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      Tromp Curve Explanation              
        1 Introduction:                
          - In order o 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.   
        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   
          - 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. Ther 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           
    All rights reserved © 2012-2016 The Mining Grinding Office