Template:Model theory (Text, Mill, Perfect Mixing, Population Balance): Difference between revisions

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* <math>s_i</math> is the mass of solids on size interval <math>i</math> in the mill load
* <math>s_i</math> is the mass of solids on size interval <math>i</math> in the mill load
* <math>R_i</math> is the breakage rate of solids on size interval <math>i</math> in the mill load  
* <math>R_i</math> is the breakage rate of solids on size interval <math>i</math> in the mill load  
* <math>A_{ij}</math> is the Appearance function, the distribution of particle mass arising from the breakage of a parent particle in size interval <math>j</math> into progeny of size interval <math>i</math>
* <math>A_{ij}</math> is the appearance function, the distribution of particle mass arising from the breakage of a parent particle in size interval <math>j</math> into progeny of size interval <math>i</math>


As the mill is perfectly mixed, the product is related to the mill contents and discharge rate as:
As the mill is perfectly mixed, the product is related to the mill contents and discharge rate as:
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:<math>p_i=\dfrac{f_i + \sum\limits_{j=1}^{i}{A_{ij}\dfrac{R_j}{D_j}p_j}}{1+\dfrac{R_i}{D_i}}</math>
:<math>p_i=\dfrac{f_i + \sum\limits_{j=1}^{i}{A_{ij}\dfrac{R_j}{D_j}p_j}}{1+\dfrac{R_i}{D_i}}</math>


The mill product can therefore be computed provided a feed rate, Appearance function and ''breakage rate per discharge rate'', <math>R_i/D_i</math>, is available. Alternatively, the <math>R_i/D_i</math> function can be determined from the feed rate, product rate and Appearance function.
The mill product can therefore be computed provided a feed rate, appearance function and ''breakage rate per discharge rate'', <math>R_i/D_i</math>, is available. Alternatively, the <math>R_i/D_i</math> function can be determined from the feed rate, product rate and appearance function.

Revision as of 05:31, 14 June 2023

The Perfect Mixing model is based on a population balance of particles entering the mill, breaking into smaller sizes, and discharging as product. For a mill operating in steady-state, the diagram in Figure 1 below represents the balance for a given size fraction:

The steady-state population balance is formulated mathematically as:

where:

  • is the index of the size interval, , is the number of size intervals
  • is the mass flow rate of solids of size interval in the mill feed
  • is the mass flow rate of solids of size interval in the mill product
  • is the mass of solids on size interval in the mill load
  • is the breakage rate of solids on size interval in the mill load
  • is the appearance function, the distribution of particle mass arising from the breakage of a parent particle in size interval into progeny of size interval

As the mill is perfectly mixed, the product is related to the mill contents and discharge rate as:

where is the rate of discharge of solids in size interval from the mill.

Substitution and rearrangement of the above equations leads to:

The mill product can therefore be computed provided a feed rate, appearance function and breakage rate per discharge rate, , is available. Alternatively, the function can be determined from the feed rate, product rate and appearance function.