Modeling of Adsorption Equilibria through Gaussian Process

Subscriber access provided by Nottingham Trent University

Thermodynamics, Transport, and Fluid Mechanics

Modeling of Adsorption Equilibria through Gaussian Process Regression of data in Dubinin's representation: Application to Water/Zeolite Li-LSX Aditya Desai, Valentin Schwamberger, Thomas Herzog, Jochen Jänchen, and Ferdinand P. Schmidt Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b03005 • Publication Date (Web): 22 Aug 2019 Downloaded from pubs.acs.org on August 29, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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

Industrial & Engineering Chemistry Research

Modeling of Adsorption Equilibria through Gaussian Process Regression of data in Dubinin’s representation: Application to Water/Zeolite Li-LSX Aditya Desai,† Valentin Schwamberger,† Thomas Herzog,‡ Jochen Jänchen,‡ and Ferdinand P. Schmidt∗,† †Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany ‡Technical University of Applied Sciences Wildau, 15745 Wildau, Germany E-mail: [email protected]

Abstract

covered by the cycle. Frequently, experimental adsorption equilibrium data for the adsorbent/adsorbate pair of interest are scarce, e.g. only a few adsorption isotherms or isobars are available. From this situation, a need arises for relatively simple equilibrium models whose parameters can be completely determined from whatever experimental data are available. One still popular approach to this problem is based on Dubinin’s phenomenological theory of volume filling of micropores. 1,2 Another approach that has the advantage of being firmly grounded in statistical thermodynamics is the GSTA approach. 3,4 However, a lot of adsorption data points are required on several adsorption isotherms for a reasonable GSTA fit, 4 and the model is not suitable in cases where fluid-fluid interactions play an important role, leading to ‘s-shaped’ isotherms (type IV/V). 5

A thermodynamically consistent modeling of adsorption equilibrium data is essential in the modeling of adsorption heat pump cycles. Especially when experimental data are scarce, the approach taken by Dubinin and co-workers to map all adsorption data onto a single ‘characteristic curve’ can be very useful. Different choices for obtaining this curve for modeling purposes are discussed and analyzed here using adsorption equilibrium data of water on zeolite Li-LSX as an example. It is shown that the fit function proposed by Dubinin and Astakhov results in a poor fit to the data. The use of arbitrary fit functions, however, leads to an overfitting of the data, as is exemplified through a cross-validation analysis. Gaussian process regression (GPR) is discussed as an alternative way to compute the ‘characteristic curve’ and is shown to result in a good fit to the data while by design avoiding overfitting.

We focus here on the Dubinin approach, at the center of which lies the reduction of the three-dimensional loading field x ≡ x(p, T ) to two-dimensional data W ≡ W (A), where W = x/ρad is the specific adsorbed volume and A = RT ln(ps /p) is Polanyi’s adsorption potential, which is the difference in Gibbs free energy between the bulk liquid and the adsorbate at the same temperature. Several functional

Introduction Modeling of cyclic adsorption processes, especially of adsorption cooling and related cycles, requires a coherent representation of adsorption equilibria across the whole loading field

ACS Paragon Plus Environment

1

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

forms have been proposed for the ‘characteristic curve’ W (A). 6–8 A key tenet of Dubinin’s theory is the temperature invariance of W (A), i.e. the temperature dependence of adsorption equilibria can be completely described by the temperature dependence of A and of the specific volume of the adsorbate. In that case, adsorption isotherms at different temperatures fall onto the same characteristic curve in a W (A) representation of the data.

Page 2 of 12

acteristic curve for an working pair. In this paper, the technique of using GPR for fitting the characteristic curve for adsorption equilibria has been applied for the adsorption of water on Li-LSX. The goodness of fit of the one obtained using GPR is compared here both to a fit according to the N´ un ˜ez approach and to a ‘classical’ Dubinin-Astakhov fit as described in literature.

While there has been some success in relating certain parameters in the ‘characteristic curve’ equations to fundamental properties of the microporous solids and their interactions with the working fluids, 9–15 the temperature invariance of W (A) remains an empirical observation 16 that is valid for some adsorbates on some microporous adsorbents (and not for others). For the modeling and numerical simulation of adsorption cooling cycles, Dubinin’s approach just provides a useful tool for representing adsorption equilibria across the whole loading field in a simple form. 17–19

Gaussian Process Regression Classical regression techniques involves assuming that the given data can be described by a certain class of functions (e.g. linear functions or polynomials). The parameters corresponding to this class of functions can then be determined by minimising the deviation from the experimentally measured data points. If the selected class of functions are too simple, they may not fit the experimental data with sufficient accuracy. If, on the other hand, the selected class of functions is too complex, then it may lead to overfitting the data, i.e. the interpolating function fits the data used to generate it very well, but in order to do so introduces features in the curve which could result in large errors for predictions at intermediate points.

T. N´ un ˜ez 20,21 introduced the approach to use arbitrary fit functions on a W (A) representation of adsorption equilibrium data. He pointed out that the underlying data should span both the full temperature range and the full adsorption potential range of the cycle to be simulated, and that this approach should be used only if the assumption of temperature invariance of W (A) was supported by the data. Such arbitrary fit functions have also been used successfully in the numerical simulation of adsorption cycles. 22–25

Gaussian processes can be considered a generalization of multivariate Gaussian distributions to an infinite number of variables. Specified by a mean and a covariance function, a Gaussian Process represents a probability distribution over functions and can be used as a prior with respect to Bayesian statistics. Then, the prior is conditioned on the provided data (here: the measurements). The resulting restricted probability distribution is called the posterior, which posterior can be used for predictions. The underlying function of the posterior is not given in closed form, but is computed using the data. 28 Both the covariance function of the GP and its free parameters, the so-called hyperparameters, have to be selected based on the assumptions about the considered function

The use of more complex function forms like rational polynomials as suggested by N´ un ˜ez 20 introduces the risk of overfitting and thus, needs manual intervention and expert knowledge at the time of determining the appropriate fit function. A new technique for finding an appropriate numerical fit based on Gaussian process regression (GPR) is proposed by V. Schwamberger. 26 This technique provides a systematic basis for the interpolation of the char-


2

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


(here: the known properties of the characteristic curve) and based on the training data. This is called the model selection process. 29 GPR has been applied in several fields like robotics, image processing and manufacturing process optimization. 27

two days in an evacuated desiccator over saturated salt solutions such as ammonium chloride at p/ps = 0.79, equal to an relative humidity, RH = 79 % or MgCl2 for a lower relative pressure (p/ps = 0.33, RH = 33 %).

For the fitting of adsorption characteristic curves, the covariance function used here is a sum of a covariance function of the Matérn class with ν = 5/2 30 and a constant σ02 which is used to represent white noise in the experimental data. This is given by: ! √ ! √ 2 5r 5r 5r + 2 exp − + σ02 , k(r) = σf2 1 + l 3l l (1) where σf , l and σ0 are hyperparameters, i.e. free parameters of the covariance function. These hyperparameters are chosen so as to lead to the ‘best guess’ of the output. This is done in the model selection process, which ensures that overfitting of the data is avoided in GPR without any restriction of the analysis to a certain class of functions. In other words, the model selected based on marginal likelikhood is the least complex fitting the data appropriately. Also highly complex dependencies can be explained, since a GP defines a distribution over an infinite number of functions. 31

Equilibrium data of the water adsorption and desorption have been generated by gravimetric isotherm measurements using a McBain-Bakr quartz spring balance 33 equipped with MKS Baratron pressure sensors covering a range of 10−5 mbar to 10−3 mbar. This allows measurements of the isotherms in the range 0.001 mbar to 25 mbar water vapor pressure. The dilation/contraction of the spring was followed with a cathetometer. The sensitivity of the quartz spring was 4 mm/mg and the resolution of the cathetometer 0.01 mm. So the resolution for the water uptake (using 100 mg sample) was 0.0004 g/g. Prior to each adsorption experiment, about 100 mg sample was degassed over 2 to 3 h at 523 K and p < 10−5 mbar. The equilibrium data can be measured in a temperature range between 253 K and 353 K. Prior to the isotherm measurements the 1.6– 2.6 mm thick Li-LSX pellets (approximately 150 mg) were calcined for at least two hours at 623 K overnight in high vacuum atmosphere (p¡10−5 mbar).

Results and Discussion

Experimental Setup

Adsorption Equilibrium Data

The adsorption equilibrium data reported here have originally been published in a project report in German. 32 We focus here on the water adsorption equilibria of zeolite Li-LSX, a lithium-exchanged low-silica X zeolite obtained from Tricat in pelletized form with 15% of clay binder. Samples of this adsorbent have been first studied by different thermogravimetric methods (TG) on a Netzsch STA 409 apparatus with a heating rate of 3 K/min up to 723 K using platinum crucibles. The purge gas flow (N2 , Air Liquide, 5.0) was 70 ml/min. Prior to the TG experiments, the samples were preconditioned at controlled atmosphere for less than

The adsorption isotherms corresponding to 313 K, 332 K, 372 K, and 392 K were measured as described above (see Figure 1a). The measured data were transformed into W -A representation, under the assumption that the adsorbate density was equal to the density of saturated liquid water at the equilibrium temperature (see Figure 1b). The density and saturated vapor pressure of water are calculated using the IAPWS IF-97 standard. 34 It can be observed that all the measured points can indeed be represented with sufficient accuracy by a single characteristic curve.


3

0.3 0.25

Loading (in g/g)

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

Specific Adsorbed Volume (m3/kg)


0.2 0.15 0.1 313 K 332 K 372 K 392 K

0.05 0 10 -6

10 -4

10 -2

10 0

Relative pressure p/p0

3.5

Page 4 of 12

×10 -4 313 K 332 K 372 K 392 K

3 2.5 2 1.5 1 0.5 0

0

500

1000

1500

2000

2500

Adsorption Potential (J/g)

(a)

(b)

Figure 1: Isotherms for the adsorption of water on zeolite Li-LSX represented as a loading vs. pressure, and b as specific adsorbed volume vs. adsorption potential. Table 1: Dubinin-Astakhov fit parameters Parameter

Value

Additionally, the W -A data were fitted to an arbitrarily chosen function as proposed by N´ un ˜ez. The commercial software package Tablecurve2D was used to this end. The best fit was obtained for a Fourier polynomial (see Equation (3)) with the fit parameters as given in Table 3.

Unit

3.227 × 10−4 m3 /kg 1037.8 J/g 1.9747 —

W0 E n

Table 2: Hyperparameters for GPR Parameter l σf2 σ02

4 X ˜ + ci sin(iA) ˜ bi cos(iA) W =a+

Value

(3)

i=1

1069.5 2.64 × 10−8 2.56 × 10−11

where A˜ = (A−Amin )/(Amax −Amin )·π, Amin = 165.48 J/g and Amax = 2491.7 J/g. Table 3: Parameters for the Fourier series polynomial

Fits The adsorption equilibrium data for the working pair of water/Li-LSX described above were fitted to the form proposed by Dubinin and Astakhov, 6 i.e. n A W = W0 exp − , (2) E where, the fit parameters are given in Table 1. An interpolated curve for the equilibrium data was also obtained using GPR, using the hyperparameters for the covariance function from Equation (1) as given in Table 2.

Parameter

Value

a b1 c1 b2 c2 b3 c3 b4 c4

0.000137272245272534983 0.000234915633837333604 -3.05701284123605818e-05 3.26004936632982228e-05 -0.000149700085022314929 -9.26987982396255557e-05 -4.32188393612230405e-06 -1.10027970015600697e-05 4.19075572740824236e-05


4

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

Industrial & Engineering Chemistry Research Specific Adsorbed Volume (m3/kg)

Page 5 of 12 3.5

×10 -4

curate description of the characteristic curve as obtained from the experimental data.

313 K

3

332 K 372 K

2.5

As discussed above, using a functional form which is more complex leads to the danger of overfitting. In order to evaluate the overfitting phenomenon, cross-validation using the ‘LeaveOne-Out’ method was performed. For this, the following steps were performed:

392 K

2

GP Fit DA Fit

1.5

Nunez Fit

1 0.5 0

0

600

1200

1800

2400

3000

1. One experimental data point was left out and the hyperparameters or coefficients for Equations (1) to (3) were determined using the reduced dataset consisting of the 38 remaining data points.

Adsorption Potential (J/g)

Figure 2: Characteristic curve data for water/Li-LSX with Dubinin-Astakhov, N´ un ˜ez (Fourier) and GPR fits

2. The W -value corresponding to the Avalue of the ‘left-out’ data point (Wfit ) as well as the deviation from the experimental value, i.e. e = Wfit − Wexp , was calculated for the different fit approaches.

The characteristic curve for the three regression methods along with the experimental data is shown in Figure 2. The interpolated characteristic curve obtained from the GP fit smoothens out the local variation from the experimental data. It is also in this case positive and non-increasing, as would be expected from Dubinin’s characteristic curve. Note, however, that monotonicity is not necessarily achieved by the GP method employed here. In some cases (for other adsorbent/adsorbate pairs) the GP fit of the W (A) data may result in a nonmonotonic characteristic curve. In such cases, it is possible to include monotonicity information in the data e.g. through adding virtual derivative observations. 35

3. Steps 1 and 2 were performed for all the 39 data points and q the RMSD was calcuP39 2 lated as RM SD = i=1 ei /39. The results of this method are shown as Method B in Table 4. The RMSD for the GP fit and DA fit is similar in magnitude to that obtained for the respective fit approach for Method A. The RMSD for the N´ un ˜ez fit, on the other hand, is significantly larger than the one for Method A, which indicates overfitting. Finally a third method for evaluating the goodness-of-fit, where the different adsorption isotherms were treated as a subsample. This reflects the point of view of an experimentalist, as the experimentalist is taking the data isotherm-by-isotherm and is interested to see how well the isotherms already taken predict the characteristic curve data of an additionally measured isotherm. The three fits were each repeated on reduced data sets, i.e. one of the measured isotherms had been removed. This method is similar to k-fold cross-validation, but has differing numbers of data points in the subsamples. Here, the removed samples are highly correlated, since they are on the same isotherm.

Goodness of fit A quantitative evaluation of the goodness-of-fit for all the fit approaches was performed using three different methods. In the first method (Method A in Table 4), the root mean square deviation (RMSD) at the points corresponding to the experimental data was calculated as a measure to determine how well the fit functions described the ‘training data’. Here it was observed that the RMSDs for the GP fit as well as the N´ un ˜ez fit are similar and approximately an order of magnitude smaller than the DA fit. This would support the hypothesis that DA fit does not describe the experimental data well enough and the other fits provide a more ac-


5


Page 6 of 12

Table 4: Root mean square deviations using different measures for goodness-of-fit RMSD [10−5 m3 /kg]

Method

DA-Fit GP-Fit N´ un ˜ez Fit (Fourier) A B C(313 K) C(332 K C(372 K) C(392 K)

1.630 1.678 1.372 1.396 2.149 2.429

0.436 0.671 0.826 0.619 0.507 0.882

For standard k-fold cross-validation, the samples are randomly partitioned. Furthermore, slight temperature dependencies also affect the results. In each case, the isotherm not used for fitting was used as test data and the RMSD for the data points in the left-out isotherm was calculated, analogous to Method B. The results of this method (Method C) are also shown in Table 4). Similar to the results from Method B, the RMSD for the test data in case of the DA and GP fits is of a similar order of magnitude as compared to that when all four isotherms were used both as training and test data. On the other hand, the N´ un ˜ez fits show a greater variation in their RMSDs. They are also larger than the RMSD for the fit using all four isotherms, which is a further indication of overfitting. In general, the GP fit shows the smallest errors across the different cases of leaving one isotherm out.

0.548 7.209 7.056 0.763 2.988 20.307 In Dubinin’s approach, the isosteric enthalpy of adsorption qst is often represented as: 1 ∂A , (4) qst = qvap + A + αA,a T ∂ ln W T where, qvap is the latent heat of vaporization and αA,a is the coefficient of thermal expansion of the adsorbate, keeping the adsorption potential A constant. The last term involving the coefficient of expansion is neglected in many cases. 21 For numerical simulations, Schwamberger and Schmidt 36 have argued for the use of the Clausius-Clapeyron equation for computing the isosteric heat of adsorption when aiming at thermodynamic consistency. Upon neglecting the volume of the adsorbed phase, the ClausiusClapeyron equation can be rearranged to: ∂p , (5) qst = vg T ∂T x

This indicates that the GP fit represents the experimental data the best out of the three fit approaches considered here. Further, it does not result in overfitting and also provides a more reliable fit when a smaller training dataset (with fewer isotherms) is available.

where vg is the specific volume of the vapor state. In this case, only the adsorption equilibrium data are fitted in the W (A) representation, and no additional approximations have to be introduced. This way, it is also easier to ensure the thermodynamic consistency of adsorbate heat capacities. 36 The calculation of the isosteric heat of adsorption using Equation (5) requires the numerical differentiation of the isosteres. The isosteres themselves can be determined using the different fits for the equilibrium data.

Isosteric heat of adsorption Apart from the equilibrium loading, the isosteric heat of adsorption is another important material property, that is used in the simulation of adsorption cycles for heat transformation applications.


6

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


The isosteric heat of adsorption as a function of loading and temperature calculated in this way using the N´ un ˜ez fit (i.e. Equation (3)) is shown in Figure 3, whereby two different temperature step sizes were chosen for the numerical differentiation. A similar calculation of the isosteric heat of adsorption using the GP fit is shown in Figure 4.

curve for the working pair, Li-LSX and water, here and equilibrium properties at intermediate points between the experimental data can be calculated. The fit obtained using GPR has been compared to the function obtained using the approach of Dubinin and Astakhov and an analytical fit function using the approach proposed by N´ un ˜ez. It is found that the GP fit has a smaller deviation from the experimental data compared to the Dubinin-Astakhov fit. It also avoids the danger of overfitting, occurring in the N´ un ˜ez fit, as has been shown for the characteristic curve as well as the isosteric heat of adsorption. Although the Gaussian Process yields a monotonically decreasing characteristic curve when applied to our dataset, this requirement of the Dubinin formalism is not automatically fulfilled. In order to make the method more universally applicable to adsorption equilibrium data, it may be necessary to enforce monotonicity of the resulting characteristic curve by adding constraints or virtual derivative observations to the Gaussian Process. 35

The isosteric heats of adsorption calculated using the N´ un ˜ez fit are more sensitive to the temperature step size. When a relatively small temperature step of 0.01 K is chosen, the heat of adsorption shows a large amount of variation, which appears as pixellation in Figure 3a. On the other hand, the temperature step size for the numerical differentiation has no discernible effect on the isosteric heats of adsorption calculated using the GP fit (compare Figure 4a with Figure 4b). This is an indication that GPR is a more robust fitting method for quantities that are relevant in the simulation of adsorption cycles, i.e. the characteristic curve as well as the derivatives of the isosteres.

Conclusion Acknowledgement The authors would like to acknowledge the funding of this work by the German Federal Ministry of Education and Research (BMBF) through the cooperative project MAKSORE (grant no. 03SF0441A).

Although Dubinin’s approach to reduce the adsorption equilibrium field to a single characteristic curve is useful in the modeling of adsorption heat pump cycles, the use of the functional form proposed by Dubinin and Astakhov may not always lead to an adequately accurate description of the equilibrium data. This has been shown here to be true for the working pair of Li-LSX and water. An alternative approach proposed by T. N´ un ˜ez allows for a broader set of functions, compared to that proposed by Dubinin and co-workers, and involves finding the one most suitable for a particular working pair. A method for arriving at the interpolating function for the characteristic curve based on Gaussian Process Regression, which extends the approach of N´ un ˜ez and provides a systematic framework for its implementation was first used by Schwamberger 26 and is applied and tested rigorously in this paper. This method is applied to fit the characteristic


7


(a)

Page 8 of 12

(b)

Figure 3: Isosteric heat of adsorption of water on zeolite Li-LSX according to Equation (5) with with isosteres from the N´ un ˜ez fit and a temperature step of a 0.01 K, and b 1 K

(a)

(b)

Figure 4: Isosteric heat of adsorption of water on zeolite Li-LSX calculated using Equation (5) with isosteres from the GP fit and a temperature step of a 0.01 K, and b 1 K


8

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


References

(10) Carrasco-Mar´ın, F.; López-Ramón, M.; C.Moreno-Castilla, Applicability of the Dubinin-Radushkevich Equation to CO 2 Adsorption on Activted Carbons. Langmuir 1993, 9, 2758–2760.

(1) Dubinin, M. M. Adsorption in Micropores. Jour. of Colloid and Interface Science 1967, 23, 487–499. (2) Dubinin, M. M. In Progress in Surface and Membrane Science; Cadenhead, D., Danielle, J., M.D., R., Eds.; Academic Press, 1975; Vol. 9; pp 1–70.

(11) Chen, S.; Yang, R. Theoretical Basis for the Potential Theory Adsorption Isotherms. The Dubinin-Radushkevich and Dubinin-Astakhov Equations. Langmuir 1994, 10, 4244–4249.

(3) Llano-Restrepo, M.; Mosquera, M. A. Accurate correlation, thermochemistry, and structural interpretation of equilibrium adsorption isotherms of water vapor in zeolite 3A by means of a generalized statistical thermodynamic adsorption model. Fluid Phase Equilibria 2009, 283, 73–88.

(12) Chen, S.; Yang, R. Theoretical Investigation of Relationship Between Characteristic Energy and Pore Size for Adsorption in Micropores. Journal of Colloid and Interface Science 1996, 177, 298–306. (13) Gil, A.; Grange, P. Application of the Dubinin-Radushkevich and DubininAstakhov Equations in the Characterization of Microporous Solids. Colloids and Surfaces 1996, 113, 39–50.

(4) Ladshaw, A.; Yiacoumi, S.; Tsouris, C.; DePaoli, D. Generalized gas-solid adsorption modeling: Single-component equilibria. Fluid Phase Equilibria 2015, 388, 169–181.

(14) Terzyk, A. P.; Gauden, P. A.; Rychlicki, G.; Wojsz, R. Fractal dimension of microporous carbon on the basis of the Polanyi-Dubinin theory of adsorption. Part 3: Adsorption and adsorption thermodynamics in the micropores of fractal carbons. Colloids and Surfaces A: Physicochemical and Engineering Aspects 1998, 136, 245–261.

(5) Goldsworthy, M. J. Measurements of water vapour sorption isotherms for RD silica gel, AQSOA-Z01, AQSOA-Z02, AQSOAZ05 and CECA zeolite 3A. Microporous and Mesoporous Materials 2014, 196, 59– 67. (6) Dubinin, M. M.; Astakhov, V. A. In Advances in Chemistry; Gould, R., Ed.; 102; American Chemical Society, 1971; pp 69– 85.

(15) Terzyk, A. P.; Gauden, P. A.; Rychlicki, G.; Wojsz, R. Fractal dimension of microporous carbon on the basis of Polanyi–Dubinin theory of adsorption. Part IV. The comparative analysis of two alternative solutions of the overall adsorption isotherm equation for microporous fractal solids. Colloids and Surfaces A: Physicochemical and Engineering Aspects 1999, 152, 293–313.

(7) M. M. Dubinin, V. V. S. Isotherm Equation for Water Vapor Adsorption by Microporous Carbonaceous Adsorbents. Carbon 1981, 19, 402–403. (8) Stoeckli, F. Dubinin’s Theory for the Volume Filling of Micropores: An Historical Approach. Adsorption, Science and Technology 1993, 10, 3–15.

(16) Jaroniec, M.; Choma, J. In Equilibria and Dynamics of Gas Adsortion on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier Science B.V., 1997; Vol. 104; Chapter

(9) Rozwadowski, M.; Wloch, J.; Erdmann, K.; Kornatowski, J. Energy Distribution Function associated with Dubinin Description of Water Adsorption on Zeolite-Y. Langmuir 1993, 9, 2661–2664.


9


Characterization of geometrical and energetic heterogeneities of active carbons by using sorption measurements, pp 715–744.

Page 10 of 12

chillers based on a validated dynamic object-oriented model. Science and Technology for the Built Environment 2015, 21, 248–257.

(17) Passos, E.; Meunier, F.; Gianola, J. Thermodynamic Performance Improvement of an Intermittent Solar-Powered Refrigeration Cycle Using Adsorption of Methanol on Activated Carbon. Journal of Heat Recovery Systems 1986, 6, 259–264.

(26) Schwamberger, V. Thermodynamische und numerische Untersuchung eines neuartigen Sorptionszyklus zur Anwendung in Adsorptionswärmepumpen und -kältemaschinen. Dissertation, Karlsruher Institut f¨ ur Technologie, 2016.

(18) Critoph, R. Performance Limitations of Adsorption Cycles for Solar Cooling. Solar Energy 1988, 41, 21–31.

(27) Rasmussen, C. E.; Williams, C. K. I. Gaussian Processes for Machine Learning; The MIT Press, 2006.

(19) Critoph, R. Activated Carbon Adsorption Cycles for Refrigeration and Heat Pumping. Carbon 1989, 27, 63–70.

(28) Ref. 27, Ch. 2. (29) Ref. 27, Ch. 5.

(20) N´ un ˜ez, T.; Mittelbach, W.; Henning, H.M. Performance Evaluation and Characterisation of Solid Sorbents for Thermal Energy Storage. IEA Annex 10 Phase Change Materials and Chemical Reactions for Thermal Energy Storage, 4th workshop, Benediktbeuren, Germany. 1999.

(30) Ref. 27, Ch. 4. (31) Ref. 27, Ch. 5.3. (32) Stach, H.; Ackermann, D.; Brösicke, W.; Jänchen, J.; Weiler, E. Erprobung der thermochemischen Wärmespeicherung mit neuen Speichermaterialien; Schlussbericht, 2003.

(21) N´ un ˜ez, T. Charakterisierung und Bewertung von Adsorption f¨ ur Wärmetransformationsanwendungen. Ph.D. thesis, Universität Freiburg, 2001.

(33) McBain, J.; Bakr, A. A NEW SORPTION BALANCE. Journal of the American Chemical Society 1926, 48, 690–695.

(22) Schawe, D. Theoretical and experimental investigations of an adsorption heat pump with heat transfer between two adsorbers. Ph.D. thesis, University of Stuttgart, 2001.

(34) The International Association for the Properties of Water and Steam, Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam. Lucerne, Switzerland, 2007; http://www.iapws. org/relguide/IF97-Rev.pdf, Accessed: 2019-07-29.

(23) Schicktanz, M.; N´ un ˜ez, T. Modelling of an adsorption chiller for dynamic system simulation. Int. J. of Refrigeration 2009, 32, 588–595.

(35) Riihimäki, J.; Vehtari, A. Gaussian processes with monotonicity information. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, AISTATS 2010, Chia Laguna Resort, Sardinia, Italy, May 13-15, 2010. 2010; pp 645–652.

(24) Riffel, D. B.; Wittstadt, U.; Schmidt, F. P.; N´ un ˜ez, T.; Belo, F. A.; Leite, A. P.; Ziegler, F. Transient modeling of an adsorber using finned-tube heat exchanger. International Journal of Heat and Mass Transfer 2010, 53, 1473–1482. (25) Lanzerath, F.; Bau, U.; Seiler, J.; Bardow, A. Optimal design of adsorption


10

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


(36) Schwamberger, V.; Schmidt, F. P. Estimating the Heat Capacity of the Adsorbate – Adsorbent System from Adsorption Equilibria Regarding Thermodynamic Consistency. Industrial & Engineering Chemistry Research 2013, 52, 16958 – 16965.


11


For Table of Contents only 82x44mm (300 x 300 DPI)


Page 12 of 12

Modeling of Adsorption Equilibria through Gaussian Process

Recommend Documents