ANÁLISIS DEL ESTADO DE FLUJOS DE EFECTIVO

especially in the food industry. Due to its low affinity for water, it is prefer-entially used to adsorb components from aqueous solutions or wet gases. It is used as a powder that, after being used, can be removed by filtration.

Molecular sieves: Also known as zeolites, they are synthetic, porous crystals of

•

metallic aluminosilicates, with a very defined structure, allowing the separation of molecules based on their molar weight, and promoting further separation by adsorption according to the polarity and insaturation of the molecule. They are used in gas and liquid drying operations and in a variety of other processes.

Polymers and resins: Substances used in water purification, recovery and

•

separation of biological compounds, and in chromatographic processes in general. They are composed of a crystalline core covered with a polymeric matrix that confers porosity to the structure, and to which specific and selec-tive chemical groups are bonded which are responsible for adsorption.

Silica gel: Together with activated carbon, is one of the most popular

adsorb-•

ents. It is used in gas drying and for the removal of saturated hydrocarbons.

Fuller earth: A natural clay that must be dryed and subjected to a fine

mill-•

ing before being used as an adsorbent. It is used in blanching, clarification, and neutralization of mineral, vegetal, and animal oils.

5.4 sORPTION EquIlIbRIuM

Phase equilibrium between the fluid and the adsorbed phase for one or more com-ponents is the most important factor for the performance of the adsorptive process.

In most cases, this factor is even more important than mass and heat transfer rates:

doubling the stoichiometric capacity of the adsorbent or significantly changing the shape of the isotherm has often a more significant impact in the unit operation than doubling the mass and heat transfer rates. The graphical description of sorption equi-librium for the adsorption of a single component is usually presented in terms of the sorption isotherm, where the relationship between the solute concentrations in the fluid and adsorbed phases, at a given constant temperature, is established.

Sorption equilibrium is a dynamic concept that describes the situation in which the rate of adsorption of a given type of molecules on the solid surface equals the rate of desorption of those molecules from that surface. The physical and chemical concepts involved in these phenomena may become very complex, and there is no single theory on adsorption describing satisfactorily all the possible systems. Fortunately, for engi-neering purposes, the only information needed is an accurate representation of sorp-tion equilibrium, and some of the first theories on adsorpsorp-tion are still widely applied.

5.4.1 s^olute a^dsorptionin d^ilute s^olutions

When an adsorbent is added to a binary solution, it is possible that either the solute or the solvent will adsorb to it. Since measurement of total adsorption is impossible, an apparent or relative adsorption is determined, instead. Total adsorption is not used because the adsorption of the solvent causes a slight, not measurable change in the enthalpy, in the volume, or in the mass of the solution. This makes it impossible

88 Engineering Aspects of Milk and Dairy Products

to differentiate the amount of solvent adsorbed from the amount of solvent simply retained in the pores of the adsorbent.

Therefore, the procedure to determine the parameters involved in the apparent adsorption of the solute consists of treating a predefined volume of solution with a known mass of the adsorbent. The ratio between these values is noted by v. As a result of the preferential adsorption of the solute, its concentration in the solution diminishes from an initial value c₀ to a final value in equilibrium, c*. If changes in the volume of the solution are disregarded, the apparent adsorption of the solute can be given by v(c₀ – c*). This relationship is satisfactory for dilute solutions when the fraction of the solvent that can be adsorbed is very small.

An increase of the initial solute concentration in the solution leads to a corre-sponding increase of the amount of solute adsorbed. However, if the solvent is also adsorbed and if the extension of this adsorption is close to that of the solute, a pref-erential adsorption of the solvent may occur from a given value of concentration onwards. This situation corresponds to an inversion of the apparent adsorptivity and, for a well-defined concentration value, apparent adsorptivity reaches the unity, in a situation similar to that of the formation of azeotropes in distillation.

The phenomenon of adsorption in liquids is less understood than in gases. In prin-ciple, the equations applicable to gases are also applicable to liquids, except in those cases in which capillary condensation occurs.

5.4.1.1 linear Isotherm

In general, for physical adsorption on a homogeneous surface and at low concentra-tions, the isotherm assumes a linear shape, with a constant slope (K), and this relation-ship may be expressed using Henry’s Law, represented by the following equation:

q = k ⋅ C (5.1)

or

q = k′ ⋅ p (5.2)

where q is the concentration of the adsorbed phase, C is the concentration of the fluid phase, and p is the partial pressure of the fluid phase (in the case of gases).

Henry’s Law is very useful at low concentrations of the adsorbate, but for higher concentration values, the interactions between the adsorbate molecules increase, and the saturation on the adsorbed phase occurs. This means that for higher concentra-tions of the adsorbate, isotherms may have more complex shapes.

5.4.1.2 Freundlich Isotherm

One of the most widespread equations representing the adsorption isotherms for liq-uids is the one proposed by Freundlich in 1926, which was developed from studies on the adsorption of organic compounds in vegetable coal, when in aqueous solution.

This equation takes the following form:

CS = a ⋅ (C*)^1/n (5.3)

Chromatographic Techniques Applied to Dairy Product Manufacturing 89

where C_S is the mass of adsorbate per unit mass of adsorbent; C^* is the concentration of solute in solution in equilibrium with the solid; a and n are constants, with n usu-ally much higher than 1.

This isotherm nearly corresponds to an exponential distribution of the adsorption enthalpy values, but in order to represent all data, it also needs to make use of the linear region of Henry’s Law. It can be used to correlate data obtained from hetero-geneous adsorbents in a wide concentration range.

5.4.1.3 langmuir Isotherm

The Langmuir model (1916) is another classical model for isotherms, which is by far the most used and frequently the first choice for the fitting of experimental data; it can be represented by the following equation:

q q K c

i is K c

i i i i

= ⋅ ⋅

+ ⋅

1 (5.4)

where q_i^s is the number of adsorption sites of the monolayer (saturation capacity), c_i is the concentration of species i, and K_i is an equilibrium constant.

This model assumes that there is a constant number of adsorption sites avail-able, only a single layer (monolayer) of adsorbed molecules is formed, adsorption is reversible, equilibrium is achieved, and interaction between the adsorbed molecules is null. Figure 5.1 shows a graphical scheme of these models.

5.4.1.4 bi-langmuir Isotherm

The Bi-Langmuir isotherm model was proposed by Graham, in 1953, in order to evalu-ate adsorption behavior in nonhomogeneous surfaces. This model considers the surface

Solute Concentration in Solution, in Equilibrium Linear

Freundlich Langmuir

Solute Concentration in the Solid Phase

FIGuRE 5.1 Adsorption isotherms: linear, Freundlich, and Langmuir.

90 Engineering Aspects of Milk and Dairy Products

as having two different types of chemical domain, which behave independently. This means that the proposed model is the result of the sum of two Langmuir isotherms:

q q b C

b C q b C

s s b C

= + +

,1 1 , +

1

2 2

1 1 2 (5.5)

This model has two saturation capacities, q_s,1 and q_s,2, corresponding to each of the two types of chemical domain. The total saturation capacity is the sum of these two capacities, and b₁ and b₂ are the equilibrium constants.

5.4.1.5 Toth Isotherm

The Toth isotherm model has three parameters and was originally derived from gas–

solid equilibrium studies. Similar to Langmuir’s model, it can be applied in the case of solid–liquid equilibrium. This isotherm is used to fit experimental equilibrium data obtained in nonhomogeneous adsorbents:

q q bC

s bC n n

= [1+( ) ]^1/ (5.6)

In this equation, q_s and b have the same meaning as q _i^s and K_i in Langmuir’s iso-therm, and n is the heterogeneity parameter (0 < n < 1). When n = 1, Toth’s isotherm equals Langmuir’s isotherm. The parameters b and n provide an independent fitting of the initial slope and of the curvature of the isotherm.

5.4.1.6 Jovanovic Isotherm

This model was derived for adsorption in a solid, homogeneous surface, consider-ing the phenomenon as nonlocalized, without lateral interactions, and allowconsider-ing for a monolayer of solute to be formed. The corresponding equation is as follows:

q q= s[1−exp(− ⋅b C)] (5.7) where qs is the surface concentration for solute saturation, and b is the appropriate binding constant.

5.4.1.7 Exponentially Modified langmuir Isotherm

In the case of molecules of biological origin, such as proteins, Langmuir’s isotherm provides a less than satisfactory description of hydrophobic adsorbents due to two main reasons:

The binding of many proteins in hydrophobic adsorbents is based in

multi-•

valent interactions.

The adsorption of proteins in a hydrophobic medium is highly influenced by

•

the concentration of other components, such as salts, but Langmuir’s model alone is unable to express this behavior despite the fact that the parameters of the model are implicit functions of the concentration of salts.

Chromatographic Techniques Applied to Dairy Product Manufacturing 91

In order to circumvent this problem, an exponentially modified Langmuir iso-therm model was proposed which contains a parameter related to the contribution of salt concentration to protein adsorption isotherms:

q b kC C

where l, b, and k are parameters of the equation; Cs is the salt concentration in the liquid phase; and C is the protein concentration in the liquid phase.

5.4.2 deterMinationoF adsorption isotherMs

There are several methodologies that can be used to experimentally determine adsorp-tion isotherms. The most widespread and best understood are the batch method and the frontal analysis method, which is used in packed-bed systems.

5.4.2.1 batch or stirred Tank Method

The procedure for the determination of adsorption isotherms in batch, or in a stirred tank system, is used when the adsorbent’s capacity and time for equilibrium are suf-ficiently high to ensure that the saturation of the adsorption sites is quickly achieved in a single step, as sought in most laboratory experiments. Due to the simplicity of the mass balances involved, the stirred tank methodology is also used for mass transfer studies between the fluid phase and the adsorbent. These data are calculated based on the values of substrate concentration in the fluid phase.

Batch adsorption experiments generate information on the amount of solute in equilibrium which cannot be obtained in continuous flow experiments. That informa-tion allows the establishment of an exact relainforma-tionship between the flux in diffusive processes and the accumulation of the adsorbate inside the adsorbing materials.

A simple way to determine an adsorption isotherm is to use a series of stirred tank reactors, as shown in Figure 5.2. The reactors are filled with predefined amounts of adsorbent which are put into contact with predefined volumes of solutions (liquid or gaseous) containing the adsorbate in increasing concentrations. The reactors are kept under constant agitation, at controlled temperature and pressure, until the equi-librium is attained.

The quantification of the amount of adsorbate in a solution is performed before and after the adsorptive process takes place. This allows determination of the corre-sponding equilibrium condition—that is, the equilibrium concentration of the adsor-bate in the adsorbent. Therefore, for each stirred tank, the concentration of adsoradsor-bate in the equilibrium is given by the following:

q V

where qi is the concentration of the adsorbate in the adorbent; Vi is the volume of solution in the stirred tank; Ci is the initial adsorbate concentration in the solution in the tank; Ceq,i is the equilibrium concentration of the adsorbate in that solution; and

92 Engineering Aspects of Milk and Dairy Products

mi is the adsorbent weight. Index i represents the tanks used (i varies from 1 to n, n being the last tank of the series).

If in each tank a solution of the adsorbate with a different initial concentration is placed, a different equilibrium concentration will be achieved, thus providing a data point for the experimental adsorption isotherm (Figure 5.3).

5.4.2.2 Frontal Analysis Method

The frontal analysis (FA) method was developed and used for the first time by James and Philips and by Schay and Szekely in the determination of adsorption isotherms.

3

1 2 4 5 n

V₁ = V₂ = V₃ = V₄ = V₅ = = V_n

...

C_eq1 C_eq2 C_eq4 ≠ C_eq5 ≠ ... C_eqn

Keep under agitation until equilibrium is reached ...

mC₁₁ =< mC₂₂ =< mC₃₃ =< mC₄₄ =< mC₅₅ =< ...... =< mC_nn

≠ ≠ C_eq3 ≠ ≠

FIGuRE 5.2 Experimental setup for the determination of adsorption isotherms using the batch methodology.

Concentration of the Adsorbent, in Equilibrium

Concentration in Solution, in Equilibrium 2

5

n

3 4

. . .

1

FIGuRE 5.3 Experimental adsorption isotherm obtained by the batch method.

Chromatographic Techniques Applied to Dairy Product Manufacturing 93

This method consists of provoking successive changes in the adsorbate concentra-tion at the inlet of a packed column and determining the rupture curves. Among the available methods to determine the adsorption isotherms of simple components, FA is the most precise. Among other applications, it has been used to determine iso-therms of peptides and proteins in different types of chromatographic techniques.

This method is adequate for adsorption works in small-diameter columns due to advantages such as a reduction in the amounts of adsorbent, adsorbate, and solvent needed in the experiments. For intensive work, it is recommended that a series of solutions of the adsorbate with known concentrations be prepared.

In FA, the amount of adsorbate, q_i+1, can be determined by

q q C C V V

i i i iV F i

a

+ = + +− + −

1 ( 1 )( _, 1 0)

(5.10)

where q_i and q_i+1 are the amounts of compounds adsorbed in a volume of adsorbent after the ith and the (i + 1)th step, when at equilibrium with the concentrations Ci and C_i+1, respectively; V_F,i+1 is the retention volume of the (i + 1)th inflexion point of the rupture curve; V₀ is the void volume of the column; and V_a is the volume of adsorbent in the column.