General balances for binary electrolytes

Solute Transport

7.3 General balances for binary electrolytes

172 Solute Transport

all cations contribute 50%, then𝑇₊=𝑇₋=½ and then𝜆=0, i.e., there is transport of current, with all cations going in one direction and all anions in the opposite direction, but there is no ‘net’ transport of ions as a whole.

As mentioned, when both the anions and cations go in the same direction (as is the case for the aforementioned RED-process), then the transport number will be<0 for the coion and>1 for the counterion, and consequently𝜆 >1. As a metric to define efficiency, numbers beyond unity are not very intuitive, and that is why in RED and similar processes, instead a different efficiency is used, which is the salt transport efficiency, 𝜗. This efficiency is defined as𝜗=1/𝜆and is a measure of how effectively the salt concentration difference is used to generate electrical current. For a well-designed RED process this ratio is close to unity. However, when membranes are used with large pores (a pore diameter of several nm’s or more), or membranes that are too thin (for instance a thickness less than 1𝜇m), then the ions flux (leakage of ions,𝐽₊+𝐽₋) is high relative to the generated current, and thus𝜗will be low. On p. 338 we discuss in more detail the low value of𝜗when very thin membranes are used, and the effect thereof on osmotic power production.

General balances for binary electrolytes 173 modelled by a refreshment term.^vi Reactions in solution are not considered – for that, see Ch. 10. The convection term can be in thex-direction towards a surface or in a direction along a surface,𝑧, and to describe both situations simultaneously we use the general formulation of∇ · (𝑐v_F)here. Thus, the ion mass balance becomes

𝑝

𝜕 𝑐_𝑖

𝜕 𝑡

=−∇ · (𝑐_𝑖v_F) + 𝜕

𝜕 𝑥

𝐷_𝑖

𝜕 𝑐_𝑖

𝜕 𝑥

+ 𝜕

𝜕 𝑥

𝐷_𝑖𝑧_𝑖𝑐_𝑖

𝜕 𝜙

𝜕 𝑥

+𝑐_∞,𝑖−𝑐_𝑖 𝜏

. (7.26)

Now we analyze Eq. (7.26) for a binary salt. We multiply each term in Eq. (7.26) by𝑧₊for the cation, and by|𝑧₋|for the anion, and define𝑐as the ‘monovalent equivalent’ (m.e.) salt concentration, which is given by𝑐=𝑧₊𝑐₊=|𝑧₋|𝑐₋. We now arrive both for the cation and for the anion at

𝑝

𝜕 𝑐

𝜕 𝑡

=−∇ · (𝑐v_F) + 𝜕

𝜕 𝑥

𝐷_𝑖

𝜕 𝑐

𝜕 𝑥

+ 𝜕

𝜕 𝑥

𝐷_𝑖𝑧_𝑖𝑐

𝜕 𝜙

𝜕 𝑥

+ 𝑐_∞−𝑐

𝜏 (7.27)

wherecwithout indexiis the m.e. salt concentration defined above. We evaluate Eq. (7.27) for both ions and equate the two results. We then arrive at the charge balance

0= 𝜕

𝜕 𝑥

(𝐷₊−𝐷₋) 𝜕 𝑐

𝜕 𝑥

+ 𝜕

𝜕 𝑥

(𝑧₊𝐷₊+ |𝑧₋|𝐷₋)𝑐

𝜕 𝜙

𝜕 𝑥

. (7.28)

Combining Eq. (7.27) (evaluated for one of the ions) with Eq. (7.28), we arrive at the salt mass balance

𝑝

𝜕 𝑐

𝜕 𝑡

=−∇ · (𝑐v_F) + 𝜕

𝜕 𝑥

𝐷hm

𝜕 𝑐

𝜕 𝑥

+ 𝑐−𝑐_∞

𝜏 (7.29)

which is a quite amazing result: for any binary salt also when it is asymmetric (such as a 2:1 salt, for instance CaCl₂), we can derive a salt balance that includes convection, diffusion, and refreshment, but not electromigration. The ‘harmonic mean’ diffusion coefficient in Eq. (7.29),𝐷

hm, is given by

𝐷hm= (𝑧₊+ |𝑧₋|)𝐷₊𝐷₋

𝑧₊𝐷₊+ |𝑧₋|𝐷₋ (7.30)

which for a binary symmetric salt (a 1:1 or 2:2 salt) simplifies to 2

𝐷hm

= 1 𝐷₊

+ 1 𝐷₋

. (7.31)

When𝐷₊=𝐷₋=𝐷, also for an asymmetric salt, it follows from Eq. (7.30) that𝐷

hm=𝐷.

viWe do not include here dispersion modelled as parallel to molecular diffusion, as discussed in §7.1.2, though the results arrived at in the present section also hold then.

174 Solute Transport

Thus Eq. (7.29) shows how also for any asymmetric binary salt, with the two ions having different diffusion coefficients and different valencies, we have transport of salt as if it is a single neutral molecule. This is only the case for a binary salt solution, i.e., with only one cation and one anion, in a phase without any other (fixed) charge. This equation can be used for electrolyte flow in a channel that is empty or filled with an uncharged mesh structure, or for flow through any other type of uncharged porous medium.^vii

Including dispersion in the general balance for binary salts. When Eq. (7.30) is solved in multiple dimensions with flow, then the refreshment term(𝑐−𝑐_∞) /𝜏is not typically used because there is no additional sideways flow that can be the origin of it (convection is already included in the∇· (𝑐v_F)-term). But instead, Eq. (7.30) can be extended with

‘dispersion parallel to diffusion’ as described in §7.1.2. Then we use a coefficient𝐷^⋄

𝑖 that combines molecular diffusion and dispersion, and this𝐷^⋄

𝑖 is used in the concentration gradient-terms in Eqs. (7.27) and (7.28) instead of just𝐷_𝑖. The electromigration-part only depends on molecular diffusion and thus𝐷_𝑖is used there. Re-analysing Eqs. (7.27) and (7.28) we then find that Eq. (7.30) is replaced by (Newman, 1983)

𝐷^⋄

hm= 𝑧₊𝐷₊𝐷^⋄₋+ |𝑧₋|𝐷₋𝐷₊^⋄

𝑧₊𝐷₊+ |𝑧₋|𝐷₋ (7.32)

where the 𝐷^⋄

𝑖’s include both molecular diffusion and dispersion. The dispersion coefficient is independent of the ion; it only depends on the hydrodynamic flow pattern near the surface and the resulting fluid mixing. For a symmetric salt (where𝑧₊=|𝑧₋|) and in case 𝐷=𝐷₊=𝐷₋, we obtain 𝐷^⋄

hm=𝐷^⋄=𝐷+𝐷

disp. If dispersion is much larger than diffusion, so the𝐷^⋄

𝑖’s become the same for both ions (because the dispersion coefficient does not depend on the ion), we end up with𝐷^⋄

hm=𝐷_dispfor any binary salt.

If this limit applies, then nothing about an ion’s charge, valency, or diffusion coefficient matters in determining salt concentration profiles, only the dispersion coefficient. This is correct in a bulk phase sufficiently far away from any surface, but certainly does not apply to the last 10s of 𝜇m’s right next to a selective interface, where mass transfer limitations most certainly are important.

The approach described in this box, where𝐷

hmdepends on a dispersion coefficient, is generally valid whichever expression is used for the dispersion coefficient, irrespective of whether it is invariant or not with position. This is the case when we use the general

viiThe presence of which, as we explain in Ch. 12, leads to an additional term𝑝/𝝉(with𝝉the tortuosity factor) in front of each factor𝐷, and to the porosity-term𝑝in front of the𝜕𝑐/𝜕𝑡-term.

General balances for binary electrolytes 175

form of Eq. (7.29) with𝐷

hm‘inside the differential’. We can only take𝐷

hmoutside the differential if there is no dependence of it on position or concentration. That will be the approach in the next box.

Simplified approaches to include dispersion in mass transfer modelling in channels. In several cases, especially for a flow channel (not so much for a surface in contact with bulk solution), it is useful to assume the dispersion coefficient is some constant, for instance we set it to a multiple of the diffusion coefficient, thus notx- orc-dependent.

We can then simplify Eq. (7.29) (also leaving out refreshment), resulting in 𝑝

𝜕 𝑐

𝜕 𝑡

=𝐷^⋄

hm

𝜕²𝑐

𝜕 𝑥2 − ∇ · (𝑐v_F) . (7.33) Even though salt transport is enhanced because of dispersion, a calculation of potentials and currents, making use of Eqs. (7.2), (7.18), and (7.28), is based on the molecular diffusion coefficients, 𝐷_𝑖, and thus is not directly affected by dispersion. This is a very useful approach to describe transport across the flow channels used in membrane- based water desalination technologies described in Chs. 11 and 12. When this model is evaluated in the limit of a very high dispersion coefficient, then flat concentration profiles are predicted in the direction across the flow channel. This limiting model should then give the same outcome as a simple model where this flat profile is a- priori assumed, i.e., where in the𝑥-direction towards an interface concentrations are assumed to be invariant. Of course in this model, all effects of mass transport across the channel that could lead to concentration profiles are gone but this may nevertheless be a realistic first order model for reverse osmosis or electrodialysis when combined with a plug flow reactor (PFR) approach in thez-direction along the membrane. An Ohmic resistance across the channels, relating current and voltage drop, depends on the moleculardiffusion coefficients, and is still part of the the model even when dispersion is so large that all concentration profiles are equalized out. For a given current, the voltage changes across a channel are less when concentrations are more equalized out, but except for this effect, more stirring does not enhance charge transport or reduce voltages.

176 Solute Transport

Convection in a model for concentration polarization (CP) in membrane transport.

When we make use of the full extended NP equation also including convection, Eq. (7.2), then a DBL model solved for a binary𝑧₊:|𝑧₋|salt for zero current leads to the result that the ‘monovalent equivalent’ (m.e.) molar flux (𝐽

m.e.=𝑧₊𝐽₊=|𝑧₋|𝐽₋) is given by

𝐽m.e.=𝑐 𝑣

F−𝐷

hm

𝜕 𝑐

𝜕 𝑥 (7.34)

where the harmonic mean diffusion coefficient 𝐷

hm is given by Eq. (7.30). For a 1:1 salt,𝐽

m.e.simply corresponds to the salt flux,𝐽_𝑖(the same molar flux for cations as for anions), andcis simply the salt concentration, at any point in the DBL. In Ch. 11 this expression is integrated to obtain an analytical expression for the DBL in front of a membrane used for water desalination.