Model Development, Reduction, and Experimental Evaluation for an Evaporator Robert B. Newelll and D. Grant Fisher2 Department o j Chemical and Petroleum Engineering, University of Alberta, Edmonton 7 , A L T , Canada
A generalized approach to the modeling of multieffect evaporators is presented which separates the development of dynamic equations from the specification of evaporator configuration. The result i s a modular approach which is effective and convenient to use. A tenth-order nonlinear, dynamic model of a doubleeffect pilot plant evaporator was derived using this approach and then simplified and linearized to produce a fifth-order state-space model which gave generally good comparisons with experimental open-loop responses. Lower-order linear models were also developed but gave satisfactory results only in specific applications. The performance of models in the design and experimental implementation of conventional, inferential, feedforward, and multivariable optimal regulatory, and state-driving control systems is also examined.
T h i s work is part of a continuing project which has been under way at the University of Alberta since 1964. The overall objective has been to develop modeling and control system design techniques that are of potential interest to industry and to evaluate them by application to the computer-controlled pilot-plant units. Initially the interest in developing mathematical models was for open-loop simulations and as a n aid for the design of conventional control systems. However, more recently the emphasis has been on state-space models for use in the design of optimal multivariable controllers, state estimators, etc. The effect of model simplifications, such as linearization and the reduction of model order, on the design and performance of control systems has been of particular interest. The first part of this paper contains a brief review of pertinent literature and presents a generalized approach to modeling industrial evaporators. The second part of the paper deals specifically with some of the models that have been developed for a double-effect evaporator and their performance in several different experimental applications. hlathematical models of evaporator systems reported in the literature have used both empirical and theoretical approaches. Johnson (1960) presented a variety of empirical models of differing complexity and fitted parameters with experimental data from his falling-film evaporator. Nisenfeld and Hoyle (1970) considered simple empirical models for feedforward control end used two first-order lags and a time delay to dynamically compensate a six-eff ect evaporation process. Theoretical derivations have also been presented in the literature. Anderson et al. (1961) derived a six-equation model for a single-effect, and simplified it to three differential equations. This was done by essentially neglecting vapor space and heat transfer dynamics. A frequency response comparison between the model and the equipment proved inconclusive. llanczak (1967) presented a n analytical procedure to determine the dynamic properties of single and Present address, MFD Division, Shell Internationale Petroleum, hlartschappy N.V., Den Haag, Netherlands T o whom correspondence should be addressed.
multiple effects. His relations were extensively linearized which resulted in a small range of applicability about the operating point. Zavorka et al. [1967] developed a general model for a single-effect of a commercial sugar evaporator. This model was extended t'o a triple-effect system after simplification and included nonlinear relations for the heat transfer coefficients in ternis of solution concentrations and liquid levels. However, their analysis omitted a general heat balance on the solution and assumed that vaporization was proportional to the heat transferred to the liquid. Andre and Ritter (1968) presented a direct derivation of a five-equation model for a n earlier configuration of the same double-effect evaporator used for experimental work in t'his study. Holzberger (1970) derived a model for a two-tank flash evaporator and obtained transient responses by simulation. Model reduction is frequently required to produce simpler models for control applications, and many different approaches have been suggested. Procedures for reducing system order while maintaining the significant dynamic modes have been presented by Xarshall (1966), Davison (1966), Chen and Shieh (1968), and Wilson et al. (1972). This approach can be intuitively approximated with state-space models by setting the derivatives of the first-order equations with sniall time constants equal to zero. Fitting methods depend on choosing a reduced-model form and fitting it to data generated from the more complex model. This is a similar problem to defining the form of a mathematical model and determining the parameters by fitting experimental data and enters the field of parameter identifi:ation which has been surveyed by Nieman et al. (1971) and Astrom and Eykhoff (1971). Development of a General Evaporator Model
There are only a few basic designs for a single-effect evaporator and these can, in general, be relatively easily modeled using overall material and energy balances and basic thermodynamic relat'ioriships. However, industrial installations are more complicated because they normally involve a number of evaporator units interconnected in series and/or in parallel. Examination of a number of typical installations, such as that shown in Figure 1, suggests t h a t the configuration and parameter values may differ widely from one installation to another, Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 2, 1972
213
CONOENSR ..
5
Figure 1. Schematic diagram of a typical six-effect evaporator showing the envelope used to write the material and energy balance equations for the ith effect (Nisenfeld and Hoyle 1970)
but the form of the dyriamic equations for each component of the evaporator system usually remains the same. Therefore the approach adopted \vas to completely separate the dynamic modeling of pieces of evaporator equipment from the specification of how these pieces Tvere interconnected. The result is a series of dynamic building blocks which can be used linked together, following a cookbook procedure, to give a complete dynansic model. This approach is ideal for evaluating different evaporator configurabions, but should be put in inore efficient form for applications involving a large number of simulations. Dynamic Models of Evaporator Components. For dynamic modeling purposes, it is convenient to break up each effect of an evaporator system, such as shown in Figure 1, into t h e following subsystems: steam chest, heat transfer surface, solution holdup, vapor space, a n d associat'ed heat capacity of vessel, piping, etc. In addition, there are auxiliary units such as the condenser(s), soap tanks, etc., that must be modeled. The following subsections coiisider the developiiient of dynamic models for the first three subsystems listed above. The derivation of equations t o describe the other units is similar and has been presented in detail b y X'ewell (1971). Steam Chest. If the density and temperature gradients are assumed negligible, the following lumped-parameter balances can be written for the steam chest of the ith effect:
d
vi - ( P i ~ i = )
'
'dt
S i H isz. -
sichic - &is - ~i s
perature, and prersure when required, for example, when it is desired to control the pressure in the vapor space. Heat Transfer Surface. Assuming negligible temperature gradients in the vapor space, in the tube u-alls, and in the solution, a lumped-parameter energy balance for the heat transfer surface between the heating medium and the solution may be written as:
The heat tiansfer coefficients may be assumed constant or expressed ab general functions of temperature, concentration, and, or condensnig or circulation rates. Tse of a n effective tube area nould allon for heat transfer through downcoiners and tube aheeti: A siniilar equation applies to the heat transfer surface in the condenser and the same approach could be used to account for the effects of heat capacity in the vessel walls and in the associated piping Solution Holdup The material and energy balances on the solution in each effect of the evaporator assume perfect mixing Khile this nould generally be close to the case, large evaporator units with viscous solutions can develop significant concentration and temperature gradients within the solution. This would be particularly true nhere the circulation rate is low compared to the feed rate and where the vessel contains dead zones. The applicable mass, solute, and energy balances are (4)
(2)
where &ZS
Ti
= =
Ci,Ai(Tic - Tim) = Si,(Ai, T is - T 3 b p
ddt
+ 6")
Tjbp= boiling point rise or superheat in connected unit
ais
=
correction for superheated steam and/or subcooled condensate
The assumption of constant volume, V s i ,implies a constant condensate level in the steam chest. When the steam chest is coniiected to the vapor space of the previous unit by a low resistance line, then Vi, could include the vapor space volume Vj-and hence reduce the total number of -Le., Vi, equat'ions required. The equations for the vapor space in each effect and for the vapor side of the condenser are very similar in form. An equation of state can be used t o relate steam density, tem-
+
214 Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1 ,
No. 2, 1972
d - (Wihhi) dt
=
(Jv'iCZ) = FiCiF
FihiF
- BiCi
- Bihi - O f H i , + Q i
(5)
- Li + + i
(6)
where Q i = U i L A i ( T i , - T I ) and $ i = heat of solution effects. This derivation has not considered boiling dynamics and assumes the vapor and solut,ion to be in phase equilibrium at all times. In practice, the effect of pressure changes can be exceedingly complex. An increase in pressure will increase the boiling point which could stop the boiling and hence decrease the heat transfer coefficient, particularly for natural circulation units, This would cause the pressure and condensing temperature in the steam chest t o rise above the normal steady-state values and hence propagate the disturbance through the heating vapor. However, except in the
event of direct, pressure control on a n effect, the pressure changes are dependent on heat transfer dynamics. Therefore, if the sensible heat in the solution is small compared to the heat load, the heating dynamics may be fast enough to prevent a total stop in boiling. However, this depends on the equipment in quest,ion. The present model will give the correct response to pressure changes provided that the attainment of phase equilibrium is more rapid t'han the pressure changes. Multieffect Model Building. Additioiial information is necessary t o construct a complete process model. First, for each evaporator effect, condenser, or other unit of equipment, t h e following is required: (1) The general model dynamic relations-e.g., equations I-6-for a n evaporator effect which define p s , H,, T,, K, C, h, and possibly p and H ( 2 ) Empirical property relationships, or equations of state, which relate the physical properties of the solution, the solvent vapor, arid of the liquid holdups to the system statee.g., h as a function of T and C (3) Physical parameters of the equipment-e.g., V , V,, Ti-, A , c,, (4) Operating variables, such as heat losses and heat transfer coefficients, which may be functions of the state of the unit.
Secondly, at' least one algebraic configuration relationship must be w i t t e n for each process stream connecting the units. .it this stage the independent variables of the process will become apparent. The complete model and configuration relations for the six-eff ect evaporat,ion process described by Yisenfeld and Hoyle (1970) have been developed by Newell (1971). A schematic flow sheet,of the process appears in Figure 1 and a n example of the configuration statements for the streams entering and leaving the second effect are given below: Stream C: S2
:= 01,
H2,,
=
HI,
Stream E : S3 = 02,M3si= H 2 , Stream D : F 2 h'p Stream B: F l
== B3, C Z F = =
C3
(B3h3- L,)/F2
+ 01Bl ( B V + aB'C')//F' (B2h2+ &'h' - L,)/F'
= B2
C1F
==
hlF
==
where 01 is the fraction of the bottoms flow, B', which is recycled and L B and Lo are the stream heat losses. Transport delays due to lorig pipes and/or low flow rates caii be included in the configuration relations. The profusion of differential and algebraic equations that' result from the modeling procedure described above can be handled directly by siuch sophisticated digital simulation programs as the Continuous Systems llodeling Program (CSLIP) on the IBlI 360 series computers. KO simplifying substitutions would be necessary although they would improve computational efficiency. T o arrive at a single ,set of differential equations i n the form of a standard state-space model, the algebraic relations must be substituted into the differential relations. These equations can then he linearized analytically. If the relationships are to be linearized iiumerically by differences about the operating conditions then these substitutions are not necessary. This cookbook approach demonstrates the power and flex-
Figure 2. Schematic diagram of the double-effect evaporator pilot plant a t the University of Alberta which was used for the experimental model evaluation and control studies reported in this paper Evaporator Variables and Steady States Feed
B = bottoms flow rate, Ib/min c = 50117 concn, % wt/wt F = feed flow rate, Ib/min h = liquid enthalpy, Btu/lb S = steam flow rate, Ib/min W = soin holdup, Ib 0 = overhead vapor flow, Ib/min T = temp, OF P = pressure, psia
1 s t Effect
3.3 3.2 5.0 162
4.85
2nd Effect
1.7 9.64
194
1.9 30
190
1.7 225
