Modeling, Simulation, and Optimization of Postcombustion CO2

Article pubs.acs.org/IECR

Modeling, Simulation, and Optimization of Postcombustion CO2 Capture for Variable Feed Concentration and Flow Rate. 2. Pressure Swing Adsorption and Vacuum Swing Adsorption Processes M. M. Faruque Hasan, Richard C. Baliban, Josephine A. Elia, and Christodoulos A. Floudas* Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, United States ABSTRACT: This paper reports studies on CO2 capture technologies and presents the mathematical modeling, simulation, and optimization of adsorption-based process alternatives, namely, pressure swing adsorption (PSA) and vacuum swing adsorption (VSA). Each technology includes feed dehydration, capture of at least 90% of CO2 from the feed, and compression to almost pure CO2 for sequestration at 150 bar. Each process alternative is optimized over a range of feed CO2 compositions and flow rates. A superstructure of alternatives is developed to select the optimum dehydration strategy for feed to each process. A fourstep process with pressurization, adsorption in multiple columns packed with 13X zeolite, N2 purging, and product recovery at moderate to low vacuum is configured. A nonlinear algebraic and partial differential equation (NAPDE) based nonisothermal adsorption model is used, which is fully discretized and solved via a kriging model. Explicit expressions for costs as functions of feed flow rate and CO2 composition are also developed for the PSA- and VSA-based CO2 capture and compression for the first time. Furthermore, a cost-based comparison of four leading CO2 capture technologies, namely, absorption-, membrane-, PSA-, and VSA-based processes, is presented over a range of flue gas compositions and flow rates. This enables selection of the most cost-effective CO2 capture and storage (CCS) technology for diverse emission scenarios. Results indicate that CO2 can be captured with the least cost using a MEA-based chemical absorption when the feed CO2 composition is less than 15−20%. For higher CO2 compositions, VSA is the preferred process.

1. INTRODUCTION With a total emission of 30.6 gigatonnes, energy-related carbon dioxide (CO2) emissions in 2010 were the highest in history.1 While the worldwide demand for energy and chemicals is projected to rise, fossil fuels are expected to continue as a predominant source of energy and petrochemicals for the next several decades; hence a rise is expected in atmospheric CO2 levels. Anthropogenic emission of CO2 has contributed measurably to man-made climate change.2 To this end, carbon capture and storage (CCS)2 is a promising route to reduce industrial CO2 emissions. This involves capture of CO2 from industrial and energy-related sources, transport to a storage location, and long-term isolation from the atmosphere. CO2 capture is the most costly step in CCS, and few technologies show promise for the commercial deployment of CO2 capture. While industrial CO2 shows diverse emission characteristics,3 most studies on CO2 capture are based on fixed feed gas compositions, flow rates, and process configurations. Contributions4−7 focusing on the evaluation and comparison of the economic performance of absorption-, adsorption-, and/or membrane-based CO2 capture are often limited by suboptimal processes and fixed feed gas characteristics. In part 18 of this series, we reported studies on monoethanolamine (MEA)-based chemical absorption and the multistage membrane technologies for industrial CO2 capture, over a wide range of feed CO2 compositions and feed flow rates. Recent theoretical6,9−−14 and pilot scale experimental15,16 studies clearly suggest that adsorption-based CO2 capture17−19 is another attractive way of mitigating industrial CO2 emissions. Major sorbents20,21 for CO2 separation include microporous/mesoporous silica or zeolites, activated carbonaceous materials, and metal organic frameworks (MOFs). Adsorption-based CO2 © 2012 American Chemical Society

capture offers three operational models, namely, pressure swing adsorption (PSA), vacuum swing adsorption (VSA), and temperature swing adsorption (TSA). One major difference between the PSA and VSA processes is the difference in their operating pressure levels. The highest operating pressure in a VSA process is atmospheric, while it can be more than atmospheric in a PSA process. The performance of these two processes can be significantly different for different feed compositions and flow rates, product specifications, and adsorbents. It is important to investigate the cost impact of PSA and VSA. To realize the full potential of PSA- and VSA-based CO2 capture processes, it is crucial to optimize their performance. However, the rigorous optimization of PSA and VSA processes is challenging. Adsorption-based CO2 capture is usually a multistep, multicomponent, and adsorbent-packed distributed process that undergoes a transient state before reaching a cyclic steady state. PSA and VSA are cyclic processes for which the performance varies with the column size, cycle configuration, step durations, and pressure levels in each step. Adsorption-based process models include nonlinear algebraic and partial differential equations (NAPDEs). Significant contributions have been made for the optimization of a PSA process. Smith and Westerberg22 presented a mixedinteger nonlinear programming (MINLP) model with time averaged mass and energy balances for the design of a PSA Received: Revised: Accepted: Published: 15665

March 7, 2012 October 19, 2012 November 1, 2012 November 1, 2012 dx.doi.org/10.1021/ie301572n | Ind. Eng. Chem. Res. 2012, 51, 15665−15682

Industrial & Engineering Chemistry Research

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process for the minimum annualized cost. Nilchan and Pantelides23 proposed a full discretization of the partial differential algebraic equations (PDAEs) describing the PSA model, in both spatial and temporal domains. The resulting large-scale nonlinear programming (NLP) model is solved using a commercial NLP solver. The efficacy of the proposed approach was demonstrated for the air separation using rapid PSA and modified PSA processes, without considering temperature effects. Ko et al.12,24 applied a sequential quadratic programming (SQP) based approach for the optimization of PSA and fractional vacuum PSA processes for CO2 capture. A direct optimization approach for determining the optimum periodic states of PSA- and TSA-based adsorption cycles was proposed by Ding et al.25 Jiang et al.26 reported a direct determination approach augmented with a hybrid trust region method for the implementation of a Newton-based approach to accelerate the convergence for cyclic steady states. Cruz et al.27,28 optimized the operating conditions and presented parametric studies on PSA and VSA processes which are based on the Skarstrom cycle with pressure equalization. A detailed modeling framework and optimization of multibed PSA processes were presented by Nikolić et al.29 Agarwal et al.30 proposed a proper orthogonal decomposition based low dimensional or reduced order model for the simulation and optimization of PSA process, and demonstrated the method for H2 recovery from a H2/CH4 mixture. Agarwal et al.13,31 also presented superstructure-based approaches to design and optimize PSA cycles for precombustion and postcombustion CO2 capture. For the precombustion CO2 capture31 from syngas (a mixture of H2 and CO2), they determined optimal cycle configurations (i.e., the duration of each step in a PSA cycle). They discretized the PDAEs in both the spatial and temporal domains, and formulated a large-scale nonlinear algebraic control problem with the step durations as the control variables which appear linearly in the model. They addressed fixed column length and diameter and feed pressure. Agarwal et al.13 extended their work on the PSA process for the postcombustion capture of CO2 by proposing a superstructure which can predict a number of different PSA operating steps. Most recently, Khajuria and Pistikopoulos32 addressed simultaneous design, operation, and control of PSA under process uncertainty. Compared to PSA, the VSA-based process for CO2 capture has not been studied extensively, especially when the adsorption heat effects on both the column bed and wall are significant. In this article, we present mathematical modeling, simulation, and optimization of a four-step pressure swing adsorption (PSA) process and a four-step vacuum swing adsorption (VSA) process for CO2 capture from industrial gas mixtures. While TSA is another attractive technological pathway for adsorption-based CO2 capture, we consider only PSA and VSA in this work. Each process is modeled and optimized over a wide range of feed gas compositions and flow rates. The emphasis is on understanding how the cost of CO2 capture changes with changing CO2 feed composition and flow rate of the gas mixtures coming from the source plant. Furthermore, we provide a quantitative approach8 toward scaling up a CO2 capture process. Our goal is to express the capture costs in terms of feed gas compositions and flow rates so as to assist in the selection of the optimal process to mitigate CO2 emissions from any given CO2 emitting source. We also investigate how we can compare among different technology options for CO2 capture and select the best capture technology for different chemical and power industries. This study also lays the foundation for (i) process synthesis studies and (ii) CO2

supply chain network studies in the carbon management domain. The novel features of this work include the following: (i) feed (flue gas) dehydration using a superstructure of dehydration alternatives (ii) study of both the PSA and VSA alternatives for adsorptionbased CO2 capture using 13X zeolite (iii) superstructure representation of PSA and VSA process configuration (iv) CO2 capture coupled with compression (v) variation in feed CO2 composition and total feed flow rate (vi) PDE-based adsorption model with heat effects and frictional pressure drop, which is fully discretized and solved via a kriging model (vii) explicit expressions for the investment and operating costs as a function of feed CO2 composition and total feed flow rate (viii) cost-based comparison of the absorption-, membrane-, PSA-, and VSA-based CO2 capture technologies We consider the feed to be a gaseous mixture of CO2, N2, O2, and H2O. However, it is not restricted to the power-plant flue gases only. Since we consider variation in the feed CO2 composition, our approach is applicable to any industrial CO2/N2/ O2/H2O mixture with predominantly CO2 and N2, including the postcombustion flue gas. The article is organized as follows. First, the process and economic basis considered in this work are presented in sections 2 and 3. Then, in section 4 a detailed description of the process configuration and the mathematical modeling of the adsorptionbased process alternatives (PSA and VSA) for CO2 capture are provided. The model implementation for each process optimization is also outlined. Next, results on the annual investment and operating costs for variable feed CO2 composition and feed flow rates for the optimized PSA- and VSA-based processes are presented. A cost-based input−output model is proposed, and explicit cost expressions as the function of feed flow rate and CO2 composition are developed. Finally, in section 5 a cost comparison between the absorption-, membrane-, PSA-, and VSAbased CO2 capture technologies with compression are presented for a range of feed CO2 compositions and feed flow rates.

2. PROCESS PARAMETERS A potential source of industrial CO2 for CCS may contain CO2, N2, O2, H2O, CO, H2, and trace amounts of NOx and SOx. In this work, we consider the feed as a gaseous mixture of CO2, N2, O2, and H2O. H2 is usually not present in the postcombustion and oxy-combustion flue gases, and CO can be converted to CO2 using a shift reaction. Furthermore, we assume that NOx and SOx are already removed using a feed desulfurization unit. The gas coming from the desulfurization unit would be usually saturated with water vapor. In this work, we consider the feed compositions to be the following: CO2, 1−70%; H2O, 5.5−15%; O2, 5.5%; balance N2. Let x (1 ≤ x ≤ 70) and y (5.5 ≤ y ≤ 15) be the percent volumetric compositions of CO2 and H2O in the feed, respectively. Therefore, the volumetric composition of N2 in the feed is given by (100 − x − y − 5.5). We fix the composition of O2 at 5.5%. The conceptual CCS structure considered in this work is described in part 1.8 We have considered feed preprocessing (mainly, dehydration), CO2 capture, and compression components of a CCS chain. We also consider the feed to be available at 1 bar and 55 °C with variable feed compositions and flow rates. The goal is to capture no less than 90% of the 15666

dx.doi.org/10.1021/ie301572n | Ind. Eng. Chem. Res. 2012, 51, 15665−15682


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4. PROCESS MODELING In this section, we describe the process models for the PSA- and VSA-based adsorption technologies with feed dehydration, capture, and compression of CO2. These models will be used to simulate and optimize for the minimum annualized cost over a range of feed CO2 compositions and feed flow rates. 4.1. Feed Dehydration. An adsorption-based process for CO2 capture may not tolerate a feed that has a high moisture content. H2O can condense during the desorption step of an adsorption-based capture process, reduce the capacity of a sorbent, and reduce the CO2 purity and recovery. A sorbent such as 13X zeolite would require a drying temperature of more than 300 °C to remove moisture from the bed, which is even higher than the minimum temperatures needed to remove CO2.11 We did not consider using the actual moisture content entering the system, as the recovery and productivity would significantly decrease,36 and the process would fail to achieve 90% recovery, especially at low CO2 concentration in the feed. Therefore, we must dehydrate the gas to reduce the moisture content to 0.1% or less before sending the gas to the adsorption column. To this end, we proposed a superstructure8 that allows multiple options for feed dehydration. The superstructure is shown in Figure 1. First, the feed which contains 5.5−15% H2O at 55 °C is cooled to 35 °C by using a direct contact cooler. The advantage is that this reduces the moisture content to 5.5%, which is the composition of H2O in a saturated flue gas at 35 °C, irrespective of the initial moisture content (5.5−15%) at 55 °C. Now, we can either bypass any further dehydration and send the gas directly to the capture process, or dehydrate the gas to 0.1% H2O. The feed dehydration can be achieved using several processes, namely, refrigeration, compression and cooling, membrane separation, and TEG (triethylene glycol) absorption. For more a detailed description of these processes and the economic evaluation of various feed dehydration alternatives, readers may consult part 18 of this series. 4.2. Adsorption-Based CO2 Capture and Compression: PSA and VSA. Pressure swing adsorption (PSA) and vacuum swing adsorption (VSA) use adsorption or desorption at two fixed pressure levels, and temperature swing adsorption (TSA) uses adsorption or desorption at two fixed temperature levels. While PSA has been widely used in oxygen enrichment, air separation, H2 purification, CO2 separation, and many other industrial applications, TSA and VSA are relatively uncommon, especially for CO2 separation. In this work, we only focus on PSA and VSA modes of adsorption technology. The 13X zeolite is selected to be the adsorbent since CO2 is selectively adsorbed in the 13X zeolite over N2. In a typical PSA process, CO2 is adsorbed at high pressure (usually higher than atmospheric). The bed is then regenerated by lowering the pressure. This is known as the desorption step. CO2 is obtained as product during this step. VSA is a cyclic process similar to PSA, and each cycle includes two or more steps. As the name suggests, a VSA process operates only between atmospheric and low vacuum pressures. In fact, the highest pressure at which adsorption takes place in VSA is atmospheric pressure. The desorption of CO2 in a VSA process is always at moderate or low vacuum. 4.2.1. Process Configuration. We use the same process configuration for both PSA and VSA processes. Figure 2 shows the process configuration. Feed gas to the adsorption process can be fed in two ways, namely, through inlets a and b. When operating in PSA mode, feed gas enters through inlet a, where a feed compressor first compresses the feed gas to a pressure

CO2 from feed and compress it to sequester at 150 bar. The adsorption-based processes are modeled, simulated, and optimized over a range of feed compositions for four total feed flow rates, namely, 0.1, 1, 5, and 10 kmol/s. For each process, we select several compositions and flow rates, and obtain the optimal investment, operating, and total costs of capture and compression. Once CO2 is captured, it is compressed to 150 bar using a six-stage compression system with intercoolers. The purity of the captured CO2 stream is also important. Note that an adsorption-based capture process may or may not achieve high purity and high recovery at the same time. However, to realize a separation, we specify the minimum recovery and purity to be 90% for the capture plant. While any CO2 purity from the adsorption-based capture unit is feasible as long as the output gas is compressed to 150 bar, low CO2 purity would mean that the compression system must compress a large volume of gas resulting in excessively high compression cost. Note that it is not possible to increase the CO2 purity during compression unless we use refrigeration for the interstage coolers.33 Therefore, achieving 90% purity during the capture is important. However, we allow the cost trade-offs between the capture and compression units as long as the minimum recovery and purity specifications are met.

3. ECONOMIC PARAMETERS Our goal is to optimize the processes for minimum total annualized cost (TAC), which is defined as TAC = AIC + AOC

(1)

where AIC and AOC are the annualized investment and operating costs, respectively. While AOC depends on the utilities, AIC is computed as follows: AIC = ϕTPC + AMC

(2)

where TPC and AMC represent the total plant cost and annual maintenance cost, respectively, and ϕ is the capital recovery factor. A value of 0.154 is used for ϕ throughout the work. TPC is the total equipment installed costs (TIC) plus the indirect cost plus the balance of plant cost. We consider indirect cost and balance of plant cost to be 32 and 20% of TIC, respectively. AMC is taken to be 5% of TPC. Lastly, the installed cost (TIC) of an equipment includes the equipment purchase cost (EPC) and the installation cost (EIC). We considered the installation costs to be 4 and 80% of the purchased costs34 for general equipment (columns, heat exchangers) and movers (compressors and vacuum pumps), respectively. The following base conditions and costs are used for all cases: 1. Cooling water is available as the cold utility. The price of cooling water is $0.001/ton. 2. Saturated steam at 5 bar is available as the hot utility. The price of hot utility is $6/ton. 3. Electricity is available at $0.07/kWh. 4. The cost of membrane5 is $50/m2. 5. Equipment costs are based on 2009 values. 6. All mover costs are taken from Uppaluri et al.35 and converted to 2009 values. 7. Each plant is operated 8000 h in a year. 8. All compressors run adiabatically with 75% thermal efficiency.11 15667



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Figure 1. Superstructure for feed dehydration.

Figure 2. Process flow diagram of the adsorption process with CO2 compression.

higher than atmospheric and then sends the gas to the column. On the other hand, inlet b is used to feed the gas at atmospheric pressure in VSA mode. Multiple columns packed with the 13X zeolite are used to maximize the plant capacity. The total number of columns is determined based on the operating conditions, which we will discuss in detail in section 4.2.3. All column inlets and outlets have valves to regulate the flows. When gas is fed to a column, the valve at the column inlet is open. Otherwise, the valve remains closed. The outflow from each column is also

maintained in a similar fashion using the valve placed at the column outlet. Each column also has three different outlets, namely, c, d, and e. Outlet c is used to vent the nonadsorbed gas to the atmosphere at atmospheric pressure, outlet d is used to purge mostly N2, and outlet e (blue line in Figure 2) is used to recover the product CO2. To enable purging at an intermediate pressure, outlet d uses a purge vacuum pump. Similarly, a second vacuum pump is used at outlet e to evacuate the column and recover CO2 at the lowest 15668



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pressure/velocity effects and heat transfer resistance across the column wall are also considered. In this section, we describe the assumptions, indices, sets, parameters, variables, and mathematical equations that constitute the mathematical model. Assumptions. The following assumptions were made while developing the model.14,16 1. Ideal gas conditions apply. 2. Constant molecular weight of the gas is assumed, and the viscosity of the gas does not depend on the pressure. 3. There are no radial variations in concentration, pressure, and temperature for gases in both the gas and solid phases. 4. A linear driving force (LDF) approximation is adequate to describe the gas transport in the adsorbent. 5. The boundary conditions for the energy balance equations follow from analogy of mass and heat transfer and from the operating conditions for the column wall. 6. Adsorption equilibrium between the gas and adsorbed phases is described by a binary Langmuir isotherm model for which the parameters are obtained using single component data. 7. The boundary conditions for the overall continuity equation follow the velocity conditions according to the mode of operation. 8. While industrial flue gases contain CO2, N2, O2, and H2O, only a CO2/N2 binary adsorption is considered. H2O will be removed using a dehydration process before the adsorption-based CO2 capture. Furthermore, it is noteworthy that a similar CO2 uptake on 13X zeolite was observed experimentally38 for a ternary mixture of CO2/ N2/O2 and for pure CO2. Therefore, we assume that O2 in the feed gas behave similarly to N2 inside an adsorption column. 9. Uniform particle size and bed voidage are assumed. 10. Gas flow in the packed bed is described by Darcy’s law. 11. The effect of Knudsen diffusion on the macropore diffusivity is neglected. 12. Each column is identical, independent, and fed equal amounts of feed gas in each cycle. Indices. i: component j: mover k: cooler n: packed column Sets. The set of all components, I, is given as follows:

pressure. Recovered CO2 is then compressed to send to the sequestration site using a six-stage compressor system with interstage cooling. PSA and VSA are cyclic processes, which means that each column undergoes a number of operational steps. Even the simplest PSA/VSA cycle requires at least two steps, namely, adsorption and desorption. For improved performance, additional cycle configurations can be used. Among these, the Skarstrom cycle (chosen for this work also) is the most popular and has been shown to perform well for CO2 capture. This cycle has four steps. The four steps in order are (i) pressurization, (ii) adsorption, (iii) forward blowdown, and (iv) reverse evacuation. Figure 3 shows these four steps in a cycle along with their usual

Figure 3. Four-step process for PSA/VSA process with pressure profiles.

pressure profiles. Feed gas is input in steps 1 and 2. In the forward blowdown step, the blowdown of N2 from the column was performed by reducing the column pressure to intermediate pressure (Pbd). In the last step, the adsorbed CO2 from flue gas on zeolite adsorbent is desorbed by further reducing the column pressure (Pevac). Once a column undergoes these four steps, a cycle is completed. The cycles are repeated one after another. A cyclic steady state is the state when the initial and final conditions for a cycle appear to be the same. Starting from an initial bed condition, the process undergoes a transient state for a number of cycles before reaching a cyclic steady state. Depending on the initial column condition, the number of cycles to reach the cyclic steady state varies. In this work, we first saturate all columns with N2 before starting the PSA or VSA process. For this condition, our experience is that the cyclic steady state is reached after 100 cycles. All results are obtained at cyclic steady state. 4.2.2. Model Development. The adsorption model used in this work involves a system of highly nonlinear algebraic and coupled partial differential equations. A detailed description of the development of the model can be found elsewhere.14,16,37 The model considers a multicomponent adsorption system in an adsorbent-packed column with nonisothermal adsorption/ desorption including frictional pressure drop. Temperature and

i ∈ I = {CO2 , N2} The set of all movers, J, is given by j ∈ J = {feed compressor, purge vacuum pump, vacuum pump for CO2 recovery, 6‐stage compressor train}

Parameters. The following parameters are defined: F: feed flow rate yi,f: mole fraction of component i in the feed gas T0: feed temperature Pf: feed flow rate uo: feed velocity Ta: outside air temperature yi,0: initial mole fraction of component i in the bed Patm: atmospheric pressure 15669



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Kz: effective heat conductivity Kw: thermal heat conductivity of column wall hin: heat transfer coefficient inside the column ho: heat transfer coefficient for outside the column Cp,w: specific heat capacity of column wall Cp,g: specific heat capacity of gas mixture Cp,a: specific heat capacity of adsorbed gas in solid Cp,s: specific heat capacity of adsorbent R: gas constant qs: saturation capacity boi : isotherm parameter for component i ΔUi: heat of adsorption for component i τp: tortuosity factor DM: molecular diffusivity of CO2−N2 mixture Dp: macropore diffusivity, Dp = DM/τp DL: axial dispersion coefficient defined by the correlation17 DL = 0.7DM + 0.5uodp UC: unit operating cost dp: particle diameter rp: particle radius, dp/2 εp: particle porosity ε: bed porosity ρw: density of column ρs: density of solid particle μ: gas viscosity ϕ: annualization factor α: unit capital cost β: exponent of power cost kp: bed permeability defined by the correlation kp = (dp2/150)(ε/(1 − ε))2 Variables. We consider the variation both in time and along the column length. To denote this, we define z: bed length t: time The following system variables are defined to describe the system. P(z,t): pressure T(z,t): temperature Tw(z,t): column wall temperature uz(z,t): velocity yi(z,t): mole fraction of component i in the gas phase xi(z,t): fractional loading of component i in the solid phase xi*(z,t): equilibrium fractional loading of component i in the solid phase ki(z,t): mass transfer coefficient of component i bi(z,t): temperature dependent parameter for component i Bi(z,t): isotherm parameter for component i ΔHi(z,t): change in internal energy for component i ρg(z,t): density of gas obtained from the ideal gas law The decision variables for process optimization are as follows: N: number of columns PH: highest pressure of the system (1 bar for VSA, 2−10 bar for PSA) L: column length (0.1−25 m) D: column diameter (0.1−25 m) Pbd: blowdown pressure (Pevac to PH) Pevac: evacuation pressure (0.001 to Pbd) tpr: duration of the pressurization step (20 s) tads: duration of the adsorption step (10−200 s)

tbd: duration of the blowdown step (10−200 s) tevac: duration of the evacuation step (10−200 s) When PH is higher than the atmospheric pressure, the adsorption-based process will be called PSA. For VSA, we fix PH at 1 atm. Constraints. We first define the following dimensionless variables. The model will be presented in terms of these dimensionless variables. P̅ =

P ; PH

Pevac ̅ =

Tw ; T0 u uz̅ = z ; uo

Tw̅ =

Pevac ; PH

Ta̅ = Z=

Ta ; T0

z ; L

Pbd ̅ = xi = τ=

Pbd ; PH

qi̅ qs

;

T ; T0 q* xi* = i ; qs T̅ =

tuo L

(3)

The sharp gradients of the CO2 isotherm, especially at low partial pressures, often make the numerical convergence very difficult while solving an adsorption-based CO2 capture model. Variable transformation as shown in eq 3 is useful for (a) numerical convergence and (b) obtaining a qualitative understanding of the system behavior. The following component mass balance is used to compute the CO2 mole fraction in the gas phase. ∂yi ∂τ

=

+

⎛ 2 ⎞ ∂y 1 ⎜ ∂ yi 1 ∂yi ∂P ̅ 1 ∂yi ∂T̅ ⎟ − uz̅ i − + ⎜ ⎟ 2 Pe ⎝ ∂Z P ̅ ∂Z ∂Z T̅ ∂Z ∂Z ⎠ ∂Z

∂x N2 ⎞ ∂x ψT ̅ ⎛ ⎜(yi − 1) i + yi ⎟ P̅ ⎝ ∂τ ⎠ ∂τ

∀ i|(i = CO2 ) (4)

where ψ = (1 − ε/ε)(RT0qs/PH), and Pe = uoL/DL is the Peclet number. Moreover, to ensure the sum of all molar fractions equals 1, we have

∑ yi = 1

(5)

i

The overall mass balance yields the following equation for pressure: ⎛ ∂u ∂P ̅ ∂P ̅ P ̅ ∂T̅ P ̅ ∂T̅ ⎞ ⎟ = ⎜ −P ̅ z̅ − uz̅ + uz̅ − ⎝ ∂Z ∂Z ∂τ T̅ ∂τ T̅ ∂Z ⎠ ⎞ ∂⎛ ⎜⎜∑ xi⎟⎟ − T̅ ψ ∂τ ⎝ i ⎠

(6)

To compute the wall temperature, the following energy balance across the column wall is used: ∂Tw̅ ∂ 2Tw̅ = π1 + π2(T̅ − Tw̅ ) − π3(Tw̅ − Ta̅ ) ∂τ ∂Z2

(7)

where π1 =

π2 = 15670

Kw ρw Cp ,wuoL

(8a)

2rh i inL ρw Cp ,wuo(ro 2 − rin 2)

(8b)


Industrial & Engineering Chemistry Research π3 =

Article

2rohoL 2

2

ρw Cp ,wuo(ro − rin )

The LDF mass transfer rate constants have been calculated using the following approximation:37

(8c)

rin and ro are respectively the inside and outside radii of the packed column. The following energy balance is used to compute the bed temperature inside the column: ∂T̅ ∂ 2T̅ ∂T̅ = π4 − π5uz̅ + 2 ∂τ ∂Z ∂Z

N

∑ (π6i + π7T̅ ) i=1

− π8(T̅ − Tw̅ )

ki = Ω

∂xi ∂τ

qsbi ⎛q⎞ ⎜ ⎟ = ⎝ c ⎠i 1 + ∑i Bi yi (9)

π5 =

π6i =

(

ε ρg Cp ,g +

Kz 1−ε (ρs Cp ,s ε

)

+ Cp ,aqs) uoL

1−ε (ρs Cp ,s ε

+ Cp ,aqs)

1−ε ( −ΔHi)qs ε 1−ε T0 ρg Cp ,g + ε (ρs Cp ,s +

(

)

(10c)

ΔHi = ΔUi − RT

π7 =

π8 =

(10a)

(10b)

Cp ,aqs)

(

(

1−ε (ρs Cp ,s ε

xi = xi*| y

)

+ Cp ,aqs)

i,0

(10e)

In a four-step cyclic process, the initial conditions for the subsequent steps are nothing but the final conditions of their previous steps, and we will not discuss the initial conditions for the other steps any further. Let Z = 0+ and Z = 1− be the two ends of an adsorption column where the boundary conditions must be applied. The boundary conditions at Z = 0+ for the pressurization step are as follows:

)

(10f)

The dimensionless interstitial velocity of gas through the porous media is related to the local pressure gradient and is computed using the following form of Darcy’s law. ⎛ k pPH ⎞ ∂P ̅ uz̅ = −⎜ ⎟ ⎝ μuoL ⎠ ∂Z

1 ∂yi = − uz̅ (yi ,f − yi ) Pe ∂Z

(11)

We consider the Langmuir type equilibrium isotherm which is given by xi* =

Bi yi 1 + ∑i Bi yi

1 ∂T̅ = − uz̅ (1 − T̅ ) PeH ∂Z

∀i

(19)

Tw̅ = Ta̅

(12)

P ̅ = f (τ ): Pevac ̅ →1

where Bi =

(17)

(18)

T̅ = 1

2h inL εru i o ρg Cp ,g +

∀i

P ̅ = Pevac ̅

(10d)

1−ε (Cp ,g − Cp ,a)T0qs ε 1−ε T0 ρg Cp ,g + ε (ρs Cp ,s + Cp ,aqs)

(16)

While eqs 3−17 provide the relationship among the system variables, appropriate boundary and initial conditions are necessary to simulate an actual PSA or VSA process. The boundary conditions are different for different steps. Since our adsorption-based process is a four-step process, we discuss the conditions for each of the four steps separately. Initial and Boundary Conditions: Pressurization. The pressurization step is the first step of the cycle. The initial condition for the adsorbent bed is assumed to be in equilibrium with flue gas having a carbon dioxide mole fraction yi,0 at pressure Pevac and temperature T0. The initial conditions (τ = 0, 0 ≤ Z ≤ 1) are given by yi = yi ,0

ρg Cp ,g ρg Cp ,g +

∀i

where Ω is an empirical parameter for which a value Ω = 15 is used.37 Furthermore

where π4 =

εDp ⎛ c ⎞ ⎜ ⎟ rp2 ⎝ q ⎠i

biP R gT

∀i

The boundary conditions at Z = 1− are given by

(13)

bi is the temperature dependent isotherm parameter which is expressed as bi = bioe−ΔUi / RT

∀i

∂yi ∂Z

(14)

∂T̅ =0 ∂Z

A linear driving force (LDF) model is used to approximate the mass transfer rate between the gas phase and solid as follows: ∂xi kL = i (xi* − xi) ∂τ uo

∀i

=0

(20)

Tw̅ = Ta̅ ∂P ̅ =0 ∂Z

(15) 15671



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Initial and Boundary Conditions: Adsorption. The boundary conditions at Z = 0 + and Z = 1 − for the adsorption step are given by eqs 21 and 22, respectively.

∂yi ∂Z

=0

∂T̅ =0 ∂Z

∂yi ∂Z

Tw̅ = Ta̅

= −Pe(yi ,f − yi )

1 ∂T̅ = −(1 − T̅ ) PeH ∂Z

P ̅ = f (τ ): Pbd ̅ → Pevac ̅

Lastly, we must ensure that at least 90% of the CO2 fed to the process is captured. To ensure this, the CO2 recovery (Re) is calculated and constrained as follows:

(21)

Tw̅ = Ta̅

t

Δ P̅ = 1 + PH ∂yi ∂Z

Re =

j

+

P̅ = 1

∑ UCkWk]

(23)

=0

(24)

Tw̅ = Ta̅

(28)

(29)

(31)

C PL, n = 300.9(3.281Dn)0.63316 (3.281Ln)0.80161

(32)

π (3.281Dn)2 (3.281Ln) 4

π C DR, n = 3·125· (3.281Dn)2 4

Initial and Boundary Conditions: Evacuation. The boundary conditions at Z = 0+ and Z = 1− for the evacuation step are given by eqs 25 and 26, respectively.

(30)

Wn = (π (39.37Dn + tS)(39.37(0.8Dn + Ln))tS ρ)

VP, n =

P ̅ = f (τ ): 1 → Pbd ̅

(33) (34)

where Ln and Dn are the length and diameter of column n, tS is the shell thickness, and ρ is the density of the carbon steel. We use FM = 2.1, CPK = 40, ρ = 0.284 lb/in.3, and tS = 2 in.39 Equations 3−34 define the nonlinear algebraic and partial differential equation (NAPDE) based model for the optimization of the adsorptionbased PSA and VSA processes for CO2 capture. 4.2.3. Model Implementation for Process Optimization. The NAPDE model involves nonlinear constraints including concave isotherm equations and a concave objective function. Furthermore, the presence of coupled and nonlinear partial differential equations, which describe the mass and energy balances, mass transfer rates, and fluid flow through porous

=0

∂T̅ =0 ∂Z

j

C V, n = exp{7.2756 + 0.18255[ln(Wn)] + 0.02297[ln(Wn)]2 }

∂T̅ =0 ∂Z

∂Z

n

where FM is the factor for materials of construction, CV,n is the free on board (f.o.b.) purchase cost of empty vessel with the weight Wn, CPL,n is the added cost for absorber platforms and ladders, VP,n is the absorber packing volume, CPK is the installed cost of the packing for unit volume, and CDR,n is the installed cost of flow distributors and redistributors of column n. These are expressed as follows:

∂P ̅ =0 ∂Z

∂yi

∑ ICCn) + ∑ UCjWj

PCn = FMC V, n + C PL, n + VP, nC PK + C DR, n

Tw̅ = Ta̅

∂Z

(27)

where Wj(k) is the power requirement of the mover (cooler) j(k), and ICCn is the investment cost of the column n. The column investment costs, ICCn, are calculated from the purchase cost (PCn) using the following expression:39

=0

∂T̅ =0 ∂Z

∂yi

≥ 0.90

k

where PeH = εuoLρgCp,g/Kz. Initial and Boundary Conditions: Blowdown. The boundary conditions at Z = 0+ and Z = 1− for the blowdown step are given by eqs 23 and 24, respectively.

∂Z

t

t

P P̅ ads dt + ∫ uz̅ yCO RT̅ dt ∫0 pr uz̅ yCO2 RT ̅ ̅ 0 2

min[∑ ϕ(αj(Wj)βj + αk(Wk)βk +

(22)

Tw̅ = Ta̅

∂yi

P̅ dt ∫0 evac uz̅ yCO2 RT ̅

Objective Function. The objective is to minimize the total annualized cost of the adsorption-based process shown in Figure 2. It is given by

=0

∂T̅ =0 ∂Z

(26)

(25)

Tw̅ = Ta̅ ∂P ̅ =0 ∂Z 15672



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media, make the NAPDE model very complex and difficult to solve, even when the decision variables are fixed. The optimization of an adsorption process, solely based on the NAPDE model, becomes intractable if all of the 10 process optimization variables (N, L, D, PH, Pbd, Pevac, tpr, tads, tbd, tevac) are optimized simultaneously. We apply the following approach to reduce the number of decision variables. First, we fix tpr, which is the time required to pressurize the column from Pevac to PH. Once the entire column is pressurized to PH, no gas would enter and any further continuation of the pressurization step would reduce the productivity of the process. Therefore, we set tpr to be the time required for the column just to reach PH from Pevac. We do this by providing a known exponential pressure profile at the column entrance such that tpr = 20 s. Next, we consider N. We assume that each column is identical and receives an equal amount of feed in each cycle. Furthermore, each column is independent; that is, the operation of a column does not have any effect on other columns. The advantage of using such identical and independent columns is that the solution of the NAPDE model for a single column can be applied to every other column. However, the number of columns should be such that the products and purge gases are handled without introducing any additional vacuum pumps and compressors. To ensure this, the maximum number of columns that are used for the process shown in Figure 2 is given by tc/tmax, where tc is the total duration of one complete cycle and tmax is the duration of the longest step as follows: tc = t pr + tads + tbd + tevac

(35)

tmax = max[t pr , tads , tbd , tevac]

(36)

Table 1. Adsorption Isotherm Parameters and Material Properties14 used for the PSA and VSA parameter

⎞⎛ Ftc ⎞ 4 ⎛ Patm To (22.4)⎟⎜ ⎟ ⎜ π ⎝ PH 273.15 ⎠⎝ uotadsN ⎠

unit

1 1 7500 0.0227

m/s bar mol/m3 cm3/mmol

isotherm parameter for N2 (boN2)

0.0507

cm3/mmol

heat of adsorption for CO2 (ΔUCO2)

−4.52

kcal/mol

heat of adsorption for N2 (ΔUN2)

−1.76

kcal/mol

bed porosity (ε) particle diameter (dp) particle porosity (εp) molecular diffusivity of CO2−N2 (DM) mixture at 298 K tortuosity factor (τp) viscosity of gas (μ) density of solid particle (ρs) density of column (ρw) heat transfer coefficient inside the column (hin) heat transfer coefficient for outside the column (ho) specific heat capacity of adsorbent (Cp,s) specific heat capacity of adsorbed gas in solid (Cp,a) specific heat capacity of gas mixture (Cp,g) specific heat capacity of column wall (Cp,w) effective heat conduction coeffecient (Kz) thermal conductivity of column wall (Kw)

0.37 0.15 0.35 0.12955 3 1.72 × 10−5 800 7.8 8.6 2.5 0.92 0.92 1.0106 0.502 0.0903 16

cm cm2/s

kg/(m·s) kg/m3 g/cm3 J/m2/s/K W/m2/K J/g/K J/g/K kJ/kg/K J/g/K J/m/s/K W/m/K

CO2 capture. We use a similar approach to optimize the PSA and VSA processes for variable feed CO2 compositions and feed flow rates. Details of the kriging method can be found elsewhere.40,43 Let us consider that we have a function y(x), where y is the output or response and x is the vector of k input (decision) variables. In the context of PSA/VSA optimization, y is the total annualized cost, and x = [L,PH,Pbd,Pevac,tads,tbd,tevac]. We approximate the NAPDE model by the following kriging model:

We take N to be the round-down integer value of this fraction. For N ≥ tc/tmax, the process would require additional movers, which we do not allow. We relate D to other variables by considering the following: (i) the total feed in each cycle is divided into N equal amounts which are then sent to each column, and (ii) the feed to the pressurization step is much smaller than the feed to the adsorption step. Therefore, the feed flow rate to the adsorption step is considered to be the rated feed flow rate for the entire cycle. With this we compute the inner diameter, D, of each column as follows: D=

value

feed velocity (uo) feed pressure (Pf) saturation capacity (qs) isotherm parameter for CO2 (boCO2)

y ̂(x new) = μ ̂ + r TR−1(y − 1μ)̂

(38)

where ŷ is the value of y at a new input variable vector xnew. Let S be the number of samples where y (response) is evaluated for known x (sample point). Then, R is a known S × S matrix and r is a S × 1 vector that contains xnew. These are defined as follows.

(37)

Finally, we have seven independent variables (L, PH, Pbd, Pevac, tads, tbd, tevac) for the PSA process and six independent variables for the VSA process (L, Pbd, Pevac, tads, tbd, tevac). We fix tpr = 20 s. For the VSA process, PH = 1 atm. N and D are related to the other variables as discussed above. With these, we optimize the PSA and the VSA processes using 13X zeolite as the adsorbent. The model parameters16 are given in Table 1. We consider identical columns with wall thickness of 0.015 m. One alternate way to optimize the NAPDE model is to use a kriging-based surrogate model40−42 of the original NAPDE model. The advantage of using such an approximate model is that commercial solvers such as CONOPT can be used for optimization. Kriging, as pointed out by Jones et al.,40 often requires the fewest function evaluations, making it computationally attractive over other competing methods. Hasan et al.14 developed a kriging-based surrogate model by synergistically fitting the NAPDE model based VSA simulation results to find the optimal VSA design and operating conditions for postcombustion

R(x(i), x(j)) = exp[−d(x(i), x(j))]

(39)

r(x(i), x new) = exp[−d(x(i), x new)]

(40)

where k

d(x(i), x(j)) =

∑ θh |x(hi) − x(hj)| p

h

θh ≥ 0, ph ∈ [1, 2]

h=1

(41)

where i (i = 1, 2, ..., S), and j (j = 1, 2, ..., S) denote sample points. Given y(i) and x(i), we can easily construct R. The only unknowns are μ̂ and θh, for which the values are to be obtained using simulations of y at different x. Our kriging-based optimization of the adsorption process involves the following steps, which we repeat for every CO2 composition and feed flow rate. Step 1. Discretization of the NAPDE Model. To develop a kriging model, we need to solve the original NAPDE model to 15673



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Step 2. Sampling. We simulate the PSA/VSA process for different values of the decision variables, and obtain the total annualized costs (eq 28). Each set of fixed decision variables for which the discretized NAPDE model is solved is a sample point, and the cost obtained at each sample point is a response. Running a PSA/VSA simulation for a particular sample point to get the response is known as “sampling”. Two important issues arise in sampling, namely, (i) where to sample and (ii) how many points to sample. The set of samples and responses to be used to fit the kriging model should cover the overall range of decision variables. We use the Latin hypercube-based space-filling method for sampling in MATLAB. While Latin hypercube design (LHD) is used to sample the independent decision variables, tpr is always 20 s and N and D are fixed by the values as independent decision variables, as discussed earlier. The accuracy of the kriging model increases with the number of sample points (S) used to construct the model. Therefore, the value of S should be chosen such that a proper balance is maintained between the time required to simulate at all sample points and the model accuracy. Jones et al.40 proposed a “10k” rule for S, where k is the number of independent decision variables (k = 7 and 6, respectively, for PSA and VSA). Following the “10k” rule, we use 70 sample points for the PSA process and 60 sample points for the VSA process to generate the response data. For each sample point, the fully discretized NAPDE model is solved. If any sample does not satisfy eq 27, it is removed from the pool and a new sample point is included which satisfies eq 27. Since the adsorption-based process described in Figure 2 is a cyclic process, we simulate the full transient state (first 100 cycles) and accept the result from the

generate input−output data. To do that, we must solve the NAPDE model as accurately as possible. Guntaka et al.16 developed an efficient technique for solving the NAPDE model numerically. They used the finite volume method44 for the discretization of the partial differential equations in space. This method, which is implemented in MATLAB with ode23s for integration is able to accurately simulate a cyclic four-step PSA/ VSA process for CO2 capture for a given design and operating condition, that is, when all the decision variables are fixed a priori. We take this version of the NAPDE model to generate input data required for the development and subsequent improvement of the kriging model. Table 2. Optimized PSA and VSA Variables at Different Feed CO2 Compositions (Feed Flow Rate = 0.1 kmol/s) feed CO2 (mole fraction)

L

PH (bar)

0.1 0.2 0.3 0.5 0.7

0.81 0.82 0.82 0.80 0.80

9.4 8.5 7.2 6.7 6.1

0.1 0.2 0.3 0.5 0.7

0.81 0.81 0.81 0.81 0.81

1 1 1 1 1

Pbd (bar)

Pevac (bar)

tads (s)

tbd (s)

tevac (s)

1 1 1 1 1

0.26 0.30 0.29 0.31 0.34

91 83 70 62 56

79 76 73 65 61

122 117 121 125 129

VSA 0.15 0.22 0.27 0.34 0.41

0.03 0.09 0.14 0.20 0.22

64 61 55 48 33

32 30 29 27 26

112 107 110 105 102

PSA

Figure 4. Optimized process flow diagram of PSA process for CO2 capture coupled with feed dehydration and CO2 compression for the feed that contains 10% CO2 and has a total flow rate of 0.1 kmol/s. 15674



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Figure 5. Optimized process flow diagram of VSA process for CO2 capture coupled with feed dehydration and CO2 compression for the feed that contains 10% CO2 and has a total flow rate of 0.1 kmol/s.

Figure 6. Optimized annual investment costs for the PSA-based process.

Figure 7. Optimized annual operating costs for the PSA-based process.

100th cycle to ensure that the costs are obtained at cyclic steady state only. We also note the sample point for which the evaluated cost is minimum for all samples as the current best solution for the PSA/VSA optimization. Step 3. Estimation of Kriging Parameters. This step involves fitting the kriging parameters using the response data generated in the previous step. We fit the parameters μ̂ and θh by applying

the maximum likelihood40 parameter estimation method for the best fit. Step 4. Optimization of the Kriging Model. The fitted kriging model given by eqs 38−41 is used to formulate the following NLP model for the optimization of the adsorption-based process. min y ̂(x new) = μ ̂ + r TR−1(y − 1μ)̂ 15675

(42)



Article

If x* satisfies eq 27, we evaluate the process cost using eq 28 at x* and compare it with the current best solution. If the value of the cost shows an improvement of 0.1% or more over the current best solution, then we add x* as a new sample point and return to step 3. Otherwise, we take x* to be the optimal solution for the PSA or VSA process.

5. RESULTS AND DISCUSSION The PSA and VSA processes were optimized over a range of feed compositions and flow rates (0.1, 1, 5, and 10 kmol/s). To maintain the distinction between the PSA and VSA processes, we set PH ≥ 2 bar and Pevac ≥ 0.1 bar for the PSA process. Table 2 shows the optimized PSA and VSA results at different feed CO2 compositions for 0.1 kmol/s. To illustrate the process configurations, we consider the case where the feed gas contains 10% CO2, 15% H2O, 5.5% O2, and 69.5% N2 and has a total flow rate of 0.1 kmol/s at 55 °C. The adsorption/desorption behavior of O2 is considered similar to that of N2. Figures 4 and 5 show the optimized PSA and VSA processes for this condition, respectively. The feed dehydration sections for both the optimized PSA and VSA processes have the same configuration. The feed is first cooled to 35 °C using a direct contact cooler and the H2O content reduces to 5.5%. Further dehydration of the feed to 0.1% H2O is achieved using the TEG absorption process. TEG absorption is selected because the total cost of dehydration is the lowest for TEG absorption among those of the dehydration alternatives shown in Figure 1. Note that we cannot use bypass, since we must dehydrate the feed to 0.1% H2O. We also cannot use compression and cooling and refrigeration, since the loss of CO2 from these two dehydration alternatives is more than the allowable limit of 10%. Therefore, the feasible alternatives for feed dehydration are sulfonated poly(ether ether ketone) (SPEEK) based membrane and TEG absorption. Now, the annualized investment and operating costs of SPEEK-based membrane process plus the direct contact cooler to dehydrate the feed from 15% to 0.1% H2O are $739,000 and $213,000, respectively, when the feed flow rate is 0.1 kmol/s. For TEG absorption, the investment and operating costs for the same are $146,000 and $67,000, respectively. Therefore, we select TEG absorption for feed dehydration. In the TEG absorption for feed dehydration, H2O in the feed gas is absorbed in liquid TEG solvent flowing countercurrently in the TEG absorber. The dry gas leaves the column from the top. The H2O-rich solvent from the bottom of the column is pumped and then heated in a heater to about 73 °C. The heated solvent is then flashed under vacuum (about 0.04 bar) for regeneration. The regenerated TEG solvent is pumped back to the absorption column after being cooled to 65 °C. While the PSA process compresses the dry feed gas to 9.4 bar, the VSA process does not use any feed compression. The gas is cooled to 25 °C before sending it to the adsorption columns. Both the PSA

Figure 8. Optimized annual investment costs for the VSA-based process.

Figure 9. Optimized annual operating costs for the VSA-based process.

subject to x L ≤ x new ≤ x U

(43)

Let x* be the optimal solution to the NLP model. We solve the fully discretized NAPDE model at x*. If x* does not satisfy eq 27, we return to step 2 to include a new sample point using LHD.

Table 3. Estimated Parameters for the Input−Output Based Cost Model [α + (βxCO2n + γ)Fm process

α

β

γ

PSA

206,010

9,601

5,731

VSA

168,128

11,531

4,793

PSA

0

4,954

7,406

VSA

0

3,992

5,857

n

m

Investment Cost ($/year) 1 0.832 1

0.82

Operating Cost ($/year) 0.93 1 0.743 15676

1

xCO2

F (mol/s)

0.01 ≤ xCO2 ≤ 0.70

100 ≤ F ≤ 10 000

0.01 ≤ xCO2 ≤ 0.70

100 ≤ F ≤ 10 000

0.01 ≤ xCO2 ≤ 0.70

100 ≤ F ≤ 10 000

0.01 ≤ xCO2 ≤ 0.70

100 ≤ F ≤ 10 000



Article

Figure 10. Annualized investment costs for CO2 capture at different feed flow rates.

Therefore, when the CO2 composition is increased, the total CO2 flow in the feed is also increased. Increased CO2 flow requires increased solvent flow and reboiler duty of the stripper in the absorption-based process. Moreover, it increases the compression cost for CO2 transportation. While increased CO2 composition reduces the operating cost of the membrane units by increasing the partial pressure driving force through the membrane materials, the saving is offset by the increased compression cost. 5.1. Input−Output Model. Our goal in this section is to provide explicit expressions8 for costs of PSA and VSA processes as a function of feed CO2 composition and feed flow rate. This is a crucial step toward a quantitative approach required for scaling up a CO2 capture process for any CO2 emitting industry. We use the same functional forms for the investment (IC) and operating costs (OC) as we used in the first part8 of this series of papers. The expressions for IC and OC as a function of feed CO2 composition (xCO2) and feed flow rate (F) are given by

and VSA processes for the example case use two adsorption columns. The results for the PSA and VSA processes over a range of feed CO2 compositions and flow rates (0.1, 1, 5, and 10 kmol/s) are shown in Figures 6−9 in logarithmic scale for clarity (as the ranges for the costs are very high). The optimal annualized investment and operating costs of the PSA-based process for variable molar composition of CO2 from 0.01 to 0.70 (i.e., feed CO2 molar content of 1−70%) and also for 0.1, 1, 5, and 10 kmol/s are shown in Figures 6 and 7, respectively. As observed for the absorption- and membrane-based processes,8 increases in costs are observed with feed flow rate, signifying the importance of the scale of a process for CO2 capture. Furthermore, for any feed flow rate, the investment and operating costs increase with CO2 composition. The optimized annualized investment and operating costs for the VSA-based process are shown in Figures 8 and 9. While the cost trends with CO2 composition are similar for the two processes, VSA is more cost-effective in general than PSA. It is clear from the plots that, for any feed flow rate, the investment and operating costs increase with CO2 composition. This is expected, since the net CO2 flow rate also increases with CO2 composition, for fixed total feed flow rate. It is important to note that each cost curve represents a constant total feed flow.

IC or OC = α + (βxCO2 n + γ )F m

(44)

where α, β, γ, n, and m are parameters which we obtain by fitting the optimized data presented earlier. The selection of 15677



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Figure 11. Annual operating costs for CO2 capture at different feed flow rates.

the maximum likelihood parameter estimation for the best fit. The resulting α, β, n, and m values are presented in Table 3. On average, the prediction errors are less than 3.4%. 5.2. Total Annualized Cost-Based Technology Comparison. One of the key objectives of our study is to compare leading technologies (absorption, membrane, PSA, and VSA) for CO2 capture, based on their costs. Furthermore, we are interested in selecting the most cost-effective technology for any given CO2 composition and feed flow rate. To this end, we combine the results of PSA- and VSA-based processes with the results presented for the absorption- and membrane-based processes in part 1.8 Figure 10 shows the results for the optimal annualized investment costs of absorption-, membrane-, PSA-, and VSA-based processes for variable molar composition of CO2 from 0.01 to 0.70 (i.e., feed CO2 molar content of 1−70%) at feed rates of 0.1, 1, 5, and 10 kmol/s. It is clear from the plots that, for any feed gas flow rate, the investment costs usually increase with the CO2 composition. The operating costs also show similar trends as shown in Figure 11. These plots provide a simple but powerful technique to compare the four major technologies for CO2 capture. Absorption is the technology of choice when the feed gas has a CO2 content of about 15−20% or less, as the investment and operating costs are the lowest for all three flow rates. For very low CO2 concentrations, the investment and operating costs for the

a capture technology t is modeled using a binary variable yt as follows: ⎧1 if technology t is selected yt = ⎨ ⎩ 0 otherwise t ∈ T |t = {PSA, VSA}

With this, eq 44 is reformulated as IC or OC = αt yt + βt xCO2 nF m + γtF m L U xCO y ≤ xCO2 ≤ xCO y, 2 t 2 t

y = [0, 1]T

(45a)

F Lyt ≤ F ≤ F Uyt , (45b)

Here, xLCO2 and xUCO2 are respectively the lower and upper bounds on the feed CO2 composition, and FL and FU are respectively the lower and upper bounds on the feed flow rate. Equation 45a ensures that when technology t is selected (i.e., yt = 1), the cost function reduces to eq 44, for a given feed CO2 composition xCO2 and feed flow rate F. Otherwise, IC and OC are zero. Equation 45b ensures that a technology is selected only when the feed CO2 composition and total feed flow rate are within their valid ranges. To obtain α, β, γ, n, and m for the PSA and VSA technologies discussed in this work, we use our optimized cost data and apply 15678



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Figure 12. CCS cost for (i) absorption-, (ii) membrane-, (iii) PSA-, and (iv) VSA-based CO2 capture processes at different feed CO2 compositions (covering major industrial and chemical sectors, namely, (A) energy and heat generation, (B) chemical industries, and (C) others) and feed flow = 0.1 kmol/s.

Figure 13. CCS cost for (i) absorption-, (ii) membrane-, (iii) PSA-, and (iv) VSA-based CO2 capture processes at different feed CO2 compositions (covering major industrial and chemical sectors, namely, (A) energy and heat generation, (B) chemical industries, and (C) others) and feed flow = 1 kmol/s.

each technology. Furthermore, the investment costs do not vary linearly with the feed flow rate. Figures 12,13, 14, and 15 show the total costs for each ton of CO2 captured and compressed, when the feed flow rates are 0.1, 1, 5, and 10 kmol/s, respectively. Interestingly, the total cost ($/ton of CO2 captured and compressed) decreases with CO2 composition. At higher compositions, the costs for the absorption-based process are high compared to those for the other three processes. However, significant cost reduction in the CCS chain is possible by capturing CO2 from highly

membrane-based process can be 2 times as high as those of the absorption-based process. However, it is also observed that the investment and operating costs of the absorption-based process increase more sharply than those of the other processes. This allows the VSA process to be the most economical at moderate and high feed CO2 compositions. As a general trend, the investment and operating costs for all the technologies, except for the membrane-based process at low CO2 compositions, show almost linear relations with feed CO2 composition. However, the incremental cost as the feed CO2 composition increases is different for 15679



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concentrated flue gases, using adsorption- and membranebased processes. Based on the feed gas CO2 composition, we note that absorption is the technology of choice for the energy and heat generation sector that includes coal and natural gas fired power plants, which exhibit CO2 compositions less than 15−20%. However, for chemical industries for which the emission scenarios are different and the CO2 compositions are higher than those of the power plants, adsorption- and membrane-based processes are more economical. Although adsorption- and membrane-based

processes are not as popular as the absorption-based process for CO2 capture, our results suggest that these can be economically feasible technological options for CO2 capture at high concentrations. Considering the diverse emission scenarios in the industrial sector in general, adsorption and membrane technologies might play a vital role in CO2 mitigation in the future. The effect of feed flow rate is also evident from the results. Costs are higher at low feed flow rates and vice versa. Interestingly, the cost increases more sharply with decreasing feed flow rate at low CO2 compositions. This highlights the benefits of economy of scale. 15680



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Even for a fixed CO2 composition, the total cost varies significantly with technology and total feed flow rate. This implies that only selecting a technology for CO2 capture is not enough; we must thrive for the economies of scale in terms of the feed flow rate, or capture CO2 from larger sources.

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6. CONCLUSION Two adsorption-based CO2 capture technology alternatives, namely, pressure swing adsorption (PSA) and vacuum swing adsorption (VSA), are modeled, simulated, and optimized based on a superstructure representation of the PSA and VSA process configuration for CO2 capture coupled with compression. A nonlinear algebraic and partial differential equation (NAPDE) based nonisothermal adsorption model was used, which is fully discretized and solved via a kriging model. The PSA- and VSAbased processes are optimized for a wide range of feed CO2 compositions and total feed flow rates. Explicit expressions for the investment and operating costs as a function of feed CO2 composition and total feed flow rate are also obtained for the PSA- and VSA-based technologies. This study was intended to investigate the optimal selection of CO2 capture technologies for various industrial, chemical, and electricity and heat generation sectors. A cost-based comparison between the absorption-, membrane-, PSA-, and VSA-based CO2 capture technologies suggests that the CO2 composition and total flow rate of the feed gas have significant effects on the CO2 capture and compression costs. Therefore, the CO2 capture costs would vary significantly from industry to industry, and even for the same industry with different CO2 emission rates. Results emphasize that we have to change our perspective from sector to sector, even from plant to plant, and our carbon mitigation effort will benefit more by early adopting carbon capture and storage technologies for the industries with higher CO2 concentrations. In addition, we may need to consider several technology options (instead of adopting the approach of “one technology suits all”) to accommodate industries with diverse CO2 emission scenarios for a comprehensive carbon management.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: (609) 258-4595. Fax: (609) 258-0211. E-mail: floudas@ titan.princeton.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge partial financial support from the National Science Foundation (NSF EFRI-0937706 and NSF CBET-1158849).

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REFERENCES

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Modeling, Simulation, and Optimization of Postcombustion CO2

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