2.1. Maintenance

Maintenance usually involves cleaning, repair, or replacement of degraded and aging components. The oldest and most common maintenance strategy is corrective maintenance (CM), or “fix it when it breaks” (Kothamasu et al., Reference Kothamasu, Huang and VerDuin2006). CM does not require any system analysis or planning effort, but at the same time, the operator has no control over the occurrence of downtime.

Preventive maintenance is also a commonly used strategy in industry (Camci, Reference Camci2009), where maintenance and repairs are performed at preestablished intervals. This strategy can effectively prevent unexpected downtimes, but because the fixed maintenance interval ignores the stochastic nature of degradation, very often maintenance is performed unnecessarily, leading to high operational cost.

CBM is an advanced maintenance strategy aimed at balancing maintenance cost, which is high in preventive maintenance, with downtime cost, which is high in CM (Dekker, Reference Dekker1996; Camci, Reference Camci2009). In CBM, system components are continuously monitored by various sensors to measure the state of the component, and computer algorithms estimate the “health” of the component, and predict the remaining useful life (RUL) using either physics-based models or data-driven methods (Wang & Vachtsevanos, Reference Wang and Vachtsevanos2001; Wang & Wang, Reference Wang and Wang2015). PHM is the research area that focuses on modeling RUL distribution and reliability in a number of different areas, including thermal and electronic components (Xing et al., Reference Xing, Ma, Tsui and Pecht2011; Yu et al., Reference Yu, Honda, Zak, Mitsos, Lienhard, Mistry, Zubair, Sharqawy, Antar and Yang2012), civil infrastructure systems (Peng et al., Reference Peng, He, Liu, Saxena, Celaya and Goebel2012; Loyola et al., Reference Loyola, Zhao, Loh and La Saponara2013), airplane maintenance (Bodden et al., Reference Bodden, Hadden, Grube and Clements2005), and battery technologies (He et al., Reference He, Williard, Osterman and Pecht2011). Information provided by PHM technologies is used in CBM to schedule maintenance accordingly.

2.2. Design–maintenance integration

There has been some research that focuses on considering reliability and fault identification during the design stage. Bodden et al. (Reference Bodden, Hadden, Grube and Clements2005) conducted a study that considers prognostics and health management as a design variable in air vehicle conceptual design. This work showed redundancies in the actuation components of air vehicles could be reduced with some knowledge of RUL. Wang et al. (Reference Wang, Huang and Du2010) presented an optimal design approach accounting for reliability, maintenance, and also warranty. Youn et al. (Reference Youn, Hu and Wang2011) proposed a framework for resilience-driven design of complex systems that integrates PHM into the design process using a reliability-based design optimization strategy. A fault identification and propagation framework has been developed for evaluating failure in the system in the early design stage (Kurtoglu & Tumer, Reference Kurtoglu and Tumer2008), as well as for integrated hardware–software systems (Mutha et al., Reference Mutha, Jensen, Tumer and Smidts2013). Previous studies have also considered methods to evaluate resilience in a system architecture (Mehrpouyan et al., Reference Mehrpouyan, Haley, Dong, Tumer and Hoyle2015) and the effect of design impulses on a system (Ghosh et al., Reference Ghosh, Devendorf and Lewis2014). Related research can also be found in disciplines outside mechanical engineering: Camci (Reference Camci2009) explored maintenance scheduling that considered the probabilistic nature of prognostics information and its effect on maintenance scheduling. Santander and Sanchez-Silva (Reference Santander and Sanchez-Silva2008) studied design and maintenance optimization for large infrastructure systems. By applying reliability-based optimization using a deterministic system model, they found that inefficient maintenance policy leads the optimization algorithm to converge to a design with higher degrees of redundancy. Monga and Zuo (Reference Monga and Zuo1998) considered both maintenance and warranty in optimal system design in their work. They compared selected system configurations with different failure rate functions, though no predictive maintenance was considered in this study.

The above-mentioned work mostly focused on the integration of design and system reliability, but it did not explicitly consider the physical degradation process nor the causal relationship between design decisions and degradation. Work done by Caputo et al. (Reference Caputo, Pelagagge and Salini2011) on joint economic optimization of heat exchanger design and maintenance policy considered the interaction between design decisions of a heat exchanger and its degradation (fouling), but only examined traditional maintenance strategy and not the uncertainty associated with degradation. Honda and Antonsson (Reference Honda and Antonsson2007) proposed the notion of grey-scale reliability to capture system performance degradation and the time dependency of reliability. They also studied design choices and their effects on system degradation. However, their study did not consider the effects of maintenance on system degradation.

There are limited studies in previous literature that link degradation to design variables, such as how the physical dimensions of a component may affect the uncertainties in its degradation. There is also a gap in studies linking system-level design decisions and advanced maintenance strategies. This paper seeks to fill these gaps through an approach that models the interaction between design and degradation. The proposed approach is demonstrated through an example of condenser (heat exchanger) design. It differs from existing reliability and maintenance based design optimization approaches (Wang et al., Reference Wang, Huang and Du2010; Youn et al., Reference Youn, Hu and Wang2011) in that the emphasis is on capturing the interdependencies between design decisions and degradation in system-level components.

3.1. Problem formulation

This work proposes a generalized modeling framework to capture the interactions between system-level design decisions and degradation, using a multiple-objective optimization approach (Wang & Shan, Reference Wang and Shan2007; Martins & Lambe, Reference Martins and Lambe2013). The problem formulation is

(1)$$\eqalign{& \hbox{minimize}\quad\lcub {C_{\rm L} - A} \rcub {\rm} = F\lpar {{\rm \tau}_{\rm m}\comma \ E_{\rm s}\comma \ {\bf x}_{{\rm cs}}} \rpar\comma \cr & \hbox{subject to}\quad{\rm \tau} _{\rm m} = f_{\rm o} \lpar {{\bf D}\lpar t \rpar\comma\ {\bf x}_{\rm m}} \rpar\comma \cr & \hskip4pc E_{\rm s} = f_{{\rm cs}} \lpar {{\bf D}\lpar t \rpar\comma\ {\bf x}_{{\rm cs}}\comma \ {\bf x}_{\rm m}} \rpar \comma \cr & \hskip4pc \,{\bf D}\lpar t \rpar {\rm} = f_{\rm D} \lpar {{\bf x}_{{\rm cs}}} \rpar\comma \cr & \hskip4pc \,g_{{\rm cs}} \lpar {{\bf x}_{{\rm cs}}\comma \ {\bf x}_{\rm m}} \rpar \le 0.} $$

where C _L is the life-cycle cost of the system and A is the availability or other metrics related to availability (such as mean time between maintenance). These are the objectives that need to be minimized and maximized in the design optimization. Life-cycle cost is a common objective for system-level design, and availability is an objective traditionally associated with degradation and maintenance. The objectives are influenced by the variables of τ_m, E _s, and x_cs. The vector x consists of all decision variables and, in this problem formulation, is subdivided into the system-level design variables x_cs and maintenance variables x_m. Examples of system-level design variables can be the types and configuration of components in the system, and examples of maintenance variables include types of maintenance strategies and thresholds for triggering a maintenance event. The τ_m variable is a sequence of times indicating maintenance occurrences during system life cycle and is computed by the operational subsystem model f _o, based on component degradation profiles D(t) and maintenance variables x_m. The E _s variable is the system-level performance, typically an efficiency or output measure, that is calculated by the physical system models f _cs based on degradation profiles, system design variables x_cs, and maintenance variables. The component degradation profiles D(t) are generated by degradation subsystem models f _D based on system design variables. It is assumed that the degradation profile only describes how the component will degrade over time from its original “clean-state” and thus will only be a function of the system-level design variables. Finally, g _cs are the system-level constraints that must be satisfied.

Figure 1 shows the system architecture model associated with this problem formulation. There are two major divisions illustrated in the figure. The first is the system design division that contains the forms and functions of the physical system and its subcomponents and disciplines. The system design model can be highly complicated with many interactions, though in this framework it is generalized into f _cs. The second division contains the degradation models f _D and the operational model f _o, and any other subsystems related to operation. It is assumed that the physical system can be modeled deterministically, while the degradation models are uncertain, and require uncertainty evaluation methods such as Monte Carlo simulation.

Fig. 1. Proposed framework for maintenance integrated at design stage.

The interface between the system design division and the maintenance division can be generalized as follows. The degradation models generate profiles that indicate how each component degrades over time. Multiple components in the system may have degradation profiles that are correlated or independent, and need to be modeled accordingly. These degradation versus time curves are used by the system model f _cs to simulate how the system-level performance (E _s) will vary over this time period. The performance profiles E _s are then fed into the operational model that computes the maintenance pattern τ_m. The operational model is simulated for N _lc years, the total duration of the system life cycle, with different degradation profiles generated over the life cycle to simulate the uncertainty in degradation.

It is critical to model the interdependencies between the subsystem components, and identify the key components and their degradation modes that contribute to system performance losses. System-level modeling techniques such as design structure matrices (Browning, Reference Browning2001; Alfaris et al., Reference Alfaris, Siddiqi, Charbel Rizk and de Weck2010; Eppinger & Browning, Reference Eppinger and Browning2012) and the engineering system matrix (Hu & Cardin, Reference Hu and Cardin2015) can be used for this purpose. Once the key degradation modes are identified, the relationship between degradation and the physical parameters associated with the component needs to be determined. This can be challenging and usually requires expert knowledge specific to the domain of the component. A survey of literature in the field may be an adequate approach to constructing a detailed model of component degradation.

3.2. Objective function definitions

This framework focuses on two design objectives: the life-cycle cost and system availability (time between maintenance). Other objectives may be considered, but cost and availability are representative for comparing optimality of system designs and maintenance strategies. The life-cycle cost C _L is defined as the total present cost of the system and contains

(2)$$C_{\rm L} = C_{\rm C} + C_{\rm M} + C_{\rm E}\comma $$

where C _C is the capital cost associated with system development. If any portion of the capital cost occurs later in the system life cycle, then the cost needs to be discounted to the present value with the discount rate r; C _M is the maintenance cost, which is the sum of all maintenance costs:

(3)$$C_{\rm M} = \sum\limits_{i = 1}^{N_{1c}} {\displaystyle{{C_{{\rm M}\comma i}} \over {\lpar {1 + r} \rpar ^i}}\comma} $$

and C _E is the efficiency cost, which is the cost due to degradation, such as the extra fuel needed:

(4)$$C_{\rm E} = \sum\limits_{i = 1}^{N_{1c}} {\displaystyle{{C_{{\rm E}\comma i}} \over {\lpar {1 + r} \rpar ^i}}\comma} $$

(5)$$C_{{\rm E}\comma i} = \int\limits_{t_{i - 1}} ^{t_i} {L\lpar {{\rm \eta} \lpar {\rm \tau} \rpar\comma\, {\rm \eta}_{\rm d}} \rpar } d{\rm \tau}\comma $$

where η is the actual system efficiency (between time t _i−1 and t _i), η_d is the designed system efficiency without degradation, and L is a function that computes the monetary loss due to degradation, dependent on the physical system.

The second objective is availability, which is generally defined in the literature as (Stapelberg, Reference Stapelberg2009):

(6)$$A = \displaystyle{{E\lpar {\hbox{up time}} \rpar } \over {E\lpar {\hbox{up time}} \rpar {\rm} + E\lpar {\hbox{down time}} \rpar }}\comma $$

where E() is the expected value operator, up time is the time that the system is operational, and down time is the time the system spent idling, including due to failure and during maintenance. In certain cases, the time taken to perform maintenance may be negligible, then an alternative measurement, mean time between maintenance (MTBM), can be used instead.

In general, the life-cycle cost of the system should be minimized, while the system availability should be maximized. Multiple-objective optimization methods can be used to find the Pareto-optimal designs. Alternatively, one of the two objectives can be treated as an intermediate variable, and single-objective optimization methods can be used.

3.3. Computational complexity

Because there are significant uncertainties associated with the life cycle analysis, uncertainty quantification methods such as Monte Carlo simulation are required to produce many different life cycle simulations to fully characterize the effects of uncertainty. The need for Monte Carlo simulation significantly increases the computational complexity of the framework.

The degradation and maintenance subdivisions of the framework contain all the stochastic models, such as the component degradation models. The physical system model can be assumed to be completely deterministic, and can be treated as a blackbox in the Monte Carlo simulation. The model only needs to be evaluated a few times to generate a surrogate model, such as a look-up table, and the Monte Carlo life cycle simulations can use the surrogate model instead of calling the full system model to reduce computing complexity.

Despite the reduction of computing complexities, the computing requirement is still significant, and thus balancing between model fidelity and complexity is a major challenge. For a multidisciplinary system, domain-specific models are usually high fidelity with very low discrepancies with reality but require significant computational time (on the order of hours or days). Furthermore, high-fidelity discipline-based models are usually represented using different software tools, making the data transfer between models complicated. Thus, high-fidelity models are not the best option for use in this framework. A common approach for model complexity reduction is to generate low-fidelity models from high-fidelity models using metamodeling methods such as Kriging or a response surface method (Wang & Shan, Reference Wang and Shan2007). Low-fidelity models can then be evaluated very quickly, but can have substantial discrepancy.

A midfidelity model is a simplified representation of a system that captures the essence of different domains by using first-order approximations of physics-based models (Lin et al., Reference Lin, de Weck, de Neufville, Robinson and MacGowan2009). Because they are physics based, no special software is needed, and this allows simple integration of different domains and subsystems. A midfidelity model has the advantages of short simulation time on the order of several seconds. Midfidelity models are commonly used in the early design phases to identify promising design strategies. Therefore, it is recommended that midfidelity models be used whenever possible.

3.4. Framework summary

Below are the proposed steps for setting up the integrated design and maintenance optimization problem:

1. 1. Identify key components and their degradation modes that contribute to system performance loss and require regular maintenance services.

2. 2. Determine the relationship between physical parameters and degradation.

3. 3. Propose feasible maintenance strategies. Determine variables associated with maintenance.

4. 4. Construct the system model. Set up domain models for the system, incorporate the degradation relationship, simulate over the lifetime operation of the system, and compute objective functions.

In sum, the proposed modeling framework is an architectural-level modeling approach aimed at capturing the interactions between the design space and the maintenance decision space in a complex engineering system. By identifying and modeling these interactions, designers can explore the trade-offs between different technology, different maintenance strategies, as well as different sets of design parameters. Generally, systems that can benefit from this framework should have the following characteristics: the degradation of key components is stochastic in nature, and the system architecture design choices can significantly affect the development of degradation. Examples include battery systems, membrane-based water production systems, and solar-powered systems.

4.1. System modeling

4.1.1. Power model

For simplicity, a Rankine cycle with a single-stage turbine is used. The power cycle only has five components: the boiler, the turbine, the condenser, the cooling water pump, and the feed water pump, as shown in Figure 2.

Fig. 2. Model schematics of Rankine cycle.

To create a numerical model, the Rankine cycle is broken down to six nodes, each representing the steam condition at the inlet and exit of a system component. The nodes are numbered 1 to 6 as shown by the subscripts in Figure 2. The variables P _i, T_i, h_i, and x _i are the pressure, temperature, enthalpy, and steam quality of the fluid at node i, respectively; ${\dot Q}_{\rm i} $ and ${\dot Q}_{\rm o} $ are for the heat (kW) input by the boiler and heat rejected by the condenser; ${\dot W}_{\rm t} $, ${\dot W}_{{\rm fp}} $, and ${\dot W}_{{\rm cp}} $ are the work (kW) extracted through the turbine, input to the feed water pump, and input to the coolant pump, respectively; and $\dot m_{\rm s} $ and $\dot m_{\rm c} $ are the mass flow rate (kg/s) of the steam/water circulated through the power cycle and cooling water through the condenser.

The power cycle model computes the cycle efficiency η as its output:

(7)$$\eqalign{& {\rm \eta} = \displaystyle{{\dot W_{\rm o}} \over {\dot Q_i}}\comma} $$

where ${\dot W}_{\rm o} $ is the net power output of the plant. Losses associated with pipe friction and inefficiency associated with pumps and turbines are neglected. The water/steam properties at each node are defined by the designer (boiler specification: P ₃, T ₃, turbine back pressure specification: P ₄), by the physical environment (T ₅), or determined through a steam table (Holmgren, Reference Holmgren2007).

4.1.2. Condenser model

The condenser model computes the condenser heat duty based on the inlet conditions and geometry using the log-mean temperature difference (LMTD) method (Kakac, Reference Kakac1991). The condenser heat load ${\dot Q}_{\rm c} $ is equal to

(8)$$\dot Q_{\rm c} = U \times \Delta T_{\rm m} \times S\comma$$

where U is the condenser's overall heat transfer coefficient [W/(m²K)], ΔT _m is the LMTD, and S is the overall surface area of the condenser (m²).

It was assumed the condenser is a one-pass X-type shell and tube heat exchanger made of 70–30 copper–nickel alloy, with the terminal conditions listed according to Figure 3, where P _atm is the atmospherical pressure, ΔP is the pressure drop in the coolant side of the condenser, and T _sat is the saturation temperature of steam at pressure P ₄.

Fig. 3. Model schematic of condenser.

An iterative process computes the condenser heat load ${\dot Q}_{\rm c} $ and coolant pressure loss ΔP based on the condenser geometry (number of tubes N_t and tube length L) and the condenser fouling resistance R _f (m²K/W). Details of the model can be found in Kakac (Reference Kakac1991).

4.1.3. Fouling model

Fouling in the condenser is measured by fouling resistance R _f. Fouling is a complicated phenomenon that has been studied extensively. A general consensus is that the overall fouling trend can be represented by this asymptotic model (Taborek et al., Reference Taborek, Aoki, Ritter, Palen and Knudsen1972a, Reference Taborek, Aoki, Ritter, Palen and Knudsen1972b; Epstein, Reference Epstein1983; Zubair et al., Reference Zubair, Sheikh, Budair and Badar1997):

(9)$$R_{\rm f} = R_{\rm f}^{\displaystyle \ast} \lpar {1 - e^{ - {\rm \beta} t}} \rpar\comma $$

where ${R}_{\rm f}^{\displaystyle \ast} $ is an asymptotic fouling resistance value related to the cooling water velocity v _c, tube inner diameter D _i, and the kind of foulant, while β (year⁻¹) is the reciprocal of the time constant. Empirical correlations used for ${R}_{\rm f}^{\displaystyle \ast} $ with calcium carbonate fouling and β are the following (Caputo et al., Reference Caputo, Pelagagge and Salini2011):

(10)$$R_{\rm f}^{\displaystyle \ast} = \displaystyle{{0.101} \over {\nu _{\rm c}^{1.33} {D}_{i}^{0.23}}}\comma $$

(11)$${\rm \beta} = {\rm} 0.0016v_{\rm c} ^{ - 0.35}. $$

Because fouling is a stochastic process, the asymptotic fouling resistance $R_{\rm f}^{\displaystyle \ast} $ can change randomly over time (Taborek et al., Reference Taborek, Aoki, Ritter, Palen and Knudsen1972a; Zubair et al., Reference Zubair, Sheikh, Budair and Badar1997). In this study, the asymptotic fouling resistance is considered a random variable (${\bf R}_{\bf F}^{\displaystyle \ast} $) that takes on a new value each time the condenser is cleaned. The distribution of ${\bf R}_{\bf F}^{\displaystyle \ast} $ is assumed to be log normal (Taborek et al., Reference Taborek, Aoki, Ritter, Palen and Knudsen1972a):

(12)$$\eqalign{{\bf R}_{\bf F}^{\displaystyle \ast} &= {\rm LN}\left( {{\rm \mu }\comma\ {\rm \sigma }} \right)\comma\cr {\rm \mu } &= \displaystyle{{0.101} \over {v_{\rm c}^{1.33} D_i^{0.23} }}\comma \cr {\rm \sigma } \big/{\rm \mu } &= 0.3.} $$

4.1.5. Operation and economics model

A 50-year lifetime is assumed for the power plant. The life-cycle cost C _L consists of the capital cost C _C, maintenance cost C _M, and efficiency cost C _E as indicated previously in Eq. (2).

For simplicity, only the capital cost of the condenser is considered for this study, which is proportional to the amount of materials used in the condenser tubes.

(13)$$C_{\rm C} = {\bf P}_{\bf m} {\rm \rho} _{\rm m} N_{\rm t} L\displaystyle{{\rm \pi} \over 4}(D_{\rm o}^2 - D_i^2 )\comma$$

where P_m is the price of raw material ($/kg), ρ_m is the density of material (kg/m³), and ${N}_{\rm t} {L}\lpar{{\rm \pi} / \hbox{4}}\rpar \lpar {{D}_{\rm o}^{\rm 2} - {D}_{i}^{\rm 2}} \rpar $ is the total material volume of the condenser tubes. Capital cost associated with implementing maintenance strategies, such as sensors and computing equipment, are not considered in this study for simplicity.

The maintenance cost is associated with the physical cleaning of the condenser tubes. The annual cleaning cost C _M,i is either zero when there is no cleaning during the year or

(14)$$C_{{\rm M}\comma i} = {\bf P}_{\bf L} + {\bf P}_{\bf s} N_{\rm t}\comma $$

when cleaning is performed. The first part is the fixed maintenance cost where P_L is the cost of labor, and the second part is the cleaning equipment cost and depends on how many tubes are in the condenser, and P_s, which is the price of each scraper.

The efficiency cost is an estimate of the productivity lost due to fouling. It was assumed the plant has zero efficiency cost when it operates at the designed efficiency η_d, which is the Rankine cycle efficiency if fouling resistance is zero. Efficiency cost increases proportionally to the amount of extra fuel consumed as a result of fouling. The annual efficiency cost C _E,i is computed as

(15)$$C_{{\rm E}\comma i} = \displaystyle{{{\bf P}_{\bf f}} \over {\hbox{HV}}}\int\limits_{t_{i - 1}} ^{t_i} {W_{\rm o} \lpar {\rm \tau} \rpar \left( {\displaystyle{1 \over {{\rm \eta} \lpar {\rm \tau} \rpar }} - \displaystyle{1 \over {{\rm \eta}_{\rm d}}}} \right)d{\rm \tau}\comma} $$

where P_f is the price of fuel ($/kg), HV is the heating value of the fuel (kJ/kg), W _o is the actual instantaneous power output during the year i, and η is the actual instantaneous thermal efficiency during the year i.

The annual maintenance cost C _M,i and efficiency cost C _E,i are discounted to present value using a discount rate r to obtain C _M and C _E, according to Eqs. (3) and (4).

4.1.6. Model integration

According to the proposed modeling structure, the system model is divided into a deterministic model, which includes the power and condenser models, and a probabilistic model, which includes the fouling and operation models.

For simplicity, boiler temperature and pressure are kept as constant parameters, and the only design variables considered are the number of condenser tubes N _t and the condenser tube length L. Constraints in the power-condenser model include the minimum turbine back pressure P _4,l and the maximum condenser coolant flow rate $\dot m_{{\rm c}\comma l} $. The deterministic system model has an input–output relationship of

(16)$${\rm \eta} = f_{{\rm sys}} \lpar {N_{\rm t}\comma\ L\comma \ R_{{\rm fs}}} \rpar .$$

In the probabilistic subsystem models, the operational model uses a Monte Carlo simulation to simulate the life cycle of the power plant randomly multiple times. The simulation process is shown in the flow chart in Figure 4.

Fig. 4. Model evaluation process.

At the beginning of the simulation, system-level design parameters are specified to determine the physical design of the condenser, and then a Monte Carlo simulation is performed to evaluate the performance of the system. Each iteration of the Monte Carlo simulation generates a life cycle simulation of the plant. At the beginning of the plant's life cycle, a fouling profile R _f,j(t) is generated with the asymptotic fouling resistance $R_{\rm f}^{\displaystyle \ast} $ randomly generated from the log-normal distribution described in Eq. (12). The operation model simulates the plant operation on a year-to-year basis and determines when maintenance is needed. After the occurrence of each maintenance event, the condenser is assumed to return to its original clean condition, and a new asymptotic fouling resistance is generated. After the entire life cycle is completed, a new life cycle is generated by the Monte Carlo simulation, and this process repeats until the desired number of iterations is reached. In this example, the Monte Carlo iteration number was set to 1000.

Some designer-defined parameter values used in the modeling process are tabulated in Table 1. The condenser cleaning equipment and labor costs are obtained from CONCO Systems, a manufacturer of heat exchanger cleaning equipment. Fuel cost is estimated based on the price of clean coal provided by the US Energy Information Administration, and the cost of condenser tube material is the average quoted price of several online vendors.

Table 1. List of user-defined parameters

4.2. Traditional design method (LMTD method)

Traditionally, the LMTD design process of choosing a condenser is based on the following three-step process (Kakac, Reference Kakac1991):

1. 1. Determine the condenser heat load $\dot Q_{\rm c} $ from the power plant's requirements.

2. 2. Select an operating fouling resistance based on empirical correlations.

3. 3. Determine the overall heat transfer coefficient U, the LMTD, and consequently calculate the condenser area, from which L and N _t are determined.

The process is similar to adding design redundancy into the condenser. The fouling resistances for different fluids are tabulated in heat exchanger design handbooks. For example, the Tubular Exchanger Manufacturers Association lists fouling resistances for seawater to be 0.352 m²K/kW and for brackish water to be 0.176 m²K/kW.

Once the fouling resistance is selected, the condenser design variables, N _t and L, can be determined using the LMTD method. For this case example, the fouling resistance was chosen to be 0.3 m²K/kW, in between the values of seawater and brackish water, and found that using the traditional design method, the condenser would require 7280 tubes and a tube length of 14.7 m.

Based on this specific condenser design, the three different maintenance strategies can be analyzed independently by simulating the life cycle with uncertainties, and a simple maintenance optimization can be solved to find the most optimal maintenance variable that results in the minimum life-cycle cost (C _L) of the system. The problem definition is shown below:

(17)$$\eqalign{&{\rm minimize}\quad{\bar C}_{\rm L} = F_{{\rm d}{\vert N_{\rm t}\comma\ {\rm L}}} \left( {{\rm Stgy}\comma\ N\comma \ R_{{\rm fs}}} \right)\comma \cr & {\rm subject}\,{\rm to}\quad g_{\vert N_{\rm t}\comma {\rm L}}\left( {{\rm Stgy}\comma \ N\comma \ R_{{\rm fs}}} \right) \le 0.} $$

where ${\rm \bar C}_{\rm L} $ is the mean value of the life-cycle cost from the Monte Carlo system model simulation, Stgy indicates one of the three maintenance strategies being considered: fixed interval, for which N is the maintenance interval in number of years, corrective and preventive, for which R _fs is the fouling resistance threshold, and F _d and g are the system models described in the section above.

The problem formulation is a mixed-integer nonlinear programming problem. However, because only three different strategies are being considered, they can be analyzed independently, and the other integer variable N is only associated with the fixed-interval strategy and can be handled through an exhaustive search, and thus reduces the problem down to a set of continuous nonlinear problems.

4.3. Optimal design method

The proposed framework allows an optimal condenser design to be found directly through optimization of the full system model. The problem definition is

(18)$$\eqalign{& \hbox{minimize}\quad\bar C_{\rm L} = F_{\rm s} \lpar {N_{\rm t}\comma\ L\comma \ \hbox{Stgy}\comma \ N\comma \ R_{{\rm fs}}} \rpar \cr & \hbox{subject}\,\hbox{to}\quad g\lpar {N_{\rm t}\comma \ L\comma\ \hbox{Stgy}\comma\ N\comma \ R_{{\rm fs}}} \rpar \le 0} $$

Similar to the maintenance optimization problem described in the previous section, this mixed-integer nonlinear programming problem can also be reduced into several continuous nonlinear problems by independently analyzing the three different maintenance strategies.

All of the programming and simulation for this case example were done in MATLAB 2011a. Built-in optimization algorithms including “ga” (genetic algorithm) were used. Because the Monte Carlo simulation resulted in long function evaluations, a population size of 30 with 20 generations were used to limit the overall computing time. In addition, “fmincon” (sequential quadratic programming) with random initial starting points was also used, and provided results with values within 2% of the genetic algorithm.