Figure 2. Raw coal washability curve.
图 2.原煤洗净性曲线。

Figure 3. Particle size distribution.
图 3.粒度分布。

Cumulative floats curve ( $\beta$$\beta$beta\beta curve), Cumulative sinks curve ( $\theta$$\theta$theta\theta curve), Densimetric curve ( $\delta$$\delta$delta\delta curve) and Near-density curve ( $\epsilon$$\epsilon$epsi\varepsilon curve). The float-sink test was conducted on the raw coal, yielding the density composition and washability curve presented in Figure 2, respectively. Assuming a cleaned coal ash content of $12\mathrm{\%}$$12\mathrm{\%}$12%12 \% and a theoretical separation density of $1.75\mathrm{g}/{\mathrm{cm}}^3$$1.75\mathrm{g}/{\mathrm{cm}}^3$1.75g//cm^(3)1.75 \mathrm{~g} / \mathrm{cm}^{3}, after deducting low-density material ( $<1.50\mathrm{g}/{\mathrm{cm}}^3$$<1.50\mathrm{g}/{\mathrm{cm}}^3$< 1.50g//cm^(3)<1.50 \mathrm{~g} / \mathrm{cm}^{3} ), the calculated $\delta \pm 0.1$$\delta \pm 0.1$delta+-0.1\delta \pm 0.1 content was determined to be $16.14\mathrm{\%}$$16.14\mathrm{\%}$16.14%16.14 \%, indicating medium washability.
累积浮点曲线 （ $\beta$$\beta$beta\beta curve）、累积汇曲线 （ $\theta$$\theta$theta\theta curve）、密度曲线 （ $\delta$$\delta$delta\delta curve） 和近密度曲线 （ $\epsilon$$\epsilon$epsi\varepsilon curve）。对原煤进行了浮沉测试，分别得到图 2 所示的密度组成和可洗性曲线。假设清洁煤灰含量为 ， $12\mathrm{\%}$$12\mathrm{\%}$12%12 \% 理论分离密度为 $1.75\mathrm{g}/{\mathrm{cm}}^3$$1.75\mathrm{g}/{\mathrm{cm}}^3$1.75g//cm^(3)1.75 \mathrm{~g} / \mathrm{cm}^{3} ，减去低密度材料 （ $<1.50\mathrm{g}/{\mathrm{cm}}^3$$<1.50\mathrm{g}/{\mathrm{cm}}^3$< 1.50g//cm^(3)<1.50 \mathrm{~g} / \mathrm{cm}^{3} ） 后，计算 $\delta \pm 0.1$$\delta \pm 0.1$delta+-0.1\delta \pm 0.1 出的含量确定为 $16.14\mathrm{\%}$$16.14\mathrm{\%}$16.14%16.14 \% ，表明介质可清洗。

In this experiment, Geldart A magnetite powder particles were primarily chosen as dense medium. The primary particle size range of magnetite powder is $58-100\mu \mathrm{m}$$58-100\mu \mathrm{m}$58-100 mum58-100 \mu \mathrm{~m}, with an average particle size of $62\mu \mathrm{m}$$62\mu \mathrm{m}$62 mum62 \mu \mathrm{~m}. The true density (material density) was measured at 4600
在本实验中，主要选择 Geldart A 磁铁矿粉末颗粒作为致密介质。磁铁矿粉末的主要粒径范围是 $58-100\mu \mathrm{m}$$58-100\mu \mathrm{m}$58-100 mum58-100 \mu \mathrm{~m} ，平均粒径为 $62\mu \mathrm{m}$$62\mu \mathrm{m}$62 mum62 \mu \mathrm{~m} 。在 4600 处测量真实密度（材料密度）
$\mathrm{kg}/{\mathrm{m}}^3$$\mathrm{kg}/{\mathrm{m}}^3$kg//m^(3)\mathrm{kg} / \mathrm{m}^{3}, while the bulk density was determined to be $2160\mathrm{kg}/{\mathrm{m}}^3$$2160\mathrm{kg}/{\mathrm{m}}^3$2160kg//m^(3)2160 \mathrm{~kg} / \mathrm{m}^{3}. The particle size distribution is illustrated in Figure 3.
$\mathrm{kg}/{\mathrm{m}}^3$$\mathrm{kg}/{\mathrm{m}}^3$kg//m^(3)\mathrm{kg} / \mathrm{m}^{3} ，而体积密度被确定为 $2160\mathrm{kg}/{\mathrm{m}}^3$$2160\mathrm{kg}/{\mathrm{m}}^3$2160kg//m^(3)2160 \mathrm{~kg} / \mathrm{m}^{3} 。粒度分布如图 3 所示。

The pulsed fluidized bed is utilized for the segregation of coal particles, with a segregation duration of 5 minutes. Subsequently, the gas supply is abruptly terminated, causing a transition from a fluidized bed to a fixed bed configuration within the particle layer. This layer is divided into five distinct strata extending from the surface to the bottom of the bed. The particles from each layer are extracted and subjected to a screening process in order to obtain distinct coal products. The coal obtained from the first and second layers is considered as fine coal, while the remaining coal is categorized as gangue. By conducting laboratory analysis to determine the ash content of each layer, it becomes possible to calculate the standard deviation of ash separation using Equation (1). This metric characterizes the efficacy of density-based coal separation within dense phase pulsed fluidized beds; higher values indicate greater divergence between ash content and that found in raw coal samples, thereby signifying enhanced density-based separation effect.
脉冲流化床用于煤颗粒的分离，分离时间为 5 分钟。随后，气体供应突然终止，导致颗粒层内从流化床过渡到固定床配置。该层分为五个不同的地层，从地表延伸到床底。从每一层中提取颗粒并经过筛选过程，以获得不同的煤炭产品。从第一层和第二层获得的煤被认为是细煤，而剩余的煤被归类为脉石。通过进行实验室分析以确定每层的灰分含量，可以使用公式 （1） 计算灰分分离的标准偏差。该指标描述了在密相脉冲流化床中基于密度的煤分离的效率;较高的值表示灰分含量与原煤样品中的灰分含量之间的差异更大，因此表示基于密度的分离效果增强。

where $n$$n$nn represents the number of layers for stratified sampling, $A(i)$$A(i)$A(i)A(i) denotes the ash content of the i-th layer, and $\overline{A}$$\overline{A}$bar(A)\bar{A} signifies the weighted average of ash content across all layers.
其中 $n$$n$nn 表示分层采样的层数， $A(i)$$A(i)$A(i)A(i) 表示第 i 层的灰分含量， $\overline{A}$$\overline{A}$bar(A)\bar{A} 表示所有层的灰分含量的加权平均值。

The hydrodynamic characteristics of a single-component fluidized bed are relatively favorable, whereas the presence of particles with different sizes and densities in a bidisperse particle fluidized bed further complicates the system.
单组分流化床的流体动力学特性相对有利，而双分散颗粒流化床中存在不同尺寸和密度的颗粒使系统进一步复杂化。

To elucidate the movement of particles within this bed, it is advantageous to consider an equivalent system comprising BL particles (characterized by big size and low density) and SH particles (characterized by small size and high density), which can be effectively represented as a single particle with equivalent properties such as density " ${\overline{\rho }}_p$${\overline{\rho }}_p$bar(rho)_(p)\bar{\rho}_{p} " and size " ${\overline{D}}_p$${\overline{D}}_p$bar(D)_(p)\bar{D}_{p}." The determination of the equivalent density is facilitated through Equation (1), while Equation (2) enables the calculation of the equivalent size. This state is referred to as an equivalent system, exemplified in Figure 4.
为了阐明颗粒在该床内的运动，考虑由 BL 颗粒（以大尺寸和低密度为特征）和 SH 颗粒（以小尺寸和高密度为特征）组成的等效系统是有利的，它们可以有效地表示为具有密度 “ ${\overline{\rho }}_p$${\overline{\rho }}_p$bar(rho)_(p)\bar{\rho}_{p} 和 尺寸 ” ${\overline{D}}_p$${\overline{D}}_p$bar(D)_(p)\bar{D}_{p} 等等效性质的单个颗粒."通过公式 （1） 有助于确定等效密度，而公式 （2） 可以计算等效尺寸。这种状态称为等效系统，如图 4 所示。

The separation fluidized bed consists of coal and magnetite powder, with the latter possessing a high density but small particle size, while the former has a low density but big particle size. This system shares common characteristics with bidisperse particle fluidized beds mentioned earlier, which also involve two types of particles. Consequently, the separation fluidized bed can be considered as an equivalent single uniform fluidized bed system. In the equivalent fluidized bed, a coal particle is examined for its force analysis. If the combined effect of drag force and buoyancy force exceeds that of gravity, the particle will ascend; otherwise, it will descend.
分离流化床由煤和磁铁矿粉组成，后者密度高但粒径小，而前者密度低但粒径大。该系统与前面提到的双分散颗粒流化床具有共同特征，后者也涉及两种类型的颗粒。因此，分离流化床可以被视为等效的单一均匀流化床系统。在等效流化床中，检查煤颗粒以进行力分析。如果阻力和浮力的综合作用超过重力，粒子将上升;否则，它将下降。

Figure 4. Equivalent diagram of bubbling fluidized bed.
图 4.鼓泡流化床的等效图。

The density of equivalent particles, denoted $as{\overline{\rho }}_p$$as{\overline{\rho }}_p$as bar(rho)_(p)a s \bar{\rho}_{p}, is determined by employing Equation (2).
等效粒子的密度（表示 $as{\overline{\rho }}_p$$as{\overline{\rho }}_p$as bar(rho)_(p)a s \bar{\rho}_{p} 为 ）由方程 （2） 确定。

where ${\rho }_{SD}$${\rho }_{SD}$rho_(SD)\rho_{S D} represents the density of magnetite powder particles, and ${\rho }_{BL}$${\rho }_{BL}$rho_(BL)\rho_{B L} denotes the density of coal particles, $x_{SH}$$x_{SH}$x_(SH)x_{S H} represents the proportion of magnetite powder particles.
其中 ${\rho }_{SD}$${\rho }_{SD}$rho_(SD)\rho_{S D} 表示磁铁矿粉末颗粒的密度， ${\rho }_{BL}$${\rho }_{BL}$rho_(BL)\rho_{B L} 表示煤颗粒 $x_{SH}$$x_{SH}$x_(SH)x_{S H} 的密度 表示磁铁矿粉末颗粒的比例。

The equivalent particle diameter, denoted as ${\overline{D}}_p$${\overline{D}}_p$bar(D)_(p)\bar{D}_{p}, can be determined using Equation (3).
等效颗粒直径（表示为 ${\overline{D}}_p$${\overline{D}}_p$bar(D)_(p)\bar{D}_{p} ）可以使用方程 （3） 确定。

where $D_{SH}$$D_{SH}$D_(SH)D_{S H} denotes the diameter of magnetite powder particles, and $D_{BL}$$D_{BL}$D_(BL)D_{B L} signifies the diameter of coal particles.
其中 $D_{SH}$$D_{SH}$D_(SH)D_{S H} 表示磁铁矿粉末颗粒的直径， $D_{BL}$$D_{BL}$D_(BL)D_{B L} 表示煤颗粒的直径。

The forces exerted on a coal particle within the fluidized bed are depicted by Equation (4).
施加在流化床内煤颗粒上的力由方程 （4） 描述。

The buoyancy of coal particles in the equivalent fluidized bed system, denoted as $F_f$$F_f$F_(f)F_{f}, is determined using Equation (5).
等效流化床系统中煤颗粒的浮力，表示为 $F_f$$F_f$F_(f)F_{f} ，使用方程 （5） 确定。

The symbol $F_g$$F_g$F_(g)F_{g} represents the gravity acting on coal particles, as depicted in Equation (6).
该符号 $F_g$$F_g$F_(g)F_{g} 表示作用在煤颗粒上的重力，如方程 （6） 所示。

Equation (7) provides an accurate estimation of the dimensionless drag force $F_0$$F_0$F_(0)F_{0}.
方程 （7） 提供了无量纲阻力 $F_0$$F_0$F_(0)F_{0} 的准确估计值。