The Synthetic Scale-up Data Generated by the Austin model and Their Estimation by the Kotake–Kanda Model

The Excel spreadsheet contains the synthetic scale-up data generated by the Austin model and their estimation using the Kotake–Kanda (KK) model, which overall yields the information presented in Table 2 of the manuscript. In view of all scale-up (SU) cases in Table 2, the Austin model, via Eqn. (12) of the manuscript, was used to produce the synthetic scale-up data, i.e., the specific breakage rate Si*, to which the Si* generated by the KK model, via Eqn. (13) of the manuscript, were compared (SU1–SU4 in Table 2). In SU1–SU4, standard values of the scale-up correction exponents, i.e., N1 = 0.5 and N2 = 0.2, were used. In SU5–SU8, N1 and N2 were estimated by fitting Eqn. (13) to the synthetic scale-up data for each diameter ratio D/DT separately. In SU9–SU12, they were fitted to the synthetic scale-up data for all D/DT data together. These two separate types of fits are presented in the Excel spreadsheet, which contains two separate spreadsheets entitled “Separately fitted” and “Simultaneously fitted.” For simplicity, Si* was replaced by Si in the Excel spreadsheet. Columns A and B yield, respectively, the ball size dB values and the particle size xi values from which the artificial Si values were calculated using the Austin model for different values of D/DT. A summary table is included at the far right side of the Excel spreadsheet, which presents the values used in Table 2 of the manuscript. Nomenclature Si,A and Si,K: Si* calculated by the Austin model and estimated by the KK model Si,Kmod: Si* estimated by the KK model with modified (fitted) N1 and N2 Sqrt(Si,A) and Sqrt(Si,K): Square root of Si,A and Si,K Sqrt(Si,Kmod): Square root of Si,Kmod SE original: square error between the Austin and the KK model for N1 = 0.5 and N2 = 0.2 SE modified: square error between the Austin and the KK model for fitted N1 and N2本Excel电子表格包含由奥斯汀模型（Austin model）生成的合成放大数据，以及采用小竹-神田（Kotake–Kanda, KK）模型对其进行的估算结果，整体涵盖了论文表2中呈现的全部信息。针对表2中的所有放大（Scale-up, SU）案例，首先通过论文公式(12)中的奥斯汀模型生成合成放大数据，即特定破碎速率Si*；随后将通过论文公式(13)中小竹-神田模型生成的Si*与之进行对比（即表2中的SU1–SU4）。在SU1–SU4中，采用了放大校正指数的标准值：N1=0.5，N2=0.2。在SU5–SU8中，则针对每组不同的直径比D/DT，通过将公式(13)拟合至合成放大数据来估算N1与N2。在SU9–SU12中，则将公式(13)拟合至全部直径比D/DT对应的合成放大数据。这两类独立的拟合结果均包含于本Excel电子表格中，该表格包含两个独立工作表，分别命名为"Separately fitted"（单独拟合）与"Simultaneously fitted"（同时拟合）。为简化起见，本表格中将Si*简写为Si。A、B两列分别给出了球径dB与粒径xi的数值，针对不同的D/DT取值，通过奥斯汀模型即可由上述数值计算得到人工生成的Si值。本Excel电子表格的最右侧附带了一张汇总表，呈现了论文表2中所使用的全部数值。符号说明：Si,A与Si,K分别代表由奥斯汀模型计算得到的Si*以及由小竹-神田模型估算得到的Si*；Si,Kmod代表经修正（拟合得到）的N1与N2代入小竹-神田模型后估算得到的Si*；Sqrt(Si,A)与Sqrt(Si,K)分别为Si,A与Si,K的平方根；Sqrt(Si,Kmod)为Si,Kmod的平方根；SE original为当N1=0.5、N2=0.2时，奥斯汀模型与小竹-神田模型之间的平方误差；SE modified为采用拟合得到的N1与N2时，奥斯汀模型与小竹-神田模型之间的平方误差。