CIMExcel Software Inc. Slide 1

CIMExcel Software Inc. Slide 1

Flotation Cell Levels – Nonlinear Model Reference Modal Control


Flotation Cell Levels - Nonlinear Model Reference Modal Control

Qair 1

Qin

Sin

L1

V1

U1

Qair i

Li

Vi

Ui

Qair n

ρn

Ln

Vn

Un

z1

zi

ρi

ρ1

note, hysteresis (backlash) is included

Vi = valve position

Ui = valve positioner


Flotation Cell Levels - Nonlinear Model Reference Modal Control

Qslurry in

÷ fi-1()

HysteresisCompensator

valve position

Vj

Feedforward

Dynamic Observer

PI

actuator positioner

Uj

Level Setpoints

• Operator Fixed• Feed Dependent• Startup/Shutdown

Leveli

Qair i

Solidsin

Leveli

Qair i

Li

+

-

++

Flotation Cell Levels - Nonlinear Model Reference Modal Controloperator cell level setpoint changes


time scale: 10 minutes per division, 100 minutes

Flotation Cell Levels - Nonlinear Model Reference Modal Controlfeed flow dependent cell level setpoint mode



Flotation Cell Levels - Nonlinear Model Reference Modal Controlfroth pushup mode



Flotation Cell Levels - Nonlinear Model Reference Modal Controlshutdown – startup mode



Flotation Cell Levels - Nonlinear Model Reference Modal Controlexcellent cell level control during variable feed flow, solids, and air flow




Flotation Cell Level Dynamicsd/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


Flotation Cell Level Dynamicssolids: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 )


Flotation Cell Level Dynamicscell air:

f = void fraction

τar i = cell average rise time (s)affi = cell averaged void fraction - air flow factor

0.5 τar 1 d/dt f1 = aff1 Qain1 - f1

0.5 τar i d/dt fi = affi Qain i - fi

0.5 τar n d/dt fn = affn Qain n - fn

Vai = Vi fi = Ai Li fi

d/dt Va1 = 2 A1 L1 (aff1Qain1 - f1) / τar 1

d/dt Vai = 2 Ai Li (affiQaini - fi) / τar i

d/dt Van = 2 An Ln (affnQainn - fn) / τar n

Flotation Cell Level Dynamics – Identification Mode valve perturbations


time scale: 1 minute per division, 10 minutes

Flotation Cell Level Dynamics – Identification Mode air valve perturbations


time scale: 1 minute per division, 10 minutes

Flotation Cell Level Dynamics – Identification Mode long term feed flow perturbations



Flotation Cell Level Dynamics – Identification Mode feed flow perturbations



Flotation Cell Level Dynamics – Identification Mode air flow perturbations



Flotation Cell Level Dynamics - Identification Mode feed solids perturbations



CIMExcel Software Inc. Slide 1

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