Optimal Cleaning Policies in Heat Exchanger ... - ACS Publications

Ind. Eng. Chem. Res. 2000, 39, 441-454

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Optimal Cleaning Policies in Heat Exchanger Networks under Rapid Fouling Michael C. Georgiadis,* Lazaros G. Papageorgiou,† and Sandro Macchietto Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, U.K.

This paper addresses the problem of short-term cleaning scheduling in a special class of heatexchanger networks (HENs). A salient characteristic of this problem is that the performance of each heat exchanger decreases with time and can then be restored to its initial state by performing cleaning operations. Because of its practical importance, a specific problem has been considered here involving decaying equipment performance due to milk fouling. A mixed-integer nonlinear-programming (MINLP) model is first presented incorporating general fouling profiles. This model is then linearized to a tight mixed-integer linear-programming (MILP) model which can be solved to global optimality. A detailed objective function is used to account for cleaning cost and energy requirements. The formulations can model serial and parallel HENs as well as network arrangements arising from the combination of these basic cases. The optimization algorithm determines simultaneously: (i) the number of cleaning operation tasks required along with their corresponding timings and (ii) the optimal utility utilization profile over time. A number of complex heat-exchanger networks examples are presented to illustrate the applicability of the proposed models together with comparative performance results between the MINLP and MILP models. 1. Introduction Fouling is the major unresolved problem of significant interest in the field of heat-exchanger design and operation. Fouling affects nearly every plant relying on heat exchangers for its operation and introduces costs which are ultimately related to the conservation of energy, operation, and capital investment. The common practice to mitigate fouling is to implement cleaningin-place (CIP) operations. This is especially applicable to processes affected by rapid fouling, such as that caused by milk. Several methods for the optimization of cleaning schedules for a single heat exchanger have been proposed in the literature. Epstein1 presented an analytical method for the calculation of the optimal evaporator cycle with scale formation. In a similar fashion, Casado2 used a detailed cost model to calculate the optimal cleaning cycle of a heat exchanger under fouling by exploring the major operating trade-offs. On the basis of this work, Sheikh et al.3 presented a reliability-based cleaning strategy by incorporating uncertainty in a linear fouling model. In the same context, Zubari et al.4,5 considered different stochastic fouling models. Ahmad6 proposed a condition monitoring system to predict cleaning schedules for a single heat exchanger under milk fouling. The main drawback of these approaches is that they are restricted to a single piece of equipment. However, in process plants complex heat-exchanger * To whom correspondence should be addressed. Current address: Chemical Process Engineering Research Institute, P.O. Box 361, Thermi 57001, Thessaloniki, Greece. Tel.: +3031-498143. Fax: +30-31-498180. E-mail: georgiad@alexandros. cperi.forth.gr. † Current address: Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, U.K.

networks (HENs) exist with many interacting processing units. In that case, analytical methods for the determination of cleaning policies cannot be applied. The state of each piece of equipment (in operating or cleaning mode) affects the overall network performance. Motivated by the above practical industrial problem, the simultaneous considerations of scheduling and process operations aspects seems to be promising. Scheduling and planning of batch and continuous plants has received considerable attention over the last decade (for example, refs 7 and 8). Several mathematical programming scheduling formulations have been presented by using either discrete (see for comprehensive reviews refs 9-11) or effective continuous time representations.12,13 The exploitation of heat integration in the operation of multipurpose plants taking into account its interactions with production scheduling was considered by Corominas et al.14 and Papageorgiou et al.15 Both works assumed that the plant equipment performance remains unchanged over the time horizon. Maintenance considerations in the scheduling of continuous and batch plants have recently received increasing attention. Dedopoulos and Shah16,17 presented a mathematical model for coordinated scheduling/production and preventive maintenance activities within a multipurpose plant. Maintenance decisions were also considered by Zhu et al.18 in the optimal design and scheduling of reverse osmosis networks under membrane fouling and scaling. Jain and Grossmann19 studied the scheduling of multiple feeds on parallel units whose performance decreases with time and therefore shutdown for maintenance after regular intervals is required. An mixed-integer nonlinear programming (MINLP) model was developed for determining the optimal cyclic schedules. Muller-Steinhagen20 proposed an integrated approach for developing alterna-

10.1021/ie990166c CCC: $19.00 © 2000 American Chemical Society Published on Web 01/07/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 2, 2000

tive fouling mitigation strategies based on both experimental and modeling work. Recently, Smaili et al.21 presented a mixed-integer nonlinear programming formulation for addressing the problem of cleaning scheduling in HENs subject to fouling with application to the sugar industry. In this paper, we consider the short-term cleaning scheduling problem of complex heat-exchanger networks under fouling. Dairy plants constitute the motivation of this work where fouling is a severe problem with dramatic consequences such as significant production losses and increased energy requirements. In addition, product quality requirements are likely to not be met when utility provision bounds are imposed. Thus, CIP operations are necessary to be carried out every 4-8 h so as to restore each heat-exchanger area back to its original state. It is evident that the appropriate cleaning synchronization (scheduling) of heat exchangers can have significant effects on the resulting process operating costs as there is a clear trade-off between a fixed cost term depending on the frequency of CIP operations and a variable cost term which is a function of the preceding heating time. The rest of this paper is organized as follows. The next section considers a motivating example emphasizing the need for using scheduling techniques to mitigate fouling effects, followed by a section where the problem is formally stated. In section 4, models for this problem are presented for the case of serial heat exchangers characterized by general fouling profiles. In sections 5 and 6, the formulations are extended to model more complex arrangements. A number of case studies are presented in section 7 to illustrate the applicability of the proposed models. Finally, some concluding remarks are drawn in section 8. 2. Motivating Example The milk sterilization process is the source of motivation for this work. This milk heat treatment is of critical importance as it is required to safeguard public health by destroying all pathogenic bacteria and ensuring good quality of final dairy products.22 When milk is heated, a deposit of milk solids is gradually formed on the heatexchanger surface as result of its instability.23 The overall performance of each heat exchanger decreases dramatically with time as the heat-transfer resistance increases because of the formation of the milk deposit. If the heating medium temperature is kept constant (e.g., high-pressure steam), then the milk outlet temperature decreases and eventually violates sterilization requirements.24 Thus, CIP operations are necessary to be applied on a short-term basis to restore equipment performance. Alternatively, to satisfy sterilization requirements (e.g., constant milk outlet temperature), the heat exchanger can be operated by increasing the heating medium temperature (e.g., steam or hot oil), resulting in higher utility provision cost. Moreover, if the exchanger operates in such a mode for a long period of time, then the cleaning cost becomes also considerable because the cleaning time increases with heating time due to a larger deposit mass. In general, the profitability of the plant decreases with time and each exchanger has to be shut down to be cleaned so as to restart operation at a higher performance level. The time between successive cleaning operations must be determined by considering the trade-

Figure 1. Schedule with frequent cleaning policy for the motivating example.

off between cleanup costs, utility costs, and also the state of each exchanger in the network. In dairy plants, plate heat exchangers (PHEs) are widely used for pasteurization and sterilization purposes. As was shown in Georgiadis et al.,25 fouling is not uniform over different parts of the PHE and only a few channels are mainly responsible for the total interruption while other channels suffer less severe fouling and thus their operation can continue for longer processing times. Because a PHE can be modeled as a complex arrangement of shell and tube heat exchangers, it is then possible to allow different parts (channels) of the PHE to operate while others are being cleaned at the same time. Conceptually, it is possible to view a PHE as a complex arrangement of individual channels, each modeled as shell and tube heat exchangers. It is then possible to think of an operation where different parts (channels) of a “smart” PHE operate in a production mode while others are being cleaned at the same time. Thus, the optimal synchronization of cleaning and heating operation of different channels subject to product quality/sterilization requirements, and utility availability level, while maintaining continuous milk production, is of great interest. Although the approach presented in this paper is implemented for the cleaning policies in PHEs, the presented models can provide the basis for extensions to various HENs subject to other types of a fouling profile (e.g., in refineries19), reactor networks where the catalyst undergoes deactivation with time, and reverse osmosis networks.18 Next, we describe a small quantitative example that highlights the underlying trade-offs as well as the need for systematic optimization approaches. Consider a small network with two heat exchangers in a serial arrangement with different fouling rates. Steam is assumed to be the heating medium with a maximum available temperature in the plant of 600 K. The steam temperature is manipulated appropriately to ensure that the milk outlet temperature is kept above a specified bound of 369.5 K (quality requirements). We examine two alternative operating policies over a time horizon of 24 h: (i) frequent and (ii) infrequent cleaning of each heat exchanger. Two feasible cleaning schedules for case (i) and (ii) are shown in Figures 1 and 2, respectively. (the single horizontal bars in the figures declare unit operation (heating) while the double horizontal bars refer to cleaning tasks.) For both cases the performance of each exchanger (i.e., the overall heattransfer coefficient) is assumed to decrease linearly with time. The cleaning cost comprises a fixed term (0.1 rmu

Ind. Eng. Chem. Res., Vol. 39, No. 2, 2000 443

Figure 2. Schedule with infrequent cleaning policy for the motivating example.

Figure 3. Optimal cleaning schedule for the motivating example.

(rmu ≡ relative money units)) and a variable term (0.11 rmu/s) which is a function of the preceding heating period. The utility cost is assumed to be a linear function of its temperature (0.7 rmu/K). The resulting total operating costs for cases (i) and (ii) are 25 500 and 26 100 rmu, respectively. However, another feasible schedule is to operate the network as shown in Figure 3 which involves a moderate number of cleaning operations. For this case, the total cost is 20 400 rmu, yielding to 21% savings in comparison to the corresponding costs of the original operating scenarios. Clearly, there is a trade-off between frequent and infrequent cleaning and related utility costs. Moreover, the timing and synchronization of cleaning operations is also very important because their cost depends not only on fixed terms but also on variable terms based on the preceding heating period. In addition, the timing of cleaning operations should be determined in such a way that the temperature of the final product is kept above a given lower value over the entire horizon of interest. The relation between the operating cost and the number of cleaning tasks is shown in Figure 4. We can notice that the operating cost decreases initially with the number of cleaning tasks because both variable cleaning and utility costs decrease, then reaches a minimum (optimal solution), and finally starts increasing as the fixed cleaning costs become dominant. The optimal solution (20 400 rmu) was obtained using the model proposed in this paper by performing 12 cleaning tasks. The above analysis involved only two exchangers. If there are more exchangers with complicated network arrangements and operating limitations, the trade-offs

Figure 4. Operating cost as a function of the number of cleaning tasks.

are considerably more complex. In particular, an upper bound on the utility temperature may exist because of its availability in the plant. For example, if the steam is available up to 405 K, then the obtained optimal schedule results in 14 cleaning tasks with a total cost of 21 100 rmu which is more expensive than the previous optimal solution without utility limitations. However, it is worth noticing that the final cost is not as high as cases (i) and (ii) because the optimal allocation of cleaning tasks is determined by the model rather than based on arbitrary decisions. In conclusion, the aim of this work is to develop mathematical models that capture these complex interactions to determine the optimal cleaning policies of special classes of HENs by optimizing a suitable economic criterion. 3. Problem Statement This work is concerned with the short-term (typically time horizon of 1 day) cleaning scheduling problem in special classes of HENs under milk fouling. It has long been recognized that fouling in heat exchangers used in the food industry is the dominant problem which imposes very frequent cleaning operations. Other aspects such as operability and uncertainty are important but the HENs’ behavior is mainly driven by the rapid fouling. This, however, may not be true for fouling in other HENs (e.g., in petrochemical plants where heat exchangers are cleaned annually). In a more integrated approach, fouling dynamics should be incorporated in the same framework with other operational aspects (cleaning scheduling, controllability, uncertainty, etc.). However, the resulting mixed-integer optimal control problem cannot be solved using current algorithms. Our approach relies on a decomposition strategy where once the optimal time-decaying profiles of each heat exchanger are defined, a suitable parameterization has been employed to approximate the behavior of the equipment and optimize the problem using integer programming techniques. The problem can be stated as Given: 1. time horizon of interest 2. structure (e.g. serial) of the HEN 3. utility bounds 4. fouling profiles of equipment 5. product quality requirements Determine: 1. cleaning schedule 2. utility utilization profile

444


Figure 5. A unit representation with its bypasses for a serial HEN.

So as to optimize an economic performance criterion. The HEN comprises a given number of heat exchangers (i ) 1...N), is assumed to operate continuously with the milk flow rate constant, and is equal to Fm. As mentioned in the previous section, the fouling effects are different from exchanger to exchanger and consequently each of them is characterized by its own fouling profile. A maximum allowable (critical) value of the deposit mass, δcrit i , is taken into account to avoid pressure drop violations. This critical value predetermines the maximum operation time for each exchanger after which no heating is allowed and a cleaning task must take place. The heating medium of the plant (e.g., steam or hot oil) is usually constrained by a maximum available temperature. The cost of cleaning operation includes two terms: a fixed term, each time a cleaning task is performed, and a variable term, as a function of the preceding heating time. A hot utility is assumed to be available within a continuous temperature range while the utility cost is a linear function of its temperature level. The overall heat-transfer coefficient, Ui(t), of each heat exchanger i can be characterized by general decaying profiles (typical examples are exponential or linear decaying). It is assumed that at the beginning of operation the heattransfer coefficients have their largest values corresponding to clean conditions. As illustrated in Figure 5, each exchanger is represented by two bypasses (one per stream) which are

Figure 6. Serial heat-exchanger network.

activated simultaneously according to the current operating mode (i.e., heating or cleaning). If the exchanger operates, then both bypasses are forced to have zero flows. Otherwise, when the exchanger is cleaned, then the bypass flow rate equals the milk flow rate. However, as discussed in section 5, this representation can only be used for serial HENs while for parallel network arrangements these bypasses should be replaced by valves or similar mechanical devices which open and close during heating and cleaning, respectively. It should be noted that the feasibility of the mechanical aspects of the implemented cleaning policies is not considered. The approach for serial HENs relies on the existence of bypasses for each processing exchanger which may not exist in practice and their installation is thus required. Moreover, the dynamics of the startup and shutdown is not considered in this work. Finally, we assume that the inlet flow rate is equally split to all immediate exchangers (see sections 5 and 6). The issue of holding time is an important one in sterilization processes. Here, an assumption made is that the holding time can be adjusted to any appropriate value by an additional downstream holding section. However, specific attention is given so as the process to meet quality (temperature) requirements. The models presented in this work, in principle, could easily be applied to longer time horizons (e.g., more than 5 days), potentially generating large-sized models. An alternative approach for problems with longer time horizons is to develop a receding horizon policy where the time horizon of interest is divided into a number of time slots. The optimal scheduling policy for the first slot can be defined by using the models developed in this work. Then, at the beginning of the next slot, the model data (e.g., fouling profiles) are updated from the plant database and finally the schedule is reoptimized. This procedure is then repeated. However, such an approach cannot guarantee optimal solutions. We adopt here a constructive approach. First, formulations for the basic building blocks related to serial and parallel arrangements of heat exchangers are studied. Then, we show how more complex networks can be


The hot utility is characterized by the following: CUp ) cost coefficient ΘLp ) minimum temperature ΘU p ) maximum temperature

The following binary variables are then introduced in the mathematical model:

{

1, if exchanger i operates over period p 0, otherwise 1, if the last cleaning of exchanger i occurred at period p - θ Xipθ ) 0, otherwise

yip )

Figure 7. Nonlinear decaying profile.

described using these basic building blocks. Serial, parallel, or combined HENs are quite common in many applications in the food industry.21 Moreover, in dairy plants most of the heat-treatment systems exhibit combinations of such networks. However, the proposed models cannot be directly applied to a network with stream splits. 4. Serial Heat-Exchanger Network Case 4.1. Mathematical Formulation. This case is concerned with a serial arrangement of heat exchangers (see Figure 6). To facilitate the mathematical description of the problem, the time horizon of interest is discretized into P periods (time intervals) with fixed duration. Piecewise constant profiles are assumed for all variables (e.g., heat-transfer coefficients, temperatures, etc.) as illustrated in Figure 7. The choice of the discretization interval depends on a reasonable approximation of the process variables as well as the minimum duration of the cleaning task. For the purpose of our analysis, the following notation is used: Notation Indices i ) heat exchanger p, θ ) time period Parameters Tinlet ) milk inlet temperature m Fm ) milk flow rate Cpm ) milk specific heat capacity TLm ) lower bound on the milk outlet temperature

Each exchanger i is characterized by the following: Ai ) heat-exchange area ∆TLi ) minimum mean temperature difference ∆TU i ) maximum mean temperature difference QU i ) maximum heating load ) minimum temperature difference between milk δTmin i and heating medium ζiθ ) value of the heat-transfer coefficient after θ periods of operation C h ip ) fixed cleaning cost coefficient over period p C ˆ ip ) variable cleaning cost coefficient over period p δcrit ) critical deposit mass i τi ) maximum heating duration

{

We assume that if exchanger i does not operate over period p (i.e., yip ) 0), then a cleaning task takes place to restore its performance. The positive values of the following continuous variables over each period p have to be determined for the following: (i) Each Exchanger i Qip ) heating load Tin ip ) milk inlet temperature Tout ip ) milk outlet temperature Uip ) heat-transfer coefficient ∆Tip ) mean temperature difference tip ) number of time intervals elapsed since the last cleaning ∆tip ) number of time intervals at period p since the last cleaning (ii) The Hot Utility Θp ) temperature

The difference between variables ∆tip and tip is clarified in section 4.1.3. 4.1.1. Energy Balance Constraints. Continuity of the milk temperature is ensured by the following equations: inlet Tin ip ) Tm out Tin ip ) Ti-1,p

∀ i ) 1, p

(1)

∀ i ) 2...N, p

(2)

is equal to 333 K for all the examples where Tinlet m considered. For each exchanger i, the following energy balance is written for the milk side over each period p: in Qip ) FmCpm(Tout ip - Tip) ∀ i, p

(3)

where Cpm is the milk heat capacity equal to 3930 J/(kg K). To take into account the effect of CIP operations on the thermal behavior, additional constraints are included:

Qip e QU i yip

∀ i, p

(4)

The above constraint guarantees that if exchanger i is cleaned during period p (i.e., yip ) 0), then the corresponding head load, Qip, is forced to zero. Consequently, the milk inlet and outlet temperatures will be equal by eq 3. On the other hand, if the exchanger i operates at period p (i.e., yip ) 1), then a specific amount of heating load should be exchanged (as determined by the model) below a specified upper bound, QU i .

446


The basic design equations for every exchanger i are as follows:

τi

∀ i, p

Qip ) UipAi∆Tip

according to the following additional constraints:

(5)

Uip )

∑ ζiθXipθ

∀ i, p

(12)

∀ i, p, θ ) 1...τi

(13)

θ)1 in (Θp - Tout ip ) + (Θp - Tip) ∆Tip ) 2

∀ i, p

(6)

The logarithmic mean temperature difference, ∆Tip, has been approximated by the arithmetic mean temperature difference. The milk outlet temperature of the last exchanger (i.e., i ) N) should always lie above a specified bound according to the dairy sterilization requirements. This is guaranteed by the following constraint: L Tout ip g Tm

∀ i ) N, p

(7)

The value of TLm is defined by sterilization requirements equal to 369 K for our analysis. The algorithm will force the milk outlet temperature close to its lower bound and not far above it, to minimize the energy consumption. Thus, no upper bound is required. The steam temperature, Θp, is a degree of freedom which can be adjusted over time to mitigate fouling effects. It is assumed that Θp is determined (in a piecewise constant mode) by the plant hot utility system with possible lower and upper bounds on the availability level:

ΘLp e Θp e ΘU p

∀p

(8)

The following feasibility constraints are added to the formulation: min Θp g Tin ip + δTi

∀ i, p

(9)

min Θp g Tout ip + δTi

∀ i, p

(10)

To guarantee that each heat exchanger operates within a small temperature range (so fouling will be mainly a function of time), a minimum number of heat exchangers, Nmin, may be imposed to be active (heating operation) over each time interval:

∑i yip g Nmin

∀p

(11)

4.1.2. Fouling Constraints. A distinguishing feature of the process is that the equipment fouling rates depend on the temperature and protein concentration in the milk processes. This dependency is expressed by imposing different fouling profiles in each heat exchanger. The heat-transfer coefficient is assumed to be only a function of time. However, other expressions including also temperature dependency can be easily used in the same modeling approach. Although in some cases the fouling rate may follow a linear profile in the general case, it will be a nonlinear profile. The most common profile is the exponential one (monotonic decreasing), as illustrated in Figure 7. In this section, a general monotonic decreasing nonlinear fouling profile is modeled. The value of the overall heat-transfer coefficient for exchanger i over period p after θ periods of continuous operation since the last cleaning is given by parameter ζiθ (see Figure 7). This parameter determines the value of the Uip variable

Xipθ e 1 - yi,p-θ τi

∑ Xipθ ) yip

∀ i, p

(14)

θ)1

where τi is the maximum number of periods over exchanger unit i which operates continuously without being cleaned. Note that if CIP takes place (i.e., yip ) 0), then constraints (14) will force all Xipθ variables to zero and consequently the corresponding Uip variables take the zero value through constraints (12). This in turn will force Qip to zero (from constraints (5)) in agreement with constraints (4). On the other hand, if exchanger i operates over period p and by assuming, first, that only one CIP task occurred at period p - θ* (θ* e τi), then all Xipθ (θ ) 1...τi, θ * θ*) variables will take the value of zero (constraints (13)) except Xipθ* which is forced to the value of 1 (constraints (14)). This is turn will determine the value of Uip being equal to ζiθ. Additionally, if exchanger i operates over period p and more than one CIP operation takes place over the last τi periods, then the most recent cleaning task (smaller θ value) should be chosen to determine the correct Uip value. This is guaranteed by the optimization algorithm as the smaller θ value (more recent) corresponds to a larger ζiθ and consequently to a larger Uip value, thus minimizing energy consumption. It should be noted that the Xipθ variables can be treated as continuous variables as they can obtain only zero or one value because of the monotonic decreasing nature of the fouling profile. Thus, the complexity of the problem can be reduced significantly. A serious problem that arises during milk heat treatment is blockage of the channels as a result of the deposit formation. This is due to extensive run times and results in significant pressure drop. One way to model this serious operating limitation is to monitor the value of deposit mass or thickness. In that case, a critical deposit thickness or mass, δcrit i , for each exchanger i can be defined, on the basis of simulation results. This critical deposit mass corresponds to a maximum time duration, τi (number of time intervals) after which exchanger i is not allowed to operate. This allows one to effectively incorporate pressure drop limitations into the mathematical model. For example, if the deposit mass, δi(t), follows a linear profile,

δi(t) ) γit where γi is the slope of the linear profile and δi(t) e δcrit i , then τi will be determined by the upper bound of deposit mass (i.e., δcrit i ) as follows:

δcrit i τi ) γi

(15)

Similarly, the values of τi can be easily determined for nonlinear deposit mass profiles.


4.1.4. Cleaning Constraints. We assume that each heat exchanger i is characterized by its own cleaning duration, di. The following constraint is then added in the formulation:

yi,p+µ e 1 - yi,p-1 + yip

Figure 8. Graphical illustration of the cleaning constraints.

4.1.3. Timing Constraints. The cost of the CIP operations represents a major cost factor in dairy plants. While heat exchangers in refineries may only be cleaned annually, in the dairy industry it is common practice to clean them every 4-8 h. The cost of cleaning comprises the cost of water, detergent, effluent disposal, and required energy and can be modeled as a function of the preceding heating time and the number of periods during which the exchanger does not operate (i.e., yip ) 0). Consequently, the effect of heating operation on the cleaning requirements is captured which is according to industrial practice since for long heating periods milk deposit requires intensive cleaning requirements to be removed. In this study, two cleaning costs are used: fixed and variable. The fixed cleaning cost represents the fixed expenses over the period of cleaning operation (i.e., periods at which yip ) 0). On the other hand, the variable cost is a function of the preceding heating period. To quantify the variable CIP cost factor, the number of elapsed time intervals, tip, since the last cleaning must be calculated for each operating exchanger i, (i.e., yip ) 1) as follows: τi

tip + yip )

∑ θXipθ

∀ i, p

(16)

θ)1

It should be noted that variables tip monitor the number of time intervals during the whole period of heating operation. During cleaning periods the above timing variables, tip, are reset to zero as both yip and the right-hand side summation of constraints (14) is zero. To capture the elapsed time since the last cleaning, in the objective function the following additional variables, ∆tip, are introduced along with the following constraints:

∆tip ) ti,p-1 + yi,p-1 - tip

∀ i,p * 1

(17)

The above variables take non-zero values only if exchanger i is cleaned at period p while it was operating at period p - 1 (i.e., yip ) 0 and yi,p-1 * 1). In all other cases, these variables are forced to zero by the optimization algorithm. The role of the above constraints and variables involved is illustrated in Figure 8.

∀ i, p, µ ) 1...di - 1: di > 1 (18)

The above constraint is only applied when exchanger i starts being cleaned at period p while it was operating at period p - 1 (i.e., yip ) 0 and yi,p-1 ) 1). Therefore, each exchanger is forced to be cleaned for di consecutive time periods. A more aggregated form of the above constraint could be derived by summing over µ, thus resulting in fewer constraints: di-1

∑ yi,p+µ e (1 - yi,p-1 + yip)(di - 1)

∀ i, p:

µ)1

di > 1 (19)

However, the above constraint exhibits looser linearprogramming relaxation. Limitations on the capacity of cleaning fluid(s) can be taken into account, in an aggregated way, by simply imposing a maximum number of cleaning tasks, M, that can be performed at each time interval. This can be expressed mathematically as follows:

∑i (1 - yip) e M

∀p

(20)

4.1.5. Objective Function. Because the network operates continuously, no term is included in the objective function to account for production losses during shutdowns. Hence, the objective function includes only utility and cleaning costs. The former term is assumed to be a linear function of utility temperature. This assumption is a simple approximation and can easily be relaxed by using more comprehensive models (taking into account the total enthalpy of utility). Thus, the utility cost is given by the following expression:

utility cost )

∑p CUpΘp

(21)

The cost of CIP operation, as mentioned in section 4.1.3, involves two terms and given by the following expression:

cleaning cost )

∑i ∑p (Ch ip(1 - yip) + Cˆ ip∆tip)

(22)

So the objective function to be minimized is given by the following expression:

min

∑p CUpΘp + ∑i ∑p (Ch ip(1 - yip) + Cˆ ip∆tip)

(23)

4.2. Summary of Formulation. The overall formulation of the optimal cleaning scheduling problem for

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serial HENs under general milk fouling profiles is summarized as follows:

min

∑p CUpΘp + ∑i ∑p (Ch ip(1 - yip) + Cˆ ip∆tip)

(P1)

subject to energy balance constraints: (1)-(11) decay constraints: (12)-(14) timing constraints: (16) and (17) cleaning constraints: (19) and (20) The formulation presented above is an MINLP model where all the nonlinearities arise from the bilinear equation (5). In general, for the solution of this problem, various two-level decomposition approaches can be employed. For example the generalized benders decomposition26 or the outer-approximation method.27 An alternative way is to use a branch-and-bound method based on the solution of a series of NLP subproblems. This method has been employed by Jain and Grossmann19 and proved to be more efficient than the outerapproximation method as implemented in DICOPT++ modeling environment.28 Another option is to linearize the above model by using standard transformation techniques as discussed in the following section. Although such a linearization scheme increases the problem size, the resulting MILP can guarantee global optimality and be solved efficiently with commercial software. 4.3. MILP Reformulation. Standard transformations can be employed to linearize the model by introducing extra variables and constraints.29 However, a key issue is to explore any special structure of the model to reduce the size of the problem. Thus, a new variable, X∆Tipθ is introduced defined as

X∆Tipθ ≡ Xipθ∆Tip

∀ i, p, θ e p

(24)

where ∆Tip is given by eq 6. The following constraints are then added in the model:

∆TLi Xipθ

e X∆Tipθ e

∆TU i Xipθ

∀ i, p, θ ) 1...τi (25)

∆Tip - ∆TU i (1 - Xipθ) e X∆Tipθ e ∆Tip ∆TLi (1

- Xipθ)

∀ i, p, θ ) 1...τi (26)

τi

∑ ζiθX∆Tipθ

θ)1

Constraints (26) can be substituted by a more aggregated form similar to those of Voudouris and Grossmann:30 τi

∆Tip )

∑ X∆Tipθ + ξip

∀ i, p

(28)

θ)1

ξip e ∆TU i (1 - yip)

∀ i, p

(29)

where ξip can be considered as a slack variable. Note that the above constraints are only defined over sets i and p and not over θ, thus reducing significantly the number of linearization constraints. 5. Parallel Heat-Exchanger Network Case

where ∆TLi and ∆TU i are lower and upper bounds of the mean temperature difference, which are calculated according to the process specifications. Constraints (25) guarantee that if Xipθ is zero, then the linearized variable is set to zero as well. On the other hand, if Xipθ ) 1, then X∆Tipθ takes the value of ∆Tip from (26). The design equation (5) can finally be rewritten as

Qip ) Ai

Figure 9. Parallel heat-exchanger network arrangement.

(27)

While the formulation presented in the previous section addresses the problem of serial heat exchangers, some modifications are required if parallel exchangers are involved (see Figure 9). This arrangement is widely used in dairy plants in the form of a one-channel per pass plate heat exchanger. The milk flow is assumed to be split equally among all the active heat exchangers. Consequently, the milk inlet flow rate as well as the inlet and outlet temperatures for each exchanger will depend on its operating mode. If exchanger i is cleaned, then the corresponding fluid flow rate and heating load are zero. Therefore, a new set k needs to be defined (k ) 1...N), denoting the number of active exchangers at period p.


A new binary variable, Wkp, is also introduced:

Wkp )

The number of active exchangers is related to the yip variables according to the following equation:

{

1, if k exchangers operate over period p 0, otherwise

During each period only one set of operating exchangers can be active so as to provide continuity in milk processing. This is expressed mathematically by the following constraints: N

∑ Wkp ) 1

∀p

(30)

k)1

If F/m is the milk flow rate processed at each active heat exchanger, the energy balance constraints (3) must be replaced by in Qip ) F/mCpm(Tout ip - Tip)

∀ i, p

(31)

where F/m is given by the following equation:

F/m )

Fm

∑k

∀ i, p

(32)

kWkp

N

∑ kWkp ) ∑i yip

N

∑

kWkp )

k)Nmin

Qip )

∑k

-

∀ i, p

Tin ip)

(33)

kWkp

and by exploiting constraints (30)

Wkp in (Tout ip - Tip) k k)1 N

Qip) FmCpm

∑

∀ i, p (34)

inlet where Tin ip is equal to the milk inlet temperature Tm . Constraint (34) involves nonlinear terms which can be linearized by introducing new variables, WTkip, defined as follows:

in WTkip ≡ Wkp(Tout ip - Tip)

∀ k, i, p

(35)

These variables are calculated by the following constraints:

WTkip e TU i Wkp

∀ k, i, p

(36)

N

in WTkip ) Tout ∑ ip - Tip k)1

∀ i, p

(37)

where TU i is an appropriate bound on the milk temperin ature difference (Tout ip - Tip) which is in the range of 40-45 K. Thus, eq 34 can be written in a linear form as follows: N

Qip ) FmCpm

WTkip

∑ k)1

k

∀ k, i, p

(38)

∑i yip

∀p

(40)

Similar changes are applied to constraint (34). Apart from the engineering perspective, the above constraints to capture the minimum number of operating exchangers result in a smaller problem size since less binary variables (Wkp) and constraints are required. The final milk outlet temperature, after mixing all streams, can be calculated from the following overall energy balance equation:

Tout i (Tout ip

(39)

Also, it may be possible, because of limitations on temperature ranges and/or pressure drop considerations, to enforce a minimum number of exchangers, Nmin, to be active (operated) over each period p. Then, constraint (39) is modified to

Combining (31) and (32), we obtain

FmCpm

∀p

k)1

)

Tinlet m

+

∑i Qip FmCpm

∀p

(41)

It should be emphasized that the milk outlet temperature of each heat exchanger i, Tout ip , will not take the zero value if the exchanger does not operate. In that case it will be equal to Tinlet through constraint (34). m However, it will not contribute to the overall energy balance because the final mixing temperature, Tout m , is calculated on the basis of the summation of heating loads of all active heat exchangers (see constraints (41)). Also, it is worth noticing that the underlying heatexchanger network will not include bypasses as the serial one. It only requires suitable valves in front of each exchanger which may open (heating) or close (cleaning), depending on the operating mode. All the fouling constraints remain the same as the ones presented in section 4.1.2. 6. Block Heat-Exchanger Network Case Once formulations for the basic serial and parallel arrangements have been described, more complex configurations can be modeled by combining these basic cases. Here, we examine a block HEN where different blocks of parallel exchangers are connected in series. Such an arrangement is widely used in industrial food and pharmaceutical processing in the form of a multipass PHE. This case utilizes the modeling issues presented in the previous section with minor modifications to account for the continuity of temperature between successive blocks. The Wkp binary variables, in the case of parallel exchangers, are now redefined for each block b (b ) 1...B). A new set Ib is defined to declare heat exchangers which belong to block b. The following energy balance

450


is then imposed to calculate the milk outlet temperature, T ˆ out bp , of block b:

∑

T ˆ out ˆ in bp ) T bp +

Qbip

i∈Ib

FmCpm

∀ b, p

(42)

where T ˆ in bp is the milk inlet temperature of block b. The following constraints are also imposed to ensure temperature continuity between blocks. in ˆ b+1,p T ˆ out bp ) T

∀ b ) 1...B - 1, p

(43)

in Clearly, for the first block T ˆ 1p ) Tinlet m . The heating load of exchanger i in block b, Qbip, is calculated by the following constraints:

∑

∀ b, i ∈ Ib, p (44)

where Nb is the total number of exchangers in block b out is the milk outlet temperature from exchanger and Tbip i in block b over period p. The resulting nonlinearities are treated in a fashion similar to that in section 5. 6.1. Fluid as a Heating Medium. So far, we have considered the case of using saturated steam as a heating medium. In general, any fluid heating medium (for example, hot oil) can be used if available in the plant. The model in such a case requires minor changes in the formulation presented in the previous sections. The thermal energy balance constraints should also include the heating medium side contribution. Thus, similar to eq 44 the following one is added: Nb

Qbip ) F0Cp0

Wkp out (Θin bip - Θbip ) k k)1

∑

∀ b, i ∈ Ib, p (45)

in out and Θbip are the inlet and outlet temperawhere Θbip tures of the heating medium, respectively, for exchanger i in block b over period p. Finally, F0 and Cp0 are given parameters declaring the heating medium flow rate and thermal capacity, respectively. The above equations can then be linearized similar to those in section 5. Furthermore, the mean temperature difference constraint (6) is updated to include heating medium temperatures:

∆Tip )

out out in (Θin bip - Tbip ) + (Θbip - Tbip) 2

exchanger

R (W/(m2 K))

β (W/(m2 K s))

γ (kg/(m2 s))

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

-0.04166 -0.045 -0.048 -0.050 -0.058 -0.063 -0.069 -0.071 -0.073 -0.068 -0.064 -0.061 -0.057 -0.054 -0.052

10-5 10-5 10-5 5 × 10-5 5.5 × 10-4 6.0 × 10-4 9.2 × 10-4 9.5 × 10-4 9.8 × 10-4 9.4 × 10-4 9.3 × 10-4 9.1 × 10-4 8.92 × 10-4 8.1 × 10-4 7.5 × 10-4

Table 2. Cost Data

Nb

Wbkp in (Tout Qbip ) FmCpm bip - Tbip) k k)1

Table 1. Fouling Data for the Serial Heat-Exchanger Network Case: General Profile

∀ b, i ∈ Ib, p (46)

It should be emphasized here that apart from the cleaning and heating timings the inlet temperature of the heating medium is an additional degree of freedom in the network which can be manipulated to meet the process thermal requirements. 7. Case Studies 7.1. Serial Heat Exchangers. Consider the serial HEN shown in Figure 6 which comprises a total number of 15 heat exchangers. Linear fouling is assumed for all exchangers whose characteristics are given in Table 1 as derived from our previous work.24 The overall heattransfer coefficient, Ui(t), and deposit mass, δi(t), profiles

CUp (rmu/K)

C h ip (rmu)

C ˆ ip (rmu/h)

2

900

360

are expressed by the following equations:

∀i

Ui(t) ) Ri + βit δi(t) ) γit

∀i

(47) (48)

where Ri, βi, and γi are given parameters for exchanger i. It is worth noting that the equipment fouling rates differ from exchanger to exchanger with the maximum value observed in exchanger 9. The critical deposit mass is 16 g/m2. The heating medium is taken to be saturated steam with a maximum temperature availability level of 550 K. The time horizon of interest is 1 day which is divided into 24 time intervals of equal duration. Milk is processed at a constant flow rate of 0.08583 kg/s while the heat-exchange area for each exchanger is 0.01101 m2. The outlet temperature of milk is constrained to be above 369 K throughout process operation. All relevant cost data are given in Table 2. The duration of the cleaning task for each heat exchanger is taken to be equal to the size of one time interval (1 h). The GAMS modeling system31 was used to implement the mathematical models. The problem was first solved using the augmented penalty outer-approximation method as implemented in DICOPT++28 for the MINLP model (P1), and the OSL MILP optimizer for the linearized model (P2). An optimality gap of 1.5% was assumed for the solution of the MILP model. Computational results are given in Table 3. This table highlights some interesting results regarding the computational features of the problem. The relaxation of model (P2) is very tight as the relative gap between the fully relaxed problem and the final MILP is 2.2%. This example could not be solved using the MINLP model (P1) in reasonable computational time (10000 CPU s). On the other hand, the problem was solved very efficiently using the MILP model (P2). Additionally, and to make a comparison between the two models, another problem with 12 exchangers was also used. Although this example was solved by using DICOPT++, the solution obtained was suboptimal (see Table 3). Once again, the MILP model (P2) performed more efficiently. Several tests have shown that the MINLP model (P1) for examples with more than 10-12 exchangers could not give any solution in reasonable time (less than 10000 CPU s).

Ind. Eng. Chem. Res., Vol. 39, No. 2, 2000 451 Table 3. Computational Results for the Serial HEN Case no. of units

d

fouling profile

relaxationa

model

major iterations in OA or branch-and-bound nodesb

CPUc

cost (rmu)

MINLP MILP MINLP MILP MINLP MILP

5 109 N/Ad 235 N/Ad 670

1910 510 N/Ad 750 N/Ad 1630

116 700 112 960 N/Ad 141 100 N/Ad 143 200

12

linear

111 604

15

linear

138 200

15

general

138 700

a Solution of the relaxed MINLP. b Termination criteria for DICOPT++: crossover of bounds. c Seconds on an ULTRA workstation. Not available solution after 10 000 CPU s.

Table 4. Fouling Data for the Complex HEN Case: General Profile Heat-Transfer Coefficient period

unit 1

unit 2

unit 3

unit 4

unit 5

1 2 3 4 5 6 7 8

2000 1790 1610 1500 1385 1170 990 815

2000 1730 1580 1440 1310 1070 890 740

2000 1700 1510 1380 1240 1050 860 660

2000 1630 1420 1300 1180 970 750 580

2000 1500 1400 1340 1110 900 750 480

period

unit 6

unit 7

unit 8

unit 9

unit 10

1 2 3 4 5 6 7 8

2000 1550 1430 1320 1170 930 750 500

2000 1500 1360 1130 1000 885 610 460

2000 1520 1320 1070 910 760 550 400

2000 1520 1280 990 805 700 510 365

2000 1600 1360 1075 910 790 620 365

period

unit 11

unit 12

unit 13

unit 14

unit 15

period

units 1,2,3

units 4,5,6

units 7,8,9

units 10,11,12

units 13,14,15

1 2 3 4 5 6 7 8

2000 1640 1450 1210 1080 910 795 680

2000 1720 1550 1310 1150 1010 895 810

2000 1790 1610 1380 1180 1050 930 860

2000 1820 1650 1400 1210 1085 960 910

2000 1850 1670 1430 1250 1135 1050 950

1 2 3 4 5 6 7 8 9

1925 1800 1670 1500 1380 1200 1120 1030 990

1870 1600 1380 1210 1030 1000 975 940 925

1900 1750 1610 1450 1300 1120 1100 1085 1050

1925 1800 1670 1500 1380 1200 1120 1010 970

1800 1730 1680 1590 1450 1360 1260 1180 1090

So far, we have considered linear fouling profiles for all exchangers. Here, the above problem, with 15 heat exchangers, has also been solved for the case of a general fouling profile (see Table 4). Again no solution was obtained in reasonable computational time using the MINLP model (P1). On the other hand, the linearized MILP model (P1) was solved with modest computational effort (Table 3). The optimal solution (schedule) is shown in Figure 10. We can notice that frequent CIP operations are performed especially in the exchangers with higher fouling rates (7 to 11 in Table 1). The temperature utilization profile of the heating medium (steam) is depicted in Figure 12. It is clear that when many exchangers are cleaned simultaneously then utility temperature reaches high levels (e.g. time interval 16). The same problem was solved for the case where utility cost is 100 times lower (CUp ) 0.02 rmu/K) than the above basic scenario and the resulting schedule is shown in Figure 11. Note that because utility cost is small, the optimization problem is mainly driven by the minimization of cleaning operations, thus requiring 42 cleaning tasks instead of 45 for the basic case. In addition, we can notice that at the end of the time horizon quite high steam temperatures are utilized as there is no penalty for leaving the exchangers dirty

Figure 10. Optimal cleaning schedule for the serial heatexchanger network case. Table 5. Fouling Data for the Complex HEN Case Heat-Transfer Coefficient

(fouled) at the end of the daily schedule. One way to tackle this deficiency is by augmenting the objective function including cost terms related only to the tiP (last interval) variables, thus reflecting the fouling state of the exchanger at the end of the day. Alternatively, a cyclic operating schedule can be applied to repeat the same pattern of operations (schedule) every day (cycle) by forcing the state of each exchanger to be the same at the start and end of the cycle. Consequently, if an exchanger is dirty at the end of a cycle, then its performance at the start of the next cycle will be decreased, thus requiring a cleaning operation in a short period. 7.2. A Complex Arrangement of Serial Blocks. Consider a HEN involving 15 heat exchangers grouped in five serial blocks of three parallel units each. The heating medium is assumed to be saturated steam at a maximum temperature of 550 K. An exponential fouling decaying behavior is assumed for all heat exchangers as given in Table 5. The critical deposit mass is 15 kg/ m2. The milk outlet temperature should be greater than 369 K. All exchangers have the same heat-exchange area equal to 0.0188 m2. The time horizon of interest is 1 day which is divided into 24 time intervals of equal duration. It is assumed that within the temperature

452


Figure 11. Optimal cleaning schedule for the serial heat-exchanger network case with small utility cost.

Figure 13. Optimal cleaning schedule for the complex heatexchanger network case. Figure 12. Optimal hot utility temperature profile for the serial heat-exchanger network case.

range of each heat exchanger, fouling is only a function of time. To guarantee that each exchanger operates (in heating mode) within a small temperature range, a minimum number of exchangers in heating operation can been imposed using constraint (38). The MILP model (P2) required 220 nodes (802 CPU s) using a 1.5% margin of optimality. The solution indicates a total cost of 142 200 rmu while the corresponding value of the relaxed LP is 139 500 rmu. Again, it is clear that the MILP results in a tight formulation (about 2% relative gap), even for this more complicated example. The optimal cleaning schedule and utility temperature profiles are depicted in Figures 13 and 14, respec-

tively, resulting in 46 cleaning tasks. Again, exchangers with high fouling rates are frequently cleaned. If an upper bound of 398 K rather than 550 K (the maximum temperature is about 400 K as shown in Figure 14) is imposed on the steam temperature, then the resulting schedule is characterized by 51 cleaning tasks while the operating cost is 145 800 rmu, yielding a 2.5% more expensive operation compared with the previous case. For the solution of this problem, 920 nodes were examined in the branch-and-bound tree with 1140 CPU s. An interesting aspect of the second case is that the cleaning scheduling policy has been changed considerably by increasing the required cleaning operation from 45 (original scenario) to 50. We can therefore conclude that the availability of the hot utility in the plant plays an important role to this problem. Furthermore, for even more limited hot utility availability, the problem may


Figure 14. Optimal hot utility temperature profile for the complex heat-exchanger network case.

trade-offs exist between the total number and timings of cleaning operations and the cost and availability of hot utility in the plant. First, an MINLP model was developed which was then linearized to an MILP model. The later model which can be solved to global optimality yields a very tight formulation, requiring a few nodes in the branchand-bound tree. On the other hand, the MINLP model cannot be used for the solution of problems with more than 10-12 exchangers over an operating horizon of 24 h. In addition, it was shown that the MINLP may lead to suboptimal solutions. Finally, a number of examples with up to 15 exchangers have been solved to illustrate the capability and efficiency of the proposed models. It was proved that the hot utility availability in the plant can have significant effect on the network cleaning policies. So far, this work has concentrated on short-term cleaning scheduling aspects. For longer time horizons, a receding horizon policy can either be applied, as discussed in section 3, or the models can be extended to account for cyclic operations, thus providing a more rigorous approach. The latter constitutes current work. Another interesting issue for future research is the development of on-line cleaning schedules techniques because the modeling of arbitrary fouling profiles enables the direct use of experimental data. Also, the low computational cost of the resulting optimization problems is a significant support for on-line applications. Furthermore, the synthesis of HENs under fouling considerations constitutes an interesting path for future work. Literature Cited

Figure 15. Hot oil inlet temperature profile.

be infeasible which is other words means that the production must be interrupted. Finally, this problem was also solved for the case where the heating medium is hot oil. The inlet temperature of the heating medium which is subject to an upper bound of 450 K is a control variable to be manipulated by the optimization algorithm. In practice, the heating medium is circulated between the HEN and an auxiliary exchanger or furnace which controls its temperature. The profile of hot oil temperature is depicted in Figure 15 while the cleaning schedule is the same as that in the previous case. The above problem was solved within 970 CPU s, examining 210 nodes during the branch-and-bound procedure. 8. Conclusions Special classes of problems dealing with the shortterm cleaning scheduling of HENs have been considered in this paper. A salient feature of this problem is that the performance of processing units decreases with time, which therefore have to be shut down for cleaning after regular time intervals. Because of its practical significance, a special application in HENs under milk fouling constituted the major motivation for this work. General fouling profiles have been considered to predict the behavior of the equipment with time. It was shown that

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Received for review March 5, 1999 Revised manuscript received October 21, 1999 Accepted November 2, 1999 IE990166C

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