May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

A TEST-BASED METHODOLOGY FOR PARAMETER ESTIMATION FOR A PILOT PLANT DISTILLATION COLUMN

´ C.-M. ASTORGA-ZARAGOZA, A. SANTIAGO, and F.-R LOPEZ-ESTRADA, V. M. ALVARADO-MART´INEZ Centro Nacional de Investigaci´ on y Desarrollo Tecnol´ ogico Interior Internado Palmira S/N, Palmira 62050, Cuernavaca, Mor., Mexico ´ ´ ´ A. HERNANDEZ-P EREZ, and D. JUAREZ-ROMERO Centro de Investigacin en Ingenier´ıa y Ciencias Aplicadas, UAEM, Av Universidad 1001 Col. Chamilpa, Cuernavaca 62209, e-mail: [email protected], Tel 777+3297084

A methodology for parameter estimation of unmeasurable parameters in a distillation process is presented. In this process the basic mass and energy balances are known, thus most of the unknowns are related to hydrodynamic equations, and with heat transfer parameters (condenser heat transfer coefficient, Weir coefficient, vapor pressure drop coefficient, Murphree efficiency, ratio of liquid-vapor in plate). As a result, instead of a black-box or a white-box, the model can be perceived as a spotted-box. This methodology consist of selective tests which are devised to isolate the effect of the unknown parameters. This is exemplified in a pilot distillation column of a two-input, two-output implementation for model predictive control.

1. Introduction The attraction of models obtained from fundamental principles is that they are globally valid, therefore adequate for optimization and control which some times require extrapolation out of the range where the data were obtained to fit the model . Lee2 has mentioned the requirements of an adequate model (1998): In nonlinear model predictive control a model should enable to predict accurately effects of both known and unknown changes on the system (output) behavior possibly in a closed-loop setting. The distillation column is instrumented with temperature sensors, feed input measurement, and actuators for reboiler heat, preheating and condenser recycle. Table 1.1 shows the operating conditions for a methanol1

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

2

Table 1.1 Property XB QB PB TB VB B ZF F QF TF YD ∆PT ot TD LD D

Operating conditions Value 0.3339 mol 100-2500 W 96.15 kPa 67.18 o C 1.6862 mol/min 0.8970 mol/min 0.50018 mol 1.25 mol/min 100-500 W 32.8 o C 0.9233 mol 3.63 kPa 63.02 o C 1.41058 mol/min 0.35265mol/min

ethanol distillation. A mathematical model was built to apply model predictive control of a two-input, two-output process, with the following assumptions: • Liquid Vapor Equilibrium can be represented by Peng-RobinsonStryjek-Vera equation of state (PRSV) 6 with 2 mixing parameters. This EOS was also used to approximate the liquid-vapor equilibrium, and the thermodynamic properties which are required in the mass, composition and energy balances. • Each plate contains two phases which affect the hydraulic behavior 1

• The heat supplied is completely absorbed by the fluid in the reboiler. • The distillate leaves the condenser as saturated liquid. The conservation equations are: Mass balance dM = Lp+1 + Fp − Lp − Vp + Vp−1 dt

(1)

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

3

Composition balance in tray p d(M x) = Lp+1 xp+1 + Fp zp − Lp xp − Vp yp + Vp−1 yp−1 (2) dt Energy balance in tray p d(Mp Ep ) = Lp+1 EL,p+1 + Fp EF,p − Lp EL,p − Vp EV,p + Vp−1 EV,p−1 + Qp dt (3) This assumptions and the associated balances produce a set of sparse algebraic-differential equations whose mathematical representation can be grouped as dx = f (t, p, τ, x, u, v) dt y = g (p, τ, x, u, v) The equations describing this model are presented in appendix A. This model contains parameters which cannot evaluate directly, thus a methodology for parameter estimation is proposed. 2. Methodology This methodology is similar to the scheme proposed for the control analysis 4 time domain analysis helps to evaluate steady state parameters and time constants, frequency domain analysis applied with set of different frequencies evaluate time delays in input-output. Some methods are complementary, for instance, while state analysis eliminates time-dependent parameters, frequency analysis 3 is more sensitive to time dependent parameters. Also, through the design of experiments we can produce a behavior whose model has a reduced number of unknown parameters. Our strategy is to isolate as much as possible the effect of every parameter by designing selective tests obtaining the parameters via model inversion; thus instead of the orthogonality of the tests, the goal of experiments is to observe the isolated effect of uncorrelated parameters. Sensitivity analysis helps us to obtain information about the value or range of the parameters and the propagation of uncertainty to the output variables. Finally, a global parameter estimation is used for the remaining parameters. The benefit of using these selective tests previous a global parameter estimation is that the problem dimension is reduced, or promising starting values for global parameter estimation are suggested. Some tests are redundant, but they help to verify the value of parameters.

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

4

2.1. Steady State Analysis Test 1. Heat transfer in condenser. The condenser acts as an inherent regulator for the column. Since the actual column is made out of glass, vibrations should be avoided during operation (see Fig. 1 ). As a result this column operates as a double pass condenser with an excess of cooling water. Thus both temperatures (cooling water and condensing mixture) have only small changes. Since input and output cooling water temperature are known, we can evaluate VC from QC = LW Cp (TI − TO ) = VC ∆HCnd,C we can estimate VC also QC = U Ah LM ∆T where at different condenser flows U can be correlated as: U ≈ Kh LγW

Figure 1.

Condenser

Test 2. Heat losses along the column. The column is not insulated, thus the glass walls are open to the environment (see Fig.2 ). Selective Conditions LF = 0; B = Ln = 0; D = 0 The main heat losses happen along the column body, since the condenser operates at atmospheric conditions, and the boiler transfer area is small (see next section ),thus a global balance for all the trays produces:

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

5

LC EL,Cond + VB EV,B = VC EV,Cond + LB+1 EL,B + QA With the QA approximated by Newton’s cooling law for the column body. NX p−1 dQA = Kp,A ∗ (Tp − TA ) dt 2

Evaluating Kp,A heat loses can be evaluated at different environmental conditions.

Figure 2.

Condenser

Test 3. Reflow valve Reflow valve is operated by a set on/off pulses with frequency variation Selective Conditions LB+1 ≈ VB ; LB = 0 Assumption VB ≈ VC Then in all the trays flows steam to maintain the operating pressure drop. LB+1 = LF + LC LC = VB − D LB+1 = LF + VB − D

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

6

then LF ≈ D With this equation the behavior of reflow valve can be correlated as: LC = f (ωR , MC , T, P ) 2.2. Time Domain analysis Time constant are static, or have a small dependence on geometry of the piece of equipment at given operating conditions. Some of this constants can be tuned by the following set of tests Test 4. Time constants Selective Conditions D=0 Assumptions The main elements for heat for heat transfer are condenser and boiler, as a result we have Then we have dMB = Ff + LB+1 − VB − B dt also QB ∆HV ap,B QC L+D = ∆HCnd,C r ρB gMB B= KvL A

VB =

dMB QC QB = Ff + − − dt ∆HCnd,C HV ap,B QB QC dMC = − dt HV ap,B ∆HCnd,C

r

ρB gMB Kv A

Here we can analyze the effect of mass holdup and the liquid discharge valve. As noticed earlier 7 , distillation process behaves like a first order dynamics. From a linearized model it is possible to obtain the time constants τ dX dt = −x Test 5. Reboiler heat holdup.

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

7

To speed the boiling response, the electric devise is inserted in a separate vessel where smaller mixture is heated (see Fig.3 ). Selective Conditions B = 0 Ff = 0

Figure 3.

Condenser

Given the small value of VB compared with MB the boiler dynamics is represented as: Mv

dHv = −VB ∆HV ap,B + QB dt

With this vessel, mass holdup and concentration have little change, but energy holdup is smaller. The estimated values of mv can be corroborated with the time constants of the linearized model. Table 2.2 shows the fitted parameters Table 3.2 Parameter QAB U Kv,L Kv,V Mv

Estimated parameters meaning Heat loses heat transfer coefficient Liquid Discharge coefficient Vapor Discharge coefficient Mass holdup in the reboiler vessel

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

8

Other parameters like φ need to be correlated globally.

3. Conclusions In building a model useful for predictive control attention was placed to the actual geometry of the distillation column The purpose of this work is to ”find ways to make identification experiment efficient to generate data that are rich in the information” 2 relevant to the model used for predictive control. Some of these test can be inferred from the incidence matrix of [ dx dt , y]vs.p; but when an experimental test is devised, the feasibility of operation is guaranteed. With this selective test we aim to tune the parameters, then we could aim to obtain a ”systematic certainty” [Balmes, El Criterio]

References 1. Benett D.L, R Agrawal, P.J. Cook, (1983), New Pressure Drop Correlation for Sieve Tray Distillation Columns, AICHE J, V 29, No 3 pp434-442 2. Lee J. H. (1998).Modeling and identificatin for nonlinear model predicitive control:Requirements, current status and future research needs, in Nonlinear Model Predictive Control, Volume 26 of Progress in Systems and Control Theory Series, Birkhauser Verlag, Basel, Switzerland. 3. Lee H, Rivera D. E. (2005) An Integrated Methodology for PlantFriendly Input Signal Design and Control-Relevant Estimation of Highly Interactive Processes, AIChE Meeting, (Cincinnati, OH) 4. Luyben W.L.(1990), Modeling, Simulation and Control for Chemical Engineers,McGraw Hill, USA. 5. Skogestad S. and Morari M. (1998), Understanding the Dynamic Behaviour of Distillations ColumnsInd. Eng. Chem. Res. 27 pp 18481862. 6. Stryjek, R. Vera, J.H. (1986), An Improved Peng-Robinson Equation of State ofr Pure Compounds and MixturesCan. J. Chem. Engn. 64, pp 334-340. 7. Wittgens and S. Skogestad (1995), Evaluation of Dynamic Models of Distillation Columns with Emphasis on the Initial Response DYCORD’95, Denmark.

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

9

Appendix A. Model Equations Equilibrium compositions are obtained by equating the fugacities. Thus for bubble point

fL,i (T, Pb , x∗ ) = fV,i (T, Pb , y ∗ ) n X Φ(T, P, x) = Pb ( Ki x∗i − 1) 1 ig

E = E (T, x∗) − E dep (T, P, x∗)

Murphree plate efficiency

yp,i = y ∗p,i ηV + (1 − ηV )y∗p−1,i

Hydraulic equations for plates

√ 3 hef f L = ρL KvL hef f = max(0, h − hwer ) h = M/Aρ L √ ∆P ρV M wV V= KvV ∆P = Pp+1 + ϕghρL − Pp

Condenser

Qc = U A LM ∆T ∆Ti = (TD − Ti ) ∆To = (TD − To ) (∆Ti −∆To ) LM ∆T = log(∆T i /∆To ) To = Ti − Qc /WH2O CpH2O

May 18, 2006

21:9

Proceedings Trim Size: 9in x 6in

P0137

10

Nomenclature B = Boiler flow C = Condenser Cp = Heat capacity E = Energy F = Feed flow H = height K = equilibrium constant L = Liquid flow M = mass holdup Mw = Molecular Weight R = Reflow t = time u = vector of manipulated variable V = Vapor flow v = vessel v = vector of measured disturbances w = vector of unmeasured disturbances. x = vector of states y = vector of measured variables Q = heat x = liquid composition y = vapor composition z = feed composition Greeks Dep =departure ∆ = increment Tot = Total ρ = density γ = fitting exponent φ = holdup effectiveness τ = time related parameters η =Murphree plate efficiency ω = frequency

Subscripts A = Atmospheric conditions a = algebraic b = bubble point B = bottom f = feed i = component D = Distillate d = differential Eff = effective f = feed h = Heat transfer I = input L = Liquid O = output p = plate number v = valve V = vapor W = Water Wer = Weir Superscripts * = equilibrium Ig = ideal gas