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Energy Procedia
Energy Procedia 4 (2011) 1435–1442
Energy Procedia 00 (2010) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/XXX
GHGT-10
The study of numerical methods and validation of a heat and mass transfer model in CO2-MEA system Xiao Luo, Ardi Hartono, Magne Hillestad, Hallvard F. Svendsen Department of Chemical Engineering, NTNU, Sem Sælands vei 4, N-7491 Trondheim, Norway Elsevier use only: Received date here; revised date here; accepted date here
Abstract The implementations of a penetration type of the heat and mass transfer model for CO2 absorption into MEA solution in MATLAB were performed and validated with the experimental data from a String of Discs Contactor (SDC). The numerical PDEs solution method and its implementation were key factors in obtaining a stable, robust and fast computation. The penetration model of heat and mass transfer was implemented for testing a variety of numerical methods. It was found that the finite differences (FDM), in general, was not a proper algorithm in the case of high reaction rates as we observed an excessive number of discretization points were needed. Both a non-uniform adaptive grid routine, and a method based on orthogonal collocation were accurate and robust, and gave relatively fast solution. However, of the two, orthogonal collocation was by far the fastest. Based on this numerical scheme, the combined heat and mass transfer model will be developed further in the next phase of this study.
c© 2010 ⃝ 2011 Elsevier Published byAll Elsevier Ltd. Open access under CC BY-NC-ND license. Ltd. rights reserved Keywords: Heat and Mass transfer; Partial differential equations; numerical methods; CO2 absorption;
1. Introduction In CO2 absorption processes large mass and heat transfer driving forces occur several places, absorber top and possibly bottom, water wash section and in the desorber. Good fundamental models for the combined heat and mass transfer including chemical reaction and convective mechanisms are still lacking. A proper description of the transfer processes is vital for improved absorption process simulation and optimization. The numerical mass transfer modeling for interpreting the absorption data of CO2 into amine solutions has been studied for many years[1,2]. Normally the film theory is used but also the penetration model is in use in case a chemical reaction occurs in the liquid phase. Numerically the solution of these two models is different as the film model, upon discretization, will give rise to a set of algebraic equations, whereas the penetration model can be discretized spatially and solved in time by integration. Which one to use may more be a matter of taste, but the
Corresponding author. Tel.: +47-73-59-41-00; fax: +47-73-59-40-80. E-mail address:
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doi:10.1016/j.egypro.2011.02.009
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penetration model may be more stable as the starting point, or initial guess, is given. Since the reaction rate between CO2 and MEA is quite high, the problem is very stiff with a very steep concentration profile for CO2 and less so for the amine constituents and for temperature. To find an appropriate method to approach this solution with reasonable computational times thus becomes very important. Numerical solution of PDEs can mainly be divided in 3 different methods: finite differences (FDM), finite elements (FEM) and finite volumes (FVM). Very often an FDM technique has been used for solving this problem, like the Barakat-Clark method[3], the Cornelisee approach[4], and using the method of lines DSS routines, Schiesser[5]. The methods above are various kinds of forward, backward and central difference methods, called uniform fixed grid methods. A fixed grid may be disadvantageous in the amine absorption case. For better solving a stiff problem of PDEs, the subsection method can be considered. It divides the interval into two or more parts, and utilize a different mesh resolution for each. Also adaptive moving grid methods were developed by Schiesser[5], and Bloom-Zegling[6]. In PDE problems with steep moving fronts, it is advantageous to concentrate the grid points in the spatial interesting regions and to move them continuously in time. In this way the number of grid points can be significantly reduced. The orthogonal collocation method on finite elements is a subclass of the Method of Weighted Residuals (MWR) and a useful method for problems whose solution has steep gradients. The method can be applied to time-dependent problems, too. The method is a bit more complicated than the others, since ordinary differential equations and algebraic equations are combined. In this study, using CO2 absorption into an MEA solution in falling film flow as example, a variety of numerical methods were tested for predicting the rate of absorption of species and the heat flux in the CO2-MEA-H2O system. It takes into account the coupling between chemical equilibrium, mass transfer, heat transfer and chemical kinetics. The experimental kinetic data are provided by a string of discs contactor (SDC) experiments[7]. The capacity comparison of different numerical methods has been given in the end. 2. Theory The overall chemical reaction taking place in the liquid phase at low CO2 loadings can be written as:
o MEACO2 MEAH CO2 2MEA m k2
(1)
k2
The reaction rate can be defined as:
k2 >CO2 @> MEA@ k2 ª¬ MEAH º¼ ª¬ MEACO2 º¼ 2
R
-
(2) -
In addition to this, the direct reaction between CO2 and OH was taken into account. The OH , water and amine were taken into account as bases in k2 using the termolecular mechanism, see Hartono et al.[7]. 2.1. Mass transfer model For the penetration model the material balances for each species yield the following set of equations:
wCCO2 wt wCMEA wt
wCCO2 w § ¨ Dco2 wx © wx
wCMEA · w § ¨ DMEA ¸ 2R wx © wx ¹
wCMEACOO wt wCMEAH wt
· ¸R ¹
wCMEACOO w § ¨ DMEACOO wx © wx wCMEAH w § ¨ DMEAH wx © wx
· ¸ R ¹
(3)
· ¸ R ¹
These four non-linear partial differential equations must be solved numerically because an analytical solution
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method is not available. Before solving this set of equations uniquely, one initial condition and two boundary conditions are necessary. As initial condition it is assumed that the system considered is in equilibrium for a given solute loading:
t 0, x t 0, CCO 2 CCO 2 0 , CMEA
CMEA0 , CMEACOO
CMEACOO , CMEAH 0
CMEAH
(4) 0
The ionic species are non-volatile and CO2 mass transfer in the film is assumed only caused by diffusion. The boundary conditions at the interface are as follow:
t ! 0, x
0,
k g CCO 2g ,bulk CCO 2g ,i CCO 2l ,i wCMEA wx
PCO 2g ,i
DCO 2
wCCO 2 wx
x 0
CCO 2 g ,i RT
(5)
H H wCMEACOO wCMEAH wx
wx
0
The boundary conditions in the liquid bulk can be expressed as:
t ! 0, x f, CCO 2 CCO 2 0 , CMEA
CMEA0 , CMEACOO
CMEACOO , CMEAH 0
CMEAH
(6) 0
2.2. Heat transfer model By taking into account the reaction heat, the overall heat transfer equation in the liquid film can be expressed as the following:
w § wT · § 'H R ¨D ¸¨ wx © wx ¹ ¨© U c p
wT wt The initial condition is:
· ¸¸ R ¹
(7)
t 0, x t 0, T T0
(8)
In the equation above the 'H R is the heat of reaction in the liquid phase, including all liquid phase reactions, but excludin the heat of solution of CO2. The heat of solution of CO2, the vaporization heat of volatile species (H2O and MEA), and sensible heat transfer of CO2 are then included in the interface boundary conditions and are given as:
t ! 0, x kO
wT wx
0,
wCCO2 § hg Ti Tg 'H S ¨ DCO2 wx ©
· ¸ c p M CO2 Ti Tg N CO2 ¹
(9)
H 2O 'HVap N H2O 'HVapMEA N MEA
And the right side boundary conditions in the liquid bulk can be expressed as:
t ! 0, x wT 0 wx
f,
(10)
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3. Comparison of numerical solutions A variety of numerical solutions were tested in this work: Uniform fixed grid methods (Central difference, DSS042, DSS044, DSS020, DSS002, DSS038), non-uniform fixed grid method (subsection), adaptive moving grid method and the Method of Weighted Residuals (orthogonal collocation). When using the uniform fixed grid method, since problem is stiff, many point mesh points are needed to get an accurate result. This results in high computation time. Fig 1 shows the effect of the number of grid points on the prediction of the CO2 concentration profile in the liquid film close to the gas-liquid interface. It can be seen that even with 500 points in the total interval, there are still a very small number of discretization points to predict the CO2 concentration trend in the steep section close to the interface. Less grid points in this area will significantly affect not only the interface value but also the integral result. 3
x 10
-5
nx=100 nx=200 nx=300 nx=400 nx=500
Concentration CO2 [kmol/m3]
2.5
2
1.5
1
0.5
0 0
0.1
0.2
0.3
0.4
0.5 x [m]
0.6
0.7
0.8
0.9
1 x 10
-6
Fig. 1 Effect of number of grid points on the free CO2 concentration profile in the liquid film in the 3 percent of total mesh closest to the gasliquid interface
From Fig.2 it can be seen that the adaptive routine can describe the CO2 trend very accurately, and that the concentration profile using a fixed grid fails in giving a proper description of the steep gradients at high reaction rates.
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7
x 10
Comparison of free CO2 concentration within 3% of film thickness
-5
Central Difference Adaptive moving grid Collocation 6
-
Concentraion of free CO2 (kmol.m3)
5
4
3
2
1
0
-1 0
0.2
0.4
0.6
0.8
1 1.2 Film thickness (m)
1.4
1.6
1.8
2 x 10
-6
Fig. 2 Concentration profiles of free CO2 close to the interface (3% of the discretized liquid film is shown)
For better predicting the CO2 flux with the stiff problem of CO2 transfer concentration profile both the subsection method, the adaptive moving grid and also the collocation method did very well. However, the subsection method spent much computation time to converge 3 boundary conditions. The adaptive moving grid method is more advanced, and it needs much fewer grid points and much less time than the sub-section method. However, it needs to generate the new and better adapted meshes, which takes time. Compared with these two, the orthogonal collocation method has an advantages on both accuracy and computation time. It uses collocation at the zeros of some orthogonal polynomial to transform the partial differential equation (PDE) to a set of ordinary differential equations (ODEs). The ODEs can then be solved by any method. It has been shown that it is usually advantageous to choose the collocation points as the zeros of the corresponding Lagrange polynomial. The collocation method will use more discretization points than the adaptive method but will still be quicker. In Fig.3, the predicted CO2 flux variation for each method is presented as a function of the number of grid points. Furthermore, the predicted CO2 flux is compared with the experimental data. As the figure shows, the calculated CO2 fluxes all converge at an approximate value which is within 10 percent deviation of the experiment data. This indicates that all the methods give the same results at high number of grid point. The Adaptive moving grid method and the orthogonal collocation method need only few mesh points to give convergence. However, the predicted results by all methods converged to a value below the experimental result 8.3*10-7 kmol/m2s. The difference, however, is reasonable, only 10-12% and this deviation does not lie in the numerical method, but in the underlying data. Focusing on the adaptive moving grid method and orthogonal collocation method in this figure, the differences between these two methods are not significant at all after convergence.
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2.5
x 10
Comparison of convergency and accuracy of each numerical method
-6
Central Difference DSS044&DSS004 DSS042&DSS020 DSS038 Subsection Adaptive moving grid Collocation Experimental data +10% deviation -10% deviation
1.5
-
-
CO2 flux (kmol.m2.s 1)
2
1
0.5
0 0
50
100
150
200 250 300 Number of grid points
350
400
450
500
Fig 3 Comparison between predicted CO2 fluxes by using different numerical methods with a different number of grid points and the experimental data
Fig. 4 is shown the predicted CO2 flux variation for each method as function of computation time spent. As can be observed, all the methods need much longer computation time to obtain convergence than the orthogonal collocation method which only needs few seconds to approach the final results. In the end, Fig. 5 gives an overview of the predicted profiles for the 4 main components concentrations and the temperature in the liquid film by using the orthogonal collocation method.
2.5
x 10
Comparison of convergency and CPU time of each numerical method
-6
Central Difference DSS044&DSS004 DSS042&DSS020 DSS038 Subsection Adaptive moving grid Collocation Experimental data +10% deviation -10% deviation
1.5
-
-
CO2 flux (kmol.m2.s 1)
2
1
0.5
0 0
10
20
30
40
50 CPU time (s)
60
70
80
90
Fig 4 The predicted CO2 flux using the different methods as function of CPU time spent
100
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-5
CO2
x 10
3 2 1 0
Concentration of MEAH+ (kmol.m-3)
4
MEAH
x 10
4.998 4.996 4.994 4.992 4.99 4.988
15
10
5
0
4.986
0
0.2
0.4 0.6 Film thickness (m)
-3
20
20
5
5
-1
-3
MEA 5.002
Concentration of free MEA (kmol.m-3)
Concentration of free CO2 (kmol.m-3)
6
0.8
4.984
1
0
2
-5
x 10
MEACOO
x 10
4 6 Film thickness (m)
8 -5
x 10
-5
0
2
4 6 Film thickness (m)
8 -5
x 10
Temperature 323.18
15
323.14 Temperature (degree C)
Concentration of MEACOO- (kmol.m-3)
323.16
10
5
0
323.12 323.1 323.08 323.06 323.04 323.02 323
-5
0
2
4 6 Film thickness (m)
8 -5
x 10
322.98
0
0.2
0.4 0.6 Film thickness (m)
0.8
1 -3
x 10
Fig 5 Concentration and temperature profiles of all the chemical species present in the liquid film. These results are taken from orthogonal collocation method
4. CONCLUSIONS The implementations of a penetration type of the heat and mass transfer model for CO2 absorption into MEA solution in MATLAB were performed and validated with the experimental data from a String of Discs Contactor (SDC). The numerical PDEs solution method and its implementation were found to be the key factor in obtaining a stable, robust and fast computation. The penetration model of heat and mass transfer was implemented for testing a variety of numerical methods. It is found that the finite differences (FDM), in general, is not a proper algorithm in the case of high reaction rates as we observed that an excessive number of discretization points were needed. Both a non-uniform adaptive grid routine, and a method based on orthogonal collocation are accurate and robust, and give relatively fast solution. However, of the two, orthogonal collocation is by far the fastest. Based on this numerical scheme, the combined heat and mass transfer model will be developed further in the next phase of this study.
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Acknowledgement
The financial support from the CCERT project is greatly appreciated. The CCERT project supported by the Research Council of Norway (NFR 182607), Shell Technology Norway AS, Metso Automation, Det Norske Veritas AS, and Statoil AS
References [1] G.F.Versteeg, J.A.M.Kuipers, F.P.H.Van Beckum, and W.P.M. Van Swaaij, “Mass transfer with complex reversible chemical reactions-I. Single reversible chemical reaction”, Chem. Eng. Sci., 44(1989), No.10, 2295-2310 [2] Ahmed Aboudheir, Paitoon Tontiwachwuthikul, Amit Chakma, Raphael Idem, “Kinetics of the reactive abosrption of carbon dioxide in high CO2-loaded, concentrated aqueous monoethanolamine solutions”, Chem. Eng. Sci., 58(2003), 5195-5210 [3] Barakat, H.Z., Clark, J.A., “On the solution of the diffusion equation by numerical methods”, Journal of Heat Transfer, Transactions of ASME, 88(1966), 421-427 [4] Cornelisse, R., Beenackers, A.A.C.M., Van Backum, F.P.H., and van Swaaij, W.P.M., “Numerical calculation of simultaneous mass transfer of two gases accompanied by complex reversible reactions”, Chem. Eng. Sci., 35(1980), 1245-1260 [5] A. Vande Wouwer, P. Saucez, W.E. Schiesser, S. Thompson. “A MATLAB implementation of upwind finite differences and adaptive grids in the method of lines”, Journal of computational and applied mathematics, 183 (2005) 245-258 [6] Blom J.G., Zegeling P.A., Algorithm 731: A Moving-Grid Interface for Systems of One-Dimensional Time-Dependent Partial Differential Equations. ACM Trans. Math Software 20(1994), 194-214 [7] Hartono, A., da Silva, E. F., and Svendsen, H. F., Kinetic of carbon dioxide absorption in aqueous solution of diethylenetriamine (DETA), Chem, Eng, Sci., 64(2009), 3205-3213.
