String To Characterize the Field Synergy during ... - ACS Publications

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String To Characterize the Field Synergy during CO2 Capture by CaCl2‑Supported MEA Adsorbent Xiao M. Wu,† De L. Mu,† Yun S. Yu,*,† and Zao X. Zhang†,‡ †

School of Chemical Engineering and Technology, Xi’an Jiaotong University, No. 28 Xianning West Road, Xi’an 710049, People’s Republic of China ‡ State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, No. 28 Xianning West Road, Xi’an 710049, People’s Republic of China S Supporting Information *

ABSTRACT: CO2 capture is quite effective for protecting the environment. A CaCl2supported MEA adsorbent was identified as an efficient adsorbent. To design this adsorbent more efficiently from the field synergy, a string method model was developed to characterize field synergistic effects. Experiments were performed to validate the string used to integrate macro and micro mechanisms of reaction and diffusion. Synergistic effects reduced the string damping time by 26%. Mole ratios of MEA/CaCl2 influenced the damping time and string mass much more than the porosity and particle diameter. The free energy and NC reaction coordinate respectively decreased by 30% and 40% as MEA/CaCl2 mole ratios decreased from 3:1 to 1:1. The addition of CaCl2 produced a 22.7% string mass decrease, which reduced the energy consumption by 30% and improved capture capacity by 27%. The string to characterize the field synergy is developed as a useful method to improve the conventional adsorbent design.

1. INTRODUCTION Effective CO2 capture is currently promising to control the greenhouse gas emission for protecting the environment.1−4 Among the varieties of CO2 capture routes, the calcium adsorbent adsorption process substantially reduces the energy penalty.5−7 The adsorbent characteristic determines the efficiency of the adsorption and desorption,8−10 where the adsorbent deactivation is identified as the main problem during the continuous adsorption and desorption cycles.5 Under this circumstance, supported amine adsorbent is developed to help adsorb and desorb CO2 at a relatively low temperature compared to conventional CO2 adsorption.11−13 However, the low CO2 capture capacity is the bottleneck of the supported amine adsorbent, which has aroused many researchers’ interest.14 Our previous work prepared a CaCl2-supported amine adsorbent to improve the CO2 capture capacity by integrating calcium adsorbent and supported amine adsorbent.15 In the preparation, monoethanolamine (MEA) and CaCl2 were mixed with the adhesive attapulgite.16 It was found that the prepared CaCl2supported MEA adsorbent improved CO2 capture capacity by 30% and reduced the energy consumption by 10%. The adsorbent performance depends greatly on the preparation conditions in the experiment. This work aims to study how to design CaCl2-supported MEA adsorbent more efficiently, by simultaneously considering the macro effects (chemical reaction and diffusion) and micro effects (molecule motion and molecule structure change) to improve the conventional design. During the conventional design, the experimental cost is normally high due to adjusting random parameters by the empirical analysis. As has been previously demonstrated, during the CaCl2supported MEA adsorbent capture CO2 process, CO2 diffusion © 2014 American Chemical Society

and chemical reaction between the adsorbent and CO2 dominate the adsorption and desorption. In this sense, the synergistic effect between chemical reaction and diffusion is important to improve the adsorbent performance. Thus, the field synergy method17−20 to describe gas and solid reaction model,21 is employed to analyze CaCl2-supported MEA adsorbent. In these researches, the synergy angle is normally used to characterize the macro synergy effects between the reaction, diffusion, and fluid flow. Additionally, field synergy has been successfully employed to optimize the multiprocess. However, it does not provide the detail of micro mechanism (molecule motion and molecule structure change) to help understand the field synergistic effects. To determine the micro mechanism, the string method, once used in micro reaction mechanism analysis, showed great ability to determine the best reaction coordinate by giving the proper committor function. The string method proceeds by evolving strings, which are smooth curves with intrinsic parametrization. This deals with the most probable transition path between two metastable regions in configuration space. Thus, the string method is introduced to indicate the molecule motion and interaction between CO2 and CaCl2-supported MEA adsorbent. Field synergy angles are used to develop the string by correlating the diffusion and reaction with molecule motion and molecule structure change inside the adsorbent particle. By the literature review, string method has been used to describe the reaction and diffusion,22,23 which has also been succeeded in computing the minimum free energy paths.24 Received: October 22, 2014 Revised: December 3, 2014 Published: December 4, 2014 473

dx.doi.org/10.1021/jp510610v | J. Phys. Chem. C 2015, 119, 473−485

The Journal of Physical Chemistry C

Article

Figure 1. Schematic of the string to characterize the field synergy. The string method and string theory are integrated to characterize the field synergy effects.

The string theory has been generally developed to offer grand unification for the micro physical phenomenon.26 In the string theory, quantum string is able to characterize all the interactions during the physical process.27 On the basis of this research, the string theory shows great potential to unify all the forces, forms of matter, and other physical phenomena into the so-called quantum string, which hence offers a great possibility to characterize the field synergy effects and strings in the string method. Different from the string in the string method, the string mass and damping time determined to characterize the quantum string28 allows us to demonstrate the dynamic characteristic of the string. Thus, the overall technical route is to characterize the field synergy by the string and then quantify the string by referencing the characterization process of the quantum string, which is schematically illustrated in Figure 1. To identify the CO2 capture process of CaCl2-supported MEA adsorbent particle, a changing grain size model, which is used for describing adsorption and desorption reactions,29,30 is first revised to obtain the synergy angle. By the changing grain size model, field synergy angle is acquired as a collective variable to quantify the string parameter and revise the molecule mean force. In this sense, the micro mechanism and macro mechanism are simultaneously quantified by the string, which analyzes the reaction, diffusion, molecule motion, and molecule structure change by the unified string parameter, string mass, and string damping time. This is expected to improve the conventional analysis (either only considering the macro parameter or just analyzing the micro parameter), guiding CaCl2-supported MEA adsorbent particle design and possibly improving the conventional adsorbent design.

These researches have shown the power of string method to analyze the complicated reaction and diffusion system, which gives interesting results between the string and free energies. This method is normally combined with a sampling technique, which usually requires the mean force and conditional expectation along the string.25 This sampling technique shows great ability to capture the transition mechanism, which is quite essential to determine the committor function of the reaction. However, interaction effects, usually occurring in reaction, are scarcely considered. Because of this trait, the field synergy angle, incorporating the interaction effects of multiprocess, is possible to offer a more real expectation and force for the string description. To achieve this, a simple way is to consider the synergy angle as one category of collective variable. Thus, this idea is to calculate the mean free path by characterizing the string with two categories of collective variables, synergy angle and dihedral angle. In this work, the mean force is redeveloped by revising the force constant with the cosines of synergy angle, which provides the synergistic effects between diffusion and reaction. After the string is characterized, the relevant flux is deduced by the string parameter accordingly, which is used to quantify the micro molecule motion, molecule structure change and the macro reaction and diffusion. This offers the possibility to compare the micro mechanism with the reaction and diffusion quantitatively. This technical route provides an alternative way to identify the reaction, diffusion, molecule motion, and molecule structure change in the same order of magnitude, which is not easy to perform by conventional method. Additionally, when CaCl2-supported MEA adsorbent adsorbs CO2 under different conditions, the string naturally produces different states and shows dynamic characteristic. Currently, there is not a good enough model to demonstrate this phenomenon, requiring more understanding about the correlation between the dynamic strings. Hence, the string is developed to describe a damping process as it varies from one state to another by referencing the analysis methods in the string theory. This is an alternative way to describe the string because of its similarities in interaction effects to the quantum string in the string theory. Damping time and mass are introduced as supplementary parameters to analyze the dynamic traits of the string.

2. STRING TO CHARACTERIZE THE FIELD SYNERGY MODEL 2.1. String Method To Describe the Field Synergy. When CaCl2-supported MEA adsorbent reacts with CO2, the free energy as a key parameter in the reaction is used to describe the complicated CO2 capture process. This free energy analysis is performed by the NVT ensemble (i.e., constant particle number N, volume V, and temperature T). The Boltzmann-Gibbs probability density function is employed to perform the statistics 474



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The diffusion flux is developed as follows:

analysis. The base of the string method is provided in Appendix A. As presented in Appendix A, the string is normally characterized by the z(α,t) and here the revision is made as follows: z(α , t ) = z(α , cos γ , cos β , t )

S 2

(1)

jM = peq

jB = peq

d 2xk

Dx

ds

+

2

dUk , z dUk , z dxk ⎛ dD ⎞ dD − x ⎟ − Dx β − x ⎜Dx β ds ⎝ ds ds ⎠ ds ds (9)

=0 kx

d 2xk ds

2

+

dUk , z dxk ⎛ dUk , z dk ⎞ dk − x ⎟ − kxβ − x =0 ⎜kxβ ds ⎝ ds ds ⎠ ds ds (10)

By referencing eqs 9 and 10, the flux caused by molecular motion and molecular structure change is supposed to follow the constraint equations below. They are used to keep the maximum flux along the molecule motion path. This is reasonable by considering the minimum resistance path in a Cartesian space, which agrees with the studied system in this work.

(3)

The free energy along the string is thus iterated according to the following: dF(z(α , cos γ , cos β , t )) ∂α N dz (α , cos γ , cos β , t ) ∂F(z(α , cos γ , cos β , t )) =∑ i dα ∂zi i=1

lcvx

d 2xk ds

−

(4)

2

+

dUk , z dUk , z dxk ⎛ d(lcvx) ⎞ − ⎜lcvxβ ⎟ − lcvxβ ds ⎝ ds ds ⎠ ds

d(lcvx) =0 ds

(11)

⎛ lclx ⎞ d 2xk dx ⎛ l l dUk , z d(lclx /t ) ⎞ lclx dUk , z ⎜ ⎟ 2 + k⎜ cxβ − β ⎟− ⎝ t ⎠ ds ds ⎝ t ds ds ⎠ t ds

The molecule interaction is predicted by the molecular dynamics, where the variations of the dihedral angle are obtained to describe the string parameter together with synergy angle. In this sense, the molecule structure and macro synergy information are integrated to determine the free energy. It is required to quantify the macro and micro mechanisms and hence the string parameter is included in the committor function q. After this, the macro and micro mechanisms are respectively identified by the reactive flux, diffusion flux, and flux due to molecule motion and its structure change. These fluxes are developed to analyze the macro and micro mechanisms in the same order of magnitude. This is improved compared to the conventional analysis, which is hard to quantify the macro and micro mechanism because of their coupled effects. The reactive flux is developed as follows:22 j

(8)

The reactive and diffusion flux satisfy the following constraint conditions to obtain the maximum flux in the process,22

In eq 2, synergy angle included in zj is used to revise the mean force and further demonstrate the synergistic effects by the string. The S is the parameter used to calculate the mean force. After these revisions, the free energy is subsequently determined as follows:

∑ Dij(∂qR /∂xj)

∑ (lclij/t )(∂qR /∂xj) j

j=1

e−βUk ,z(x)dx)

(7)

The flux caused by a molecular structure change is developed as follows:

(2)

jR = peq

∑ (lcvij)(∂qD/∂xj) j

∑ (θj(x) − zj(α , cosγ , cosβ , t ))2

∫

(6)

The flux caused by molecular motion is developed as follows:

N

Fk(z) = −kBT ln(Z −1

kij(∂qD/∂xj)

j

In previous studies, the string parameter α mainly considers the dihedral angle of the typical molecule in the reaction. The improvement here is adding the synergy angle γ and β in the string parameter. The synergy angle reflects the macro diffusion and chemical reaction, which complements the dihedral angle that only reflects the micro molecule information. Thus, the string here represents the macro and micro information simultaneously. To calculate the string, the mean force is correlated with the potential below: Uk , z(x) = V (x) +

∑

jD = peq

−

d(lclx /t ) =0 ds

(12)

2.2. Diffusion and Reaction Model To Obtain the Synergy Angle. The diffusion and reaction influence the synergistic effects in CO2 capture process. The synergy angle given in eq 1 is determined by the equations below: cos γ =

(D/l(B , ψ )) ·∇C |D/l(B , ψ )||∇C|

(13)

cos β =

(k /l(B , ψ )) ·∇C |k /l(B , ψ )||∇C|

(14)

Here, the diffusion and chemical reaction (here k is the reaction rate, m2·s−1) are characterized by their radial directions B and peripheral directions ψ, which are directly transformed to vector by the length position vector of l(B, ψ).To obtain the concentration distribution, a changing grain size model is introduced in Appendix B. On the basis of eqs 13 and 14 and models in Appendix B, synergy angle is incorporated in the string z(α,cos γ,cos β, t), which indicates the micro mechanism to the synergistic effects.

(5)

where Dij is the diffusivity component. This reactive flux reflects the flux trajectories that arrives at the product state and is a subset of the total flux. Therefore, the diffusion flux, flux caused by molecule motion and molecule structure change are developed by replacing the relevant parameters Dij and qR. This will provide the other subsets of the total flux. 475



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Figure 2. Integration of the submodel. The diffusion and reaction model are incorporated in the string method model. The string theory model is used to offer the physical meaning of the string.

2.3. String Theory To Physically Quantify the String. On the basis of the equations above, the string is obtained at a given temperature, which can be simply taken as a curve. For CO2 adsorption with CaCl2-supported MEA adsorbent, the temperature changes during the reaction process obviously produce multistates for the string. In other words, the string is supposed to vary as the CO2 adsorption or desorption occurs. It is required to understand the relation between multistates of strings, namely, to quantify the dynamic characteristic. However, the string parameter provides insufficient information to describe the dynamic characteristic of the string. Hence, the string is reasonably considered to dampen during CO2 adsorption and desorption process. The mass and damping time are deduced to characterize the string. The string deviation from one state to another is developed by referencing the quantum string deviating its equilibrium state,31,32 ∂ 2ξ 2 ∂ξ 2 + − Δξ − 2 ξ = 0 ∂t td ∂t R

By solving the equations above, synergistic effects for the CO2 capture by CaCl2-supported MEA adsorbent are characterized by the string numerically. The integration of the submodel in sections 2.1, 2.2 and 2.3 is illustrated in Figure 2, which shows the logics between the submodels. The damping time and mass of the string are identified simultaneously, which offers the micro mechanism to the synergistic effects and the string evolution.

3. MODEL VALIDATION To test the reaction and diffusion model, experiments were performed as that in our previous work.15 Because the micro performance is focused on here, 0.25 mm particle is first used in the experiments in Figure 3(a). The simulated conversions under different temperatures are compared with the experiments. Moreover, the case studies are performed under different particle sizes from 0.5 mm to 3 mm as shown in Figure 3(b). According to the reaction and diffusion model, the simulated adsorption and desorption conversions fit well with the experiment data (see Figure 3). It demonstrates that the reaction and diffusion model is effective to predict the adsorption and desorption of CO2 by the prepared CaCl2-supported MEA adsorbent. As free energy is a key factor in the adsorption and desorption process, the calculated free energies of MEA and MEA adsorption on HCl show good agreement with the literature data33−35 (see Table 1). In Figure 4, the free energy of the MEA and CO2 system shows negligible difference with literature,36 which suggests that the string method model predicts precise data. Meanwhile, the free energy of the resultant Ca2+ and CO32− under water environment is offered in Figure 5, which shows good agreement with literature data.37 The results above suggest that the overall model is accurate enough to perform the string analysis for field synergy. During CO2 capture by CaCl2-supported MEA adsorbent, there are two categories of performance, including macro performance (conformer transformation and diffusion) and micro performance (carbon (C) atom position movement, molecule motion and NC reaction coordinate change), which are significantly affected by the string. The following section will discuss the impacts of the string on the two categories of performance.

(15)

The ξ is the normal displacement of the string and td is the damping time. R is the radius curvature of the string. R = f (z(α , cos γ , cos β , t ))

ξ=

∂z(α , cosγ , cosβ , t ) dt ∂t n

(16)

(17)

By the energy conservation, the mass of the string m is theoretically deduced as follows: ⎧ m=⎨ ⎩

∫0

∂F(z(α , cos γ , cos β , t )) ⎫ dt ⎬ ⎭ ∂t

td

2 ⎧⎡ ⎛ ∂z(α , cosγ , cosβ , t ) ⎞ ⎨⎢0.5⎜ ⎟ ⎪ ⎠ ∂t ⎩⎢⎣ ⎝ ⎪

+g

∫0

∫0 td

td

∂z(α , cos γ , cos β , t ) ⎤ dt ⎥ ∂t ⎦⎥

⎫−1 ∂z(α , cos γ , cos β , t ) ⎪ dt ⎬ ⎪ ∂t ⎭

4. RESULTS The string characteristic here is related to free energy, dihedral angle, and synergy angle. The string parameter α is determined

(18) 476



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Figure 4. Free energy of the MEA and CO2 system. The literature data is obtained at the B3LYP/6-311++G(d,p) QM/MM level of theory. The N and C respectively refer to the N atom of MEA and C atom of CO2.

Figure 3. Adsorption and desorption conversion (a) and the conversion under different particle size (b). The 0.25 mm particle adsorbs and desorbs CO2 at 10% and 15% CaCl2 in (a). The 0.5 mm to 3 mm particle adsorbs CO2 at 313 K in (b). Figure 5. Free energy of carbonation in water. The effects between Ca2+ and CO32− are focused on in aqueous environment.

Table 1. Free Energy of MEA and MEA Adsorption on HCl free energy/kJ/mol system

literature data

this work

MEA MEA+HCl

5.17−12.8733,35 13.79−16.7234

6.5−11.78 13.28−17.11

with and without synergy angle. The mass and damping time of the string are offered below. All the results are given by analyzing the typical 1:1 CaCl2-supported MEA adsorbent to capture CO2. The effects of different mole ratio of CaCl2 will be discussed in Section 5. 4.1. String Characteristic. The free energy has been found to be of great importance to characterize the string,25 as it reflects the molecular structure. This is proven by the results shown in Figure 6. Here, the molecular structure is characterized by two dihedral angles. It is clearly discerned that the free energy of MEA varies from 0.01 kJ/mol to 30.8 kJ/mol. The minimum free energy typically exists at the zone with one dihedral angle around 40° and the other dihedral angle around 45°. At the extreme conditions, i.e., one of the dihedral angle close to 90°, there are still some clear minimum free energy zones in Figure 6. By the results in Figure 6, the profile of the important string parameter α is given in Figure 7(a), in which the string parameter

Figure 6. Free energy versus the dihedral angle of amine molecule. This deals with different amine molecule structures.

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8(a). As clearly presented, the larger string parameter produces the conformers with larger free energy. At a string parameter

Figure 7. String affected by synergy angle (a) and damping time and mass of the string (b). The diffusion and reaction influence the string parameter and further have impacts on damping time and string mass.

is identified as a function of synergy angle. As clearly seen, the increase of synergy angle produces the string parameter ascent up to 0.9. When the synergy angle is used to characterize the string, the synergistic effects improve the string characteristic, with an average of 45% reduction of string parameter compared to the dihedral angle as the only collective variable for characterizing the string. This demonstrates that synergy angle plays a significant role in the string characteristic. The damping time and mass of the string are illustrated in Figure 7(b). As is apparently shown, the damping time of the string reaches the maximum value of 0.011s at the string parameter α around 0.8. The damping time increases as the string parameter increases from 0 to 0.8 and then decreases as the string parameter goes up to 1.0. The string mass is a key factor that affects the field synergy, and hence its variation trend is presented in Figure 7(b). It is clear that the string mass decreases as the string parameter increases from 0 to 0.58, and thereafter the string mass increases up to 2.4 × 10−4g/m. Thus, there is a minimum string mass of 1.92 × 10−4g/m at the string parameter of 0.58. After the string is characterized, the macro and micro performance are discussed in Sections 4.2 and 4.3 to find out how the string affects CO2 capture. This will probably offer some new insights in CO2 capture adsorbent design optimization. 4.2. Macro Performance Affected by the String. Usually, different MEA conformers influence the capture capacity greatly.38 On account of this, the typical MEA conformer transformation versus the string parameter α is given in Figure

Figure 8. Conformer of amine (a), diffusion (b), and reaction kinetics (c) affected by string. Synergy effects are implemented by incorporating synergy angles as another category of variables besides dihedral angles.

equal to 1, the conformer of MEA is g’Gg with the largest free energy of 25.08 kJ/mol. When the synergy angle is coupled with the dihedral angle to characterize the string, the conformer transforms from the higher free energy one to the lower free energy one. For example, at the string parameter 0.6, the MEA 478



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two fluxes decrease as the mass of the string increases. This is due to the fact that the smaller mass of the string probably consumes less energy during the string evolution process in CO2 capture process. Thus, it is quite clear that the smaller mass of the string favors CO2 adsorption by CaCl2-supported MEA adsorbent. As for the damping effects shown in Figure 9(b), the two fluxes also decrease as the damping time increases. Shorter damping time benefits CO2 adsorption because random energy losses are controlled as low as possible. To match the reaction with the diffusion, Figure 9 offers some alternative insights. The reactive flux is larger than the diffusion flux when the mass and damping time of the string are respectively below 1.7 × 10−4 g/m and 0.0112 s. However, when the mass and damping time are greater than the two values, the reactive flux turns to be below the diffusion flux. This suggests that reaction and diffusion show dynamic traits to dominate CO2 adsorption by CaCl2-supported MEA adsorbent, which also allows that it is possible to balance the reaction and diffusion by controlling the mass and damping time of the string. For example, through adjusting the temperature changes, the synergy angle improves the string parameter and finally changes the mass and damping time of the string. 4.3. Micro Performance Affected by the String. Carbon (C) atom position is regarded as an important parameter that affects the molecular force during adsorption and desorption. The effect of the string parameter α on the position is shown in Figure 10(a). It is clearly discerned that C atom position normally increases as the string parameter α increases during the adsorption and desorption (Figure 10(a)). However, at the string parameter around 0.8, the C atom position drops to about 5 Å and then increases up to 8 Å. It is apparently found that the included synergy angle in characterizing the string produces 12% longer C atom position compared to what typical dihedral angle does. As shown in Figure 10(b), molecular motion is affected by the string. Here, the cosine of the synergy angle between the molecule velocity and its gradient is used as an index to quantify the molecular motion. As clearly presented, the cosine value increases as the string parameter α increases from 0 to 1. When the synergy angle and dihedral angle are both employed to characterize the string, the cosine value is increased by 15% averagely, which provides that the molecular motion is improved by synergistic effects occurring in CO2 capture process. The NC reaction coordinate significantly influences the reaction process and its relation with the string parameter α is given in Figure 10(c). It is quite clear that the NC reaction coordinate increases from about 1.0 Å to above 5.0 Å as the string parameter increases. When synergy angle is used as a variable to characterize the string, the synergistic effect reduces the NC reaction coordinate by an average of 18%. Hence, the shorter NC reaction coordinate improves the reaction rate and possibly reduces the cost by controlling the consumption of adsorbent. The molecular motion and its structure change influence CO2 capture. To demonstrate this point from the string view, the flux caused by molecule motion and molecule structure change is provided in Figure 11. Apparently, the two fluxes caused by molecule motion and molecule structure change decrease as the mass of the string increases in Figure 11(a). The reason is that the evolution of the string with larger mass consumes more energy, which reduces the drive force for the molecule motion. Compared with the reactive flux and diffusion flux in Figure 9(a), the fluxes caused by molecular motion and molecular

conformer tTt with 9.6 kJ/mol transforms to gTt with 7.52 kJ/ mol. Diffusion is quite important for CO2 adsorption and desorption and here its relation with the string is given in Figure 8(b). As expected, there is a little increase in diffusivity during the string parameter α varying from 0.3 to 0.7. However, at the extreme conditions, either string parameter ranging from 0 to 0.2 or from 0.7 to 1.0, has a quite clear diffusivity increase. When synergy angle is used to characterize the string, there is an average of 10% increase of diffusivity because of synergistic effects improving the diffusion. The adsorption and desorption kinetics are both affected by the string. The ratio of desorption kinetics to adsorption kinetics versus the string parameter is given in Figure 8(c). The ratio decreases first then increases as the string parameter α varies from 0 to 1. Thus, there is a minimum ratio at the string parameter around 0.58, which suggests that desorption and adsorption are synergized better at this point. When the synergy angle is included as a variable for characterizing the string, the ratio is decreased by 8%, which greatly supports that the synergistic effects improve the kinetics by reducing the resistance between the adsorption and desorption. To demonstrate the macro performance more clearly, reactive and diffusion fluxes are given in Figure 9. It is interesting that the

Figure 9. Reactive and diffusion flux affected by mass (a) and damping time (b) of the string. The two fluxes reflect the macro performance and show routes to improve the reaction and diffusion processes. 479



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Figure 11. Flux affected by mass (a) and damping time (b) of the string. The molecular structure change and molecular motion are improved by the string to reduce the resistance.

by molecule motion is larger than the flux caused by molecule structure change as the mass of the string increases from 1.8 × 10−4 to 2.4 × 10−4 g/m. This result provides that the string dynamically affects the molecular motion and its structural change during CO2 capture. If one anticipates to synergize the molecular motion and molecular structure change, it is better to control the mass of the string around 1.8 × 10−4g/m. As for the effects of the damping time, the corresponding results are given in Figure 11(b). The fluxes caused by molecular motion and molecular structure change are decreased as the damping time increases from 0.006 to 0.02 s. This is due to the fact that the string with longer damping time dissipates more during the CO2 adsorption and desorption process. The flux caused by molecular structure change normally is larger than the flux caused by molecular motion. However, this phenomenon is inverse as the damping time of the string is in the range of 0.011 to 0.0172 s. This suggests that there is chance to balance the molecule motion and its structure change by controlling the damping time. Compared with results given in Figure 9(b), the fluxes caused by molecular motion and molecular structure change are averagely 48% lower than those reactive and diffusion fluxes. This result shows that the string affects molecular motion and its structural change, but not as significantly as what the reaction and diffusion do.

Figure 10. C atom position (a), molecule motion (b) and NC reaction coordinate (c) affected by string. The C atom position reflects the molecule structure. NC reaction coordinate is determined by N atom of MEA and C atom of CO2.

structure change are respectively decreased by 50% and 45%. This suggests that the string does influence molecule motion and its structure change. However, it influences the reaction and diffusion much more. Additionally, it is clear that the flux caused 480



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5. DISCUSSION As analyzed in Section 4, the string is successfully employed to characterize the field synergistic effects during CO2 capture by CaCl2-supported MEA adsorbent. Even though the string method, field synergy, and string theory have developed rapidly, it is scarce in the research to integrate them for describing the complicated CO2 adsorption process. The recent development of string field theory shows some improvement for the string theory.39 However, the key is tackling the theoretical solutions for the open string, which mainly focuses on the open bosonic string field theory. By the integration of the string method, field synergy, and string theory, our work affords to optimize the CO2 adsorption process by CaCl2-supported MEA adsorbent. Different mole ratios of MEA/CaCl2, porosity, and particle diameter are supposed to affect the string analysis results. These results are given in Figure 12, which considers the 30% relative

Figure 12. Effect of mole ratio of MEA/CaCl2, porosity, and particle diameter on the string. The relative 30% change of the parameters are considered against the baseline case of 1:1 MEA/CaCl2, porosity of 0.5, and particle diameter of 1 mm. Figure 13. Effect of CaCl2 on the free energy (a) and the NC reaction coordinate (b) by the string. The effects of 3:1, 2:1, and 1:1 MEA/CaCl2 are compared by considering synergy angle and dihedral angle as the two categories of variables for the string parameter.

change of the parameters above. It is found that the mole ratio of MEA/CaCl2 influences the damping time and string mass much more than the porosity and particle diameter. Hence, the following section will provide a detail string analysis for CO2 capture by 3:1, 2:1, and 1:1 CaCl2-supported MEA adsorbent. Additionally, CO2 capture by CaCl2-supported MEA adsorbent is compared with the typical MEA absorption and supported amine adsorbent adsorption, by offering their string characteristics, energy consumptions, and capture capacities. These discussions cover the results given in Figures 13 and 14 and Table 2. As apparently seen in Figure 13(a), under the respective MEA/CaCl2 ratios of 3:1, 2:1, and 1:1 the free energy peaks at the string parameter around 0.5. The maximum free energy reaches as high as 108 kJ/mol, with the minimum free energy being about 60 kJ/mol at a MEA/CaCl2 ratio of 1:1. The free energy clearly decreases by 30% as the MEA/CaCl2 ratio decreases from 3:1 to 1:1. Additionally, as shown in Figure 13(b), the ratio decreasing from 3:1 to 1:1 reduces the NC reaction coordinate by 40%. This suggests that addition of CaCl2 intensifies CO2 capture process. The reason is that the produced HCl enhances the desorption process. As discussed in Section 4, the synergistic effects improve the string characteristic. The detail comparison of string characteristic between MEA and CaCl2-supported MEA adsorbent is

Figure 14. Energy consumption and capture capacity affected by the string. The 1:1 MEA/CaCl2 adsorbent is chosen to compare with the typical 30 wt % MEA absorbent and conventional supported amine adsorbent. 481



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On the basis of the results and discussions, the overall principle to design the CaCl2-supported amine adsorbent is summarized by the string method (shown in Figure 15). The string with moderate string parameter, small mass, and short damping time achieves the optimum adsorbent design. It means that whatever the macro and micro effects change, the one that produces the small mass, short damping time, and moderate string parameter favors the CO2 capture process. This principle synergizes the macro and micro effects which are normally hard to obtain using the conventional parameter optimization method. The intensification effects of adding CaCl2 are identified by the string. In the future, it will be possible to perform the string analysis for the fluid flow field, which usually occurs in the practical reactor for industrial CO2 capture.

Table 2. String Characteristic Comparison with MEA and CaCl2-Supported MEA Adsorbent string characteristic system

damping time (s)

mass (10−4g/m)

MEA with synergy angle MEA without synergy angle 1:1 MEA/CaCl2 with synergy angle 1:1 MEA/CaCl2 without synergy angle

0.011 0.015 0.0078 0.0095

2.2 2.5 1.7 1.9

provided in Table 2. It is quite clear that in the MEA absorption process (30% MEA weight fraction), a 0.011 s damping time with synergy angle to characterize the string is 26% below that without synergy angle, which shows the same tendency for CO2 capture by CaCl 2-supported MEA adsorbent. During the MEA absorption process, the mass of the string is characterized as 2.2 × 10−4g/m when the synergy angle is included, providing a 12% lighter result compared to that without the synergy angle. In the same sense, 10% string mass reduction is identified in CO2 capture by CaCl2-supported MEA adsorbent. This suggests that the string is straightened out in shorter time as the synergistic effects are considered. The lighter string benefits the CO2 capture process because of good contacts between the fields. When CaCl2 is added, the damping time is reduced by 29% compared to MEA capture CO2. Moreover, the mass is determined as 1.7 × 10−4g/m for the 1:1 CaCl2-supported MEA adsorbent, 22.7% lighter than that of the MEA system. It is expected that the string affects the energy consumption since it is regarded as the bridge between fields. As shown in Figure 14, the energy consumption increases as the string mass increases from 1.4 × 10−4g/m to 2.4 × 10−4g/m, whatever the MEA or the 1:1 CaCl2-supported MEA adsorbent is used in CO2 capture. The most interesting thing is that the 1:1 MEA/CaCl2 adsorbent reduces the energy consumption by 30% compared to the conventional MEA absorption process. At the string mass below 1.8 × 10−4g/m, the string mass has a light influence on the energy consumption. Additionally, the capture capacity decreases as the mass of the string increases, with 1:1 MEA/ CaCl2 producing about 27% higher capture capacity compared to that of the conventional supported amine adsorbent.40,41

6. CONCLUSIONS To design CaCl2-supported MEA adsorbent more efficiently to capture CO2, a string method model was precisely developed to characterize the field synergy. The technical route was to use the synergy angle and dihedral angle to determine the string parameter. By the model, the string was developed to simultaneously quantify the macro and micro mechanism by the relevant fluxes, which was usually hard to perform in the conventional analysis. The effects of the string parameter on the MEA conformer, diffusion, chemical kinetics, C atom position, molecule motion, and NC reaction coordinate were identified very clearly. The analysis results demonstrated that there was a respective 30% free energy reduction and 40% NC reaction coordinate decrease as the MEA/CaCl2 ratio decreased from 3:1 to 1:1. Moreover, the synergistic effects reduced the string damping time by 26%. The decrease of the string mass reduces the energy consumption. The addition of CaCl2 reduced the string mass by 22.7% and energy consumption by 30% compared to CO2 capture by MEA process. The 1:1 MEA/CaCl2 produced a 27% higher capture capacity compared to that of conventional supported MEA adsorbent. Finally, the string to characterize the field synergy is developed as an alternative way to design the adsorbent, which is suggested as a useful way to improve the conventional adsorbent design method.

Figure 15. Improved adsorbent design process by the string analysis. The improvement affords determination of key parameters in the adsorbent design. 482



■

Article

APPENDIX A This section describes the base of the string method. In the Cartesian coordinates x ∈ Rn, when there is Z = ∫ Rne−βv(x)dx, the free energy F(z) is normally associated with collective variables θi (x) in the landscape V(x) at the temperature T, which is given as follows: ⎡ F(z) = −kBT ln⎣⎢Z −1

∫e

−βV (x)

S0 =

f=

rg 3 = Wr0 3 + (1 − W )ri 3

∂zi(α , t ) ∂F(z(α , t )) = − ∑ Pij(α , t )Mjk(z(α , t )) ∂zk ∂t j,k=1

W=

where,

∫∑ k=1

■

■

∂θi(x) ∂θj(x) −βV (x) e ∂xk ∂xk

∂t

■

AUTHOR INFORMATION

*Tel.: +86-29-8266 8566; fax: +86-29-8266 0689; e-mail: cloud. [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The financial support of the National Natural Science Foundation of China (nos. 51276141 and 20936004) are gratefully acknowledged. This work is also supported by the China Postdoctoral Science Foundation funded project (no. 2013M530422) and “Fundamental Research Funds for the Central Universities”.

(B1)

■

1 ∂ ⎛⎜ 1 ∂T ⎞ ∂ ⎛ ∂T ⎞ ∂T = 2 λef B2 ⎟ + 2 ⎜λef sin ψ ⎟ ⎝ ⎠ ∂t ∂B ∂ψ ⎠ B ∂B B sin ψ ∂ψ ⎝

+ rCO2ΔHr

ASSOCIATED CONTENT

Corresponding Author

where De is the effective diffusivity, m2·s−1. For a spherical particle, the unsteady-state heat transfer is written as follows: ρcp

(B8)

Numerical simulation details implemented in string analysis. This material is available free of charge via the Internet at http:// pubs.acs.org.

(A4)

1 ∂ ⎛ 2 ∂CCO2 ⎞ 1 ∂ = 2 ⎜DeB ⎟+ 2 B B ∂ ∂ ∂ B ⎝ ⎠ B sin ψ ψ

⎛ ∂CCO2 ⎞ ⎜De sin ψ ⎟ + rCO2 ∂ψ ⎠ ⎝

Vm,rt

S Supporting Information *

APPENDIX B This section offers the details of the changing grain size model. By the model, the governing equation for CO2 concentration inside the particle is revised by considering the radial and peripheral directions simultaneously, given as follows: ∂CCO2

Vm,pt

where Vm,pt and Vm,rt refer to molar volume of the product and reactant.

(A3)

× δ(z1 − θ1(x))··· δ(zN − θN (x))dx

(B7)

Here, (A2)

Mjk(z(α , t )) = Z e

(B6)

The grain size rg can be calculated as follows:

N

n

(B5)

⎛ r r ⎞ dri = −⎜ a − d ⎟ dt Cdad ⎠ ⎝ Caaa

The path zi evolution equation for the string is typically correlated with string parameter α and time t, which gives the following:

−1 −βV (x)

Vrt Vpt

Here, Vrt and Vpt refer to volume of the reactant and product, respectively. The change of unreacted MEA/CaCl2 radius is calculated in eq B6, (A1)

∂zi /∂α ∂zj/∂α Pij(α , t ) = δij − |∂zi /∂α| |∂zj/∂α|

(B4)

with

× δ(z1 − θ1(x))

⎤ ··· δ(zN − θN (x))dx ⎥⎦

3(1 − ε0) f r0

a B C cp D F f g jB jD jM jR ka kB kd

(B2)

The reaction rate rCO2 is obtained by revising the changing grain size model, which is developed as ⎡ ⎛ r ⎞2 ⎛ r ⎞2 ⎤ rCO2 = −⎢ra·S0·⎜ i ⎟ ·VR ·CCO2 − rd·S0·⎜ i ⎟ ⎥(1 + sin ψ ) ⎢⎣ ⎝ r0 ⎠ ⎝ r0 ⎠ ⎥⎦ (B3)

where rCO2 is the local reaction rate, which is obtained by introducing both adsorption and desorption reactions. The initial specific surface area of S0 is as follows: 483

NOMENCLATURE specific surface, m2 m−3 particle radius, m molar concentration, mol m−3 thermal capacity, kJ kg−1 K−1 diffusivity, m2 s−1 free energy, kJ mol−1 weight fraction acceleration of gravity, m s−2 flux due to molecule structure change diffusion flux flux due to molecule motion reaction flux adsorption reaction kinetic constant, s−1 Boltzmann constant, J K−1 desorption reaction kinetic constant, s−1 dx.doi.org/10.1021/jp510610v | J. Phys. Chem. C 2015, 119, 473−485

The Journal of Physical Chemistry C lc lij Mjk m n P Peq peq Pij q r R s T t td U V V(x) vij x Y W z ρ θ β γ λ ξ ψ ΔHr

Article

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molecule characteristic length, m molecule structure length, m tensor, Pa string mass, g m−1 normal direction pressure tensor, Pa equilibrium pressure, kPa equilibrium probability distribution projector on the plane perpendicular to the path committor function reaction rate, mol m−3 s−1 or grain size, m radius curvature of the string, m arc length of the path temperature, K time, s damping time of the string, s potential energy, kJ mol−1 volume, m3 energy, kJ mol−1 molecule velocity, m s−1 fraction or position, m concentration volume ratio string symbol density, kg m−3 variable inverse temperature with the Boltzmann’s constant synergy angle, degree thermal conductivity, W m−1k−1 normal displacement of the string, m peripheral direction reaction heat, kJ mol−1

Subscript

a adsorption d desorption

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