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Article(id=1322909533445694179, tenantId=1146029695717560320, journalId=1225396423026438145, issueId=1322909380756243363, articleNumber=null, orderNo=null, doi=null, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1715875200000, receivedDateStr=2024-05-17, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793335811243, onlineDateStr=2026-10-30, pubDate=1745510400000, pubDateStr=2025-04-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793335811243, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793335811243, creator=13701087609, updateTime=1793335811243, updator=13701087609, issue=Issue{id=1322909380756243363, tenantId=1146029695717560320, journalId=1225396423026438145, year='2025', volume='45', issue='4', pageStart='1', pageEnd='288', issueExtLink='null', onlineDate='null', pubDate='1745510400000', pubDateStr='2025-04-25', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793335774840, creator='13701087609', updateTime=1793335774840, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=204, endPage=211, ext={EN=ArticleExt(id=1322909535219884772, articleId=1322909533445694179, tenantId=1146029695717560320, journalId=1225396423026438145, language=EN, title=Research on Grinding Energy Consumption of Ball Mill Based on Response Surface Methodology, columnId=null, journalTitle=Mining Research and Development, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In order to reduce the energy consumption of ball mill, ZQM250×100 ball mill was used to study the grinding energy consumption of an iron ore after high-pressure roller grinding, and the energy required for the newly generated －0.074 mm grade product after grinding was used as the evaluation standard of grinding energy consumption. The effects of medium filling rate, ore addition quantity, water addition quantity and mill frequency on the energy consumption of the newly generated －0.074 mm product, the significance of the interaction between the factors and the optimal test conditions were analyzed and predicted by the response surface methodology, and the results were compared with the single factor test results. The optimum conditions obtained by single factor test are as follows: medium filling rate is 40%, ore addition quantity is 700 g, water addition quantity is 300 mL (mass concentration is 70%), mill frequency is 50 Hz, and energy consumption of the newly generated －0.074 mm grade product is 169.54 (kW·h)/t. The response surface methodology analysis show that the mill frequency has the greatest influence on the energy consumption of the newly generated －0.074 mm grade product, followed by the medium filling rate, water addition quantity, and ore addition quantity. The interaction between medium filling rate and mill frequency is the most obvious, followed by water addition quantity and mill frequency, ore addition quantity and water addition quantity, medium filling rate and ore addition quantity, ore addition quantity and mill frequency, medium filling rate and water addition quantity. The model prediction based on the response surface methodology shows that the optimum test conditions are: 39.71% medium filling rate, 698.15 gore addition quantity, 339.65 mL (mass concentration 67.27%) water addition quantity, and 49.73 Hz mill frequency. Under this optimal condition, the predicted energy consumption of the newly generated －0.074 mm product is 120.53 (kW·h)/t, and the average energy consumption of the verification test is 118.97 (kW·h)/t, which is very close to the predicted value, indicating that the response surface prediction model is reasonable and effective.

, authors=Chen LI, Ying HOU, Cuncun ZOU, Ziyang WANG, Peng CHEN, Hongxu LYU, authorsList=Chen LI, Ying HOU, Cuncun ZOU, Ziyang WANG, Peng CHEN, Hongxu LYU, authorCompany=null, correspAuthors=null, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1322909540185940749, articleId=1322909533445694179, tenantId=1146029695717560320, journalId=1225396423026438145, language=CN, title=基于响应曲面法的球磨机磨矿能耗研究, columnId=1226462295946085268, journalTitle=矿业研究与开发, columnName=矿山机电与矿业智能化, runingTitle=null, highlight=null, articleAbstract=

为了降低球磨机的能耗，采用ZQM250×100型球磨机对高压辊磨后的某铁矿的磨矿能耗进行了研究，并使用磨后新生成的－0.074 mm粒级产品所需的能量作为磨矿能耗的评估标准。通过响应曲面法分析并预测了磨矿过程中，球磨机的介质充填率、加矿量、加水量和磨机频率对新生成－0.074 mm粒级产品的能耗影响、因素之间交互作用的显著性和最佳试验条件，并与单因素试验结果进行了比较。单因素试验得到的最佳条件：介质充填率为40%，加矿量为700 g，加水量为300 mL（质量浓度70%)，磨机频率为50 Hz，对应的新生成－0.074 mm粒级产品的能耗为169.54 (kW·h)/t。响应曲面法分析得出：磨机频率对新生成－0.074 mm粒级产品的能耗影响最大，然后依次是介质充填率、加水量，加矿量；介质充填率与磨机频率的交互作用影响最为明显，随后依次是加水量与磨机频率、加矿量与加水量、介质充填率与加矿量、加矿量与磨机频率、介质充填率与加水量。基于响应曲面法的模型预测显示，最佳试验条件为：39.71%的介质充填率，698.15 g的加矿量，339.65 mL（质量浓度67.27%)的加水量，以及49.73 Hz的磨机频率。在此最佳条件下，新生成－0.074 mm粒级产品的能耗预测值为120.53 (kW·h)/t，验证试验的能耗平均值为118.97 (kW·h)/t，与预测值非常接近，说明响应曲面法预测模型是合理且有效的。

, authors=李晨, 侯英, 邹存存, 王子扬, 陈鹏, 吕虹序, authorsList=李晨, 侯英, 邹存存, 王子扬, 陈鹏, 吕虹序, authorCompany=null, correspAuthors=null, authorNote=

李晨（1999—），男，辽宁本溪人，硕士研究生，主要研究方向为难选矿物高效碎磨分选。E-mail:

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李晨（1999—），男，辽宁本溪人，硕士研究生，主要研究方向为难选矿物高效碎磨分选。E-mail:

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李晨（1999—），男，辽宁本溪人，硕士研究生，主要研究方向为难选矿物高效碎磨分选。E-mail:

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Journal of Food Engineering, 2007, 78(3):836-845., articleTitle=Modeling and optimization I: usability of response surface methodology, refAbstract=null), Reference(id=1322909553624490875, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, doi=null, pmid=null, pmcid=null, year=2010, volume=31, issue=4, pageStart=1768, pageEnd=1775, url=null, language=null, rfNumber=[16], rfOrder=26, authorNames=DEHGHANI K, NEKAHI A, MIRZAIE M A M, journalName=Materials & Design, refType=null, unstructuredReference=DEHGHANI K, NEKAHI A, MIRZAIE M A M. Optimizing the bake hardening behavior of Al7075 using response surface methodology[J]. 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mm粒级产品能耗的交互影响, figureFileSmall=LEikWcChLC0/sBGX8DcOrQ==, figureFileBig=gZ0sCUfKS3n9mUwym3GWmw==, tableContent=null), ArticleFig(id=1322909548725543766, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, language=EN, label=Table 1, caption=

Design factors and levels of response surface methodology

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水平介质充填率x1/%加矿量x2/g加水量x3/mL磨机频率x4/Hz
－130600376.9240
040700300.0050
150800233.3360
), ArticleFig(id=1322909548851372887, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, language=CN, label=表1, caption=

响应曲面法设计因素和水平

, figureFileSmall=null, figureFileBig=null, tableContent=
水平介质充填率x1/%加矿量x2/g加水量x3/mL磨机频率x4/Hz
－130600376.9240
040700300.0050
150800233.3360
), ArticleFig(id=1322909548922676056, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, language=EN, label=Table 2, caption=

Test design and results

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序号充填率/%加矿量/g加水量/mL磨机频率/Hz能耗/(kW·h·t－1)
130500348.550189.55
230900348.550191.85
330700348.540178.41
430700348.560190.31
530700124.050185.38
630700573.050187.33
740700124.040182.87
840700573.040186.56
940700124.060190.93
1040700573.060185.73
1140500124.050176.22
1240900124.050172.92
1340500573.050175.28
1440900573.050179.76
1540500348.540183.98
1640900348.540180.43
1740500348.560190.35
1840900348.560185.11
1940700348.550120.55
2040700348.550120.55
2140700348.550120.55
2250500348.550195.80
2350900348.550192.48
2450700348.540187.18
2550700348.560185.76
2650700124.050186.31
2750700573.050189.36
), ArticleFig(id=1322909549283386201, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, language=CN, label=表2, caption=

试验设计及结果

, figureFileSmall=null, figureFileBig=null, tableContent=
序号充填率/%加矿量/g加水量/mL磨机频率/Hz能耗/(kW·h·t－1)
130500348.550189.55
230900348.550191.85
330700348.540178.41
430700348.560190.31
530700124.050185.38
630700573.050187.33
740700124.040182.87
840700573.040186.56
940700124.060190.93
1040700573.060185.73
1140500124.050176.22
1240900124.050172.92
1340500573.050175.28
1440900573.050179.76
1540500348.540183.98
1640900348.540180.43
1740500348.560190.35
1840900348.560185.11
1940700348.550120.55
2040700348.550120.55
2140700348.550120.55
2250500348.550195.80
2350900348.550192.48
2450700348.540187.18
2550700348.560185.76
2650700124.050186.31
2750700573.050189.36
), ArticleFig(id=1322909549371466587, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, language=EN, label=Table 3, caption=

Model statistics

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方差来源平方和自由度均方值F值P值
线性模型5.340 031.780.550.659 2
2FI模型1.930 030.640.140.930 8
二次方程26.880 038.96159.190.000 8
三次方程0.168 930.05
), ArticleFig(id=1322909549446964060, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, language=CN, label=表3, caption=

模型统计

, figureFileSmall=null, figureFileBig=null, tableContent=
方差来源平方和自由度均方值F值P值
线性模型5.340 031.780.550.659 2
2FI模型1.930 030.640.140.930 8
二次方程26.880 038.96159.190.000 8
三次方程0.168 930.05
), ArticleFig(id=1322909549698622301, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, language=EN, label=Table 4, caption=

Analysis of variance of the quadratic model of energ consumption for the generation of –0.074 mm particle size products

, figureFileSmall=null, figureFileBig=null, tableContent=
项目变差平方和自由度均方F值P值
模型11 659.7914832.8429.72<0.000 1
x116.47116.470.580.450 0
x26.2016.200.220.640 0
x37.3417.340.260.610 0
x468.92168.922.450.140 0
x1x27.8917.890.280.600 0
x1x30.3010.300.010.910 0
x1x444.35144.351.580.230 0
x2x315.13115.130.530.470 0
x2x40.7110.710.020.870 0
x3x419.75119.750.700.410 0
7 214.4317214.43257.44<0.000 1
5 135.3915 135.39183.25<0.000 1
4 530.4214 530.42161.66<0.000 1
5 726.1515 726.15<0.000 1
残差336.271228.02
总和11 996.0726
), ArticleFig(id=1322909549782508382, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1322909533445694179, language=CN, label=表4, caption=

新生成–0.074 mm粒级产品的能耗二次模型方差分析结果

, figureFileSmall=null, figureFileBig=null, tableContent=
项目变差平方和自由度均方F值P值
模型11 659.7914832.8429.72<0.000 1
x116.47116.470.580.450 0
x26.2016.200.220.640 0
x37.3417.340.260.610 0
x468.92168.922.450.140 0
x1x27.8917.890.280.600 0
x1x30.3010.300.010.910 0
x1x444.35144.351.580.230 0
x2x315.13115.130.530.470 0
x2x40.7110.710.020.870 0
x3x419.75119.750.700.410 0
7 214.4317214.43257.44<0.000 1
5 135.3915 135.39183.25<0.000 1
4 530.4214 530.42161.66<0.000 1
5 726.1515 726.15<0.000 1
残差336.271228.02
总和11 996.0726
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基于响应曲面法的球磨机磨矿能耗研究
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李晨 , 侯英 , 邹存存 , 王子扬 , 陈鹏 , 吕虹序
矿业研究与开发 | 矿山机电与矿业智能化 2025,45(4): 204-211
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矿业研究与开发 |矿山机电与矿业智能化 2025 , 45 (4) : 204 -211
基于响应曲面法的球磨机磨矿能耗研究
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李晨 , 侯英, 邹存存, 王子扬, 陈鹏, 吕虹序
作者信息
  • 辽宁科技大学 矿业工程学院，辽宁　鞍山市　114051
作者简介:

李晨（1999—），男，辽宁本溪人，硕士研究生，主要研究方向为难选矿物高效碎磨分选。E-mail:

Research on Grinding Energy Consumption of Ball Mill Based on Response Surface Methodology
Chen LI , Ying HOU, Cuncun ZOU, Ziyang WANG, Peng CHEN, Hongxu LYU
Affiliations
  • School of Mining Engineering, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China
出版时间: 2025-04-25
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摘要
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为了降低球磨机的能耗，采用ZQM250×100型球磨机对高压辊磨后的某铁矿的磨矿能耗进行了研究，并使用磨后新生成的－0.074 mm粒级产品所需的能量作为磨矿能耗的评估标准。通过响应曲面法分析并预测了磨矿过程中，球磨机的介质充填率、加矿量、加水量和磨机频率对新生成－0.074 mm粒级产品的能耗影响、因素之间交互作用的显著性和最佳试验条件，并与单因素试验结果进行了比较。单因素试验得到的最佳条件：介质充填率为40%，加矿量为700 g，加水量为300 mL（质量浓度70%)，磨机频率为50 Hz，对应的新生成－0.074 mm粒级产品的能耗为169.54 (kW·h)/t。响应曲面法分析得出：磨机频率对新生成－0.074 mm粒级产品的能耗影响最大，然后依次是介质充填率、加水量，加矿量；介质充填率与磨机频率的交互作用影响最为明显，随后依次是加水量与磨机频率、加矿量与加水量、介质充填率与加矿量、加矿量与磨机频率、介质充填率与加水量。基于响应曲面法的模型预测显示，最佳试验条件为：39.71%的介质充填率，698.15 g的加矿量，339.65 mL（质量浓度67.27%)的加水量，以及49.73 Hz的磨机频率。在此最佳条件下，新生成－0.074 mm粒级产品的能耗预测值为120.53 (kW·h)/t，验证试验的能耗平均值为118.97 (kW·h)/t，与预测值非常接近，说明响应曲面法预测模型是合理且有效的。

球磨机  /  能耗  /  影响因素  /  铁矿石  /  响应曲面法

In order to reduce the energy consumption of ball mill, ZQM250×100 ball mill was used to study the grinding energy consumption of an iron ore after high-pressure roller grinding, and the energy required for the newly generated －0.074 mm grade product after grinding was used as the evaluation standard of grinding energy consumption. The effects of medium filling rate, ore addition quantity, water addition quantity and mill frequency on the energy consumption of the newly generated －0.074 mm product, the significance of the interaction between the factors and the optimal test conditions were analyzed and predicted by the response surface methodology, and the results were compared with the single factor test results. The optimum conditions obtained by single factor test are as follows: medium filling rate is 40%, ore addition quantity is 700 g, water addition quantity is 300 mL (mass concentration is 70%), mill frequency is 50 Hz, and energy consumption of the newly generated －0.074 mm grade product is 169.54 (kW·h)/t. The response surface methodology analysis show that the mill frequency has the greatest influence on the energy consumption of the newly generated －0.074 mm grade product, followed by the medium filling rate, water addition quantity, and ore addition quantity. The interaction between medium filling rate and mill frequency is the most obvious, followed by water addition quantity and mill frequency, ore addition quantity and water addition quantity, medium filling rate and ore addition quantity, ore addition quantity and mill frequency, medium filling rate and water addition quantity. The model prediction based on the response surface methodology shows that the optimum test conditions are: 39.71% medium filling rate, 698.15 gore addition quantity, 339.65 mL (mass concentration 67.27%) water addition quantity, and 49.73 Hz mill frequency. Under this optimal condition, the predicted energy consumption of the newly generated －0.074 mm product is 120.53 (kW·h)/t, and the average energy consumption of the verification test is 118.97 (kW·h)/t, which is very close to the predicted value, indicating that the response surface prediction model is reasonable and effective.

Ball mill  /  Energy consumption  /  Influencing factor  /  Iron ore  /  Response surface methodology
李晨, 侯英, 邹存存, 王子扬, 陈鹏, 吕虹序. 基于响应曲面法的球磨机磨矿能耗研究. 矿业研究与开发, 2025 , 45 (4) : 204 -211 .
Chen LI, Ying HOU, Cuncun ZOU, Ziyang WANG, Peng CHEN, Hongxu LYU. Research on Grinding Energy Consumption of Ball Mill Based on Response Surface Methodology[J]. Mining Research and Development, 2025 , 45 (4) : 204 -211 .
正文
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0　引言
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在对铁矿进行选矿之前，通常需要先对矿石进行破碎和研磨处理。在选矿生产流程中，磨矿是一个能耗高的环节，所消耗的能量占选矿厂总能源消耗的40%~60%[1]。降低选矿能耗已经成为了众多矿山企业所面临的主要问题之一。为追求“双碳”目标，矿山生产企业急需开展“节能降耗、降本增效”的相关工作[2]。实施“多碎少磨”的策略是一种降低选矿能源消耗的高效方法[3-4]。高压辊磨具有能耗低、效率高、生产能力大等特点，已逐渐成为国内外选矿行业的主流设备之一。高压辊磨机作为一种高效且节能的破碎设备，已在各选矿厂的超细碎操作中得到应用，并在提高磨矿效率和分选标准方面表现出色[5-6]。高压辊磨机产品的性能直接影响到后续工艺的效果，因此对细颗粒物料的粉碎机理以及粒度分布规律等有必要深入研究。现阶段关于高压辊磨产品的磨矿特性的研究，主要是依赖于使用球磨机来完成[7-8]。因此开展降低球磨机能耗的研究具有重要意义。
1　试验原料及方法
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1.1　试验矿样
试验矿样为某铁矿山高压辊磨后的预选精矿，XRD分析结果如图1所示。
1.2　试验步骤
在球磨机筒体中加入特定质量的研磨介质、矿样和适量的水，在球磨机变频器上调整磨机频率，并使其跳转至功率显示界面。在此条件下进行磨矿试验，当磨机达到额定速度时开始工作。使用秒表进行200 s的计时后启动磨机，在设定时间内完成试磨。在磨矿的过程中，需要观察液晶显示屏上的功率数据，记录磨机频率的最高值和最低值，在磨矿结束后计算出平均值作为试验功率。
能耗采用新生成－0.074 mm粒级产品所需的能量来评价，其计算公式如下：
式中：E为新生成－0.074 mm粒级产品的能耗，(kW·h)/t；M为新生成－0.074 mm粒级产品的质量，t；T为磨矿时间，h；P为磨机的功率，kW。
1.3　单因素试验
在加矿量为500 g、磨机频率为50 Hz（磨机变频器上的表显频率）、加水量为214.3 mL（质量浓度70%）的条件下，通过改变介质充填率获得的最低能耗为198.87 (kW·h)/t，试验结果如图2所示。
当介质充填率取图2中的最佳值40%、磨机频率为50 Hz、质量浓度70%（由于随着加矿量改变同质量浓度的加水量也随之改变，所以这里用质量浓度作为定值）时，通过改变加矿量获得的最低能耗为169.54 (kW·h)/t，如图3所示。
当介质充填率取图2中的最佳值40%、加矿量取图3中的最佳值700 g、磨机频率为50 Hz时，通过改变加水量获得的最低能耗为169.54 (kW·h)/t，如图4所示。
当介质充填率取图2中的最佳值40%、加矿量取图3中的最佳值700 g、加水量取图4中的最佳值300 mL（质量浓度为70%)时，通过调整磨机频率获得的最低能耗为169.54 (kW·h)/t，如图5所示。
根据图2、图3、图4、图5可得到单因素最佳试验条件及结果分别是：介质充填率为40%，加矿量为700 g，加水量为300 mL（质量浓度70%），磨机频率为50 Hz，对应新生成的－0.074 mm粒级产品的能耗为169.54 (kW·h)/t。
1.4　响应曲面试验设计
响应曲面法是一种能够根据单因素试验结果进行试验设计和优化的方法。该方法通过在特定数据点上进行局部试验，并采用回归分析来模拟全局范围内各因素与试验结果之间的函数关系。因此响应曲面法可以量化分析各因素之间的相互作用对试验结果的影响，并通过优化分析以确定各因素的最优水平值[9]。影响球磨机能耗的主要因素为介质充填率、加矿量、加水量以及磨机频率，因此对这4个关键因素通过响应曲面法进行研究，并分别用x1，x2，x3，x4来表示。响应曲面法的试验条件和编码方式见表1，选择新生成的－0.074 mm粒级产品的能耗作为评定响应曲面优化结果的指标。
利用软件Design-Expert 13.0进行试验设计，将选择的因素和水平值输入系统，按 Box-Behnken组合方法设计出试验方案，见表2。
按照设计方案进行试验，结果见表2。新生成－0.074 mm粒级试验所消耗的能量为120.55~195.8 (kW·h)/t。
2　选择响应模型
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使用Design-Expert 13.0软件对试验数据进行分析，结果见表3。鉴于试验中新生成－0.074 mm粒级产品的能耗受到多种因素的影响，因此对表3中多个模型的显著性、失拟项和相关性数据进行综合分析，并使用二阶模型近似模拟。
2.1　响应模型的建立及方差分析
通过软件分析计算，得出能耗关于变量的拟合方程为：
对新生成－0.074 mm粒级产品所需能耗的二阶模型进行方差分析和显著性检验，结果见表4。表4中的F值和P值可以反映出各个因素或因素之间交互作用的显著性（F值越高或P值越低，表明该因素或因素之间的交互作用显著性越好），如果P值低于0.05，则认为该模型具有显著性[10]。多项式模型的拟合效果可以通过R2来评估，多元相关系数R2的值越高，意味着模型的相关性越强，因此模型的拟合相关系数不应低于0.80[11-12]。在此模型中，相关系数(R2=0.972 0)揭示了模型方程与试验数据之间的一致性较好，而调整后的确定系数(R2=0.939 3)进一步表明了该模型的显著性。变异系数(CV)被用于评估试验的精确度，并被视为评定试验可靠性的核心标准。CV的数值越小，试验的稳定性就越强[13]。在此次试验中，CV的数值相对偏低，仅为2.97%，这意味着试验的结果是相对可靠的。通常情况下，信噪比超过4的测量方法被视为非常精确[14]，在此模型中，信噪比已经达到了18.78，这进一步验证了该模型的可靠性。
新构建的－0.074 mm粒级产品能耗模型的残差正态概率分布，以及预测值与试验值的比较见图6和图7，其中图7的横纵坐标代表介质充填率、加矿量、加水量、磨机频率4个因素所对应的能耗试验值和模型预测值。从图6可以看出，新生成－0.074 mm粒级产品的能耗模型的残差散点基本上呈直线分布，但是在模型残差的正态概率图中，中间概率上的残差散点分布比两侧的更密集。残差超出区间(－3,2)的概率是0，这证明该模型的拟合能力是非常优秀的。从图7可以看出，能量消耗的预测值和试验值的散点分布大致呈直线关系(y=x)，这表明试验值和预测值之间存在很高的一致性。
2.2　响应曲面分析
图8至图13展示了自变量和因变量之间的响应曲面和等值线图。响应曲面是由多个相互影响的因素产生的响应值所组合而成，能直观地展示各种试验变量如何影响响应值。如果一个曲面的坡度比较陡峭，那么这个因素对响应值的影响会更加明显；反之，这个因素对响应值产生的影响会比较微小。通过等值线图，可以清楚地看到因素之间交互作用的显著性。当等值线的形状更接近圆形时，这意味着这些因素之间的交互作用相对较弱[15-16]。
由图8(a)可知，当介质充填率一定时，新生成－0.074 mm粒级产品的能耗随着加矿量的增大先减小后增大，总体变化幅度较大，可见加矿量对新生成－0.074 mm粒级产品的能耗影响较大，同理介质充填率的影响也相对较大。当加矿量一定时，新生成－0.074 mm粒级产品的能耗先减少并达到最小值后增加，该极值出现在中等介质充填率(35%~45%)及中等加矿量(600~800 g)范围内。由图8(b)所示的等值线形状近似为圆形，说明介质充填率与加矿量的交互作用不显著。
从图9(a)可以看出，当介质充填率一定时，新生成－0.074 mm粒级产品的能耗随着加水量的增大先减小后增加，变化幅度也相对较大，说明加水量对新生成－0.074 mm粒级产品的能耗影响较大。由图9(b)可知，等值线轮廓呈圆形，介质充填率与磨机频率的交互作用影响不显著。
由图10(a)可知，当介质充填率一定时，新生成－0.074 mm粒级产品的能耗随着磨机频率的增大先减小后增加，变化幅度相对较大。说明磨机频率对新生成－0.074 mm粒级产品的能耗影响较大。由图10(b)可知，等值线图轮廓近似圆形，说明介质充填率和磨机频率的交互作用对新生成－0.074 mm粒级产品的能耗影响较小。
由图11(a)可知，极值出现在中等加矿量(600~800 g)及中等加水量(303.6~393.4 mL)范围内。图11(b)所示的等值线形状近似圆形，说明加矿量与加水量的交互作用影响不显著。
从图12(a)可以看出，极值出现在中等加矿量(600~800 g)及中等磨机频率(45~55 Hz)范围内。由图12(b)可知，等值线轮廓近似圆形，说明加矿量与磨机频率的交互作用影响不太显著。
图13(a)可知，磨机频率对新生成－0.074 mm粒级产品能耗的影响大于加水量的影响，最小值所在区域对应的加水量为303.6~393.4 mL，磨机频率为45~55 Hz。由图13(b)可知，加水量和磨机频率的交互作用对新生成－0.074 mm粒级产品的能耗影响不显著。
综合方差和响应面分析可知，4个因素对新生成－0.074 mm粒级产品能耗的影响从大到小依次为磨机频率、介质充填率、加水量、加矿量；介质充填率与磨机频率的交互作用影响最显著，随后是加水量与磨机频率、加矿量量与加水量、介质充填率与加矿量、加矿量与磨机频率，而介质充填率与加水量的交互作用影响最不明显。
2.3　过程优化
利用软件优化功能得出模型预测的最优条件如下：介质充填率39.71%，加矿量698.15 g，加水量339.65 mL（质量浓度67.27%），磨机频率49.73 Hz，该条件下新生成－0.074 mm粒级产品的能耗预测值为120.53 (kW·h)/t。在此条件下进行验证试验，3次试验得到新生成－0.074 mm粒级产品的能耗分别为121.56 (kW·h)/t、117.90 (kW·h)/t及117.45 (kW·h)/t，平均值为118.97 (kW·h)/t，与模型预测值接近，说明该能耗预测模型在本试验的研究范围内是合理的，证明应用响应面法能够快速有效地对球磨机磨矿工艺参数进行优化。
3　结论
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（1）球磨机磨矿能耗单因素试验量佳结果为：介质充填率40%，加矿量700 g，加水量300 mL（质量浓度70%），磨机频率50 Hz。在该条件下新生成－0.074 mm粒级产品的最低能耗为169.54 (kW·h)/t。响应曲面法的最佳试验结果为：介质的充填率39.71%，加矿量为698.15 g，加水量339.65 mL（质量浓度67.27%），磨机频率49.73 Hz。在该条件下新生成－0.074 mm粒级产品的能耗预测值为120.53 (kW·h)/t，验证试验平均值为118.97 (kW·h)/t，相对单因素试验降低了29.83%。
（2）磨机频率对新生成－0.074 mm粒级产品的能耗影响最大，随后是介质充填率、加水量，加矿量对新生成－0.074 mm粒级产品的能耗影响最小。介质充填率与磨机频率的交互作用影响最显著，随后是加水量与磨机频率、加矿量与加水量、介质充填率与加矿量、加矿量与磨机频率，而介质充填率与加水量的交互作用影响最不明显。
（3）通过模型预测得到的最优条件为：介质的充填率39.71%，加矿量698.15 g，加水量339.65 mL(质量浓度67.27%)，磨机频率49.73 Hz。在此最佳条件下，新生成－0.074 mm粒级产品的能耗预测值为120.53 (kW·h)/t，验证试验的能耗平均值为118.97 (kW·h)/t，两者相差1.29%，验证了在本试验研究范围内该模型的合理有效性。
基金
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  • 辽宁省自然科学基金项目(20180550198)
  • 辽宁省教育厅项目(LJKFZ20220195; 2019LNQN09; 2016TSPY12)
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2025年第45卷第4期
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  • 接收时间：2024-05-17
  • 首发时间：2026-10-30
  • 出版时间：2025-04-25
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  • 收稿日期：2024-05-17
基金
辽宁省自然科学基金项目(20180550198)
辽宁省教育厅项目(LJKFZ20220195; 2019LNQN09; 2016TSPY12)
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    辽宁科技大学 矿业工程学院，辽宁　鞍山市　114051
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