The energy-minimization multi-scale (EMMS) model has succeeded in describing the mesoscale structures in gas–solid fluidization by introducing a stability condition to complete the model, viz., the mass specific energy consumption for suspending and transporting the solids, N_st tends to be a minimum. However, direct verification of this condition is very limited so far. This work, therefore, monitors the evolution of N_st in particle-resolved direct numerical simulation (PR-DNS) of a small-scale fluidized bed. Compared with the earlier simulation study on this issue based on pseudo-particle modeling (PPM), this work is more comprehensive thanks to the higher computational efficiency of PR-DNS. As a result, the previously neglected acceleration effect is now included in the force balance and the fluidization system is simulated under different initial distributions. The simulation and analysis demonstrate that, the stability condition is satisfied in all the cases, which further verifies the EMMS model. With PR-DNS as a powerful tool, deeper understanding of the EMMS model and gas–solid fluidization can be achieved in the future.

能量最小化多尺度（EMMS）模型通过引入稳定性条件来完善模型，成功地描述了气固流化中的中尺度结构，即悬浮和运输固体的质量比能耗N _st往往是最小的。但是，到目前为止，对该条件的直接验证非常有限。因此，这项工作监测了N _st的演变。小型流化床的粒子解析直接数值模拟（PR-DNS）中的应用。与早期基于伪粒子建模（PPM）的仿真研究相比，由于PR-DNS的计算效率更高，这项工作更加全面。结果，以前忽略的加速效果现在包括在力平衡中，并且在不同的初始分布下模拟了流化系统。仿真分析表明，在所有情况下都满足稳定条件，从而进一步验证了EMMS模型。使用PR-DNS作为强大的工具，将来可以对EMMS模型和气固流化技术有更深入的了解。