Scheduling the cleaning actions for a fouled heat exchanger subject to ageing: MINLP formulation
Highlights
► The heat exchanger cleaning scheduling problem is revisited. ► Two cleaning methods are available with different cost and effectiveness. ► Ageing determines cleaning effectiveness. ► A new MINLP model is presented including these aspects. ► Results for a single heat exchanger show patterns called super-cycles.
Introduction
Fouling in heat exchangers (HEXs) is a long-standing problem in numerous industries (Steinhagen, Müller-Steinhagen, & Maani, 1993). The deposition of unwanted materials on the heat transfer surface dominates the performance of the devices causing acute financial loss. The deposits have different physical properties, their nature depending on the formation mechanism. The common characteristic is their low thermal conductivity, which causes a decrease in the heat transfer coefficient in a HEX as more material is deposited. As a result, the heat transfer rate of the unit decreases. One effective mitigation strategy is the regular cleaning of fouled heat exchangers. As outlined by Wilson (2005), fouling and cleaning are symbiotic processes. However, along with choosing regular cleaning as a mitigation strategy for fouling, the challenge arises of deciding when to clean a unit in order to minimize energy losses. Furthermore, when dealing with a heat exchanger network (HEN) a second question needs to be answered, namely which unit is to be cleaned?
Following the heuristic optimization strategies proposed by Ma and Epstein (1981), Casado (1990) and Sheikh, Zubair, Haq, and Budair (1996), in recent years, the scheduling problem of a heat exchanger subject to fouling has drawn the attention of the Process Systems community. Smaïli et al. (1999) studied the problem for a HEN of 11 units and proposed a mixed-integer nonlinear programming formulation (MINLP) for the maximization of the target temperature over a time horizon of 120 days. The proposed formulation was extended to larger networks in Smaïli, Vassiliadis, and Wilson (2001) where the goal was to minimize the operating cost over a 3-year horizon. A different MINLP scheduling model was presented by Markowski and Urbaniec (2005). There, the case where the length of the time intervals between cleaning actions was not equal (e.g. these were decision variables for the problem) was examined. Such a formulation is highly nonconvex as the authors concluded. All the aforementioned MINLP models are nonconvex and, therefore, the global optimality of the solution is not guaranteed.
The same shortcoming is also true for a number of heuristic approaches appearing in the literature, e.g. Smaïli, Vassiliadis, and Wilson (2002), Rodriguez and Smith (2007) and Ishiyama, Paterson, and Wilson (2009). Moreover, the latest trend in the development of heuristic approaches is to use genetic algorithms/simulated annealing (GA/SA) to obtain, simultaneously, an optimal design and a maintenance policy for a single unit (Caputo, Pelagagge, & Salini, 2011) or an optimal synthesis and a cleaning schedule for a network (Xiao, Du, Liu, Luan, & Yao, 2010).
Trying to avoid the drawbacks associated with an MINLP model, Georgiadis and co-workers proposed a mixed-integer linear programming (MILP) formulation for a milk industry case study Georgiadis et al., 1999, Georgiadis and Papageorgiou, 2000. There, linearization of the model was achieved by replacing the log-mean temperature difference (LMTD) with the arithmetic mean average in the heat transfer performance equation (1). Such an assumption is not valid for large networks which feature feedback streams (Smaïli et al., 2001). In the same fashion, Lavaja and Bagajewicz (2004) successfully formulated the scheduling problem into an MILP without introducing any linear approximations to the nonlinear equations related to the heat transfer process or the fouling model. They achieved this by creating history profiles for the overall heat transfer coefficient, U. As a result, the nonlinear equations contain only bilinear products of integers with continuous variables which can be readily linearized by adding new continuous variables and supplementary constraints. The importance of this accomplishment lies in the fact that MILP models can be solved to yield globally optimal solutions.
These different approaches demonstrated the potential benefits of using optimization tools rather than heuristic rules to develop cleaning schedules. It is obvious, that by performing cleaning actions according to such schedules, even if these are sub-optimal solutions, the energy losses and the operating cost are significantly lower (Lavaja & Bagajewicz, 2004). The approaches must, however, be able to incorporate more detailed fouling models, if additional gains are to realized and describe real behaviours.
In this work, the impacts of ageing on fouling and cleaning dynamics are considered. Ageing is the transformation of the initial soft deposit into a more cohesive form over time, due to extended exposure to conditions at the heat exchanger wall. Epstein (1983) identified ageing as one of the primary mechanistic steps in the process of fouling. A detailed description of this phenomenon and its impact on heat transfer for the case of chemical reaction fouling was considered by Ishiyama, Paterson, and Wilson (2011a).
The extent of ageing determines the state of the deposit and therefore the ease with which the layer can be removed. In practice, an operator must make a choice between different cleaning methods (Müller-Steinhagen, 2000), and the primary factor considered by an operator will be whether the method to be used can remove the deposit. However, a cleaning method may be chosen even if it is not 100% effective if it can be performed quickly. This element of choice is the subject of this work, and revisits the scheduling problem for a single HEX when two cleaning methods are available. The first technique, typically a cleaning-in place (CIP) one, can only remove fresh deposit while the second, ex situ cleaning (ESC) removes the harder aged layer as well. In practice, there is likely to be a continuous variation in cleanability (linked to rheology and/or chemical activity) across the deposit layer. For the purpose of this work, the demarcation between aged and fresh deposit would then be set by the nature of the CIP process. Performing a number of cleanings using the CIP method can extend the operating time between ex situ cleanings considerably. This mixed cleaning scheme, called a super-cycle, was proposed by Ishiyama, Paterson, and Wilson (2011b) and Pogiatzis, Ishiyama, Paterson, Vassiliadis, and Wilson (2012).
Following the above, we propose an MINLP formulation for scheduling the cleaning for a single HEX. The decision variables are: (i) the time interval at which the cleaning actions will take place; and (ii) the cleaning mode at each chosen time interval.
After reviewing the scheduling approaches reported in the literature in the previous section, we concluded that the most successful one, in our opinion, is the one presented by Lavaja and Bagajewicz (2004). Their strategy features two very important properties:
- (i)
the optimization model proposed is an MILP
- (ii)
no linear approximations were made in the nonlinear equations describing the process.
Consequently, the solution of this model will render the global solution for the scheduling problem. Nonetheless, the proposed methodology has some important drawbacks. First of all, in order to preserve the linear characteristics, a number of extra variables and constraints must be added to the model. In particular, the number of additional constraints grows rapidly as the number of time intervals increases, making the solution of large problems computationally unaffordable. To overcome this difficulty, the authors suggested a decomposition method based on the assumption that the cleaning schedule of an individual unit is not affected by the schedules of the rest of the units. Using this decomposition technique the computational effort for obtaining a solution is reduced significantly. However, such an assumption would not be valid for large networks with strong couplings between heat exchangers arising from hot streams passing through units in series.
In our approach (this will be presented later) the deposit on the heat transfer surface is modeled as two discrete layers due to ageing. Correspondingly, a choice has to be made between two cleaning modes depending on the energy losses associated with the two layers. Thus, to our understanding, the formulation proposed by Lavaja and Bagajewicz (2004) is very difficult to be adapted to this problem. Furthermore, even if one succeeds in adapting such a formulation to this problem, the resulting models will require a prohibitively large computational effort to be solved.
Taking all the above into account and bearing in mind all the difficulties associated with nonconvex models, we chose to proceed with an MINLP formulation.
Section snippets
Heat transfer analysis
The scheduling problem is formulated for a single pass shell-and-tube HEX operating in counter-current mode. It can be easily reformed to accommodate different types of heat exchangers.
It is assumed that the cold stream flows on the tube side and the hot stream on the shell side. Also, there is no phase change happening during the process. The heat performance is modeled using the log-mean temperature difference (LTMD) method. As a result, the heat duty is given by
Results
The scheduling model is coded in GAMS (Brooke, Kendrick, & Meeraus, 1992) and solved using DICOPT on an ASUS Chassis AMD Athlon Processor 2.21 GHz PC. In DICOPT, the MINLP problems are solved by implementing an outer approximation/equality relaxation (OA/ER) algorithm (Kocis & Grossmann, 1989).
The OA/ER algorithm was developed by Duran and Grossmamn (1986) and Kocis and Grossmann (1987) and solves convex MINLPs. The OA method is based on the decomposition of the initial problem to: (i) a primal
Conclusions
The optimization problem of scheduling the cleaning actions for an individual heat exchanger subject to fouling was revisited. The new MINLP formulation takes into account the effects of ageing on fouling and cleaning dynamics using a simple hypothetical model. Furthermore, an extra decision variable is added to the scheduling model, specifically, the decision between two cleaning methods which differ in their effectiveness to remove aged material. When ageing is slow, the mixed-cleaning scheme
Acknowledgment
Funding for Thomas Pogiatzis from the Onassis Foundation is gratefully acknowledged.
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