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К ВОПРОСУ ИСПОЛЬЗОВАНИЯ УРАВНЕНИЯ ДУБИНИНА-РАДУШКЕВИЧА-АСТАХОВА
В РАСЧЕТАХ ИЗОТЕРМ H2, CO2, CO ДЛЯ ПРОЦЕССА КОРОТКОЦИКЛОВОЙ БЕЗНАГРЕВНОЙ АДСОРБЦИИ ИЗВЛЕЧЕНИЯ ВОДОРОДА НА БЛОЧНЫХ ЦЕОЛИТАХ 13X
О.О. Голубятников, Е.И. Акулинин
Олег Олегович Голубятников (ORCID 0000-0002-0530-0964)*, Евгений Игоревич Акулинин (ORCID 0000-0003-0848-4380)
Кафедра Технологий и оборудования пищевых и химических производств, Тамбовский государственный технический университет, ул. Советская, 106/5, Тамбов, 392000, Россия E-mail: golubyatnikov_ol@mail.ru*, akulinin-2006@yandex.ru
Расчет изотерм адсорбции компонентов синтез-газа в процессе короткоцикловой безнагревной адсорбции (КБА) для извлечения из него водорода на цеолитовых адсорбентах традиционно осуществляется с использованием уравнения Ленгмюра или производных от него уравнений. Недостатком его использования является необходимость получения экспериментальных изотерм адсорбции по каждому газу, входящему в газовую смесь на используемом адсорбенте. Такой подход является затратным, особенно в случае многокомпонентных смесей, таких, как синтез-газ. В статье продемонстрирован способ расчета изотерм адсорбции газов H2, CO2, CO в процессе КБА извлечения водорода из синтез-газа на перспективных блочных цеолитах 13X с использованием уравнения Дубинина-Ра-душкевича-Астахова (ДРА). Получены экспериментальные изотермы адсорбции H2, CO2, CO при температурах 293 и 323 К, изотермы адсорбции N2 при температуре его кипения 77,35 К на цеолитах 13X трех образцов. С использованием экспериментальных изотерм решены задачи параметрической идентификации и определены параметры уравнения ДРА для газов H2, CO2, CO на блочных цеолитах 13X. Установлено, что расчет изотерм адсорбции газов H2, CO2, CO на блочных цеолитах 13X, характеристики которых попадают в интервалы предельного адсорбционного объема 0,258-0,296 см3/г и характеристической энергии адсорбции 10279-13395 Дж/моль, можно корректно осуществлять с использованием уравнения ДРА в диапазоне температур 293-323 К и давлений 0-30 атм. При этом максимальная ошибка в рассогласовании между экспериментальными изотермами газов и рассчитанными по уравнению ДРА составила 14,69% по H2; минимальная - 1,81% по CO2. В отличие от традиционного способа, полученные коэффициенты аффинности позволяют осуществлять расчет изотерм адсорбции газов H2, CO2, CO в найденных интервалах предельного адсорбционного объема и характеристической энергии адсорбции всего лишь по одной изотерме стандартного газа (N2) на используемом цеолите 13X. Универсальность способа расчета по предложенной методике может быть использована для расчетов изотерм адсорбции различных газов, смесей и адсорбентов в циклических адсорбционных процессах.
Ключевые слова: короткоцикловая безнагревная адсорбция, цеолит 13X, водород, синтез-газ, изотерма, уравнение Дубинина-Радушкевича-Астахова
ON THE ISSUE OF USING THE DUBININ-RADUSHKEVICH-ASTAKHOV EQUATION IN CALCULATING ISOTHERMS OF H2, CO2, CO IN THE PRESSURE SWING ADSORPTION PROCESS FOR HYDROGEN RECOVERY ON BLOCK ZEOLITES 13X
O.O. Golubyatnikov, E.I. Akulinin
Oleg O. Golubyatnikov (ORCID 0000-0002-0530-0964)*, Evgeny I. Akulinin (ORCID 0000-0003-0848-4380)
Department of Technologies and Equipment for Food and Chemical Industries, Tambov State Technical University, 106/5, Sovetskaya St., Tambov, 392000, Russia E-mail: golubyatnikov_ol@mail.ru *, akulinin-2006@yandex.ru
The calculation of adsorption isotherms of synthesis gas components in the process ofpressure swing adsorption (PSA) for hydrogen extraction on zeolite adsorbents is traditionally carried out using the Langmuir equation or equations derived from it. Its disadvantage consists in the necessity to obtain experimental adsorption isotherms for each gas in the gas mixture on the used adsorbent. This approach is expensive, especially in the case of multicomponent mixtures, such as synthesis gas. The paper demonstrates the prospects of an alternative method for calculating adsorption isotherms of H2, CO2, CO gases in the PSA process for hydrogen extracion from synthesis gas on block zeolites 13X using the Dubinin-Radushkevich-Astakhov (DRA) equation. Using zeolites 13Xmade by three samples, experimental adsorption isotherms of H2, CO2, CO at temperatures of293 and 323 K, and adsorption isotherms of N2 at a boiling point of 77.35K were obtained. Using experimental isotherms, the problems ofparametric identification were solved, and the parameters of the DRA equation for H2, CO2, CO gases on block zeolites 13X were determined. It has been found that the calculation of adsorption isotherms of H2, CO2, CO gases on zeolites 13X, the characteristics of which fall within the ranges of the limiting adsorption volume 0.258-0.296 cm3/g and the characteristic adsorption energy 10279-13395 J/mol, respectively, can be correctly carried out using the DRA equation in the temperature range of293-323 K and the pressure range of 0-30 atm. In this case, the maximum value of the mismatch between the experimental gas isotherms and those calculated by the DRA equation is 14.69% for H2; the minimum value is 1.81% for CO2. Certain affinity coefficients make it possible to calculate the adsorption isotherms of H2, CO2, CO gases in the obtained ranges of the limiting adsorption volume and characteristic adsorption energy using only one isotherm of the standard gas (N2) on the zeolite 13X used. The versatility of the DRA equation can be used to calculate the adsorption isotherms of various gases, mixtures, and adsorbents in cyclic adsorption processes.
Key words: pressure swing adsorption, zeolite 13X, hydrogen, synthesis gas, isotherm, Dubinin-Asta-khov equation
Для цитирования:
Голубятников О.О., Акулинин Е.И. К вопросу использования уравнения Дубинина-Радушкевича-Астахова в расчетах изотерм H2, CO2, CO для процесса короткоцикловой безнагревной адсорбции извлечения водорода на блочных цеолитах 13X. Изв. вузов. Химия и хим. технология. 2022. Т. 65. Вып. 8. С. 39-47. DOI: 10.6060/ivkkt.20226508.6625.
For citation:
Golubyatnikov O.O., Akulinin E.I. On the issue of using the Dubinin-Radushkevich-Astakhov equation in calculating isotherms of H2, CO2, CO in the pressure swing adsorption process for hydrogen recovery on block zeolites 13X. Chem-ChemTech [Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol.]. 2022. V. 65. N 8. P. 39-47. DOI: 10.6060/ivkkt.20226508.6625.
INTRODUCTION main components accompanying hydrogen in synthesis gas are CO2, CO, CH4, and N2 [3]. The adsorption
Hydrogen is a promising source of ^rg^ and separation process is usually organized according to its consumption in the world doubles every fitted the principle of pressure swing adsorption (PSA) [4]. years. For 2020, more than 75 million tons of hydrogen The PSA unit for hydrogen extraction is a complex are consumed in the world. M°re than 50% of hydro- technological object that includes four or more ad-gen is consumed in the production of ammonia and sorbers, shut-off valves, receivers, and instrumenta-more than 25% in the oil refmmg m^r^ aM there is tion. The principle of the unit operation consists in sea constant increase in hydrogen consumption in the en- quential alternation of adsorption and desorption pro-ergy sector [1, 2]. cesses in adsorbers filled with an adsorbent. This is
То^ the main method of hydrogen produc- achieved by providing different gas pressures at the ad-
tion is the steam refommg of natural gas which occu- sorption and desorption stages (for implementing the
pies 75% of the production volume and makes it pos- psa process, pressure at the adsorption stage must be
sible to obtain production hydrogen at the lowest cost higher than at the desorption stage). As a result, a con-
price of 2.5-5 $ per 1 kg [1, 2]. One of the key stages tinuous flow of product hydrogen with a purity of
is the process of adsorption separation of steam reform- 99.99 vol.% and higher is formed at the unit outlet [3].
ing products in order to extract hydrogen of high purity Both various brands of active carbons [5, 6] and indus-
("."% vo1. and higher) from synthesis gas [3]. The trial zeolites 13X [7, 8] and 5A [9, 10], which possess
high selectivity for hydrogen-accompanying synthesis gas components CO2, CO, CH4, N2, are most often used as microporous adsorbents for hydrogen extraction from synthesis gas. In addition, in recent years, new promising adsorbents for hydrogen extraction, such as metal-organic framework structures MOF, Cu-BTS, Cu-TDPAT, new block adsorbents and a number of others, have been developed [11-13].
The complexity and scale of the PSA unit and the processes occurring in it require high economic costs for their study by experimental methods. The computational capabilities of the modern computer technology allow for the study and optimal design of PSA units using mathematical modeling methods [3, 10, 14]. Mathematical models are systems of partial differential equations describing the fields of velocities, concentrations and temperatures [3, 15, 16]. Such systems are complex, since the variables of the model are in complex relationships with each other and with the transfer coefficients, and the solution of the systems is carried out before the onset of the cyclic steady state [17]. Currently, several approaches have been developed to solve the PSA models [14].
One of the main tasks in the construction of mathematical models is the correct calculation of adsorption - desorption isotherms. In the majority of works on modeling the PSA process for hydrogen extraction from multicomponent mixtures, the equations of Langmuir isotherms and derivative equations (extended Langmuir, dual-site Langmuir, Langmuir-Freundlich) are used [3, 7, 9, 10, 17-19]. The coefficient values of isotherm equations for specific gas mixtures are found by experimentally determining the isotherms of gases included in the mixture in the temperature range of 293-323 K (the range at which the PSA process for hydrogen extraction from synthesis gas is carried out) and then solving the problem of parametric identification of the coefficients. The analysis of the works shows that the values of the Langmuir isotherm coefficients for the same gases and zeolites can differ by whole orders of magnitude [3, 7, 9, 10, 17-19]. This discrepancy can be explained by the fact that the properties of industrial zeolites of the same type (for example, 13X or 5A) made by different manufacturers differ from each other due to the peculiarities of the production technology (the use of various binding components in different amounts, different production methods). Moreover, the properties of the zeolite adsorbent can also vary from the same manufacturer in different batches. Therefore, in order to achieve high accuracy of the PSA process model, it is necessary to experimentally determine the Langmuir isotherm coefficients for each gas included in the initial gas mixture, and not to
use the literature data. This approach is expensive, especially in the case of multicomponent mixtures, such as synthesis gas (which can include up to 7 components).
An alternative method for calculating isotherms is the Dubinin-Radushkevich-Astakhov (DRA) equation based on the theory of volumetric filling of micropores (TVFM) [20]. According to TVFM, for any nature of adsorption interactions (i.e., under the action of dispersion, electrostatic, or other forces) that cause physical adsorption, an adsorption field created by a solid appears in the entire micropore space. Adsorption in micropores is characterized by volumetric filling of the adsorption space, while the molecules sequentially adsorbed in micropores do not form adsorption beds. Therefore, the main geometric parameter characterizing a microporous adsorbent is the volume of micropores, and not their surface. This theory gave birth to the Dubinin-Radushkevich-Astakhov (DRA) isotherm equation which was originally derived to describe adsorption isotherms at temperatures below critical for adsorptives using equilibrium standard states of components in equilibrium phases: limiting adsorption at the equilibrium saturated vapor pressure of the adsorptive [20]. The use of TVFM in the region of supercritical temperatures is usually limited to an interval extending to 100-150 K above the critical temperature [21].
However, a number of works [12, 21-32] shows that the DRA equation can be used to quantitatively describe experimental isotherms also at temperatures above critical by more than 100-150 K on active microporous carbons, zeolites, silica gels, and molecular sieves. In particular, the DRA equation can be used to describe the adsorption of CH4, N2, H2 on Zeo-5A@MOF-74-1 zeolite at temperature T = 298 K and pressure P = 0-22 atm [12]; H2 on active carbons [23, 24]; Ar on zeolite 13X [32]; CH4 on zeolites 13X, CaLSX, KLSX, LiLSX at T = 298-343 K and P = 0-1 atm [31]; CH4, N2 on zeolites Cu-BTC and MaxSorb at T = 298303 K and P = 0-40 atm [30]; CO on zeolite 13X at T = 293-343 K and P = 0-1 atm [27]. Works [22, 25] propose to replace the saturation pressure with standard pressures. With the help of molecular dynamics calculations, it is shown that this replacement leads to an increase in the accuracy of calculating isotherms using the DRA equation in subcritical temperature ranges for microporous carbons [22].
The application of the DRA equation for calculating adsorption isotherms is carried out according to the following procedure. First, it is necessary to experimentally determine the isotherm of the standard substance (for zeolites, the standard gas is usually N2).
Then, with its help, the coefficient values of the limiting adsorption volume of the adsorbent Wo and the characteristic energy of adsorption E are restored. To calculate the isotherms of other gases included in the gas mixture using the DRA equation, the affinity coefficients P (for a standard gas P = 1), which are determined experimentally or calculated using approximate equations, are used on this adsorbent. The most accurate method for determining the affinity coefficients P is experimental. The affinity coefficient P for the corresponding gas is thus determined from the ratio of the ordinates of the characteristic curves [20]. There are also several approximate equations for calculating the affinity coefficient, such as the parachor relation equation, molar heat capacity equation of the considered and standard substances; equations that take into account the polarity and non-polarity of adsorptive molecules, the multiplicity of bonds, the presence of quad-rupole and dipole moments and a number of other effects [33]. However, in practice, all these equations are often inferior in accuracy to the experimental method, when commercial adsorbents are used as adsorbents suitable for the equation.
It should be noted that one experimental isotherm of the standard (model) gas, from which the coefficients Woo and E are determined, is sufficient for using the DRA equation. The remaining isotherms of gases entering the gas mixture can be calculated using the found Wo and E for the model gas and the affinity coefficient P for the gas under consideration.
The purpose of this work is to study the possibility of using the DRA equation for calculating the adsorption isotherms of synthesis gas components in the PSA process for hydrogen extraction (using perspective block zeolites 13X as adsorbents) and determining the ranges of coefficient values of the DRA equation for block zeolites 13X, which ensure the correct calculation of adsorption isotherms.
METHODS AND MATERIALS
To determine the equilibrium conditions in the gas mixture-adsorbent system, studies were carried out to obtain adsorption-desorption isotherms of H2, CO2, CO (which are the main components of synthesis gas -the initial gas mixture). Three samples of block zeolites 13X (Roskhimzashchita Corp.) made of zeolite powder from different manufacturers were selected as adsorbents. This is necessary for determining the ranges of coefficients Wo and E of the DRA equation, where the correct calculation of adsorption isotherms is provided. The study used the following samples of zeolites: 13X-1 - block with equivalent diameter of pore channels d = 0.20 mm made from zeolite powder by Zeochem
(Switzerland); 13X-2 - block with d = 0.26 mm made from zeolite powder by HKC Corp. (Hong-Kong, Chine); 13X-3 - block with d = 0.33 mm made from zeolite powder by KNT Group (Russia).
At the first stage, the process of obtaining adsorption-desorption isotherms for a model gas (N2) was studied. Using an Autosorp IQ gas sorption meter (Quantachrome, USA), the complete N2 isotherm was obtained at a boiling point of 77.35 K for each sample of zeolite 13X.
Further, according to M.M. Dubinin's methodology, the obtained isotherms were plotted in semi-logarithmic coordinates for the working pressure range of 0-30 atm, at which the PSA process for H2 extraction from synthesis gas proceeds. For this purpose, the dependencies of the equilibrium concentration aN2(P/Ps) are plotted in semi-logarithmic coordinates, where the
values lg2 are plotted along the x-axis, and the
values lg(aN2) are plotted along the y-axis. Next, according to the experimental points of the graph
lg(aN2) from lg2 i-^2), the regression equation
^ PN2 /
y = Dx + C is constructed, where the values of the regression coefficients D, C are determined.
The linearized isotherms for each zeolite sample are used to calculate the values of two coefficients of the DRA equation (1), i.e. the limiting adsorption volume W0 (cm3/g) and the characteristic adsorption energy E (J/mol):
* Wo
«fe = U-exP
ufc
RTglgl
—
P,fcy
D^k
W0 = u" 10С,В = = 4.184 X 4.574
0 "2 0.434Г2
(1) 1
B'
_ 10-3Mg,fc _ Pit
Ufe = —^ Pfe = Pb'felO-0-434«^-^)'
a = Vpcr,fc/ P* = -ё,« ¿ =
fe 0.434(Tcr,fc-Tb,fc)' Pcr'fe 1000bfc' fe 8P„,fc
-g,fc и - ^cr'fc
Ps'fe =
exp
g-Lfc
760
7fc
100
where k is the initial gas mixture component, k = H2, CO2, CO; is the equilibrium concentration in the adsorbent, mol/kg; u^ is the molar volume of the adsorbate, cm3/mmol; 7g is gas temperature; p^ is the affinity coefficient to standard gas (N2); B is a parameter characterizing the predominant size of micropores, 1/K2; R is a universal gas constant, J/(mol K), R = 8.31; Ps,k is saturation pressure, atm; Pp,fc is partial pressure, atm; n is the exponent of the DRA equation; M&fc is the molar mass of the k-th component of the gas mixture,
n
p
s.fc
kg/kmol; pk is the density of the adsórbate, g/cm3; pb,k is the gas density at a boiling point, g/cm3; T^ is the boiling point of gas, K; afe is the thermal coefficient of limiting adsorption; rcr,fe is the critical temperature of the k-th component in the gas mixture; pCr,k is the density of the adsorbate at the critical temperature, g/cm3;
is the molar volume constant, l/mol; fi* is the universal gas constant, l atm/(mol K), fi* = 0.0821; Pcr,fe is the critical pressure of gas, atm; Ffe, Cfe are Antoine equation constants; P is the current pressure of gas, atm; yfe is the volumetric gas concentration, vol.%. The initial data for calculating isotherms are taken from [34]. For n = 2, the DRA equation (1) is also known in the literature as Dubinin-Radushkevich (DR) equation, and for n = 3, it is called the Dubinin-Astakhov (DA) equation.
It should be noted that obtaining the complete isotherm of the model gas is an optional condition; a fragment of its isotherm is sufficient for the operating pressure range, at which the PSA process is carried out.
Since the PSA process for hydrogen extraction from synthesis gas proceeds in the temperature range of 293-323 K, then at the second stage, using an iSorb gas sorption analyzer (Quantachrome, USA), adsorption isotherms of H2, CO2, CO are obtained at gas temperatures of 293 and 323 K and pressures of 0-30 atm, which correspond to the operating range of temperatures and pressures of the PSA process under consideration.
Next, the affinity coefficients of H2, CO2, CO gases included in the synthesis gas were determined. The p k values for H2, CO2, CO gases were determined by solving parametric identification problems (2) - (4) presented below.
The problem was formulated as follows: for each k-th gas entering the synthesis gas (fc = H2, CO2, CO), it is necessary to find such values of the affinity coefficient Pk and the exponent n of the DRA equation, for which the mismatch RMS ** (root-mean-square error) between the adsorption value afe determined experimentally and the equilibrium concentration a** determined by the DRA equation will be the smallest.
case 1
RMS** = minQn, n = 1,2,3,4,5; (2)
Qn =
n
min
0.01<Pfc<10
RMS
k,n;
(3)
RMS fc,n = M
case 2
!yW /Vi(Pi)-afc,i(pi,n)
afc,í(Pí)
RMSt = min RMSk;
k 0.01<вс<10 k
0.01<n<5
100%; (4)
(2')
RMS k =
M
1 vw
N /W,¿(P¿)-afc,¿(p¿)
. 100%; (4')
where RMS k is the root mean square error; ak is the adsorption value found experimentally, mol/kg; ak is the equilibrium concentration found by the DRA equation (1), mol/kg; P is gas pressure, atm; i is a measurement number; N is the total number of measurements of the adsorption value for three samples of the same type of zeolite 13X.
In the optimization problem (3), case 1, the value of the affinity coefficient p k for the k-th gas is determined, where the minimum value of the root-mean-square error RMS k,n is achieved for a given integer value of the exponent n of the DRA equation (1). In the optimization problem (2), case 1, among the five values of RMS k,n (n = 1,2,3,4,5) defined in problem (3), the smallest value RMS k n and the corresponding value of the exponent n of the DRA equation (1) is chosen for the k-th gas. If the final value of RMSk is more than 15%, then another optimization problem is solved - (2'), where n is the optimized continuous variable 0.01 < n < 5.
Problems (2)-(4) were solved in the MatLab software environment using the fmincon solver. The SQP sequential quadratic programming algorithm was chosen as the solution algorithm.
RESULT AND DISCUSSION
The complete isotherms of N2 at a boiling point of 77.35 K on zeolites 13X are shown in Fig. 1. Table 1 shows W0 and E coefficients determined by Dubinin's method for all samples of industrial zeolites 13X. Table 1 also shows the values of W0 and E coefficients for zeolites 13X found in the works of other authors.
Fig. 1. Complete nitrogen isotherms at a nitrogen boiling point of 77.35 K on zeolites 13X: 1 - 13X-1; 2 - 13X-2; 3 - 13X-3 Рис. 1. Полные изотермы азота при температуре кипения азота 77,35 К на цеолитах 13X: 1 - 13X-1; 2 - 13X-2; 3 - 13X-3
2
2
Table 1
Coefficient values of the DRA equation for zeolites 13X (standard gas N2) Таблица 1. Коэффициенты уравнения ДРА
Source Wo, cm3/g E, J/mol n
[20] 0.235 7473 2
[13] 0.230 15750 4.52
[32] 0.269 13638 2
13X-1 0.296 13395 2
13X-2 0.262 12916 2
13X-3 0.258 10279 2
The results obtained in the course of experimental studies show that the values of the limiting adsorption volume and characteristic adsorption energy for block zeolite 13X are in the ranges of W0 = 0.258 — 0.296 cm3/g and E = 10279 — 13395 J/mol. The spread of W0 and E values is as follows: the spread of W0 values is 14.73%, the spread of E values is 30.31%. This indicates that in order to calculate the PSA process for hydrogen extraction on zeolites 13X, it is recommended to clarify the values of these coefficients using experimental nitrogen isotherms (as a model gas) in the operating pressure range of the PSA process on the industrial zeolite 13X used (if the manufacturer differs from those presented in Table 1). If it is not possible to obtain an experimental isotherm on a specific zeolite 13X, it is necessary to take into account the obtained spread of W0 and E coefficient values while calculating the isotherm using the DRA equation.
Fig. 2. Isotherms of H2 on zeolites 13X at temperatures: 293 K (curve 1), 323 K (curve 2) and calculation using the DRA equation (solid line)
Рис. 2. Изотермы H2 на цеолитах 13X при температурах 293 K (кривая 1), 323 K (кривая 2) и расчет по уравнению ДРА (сплошная линия)
Figs. 2-4 show the obtained isotherms of H2, CO2, CO at temperatures of 293, 323 K and pressures of 0-30 atm, as well as the calculation of the equilibrium gas concentration using the DRA equation, where
the coefficients and n were determined for H2, CO2, CO by solving parametric identification problems (2) -(4). The results of solving the problems are presented in Table 2 (the line "RMS*, %" contains the obtained minimum and maximum RMS).
15 P, atm
Fig. 3. Isotherms of CO2 on zeolites 13X at temperatures: 293 K (curve 1), 323 K (curve 2) and calculation using the DRA equation (solid line)
Рис. 3. Изотермы CO2 на цеолитах 13X при температурах 293 K (кривая 1), 323 K (кривая 2) и расчет по уравнению ДРА (сплошная линия)
15 Р, atm
Fig. 4. Isotherms of CO on zeolites 13X at temperatures: 293 K (curve 1), 323 K (curve 2) and calculation using the DRA equation (solid line)
Рис. 4. Изотермы CO на цеолитах 13X при температурах 293 K (кривая 1), 323 K (кривая 2) и расчет по уравнению ДРА (сплошная линия)
Table 2
Results of solving the problem of identifying the affinity coefficients в and the exponent n of the DRA equation for zeolite 13X Таблица 2. Результаты решения задачи идентификации коэффициентов аффинности в и
Gas в* n* RMS*, %
H2 0.13 0.81 6.74-14.69
CO2 1.87 2 1.81-10.08
CO 1.06 2 7.95-10.59
As a result of solving the problem of identifying the affinity coefficients of the DRA equation, it can be concluded that the DRA equation allows to correctly describe the isotherms of H2, CO2, CO on block zeolite adsorbents 13X in the temperature range of 293-323 K and pressure range of 0-30 atm. The maximum mismatch does not exceed 14.69%, which is achieved when calculating the H2 isotherm (Table 2). Thus, the calculation of adsorption isotherms of the PSA process for hydrogen extraction from synthesis gas (with the composition H2, CO2, CO) on block zeolites 13X can be carried out using the DRA equation.
It is important to illustrate the achieved advantage of calculating isotherms using the DRA equation. In the PSA process for H2 extraction from synthesis gas with the composition H2, CO2, CO the number of isotherms that must be obtained experimentally will be:
using the Langmuir equation or its derivatives - six experiments (two adsorption isotherms for each H2, CO2, CO gas at two temperatures of293 and 323 K);
using the DRA equation - one experiment for the operating pressure range of 0-30 atm (1 isotherm of N2 obtained at room temperature or at a boiling point of N2).
As it can be seen, the advantage is achieved by reducing the number of experimental isotherms that must be obtained for calculations.
It should be noted that for all gases, except H2, the optimal exponent n of the equation was n = 2 (Table 2), which indicates the correctness of applying the particular case of the DRA equation for calculating adsorption isotherms, i.e. the Dubinin-Radushkevich equation. For hydrogen, the use of the Dubinin-Ra-dushkevich equation for one of the samples gave the value of RMS = 45.64% for 13X-2, which is too large an error. The average RMS for all H2 samples for the Dubinin-Radushkevich equation were RMS = 35.15% for 13X. The RMS values for integer values of the exponent n > 2 turned out to be even higher. The best result was obtained at n = 1 . The average RMS for all samples at n = 1 were: RMS = 24.68% for 13X. Therefore, the identification problem for H2 was also solved according to the second scenario, case 2, problem (2'), (4'), which made it possible to continuously vary the exponent of the DRA equation. This difference from other gases is explained by the fact that H2 is rather poorly sorbed on zeolites 13X and 5A in contrast to other gases CO2, CO, which is clearly seen from the isotherms shown in Fig. 2-4. Hydrogen is in the region of low fillings, and its isotherm in the pressure range of 0-30 atm and temperature range of 293-323 K is close to linear; therefore, it is difficult to use the Du-binin-Radushkevich equation (n = 2) . The DRA
equation is a more general case and, using this equation, it is possible to find the values of the affinity coefficients P and the exponent n, at which acceptable RMS values are achieved (Table 2).
Table 3 shows the values of the adsorption heats AH for H2, CO2, CO gases on block zeolites 13X calculated on the basis of the obtained adsorption isotherms (Figs. 2-4), as well as the balance of the PSA process for hydrogen recovery using a mathematical model [3, 15, 16, 37].
Table 3
Values of adsorption heat AH (kJ/mol) for zeolite 13X Таблица 3. Теплоты адсорбции AH (кДж/моль) для _цеолита 13X_
Source H2 CO2 CO
13X-1 7.28 31.56 22.95
13X-2 8.81 32.84 24.76
13X-3 10.17 43.22 19.53
[33] - 46.0 23.6
[33] 11.2 43.3 26.1
[7] 9.5 31.6 23.3
As in the case with the coefficients W0, F, the obtained values of the adsorption heat ДН are also in the ranges of ДЯ (kJ/mol): 7.28-10.17 for H2, 31.56-43.22 for CO2, 19.53- 24.76 for CO. The obtained intervals are valid for the temperature range T = 293 — 323 K and pressure range P = 0 — 30 atm. The spread in the adsorption heat ДН values is: 39.70% for H2, 36.95% for CO2, 26.78% for CO. The resulting spread in the heat ДН values is confirmed by a comparative analysis of the heat ДН values presented in other works (Table 3). This spread must be taken into account when calculating the heat balance of the PSA process for hydrogen extraction from synthesis gas on industrial zeolites 13X.
Table 4 shows the values of the affinity coefficients p k of the DRA equation for H2, CO2, CO gases using zeolites 13X presented in other works.
Table 4
Values of the affinity coefficients в of the DRA equation
for zeolite 13X (nitrogen as a standard gas) Таблица 4. Значения коэффициентов аффинности в уравнения ДРА для цеолита 13X
Source H2 CO2 CO
This work 0.13 1.87 1.06
[33] - 1.89 1.38
[33] - 1.86 1.37
[33] - 2.31 1.30
[38] - 2.43 1.39
In the comparative analysis of the affinity coefficients P of the DRA equation obtained in this work, some discrepancy with other works can be identified (Table 4). The discrepancy is explained as follows. Apparently, each of the works in Table 4 [33, 38] introduced the experimental studies carried out for zeolites 13X made only by one manufacturer. Their names are not given in the works presented in Table 4. In this study, the problem of identifying the affinity coefficients P and the exponent n (2)-(4) was solved using experimental data from several manufacturers. In fact, the found values of the affinity coefficients P and the exponent n cover the spread range of the coefficient values W0 = 0.258 — 0.296 cm3/g, E = 10279 — 13395 J/mol for zeolites 13X (Table 1). This range is the confidence interval for using the obtained affinity coefficients P and the exponent n (Table 2). Thus, in this work, the refined values of the affinity coefficients P and the exponent n of the DRA equation were obtained for calculating the adsorption isotherms of the PSA process for hydrogen extraction from synthesis gas on industrial zeolites 13X.
CONCLUSION
Using experimental methods for obtaining adsorption isotherms of gases and the theory of volumetric filling of micropores, the ranges of the coefficients of the DRA equation have been determined, where the correct calculation of sorption isotherms of H2, CO2, CO on block zeolites 13X is ensured (using three samples from different manufacturers): limiting adsorption volume W0 = 0.258 — 0.296 cm3/g and characteristic adsorption energy £ = 10279 — 13395 J/mol. It has been established that the calculation of adsorption isotherms of H2, CO2, CO gases on industrial zeolites 13X, the characteristics of which fall within the obtained ranges for W0 and E, can be correctly performed using the DRA equation in the temperature ranges of 293-323 K and pressure ranges of 0-30 atm, where the PSA process for H2 extraction from synthesis gas with the composition H2, CO2, CO is carried out.
The use of the DRA equation for calculating the adsorption isotherms of H2, CO2, CO in the PSA process for hydrogen extraction from synthesis gas will eliminate costly experiments to determine the isotherms of H2, CO2, CO (included in the synthesis gas) in the temperature range of 293-323 K and pressure range of 0-30 atm. It is enough to be limited to only one experiment on the removal of the sorption isotherm of N2 (as a model gas) on the selected zeolite 13X to refine the coefficients W0 and If the obtained coefficients W0 and E fall into the intervals for W0 and E found in this work, then to calculate the isotherms of the remaining gases H2, CO2, CO it is necessary to use the obtained affinity coefficients and the exponent n of
the DRA equation, which were determined as a result of solving parametric identification problems. The maximum value of the mismatch between the obtained experimental isotherms of H2, CO2, CO gases and the isotherms calculated using the DRA equation was 14.69% for H2, and the minimum value was 1.81% for CO2.
The approach proposed in the paper can be used to assess the applicability of the DRA equation for calculating adsorption isotherms of gases in various cyclic adsorption processes with various gas mixtures and adsorbents. The applicability of the DRA equation for calculating the adsorption isotherms of various gases in the PSA process on other zeolite adsorbents requires additional research.
The study was carried out at the expense of a grant from the Russian Science Foundation (Project No. 21-79-00092, https://rscf.ru/en/project/21-79-00092/).
The authors declare the absence a conflict of interest warranting disclosure in this article.
Исследование выполнено за счет гранита РНФ (проект № 21-79-00092, https:/rscf.ru/project/21-7900092/).
Авторы заявляют об отсутствии конфликта интересов, требующего раскрытия в данной статье.
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Поступила в редакцию (Received) 30.03.2022 Принята к опубликованию (Accepted) 27.05.2022
