The governing equations of gas motion including the continuity and momentum equations obtained from the mean Navier-Stokes equation are expressed by [21]:(1) ∂(εgρg)∂t+∇⋅(εgρgug)=0 (2) ∂(εgρgug)∂t+∇⋅(εgρgugug)=−εg∇p−1Vcell∑n=1NpVpnβ1−εg(ug−vpn)−∇⋅(εgτg)+εgρgg where ε_g is the void fraction, -; ρ_g is the gas density, kg/m³; u_g is the gas velocity, m/s; g is the gravity, m/s²; p is the gas pressure, Pa; τ_g is the gas viscous stress tensor, kg/(m∙s²); V_cell is the computing cell volume, m³; N_p is the number of particles in a computing cell, -; V_p is the particle volume, m³; v_p is the particle velocity, m/s; β is the interphase momentum exchange coefficient between gas and solid, kg/(m³∙s), which is expressed by the Gidaspow correlation [22] as:(3) β={34Cdεg(1−εg)dpρg|ug−vp|εg−2.65, εg≥0.8150(1−εg)2εgμgdp2+1.75(1−εg)ρgdp|ug−vp|, εg<0.8 (4) Cd={24Rep(1+0.15Rep0.687),Rep<10000.44,Rep≥1000 (5) Rep=εgρg|ug−vp|dpμg where d_p is the particle diameter, m; C_d is the drag coefficient, -; Re_p is the particle Reynolds number,-.

The particle motion in discrete phase (DEM) follows Newton’s second law, and the translational and rotational equations [23,24] are given by:(6) mpdpvpdt=−Vp∇p+Vpβ1−εg(ug−vp)+mpg+Fc+Flb (7) Ipdωpdt=Tp=rp×(Fc,t+Flb,t)+Tr where m_p is the particle mass, kg; F_c is the contact force, N; F_lb is the liquid bridge force, N; ω_p is the particle rotational velocity, rad/s; I_p is the particle moment of inertia, kg∙m²; T_p is the torque, N∙m; r_p is the position of particle center, m; T_r is the rolling friction torque, N∙m. The subscripts n and t are the normal and tangential parameters, respectively.

In this paper, the linear-spring-damping (LSD) soft-sphere model is applied to calculate the contact force, and the expressions of the normal and tangential contact forces are given by:(8) Fc,n=−knδn−ηnvr.n (9) Fc,t={−ktδt−ηtvr,t⁡|Fc,t|≤μs|Fc,n|−μs|Fc,n|vr,t|vr,t|⁡|Fc,t|>μs|Fc,n| where δ is the elastic deformation, m; μ_s is the sliding friction coefficient, -; v_r is the relative collision velocity, m/s; k is the spring stiffness, N/m; η is the damping coefficient, N∙s/m.

The liquid bridge force consists of the static liquid bridge force (capillary force) and the dynamic liquid bridge force (viscous force), which collaborate to regulate the motion of wet particles. The capillary force is expressed as follows:(10) Fcap=πγR[exp⁡(AH/AHR+BR+B)+C] where R is the particle radius, m; γ is the surface tension coefficient, N/m; H is the distance between a particle and the wall or between two particles, m; and the parameters A, B and C are calculated as follows [22]:

Between two particles(11) A=−1.1V^lb−0.53 (12) B=(−0.34ln⁡V^lb−0.96)θ2−0.019ln⁡V^lb+0.48 (13) C=0.0042ln⁡V^lb+0.78

Between a particle and the wall(14) A=−1.9V^lb−0.51 (15) B=(−0.016ln⁡V^lb−0.76)θ2−0.12ln⁡V^lb+1.2 (16) C=0.013ln⁡V^lb+0.18 where V^lb is the dimensionless liquid bridge volume; θ is the contact angle, rad.

The expressions of tangential and normal viscous forces [25] are, respectively:(17) Fv,t=6πμlbRvr,t(815lnRH+0.9588) (18) Fv,n=6πμlbRvr,nRH where μ_lb is the liquid viscosity, Pa∙s.

When the liquid bridge breaks, the distance between particles is called the critical rupture distance. The critical rupture distance H_cr is expressed as:(19) Hcr=(1+0.5θ)Vlb∗1/3

The heat transfer energy equations for gas and particles are as follows [26–28]:(20) ∂∂t(εgρgCp,gTg)+∇⋅(εgρgCp,gTgug)=−∇⋅((1−1−εg)λg∇(Tg))+1Vcell∑n=1Nphg,pApn(Tg−Tpn)+Sq (21) mpCpdTpdt=Qp,p+Qg,p+Qp,w+Qeva=Tcπrc2tc−1/2(ρp1Cp1λp1)−1/2+(ρp2Cp2λp2)−1/2+1Vcell∑n=1Nphg,pApn(Tg−Tpn)+hp,wAp,w(Tw−Tp)tcw+LΔml where C_p,g is the gas specific heat capacitys, J/(kg∙K); T_g is the gas temperature, K; T_p is the particle temperature, K; A_p is the particle surface area, m²; λ_g is the thermal conductivity of gas, W/(m∙K); S_q is the source terms; Q_g,p is the thermal exchange between the gas and solid phases, J/s; Q_p,p is the heat transfer between particles, J/s; Q_p,w is the heat transfer between the particle and a wall, J/s; Q_eva is the heat taken away by evaporation, J/s; T_c is the relative temperature between particles, K; r_c is the contact surface radius, m; t_c is the particle contact time, s; λ_p is particle conduction coefficient, W/(m∙K); ρ_p is particle density, kg/m³; C_p is the specific heat capacity for the particle, J/(kg∙K); h_p,w is the heat transfer coefficient between particle and wall, W/(m²∙K); A_p,w is the contact area between particle and wall, m²; T_w is the wall temperature, K; t_cw is the contact time when the particle collides with the wall, s; L is the latent heat of liquid evaporation, J/kg; m_l is the evaporated liquid mass; h_g,p represents the convective heat transfer coefficient between the gas and the particle, which the empirical correlation is given by Gunn [29]:(22) hg,p=[(7−10εg+5εg2)(1+0.7Rep0.2Pr0.33)+(1.33−2.4εg+1.2εg2)Rep0.7(μgCp,g/μgCp,gλgλg)0.33]λgdp where Nu and Pr are the particle nusselt and Prandtl number, respectively.

Fig. 5 illustrates the correlation between the latent heat of liquid vaporization and the particle temperature in a wet particle system under u_g = 0.5 m/s, T_g = 413.15 K, and G_p = 70 kg/h. It can be obtained that as the bed height increases (from particle inlet to outlet), the latent heat of liquid evaporation decreases, and the particle temperature increases. This also indicates that the volume of liquid on the particle surface decreases along with the particles downward direction, resulting in a gradual reduction in the impact of the liquid on the flow characteristics of the particles. The reason may be that as the bed height increases, the wet particles are affected by the gas temperature for a longer period, leading to reduced liquid on the particle surface. Moreover, as the moisture content increases, the particle temperature also increases, while the latent heat of liquid evaporation decreases. This is mainly due to the fact that the higher the moisture content, the closer the distance between particles, and the greater the number of liquid bridges, leading to a larger neck radius of the liquid bridge and a more pronounced thermal conductivity (indirect thermal conductivity) of the liquid bridge [31]. In a word, the latent heat of liquid evaporation decreases as the particle temperature increases.

Liu et al. [20] found that flooding is a gradual process usually accompanied by drastic changes in bed kinetic parameters. In the dry particle system, flooding occurs when the gas velocity exceeds the critical value, causing numerous particles moving out of the bed. Furthermore, when the particle mass flow rate exceeds the critical value, a bridging phenomenon occurs that hinders the upward flow of gas, leading to flooding. In other words, flooding results in the blockage of the inlet and a sudden change in the bed’s porosity to 1.

This study pays special attention to the influence of moisture content on flood phenomena. Fig. 6 shows the correlation between moisture content and flooding under u_g = 0.5 m/s, T_g = 413.15 K, and G_p = 70 kg/h. It can be obtained that the solid holdup of wet particles is primarily controlled by liquid bridging force. Subsequently, because the wet particles easily adhere to the wall due to the liquid bridging force effect on them, there is increased friction between the particles and the wall, leading to a relatively long drying time of the particles in the drying unit. However, the liquid bridge force and the bed mean solid holdup experiences a sudden decrease at time t = 2.7 s when the moisture content is 0.3%, indicating the occurrence of flooding.

Figs. 7 and 8 show the instantaneous velocity contour of particles and gas under the conditions of u_g = 0.5 m/s, T_g = 413.15 K, G_p = 70 kg/h, and t = 6 s. It can be observed that when the moisture content is 0% and 0.2%, an increase in moisture content results in higher particle agglomeration number and gas velocity, and a further strengthening of the heterogeneity of gas-solid flow structure. When the moisture content reaches 0.3%, some particles will agglomerate at the top of the drying unit, resulting in a blockage of the top of the drying unit and hindering the inflow of wet particles. Moreover, the particles have a high initial velocity and the bed width is narrow, while the gas flows upward against gravity, leading to the bias phenomenon and a localized high-speed gas region.

The flow characteristics of particles are studied under different particle mass flow conditions, as shown in Fig. 9, when u_g and T_g are 0.5 m/s and 413.15 K, respectively.

Fig. 9a shows the variation of mean particle velocity with particle mass flow rate. When the moisture content is 0%, the mean particle velocity increases with the particle mass flow rate’s increasing. The reason is that particles’s number in the drying unit increases as the particle mass flow rate increases, leading to an increase in particle gravity. When the moisture content is 0.1% and 0.2%, the mean particle velocity initially increases and then decreases with the particle mass flow rate’s increasing (the inflection point is at 50 kg/h). However, when the moisture content is 0.3%, the mean particle velocity decreases with the particle mass flow rate’s increasing. Moreover, it is worth noting that under the same particle mass flow rate, the mean particle velocity decreases with the moisture content increasing, the possible cause is that the contact rate, friction force, and liquid bridge force increase as the moisture content increases.

Fig. 9b shows the variation of bed mean solid holdup with particle mass flow rate under different moisture contents. When the moisture content is not larger than 0.2%, it is positively correlated with the bed mean solid holdup, at the same time, there is a positive correlation between the bed mean solid holdup and the particle mass flow rate. However, when the moisture content is at 0.3%, the bed mean solid holdup initially increases and then suddenly drops to a very low level (the inflection point is at 50 kg/h), the possible cause is that under higher moisture content and particle mass flow rate, the drying unit experiences flooding, resulting in a significant decrease in the number of particles in the unit.

Fig. 9c shows the variation of particles’MRT with particle mass flow rate, that is the MRT of particles increases as the particle mass flow rate increases when the moisture content is under 0.2%. Once the moisture content exceeds 0.3%, the MRT of particles initially increases and then decreases with the increase of particle mass flow rate. Furthermore, at V_lb ≤ 0.3% and G_p ≤ 50 kg/h,, the MRT of particles increases with the increase of moisture content since higher moisture content leads to a decrease in the distance between particles and an increase in the liquid bridge force. Consequently, the drying space is limited resulting in a larger friction force and relatively slower particle velocity.

The flow characteristics of particles with particle mass flow rate of 50 kg/h and superficial gas temperature of 413.15 K under various superficial gas velocities are given in Fig. 10.

It can be obtained in Figs. 10a and 10c that the gas-solid drag force increases since the superficial gas velocity increases which results in a decrease in the mean particle velocity and a relative increase in the MRT of particles. At the same time, the liquid bridge force increases with increasing moisture content, which leads to the contact frequency, and friction force between a particle and a wall increase, further a decrease in mean particle velocity and an increase in MRT of particles. Fig. 10b illustrates the different bed mean solid holdup changing tendency under variational moisture content. That is, in the dry particle system the bed mean solid holdup increases with an increase in superficial gas velocity, while in the wet particle system, the bed mean solid holdup initially increases and then decreases as the superficial gas velocity increases.

Moreover, when the superficial gas velocity is larger than 2.5 m/s and the moisture content larger than 0.1%, there occur fluctuations in the mean particle velocity due to the flooding of the drying unit inlet and a decrease in the number of particles in the unit.

The flow characteristics of particles under different superficial gas temperature with particle mass rate of 50 kg/h and superficial gas velocity of 0.5 m/s is shown in Fig. 11.

It can be obtained in Figs. 11a and 11c that as the superficial gas temperature increases, the mean particle velocity also increases, while the MRT of particles decreases overall. This is because higher superficial gas temperature and lower gas density lead to an increase in convective heat transfer coefficient, a decrease in particle resistance and a reduction in the liquid bridge force between two particles or between a particle and the wall. However, as the moisture content increases, the heat taken away by evaporation also increases, leading to a decrease in gas temperature and a relative increase in gas density. Consequently, the mean particle velocity and MRT of particles change less.

According to Fig. 11b, the bed mean solid holdup initially decreases and then increases with the increase in superficial gas temperature when the moisture content is low (V_lb ≤ 0.1%). However, when the moisture content is high (V_lb = 0.3%), the bed mean solid holdup initially increases, then decreases, and then increases again with the rise in superficial gas temperature, indicating that the gas temperature also influences the particle flow. The reason is that at low superficial gas temperature and moisture content, the liquid bridge force decreases as the gas temperature increases, which leads to a smaller particle agglomeration number and reduced updraft resistance, resulting in a decrease in bed mean solid holdup. Conversely, at high superficial gas temperature, a larger evaporation rate of water on the particle surface causes water vapor to be unable to be discharged quickly from the drying unit, resulting in the formation of a large agglomeration number at the inlet of the drying unit (with a size smaller than the drying space size), and finally, an increase in the number of particles.

In order to compare the simulation results from the previous study and discuss the radial (or x-direction) distribution characteristics of wet particles under different bed height conditions, a series of chosen working conditions are presented in Table 3.

Fig. 12 shows the effect of multi-factor conditions on the radial distribution of local particle velocity for d_p = 2.5 mm. It can be obtained that the local particle velocity increases with an increase in particle mass flow rate and superficial gas temperature at different bed heights, while it decreases with the increase in superficial gas velocity. As the bed height (from particle inlet to outlet) increases, the mean particle velocity gradually increases. Under all operating conditions, the local particle velocity decreases with an increase in moisture content. The particle flow structure exhibits local non-uniformity, leading to a skewed flow and a higher concentration of particles in the near-wall region. Moreover, the change in local particle velocity is more pronounced at the outlet, this is because the superficial gas temperature increases along the bed height, resulting in a decrease in the moisture content of wet particles. In conclusion, within a certain range, increasing the particle diameter can reduce the occurrence probability of flooding phenomenon, as shown in Figs. 9a, 10a, and 11a.

Fig. 13 shows the effect of multi-factor conditions on the local solid holdup radial distribution for d_p = 2.5 mm. It can be obtained that the local solid holdup increases with the increase of particle mass flow rate, superficial gas velocity, and moisture content at different bed heights. In most operating conditions, the radial profile of solids holdup exhibits heterogeneity in the upper section of the drying unit. At all bed heights, the solid holdup is higher in the near-wall region compared to the near-fluid region. This phenomenon occurs because the particles tend to align near the wall, where the gas velocity is close to zero. Furthermore, the solid holdup is highest at the particle inlet of the drying unit and decreases along the bed height (from particle inlet to outlet).

In a word, when the superficial gas velocity and particle mass flow exceed a certain critical value, flooding occurs. However, in the case of high moisture content, the probability of flooding can be reduced by decreasing the particle mass flow rate, the superficial gas velocity or increasing particle diameter.

Response surface methodology (RSM) is an empirical modeling approach in which data is obtained through finite order orthogonal experiments, and the response variables fitted can globally approximate the actual response values, and the optimal set of parameters is determined to solve the multivariate optimization problem [32]. The Box-Behnken design (BBD) is a faster and more efficient method that requires the fewest number of experiments, and can optimize the analysis conditions and reduce computational costs.

The combination of RSM and the BBD is advantageous for predicting the flow characteristics of wet particles and reducing the likelihood of flooding. In addition, the approach is divided into three steps: the experimental design, the regression models, and the optimization [33]. The regression coefficient R² is used to evaluate the performance of the model, and the closer the R² value is to 1.0, the more accurate the response surface model is.

In this work, the RSM-BBD method is used to generate a design consisting of 29 runs. The RSM-BBD is used to analyze the effects of particle mass flow rate (G_p, kg/h), moisture content (V_lb, %), superficial gas velocity (u_g, m/s), and superficial gas temperature (T_g, K) on the mean particle velocity (v_p, m/s) and bed mean solid holdup (ε_p,bed, %). The recommended range for the operating condition parameters is as follows: 30 kg/h ≤ G_p ≤ 50 kg/h, 0 m/s ≤ u_g ≤ 1 m/s, 388.15 K ≤ T_g ≤ 428.15 K, and 0% ≤ V_lb ≤ 0.3%.

The RSM-BBD design and simulated results are given in Table 4.