4.1. $\textrm {CO}_2$ desublimation and sublimation on a single packing grain

The initial objective of this study is to identify the $\textrm {CO}_2$ desublimation and sublimation properties on a single packing grain. A test with the subcooling degree $\Delta T_s=0.185$ and the Péclet number ${{Pe}}=15.57$ is simulated. The obtained distributions of solid $\textrm {CO}_2$, temperature $T$ and $\textrm {CO}_2$ mass fraction $Y$ at five time instants are provided in figure 4.

Figure 4.

The $\textrm {CO}_2$ desublimation and sublimation properties on a single packing grain with the subcooling degree $\Delta T_s=0.185$ and the Péclet number ${{Pe}}=15.57$. Contours of (a) solid $\textrm {CO}_2$, (b) temperature ($T$) and(c) $\textrm {CO}_2$ mass fraction ($Y$) at five time instants $t=4.83, 14.51, 29.01, 58.02, 101.77\, \textrm {s}$.

In the early stage (figure 4 at $t=4.83, 14.51\, \textrm {s}$), the injected flue gas is cooled by the grain and its component $\textrm {CO}_2$ is desublimated to generate SCL on the grain surface. Meanwhile, the incoming warm flue gas and the heat $Q_d$ released from $\textrm {CO}_2$ desublimation raise the temperature of the packing grain, which gradually brings about the sublimation of the previously generated SCL (figure 4 at $t=29.01\, \textrm {s}$). Following the continuous increase in grain temperature, the desublimation strength of $\textrm {CO}_2$ on the grain surface starts to decrease. As a result, the evident sublimation of SCL is observed compared with its generation (figure 4 at $t=58.02\, \textrm {s}$). Finally, the grain becomes warm and SCL is completely sublimated to gaseous $\textrm {CO}_2$ at $t_e=101.77\, \textrm {s}$, signaling the cessation of both $\textrm {CO}_2$ desublimation and sublimation.

Given the vital role of the captured SCL in assessing the performance of CCC, the volume fraction of solid $\textrm {CO}_2$ ($\phi _c$) on the packing grain is quantified by

(4.2)\begin{equation} \phi_c= \frac{1}{l_xl_y}\int_0^{l_x}\int_0^{l_y}V_s\left(x,y\right)\,{\rm d}y\,{\rm d}\kern0.06em x. \end{equation}

The calculated $\phi _c$ is recorded versus time in figure 5(a), whose behaviour is explained with the help of the averaged temperature ($\bar {T}_a$) and overall mass transfer rates via desublimation and sublimation ($m_r^*$) in figure 5(b–d). Here $\bar {T}_a$ is defined as the averaged temperature for the active gas–solid interface $I_{d,s}$, where $\textrm {CO}_2$ desublimation and sublimation take place:

(4.3)\begin{equation} \bar{T}_a= \frac{1}{I_{d,s}}\sum_{I_{d,s}}T\left(x,y\right). \end{equation}

The overall mass transfer rates are calculated as

(4.4)\begin{equation} \left. \begin{gathered} m_{rd}^*= \frac{1}{\rho_g u_0}\sum_{I_{d}}m_r\left(x,y\right), \quad m_{rs}^*={-}\frac{1}{\rho_g u_0}\sum_{I_{s}}m_r\left(x,y\right), \\ m_{r}^* = \frac{1}{\rho_g u_0}\sum_{I_{d,s}}m_r\left(x,y\right)= m_{rd}^*+({-}m_{rs}^*). \end{gathered} \right\} \end{equation}

Here, $m_{rd}^*$ and $m_{rs}^*$ represent the mass transfer rates through desublimation at $I_d$ and sublimation at $I_s$, respectively; $m_{r}^*$ stands for the combined mass transfer rate by both desublimation and sublimation at $I_{d,s}$.

Figure 5.

Analyses of $\textrm {CO}_2$ desublimation and sublimation on a single packing grain with the subcooling degree $\Delta T_s=0.185$ and the Péclet number $Pe =15.57$. Temporal evolutions of (a) volume fraction of the solid $\textrm {CO}_2$ captured ($\phi _c$), (b) averaged temperature of active boundaries ($\bar {T}_a$) and (c,d) overall mass transfer rate via desublimation and sublimation ($m_r^*$, $m_{rd}^*$ and $m_{rs}^*$). Contours of solid $\textrm {CO}_2$, temperature ($T$) and $\textrm {CO}_2$ mass fraction ($Y$) at (e) peak point $t_m$ and (f) inflection point $t_i$.

Initially, $\phi _c$ in figure 5(a) increases with time due to the stronger desublimation rate compared with the sublimation rate (i.e. $m_r^*>0$ in figure 5c). The growth rate of $\phi _c$ is observed to decrease over time. As shown in figure 5(b–d), this is driven by the fact that the ascending $\bar {T}_a$ decelerates desublimation ($m_{rd}^*$) while accelerating sublimation ($m_{rs}^*$). Subsequently, the desublimation rate gradually drops to an equilibrium level with the sublimation rate at $t_m$ (i.e. $m_r^*=0$), where $\phi _c$ reaches a peak value $\phi _{cm}$. Scalar distributions at this peak point $t_m$ are shown in figure 5(e). After $t_m$, $\phi _c$ decreases until it reaches zero at $t_e=101.77\, \textrm {s}$, wherein the desublimation rate decreases steadily and becomes weaker than the sublimation rate (i.e. $m_r^*<0$). The decrease in $\phi _c$ experiences an inflection point at $t_i$, after which the drop rate of $\phi _c$ slows down. This is because the SCL on the front part of the grain is completely sublimated at $t_i$, leading to the diminished sublimation rate. For illustration, figure 5(a) plots the temporal evolutions of $\phi _c$ for the front and back grain areas, and figure 5(f) provides the scalar distributions at $t_i$. Besides, the warm gas stream and $\textrm {CO}_2$ desublimation contribute to raising $\bar {T}_a$ towards an equilibrium value of approximately $189.3\, \textrm {K}$, which is in line with experimental findings (Tuinier et al. Reference Tuinier, van Sint Annaland, Kramer and Kuipers2010, Reference Tuinier, van Sint Annaland and Kuipers2011b). Taken together, these quantitative analyses corroborate the above qualitative observations in figure 4 and also provide insights into the $\textrm {CO}_2$ desublimation and sublimation processes.

4.2. Effects of $\Delta t_s$ and $Pe $ in the single-grain case

In the single-grain case, after the discussion of general $\textrm {CO}_2$ desublimation and sublimation properties, a parametric study is set out to explore the impact of subcooling degree $\Delta T_s$ and Péclet number $Pe $ on the carbon capture performance of the cold grain. The $\textrm {CO}_2$ desublimation and sublimation processes are simulated at Péclet numbers $Pe \in [1.55, 46.70]$, and each $Pe $ contains a subsection of subcooling degrees $\Delta T_s \in [0.049,\ 0.389]$. From the above discussions on $\textrm {CO}_2$ desublimation and sublimation properties in § 4.1, two important metrics stand out to quantify the carbon capture performance of the grain, namely, the maximum volume fraction of solid $\textrm {CO}_2$ captured by the packing grain ($\phi _{cm}$) and the operating time of the $\textrm {CO}_2$ desublimation and sublimation processes ($t_e$). Based on these two metrics, the $\textrm {CO}_2$ capture rate ($v_c$) is calculated as

(4.5)\begin{equation} v_c = \frac{\phi_{cm}}{t_e}. \end{equation}

A grain with a larger $v_c$ demonstrates a higher efficiency in capturing the $\textrm {CO}_2$ component, making it desirable.

Figure 6 depicts values of ($\phi _{cm}$, $t_e$, $v_c$) for a wide range of $\Delta T_s$ and $Pe $, where each dot represents a simulation test at a given operating condition ($\Delta T_s$, $Pe $). For a fixed $Pe $, effects of $\Delta T_s$ on the $\textrm {CO}_2$ capture performance of the grain share a similar trend: the successive rise in $\Delta T_s$ introduces the continuous increase in both $t_e$ and $\phi _{cm}$. For example, as $\Delta T_s$ varies from $0.049$ to $0.389$ at $Pe =15.57$, $\phi _{cm}$ increases from $0.005$ to $0.138$ and $t_e$ grows from $48.59\, \textrm {s}$ to $150.84\, \textrm {s}$. As supported by figure 7(a), a possible explanation is that a packing grain with a higher $\Delta T_s$ holds a lower temperature $\bar {T}_p$. Under the fixed gas feed rate condition (i.e. constant $Pe $), the convective heat transfer strength is similar and, thus, a colder grain requires a lengthier duration ($t_e$) for the incoming warm flue gas to heat it up. This subsequently produces an increased amount of solid $\textrm {CO}_2$ captured ($\phi _{cm}$). On the other hand, $v_c$ in figure 6(c) is seen to increase with the ascending $\Delta T_s$, indicating that a colder packing grain is beneficial to improving the $\textrm {CO}_2$ capture performance of the grain. Meanwhile, based on the value of $Pe $, two distinct regimes arise from the comparison of $v_c$ evolutions in response to changes in $Pe $ and $\Delta T_s$.

Figure 6.

Analyses of $\textrm {CO}_2$ capture performance in single-grain tests with subcooling degrees $\Delta T_s \in [0.049, 0.389]$ and Péclet numbers $Pe = 1.55, 3.89, 7.78, 15.57, 31.14, 46.70$. (a) The maximum volume fraction of solid $\textrm {CO}_2$ captured by the grain ($\phi _{cm}$). (b) The operating time for $\textrm {CO}_2$ desublimation and sublimation ($t_e$). (c) The $\textrm {CO}_2$ capture rate ($v_c$). (d) Contours of solid $\textrm {CO}_2$, temperature ($T$) and $\textrm {CO}_2$ mass fraction ($Y$) in a convection-limited test with $\Delta T_s=0.185$ and $Pe = 1.55$ at two time instants $t=24.17, 48.34\, \textrm {s}$.

Figure 7.

Analyses of $\textrm {CO}_2$ capture performance in single-grain tests with subcooling degrees ${\Delta T_s = 0.117}, 0.185, 0.253, 0.321$ and Péclet numbers $Pe = 7.78, 15.57, 23.35, 31.14$. Temporal evolutions of (a,c) averaged temperature of the packing grain ($\bar {T}_p$) and (b,d) overall mass transfer rate via desublimation and sublimation ($m_r^*$).

First, in large-$Pe $ tests (e.g. $Pe > 7.8$ at $\Delta T_s=0.185$), the increasing $\Delta T_s$ accelerates $v_c$, while $Pe $ has minimal or no effects on it. Accordingly, a correlation is fitted as $v_c=22.56\Delta T_s-0.76$. As shown in figure 7(a,b), for a fixed $Pe $ in this regime, the larger $\Delta T_s$ brings about the colder packing grain ($\bar {T}_p$) and the faster mass transfer rate ($m_r^*$), introducing the accelerated $\textrm {CO}_2$ capture rate $v_c$. On the other hand, the negligible impact of increasing $Pe $ on $v_c$ can be explained by effects of convection between the warm gas stream and solid phases. The rapid convective gas flow augments the $\textrm {CO}_2$ delivery and amplifies the $\textrm {CO}_2$ desublimation rate. At the same time, however, the fast convective heat transfer rapidly heats up the grain and consequently slows down the desublimation of $\textrm {CO}_2$. Figure 7(c,d) is presented to corroborate the aforementioned effects of $Pe $. Following the growing $Pe $ in this regime, the $\textrm {CO}_2$ supply is abundant and $m_r^*$ is amplified at first ($t<6\, \textrm {s}$ in figure 7d). After a short period, the ascending $Pe $ intensifies the convective heat transfer, yielding a significant increase in $\bar {T}_p$ and a rapid slowdown in $m_r^*$ ($t>6\, \textrm {s}$ in figure 7c,d). As a result, the high $Pe $ leads to a reduction in both $t_e$ and $\phi _{cm}$ (figure 6a,b). Since $t_e$ and $\phi _{cm}$ exhibit a similar decrease rate, $v_c$ is barely or not affected by $Pe $. Considering that $v_c$ is primarily influenced by $\Delta T_s$ (or desublimation) rather than $Pe $, the $\textrm {CO}_2$ desublimation and sublimation processes in the large-$Pe $ range are denoted as the desublimation-limited (or subcooling-limited) regime.

Second, as $Pe $ decreases to a rather small value (e.g. $Pe < 7.8$ at $\Delta T_s=0.185$), the increase in $\Delta T_s$ leads to the growing $\phi _{cm}$, $t_e$ and $v_c$, which is in line with the above desublimation-limited regime. However, the decline in $Pe $ induces a growth in $t_e$ but a slight variation in $\phi _{cm}$, giving rise to the diminished $v_c$. Therefore, both $\Delta T_s$ and $Pe $ play roles in determining $v_c$, making $v_c$ deviate from the correlation $v_c=22.56\Delta T_s-0.76$. This inconsistency from the desublimation-limited regime stems from the weak convection under small $Pe $. It is clear from figure 7(c,d) that the weak convective heat transfer extends the time period for heating up the packing grain, resulting in the prolonged $t_e$. For example, the decreased $Pe $ from 15.57 to 7.78 leads to the augmented $t_e$ from $101.77\, \textrm {s}$ to $123.04\, \textrm {s}$. In addition, the inadequate $\textrm {CO}_2$ supply via weak gas convection, along with the fixed $\Delta T_s$, results in the unchanged or even marginally diminished $\phi _{cm}$. The combination of these two factors is responsible for the obvious departure of $v_c$ from the correlation. These small-$Pe $ tests are classified as the convection-limited regime due to the weak convection. As an example, figure 6(d) provides scalar distributions in a convection-limited test with $\Delta T_s=0.185$ and $Pe = 1.55$. The insufficient $\textrm {CO}_2$ supply via weak convection is illustrated clearly by contours of the $\textrm {CO}_2$ mass fraction.

To directly demonstrate impacts of $Pe $ on the $\textrm {CO}_2$ capture performance of the cold grain, simulation results for $Pe \in [1.55, 46.70]$ and $\Delta T_s=0.185$ are presented in figure 8. With the growing $Pe $, there is an obvious drop in both $\phi _{cm}$ and $t_e$, while $v_c$ initially experiences a sharp increase and then stabilizes after reaching the threshold of $Pe _c\approx 7.8$ (at $\Delta T_s=0.185$). Accordingly, the $\textrm {CO}_2$ desublimation and sublimation processes shift from the convection-limited regime to the desublimation-limited regime. These tendencies corroborate findings in figure 6, which are driven by changes in the relative strength between convection and desublimation.

Figure 8.

Analyses of $\textrm {CO}_2$ capture performance in single-grain tests with the subcooling degrees $\Delta T_s=0.185$ and the Péclet numbers $Pe \in [1.55, 46.70]$. (a) The maximum volume fraction of solid $\textrm {CO}_2$ ($\phi _{cm}$) captured by the grain. (b) The operating time for $\textrm {CO}_2$ desublimation and sublimation ($t_e$). (c) The $\textrm {CO}_2$ capture rate ($v_c$).

Finally, a comprehensive map of the simulated $v_c$ is plotted against $\Delta T_s \in [0.049, 0.389]$ and $Pe \in [1.55, 46.70]$ in figure 9. For comparison, the calculated $v_{cr}$ via the correlation $v_c=22.56\Delta T_s-0.76$ is presented as a grey surface. Here $v_c$ aligns closely with $v_{cr}$ in the large-$Pe $ region but exhibits notable deviations in the small-$Pe $ space. Thereby, distributions of the convection-limited (I) and desublimation-limited (II) regimes are obtained in the $\Delta T_s$–$Pe $ parameter space (figure 10). The boundary line between the two regimes is established (i.e. solid white line in figure 9 and grey dash line in figure 10), with threshold values ($\Delta T_{sc}$, $Pe _c$) situated upon this boundary. As the $\Delta T_{s}$ (or $Pe $) increases, there is a consequent growth in the threshold value of $Pe _c$ (or $\Delta T_{sc}$). This relationship reflects that the augmented convection strength is necessary to counterbalance the increased desublimation rate. According to the regime diagram, it is essential that $\Delta T_s$ stays below $\Delta T_{sc}$ and $Pe $ exceeds $Pe _c$ in order to ensure the desublimation-limited regime and prevent weak convection limitations. Note that, the continuous increase in $Pe $ beyond $Pe _c$ is unnecessary as it no longer accelerates $v_c$. Although a large $\Delta T_s$ helps accelerate $v_c$, $\Delta T_s$ should be increased with caution because it has the potential to significantly boost the cooling duty. This aspect is beyond the scope of the present work and will be considered in our future work. In general, the regime diagram can provide guidance on how to select operation conditions for an optimal CCC system.

Figure 9.

Analyses of the $\textrm {CO}_2$ capture rate $v_c$ in single-grain tests with subcooling degrees $\Delta T_s \in [0.049, 0.389]$ and Péclet numbers $Pe \in [1.55, 46.70]$. The grey surface represents the correlation $v_c=22.56\Delta T_s-0.76$. The white solid line shows the boundary between the convection-limited (I) and desublimation-limited (II) regimes.

Figure 10.

Analyses of $\textrm {CO}_2$ capture performance in single-grain tests with subcooling degrees $\Delta T_s \in [0.049, 0.389]$ and Péclet numbers $Pe \in [1.55, 46.70]$. Simulation data points are plotted against $\Delta T_s$ and $Pe $. The grey dashed line divides the plane into the convection-limited (I) and desublimation-limited (II) regimes.

The thematically aligned research we conducted earlier explored $\textrm {CO}_2$ desublimation on an isothermal grain during the capture of CCC (Lei et al. Reference Lei, Luo, Hernández Pérez, Wang, Wang, Restrepo Cano and Im2023). In contrast, this work extends to investigate both $\textrm {CO}_2$ desublimation and sublimation on a non-isothermal grain, encompassing the capture and recovery of CCC. These considerations make the present study incorporate a more comprehensive thermal interaction between grains and flow, leading to significantly distinct results that are more realistic in an actual CCC process. Some key differences are summarized as follows. (1) The diffusion-controlled and join-controlled regimes are no longer as distinct as they were predicted in the previous study (Lei et al. Reference Lei, Luo, Hernández Pérez, Wang, Wang, Restrepo Cano and Im2023), because the increase in the grain temperature diminishes the desublimation rate over time. (2) The previous study predicted that the solid $\textrm {CO}_2$ captured ($\phi _{cm}$) continued to increase with $Pe $, while the present model shows that an increase in $Pe $ results in a decrease in $\phi _{cm}$ due to the diminished desublimation rate. (3) The previous study demonstrated an unrealistically high desublimation rate as $\Delta T_s$ increases. This results in the formation of cluster-like SCLs with highly porous structures, leading to a reduced desublimation duration and $\phi _{cm}$. By contrast, the present study correctly predicts that the ascending $\Delta T_s$ results in the growing $\phi _{cm}$ and the extended desublimation duration. (4) The sublimation of SCL and the $\textrm {CO}_2$ capture rate $v_c$ are newly analysed here, producing new and valuable insights into $\textrm {CO}_2$ desublimation and sublimation during CCC. Therefore, the present study provides more realistic and valuable insights into the CCC process and allows an improved understanding compared with our previous approach (Lei et al. Reference Lei, Luo, Hernández Pérez, Wang, Wang, Restrepo Cano and Im2023). In the following subsection, $\textrm {CO}_2$ desublimation and sublimation in a cryogenic packed bed are discussed.

4.3. The $\textrm {CO}_2$ desublimation and sublimation in a packed bed

In the packed-bed case, simulation tests for a wide range of subcooling degrees $\Delta T_s$ and Péclet numbers $Pe $ are performed, which successfully reproduce $\textrm {CO}_2$ desublimation and sublimation properties during the capture and recovery steps of CCC. For illustration, figure 11(a,b) depicts numerical results for the test with $\Delta T_s=0.185$ and $Pe =15.57$, including contours of solid $\textrm {CO}_2$ and temperature distributions at five time instants. When feeding the warm flue gas to the packed bed, the flue gas is cooled until its component $\textrm {CO}_2$ starts to desublimate and generates an SCL on the packing grains. As time goes on, the SCL grows on the grains and forms a desublimation front ($l_d$), which expands toward the outlet. In the meantime, under effects of the incoming warm flue gas and the exothermic $\textrm {CO}_2$ desublimation process, packing grains close to the inlet are significantly heated from $T_w$ to $T_0$. Owing to such a rise in temperature, the previously formed SCL is sublimated and a sublimation front ($l_s$) develops. Following $l_d$, $l_s$ also progresses outwards but at a smaller velocity. In the packed bed, grains from $l_s$ to $l_d$ are coated by solid $\textrm {CO}_2$ and, thus, they are denoted as the SCL area.

Figure 11.

The $\textrm {CO}_2$ desublimation and sublimation properties in a packed bed with the subcooling degree $\Delta T_s=0.185$ and the Péclet number $Pe =15.57$. Contours of (a) solid $\textrm {CO}_2$ and (b) temperature ($T$) at five time instants $t=24.78, 86.74, 173.48, 322.17, 589.82\, \textrm {s}$. Vertically averaged (c) volume fraction of solid $\textrm {CO}_2$ ($\phi _{cx}$) and temperature ($\bar {T}_x$) at two time instants $t=86.74, 322.17\, \textrm {s}$.

For quantification, vertically averaged scalars ($\bar {\zeta }_x$) are introduced as

(4.6)\begin{equation} \bar{\zeta}_x= \frac{1}{l_y}\int_0^{l_y}\zeta\left(x,y\right)\,{{\rm d} y}, \quad \text{with}\ \zeta = V_s, T, Y. \end{equation}

Figure 11(c,d) presents the vertically averaged volume fraction of solid $\textrm {CO}_2$ ($\phi _{cx}$) and temperature ($\bar {T}_x$) at two time instants. The calculated profiles of $\bar {T}_x$ for both grains (dashed lines) and gas (solid lines) exhibit a declining trend from the inlet to the outlet. On the other hand, ignoring fluctuations introduced by variations in the grain surface along the $x$ direction, curves of $\phi _{cx}$ change non-monotonically from the inlet to the outlet. For instance, at $86.74\, \textrm {s}$, starting from the inlet $\phi _{cx}$ remains zero at first, then it rises to a maximum value, subsequently it decreases back to zero and finally it remains constant until the outlet. From each profile of $\phi _{cx}$, the desublimation and sublimation fronts are identified as the two positions with $\phi _{cx}=0.01$. The region between these two fronts corresponds to the SCL area. From the comparison of $\phi _{cx}$ profiles at $86.74\, \textrm {s}$ and $322.17\, \textrm {s}$, it is evident that the SCL area together with the two fronts propagate outwards. These findings quantitatively support the observations in figure 11(a,b).

As $l_d$ arrives at the bed exit, the component $\textrm {CO}_2$ starts to break through the packed bed and the outgoing $\textrm {CO}_2$ mass fraction gradually grows to the inlet value, i.e. $Y(l_x)=Y_0$. This phenomenon is termed the operational saturation and the CCC system is thus switched to the recovery step. To clarify the saturation condition of the packed bed, figure 12 displays distributions of $\textrm {CO}_2$ mass fraction ($Y$) and profiles of the vertically averaged values ($\bar {Y}_x$) at three time instants. At the early stage (i.e. $t=39.57\, \textrm {s}$), packing grains are cold enough to fully capture the injected $\textrm {CO}_2$. Over time, grains are gradually heated by the incoming warm flue gas and the exothermic desublimation, and thus, the injected $\textrm {CO}_2$ passes warm grains and moves toward the exit. With the continuous rise in grain temperature, part of the injected $\textrm {CO}_2$ starts to leave the packed bed without desublimation (i.e. $t=105.96\, \textrm {s}$). The outlet $\textrm {CO}_2$ mass fraction progressively increases to a critical value (i.e. 10 % in this study), indicating that the bed reaches the saturation point $t_{sat}=136.34\, \textrm {s}$. After $t_{sat}$, the injected $\textrm {CO}_2$ is less efficiently captured and the CCC system enters the recovery step to collect the captured SCL. To quantify $t_{sat}$, profiles of $\bar {Y}_x$ are utilized to define the saturation front ($l_{sat}$) as the position with $\bar {Y}_x=0.1$. Saturation point $t_{sat}$ is then determined as the time instant when $l_{sat}$ moves to the outlet (i.e. $t_{sat}=136.34\, \textrm {s}$ in figure 12b).

Figure 12.

The $\textrm {CO}_2$ desublimation and sublimation properties in a packed bed with the subcooling degree $\Delta T_s=0.185$ and the Péclet number $Pe =15.57$. (a) Contours of $\textrm {CO}_2$ mass fraction ($Y$) and (b) vertically averaged $\textrm {CO}_2$ mass fraction ($\bar {Y}_x$) at three time instants $t=39.57, 105.96, 136.34\, \textrm {s}$.

Following the same procedure as in the single-grain case, temporal evolutions of the solid $\textrm {CO}_2$ volume fraction ($\phi _c$) and the mass transfer rate ($m_r^*$) are calculated and plotted in figure 13 to quantify the $\textrm {CO}_2$ capture performance of the packed bed. Initially, the curve of $\phi _c$ experiences a sharp increase. This is because the packing grains are sufficiently cold and the $\textrm {CO}_2$ desublimation dominates the system. However, as grains are heated by the incoming warm gas and the released heat from desublimation $Q_d$, the mass transfer rate $m_r^*$ drops to be negative. Consequently, the $\textrm {CO}_2$ sublimation controls the system and $\phi _c$ starts to decrease until all the captured SCL is recovered at $t_e$. From this nonlinear profile of $\phi _c$, the maximum solid $\textrm {CO}_2$ capture capacity of the packed bed is determined as $\phi _{cm}$ at $t_m$. The non-monotonic variation of $\phi _c$ aligns with that observed in the single-grain case in figure 5. In contrast, no inflection point is detected during the decreasing stage of $\phi _c$ here. This stems from the fact that the multiple packing grains help to average out the temperature difference between the front and back areas of each single grain.

Figure 13.

Analyses of $\textrm {CO}_2$ desublimation and sublimation in a packed bed with the subcooling degree $\Delta T_s=0.185$ and the Péclet number $Pe =15.57$. Temporal evolutions of (a) volume fraction of the solid $\textrm {CO}_2$ captured ($\phi _c$), (b) position of the saturation front ($l_{sat}$) and (c,d) overall mass transfer rate via desublimation and sublimation ($m_r^*$, $m_{rd}^*$ and $m_{rs}^*$).

For the operation of CCC in a packed bed, a key issue is the timely detection of the operational saturation point $t_{sat}$, at which the system should initiate the recovery step to gather the captured SCL. Otherwise, any delayed actions could result in detrimental consequences, such as CCC malfunction and flue gas breakthrough. For this purpose, the position of the saturation front ($l_{sat}$) is identified based on the vertically averaged $\textrm {CO}_2$ mass fraction ($\bar {Y}_x$) and set out as a function of time in figure 13(b). There is a clear ascending trend in the temporal evolution of $l_{sat}$ from the inlet ($l_{sat}=0$) to the outlet ($l_{sat}=l_x$), which corroborates the outward movement of $l_{sat}$ in figure 12. As marked by the red dash circle in figure 13(b), $t_{sat}$ is quantitatively identified as the earliest time instant when $l_{sat}$ advances to the outlet. One unexpected finding is that $t_{sat}$ is earlier than $t_m$. During the operation of CCC, however, the system is desired to start the recovery of the SCL at $t_m$, so as to capture the maximum amount of $\textrm {CO}_2$. To quantify the performance degradation, two metrics associated with the time delay are defined. One is the delayed time period, $t_{d} = t_m - t_{sat}$, and the other one is the capacity loss

(4.7)\begin{equation} \eta_d= \frac{\phi_{m}-\phi_{sat}}{\phi_{m}}. \end{equation}

It is apparent that, as $t_d$ and $\eta _d$ approach zero, the carbon capture performance of the packed bed is optimal.

These numerically obtained $\textrm {CO}_2$ desublimation and sublimation properties in a packed bed accord closely with experimental observations during the capture and recovery steps (Tuinier et al. Reference Tuinier, van Sint Annaland, Kramer and Kuipers2010, Reference Tuinier, van Sint Annaland and Kuipers2011b; Ali et al. Reference Ali, Maqsood, Syahera, Shariff and Ganguly2014). Moreover, the detailed spatio-temporal evolution of scalar distributions and quantitative analyses, which are difficult to obtain from experiments, contribute to a better understanding of CCC. After that, parametric analyses are set out to elucidate the key operating factors. Compared with the single-grain case, the same range of $\Delta T_s \in [0.049, 0.389]$ is considered, while the range of $\textrm {Pe}$ is extended to be $[6.23, 77.84]$ due to the resistance of multiple grains. In addition, the porous structure of the packed bed is expected to impact its flow channels and $\textrm {CO}_2$ capture performance. Therefore, the following packed-bed tests aim to examine effects of the gas velocity, bed temperature and bed structure.

4.4. Effects of $\Delta t_s$ and $Pe $ in the packed-bed case

From the above $\textrm {CO}_2$ desublimation and sublimation properties, four key metrics emerge for evaluating the $\textrm {CO}_2$ capture performance of the packed bed, including the maximum volume fraction of solid $\textrm {CO}_2$ captured ($\phi _{cm}$), the operating time of $\textrm {CO}_2$ desublimation and sublimation ($t_e$), the $\textrm {CO}_2$ capture capacity loss due to the delay between the maximum point and the saturation point ($\eta _d$), and the $\textrm {CO}_2$ capture rate ($v_c$). These four metrics are calculated and analysed for packed-bed tests with varying $\Delta T_s$ and $Pe $.

Figure 14 compares values of metrics ($\phi _{cm}$, $t_e$, $\eta _d$, $v_c$) for tests with $\Delta T_s \in [0.049, 0.389]$ and $Pe = 15.57, 31.14, 46.70, 62.27, 77.84$. As $\Delta T_s$ increases at a fixed $Pe $, the results follow a similar trend. On the one hand, a rise in $\Delta T_s$ increases $\phi _{cm}$ and $t_e$ (figure 14a,b). As explained in the single-grain case, this stems from the lower temperature in grains and the enhanced desublimation intensity ($m_r$). On the other hand, the ascending $\Delta T_s$ yields the growing $v_c$ but at a diminished growth rate (figure 14d). From comparisons of results at different $Pe $ numbers, it is evident that the large $Pe $ (or fast gas convection) amplifies $v_c$. Incorporating impacts of both $\Delta T_s$ and $Pe $, a correlation is established to fit the simulation data as $v_c=(0.02Pe -0.19)\ln (\Delta T_s-0.024)+(0.083Pe -0.37)$. Therefore, $v_c$ is determined by both $\Delta T_s$ and $Pe $ in the packed-bed case, suggesting all the packed-bed tests operate in the convection-limited regime. Unlike the single-grain case, no linear correlation between $\Delta T_s$ and $v_c$ appears and the desublimation-limited regime is not identified. This difference stems from the presence of multiple grains in the packed bed, which lead to the insufficient $\textrm {CO}_2$ supply and the convection-limited mechanism.

Figure 14.

Analyses of $\textrm {CO}_2$ capture performance in packed-bed tests with subcooling degrees $\Delta T_s \in [0.049, 0.389]$ and Péclet numbers $Pe = 15.57, 31.14, 46.70, 62.27, 77.84$. (a) The maximum volume fraction of solid $\textrm {CO}_2$ ($\phi _{cm}$) captured by the bed. (b) The operating time for $\textrm {CO}_2$ desublimation and sublimation ($t_e$). (c) The $\textrm {CO}_2$ capture capacity loss due to the time delay between the maximum and saturation points ($\eta _d$). (d) The $\textrm {CO}_2$ capture rate ($v_c$).

Figure 14(c) displays results of $\eta _d$, a parameter absent in the single-grain case. An ascending $\Delta T_s$ is found to yield a consistent drop in $\eta _d$ (or diminished $\textrm {CO}_2$ capture capacity loss). For instance, when $Pe $ is set at 15.57, a relatively small $\Delta T_s \approx 0.09$ results in a $\eta _d$ approaching 100 %, and the CCC system ceases to function. As $\Delta T_s$ grows to a significant level $\Delta T_s > 0.14$, however, a noticeable decrease in $\eta _d$ to approximately 10 % becomes evident. It is because the higher $\Delta T_s$ brings about the stronger desublimation intensity and subsequently the smaller $\eta _d$. To prevent the weak desublimation and the significant $\eta _d$, $\Delta T_s$ needs to be maintained above a threshold $\Delta T_{sc}$ at a given $Pe $. A critical value of $\eta _{dc}=0.2$ is set in this study to determine the value of $\Delta T_{sc}$ at each $Pe $ (e.g. $\Delta T_{sc}\approx 0.15$ at $Pe = 15.57$). Small-$\Delta T_s$ tests (i.e. $\Delta T_s < \Delta T_{sc}$) featuring $\eta _{d}>0.2$, are categorized into the desublimation-limited regime. In addition, the growing $Pe $ amplifies the convective gas flow, yielding the severe $\eta _d$ and the increased $\Delta T_{sc}$. By combining results of $v_c$ and $\eta _d$, both the convection-limited and desublimation-limited regimes are identified.

As mentioned above, the increasing $Pe $ accelerates the $\textrm {CO}_2$ capture rate ($v_c$) but deteriorates the capture capacity ($\eta _d$). To clarify effects of $Pe $ on the CCC system, figure 15 presents metrics ($\phi _{cm}$, $t_e$, $\eta _d$, $v_c$) for tests with $\Delta T_s = 0.185$ and $Pe \in [6.23, 77.84]$. The ascending $Pe $ (or intensified convection) results in the decreasing $\phi _{cm}$ and $t_e$ (figure 15a,b). As for simulated $v_c$ values, they fit well with the correlation obtained from figure 14(d), namely, $v_c=(0.02Pe -0.19)\ln (\Delta T_s-0.024)+(0.083Pe -0.37)$ at $\Delta T_s = 0.185$. In contrast to the single-grain case, $v_c$ grows continuously with $Pe $ because the multiple cold grains are sufficient to capture $\textrm {CO}_2$. According to results of $v_c$, all tests are not constrained by $\Delta T_s$ and fall into the convection-limited regime. On the other side, $\eta _d$ in figure 15(c) exhibits two distinct stages with the ascending $Pe $: $\eta _d$ initially remains almost unchanged at around 10 % but it starts to increase sharply after a critical value of $Pe _c\approx 39$ (at $\Delta T_s = 0.185$). The intensified convection induces a shift from the convection-limited regime to the desublimation-limited one.

Figure 15.

Analyses of $\textrm {CO}_2$ capture performance in packed-bed tests with the subcooling degree $\Delta T_s = 0.185$ and the Péclet numbers $Pe \in [6.23, 77.84]$. (a) The maximum volume fraction of solid $\textrm {CO}_2$ ($\phi _{cm}$) captured by the bed. (b) The operating time for $\textrm {CO}_2$ desublimation and sublimation ($t_e$). (c) The $\textrm {CO}_2$ capacity loss due to the time delay between the maximum point and the saturation point ($\eta _d$). (d) The $\textrm {CO}_2$ capture rate ($v_c$).

From results in figures 14 and 15, either a small $\Delta T_s$ or a large $Pe $ is found to cause the desublimation-limited regime with remarkable $\textrm {CO}_2$ capture capacity loss. To show the $\textrm {CO}_2$ desublimation and sublimation properties under these two conditions, results for two tests with ($\Delta T_s = 0.006$, $Pe = 15.57$) and ($\Delta T_s = 0.185$, $Pe = 77.84$) are provided in figure 16, including distributions of solid $\textrm {CO}_2$, temperature ($T$), $\textrm {CO}_2$ mass fraction ($Y$) and their vertically averaged profiles. On the one hand, in the small-$\Delta T_s$ test ($\Delta T_s = 0.006$, $Pe = 15.57$), the desublimation rate is slow. Both the distribution of $Y$ and the profile of $\bar {Y}_x$ support that the injected $\textrm {CO}_2$ exceeds the desublimation rate and $\textrm {CO}_2$ breaks through the bed early. On the other hand, in the large-$Pe $ test ($\Delta T_s = 0.185$, $Pe = 77.84$), the abundant $\textrm {CO}_2$ is injected into the bed, and thus, the desublimation rate is insufficient to fully capture the injected $\textrm {CO}_2$. In these two tests, either the slow desublimation rate or the fast gas injection (or convection) make the CCC system operate in the desublimation-limited regime. Under this regime, the system reaches the saturation point too early and the cold grains are not efficiently heated up.

Figure 16.

The $\textrm {CO}_2$ desublimation and sublimation properties in a packed bed. Contours of solid $\textrm {CO}_2$, temperature ($T$), $\textrm {CO}_2$ mass fraction ($Y$) and the corresponding vertically averaged profiles in (a) the test with the subcooling degree $\Delta T_s = 0.006$ and the Péclet number $Pe = 15.57$ at $t=6.20\,\textrm {s}$, and (b) the test with the subcooling degree $\Delta T_s = 0.185$ and the Péclet number $Pe = 77.84$ at $t=8.67\,\textrm {s}$.

To comprehensively illustrate effects of $\Delta T_s$ and $Pe $ on the $\textrm {CO}_2$ capture performance of CCC, values of $\eta _d$ and $v_c$ from simulations are plotted in the $\Delta T_s$–$Pe $ space (figure 17). For comparison, two grey surfaces are included for the critical value $\eta _{dc}=0.2$ and the calculated $v_{cr}$ from the correlation $v_c=(0.02Pe -0.19)\ln (\Delta T_s-0.024)+(0.083Pe -0.37)$. The agreement between $v_c$ and $v_{cr}$ across all tests indicates the dominance of the convection-limited regime. Moreover, through a comparison between $\eta _d$ and $\eta _{dc}$, threshold values ($\Delta T_{sc}$, $Pe _c$) are obtained to delimit the domain for the desublimation-limited regime. The joint analysis of $v_c$ and $\eta _d$ produces a regime diagram in figure 18, showing distributions of the convection-limited (I) and desublimation-limited (II) regimes. The regime diagram suggests to control the gas feed rate below $Pe _c$ and raise the subcooling degree of grains beyond $\Delta T_{sc}$. This ensures that the CCC system operates within the convection-limited regime rather than the desublimation-limited one. Otherwise, the severe $\textrm {CO}_2$ capture capacity loss, coupled with the limited desublimation, will degrade the performance of CCC. Within the suggested parameter range, a large $Pe $ contributes to increasing $v_c$ while affecting $\eta _d$ slightly, hence, the improved $\textrm {CO}_2$ capture performance. Besides, as $\Delta T_s$ increases, the CCC system becomes more efficient in capturing $\textrm {CO}_2$ with larger $v_c$ and smaller $\eta _d$. However, considering the diminished improvement in the $\textrm {CO}_2$ capture performance and the escalated requirement for cooling duty, the continued rise in $\Delta T_s$ after $\Delta T_{sc}$ should be implemented with care.

Figure 17.

Analyses of (a) the $\textrm {CO}_2$ capacity loss ($\eta _d$) due to the time delay between the maximum point and the saturation point and (b) the $\textrm {CO}_2$ capture rate ($v_c$) in packed-bed tests with subcooling degrees $\Delta T_s \in [0.049, 0.389]$ and Péclet numbers $Pe \in [15.57, 77.84]$. Grey surfaces represent the threshold $\eta _{dc}=0.2$ and the correlation $v_c=(0.02Pe -0.19)\ln (\Delta T_s-0.024)+(0.083Pe -0.37)$. The white solid line shows the boundary between the convection-limited (I) and desublimation-limited (II) regimes.

Figure 18.

Analyses of $\textrm {CO}_2$ capture performance in packed-bed tests with subcooling degrees $\Delta T_s \in [0.049, 0.389]$ and Péclet numbers $Pe \in [15.57, 77.84]$. Simulation data points are plotted against $\Delta T_s$ and $Pe $. The grey dashed lines divide the plane into the convection-limited (I) and desublimation-limited (II) regimes.

The comparison between single-grain and packed-bed cases reveals their unique regime distributions and optimal operations. The desublimation-limited regime, featuring rapid $\textrm {CO}_2$ capture rate $v_c$, is more suited for the single-grain configuration. However, in a packed bed, the CCC system operates more efficiently in the convection-limited regime, despite the high $v_c$ in the desublimation-limited regime. This arises from the fact that the packed-bed case considers the potential loss in $\textrm {CO}_2$ capture capacity ($\eta _d$), and the limited desublimation significantly boosts $\eta _d$. On the other hand, the regime diagram for the packed-bed scenario (figure 18) illustrates an extended distribution of the convection-limited regime, when compared with the diagram for the single-grain situation (figure 10). This is attributed to the collective influence of multiple grains. These grains offer ample surface area and low temperature space, which necessitate the substantial convective $\textrm {CO}_2$ supply to sustain $\textrm {CO}_2$ desublimation. Consequently, a high $Pe _c$ at each $\Delta T_s$ is detected in the regime diagram for the packed-bed case, forming a broad range of the convection-limited regime.

4.5. Effects of bed structure in the packed-bed case

In the above simulations only one porosity of the packed bed is considered as $\psi =0.64$. In practical applications, however, $\psi$ varies among different packed beds by adjusting the packing grain number. Attention is therefore turned to the influence of bed porosity on the $\textrm {CO}_2$ capture performance of CCC. As constructed in Appendix D, a set of bed structures with porosities $\psi \in [0.53, 0.71]$ are utilized. A test with $\Delta T_s = 0.185$ and $Pe = 15.57$ is simulated in these cryogenic beds. The obtained four metrics ($\phi _{cm}$, $t_e$, $\eta _d$, $v_c$) are plotted in figure 19.

Figure 19.

Analyses of $\textrm {CO}_2$ capture performance in packed-bed tests with the subcooling degree ${\Delta T_s = 0.185}$, Péclet number $Pe = 15.57$ and porosities $\psi \in [0.53, 0.71]$. (a) The maximum volume fraction of solid $\textrm {CO}_2$ ($\phi _{cm}$) captured by the bed. (b) The operating time for $\textrm {CO}_2$ desublimation and sublimation ($t_e$). (c) The $\textrm {CO}_2$ capacity loss due to the time delay between the maximum point and the saturation point ($\eta _d$). (d) The $\textrm {CO}_2$ capture rate ($v_c$).

As $\psi$ increases, there is a corresponding reduction in both the solid $\textrm {CO}_2$ captured ($\phi _{cm}$) and the operating time ($t_e$). The primary explanation for this trend is that, in a bed with large $\psi$, the fluid mobility is enhanced and the gas stream is accelerated. Similar to a large-$Pe $ test (figure 15), the rapid gas stream leads to the drop in both $\phi _{cm}$ and $t_e$. In the meantime, while comparing to the growing-$Pe $ scenario in figure 15, two distinct findings arise from figure 19. First, with the ascending $\psi$, the $\textrm {CO}_2$ capture rate ($v_c$) increases at first but changes to decrease after a critical value $\psi _c \approx 0.61$. Different from the continuously growing $v_c$ in figure 15(c), the non-monotonic variations in $v_c$ here are driven by the two competing factors, the optimized flow channels and the diminished cold grain volume. Specifically, in the small-$\psi$ range (i.e. $\psi <\psi _c$), the slow gas stream constrains the $\textrm {CO}_2$ capture rate $v_c$, and the increased $\psi$ accelerates the flue gas flow and $v_c$. In the large-$\psi$ range (i.e. $\psi >\psi _c$), however, the gas stream has been sufficiently enhanced and the limited cold grains turn to dominate the $\textrm {CO}_2$ capture performance of CCC. The continuous rise in $\psi$ diminishes the available cold grains for capturing $\textrm {CO}_2$ and thereby decelerates $v_c$. The second distinct observation from figure 15 is that the $\textrm {CO}_2$ capture capacity loss ($\eta _d$) changes non-monotonically with $\psi$ and remains within a relatively small level as $\eta _d<15\,\%$. This is because the injected flue gas is fixed at a given $Pe $ and the capacity loss is limited to a certain range. These results help to draw a conclusion that a moderate bed porosity $\psi _c \approx 0.61$ is favourable for the optimal $\textrm {CO}_2$ capture performance of CCC. Compared with $Pe $ and $\Delta T_s$, $\psi$ exerts marginal impacts on the $\textrm {CO}_2$ capture performance. Furthermore, effects of $Pe $ and $\Delta T_s$ are examined at a porosity 0.64 in this study, which aligns with the recommended $\phi _c$. Given these considerations, explorations into impacts of $Pe $ and $\Delta T_s$ at other porosity values are deemed unnecessary for the scope of this study.

Overall, for a wide range of operating parameters (i.e. gas velocity, initial bed temperature and bed structures), $\textrm {CO}_2$ desublimation and sublimation during CCC have been investigated both on a single packing grain and in a packed bed. The general $\textrm {CO}_2$ desublimation and sublimation properties during the capture and recovery of CCC are successfully reproduced, which corroborate experimental observations (Tuinier et al. Reference Tuinier, van Sint Annaland, Kramer and Kuipers2010; Ali et al. Reference Ali, Maqsood, Syahera, Shariff and Ganguly2014). Furthermore, the quantitative analyses and parametric studies shed light on the underlying physics of CCC and also suggest optimal operating conditions.