Adsorption is a spontaneous process and when the gas is putted in contact with the adsorbent, a new equilibrium state will be established, depending on the partial pressure of each of the gases and on the total temperature of the system. After achieving such equilibrium, no more adsorption takes place and the adsorbent should be regenerated. For this reason, a PSA column should be regenerated periodically to be able to absorb CO2 in different cycles. In order to keep constant feed processing, more than one column are employed in parallel: when biogas is fed for selective removal of CO2, the other column(s) are being regenerated.

The operation of a PSA process for biogas upgrading can be explained by showing what happens when a mixture of CH4-CO2 is fed to a column filled with adsorbent. For simplicity, the column will be considered to be at the same pressure of the biogas stream and filled with an inert gas (helium). An example of such behaviour is normally termed as "breakthrough experiments". An example of a breakthrough curve of CH4 (55%) — CO2 (45%) mixture in CMS-3K is shown in Figure 4 (Cavenati et al., 2005). It can be observed that in the initial moments, methane molecules travel across the column filling the gas phase in the inter-particle space, but also in the intra-particle voids (macropores), replacing helium. Due to the very large resistance to diffuse into the micropores, CH4 adsorption is very difficult, reason why it breaks through the column very fast. On the other side, CO2 takes a very long time to break through the column since it is being continuously adsorbed. Note that before CO2 breakthrough, there is a period of time where only methane is obtained at the column product end. In Figure 4(b) also the temperature increase on the different positions of the column is shown. Note that in this experiment, temperature increase is due solely to CO2 adsorption. This experiment was carried out under non-isothermal and non­adiabatic conditions. In the case of larger adsorbers where adiabatic conditions can be found, temperature increase should be higher having a stronger negative impact in the adsorption of CO2 (faster breakthrough).

Another important thing that can be observed in Figure 4 is the dispersion of the CO2 curve. The perturbation in the feed stream was a step increase in CH4 and CO2 partial pressure and the breakthrough result indicates that the response to that input after passing through the column is quite spread. The shape of the adsorption breakthrough curves is associated to diverse factors:

1. Slope of the adsorption isotherms: comprise the concentration wave if isotherm is favourable (Langmuir Type) and dispersive if the adsorption equilibrium is unfavourable (desorption for Langmuir-type isotherms). No effect if the isotherm is linear,

2. Axial dispersion of the adsorption column: disperse the concentration wave,

3. Resistance to diffusion within the porous structure of the adsorbent: disperse the concentration wave.

4. Thermal effects: normally in gas separations the thermal wave travels at the same velocity as the concentration wave (Yang, 1987; Ruthven et al., 1994; Basmadjian, 1997) and its effect is to disperse the concentration wave. Thermal effects can control the shape of the breakthrough curve.

To compare the performance of different adsorbents, the thermal effects associated to adsorption of CO2 in zeolite 13X extrudates can be observed in Figure 5 where a breakthrough of CO2 was carried out (Cavenati et al., 2006). The experiment was conducted at 299 K and a total pressure of 3.2 bar. It can be observed that CO2 breaks through the bed quite sharply due to the strong non-linearity of the CO2 adsorption isotherm that tends to compress the concentration front. After the initial sharp breakthrough, the shape of the curve gets quite dispersed due to thermal effects. It can be seen in Figure 5(b) that the temperature increase in certain points of the column is quite high, reducing the loading of CO2 and making breakthrough quite faster than it should be if carried out at isothermal conditions. The opposite effect will take place in desorption of CO2: the temperature in the bed will drop increasing the steepness of the adsorption isotherm, making desorption more unfavourable.

Due to the thermal effects and the steepness of the CO2 isotherm on zeolite 13X, it was concluded that using a similar PSA cycle, if the temperature of the biogas stream is close to ambient temperature, it is better to use the Carbon Molecular Sieve (CMS-3K) than zeolite 13X (Grande and Rodrigues, 2007).

The solid lines shown in Figures 4 and 5, represent the prediction of a mathematical model, based on pure gas adsorption equilibrium and kinetics (Cavenati et al., 2004; Cavenati et al., 2005). The resulting equations for the prediction of the fixed-bed behaviour are (Da Silva, 1999):

i. mass balances in the column, particle and micropores (crystals) of the adsorbent.

ii. Energy balances in the gas and solid phases and column wall

iii. Momentum balance (simplified to the Ergun equation)

iv. Multicomponent adsorption isotherm model.

Note that the mass, energy and momentum balances are partial differential equations linked by a (generally) non-linear equation (isotherm model). The mathematical model was tested under diverse adsorbents and operating conditions for CH-CO2 separation as well as for other gas mixtures. The mathematical model employed is termed as "homogeneous model" since it considers mass and heat transfer in different phases using different equations. Heterogeneous models (single energy balance) and also more simplified mass transfer models can also be employed to predict column behaviour with good accuracy (Ruthven, 1984; Yang, 1987; Ruthven et al., 1994).