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      Tromp Curve Explanation              
                               
        1    Introduction:                
            - In order to determine the performances of a separator, one uses generally the Tromp curve  
              so called separation curve or selectivity curve.          
            - Of this Tromp curve are generated various parameters allowing to compare the separator with  
              other generations of separators. We shall see in detail the Tromp curve further in the presentation.  
            - Another tool of analysis is the efficiency of the separator. Contrary to the Tromp curve which is based  
              on the recovery in the rejects, the efficiency is based on the recovery in the fines of the separator.  
            - To establish a Tromp curve, a sampling campaign in good conditions must be organized . The following link 
              allows to know better the procedure to obtain analyzable samples:      
                 
     
         
        2    Background of the theory:            
             
     
               
                         
                         
                         
                         
                         
                         
                         
                         
                         
                         
                         
            - We have:                
              F = throughput of the separator feed in tons/hour          
              R = throughput of the separator rejects in tons/hour        
              P = throughput of the separator fines in tons/hour          
            - and:                
              f = passing at a certain sieve of the separator feed in %        
              r = passing at a certain sieve of the separator rejects in %        
              p = passing at a certain sieve of the separator fines in %        
              NB: These percentages of passing are generally cumulated and are expressed in %     
              Warning: In some cases, there can also be specified in mass unit (as grams or kg)    
            - and:                
              Δf = fraction of f for a defined interval in %          
              Δr = fraction of r for a defined interval in %          
              Δp = fraction of p for a defined interval in %           
              Warning: In some cases, there can also be specified in mass unit        
            - In a steady state, let's pose the following two equations:        
             
     
                 
                           
             
     
               
                         
            - From these 2 equations, we define the percentage of fines leaving the separator    
              in relation to the quantity fed to the separator (Vf):          
             
     
               
                         
                         
              From Vf, we define Vr:              
             
     
                 
                           
        3    Circulation factor calculation:            
            - The circulation factor can be calculated in various ways and is also called circulating load.    
              The most important thing is to know what we're talking about regardless of how it is expressed.  
            - In this presentation, the circulation factor is the ratio feed/fines and is defined by:    
             
     
                 
                           
                           
            - We can notice that it is the reverse of Vf          
            - The circulating load will be the ratio tails/fines (often expressed in %) is defined by:    
             
     
             
                       
                       
            - Theorically, the circulation factor must be equal for all sieves but it is not the case in reality.  
              A well accepted method is to calculate the average.        
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        4    Separator's efficiency:              
            - The separation efficiency is the proportion of passing at a certain sieve which passes from    
              the feed to the fines in %.            
            - Its equation is:              
             
     
             
                       
                       
            - The separation's efficiency doesn't give any idea of what proportion of fines particles go back to the   
              rejects. It is why the Tromp curve has been developed.        
        5    Tromp curve:                
            - The Tromp curve is an effective tool to evaluate separator performance.      
            - The Tromp curve is a chart showing the probability of a given size of particle in the separator  
              feed that will go to the rejects.             
            - This probability is also called "degree of selectivity".        
            - This probability is calculated on each size fraction of the sample.      
              A size fraction is for example the passing in % between 4µ and 8µ of the feed      
            - The general formula to expressed the Tromp curve for each fraction size is:      
             
     
               
                         
                         
            - The expression we used in the previous presentation was:        
             
     
               
                         
                         
            - Both expression are identical. See the small infography here:
     
         
            - One can calculate the Tromp curve using each circulation factor        
              or using the average circulation factor. In fact, results are not so        
              different like we can observe on the graph below, but we prefer      
              to calculate it with the average of the circulation factors. The main important is to have a good   
              correlation in order to well interpret the different parameters.         
            - To be representative, the Tromp Curve has to be a correlation of at least 0,99.      
            - The correlation is calculated by the least squares method.        
            - The graph below show that it is only on the left of the minimum of the Tromp curve (in the area where  
              nobody knows what is happening) that there is a notable difference.      
             
     
       
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
        6    Tromp curve parameters:            
          6.1 Cut size:                
            - D50 is the cut size. Dimension for which there are (from the separator feed)      
              50% of passing going in the rejects. In other words, the size at which the quantities    
              of fine and coarse material are equal.            
          6.2 D limit (Limit dimension):              
            - On the left of this point, the curve rises again. It means that there is no      
              more selective separation below D limit.          
          6.3 By-pass:                
            - Minimum percentage which returns to the rejects at a certain dimension (often the Dlimit).   
            - The by-pass has to be the lowest possible.          
          6.4 Sharpness of separation:              
            - An other mean to analyse a separator is the sharpness of separation defined by the formula below:  
             
     
                 
                           
                           
            - Where D75 is and D25 are the dimensions in µm at 75% and 25% in the y-axis of the curve.     
            - An ideal separator has a Sh of 1.            
              Warning: if the by-pass is higher than 25%, which is mostly the case for first generation separators,  
              it is imperative to extrapolate the value of D25          
          6.5 Imperfection factor:              
            - Last but not least, the imperfection factor gives a excellent idea of the separator behaviour. It is also good to
              compare separators between them.            
            - It is given by the following formula:            
             
     
                 
                           
                           
            - Where D75, D50 is and D25 are the dimensions in µm at 75% and 25% in the y-axis of the curve.   
            - It should be as small as possible.            
            - Usual figures and grades:              
              I < 0,2 Excellent separator            
              0,2 < I < 0,3 Good separator            
              0,3 < I < 0,4 Medium separator            
              0,4 < I < 0,6   Poor separator            
              0,6 < I < 0,7 Bad separator            
              I > 0,7 Excecrable separator            
          6.6 Illustration of the parameters:            
             
     
         
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
            - See the complete infography here below:          
                 
     
         
          6.7 Tromp curve reduced:              
            - On the left of the Dlimit, the finest particles are following the flow and are splitted between  
              the rejects and the fines in a total incertitude. There is no more separation in this zone.    
            - In order to remove the effect  of that zone, the Tromp curve is reduced by the following equation:  
           
     
                     
                               
                               
              Where Tr = Tromp reduced            
              and Bp = By-pass              
            - The result is showed on this graph:            
             
     
         
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
        Warning: Results of interpretation may be different in function of the type of graph, the scales and other extrapolations
        Note: Example and Exercise pages didn't need to be updated           
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