Principles and Theory

The principle of the IGS method was described in detail by Laroi et al. (1977) and in Chapter 4. Laroi et al. (1977) also derived a very simple equation for the determination of limiting activity coeffiCients. These equations are however limited to a few types of systems based on the nature of the components. Beyond that, equations were derived taking into account the vapour phase correction that is important for solutes with higher volaWities (Duhem and Vidal (1978)). This chapter summarizes the new1y derived equation by Krummen et al. (2000) which requires the use of a saturation cell. Hovorka and Dohnal (1997) also derived a set of equations for use with the inert gas stripping technique. All significant developments in the determination of limiting activity coefficients are outlined in this chapter.

The limiting activity coefficient is related to other well known parameters used for designing highly specialized equipment. Its relation to these parameters is outlined to stress the importance of accurately knowing limiting activity coefficients. Thereafter the derivations of the various equations for the determination of limiting activity coefficients are outlined. These equations can only be used with the inert gas stripping technique.

6.1 Activity Coefficients, Selectivity. Capacity and Selection Factor

The limiting activity coefficient is related to other well known parameters namely selectivity, capacity and selection factor. These factors are very important for the separation of high purity chemicals because removal of the last traces of impurities requires the greatest separation effort. The limiting activity coefficient is required to select entrainers for separation processes and to check for separation problems such as azeotropes and miscibility gaps, aiding in the design of various separation units. Accurate knowledge of the limiting activity coeffiCient makes these parameters, which are important for the economics of a separation process, easy to calculate.

The measurement of limiting activity coefficients (r~) in multi-component systems are of great interest, because the addition of small amounts of a solvent to an entrainer has a considerable effect on the activity coefficient at infinite dilution and thus on the selectivity

ls;

⁼

% )

^and

capacity

\k /'"

⁼

X i)

of the entrainer. This is useful knowledge for the separation of mixtures.

, ,

The addition of an entrainer or of a solvent mixture can simplify the separation considerably. The selectivity can be increased by the addition of a second solvent. An increase in the selectivity often leads to a decrease in the capacity of a solvent or solvent mixture; however it is important for the economic efficiency of a separation process.

The activity coefficient at infinite dilution (r;<G) is an important parameter, particulany for the reliable design of thermal separation processes such as extractive distillation. Thus the synthesis, simulation and optimization factors (aij) which, depending on pressure, temperature and the composition of the mixture, can be calculated across the complete concentration range using the following simplified equation:

r p,sor

a - I I

i} - r

,

.P~1

,

^6.1

where j is the low boiling component and j is the high boiling component and

P

^S01 is the saturated pure component pressure. It is seen that the separation of the final traces of a component requires the greatest effort because the least favourable values of the separation factor occur at high dilution. In the case of positive deviations from Raoult's Law (y I

>

I) the greatest separation effort is required at the top of the column (xI

--+

1). In such cases the relation below applies.

6.2

At the bottom of the column

(x

_j

--+

1) the effort involved in the separation is largest for negative deviations from Raoult's Law (ri < 1). In such cases the relation below applies.

6.3

The effect necessary for the separation is determined by the value of a. -1, To avoid an over design of a distillation column and to minimize the investment and operating costs, reliable

Chapter 6

knowledge of the separation factor at high dilution (a; ) is important. Taking into account limiting activity coefficients also improves the reliability of the description in the dilute region when reliable g E model parameters are to be fitted or in the development and improvement of group contribution methods. In addition, it is possible to obtain reliable values for Henry's constants and partition coefficients as shown later.

Some very basic equations were formulated by Leroi et at (1977) in order to determine activity coefficients at infinite dilution for solutes that are volatile in nature. These equations, depending on the nature of the solvent, can be used to calculate limiting activity coefficients for most systems. Duhem and Vidal (1978) and Boa and Han (1995) modified the Leroi et at. (1977) equations taking into account some of the simplifying assumptions that are usually not valid for most systems. The derivations of all these important equations together with assumptions are outlined below.

6.2 Thermodynamic Formulations for the Laroi et al. (1977), Duhem and Vidal (1978) and Boa and Han (1995) Equations

Assuming that the 9as phase is in equilibrium with the liquid phase, it is possible to write the equilibrium equations for each component: solute (sol ), solvent (s ) and carrier gas (CG ) as follows.

6.4

6.5 6.6

where x is the mole fraction in the liquid phase, y is the mole fraction in the vapour phase, / is the fugacity, /OLO is the reference fugacity for a liquid at pure state and zero pressure, I is the Poyntin9 correction, <p is the fugacity coefficient, P is the pressure, y is the activity coefficient and H is the Henry's constant. For both solvent and solute, the reference state is the pure liquid at zero pressure. For equilibrium at low pressure, which is mainly studied here, vapour phase corrections <p / can be derived from second virial coefficients.

6.7 6.8

BIf, is the virial coefficient characterizing bimolecular interaction between molecule j and molecule j ,while BM is the mixture second virial coefficient, T is the temperature and R is the Universal gas constant. Reference fugacity's

It

^OL' are obtained from the equation

J,

^{I 0L' = PVI rn'1', ~(T •}

r)ex _

^I

{ O'·r)

^{VI RT I 6.9}

where ^VIOL' is the molar volume,

t.

^0LO is the reference fugacity for the i^1lo component of the pure liquid at zero pressure,

P /

is the vapour pressure and rp/o is the fugacity coefficient in the vapour phase at saturation. If the solute is highly dilute in the solvent and if the solubility of the carrier gas in the liquid phase is negligible, the solute activity coefficient may be approximated by its value at infinite dilution. It can be shown that in most cases this approximation is valid if the mole fraction of the solute xsol is less than 10-³ (Leroi et al. (1977)). The solvent mole fraction in the liquid phase and the activity coefficient

r

^s may be taken equal to 1 in Equation 6.5, under the same conditions. If the vapour phase corrections are neglected then the following equations apply.

6.10 6.11

If nand N are respectively the total number of moles of solute and solvent in the equilibrium cell at time l, the quantities ( - dn ) and ( - dN ) withdrawn from the solution during dt by the carrier gas flow are

D dt dn = -y

P - ' -

"" RT

^6.12

dN= _y pD, dt , RT

Chapter 6

6.13

Dz is the total volumetric rate of gas flowing out of the still converted to pressure (P) and temperature (T). From Equations 6.10 to 6.13 it can be deduced that

dN= _r D , dt ' RT

An overall mass balance around the dilution still gives

D

=

D - RT (dn + dN )

, P dt dt

6.14

6.15

6.16

where D is the pure carrier gas flow rate measured at system temperature (T) and system pressure (P). Combining Equations 6.16, 6.14 and 6.15 yields

6.17

If

D2

is replaced by Equation 6.17 in Equations 6.14 and 6.15 and if xsol is replaced by Equation 6.18 at infinite dilution

n n

x = - - = -

10/

n+N- N

^6.18

where ¹¹ is the molar amount of solute in the still and N is the molar amount of solvent in the still, as mentioned before. If the vapour and liquid phases are in equilibrium in the dilutor cell, neglecting vapour phase corrections and carrier gas solubility in the liquid phase and in the highly diluted range, then the basic differential equations relating the variations of the amounts of solute and solvent with time are the following:

6.19

dN

r'

^D

- = - - '

---=-=----,-

dl RT p~' p~'

1- ~Y:/-;---~-

6.20

where ~S41 is the pressure of component

i

at saturation.

The equations used in the activity coefficient calculation procedure are based on the following

two assumptions:

»

Ideal vapour phase

}> Negligible solubility of inert gas in the liquid

A further assumption is that the solution of these equations depends on the type of solvent.

6.2.1 Leroi et al. (1977) Equations

The above equations lead to two very simple equations derived by Leroi et al. (1977). These equations are the simplest equations to date and give accurate results. The assumptions made when deriving these equations are justified by experimental conditions.

6.2.1.1 Non-volatile Solvent

A simplifying assumption is made in the next development by neglecting the term

n ^OD p~' ^sol