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Cryogenic carbon capture (CCC) is an innovative technology to desublimate 
$\text {CO}_2$ out of industrial flue gases. A comprehensive understanding of 
$\text {CO}_2$ desublimation and sublimation is essential for widespread application of CCC, which is highly challenging due to the complex physics behind. In this work, a lattice Boltzmann (LB) model is proposed to study 
$\text {CO}_2$ desublimation and sublimation for different operating conditions, including the bed temperature (subcooling degree 
$\Delta T_s$), gas feed rate (Péclet number 
$Pe $) and bed porosity (
$\psi$). The 
$\text {CO}_2$ desublimation and sublimation properties are reproduced. Interactions between convective 
$\text {CO}_2$ supply and desublimation/sublimation intensity are analysed. In the single-grain case, 
$Pe $ is suggested to exceed a critical value 
$Pe _c$ at each 
$\Delta T_s$ to avoid the convection-limited regime. Beyond 
$Pe _c$, the 
$\text {CO}_2$ capture rate (
$v_c$) grows monotonically with 
$\Delta T_s$, indicating a desublimation-limited regime. In the packed bed case, multiple grains render the convective 
$\text {CO}_2$ supply insufficient and make CCC operate under the convection-limited mechanism. Besides, in small-
$\Delta T_s$ and high-
$Pe $ tests, 
$\text {CO}_2$ desublimation becomes insufficient compared with convective 
$\text {CO}_2$ supply, thus introducing the desublimation-limited regime with severe 
$\text {CO}_2$ capture capacity loss (
$\eta _d$). Moreover, large 
$\psi$ enhances gas mobility while decreasing cold grain volume. A moderate porosity 
$\psi _c$ is recommended for improving the 
$\text {CO}_2$ capture performance. By analysing 
$v_c$ and 
$\eta _d$, regime diagrams are proposed in 
$\Delta T_s$–
$Pe $ space to show distributions of convection-limited and desublimation-limited regimes, thus suggesting optimal conditions for efficient 
$\text {CO}_2$ capture. This work develops a viable LB model to examine CCC under extensive operating conditions, contributing to facilitating its application.
Cryogenic carbon capture (CCC) can preferentially desublimate 
$\text {CO}_2$ out of the flue gas. A widespread application of CCC requires a comprehensive understanding of 
$\text {CO}_2$ desublimation properties. This is, however, highly challenging due to the multiphysics behind it. This study proposes a lattice Boltzmann (LB) model to study 
$\text {CO}_2$ desublimation on a cooled cylinder surface during CCC. In two-dimensional (2-D) simulations, various 
$\text {CO}_2$ desublimation and capture behaviours are produced in response to different operation conditions, namely, gas velocity (Péclet number 
$\textit {Pe}$) and cylinder temperature (subcooling degree 
$\Delta T_{sub}$). As 
$\textit {Pe}$ increases or 
$\Delta T_{sub}$ decreases, the desublimation rate gradually becomes insufficient compared with the 
$\text {CO}_2$ supply via convection/diffusion. Correspondingly, the desublimated solid 
$\text {CO}_2$ layer (SCL) transforms from a loose (i.e. cluster-like, dendritic or incomplete) structure to a dense one. Four desublimation regimes are thus classified as diffusion-controlled, joint-controlled, convection-controlled and desublimation-controlled regimes. The joint-controlled regime shows quantitatively a desirable 
$\text {CO}_2$ capture performance: fast desublimation rate, high capture capacity, and full cylinder utilization. Regime distributions are summarized on a 
$\textit {Pe}$–
$\Delta T_{sub}$ space to determine operation parameters for the joint-controlled regime. Moreover, three-dimensional simulations demonstrate four similar desublimation regimes, verifying the reliability of 2-D results. Under regimes with loose SCLs, however, the desublimation process shows an improved 
$\text {CO}_2$ capture performance in three dimensions. This is attributed to the enhanced availability of gas–solid interface and flow paths. This work develops a reliable LB model to study 
$\text {CO}_2$ desublimation, which can facilitate applications of CCC for mitigating climate change.
