Template:Model theory (Text, Ball Mill, Perfect Mixing, Breakage Scaling)

The scaling factors are defined as:

${\begin{array}{lll}f_{\rm {D}}={\sqrt {\dfrac {D_{\rm {Sim}}}{D_{\rm {Orig}}}}},&&&f_{\rm {LF}}={\dfrac {(1-{\rm {LF}}_{\rm {Sim}}).{\rm {LF}}_{\rm {Sim}}}{(1-{\rm {LF}}_{\rm {Orig}}).{\rm {LF}}_{\rm {Orig}}}},\\f_{\rm {CS}}={\dfrac {(C_{\rm {s}})_{\rm {Sim}}}{(C_{\rm {s}})_{\rm {Orig}}}},&&&f_{\rm {WI}}=\left({\dfrac {{\rm {WI}}_{\rm {Orig}}}{{\rm {WI}}_{\rm {Sim}}}}\right)^{0.8},\\\end{array}}$

$f_{\rm {Db}}={\begin{cases}{\dfrac {Db_{\rm {Orig}}}{Db_{\rm {Sim}}}}&{\text{for }}x<x_{\rm {m(small)}},\;\;\;x_{\rm {m(small)}}=\min {\left(K.Db_{\rm {Orig}}^{2},K.Db_{\rm {Sim}}^{2}\right)}\\\left({\dfrac {Db_{\rm {Orig}}}{Db_{\rm {Sim}}}}\right)^{2}&{\text{for }}x\geq x_{\rm {m(large)}},\;\;\;x_{\rm {m(large)}}=\max {\left(K.Db_{\rm {Orig}}^{2},K.Db_{\rm {Sim}}^{2}\right)}\\\end{cases}}$

where:

• $D$ is mill diameter (m)
• ${\rm {LF}}$ is load fraction, the load volume as a fraction of mill volume (v/v)
• $C_{\rm {s}}$ is the fraction critical speed of the mill (frac)
• ${\rm {WI}}$ is the Bond Ball Work Index of the ore (kWh/t)
• $Db$ is the ball diameter (mm)
• $x$ is the diameter of a particle of size interval $i$ (mm)
• $K$ is the maximum breakage rate factor which relates ball size and the size at which $R_{i}/D_{i}^{*}$ is maximum, i.e. $x_{m}=K.Db^{2}$
• $f_{\rm {Db}}$ is interpolated for $x_{\rm {m(small)}}<x<x_{\rm {m(large)}}$

and the ${\rm {Orig}}$ subscript refers to the original mill from which $R_{i}/D_{i}^{*}$ was derived and ${\rm {Sim}}$ refers to the mill being simulated (scaled).