Optimal Design of a Postcombustion CO2 Capture Pilot-Scale Plant

Article pubs.acs.org/IECR

Optimal Design of a Postcombustion CO2 Capture Pilot-Scale Plant under Process Uncertainty: A Ranking-Based Approach Sami S. Bahakim and Luis A. Ricardez-Sandoval* Department of Chemical Engineering, University of Waterloo, N2L 3G1 Waterloo, Canada ABSTRACT: This article presents a study of the effect of process uncertainty on the optimal design of a CO2 capture pilot-scale plant for coal-based power plants. The presented work employed a novel method in the optimal design of large-scale chemical processes (such as the CO2-capture plant) under uncertainty, which uses a power series expansion (PSE) approximation to the actual nonlinear process in computing the output distribution of the process constraints due to uncertainty. A ranking-based approach is employed here where priorities or probabilities of satisfaction are assigned to the process constraints considered in the analysis. In this work, uncertainty is assumed in three input variables affecting the operation of a CO2-capture pilot plant, namely, the CO2 content and the temperature and flow rate of the flue gas stream. The design of the optimal plant aims to specify the sizes of the key process units included in the CO2-capture pilot plant, such as the packed column height and diameters and heat exchanger and condenser areas, that minimize the process economics in the presence of uncertainty in the flue-gas stream conditions. The results of the study using the proposed method show that, to ensure a desired target CO2 removal rate in the presence of process uncertainties in the flue gas stream, larger designs for both the absorber and stripper towers and a higher reboiler heat duty are required. Although the present method yields larger and, thus, more expensive designs, it ensures that the environmental and operating constraints are satisfied according to the user-defined probability of satisfaction, whereas the original pilot-plant base-case design violates the target for the CO2 removal rate most of the time when operating under uncertainty.

1. INTRODUCTION The effects of greenhouse gases on global climate change, also known as global warming, have been more drastic in recent decades, causing more awareness among both scientists and the general public about their possible threats to the environment.1 Carbon dioxide (a greenhouse gas) has a significant impact on global warming2,3 and is sometimes considered the main contributor among all greenhouse gases.4 Emissions of CO2 to the environment are generated from power plants using fossilfuel combustion sources such as coal and natural gas; coalbased power plants release twice the amount of CO2 per unit of electricity generated than natural gas-based plants.5,6 Currently, there is a significant dependence on fossil fuels as a source of energy because of their availability, abundance, energy density, and existing infrastructure for distribution and delivery, making them a more reliable and economically attractive option over other relatively new alternative sources such as nuclear or renewables.7−9 Therefore, the need to develop and implement approaches that mitigate and control CO2 emissions is highly motivated by the interest in continuing to use fossil-fuel-based energy. CO2 capture and storage is considered to be an effective option for reducing the amount of CO2 released to the environment10−13 and has been implemented in various industrial processes, such as coal gasification, natural gas production, and fertilizer manufacturing.14−16 Several approaches are available for capturing CO2, including precombustion,17,18 postcombustion,19−24 and oxy-combustion.25,26 Postcombustion CO2 capture using chemical absorption with amine solvents is by far the most common and developed technique for capturing CO2 from flue gas with low CO2 concentrations. This method is preferred over the other two approaches because it can be retrofit into existing power plants without © 2015 American Chemical Society

major changes in equipment configurations that would otherwise be more costly.1,21,27,28 In chemical absorption, the CO2 in the flue gas reacts with the amine solvent to form an intermediate compound that decomposes upon the application of heat to regenerate the solvent. Monoethanolamine (MEA) is the most widely used amine solvent for this purpose because of its high reactivity with CO2.29 An MEA-based carbon-capture unit includes an absorption column to capture CO2 in the entering flue-gas stream using the MEA solvent, a stripping column with a reboiler at the bottom of the unit to heat the CO2-rich solvent for its regeneration, and a condenser typically located at the top of the stripper to recover a CO2-rich gas stream; heat exchangers are also included in the plant layout to maintain the temperature requirements for the process. Several studies have been performed to optimize the design and operation of MEAbased CO2-capture plants.27,28,30−35 The focus of those studies was to identify the optimal process operating parameters, such as the inlet concentration and temperature of the amine solvent, stripper operating pressure, and CO2 loading, and design parameters, such as the numbers of stages for the absorber and stripper columns, with the objective of minimizing an economic cost function. The resulting optimal CO2-capture plant design is expected to satisfy its process and target constraints (e.g., CO2 emissions, CO2 removal) under nominal operating conditions. However, in the presence of uncertainties Received: Revised: Accepted: Published: 3879

December March 20, March 31, March 31,

10, 2014 2015 2015 2015 DOI: 10.1021/ie5048253 Ind. Eng. Chem. Res. 2015, 54, 3879−3892

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Industrial & Engineering Chemistry Research or fluctuations in the process inputs, these designs might fail to comply with the desired targets or process constraints. In a practical context, uncertainty is almost inherent in every process. Hence, efforts to account for uncertainty at the design stage have been suggested and studied in the field of process systems engineering,36−41 as opposed to steady-state process optimization (without consideration of process uncertainty).42,43 In the context of CO2-capture plants, several studies have investigated the effects of uncertainty in the prices of key economic parameters (e.g., fuel, electricity, CO2) on the optimal planning of power generation plants coupled with CO2 capture when considering different technologies or timing of investments.44−51 Nonetheless, those studies targeted higherlevel decisions of planning, investment, and technology selection with financial and risk uncertainties, rather than the effects of process-level uncertainties on the process operation and equipment design. To the authors’ knowledge, no study in the literature has investigated the latter case. Several approaches have been proposed in the literature to account for uncertainty at the design stage of a process. Traditionally, a brute-force approach would be to overdesign the nominal steady-state process designs by a reasonable safety factor such that the process remains feasible under the uncertain conditions. However, such a practice can be costly as it can lead to overly conservative (and unnecessary) designs, especially if the critical realizations in the uncertain process variables are rare occurrences during normal plant operation. Therefore, efforts have been directed toward the development of systematic methods to address the optimal design of chemical systems under uncertainty. Stochastic programming is one such method that simulates the process model multiple times within the domains of the uncertain variables to determine their effects on the process outputs.52−54 For the case of continuous uncertain space regions, there is an infinite set of realizations for the uncertain variables, which demands the use of appropriate sampling techniques, such as the Monte Carlo (MC) sampling method, to make the problem tractable and prevent prohibitive calculations. Nonetheless, the accuracy of the solution improves with increasing sampling size, where the computational burden can sometimes limit its use for largescale systems such as CO2-capture pilot plants. Chanceconstrained programming is another promising approach that has been proposed to handle uncertainty for optimal design or planning problems.55−58 Instead of searching for an optimal solution that will satisfy its process constraints for every realization or scenario in the uncertain variables, this method requires a user-defined input of the minimum probability of satisfaction for each process constraint and seeks an optimal solution that satisfies each constraint with a probability at least equal to the user-defined value. This feature of ranking process constraints (according to their significance) by assigning high or low minimum probabilities of satisfaction allows the specification of optimal solutions subject to violations in the constraints that the user has preauthorized and agreed to be acceptable. In a practical sense, the bounds on the temperature in a catalytic reactor (which is important for the catalysts) should be defined as a constraint with a high probability of satisfaction, thus allowing few or even no violations of this constraint during operation; otherwise, exceeding the reaction temperature range often will have a direct effect on the product’s quality, which has a direct implication on the plant’s economics. Meanwhile, the level constraints of a water storage tank can sometimes be violated with a much lower impact on

the economics of the plant, and thus, assigning lower probabilities to such constraints could yield more optimal solutions. In chance-constrained programming, the expected values and variances for the objective function and constraints at a given probability limit need to be computed for processes under the influence of uncertainty. This requires the evaluation of multiple integrals that proved to be the main computational challenge for this approach. In a recent work, Bahakim et al.59 proposed a systematic ranking-based approach for the optimal design of process systems under uncertainty using a power series expansion (PSE) approximation. This PSE-based approach does not require the simulation of the nonlinear process model for each sampled realization of the uncertain variables; instead, this method makes use of a reduced PSE-model. This aspect of the proposed method makes the approach computationally attractive, especially for large-scale chemical processes. The evaluation of multiple integrals is avoided in this method. Hence, the goal of this study is to present the implementation of the ranking-based approach to search for the optimal design of a CO2-capture pilot plant under uncertainty. Because a commercial-scale CO2-capture unit for coal-based power plants has just recently been put into operation, no data are available in the open literature. Therefore, this work focuses on analyzing the effects of fluctuations in the key input variables for the optimal design of a CO2-capture pilot-scale plant for coal-based power plants. A previous study by Dugas60 on a CO2-capture pilot plant using MEA was used as the design basis for this study. The effects of uncertainty in three key process variables, namely, the CO2 content and the temperature and flow rate of the flue-gas stream, on the optimal design of the pilot plant are investigated using the PSE-based method. The conclusions drawn from the present study can be extended to larger commercial-scale CO2-capture plants. The organization of this article is as follows: Section 2 describes the methodology of the PSE-based approach in handling uncertainty for optimal process design. The process description of the CO2-capture process and the implementation and formulation of the problem are presented in section 3. Two case studies involving the optimal design of a CO2-capture pilot plant under single and multiple process uncertainties are presented in section 4. Concluding remarks are presented in section 5.

2. PSE-BASED OPTIMAL PROCESS DESIGN UNDER UNCERTAINTY In this section, the methodology recently proposed by Bahakim et al.59 for the optimal design of process systems under uncertainty is presented. A process model J describing the behavior of the system under analysis is assumed to be available for simulations and is described as follows J(d, κ , x , γ , u , δ) = 0

(1)

where d is the vector of design variables, κ is the model parameters, and x represents the state variables. The model outputs and inputs of the process are denoted by γ and u, respectively. Process uncertainty δ can be inherent in the model inputs (u′), because of fluctuations in certain input process variables such as the raw material flow rate, or in the model parameters (κ′), because of a lack of knowledge of the actual modeled process. Hence δ = [u′, κ′] 3880

u′ ∈ u , κ ′ ∈ κ

(2)

DOI: 10.1021/ie5048253 Ind. Eng. Chem. Res. 2015, 54, 3879−3892

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Industrial & Engineering Chemistry Research The uncertainties are assumed to follow known probability distribution functions (PDFs) with known distribution parameters ψ δc ∼ PDF(ψc)

c = 1, 2, ..., C

hPSE(δ) = h(δ ̅ ) + S(1)(δ − δ ̅ ) + + ... S(1) =

(3)

where each of the C uncertain variables might follow a different PDF with specific distribution parameters ψc. The description of the uncertain parameters can be obtained from process experience (e.g., from other existing similar processes) or from historical data where the PDF can be heuristically or mathematically approximated. However, if no prior information is available, Gaussian or uniform probability distributions can typically be assumed for the uncertain parameters, as such distributions fit most engineering applications reasonably well.61 However, the choice of another symmetric or nonsymmetric PDF, such as a log-normal or exponential distribution, is also possible, so this does not represent a limitation for the present method. Therefore, the present method can be applied to design new chemical plants or to retrofit existing plants. From the uncertainty description in eq 3, a large finite number N of possible realizations of the uncertainty set can be generated using appropriate sampling techniques, such as the MC sampling method, to capture the uncertainty behavior of the system. In the context of an optimization problem, the effect of uncertainty on the constraints plays a key role in determining the feasibility of the selected designs when operating under uncertainty. The process constraints are typically described as functions of the model parameters, state variables, and model inputs and outputs, as follows h(κ , x , γ , u , δ) ≤ 0

1 (δ − δ ̅ )T S(2)(δ − δ ̅ ) 2

S(2) =

∂h ∂δ ∂ 2h ∂δ2

E[δ] = δ ̅ ,

δ=δ̅ ,η

δ=δ̅ ,η

h ∈ h,

hPSE ∈ hPSE (5)

(i)

where S refers to the ith sensitivity term of the processconstraint function h; that is, S(1) and S(2) represent the Jacobian and Hessian matrices of the process-constraint function h. Note that the S(i) terms are evaluated around a nominal point specified by δ̅, which represents the nominal values of the uncertain variables, and η, which denotes the current values assigned to the decision (design) variables considered in the optimal design under uncertainty problem. Therefore, the PSE expansions are suitable approximations around that nominal point because, for a given set of values of the decision variables, the only parameters that are assumed to change are the uncertain parameters, which represent the inputs to the PSE expansion, as shown in eq 5, and are used to evaluate the probability of satisfaction of the constraints under the effects of parameter uncertainty. The reduced constraint function in eq 5 was expanded up to second order for brevity; however, it can be expanded to any higher order q as required. The sensitivity terms S(i) are computed either analytically (if such an expression is available) or using numerical approximations.62 In the latter case, the simulation of the process model J is needed because the computation of these terms requires the evaluation of the actual nonlinear process constraint h for several realizations in δ. Even though this step represents the highest computational cost for this approach, it requires only a few simulations of the process model J (depending on the numerical technique and expansion order q used in the analysis) to obtain the process-constraint expression hPSE. To determine the variability of the process constraints in the presence of process uncertainty, which follows the descriptions shown in eq 3, the analytical expression hPSE is evaluated at each of the N sampled realizations δN ∈ 9 N × C ; this returns an output vector hPSE(δN ) ∈ 9 N × 1 that captures the distribution of the process constraint due to uncertainty in δ. Other methods, such as two-stage stochastic programming approach, require N simulations of the actual process model J to obtain the output variability of the actual process constraints h due to δN, which can be computationally intensive if N is large. Regardless of the sampling size N, the presented PSE-based method is fast to evaluate and therefore computes the variability in the process constraints using the same number of simulations as employed when using the process model J but in shorter CPU times, giving this method a promising computational advantage over other methods proposed in the literature.59 Because the PSE-based process-constraint expression hPSE is an approximation, its variability or distribution hPSE(δN) is also an approximation to the actual PDF of the system. To improve the accuracy of the estimated histogram for hPSE(δN), higherorder PSE-based expressions can be used to capture the true behavior of the process, especially if the system under analysis

(4)

where h represent the vector of process (nonlinear) constraints, which can include any safety, environmental, operating, or physical limitation imposed on the process. Assessing the effects of the N different uncertainty realizations on the system outputs, and therefore on the constraints h, requires the simulation of the process model N times. However, unlike other methods that use the actual nonlinear process model J to evaluate the process constraints h for each of the N uncertain realizations, the present methodology employs a reduced model to represent the process constraints (hPSE), where the only inputs are the uncertain process variables (δ). These reduced expressions of the process constraints (hPSE) are derived using power series expansion (PSE) approximations to the actual process constraints (h). The next subsection describes the method used to obtain hPSE using the PSE approximation method. 2.1. PSE Approximation of Nonlinear Process Constraints. The key idea of this method is to compute analytical expressions for each of the process-constraint functions h due to potential realizations in δ using power series expansion (PSE) functions. Therefore, an actual nonlinear constraint function h is approximated in the present analysis by the reduced expression hPSE as follows 3881


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can be accomplished by assigning lower probabilities to these constraints. To use this ranking-based feature, the constraint condition in eq 4 is replaced by the probabilistic equivalent PSE-based approximation as follows

is highly nonlinear. However, increasing the expansion order (q) requires the evaluation of additional (higher-order) sensitivity terms, which increases the computational costs. Accordingly, there is a tradeoff between accuracy and efficiency. As a result, the selection of the expansion order (q) in the PSEbased expression is problem-specific, relying on several aspects of the system under analysis, such as the degree of nonlinearity of the system, the size of the process model, and the method used to compute the terms in the expansion (i.e., analytic or numerical). Another way of improving the accuracy of the estimated output variability hPSE(δN) is by increasing the sample size N, as this improves the accuracy of the hPSE output distribution. 2.2. Probabilistic-Based Ranking of Process Constraints. The distribution or variability of the process constraints due to process uncertainty is obtained from the set of values hPSE(δN), often depicted as a histogram (Figure 1).

P(h(κ , x , γ , u , δ) ≤ 0) ≥ Prh → P(hPSE(δN ) ≤ 0) ≥ Prh

From eq 6 P[hPSE(δN ) − ωhPSE ≤ 0] = Prh

(7)

Then, it follows that

ωhPSE ≤ 0 where the expression on the left-hand side represents the actual constraint function (h) evaluated using the process model J at a given probability of occurrence Prh. This expression is replaced in the present analysis by the corresponding PSE approximation, represented by ω h P S E . Note that ω h P S E = ωhPSE(F,hPSE(δN),Prh). Setting Prh to the extreme probability (Prh → 1) results in the worst-case-scenario approach, which often yields conservative (expensive) designs (see Figure 1).40,64 2.3. Optimal Design under Uncertainty Framework. The optimal design of a process system under uncertainty can be formulated based on the developments in the previous subsections as follows59 min ΦPSE = CAP(d, δ) + OP(κ , x, γ , u , δ)

η= [d, u]

s.t. J(d, κ , x , γ , u , δ) = 0

Figure 1. Schematic representation of the distributional analysis of the process constraints.

ωhPSE(F , hPSE(δN ), Prh) ≤ 0

hPSE ∈ hPSE

dl ≤ d ≤ du

In the present approach, a user-defined value for the minimum probability of satisfaction, Prh, is assigned to each process constraint h in the problem. This user-defined parameter defines the importance or weight of each constraint in the process design. The process-constraint value that covers (100 × Prh)% of the distribution under uncertainty shown in Figure 1 is called the extreme value, ωhPSE, at the specified probability limit and is calculated as follows63

ul ≤ u ≤ uu (8)

This framework minimizes the objective cost function ΦPSE, typically described in terms of the capital (CAP) and operating (OP) costs, by selecting feasible process designs d and operating conditions or model inputs u. The process constraints are satisfied with individual minimum probabilities of satisfaction Prh in the presence of uncertainty, thus making the approach a ranking-based method; that is, different Prh values can be assigned to each constraint considered in problem 8. Figure 2 summarizes the main features of the proposed approach in comparison to the traditional stochastic programming method. The most computationally demanding part of each algorithm is represented by the blocks within the dashed box in Figure 2, which requires the simulation of the actual nonlinear process model J. Whereas the stochastic programming approach (Figure 2b) requires of N simulations of the actual nonlinear process model J, each corresponding to a particular realization of the uncertain parameters, the present PSE-based approach requires only a few simulations of the process model J to calculate the sensitivity terms S(i) in eq 5 that are required to build the PSE functions hPSE and ΦPSE (Figure 2a). For example, only two simulations per uncertain parameter are needed to define first-order PSE functions for hPSE and ΦPSE. To obtain an accurate representation of the distribution of constraints and the cost function, a large number of realizations of the uncertain parameters (N) must be

Prh = F(ωhPSE , hPSE(δN )) = P(hPSE ≤ ωhPSE) ωhPSE = {ωhPSE : F(ωhPSE , hPSE(δN )) = Prh} = F −1(Prh, hPSE(δN ))

(6)

where F is the cumulative probability function of the processconstraint function. If the extreme value ωhPSE for a given probability limit Prh satisfies the restriction condition in eq 4, then the selected process design complies with that process constraint a minimum of (100 × Prh)% of the time for all possible realizations of the uncertainty (δ). For critical constraints such as those that affect the safe operation of the process, higher probabilities close to unity (Prh → 1) can be assigned to ensure the feasibility of the process over a larger uncertainty space range. On the other hand, with less critical (or low-ranked) constraints, the choice of probability limits is an engineering decision that depends on how much violation of this constraint is acceptable and how much savings or profits 3882


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Figure 2. Schematic representation: (a) PSE-based approach; (b) traditional stochastic programming method.

evaluated. As described in section 2.1, setting N to a large value is critical and can result in intensive calculations in the stochastic programming approach because the evaluation of each realization might be computationally intensive. However, the key advantage of the present approach is that the uncertain parameters δ are the only inputs to the PSE functions hPSE and ΦPSE, and therefore, these functions can be evaluated at relatively low cost, thus allowing the use of large sample sizes (N) in the PSE-based approach. Accordingly, N can be set to large values in the present approach (e.g., 1 × 105 or 1 × 106) without significantly affecting the overall computational costs. Therefore, the present ranking-based approach is computationally attractive and suitable for addressing the optimal design of large-scale systems such as CO2-capture plants. As shown in Figure 2a, the evaluations of hPSE and ΦPSE at a user-defined probability limit Pr determine the feasibility of the current design, specified by the current set of values in the decision variables. If the constraint functions hPSE are feasible and a stopping criterion, specified by the optimization algorithm used to solve the optimization formulation 8, is met, then STOP, an optimal design has been identified; otherwise, the optimization algorithm will select new values in the decision variables and the entire procedure is repeated. More details about this methodology can be found elsewhere.59

Figure 3. Schematic diagram of the main units of a typical aminebased carbon-capture unit.

of the absorber column, selectively absorbing CO2 from the flue gas. The treated flue gas leaves the top of the absorber column and is discharged from the process in the vent gas stream; the bottoms of the absorption process represent rich amine solvent with all the absorbed CO2. This rich amine solvent is then preheated in a cross heat exchanger using the recycled lean amine solvent stream coming from the stripping section of the plant. The heated rich amine stream enters the stripper column for solvent regeneration (removal of absorbed CO 2 ). Desorption of CO2 from the amine solvent is an endothermic process, requiring additional heat supplied by the reboiler steam unit located at the bottom of the stripper column (Figure

3. CO2-CAPTURE PROCESS DESIGN PROBLEM Figure 3 presents a schematic diagram of a typical amine-based postcombustion CO2-capture unit, consisting mainly of an absorber and a stripper column with the required heating and cooling equipment. The flue gas enters through the bottom of the absorption column and comes in contact with the lean amine solvent (such as MEA) flowing downward from the top 3883


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Industrial & Engineering Chemistry Research 3). Desorbed CO2 leaves the top of the stripper in a vapor stream, which is then passed through a reflux condenser to obtain a CO2-rich product gas. On the other end, regenerated

Table 1. Validation of the Developed Pilot-Plant Model in Aspen HYSYS pilot-plant model

lean amine solvent is cooled before being recycled back to the absorber column to remove the incoming CO2 in the flue-gas stream. Because solvent is lost in the regeneration process, makeup streams consisting of water and MEA are needed to maintain the operation of this plant. A previous study by Dugas60 on a CO2-capture pilot plant using MEA was used as the design basis for this work because of the availability of data on the design and operating conditions. In addition to the process flowsheet reported by Dugas,60 the present work adds a condenser unit (as shown in Figure 3) at the top of the stripper column to represent the

temperature (K) molar flow rate (mol/s) mole fractions CO2 H2O N2

Flue-Gas Flow Rate 319.70 4.01

0.175 0.025 0.8 Absorber height (m) 6.1 internal diameter (m) 0.43 temperature (K) 325 pressure (kPa) 102 Stripper height (m) 6.1 internal diameter (m) 0.43 temperature (K) 358 pressure (kPa) 160 Process Variables reboiler temperature (K) 386.9 reboiler pressure (kPa) 160 condenser temperature (K) 314.3 condenser pressure (kPa) 159.5 CO2 recovery (mol %) 95.04 CO2 product (mol %) 95 lean solvent temperature (K) 314 vent gas CO2 content (mole 0.0010 fraction)

actual operation of a CO2-capture plant. The steady-state process was modeled in Aspen HYSYS using the base-case operating conditions of the pilot-plant process data reported by Dugas.60 A rate-based model was employed to model the absorption/stripping columns, as opposed to an equilibriumbased model. The assumptions of theoretical stages and phase equilibrium are insufficient to describe the behavior of the absorption process where reactions are taking place inside the packed column. Thus, the rate-based method, which uses a reaction mechanism model, has gained more acceptance over

specification

source

319.71 4.01

ref 60 ref 60

0.175 0.025 0.8

ref 60 ref 60 ref 60

6.1 0.43 314−329 101.3−103.5

ref ref ref ref

60 60 60 60

6.1 0.43 350−380 159.5−160

ref ref ref ref

60 60 60 60

383−393 160 312−315 159 95.9 95 312.8 0.0055−0.0085

ref ref ref ref ref ref ref ref

10 10 11 11 11 11 11 10

ΦCC = Cabs + Cstrp + C HX + Ccond + Creb

traditional equilibrium-based approaches4,5,65 and was employed in the present work. The mechanism that describes the reaction between CO2 and MEA used in the present model is the zwitterion mechanism, which is the most accepted kinetic model for the absorption of CO2 in aqueous MEA.66 For this model, the kinetic data presented in refs 67 and 68 were used.

⎛ M&S ⎞ 1.066 ⎟D Cabs = 290.82 × ROR⎜ Habs 0.82 ⎝ 280 ⎠ abs ⎛ M&S ⎞ 1.066 ⎟D Cstrp = 290.82 × ROR⎜ Hstrp 0.82 ⎝ 280 ⎠ strp ⎛ M&S ⎞ 0.65 ⎟A C HX = 97.068 × ROR⎜ ⎝ 280 ⎠ HX

The nonrandom two-liquid (NRTL) model was implemented as the base equation of state for this process, whereas the Kent−Eisenberg thermodynamic model was used for the aqueous amine solutions. Table 1 presents the validation of the developed plant model in Aspen HYSYS. As shown in this

⎛ M&S ⎞ 0.65 ⎟A Ccond = 97.068 × ROR⎜ ⎝ 280 ⎠ cond ⎛ M&S ⎞ 0.65 ⎟A Creb = 97.068 × ROR⎜ ⎝ 280 ⎠ reb + 0.42857(Q reb/ΔH vap)Csteam

table, the developed model is in reasonable agreement with the experimental data reported in the literature for this pilot-scale CO2-capture plant.10,11,60 Note that some of the operating conditions and equipment sizes were compared with those in other references in the literature because those parameters were

(9)

where the capital costs include the costs of the main process equipment, namely, the absorber (Cabs), stripper (Cstrp), cross heat exchanger (CHX), and condenser (Ccond), whereas Creb denotes the capital and operating costs associated with the reboiler. In this work, only the cost of the reboiler heat duty is considered in the operating costs because the heating consumption for solvent regeneration dominates all other operating costs.7 The above costs are annualized using a 20% rate of return (ROR = 0.2). M&S is the Marshall & Swift equipment cost index (1536.5 for the fourth quarter of 2011).70The diameter D (m) and height H (m) are used to calculate the cost of the columns, whereas the heat-transfer area A (m2) determines the cost of the heating equipment. The reboiler operating cost is calculated from the cost of steam

not reported by Dugas.60 3.1. Objective Function. The aim of this work is to optimize the design of the CO2-capture pilot plant based on an economic objective function. The annualized objective function (ΦCC) for this plant is defined in terms of the capital and operating costs calculated using Guthrie’s69 correlations as follows 3884


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Industrial & Engineering Chemistry Research (Csteam = $0.003/kg)1 needed to provide the necessary reboiler duty Qreb (kW). ΔHvap is the heat of vaporization of water (2257 kJ/kg). 3.2. Process Constraints. The optimal process design of a CO2-capture plant is subject to operating and performance constraints that need to be satisfied in the presence of process uncertainties. The percentage of CO2 removed or captured (φ) from the flue-gas stream is a metric typically used to measure the performance of these plants and is usually expected to be high enough for the process to be economically viable, with a recent study showing a 95% CO2-capture rate to be near optimal.7 The percentage of CO2 captured, φ, is defined as follows: φ=1−

number of moles of CO2 in vent gas number of moles of CO2 in flue gas

were equally spaced around the nominal (mean) values specified for these uncertain parameters. Once calculated, the PSE function hPSE was built and used to perform the uncertainty propagation on the process constraints; that is, the PSE-based function hPSE was simulated using N MCsampled realizations of the uncertain variables. The simulation results were then used to build a histogram of the distribution of each process constraint. Using a user-defined probability value Prh, along with the resulting histograms, the extreme value ωhPSE for each constraint was computed as shown in Figure 1. For example, the PSE-based constraint formulation for constraint 11 is P(0.95 − φ ≤ 0) ≥ Prconstraint 11 → ωconstraint 11 ≤ 0

The rest of the constraints were formulated in the same fashion and are not shown here for brevity. 3.3. Optimization Variables. The set of design and operating variables for the CO2-capture (CC) plant (ηCC) that were considered included the heights and diameters of both packed columns, the heat-transfer areas of the cross heat exchanger and condenser, and the heat duty and heat-transfer area of the reboiler

(10)

According to the optimization framework shown in problem 8, this performance metric is used as a process constraint that targets the optimal design to achieve at least 95% CO2 capture 0.95 − φ ≤ 0 (11) The CO2 product stream leaving the top of the stripper is desired to have a high concentrations of CO2; hence, a minimum of 95% CO2 purity (ς) in the product stream is defined as the constraint 0.95 − ς ≤ 0 (12)

d CC = [Habs , Hstrp , Dabs , Dstrp , AHX , Acond , A reb] uCC = [Q reb] ηCC = [d CC, uCC]

In addition to the constraints in eqs 11 and 12, the following operating constraints on the temperature in the reboiler and the lean solvent entering the absorber are included to ensure the feasible operation of this process

383 − Treb ≤ 0

(13)

Treb − 393 ≤ 0

(14)

313 − Tlean ≤ 0

(15)

Tlean − 315 ≤ 0

(16)

(17)

(18)

The base-case design and operation of the CO2-capture plant are presented in Table 2.10,11,60 Table 2. Base-Case Plant Design and Optimal Steady-State Plant Design (Scenario A) base-case design10,11,60 Decision Variables 153.6 reboiler duty, Qreb (kW) absorber height, Habs (m) 6.1 absorber diameter, Dabs (m) 0.43 stripper height, Hstrp (m) 6.1 stripper diameter, Dstrp (m) 0.43 heat-transfer area, AHX (m2) 22.47 heat-transfer area, Acond (m2) 14.40 Annualized Costs CAP ($/year) 4.66 × 104 OP ($/year) 6.17 × 103 total costs ($/year) 5.27 × 104

When the bottom stream of the stripper in the reboiler is heated to regenerate the amine solution, degradation of the MEA solvent can occur at high temperatures.14 Hence, constraints 13 and 14 are aimed at maintaining the operating temperature in the reboiler Treb within the range 383−393 K.10,11 The temperature of the lean amine solvent entering the absorber has a direct effect on the amount of CO2 captured,71 and thus, its operating temperature is maintained at approximately 314 K.72 To ease the optimal search, the formulation in eqs 15 and 16 allows for a deviation of 1 K from the target value of 314 K. To implement the ranking-based optimization framework proposed above, the constraints considered in the CO2-capture pilot plant were reformulated using the PSE-based approach shown in eq 7. Given the order q of the PSE approximation, the PSE-based function hPSE is constructed first using the actual nonlinear CO2-capture pilot-plant model described above. The sensitivity coefficients, namely, S(i) in eq 5, are the key parameters required to build the PSE function hPSE at each step in the optimization algorithm. In this work, these coefficients were calculated using centered finite differences. That is, for each set of values specified for the decision variables by the optimization algorithm, depending on the order q specified for the PSE function, the process model J was simulated for a set of forward and backward values in the uncertain parameters that

scenario A 172.12 6.1 0.3005 3.05 0.3011 19.80 11.10 3.58 × 104 7.42 × 103 4.33 × 104

The cost of this base-case design was evaluated using the capital and operating cost functions in eqs 9. Additional equipment specifications and operating conditions of the CO2capture pilot plant can be found elsewhere.10,11,60 In the next section, the application of the present approach to address the optimal process design and operation of the CO2-capture pilot scale plant is presented. The optimization problems formulated in this study were solved using fmincon, which is a numerical subroutine available in MATLAB that implements a modified version of the interior-point optimization algorithm.73,74 In this work, we implemented the PSE approximations and optimization in MATLAB, whereas the actual nonlinear process model was implemented in Aspen HYSYS. The framework of data linking between MATLAB and Aspen HYSYS is illustrated in Figure 4, where the dashed lines 3885


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optimal design specified for this scenario was about 18% lower in total costs than that estimated with the base-case design. 4.2. Scenario B: Uncertainty in the CO2 Composition of the Flue-Gas Stream. This scenario aimed to search for the optimal design of a CO2-capture pilot plant that remained feasible in the presence of uncertainty in the CO2 content of the flue-gas stream. For the purpose of this study, the uncertainty in this input variable was assumed to follow a Gaussian (normal) distribution with the following mean and standard deviation %CO2 ∼ N (%CO2 , 0.175 mol %) Figure 4. Data communication framework between MATLAB and Aspen HYSYS used in this work.

As described in section 2.2, the ranks and the corresponding probabilities of satisfaction were assigned based on the importance (relevance) of each of the constraints considered for the optimal design; that is, high probability limits (high ranks) were set for those constraints that were more critical to the process safety and economics of the plant. Therefore, the selection of ranks is problem-specific. Accordingly, the probability limits specified for the different scenarios presented in this study were arbitrarily chosen and reflect typical values that can be assigned to the probabilities of satisfaction of the constraints during the optimal design of a chemical system under uncertainty. Based on the ranking-based approach, the user-defined minimum probability of satisfaction Prh was set to 85% for each of the constraints as shown in Table 3.

represent the data communication between the two platforms. Most of the numerical calculations in the algorithm were executed in MATLAB with the PSE-based model equations used by the optimizer to obtain the evaluations of the cost functions and process constraints. However, with each set of design variables analyzed at each optimization step, the nonlinear Aspen HYSYS model was simulated several times (according to the PSE order selected) to obtain estimates for the process outputs, which were used in the sensitivity analysis to compute the PSE-based models shown in eq 5. The studies presented in the next section were performed on an Intel Core i7 3770 CPU at 3.4 GHz (8 GB of RAM).

Table 3. Input Probability Limits for the Process Constraints probability of satisfaction (%)

4. RESULTS AND DISCUSSION In this section, the formulation proposed to address the optimal design of the CO2-capture pilot plant was tested under various scenarios. In this work, uncertainty was assumed in three input variables, namely, the CO2 content of the entering flue gas (% CO2) and the temperature (Tin) and flow rate (Fin) of this stream. The results obtained for each scenario considered are presented next. 4.1. Scenario A: Steady-State Optimization without Uncertainty. The first scenario considered the design of the CO2-capture pilot plant under the assumption of perfectly known process parameters; that is, all three input uncertain variables were assumed to be perfectly known and equal to their nominal steady-state values

constraint eq eq eq eq eq eq

11 12 13 14 15 16

scenario B

scenario C

scenario C1

scenario D

85 85 85 85 85 85

85 95 75 75 90 90

85 60 75 75 90 90

95 85 75 75 90 90

This means that the optimal design will need to satisfy each constraint 85% of the time or more when subjected to the process uncertain description shown in eq 20. As discussed in section 2, the method used in this work approximates the actual distribution of the process constraints due to the realizations of the uncertain process variables using a qth-order power series expansion (PSE). Increasing the expansion order q improves the distribution approximation and must be done when dealing with highly nonlinear systems. However, the higher the expansion order, the more computationally intensive the problem becomes. Thus, an increase in the expansion order is justified only if it yields a significant improvement in the resulting probability distribution of the process constraints. To illustrate the convergence characteristics of the PSE-based method and to also select the order q to be used in this problem, the present scenario was solved using different expansion orders (Table 4). Figure 5 shows the convergence characteristics of the distribution for different orders of the PSE expansion, namely, from q = 1 to q = 6. Whereas Figure 5 might suggest that q = 6 provides the best approximation, Table 4 shows that the optimal solution converges at or near q = 3, where the maximum difference with the solution for q = 6 in both the process design and the plant costs is less than 1%. As shown in Table 4, the computational effort for solving the

%CO2 = 17.5 mol % Tin = 319 K Fin = 4.01 mol/s

(20)

(19)

where the overbars denote the nominal values of those variables. As shown in Table 2, the optimal design obtained for scenario A was one-half the height of the stripper specified for the base-case design; also, the diameters of both packed columns were slightly smaller than those specified in the basecase design. Both the heat exchanger and the condenser were also slightly smaller than in the actual design of the pilot plant specified by Dugas.60 To maintain the performance specifications and still satisfy the plant process constraints, a higher reboiler duty was required for use with the smaller equipment. This tradeoff, namely, a higher reboiler duty for lower equipment sizes, resulted in a more economical design because the plant’s capital costs dominate the process economics. Thus, although the present scenario had higher operating costs, the 3886


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Industrial & Engineering Chemistry Research Table 4. Optimal Steady-State Plant Designs under Uncertainty: Scenarios B, C, and D scenario B q=1

q=2

q=3

Qreb (kW) Habs (m) Dabs (m) Hstrp (m) Dstrp (m) AHX (m2) Acond (m2) Areb (m2)

184.50 6.1 0.3950 3.05 0.3150 10.80 19.80 28.96

195.50 6.1 0.3794 5.3375 0.4322 10.83 19.80 30.69

194.25 6.1 0.3401 5.3375 0.6365 10.86 20.39 30.49

CAP ($/year) OP ($/year) total ($/year) CPU time (h)

4.23 × 104 7.42 × 103 4.98 × 104 1.489

4.61 × 104 7.86 × 103 5.40 × 104 2.031

4.92 × 104 7.81 × 103 5.70 × 104 2.934

q=4

q=5

Decision Variables 194.53 197.29 6.1 6.1 0.3390 0.3345 5.3375 5.3375 0.6377 0.6382 10.83 10.83 20.39 20.39 30.54 30.97 Costs 4.93 × 104 4.93 × 104 3 7.82 × 10 7.93 × 103 5.71 × 104 5.72 × 104 3.832 5.089

q=6

scenario C

scenario C1

scenario D

q=3

q=3

q=4

196.15 6.1 0.3371 5.3375 0.6379 10.82 20.39 30.79

180.50 6.1 0.3900 6.1 0.8722 10.89 20.42 28.34

182.62 6.1 0.3100 6.1 0.8100 10.89 20.43 28.67

252.00 7.625 0.8370 6.1 0.5550 10.89 21.25 39.56

4.93 × 104 7.88 × 103 5.72 × 104 6.394

5.61 × 104 7.26 × 103 6.33 × 104 2.943

5.33 × 104 7.34 × 103 6.07 × 104 2.878

6.34 × 104 1.01 × 104 7.35 × 104 3.901

Figure 5. PSE fitting for the distribution of the constraint on the CO2-capture rate using different expansion orders.

optimal design problem is directly correlated with the expansion order used. Therefore, q = 3 is sufficient to yield good accuracy at lower computational costs (23% faster than q = 4 and 54% faster than q = 6). The sole purpose of the present analysis was to illustrate the convergence properties of the PSEbased approach proposed in this work. It is not expected that the user will perform this analysis in practice to select the order of the PSE model. Although the order of the PSE can be selected based on the size of the problem under consideration, the desired accuracy of the results, and the computational resources available (i.e., as was done here), methods such as Akaike’s information criterion (AIC),75 Rissanen’s minimum description length (MDL) criterion, 76 or the Bayesian information criterion (BIC)76 can also be used to select the most compact PSE, that is, the PSE model with the lowest number of coefficients that provides a suitable description of the process. Implementation of these methods for the selection of the number of terms in the PSE is expected to provide

sufficiently accurate results at the lowest possible computational cost. As shown in Table 4 (scenario B, q = 3), uncertainty in the CO2 composition of the flue-gas stream affects the optimal design, requiring larger columns (absorber diameter and stripper height and diameter) as well as a larger heat duty than required for scenario A. Because of the presence of uncertainty, there will be instances where the CO2 composition in flue gas will be higher than its nominal value, which demands larger columns and a higher reboiler duty to capture the extra CO2 contained in the flue-gas stream and to regenerate the MEA from the rich amine solvent stream. As a result, this scenario yielded optimal designs that were 5% and 27% higher in operating and capital costs, respectively, than those obtained from the scenario A design; however, the scenario B design satisfied the process constraints under uncertainty in CO2 content at least 85% of the time, which is the minimum probability of constraint satisfaction specified for this scenario 3887


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(eq 11). The optimal design for this scenario had a larger absorber diameter and stripper height than that obtained for the scenario B design; nevertheless, the stripper diameter was the design parameter that changed the most, with a 37% increase in size with respect to that of the scenario B design. Even though the larger-sized plant required a lower heat duty in the reboiler unit during the regeneration of the amine solvent, the capital costs associated with the size of the columns (diameter and height) dominated the process economics and thus resulted in an overall increase of 10% in the total costs for this scenario as compared to the scenario B design. To provide further insight, the sensitivity of the design to changes in the probability of satisfaction of one of the constraints was considered for the present scenario. As shown in Table 3, scenario C1, the probability of satisfaction of the minimum CO2 product purity constraint was changed from 95% to 60%, whereas the rest of the constraints remained at the same limits as specified for scenario C. This constraint is key for the feasible operation of this process because a high CO2 product purity is needed (typically above 95%) for downstream operations, namely, compression, transportation, and sequestration of the CO2. As shown in Table 4, scenario C1, a low sensitivity of the probability of satisfaction for this constraint to the overall design of the CO2-capture plant was observed. The optimal design obtained for this scenario specified slightly smaller diameters for the stripper and absorber columns than obtained for a higher probability of satisfaction of the CO2 product purity constraint (scenario C). Conversely, a slightly higher heat duty and a larger heat-transfer area for the reboiler unit were specified for scenario C1, whereas the heat-transfer areas for the condenser and the heat exchanger units remained unchanged. These changes resulted in an annual cost that was approximately 1% lower than that obtained for scenario C. 4.4. Scenario D: Multiple Process Uncertainties. Scenario D extended scenario B and considered the simultaneous occurrence of three uncertain realizations in the flue-gas stream, namely, the CO2 content, flow rate, and temperature. The uncertainty description for this scenario is defined as

(see Table 3). Process constraint 11, which is associated with the percentage of CO2 captured, φ, was found to be the only active constraint for the solution of scenario B. The optimal design obtained for scenario B was validated by performing 1000 simulations of the actual plant model using sampled uncertain realizations in CO2 content for each simulation. As shown in Figure 6a, the scenario B design was able to satisfy

Figure 6. Frequency histograms for the constraint on the CO2-capture rate under single uncertainty for the (a) scenario B design and (b) the base-case design.

active process constraint 11 according to the user-defined minimum probability limit, Prh = 0.85. On the other hand, Figure 6b shows that the original base-case design was inoperable under uncertainty in CO2 content because process constraint 11 was found to be violated almost 82% of the time for that design. The scenario A design yielded similar high violations in active constraint 11 when operating under the uncertainty description provided in eq 20, and it is not shown here for brevity. Note that the areas of the cross heat exchanger and condenser were not significantly affected by the presence of uncertainty in this scenario. 4.3. Scenario C: Ranking-Based Designs. In this scenario, the ranking-based feature of the optimal process design approach described in section 2 was explored for this case study. Accordingly, each process constraint was assigned to different probability limits Prh, as shown in Table 3. Instead of the assumption of a Gaussian distribution as in eq 20, a uniform distribution was assumed for the uncertain parameters in this scenario. Thus, the uncertainty in CO2 content was assumed to follow a uniform distribution with upper and lower bounds defined as follows %CO2 = %CO2 ± 0.21 mol %

%CO2 ∼ N (%CO2 , 0.175 mol %) Tin ∼ N ( Tin , 16K ) Fin ∼ N ( Fin , 0.2 mol/s)

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The uncertainty descriptions used in the present study were defined according to reasonable assumptions that were based on experimental data reported in the literature for typical coalfired power plants. For example, the uncertainty description specified for the CO2 composition in the flue-gas stream was defined on the basis of the typical variability observed for this process variable (i.e., approximately ±1%).77 The input probability limits assigned for the process constraints are listed in Table 3. Because of the increased degree of nonlinearity from the interaction of the simultaneous occurrence of multiple uncertainties, an order of q = 4 was found to be suitable to obtain reasonably good approximations to the output distributions of the process constraints. Also note in Table 3 that a higher probability limit (compared to the previous scenarios) of 95% was assigned to the active constraint in eq 11. This aspect, along with the simultaneous occurrence of multiple process uncertainties, made this scenario even more challenging and computationally demanding. The simultaneous occurrence

(21)

Based on the results from the previous scenario, the order in the PSE expansion was set to q = 3 for this scenario. Table 4, scenario C, shows the optimal design obtained for this scenario. With a uniform (rather than Gaussian) probability distribution describing the uncertainty, a more conservative (expensive) design was obtained; this was mainly due to the 16% increase in the plant’s capital costs with respect to the scenario B design. The ranking-based structure of the input probability limits (shown in Table 3) did not change the active constraint of the problem, which remained the percentage of CO2 captured, φ 3888


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though additional uncertain parameters were considered in this scenario.

of multiple process uncertainties can lead to higher nonlinearities in the output of the process constraints and, thus, can require higher PSE orders. From the computational point of view, this scenario required the computation of higher-order sensitivity terms of the process constraints with respect to three different uncertain variables, as opposed to just one as in the previous scenarios. In addition, the higher probability limit assigned to the active constraint in this scenario meant that the optimal design had to satisfy the same constraint more frequently while subjected to the simultaneous occurrence of multiple uncertainties. The optimal design obtained for scenario D is included in Table 4 (scenario D). For this scenario, a 25% increase in the absorber height and a more than 100% increase in the diameter compared to the values reported for scenario B were specified. Likewise, the overall volume of the stripper column was reduced by about 13% because of its smaller diameter and only slight increase in height, when compared to the scenario B design. As shown in Table 4, scenario D, a 30% increase in the reboiler heat duty compared to that obtained for scenario B was needed to achieve a feasible design. Overall, both the operating and capital costs were higher for this scenario, which can be directly justified by the need to accommodate additional uncertainties; also, the present scenario was required to meet a higher demand of satisfying the active constraint with a minimum probability of 95% (rather than 85% as in the previous scenarios). Moreover, the areas of the cross heat exchanger and condenser were not significantly affected by the presence of multiple uncertainties. However, a higher heat-transfer area was required for the reboiler unit to maintain the operation within its feasible limits. Although the plant’s costs for this scenario were 22% more than those obtained for scenario B, this scenario specified a CO2-capture plant that can meet the desired specifications at the predefined minimum probability of satisfaction while using the uncertainty descriptions given in eq 22 (see Figure 7). Because a higher

5. CONCLUSIONS The optimal design of a CO2-capture pilot plant in the presence of process uncertainty was studied in this work. The present work employed a recently developed method for the optimal design of large-scale chemical processes under uncertainty that uses a power series expansion (PSE) approximation of the actual nonlinear process to obtain the output distribution of the process constraints. The need to incorporate uncertainty in the optimal design procedure was justified by the fact that process constraints are usually violated more often for those designs that did not consider the effects of process uncertainty at the design stage. The ranking-based approach presented in this work allows the user to specify how often each process constraint must be satisfied (under process uncertainty) according to its significance to plant performance or safety. At steady state, without consideration of uncertainty, the optimal feasible design specified an absorber that is twice the height of the stripper column. In the presence of uncertainty in the CO2 content of the flue-gas stream, the size of the stripper column (both diameter and height) increased significantly to satisfy the process constraints; the absorber size remained unchanged. The reboiler heat duty, which is an operating variable in the stripping section of the plant, was also higher when uncertainty in the CO2 content of the flue-gas stream was considered. With the introduction of more uncertainties in the flue-gas input stream, the plant’s absorber column increased more than its stripper. Although the addition of uncertainties required a higher reboiler heat duty, and therefore higher operating costs, the areas of the cross heat exchanger and condenser did not seem to be affected by the process uncertainty considered in this analysis. The optimal designs obtained under uncertainty generally yielded larger plants and required more utility (i.e., reboiler duty). As a result, these designs were more expensive than the steady-state design (without considering uncertainty), with higher operating and especially capital costs, where the latter represents the dominant term in the economic cost function. However, these designs were found to satisfy the process performance and operating constraints under uncertainty in the flue-gas stream. Although the present method yielded larger and therefore more expensive designs, it ensured that the environmental and operating constraints were satisfied according to the user-defined probability of satisfaction, whereas the original plant base-case design did not meet the target for the CO2 removal rate most of the time when operating under uncertainty. Therefore, the designs presented in this study will potentially lead to economic savings because the plant’s CO2 removal rate might not need to be reduced, or the plant itself might not need to be shut down, when changes in the conditions of the flue-gas stream occur. Instead, the proposed designs will ensure that the plant can operate continuously at its design specifications because it can accommodate the potential changes that can occur in the operation of a fossil-fired power plant due to varying changes in the electricity demands.

Figure 7. Frequency histogram for the constraint on the CO2-capture rate under multiple uncertainty for the scenario D design.

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expansion order (q = 4) was used in this scenario, a higher CPU time was needed to solve the optimal design problem under uncertainty than for scenarios B and C, for which a lower order (q = 3) was employed. Note that the CPU time for this scenario was comparable to that for scenario B with q = 4, even

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the financial support provided by the United Mining Investments Co. (UMIC) to carry out the research work presented in this article.

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NOMENCLATURE C = number of uncertain parameters d = design variables dl,du = lower and upper bounds, respectively, on design variables F = cumulative distribution function h = process constraints J = steady-state process model N = number of sampled realizations in system’s uncertainty Prh = probability of satisfaction of constraint h S(i) = ith-order sensitivity term u = model inputs ul,uu = lower and upper bounds, respectively, on model inputs u′ = uncertain model inputs x = state variables δ = system’s uncertain variables γ = model outputs η = set of decision variables κ = model parameters κ′ = uncertain model parameters ψc = probability distribution parameters ωh = extreme possible value at user-defined Prh

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