3.2. Result of the Measured CO2 Adsorption Rate in Early Stage of Adsorption
The
Figure 7a shows the pressure variation after introducing the mixed gas into the sample column of
Figure 5. In the results with the
d = 1.5 mm adsorbent pellets, a rapid pressure decrease was observed within a few seconds after gas introduction. However, the pressure decrease within the same time range was relatively small in the case of the
d = 3.0 mm adsorbent pellets. The
Figure 7b shows the variation of equilibrium achievement, which is defined as the ratio of the maximum pressure decrease after reaching equilibrium ∆
pe and the temporal pressure decrease ∆
p. The effect of the adsorbent size could be observed in the early stage of gas adsorption, as the gradient of the graph of the
d = 1.5 mm adsorbent pellets was much larger than that of
d = 3.0 mm adsorbent pellets. The gradient of equilibrium achievement became smaller as it approached 1.0 for the equilibrium state. The retention time for equilibrium achievement varied depending on the type of adsorbent and the adsorbed gas species.
Based on the results shown in
Figure 7, the overall volumetric mass transfer coefficient in this experiment was evaluated by the following calculation. The adsorbed gas amount in every second Δ
n can be calculated by the gas state Equation (1) from the measured pressure drop ∆
p and the known volume
v0.
In the case of Zeolite 13X, adsorption of CO
2 is much stronger than that of N
2 owing to the higher quadrupole moment of CO
2 molecules. Therefore, the measured pressure drop in this experiment can be associated with the adsorbed CO
2 amount under the assumption that the pressure drop occurred mainly by CO
2 adsorption in this case. The driving force for CO
2 adsorption is the pressure difference between the equilibrium CO
2 pressure for the adsorbed CO
2 amount and the actual CO
2 pressure in the gas phase. The equilibrium CO
2 pressure can be obtained from the measured CO
2 isotherms, which were measured previously at 298 K. After converting the CO
2 pressure to the CO
2 concentration by the gas state equation, the overall volumetric mass transfer coefficient
K′CO2 can be calculated by the following Equation (2). The increment of the adsorbed CO
2 amount ∆
q in each time step Δ
t can be calculated by unit conversion of Δ
n. In this experimental setup, the overall volumetric mass transfer coefficient is defined as a transient value during the progress of gas adsorption.
Figure 8 shows the time variations of
K′CO2 derived from the measured pressure for both the
d = 1.5 mm smaller pellet and the
d = 3.0 mm larger pellet. The larger difference between these two graphs was observed in the early stage of gas adsorption. However, this gap decreased drastically within seconds, and these two graphs converged to the lower level of around 0.5 and showed almost no difference in the later stage of gas adsorption. This result means that the adsorbent size effect appears only when initial gas diffusion occurs and then diminishes rapidly as gas adsorption proceeds.
3.3. Discussion
From the results in
Figure 8, it is estimated that the gas diffusion inside an adsorbent pellet occurs in accordance with the effect of a hierarchal pore structure consisting of the macro-pores between crystallites and micro-pores in the crystallite, which creates sequential gas diffusion steps. In addition, an external gas diffusion layer outside the pellet surface is also created and has some effect on the overall gas diffusion behavior.
Figure 9 shows a schematic image of the hierarchal pore structure and the gas diffusion into an adsorbent pellet having a pellet radius of
rp. The gas diffusion stages are indicated by the different colors of the bold arrows. In
Figure 9,
K1 is the volumetric mass transfer coefficient in the external gas diffusion layer, and its value depends on the gas flow conditions around the outer area of the pellet.
K2 is the volumetric mass transfer coefficient in the macro-pores between crystallites, which is closely related to the macro-pore size inside the pellet.
K3 is the volumetric mass transfer coefficient in the micro-pores inside a crystallite having a radius of
rc. From the SEM image in
Figure 9, the size of the crystallite seems not to be uniform; therefore,
rc can be roughly characterized by the average size of the crystallites inside the pellet.
The gas adsorption rate in an adsorbent pellet is generally defined by the following Equation (3) based on the linear driving force model. The difference between the equilibrium gas concentration for the adsorbed gas amount and the actual gas concentration in the gas phase works as a driving force of the incremental gas adsorption.
K′ is the overall volumetric mass transfer coefficient and includes the effects of the gas diffusivities of all these gas diffusion steps and the relevant pellet surface area per unit volume. According to the hierarchical pore structure of the adsorbent,
K′ can be defined by the following Equation (4), which means that the gas adsorption rate is governed by the smallest value among
K1,
K2, and
K3.
The volumetric mass transfer coefficient in the outer layer of the pellet
K1 can be estimated by Yoshida’s Equation (5), which includes both the Schmidt number
Sc and the Reynolds number
Re [
26]. By definition, the Reynolds number is proportional to the gas velocity
u. Therefore,
K1 becomes large in case of a higher gas velocity, which has a negligibly small effect on the overall gas adsorption rate according to Equation (5). The gas molecular diffusion coefficient can be derived by Chapman–Enskog’s equation, shown as Equation (6) [
27]. For simplicity, only CO
2 and N
2 were considered as the diffusing gas species in the following calculation. The feed gas of the PSA experiment described above included CO in coexistence with CO
2 and N
2. Because the molecular weight of CO is almost equal to N
2, the effect of CO on CO
2 diffusion behavior is expected to be similar to that of N
2, especially in gas phase diffusion.
The volumetric mass transfer coefficient in the macro-pores of the pellet
K2 can be estimated by the following Equation (7), in which the effective diffusion coefficient
Dep is expressed as the combined form with the molecular diffusion coefficient
Dm and the Knudsen diffusion coefficient
Dkp [
28].
τp is the tortuosity factor of the porous structure and changes depending on the porosity and shape of the particle. Many reports have been published on the tortuosity of various porous materials investigated by experiments or simulations. Sun et al. reported the relationship between porosity and tortuosity for artificial particles of different shapes obtained from experimental data [
29]. In the following Equation (7), tortuosity was estimated by the relationship in Sun’s report based on the measured porosity of the adsorbent pellet. The Knudsen diffusion coefficient of CO
2 in the macro-pore can be calculated by the following Equation (8), which includes the macro-pore diameter
dmacro, the gas temperature
T and the molecular weight of the diffusing gas species
Mi. The average macro-pore diameter measured by the mercury penetration method in
Table 2 was used as the
dmacro value in this calculation.
The volumetric mass transfer coefficient in the micro-pores of the crystallite
K3 was defined by the following Equation (9) under the assumption that the size of the crystallite is represented as the average value. The gas phase diffusion in the micro-pores can be expressed simply as the Knudsen diffusion owing to the restricted space for molecular diffusion. The Knudsen diffusion coefficient of CO
2 in the micro-pores can be calculated by Equation (10), which is the same expression as Equation (8) except that the pore diameter is the micro-pore diameter of the crystallite. The differential term on the right side of Equation (9) is related to the mass transfer of the adsorbed phase in the crystallite that also occurs due to the effect of the gradient of the adsorbed gas concentration, and is the so-called surface diffusion, which is expressed as the differential of the adsorbed gas amount against the displacement in the crystallite.
Table 4 shows the calculation conditions. As the focus here is the gas phase diffusion in the early stage of adsorption in the fresh adsorbent, only the first term of the right side of Equation (9) was considered in the following calculation. The surface area per unit volume
av was estimated from the packing density of the adsorbent and the average pellet shape having a representative volume and surface area. The crystallite diameter was defined to be 5 μm from the appearance in the SEM image in
Figure 9. The macro-pore diameter was the measured value in
Table 2. The micro-pore diameter of 0.735 nm is the maximum pore diameter of Zeolite 13X (Faujasite) obtained from the database of zeolite structures, in which the light gas molecules are diffusible. The gas pressure of 151 kPa is the gas adsorption pressure in the CO
2-PSA experiment in
Table 3, which is close to result of the initial gas pressure in the gas adsorption rate measurement in
Figure 7. The temperature of the adsorbent was assumed to be constant at 298.15 K. In the laboratory-scale measurement of CO
2 adsorption rate in
Figure 5, the constant temperature was achieved by using a thermostat, though some degrees of temperature increase by the gas adsorption heat could possibly occur at the local adsorption site. The total porosity inside a pellet was measured previously by a water immersion test in which the total pore volume was quantified as the water volume added to an adsorbent packed bed in a glass column.
Table 5 shows the calculated gas diffusion coefficients
D which were obtained as common values for both the
d = 1.5 mm smaller pellet and the
d = 3.0 mm larger pellet. The Knudsen gas diffusion coefficient in the macro-pores of the pellet
Dkp is larger than the gas molecular diffusion coefficient
Dm, which means that the size of the macro-pores inside the pellet is large enough to consider the gas diffusion inside the macro-pores as bulk phase diffusion. The Knudsen gas diffusion coefficient in the micro-pores of the crystallite
Dkc, on the other hand, is much smaller than
Dm, which means that the gas diffusion inside the micro-pores is substantially Knudsen diffusion due to the smaller pore size against the mean free path of the gas molecules.
Table 6 shows the volumetric mass transfer coefficients
K calculated for both adsorbent pellets. The volumetric mass transfer coefficient in the macro-pores of the pellet
K2 is the smallest among
K1,
K2, and
K3 in both cases, meaning that the gas diffusion in the macro-pores is the rate-determining process in the gas phase. In the experimental results in
Figure 8, the initial values of
K′CO2 were nearly equal to the
K′CO2_calc in
Table 6, which means that the gas diffusion in the macro-pores is the most dominant process in the early stage of gas adsorption. However, the values of
K′CO2 in
Figure 8 decreased rapidly after 10 s. It was estimated that the diffusion of the adsorbed phase, which can be described as surface diffusion inside the micro-pores in the crystallite in Equation (9), becomes more prominent than gas phase diffusion as adsorption proceeds. The volumetric mass transfer coefficients of surface diffusion were estimated to be about 0.5 from the converged value in
Figure 8. In the CO
2-PSA experiments introduced above, the total gas adsorption time including both the gas adsorption step and the rinse step was 200 s in one cycle. This time was much longer than the initial gas diffusion time when the effect of the pellet size was prominent. Therefore, surface diffusion was expected to be the most dominant process for CO
2 adsorption in CO
2-PSA with the Zeolite 13X adsorbent. The similarity of the performance of the adsorbents with various shapes in the laboratory-scale CO
2-PSA in
Figure 6 is considered to be a clear verification of this gas adsorption mechanism.