2.3.1
Equilibrium Loading and Isotherms
A solute that is adsorbed is known as the adsorbate and the amount of solute adsorbed is known as the loading, capacity or uptake. An adsorption system is said to be at equilibrium when the mass transfer of a solute to the surface of an adsorbent is equal to the mass transfer from the adsorbent surface. The amount
CHAPTER 2. LITERATURE REVIEW 18 adsorbed at equilibrium depends on the temperature and the concentration of the solutes in the fluid phase.
If the amount of a solute adsorbed at equilibrium per unit mass of adsor- bent is measured, at the same temperature, for various solute concentrations in the fluid phase (or solute partial pressures if the fluid is a gas) then an important curve known as the adsorption isotherm is obtained. Typically, as the concentration of the solute in the fluid is increased, the amount adsorbed also increases. As the adsorbent reaches its maximum capacity, it becomes saturated.
For a gas-solid system, the loading of a component i at equilibrium (q∗ i) is a function of temperature (T ) and its partial pressure in the fluid (pi):
q∗
i = f (T, pi) (2.3)
For a single solute i, the total system pressure, P , is equal to pi. A plot of q∗
i as a function of pi at a fixed temperature, T , is known as an isotherm. Generally, as pi increases at a given temperature, T , q∗
i increases. Additionally, if T is increased, q∗
i is reduced.
2.3.2
Types of Isotherms
There are five types of isotherms that can be found in detailed literature about adsorption (Yang (1987), Ruthven (1984)). It is known as the BET (Brunauer, Emmett and Teller) classification. Three adsorbent types that have been en- countered in the following work are types I, III and V as shown in Figure 2.5.
Figure 2.5: Isotherm Types
Type I is the most common type and CO2 adsorption on most adsorbents is generally of this type. It is commonly represented by the Langmuir model (described in section 2.3.3). Type III and V are common for water adsorption on certain adsorbents. for example, the type V isotherm is observed for water on activated carbon.
2.3.3
Isotherm Models
The isotherm is important for assessing the performance of an adsorbent. A suitable isotherm model is also fundamental for modelling adsorption pro- cesses. To develop an isotherm model, equilibrium loadings need to be measured over the required range of temperatures and adsorbate partial pressures. An isotherm model is then fit to this data. To facilitate calculations and to re- duce computational time, relatively simplistic isotherm models are desirable. However, the error between experimental equilibrium loadings and those found from the model should be small for greater accuracy.
In the literature for adsorption, there are many isotherm models which are more or less suited for specific adsorbate and adsorbent pairs. Kinetic or thermodynamic equations are mostly used to obtain expressions for the loading at equilibrium as a function of temperature and adsorbate concentration in the gas. In some cases, empirical models derived from fundamental isotherm models are formulated to improve the fit of the predictive model to the experimental data. As well as allowing better understanding of the adsorption behaviour of an adsorbate onto an adsorbent, a model with physical significance can avoid the errors associated with interpolation/extrapolation of experimental data.
The most common isotherm models for a single adsorbed component are shown next along with a model for multiple adsorbed components.
2.3.3.1 Langmuir Model
Many type I isotherms can be represented by the Langmuir model. For initial increases in partial pressure of the adsorbate in the fluid, the equilibrium loading increases steeply but as more of the adsorption sites get filled with adsorbate, the loading levels off. The assumptions for the Langmuir isotherm are that molecules are adsorbed at a fixed number of adsorption sites and that each site can only hold one molecule. In addition to all sites being equivalent, there is no interaction between adsorbed molecules on neighbouring sites.
The equation for the Langmuir model for a single component is: q∗
i =
qmax,ibipi
1 + bipi (2.4)
The loading at saturation should be the same for all temperatures, however the equilibrium constant, bi, is temperature dependant and it can follow a van ’t Hoff expression (Yang (1987)):
bi = bi0exp −∆Hads,i RT
CHAPTER 2. LITERATURE REVIEW 20 with:
q∗
i: equilibrium loading of adsorbate i (mol.kg−1)
qmax,i: maximum/saturated loading of adsorbate i (mol.kg−1) bi: Langmuir equilibrium constant of i (Pa−1)
pi: partial pressure of adsorbate i in fluid (Pa) bi0: pre-exponential factor (Pa−1)
∆Hads,i: heat of adsorption of i (J.mol−1) R: ideal gas constant (J.mol−1.K−1) T : temperature (K)
From Equation 2.5, bi increases as temperature, T , decreases and therefore the loading at a given partial pressure increases.
For very low partial pressures of adsorbate, the Langmuir equation follows the Henry’s Law and it is equivalent to a linear isotherm with a slope of qmax,ibi:
q∗
i = qmax,ibipi (2.6)
For very high partial pressures of adsorbate, the equilibrium loading reaches the maximum adsorption capacity for the adsorbent:
q∗
i = qmax,i (2.7)
As bi tends to infinity, the equilibrium loading is constant for all partial pres- sures of solute in the gas. This corresponds to a rectangular or irreversible isotherm for which the loading for all partial pressures above zero is qmax,i. 2.3.3.2 Extended Langmuir Model
The Langmuir isotherm for a single component can be extended for multiple components if the isotherms of the single components can be represented by the Langmuir model. The derivation of the extended Langmuir model follows a similar approach to the single component model and can be found in the literature such as Yang (1987). The equilibrium loading of a component i in a mixture of N adsorbed components is given by:
q∗ i = qmax,ibipi 1 + N P j=1 bjpj (2.8)
with: q∗
i: equilibrium loading of i (mol.kg−1)
qmax,i: maximum/saturated loading of i (mol.kg−1)
bi: Langmuir equilibrium constant of i (Pa−1) which can be found by Equation 2.5
bj: Langmuir equilibrium constant of j (Pa−1) pi: partial pressure of adsorbate in fluid of i (Pa) pj: partial pressure of adsorbate in fluid of j (Pa)
From the equation of this model it can be seen that the maximum adsorbed ca- pacity is different for each component. The term, qmax,i, is more of an empirical term than a parameter of physical significance as the molecules of components adsorbed have different sizes and therefore there would be different saturation loadings for each component. In the derivation of Equation 2.8, qmax,i should be the same for all components.