rep-resenting an adsorptive process. Balance equations can be presented in several ways:
as a function of the specific application and of the accuracy needed. Examples of the use of such equations are given below, where mass balances for two types of adsorp-tive processes and an energy balance for a fixed packed bed are presented.
5.5.1 Mass Balances
At the microscale, a given adsorbate can exist in three locations: at the adsorbed phase, in the fluid inside the pores, or in the fluid phase out of the adsorbent particles.
As a consequence of this, a mass balance must consider terms involving ni (adsorbed weight of the adsorbate per unit weight of the adsorbent), cpi (concentration of the adsorbate inside the pores), and ci (concentration of the adsorbate in the fluid outside the adsorbent particles). Once it is impossible to determine the local concentration inside the particles of the adsorbent, the terms qi and cpi will be used as average con-centration values. Figure 5.5 shows the regions involved in this mass balance where the adsorbate may be present.
Solute’s Concentration at Equilibrium
Adsorbed Weight/Unit Weight of the Adsorbent
Desorption Adsorption
FIGuRE 5.4 Sorption isotherm with hysteresis.
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For batch processes, or in stirred tanks, the mass balance for a given component i is given by
where m is the adsorbent mass; V is the fluid volume external to the adsorbent par-ticle; V.
is the volumetric flow rate entering (in) and exiting (out) the particle; rp and rb are the densities of the particle and of the fluid, respectively; and e and ep are the volume fraction outside the particle and the particle’s porosity, respectively.
For a process in a fixed bed, the mass balance for component i is given by
ρb i ε i ε i ε where v is the interstitial velocity of the fluid; DL is Fick’s axial dispersion coeffi-cient; and yi is the molar fraction of component i in the fluid phase.
5.5.2 energy Balance
The energy balance for a fixed bed, ignoring dispersion, is given by
ρ ε ε
FIGuRE 5.5 Scheme of the structure of an adsorbent particle.
96 Engineering Aspects of Milk and Dairy Products
where Us and Uf are the internal energies of the stagnant phase and of the fluid phase, respectively; Hf and Hw are the enthalpy of the fluid phase and the heat transfer coef-ficient at the wall of the column, respectively; and Rc is the radius of the column.
The second term in Equation 5.14 expresses the contributions in terms of energy, both for the fluid in the pores and for the fluid outside the particle. It is necessary to use thermodynamics to evaluate the enthalpy (or the internal energy) of the fluid phase and the internal energy of the stagnant phase. For a process in which the fluid is a gas at low or moderate pressures, the contribution of this second term for enthalpy calculation may be ignored. Therefore, it is necessary to define the reference state for each pure component as being that of an ideal gas at a given temperature T°, and the reference state for the stagnant phase as being that of the adsorbent free from the adsorbate, also at T°. Therefore, it holds for the gaseous phase that
Hf y Hi f i y H C dT
where Cp0 is the heat capacity of the ideal gas, and P is the pressure in the gas phase.
And for stagnant phase,
where Cp,s is the heat capacity of the adsorbent, and q is the weight of the adsorbate per unit weight of the adsorbent.
The enthalpy of the adsorbed phase, Ha, is evaluated considering that each com-ponent of the gas phase goes through a temperature shift from T° to T, followed by an isothermal adsorption, resulting in
where xi is the molar fraction of the adsorbed phase.
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For the condition of isosteric energy of adsorption (the isosteric condition is given by the change of pressure with temperature, for a constant adsorptive capacity, ni), qist depends on the system’s composition. The sum of integrals in Equation 5.19 becomes difficult to evaluate for multicomponent adsorption if the isosteric energy of adsorption of each component is, in fact, dependent on the weights (contributions) of each of them.
Once the isosteric energy of each component depends on the contribution of all the other components, the sum must be evaluated starting from a clean adsorbent condition and finishing with the contributions of all components. If the isosteric energy is constant, as it is usually considered to be, then the energy balance in Equation 5.14 becomes
ρb s i p i ε ρ
where Equation 5.13 has been used assuming DL equals zero. Equation 5.20 is a com-mon expression for the energy balance in a fixed bed. Often, the first sum of the left-hand side of Equation 5.20, involving gas phase heat capacities, is neglected, or the gas phase heat capacities are replaced by the heat capacities of the adsorbent phase.
Nonisothermic processes with a liquid phase involved may be conducted by changing the temperature at the feed stream, or by heating or cooling the column through its wall. This means that adsorption energies and pressure effects are minor influences in this case, and the energy balance becomes
ρb sc εbcCp T ε p w w