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      Tromp Curve Explanation              
                             
        1 Introduction:                
          - In order to determine the performances of a hydrocyclone, 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 hydrocyclone with  
            other cyclones.              
          - We shall see in detail the Tromp curve further in the presentation.      
          - Another tool of analysis is the efficiency of the hydrocyclone.        
          - Contrary to the Tromp curve which is based on the recovery in the underflow, the efficiency is based on
            the recovery in the overflow of the hydrocyclone.          
          - To establish a Tromp curve, a sampling campaign in good conditions must be organized .     
     
        2 Background of the theory:            
           
     
             
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
                     
          - We have:                
            F = throughput of the hydrocyclone feed in tons/hour        
            U = throughput of the hydrocyclone underflow in tons/hour        
            O = throughput of the hydrocyclone overflow in tons/hour        
          - and:                
            f = passing at a certain sieve of the hydrocyclone feed in %        
            u = passing at a certain sieve of the hydrocyclone underflow in %      
            o = passing at a certain sieve of the hydrocyclone overflow 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 %          
            Δu = fraction of u for a defined interval in %          
            Δo = fraction of o 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 cyclone    
            in relation to the quantity fed to the cyclone (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/overflow 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 Cyclone efficiency:              
          - The separation efficiency is the proportion of passing at a certain sieve which passes from the feed to the
            overflow in %.              
          - Its equation is:              
           
     
             
                     
                     
          - The separation's efficiency doesn't give any idea of what proportion of fines particles go back to the underflow.
          - It is why the Tromp curve has been developed.          
        5 Tromp curve:                
          - The Tromp curve is an effective tool to evaluate the hydrocyclone performance.    
          - The Tromp curve is a chart showing the probability of a given size of particle in the hydrocyclone feed that will go 
            to the underflow.               
          - 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 expressions 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 underflow.
          - 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 underflow 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 cyclone would be a Sh of 1.            
            Warning: if the by-pass is higher than 25%, which is mostly the case for high circulatng load,    
            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 cyclone behaviour.    
          - It is also good to compare hydrocyclones between them.        
          - I 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 < 0,4  Good hydrocyclone            
            * 0,4 < I < 0,6 Medium hydrocyclone            
            * 0,6 < I < 0,8  Poor hydrocyclone             
            * I > 0,8 Bad hydrocyclone            
        6.6 Illustration of the parameters:            
           
     
                   
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
          - See the complete infography here below:          
               
     
           
        6.7 Tromp curve reduced:              
          - On the left of the D limit, the finest particles are following the flow and are splitted between the underflow 
            and the overflow.              
          - There is no more separation in this zone.          
          - This is often called the flow splitting or "dead flux".        
          - In order to remove the effect  of the flow splitting from the separation definition, 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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