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CIMExcel Software Inc. Slide 1

Flotation Cell Levels – Nonlinear Model Reference Modal Control. CIMExcel Software Inc. Slide 1. Flotation Cell Levels - Nonlinear Model Reference Modal Control. Q air 1. L 1. Q air i. ρ 1. U 1. L i. Q air n. Q in. V 1. ρ i. S in. U i. z 1. L n. V i. ρ n. U n. z i. V n.

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CIMExcel Software Inc. Slide 1

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  1. Flotation Cell Levels – Nonlinear Model Reference Modal Control CIMExcel Software Inc. Slide 1

  2. Flotation Cell Levels - Nonlinear Model Reference Modal Control Qair 1 L1 Qair i ρ1 U1 Li Qair n Qin V1 ρi Sin Ui z1 Ln Vi ρn Un zi Vn • note, hysteresis (backlash) is included • Vi = valve position • Ui = valve positioner CIMExcel Software Inc. Slide 2

  3. Flotation Cell Levels - Nonlinear Model Reference Modal Control Qslurry in Feedforward Leveli Qair i valve position Vj actuator positioner Uj • Level Setpoints • Operator Fixed • Feed Dependent • Startup/Shutdown + + Hysteresis Compensator + ÷ PI fi-1() - Li Solidsin Dynamic Observer Leveli Qair i CIMExcel Software Inc. Slide 3

  4. Flotation Cell Levels - Nonlinear Model Reference Modal Control • operator cell level setpoint changes time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 4

  5. Flotation Cell Levels - Nonlinear Model Reference Modal Control • feed flow dependent cell level setpoint mode time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 5

  6. Flotation Cell Levels - Nonlinear Model Reference Modal Control • froth pushup mode time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 6

  7. Flotation Cell Levels - Nonlinear Model Reference Modal Control • shutdown – startup mode time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 7

  8. Flotation Cell Levels - Nonlinear Model Reference Modal Control • excellent cell level control during variable feed flow, solids, and air flow time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 8

  9. Flotation Cell Level Dynamics • d/dt V1 = A1 d/dt L1 = d/dt Va1 + d/dt Vp1 • A1 d/dt L1 = d/dt Va1 + Qin - Q1out • = d/dt Va1 + Qin - α1 f1(V1)(ρ1L1 + ρ1z1 - ρ2L2 )0.5 • Ai d/dt Li = d/dt Vai + Qi-1out - Qiout • = d/dt Vai + αi-1 fi-1(Vi-1)(ρi-1Li-1 + ρi-1zi-1 - ρiLi )0.5 - αi fi(Vi)(ρiLi + ρizi - ρi+1Li+1 )0.5 • An d/dt Ln = d/dt Van + Qn-1out - Qnout • = d/dt Van + αn-1 fn-1(Vn-1)(ρn-1Ln-1 + ρn-1zn-1 - ρnLn )0.5 - αn fn(Vn)(ρnLn )0.5 • α1 , αi , αn constants. • L1 , Li , Ln from bottom of tank. • z1 = tank 1 – tank 2 elevations. zi = tank i - tank i+1 elevations. zn = 0. • Q1out = Qin + d/dt Va1 - A1 d/dt L1 • = α1 f1(V1)(ρ1L1 + ρ1z1 - ρ2L2 )0.5 • Q2out = Q1out + d/dt Va2 - A2 d/dt L2 = Qin + d/dt Va1 + d/dt Va2 - A1 d/dt L1 - A2 d/dt L2 • Qiout = Qin + ∑j=1i (d/dt Vaj - Aj d/dt Lj) • = αi fi(Vi)(ρiLi + ρizi - ρi+1Li+1 )0.5 • Qnout = Qin + ∑j=1n (d/dt Vaj - Aj d/dt Lj) • = αn fn(Vn)(ρnLn )0.5 CIMExcel Software Inc. Slide 9

  10. Flotation Cell Level Dynamics • solids: • τs1 d/dt S1 = Sin - S1 • τsi d/dt Si = Si-1 - Si • τsn d/dt Sn = Sn-1 - Sn • solids time constant: • τs1 = ρ1 V1 / ρin Qin • τsi = ρi Vi / ρs i-1 Qi-1 • τsn = ρn Vn / ρs n-1 Qn-1 • slurry density from solids: • ρs1 = ρore/ ( S1 + (ρore/ρwater)(1 - S1) ) • Ρs i = ρore/ ( Si + (ρore/ρwater)(1 - Si) ) • ρsn = ρore/ ( Sn + (ρore/ρwater)(1 - Sn) ) • cell density: • ρ1 = ρs1(1 - f1 ) • ρi = ρsi(1 - fi ) • ρn = ρsn(1 - fn ) CIMExcel Software Inc. Slide 10

  11. Flotation Cell Level Dynamics • cell air: • f = void fraction • τar i = cell average rise time (s) • affi = cell averaged void fraction - air flow factor • 0.5τar 1 d/dt f1 = aff1 Qain1 - f1 • 0.5τar i d/dt fi = affi Qain i - fi • 0.5τar n d/dt fn = affn Qain n - fn • Vai = Vi fi= Ai Li fi • d/dt Va1 = 2A1 L1 (aff1Qain1 - f1) / τar 1 • d/dt Vai = 2Ai Li (affiQaini - fi) / τar i • d/dt Van = 2An Ln (affnQainn - fn) / τar n CIMExcel Software Inc. Slide 11

  12. Flotation Cell Level Dynamics – Identification Mode • valve perturbations time scale: 1 minute per division, 10 minutes CIMExcel Software Inc. Slide 12

  13. Flotation Cell Level Dynamics – Identification Mode • air valve perturbations time scale: 1 minute per division, 10 minutes CIMExcel Software Inc. Slide 13

  14. Flotation Cell Level Dynamics – Identification Mode • long term feed flow perturbations time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 14

  15. Flotation Cell Level Dynamics – Identification Mode • feed flow perturbations time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 15

  16. Flotation Cell Level Dynamics – Identification Mode • air flow perturbations time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 16

  17. Flotation Cell Level Dynamics - Identification Mode • feed solids perturbations time scale: 10 minutes per division, 100 minutes CIMExcel Software Inc. Slide 17

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