Thermal Swing Adsorption Process for Carbon Dioxide Capture and

Article pubs.acs.org/IECR

Thermal Swing Adsorption Process for Carbon Dioxide Capture and Recovery: Modeling, Simulation, Parameters Estimability, and Identification M. Lei, C. Vallieres, G. Grevillot, and M. A. Latifi* Laboratoire Réactions et Génie des Procédés CNRSENSIC, BP20451, 1 Rue Grandville, 54001 Nancy Cedex, France ABSTRACT: In one of our previous studies, several experimental measurements were carried out on a thermal swing adsorption (TSA) process (adsorption and desorption steps) used for the capture of CO2 from a CO2/N2 mixture. The present study deals with a two-dimensional modeling and simulation of the aforementioned TSA process. The models are described by partial differential equations (PDEs) including conservation equations, models for equation of state, equilibrium, thermodynamic, and transport properties. The resulting models involve different unknown parameters to be estimated from the available experimental measurements. An estimability analysis was carried out in order to determine which parameters are able to be estimated from the available experimental results. The most estimable parameters are then identified, and the less estimable parameters are fixed from previous works or from literature. The resulting models are implemented and solved within Comsol Multiphysics software. The results show that the model predictions fit quite well with the experimental results. This can be carried out by means of a VSA process.2 Another alternative to rapidly increase the sorbent temperature is to heat the bed with superheated water vapor. This is widely used in VOC recovery, but only when the VOC is immiscible with the condensed water. This alternative could be appropriate for carbon dioxide recovery, but the resulting product will be humid and will need large amounts of heat to be dried. It clearly appears that in carbon dioxide recovery the percolation of hot gas through the bed is not suitable since it will lead to poor carbon dioxide purity or involve large amounts of heat. The most suitable technique consists therefore in heating the sorbent through the wall. This is mainly carried out by means of (finned) tubular heat exchangers embedded in the bed.3−7 Water vapor or hot gas flows then inside the tubes to heat the bed in the regeneration step, whereas water is used to cool down the bed in the adsorption step. This technique is widely used in industry, and among the examples Knaebel8 from Adsorption Research Inc. proposed a moving bed technology for CO2 capture. CO2 is desorbed by heating the adsorbent by hot flue gas in a heat-exchanger-type contactor, thus by the wall of a column. More recently ADA Environmental Company (USA)9 is currently building a quite large pilot of 1 MW using solid sorbents in a temperature swing process where the bed is heated indirectly using water vapor. The TSA considered in the present paper10 is actually equivalent to an adsorption process heated by a tubular heat exchanger through the wall. The objective of the present work is the study of a TSA process for CO2 capture and recovery. More specifically, modeling, simulation, parameters estimability, and identification for our previous experimentally studied process10 are

1. INTRODUCTION It is well-known that most of the world energy needs come from fossil fuels. It is also well-known that their combustion is one of the major sources of the greenhouse gas carbon dioxide. Technologies that will allow the use of fossil fuels while reducing greenhouse gas emissions are therefore needed. Current available technologies for CO2 capture are expensive and energy costly. Improved technologies for CO2 capture are necessary in order to respect more and more strict regulations on CO2 emissions with reduced energy consumption. The potential techniques to be used for CO2 capture are pressure swing adsorption (PSA) and temperature swing adsorption (TSA). These techniques are widely used in industry for purification and separation. However it is important to notice that carbon dioxide recovery is a particular case in adsorption. As a matter of fact, carbon dioxide has to be recovered as the purest possible and then compressed at high pressure (50−100 bar) in order to be transported to an underground site for storage. In many TSA processes the sorbent is heated by percolating a hot gas through the bed. In this case, the recovered adsorbate is always diluted in the hot gas. This is due to two phenomena: (i) on the one hand the temperature rise and consequently the isotherm decrease occur slowly and then result in lower desorbed quantities and (ii) on the other hand the hot gas plays the role of carrier gas hence leading to a dispersive type of desorption front, i.e. the desorbed component concentration decreases progressively (equilibrium theory of adsorption columns1). To rapidly increase the sorbent temperature, a pressurized gas can be used. In this case of natural gas desulfurization the natural gas itself is used since it comes out of the well under pressure (e.g., 80 bar). This is not suitable for carbon dioxide recovery, otherwise one should use a pressurized gas where carbon dioxide would be diluted. However in order to avoid the dilution of carbon dioxide, one should not have a carrier gas. © XXXX American Chemical Society

Received: October 24, 2012 Revised: March 25, 2013 Accepted: April 26, 2013

A

dx.doi.org/10.1021/ie3029152 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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• Only carbon dioxide is adsorbed. Here we assumed that nitrogen does not adsorb although it is well-known that it does. However the adsorbed quantities could be considered as negligible without a big change in results. Moreover, a recent study18 where adsorbed quantities on zeolite 5A of both nitrogen and carbon dioxide were measured for different values of pressure and temperature, showed that the nitrogen adsorbed does not exceed 7% of the adsorbed carbon dioxide. • The resistance of mass transfer in the gas phase is negligible; the kinetics of mass transfer within a particle is approximated by the linear driving force (LDF) model. • The gas phase is in equilibrium with the adsorbent. • Isosteric heat of adsorption (ΔH) does not vary with temperature. • The adsorbent is considered as a homogeneous phase. • The physical properties of adsorbent are assumed to be constant. 2.2. Model Equations. On the basis of the above assumptions, two-dimensional models describing both adsorption and regeneration steps of the TSA process considered are developed. The models are described by partial differentialalgebraic equations (PADEs) including conservation equations, models for equation of state, equilibrium, thermodynamic, and transport properties. The main differences between the models of the two steps are the initial and boundary conditions which define the operating conditions. The resulting model equations may be written as follows. The mass balance for the adsorbent component (CO2) can be written as

considered. In fact, in Tlili et al., 2009,10 a TSA process was designed and used to capture carbon dioxide from a gas mixture of carbon dioxide and nitrogen. The mixture feeds an adsorption column of 10 cm length and 3.5 cm diameter. The column is packed with particles of zeolite 5A where carbon dioxide is adsorbed. The adsorbent is then regenerated by heating the column through the wall. Experimental adsorption isotherms of the adsorbent have also been measured. Since the column is not adiabatic, the radial temperature is not uniform, especially for the desorption step. It is therefore necessary to take into account the influence of radial temperature difference within the bed of the column. Even though many studies on TSA processes have been carried out, very few research contributions in the open literature have been devoted to the two-dimensional (2D) cyclic modeling and simulation11−14 and how to determine the unknown parameters involved in the model from the available experimental measurements. In this paper, a two-dimensional model was established and solved using the software Comsol Multiphysics. The model consists of partial differential equations representing heat balance, mass balance of CO2, overall mass balance, and momentum balance and equations for equilibrium, thermodynamic, and transport properties. Heat transfer to the wall of the column is taken into account. Since the model involves many unknown parameters whose values cannot be obtained accurately from literature, an estimability analysis method was used in order to determine those which are estimable from the available experimental measurements. The resulting set of estimable parameters was then identified by the combination of Matlab and Comsol Multiphysics. With the optimized parameters, the TSA processes (adsorption and regeneration steps) were simulated and compared to the experimental measurements

∂y ∂y 1 − ε RT ∂q1 +u 1 +v 1 P ∂t ∂z ∂r ∂t ε ⎡ 1 ⎛ ∂y ∂P ⎞ ∂y ∂P = ∇(D∇y1) + D⎢ ⎜ 1 + 1 ⎟ ⎢⎣ P ⎝ ∂z ∂z ∂r ∂r ⎠

∂y1

2. TSA PROCESS MODEL One of the important issues in process modeling is the choice of the appropriate model dimension, that is, one, two, or threedimensional model. In the modeling of the adsorption step, although a twodimensional model predicts better the profiles of the most important variables, that is, temperature, concentration, and flow rate,15 a one-dimensional model (radial variations are negligible) is able to correctly predict the same profiles16 with a reasonable accuracy (mainly when the heat of adsorption involved is not large or the bed diameter is relatively small resulting in an insignificant radial temperature difference between the center and the inner wall). In the desorption step however, a one-dimensional model is not able to describe the phenomena involved since the column is heated through the wall at regeneration (wall) temperatures ranging from 130 to 210 °C.10 The column is thus neither adiabatic nor isothermal, and the radial temperature profile is no longer uniform. A two-dimensional model is therefore needed. So the use of a one-dimensional model for the adsorption step and a two-dimensional model for the desorption step can be easily handled.17 However, for coherence reasons we chose to use a two-dimensional model for both steps and throughout the paper. 2.1. Model Assumptions. The process model developed is based on the following assumptions:

+ (1 − y1)

−

∂y ∂T ⎞⎤ 1 ⎛ ∂y1 ∂T + 1 ⎟⎥ ⎜ T ⎝ ∂z ∂z ∂r ∂r ⎠⎥⎦

(1)

where y1 is the mole fraction of CO2 in the gas phase, D is the effective dispersion coefficient, ε the bed porosity, q1 is the adsorbed phase concentration averaged over the adsorbent particle, u the axial interstitial bulk fluid velocity, v is the radial interstitial bulk fluid velocity, and P the total pressure. The overall mass balance is given by the following equation: ∂u ∂v ∂T ⎞ v 1 − ε RT ∂q1 1 ⎛ ∂T + + + − ⎜u +v ⎟ ⎝ ∂z ∂r ε ∂r ⎠ r P ∂t T ∂z ∂P ⎞ 1 ∂T 1 ⎛ ∂P 1 ∂P − + ⎜u +v ⎟+ =0 ⎝ ⎠ (2) ∂r T ∂t P ∂z P ∂t

This equation will allows us to determine the pressure. The heat balance is ⎛ ∂T ⎤ ∂T 1 − ε RT ∂T ⎞ ⎡ (ρ Cps + q1Cpg )⎥ C pg ⎜u + v ⎟ + ⎢C pg + ⎦ ∂t ⎝ ∂Z P s ∂r ⎠ ⎣ ε ⎛ ⎞ 1 RT 1 − ε RT ∂q1 ⎜ ∂ΔH ⎟ = ∇(λ∇T ) − ΔH + q1 P ∂t ⎜⎝ ε P ε ∂q1 ⎟⎠ (3)

where ΔH is the isosteric heat of adsorption. It is defined by a Clausius−Clapeyron equation type as19

• The gaseous mixture obeys the perfect gas law. B


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∂(ln p) ΔH = 2 ∂T RT

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Table 1. Boundary Conditions (4)

where p is the adsorbent partial pressure. ΔH is then calculated for different values of adsorbed phase concentration q, and fitted by means of a third order polynomial as ΔH = 842.3q3 − 447.2q2 + 1577q + 17847

(5)

The heat capacity of gas is given by the following equation which is the composition-weighted sum of temperature dependent heat capacity of pure components, carbon dioxide and nitrogen:

(6)

The heat capacities of pure components are taken from the databank SIMSCI of the flowsheeting software PRO/II. The momentum balance equations were simplified and reduced to the following two Ergun equations:20,21

P = P1

∂P/∂r = 0

parameter ε R Cps23 Rc ρs

(9)

(14)

unit

value

J mol−1 K−1 J kg−1 K−1 m kg m−3

0.4 8.314 1046 0.0175 1212

3. PARAMETERS ESTIMABILITY ANALYSIS AND IDENTIFICATION The TSA process model developed involves many unknown parameters that strongly influence the predictions. On the other hand, only some experimental measurements carried out in the center and at the exit of the column and at limited time samples are available. The question is whether or not the measurements contain the necessary information to identify all the unknown parameters. If not, which ones are the most estimable and in which order. To answer these questions, an estimability analysis was carried out, followed by a parameter identification study. 3.1. Parameters Estimability Analysis. A first step in the development of a reliable mathematical model, prior to the parameter identification problem, is to evaluate the structural identifiability and estimability (i.e., practical identifiability) of the model parameters. The first approach is exclusively based on the structure of the mathematical model and aims at investigating whether or not the model parameters are globally or locally structurally identifiable.24,25 The objective of estimability analysis is to determine the subset of parameters with the highest estimabililty potential, based on a predefined experimental design or available data. Owing to the lack of

(10)

(11)

and (12)

2.3. Initial and Boundary Conditions. The boundary conditions for both adsorption and regeneration steps are summarized in Table 1. For the regeneration step, uin = 0, and the time-varying experimental data of the outside wall temperature of the column are assumed to be represented by the following equation: ⎛ 120 ⎞ Tout = (Tregen − T0) exp⎜ − 1.5 ⎟ + T0 ⎝ t ⎠

∂P/∂z = 0

Table 2. Basic Parameters for Numerical Computation

where

⎛ 13904 ⎞ ⎟ K = 6.53 × 10−7 exp⎜ ⎝ RT ⎠

P

∂u/∂r = 0 v=0 −λ∂T/∂r = kc(T − Tout) ∂P/∂r = 0

The basic parameters values used for numerical computation are given in Table 2.

qmKP

qm = ( −0.0145T + 7.531)ρs

∂u/∂r = 0 v=0 ∂T/∂r = 0

P(r , z)|t = 0 = P0

where k1 is the overall mass transfer coefficient and qe is the amount of adsorbed component at equilibrium The adsorption isotherm is represented using the Langmuir isotherm as10 1 + KP

∂u/∂z = 0 ∂v/∂z = 0 ∂T/∂z = 0

T (r , z)|t = 0 = T0

These two equations will allow us to determine the axial and radial fluid velocities. The kinetics of mass transfer within a particle is approximated by the linear driving force (LDF) model as22

qe =

u v T

q1(r , z)|t = 0 = q0

(8)

= k1(qe − q1)

r = Rc

v(r , z)|t = 0 = 0

2 2 (1 − ε)2 μF v ∂P 1 − ε ρF u + v v = 150 + 1.75 ∂r ε dp ε2 dP2

∂t

∂y1/∂r = 0

u(r , z)|t = 0 = 0

(7)

∂q1

r=0 ∂y1/∂r = 0

y1(r , z)|t = 0 = y0

2 2 (1 − ε)2 μF u ∂P 1 − ε ρF u + v u − = 150 + 1.75 ∂z ε dp ε2 dP2

−

z=L ∂y1/∂z = 0

Here Tregen is the regeneration temperature, T0 is the initial temperature, and t is the elapsed time from the starting time point of the regeneration step. For the adsorption step, the outside wall temperature is assumed to be constant and equals to the room temperature. The initial conditions of the unknown variables are given as follows for 0 ≤ z ≤ L and 0 ≤ r ≤ Rc:

Cpg = (15.51 + 0.095T − 7.10−5T 2)y1 + (29.87 − 0.005T + 9.10−6T 2)(1 − y1)

z=0 −D∂y1/∂z = (yin − y1)uin u = uin ∂v/∂z = 0 ∂T/∂z = 0

y1

(13) C



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measurements, accurate estimation of some parameters may be impossible. The main limitation to the parameters estimability is their weak influence on the measured outputs and/or the correlation between the parameters effects. Owing to their poor accuracy, the estimation of these parameters can lead to significant degradation in the predictive capability of the model. Different selection methods based on the estimability principle have been developed as reported in the literature, including the principal component method,26,27 the singular value decomposition,28 the correlation methods29 and the eigenvalue method.17 A good review of the most important methods can be found in refs 25 and 30. The orthogonalization-based methods developed in refs 31, 32 and 33 are particularly efficient to rank the parameters with the highest estimability potential. Those methods have been widely used over the last five years in different research areas.34−39 In addition, the orthogonalization-based methods are relatively straightforward and can be used prior to the experimental campaign to identify the best parameter candidate to be estimated, based on the model predictions and an initial guess of the parameters vector, obtained generally from literature or a previous identification procedure. When a parameter of interest is not estimable according to the experimental design adopted, the latter may be modified to include sampling times corresponding to the highest influence of the parameter, which may potentially improve the estimability of this parameter if it is not correlated with another parameter. In this paper, we implement the sequential orthogonalization method developed in ref 32 as follows: A sensitivity coefficient matrix Z is built, from the individual coefficient as ⎡ S11|t = t 1 ⎢ ⎢⋮ ⎢ ⎢ Sny1|t = t1 Z=⎢ ⎢ S11|t = t2 ⎢ ⎢⋮ ⎢ ⎢⎣ Sny1|t = tns

··· S1np|t = t1 ⎤ ⎥ ⎥ ⋱ ⋮ ⎥ ··· Snynp|t = t1 ⎥ ⎥ ··· S1np|t = t2 ⎥ ⎥ ⎥ ⋱ ⋮ ⎥ ··· Snynp|t = tns ⎥⎦

2. The parameter whose column in Z has the highest magnitude is selected as the first parameter. 3. The corresponding column is marked as XL (L = 1 for the first iteration). 4. ZL, prediction of the full sensitivity matrix Z, is computed using the subset of columns XL as ZL = XL(XL TXL)−1XL TZ

5. The residual matrix RL is built as RL = Z − ZL

pj̅ si̅ |t = tk

∂sî ∂pj

t = tk

(18)

6. The quadratic sum of the residuals is calculated in each column of RL. The column with the highest magnitude corresponds to the next estimable parameter (among the remaining ones) having the largest effects on investigated data and not correlated with the effects of the already selected parameters. 7. The corresponding column in Z is selected and included into XL. The new matrix is called XL+1. 8. The iteration counter is incremented and steps 4 to 7 are repeated until the last parameter was treated. The algorithm starts with the choice of the most estimable parameter p1 (steps 1, 2). But p1 may be correlated to the others. Orthogonalization is required to rank individual parameter influences on investigated data: the influence of p1 on the other parameters has to be adjusted for any possible correlation between the column of p1 in Z and the others so that the “net influence” of each of the remaining parameters could be assessed. This adjustment is done by regressing all of the original columns of the Z matrix on the column associated with p1. This regression results in a matrix of the same dimensions as Z, and the column of this residual matrix associated with p1 is a column of zeros (steps 3−5). Orthogonalization is followed by identifying the largest column in the residual matrix in order to find the parameter p2 having the second strongest “net influence” on investigated data, after the influence of p1 has been eliminated (steps 6, 7). Orthogonalization is carried on until the last model parameter is treated. Yao et al. (2003)32 used 0.2 as a cutoff criterion for considering whether a model parameter was estimable or not. They considered thus that information is sufficient to estimate a parameter if a 10% change in its value implies a 2% change in output. It is worth noting that the choice of a cutoff criterion is arbitrary and strongly depends on the process considered. In our case the cutoff criterion was set to 0.4 meaning that a 10% change in an estimable parameter value implies a 4% change in output.15 3.2. Parameters Identification. According to the parameter estimability analysis, we know the set of the most estimable parameters to be identified from the available experiments. The nonestimable parameters can be fixed either from previous studies or from literature. To identify the estimable parameters p, the following objective function F is defined:

(15)

where Sij|t = tk =

(17)

(16)

is the normalized sensitivity of the model prediction ŝi of output variable si with respect to parameter pj at time tk. p̅j is the nominal value of the jth parameter and si̅ |t=tk is the model prediction of the ith output evaluated at a sampling time tk using the nominal vector of parameters (p). ̅ The number of columns, np, is equal to the number of model parameters and each row corresponds to measured points of investigated data. The Z matrix elements are calculated by means of the centered finite differences method. The subset of estimable parameters is identified applying the following 8-step algorithm:32

nm

1. The magnitude of each column is computed. It is defined as the square root of quadratic sum of the elements.

F(p) =

∑ Ec (p) r

i=1

D

(19)



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Table 5. Optimized Values of Parameters in the Adsorption Step

The term Ecr corresponds to the least-squares between the experimental measurement and its corresponding predicted value. The identification problem is then defined as the following constrained nonlinear programming (NLP) problem: Find p* such that: F(p*) = min F(p)

optimized value

subject to pmin ≤ p ≤ pmax

k1 (s−1)

λ (W/m/K)

1.9732

0.0051

0.060

The results are in a quite good agreement with the common values for the process under consideration. The computed predictions of the mole fraction of CO2 at the exit and temperature at the center of column were compared to the experimental measurement using the optimized values of the estimable parameters. The temperature measurements were collected at the center of the column at 7 cm from the inlet for all experimental runs. Figures 1 and 2 show the breakthrough

(20)

p

D × 105 (m2/s)

(21)

In this work, the identification problem is solved using the function “fmincon” available in Matlab environment.

4. RESULTS AND DISCUSSION 4.1. Adsorption Step. The operating conditions for the adsorption step are presented in Table 3.10 Table 3. Operating Conditions for the Adsorption Step parameter

exp1

exp 2

exp 3

exp 4

y0 yin (%) P0 (Pa) feeding flow rate (mL/min) T0 (K) Tc (K) Tout (K)

0 12.97 111102 218 302 302 300

0 22.05 108691 210.5 292.5 292.5 289.5

0 29.3 110142 252 300 300 294.5

0 36.2 112419 268.5 297 297 293

Figure 1. Comparison of predicted breakthrough curves with experimental data in the adsorption step for different feed concentrations.

(a). Parameter Estimability Analysis. For the adsorption step, the temperature in the center of column (T) and the mole fraction of CO2 at the exit (y1) were measured. Their experimental values were compared to the corresponding model predictions. The unknown parameters are p = [D,k1,kc,λ]. The estimability analysis algorithm presented in Section 3.1 was applied to the adsorption step and led to the results shown in Table 4.

curves and the bed temperature, respectively, for different feed concentrations. Both numerical (solid lines) and experimental (symbols) results are reported. It can be seen that for breakthrough curves (Figure 1) the agreement is quite good except for the experiment with yin = 0.1297 where the model appears earlier than the measurements. This is probably due to the inaccuracy of the adsorption isotherm at low concentrations. For the bed temperatures (Figure 2), the predictions agree well with the experimental measurements for all feed concentrations analyzed. 4.2. Regeneration Step. In the regeneration step, the operating conditions are presented in Table 6.10 (a). Parameter Estimability Analysis. Here the outputs of the model are the outlet gas flow rates, Q, and the temperature at the center of the column. On the other hand, in the model equations of this step, the dispersion term is neglected since the concentration of desorbed CO2 becomes very rapidly uniform in the gas phase leading hence to negligible spatial gradients. The unknown parameters are therefore p = [k1, kc, λ] The same way as in adsorption step, the estimability analysis algorithm presented in section 3.1 was applied to the regeneration step and led to the results shown in Table 7. It can be seen here that the order of estimability of is λ > k1 > kc. On the other hand, since the threshold value of the column magnitude of Z matrix was fixed to 0.4, the last parameter, that is, kc, is considered as nonestimable from the available experimental measurements. Its value was fixed from literature to 10 W m−2 K−1.37 (b). Parameter Identification and Validation. The experimental measurements used for identification of the most estimable parameters are time-varying profiles of outlet

Table 4. Estimability Analysis in the Adsorption Step rank 1

rank 2

parameters

λ (W/m/K)

k1 (s−1)

D

rank 3

initial values column magnitude of Z matrix iteration 1 iteration 2 iteration 3 iteration 4

0.05 5.6417

0.004 1.5265

1.10−5 0.9516

15 0.2960

5.6417 0 0 0

1.5265 1.4735 0 0

0.9516 0.9512 0.8404 0

0.2960 0.2680 0.2655 0.2650

(m2/s)

rank 4 kc

(W/m2/K)

It can be seen that the order of estimability of parameters is λ > k1> D > kc. On the other hand, since the threshold value of the column magnitude of Z matrix was fixed to 0.4, the last parameter, that is, kc, is considered as nonestimable from the available experimental measurements. Its value was fixed from literature to 10 W m−2 K−1.37 (b). Parameter Identification and Validation. The data used correspond to a set of measurements with four different concentrations of CO2. To identify the most estimable parameters, the resulting constrained NLP problem was solved and the optimized values are presented in Table 5. E



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Table 6. Operating Conditions in the Regeneration Step parameter

Exp 5

Exp 6

Exp 7

Exp 8

y0 (%) P0 (Pa) P1 (Pa) T0 (K) Tregen (K)

12.87 109473 99414 299 403

13.03 109961 98733 300 424.2

13.0 111621 100879 295.5 447

12.79 111816 102246 293 483

Table 7. Estimability Analysis in the Regeneration Step parameter initial values column magnitude of Z matrix iteration 1 iteration 2 iteration 3

rank 1

rank 2

rank 3

λ (W/m/K) 0.0278 2.2526 2.2526 0 0

k1 (s−1) 0.0065 0.5626 0.6415 0.5626 0

kc (W/m2/K) 13.7501 0.0717 0.1448 0.0719 0.0717

Table 8. Optimized Values of Parameters in the Regeneration Step optimized values T (°C)

λ (W/m/K)

k1 (s−1)

130 150 170 210

0.0664 0.0687 0.0608 0.0649

0.0015 0.0019 0.0018 0.0022

It is worth noticing that λ seems to be constant when the operating (regeneration or wall) temperature increases. However k1 shows a substantial increase. The computed predictions of the gas flow rate and temperature are compared to the experimental measurements using the optimized values of the estimable parameters. Figures 3 and 4 show the gas flow rate curves and the bed temperature,

Figure 3. Comparison of predictions and measurements of the gas flow rate in the regeneration step for different regeneration temperatures.

respectively, for different temperatures. Both numerical (solid lines) and experimental (symbols) results are reported. The agreement is quite good for the two variables. The agreement between the experimental measurements and their corresponding model predictions for both adsorption and regeneration steps shows the importance of parameter estimability analysis prior to any identification of the unknown parameters from the available experimental data. An attempt to identify the whole unknown parameters by including kc in the set of unknown parameters for both steps resulted in numerical

Figure 2. Comparison of predicted temperature profiles with experimental results in the adsorption step for different feed concentrations.

gas flow rate and temperature at the center of the column obtained for four regeneration (wall) temperatures. The resulting constrained NLP problem was solved, and the optimized values are presented in Table 8. F



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nm = number of experimental measurements P = pressure (Pa) q = amount adsorbed of CO2 in a pellet or particle (mol m−3) R = perfect gas constant (J mol−1 K−1) Rc = radius of the column (m) s = model outputs S = sensitivity coefficients T = temperature (K) u = axial interstitial fluid velocity (m s−1) v = radial interstitial fluid velocity (m s−1) y = mole fraction of CO2

Figure 4. Comparison of computed and measured temperature profiles rate in the regeneration step for different regeneration temperatures.

Greek Letters

ε = Bed porosity ρs = Solid density (kg m−3) ρF = Fluid density (m s−1) μF = Fluid viscosity (Pa s) λ = Heat conductivity (W m−1 K−1) θ = A set of unknown parameters

convergence problems of the NLP optimizer and/or poor estimations due in particular to large parameter correlations. Additional and appropriate experiments are needed in order to accurately estimate kc.

Superscript

5. CONCLUSIONS A two-dimensional nonadiabatic model was developed to simulate a temperature swing adsorption (TSA) process (temperature and concentration for adsorption step, temperature and flow rate for regeneration step). Since many unknown parameters are involved in the model, prior to their identification, an estimability analysis was carried out in order to determine the set of the most estimable parameters from the available experimental measurements. The resulting estimable parameters were then identified from the measurements of temperature and CO2 concentration in the adsorption step, and from the measurements of temperature and gas flow rate in the regeneration step. The values of nonestimable parameters were taken from previous works and/ or from literature The model predictions computed using the optimized parameters were compared to the experimental measurements. The agreement was quite good in both adsorption and regeneration steps. However a slight deviation appeared for small values of the feed concentration. This deviation could certainly be reduced by means of accurate measurements of adsorption isotherms.

■

calc = calculated exp = experimental Subscript

■

0 = initial 1 = exit, average, CO2 e = equilibrium in = inlet m = saturated max = maximum min = minimum out = outside the wall regen = regeneration ti = measurement time

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: Abderrazak.Latifi@univ-lorraine.fr. Tel. +33 3 83 17 52 34. Fax +33 3 83 17 53 26. Notes

The authors declare no competing financial interest.

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NOTATIONS Cpg = gas heat capacity (J mol−1 K−1) Cps = solid heat capacity (J mol−1 K−1) D = effective dispersion coefficient (m2 s−1) dP = particle diameter (m) ΔH = isosteric heat of adsorption (J/mol) k1 = overall mass transfer coefficient (LDF), (s−1) kc = heat transfer coefficient (W m−2 K−1) L = adsorbent bed length (m) np = number of unknown parameters ny = number of model outputs ns = number of time samples G



Article

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Thermal Swing Adsorption Process for Carbon Dioxide Capture and

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