**Shicheng Wang, Chenyi Xu, Wei Liu and Zhichun Liu \***

School of Energy and Power engineering, Huazhong University of Science and Technology, Wuhan 430074, China; shichengwang@hust.edu.cn (S.W.); cy\_xu@hust.edu.cn (C.X.); w\_liu@hust.edu.cn (W.L.)

**\*** Correspondence: zcliu@hust.edu.cn

Received: 17 December 2018; Accepted: 22 January 2019; Published: 28 January 2019

**Abstract:** Packed beds are widely used in industries and it is of great significance to enhance the heat transfer between gas and solid states inside the bed. In this paper, numerical simulation method is adopted to investigate the heat transfer principle in the bed at particle scale, and to develop the direct enhanced heat transfer methods in packed beds. The gas is treated as continuous phase and solved by Computational Fluid Dynamics (CFD), while the particles are treated as discrete phase and solved by the Discrete Element Method (DEM); taking entransy dissipation to evaluate the heat transfer process. Considering the overall performance and entransy dissipation, the results show that, compared with the uniform particle size distribution, radial distribution of multiparticle size can effectively improve the heat transfer performance because it optimizes the velocity and temperature field, reduces the equivalent thermal resistance of convection heat transfer process, and the temperature of outlet gas increases significantly, which indicates the heat quality of the gas has been greatly improved. The increase in distribution thickness obviously enhances heat transfer performance without reducing the equivalent thermal resistance in the bed. The result is of great importance for guiding practical engineering applications.

**Keywords:** Discrete Element Model; gas–solid flow; heat transfer enhancement; entransy dissipation; numerical simulation; optimization

#### **1. Introduction**

Packed beds are widely used in various industry process, such as catalytic reactors, high temperature gas-cooled nuclear reactors, absorption towers, and so on [1,2]. The structure of the packed reactor is simple and the efficiency is high, so it is the most commonly used reactor in industrial production and scientific research. However, the heat transfer coefficient of the packed bed is relatively low, which is very detrimental for high temperature reactors, such as nuclear reactors. Therefore it is very import to improve the heat transfer performance, cool the bed, and raise the outlet gas temperature of the packed bed reactors.

The flow structure inside the packed bed is complex, so flow maldistribution exists, especially when the tube-to-particle *dt/dp* ratio is small. The ratio *dt/dp* affects the properties near the wall region because the porosity is large here [3–6], and the changes of porosity and velocity may cover the core area of the bed, where the heat transfer mainly occurs [7,8]. The flow maldistribution is obvious with low tube-to-particle ratio (*dt/dp* < 15) and will seriously affect the heat transfer or reaction in the bed [9], which determines the design of the bed. When the bed is packed with uniform size particles, there exists a large void fraction near the wall region and it can be called wall effects [10]. Several studies aim to reveal the flow and heat transfer characteristics in the packed bed. However, the correlations of heat transfer in the packed bed with small value of *dt/dp* is hardly to satisfy all the packed beds [4,11]. There are no standard empirical correlations which can be applied to all the

range of tube-to-particle ratios. So, various effective parameters under steady state are proposed, including effective thermal conductivities [12], overall heat transfer coefficient, and effective transport parameters [13–15]. Recently, with the development of computer technology, the Computational Fluid Dynamics (CFD) method is adopted to save the time and economic cost. Nijemeisland [16] found that local heat transfer rates did not correlate statistically with the local flow field but related to large scale flow structures—the CFD method is adopted to study the velocity and temperature distribution. The method of composite packing can improve the heat transfer efficiency and heat flux of packed bed with low tube-to-particle ratio and restrain the wall effects [17], compared with the randomly packing. By studying the creeping, transition, and turbulent flow with tube-to-particle ratio *dt/dp* equals to 3 and 10, Reddy [11] concluded that the wall effects decrease as the ratio increases in the creeping and turbulent regimes. Many researches focus on the flow and heat transfer characteristics with low tube-to-particle ratio using CFD methods [18,19], some of them compared the effects of different computational models on the result. The CFD results show that the wall effects and flow maldistribution are obvious. However, the CFD modeling ignores the interaction among particles, which is important in dense gas–solid flow.

Theoretically, the flow in the packed bed includes particle motion, fluid flow, and the interactions between particles and fluid, between particles and particles, so particle scale study is necessary to get the information of particles and has been a research focus in the past decades [20]. The Computational Fluid Dynamics coupled with Discrete Element Method (CFD-DEM) approach has been fully developed and widely used in granular flows and fluidized beds [21,22]. The motion of discrete particles is solved by the Newton's second law, which is known as Discrete Element Method (DEM [23]), while the flow of continuum fluid is solved by the locally averaged Navier-Stokes equations, which is known as CFD. It combines the CFD for the continuum fluid and DEM for the discrete particles, and this method is able to capture the particle physics compared with CFD methods [21]. The coupling between fluids and particles is performed by Eulerian–Lagrangian framework for dense flow [24]. Some researchers apply this method to study the flow behavior of fluidized beds [24–26], and find that the simulation results agree well with experimental results, which means this method is reliable enough; the CFD-DEM methods also have some other applications in industrial [27–30]. However, in terms of packed beds, few studies apply CFD-DEM approach to research the flow and heat transfer characteristics and enhance the heat transfer inside the bed. J. Yang [17] used the discrete element model to generate the randomly packed bed, but not involved in the calculation for the flow and heat transfer. H. Wu [31] studied the thermal radiation of high temperature gas-cooled reactor using CFD-DEM approach.

In order to better understand the nature of the convective heat transfer, Guo el al. [32,33] proposed the field synergy principle, then further developed by Liu et al. [34,35], to guide the design of convective heat trnsfer process with higher heat transfer efficiency and lower flow resistance. With the development of the computation techonology, there are a tendency to design heat transfer unit or system based on the combination of the CFD and optimization algorithm [36–39]. In the current work, to better compare the performance of different parameter configurations, a new physical quantity entransy is introduced, which represents the heat transfer ability of an object [40], and the expression of entransy dissipation is derived from the entransy balance equation. It concluded that the entransy dissipation can be used to measure the irreversibility of the heating or cooling process [41]; moreover, the entransy dissipation extremum principle is proposed as a criterion to guide the optimization of heat transfer process [42]. However, this criterion is different from minimum entropy production principle. Chen et al. [43] pointed that the minimum entropy production principle should be adopted to minimize the usable energy dissipation, while the entransy dissipation extremum principle should be adopted to maximum the heat transfer ability. For heat transfer process only to heat or cool fluids, entransy dissipation is more suitable as a measure of irreversibility. Actually, entransy is an unconserved quantity during the heat transfer process, the entransy dissipation is inevitable, so based on the concept of entransy dissipation, the equivalent thermal resistance of the multidimensional

problem is defined, and the goal of the heat transfer optimization is to minimize the equivalent thermal resistance [42].

From all available literatures about packed beds, most focus on the uniform size particles with CFD methods, and aimed to reveal the flow and heat transfer characteristics. However, the effects of multisizes particle mixing and its distribution are still not clear, and few studies have optimized the heat transfer performance inside the bed. J. Yang [17] pointed that the method of composite packing of different particle size can restrain the wall effects. Therefore, based on the CFD-DEM methods, a full numerical simulation is carried out for packed beds with low tube-to-particle ratios in this work. When the wall effects are obvious, the fluid will flow away from the wall region and heat transfer in the center is poor, so the optimal objective is to restrain wall effects, reduce the porosity near the wall, and strengthen the heat transfer in the core area. Considering particle motion and heat conduction between particles, the effects of radial distribution of particle size and the distribution thickness on the heat transfer and fluids flow are discussed. Entransy dissipation is used as criteria to evaluate the performance of different parameter configurations. The result is of great significance to design the packed beds reactors and reduce the volume of beds, especially for high temperature gas-cooled reactors.

#### **2. Mathematical Model**

In the CFD-DEM coupling approach, the discrete element model is based on the so called soft sphere model, and the gas phase is modeled as a continuum. The force and motion of particles are tracked at particle-scale level, and the key assumption is that the time step is small enough that the disturbances propagation distance is no more than one particle, so the velocity and acceleration of each individual particle are constant in one time step, so the interaction between particles within each time step can be ignored, and, after the end of one time step, the information of the interaction between particles will be updated and will be the start of the next step [44]; the macroscopic behavior of particles clusters is the cumulative result of the particle-scale behavior. The interaction of solid particles on gas phase is considered as the source of mass, momentum and energy equations. In current work, the mass exchange between particle and gas is neglected. The model description is given below.
