When dealing with ideal gas mixtures one can express the driving force for the diffusion of component / as:
n \
.P y
Vp, (4.2)
In all the situations to be analysed in this work the system pressure will be considered constant along the diffusion path. This will allow us to rewrite equation (4.1) combined with equation (4.2) as:
(4.3) y=i ^ i j j=^ ^i.j
It is important to stress that only nc-1 equations (4.3) are independent since the
nc
summation of the component gradients equals zero ( ^ V y , = 0 ) . /=i
Equations (4.3) are more useful when written in a matricial form (dimension
no-1) as:
Chapter 4 - Multicomponent Mass Transfer 72 with the elements of matrix B defined by
(4.5) ^i,nc k^^ ^i.k i*k ^ i j - - y , V ^ / . y ^i,no J (4.6)
4.2.2- Nonideal Fluids
When analysing the diffusion in nonideal fluids the driving force d, will be defined by
(4.7)
As with the ideal gas analysis, the sum of the nc driving forces equals zero, therefore only nc-1 equations (4.1) are independent
To avoid dealing with chemical potential gradients when analysing the diffusion of nonideal liquids the driving force will be expressed employing mole fraction gradients instead. n c -1 d, = I n . V x , (4.8) y=i where (4.9) T, PX
and where S indicates that the derivatives must be evaluated keeping constant all the mole fractions but that of components y and nc. To correct for deviations from ideal gas behaviour when working with dense gas mixtures the activity coefficient in equation (4.9) can be replaced by the fugacity coefficient.
Chapter 4 - Multicomponent Mass Transfer 73 For the nonideal case the result of the combination of equations (4.1) and (4.8), cast in a nc-1 dimensional matrix form, is
(J) = - o ^ [B ]-\r ]{V x ) (4.10)
4 .3 - Pic k’s La w fo r Mu l t ic o m p o n e n t Sy s t e m s
The analysis of diffusion in binary mixtures led Adolf Fick to establish in the late 1800’s that molecular diffusion is proportional to the decrease in the concentration gradient [Henley & Seader (1981)]. The molar flux of component
1 can be calculated using one of the forms of his ‘law’:
J ^ = —CfD^fVx^ (4.11)
Equation (4.11) can be seen as the definition of the Fick diffusion coefficient. Since the sum of the diffusion fluxes, Ji and J2, is zero and the summation of the mole fractions, x^ and X2, is equal to unity, it follows from equation (4.11) that D i,2 = 0 2,1.
The generalisation of equation (4.11) to encompass diffusion in a system with
/7 C components is given by equation (4.12).
nc -1
i - (4-12)
k=^
The matricial form (dimension nc-1) of equation (4.12) is
(J) = -c ^ [D ]V x (4.13)
4 .4 - In t e r a c t io n Ef f e c t s
It is clear from the generalised Fick’s law [equation (4.13)] that the existence of nonzero cross-coefficients (or off diagonal elements) in the Fickian matrix [0] will allow the multicomponent systems to have a behaviour quite different from the binary systems.
Chapter 4 - Multicomponent Mass Transfer 74
Toor (1957) investigated this different behaviour of multicomponent systems showing that it is possible to have a diffusion flux of one component even in the absence of a composition gradient for this component (osmotic diffusion). The phenomenon known as diffusion barrier will happen when, despite the presence of a composition gradient of a component, this component does not diffuse at all. It is also possible for a component to diffuse against its own composition gradient, a condition known as reverse diffusion.
Taylor and Krishna (1993) presented a picture where these effects are easily observed. In Figure 4.1 the diffusion flux of component 1 was plotted against the negative of its composition gradient. For binary systems the line representing Fick’s law passes through the origin [Figure 4.1(a)] but this may not happen in a multicomponent system. The general rule for multicomponent systems is to have an intercept different from zero (either positive or negative) leading to the situation where the interaction effects described in the previous paragraph may occur [Figure 4.1(b)].
osmotic diffusion “normal” diffusion behaviour diffusion barrier "normal" diffusion behaviour “normal” diffusion behaviour reverse diffusion
Figure 4.1. Interaction effects on diffusion in multicotnponent systems [adapted from Taylor and Krishna (1993)].
Chapter 4 - Multicomponent Mass Transfer 75 It must be clear that in the regions labelled ‘normal’ diffusion behaviour the diffusion flux of component 1 may be affected by the composition gradients of the other components.
4 .5 - Dif fu s io n Co e f f ic ie n t s
4.5.1- Binary Mixtures
Writing down the Maxwell-Stefan equation (4.10) for a binary mixture as
J, = (4.14)
and comparing it with Pick’s ‘law’, equation (4.11), one can get the relationship between the Fickian diffusion coefficient and the Maxwell-Stefan diffusivity for a binary mixture as:
= (4.15)
For ideal systems the two diffusion coefficients are the same since r becomes equal to unity.
There are several methods for the estimation of the binary diffusion coefficients in binary gas mixtures [Reid et a!. (1987)]. The reader is directed to Chapter 5 where the matter is further discussed.
Some extra care must be taken when estimating the binary diffusion coefficients for liquid mixtures because they can be strongly affected by the mixture composition. Normally the estimation is made in two steps: first the infinite dilution diffusion coefficients are calculated and secondly they are combined to take into account the mixture composition.
When one of the mole fractions approaches unity the thermodynamic factor r also tends to unity and the two diffusivities become equal to what is known as the infinite dilution diffusion coefficient (e.g., is the diffusion coefficient of
Chapter 4 - Multicomponent Mass Transfer 76 component 1 infinitely diluted in solvent 2). In Chapter 5 the method used in this work for the evaluation of the infinite dilution diffusion coefficients will be presented.
There are in the literature many methods to predict the composition dependence of the Maxwell-Stefan diffusion coefficients in binary systems. Reid et al. (1987) recommend the procedure presented by Vignes (1966) and that is given in the following equation.
^1,2 (416)