Design and Multiobjective Dynamic Optimization of Superheaters for Load-Following Operation in Pulverized Coal Power Plants

Pulverized coal power plants are increasingly participating in aggressive load-following markets, therefore necessitating the design and optimization of primary superheaters for flexible operations. These superheaters play a critical role in maintaining the final steam temperature of the steam turbine, but their high operating temperatures and pressures make them prone to failure. This study focuses on the optimal design of future-generation primary superheaters for a fast load-following operation. To achieve this, a detailed first-principles model of a primary superheater is developed along with submodels for stress and fatigue damage. Two single-objective optimization problems are solved: one for minimizing metal mass as a measure of capital cost and another for minimizing pressure drop on the steam side as a measure of operating cost. Since these objective functions conflict, a multiobjective optimization problem is executed using a weighted metric methodology. Results from these optimization studies show that the base case design can violate stress constraints during the aggressive load-following operation. However, by optimizing the design variables, it is possible to not only satisfy tight stress constraints but also achieve the desired number of allowable cycles and adhere to the steam outlet temperature constraint. In addition, the optimized design reduces either the metal mass or the steam-side pressure drop compared to that of the base case design. Importantly, this approach is not limited to primary superheaters alone but can also be applied to similar high-temperature heat exchangers in other applications.


■ INTRODUCTION
Fossil fuel-fired power plants are expected to continue to generate a considerable amount of electricity for the foreseeable future.However, the increasing penetration of renewable energy sources to the grid poses challenges for these power plants, including the need for demanding load-following operations such as fast ramp-up and ramp-down, low-load operation, and frequent start-up and shutdowns.These operational demands affect the health and life span of critical components, especially high-temperature boiler components, often leading to undesired outages and forced shutdowns.One critical requirement for efficient load-following operation by power plants is ensuring the reliability of boiler components.To achieve the desired reliability of these high-temperature boiler components, it is essential not only to optimize the operating conditions but also to design these critical components in an optimal manner for load-following operations.Since most existing boilers are decades old, they were not necessarily originally designed to handle these newer and unforeseen operating challenges.
−3 For flexible operation, fast start-ups/shutdowns and steep ramp changes in load must be done more rapidly and more cost-effectively. 4The impact of load-following operations on the life consumption of critical steam generation components is not yet fully understood.However, studies have reported that these undesired operations lead to creep and fatigue damages, as well as damages caused by the synergistic effects of creep and fatigue, significantly shortening the life of the boiler components. 5Among the critical components, superheaters operating at the highest temperatures are particularly affected by cycling operations.Superheaters not only suffer from fatigue damage but also face the risk of exceeding their design temperature during low-load and cycling operations, leading to creep damages. 6It has been reported that superheater damage contributes to about 40% of all boiler unplanned outages, 7 which, in turn, increases the plant equivalent forced outage rate and/or operation and maintenance cost.Thermal fatigue and creep-fatigue interaction can be considerably damaging and can lead to highly localized damages, such as ligament cracking. 8dentification and quantification of the localized damage are challenging without inspection during the plant shutdown.Even with inspections, it can be cost-prohibitive to inspect all possible locations for damage. 5One way of reducing the probability of failure occurrences is to optimally design the superheater components for flexible operations.Traditionally superheaters have been designed neither for rapid ramp-up/down operation nor for low-load operations.Therefore, it is highly desirable to design future superheaters while considering creep and fatigue damages resulting from expected increased cycling operations.
While simple models of superheaters are available in the literature, 9 these models do not account for the complex flow arrangement and heat transfer mechanisms in superheaters.In addition to the convective heat transfer, radiative heat transfer can also take place in superheaters due to the high temperature of the flue gas. 10 The presence of oxide scale on the inner tube surface and ash accumulation on the exterior of tube surfaces make it difficult to accurately quantify heat transfer resistances along the boiler tubes at any instant of time. 11Distributedparameter dynamic models of superheaters are desirable for capturing the spatial and temporal variabilities of these resistances, as well as the effect of flow arrangement. 12A quasi-two-dimensional (2D) radiant pendant superheater model was developed by Rousseau and Gwebu 13 to quantify the distribution of flows and temperatures on the steam and flue gas sides during ramp changes.Granda et al. 14 developed a computational fluid dynamics model of a steam superheater to investigate the transient variation of transport profiles due to attemperator activation.However, the works noted above did not evaluate thermo-mechanical stress during transient superheater operation.
During the flexible operation of power plants, the superheaters usually experience elevated temperature gradients and steam pressure variability leading to creep damage and thermomechanical fatigue.Madejski and Taler 15 investigated the temperature and stress distributions in "double-omega" tubes due to the spraying of cold water in the attemperator and during sootblowing for removing ash deposits from superheater tube surfaces.Several authors have studied stress transients in superheater headers. 16,17Yasniy et al. 18 reported that there can be significant damage and multiple cracks due to the nonuniform distribution of temperature through the wall of the headers.Farragher et al. 19 compared the damage between the inner and outer surfaces and observed that the cracking of the inner side was more pronounced due to steam pressure and steam oxidation.Dynamic modeling and lifetime estimation of boiler components were presented by Benato and co-workers. 20,21Their dynamic simulation results showed that a 52.9% reduction of the high-pressure superheater header lifetime was observed when the load change was 50% faster than their reference case while achieving an increase of 35.8% life if the transient was 50% slower.Furthermore, a 7.2% reduction and 10.6% increase in the lifetime of the superheater tube bank were estimated when the equipment ramped 50% faster and 50% slower than the reference case, respectively.In their studies, Benato and co-workers used the dynamic superheater model to predict the tube wall temperatures along both the tube length and the radius.Given the significant temperature transients experienced by the tube wall, particularly during fast loadfollowing operations, it is also crucial to consider the throughwall stress profile to assess damage accumulation accurately.
Dynamic optimization has been widely used for chemical and integrated-energy processes.In a study by Kim et al., 22 a MILPbased dynamic optimization was undertaken to optimize dispatch profiles of supercritical pulverized coal (SCPC), natural gas combined cycle (NGCC) plants, and sodium sulfur batteries for varying renewable penetration levels.Pattison et al. 23 proposed an approach for optimal scheduling of the continuous processes subjected to high market variability.Dering et al. 24 presented a dynamic optimization approach for the basic oxygen furnace leading to a 6.7% decrease in the amount of carbon emission and a 2.9% decrease in the production cost compared to the base case.Yancy-Caballero et al. 25 undertook constrained optimization for the oxidative coupling of methane to improve C 2 H 4 and C 2 H 6 yield.Leipold et al. 26 focused on optimizing the periodic operation of methanol synthesis in an isobaric and isothermal fixed-bed reactor.Bremer et al. 27 undertook dynamic optimization for a fixed-bed reactor used for CO 2 methanation for determining control trajectories that can prevent distinct hot spot formation during a time-optimal start-up.
Similar to chemical and process plants, dynamic optimization can be applied to improve the operational efficiency of power plants, especially under load-following operation. 1Quantifying thermo-mechanical stresses for plant equipment and including them in dynamic optimization as constraints can help to reduce damage of critical assets.A comprehensive review of the literature on dynamic power plant modeling and optimization can be found in Alobaid et al. 28 Kruger et al. 29 developed the optimum start-up strategy without violating the stress limitation for both drum and superheater outlet headers in power plant boilers.Kim et al. 30 analyzed the thermal stress evolution in the steam drum of a heat-recovery steam generator under three different start-up strategies.Ruá et al. 31,32 developed model predictive control (MPC) to ensure that the stress constraints in the steam drum and the turbine rotor in an NGCC plant were not violated during plant transients.In our previous study, 33 dynamic optimization was undertaken for maximizing the efficiency of a coal-fired power plant while maintaining the main steam temperature during transient operation.A comprehensive dynamic optimization study was conducted by some of the authors of this paper 5 for load-following operation of an NGCC plant under stress constraints for the steam drum.It was observed that for certain desired ramp rates and stress constraints, satisfying the stress constraints is not feasible without relaxing the ramp rate.Therefore, multiobjective dynamic optimization was undertaken for minimizing ramp rate relaxation while maximizing plant efficiency under stress constraints.However, that study optimized only plant operating conditions and focused only on the steam drum.
To the best of our knowledge, there is currently no work in the literature on the optimal design and operation of superheaters under transient operations with considerations of thermal stress and fatigue damage.Optimal superheater design is critical for achieving higher power plant performance since superheaters operate at elevated temperatures and need to transfer large amounts of heat without causing a greater pressure drop than desired (only a few psi).Minimizing metal mass can be considered as an objective function for superheater design. 34,35ess metal mass not only reduces capital cost but also enables lower thermal hold-up and faster transients, making the superheater more desirable for fast load-following operations.In addition, a lower pressure drop through the superheater tubes leads to higher-pressure steam entering the steam turbine, thereby increasing the power production.Thus, it is desirable to minimize the pressure drop as a measure of the operating cost.However, the optimization objectives mentioned above are conflicting; therefore, a multiobjective optimization problem is solved in this work for optimal design of the superheater.

■ SUPERHEATER CONFIGURATION AND MODEL
A superheater is a cross-flow shell-and-tube heat exchanger where the hot flue gas flows on the shell side, while the cold steam flows inside the tubes.Because the fluid temperature changes along the flow path on both the shell and tube sides, the velocity of the fluid also changes from the inlet to the outlet.There is also spatial variation of parameters such as the heat transfer coefficients and friction factors on both the shell and tube sides.To model the spatial variation of transport variables, a two-dimensional (2D) model is constructed with discretization along the flow path and across the tube thickness.Figure 1 illustrates the schematic diagram of the typical superheater with a parallel-serpentine tube bundle configuration that is modeled in this paper.The steam enters the tubes of multiple parallel rows with each row containing multiple columns to form a tube bundle.For simplicity, as seen in Figure 1, the superheater configuration shows just two tube rows, but a bundle with 2−8 tube rows is very common.
Due to the high metal mass and heat capacity of the tube wall, its thermal holdup should not be ignored especially for a dynamic superheater model intended for simulating transient processes during load-following operations.Therefore, in addition to modeling the temperature profiles of steam and flue gas along their flow paths, spatial and temporal variations of the temperature along the tube wall thickness need to be modeled as well.Evaluation of the spatiotemporal temperature profiles is particularly important for a model intended for studying local stress evolution and its impact on equipment health.Those considerations lead to the development of a 2D partial differential equation (PDE) first-principles model.Although the flue gas flow direction and the overall steam flow direction are perpendicular to one another, the energy balances are still one-dimensional for each fluid. 36The energy conservation for the flue gas side is given by eq 1 In eq 1, ρ,v, and H ̂denote the density, velocity, and specific enthalpy of the flue gas, respectively, and q wall denotes the heat loss from the flue gas to the shell-side boundary of the tube wall.Note that for most engineering applications, the convective term (the first term on the right side of the equation) is much larger than the conduction term, and hence the conduction term is ignored in the energy equation.A similar equation is written for the steam side.Both the flue gas and steam sides are discretized.The shell-side flow is divided into multiple segments as shown in Figure 1.Due to the parallel-serpentine-tube configuration, the flow direction inside the tube is altered between two neighboring tube segments.Flow properties are calculated based on the transport variables, as appropriate, in individual discrete elements.Figure 2 illustrates the discretization of the tube wall temperature along the tube radius direction.The tube wall temperature T wall,r at radius r is calculated based on the transient heat conduction equation as shown below where α is the thermal diffusivity of the tube material.
The boundary condition at the inner tube wall, i.e., r = r i , is given by where k is the thermal conductivity of the tube metal, T f,tube is the temperature of the fluid inside the tube, and h in is the equivalent convective heat transfer coefficient between the tube inner wall

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and the steam while accounting for the heat transfer resistance due to fouling.
The boundary condition at the outer tube wall, i.e., r = r o , is given by where T f,shell is the shell-side flue gas temperature and h out is the equivalent heat transfer coefficient by considering the convective heat transfer, radiative heat transfer, if any, and the fouling resistance on the shell side.
The convective heat transfer coefficient h in,conv on the tube side is calculated as follows where k f,tube is the thermal conductivity of the fluid inside the tube.The Nusselt number, Nu tube , for turbulent flow in a tube is calculated by relating it to the Reynolds number Re tube and Prandtl number Pr tube 37 as follows The Reynolds number is given by where υ f,tube is the kinematic viscosity of the fluid inside the tube and V f,tube is the fluid velocity.
h in is calculated by where f htc,tube is a correction factor for heat transfer coefficient due to a nonuniform distribution of the flow inside multiple tubes and r foul,tube denotes the fouling resistance at the inside of tubes.If the tube-side distribution is uniform, then f htc,tube = 1 h out,conv is calculated using where k f,shell is the thermal conductivity of the fluid on the shell side and Nu shell is the shell-side Nusselt number.
where Re shell and Pr shell are the shell-side Reynolds number and Prandtl number, respectively, and f arr is a factor that depends on the tube arrangement.For the staggered tube arrangement, f arr = 1, and for the in-line tube arrangement, f arr = 0.788.Re shell is calculated as follows where υ f,shell is the kinematic viscosity of the fluid on the shell side and V f,shell is the flue gas velocity on the shell side.
When the temperature of the fluid on the shell side is significantly high (i.e., above 800 K), heat transfer through radiation on the shell side must be considered.An algebraic surrogate model for calculating gas emissivity is developed using ALAMO 39 from a narrow band high-fidelity model RADCAL, developed at the National Institute of Standards and Technology (NIST). 40The gas emissivity is expressed as a function of the mole fractions of CO 2 , H 2 O, O 2 , and N 2 , the pressure on the shell side, and the mean beam length estimated based on the tube outside diameter and pitches between neighboring tubes. 41he mean beam length is calculated based on the ratio of volume to surface area of the shell-side cavity 42 where P x and P y are the pitches (distance between two neighboring tubes) in the parallel and perpendicular directions to the shell-side flow, respectively.The gas-surface radiation exchange factor F rad for the shellside wall is calculated as 43 where f gray is the fraction of gray gas in the entire spectrum, which is calculated from the gas emissivity surrogate model, ε gas is the gas emissivity at the shell-side fluid temperature, and ε w is the emissivity of the tube wall considering any fouling.The radiation contribution to the shell-side heat transfer coefficient after linearizing the radiation heat transfer rate expression is 43 where σ is the Stefan−Boltzmann constant.Finally, by considering the fouling resistance on the shell side r foul,shell , the equivalent heat transfer coefficient on the shell side is calculated as where f htc,shell is a correction factor for convective heat transfer coefficient due to nonuniform flow distribution on the shell side.f htc,shell = 1 for a uniformly distributed flow.
For calculating the tube-side pressure drop, the Darcy friction factor is calculated based on the turbulent flow 37

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where f dp,tube is a correction factor accounting for the roughness of the tubes.The pressure drop per unit length of the tube is calculated as where ρ f,tube is the fluid density on the tube side.
For the staggered tube arrangement, the friction factor for the shell side is calculated by 44  where P y is the pitch in the direction perpendicular to the shellside flow and f dp,shell is the correction factor due to nonuniform flow distribution on the shell side.For in-line tube arrangement, the friction factor is calculated as 44

=
where P x and P y are the pitches in the parallel and perpendicular directions to the shell-side flow, respectively.The pressure drop for each tube row is calculated as 44 where ρ f,shell is the fluid density on the shell side.
■ MODELS FOR SUPERHEATER STRESS AND FATIGUE DAMAGE Stress Model.Using the classic elasticity theory, 45 thermomechanical principal stresses are calculated for a radial tube location r i .For computing principal mechanical stresses in the radial, tangential, and axial directions, the following equations are used, respectively 46 where P in , r in and P out , r out are the pressure and radius at the inside and outside surfaces, respectively.Thermal stresses are computed using the following equations 47,48 = [ ] where E is the Young's modulus, β is a linear temperature expansion coefficient, and ν is the Poisson's ratio of the material of construction.
The mean temperatures T m and T r,m are calculated using the following equations In this study, the integrals in eqs 23 and 24 are calculated by using the trapezoidal rule.The von Mises stress, which is a scalar measure capturing the overall effect of the principal stress components, is computed as follows where σ r , σ θ , and σ z are the principal structural stresses including mechanical and thermal stresses that are calculated as follows Fatigue Damage Model.The maximum number of allowable cycles is evaluated using the international standard EN 13445. 46The allowable number of fatigue cycles N is computed as where R m is the material tensile strength at room temperature.The reference stress range Δσ Rd i depends on the stress range Δσ i and is calculated as follows where K f indicates the effective stress concentration factor.K f depends on the theoretical stress concentration factor K t , the stress range Δσ Ri , the plasticity correction factors k e and k v , and the material endurance limit Δσ D .
The overall correction factor f u is dependent on the surface finish correction factor (f s ), the thickness correction factor (f e ), the temperature correction factor (f t* ), and the mean stress Industrial & Engineering Chemistry Research correction factor (f m ).Because the reference stress range Δσ Ri depends on N and N is computed as a function of Δσ Ri , an iterative calculation is used to determine N.
Dynamic Optimization.The dynamic optimization of a superheater in a subcritical pulverized coal power plant is undertaken to minimize the metal mass and/or the total pressure drop while satisfying the outlet steam temperature and the equipment stress constraint under load-following operation.In this study, a load ramp rate of 10% is evaluated, which is about the maximum rate desired in a power plant operation.For the dynamic optimization, the load is ramped down from 100, to 50% and back up to 100%.The design variables optimized for improved performance include tube thickness, tube inner diameter, tube length per segment, number of tubes in each row, and number of inlet tube rows.Single-objective optimization problems are initially solved, followed by subsequent solution of a multiobjective optimization.
Single-Objective Optimization.Single-Objective Optimization of Metal Mass.One of the single-objective optimizations is to minimize the metal mass, M metal_tube .The optimization problem is given in eq 29.It should be noted that while the normalization of the metal mass as given in the objective function J 1 is not necessary for the single-objective optimization, it is done mainly for the multiobjective optimization.
The first constraint includes the equality and inequality constraints related to the superheater process models and stress and fatigue models described before.The second constraint ensures that the heat duty of the superheater under nominal conditions, Q nom , remains within the allowable heat duty deviation α.The third constraint bounds the stress, σ, at the location with the highest stress below the upper bound σ max by considering the entire temporal profile.The fourth constraint bounds the allowable number of cycles N at the location, with the minimum number of allowable cycles to be higher than the desired minimum number of cycles by considering the entire temporal trajectory.It should be noted that satisfying the third and fourth constraints is challenging since neither the location nor the time of the maximum stress and minimum number of allowable cycles is known a priori.In addition, both location and time can vary as the design variables are optimized.The fifth constraint ensures that the outlet steam temperature remains bounded during the entire operation.It should be noted that, like the third and fourth constraints, satisfying the fifth constraint is challenging due to the uncertainty of when the outlet steam temperature may potentially violate the temper-ature constraint.Finally, the sixth constraint keeps the decision variables bounded.
Single-Objective Optimization of Pressure Drop.Another single-objective optimization is to minimize the integral pressure drop through the tube (i.e., the steam side) over the entire time of operation.The optimization problem is given by eq 30, where ΔP total is the instantaneous pressure drop.Constraints are the same as those for the previous single-objective optimization.
Multiobjective Optimization.For the multiobjective optimization, a weighted metric approach is used.A general formulation is given below If the minimization problem is completely convex, provided that the weights ω i > 0, any solution found using the weighted metric technique is a Pareto-optimal solution.
For this problem, there are two weighting factors, ω 1 and ω 2 .
The value of ω 1 is varied from 0.1 to 0.9 with an interval of 0.1.p is the metric used to measure the distance between the reference point and the objective feasible region, and it is set to 2. The resulting multiobjective optimization problem is given by In eq 32, optimal values of the objective functions from the single-objective optimizations described above are denoted by J 1 * and J 2 *, respectively.In summary, Figure 3 illustrates a general procedure for solving the multiobjective dynamic optimization in this study.
The time domain is fully discretized 49 to facilitate satisfying the constraints for stress, minimum number of allowable cycles, and the main steam temperature.However, one of the difficulties when solving the fully discretized problem is the generation of good initial guesses for all variables as the decision variables are changed.If the single-variable optimization problems fail to converge for given values of design variables, then the initialization problem is resolved and an element-by-element approach is employed to simulate the model for the entire time range.The element-by-element solution approach is similar to the typical dynamic integrator approach but adapted for a fully discretized approach.In this approach, the entire time range is divided into multiple time windows, and only 1 time window is solved at a time.The values of all variables at the end of a window used not only for initializing the appropriate variables for the next window but also as the initial guess for all variables for the next entire time window.This continued until the end of the time range is reached.The element-by-element approach is found to be highly effective in reliably solving singlevariable optimizations.
Design Constraints and Optimization.Table 1 lists the base case values and bounds for the decision variables for the design of the primary superheater.

■ RESULTS AND DISCUSSION
Simulation of an Existing Superheater.The process model for the primary superheater is developed in the Pyomobased IDAES software. 50IAPWS-95 is used for the steam properties. 51PYOMO.DAE is used for discretizing the differential equations. 52The fully discretized model yields a set of nonlinear algebraic equations, which is solved using IPOPT. 53ables 2 and 3 list the base case specifications and boundary conditions for the superheater, respectively.The base case    conditions, specifications, and boundary conditions listed in Tables Tables 1−3, respectively, are the same as those used in our earlier work. 33igure 4 shows the dynamic simulation of two load changes at ramp rates of 10%/min.The dynamic model of the power plant used for simulating the load change has been presented in our earlier works. 33,51The load is changed from the full load to 50% and maintained at 50% load for 10 min before ramping back up to the full load.Figure 4 also shows the corresponding change in the coal flow rate.The undershoot in the coal flow rate at the end of the downward ramp is due to the release of stored thermal energy in the boiler, mainly from the drum and waterwalls, and the overshoot above the steady-state value at the end of the upward ramp is because extra fuel is fired to build up the stored energy.It is obvious that the flue gas flow rate will have characteristics similar to those of the coal flow rate.
Figure 5 presents the transient response in the superheater inlet steam temperature and pressure, flue gas temperature, and flow rates of the steam and flue gas.The undershoot and overshoot in the flue gas temperatures correspond to the similar undershoot and overshoot in coal flow rates.Figure 6 presents the distributed profiles of the von Mises stress in the superheater.The highest stress is found to occur at the steam inlet location.
Impact of Different Ramp Rates on Fatigue and Creep Damages.The impacts of ramp rates on superheater fatigue and creep damages are studied by simulating 1, 3, and 5%/min ramp changes in load with the load variation as in Ma et al. 33 Figure 7 presents the effect of different ramp rates on the total  Von Mises stress at the steam inlet and outlet locations.The stress remains slightly higher at the inlet location.It is found that while the rate of change of the stress does differ with the ramp rate as expected, the highest and lowest values of stress do not differ with the ramp rate.Figure 8 shows the change in the fatigue damage, including the relative allowable number of cycles at the steam inlet and outlet locations.The relative allowable number of cycles is calculated by scaling the allowable number of cycles for any given ramp rate with respect to the allowable number of cycles at a 5%/min ramp rate.It is found that at the steam inlet location, a ramp rate of 1%/min results in approximately 5% more allowable cycles compared to a ramp rate of 5%/min.At the steam outlet location, a ramp rate of 1%/ min yields approximately 11% more allowable cycles than a ramp rate of 5%/min.
Optimization Results.Single-Objective Optimization of Metal Mass.For the single-objective optimization that seeks to minimize the metal mass, two cases are considered where the only difference is in the constraint for maximum stress, i.e., σ max .The value of σ max can vary significantly depending on the material of construction, desired lifetime, expected operational profile, level of conservatism by the utility, etc.Based on sensitivity studies, it was found that the maximum stress reached at any location at any time instant remains below 80 MPa.Therefore, in Case 1, the σ max value is set to 150 MPa, significantly higher than the maximum stress experienced at any location or time instant, serving as a case where the stress constraint remains inactive.In Case 2, σ max is set to be 75 MPa, making the stress constraint active for both single-objective and multiobjective optimizations.For both cases, the fatigue damage constraint N min is set to 0.57N min * , where N min * is the minimum number of allowable cycles obtained in the base case simulation.
Table 4 lists the results for the superheater design variables for the unconstrained (Case 1) and constrained (Case 2) optimizations as well as the base case values from Table 1.The optimized tube thickness is slightly higher for Case 2 compared to Case 1 since the stress constraint is tighter for Case 2. However, the tube thickness for both optimization cases is still lower than the thickness in the base case.Tube diameter shows the opposite trend and keeps decreasing as the stress constraint is tightened.Tube length for both Cases 1 and 2 does not change but is found to be lower than the base case.The maximum number of tubes in a row for Cases 1 and 2 does not change but is found to be higher than the base case.
Figure 9 shows the % change in the metal mass, pressure drop, and the allowable number of cycles for Cases 1 and 2 with respect to the base case.In Figure 9, a negative value indicates that the value of the corresponding variable is lower than that in the base case.The required metal mass is smaller for both Case 1 and 2 compared to the base case.However, the metal mass is reduced more in Case 1 since the maximum stress is unconstrained.In Case 1, the pressure drop and allowable number of cycles are lower than the base case, while the opposite trend is observed for Case 2. The transient profile of the steam outlet temperature, maximum stress, and pressure drop for Case 2 versus the base case can be found in the Supporting Information in Figures S1.1−S1.3.As expected, the steam outlet temperature constraint is always satisfied.It is also observed that there is a considerable transient change in the maximum stress mainly because of the transient change in the steam pressure.
Single-Objective Optimization of Pressure Drop.Table 5 lists the results for the superheater design variables when minimizing pressure drop for the same inactive stress (Case 1) and constrained active stress (Case 2) cases used in the previous

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single-objective optimization, respectively.Tube thickness in Case 2 is higher than Case 1 as the stress constraint is tightened like in single-objective optimization of metal mass; however, unlike in the single-objective optimization of metal mass, tube thickness for both Cases 1 and 2 is higher than that in the base case.Tube diameter remains the same for both Cases 1 and 2 and is about 10% higher than the base case, as would be expected.These results are in contrast to the single-objective optimization of metal mass, where the tube diameter was 5.3 and 6.6% lower for Cases 1 and 2, respectively, compared to the base case.Tube length for both Cases 1 and 2 does not change but is found to be lower than that of the base case and at the lower bound as in the single-objective optimization of metal mass.The maximum number of tubes in a row for Cases 1 and 2 does not change but is found to be lower than the base case, unlike in single-objective optimization of metal mass.Overall, these results show the opposing trends obtained in the two singleobjective optimizations for tube thickness, tube diameter, and the maximum number of tubes in a row.
Figure 10 shows the % change in the metal mass, pressure drops, and the allowable number of cycles for Cases 1−2 with respect to the base case.In contrast to the single-objective optimization of metal mass, the pressure drop is much lower than the base case and the single-objective optimization of metal mass, as would be expected.The metal mass is higher for both Cases 1 and 2 compared to the base case and the single-objective optimization of the metal mass.For both Cases 1 and 2, the allowable number of cycles is lower than the base case and obviously much lower than Cases 1 and 2 for the single-objective optimization of metal mass.In general, these results show the conflict between the objective functions for metal mass and pressure drop minimization, thus motivating the multiobjective optimization.Transient profiles of the steam outlet temperature, maximum stress, and pressure drop for the base case versus Case 2 can be found in the Supporting Information in Figures S1. 4     so is not shown here.Instead, Table 6 lists the optimal values of the decision variables for both cases when ω 1 = 0.5.Interestingly, the trends for tube thickness and tube diameter for Cases 1 and 2 are higher than the base case, as in the results for the singleobjective optimization of pressure drop, but the maximum number of tubes in a row is higher than the base case, as in results for the single-objective optimization of metal mass.Corresponding to the optimal design variables presented in Table 6 for Case 1, the pressure drop, metal mass, and allowable number of cycles have reduced by 31.5, 0.52, and 31.9%compared to the base case as shown in Figure 12.Corresponding to the optimal design variables presented in Table 6 for Case 2, the pressure drop has reduced by 37.35% compared to the base case, while the metal mass and allowable number of cycles have increased by 5.25, and 16%, respectively, compared to the base case.For both cases, the optimal results demonstrated that a significant decrease in pressure drops and an increase in the allowable number of cycles can be obtained with a smaller increase in the superheater metal mass.The multioptimization results also show that when the metal mass is decreased, the fatigue damage is increased and the allowable number of cycles is reduced; however, the pressure drop is significantly lower.Transient profiles of the steam outlet temperature, maximum stress, and pressure drop for baseline versus Cases 1 and 2 can be found in the Supporting Information in Figures S1.7−S1.9.

■ CONCLUSIONS
This study has presented an approach to the optimal design of the primary superheater for power plants increasingly subjected to load-following operations.Two single-objective optimization problems are solved: one for minimizing metal mass and another for minimizing the pressure drop.Because these two objective functions are conflicting, a multiobjective optimization problem is also solved.The same constraints for maximum stress, minimum allowable number of cycles, and main steam temperature are considered for all three optimization problems.
It is observed that even with a ramp change in the load as high as 10%, all constraints are consistently satisfied by using the design decision variables as degrees of freedom.Furthermore, a notable observation is that the stress constraint significantly affects the results from the optimization problem.For example, for the single-objective minimization of metal mass, tightening the maximum stress constraint results in a reduction in metal mass and an increase in pressure drop compared to the base case.However, the minimum allowable number of cycles is higher    when the metal mass is minimized.In addition, it is also observed that in Case 2, where the stress constraint is active, a slight increase in the metal mass leads to a decrease in the pressure drop and an increase in the minimum allowable number of cycles.For example, when the pressure drop is minimized, the pressure drop decreases by about 45%, but the metal mass increases by about 10%.Similarly for the multiobjective optimization, the pressure drop decreases by about 37% at the expense of a 5% increase in metal mass in another specific scenario.To summarize, this study demonstrates the feasibility and benefits of optimizing the design of superheaters to withstand high ramp rates without compromising stress constraints and/or the allowable number of cycles, providing a generic approach that can be applied to other hightemperature heat exchangers susceptible to thermo-mechanical damage caused by temperature and pressure fluctuations during load-following operations.

Figure 1 .
Figure 1.Schematic diagram of a typical superheater and discretization along the flow path of the shell-side fluid.

Figure 2 .
Figure 2. Schematic diagram showing the discretization of tube wall temperature along the tube radius and various variables for the tube wall model.

Figure 3 .
Figure 3. Workflow for the dynamic optimization of the primary superheater.
tube pitch between rows (i.e., longitudinal pitch parallel to the flue gas flow direction) mm 95.25 tube pitch between columns (i.e., longitudinal pitch perpendicular to the flue gas flow direction

Figure 4 .
Figure 4. Transient response for the simulated dynamics in load.

Figure 5 .
Figure 5. Transient response in steam and flue gas conditions at the inlet of the primary superheater.

Figure 6 .
Figure 6.Transient response in the von Mises stress distribution profiles at different locations in the superheater tube.

Figure 7 .
Figure 7. Sensitivity analysis of different ramp rates on total Von Mises stress at the (a) steam inlet location and (b) steam outlet location.

− S1. 6 .
Multiobjective Optimization.Figure 11 shows the Pareto plot for Case 1. Due to a slightly tighter active stress constraint for Case 2, its Pareto plot looks very similar to that of Case 1 and

Figure 8 .
Figure 8. Impact of different ramp rates on fatigue damage on (a) steam inlet location and (b) steam outlet location.

Figure 9 .
Figure 9. Percentage change in metal mass, pressure drop, and allowable number of cycles for Cases 1 and 2 with respect to the base case for the single-objective optimization of metal mass.

Figure 10 .
Figure 10.Percentage change in metal mass, pressure drop, and allowable number of cycles for Cases 1 and 2 with respect to the base case for the single-objective optimization of pressure drop.

Figure 11 .
Figure 11.Pareto curve for multiobjective optimization for the inactive stress constraint case (Case 1).

Figure 12 .
Figure 12.Percentage change in metal mass, pressure drop, and allowable number of cycles for Case 1 and 2 with respect to the base case for the multiobjective optimization corresponding to ω 1 = 0.5.
The following equation is used for calculating Nu shell

Table 1 .
Base Case Values and Bounds for the Design Decision Variables

Table 2 .
Base Case Specifications of the Superheater

Table 3 .
Steam and Flue Gas Boundary Conditions

Table 4 .
Optimal Design of Primary Superheater for the Single-Objective Optimization of Metal Mass

Table 5 .
Optimal Design of the Superheater for the Single-Objective Optimization of Pressure Drop

Table 6 .
Optimal Design of the Superheater for the Multiobjective Optimization Corresponding to ω 1 = 0.5 The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.iecr.3c02130.Additional results for base case vs Case 2 for the singleobjective optimization problems and additional results for base case vs multiobjective optimization problems (PDF) Debangsu.Bhattacharyya@mail.wvu.eduThiswork was conducted as part of the Institute for the Design of Advanced Energy Systems (IDAES) with support through the Simulation-Based Engineering, Crosscutting Research Program within the U.S. Department of Energy's Office of Fossil Energy and Carbon Management.D.B. and Q.L. acknowledge funding from the U.S. Department of Energy's National Energy Technology Laboratory under the Mission Execution and Strategic Analysis contract (DE-FE0025912) for support services through KeyLogic Systems, Inc.This report was prepared as an account of work sponsored by an agency of the United States Government.Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.The authors declare no competing financial interest.