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      Laws of Comminution                
                             
      1 Introduction                  
        - Particle size reduction, or comminution is an important step in many technological operations.    
        - In the cement industry, the grinding of the clinker and raw material is the process which requires the highest percentage
          of energy.                  
        - Consequently, the high costs of energy drives us to optimize the processes in the plant, and among them the milling process.
        - Various methods to study the grinding process and to know the energy required have been developed (like the work index).
        - All these methods are coming from the comminution laws.          
                             
      2 The three laws of comminution              
      2.1 First law - RITTINGER (1867):                
        - According to this law, the area of the new surface produced by crushing or grinding is directly proportional to the useful 
          work consumed, is to say that the fragmentation work is proportional to the sum of new surfaces produced.  
        - Mathematically, it is expressed by the following relation:          
         
     
                 
          Where:                  
          W is the energy consumed,              
          d is the dimension of the particles of the product,            
          D is the dimension of the particles of the feed,             
          Si is the initial area,                
          Sf is the final area,                
          and K is a coefficient                
        - K coefficient depends on the shape of the particle, the type of material, the number of defects and the efficiency of the forces
          applied for the comminution work.              
        - This law only applies to the fragmentation of fine particles (< 100μ).        
      2.2 Second law - KICK (1885):                
        - This law says that the energy required is directly proportional to the volume reduction between particles before and after
          the fragmentation operation (crushing), ie proportional to the variation of volume of the particles.    
        - Mathematically, it is expressed by the following relation:          
         
     
                   
          Where:                  
          W is the energy consumed,              
          d80 is the size for 80% of passing of the product,            
          D80 is the size for 80% of passing of the feed            
          and K is a coefficient                
        - This law only applies for the crushing, i.e coarse fragmentation (>10 cm).        
      2.3 Third law - BOND (1952):                
        - This law says that the work consumed is proportional to the new crack length produced by the breakage of particles,  
          because once created the crack, the rock breaks.            
        - Mathematically, it is expressed by the following relation:          
         
     
                 
          Where:                  
          W is the energy consumed,              
          d80 is the size for 80% of passing of the product,            
          D80 is the size for 80% of passing of the feed            
          and Wi is the Bond Index                
        - The Bond Index is defined by a grindability test.            
        - Concerning the Bond Index, see the following links:            
          http://www.thecementgrindingoffice.com/grindabilitytests.html        
          http://www.thecementgrindingoffice.com/cemballmillsizingexpl.html        
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      3 Summary chart                
         
     
         
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
      4 General law                  
        - Charles, Holmes (1957) and Hukki (1961) attempted to unify the laws described here above.      
        - These attempts concluded that these three proposals can be considered as being the integrals of the same differential equation.
        - This differential equation is the following:            
         
     
                   
          Where:                  
          dW is the variation of energy consumed in comminution,          
          K is a constant related to the material,              
          dx is the variation in dimension that needs a work dW per unit of volume,        
          x is the characteristic dimension of the product and            
          n is the exponent (depending on the shape of crushing, tests on samples)        
        - And the summary chart becomes:              
         
     
               
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                             
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