Intraparticle Diffusion Kinetics of Carbon Dioxide Adsorption on Zeolites

E. P. Abakumov, A. S. Guzenberg, A. M. Ryabkin, G. G. Savel'ev

(Presented by the scientific seminar of the Department of Radiation Chemistry)

Questions concerning the kinetics of carbon dioxide on zeolites, in addition to their scientific importance, are of great practical significance for the use of zeolites in purifying air from carbon dioxide.

We carried out experiments to obtain kinetic curves on NaX zeolites during the sorption of carbon dioxide from a mixture with nitrogen and during the sorption of pure carbon dioxide.

The experiments were conducted on an apparatus designed on the basis of a McBain balance.

To achieve rapid and continuous recording of the change in weight, the tracking process was automated. The automatic recording system consisted of a light source, a system of lenses and mirrors, by means of which the image of the lower end of the spiral was projected onto the carriage of a recorder. Two photoresistors of the FSK type were mounted on the recorder carriage and connected to the recorder amplifier in a bridge circuit, so that, with different illumination of the photoresistors, a signal appeared at the recorder input and the carriage moved. The polarity of the emf source was selected so that, when the right photoresistor was darkened, the carriage deflected to the right, and when the left one was darkened, to the left; i.e., the carriage was positioned so that the darkest point lay between the photoresistors.

Experiments on the sorption of pure carbon dioxide were carried out on NaX-929 zeolite with granule diameters of 2 mm, 1 mm, and 4 mm at a pressure of 7.6 torr; and with a granule diameter of 3 mm at ...

at a pressure of 190 torr. Experiments on sorption from a mixture of nitrogen and carbon dioxide were carried out on zeolites NaX-929 and NaX-740 with a granule diameter of 2 mm, at a mixture pressure of 760 torr and at partial pressures of carbon dioxide in it of 3.8 torr, 7.6 torr, and 190 torr. The gas temperature in the experiments was \(25 \pm 1^\circ\mathrm{C}\). In sorption from mixtures, a flow rate was selected at which external mass transfer practically had no effect on the sorption kinetics. The sorbents used in the experiments were industrial zeolites of serial production with cylindrical granules. The weight of the experimental sample in the cup did not exceed 200 mg. The experimental curves are shown in the graphs in Figs. 1 and 2.

Fig. 1.
Kinetics of sorption of \(\mathrm{CO_2}\) on NaX zeolites at
\(P_{\mathrm{CO_2}} = 12.8\) torr.

— experimental;

--- calculated from the equation

\[ a = a_0 \left(1 - e^{-k_1 t^{k_2}}\right); \]

—×— calculated from the equation

\[ a = k_1 t + k_2 t^2 + k_3 t^3 + k_4 t^4 + k_5 t^5. \]

Axes in the graph: \(a\,[\mathrm{mg}/\mathrm{g}]\); \(t\,[\mathrm{min}]\).

For kinetic experiments with a grain or a group of grains, the equation for sorption kinetics has the form:

\[ \frac{dq}{dt} = \beta_{\mathrm{on}}(C_0 - y) \tag{I} \]

We tested the possibility of applying this equation to the experiments performed.

As Yu. S. Lezin and M. M. Dubinin [1] pointed out, determination of \(\frac{dq}{dt}\) by graphical differentiation of experimental curves is associated with significant errors. To obtain na-

...of the current values of $\dfrac{da}{dt}$, the experimental kinetic curve must be described by the corresponding equations. For the description we used an equation close in form to the solution of the diffusion equation for a linear isotherm [2, 3]:

\[ \gamma = 1 - e^{-K_1 t^{K_2}} \tag{2} \]

where $\gamma = \dfrac{a}{a_0}$, $t$ is time, and $K_1$ and $K_2$ are constants.

Processing of the experimental data by equation (2) was carried out using the M-220 computer by the method for determining the parameters of the empirical formula (4), in order to determine the coefficients $K_1$ and $K_2$. Table 1 gives the best values obtained for the coefficients $K_1$ and $K_2$ of formula (2) for describing each experimental curve.

As can be seen from Table 1, the coefficient $K_2$ for all experiments is close to 0.5; therefore, approximately, for the kinetics of carbon dioxide sorption on NaX zeolites one may use the formula

\[ \gamma = 1 - e^{-K_1 \sqrt{t}} \tag{3} \]

the coefficient of which can be obtained by plotting

\[ \ln(1-\gamma) = f\left(\sqrt{t}\right). \]

Using formula (2), with the aid of a computer, the values of $a$ were calculated from the coefficients given in Table 1 over the interval from 2 sec to 180 sec. As can be seen from the graphs in Fig. 2, the calculated data describe the experimental results well.

From the obtained values of $a$, the values of $\Delta a/\Delta t$ and the values of the equilibrium concentration $y$ were calculated, using the virial adsorption equation

\[ y = a e^{B_1 + B_2 a + B_3 a^2} \tag{4} \]

where $B_1, B_2, B_3$ are constants calculated from the experimental data: for NaA-929, $B_1 = 0.3958$, $B_2 = 1.618$, $B_3 = -0.1256$; for NaX-740, $B_1 = 0.313$, $B_2 = 1.516$, $B_3 = -0.142$.

From the calculated data, dependences $\dfrac{\Delta a}{\Delta t} = f(a-y)$ were constructed, which should have given straight lines in the case of describing the ki-

Table 1

Fig. 2. Processing of the experimental results according to the equation

\[ \frac{\Delta a}{\Delta t}=\beta_0(c_0-y)\frac{\Delta a}{\Delta y}. \]

\[ +\;-\;P=660\ \text{torr}, \]

\[ \circ\;-\;P=165\ \text{torr}, \]

\[ \bullet\;-\;P=12{,}8\ \text{torr}. \]

Plot labels visible in the figure:

\[ \frac{\Delta a}{\Delta t}\cdot 10^{-3}\ [\ldots] \]

\[ (c_0-y)\frac{\Delta a}{\Delta y}\cdot 10^{4}\ [\ldots] \]

...by formula (I). However, the obvious curvilinearity of these dependences indicates that the coefficient \(\beta_{\mathrm{in}}\) is variable. The coefficient \(\beta_{\mathrm{in}}\) is directly proportional to the diffusion coefficient and, for a linear isotherm, can be expressed by formula (5)

\[ \beta_{\mathrm{in}} = K D^{e}\cdot \Gamma \tag{5} \]

where \(K\) is a constant of the geometrical form of the sorbent. Then, for a curvilinear isotherm, introducing the notation \(K D^{e}=\beta_{0\mathrm{in}}\), one may write:

\[ \beta_{\mathrm{in}} = \beta_{0\mathrm{in}}\cdot \frac{dq}{dy} \tag{6} \]

Taking equation (6) into account, equation (I) is reduced to the form:

\[ \frac{dq}{dt}=\beta_{\mathrm{in}}(c_{0}-y)\frac{dq}{dy} \tag{7} \]

From the calculated values of \(q\) and \(y\) /see above/, the quantities \(\Delta q/\Delta y\) and \((c_{0}-y)\Delta q/\Delta y\) were calculated. In Fig. 2, graphs of the dependence

\[ \frac{\Delta q}{\Delta t}=f\left[(c_{0}-y)\frac{\Delta q}{\Delta y}\right] \]

are presented.

As is evident from these graphs, the dependences obtained are close to rectilinear for the sorption of carbon dioxide from the mixture and for the sorption of pure carbon dioxide at a pressure of 190 torr. For the sorption of pure carbon dioxide at low pressures, the dependence remains somewhat curvilinear, which may be explained by the different character of flow in the secondary porosity at absolute pressures and by the necessity, in this case, of separately taking into account the influence of the secondary porosity. In addition to the indicated point, the absence of a completely rectilinear dependence may be connected with the fact that the coefficient \(\beta_{\mathrm{in}}\) may depend on the degree of filling of the sorbent, which is not taken into account in the present work. The experimental data lie on a single straight line, the slope of which corresponds to \(\beta_{0\mathrm{in}}\), and for the curves shown \(\beta_{0\mathrm{in}}=0.14\ 1/\mathrm{min}\).

LITERATURE

1. D. S. Lezin, M. M. Dubinin, DAN SSSR, 171, No. 2, 382 (1966).

2. D. P. Timofeev, Kinetics of Adsorption, Moscow, Publishing House of the Academy of Sciences of the USSR, 1962.

3. A. I. Kharin, I. E. Antilagov, Collection “Chromatography, Its Theory and Application,” Moscow, Publishing House of the Academy of Sciences of the USSR, 267, 1960.