Binary Distillation Column Startup Using Two Temperature

ARTICLE pubs.acs.org/IECR

Binary Distillation Column Startup Using Two Temperature Measurements† Eduardo Castellanos-Sahagun*,‡ and Jesus Alvarez§ ‡

Centro de Investigacion en Polímeros, SA de CV, Marcos Achar Lobaton #2, Tepexpan, Mpo. De Acolman, Edo. Mexico, Mexico Departamento de Ingeniería de Procesos e Hidraulica, Unidad Iztapalapa, Universidad Autonoma Metropolitana, Apdo. Postal 55534, 09340, Mexico D. F., Mexico

§

ABSTRACT: The problem of designing the startup (SU) control scheme for a binary distillation column with two temperature and two level measurements is addressed. The objective is the quick attainment, with minimum off-spec production and heat consumption, of closed-loop stable operation in continuous regime. The combination of advanced constructive control and conventional distillation dynamics and control tools leads to a variable control structure scheme with: (i) an event controller which, on the basis of the two temperature measurements, decides the duration of an initial open-loop batch operation period at total reflux, and (ii) a set of four (two level and two temperature) linear-decentralized controllers activated in a sequential manner, which steers the column toward its nominal steady-state in a continuous regime, by manipulating the heat duty, reflux, distillate, and bottom flow rates. The proposed SU control scheme is tested through simulations with three high-purity column examples with ideal and nonideal thermodynamics, yielding a behavior that matches or improves the ones obtained in previous studies with detailed modelbased schemes.

1. INTRODUCTION The production of many intermediate and final products in chemical and petrochemical industries depends heavily on energy intensive distillation columns, and their efficient operation requires the simultaneous regulation of distillate and bottoms compositions.1 While the corresponding regulation control problem has been extensively studied with a diversity of multivariable controllers, the startup (SU) control problem has received less attention.2 In industrial practice, most industrial distillation columns are started up with the so-called conventional approach:3,4 (i) first, the column trays are hydraulically sealed, and then (ii) the feed, reflux, and vapor flow rates are set at their nominal continuous values, with reboiler and condenser inventories regulated with level loops, inducing the column to reach, in an open-loop manner, its nominal continuous steadystate (SS). The preceding open-loop SU operation has been improved by the so-called minimum time algorithm3 (MTA), by adding, in a way that resembles the optimal operation of batch columns,5,6 an initial period at total-reflux which lasts until a measure of the actual minus SS temperature profile reaches its minimum value. The implementation of this scheme requires the measurement of the entire temperature profile or its model-based estimation from a few (at least two) temperature measurements. This MTAbased SU procedure has been improved by steering the batch-tocontinuous open-loop transition with optimal reflux-heat load control input pair policies.4,7 The implementation of these openloop SU procedures requires model-based estimates of the column state at the total reflux-to-extraction period switching time and the model-based precomputation of the optimal control input over the extraction period. On the other hand, model-based nonlinear geometric state-feedback (SF) control has been applied to steer the column directly from the initial condition to the r 2011 American Chemical Society

prescribed continuous SS,812 and the implementation of this closed-loop SU scheme needs a model-based nonlinear observer driven by temperature measurements. Even though the preceding model-based open- and closedloop SU schemes demonstrate the feasibility of speeding up the conventional SU procedure and provide valuable insight into the problem, their implementation still raises complexity, reliability, and cost concerns among practitioners, due to the strong dependency of the resulting OF control schemes on the detailed column model. These considerations motivate the scope of the present study: the speeding up of the SU procedure on the basis of simpler and less model dependent control scheme. On one hand, it is known13,14 that: (i) for regulation about the prescribed SS, the behavior of exact model-based nonlinear robust two-point temperature (TPT) controllers can be adequately recovered with a pair of linear-decentralized PI temperature controllers, and (ii) this approach has been applied successfully for columns with ideal or nonideal thermodynamics and input uncertainties. Moreover, according to the wave-model based control approach,12 the column effluent pair can be controlled by fixing two wave fronts (located at two sensitive trays, one per section) of the column temperature (i.e., composition) profile. These considerations suggest to us (i) the possibility of simplifying Yasuoka’s MTA criteria3 to stop the total-reflux period, by employing only two sensitive-tray measurements, and (ii) the employment of linear-decentralized feedback controllers to speed-up the batch-to-continuous column transition. Received: August 3, 2010 Accepted: March 8, 2011 Revised: January 4, 2011 Published: April 12, 2011 6187

dx.doi.org/10.1021/ie101659w | Ind. Eng. Chem. Res. 2011, 50, 6187–6204

Industrial & Engineering Chemistry Research

ARTICLE

In this work, the problem of starting up a binary distillation column with temperature and level measurements is addressed, with the objective being the quick attainment, with minimum offspec production, of closed-loop operation in continuous regime. The combination of ideas from continuous and optimal batch distillation operation and control design yields a simple SU control scheme with: (i) an event controller which, on the basis of two temperature measurements, decides the duration of an initial period at total reflux and (ii) a set of four (two level and two temperature) linear-decentralized controllers activated in sequential manner, to steer the column toward its nominal steady-state in the continuous regime. The proposed approach is illustrated and tested via numerical simulations with three case examples, including runs with ideal and nonideal thermodynamics and linear and nonlinear hydraulic dynamics, as well as manipulated control inputs in a molar or volumetric flow basis. According to the results, the proposed approach yields behaviors that match or improve the ones of previous schemes, in terms of online modeling and measurement requirements as well as settling time, off-spec production, and heat consumption.

2. STARTUP PROBLEM Consider the N-tray binary distillation column (shown in Figure 1), where a binary mixture with molar flow F and composition cF is fed at the tray nF, yielding the effluent pair (B, D) with compositions (cB, cD) respectively. From standard assumptions15 (constant pressure, equilibrium in all trays, equimolal overflows, etc.), a model with composition and tray holdup dynamics can be written as follows: : ci ¼ ½Lðmi þ 1 ÞΔþ ci VΔ Eðci Þ þ δi, nF FðcF ci Þ=mi 0 e i e N1 ð1aÞ : cN ¼ ½RΔþ cN VΔ EðcN Þ=mN : cN þ 1 ¼ V½EðcN cN þ 1 Þ=mD

mi ð0Þ ¼ moi ,

1 e i e N1

: mD ¼ V R D

ð1f,gÞ

δ ¼ ðF, cF Þ0 , ψ ¼ ðTs , Te , mB , mD , Þ0 ,

cN þ 1 ¼ cD ,

cB ¼ c0 ,

Ts ¼ βðcs Þ,

Lðmi Þ ¼ R þ F þ ðmi mi Þ=τh , Lðmi Þ ¼ R þ ðmi mi Þ=τh

Te ¼ βðce Þ

1 e i e nF

ð1hÞ

nF þ 1 e i e N

ð1iÞ

where δi,nF is Kronecker’s delta, ci (or mi) is the light component mole fraction (or holdup) at the ith tray, E and β are the liquidvapor equilibrium and bubble point nonlinear functions, respectively, and L is the linearized version of Francis weir’s equation,16 with hydraulic time constant τh. There are two level (mB and mD) and two temperature measurements in trays s and e (Ts and Te). Since the holdup dynamics are stable17 and considerably (typically 50 to 200 times) faster than the composition dynamics,18

ψ ¼ hðcÞ

ð2aeÞ

ν ¼ ðV, R, D, BÞ0 , hðcÞ ¼ ½βðcs Þ, βðce Þ, mB , mD 0

( 3 )0 denotes the transpose of matrix ( 3 ), and (h3 ) denotes the nominal steady-state value of ( 3 ) in the continuous regime, and moi is the solution for mi of the holdup system in QSS regime, according to the equations: 0 ¼ Fmi ðmoi , δ, νÞ, i ¼ 1, :::, N,

where Δþ ci Z ci þ 1 ci , Δ Eðci Þ Z Eðci Þ Eðci 1 Þ; Eðc1 Þ Z c0

i ¼ 1, :::, N,

where c ¼ ðc0 , :::, cN þ 1 Þ0 , m ¼ ðmB , m1 , :::, mN , mD Þ0 , mB ð0Þ ¼ mB , mD ð0Þ ¼ mD

ð1dÞ

ð1eÞ : mN ¼ R LðmN Þ

according to the singular perturbation control approach:19 (i) the controller construction step should be performed by enforcing the quasi steady-state (QSS) assumption upon the holdup dynamics, and (ii) the effect of the holdup dynamics should be accounted for in the closed-loop behavior assessment, in the understanding that the presence of the parasitic holdup dynamics limits the column control performance.20 In compact vector notation, the column dynamics are given by: : : c ¼ Fc ðc, m, δ, νÞ, cð0Þ ¼ cF ; m ¼ Fm ðm, δ, νÞ,

ð1b,cÞ

: mB ¼ Lðm1 Þ V B : mi ¼ Lðmi þ 1 Þ Lðmi Þ þ δi, nF F

Figure 1. Distillation column.

ν ¼ ðV, R ¼ V, 0, 0Þ0 ,

δ ¼ ð0, 0Þ0 , :::, N þ 1

ð3aÞ ð3bcÞ

The startup control problem consists in designing a SU feedback control scheme to steer the column, toward its nominal operation in the continuous regime, by deciding: (i) the duration ts of the initial total reflux operation period and (ii) the online adjustment of the feed (F), internal (R and V), and extraction (D and B) flows on the basis of two level measurements (mB, mD) and two temperature measurements (Ts, Te) at locations to be determined (see Figure 1), in such a way that the effluent composition pair (cB, cD) reaches its prescribed value in minimum time, with reduced off-spec distillate and bottoms production. We are interested in a simple SU control scheme, with linearity, decentralization, reduced model-dependency features, sensor location criteria, and a systematic construction-tuning procedure. 6188

dx.doi.org/10.1021/ie101659w |Ind. Eng. Chem. Res. 2011, 50, 6187–6204


ARTICLE

Table 1. Steady-State Data for Column B benzene-toluene column thermodynamic model

characteristic Raoult’s law; vapor pressure from Patel et al.47

pressure, atm

1

cf, feed composition

0.5

cd, distillate composition

0.995

cb, bottoms composition

0.01

nF, feed tray

9

N, number of trays (not including the reboiler and condenser drum)

18

F, mol/s

10

V/F

1.4516

R/F

0.9542

condenser and reboiler holdups, kmol

10

tray holdup, kmol

0.13

tray time constant

3.5

control trays, stripping/rectifying section (counting from below)

5/12

Without restricting the approach, the control-oriented methodological developments will be performed under the assumption that measured molar (i.e., reboiler and condenser drum) level holdups and tray temperatures are controlled with molar flow rate inputs, in the understanding that the control scheme can be applied to the case where measured volume holdups and tray temperatures are regulated with control inputs adjusted in a volumetric flow rate basis. The approach will be tested with three representative high-purity column case examples: (i) Column A (employed in a previous SU control study12 with nonlinear wave-model based techniques), with 39 trays, ideal thermodynamics, equimolal feed mixture, (99, 1)% distillatebottoms split, molar level holdup measurements, and adjustable molar flow rates. (ii) Column B, with 18 trays, benzenetoluene ideal thermodynamics, equimolal feed, (99.5, 1)% distillatebottoms split, molar level holdup measurements, adjustable molar flow rates, and characteristics listed in Table 1. (iii) Column C, with 12 trays, methanolwater equimolal feed, 1% impurity products, nonideal thermodynamics, nonlinear hydraulic dynamics according to Francis’ weir formula, molar or volumetric level holdup measurements, and adjustable molar or volumetric flow rates. Model details are listed in Appendix D.

3. OF EVENT CONTROL FOR INITIAL PERIOD AT TOTALREFLUX In this section, the combination of the minimum time algorithm (MTA)3 with wave model-based control12 and sensor location criteria13,21,22 for distillation columns yields an event controller which, on the basis of two temperature measurements, decides the duration of the initial operation period at total reflux. In a notation suited for the purpose at hand, let us recall the Yasuoka et al.’s3 event controller, denoted as the minimum time algorithm (MTA) which determines the duration ts ¼ arg½min MT ðtÞ, t

MT ðtÞ ¼

Nþ1

_

∑ jTi ðtÞ T i j i¼0

ð4a,bÞ

of the initial total reflux period as the time ts when the measure _ transient [T0(t), ..., TNþ1_ the MT(t) of the difference between (t)] and continuous SS [T 0, ..., T Nþ1] temperature profiles reaches its minimum value. It must be pointed out that the implementation of this controller requires the online measurement of the entire temperature profile, its interpolation-based approximation on the basis of a sufficient number of adequately located measurements, or a model-based extended Kalman filter (EKF)23 or geometric24 estimator with at least two measurements. The argument that the difference (or sum) of two column sensitive temperatures, one per section, manifests the column separation (or temperature profile position) has led to the employment of temperature-based synthetic outputs in continuous distillation column control schemes.2527 Moreover, the wave modelbased control approach12,28 is based on the idea that column separation can be regulated by fixing the position of the two-wave fronts related to the stripping and rectifying temperature (or composition) profiles. In binary continuous columns, these fronts: (i) coincide with the trays that exhibit the largest sensitivity with respect to disturbance and control inputs, that is, with the largest stage-to-stage temperature gradient, and (ii) are related to the well established criteria “of placing the sensors at sensitive trays” employed in SISO and MIMO control design methodologies.13,21,22 This connection between column separation and sensitive trays suggests: (i) that the temperature profile-based measure MT 4a,b of the MTA can be replaced by the simpler separationbased MW measure 5a,b, where Ts (or Te) is a sensitive tray in the stripping (or rectifying) section, and (ii) consequently, that the event controller (eqs 4a,b) of the MTA can be replaced by its simpler counterpart ts ¼ arg½min MW ðtÞ Z μs ½ys ðtÞ, ye ðtÞ, t _ _ MW ðtÞ ¼ jj½Ts ðtÞ Te ðtÞj jðT s T e Þjj ð5a,bÞ The evolutions of the temperature profiles for the three column examples in total reflux regime are shown in Figure 2, the related (actual minus SS) temperature profile deviations are presented in Figure 3, and the corresponding evolutions of the MTA (eqs 4a,b) and proposed (eqs 5a,b) profile deviation measures are shown in Figure 4. From the examination of these figures the next observations and conclusions follow: (i) According to Figures 2a and 3a (or Figures 2c and 3c), column A (or column C) with ideal thermodynamics (or nonideal methanolwater mixture), and trays (13, 24) [or (1, 4)] as the sensitive ones, exhibits the least (or most) asymmetric profile approach to the SS profile in continuous regime, and column B with ideal benzenetoluene mixture (Figures 2b and 3b), with trays (5, 12) as the control trays exhibits an in-between profile evolution. (ii) In the case of Column A (see Figure 4a), the MTA (eqs 4a, b) and proposed measures (eqs 5a,b) reach their minimum value at about the same time ts ≈ 24 min. (iii) In the case of Column B (see Figure 4b), the MTA (or proposed) measure MT (eqs 4a,b) [or MW (eqs 5a,b)] reaches its minimum at ts ≈ 53 (or 40) min, meaning that MW reaches its minimum in a time ts that is ≈ 25% faster than the one of MT. (iv) In the case of Column C (see Figure 4c), the MTA (or proposed) measure MT (eqs 4a,b) [or MW (eqs 5a,b)] reaches its minimum at ts ≈ 80 (or 64) min, meaning that MW reaches its minimum in a time ts that is ≈ 20% faster than the one of MT. 6189



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Figure 2. Evolution of temperature profile at total reflux for columns A, B, and C.

According to these results for three different columns, the proposed measure Mw (eq 5) computed with the sensitive measurement pair (Ts, Te), yields durations (ts) of total reflux periods which are equal or (20 to 25%) faster than the durations given by the measure MT (eq 4) computed with the entire temperature profile [T0(t), ..., TNþ1(t)], of the MTA.

4. OF CONTROLLER FOR THE BATCH-TO-CONTINUOUS TRANSIENT PERIOD Following the OF constructive control approach employed in previous continuous and batch distillation columns,6,13,14,20 as well as chemical29,30 reactors, in this section the output-feedback (OF) control problem for the total reflux-to-continuous operation is addressed. 4.1. Solvability Assessment. From constructive control arguments31 we know that optimal feedforward-state feedback (FFSF) controllers are: (i) inherently robust, (ii) passive (i.e., with relative degrees equal to one and stable zero-dynamics) with respect to the regulated outputs, and (iii) nonwasteful in the sense of efficiently coordinated control action. Moreover, optimal controllers can be tractably designed via inverse optimality,31 by starting with a passive controller and verifying with respect to which objective function the controller is optimal. The discussion of these methodological tools in the context of a dual composition and two-point temperature control problems in distillation columns can be seen elsewhere,13,14,20 and here it suffices to mention that: (i) as it stands, the four control-four measured regulated concentration-holdup column dynamics (eq 1) are not passive, and (ii) the column dynamics can be passivated

by enforcing the quasi steady-state (QSS) assumption upon the holdup dynamics, as they are stable17 and 2 orders of magnitude (say, 50 to 200 times) faster than the composition dynamics.18,20 The enforcement of this QSS assumption upon the column dynamics (eqs 2ae) yields the reduced-order model, in deviation coordinates form about the nominal SS (fI and fII are defined in Appendix A): : : x I ¼ f I ðx I , x II , u, dÞ, x II ¼ f II ðx I , x II , u, dÞ, ð6acÞ y ¼ xI where

x ¼ ðx I , x II Þ0 , 0

0

x I ¼ ðx s , x e , x B , x D Þ0 ,

x II ¼ cII cII

ð6dwÞ

x s ¼ βðcs Þ βðcs Þ, x e ¼ βðce Þ βðce Þ, x B ¼ mB mB , x D ¼ mD mD cII ¼ ðcB , c1 , c2 , :::, cs 1 , cs þ 1 , :::, ce 1 , ce þ 1 , :::, cN , cD Þ0 u ¼ ðuV , uR , uB , uD Þ0 ; uV ¼ V V, uR Z R R, uB Z B B, uD Z D D d ¼ ðdF , dc Þ0 , y s ¼ Ts Ts ;

6190

dF ¼ F F,

dc ¼ cF cF

_ y e ¼ T e T e , y B ¼ mB mB , y D ¼ mD mD



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Figure 3. Evolution of temperature profile deviation (with respect to SS profile) at total reflux for columns A, B, and C.

xs (or xe) is the concentration-to-temperature coordinate change at the sensitive temperature measurement tray in the stripping (or rectifying) section, xB (or xD) is the bottoms (or distillate) measured concentration, and cII are the remaining compositions. This model is passive32 if and only if: (i) eq 6a has the unique solution : ð7Þ u ¼ f I 1 ðx I , x II , x I , dÞ

μI ðx II , dÞ ¼ f I ð0, x II , 0, dÞ

ð8bcÞ

are stable. According to the ratio test condition,33 the relative degree (RD = 1) condition 7 is met after the hydraulics transient (induced by the sudden feed-extraction event) settles at a short time interval θh (of the same magnitude order as the hydraulic dynamics), because the determinants (γ1 and γ2) of the first two leading minors of the (state-dependent) Jacobian matrix ∂ufI meet the conditions: " t ∈ ½ts þ θh , ¥Þ :

where

3 0 7 0 7 7, 0 7 5 1

0 0 1 0

i ¼ s, e

mi > 0,

Ri ¼ β0 ðci ÞðΔþ ci Þ < 0, pi ¼ Δ Eðci Þ=ðΔþ ci Þ > 0,

θi ¼ 1 þ ðpi V Li Þτh =mi , Ri Δþ Ti , γ2

θi 1, ½ Rs θs =ms ½ Re θe =me ðLs

i ¼ s, e

ð10adÞ

Le Þ=V

ð11acÞ

pi Li =V,

m*s (or m*e ) is the quasi-static holudp in tray s (or e) [see Appendix A, eq A.3a], Ls (or Le) is the quasi-static liquid flow in the stripping (or rectifying) section, Rs (or Re) is approximately the tray-to-tray temperature change at stage s (or e), and ps (or pe) is approximately the slope of the (time-varying) 5,34 operating line in the stripping (or rectifying) section. In fact, at the continuous SS (x, u) = (x, u), the solvability condition (eqs 9a,b) becomes the local one

Rs θs =ms Re θe =me 1 1

ð9c,dÞ

γ1 ¼ ½β0 ðcs ÞΔþ cs ½ðF þ RÞ=V=ms < 0,

γ1 ¼ Rs ps =ms > 0,

γ2 ¼ ½Rs =ms ½Re =me ðθe ps θs pe Þ < 0

Rs ps =ms 6 6 Re pe =me ¼6 6 1 4 1

for the control input u, or equivalently, the system has its four relative degrees (RDs) equal to one, and (ii) the corresponding zero-dynamics (ZD) : x II ¼ f II ½0, x II , μI ðx II , dÞ, d Z Φðx II , dÞ, ð8aÞ 1

Jðx I , x II , u, dÞ Z Du f I

2

γ2 ¼ ½β0 ðcs ÞΔþ cs =ms ½β0 ðce ÞΔþ ce =me =F=V < 0

ð9a,bÞ

reported before for continuous TPT control schemes.13 6191



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Figure 4. Evolution of temperature profile deviation measures (MT and MW) at total reflux for columns A, B, and C.

The zero-dynamics (ZD) (eqs 8a): (i) are an inventory-based stable restriction of the system dynamics, with a material balance controller13,15,20 that maintains the effluent concentrations (cs and ce) and the holdups (mB and mN) fixed at their nominal values in continuous regime, by exactly balancing the mass an energy delivered to the column by the controller against the demand of the load disturbance(s), and yields a unique, stable steady state, and (ii) constitute the limiting closed-loop attainable behavior with feedback control. This closed-loop stability property associated with the ZD: (i) is different from the one observed during open-loop startup operations,37 and (ii) explains the stabilization, with a unique attractor, via feedback control of open-loop columns with multiple SS.35,36 More on the key subject of stability of the SU operation (over the total reflux plus extraction periods) will be discussed in Section 5.2 on “Stability”. Summarizing, the static-passive control problem is solvable for the reduced model over the extraction period. Moreover, the same sensitive-tray arguments employed in the TPT control design for distillation columns13 lead us to conclude that: (i) there must be one temperature sensor per section, and (ii) the singularity measure of J is minimized when the temperature sensors are placed in the sensitive trays (one per section). The facts that the TPT control problem is not solvable over the total-reflux period, because6 " t ∈ ½0, ts :

Ls ¼ Le ¼ R ¼ V w γ2 ¼ 0

batch-to-continuous regime excites the hydraulics dynamics with settling time θH, suggest us the following control scheme: (i) In the interval [ts, ts þ θH], the feed-reflux-vapor flow triplet is fixed at its nominal SS value (F, R, V) in continuous regime, and the level control pair is set to maintain the inventory by setting the distillate and bottoms extraction. (ii) In the interval [ts þ θH, ¥), the level control pair is maintained and the temperaure control pair is set. This in the understanding that the fulfillment of condition (eqs 9a,b) guarantees: (i) the solvability of the four-input four-output level-temperature passive control problem over [ts þ θH, ¥), and (ii) the solvability of the twoinput two-output level passive control problem over the interval [ts, ts þ θH], as the 2 2 bottom-right submatrix of J is nonsingular, are well conditioned and independent of the model reduction procedure. 4.2. Feedforward-State Feedback (FF-SF) Control. Recall the solvability of the passive nonlinear FF-SF control problem, and enforce the closed-loop output linear-decoupled dynamics (bd = block diagonal matrix) : ð12aÞ x I ¼ Kx I , y ¼ x I , K ¼ bdðk s , k e , k B , k D Þ upon the reduced column model (eqs 6ac) to obtain the nonlinear FF-SF stabilizing controller u ¼ f 1 I ðx I , x II , Kx I , dÞ Z μðx I , x II , dÞ,

and that the model reduction assumption is not valid over the interval [ts, ts þ θH], when the sudden change from

μI ðx II , dÞ ¼ μð0, x II , dÞ 6192

ð12bÞ



ARTICLE

or, equivalently, in detailed form

1 ks ½βðcs Þ βðcs Þτh =Rs 1 1 ke ½βðce Þ βðce Þτh =Re ks ½βðcs Þ βðcs ÞðdF RÞτh þ ms =Rs F ke ½βðce Þ βðce Þðme Rτh =Rs 1 1 uV 1 kB ðmB mB Þ B uB ¼ þ Fþ 1 1 uR 0 uD D kD ðmD mD Þ uV uR

¼

V ps þ R pe

(eq 6), and rewrite the xI-dynamics (eq 6a) in the form : x I ¼ Au þ b, y ¼ x I ; : b ¼ ηðx I , x II , u, dÞ; x II ¼ f II ðx I , x II , u, dÞ ð15adÞ where A ¼ diagðj11 , j22 , j33 , j44 Þ, _ as ¼ ðΔþ T s Þ=ms Rs ,

ð12cÞ The application of this controller to the actual column in singular perturbation form (eqs B.2acB.2 in Appendix B) yields the closed-loop dynamics (~ ~pI, ~pII, and ~pz are defined in Appendix C): : pI ðx I , x II , d, u; zÞ, y ¼ x I ð13a,bÞ x I ¼ Kx I þ ~ : x II ¼ f II ½x I , x II , μðx I , x II , dÞ, d þ ~pII ½x I , x II , d, μðx I , x II , dÞ; zÞ ð13cÞ : : z ¼ θðx, d; zÞ þ ~pz ðx, z, d, dÞ

k ¼ minðk s , k e , k B , k D Þ e k þ ð1=λh Þ, λh 4=ðNτh Þ

ð13e,f Þ

where eq 13f approximates λh in terms of the tray hydraulic time constants38 τh. By construction, the nonlinear passive (NLP) FF-FB controller (eq 12) is a a nonlinear combination of a material balance feedforward plus feedback components, which according to industrial practice is the most powerful control scheme to handle difficult processes:15,39 the feedforward (FF) component (eq 12a with K = 0 or, equivalently, the material balance controller in eq 8b) performs most of the disturbance rejection task, and the feedback (FB) (active when K 6¼ 0 in eq 12a) is just dedicated to compensate the modeling errors of the feedforward component. The resulting closed-loop behavior (CL) (eq 13) associated with the FF-SF controller (eq 12), built on the basis of the exact reduced model (eq 7), represents the limiting performance attainable with any OF robust controller, and such behavior will be the recovery target for the OF control design in the next section. In compact notation the closed-loop stable column dynamics (eq 13) are given by [φ is defined in Appendix C]: : _ χ ¼ φðχ, d, dÞ,

χ ¼ ðx 0 , z0 Þ0

ð14Þ

j22 ¼ ae ,

j44 ¼ 1

_ ae ¼ ðΔþ T s Þ=me Re

ηðx I , x II , u, dÞ ¼ f I ðx I , x II , u, dÞ Au,

rdðu, yÞ ¼ ð1, 1, 1, 1Þ

and the entries of the nonsingular matrix A are approximations of diagonal entries of the SS jacobian matrix J (eq 9b). This system has three components: (i) a linear-decentralized dynamical system (eqs 15a,b) driven by the load input b and measurement y, a nonlinear-static system (eq 15c) which determines b according to (xI, xII, u, d), and (ii) the internal dynamics (eq 15d) driven by (xI, u, d). In terms of the pair (A, b), the nonlinear FF-SF stabilizing controller (eqs 6ac) is written as follows:

ð13dÞ

where ~pI (which vanishes at z = 0) manifests the effect of the QSS holdup dynamics assumption on the closed-loop dynamics. According to the stability proof for interconected systems in singular perturbation form20 via the small gain theorem arguments (refs 30 and 32 and the Appendices therein), the closed-loop column is stable if the control gains are chosen sufficiently slower than an upper limit kþ determined by the dominant frequency of the holdup dynamics λh, according to the inequality:

j11 ¼ as ps ,

j33 ¼ 1,

u ¼ A 1 ðKx I þ bÞ,

b ¼ ηðx I , x II , u, dÞ

ð16a,bÞ

On the other hand, the solution for b of eqs 15a,b is given by : b ¼ y Au ð16cÞ meaning that, at each time, the value b of the nonlinear map η is uniquely determined by the known control-output pair (u, y), implying that, without knowing η, a quick estimate ^b of b can be generated by the battery of linear-decentralized reduced-order observers:30,40 : w s ¼ ωT ðw s þ ωT ys þ j11 uV Þ, ^bs ¼ w s þ ωT y , w s ð0Þ ¼ ^bs ð0Þ ωT y ð0Þ ð17aÞ s

s

: w e ¼ ωT ðw e þ ωT y e þ j22 uR Þ, ^be ¼ w e þ ωT y , w e ð0Þ ¼ ^be ð0Þ ωT y ð0Þ ð17bÞ e e : w B ¼ ωT ðw B þ ωL y B þ j33 uB Þ, ^bB ¼ w B þ ωT y , w B ð0Þ ¼ ^bB ð0Þ ωT y ð0Þ B B ð17cÞ : w D ¼ ωL ðw D þ ωL yD þ j44 uD Þ, ^bD ¼ w D þ ωL y , w D ð0Þ ¼ ^bB ð0Þ ωT y ð0Þ D D ð17dÞ where ωT (or ωL) is the temperature (or level) observer gain. The combination of this observer with the NLP SF controller (eqs 16a,b) (with b replaced by its estimate ^b) yields the OF linear decentralized controller: : w s ¼ ωT ðw s þ ωT ys þ j11 uV Þ, uV ¼ ½ðk s þ ωT Þys þ w s =j11 ð18aÞ : w e ¼ ωT ðw e þ ωT y e þ j22 uR Þ, uR ¼ ½ðk e þ ωT Þy e þ w e =j22

4.3. OF Control. Following the idea which underlies a pre13

vious TPT linear-decentralized control design, recall the passive structure of the four-input four-output reduced column model 6193

ð18bÞ



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: w B ¼ ωL ðw B þ ωT y B þ j33 uB Þ, uB ¼ ½ðk B þ ωL ÞyB þ w B =j33

ð18cÞ

: w D ¼ ωL ðw D þ ωT y D þ j44 uD Þ, uD ¼ ½ðk D þ ωL Þy D þ w D =j44

ð18dÞ

or equivalently, in vector level-temperature form: : w T ¼ ΩT ðw D þ Ωy T þ A T uT Þ, uT ¼ A T 1 ½ðK T þ ΩT ÞyT þ w T ,

A T ¼ diagðj11 , j22 Þ

: w L ¼ ΩL ðw L þ ΩyL þ A L uL Þ, uL ¼ A L 1 ½ðK L þ ΩL Þy L þ w L ,

A L ¼ diagðj33 , j44 Þ

where w T ¼ ðw s , w e Þ0 ,

ΩT ¼ ωT I22 ,

K T ¼ diagðk s , k e Þ,

uT ¼ ½uV , uR 0 w L ¼ ðw B , w D Þ0 ,

ΩL ¼ ωL I22 , uL ¼ ½uB , uD 0

K L ¼ diagðk B , k D Þ,

This internal model control (IMC) realization (made by a controller and an open-loop observer) has anti-windup protection because the load observer works with and without control saturation.41 4.4. Closed-Loop Dynamics and Tuning. The column (eq 2) with the preceding controller (eq 18) yields the closed-loop dynamics (the nonlinear functions are defined in Appendix C): : : _ ð19aÞ χ ¼ φðχ, d, dÞ þ qðχ, d, d; eÞ, χ ¼ ðx 0 , z0 Þ0 : : e ¼ Ωe þ qe ðχ, e, d, dÞ,

e ¼ ^b b,

Ω ¼ bdðΩT , ΩL Þ

ð19bÞ

3 Set the temperature control gains somewhat faster then the open-loop composition dynamics: ks = ke = kT = nλx, n ∈ [11.5]. 4 Set the level control gains similar to the open-loop composition dynamics λx: kB = kD = λx. 5 Set the observer gain (at least) three times slower than the holdup dynamics: ωL = ωB = ω = λh/3. 6 Make a small set point or load change and assess the control and output responses while gradually increasing the observer gain up to its ultimate value ωþ where the response becomes oscillatory, and back off by setting ω e ωþ/3. 7 Gradually increase kT, determine the ultimate control gain kþ that makes the response oscillatory, and back off by setting kT e kþ/3. If necessary, adjust ks and/or ke to improve response versus input behavior. 8 Set the level controller gains smaller than the ones of the temperature loops,35 say at kB = kN = kT/q with q ∈ [3, 10]. The proposed linear-decentralized (LD) OF controller (eq 18) recovers the behavior (up to load estimation convergence) of its exact reduced model-based nonlinear FF-SF passive counterpart (eq 12), and the closed-loop form (eq 19) of the controller can be realized as a set of four decentralized PI loops, as it has been done in previous works.20,30 The advantage of the observercontroller realization [in IMC form, eq 18] is that it has inherent anti-windup protection, as the observer functioning is independent of actuator saturation. 4.5. Control Loops in Terms of Volume Level Measurements and Actuators. In an industrial application the common situation is to measure volume level holdups and manipulate volumetric flow rates. In this case, the passive model has the same relative degrees as the ones of the model with molar holdup outputs and flow rate inputs, and the application of the same methodology yields the OF controller : w s ¼ ωT ðw s þ ωT y s þ ξ11 uFV Þ, ð21aÞ uFV ¼ ½ðk s þ ωT Þys þ w s =ξ11

30,32

to the resulting The application of the small gain theorem closed-loop system (eq 19) yields that the closed-loop system is stable if: (i) the observer gains are sufficiently slower (or faster) than a high (or low) limit ωþ (or ω) imposed by the highfrequency (i.e., parasitic) holdup dynamics λh (or the low frequency modeling errors, of the order of the open-loop composition characteristic frequency λx), and (ii) the control gains are set sufficiently slower than the observer gain. Technically speaking, the closed-loop system is stable if the control and observer gains are chosen so that 0 < k < k þ ðω, λh Þ,

ω ðkÞ < ω < ωþ ðλh Þ

ð20Þ

where

: w e ¼ ωT ðw e þ ωT ye þ ξ22 uFR Þ, uFR ¼ ½ðk e þ ωT Þye þ w e =ξ22

ð21bÞ

: w B ¼ ωL ðw B þ ωL yB þ ξ33 uFB Þ, uFB ¼ ½ðk B þ ωL Þy B þ w B =ξ33

ð21cÞ

: w D ¼ ωL ðw D þ ωL yD þ ξ44 uFD Þ, uFD ¼ ½ðk D þ ωL Þy D þ w D =ξ44

ð21dÞ

_

_

y s ¼ Ts T s , y e ¼ Te T e , y B ¼ SB SB , y D ¼ SD SD

where k ¼ minðk s , k e , k B , k D Þ,

ω ¼ minðωT , ωL Þ

From the previous CL stability assessment, conventional-like tuning guidelines can be established. Furthermore, these guidelines can be applied during the column’s operation around its SS, to draw the tuning for the SU period: 1 From tests and/or simulations, estimate the open-loop composition (λx) and holdup (λh) characteristic frequencies with λh ≈ 4/τL, where τL = Nτh is the overall liquid lag, τh is the tray time constant, and N is number of trays.38,42 2 Set the batch-to-continuous transition time θH from three to then times the overall liquid lag τL: θH = ντL, ν ∈ [310].

ξ11

uFV ¼ V V V V , uFR ¼ R V R V , uFB ¼ BV BV , uFD ¼ DV DV ¼ as ps σB , ξ22 ¼ ae σ D , ξ33 ¼ 1, ξ44 ¼ 1

SB (or SD) is the volume in the reboiler (or condenser) drum, and (VV, RV, DV, BV)0 is the control vector υ = (V, R, D, B)0 expressed in volumetric flow units, σ hD), to the SS density of the vapor hB (or σ in the reboiler (or the liquid in the condenser). Basically, this volume-based controller has the same structure and parameters as the ones of the mole-based counterpart, in the sense that: (i) 6194



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Figure 5. Output responses for column A with conventional, MTA, and proposed SU approaches.

temperature control

both controllers share the same relative degree structure, and (ii) the controller functioning after tuning is independent of the exact numerical values of the static parameters, as long as the sign of the static parameter is correct.13,14,20

5. OF STARTUP CONTROL In this section the event and temperature-level OF controllers are put together to yield the control scheme for the closed-loop startup operation scheme. 5.1. OF Startup Controller. In the case of molar holdups and flow rates, the combination of the event controller (eqs 5a,b) (which decides the duration of the total reflux period) with the linearar-decentralized controller, eq 17 [or eq 21] (which steers the column to the prescribed SS in continuous regime) yields the variable-structure SU controller: • Total Reflux period (0 e t e ts): event control ðF, D, BÞ ¼ ð0, 0, 0Þ,

V ¼ R ¼ V,

ts ¼ μs ½ys ðtÞ, ye ðtÞ

ð22aÞ

: w T ¼ ΩT ðw T þ Ωy T þ A T uT Þ, uT ¼ A T 1 ½ðK T þ ΩT ÞyT þ w T

In the total-reflux period [0, ts], the vapor and reflux flows are equal (V = R = V), there is neither feed nor extraction (F = B = D = 0), and the duration ts of the total reflux period is online decided by the OF event controller (eq 22a) on the basis of the sensitive temperature measurement pair (ys, ye). At the end of the total reflux period, the reboilercondenser holdups (mB, mD) are regulated at their prescribed nominal values by manipulating the effluent flow rate pair (B, D), and the feed, reflux, and vapor flow rates are set at their SS values for a short period of time in order to ensure CL stability during the extraction period. Over the extraction period, the reboilercondenser holdup levels (mB, mD) and the temperature pair (ys, ye) are regulated at their prescribed nominal values by manipulating the extraction (B, D) and internal vaporreflux (V, R) flow rate pairs. 5.2. Stability. The column state motion during the initial total reflux operation is denoted by x b ðtÞ ¼ Ξb ½ts , 0,

• Extraction period (ts e t < ¥): For

ts < t e ts þ θH :

0

ðF, R, VÞ ¼ ðF, R, VÞ ð22bÞ

level control : w L ¼ ΩL ðw L þ ΩyL þ A L uL Þ, uL ¼ A L 1 ½ðK L þ ΩL ÞyL þ w L

ð22cÞ

For ts þ θH < t < ¥ : level control : w L ¼ ΩL ðw L þ ΩyL þ A L uL Þ, uL ¼ A L 1 ½ðK L þ ΩL Þy L þ w L

ð22dÞ

ð22eÞ

0 0

x b ¼ ðc , m Þ ,

x bo , db ð 3 Þ,

0 e t e ts ¼ μs ½ys ðtÞ, ye ðtÞ

ð23Þ

where xbo is the initial condition and db is the exogenous input disturbance which includes parameter and flow errors. From standard arguments in distillation column engineering,43 the composition profile at total reflux reaches asymptotically a unique SS denoted by xb*, that is, xb(t) f xb*, meaning that the motion xb(t) is stable over the period [0, ts],29 with final state xbf = xb(ts). The column state motion of the closed-loop column with the level controller (eq 22c) (including column and level controller dynamic states wL) over the initial interval [ts, tc] of the extraction period is denoted by 6195



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Figure 6. Input and off-spec production responses for column A with conventional, MTA, and proposed SU approaches.

x σ ðtÞ ¼ Ξσ ½ts , tc , x bf , dσ ð 3 Þ, 0

x σ ¼ ðc0 , m0 , w L Þ ,

ts e t e tc ¼ ts þ θH

j~x so j e δo , ð24Þ

where xbf is the initial condition and dσ is the exogenous input disturbance. Assuming that the level controller and observer gains were chosen so that the system (eq 20) is stable, and from standard arguments in distillation column engineering,3,44 this closed-loop motion reaches asymptotically the nominal SS x σ = (x, m), that is, xσ(t) f x σ, meaning that the motion xσ(t) is stable over the period [ts, tc],29 with final state xσf = xσ(tc). Recall the closed-loop column (including column and controller dynamic states) over the batch-to-continuous period, eq 20, assuming that the controller and observer gains were chosen so that the system (eq 20) is stable, according to the guidelines of Section 4.5, and write the corresponding motion: x c ðtÞ ¼ Ξc ½tc , t, x σf , dc ð 3 Þ,

x cs ¼ ðx 0bf , 0Þ,

x c ¼ ðc0 , m0 , W 0 Þ0 , tc < t < ¥

ð25Þ

with initial condition xσf and disturbance dc( 3 ). By construction, this motion reaches asymptotically a compact neighborhood of the prescribed SS, that is, xc(t) f x c. Consequently, the combined motion x s ðtÞ ¼ Ξs ½ts , t, x so , ds ð 3 Þ, xs ¼ xb

if

0ete¥:

t ∈ ½0, ts ; x s ¼ x σ if t ∈ ðts , tc ; x s ¼ x c if t ∈ ðtc , ¥Þ ð26Þ

of the variable-structure SU control system is practically exponentially (PE) stable, in the sense that, for given disturbance size set (δo, δd, εx) > 0 there is a set (a, λ, bd, bp) of positive constants so that the perturbed motions ~x (t) are bounded as follows:29

jj~ds ðtÞjj ¼ sup j~ds ðtÞj e δd

w j~x ðtÞj e aeλt j~x so jλt þ bd jj~ds ðτÞjj e εx " t ∈ ½0, ¥Þ It must be pointed out that the possibility of having open-loop instability due to mass (rather than molar) flow manipulation has been reported,35,36 and its implications for open-loop startup column policies have been assessed: depending on the open-loop startup policy, different steady states may be reached.37 In our SU approach this is not a problem, as the column state motion reaches a unique steady-state in continuous regime because: (i) the steady-state at total reflux, toward which the column tends in the initial period at total reflux, is unique, and (ii) in the extraction period the column motion is steered toward a unique steady-state in continuous regime, provided sufficiently tight feedback level in conjunction with single or two-point temperature control is used,36 which in our nonlinear SF control design terminology means that the closed-loop zero dynamics for the extraction period is stable with a unique steady state. 5.3. Concluding Remarks. The proposed SU control scheme functions as follows: (i) the duration ts of the total reflux period is decided by the event controller (eqs 5a,b) driven by the sensitive temperature pair (Ts, Te), (ii) over the short feed-extraction period [ts, ts þ θH] the feed is injected at its nominal value (F = F) and the reboiler and condenser decentralized loops are closed to maintain the corresponding holdups at the prescribed SS (mD ≈ mD, mB ≈ mB) by adjusting the extraction flows (D and B), and (iii) over the extraction period [ts þ θH, ¥), the feed is maintained at its nominal _ maintained _ close to their set value (F = F), and the four outputs are points (mD ≈ mD, mB ≈ mB, Ts ≈ T s, Te ≈ T e) by means of four decentralized SISO loops which adjust the two internal (V and R) and two external (B and D) flow rates. In the closed-loop extraction period, the temperature controllers are conventional loops in the 6196



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Table 2. Regulation Times for Distillation Column A startup approach conventional

distillate, min

% savings

bottoms, min

101

% savings

117

wave model based control (Han and Park12)

80

20.79%

100

14.53%

minimum time algorithm (Yasuoka et al.3)

77.5

23.27%

104.30

10.85%

proposed approach

66.5

34.16%

56.20

51.97%

Table 3. Off-Spec Products for Distillation Column A startup approach

distillate, kmol

% savings

bottoms, kmol

conventional

50.34

minimum time algorithm (Yasuoka et al.3) proposed approach

% savings

25.94

48.47%

39.33

32.75%

21.49

57.31%

15.6

73.32%

58.48

Figure 7. Output responses for column B with conventional, MTA, and proposed SU approaches.

sense that each loop is driven by one sensitive temperature per section and not by a synthetic (combined) temperature signal. The proposed SU algorithm (eq 22) is a rather simple variable structure controller, with a straightforward implementation which requires approximated values of three SS parameters: _ _ ð27Þ Σ ¼ ðps , Δþ T e =ms , Δþ T e =me Þ which are typically available in industrial settings and online (two temperature and two level) measurements, without appreciable delays and/or deadtime. Since the control performance is limited by noise-like effect of the parasitic holdup dynamics, a simulationbased tuning must be executed with a model with holdup dynamics. Otherwise, unrealistically large control and observer gains with fast closed-loop response will be obtained.20 Once the column has reached a prescribed neighborhood of its prescribed SS, the column regulation about its SS can be done with either the MIMO control scheme of the extraction period of the proposed SU scheme or with any of the existing one or two point (composition, temperature, cascade, etc.) control algorithms, depending on the particular column’s control objectives, the desired disturbance rejection capabilities, and the available information.45

6. CASE EXAMPLES In this section the proposed SU scheme is illustrated and tested with the three (two ideal and one nonideal) column case examples, including: (i) duration, off-spec production, and heat consumption assessments, (ii) comparisons with the conventional and MTA3 SU approaches, and (iii) implementation for the nonideal column with nonlinear tray holdup dynamics for two cases: (a) molar holdup (i.e., level) measurements in the reboiler and condenser drum using molar control inputs and (b) volume holdup measurements and volumetric flow rate inputs. Following Shinskey’s recommendation,46 the comparison of response times will be performed in terms of the natural settling time units (Nstu). For the startup, the natural settling time corresponds to that of the conventional startup (see Section 1). The convergence criterion requires that the effluent purity stays within ( 0.001 of the SS value. 6.1. Column A. From the SS temperature profile shape (Figure 2a) in the light of the sensitive tray-based measurement 6197



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Figure 8. Input and off-spec production responses for column B with conventional, MTA, and proposed SU approaches.

Table 4. Regulation Times for Distillation Column B startup approach

distillate, min

% savings

bottoms, min

conventional

120.2

minimum time algorithm (Yasuoka et al.3) proposed approach (molar flows)

% savings

116.0

3.46%

121.7

17.17%

118.0

1.80%

76.0

48.29%

% savings

bottoms, kmol

% savings

147.0

Table 5. Off-Spec Products for Distillation Column B startup approach

distillate, kmol

conventional

35.9


18.9

47.30%

20.7

53.23%

proposed approach (molar flows)

23.3

35.03%

10.7

75.97%

location criteria,13,21,22 the proposed SU scheme was set with the sensitive temperatures in trays 13 and 24 in the stripping and rectifying sections, respectively. The transition time θH = 10 min was set 4 times the overall liquid lag τL = 2.5 min, and the application of the tuning guidelines (given in Section 4.4) yielded the control and observer gains: ωo ¼ 3:33 min1 ,

ωs ¼ ωe ¼ 10ωB ¼ 10ωN ¼ 0:267 min1

On the basis of the conventional open-loop SU scheme,3 the bottoms (or distillate) settling time is Nstu =117 (or 101) min. The output (Figure 5) and input (Figure 6) responses with the conventional, MTA, and proposed SU approaches are presented in Figures 5 and 6, and the corresponding time (or off-spec product) assessments are listed in Table 2 (or 3). According to these results, with a reasonable control action (Figure 6), the proposed SU technique yields: (i) the shortest duration of 56.2 (or 66.5) min for the bottoms (or distillate) composition (Figure 5a,b and Table 2) and (ii) the least bottoms and distillate

44.3

off-spec production (Figures 6e,f and Table 3). For the sake of comparison with an optimization-based SU applied to a different column,4 only the bottoms product composition regulation time was reduced (40.91%), while the distillate product took 112.5% longer time to attain its SS composition. 6.2. Column B (BenzeneToluene). From the examination of the SS temperature profile shape (Figure 2b) in light of the sensitive tray-based measurement location criteria,13,21,22 the SU scheme was set with the sensitive temperatures in trays 5 and 12 in the stripping and rectifying sections, respectively. The short extraction time θH = 5 min was set equal to 4 times the overall liquid lag τL = 1.05 min. The application of the tuning guidelines (given in Section 4.4) yielded the OF control gains: ωo ¼ 3:33 min1 ,

ωs ¼ ωe ¼ 2ωB ¼ 2ωN ¼ 0:133 min1

According to the conventional open-loop SU scheme, the bottoms (or distillate) settling time is Nstu = 147 (or 120) min. The output (Figure 7) and input (Figure 8) responses with the 6198



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Figure 9. Output responses for column C with conventional, MTA, and proposed SU approaches implemented with molar level holdup measurements and control inputs.

Table 6. Regulation Times for Distillation Column C startup approach

distillate, min

% savings

bottoms, min

% savings

conventional minimum time algorithm (Yasuoka et al.3)

154.0 136.0

11.69%

95.0 80.0

15.79%

proposed approach, molar flows

141.0

8.44%

76.5

19.47%

proposed approach, volumetric flows

135.0

12.34%

79.5

16.32%

% savings

bottoms, kmol

% savings

Table 7. Off-Spec Products for Distillation Column C startup approach

distillate, kmol

conventional

46.1


17.1

62.90%

0.3

98.87%

proposed approach, molar flows

22.9

50.32%

3.1

89.13%

proposed approach, volumetric flows

20.4

55.79%

4.2

85.38%

conventional, MTA, and proposed SU approaches are presented in Figures 7 and 8, and the corresponding regulation times (or off-spec product) assessments are listed in Table 4 (or 5). With rather overall smooth control action (Figure 8), the proposed SU technique yields: (i) the shortest duration of 76 min for the bottoms composition (Figure 7a and Table 4) response, and (ii) a duration of 118 min for the distillate regulation, which is similar to the ones obtained with the conventional and MTA approaches (Figure 7b and Table 4). Compared with the conventional approach (Figure 6e,f and Table 5): (i) the proposed SU scheme attains the largest reduction (76%) in the off-spec bottoms production, followed by the MTA approach (53.2%), and (ii) the MTA approach yields the largest reduction (47.3%) in the off-spec distillate production, followed by the proposed approach (35%). 6.3. Column C (MethanolWater). In this subsection, the proposed SU approach is tested with the nonideal methanol water column case example, including system detailed modeling with energy balances and nonlinear tray holdup dynamics, as well

28.7

as molar and volumetric flow-based control implementations, in the understanding that the regulation problem of this column has been tested previously with linear and nonlinear two-point temperature controllers endowed with set point compensation schemes.13,14 The purpose is twofold: (i) the testing of the proposed SU approach with a situation that is more realistic than the ones of the ideal columns (A and B) with linear tray holdup dynamics and molar flow-based control implementation, and (ii) the verification of the SU algorithm robust functioning, with simplified model-based control scheme and detailed column model-based testing. On the basis of the SS temperature (rather asymmetric) profile (Figure 2c) and the application of the sensitive tray location criteria,13,21,22 the SU scheme was set with the sensitive temperatures in trays 1 and 4 in the stripping and rectifying sections, respectively. The short extraction time θH = 4 min was set equal to 4 times the overall liquid lag τL = 0.6 min. From the application of the tuning guidelines (given in Section 4.4) the next OF control gains follow: ωo = 2.00 min1, 3ωs = 3ωe = ωB = ωN = 6199



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Figure 10. Input responses for column C with conventional, MTA, and proposed SU approaches implemented with molar holdup measurements and control inputs.

Figure 11. Behavior of column C with proposed approach implemented with volumetric flow control inputs and holdups.

0.4 min1. From the examination of the response with the conventional open-loop SU scheme (Figure 9a,b), the bottoms (or distillate) settling time is Nstu = 95 (or 154) min. As depicted in Figure 4C, for this column, the SS deviation measures behaves as follows: (i) Mw shows only one minimum, at 79 min, and (ii) MT has its minimum at approximately 64 min. For the startup, the settling time corresponds to that of the conventional startup. For the case of molar flow control inputs, the column SU behavior with the conventional, MTA, and proposed approaches are presented in Figure 9 and Tables 6 and 7, showing that: (i)

the proposed and MTA approaches yield similar response times for bottoms (80 min) and distillate (140 min) compositions (Figure 9 and Table 6) that, as expected, are faster than the ones obtained with the conventional approach, and (ii) when compared with the conventional SU approach, the MTA yields the largest reduction, of 98.87 (or 62.9) %, in off-spec bottoms (or distillate) product, closely followed by the proposed approach with 89.13 (or 50.32)%, in off-spec bottoms (or distillate) product. In terms of performance versus scheme simplicity and amount of online information required, the comparative results for the proposed scheme are rather favorable. 6200



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Table 8. Total Heat Requirement for Startup column A R

V dt, kmol

benzenetoluene column

% savings

R

V dt, kmol

methanolwater column R

% savings

V*ΔHB dt, kJ

% savings

conventional

375.14

0.00%

128.00

0.00%

2.34 10

minimum time algorithm

334.42

10.85%

106.00

17.19%

1.97 106

15.81%

proposed approach, molar inputs

185.22

50.63%

67.80

47.03%

1.99 106

14.96%

2.02 106

13.68%

proposed approach, volumetric inputs

The column behavior with the proposed volumetric flow inputbased SU approach (eq 21) is presented in Figure 11, showing that the output, input, and off-spec responses are basically similar to the ones obtained with the molar flow-based control implementation (Figures 9 and 10). This comparison verifies the theoretically drawn claim that the SU molar and volumetric flowbased controllers are structurally equivalent, in the sense that both implementations are supported by models with the same relative degree structure and that the control functioning is robust with respect to the values of the static model parameters (provided the gain sign is the correct one). 6.4. Heat Consumption. As a refinement of the input behavior results presented in the preceding subsections, in this subsection the dependency of the heat consumption on the (ideal or nonideal) column type and (conventional, MTA, and proposed) SU scheme is assessed. In the ideal cases (columns A and B), the heat consumption is just the integral of the vapor flow over the SU duration multiplied by a constant (the heat of vaporization), with the duration being determined by the bottoms composition response within prescribed tolerance (see Tables 2, 4, and 6). In the nonideal methanolwater case (column C), the heat consumption is computed by integrating, over the SU duration, the product of the vapor molar flow rate with the corresponding composition-dependent heat of vaporization. The results for the three columns with the conventional, MTA, and proposed SU approaches are listed in Table 8, showing that: (i) in the ideal column A (or B) cases, the proposed SU approach yields the largest 50.63 (or 47.03)% reduction in heat consumption with respect to the one (100%) of the conventional approach, followed by the MTA approach with 10.85 (or 17.19)% reduction, and (ii) in the nonideal column C with molar (or volumetric) flow-based control, the proposed SU approach yields 20 (or 13.65)% reduction in heat consumption with respect to the one (100%) of the conventional approach. According to these results, the proposed SU approach yields: (i) with respect to the conventional SU approach, a considerable ≈50 (or 20)% reduction in energy for the ideal (or nonideal) column case and, (ii) with respect to the mole-based control implementation, a moderate difference (≈6.5%) in energy consumption with respect to the one with volume-based control implementation.

0.00%

initial open-loop operation at total reflux, and (ii) a linear-decentralized MIMO (4 4) controller with two holdup (reboiler and condenser) loops and two temperature loops (one per section, at a senstivie tray) that over the extraction period steer the columns toward its prescribed SS in continuous regime. The temperature sensors are located according to well-known sensitive tray criteria employed in temperature control designs for distillation columns, the control scheme has a reduced model dependency in the sense that its implemention requires only estimates of static parameters, and the control gains are set according to easily apply tuning guidelines drawn from a formal closed-loop stability assessment. By virtue of the structure-oriented approach employed, the control loops can be implemented either in a molar or in a volumetric flow basis. The proposed SU control scheme was tested through simulations with three high-purity columns (two ideal ones with holdup-flow mole control, and one nonideal with holdup-flow volume control), finding that, with considerably less model dependency and online measurements, the proposed SU approach matches or improves the behavior of previous approaches, in terms of duration, off-spec production, and heat consumption. In principle, the proposed approach can be extended to the case of multicomponent columns.

’ APPENDIX A: REDUCED ORDER PASSIVE MODEL Here, a passive reduced order model for control design purposes is drawn. As it is known in distillation column control, the hydraulic dynamics are faster than composition dynamics18 so that they can be assumed in quasi-steady state in the design stage, and their effect must be accounted for in the tuning stage.13,20 Then, eqs 1ef can be set as eq A.1 with liquid flows given by eq A.2: : ðA.1Þ m ¼ Fm ðm, δ, νÞ 0 Stripping section Lðmi Þ R þ F,

1 e i e nF

ðA.2aÞ

nF þ 1 e i e N

ðA.2bÞ

Rectifying section Lðmi Þ R,

The unique root of eq A.1 around the SS (c, m , δh, νh) is given by

7. CONCLUSIONS A control scheme for the startup (SU) of a binary distillation column with two temperature and two level measurements has been developed, for the quick attainment of closed-loop stable operation in the continuous regime with minimum off-spec production and heat consumption. The results are a variable structure scheme with: (i) an event controller which, driven by two temperature measurements located at sensitive trays (one per section), decides the duration of an

6

mi ¼ Gi ðδ, νÞ ¼ ðR þ F R FÞτh þ mi ,

1 e i e nF ðA.3aÞ

mi ¼ Gi ðδ, νÞ ¼ ðR RÞτh þ mi ,

nF þ 1 e i e N ðA.3bÞ

where τh is the tray hydraulic time constant for the linearized Francis weir equation for the holdup dynamics (eq 1hi). 6201



ARTICLE

Substitute eqs A.2 and A.3 in eq 2a to obtain the composition dynamics reduced-order model: : : c0 ¼ ½ðR þ FÞðc1 c0 Þ VΔ Eðc0 Þ=mB Z f 0 ðc, ν, δÞ

where the nonlinear functions f, g, F, Fz are given by

ðA.4aÞ

jðx, m, d, uÞ ¼ Fc ½ΦðxÞ, m, d þ δ, u þ ν

: ci ¼ ½ðR þ FÞΔþ cþi VΔ Eðci Þ= ½ðR þ F R FÞτh þ mi Z f i ðc, ν, δÞ,

1 e i e nF 1

ðx I , x II Þ0 ¼ ΦðxÞ; ΦðxÞ ¼ fh1 ½hðcÞ þ x I 0 ; ½cII þ x II 0 g0 0

f ðx, d, uÞ ¼ j½x, Gðd, uÞ, d, u,

F ¼ ðFI , FII Þ0 ; 0

: cnF ¼ ½RΔþ cnF þ FðcF cnF Þ VΔ Eðci Þ= ½ðR þ F R FÞτh þ mnFl Z f nF ðc, ν, δÞ ðA.4cÞ

ψ ¼ hðcÞ ¼ ½βðcs Þ, βðce Þ, mB , mD

0

ðA.4eÞ where ðA:4fgÞ ðA.4hÞ

Am

Now consider the time derivative of the output in deviation coordinates: : x s ¼ β0 ðcs Þ½ðR þ FÞΔþ cs VΔ Eðcs Þ= ½ðR þ F R FÞτh þ me Z f s ðx I , x II , u, dÞ ðA.5aÞ

ðA.5c,dÞ

Then rewrite the reduced system column dynamics (eq A.4) in terms of the deviation coordinate change (eqs 6dw) and rewrite them in compact notation to obtain eq 6a, where f II 0

f I 0 ¼ ðf s , fe, f B , f D Þ½x I , x II , u, d, ¼ ðf 1 , f 2 , :::, f s 1 , f s þ 1 , :::, f e 1 , f e þ 1 , :::, f N , f D Þ½xI , xII , u, d

’ APPENDIX B. COLUMN DYNAMICS IN SINGULAR PERTURBATION FORM Consider the coordinate change (eqs 6ad), for the slow system (composition) dynamics, and the following coordinate change for the fast holdup dynamics:

zi ¼ mi mi ¼ mi Gi ðδ, νÞ

ðB.1Þ

where Gi(δ, υ) is given by eq A.3, and express the distillation column (eq 2) in the following singular perturbation form:19 : x ¼ f ðx, d, uÞ þ Fðx, d, u; zÞ, y ¼ x I ,

x ¼ ðx I , x II Þ0 0

: : : z ¼ gðd, u; zÞ þ Fz ðd, u; d, uÞ

0

2

1=τh 6 6 0 6 6 0 ¼6 6 l 6 6 0 4 0

1=τh 1=τh 0 l 0 0

0 1=τh 1=τh l 0 0 2

3 τh 6 6 τh 7 7 7 Au ¼ 6 6 l 7, 4 5 τh

: x e ¼ β0 ðce Þ½RΔþ ce VΔ Eðce Þ=½ðR RÞτh þ me Z f e ðx I , x II , u, dÞ ðA.5bÞ : x B ¼ R þ F V B Z f B ðu, dÞ, : x D ¼ V R D Z f D ðu, dÞ

0

0

f ð0, 0, 0Þ ¼ 0

Fðx, d, u; 0Þ ¼ 0

gðd, u; zÞ ¼ Fm ½z þ Gðd, uÞ, d þ δ, u þ ν ¼ A m z, gðd, u; 0Þ ¼ 0 : : : : Fz ðd, u; d, uÞ ¼ ½Dd Gðd, uÞd ½Du Gðd, uÞu : : ¼ A uuR þ A ddF , Fz ðd, u; 0, 0Þ ¼ 0

: ci ¼ ½RΔþ ci VΔ Eðci Þ=½ðR RÞτh þ mi Z f i ðc, ν, δÞ, nF þ 1 e i e N ðA.4dÞ

: mB ¼ R þ F V B Z qB ðν, δÞ, : mD ¼ V R D Z qD ðν, δÞ

f ¼ ðf I , f II Þ0

Fðx, d, u; zÞ ¼ j½x, z þ Gðd, uÞ, d, u j½x, Gðd, uÞ, d, u,

ðA.4bÞ

: cN þ 1 ¼ V½EðcN Þ cN þ 1 =mD Z f N þ 1 ðc, ν, δÞ

0

0 0 1=τh 3: 0 0

0

333 3: 3: 1=τh 0 2 3 τh 6 7 6 l 7 6 7 6 τh 7 7 Ad ¼ 6 607 6 7 6 : 7 4 5 0

3 0 7 0 7 7 0 7 7, 0 7 7 1=τh 7 5 1=τh

_ u) Note that the functions g(d, u; z), F(x, d, u; z), and Fz(d, u; d, _ vanish with the arguments after the semicolon. Equations B.2ac correspond to the slow composition dynamics in the new coordinates x, and eq B.2d corresponds to the fast holdup dynamics. The holdup dynamics z are stable because the bidiagonal matrix A m has negative values in the principal diagonal. Furthermore, the variable z always converge to zero, since the roots of the QSS eq A.1 are unique and are given by eq A.3. The only mechanism that can destabilize the holdup dynamics z corresponds to the case where the reflux and/or feed rate changes very quickly, that is, their derivatives have very large _ u) _ = and sustained changes, as shown by the function Fz(d, u; d, _ In other words, the singular perturbation model A uu_ R þ A dd. (eq B.2) can be used as long as not very large and sustained changes in the column flows are made or required.

’ APPENDIX C. NONLINEAR MAPS ~pI ðx I , x II , d, u; zÞ ¼ FI ½x, d, μðx I , x II , dÞ; z; ~pI ðx I , x II , d, u; 0Þ ¼ 0

ðB:2acÞ

~pII ðx I , x II , d, u; zÞ ¼ FII ½x, d, μðx I , x II , dÞ; z; ~pII ðx I , x II , d, u; 0Þ ¼ 0

ðB.2dÞ 6202



ARTICLE

θðx, d; zÞ ¼ g½d, μðx I , x II , dÞ; z; : : : ~pz ðx, z, d, dÞ ¼ Fz ½d, μðx I , x II , dÞ; d, pu ðx, z, d, dÞ φx ðx, dÞ ¼ f½ Kx I 0 ; f II ½x I , x II , μðx I , x II , dÞ, d0 g0 ; ~pðx, d; zÞ ¼ F½x, d, μðx I , x II , dÞ; z : pu ðx, z, d, dÞ ¼ ½ðDf c =DuÞ1 fK c ½ K c x c þ pc ðx, d; zÞ : þ ðDf c =DxÞ½φx ðx, dÞ þ pðx, d; zÞ þ ðDf c =DdÞdg : : φðχ, d, dÞ ¼ f½φx ðx, dÞ þ ~pðx, d; zÞ0 , ½θðx, d; zÞ þ ~pz ðx, z, d, dÞ0 g0 : qðχ, d, d; eÞ ¼ f½qx ðχ, d; eÞ þ rx ðχ, d; eÞ0 , ½qz ðχ, d; eÞ : þ rz ðχ, d, d; eÞ0 g0 qx ðχ, d; eÞ ¼ f ½x, d, μðx I , x II , dÞ A 1 e f ½x, d, μðx I , x II , dÞ; qx ¼ ðqc , qI Þ0 0

0

rx ðχ, d; eÞ ¼ F½x, d, μðx I , x II , dÞ A 1 e; z F½x, d, μðx I , x II , dÞ; z,

rx ¼ ðrc , rI Þ0 0

0

qz ðχ, d; eÞ ¼ g½d, μðx I , x II , dÞ A 1 e; z g½d, μðx I , x II , dÞ; z : : : rz ðχ, d, d; eÞ ¼ Fz fd, μðx I , x II , dÞ A 1 e; d, pu ðx, z, d, dÞg : : Fz ½d, μðx I , x II , dÞ; d, pu ðx, z, d, dÞ : qe ðχ, e, d, dÞ ¼ fðDf c =DxÞ½φx ðx, dÞ þ pðx, d; zÞ : þ qx ðχ, d; eÞ þ rx ðχ, d; eÞ þ ðDf c =DdÞdg

þ ½ðDf c =DuÞ AA 1 f K½ Kx I þ pc ðx, d; zÞ þ qc ðχ, d; eÞ þ rc ðχ, d; eÞ þ Ωeg

’ APPENDIX D. NONIDEAL MODEL FOR COLUMN C The SS parameters for this methanolwater column are given in ref 13. The model for this column should include energy balances, as described in ref 15. The thermodynamic model is composed of vapor pressure correlations,47 the Wilson’s equation for computing activity coefficients,48 and liquid and gas enthalpy calculations.49 The Francis weir equation16 FL ¼ 3:33Lw ðhow Þ1:5

ðD.1Þ

where FL is the liquid flow rate over the weir (ft3/s) Lw is the length of the weir (0.623 ft) how is the height of the liquid over weir (ft) is modified to obtain the molar liquid flow exiting form tray i Li ¼ Ri ðmi mi0 Þ1:5 Ri ¼ 3:33Lw ½30:48=σi 1=2 =A 3=2 ,

ðD.2Þ

mi0 ¼ σi ðAhw Þ

where Li: liquid molar flow exiting from tray i, mol/s σi: molar density of liquid in tray i, mol/cm3 (ref 50) A: tray active surface (540 cm2)

Lw: weir length (0.623 ft = 19 cm) hw: weir height (3.2 cm) mi: total molar holdup in tray i (mol) mi0: molar holdup at the weir height (mol) Equation D.2 is the function used to simulate nonlinear tray hydraulics dynamics for the methanolwater system.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Fax: 52 (55) 16691505. Notes †

A preliminary version of this paper was presented at International Symposium on Advanced Control of Industrial Processes 2008, (ADCONIP 2008).

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Binary Distillation Column Startup Using Two Temperature

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