A new mathematical programming formulation was presented in this chapter for the scheduling problem of cleaning heat exchanger networks subject to chemical reaction fouling. The scheduling problem takes into account the effects of ageing on fouling and cleaning dynamics. Moreover, it includes the selection between two cleaning methods, one mechanical and one chemical, which differ in their ability to remove the aged deposit.
A two-layer fouling model is used to track the effect of fouling on heat recovery and on cleaning effectiveness. It is assumed that the growth rate of both the gel layer (fresh deposit) and the coke layer (aged deposit) is constant and independent of any other parameters. The aged layer is considered to be more conductive than the fresh deposit and also not susceptible to removal by the chemical cleaning method.
The proposed formulation is reliant on the availability of the parameters of the two-layer fouling model. It is very likely that the resulting scheduling problems
5. Chemical reaction fouling: formulation & solution methods
will be highly sensitive to these parameters and reliable estimation of their values is going to be crucial in the application of the described scheduling formulation.
The scheduling problem is formulated for networks of single pass shell-and-tube exchangers operating in counter-current mode. It can be easily extended for other types of heat exchangers and different flow configurations. It is assumed that deposition of foulant occurs only in the tubes of the units where the cold stream flows.
The need to simulate the operation of heat exchanger networks in order to calculate the process costs due to fouling favours a discrete representation of time.
An orthogonal collocation scheme is included in the problem formulation and is used to obtain numerical solutions for the differential equations, albeit these are of zero order (the future direction of the work is to replace the simple two-layer model with a more detailed one, e.g. the first order model described by Ishiyama et al.[2011a]), and to estimate the integral of the process costs over the examined time horizon.
The mathematical programming formulation of the scheduling task corre-sponds to a non-convex MINLP problem. Due to the non-convex characteristics of the problem it is very difficult to guarantee that a local solution is the globally optimal point. The non-convexity of the problem arises from the sets of equality constraints (5.37) – (5.39) and from the constraints defined by (5.23) – (5.26) (can be replaced by linear constraints if required / can be treated explicitly by Generalised Benders Decomposition).
The objective function of the scheduling problem is formulated for a special class of heat exchanger networks called preheat trains. Nonetheless, the schedul-ing problem may be extended to other types of heat exchanger networks after minor modifications. A preheat train is used to raise the temperature of a cold stream to a certain value before it enters some other process. Alas this target temperature is not achieved due to fouling. The objective function includes the energy losses due to fouling, the lost-production opportunity during the cleaning intervals and the maintenance costs.
The standard Outer Approximation/Equality Relaxation decomposition algo-rithm is deemed as unsuitable to attack large instances of the non-convex schedul-ing problem. In that regard, two alternative solution methods are proposed.
5. Chemical reaction fouling: formulation & solution methods
The first algorithm applies Generalised Benders Decomposition, a well-known exact solution method for convex MIP problems. For non-convex problems such as the one studied here there is a high probability that the global solution will be excluded by the search procedure at some iteration. Nonetheless, bearing in mind the unavoidable difficulties associated with non-convex problems, the goal here is to obtain ‘good’ local solutions with moderate computational cost. For that purpose, the use of Generalised Benders Decomposition is favourable.
The second solution approach is inspired by Model Predictive Control. The advantage of this heuristic solution procedure lies in the fact that the scheduling problem is solved over a short time horizon (instead of the whole time horizon) at each iteration. Thus, it is expected that a cleaning schedule can be obtained with relatively small computational effort even for large instances of the scheduling problem.
Chapter 6
Chemical reaction fouling:
computational studies
In Chapter 5, a new MINLP formulation was presented for the problem of schedul-ing the cleanschedul-ing actions for heat exchanger networks subject to chemical reaction fouling and ageing. The proposed formulation is evaluated in the current chapter through a series of computational studies. At first, the scheduling formulation is implemented for an isolated heat exchanger and the resulting model is solved using the Outer Approximation/Equality Relaxation algorithm. Subsequently, cleaning schedules are obtained for two heat exchanger networks of different size using the Generalized Benders Decomposition algorithm and the Receding Hori-zon heuristic procedure. An assessment is presented at the end of the chapter regarding the produced results and the computational performance of the differ-ent solution procedures.