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Article(id=1226462295660872589, tenantId=1146029695717560320, journalId=1225396423026438145, issueId=1226462293408531329, 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=1742054400000, receivedDateStr=2025-03-16, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1770340997852, onlineDateStr=2026-02-06, pubDate=1761321600000, pubDateStr=2025-10-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770340997852, onlineIssueDateStr=2026-02-06, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770340997852, creator=13701087609, updateTime=1770340997852, updator=13701087609, issue=Issue{id=1226462293408531329, tenantId=1146029695717560320, journalId=1225396423026438145, year='2025', volume='45', issue='10', pageStart='1', pageEnd='288', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1770340997315, creator=13701087609, updateTime=1770341205851, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1226463168155792201, tenantId=1146029695717560320, journalId=1225396423026438145, issueId=1226462293408531329, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1226463168155792202, tenantId=1146029695717560320, journalId=1225396423026438145, issueId=1226462293408531329, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=215, endPage=223, ext={EN=ArticleExt(id=1226462295862199187, articleId=1226462295660872589, tenantId=1146029695717560320, journalId=1225396423026438145, language=EN, title=Energy Consumption Law of Irregular Ore Particles Under Impact Crushing, columnId=null, journalTitle=Mining Research and Development, columnName=null, runingTitle=null, highlight=null, articleAbstract=

To reveal the energy consumption patterns of irregular ore particles under impact crushing, impact crushing tests were conducted on six types of iron ore with different properties, and the fractal characteristics of irregular iron ore fragmentation were analyzed, as well as the size effect on average particle size of fragments, fractal dimension, and unit absorption energy. Then, the energy consumption model of irregular single particle crushing was established. The results show that the average particle size of fragments, fractal dimension, unit absorption energy all exhibit a power function relationship with initial isosphere diameter under the same impact conditions. As the isosphere diameter of the particles increases, the average particle size of the fragments gradually increases, and the fractal dimension and unit absorption energy gradually decrease. There is an increasing relationship with power function between unit absorption energy and average particle size of fragments. The relationship between the logarithm of unit absorption energy and fractal dimension shows a linear increase. Through the method of ore crushing tests of irregular single ore particle, relationship models of unit crushing energy consumption of irregular iron ore particles changing with the initial size, average particle size of fragments, and fractal dimension were established, with an average correlation coefficient of 0.789, which effectively describes the energy consumption pattern of ore crushing.

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为揭示不规则矿石颗粒冲击破碎能耗规律，采用自主改进的落锤冲击试验机，对6种不同性质的铁矿石开展单颗粒冲击破碎试验，分析不规则铁矿石破碎的分形特征，以及碎块平均粒度、分形维数和单位吸收能的尺寸效应，建立不规则单颗粒矿石冲击破碎的能耗模型。结果表明，在相同的冲击条件下，碎块平均粒度、分形维数、单位吸收能与等球直径均呈幂函数关系，随着颗粒等球直径的增大，碎块平均粒度逐渐增大，分形维数与单位吸收能逐渐降低；单位吸收能与碎块平均粒度呈幂函数增大的变化关系，单位吸收能对数与分形维数呈线性增长的关系；通过不规则单颗粒矿石破碎试验的方法，建立了不规则颗粒单位破碎能关于初始尺寸、碎块平均粒度与分形维数的关系模型，平均相关系数为0.789，较好地描述了矿石破碎能耗规律。

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郭君（1983一），男，陕西神木人，博士，讲师，主要从事岩石破碎学、爆破工程方面的研究。E-mail:
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高锋（1988一），男，安徽淮北人，博士，讲师，主要从事岩石破碎节能降耗工艺与理论研究。E-mail:

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高锋（1988一），男，安徽淮北人，博士，讲师，主要从事岩石破碎节能降耗工艺与理论研究。E-mail:

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高锋（1988一），男，安徽淮北人，博士，讲师，主要从事岩石破碎节能降耗工艺与理论研究。E-mail:

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journalId=1225396423026438145, articleId=1226462295660872589, companyId=1226462302766023046, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=4.山西工程技术学院矿业工程系，山西　阳泉市　045000)])], figs=[ArticleFig(id=1226462305777533612, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Fig.1, caption=Irregular iron ore samples, figureFileSmall=Ot/mdGzLNXuwZofEnkvfpg==, figureFileBig=EcLfNXjJ5an8qoHuqqGJdw==, tableContent=null), ArticleFig(id=1226462305907557046, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=图1, caption=不规则铁矿石试样, figureFileSmall=Ot/mdGzLNXuwZofEnkvfpg==, figureFileBig=EcLfNXjJ5an8qoHuqqGJdw==, tableContent=null), ArticleFig(id=1226462306184381136, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Fig.2, caption=Mass and isosphere diameters of iron ore samples, figureFileSmall=CtbwJaa2G/uDpHfPqgs6cg==, figureFileBig=j0MOI2QRSiZgEu1KaASsvA==, tableContent=null), ArticleFig(id=1226462306301821656, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=图2, caption=铁矿石试样质量与等球直径, figureFileSmall=CtbwJaa2G/uDpHfPqgs6cg==, figureFileBig=j0MOI2QRSiZgEu1KaASsvA==, tableContent=null), ArticleFig(id=1226462306398290656, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Fig.3, caption=Grading curves of typical iron ore fragments under drop hammer impact, figureFileSmall=XHLQhDweTGPVvROKoX8Z1A==, figureFileBig=ll4QmjWvHDAKPHdTt/NJmw==, tableContent=null), ArticleFig(id=1226462306519925479, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=图3, caption=典型铁矿石试样落锤冲击破碎的级配曲线, figureFileSmall=XHLQhDweTGPVvROKoX8Z1A==, figureFileBig=ll4QmjWvHDAKPHdTt/NJmw==, tableContent=null), ArticleFig(id=1226462306624783086, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Fig.4, caption=Fractal dimension of fragment size, figureFileSmall=4Csvo0PfBJ8tsyyjzGqMEQ==, figureFileBig=aLbBpqH35CQVx54U0kUX8g==, tableContent=null), ArticleFig(id=1226462306700280564, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=图4, caption=碎块粒度分形维数, figureFileSmall=4Csvo0PfBJ8tsyyjzGqMEQ==, figureFileBig=aLbBpqH35CQVx54U0kUX8g==, tableContent=null), ArticleFig(id=1226462306809332475, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Fig.5, caption=Average particle sizes and crushing ratios of fragments, figureFileSmall=MY8v2qoFyvJS+orZ14sZLg==, figureFileBig=6x8O4Z5DtL4g/THpAQCuCw==, tableContent=null), ArticleFig(id=1226462306901607171, tenantId=1146029695717560320, 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caption=Relationship between unit absorption energy and isosphere diameter of samples, figureFileSmall=VT0xTYp1oBY+SvBBVI0ZoQ==, figureFileBig=KR7IukrzrJ7G6Am4cGxqMw==, tableContent=null), ArticleFig(id=1226462307312648988, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=图7, caption=单位吸收能与试样等球直径的关系, figureFileSmall=VT0xTYp1oBY+SvBBVI0ZoQ==, figureFileBig=KR7IukrzrJ7G6Am4cGxqMw==, tableContent=null), ArticleFig(id=1226462307396535074, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Fig.8, caption=Relationship between unit absorption energy and average particle size, figureFileSmall=V2W79AEUFtCXcssTgSMEfQ==, figureFileBig=3XeGgWLpazHHfF468wS8og==, tableContent=null), ArticleFig(id=1226462307509781287, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=图8, caption=单位吸收能与平均粒度的关系, figureFileSmall=V2W79AEUFtCXcssTgSMEfQ==, figureFileBig=3XeGgWLpazHHfF468wS8og==, tableContent=null), ArticleFig(id=1226462307581084461, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Fig.9, caption=Relationship between the logarithm of unit absorption energy and fractal dimension, figureFileSmall=aN1utwt/Ymt/WHJd5iyU5g==, figureFileBig=gVtMQL4TU6vTVLb6tKm2nw==, tableContent=null), ArticleFig(id=1226462307694330677, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=图9, caption=单位吸收能对数与分形维数的关系, figureFileSmall=aN1utwt/Ymt/WHJd5iyU5g==, figureFileBig=gVtMQL4TU6vTVLb6tKm2nw==, tableContent=null), ArticleFig(id=1226462307811771194, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Fig.10, caption=Coupling relationship surfaces of unit absorption energy with average fragment size and fractal dimension, figureFileSmall=MYF0eXZ9SlnimWZGPawgog==, figureFileBig=rpI5ROm0hQAOJbDJ8U0cgw==, tableContent=null), ArticleFig(id=1226462307899851581, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=图10, caption=单位吸收能关于碎块平均粒度、分形维数的耦合关系曲面, figureFileSmall=MYF0eXZ9SlnimWZGPawgog==, figureFileBig=rpI5ROm0hQAOJbDJ8U0cgw==, tableContent=null), ArticleFig(id=1226462308214424387, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Table 1, caption=

Density and uniaxial compressive strength of iron ores

, figureFileSmall=null, figureFileBig=null, tableContent=
样品编号密度/(g/cm3)单轴抗压强度/MPa
CL12.50368.257
CL22.63475.781
CL32.64237.890
CL42.751128.410
CL52.64162.790
CL63.15963.890
), ArticleFig(id=1226462308352836423, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=表1, caption=

铁矿石密度与单轴抗压强度

, figureFileSmall=null, figureFileBig=null, tableContent=
样品编号密度/(g/cm3)单轴抗压强度/MPa
CL12.50368.257
CL22.63475.781
CL32.64237.890
CL42.751128.410
CL52.64162.790
CL63.15963.890
), ArticleFig(id=1226462308461888333, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Table 2, caption=

Fitting results of the relationships between isosphere diameter and average particle size, fractal dimension

, figureFileSmall=null, figureFileBig=null, tableContent=
序号等球直径与碎块平均粒径等球直径与碎块粒度分形维数
a1b1相关系数a2b2相关系数
CL10.0072.1260.8303.6230.1450.706
CL20.3801.0490.4603.5980.1480.556
CL30.0671.5350.6024.0800.1870.649
CL40.0102.1110.6733.9220.1720.549
CL56.72×10－42.8490.7832.9490.0890.361
CL60.0082.1630.7133.1750.1120.652
), ArticleFig(id=1226462308608688980, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=表2, caption=

等球直径与平均粒度、分形维数关系的拟合结果

, figureFileSmall=null, figureFileBig=null, tableContent=
序号等球直径与碎块平均粒径等球直径与碎块粒度分形维数
a1b1相关系数a2b2相关系数
CL10.0072.1260.8303.6230.1450.706
CL20.3801.0490.4603.5980.1480.556
CL30.0671.5350.6024.0800.1870.649
CL40.0102.1110.6733.9220.1720.549
CL56.72×10－42.8490.7832.9490.0890.361
CL60.0082.1630.7133.1750.1120.652
), ArticleFig(id=1226462308692575064, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Table 3, caption=

The fitting results of the relationship between unit absorption energy and isosphere diameter

, figureFileSmall=null, figureFileBig=null, tableContent=
组号c1k相关系数
CL14.674×1073.0230.982
CL22.269×1072.8290.973
CL33.953×1072.9830.967
CL43.071×1072.9210.971
CL52.298×1072.8150.995
CL63.070×1072.9520.996
), ArticleFig(id=1226462308797432671, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=表3, caption=

单位吸收能与等球直径关系的拟合结果

, figureFileSmall=null, figureFileBig=null, tableContent=
组号c1k相关系数
CL14.674×1073.0230.982
CL22.269×1072.8290.973
CL33.953×1072.9830.967
CL43.071×1072.9210.971
CL52.298×1072.8150.995
CL63.070×1072.9520.996
), ArticleFig(id=1226462308885513057, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Table 4, caption=

Fitting results of the relationship between unit absorption energy and parameters of crushing characteristic

, figureFileSmall=null, figureFileBig=null, tableContent=
序号单位吸收能与碎块平均粒径单位吸收能对数与分形维数
mt相关系数sp相关系数
CL12627.6－0.7520.8561.4030.1100.723
CL2152.9－0.3330.4751.3420.1140.576
CL3390.6－0.4700.6031.1720.1340.694
CL41599.5－0.6510.7001.3130.1200.558
CL515784.1－0.9870.7921.6900.0660.350
CL62432.8－0.7300.7021.5830.0820.643
), ArticleFig(id=1226462309015536489, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=表4, caption=

单位吸收能与破碎特征参数关系的拟合结果

, figureFileSmall=null, figureFileBig=null, tableContent=
序号单位吸收能与碎块平均粒径单位吸收能对数与分形维数
mt相关系数sp相关系数
CL12627.6－0.7520.8561.4030.1100.723
CL2152.9－0.3330.4751.3420.1140.576
CL3390.6－0.4700.6031.1720.1340.694
CL41599.5－0.6510.7001.3130.1200.558
CL515784.1－0.9870.7921.6900.0660.350
CL62432.8－0.7300.7021.5830.0820.643
), ArticleFig(id=1226462309120394093, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Table 5, caption=

Fitting results of the coupling model between crushing energy consumption and crushing degree

, figureFileSmall=null, figureFileBig=null, tableContent=
样品编号C相关系数
CL19 803.80.837
CL211 845.30.727
CL312 622.70.841
CL411 701.50.713
CL511 154.10.769
CL611 183.30.898
), ArticleFig(id=1226462309288166264, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=表5, caption=

破碎能耗与破碎程度耦合模型的拟合结果

, figureFileSmall=null, figureFileBig=null, tableContent=
样品编号C相关系数
CL19 803.80.837
CL211 845.30.727
CL312 622.70.841
CL411 701.50.713
CL511 154.10.769
CL611 183.30.898
), ArticleFig(id=1226462309384635265, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=EN, label=Table 6, caption=

Fitting results of energy consumption model for irregular particle crushing of iron ore

, figureFileSmall=null, figureFileBig=null, tableContent=
组号c3kc4e0/(J/kg)相关系数
CL11.07×1010－4.349×1088 407.3261.400.831
CL23.92×108－1.745×10710 074.5214.300.699
CL31.05×106－3.694×10411 392.8160.100.823
CL43.20×106－1.817×1058 473.3406.500.750
CL57.10×108－2.892×1079 568.2252.860.744
CL63.79×1010－1.573×10910 366.6155.100.889
), ArticleFig(id=1226462309476909961, tenantId=1146029695717560320, journalId=1225396423026438145, articleId=1226462295660872589, language=CN, label=表6, caption=

不规则铁矿石颗粒破碎能耗模型拟合结果

, figureFileSmall=null, figureFileBig=null, tableContent=
组号c3kc4e0/(J/kg)相关系数
CL11.07×1010－4.349×1088 407.3261.400.831
CL23.92×108－1.745×10710 074.5214.300.699
CL31.05×106－3.694×10411 392.8160.100.823
CL43.20×106－1.817×1058 473.3406.500.750
CL57.10×108－2.892×1079 568.2252.860.744
CL63.79×1010－1.573×10910 366.6155.100.889
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不规则矿石颗粒冲击破碎能耗规律
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高锋 1 , 郭君 2 , 甘德清 3 , 许佳宁 3 , 邵静静 4
矿业研究与开发 | 矿山机电与矿业智能化 2025,45(10): 215-223
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矿业研究与开发 | 矿山机电与矿业智能化 2025, 45(10): 215-223
不规则矿石颗粒冲击破碎能耗规律
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高锋1 , 郭君2 , 甘德清3, 许佳宁3, 邵静静4
作者信息
  • 1.山西工程技术学院科研工作部，山西　阳泉市　045000
  • 2.唐山学院土木工程学院，河北　唐山市　063000
  • 3.华北理工大学矿业工程学院，河北　唐山市　063000
  • 4.山西工程技术学院矿业工程系，山西　阳泉市　045000

    • 高锋（1988一），男，安徽淮北人，博士，讲师，主要从事岩石破碎节能降耗工艺与理论研究。E-mail:

通讯作者:

郭君（1983一），男，陕西神木人，博士，讲师，主要从事岩石破碎学、爆破工程方面的研究。E-mail:
Energy Consumption Law of Irregular Ore Particles Under Impact Crushing
Feng GAO1 , Jun GUO2 , Deqing GAN3, Jianing XU3, Jingjing SHAO4
Affiliations
  • 1.Department of Scientific Research Work, Shanxi Institute of Technology, Yangquan, Shanxi 045000, China
  • 2.College of Civil Engineering, Tangshan University, Tangshan, Hebei 063000, China
  • 3.College of Mining Engineering, North China University of Science and Technology, Tangshan, Hebei 063000, China
  • 4.Department of Mining Engineering, Shanxi Institute of Technology, Yangquan, Shanxi 045000, China
出版时间: 2025-10-25
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摘要
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为揭示不规则矿石颗粒冲击破碎能耗规律，采用自主改进的落锤冲击试验机，对6种不同性质的铁矿石开展单颗粒冲击破碎试验，分析不规则铁矿石破碎的分形特征，以及碎块平均粒度、分形维数和单位吸收能的尺寸效应，建立不规则单颗粒矿石冲击破碎的能耗模型。结果表明，在相同的冲击条件下，碎块平均粒度、分形维数、单位吸收能与等球直径均呈幂函数关系，随着颗粒等球直径的增大，碎块平均粒度逐渐增大，分形维数与单位吸收能逐渐降低；单位吸收能与碎块平均粒度呈幂函数增大的变化关系，单位吸收能对数与分形维数呈线性增长的关系；通过不规则单颗粒矿石破碎试验的方法，建立了不规则颗粒单位破碎能关于初始尺寸、碎块平均粒度与分形维数的关系模型，平均相关系数为0.789，较好地描述了矿石破碎能耗规律。

矿石破碎  /  落锤冲击  /  不规则颗粒  /  尺寸效应  /  能耗模型

To reveal the energy consumption patterns of irregular ore particles under impact crushing, impact crushing tests were conducted on six types of iron ore with different properties, and the fractal characteristics of irregular iron ore fragmentation were analyzed, as well as the size effect on average particle size of fragments, fractal dimension, and unit absorption energy. Then, the energy consumption model of irregular single particle crushing was established. The results show that the average particle size of fragments, fractal dimension, unit absorption energy all exhibit a power function relationship with initial isosphere diameter under the same impact conditions. As the isosphere diameter of the particles increases, the average particle size of the fragments gradually increases, and the fractal dimension and unit absorption energy gradually decrease. There is an increasing relationship with power function between unit absorption energy and average particle size of fragments. The relationship between the logarithm of unit absorption energy and fractal dimension shows a linear increase. Through the method of ore crushing tests of irregular single ore particle, relationship models of unit crushing energy consumption of irregular iron ore particles changing with the initial size, average particle size of fragments, and fractal dimension were established, with an average correlation coefficient of 0.789, which effectively describes the energy consumption pattern of ore crushing.

Ore crushing  /  Drop hammer impact  /  Irregular particle  /  Size effect  /  Energy consumption model
高锋, 郭君, 甘德清, 许佳宁, 邵静静. 不规则矿石颗粒冲击破碎能耗规律. 矿业研究与开发, 2025 , 45 (10) : 215 -223 .
Feng GAO, Jun GUO, Deqing GAN, Jianing XU, Jingjing SHAO. Energy Consumption Law of Irregular Ore Particles Under Impact Crushing[J]. Mining Research and Development, 2025 , 45 (10) : 215 -223 .
正文
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0　引言
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采矿是钢铁、有色金属、建材、水泥等高耗能行业的上游产业，矿石破碎是矿山企业的主要作业形式，包括钻孔爆破、三段破碎和两段磨矿等，消耗能量巨大，据统计，2019年矿山企业能耗占我国工业能耗的5.96%[1]。国务院发布《2030年前碳达峰行动方案》，要求严格控制能耗强度，实施重点行业节能降碳工程。理论研究与生产实践表明，随着矿石粒度的降低，破碎能耗呈指数增长，但能量利用效率逐渐降低，如碎矿阶段能效仅为3%~5%，磨矿阶段能效不足1%，造成了电能的大量浪费[2]。矿石破碎节能降耗一直是采矿领域研究的热点，虽然近年来国内发展了一批先进碎磨矿技术与装备，但面向碳达峰目标矿山领域节能降耗的需求依然迫切[3-4]。因此，需要继续深入研究矿石破碎能耗规律，掌握矿石破碎粒度与能耗控制，为实现节能降碳改造升级提供依据。
矿石破碎能耗规律的研究已有150多年的历史，发展了经典的Rittinger面积假说(1867年)、Kick体积假说(1885年)与Bond裂隙假说(1961年)及基于三大假说研究提出的能耗与矿石粒度的关系模型[5-7]。相关文献研究表明，矿石与岩石的破碎具有分形特征，谢和平等[8]、XU等[9]基于分形几何理论，提出了破碎能耗关于破碎特征粒度、碎块分形维数的关系模型。矿石破碎是在能量作用下粒度不断减小的过程，在一定的能量作用下破碎特征存在尺寸效应。TAVARES等[10]总结了6种矿石、岩石的破碎特征，发现试件初始尺寸越大，单位体积断裂能越低，逐渐趋近于特定的常数。彭相愿[11]研究得出煤冲击破坏特征与能量演化存在尺寸效应，初始尺寸增加，单位体积吸收能降低。KING等[12]和SHI等[13]相关研究表明，初始尺寸较大的岩石或矿石试件破碎至相同程度，消耗的单位能量较低。李晓彤等[14]通过室内试验与数值模拟发现矸石块的破碎程度与试件尺寸存在相关性。XIAO等[15]研究了堆石料的破碎与变形特征的尺寸效应，发现单轴压缩的屈服应力随着颗粒尺寸的减小而增大。
单颗粒破碎可以避免颗粒摩擦、设备运动等因素产生的能量损失，能够准确模拟颗粒粒度减小的复杂过程，是研究破碎特征的有效方法[16]。LI等[17]通过单颗粒冲击破碎试验，研究铁矿石烧结球团的断裂行为。CAMPOS等[18]指出单颗粒破碎试验长期应用于表征材料破碎过程中的内部响应，提出了细颗粒铁精矿粉碎测试方法。FARAMARZI等[19]扩展了JK落锤试验装置的测试功能，用于分析单颗粒破碎过程中断裂异质性。
受矿岩石结构、尺寸和形状等因素的影响，不规则颗粒冲击破碎特征具有较大的离散性[20]，目前关于矿石颗粒冲击破碎程度和破碎能耗的初始尺寸效应研究较少，基于初始尺寸效应和碎块分形特征的矿石颗粒破碎能耗规律的认识尚不充分。我国铁矿石是仅次于煤炭的大宗矿产资源，2024年铁矿石原矿产量约14　亿　t，但以贫矿为主，铁矿石破碎作业规模巨大。以磨矿前的不规则铁矿石颗粒为研究对象，改进落锤冲击试验装置测量功能，研究不规则铁矿石破碎的尺寸效应和能耗规律，构建不规则铁矿石单颗粒破碎能耗模型，进一步丰富矿石破碎能耗理论体系研究，为铁矿山改进矿石粉碎工艺、实现节能增效提供理论依据。
1　不规则铁矿石落锤冲击破碎试验
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1.1　试样
采集首都钢铁集团水厂铁矿、马兰庄铁矿条带状和片麻状磁铁矿石，测试矿石密度与单轴抗压强度，见表1。将矿石样品破碎成6组20~110 g的不规则试样，每组35~40个质地均匀的颗粒，如图1所示。通过矿石密度和每个试样质量，计算等体积球直径作为初始尺寸，计算结果如图2所示。
1.2　落锤冲击试验
使用自主改造的落锤冲击试验机开展冲击粉碎试验。该试验系统由提升系统、落锤冲击系统、数据采集与分析系统、操作控制系统和制动与防二次冲击系统组成，采用全自动化控制，通过软件实现挂锤、落锤、冲击试验的全过程管控，可直观显示力-能量-位移等7种相关特征曲线，具有防二次冲击功能。落锤质量37.5 kg，最大冲击高度1.5 m，锤头内部置入冲击力传感器，量程100 kN，标定后测试冲击过程中锤头与矿石相互作用力。
试验时设定CL1~CL6共6组矿石的冲击高度为0.25 m，每组有效破碎的试样不少于25个。每个试样破碎后采用筛分法绘制碎块级配曲线，分析破碎分形特征，计算碎块粒度分形维数与平均粒度作为破碎特征参数，研究矿石试样初始尺寸（等球直径）对破碎特征的影响；每次冲击时实时采集冲击力与位移数据，计算矿石破碎的单位吸收能作为破碎能耗，分析破碎能耗与破碎特征参数的变化关系。
2　矿石冲击破碎特征分析
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2.1　矿石破碎分形理论
矿石在一定的能量作用下破碎成不同粒度的碎块，在不同的粒度范围内总能找到与初始矿石形状相近的碎块；同岩石一样，矿石的宏观破碎是其内部微裂隙不断萌生、发育、扩展、聚集和贯通的结果，裂隙发展变化的这一过程具有分形的性质；大的碎块由规模较大的裂隙切割分离，小碎块又由较小的裂隙切割分离，因此矿石破碎是一种自相似行为，碎块粒度也有自相似特征，可用标准筛筛分试验数据计算碎块分形维数[6]，见式(1)。
式中：d为碎块的特征粒度，mm；dmax为碎块的最大粒度，mm；M(r<d)为粒度小于特征粒度的碎块质量，g；MT为碎块的总质量，g；Ds为碎块粒度的分形维数。
2.2　不规则铁矿石破碎的级配特征
采用网孔尺寸为0.075 mm、0.16 mm、0.63 mm、1.25 mm、2.5 mm、5 mm、10 mm、16 mm、20 mm、25 mm的标准筛，以恒定的时间振动筛分落锤冲击破碎后的矿石碎块，对筛下磁铁矿石精准称重，得到筛下累积质量百分数，绘制代表性试样冲击破碎后碎块累积质量分布曲线，见图3（以CL1~CL3为例）。规则铁矿石落锤冲击破碎受矿石初始尺寸等因素的影响，级配曲线的曲率变化显著。
2.3　不规则铁矿石破碎的分形维数
将铁矿石碎块筛下累积质量百分数和筛孔特征尺寸对数进行拟合，得到拟合斜率n和拟合度，通过(3−n)得到每个试样落锤冲击破碎后的碎块粒度分形维数，见图4。分形维数计算的拟合度均大于0.90，分形维数范围为2.015~2.476，表明不规则铁矿石破碎分形特征显著。
2.4　不规则铁矿石破碎的平均粒度与破碎比
相同的冲击高度下，矿石颗粒破碎产生的碎块有较大的粒度差异性，但不同大小的碎块密度是相近的，对每个矿石试样的破碎结果进行分析时，可以使用加权平均粒度反映碎块整体的尺寸，计算式见式（2）。
式中：dp为碎块的平均粒度，mm；di为碎屑的特征粒度，采用相邻筛孔孔径的平均值表达，mm；mi为每组筛分过程中每个标准筛筛上的颗粒质量，g。
根据铁矿石冲击破碎后碎块的累积质量分布，计算每个试样冲击破碎后的平均粒度，得到每个试样破碎前后的破碎比，即铁矿石破碎之前初始尺寸与破碎后碎块平均粒度的比值。计算结果如图5所示，铁矿石试样落锤冲击破碎后平均粒度范围为3.571~26.439 m，破碎比的范围为1.259~7.698，发生有效破碎。
3　不规则铁矿石破碎特征的尺寸效应
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为探究不规则铁矿石落锤冲击破碎的尺寸效应，以等体积球直径为初始尺寸，建立相同冲击高度条件下破碎程度参数与初始尺寸的相关关系。绘制试样等球直径与平均粒度、分形维数的关系图，如图6所示。
在冲击高度一定的情况下，试样破碎前的等球直径对破碎程度的影响显著，随着不规则铁矿石试样等球直径的增大，落锤冲击破碎后的碎块平均粒度呈幂函数的形式增大，粒度的分形维数呈负幂函数的形式减小，相关关系见式(3)、式(4)。
式中：dp为碎块平均粒度，mm；d1为不规则铁矿石颗粒等球直径，mm；Ds为碎块粒度的分形维数；a1、a2、b1、b2为相关关系的拟合系数。
表2为等球直径与平均粒径、分形维数幂函数关系的拟合结果。铁矿石初始尺寸越大，内部包含的随机分布裂隙越多，破碎强度降低，在相同冲击高度下，铁矿石破碎时裂隙扩展切割块体的随机性更大，产生的碎块的块度大，粒度分布不均匀。
4　不规则铁矿石破碎能耗规律
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在一定的冲击条件下，矿石破碎能耗受初始形状、尺寸等因素的影响，与破碎程度参数同样存在相关关系。为研究不规则铁矿石破碎能耗规律，本文以矿石单位质量吸收能为破碎能耗，分析矿石破碎能耗随试样等球直径的变化特征，建立破碎能耗与破碎特征参数的量化关系。
4.1　破碎能耗的尺寸效应
对落锤冲击试验冲击力与矿石变形数据累积计算，得到铁矿石破碎的吸收能，计算吸收能与磁铁矿试样质量的比值，得到铁矿石破碎的单位吸收能，作为矿石单位质量破碎能耗，见式(5)。
式中：E为单位吸收能，J/kg；i为载荷变形数据组序号；li为第i组数据铁矿石的变形值，mm；li−1为第i−1组数据铁矿石的变形值，mm；Fi为第i组数据铁矿石载荷，kN；Fi−1为第i−1组数据铁矿石载荷，kN；mc为铁矿石质量，kg。
单位吸收能是冲击力对铁矿石做功时铁矿石变形吸收的单位能量，是促使铁矿石破碎的有效能量，使用单位吸收能作为单位能耗可避免其他破碎方式下产生的热能与动能损耗。以等球直径为初始尺寸，使用负幂函数拟合相同冲击高度下单位吸收能与等球直径的关系，结果如图7所示。从图7可以看出，矿石破碎单位能耗与矿石初始尺寸符合幂函数关系，随着初始尺寸的增大，单位破碎能呈现逐渐降低的变化趋势，且减小的速率降低，与文献[21]中用规则圆柱体试件的研究结果基本一致。
磁铁矿石试件内随机分布着裂隙、孔隙等缺陷，根据Griffith断裂准则，磁铁矿石试件内存在的裂隙、孔隙等缺陷是诱导破碎的主要因素[22]；试件发生破碎的难易程度取决于内部缺陷的密度，小尺寸试件的质地均匀程度较高，缺陷密度较小，断裂时需要较高的破碎能密度；随着试件尺寸的增大，缺陷数量增多，如果缺陷数量增长的速率超过体积增长速率，试件断裂需要的破碎能密度降低[23]。TAVARES等[10]认为矿石和岩石的破碎能与试件内部高应力区域的缺陷有关，随着矿石或岩石初始尺寸的增大，内部最大微裂隙的尺寸增大，导致单位破碎能降低，单位质量破碎能与矿石、岩石初始尺寸的关系符合幂函数规律。上述研究说明了图7中曲线关系的正确性，二者变化关系可用式(6)表示。
式中：e为单位吸收能，J/kg；c1为拟合系数；k为初始尺寸的幂指数，k>0。
矿石单位吸收能与等球直径关系的拟合结果见表3，二者有较好的相关性。
4.2　破碎能耗与破碎特征参数的关系
4.2.1　单位吸收能与平均粒度的关系
绘制并拟合单位吸收能与平均粒度的散点图（见图8），分析单位吸收能与平均粒度的相关关系。从图8可知，随着单位吸收能的增大，碎块平均粒度逐渐降低，但降低的速率减慢，表明矿石破碎的粒度越小，消耗能量越大。二者呈负幂函数的变化关系，可用式(7)表达。
式中：m为能量关系系数；t为能量指数。
铁矿石破碎单位吸收能与破碎特征参数关系的拟合结果见表4。矿石破碎后的碎块平均粒度越小需要的单位破碎能越高，矿石破碎至较小的碎块平均粒度后，单位破碎能耗迅速增加。
4.2.2　单位破碎能与粒度分形维数的关系
根据理论分析结果，拟合单位破碎能对数与粒度分形维数的线性关系，如图9所示。从图9可以看出，随着单位吸收能对数的增大，分形维数呈现线性增大的变化规律。单位吸收能越大，矿石破碎越充分，碎块粒度分布越均匀。
根据图9，单位破碎能与粒度分形维数的关系可用式(8)表达：
式中：s为相关关系系数；p为拟合常数。
4.3　破碎能耗与破碎特征参数的变化规律
Nagahama基于Griffith能量平衡理论、岩石粒度与颗粒数量分形关系和断面分形维数与粒度分形维数的数值关系，提出了破碎能量与粒度及分形维数的理论关系[24]，见式（9）：
式中：E为破碎能量密度，J/mm3，岩石密度为定时单位可转化为J/kg；r为破碎产品粒度，mm；φ为能量粒度关系指数。
谢和平等[8]在研究煤岩破碎分形模型时在理论上提出了能量分形公式，用破碎过程的自相似维数作为能量粒度关系的幂指数，见式（10）：
式中，C为能量系数。目前，式(10)尚没有经过单颗粒矿石破碎物理试验验证。
闫铁等[25]提出了能量粒度的分形关系公式，公式中用最大碎屑粒度rmax代替r，见式(11)，进而推导出钻井破碎岩石能效评价模型，并进行了应用。
如上所述，岩石破碎过程中累积单位能量与碎块粒度、分形维数之间有明确的数学关系，认为单颗粒矿石落锤冲击破碎能量与碎块粒度的理论关系同样符合上述关系式。
根据试验结果，以CL1~CL3组铁矿石样品为例，分别绘制单位吸收能关于碎块平均粒度、分形维数的耦合变化曲面图，如图10所示。可以看出，单位吸收能与碎块平均粒度、分形维数的综合相关性显著，随着碎块平均粒度和分形维数的增大，单位吸收能显著升高。使用公式(10)拟合铁矿石各级破碎后的碎块粒度、分形维数和单位破碎能量，能量系数和相关系数见表5。
由表5可知，铁矿石单颗粒破碎的单位破碎能与碎块平均粒度、分形维数的综合相关关系显著，可用式(12)表示。
式中，et为铁矿石单颗粒破碎的单位破碎能，J/kg.
从铁矿石单颗粒冲击破碎试验的角度验证了矿岩石破碎能耗关于碎块粒度、分形维数的理论关系的正确性。
5　不规则矿石单颗粒破碎能耗模型
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5.1　最低破碎能理论分析
岩石破碎存在能耗密度门槛值，对岩石施加外载荷时，岩石中会有一部分能量以弹性波的形式耗散，当施加的能量作用密度低于门槛值时，弹性波只能引起岩石内部损伤而不会使岩石断裂，Steverding-Lehnigk动态断裂准则显示，当应力波能量密度达到断裂准则的门槛值，裂纹就会扩展使岩石断裂或者破碎，见式(13)[6,26]。
式中：σ(t)为应力波幅值，Pa；t1为应力波作用时间，s；γ为岩石的比表面能，J/m2；Em为岩石弹性模量，GPa；v为岩石纵波波速，m/s。
SHI[13]考虑了初始粒度对破碎结果的影响，并提出了新的JK破碎模型，见式(14)。当矿石类型、密度、形状和初始尺寸一定时，模型中k的计算结果为常数。只有当破碎能大于矿石破碎所需的能量阈值时，矿石才会发生有效破碎[6,13]。
式中：t10为破碎产品通过粒度为初始粒度10%的累计质量百分数，%；Mm为材料破碎的最大程度指标，%；β为材料特征系数，(kW·h)−1·(mm)q−1；d1为矿石初始粒度，mm；q为拟合指数，一般情况下q的取值为(0, 1)；Ecs为单位破碎能量，(kW·h)/t或J/kg；E0为破碎所需的能量阈值，低于这一能量试件不能破碎，(kW·h)/t或J/kg。
5.2　铁矿石单颗粒破碎能耗模型构建
根据上述研究，不规则铁矿石颗粒破碎在理论上同样存在最低破碎能，结合不规则铁矿石单颗粒冲击破碎单位吸收能与初始尺寸（等球直径）、碎块平均粒度和分形维数的相关关系，构建不规则铁矿石单颗粒破碎的能耗模型，见式(15)。
式中：em为矿石颗粒从初始尺寸d1破碎到平均粒度d2的单位能耗，J/kg；c3、c4为拟合系数；e0为矿石颗粒的最小破碎能，J/kg。
使用式(15)拟合6组铁矿石颗粒落锤冲击破碎的单位吸收能、等球直径、碎块平均粒度与分形维数，计算结果见表6，其平均相关系数为0.789，表明式(15)可以较好地描述铁矿石颗粒破碎能耗随初始尺寸、碎块平均粒度、分形维数演化的综合规律。
6　结论
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针对6种不同性质的不规则铁矿石颗粒，使用落锤试验机开展单颗粒冲击破碎试验，分析相同冲击条件下铁矿石单颗粒破碎特征的尺寸效应，研究单位吸收能与碎块平均粒度、分形维数的变化规律，从物理试验的角度构建了不规则矿石颗粒从初始尺寸破碎到控制粒度与分形的能耗模型，拓展了矿石破碎能耗理论。

（1）铁矿石单颗粒冲击破碎的平均粒度与颗粒等球直径呈幂函数关系，分形维数与等球直径呈负幂函数关系；随着颗粒初始尺寸的增大，碎块平均粒度逐渐增大，分形维数逐渐减小；初始尺寸越大，破碎后产生的大块越多，粒度分布越不均匀。

（2）落锤冲击破碎能耗存在显著的尺寸效应，随着初始尺寸的增大，单位破碎能耗降低，但变化速率减小，在小尺寸颗粒范围内，破碎能耗的变化较快。

（3）随着矿石单位吸收能的增大，碎块平均粒度以负幂函数的形式减小；粒度分形维数随单位吸收能对数的增大线性增大。单位吸收能与碎块平均粒度、分形维数之间的综合相关性显著。

（4）不规则铁矿石颗粒破碎存在最低破碎能，构建了铁矿石颗粒单位破碎能耗关于初始尺寸、碎块平均粒度与分形维数的数学模型，其平均相关系数为0.789，可以较好地描述变化规律。

基金
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  • 山西省基础研究计划资助项目(20210302124135)
  • 山西省高等学校科技创新项目(2021L581)
  • 山西工程技术学院科研启动项目(2021QD-10)
文献
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2025年第45卷第10期
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  • 接收时间：2025-03-16
  • 首发时间：2026-02-06
  • 出版时间：2025-10-25
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  • 收稿日期：2025-03-16
基金
山西省基础研究计划资助项目(20210302124135)
山西省高等学校科技创新项目(2021L581)
山西工程技术学院科研启动项目(2021QD-10)
作者信息
    1.山西工程技术学院科研工作部，山西　阳泉市　045000
    2.唐山学院土木工程学院，河北　唐山市　063000
    3.华北理工大学矿业工程学院，河北　唐山市　063000
    4.山西工程技术学院矿业工程系，山西　阳泉市　045000

通讯作者:

郭君（1983一），男，陕西神木人，博士，讲师，主要从事岩石破碎学、爆破工程方面的研究。E-mail:
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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