MODELLING AND CONTROL OF TWO-PHASE SYSTEMS

Daniel R. Lewin", Miriam Faigon, Avraham Fuchs and Raphael SemiatDepartment of Chemical Engineering, Technion I.I.T., Haifa 32000, Israel

ABSTRACT

This paper outlines a linear model-based control design procedure, suitable for processes involving two­phase phenomena, where the transient response is often subject to nonminimum phase characteristics:inverse response and time delays. We show that in order to capture these features in the process modelsimplicit in the control scheme, it may be necessary to model the transients in each phase separately,thus precluding simple homogeneous approaches. Furthermore, as shown by two examples, combina­tions of nonminimum phase characteristics can be accurately modelled by the Parallel Linear Structure,introduced here, which can be the basis for improved linear control design.

KEYWORDS:Process Dynamics, Two-Phase Systems, Inverse Response, Evaporators, Boiling, Feedback Control

1. INTRODUCTION

Inverse response phenomena are commonly found in chemical processing equipment. They may occurwhen a single input perturbation affects the process output in two opposing directions, with onedirection being the dominant steady state effect, and the other being the dominant transient effect.Consider the simple example of a liquid flowing in a heated pipe. Since the thermal expansion coeffi­cient is usually positive, a sudden temperature increase at the inlet will soon cause the first part of thepipe contents to heat up and also to expand. Therefore, the residence time of the fluid willprogressively decrease, so that the outlet temperature will also decrease before the expected (positive)thermal wave reaches the outlet (Fuchs et al, 1991).

Since the mean density of a two-phase mixture is usually a very sensitive function of intensivevariables, one would expect that inverse response phenomena should be common in two-phase systems.Selected examples amongst the many reported cases in the literature are:

1) Distillation Column Tray Dynamics: Luyben (1969), analyzed the response of a binary distillationcolumn to a step increase in the internal vapor flow rate. This change affects the transient of thelight component concentration on a given column tray through two mechanisms. The first,dominated by the tray hydrodynamic time constant, will cause the concentration to rise because theadditional vapor, rising through the tray and rich in the lighter component, will displace fluid heldon the tray, previously richer in the heavier component. The long-term effect is controlled bythermodynamics and will therefore reflect the overall tray steady state gain: as the temperatureincreases, the concentration of the lighter component will decrease. The overall transient thereforeexhibits inverse response.

2) Azeotropic Distillation Column Dynamics: Andersen et al (1991) analyzed the dynamics ofhomogeneous azeotropic distillation, and found that the effect of internal flows in the azeotropiccolumn affect the overall separation in two directions, each dominated by different time constants.Thus, depending on the chosen operating point, both overshoot and inverse response phenomena arepossible.

3) Two-phase Reactor Dynamics: Tsai and Tsao (1991) describe the dynamics of a solid-catalyzed,liquid-phase reaction in a CSTR. The existence of inverse response in the bulk fluid concentrationtransient for such a system is shown to be caused by an appropriate temporal balance betweenreactant supply rate to the bulk fluid and the mass transfer rate to the solid catalyst.

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Bearing in mind that inverse response is usually not a desired property, it is of interest to be able topredict under which conditions such a characteristic will occur. This is the justification for detailedmodelling of dynamical systems which potentially exhibit this phenomenon. Given a dynamic modelwhich captures the process' true nonminimum phase nature, the unified approach offered by InternalModel Control design principles (Morari and Zafiriou, 1989) can be utilized to perform adequatefeedback control design. In this paper, we shall consider two common chemical processing units, bothfeaturing two-phase flow: an evaporator and a crude oil preheating furnace. We shall show that carefulmodelling of these systems generates uncertain, linear approximations of the process dynamics whichincludes nonminimum phase components. Control designed on the basis of these uncertain models canbe guaranteed to satisfy robust performance specifications.

2. NONMINIMUM PHASE BEHAVIOR IN LINEAR PROCESS DYNAMICS

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Processes which exhibit inverse response and delay time are potentially difficult to control. These twophenomena are termed nonminimum phase because they impart additional phase lag to the open loopresponse. Consider for example, the net first order processes:

k(T3S+1) d k( - T3s+1)p (s) - an p (s) - ~-'--:-:-"":;---':-",1 - (T1S+1)(T2s+1) 2 - (T1s+1)(T2s+1)

The two processes have the same frequency-dependent magnitude. The expected high-frequency phaselag is therefore ;'moo<p(pj(iw)) = - I' which is the case for Pl' However, the presence of the positivezero (s= I/T3) in P2 adds an additional phase lag of -1r radians at high frequency.

<b>

15

---

I

105o-1.0 -t-----.---r--~--y---.__1

15

<a>

105

1.5

1.0

0.5

0.0

-0.5

-1.00

Time Time

Fig 1: The open loop response of pds) and P2(s) to unit step changes.Parameters are: k= 1, T}=l , T2=5 and (a)T3=O.5; (b) T3=5.

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The presence of the delay time in the process transfer function also has serious consequences regardingachievable closed loop performance. The relationship between the two nonminimum phase componentscan be gaged by the use of a Pade approximation:

(J -~s + 1e - s - 2 (2)- £8 + 1

Thus, as shown by the approximation (2) and by the simulated responses in figure I, the inverseresponse caused by the positive zero depends on its magnitude relative to that of the dominant processpole. The larger the values of T 3 in P2(s), equivalent to large (J in (2), the more problematic will be thecontrol problem . Mathematically, the transfer function form given as P2(s) can result from twoparallel first order responses to the same input signal affecting the same output in different directions:

k} k2 k( - T3s+1)P2(s) = T}s+1 - T2s+1 = (T}s+1)(T2s+1)

k 21"}-k}1"2 . ..where k = k} - k2 and T3 = k k . For Inverse response, T3 must be positive,

1 2

3. MODEL-BASED CONTROL DESIGN

3.1 Overview.In order to successfully design model-based feedback control, the following items of information mustbe available: (a) a model describing the response of the output to control variable perturbations; (b) aquantitative estimate of the reliability of the model; (c) a definition of the disturbances to be rejected

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(b)

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and/or the types of setpoint changes to be tracked; (d) specifications concerning the closed loopperformance.

r-r q ~ : t=:;r'(a)

Fig 2: (a) The fMC feedback structure; (b) The classical feedback structure.

The concept of Internal Model Control (Morari and Zafiriou, 1989), enables systematic control syn­thesis for processes with uncertain linear models. IMC design utilizes the structure given in figure 2(a),which explicitly includes the nominal process model, p(s), and a model-based controller, q(s). Followingthe IMC design procedure, the controller is generated in two steps. Firstly, an ISE (Integral SquareError) optimal controller, q(s), is computed based on the nominal process model, disregarding bothmodel uncertainty and any constraints. In its simplest form, the optimal controller is the inverse of theinvertible part of the nominal model. This is then used to form the IMC controller, by augmenting itwith a low-pass filter, q(s) = q(s)f(s). This ensures that the controller is realizable (the number of polesmust at least equal the number of zeros), and that it can be detuned to compensate for thedestabilizing effect of process uncertainty. The IMC and classical feedback structures are related,making it possible to use the IMC procedure to carry out the design and revert to the classical struc­ture for implementation. The classical controller, c(s), can be expressed as: c(s) = q(s)(l- p(s)q(s))-I.

A prerequisite to the design of linear model-based controllers is the generation of linear approximationsfor the process dynamics. These can be obtained by either linearizing a non-linear process model or byempirically fitting linear transfer functions to plant data or simulation results. As a result, the linearmodel parameter values may have significant associated uncertainty, which can be expressed as abounded range. This defines the set p(s) E 7I"(s), where 7I"(s) defines the set of processes computed fromall possible combinations of extreme parameter values. A multiplicative uncertainty bound, lm(s), canbe defined, on the basis of a nominal process model, p(s):

Ip(iw) - p(iw) I -

lm(w) = P(iw) ~ Im(w) 'tIw, 'tipE 71"

Dependent on the efficacy of this uncertainty description, a necessary and sufficient condition can bederived for robust stability. Assuming that the controller stabilizes the nominal process model, thenclosed loop stability is guaranteed on the uncertain process p(s), if and only if

lii(iw)llm(w)<l 'tIw, 'tipE 71" (5)

where ii = pc(l+pcrl is the nominal complementary sensitivity function, which can also be related theIMC controller, ii = pqf. The form of the complementary sensitivity function is a design decision. Twoimportant factors playa role in its correct definition:1) Nonrninimum phase process characteristics: If the process dynamics includes noninvertible nonmini­

mum phase components, the effect of these characteristics will also appear in the closed loopresponse, and also form part of the complementary sensitivity function.

2) Input type: The nominal loop transfer function, pc, must have as many integrators as the number ofintegrators in the input signal (disturbances to be rejected or setpoints to be tracked). Thus, forexample, if the input is effectively a ramp, jic must contain two integrators. This has consequencesregarding the form of ii. In the IMC design procedure, the requirements of input type are met bychoosing a filter of appropriate form. Thus, for step inputs, the simplest form of the filter is:f(s) =(As+lrn while for ramp inputs, the simplest form is: f(s) = (nAs+l)(As+lrn. In both cases,the number of filter poles, n, must be equal to or greater than the net order of the open loop process.

3.2 Design Methodology.

A. Modelling Process Dynamics.Often heterogeneous models will be required in order to capture inverse response characteristicsobserved in two-phase systems. In addition, the thermal feedback associated with heat integratedsystems will result in delayed shock wave effects, modelled by delay times. A generalized ParallelLinear Structure (PLS) which will allow both inverse response and delayed shock waves to be modelledis shown in figure 3. In this structure, PI' P2 and P3 are all minimum phase transfer functions. The

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presence of the delay terms, permit delayed shock waves to be modelled. The parameters for the PLSmodel can be derived by an analytical approach involving linearizing a lumped parameter non-linearmodel or by empirically fitting of the parameters in order that the PLS prediction matches thesimulated response of the non-linear model to disturbances in the input variables. This is illustrated infigure 4, where the simulated response of the effluent temperature of a crude preheating furnace to astep change in the fuel feed rate, as predicted by a detailed , distributed parameter, non-linear model iscompared with the response using a PLS model with a significant delay on the third term (Fuchs ei ai,1991). In general, the analytical approach is preferred since this will yield condit ions for the existence ofinverse response phenomena.

Fig 3: The Parallel Linear Structure

3

7r--------------=""'"------.1'_.

I :

100 200 300 400 500 600 700 800- Time (secollds)

Fig 4: Fitting the PLS model to the response of the furnaceexit temperature to a step change in the fuel flow:- nonlinear simulat ion; -- PLS model.

yu

In order to design adequate control, it is imperative that the numerical values of the linear modelparameters used in the design be bounded within a predefined range. Only then can closed loopstability and performance be guaranteed. The analytical linearization approach can be used to definethe extreme parameter limits given the process' operating range. If analytically defined coefficients arenot available, the range of parameter limits can be estimated by performing a series of random changeson each process input in the vicinity of a given operating point , and fitting the parameters of the PLSto each simulated response. The result will be a set of possible numerical values for each coefficient ofthe model, which will delineate extreme values.

B. Control Design.The modelling step will have produced a PLS approximation for the process dynamics. The bestpossible feedback performance will be obtained if the same structure is conserved and integrated into anIMC controller structure. Conventional, classical feedback can also be designed, but this will requiremodel reduction to generate a single rational transfer function approximation for the PLS. In general,the price paid for this simpler control implementation is reduced performance, although if there issignificant uncertainty in the PLS coefficients, the loss of performance relative to the full-order PLS­based controller may be negligible.

If required, model reduction should be carried out while taking closed loop performance specificationsinto account, thus simultaneously defining the controller's tunable parameters. For example, for thecontrol of furnace effluent temperature, the design of a PI controller required that the three block PLSmodel be approximated by a first order lag. This approximation should be carried out by matching theclosed loop transfer functions obtained with the reduced order model, with that obtained with the PLSmodel. This allows the simultaneous tuning of reset time and controller gain , in a control-relevantmanner, and will maximize the closed loop bandwidth . In general, the IMC parametr ization simplifiesthe implementation of this approach because model-based control synthesis reduces the number oftunable parameters.

4. CASE STUDY: A PILOT SCALE-EVAPORATOR

4.1Modelling.Figure 5 shows a schematic representation of the single stage pilot-scale vertical evaporator used in thisstudy. The subcooled liquid feed enters the riser tubes from below, while steam enters the shell aroundthe tubes and condenses on their outer surfaces. The two phase mixture flows upward into the overheaddrum, which is a large separation vessel, where the vapor produced separates out of the fluid phaseand exits to a condenser. The concentrated solution is withdrawn from the bottom of the separationvessel as product.

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steam chest

-+vapor

~zQ~'if:.-L 'ovorb.ad drum

:: 0- b o 00"': .f:'" /0 -+concentrate

A mathematical model was developedto describe the transient response of Fig 5: A single evaporator stage.the evaporator overhead drum fluidlevel as a function of perturbations in the process inputs: the steam feed pressure, Pst, the feedtemperature, Tin' and volumetric flow rate, Lt , and the concentrate flow, B. The following assumptionswere made:1) The dynamics of the boiler tube fluid can be treated as pseudo steady state. This is justified because

of the large fluid velocities associated with the two-phase mixture rising in the tubes and itsrelatively small capaci ty. Thus, a steady state model relat es th e current value of the process inputsto the volumetric vapor fraction exiting the tubes and entering the overhead drum, x.

2) Inverse response phenomena, if present , will be observed in the overhead drum fluid level. Thischaracteristic will therefore be investigated in the absence of solute in the feed. Thus, a mass balanceon the solute phase will not be required in this study.

3) In the pilot plant , the steam pressure is measur ed immediately downstream of a pressure regulationsystem, while the pressure value of interest is actually that of the steam chest shell. Thus, a changein the measured pressure will only have an effect on the vapor fraction exiting the tubes after adelay associated with mass transport to the steam chest. This delay time will be determinedempirically.

4) The steam chest condensate is assumed to have a constant radial temperature profile. The steamchest is assumed to be perfectly insulated.

5) The two-phase fluid in the overhead drum is assumed to be a homogeneous mixture of vapor andliquid, with volumes Vg and VI respectively.

Boiler Tubes

Most published work on dynamicmodelling of evaporators (e.g. Andreand Ritter (1968) and Newell andFisher (1972)) , has consisted of lumpedparameter models , derived from massand energy balances for steam, for themetal walls and for the heated fluidand for the solute. These models steam feed--..cannot exhibi t inverse response, sincethe process fluid accumulated in theevaporator drum is represented as ahomogeneous mixture, thus disregard-ing its true two-phase nature.

Overhead Drum....... .. ..... . -- ..... --- .1o;.j.--:x=-+!, ~:~: ~ :~,~ :~: ~: ~ :~ : ~:~ :

h

B

Fig 6: A schematic representation of the model structure.

The model therefore consists of two parts as shown schematically in figure 6. First, a change in theprocess disturbance vector, if= [Lt , Tin' psJT will cause an immediate change in the vapor fractionentering the overhead drum. Modelling of heat transfer in the evaporator riser tubes involvescomputation of single phase heat transfer in the inlet section of the tubes, and of heat transfer to thetwo-phase process fluid thereafter. The former computation yields the location along the tubes whereboiling begins (Serizawa, 1979). Chen 's correlation (Chen, 1963) was utilized for computation of heattransfer to the two-phase fluid, with modifications to account for different operating conditions used inour system. The contribution of steam condensate on the external surfaces of the riser tubes was alsoconsidered. Linearization of the effects of perturbations on each component of if on the change in thevapor fraction will define the sensitivity to each of them, as indicated by the gains K1, K2 and K3

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marked in the figure. As explained, the system topology requires that pressure changes be modelledwith a delayed effect on x, Then, given the current value of the vapor fraction entering the overheaddrum, as well as all process inputs, mass balances on the overhead drum vapor and liquid phases willconstitute the state equations. With current volumes of gas and liquid in the overhead drum fluid, theoutput of interest, the evaporator level, can be easily computed.

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The accumulation of liquid in the drum depends on a balance between the unevaporated feed enteringvia the evaporator tubes and the exiting concentrate flow:

dVIdt =Lt(l - x) - B

The accumulation of vapor in the fluid depends on a balance between the gas-phase feed entering fromthe riser tubes, and the vapor escaping from the fluid, whose rate we assume to be proportional to thevapor fraction in the fluid, to the superficial vapor velocity, ug, and to the cross-sectional area of thedrum, At: dV V

g PI g<It = Ltxpg -ugAt VI + V

g(7)

A value for the superficial vapor velocity will be determined by empirical fit with plant data. Theinstantaneous level in the reflux drum depends of the total volume of fluid:

A VI + Vgh = A (8)

t

.7"

........

....

a..

D • 30 lee uc=3.0 01/'

..i-~::...---"'::'::---"'::""::::..J~~~J .,. 1:,at.::I:::

.a. I.'

.1. I.'

.... 0.0

Experimental data were collected following a B,~

change in the steam supply pressure. The liquid h,b (m) lal/miD

level of the evaporator hold-up drum was l._r----------------Y'.-measured using a conductivity cell. The recordedprocess inputs were used as data for the simula­tion. The data were reconciled in order to satisfythe mass balance at steady state, generating acorrected concentrate flow rate. Empirical valuesfor ug, the superficial vapor velocity, and D, the ./delay between the measured steam supply pre- LJ2ssure and its effect on x, were selected in order to 8 .8 '" ..8 a.. a••

fit the predicted level with that experimentally . Time (lee) .

d Th It ' fit' h . fi 7 Fig 7: Response to a pressure disturbance.measure. e resu mg I IS sown in igure ,with ug = 3 m/sec and D = 30 sec. The empirical value for ug is close to that predicted by Nicklin'sequation, which computes the terminal velocity for piston flow. As shown in the figure, the modelprediction, h, is close to the the experimentally observed inverse response, h.

Table I: The Sensitivity of x to d.

Ki Minimum Value Maximum Value Units3 -1

K1 = J8x/8LtJ - 4.67 x 103 - 2.99 X 103 (~c)Tin,Pst

K2 = 18x/ 8Tjn l :::0 4.2 x 10-4 (~rPst' Lt

K3 = 18x/ 8Pst l 6.47 x 10-4 26.4 X 10-4 (:SlTjn,Lt

Having fixed the model parameters in such a way that the experimental vapor fraction data ispredicted for the range of operating conditions used, our dynamic modelling approach requires that thesensitivities of vapor fraction with respect to changes in process inputs be determined. This wasaccomplished by solving the model describing conditions in the evaporator tubes at steady state, asdescribed by Faigon (1991). The numerical results are summarized in Table I. Bearing in mind that thetypical disturbance perturbations in the three inputs will be: IilLt 1= 2 X 10-5 m3/sec, IilTin I=20·c andIilPatl = 100 kN/m2, this means that the sensitivity to the average perturbation in the inlet tempera­ture will be 2% of the sensitivity to the average pressure perturbation. This is the justification forignoring the effect of inlet temperature on the evaporator dynamics.

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Linearization and Laplace transformation of the model given by equations (6) - (8) yields transferfunctions relating the evaporator level to perturbation in the control variable, B, and in the threedisturbances (Pst. Tin and Lt). The response of the evaporator level to perturbations in the concentrateflow rate, B, is given by:

((lL)s+ 1)H(s) ( h) ug ssB(s) = p(s) = - AtVl 5S ((A h2) ) (9)

s r s+1lUg

ss

The transfer function relating perturbations in the total feed rate to the evaporator to perturbations in

the evaporator level is: (() )h(l- b+ab) 1

s+H(s) _ _ (h(l- b)) ug(l- b) 55Lt(s) - Pdl(s) - VI ((A h2) ) (10)

5S s -Vt S + 1lUg

where a - PI and b - x-l-L [ox] S5- Pg - t oLt

Tin' Pst

Noting that a ~ 1, and that Ibl=o(I), the expression (10) can be simplified to:

H(s)Lt(s)

(I( hab )S+I)

(h(l - b)) \: ug(1 - b) ss

VI ((A h2) )5S s Vl~g s + 1ss

(11)

(12)H(s)

P5t(s)

= Pd3(s) =

The sign of the process zero depends of the sign of b. For the operating range of the evaporator, thelimits for x and Lt(8X/ 8Lt) are: 0.1 :S x :S 0.6 and - 0.375 :S Lt(8X/ 8Lt) :S - 0.075. As a consequence,the parameter b is limited by the values: - 0.275 :S b :S 0.525. Note that the steady state gain is alwayspositive. This result is interesting because it indicates that inverse response is possible as a result of aperturbation in the flow rate. Analysis of the parameter values which give a positive zero, we find thatthey occur when x is minimum (i.e. Pst is minimized) and the term Lt l8x/ 8Lt Iis maximized (i.e. Lt ismaximized), which makes sense physically.The transfer function relating perturbations in pressure toperturbations in the evaporator level is:

Here, the steady state gain is always negative, while the process zero is always positive. From analysisof the relative sensitivity of the evaporator hold-up to the two disturbances considered, it is apparentthat the pressure disturbance is the more significant of the two. We shall now consider the design ofcontrol to reject this disturbance.

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4.2 Control Design.From the linear approximations describing the response of the evaporator level to changes in theconcentrate (eq.9) and steam pressure (eq. 12), it is apparent that a step change in Pst is seen as aramp (l/s2) in the process output. Thus, the IMe controller that will guarantee set point tracking forsuch a disturbance will be:

_ -12.\s+1 s(rp2s+1)(2.\s+1)q(s) = p(s) --=-'"''-------:--:----,-;;­

(.\s+ 1)2 kp(rp l s+ 1)(.\s+ 1)2

A h2where kp=(A Jy ), Tpl=(~) and TpZ=(Vtu ). This is equivalent to the classical feedback controller:

tlss gss Igss

q(s) (Tp2s+1)(2.\s+l)c(s) = I-p(s)p(s) = kp.\2s(Tp1S+1)

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..........which can be readily identified as a PID Lr .1Yzcontroller, though we note that in this case, h (m) aal/min

IMC and PID control are completely equiva- r.-r-----'--------------.....-lent. The only tunable parameter, A, must bechosen to satisfy the robust stability criterion(eq.4). The multiplicative uncertainty bound .........--Lr-<,-----~Jo-.........---i ...is computed using extreme values for the -i-H---p~~-----'''-'-~~-.r-"'....,...-'---_vt_ ~~

bl p.......... '_1 4

: :':"model parameters within the possi e opera- ...ting range of the evaporator. As a result, ...closed loop stability of the system can be .1'" 1."guaranteed. The performance of the control • .. 1.. n. .... n. •....- •.•system is illustrated in figure 8. We note that. Time(.ec)..

the level can be kept almost constant, at the FIg 8: Closed loop response to a dIsturbance In Pst·

cost of a rather violent manipulation of B. The large control action is a result of the relatively lowsensitivity of the level to concentrate flow.

5. CONCLUSIONS

This paper has outlined a control design methodology for processes featuring two-phase flow. Theevaporator example has shown the need for accounting for mass accumulation in each phase separately,which is the cause of potential inverse response effects which would be absent in a homogeneous model.The Parallel Linear Structure (PLS) has been introduced as a means of modelling complex dynamicresponse. Such a structure can be used to model both inverse response (as brought by positive zeros),and transportation time delay, which can enter in a complex fashion. In the furnace model, the effect ofall inputs considered (feed rate, inlet temperature and fuel feed rate) can accurately modelled by threefirst order transfer functions in parallel, appropriately delayed to account for shock-wave effects. In theevaporator, two at most are required. Finally, the concept of Internal Model Control has been shown toprovide a suitable framework for robust control design.

ACKNOWLEDGEMENT

We are thankful to Prof. Samuel J. Wajc for his helpful comments. This research was partiallysupported by the Henri Gutwirth Fund for Advancement of Research.

REFERENCES

Andersen, H. W., Laroche, 1. and Morari, M. (1991): "Dynamics of Azeotropic Distillation Columns",Ind. Eng. Chern. Res., Jill, 1846-1855

Andre, H. and Ritter, R. A. (1968): "Dynamic Response of a Double Effect Evaporator, Can. Jnl. ofChern. Eng.,1§., 1259-1264

Chen, J. C. (1963): "A Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow",ASME Paper 63-HT-34

Faigon, M. (1991): "Simulation and Robust Control of a Two Stage Evaporator", M.Sc. Thesis (inHebrew), Technion

Fuchs, A., Lewin, D. R. and Wajc, S. J. (1991): "Modelling, Simulation and Control of a Crude Pre­heating Furnace: I - Approximate Crude Description", submitted to Chern. Eng. Sci.

Luyben, W. 1. (1969): "Feedback and Feedforward Control of Distillation Columns with Inverse Res­ponse", I. Chern. E. Symp. Series.al, 6:39-48

Morari, M. and Zafiriou, E. (1989): Robust Process Control, Prentice-Hall Inc., Englewood Cliffs, NJNewell, R. B. and Fisher, G. D. (1972): "Model Development, Reduction and Experimental Evaluation

of an Evaporator", Ind. Eng. Chern. Proc. Des. Dev., 11(2), 213-221Serizawa, A. (1979): "A Study of Forced Convective Subcooled Flow Boiling", in Two Phase

Momentum, Heat and Mass Transfer, ed. Durst et ai, Hemisphere Publishing Corporation.Tsai, J-G. and Tsao, G. T. (1991): "Dynamic Response of a Two-phase Continuous Flow Reactor",

Chern. Eng. Sci., 12(3), 881-888