Investigation of CO2 Sorption Mechanisms in Isothermal Columns via

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Kinetics, Catalysis, and Reaction Engineering

Investigation of CO2 sorption mechanisms in isothermal columns via transient material and energy balance PDE models. Manda Yang, Linxi Wang, Seyed Mehdi Kamali Shahri, Robert M. Rioux, and Antonios Armaou Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02176 • Publication Date (Web): 09 Jul 2018 Downloaded from http://pubs.acs.org on July 16, 2018

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Industrial & Engineering Chemistry Research

Investigation of CO2 sorption mechanisms in isothermal columns via transient material and energy balance PDE models. Manda Yang,† Linxi Wang,† Seyed Mehdi Kamali Shahri,† Robert M. Rioux,∗,†,‡ and Antonios Armaou∗,†,¶,§ †Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, USA ‡Department of Chemistry, The Pennsylvania State University, University Park, PA 16802, USA ¶Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA 16802, USA §Department of Mechanical Engineering, Wenzhou University, Zhejiang 325005, China E-mail: [email protected]; [email protected]

Abstract The behavior of an isothermal packed bed sorption column for CO2 is investigated based on combined mass spectrometry and calorimetry temporal measurements at different temperatures. The inclusion of the calorimetry data stream at multiple temperatures accentuates the limitations of our previous spatiotemporal model with a simple sorption mechanism to describe the experimental observations. We propose four models, which consider dispersion and convection with different descriptions of the sorption mechanism. The unknown parameters in the models are determined by

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minimizing the integral in time of the squared difference between model prediction and the experimental measurements. Analyzing the simulation results for thermodynamic consistency and site density, we conclude physical sorption followed by chemical sorption is a probable mechanism to describe the specific experiment data; we also conclude that a combination of multiple measurement streams is required to verify the consistency of proposed mechanisms and establish a clear(er) macroscopic description of the underlying physicochemical phenomena.

1

Introduction

In recent years, extensive research effort has been dedicated to CO2 capture technology due to the contribution of CO2 emissions to global climate change. 1–3 Technologies to mitigate CO2 include pre-combustion carbon capture, post-combustion carbon capture and oxy-combustion. 4–6 Compared with other technologies, post-combustion carbon capture has the advantages of easier implementation in existing plants and maintenance operations don’t require main plant operation cycle to shut down. 7 Post-combustion capture materials include but not limited to physical solvent (Pursiol, Selexol, Rectisol), 8 composite membranes, 9 CO2 capture sorbents, 10,11 metal organic frameworks, 12 enzyme-based system, 13 aqueous ammonia 14,15 and Higee technology. 16 While ammonia based CO2 scrubbing is a mature technology, solid CO2 sorbents are less caustic and consume less energy during regeneration. 16 To predict the dynamics of CO2 sorption in a fixed bed column, many models have been proposed in the literature. 7,17–24 In work by Heydari-Gorji, 19 an adsorption model based on Avrami’s equation with no consideration of transport phenomena was proposed. In Monazam’s work, 25 different adsorption models were compared to determine the equilibrium relationships between sorbent and sorbate; the adsorption rate was considered to be a function of CO2 concentration, surface coverage of sorbed CO2 and temperature. However, axial dispersion in the column wasn’t considered. Knox’s work 26 considered axial dispersion in the column, but an unphysical assumption of linear driving force was made. Bollini and 2


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coworkers considered both heat and mass transfer in their model, 27 but were unable to capture the experimental temperature profile in their model. In previous work, 28 we proposed a model to circumvent the unphysical assumptions of linear driving force and uniform adsorption rates enabling the adsorption rate constants and capacity of the bed to be determined from the breakthrough curves. However, the released heat due to adsorption as measured by microcalorimetry was not accurately captured by the previous model. The current work improves the accuracy of our previous model by relaxing some assumptions, and then considers different reaction mechanisms to describe the experimental observations. Physical quantities are estimated (including concentration of accessible amine sites, relevant rate constants and heat of sorption) by minimizing the deviation of the model prediction from the experiment measurements in time. In our previous model, we only examined the concentration of CO2 during sorption. To provide a richer behavior for estimation, in this work, we utilized the entire pulse of CO2 and the associated calorimetry profiles at various temperatures.

2 2.1

Experimental apparatus Materials

The material studied in this work was received from National Energy Technology Laboratory (NETL) and used as-is. The material consists of 40% polyethyleneimine (PEI) supported on SiO2 (Cariact) which was prepared by wet impregnation. PEI was dissolved in methanol and mixed with silica in a weight ratio of 0.67. The PEI was a linear type with an average molecular weight of 423 g/mol. We conducted physical sorption characterization on the 40%PEI/silica sample with N2 at 77 K. The sample has a BET surface area of 69 m2 /g and pore volume of 0.32 cm3 /g.

3



2.2

CO2 breakthrough reactor (BTR) measurements

The sorption behavior of CO2 was studied in a home-built breakthrough reactor (BTR). A schematic of the BTR with a description is provided in Supporting information §S1. We weighed 40% PEI-impregnated SiO2 (65 mg) and packed it into the 1/4” O.D. quartz tube positioned between two pieces of quartz wool. A pulse of CO2 was introduced to the BTR through a 10-way valve which switches between the purge gas (Helium, AirGas, UHP, 99.9999%) and sorption gas (10% CO2 /1% Ar/He, AirGas, UHP, 99.9999%). Ar served as a tracer gas in order to accurately determine the total CO2 capacity of the BTR by correcting for valve dynamics. The effluent gas was monitored with a Hiden HPR-20 QIC R&D mass spectrometer. In every run, CO2 (m/z = 44), Ar (m/z = 40), H2 O (m/z = 18) were followed. The total capacity of CO2 was determined via integration of the area between the breakthrough curve for CO2 and the tracer gas, Ar. Our breakthrough reactor was coupled with a differential scanning calorimeter (Setaram, Sensys Evo TG-DSC) to measure the transient heat flow necessary to maintain the BTR at isoperibol conditions. BTR curves were measured at 25°C, 40°C, 50°C, 60°C, 70°C and 80°C. Prior to each measurement, we pretreated the sample under He flow at 105°C for 1 h, and monitored the H2 O and CO2 concentrations in the effluent stream until they became negligible. The bed was subsequently cooled to the sorption temperature and held for 1 h. We switched the 10-way valve to direct the sorption gas through the packed bed, and tracked the concentration profile of CO2 and Ar with time. Ar does not sorb to the sample and acted as an inert tracer indicating the time CO2 entered the packed bed. The Ar and CO2 signals were normalized by their maximum ion count value (plateau region of the BTR curve) to determine the normalized concentration (c/c0 ). We also considered the change of volumetric flow rate based on the missing CO2 (≤ 10% of the total gas flow), and multiplied the corrected volumetric flow rate by the normalized concentrations to obtain the normalized molar flow rate. The flow rates of Ar and CO2 were plotted with time and the sorption capacity was determined by integrating the area enclosed by the Ar and CO2 curves, and multiplied with the inlet molar 4


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flow rate calculated from the ideal gas law (Eq. 1).

F0 =

γ0 P0 v0 RT

(1)

where v0 is the inlet volumetric flow rate (30 sccm), γ0 is the percentage of CO2 /Ar in inlet (γCO2 ,0 = 0.1 and γAr,0 = 0.01), P0 is the total pressure of the gas (13.9-14.1 psia). The heat generated from CO2 sorption was measured by the DSC, which recorded the transient heat flow during the entire experiment. As CO2 sorbed to the sample and later saturated, the heat flow increased and then returned to the initial baseline. The total exothermic heat was calculated by integrating the area under the heat flow curve from the baseline. Enthalpy values normalized to the total number of moles of CO2 sorbed were calculated from the integrated heat flow and the total column capacity. The length of the column was L = 10 mm. The superficial gas velocity was calculated based on the ideal gas assumption at different temperatures. The value of the skeletal density of the column was ρb = 1.411 × 103 kg/m3 as measured by He pycnometry. Void fraction was calculated using the average bulk density ρ and the skeletal density ρb , = 1 −

ρ ρb

= 0.5716.

Due to the time constant of DSC, the delay in the transient heat flow signal was corrected. To account for this dynamic delay between the onset of heat flow and the recorded heat flow, we employed a first order dynamic model (Eq. 2) and identified the time constant τ of the calorimeter τ

dq + q = q0 dt

(2)

where q is the measured heat flow profile and q0 is the heat flow produced by sorption. The measured time constant value for the apparatus used in this experiment was computed to be 42.1 s. A detailed discussion of the experimental procedure used to determine the time constant is presented in Supporting information §S.1.

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3

Derivation of BTR spatiotemporal model

In order to derive a physically relevant model we avoided implementing unphysical assumptions made in other reports. 19,26,27 However, to reduce model complexity we did make the following assumptions 1. The process is isothermal. The experiment measurement showed that the offset of the temperature at the wall of the column was less than 0.3°C and the amplitude of the oscillation was less than 0.02°C. Therefore, assumption 1 is valid. 2. CO2 concentration gradient in the axial direction is considered; radial gradients can be neglected. This assumption is validated by the fact that the Reynolds number is around 3 and L/Rp > 100 in the experiment. 29 3. Helium and Argon do not sorb. 4. The transport behavior (dispersion, convection) of Helium and Argon are identical. 5. The thermodynamic behavior of all gaseous species can be described by the ideal gas equation of state. The compressibility factors of CO2 , He and Ar 30,31 in the experiments are very close to 1. Therefore, assumption 5 is valid. To extend our previous results 28 and improve the accuracy of the proposed model, we relaxed the following assumptions (i) The total concentration of accessible sites on the sorbent column remains constant. (ii) The superficial velocity is uniform in space. To justify relaxing Assumption (i) we considered that although the total concentration of the active sites should be the same at different temperatures, the higher kinetic barrier for the diffusion of CO2 at lower temperatures reduces the number of accessible sites. 32 Based on Assumption 3 and 5, the concentration of Ar will increase when CO2 is sorbed, therefore the superficial velocity will decrease (Assumption ii). 6


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3.1

Transport behavior in BTR spatiotemporal model

Applying material conservation, we can obtain 28 ∂CCO2 ∂t

=−

∂FCO2 z ∂z

−

1− ρb rsor

(3)

where z is the distance in axial direction of the column, t denotes time, CCO2 refers to the concentration of CO2 in the gas phase. FCO2 z is the molar flow rate at position z. ρb [kgsor /m3 ] denotes the density of the sorbent, rsor is the sorption rate of CO2 , and refers to the void fraction. Applying Fick’s first law, we obtain

FCO2 z = −DL where ui =

us

∂CCO2 ∂z

+ ui CCO2

(4)

denotes the interstitial velocity and us is the superficial velocity. The axial

dispersion coefficient DL is determined by

DL = γ1 Dm + 2γ2 Rp ui

(5)

where γ1 = 0.45 + 0.55, Rp = 5 : 625 × 10−2 mm is the sorbent pellet radius and γ2 = 0.5. 33 To consider the dependence of diffusion coefficient Dm on temperature, Chapman-Enskog theory 34 is applied. The values of Dm are given in Supporting Information §S.2 Table S1. By substituting FCO2 z ( Eq.4) in Eq. 3, we obtain ∂CCO2 ∂t

= DL

∂ 2 CCO2 ∂z 2

−

CCO2 ∂us 1 − us ∂CCO2 − − ρb rsor ∂z ∂z

(6)

The superficial velocity varies in space because of the sorption and desorption of CO2 . Our measurement indicates pressure drop is negligible; therefore

CCO2 + CHe + CAr = const.

7


(7)


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We assume the time derivative of the concentration of He and Ar is negligible; therefore the molar flow rates of He and Ar are uniform. The superficial velocity over space becomes

us |0 (CHe |0 +CAr |0 ) = us (CHe |0 +CAr |0 +CCO2 |0 −CCO2 )

(8)

where |0 refers to the value of the variable at the inlet of the column (z = 0).

3.2

Reaction mechanism incorporated in BTR spatiotemporal model

In this section, we modified the reaction mechanism used in our previous work, 28 then introduced four additional sorption models, including a model based on the ammonium carbamate reported in the literature, two models based on physical sorption 35 and a model based on simultaneous single-site sorption and dual-site sorption. 36

3.2.1

Dual-Site Chemisorption Scheme (DSCS)

This is the common carbamate mechanism 37,38 with no distinction as to the type of amine sites. Since the elementary reactions are lumped into one reaction, the number of parameters in the model is reduced

k1

−− CO2 (g) + 2 S ) −* − S(CO2 )S ∆H k−1

(9)

where S denotes site and S(CO2 )S refers to one molecule of CO2 being sorbed to two sites. The net sorption rate is

rsor = k1 PCO2 CS2 − k−1 CS(CO2 )S

8


(10)

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and the rate of heat released by the reactions at each point in space is

rh = (k1 PCO2 CS2 − k−1 CS(CO2 )S )∆H1 ρb (1 − )

(11)

The unit of rh is W/m3 . Previously, 28 the equilibrium sorption constant was used to reduce the number of unknown parameters in the model. Presently, we no longer use such constant because the reactions are not at equilibrium after breakthrough. Another reason is using equilibrium sorption constant requires calculating the total amount of sorbed CO2 beforehand. However, the calculated total amount of sorbed CO2 may not be accurate because of the baseline change and the long tail of the breakthrough curves.

3.2.2

Amine Carbamate Configuration Specific (ACCS) scheme

Under dry conditions, the main reactions between PEI (primary and secondary amines) and CO2 25 are k

1 + − −− 2 RNH2 + CO2 ) −* − RNH3 + RNHCOO ∆H1 primary − primary

k−1

k

2 + − R2 NH + CO2 + RNH2 − )− −* − R2 NH2 + RNHCOO ∆H2 primary − secondary k−2

(12)

k

3 + − 2 R2 NH + CO2 − )− −* − R2 NH2 + R2 NCOO ∆H3 secondary − secondary k−3

Since only the beginning and the end of the long polymer chain have a primary amine, the probability a primary amine reacts with another primary amine is low compared with the other two reactions. We believe intermolecular carbamate formation is a minor reaction pathway. Therefore, we only consider the second and third reactions.

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Based on reaction scheme shown in (12), we obtain the following reaction rates:

r2a = k2 CR2 NH CRNH2 PCO2 r2d = k−2 CR

2 NH2

r3a =

+

CRNHCOO−

(13)

k3 CR2 NH PCO2 2

r3d = k−3 CR

2 NH2

+

CR

2 NCOO

−

where PCO2 [P a] is the partial pressure of CO2 . The net sorption rate is then rsor = r2a − r2d + r3a − r3d

(14)

and the heat released by the reactions at each point in z is

rh = [(r2a − r2d )∆H2 + (r3a − r3d )∆H3 ]ρb (1 − )

3.2.3

(15)

Dual-site Physisorption-Chemisorption Scheme (DPCS)

This scheme is proposed based on the hypothesis that CO2 is sorbed by physical sorption and then converted to chemically sorbed CO2 . In the work by Ebner et al., it is assumed physically sorbed CO2 reacts with an adjacent amine. 35 Compared to their work, we relax the linear driving force assumption and the assumption that physically sorbed CO2 is in equilibrium with CO2 in the gas phase. The heat of physisorption of CO2 is smaller than chemisorption (around 25 to 40 kJ/mol); 39 to simplify the model, we don’t consider the heat of physisorption in our model. To reduce the number of parameters in the model, we don’t differentiate between primary and secondary amines. They are both denoted by S. In this scheme, one molecule of CO2 is physically sorbed on 2 unoccupied sites. The sorption is reversible. Physically sorbed CO2 converts to

10


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chemically sorbed CO2 . k1 −− CO2 (g) + 2 S ) =0 −* − CO2 (p) ∆H0 ∼ k−1 k

(16)

CO2 (p) −−2→ CO2 (c) ∆H k−2

CO2 (c) −−→ CO2 (g) + 2 S

− ∆H

The net sorption rate is

rsor = k1 PCO2 CS2 − k−1 CCO2 (p) − k−2 CCO2 (c)

(17)

and the heat released by the reactions at each point in space is

rh = (k2 CCO2 (p) − k−2 CCO2 (c) )∆Hρb (1 − ) 3.2.4

(18)

Single-site Physisorption-Chemisorption in Series (SPCS) scheme

This scheme is also based on physical sorption. The difference between DPCS and SPCS lies in the (stoichiometric) amine requirements by each CO2 in the physisorption process, where SPCS requires one amine while DPCS requires two. k1 −− CO2 (g) + S ) =0 −* − CO2 (p) ∆H0 ∼ k−1 k

CO2 (p) + S −−2→ CO2 (c) ∆H k−2

CO2 (c) −−→ CO2 (g) + 2 S

(19)

− ∆H

The net sorption rate is

rsor = k1 PCO2 CS − k−1 CCO2 (p) − k−2 CCO2 (c)

11


(20)


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rh = (k2 CCO2 (p) CS − k−2 CCO2 (c) )∆Hρb (1 − ) 3.2.5

(21)

Competitive Single-/Dual-site Sorption (CSDS) scheme

Serna-Guerrero argues “carbamate is not a reaction intermediate in the generation of bicarbonate” and there might be two competitive reaction pathways. 40 Based on this hypothesis, we proposed a two reaction scheme, which is a combination of single-site sorption and dualsite sorption. k1

−− CO2 (g) + S ) −* − S(CO2 ) ∆H1 k−1

(22)

k2

−− CO2 (g) + 2 S ) −* − S(CO2 )S ∆H2 k−2 where S(CO2 ) denotes one molecule of CO2 being sorbed to a single site. Primary and secondary amines are not differentiated because in this model, the competitive reactions are considered to be more important and a more complicated model may result in overfitting. The net sorption rate is

rsor = k1 PCO2 CS + k2 PCO2 CS2 − k−1 CS(CO2 ) − k−2 CS(CO2 )S

(23)


rh = [(k1 PCO2 CS − k−1 CS(CO2 ) )∆H1 + (k2 PCO2 CS2 − k−2 CS(CO2 )S )∆H2 ]ρb (1 − )

3.3

(24)

PDE model of BTR mass and energy spatiotemporal behavior

Combining the various sorption schemes (Eq. 10, 14, 17, 20, 23) and Eq. 6, we can obtain PDEs models which have the following general form (the specific forms are given in

12


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Supporting Information §S.2). ∂CCO2 ∂t

=DL

∂ 2 CCO2 ∂z 2

−

h CCO

2

us i ∂CCO2 us |0 (CHe |0 +CAr |0 ) + ∂z (CHe |0 +CAr |0 +CCO2 |0 −CCO2 )2

(25)

1− − ρb rsor ∂Cpro = A C(Cpro , CCO2 , Ct ) ∂t us =

us |0 (CHe |0 +CAr |0 ) CHe |0 +CAr |0 +CCO2 |0 −CCO2

(26) (27)

where Ct denotes the total number of sites; Cpro is a vector of products of the reactions; A is a matrix of unknown rate constants; and each element in C ∈ Rm×1 is a function of Cpro , CCO2 and Ct . Eq. 26 describes how the concentration of the products in each scheme change over time and space. The value of m depends on the reaction rates in each scheme. In DSCS, Cpro ∈ R1×1 , and A ∈ R1×m . In other four schemes, Cpro ∈ R2×1 , and A ∈ R2×m . This is because DSCS has fewer reactions than other schemes, and is subject to initial conditions:

CCO2 (z, 0) = 0

(28)

Cpro (z, 0) = 0

(29)

CCO2 (0, t) = CCO2 |0

(30)

∂CCO2 (z, 0)

(L, t) = 0

(31)

πd2 4

(32)

and boundary condition

∂z The total heat released by the column is Z Q=

L

rh dz 0

where L and d are the length and the diameter of the column.

13



3.4

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Parameter estimation from BTR PDE modeling

Prior to each breakthrough experiment, we pretreated the sample under pure He flow to remove moisture and sorbed CO2 . The sorption experiment begins at t = 0, when sorption gas containing CO2 , Ar and He enters the reactor; at t = t1 , the sample is saturated and the concentration of exiting CO2 stabilizes at a maximum; at t = t2 the gas is switched back to pure He to initiate pressure swing desorption. In the whole scenario, [0 t1 ] is called the “upswing”, [t1 t2 ] the “plateau period” and [t2 t3 ] is called the “downswing”. An illustrative breakthrough process is displayed in Fig. 1, where t1 = 10 min and t2 = 150 min. In this illustration, the first 10 min is the upswing, 10-150 min is the plateau period and 150-160 min is the downswing. Since the length of plateau period (∼9000 s) is much longer than upswing (∼300 s) and downswing (∼1000 s), we only consider the deviation of the estimated breakthrough curve and heat profile from the experiment result during the upswing and the downswing. The integral of the squared deviation over time is balanced by the length of each period so that the data in the upswing and the downswing is equally weighted in the cost function (Eq. 33). In each of the five reaction schemes, the unknown parameters are displayed in Table 1, where dt is used to align the heat flow data stream and the breakthrough curve data stream. We denote the set of unknown parameters by Ui , where i indicates a different experimental temperature Ui∗

=arg min α Ui

Z

1 exp t1 (CCO

1

)2 2 ,max,up Z t3

0

t1

exp (CCO2 (L, t) − CCO (L, t))2 dt+ 2

exp CCO (L, t))2 dt 2

(CCO2 (L, t) − exp (t3 − t2 )(CCO )2 t2 2 ,max,down Z t1 1 2 (QTi (t) − Qexp exp Ti (L, t)) dt+ 2 t1 (QTi ,max,up ) 0 Z t3 1 2 exp (QTi (t) − QTi (t)) dt 2 (t3 − t2 )(Qexp t2 Ti ,max,down )

+ (33)

where CCO2 max and QTi max denote the maximum of the breakthrough curves and heat profiles 14


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at T = Ti . The superscript exp refers to the experiment result and superscript ∗ denotes optimal values. When α = 1, the deviation of the breakthrough curve and heat flow profile are penalized equally in the optimization problem. When the concentration of CO2 in the outlet increases to the maximum, the error of all the models becomes negligible; yet the model prediction of heat flow does not match the experiment result. Because of this, the values of the last 2 terms in Eq. 33 are larger than those of the first 2 terms if α is 1, which will result in the better fit of heat flow profile than the breakthrough curves because the deviation of heat flow profile are penalized more compared with the deviation of breakthrough curves. To balance the two different data streams we choose α = 10.

3.5

Estimation of CO2 concentration in BTR inlet

In our previous work, 28 the CO2 inlet concentration profile was obtained by considering the spatiotemporal PDE model of the tracer (Ar) concentration. us ∂ ∂2 ∂ CAr (z, t) = DAr 2 CAr (z, t) − CAr (z, t) ∂t ∂z ∂z

(34)

Using the Ar concentration in the outlet, the Ar concentration in the inlet can be obtained, enabling the determination of the CO2 inlet concentration CCO2 ,in (t) CCO2 ,0

=

CAr,in (t) CAr,0

(35)

However, the residence time of Ar is very short compared with the timescale of the whole process; as a result, the difference between the profile of inlet concentration and outlet concentration is negligible, which was confirmed by simulation. The simulation result is given in Supporting Information §S.1.2, which demonstrates the effect of dispersion and convection of Ar inside the column is negligible. Another issue with this approach is it was based on the assumption that volumetric flow rate of the gas mixture remains constant through the column, yet the volumetric flow rate decreases by 10% when CO2 is completely 15



Page 16 of 43

sorbed. Therefore, in this work, we relax the uniform flow rate assumption and Eq. 34 is no longer applicable. Based on assumptions 1 and 5, the total concentration of all the gases in inlet and outlet is constant. CCO2 ,in + CAr,in + CHe,in = CT

(36)

CCO2 ,out + CAr,out + CHe,out = CT

(37)

Based on assumption 4, we obtain CAr,out CAr,in = . CHe,in CHe,out

(38)

Combining Equations 35, 36, 37 and 38, we can solve for CCO2 ,in using CCO2 ,out and CAr,out . Considering that

CCO 0 2 CT

= 10% and

CCO2 inlet = where CCO2 inlet =

CCO ,in (t) 2 , CCO ,0 2

CAr0 CT

= 1%, we obtain

CAroutlet 1 − 0.1CCO2 outlet + 0.1CAroutlet

CAroutlet =

CAr,out (t) CAr,0

and CCO2 outlet =

(39)

CCO ,out (t) 2 . CCO ,0 2

We first use a low pass filter to reduce the noise in the raw data for both Ar and CO2 . The filtered Ar signal (

CAr,out (t) ) CAr,0

is displayed in Fig 2; it can be observed that because of the

sorption of CO2 , the concentration of Ar increases to a value greater than the inlet CAr,0 and then returns to CAr,0 . We use Eq. 39 to obtain the concentration of CO2 in the inlet, which is displayed in Fig 2. The concentration of CO2 should not be more than CCO2 ,0 , but during t ∈ [100, 300], the calculated CO2 concentration in the inlet is larger than CCO2 ,0 . To deal with this problem, CCO2 inlet is forced to be less than or equal to 1 at all temperatures. To check how this will affect the value of Ct , we calculate the amount of sorbed CO2 using the spatial integration of CCO2 inlet − CCO2 outlet . As temperature increases, the error introduced by forcing CCO2 inlet to be less than 1 decreases; at T = 25°C, the amount of sorbed CO2 is 5% less than the real value. Therefore, we consider this data preprocessing to be acceptable. 16


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4

Results and Discussion

4.1

Experimental BTR determination of CO2 sorption capacity and heat of sorption

We determine the CO2 sorption capacity and heat of sorption by calculating the total amount of sorbed CO2 and released heat during the breakthrough experiment. CO2 capacity is equal to total amount of sorbed CO2 divided by total mass of the sorbent. ∆Hsor is equal to released heat during the breakthrough experiment divided by total amount of sorbed CO2 . As listed in Table 2, the CO2 sorption capacity increases with temperature at low temperatures, maximizes at 50°C, and decreases at high temperatures. Similar results have been observed in other PEI-impregnated materials at high loadings, and this behavior has been attributed to a balance between diffusional limitations at low temperature and LeChatelier’s Principle at high temperature. 41,42 The calculated heat of sorption lies within a narrow range between 67 and 73 kJ/mol, which suggests CO2 sorbs to sites with similar binding energies regardless of the temperature.

4.2

Results of BTR spatiotemporal simulations

Initially, we used a moving average filter to smooth the temporal profile of CO2 and Ar, then Eq. S-1 to obtain the temporal profile of the heat released due to sorption/desorption. We used Eq. 39 to calculate the concentration of CO2 in the inlet. With the concentration of CO2 in the inlet as a boundary condition, we can solve the optimization problem of Eq. 33 at each temperature. We obtained the rate constants, total accessible sites and heat of sorption (the results are given in Supporting Information §S.2 in Table S2, S4, S7, S9, and S11 for each mechanism). We started our investigation considering the DSCS mechanism. DSCS is the same reaction scheme as that in our previous work, 28 but with modifications to the transport phenomena and relaxed assumptions. In DSCS, one molecule of CO2 is chemically sorbed by 17



2 molecules of amine via a reversible reaction. The temporal profiles of dimensionless outlet concentration of CO2 and heat flow at different temperatures and the model prediction based on the updated DSCS are presented in Fig. 3. The model prediction provides a good fit to the experiment result. To describe the temperature dependence of the rate constants, we employed the Arrhenius equation k = A0 exp(−Ea /R T ). Parameter values and the corresponding Arrhenius plot are presented in Table 3 and Fig. 4. Although the rate constants can be described well by the Arrhenius equation (note the high values of R2 in Table 3), the activation energy of k1 is negative. One possible explanation is the dynamics of the forward reaction in our model may result from a more complicated mechanism involving more reactions, and the activation energy of a reverse reaction is larger than that of the forward reaction. 43 Another possible reason is CO2 has to be physically sorbed first before it reacts with amine. 35 Physical sorption rate decreases as temperature increases, which results in an apparent negative activation energy. As DSCS is kinetically inconsistent, we concluded DSCS is not descriptive of the phenomena that take place during the experiment. Based on the first possible explanation, we hypothesized the forward reaction is not an elementary reaction and we proposed Amine Carbamate Configuration Specific scheme (ACCS) and Competitive Single-/Dual-site Sorption scheme (CSDS); based on the second possible explanation of our observation, we also proposed Dual-site Physisorption-Chemisorption Scheme (DPCS) and Single-site Physisorption-Chemisorption in Series scheme (SPCS). These mechanisms can potentially eliminate the problem of negative activation energy in the reaction rate constant at the expense of more elaborate mechanisms that include, instead of 2 reactions, 4 elementary reactions. This results in more complicated reaction rate expressions, but the reaction rates are consistent with the theoretical observation that activation energy has be to positive. In DPCS, one mole of CO2 is physically sorbed by 2 moles of amine. The amount of physically sorbed CO2 increases during the beginning of the upswing and then decreases 18


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when most of the physically sorbed CO2 is converted to chemically sorbed CO2 . When the concentration of CO2 decreases during the downswing, the chemically sorbed CO2 will desorb into gas phase directly. SPCS is similar to DPCS but requires an amine in both the physisorption and chemisorption steps, while the latter only requires amines in the physisorption step. ACCS considers CO2 reaction with a secondary amine and a primary amine, and with two secondary amines. Both reactions are reversible. Compared to ACCS, CSDS doesn’t differentiate between primary amine and secondary amine. Because of space limitations, we only present results for DSCS and DPCS here. A more detailed analysis and parameter values of SPCS, ACCS, and CSDS are given in Supporting Information §S2. 4.2.1

Simulated fitting results of breakthrough and corresponding heat profile behavior

The breakthrough curves and heat profile as predicted by the DPCS model are depicted in Fig. 5 with the corresponding experimental result. All the schemes can capture the behavior of breakthrough curves and heat profiles except for SPCS, which poorly fits the upswing of breakthrough curves. A graphical comparison of all five reaction schemes at each temperature is given in Supporting Information §S4 Fig. S21-S44. We also calculated the total amount of sorbed CO2 in the upswing and the desorbed CO2 in the downswing based on the experiment result and simulation prediction. For brevity, the relative error is given in Table S13 in supporting information §S4. We observe the error of the upswing is smaller than the downswing. This is because in the cost function (Eq. 33), the average absolute errors over time of upswing and downswing are penalized equally while the average value of downswing is smaller than the upswing. As a result, the relative error of downswing is larger than the upswing.

19



4.2.2

Estimation of accessible amine sites concentration during transient breakthrough

The total number of sites optimized from different schemes are compared in Fig. 6. We calculated the total amount of possible sites to be 9.3 ∼ 10.6 mol/kgsor based on the proposed structure of the PEI polymer, assuming every two amines sorb one CO2 . The total number of sites predicted by all 5 schemes is smaller than 10.6 mol/kgsor , since not all of amines are accessible. Compared with the other 4 schemes, the total number of sites predicted by CSDS is much smaller. This is because CSDS predicts most of CO2 is sorbed by a single site, which is contrary to the consensus that most CO2 is sorbed through dual-site interaction. 36 Therefore, CSDS is unlikely the best scheme for our system. In ACCS, SPCS, DSCS, and DPCSschemes, the number of accessible sites increases with increase in temperature within some range and decreases with further temperature increase. This is due to a compromise between CO2 diffusion and sorption thermodynamics, where the increasing temperature prompts the CO2 diffusion into the bulk polymer, but decreases the stability of sorbed CO2 species. 41,42,44 4.2.3

Estimation of the heat of CO2 sorption during transient breakthrough

The optimized heat of sorption (∆H) based on different schemes are compared in Fig 7. The ∆H values of DSCS, DPCS, and SPCS show good agreement with experiment results, while ∆H in CSDS and ACCS shows random variation and inconsistent with experimental results. The heat of CO2 sorption to solid amine materials often spans from 60-90 kJ/mol 45–47 and is a function of amine density, 48 CO2 coverage 48,49 and other experimental conditions. 50 For example, Li et al. found the ∆H decreases with increasing PEI molecular weight, and branched PEI have higher ∆H than linear PEIs. 45 Zhang et al observed increasing ∆H at elevated temperature, 50 which was in agreement with the ∆H for CO2 absorption in aqueous 20


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amine solutions. 51 Alkhabbaz et al. calculated the isosteric heat at different CO2 coverages and found the ∆H decreases with CO2 capacity. 48 The ∆H values for DSCS, DPCS, and SPCS lie between 60-90 kJ/mol (expt. Data), which is comparable to those reported by other groups. Some ∆H values in CSDS and ACCS are out of this range.

4.2.4

Evaluation of simulated rate constants for kinetic consistency

Even though the reaction rate constants in DSCS and DPCS lead to a good fit with the experimental breakthrough curves, an important question is if they are theoretically consistent. The Arrhenius plot for DPCS is depicted in Fig. 8 and the Arrhenius parameter values are presented in Table 4. We observe that k1 still has a negative activation energy. Note that in DPCS, k1 is the rate constant of the forward reaction of physisorption; it makes sense that the physical sorption rate decreases as temperature increases. However, the second reaction (k2 ) is a chemical reaction and shows negative activation energy. Considering DPCS is a more elaborate mechanism, we will investigate possible other reasons for this lack of theoretical consistency with simulation result. In all these four schemes, only 2 out of 4 rate constants obey the Arrhenius equation with any appreciable confidence in their fit. For example, in Table 4, rate constants k−1 and k2 have an R2 value fit of 0.55 which is in stark contrast to k1 and k−2 with R2 value fit of 0.99. This suggests we don’t have enough data to accurately determine all the rate constants in the model or the experimental system is insensitive to their value and as such from a system’s theoretical perspective, those parameters are difficult to estimate due to the structure of the model itself. To answer this question, we investigated the sensitivity of the PDE model to the reaction parameters. To analyze the model sensitivity, we increased or decreased the value of parameters determined by optimization by 20%, then calculate the value of the cost function V (Eq. 33). The value of the cost function at the minimum is denoted by V0 .

V −V0 V0

indicates

how sensitive the model is to the change in each parameter. The model sensitivity to the

21



rate constants in DSCS and DPCS is presented in Fig. 9 and 10. We can see that in the Dual-Site Chemisorption Scheme, the model is sensitive to both rate constants, while in the Dual-site Physisorption-Chemisorption Scheme, the model is sensitive to 2 out of 4 rate constants. Therefore, we do not have confidence in the other 2 rate constants. In Supporting Information §S.2 all five reaction schemes are similarly analyzed and and we observe that for all four schemes the models are sensitive to only 2 out of 4 rate constants. The sensitivity results suggest that there is a structural observability problem for the proposed reaction mechanisms and the experiments need to be further enriched in order for an observer to verify or disprove their viability to describe the reaction outcomes. To verify the model is not sensitive to k2 , we use a constant value of k2 = 100 s−1 for all temperatures (i.e., activation energy is zero) and keep the other parameters the same as those of the optimization result, then we check how the objective function (Eq. 33) changes. The result is given in §S.2 Table S6. We observe forcing k2 to be constant has a negligible effect on the objective function. This observation allows us to still consider this mechanism as a viable description of the sorption that takes place in the column. After comparing the four schemes in fitting performance, accessible sites, heat of sorption and rate constants, we conclude the DPCS has the highest potential to be the correct reaction mechanism among those investigated. However, the model was insensitive to two reaction parameters, prohibiting us from making a rigorous argument for or against this mechanism. Given the model complexity, richer data in an information theory 52 sense are needed to obtain a “unique” result of the optimization problem and argue for the correctness of the proposed mechanism.

5

Summary

We relaxed several assumptions in our previous work and proposed four reaction schemes to improve the accuracy of the model for CO2 sorption. We then determined the unknown

22


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parameters by minimizing the integral in time of the squared deviation between model predictions and experiment results. The DSCS model fits the experimental data well, but has a negative activation energy in the forward reaction, implying the underlying reaction mechanism is more complicated. By comparing the breakthrough curves and heat profiles at different temperatures and analyzing the concentration of accessible sites, heat of sorption and Arrhenius equation parameters, we concluded the Dual-site Physisorption-Chemisorption Scheme is the most probable mechanism that describes the data in a physically consistent manner. Since the model is insensitive to 2 parameters, further study is needed to investigate the validity of the scheme.

Acknowledgement M.Y. and A.A. acknowledge the financial support from the National Science foundation, CMMI award #13-00322. A.A. acknowledges the financial support from Zhejiang Provincial Thousand Foreign Talents Program, and Chinese National Fund Award, High-end Foreign Experts Program. L. W. and R. M. R. acknowledge the US National Science Foundation (NSF grant # CBET-1551119) for financial support of this work. Additional support of this work was provided to S. M. K. S. and R. M. R. by the Institutes for Energy and the Environment at the Pennsylvania State University. We acknowledge McMahan Gray and James Hoffman from the National Energy Technology Laboratory for the PEI423 material.

Supporting Information Apparatus schematic, DSC measurement time constant & Ar inlet profile computations; detailed PDE models of Competitive Single-/Dual-site Sorption, Single-site PhysisorptionChemisorption in Series, Dual-site Physisorption-Chemisorption, Amine Carbamate Configuration Specific, Dual-Site Chemisorption reaction schemes with simulation results & identified parameter values; presentations of detailed comparison results between reaction 23



mechanisms.

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Nomenclature A

matrix of unknown rate constants

Rp

sorbent pellet radius (m)

A0

pre-exponential factor

S

amine site

temperature (K) matrix: each element of which is a T function of Cpro , CCO2 and Ct Ut set of unknown parameters CT total concentration of all the gases V the value of objective function (mol/m3 ) V0 the minimum of objective function Ct total concentration of sites (mol/kg) ∆H1 , ∆H2 , ∆H3 , ∆H heat of sorption CAr,0 concentration of Ar in the inlet (kJ/mol) (mol/m3 ) d diameter of the column (m) CAr concentration of Ar (mol/m3 ) k1 , k−1 , k2 , k−2 , k3 , k−3 rate constants CCO2 ,0 concentration of CO2 in the inlet q heat flow profile we measured (W) (mol/m3 ) q heat flow produced by sorptions (W) CCO2 concentration of CO2 in the gas 0 rh (z, t) rate of heat released by the reacphase (mol/m3 ) tions (W/m3 ) CHe,0 concentration of He in the inlet rsor sorption rate of CO2 (mol/kg s) (mol/m3 ) t time (s) CHe concentration of He (mol/m3 )

C

Cpro

vector of products of the reactions

ui

interstitial velocity (m/s)

DL

axial dispersion coefficient (m2 /s)

us

superficial velocity (m/s)

Dm

diffusion coefficient (m2 /s)

v0

inlet volumetric flow rate (sccm)

DAr Ea

distance in axial direction of the colaxial dispersion coefficient for Ar z 2 umn (m) (m /s) Greek Symbols activation energy (J/mol)

F0

molar flow rate (mol/m2 s)

α

FCO2 z molar flow rate of CO2 at position z (mol/m2 s) γ1 L length of the column (m) P0

total pressure of the gas (Pa)

γ2

a coefficient used to balance different data streams void fraction coefficient γ1 = 0.45 + 0.55 coefficient γ2 = 0.5

γ percentage of certain gas in inlet PCO2 pressure of CO2 in the gas phase 0 γCO2 ,0 percentage of CO2 in inlet (Pa) Q(t) total heat released by the column (J) ρb

density of the column (kg/m3 )

R

time constant (s)

gas constant (J/mol K)

τ

31



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Table 1. Unknown parameters in each of the five reaction schemes scheme acronym DSCS ACCS DPCS SPCS CSDS

reaction rate k1 , k2 , k1 , k1 , k1 ,

k−1 k−2 , k−1 , k−1 , k−1 ,

k3 , k2 , k2 , k2 ,

k−3 k−2 k−2 k−2

unknown parameters heat of total number sorption of sites ∆H1 Ct ∆H2 , ∆H3 Ct ∆H Ct ∆H Ct ∆H1 , ∆H2 Ct

data streams alignment factor dt dt dt dt dt

Table 2. CO2 sorption capacity and heat of sorption at different temperatures measured in the breakthrough experiments Temperature (°C) CO2 capacity (mol/kgsor ) ∆H (kJ/molCO2 ) 25 1.99 72.6 40 2.10 69.8 50 2.21 67.2 60 2.13 67.6 70 1.81 69.0 80 1.45 72.8

Table 3. Arrhenius dependence of the DSCS mechanism reaction rate constants k1 k−1 1 2

R2 A0 Ea (kJ/mol) 1 0.9921 5.88E-12 -35.3 0.9940 2.04E+092 73.6

units are kgsor /mol Pa s units are 1/s

Table 4. Arrhenius dependence of the DPCS mechanism reaction rate constants k1 k2 k−1 k−2 1 2

R2 0.9888 0.5718 0.5229 0.9899

A0 5.46E-121 9.26E-052 3.46E-022 3.85E+092

Ea (kJ/mol) -35.4 -43.6 36.0 75.2

units are kgsor /mol Pa s units are 1/s

32


Page 33 of 43

120

plateau region

upswing

downswing

1.2 80

0.8 40

0.4

Heat flow (mW)

Normalized flow rate (F/F0)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

0.0

-40 0

5

80

85

150

155

160

Time (min)

Figure 1. Upswing, plateau period and downswing of the breakthrough experiment. The black line denotes the heat flow profile, the blue line is the normalized flow rate of Ar and the red line is the normalized flow rate of CO2 .

33


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

normalized concentration


Page 34 of 43

1

0.5

0 0

Ar outlet CO2 inlet 200

400 t(s)

Figure 2. Temporal profile of the normalized concentration of Ar and CO2 . The values of C become larger than one as a result of the sorption of CO2 .

34


Page 35 of 43

1

C/C0

C/C0

1

0.5

0 0

100

200

0.5

0 0

300

500

t(s)

1000

t(s)

(a) breakthrough curves of the upswing

(b) breakthrough curves of the downswing

150

20 0

Q(mW)

100

Q(mW)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-40

0 -50 0

100

200

-80 0

300

500

t(s)

1000

t(s)

(c) heat flow of the upswing

(d) heat flow of the downswing

Figure 3. Temporal profiles of simulation predictions based on DSCS of normalized concentration of CO2 and respective heat flow during upswing and downswing for different temperatures presented with solid lines, and the respective experimental observation presented with dashed red lines (magenta line: 25°C; cyan line: 40°C; green line: 50°C; blue line: 60°C; black line: 70°C; yellow line: 80°C)

35



-2

-6 k-1

ln(ki)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 43

-10 k1

-14 2.8

3

3.2

3.4

1000/T Figure 4. Temperature dependence (expressed as 1000/T ) of reaction rate constants (expressed as ln(k)) for DSCS(k1 [kgsor /mol Pa s]; k−1 [1/s]; T [K]).

36


Page 37 of 43

1

C/C0

C/C0

1

0.5

0 0

100

200

0.5

0 0

300

500

t(s)

1000

t(s)

(a) breakthrough curves of the upswing.

(b) breakthrough curves of the downswing.

150

20 0

Q(mW)

100

Q(mW)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-40

0 -50 0

100

200

-80 0

300

500

t(s)

1000

t(s)

(c) heat flow of the upswing.

(d) heat flow of the downswing.

Figure 5. Temporal profiles of simulation predictions based on DPCS of normalized concentration of CO2 and respective heat flow during upswing and downswing for different temperatures presented with solid lines, and the respective experimental observation presented with dashed red lines (magenta line: 25°C; cyan line: 40°C; green line: 50°C; blue line: 60°C; black line: 70°C; yellow line: 80°C)

37



10.0

(a)

7.5 5.0 10.0

(b)

7.5 5.0 10.0

Ct mol∘kgsor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 43

7.5

(c)

5.0

10.0

(d)

5.0 10.0

(e)

7.5 5.0 30

40

50

60

70

80

T ( ∘ C) Figure 6. Total sites concentration per sorbent weight as a function of temperature for the different reaction mechanisms (a. ACCS; b. DPCS; c. SPCS; d. CSDS; e. DSCS), as represented by blue lines. The theoretical amine concentration per sorbent weight is estimated from the PEI structure and lies between 9.32∼10.55. Orange lines denote 10. 38 ACS Paragon Plus Environment

Page 39 of 43

100 80

(a)

ΔH3 ΔH2

60 40 90 80

(b)

70 heatΔofΔso ptionΔkJ∘molCO2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


60 90 80

(c)

70 60 100 75

(d)

ΔH1

50 25 90 80

ΔH2 (e)

70 60

30

40

50

60

70

80

TΔ( ∘ C) Figure 7. Heat of sorption as a function of temperature for different reaction mechanisms (a. ACCS; b. DPCS; c. SPCS; d. CSDS; e. DSCS). ∆H2 of CSDS oscillates too strongly. Black lines denote the experiment result (Table 2). 39



10 k2

0 ln(ki)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 43

k-2 k1

-10

k-1

-20 2.8

3

3.2

3.4

1000/T Figure 8. Temperature dependence (expressed as 1000/T ) of reaction rate constants (expressed as ln(k)) for DPCS. k2 and k−1 don’t fit well, with Arrhenius dependence fit R2 ≈ 0.57 (k1 [kgsor /mol Pa s]; k2 , k−1 , k−2 [1/s]; T [K]).

40


Page 41 of 43

0.6

(V-V0)/V0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20% -20%

0.3

0

k1

k-1

Figure 9. Normalized sensitivity of objective function of Eq.33 to change in the value of the kinetic parameters from their optimal value for DSCS.

41



20% -20%

0.6

(V-V0)/V0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 43

0.3

0 k1

k2

k-1

k-2

Figure 10. Normalized sensitivity of objective function of Eq.33 to change in the value of the kinetic parameters from their optimal value for DPCS.

42


Page 43 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


46x26mm (600 x 600 DPI)


Investigation of CO2 Sorption Mechanisms in Isothermal Columns via

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