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      Mass Balance of grinding circuits        
                   
      1   Introduction        
        - Like the heat balance of a grinding plant, the mass balance responds to a very simple rule: anything that comes
          in the circuit will go out of the circuit.        
        - This is illustrated by the small picture below:      
         
     
         
               
               
               
               
               
        - This short presentation introduces the mass balance calculators that were developed for 7 different grinding circuits.
        - The seven circuits are the following:        
        - Open circuit:        
         
     
         
               
               
               
               
               
               
               
        - Closed circuit (separator of the first generation, product of the filter in the finished product)  
         
     
         
               
               
               
               
               
               
               
               
               
               
               
        - Closed circuit (separator of the 1st generation, filter product to the elevator)    
         
     
       
             
             
             
             
             
             
             
             
             
             
             
        - Closed circuit (airswept mill with static separator)      
         
     
         
               
               
               
               
               
               
               
               
               
               
        - Closed circuit (airswept mill with static separator and cyclone)      
         
     
       
             
             
             
             
             
             
             
             
             
             
        - Closed circuit (separator of the 1st generation with static separator)    
         
     
         
               
               
               
               
               
               
               
               
               
               
               
               
        - Closed circuit (birotateur mill, separator of the 2nd or 3rd generation with cyclones, with static separator)
         
     
       
             
             
             
             
             
             
             
             
             
             
             
             
             
             
             
             
             
        - Below, we develop a method to determine the flow rates in the different parts of the circuit.  
                   
      2   Necessary data        
       2.1   Production - output:        
        - We assume that we only know the flow rate given by the weighfeeders (clinker, gypsum ... etc).
        - One does not consider other possible rates that may be encountered, such as rejects flow meters for example.
         
     
       
             
             
       2.2   Results of a circuit sampling:        
        - A sampling campaign should be organized when the mill is steady.      
        - Samples are to be taken according to the type of circuit:      
        - Open circuit:        
          * Mill output        
          * Filter product        
          * Finished product        
        - Closed circuit (separator of the first generation, product of the filter in the finished product):  
          * Mill output (= separator feed)        
          * Separator rejects        
          * Separator fines        
          * Filter product        
          * Finished product        
        - Closed circuit (separator of the 1st generation, filter product to the elevator):    
          * Mill output        
          * Filter product        
          * Separator feed        
          * Separator rejects        
          * Separator fines        
          * Finished product        
        - Closed circuit (airswept mill with static separator):      
          * Mill output (in fact, it is a sample to be taken inside the mill at the exit!)    
          * Static separator rejects        
          * Filter product (= finished product)        
        - Closed circuit (airswept mill with static separator and cyclone):      
          * Mill output (in fact, it is a sample to be taken inside the mill at the exit!)    
          * Static separator rejects        
          * Cyclone(s) fines        
          * Filter product        
          * Finished product        
          NB: Please note that there is still an impossible point to collect: fines of the static separator!  
        - Closed circuit (separator of the 1st generation with static separator):    
          * Mill output        
          * Static separator rejects        
          * Separator feed        
          * Separator rejects        
          * Separator fines        
          * Filter product        
          * Finished product        
          NB: Like in the previous case, there is still an impossible point to collect: fines of the static separator!
        - Closed circuit (birotateur mill, separator of the 2nd or 3rd generation with cyclones, with static separator):
          * Mill output        
          * Static separator rejects        
          * Separator feed        
          * Separator rejects        
          * Separator fines        
          * Mill filter product        
          * Separator filter product        
          * Finished product        
          NB: Please note that there is still an impossible point to collect: feed of the static separator!  
        - Moreover, one can also analyze the fresh material, not because it helps in the calculation of mass balance,
          but to get an idea of the reduction rate in the mill.      
        - This additional work is, of course, not mandatory.      
                   
      3   Equations used        
       3.1   Separation systems:        
        - Separation systems present in the 7 circuits described here are:      
          * Dynamic Separator        
          * Static Separator        
          * Cyclone        
        - Other systems may also be included in this list:      
          * Vibrating screens        
          * V-separators        
        - These systems normally have one input and two outputs      
        - They can be represented by this simple flowsheet:      
         
     
           
                 
                 
                 
                 
                 
                 
                 
                 
        - Moreover, all kinds of filters are also included in these systems.      
        - But in the case of the calculators of the website, filters are considered to have an efficiency of 100%.
        - That means that all the material will go out on one side and the air (or gas) will go out on the other side.
        - The equations are:        
         
     
           
                 
         
     
           
                 
        - With:        
          F = System feed output in tons/hour        
          R = System rejects output in tons/hour        
          P = System fines output in tons/hour        
        - And:        
          f = passing at a certain sieve of the system feed in %      
          r = passing at a certain sieve of the system rejects in %      
          p = passing at a certain sieve of the system fines in %      
        - NB: These percentages of passing are cumulated.      
        - In the case of static separator or cyclone, it is almost always impossible to take samples from the fines and/or feed.
        - Then we must use the following tricks:        
          * In the case of an airswept mill where all the material goes to the static separator, we have to take a sample
          inside the mill and at the exit of it.        
          This sample will serve as reference for the static feed.      
          * In the other cases, it is necessary to define a hypothesis (for what it's worth).    
          In our case, we choose the efficiency of the static separator or cyclone as chosen value.  
        - Keep in mind that the efficiency on a given sieve is defined by the following formula:  
         
     
           
                 
                 
       3.2   Finished product:        
        - The finished product is often the sum of products coming from different systems.  
        - The "finished product" area of the grinding circuits has normally two or more inputs and one output.
        - They can be represented by this simple flowsheet:      
         
     
           
                 
                 
                 
                 
                 
                 
                 
                 
        - The equations are similar to those of separation systems:      
         
     
           
                 
         
     
         
               
        - With:        
          PF = finished product output in tons/hour        
          P1 = product 1 output in tons/hour        
          P2 = product 2 output in tons/hour        
          P3 = product 3 output in tons/hour        
        - And:        
          pf= passing at a certain sieve of the finished product in %      
          p1= passing at a certain sieve of the product 1 in %      
          p2 = passing at a certain sieve of the product 2 in %      
          p3 = passing at a certain sieve of the product 3 in %      
        - NB: These percentages of passing are cumulated.      
       3.3   Systems of equations:        
        - Of course, and depending on the complexity of the circuit one can have to solve systems of m linear equations
          with n unknowns.        
        - We must therefore make sure that the following condition is respected:    
         
     
           
                 
                   
      4   Sieves used in the calculation        
        - To take representative samples in a grinding circuit is and will always be difficult.  
        - We must take into account factors such as:        
          * Level of stability of the system        
          * Diverse hazards as equipment problems or variations of the fresh feed    
          * Human error        
        - The ideal would be to achieve the results from a maximum range of sieves but this is not a realistic option.
        - There is also the measurement of the Blaine which could be a solution, but the risk of error is too large.
        - The calculators of the website are therefore based on the sieve of reference used in almost all cement plants.
        - The sieves available are: 32, 45, 63, 90, 150 and 210 microns.      
        - NB: The last two sieves are rather reserved for raw meal grinding circuits.    
                   
      5   Link to calculators        
                   
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