A hybrid methodology for optimization of multi-stage flash-mixer desalination systems
S. F. Mussati1, P. A. Aguirre1 and N. J. Scenna1
1 INGAR - Instituto de Desarrollo
y Diseño -CONICET
{mussati; paguir; nscenna}@ceride.gov.ar
Abstract- In this paper, a new strategy involving an evolutionary algorithmic procedure for the optimization of Multiple Stage Flash (MSF-M) Systems is presented. A "detailed model" of an MSF-M System is developed according to rigorous material, momentum and energy balances for each stage. The model of the MSF-M System is represented as a complex NLP, which incorporates a high number of nonlinear constraints that difficult the global optimum determination. Here we present a hybrid methodology that uses optimal solutions obtained from a "thermodynamic method" to find the "economic global optimal solution". A pre-processing stage (solving successive NLPs) is used to initialize the final NLP problem. A Case Study and a discussion of the results are presented.
Keywords- MSF-Mixer Desalination System. Optimal Synthesis and Design. Hybrid Methodology.
I. INTRODUCTION
In previous papers (Scenna, 1987a,b; Scenna et al., 1993), a simplified
model was presented using a Thermodynamic Based Methodology, which allows
finding optimal solutions involving both structural and operational conditions
of MSF-M systems. In practical designs, models with a more detailed description
become necessary (for example pressure drops, inter-stage flow-rates and liquid
levels are critical due to operative conditions), but generally they are very
difficult to solve using the above-mentioned approach.
From the Mathematical Programming point of view, we can use NLP models (Mussati et
al. 2001) for system description. In this paper, a strategy to determine
the optimal design for the MSF-M System is presented. It involves solutions
obtained in a first step involving a thermodynamic model. Thermodynamic solutions
are used to initialize an MSF-M rigorous model when an economic objective function
is considered (which includes the capital and operation costs).
This idea is based on previous work (Scenna and Aguirre, 1993) to solve the
synthesis of dual-purpose desalination plants. Indeed, several researchers
(Gundersen and Grossmann, 1990; Bek-Pedersen et al., 2000) have used
thermodynamic functions in different ways to make the MINLP or NLP solution
easier, proposing a "physical insight" to solve complex synthesis
problem.
Here, we introduce a formal context for using of thermodynamic models for solving
a complex optimization problem based on previous experiences and results. Also,
a discussion about the global optimality of the achieved solution is presented.
In addition, a general procedure to handle the link between "thermodynamic" and "economic" based
solutions will be approaches.
II. RIGOROUS NLP FORMULATION FOR THE MSF-M SYSTEM OPTIMIZATION
Figure 1.a shows the flow-sheet of an MSF-M System. A typical flash stage is shown in Fig. 1b. The evaporator is divided into stages; each stage has a seawater condenser, a brine flash chamber, a demister, distillate collecting and a transfer system.
Figure 1a. Diagram of MSF-M Desalination Process.
Figure. 1b. Schematic Representation of the jth stage.
The seawater (feed) enters at temperature T0 and it is heated at the maximum
temperature (Tmax) as it flows in series through the condenser tubes of each
stage and brine heater. It then flows into the first stage inlet box and is
evenly distributed across the width of the evaporator. It enters the first
stage through an orifice that controls its flow rate and flashing contour
characteristics. The flashed vapor flows through a demister, which removes
any entrained brine, then over a condenser that condenses it. The condensate,
referred to as distillate, is collected in a distillate collection. The seawater,
now referred to as brine, because its salinity has increased, then flows through
the evaporator stages in turn, releasing flashed vapor in each stage in the
same manner. The brine is rejected from the last stage evaporator by a brine
blow down pump to the sea. Part of this stream is recycled and mixed with
the feed to enter on the pre-heater tubes as was described previously.
The distillate formed in each stage passes through the transfer system into
the succeeding lower temperature stage, where a proportion is flashed to vapor
similarly to the seawater. This vapor flows over the condenser of the stage,
together with the vapor flashed from the sea water and is condensed and transferred
to the next stage. The distillate accumulates as it flows through succeeding
stages and is discharged from the last stage to a product water tank by the
distillate pump.
The model (see Appendix 1) considers rigorous material, momentum and energy
balances for each stage. The stage area is calculated taking into account chamber
length and width, gate height and total chamber height. The flow between two
adjacent chambers is given by the pressure gradient, which is related with
the pressure in the vapor space, the liquid level in the two adjacent chambers
and the pressure drop along the orifice. Also, the Boiling Point Elevation
(BPE) that depends on the salinity and flashing temperature at each stage,
and the Non-Equilibrium Allowance (NEA) that represents a measure of the thermal
flashing process efficiency are taken into account.
Finally, the variation of the overall heat transfer coefficient is considered
according to brine velocity in the condenser tubes, tube diameters, fouling
factor and condensation temperature.
The modeling system General Algebraic Modeling System GAMS (Brooke et al.,
1988) is used to implement the model and the solution method. The generalized
reduced gradient algorithm CONOPT 2.041 is employed.
III. THE PROPOSED METHOD FOR SOLVING THE NLP PROBLEM
In order to determine the optimal design for the MSF-M Systems (minimum Total Annual Cost TAC), we propose the use of the optimal solutions obtained from the formulation of an associated "thermodynamic model" as a guide to solve the original NLP problem ("economic" problem). Before describing the methodology, the two main NLP models will be presented.
A. Cost based Model (P1)
The cost associated to an MSF-M process consists of operating and capital
costs. Operating costs include steam cost, pretreatment of the saline feed
stream and pumping. Capital cost includes condenser and heat exchanger tubes
and the flashing chambers with the associated piping cost. The most representative
costs are steam cost and the capital costs associated to the heat transfer
area and the flashing chambers.
In this way, the problem is defined as follows:
subject to:
model constraints (1-30) presented in Appendix I.
Factor F can be calculated (the minimum value) as a ratio between the tube
and the wall-chamber thickness. It depends on the resistance corrosion, tendencies
to form scale and mechanical strength of the material used for the construction
and the chamber geometry. Also F must consider the necessary instrumentation
and others factors charged to chamber cost (Genthner et al., 2000).
Assuming the above-mentioned hypothesis the minimum value for factor F is
25.
B. Thermodynamic based Model (P2)
Here, we consider an NLP problem, similar to the above-defined one, with a "thermodynamic" objective function. In this way the optimization problem is:
Prob. 2.1 (P.2.1)
subject to:
and the constraints of the model (1-30) presented on the Appendix I. A total
is a model parameter.
The above-defined problem is equivalent to the following:
Prob. 2.2 (P.2.2)
subject to:
and model constraints (1-30) presented in Appendix I. Q0 is a model parameter.
The basic idea of the methodology is to use the solution of the thermodynamic based NLP models P.2.1 or P.2.2 as a starting point for solving P1. Since both problems differ from each other only in the objective function, they are generally equally complex to solve. The advantage of using a thermodynamic model is that it always allows us to find physical insight of the process Mussati et al. (2001, 2002). As it will be described in the next section, it also provides relationships associated to the original problem (P1) and the "thermodynamic" based one (P2).
IV. RESOLUTION METHODOLOGY
In previous works Scenna (1987b) and Mussati et al. (2001), some properties
of the solution of this problem obtained from a thermodynamic point of view
and some physicals insights were analyzed. In Mussati et al. (2001)
using mathematical programming, a detailed model considering mass and energy
balances and the physicochemical properties was presented. The Hydraulic calculations
and the chamber geometry, which in turn permit an exhaustive evaluation of
the chamber cost, were not considered in that model. From this point of view,
the previous model (Mussati et al., 2001; 2002) is a special case of
our model presented in Section 2) of this paper (considering F = 0). For example,
if we not consider the chamber cost (F=0) the solution of P.2.1 and P.2.2
stated in section 2 corresponds to a number of stages tending to ¥ .
According to the methodology considered in this paper, and due to consider
an implicitly chamber cost considering F, at difference with that paper, the
optimal stage number can be achieved.
In order to solve the problem P1, we can summarize our procedure in the following
steps:
1) Obtain QDes vs. Atotal curves solving the optimization problem
P.2.1 or P.2.2. Figures 2a and 2b show the optimal solution achieved solving
the P.2.2 problem. Given the Q0 value (QDes) for each number of
stage NS the minimum total heat transfer area value has been determined. For
example, Fig. 2a shows Atotal vs NS for QDes= 140 Gcal/hr.
Figure 2a. Atotal vs. NS for QDes=140 Gcal/h
Figure
2b. Atotal vs. NS for QDes=78 Gcal/h
2) Determine from 1) the values NSoptim and Atotalmin corresponding to a minimum for each QDes. Obtain the curve QDes and NSoptim in terms of total area Atotalmin (Fig. 2c).
Figure 2c. Minimum total area corresponding to Noptim vs QDes.
3) Given the data costs (CA, CQDes, CRF) solve the NLP problem P1 using the relation dAtotal / dQDes = - RCF CQDes/CA and Figs. 3 and 2c to determine the initial point. So, we determine QDes*, the tubing area and the chamber area associated to each stage and NS* to be used as initial values to solve the problem P1.
Figure 3. dAtotal / dQDes vs. Atotal.
4) Solve the objective function value for NS*-1, NS* and NS+1* (according
to 3), and then determine the optimal number of stages according to the objective
function values.
In order to obtain the QDes vs. Atotalmin curve for the
P2 (Fig. 2c), first we must solve successive NLP problems to determine the
values of NSoptim and Atotalmin for each QDes. As it
can be seen in Figs. 2a and 2b, for a fixed value of external heat consumption
QDes , there exists a solution which characterizes the number of
stage (NSoptim) for which the total area (Atotal) is the minimum (Atotalmin).
It is then possible to draw (Fig. 2c) the relationship QDes, Nmin
in terms of the minimum total area (Atotalmin).
Each point of this curve represents an optimal solution of P2 and allows us
to determine a cuasi-optimal initial point to solve P1. Then, given the complete
space of "thermodynamic" optimal solutions, we must relate them with
the optimal solutions of the "equivalent" NLP problem P1. According
to the Kuhn-Tucker conditions for both problems it follows that the optimal
solution of the cost based problem represents one of the optimal solutions
of the "thermodynamic problem". The thermodynamic solution that corresponds
to the cost based optimal solution can be found when the values CQDes,
CArea and CRF are given. Selecting the dual multiplier u*=CA CRF/CQDes,
it can be shown that the Kuhn-Tucker conditions for both problems are identical
(Mussati et al., 2002). Furthermore, from the duality theory, the following
condition is established:
Since for a given problem we know u*, it is possible to relate the optimal values for QDes and Atotal according to the optimal thermodynamic curve (see Fig. 2c and Fig. 3). In turn, these values allow us to specify "optimal critical variables" at each stage by applying simple relationships. These variables are a sort of optimal initialization for the problem P1.
V. CASE STUDY
Given the following data and parameters: CA= 50$/m2, CQDes=0.076/106$/Kcal
,RCF=0.182$/year, we will apply the above-describe procedure in order to determine
the optimal MSF-M conditions representing the minimum total annual cost. Solving
the first problem according to step 1) and step 2) of the above-proposed method,
the optimal solution family indicated in Fig. 2c was obtained. For this case
the following parameters have been considered: Tmax = 387 K, Tfeed = 295 K,
F = 25, production = 1,000 tn/hr, xi=45,000 ppm. In order to limit scale formation,
the brine concentration, which is the ratio of the recirculating brine concentration
to the feed concentration is kept at 1.5.
Using eq. (A) and Figs. 3 and 2c (that are universal in the sense that their
have general validity), the critical variables to initialize P1 were established.
It is important to stress that P1 was solved in few iterations using this initialization.
Also the "optimal economic solution" was checked satisfying eq. (A)
and the appropriate point in Fig. 3. Finally, according to step 4) the optimal
number of stage NS value is determined. A good agreement is found between the
data of a real plant in operation and those obtained from the model.
Figure 4. Dif. between initial values and opt. solution
Some of the results of the optimal solution are shown in Figs. 5, 6, 7 and 8.
Figure 5. Stage Length and Width Stage distribution through the flashing
chambers
Figure 6. Tube Number and U distribution through the flashing chambers.
Figure 7. Temp. distribution for brine flowing in the pre-heaters and
through the flashing chambers.
Figure 8. Vbrin j, Dtmlj
and Velvap j distribution through the flashing chambers
It is important to mention that our procedure determines a good (quasi-optimal) initial feasible point (IP) to be used for the resolution of P1. Figure 4 shows that the difference between the initial point and the optimal solution is only 0.5%. The initial point determined by this procedure is always in the neighborhood of the real optimal solution. This is very important in order to facilitate convergence, generally obtaining the optimal solution in few iterations.
VI. CONCLUSIONS AND FUTURE WORK
In this paper an algorithm for the optimal design of an MSF-M System is presented.
A rigorous NLP model has been developed based on a thermodynamic preliminary
approach; it is shown that it is possible to establish a link between our
called "thermodynamic solution" and the solution considering "real
costs". From this relationship, we can estimate the optimal solution
when the total annualized costs are given. As was depicted in Fig. 4 the difference
between the initial point and the estimated optimal solution is only 0.5%.
The optimal solution family reported on Fig. 2c and Fig. 3 are universal because
this is not a function of geographical and contingent factors, having a general
validity. Then, these solutions constitute a basis for the designer´s selection
when the economic objective function is considered.
Although it is necessary to solve a set of parametric thermodynamic problems
(solving P2 for different area values) to obtain Fig. 2c, this task can be
easily done using a continuation algorithm (Mussati et al., 2002).
Moreover, we can state that by using universal thermodynamic properties for
heat transfer operations (Scenna and Aguirre, 1993) for this case, we can find
a set of very useful thermodynamic heuristics to make P2 problem solution easier.
In this way, the construction of the optimal curve will be also easier. In
fact, we have achieved an appropriate approximation to the optimal curve employing
these heuristics by using only a hand calculator (only solving algebraic equations),
thus achieving a tremendous simplification for implementing the proposed methodology.
The explanation of this procedure will be the object of a future work.
Figure 9. Superstructure for Multi-Stage Flash Desalination System.
In order to develop new configurations for the MSF System, we will propose in a future work a superstructure in which the flow-pattern of the distillate and the recycle are variables of the problem (see Fig. 9).
Acknowledgements
The authors gratefully acknowledge the financial support of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas de Argentina), UNL (Universidad Nacional del Litoral), ICWES (International Center for Water and Energy Systems) and IFFWSAT (Institute Foundation for Water Science and Technology, ABU DHABI).
Notation
Parameters:
Variables:
Appendix 1
With the notation above mentioned, the model formulation can be presented.
Mass conservation equations for primary stage:
j = 1,….NS (1)
j = 1,….NS (2)
Energy balance for primary stage:
Mass and energy balances for secondary stage:
j = 1,….NS (4)
j = 1,….NS (5)
(6)
Mass balance in preheater:
j = 1,….NS (7)
j = 1,….NS (8)
Energy balance in preheater:
(9)
Relations associated to brine recirculation:
For j = NS+1, we have:
(10)
(11)
(12)
(13)
(14)
(15)
Design equations
Heat exchange area for each stage:
(16)
Chamber length:
![]() |
j = 1,…..NS (17) |
Gate height (from momentum balance):
![]() |
(18) |
Number of tubes for each pre-heater:
![]() |
j = 1,…..NS (19) |
Total number of tubes (from momentum balance):
![]() |
(20) |
For an equilateral triangular pitch arrangement, the equivalent number of tubes in a vertical row (N) can be predicted from the equation (El-Dessouky et al. (1995)):
j = 1,…..NS
(21)
Also, the number the tubes in the vertical direction (N) is related to the shell diameter and pitch by:
![]() |
j = 1,…..NS (22) |
Approximate stage area:
(23)
where:
j = 1,…..NS
(24)
Temperature constraints:
j = 1,…..NS
(25)
j = 1,….NS (26)
j = 1,….NS (27)
Property correlations:
For the Boiling Point Elevation [K]:
![]() |
(28) |
where:
For the overall heat transfer coefficient [BTU/(hr ft2 °F)] (Griffin and Keller, 1965):
![]() |
; where: | ![]() |
(29) |
![]() |
and | ![]() |
where a0 = 0.1024768 x 10-2, a1 =-0.7473939
x 10-5, a2 = 0.999077 x 10-7, a3 =-0.430046
x 10-9, and a4 = 0.6206744 x 10-12. z is
the sum of vapor side resistances, y is the brine-side film resistance. The
Fouling Factors FF [(hr ft2 °F/BTU] is considered as: 8.54439 x
10-4 .
For the Non-Equilibrium Allowance [K], there are several empirical correlations
developed, we adopted for our model the correlations developed for Oak Ridge
National Laboratory (ORNL):
(30)
![]() |
(31) |
where D TS is the flashing range, W is the mass flow rate of recirculated brine per unit of chamber width, H is the brine level inside the flashing chamber.
REFERENCES
Bek-Pedersen Eric, Hostrup Martín and Gani Rafiqul, "Using
driving force based separation efficiency curves within an integrated system
for process synthesis/design", European Symposium on Computer Aided
Process Engineering-10, 955-960 (2000).
Brooke A., Kendrick D. and Meeraus A., "GAMS: A User's Guide". Scientific
Press (1988).
El-Dessouky, H.; Shaban, H.; Al-Ramadan, H., "Steady-state analysis of
multi-stage flash desalination process". Desalination J. 103,
271-287 (1995).
Genthner K., Al-Gobaisi D., Wangnick K., Bodendieck F., "The Next Size
Generation of MSF Evaporators: 100,000 cu-m/day". International Desalination
Association IDA Volume I, 271-324 (1997).
Griffin W. L. and Keller R. M., Oak Ridge National Laboratory, Technical Report
ORNL/TM-1299, Tennessee, November (1965).
Gundersen T. and Grossmann I., "Improved Optimization Strategies for Automated
Heat Exchanger Network Synthesis Through Physical Insight", Comp. Chem.
Eng. 14, 925-944 (1990).
Mussati S., Aguirre P. and Scenna N., "Optimal Synthesis of Multi-Stage
Flash Desalination System", Latin American Applied Research, 31,
Nº 4, 293-301 (2001).
Mussati S., Aguirre P. and Scenna N., "Synthesis and Optimization of Dual
Purpose Thermal Desalination Plants, Dual Purpose Power-Desalination Plants",
Book Chapter Nº 1; EOLSS Publisher Co Ltd, Oxford - United Kingdom, (2002).
Scenna N.J., "Synthesis of thermal desalination processes. Part I. Multistage
flash distillation systems", Desalination J. 64, 111-122
(1987a).
Scenna N.J. "Synthesis of thermal desalination processes", PhD
Thesis, Universidad Nacional del Litoral, Argentina (1987b).
Scenna N. and Aguirre P., "Thermodynamic Synthesis of Dual Purpose Desalination
Systems. Back-pressure vs. Condensation turbines", TransIChem,
Part A, 71, 77-84 (1993).
Received: September 16, 2001.
Accepted for publication: August 13, 2002.
Recommended by Guest Editors: J. Cerdá, S. Díaz and A. Bandoni.