1=\left(\frac{p_b+p_c}{2}-p_i\right)/H_1$$

(1)

$$\Delta P_2/H_2=\left(p_d-p_{12}\right)/H_2$$

(2)

Solid circulation flux, G_s, represents the solid circulation ratio (kg/s) per unit area of the FR, which was estimated by [20,21]

$$G_s=\frac{{\rho }_b{u}_s{A}_{ud}}{A_f}=\frac{{\rho }_b{A}_{ud}}{A_f}(\Delta H/t)$$

(3)

The solids holdup in the FR, ε_s, could be estimated according to the local pressure drop [13,14,21,22,23,24].

$$\Delta P_Z/\Delta Z\approx [{\rho }_s{\epsilon }_s+{\rho }_g(1-{\epsilon }_s)]g$$

(4)

The FR leakage ratio, f₁, represents the gas leakage ratio from the FR to the AR. During the experimental process, the FR leakage ratio was measured by using tracer gas 1 [14,20].

$$f_1=-\frac{Q_{b,sta}{x}_{b,CO}}{Q_{a,sta}{x}_{a,CO}+Q_{b,sta}{x}_{b,CO}}$$

(5)

In this study, the upward direction is defined as the positive direction, and hence the FR leakage ratio should be negative.

The AR leakage ratio, f₂, represents the gas leakage ratio from the AR to the FR, which could be measured by using tracer gas 2 [14,20].

$$f_2=\frac{Q_{a,sta}{x}_{a,CO}^{\prime }}{Q_{a,sta}{x}_{a,CO}^{\prime }+Q_{b,sta}{x}_{b,CO}^{\prime }}$$

(6)

The J-valve leakage ratio, f₃, represents the gas leakage ratio of the J-valve aeration air into the AR, which was measured by using tracer gas 3 [20].

$$f_3=\frac{Q_{b,sta}{x}_{b,CO}^{″}}{Q_{a,sta}{x}_{a,CO}^{″}+Q_{b,sta}{x}_{b,CO}^{″}}$$

(7)

The detailed meanings of the symbols can be found in the nomenclature.

3. Results and Discussion

3.1. Gas–Solid Flow Characteristics

In typical circulating fluidized bed (CFB) reactors, the downcomer dipleg plays a critical role in solid circulation and gas seal. In order to drive particles, the whole dipleg is usually kept in a state of negative pressure gradient (i.e., a decrease of pressure with the increase of downcomer height) [25,26,27,28,29,30,31]. However, the coupling of the AR into the downcomer of our high-flux CDCLC system together with the special requirements of gas leakages makes the operation mechanism of the dipleg and even the whole system much more complicated. As shown in Figure 2, the existence of the AR divides the dipleg into the upper dipleg and the lower dipleg. During the CDCLC process, the lower dipleg stays at a negative pressure gradient state with the J-valve owning the maximum pressure so as to drive the solids for circulation. But the upper dipleg can situate at a pressure region across the positive and negative pressure gradient states, which has crucial effects on the circulation stability and gas leakage ratios. Therefore, a systematic study is necessary to improve the understanding of AR coupling effects on the hydrodynamic mechanism of this CDCLC system in terms of solid circulation stability and gas leakage controllability.

Figure 3 shows the possible gas–solid flow states in the upper dipleg during the high-flux CDCLC process, where the upward direction is defined as the positive direction [32]. As the abovementioned discussions, the upper dipleg flow can be categorized into seven flow states, according to the differential pressure between the two ends of the upper dipleg. In the first state, the top pressure of the upper dipleg is much larger than that at the bottom, and the gas flow moves downward with a much higher velocity than that of the solids. The big velocity difference between the gas–solid phases means the dramatic CO₂ leakage from the carbon stripper into the AR, and hence the great reduction of CO₂ capture efficiency. In the second state, the positive differential pressure between the top and bottom becomes much smaller, and the gas velocity is only slightly higher than the solids velocity, indicating a modest gas leakage from the FR to the AR. In the third state, when the differential pressure between the two ends of the upper dipleg becomes zero, the solids downward flow is controlled by gravity and the gas–solid relative velocity becomes zero. Then in the fourth state, the bottom pressure of the upper dipleg starts to outpace the top pressure, and the gas–solid flow becomes negative pressure gradient flow, leading to a further reduction of the downward velocity of gas phase. When the downward gas velocity further decreases to zero, it comes to the fifth state, so-called the ideal sealing state. At this point, the gas–solid relative velocity is equal to the absolute value of the solids descending velocity, indicating the ideal suppression of the gas leakages between the FR and AR. In the sixth state, with the further enhancement of the negative pressure gradient, the gas begins to flow upward with a low velocity, indicating a small amount of gas leakage from the AR to the FR. Finally, when a large amount of gas flow moves upward in terms of visible large bubbles, the upper dipleg will enter into the last state, so-called the critical sealing state, meaning that the whole-system particle circulation is about to be broken together with a dramatical leakage of N₂ from the AR into FR. In general, States 1 and 2 belong to the positive pressure gradient flow, State 3 belongs to the zero pressure gradient flow, and States 4 to 7 belong to the negative pressure gradient flow.

Our previous experimental studies found that with the increase of the upper pressure gradient ΔP₁/H₁, the FR leakage ratio f₁ had a linear drop until extinction while the AR leakage ratio f₂ firstly stayed at zero and then had a linear increase [20]. Thus, the variations of gas–solid flow state in the upper dipleg corresponding to the upper pressure gradient could be further deduced, as shown in Figure 4. It can be found that the gas flow direction in the upper dipleg changed from downward to upward. By referring to Figure 3, it can be concluded that the gas–solid flow in the upper dipleg had gone through States 2 to 6, demonstrating the feasibility of the selection of optimal operation region for the gas leakage control and solid circulation by means of the adjustment of the upper pressure gradient ΔP₁/H₁. Consistent with the experimental studies [20], we set −3% and 3% as the limit values of the two gas leakages f₁ and f₂, respectively, and as the selection criteria of the upper pressure gradient. Thus, we can get the optimal region of ΔP₁/H₁ corresponding to States 2 to 6 under the involved operation conditions: State 2 (−2.1 kPa/m < ΔP₁/H₁ < 0 kPa/m), State 3 (ΔP₁/H₁ = 0 kPa/m), State 4 (0 kPa/m < ΔP₁/H₁ < 1.6 kPa/m), State 5 (ΔP₁/H₁ = 1.6 kPa/m), and State 6 (1.6 kPa/m < ΔP₁/H₁ < 3.0 kPa/m).

On the other hand, as mentioned above, because the J-valve is the driving source for solid circulation, it has the maximum pressure of the whole system. Hence, the lower dipleg necessarily stays at a negative pressure gradient region, i.e., States 4 to 7 as shown in Figure 3. However, considering that the gas leakage from the AR to the J-valve will result in the mixing of N₂ into the FR, and further the reduction of CO₂ capture concentration, State 4 should also better be avoided during the CDCLC process. Moreover, an excess gas leakage (i.e., State 7 shown in Figure 3) will cause serious damage on the stability of the solids downward flow, and further the solid circulation. Therefore, only States 5 and 6 were regarded as the preferred gas–solid flow states in the lower dipleg. Consistent with our previous experimental studies [20], we set 20% as the upper limit value of the gas leakage f₃, and as the selection criterion of optimal region under the involved operation conditions.

3.2. Theoretical Methodology for Gas Leakage Restraint

3.2.1. Semi-Theoretical Formulas of the Upper Pressure Gradient

From the analyses made above, we can find that the coupling of the CFMB AR makes the gas–solid flow in the upper dipleg very complicated, which covers a diversity of flow structures from positive pressure gradient to negative pressure gradient states. In a real CDCLC application, the optimal operation region should better locate among States 2 to 6 in order to acquire stable solid circulation and favorable restraint of gas leakages. Fortunately, we found that the optimal region exhibited a relatively symmetrical distribution rule with the ideal sealing state (i.e., State 5) as the core. Thus, it provides a possibility for us to propose an empirical equation applied to the high-flux CDCLC system based on the ideal sealing theory.

The modified Ergun equation was attempted to be applied in the stable moving bed flow of the upper dipleg under the ideal sealing state, which had the form of Equation (8) [33,34,35]. Meanwhile, according to the relationship between the solids velocity and solid circulation flux (see Equation (9)), we further got the correlation equation of the upper pressure gradient under the ideal sealing state with the solid circulation flux (i.e., Equation (10)).

$${\left(\Delta P_1/H_1\right)}_i=150\frac{{\left(1-\epsilon \right)}^2{\mu }_g\left|U_s\right|}{{\left(\epsilon {\varphi }_s{d}_s\right)}^2}+1.75\frac{(1-\epsilon ){\rho }_g{U}_s^2}{\epsilon {\varphi }_s{d}_s}$$

(8)

$$\left|U_s\right|={\left(\frac{D_f}{D_{ud}}\right)}^2\frac{G_s}{{\rho }_s\left(1-\epsilon \right)}$$

(9)

$${\left(\Delta P_1/H_1\right)}_i=\left[150\frac{\left(1-\epsilon \right){\mu }_g}{{\rho }_s{\left(\epsilon {\varphi }_s{d}_s\right)}^2}{\left(\frac{D_f}{D_{ud}}\right)}^2\right]G_s+\left[1.75\frac{{\rho }_g}{\epsilon {\varphi }_s{d}_s(1-\epsilon ){\rho }_s^2}{\left(\frac{D_f}{D_{ud}}\right)}^4\right]G_s^2$$

(10)

Figure 5 illustrates the comparison of predicted and experimental upper pressure gradients under the ideal sealing state (i.e., State 5).

Table 1. Parameters for the calculation of the ideal upper pressure gradient (OC: oxygen carrier; FR: fuel reactor).

Description	Value
Density of air ρg (kg/m3)	1.29
Dynamic viscosity of air μg (Pa·s)	1.78 × 10−5
Apparent density of the OC ρs (kg/m3)	3015
Void fraction in the downcomer ε (-)	0.477
Mean diameter of the OC ds (mm)	0.43
Sphere coefficient of the OC φs (-)	0.7
Diameter of the FR Df (m)	0.06
Diameter of the upper downcomer Dud (m)	0.10

lists the main parameters required for the calculation of the ideal upper pressure gradient. It can be seen that, with a solid circulation flux of 200 kg/m²·s, the value of the theoretical ideal pressure gradient was about 1.6 kPa/m which was almost the same with the experimental value. Then, when the solid circulation flux increased to 300 kg/m²·s, the theoretical and measured values of the ideal pressure gradient were increased to about 2.5 and 2.8 kPa/m, respectively. In general, the relative errors between the measured and predicted values of the ideal pressure gradient were kept to be lower than 15%, demonstrating the application feasibility of the modified Ergun equation in the prediction of the ideal pressure gradient of the high-flux CDCLC system.

In addition, from Figure 4, we can get the optimal region of the upper pressure gradient ΔP₁/H₁ under a high-flux condition of 200 kg/m²·s, which ranged between −2.1 kPa/m and 3.0 kPa/m. Thus, by associating the optimal region with the ideal pressure gradient (1.6 kPa/m), a semi-theoretical formula of gas leakages between the two reactors (i.e., Equation (11)) could be deduced, which includes two conterminal linear equations with the ideal pressure gradient chosen as the boundary point. This formula successfully established the important mapping relationships between the gas–solid flow states in the upper dipleg and the upper pressure gradient, which should be important coupling criteria of selecting design parameters and operating conditions.

$$f_i=\{\begin{array}{l}{f}_1={\alpha }_1\left[{\left(\Delta P_1/H_1\right)}_t-{\left(\Delta P_1/H_1\right)}_i\right]\left({\left(\Delta P_1/H_1\right)}_t<{\left(\Delta P_1/H_1\right)}_i,{\alpha }_1\approx 0.8\right)\\ f_2={\alpha }_2\left[{\left(\Delta P_1/H_1\right)}_t-{\left(\Delta P_1/H_1\right)}_i\right]\left({\left(\Delta P_1/H_1\right)}_t\ge {\left(\Delta P_1/H_1\right)}_i,{\alpha }_2\approx 2.2\right)\end{array}$$

(11)

3.2.2. Semi-Theoretical Formulas of the Lower Pressure Gradient

From the analyses shown in Section 3.1, we can find that the optimal operation region of the lower dipleg should better locate between State 5 (i.e., the ideal sealing state) and State 6 in order to ensure stable solid circulation with acceptable gas leakage.

The correlation equation of the lower pressure gradient under the ideal sealing state was also derived from the modified Ergun equation [33,34,35], as shown in Equation (12). Here, it should be noted that the lower downcomer was designed to be cuboid shaped (0.1 m length × 0.1 m width) and the upper downcomer was cylinder shaped, which made the form of Equation (12) a bit different from that of Equation (10). Thus, the theoretical value of the ideal lower pressure gradient could be deduced to be about 1.3 kPa/m. Further, according to our previous experimental results [20], the optimal region of ΔP₂/H₂ under a high-flux condition of 200 kg/m²·s should be limited within 6.0 kPa/m in order to guarantee the J-valve leakage ratio lower than 20%. Thus, by associating the optimal region of the lower pressure gradient with the ideal pressure gradient, a semi-theoretical formula of J-valve gas leakage (i.e., Equation (13)) could be deduced, in which the coefficient β was used as the slope. Similarly, with Equation (11), this formula established the mapping relationships between the J-valve gas leakage and the lower pressure gradient, enabling a coupling criterion of selecting design parameters and operating conditions during the CDCLC process.

$${\left(\Delta P_2/H_2\right)}_i=\left[150\frac{\pi }{4}\frac{\left(1-\epsilon \right){\mu }_g}{{\rho }_s{\left(\epsilon {\varphi }_s{d}_s\right)}^2}{\left(\frac{D_f}{L_{ld}}\right)}^2\right]G_s+\left[1.75{\left(\frac{\pi }{4}\right)}^2\frac{{\rho }_g}{\epsilon {\varphi }_s{d}_s(1-\epsilon ){\rho }_s^2}{\left(\frac{D_f}{L_{ld}}\right)}^4\right]G_s^2$$

(12)

$$f_3=\beta \left[{\left(\Delta P_2/H_2\right)}_t-{\left(\Delta P_2/H_2\right)}_i\right]\left({\left(\Delta P_2/H_2\right)}_t\ge {\left(\Delta P_2/H_2\right)}_i,\beta \approx 4.3\right)$$

(13)

3.3. Theoretical Methodology for Circulation Stability

In a real CDCLC application, a critical sealing state (i.e., State 7 shown in Figure 3) can be used to identify the advent of circulation collapse. Therefore, it is also necessary to understand the critical sealing ability of the CDCLC system so as to prevent the emergency situation of operation instability. Here, an empirical formula of critical sealing proposed by Chang et al. [32] (see Equation (14)) was attempted to be applied in this high-flux CDCLC system, in which the coefficient γ was between 0.6–0.7.

$${\left|\frac{\Delta P}{H}\right|}_c=\gamma g_c\left[{\rho }_s\left(1-\epsilon \right)-136\right]$$

(14)

Figure 6 exhibits the comparison of predicted and experimental upper pressure gradients under the critical sealing state. It can be found that, with a solid circulation flux of 250 kg/m²·s, the experimental value of the critical sealing gradient was 10.7 kPa/m under an upper dipleg height of 1.07 m [20]. In order to ensure the accuracy of test measurement, another dipleg height (0.87 m) was adopted for the measure of the critical sealing gradient while the other operating conditions were kept constant. It can be seen that these two experimental results (10.9 kPa/m for 0.87 m height, and 10.7 kPa/m for 1.07 m height) were very close to each other, demonstrating the constancy of the critical sealing gradient. On the other hand, the calculation value of the critical sealing gradient based on Equation (14) was between 8.5 kPa/m and 9.9 kPa/m. Thus, the relative error between the measured and predicted values of the critical sealing gradient could be further calculated to be lower than 21% for γ of 0.6, and 8% for γ of 0.7, demonstrating the application feasibility of Chang et al. [32] equation in the prediction of the critical pressure gradient of the high-flux CDCLC system. Moreover, the value of 0.7 for the coefficient γ seems to be more suitable for this system, in view of the least relative error with the experimental values. Therefore, the semi-theoretical formula for the circulation stability of this high-flux system can be finally expressed as Equation (15).

$${\left(\Delta P/H\right)}_c=0.7g_c\left[{\rho }_s\left(1-\epsilon \right)-136\right]$$

(15)

3.4. Theoretical Methodology Application to Condition Designs of the Cold System

Based on the theoretical methodology for the gas leakages and solid circulation, we proposed an optimal operation condition of the cold CDCLC system. Firstly, considering the feature and requirement of the high-flux operation, we selected a higher value of 300 kg/m²·s as the solid circulation flux G_s while the corresponding FR superficial gas velocity U_f,sta and the inlet air flow rate of the AR Q_4,sta were set to be 10.7 m/s and 44 m³/h, respectively. Thus, according to Equations (10) and (12), the theoretical ideal pressure gradients of the upper dipleg and the lower dipleg should be about 2.5 kPa/m and 1.9 kPa/m, respectively. Then, on the basis of the above semi-theoretical formulas (i.e., Equations (11) and (13)), we could further deduce that the optimal regions for gas leakage restraint were about −1.3 to 3.9 kPa/m for the upper pressure gradient ΔP₁/H₁, and about 1.9 to 6.6 kPa/m for the lower pressure gradient ΔP₂/H₂. Under this premise, we selected 3.8 kPa/m and 5.2 kPa/m as the proposed values of ΔP₁/H₁ and ΔP₂/H₂, respectively.

Figure 7 exhibits the whole-system pressure profile and the apparent solids holdup along the FR height under the proposed operation condition. As shown in Figure 7a, the pressures of each part under the high-flux condition were smoothly connected with each other, demonstrating the operation stability and the favorable coupling between each component. Besides, the high-pressure drop in the FR and low-pressure drop in the AR, they successfully exhibited the operation features of HFCFB and CFMB. From Figure 7b, we further observed the high solids holdup along the whole FR height, indicating the positive effect of high solid circulation flux on the efficiencies of gas–solid contact and reaction [14]. In addition,

Table 2. Pressure gradients and gas leakage ratios under the proposed operation condition (AR: air reactor).

Description	Parameter	Measured Value (%)	Calculation Value (%)	Relative Error (%)
ΔP1/H1 = 3.8 kPa/m	FR leakage ratio f1	−0.1	0	-
	AR leakage ratio f2	2.5	2.9	14
ΔP2/H2 = 5.2 kPa/m	J-valve leakage ratio f3	15.2	14.2	7

summarizes the pressure gradients and gas leakage ratios under the proposed operation condition. It can be found that, although the upper pressure gradient ΔP₁/H₁ close to the upper limit of the optimal region, the FR leakage ratio f₁ (−0.1%) and the AR leakage ratio f₂ (2.5%) can still be limited within their limits (i.e., −3% for f₁ and 3% for f₂), demonstrating the feasibility of the semi-theoretical Equation (11) for the prediction of the optimal region for gas leakage control. On the other hand, the lower pressure gradient ΔP₂/H₂ was located at a value of 5.2 kPa/m, in which the J-valve leakage ratio f₃ (15.2%) could be kept within a proposed region (<20%) in order to ensure a favorable solid circulation.

In general, the system operation stability, the high-flux feature, and particularly the gas leakage restraint were successfully achieved in this proposed operation condition, indicating the application feasibility of the semi-theoretical methodology to the system optimal design.

3.5. Hot State Application Assessment of the Theoretical Methodology

From the theoretical methodology of AR coupling principle with the high-flux CDCLC system, we could further carry out a capability assessment of the system in terms of the restraint of gas leakages and the stability of solid circulation under hot states. To facilitate the analysis and comparison, the structure parameters and OC material of the cold system were also adopted in the hot system. Besides, the solid circulation flux in the hot state was also selected as 300 kg/m²·s so as to keep consistent with the proposed cold-state operation condition mentioned above. The only difference was that under the hot state, the operating temperature was as high as 1243 K with the corresponding dynamic viscosity μ_g,hot = 4.7 × 10⁻⁵ Pa·s.

Table 3. Parameters for the calculation of the ideal pressure gradients under the hot state.

Description	Value
Temperature (K)	1243
Solid circulation flux Gs (kg/m2·s)	300
Gas dynamic viscosity under the hot state μg,hot (Pa·s)	4.7 × 10−5
Apparent density of the OC ρs (kg/m3)	3015
Void fraction in the downcomer ε (-)	0.477
Mean diameter of the OC ds (mm)	0.43
Sphere coefficient of the OC φs (-)	0.7
FR diameter Df (m)	0.06
Upper downcomer diameter Dud (m)	0.10
Side length of the lower downcomer Lld (m)	0.10

details the parameters for the calculation of ideal pressure gradients under the hot state.

According to Equation (10), the theoretical ideal pressure gradient of the upper dipleg under the hot state should be about 6.4 kPa/m. Similarly, according to Equation (12), the theoretical ideal pressure gradient of the lower dipleg under the hot state was about 5.0 kPa/m. Figure 8 shows the comparison of the ideal pressure gradients between the hot and cold states. We can observe the ideal pressure gradients of the two diplegs under the hot state were about 2.6 times of those under the cold state. This implies a lower requirement of the sealing height in the hot state, which should be beneficial for the spatial arrangement of the hot-state system. In addition, on the basis of the semi-theoretical formulas for gas leakage restraint (i.e., Equations (11) and (13)), the optimal regions were further calculated to be about 2.7 to 7.8 kPa/m for ΔP₁/H₁, and about 5 to 9.7 kPa/m for ΔP₂/H₂. On the other hand, according to Equation (15), the critical pressure gradient for the circulation stability could be deduced to be about 9.9 kPa/m. It can be found that the upper limit of the optimal region of ΔP₂/H₂ (9.7 kPa/m) for gas leakage restraint was very close to the critical pressure gradient for the circulation stability (9.9 kPa/m), demonstrating the rationality of the choice of 20% as the upper limit standard of the J-valve leakage. Certainly, it should be noted that the approach of the optimal pressure gradients for gas leakages to the critical pressure gradient for circulation stability also means the increase in the risk of circulation collapse during the hot-state operation process.

For real high-flux CDCLC applications, we can first get the optimal sizes of the reactors and the downcomer based on the ideal pressure gradient equations and the solid circulation flux range in the design process. Then during the operating process, the relevant parameters (i.e., the solid-seal heights in the downcomer, the pressures of the FR and AR, the solid circulation flux, and the gas flow rates) can be adjusted flexibly and optimally to make sure the pressure gradients within the optimal regions for a favorable performance of operation and reaction.

4. Conclusions

Built upon the previous experimental studies of a high-flux CDCLC system, the objective of this study is to further investigate the fundamental coupling mechanism of the AR, and develop a comprehensive theoretical methodology to the system optimal design with favorable operation stability and low gas leakages. The following conclusions can be drawn from the present study:

(1)

During the CDCLC process, the dipleg flow can situate at a pressure region across the positive and negative pressure gradients, which can be categorized into seven flow states. Considering the gas leakages and the circulation stability, the upper dipleg of the AR was recommended to be operated among State 2 to 6 while the lower dipleg of the AR should better run between States 5 and 6.

(2)

The gas leakages between the two reactors were expressed as two conterminal linear equations with the ideal pressure gradient chosen as the boundary point, which can be used to predict the optimal regions of the upper pressure gradient. Similarly, the J-valve leakage within the optimal region was expressed as a linear function of the lower pressure gradient of the AR. In addition, an empirical formula of critical sealing was developed for this high-flux CDCLC system, which can be used to identify the advent of circulation collapse so as to guarantee the operation stability.

(3)

The theoretical methodology for gas leakages and solid circulation was successfully applied to the condition design and operation of the cold system, achieving favorable gas–solid flow and circulation together with good control of gas leakages in the whole system.

(4)

The theoretical methodology was adopted to carry out a capability assessment of the high-flux CDCLC system under a hot state in terms of the restraint of gas leakages and the stability of solid circulation. The ideal pressure gradients under the hot state of 1243 K were about 2.6 times than those under the cold state, implying a lower requirement of sealing height in the hot state. However, on the other hand, the increase of the ideal pressure gradients also led to the approach of the optimal pressure gradients for gas leakages to the critical pressure gradients for circulation stability, which would increase the risk of circulation collapse during the operation process.