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      Particle Size Distribution - Representation          
                             
      1    Introduction:                
            Analysis of samples of material in the mining industry, either by dry or wet sieving, or by laser are very extensive.
            The results arrived at the office of the Process Engineer will be subject to interpretation.    
            Five methods are often used to interpret the results of a particle size analysis: Linear, Linear + x-log scale,
            Log - Normal, Gaudin - Shuhmann and Rosin - Rammler distributions.      
            The five methods are described below.          
            This list is not exhaustive.            
                             
      2    Linear distribution:              
          - It is the simplest representation.            
          - Example:
     
                 
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
          - X and Y axis are linear.              
          - This linear distribution is not very used in the mining industry.        
          - It is difficult to interpret something with this kind of curve.        
          - It is OK to compare various samples.            
          - This method is the one used in the PSD Calculator (http://www.themininggrindingoffice.com/psd.html) 
            because the HTML converter is not able to represent more complicated graphs.    
                             
      3    Linear distribution with x-log scale axis:          
                             
          - This representation is very common and widely used.        
          - Example:                
             
     
                 
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
          - X-axis has a log scale.              
          - Y-axis has a linear scale.              
          - It is difficult to interpret something with this kind of curve.        
          - It is OK to compare various samples.            
                             
      4    Log - Normal distribution:              
                             
          - The Y-axis is a probability of passing cumulated in %, then it uses a probability scale.    
          - See the complete explanation of a Log - Normal distribution in Wikipedia:      
            http://en.wikipedia.org/wiki/Log-normal_distribution#Characterization  
          - Example:
     
                 
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
          - X-axis is logarithmic.              
          - A special paper is needed to plot a graph.          
          - Also, it is possible to create such distribution in a software like Excel with some effort.  
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      5    Gaudin - Shuhmann distribution:            
          - The Gaudin - Shuhmann distribution is also called power fit distribution.      
          - The Gaudin-Schumann equation used to determine the size distribution is as follows:     
           
     
                 
                         
                         
            Where:                
            P = mass passing (%)               
            x = particle size in microns            
            k = size modulus - size when P = 100             
            m = distribution modulus = slope of the log-log plot P vs x         
          - Considering that we take the logs on both sides of the equation, we have:      
           
     
               
            and          
                       
           
     
             
                     
            and:                 
           
     
               
                       
            Where:                
            C = intercept of the line              
          - In a log-log graph, the Gaudin-Shuhmann distribution should be a straight line.    
          - Example:                
                             
           
     
                   
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
          - When the Gaudin - Shuhmann distribution is plotted (blue line in the graph hereabove), it is still necessary to
            calculate a trendline (or a linear regression) in order to know the slope of the PSD (particle size distribution).
          - In the example, m = 0,325.            
          - Note that the calculation of a linear regression is explained below.      
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      6    Rosin - Rammler - Bennett distribution (RRB):          
          - Also called Rosin - Rammler - Sperling - Bennett (RRSB) distribution.      
          - Also called Weibull distribution.            
          - It is probably the most well-known distribution in the cement and the mining industry.    
          - RRB is widely used to analyse all types of materials, crushed or not, ground or not.    
          - The conventional RRB function is described by:           
           
     
                 
                         
                         
            Where:                
            R = mass retained in %              
            x = size in microns              
            m = slope of the plot              
            a = size at 36,8% of particles retained (36,8 is the ratio 100/e and e is the Neper number - 2,718)  
          - From the equation hereabove, we obtain:          
           
     
                 
                         
                         
          - Considering that we take the logs on both sides of the equation, we have:      
             
     
             
                       
                       
             
     
           
            and:        
                     
             
     
         
            and:      
                   
             
     
           
            and:        
                     
          - The size parameter a can be determined by classifying a given material on a mesh size a = x.  
            This substitution in the main equation hereabove will produce a constant of about 36.8% material retained.
          - The RRB representation with a log.log vs log should be a straight line.      
          - Example:                
                             
           
     
                   
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
          - We can see that it is necessary to modify the Y axis in order to get a readable and intelligible representation.
          - On the Y-axis, -1,0 must be replaced by 20,6% of residue cumulated, -0,800 by 30,6%, etc...    
          - When the RRB distribution is plotted (blue line in the graph hereabove), it is still necessary to  
            calculate a trendline (or a linear regression) in order to know the slope of the PSD (particle size distribution).
          - In the example, m = 0,9027 with a normal correlation.        
          - The slope of the PSD of a cement, for example, is an important factor. More the slope is higher and tighter is the PSD.
          - The RRB calculator of the website doesn't have the right scale due to the limitations of the Excel converter.
              http://www.themininggrindingoffice.com/psd.html      
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      6    Linear regression and correlation:            
          - A linear regression (or trendline) has the following equation:        
           
     
                 
                         
            Where:                
            m is the slope of the line              
            b is the intercept at the origin            
          - m and b are defined with the method of least squares.        
          - The equations for calculating m is the following:          
           
     
               
                       
                       
          - The equations for calculating b is the following:          
           
     
                 
                         
                         
            where n is the sieve number            
          - To check if the trendline is in phase with the real data (particle size, residue cumulated in %), it is necessary to calculate
            the correlation with the following formula:          
           
     
             
                     
                     
                     
          - Example:                
           
     
         
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
           
     
           
                   
                   
           
     
           
                   
                   
          - See that results are the same as those of the Excel graph trendline hereabove    
          - Now, we have to check if the correlation r is OK:          
           
     
       
               
               
           
     
                 
                         
          - The trendline is OK.              
                             
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