User:Dhatfield/Sandbox Two - product formula

Two product formula
To derive the two-product formula, define mass balances and recovery. 1 F= C+U Feed (tph) = Coarse (tph) + Underflow (tph) 2 Ffi=Cci+Uui Mass balance for each size fraction i.  3 Reci=Cci/Ffi Recovery defined for each size fraction i.   Multiply (1) by ui and subtract from (2): 5 Fui=Cui+Uui 2 Ffi=Cci+Uui 6 Ffi-Fui=Cci-Cui+Uui-Uui Factor (6): 7 F(fi-ui)=C(ci-ui) Solve (7) for C/F: 8 C/F=(fi-ui)/(ci-ui) Substitute (8) into (3): 9 Reci=ci(fi-ui)/fi(ci-ui) Two-product formula

Example equations
The derivative term is given by:
 * $$D_{\mathrm{out}}=K_d\frac{de}{dt}(t)$$

where
 * $$D_{\mathrm{out}}$$: Derivative term of output
 * $$K_d$$: Derivative gain, a tuning parameter
 * $$e$$: Error $$ = SP - PV$$
 * $$t$$: Time or instantaneous time (the present)

Slurry calculations
To determine the percent solids (or solids fraction) of a slurry from the density of the slurry, solids and liquid :
 * $$\mathrm{\% solids}=100 * \frac{\rho_{solids}(\rho_{slurry} - \rho_{liquid})}{\rho_{slurry}(\rho_{solids} - \rho_{liquid})}$$

where
 * $$\rho_{solids}$$ is the solids density
 * $$\rho_{slurry}$$ is the slurry density
 * $$\rho_{liquid}$$ is the liquid density.

Flow from % solids

 * $$\rho_{slurry}=\frac{M_{slurry}}{Q_{slurry}}$$
 * $$\phi_{sl}=\frac{M_{s}}{M_{sl}}$$