Advanced Numerical Analysis of Adsorption Processes

Transcription

1 Advanced Numerical Analysis of Adsorption Processes Ben Williamson, 1 Kimberly Nelson, 2 Shannon M. Mahurin, 3 and Craig M. Teague 1 1 Department of Chemistry, Cornell College, Mount Vernon, IA 2 Department of Chemistry, University of Tennessee, Knoxville, TN 3 Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN

2 Pure Component Adsorption Equilibria Adsorption is analyzed primarily from pure gas isotherms Isotherms are obtained by measuring pure gas loading at varying pressures An adsorption isotherm equation is then fit to the experimental data to help characterize the process CO 2 Uptake (Weight %) Pressure (kpa) Figure 1: Experimentally measured loading of CO 2 onto mesoporous carbon.

3 Mixed gas Adsorption While pure gas adsorption equilibria are easily measured, mixed gas data can be difficult and expensive to collect Alternatively, Ideal Adsorbed Solution Theory (IAST) 1 can be used to model and predict mixed gas adsorption isotherms from pure component data We created a MATLAB program to model binary adsorption equilibria from pure component data using IAST

4 IAST Overview To perform this calculation, adsorption isotherms must first be fit to the pure component data The quality of fit of these models is a key factor in the prediction accuracy We have chosen to fit the pure component data to one of the four following isotherm expressions:

5 IAST Overview (cont.) IAST centers around the following two equations: Once the values of P 0 and π are obtained at a given pressure, the mixed gas loading can be calculated

6 Preliminary Testing Predictions from pure component data were compared to experimental mixed gas adsorption obtained from the literature 2 Our IAST predictions show reasonable agreement with experimental data Mixed Gas Loading of Ethane and Methane on Templated Carbon Figure 2: Experimental results of mixed gas loading (markers) are graphed against the IAST results obtained from the Toth isotherm (lines).

7 Adsorption on Heterogeneous Materials While the Langmuir isotherm effectively models adsorption equilibria on a number of materials, it assumes a completely homogenous adsorbing surface Other isotherms (Sips, Toth, and many more) attempt to account for heterogeneity by adding more parameters These equations result in better fitting models but provide little information regarding the nature of the heterogeneity The integral adsorption equation can be used to gain more detailed insight into surface heterogeneity

8 The Integral Adsorption Equation The integral adsorption equation always has the same general construction:, θ t (p) : experimentally measured fractional loading x: represents some type of heterogeneity (commonly adsorption energy or pore size) f(x): distribution function normalized to 1 θ l (p,x): local isotherm equation written as a function of p and x (commonly called the kernel) A more detailed picture of heterogeneity can be obtained by finding the distribution function f(x) The calculation of the distribution function is an ill posed problem but a number of algorithms exist to compensate The CAESAR (Computed Adsorption Energies, SVD Analysis Result) algorithm 4 can be used to solve for f(x)

9 The CAESAR Algorithm The integral adsorption equation can be approximated as:, 1,2, n: number of subintervals m: number of experimental points This expression is mathematically equivalent to the following matrix vector equation: A: a matrix with elements A j,i = θ l (x i,p j ) x θ t : a vector with experimental isotherm The distribution function f can be found if A 1 is known A is a non invertible matrix but a careful approximation can be used to obtain stable solutions for f

10 The CAESAR Algorithm (cont.) Singular Value Decomposition (SVD) is used to factor A giving: U and V are orthogonal matrices and Q is a diagonal matrix containing the singular values of A The rank r of A is calculated and the first r singular values of Q are kept with the rest being set to zero The matrices are recombined giving an invertible matrix A p approximately equal to A Finally, the non negative least squares method 5 is used to find non negative solutions for f

11 Implementation We wrote a MATLAB program to calculate f(x) using the CAESAR algorithm The Langmuir equation was used as the kernel: 1 where b can be written as a function adsorption energy: 3 / E: adsorption energy b 0 : pre exponential factor R: gas constant T: temperature

12 Testing The program was tested on data for CO 2 loading onto mesoporous carbon The two peaks in the distribution function suggest the presence of two adsorption sites with different adsorption energies

13 Other Kinds of Heterogeneity Although our program currently analyzes energetic heterogeneity, it can be easily applied to analyze other types One type of heterogeneity that is of particular interest in nanomaterials research is pore size Non local density functional theory (NLDFT) has been used to find a kernel based on pore size 6 The use of such a kernel will allow for the pore size distribution of a material to be calculated from a pure component isotherm

14 Future Work Continued testing of IAST program with other types of experimental data Additional isotherm fitting options in the IAST program Extension of the integral adsorption program to other local isotherms (possibly Hill de Boer or Frunking Fowler Guggenheinm equations)

15 Acknowledgments This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences: Chemical Sciences, Geosciences, and Biosciences Division This work was supported by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists under the Visiting Faculty Program We thank Cornell College and the University of Tennessee for funding specific aspects of this work

16 References 1. Myers, A.L. and Prausnitz, J.M.; Thermodynamics of mixed gas adsorption, AlChe.J., 1965, 11, Chen, Jiahui; Loo, Leslie; Wang, Kean; An Ideal Absorbed Solution Theory (IAST) Study of Adsorption Equilibria of Binary Mixtures of Methane and Ethane on a Templated Carbon, Journal of Chemical & Engineering Data (4), Do, D., Adsorption analysis: equilibria and kinetics, Series on Chemical Engineering, Koopal, L. and Vos, C., Surface heterogeneity analysis by gas adsorption: Improved calculation of the adsorption energy distribution function using a new algorithm named CAESAR. J. Col. and Int. Sci., 1985, 105: Lawson, C. and Hanson, R., Solving Least Squares Problems, Prentice Hall, Chapter 23, p. 161, Landers, J.; Gor, G.; and Neimark, A., Density functional theory methods for characterization of porous materials. Colloids and Surfaces A: Physiochem. Eng. Aspects, 2013, 437: 3 32.