Dynamic prediction method of particle size distribution in ternary precursor crystallization process based on population balance equations

doi:10.11949/0438-1157.20250094

Abstract

Abstract:

In the preparation of high-performance ternary cathode materials, the ternary precursor serves as a core raw material, and its quality—especially the particle size distribution—has a decisive impact on the physicochemical properties of the sintered cathode product. The preparation of ternary precursors involves various process parameters, among which ammonia concentration, reaction temperature, pH value during the reaction, stirring rate, and reaction time all significantly influence particle size. Among these, reaction time has the most pronounced effect on the particle size of the ternary precursor. This study proposes a dynamic prediction model for the particle size distribution during the crystallization process of ternary precursors, based on a particle population balance equation. First, considering the growth characteristics of the co-precipitation process of ternary precursors, a particle size prediction model based on the population balance equation is established, taking into account both the growth rate described by the ASL equation and a generalized growth rate expression. Secondly, the population balance equation was discretized in space and time dimensions using the collocation method, and the unknown parameters in the model were identified by constructing an optimization problem. Finally, simulation results show that the proposed prediction method can accurately simulate the particle size evolution of ternary precursors with different initial sizes during the crystallization process, achieving precise prediction of particle size distribution. This provides a solid theoretical foundation and reference for optimizing the sintering process of ternary cathode materials.

Key words: ternary precursors, crystallization, population balance equations, chemical reaction, particle size distribution

摘要：

在高性能三元正极材料的制备过程中，三元前体作为核心原料，其品质尤其是粒度分布对正极烧结产物的理化性能具有决定性影响。三元前体的制备涉及多种工艺参数，氨水浓度、反应温度、反应过程pH、搅拌速率和反应时间均对粒径产生显著影响，其中反应时间对三元前体的粒径影响最为显著。提出了一种基于粒数衡算方程的三元前体结晶过程粒度分布动态预测模型。首先，针对三元前体共沉淀过程的生长特性，建立了基于粒数衡算方程的粒径预测模型，分别考虑了符合ASL方程的生长速率和具有普适性的生长速率表达式。其次，运用配点法对粒数衡算方程进行空间和时间维度的离散化求解，并通过构建优化问题对模型中的未知参数进行辨识。最终，仿真结果表明，所提出的预测方法能够准确模拟不同初始粒径的三元前体在结晶过程中的粒径变化，实现对粒度分布的精确预测，为优化三元正极材料的烧结工艺提供了重要的理论依据和参考。

关键词: 三元前体, 结晶, 粒数衡算方程, 化学反应, 粒度分布

CLC Number:

TQ 021.8

Yuanshen DAI, Zhijiang SHAO, Weifeng CHEN, Ning CHEN. Dynamic prediction method of particle size distribution in ternary precursor crystallization process based on population balance equations[J]. CIESC Journal, 2025, 76(8): 4119-4128.

戴元燊, 邵之江, 陈伟锋, 陈宁. 基于粒数衡算方程的三元前体结晶过程粒度分布动态预测方法[J]. 化工学报, 2025, 76(8): 4119-4128.

Figures/Tables 8

Fig.1 Schematic diagram of batch production preparation of ternary precursors

Table 1 Parameter estimation results based on ASL equation

批次	K_G	γ	b
1st (r₀=2.4 μm)	0.00797388	1006.63	0.499995
2nd (r₀=2.48 μm)	0.00792696	1019.0	0.49995
3rd (r₀=2.56 μm)	0.0080318	991.77	0.50002

Fig.2 Under the condition that the growth rate satisfies the ASL equation, the actual particle size distribution curves (blue solid lines—45 h, green dashed lines—50 h) and predicted particle size distribution curves (red circles—45 h, black plus sign—50 h) corresponding to t=45 h and t=50 h for r0=2.4,2.48,2.56 μm

Table 2 Relative error of parameter estimation

批次	相对误差/%
批次	K_G	γ	b
1st (r₀=2.4 μm)	0.327	0.663	0.001
2nd (r₀=2.48 μm)	0.913	1.907	0.008
3rd (r₀=2.56 μm)	0.399	0.822	0.005

Table 3 The values of rd corresponding to t=45 h and t=50 h when the growth rate satisfies the ASL equation

批次	r_d
批次	t=45 h	t=50 h
1st (r₀=2.4 μm)	5.10×10^-6	5.62×10^-6
2nd (r₀=2.48 μm)	8.89×10^-6	9.43×10^-6
3rd (r₀=2.56 μm)	4.07×10^-6	4.45×10^-6

Table 4 Parameter estimation results based on approximate growth rate equation

参数	1st (r₀=2.4 μm)	2nd (r₀=2.48 μm)	3rd (r₀=2.56 μm)
g₁	0.0251262	0.0251169	0.0251549
g₂	0.0557397	0.0556703	0.0556456
g₃	0.0861399	0.0861498	0.0861704
g₄	0.115193	0.115193	0.115192
g₅	0.142446	0.142449	0.142449
g₆	0.167488	0.167494	0.167497
g₇	0.189929	0.189931	0.189943
g₈	0.209357	0.209393	0.209465
g₉	0.228366	0.22766	0.227164
g₁₀	0.233492	0.233729	0.234133
g₁₁	0.237959	0.240243	0.242872
g₁₂	0.301832	0.297165	0.295236

Fig.3 Under the condition that the growth rate is universal, the actual particle size distribution curves (blue solid lines—45 h, green dashed lines—50 h) and predicted particle size distribution curves (red circles—45 h, black plus sign—50 h) corresponding to t=45 h and t=50 h for r0=2.4,2.48,2.56

Table 5 The values of rd corresponding to t=45 h and t=50 h when the growth rate model has universality

批次	r_d
批次	t=45 h	t=50 h
1st (r₀=2.4 μm)	3.20×10^-4	3.69×10^-4
2nd (r₀=2.48 μm)	2.66×10^-4	3.15×10^-4
3rd (r₀=2.56 μm)	2.14×10^-4	2.62×10^-4

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