Simplified solutions for a symmetric 1:1 salt

Solute Transport

7.5 Simplified solutions for a symmetric 1:1 salt

180 Solute Transport

equally valid for a 1:1 and a 2:2 salt. Note that for a 2:2 salt, concentrationcis twice higher than the salt concentration.

As explained, an important application of the above expressions for 𝐽

ch as function of 𝜕 𝑐/𝜕 𝑥|^∗ is that it allows us to generalize results from §7.1. Thus, for instance, we can use Eq. (7.13) for the convection-along-the-surface-with-constant-𝜏-model, implement 𝐽=−𝐷

hm·𝜕 𝑐/𝜕 𝑥|^∗, and have 𝜕 𝑐/𝜕 𝑥|^∗replaced by an expression involving𝐽

chand the𝑇_𝑖’s based on any of the expressions just discussed, such as Eq. (7.41) in the general case.

Here let us analyze the example of a solution of a 1:1 salt, with equal diffusion coefficients, 𝐷=𝐷_hm, based on Eq. (7.44). We use Eq. (7.13) and then arrive at

𝐽ch=

√︂

𝐷 𝜏

· 2

𝑇₊−𝑇₋

· (𝑐_∞−𝑐^∗) . (7.45)

In a different example, again for a 1:1 salt, but now with unequal diffusion coefficients, for a perfectly selective interface (only allowing cations through, thus𝑇₊=1), combination of Eqs. (7.13) and (7.42) leads to

𝐽_ch =2

√︂

𝐷₊ 𝜏

√︂

𝐷₊ 𝐷hm

· (𝑐_∞−𝑐^∗) (7.46)

which interestingly still includes a dependence on the coion (anion) diffusion coefficient (via 𝐷hm). There will not be such an influence of the coion when we use the classical film model without a refreshment effect from §7.1.1. The two last expressions lead to the same result when we set𝑇₊=1 and𝑇₋=0 in the first, and𝐷

hm=𝐷₊=𝐷in the second expression.

Note that when expressions similar to Eqs. (7.45) and (7.46) are derived for 2:1, 2:2, etc., salts, that𝑐_∞and𝑐^∗again refer to bulk and surface, but they are not the salt concentration, in the way that throughout this book𝑐_∞is defined as a salt concentration of a 1:1 salt. Instead they are monovalent equivalent salt concentrations.

Simplified solutions for a symmetric 1:1 salt 181 where we implemented Eq. (7.25) for current efficiency,𝜆. And thus according to Eq. (7.24), the ions flux across the DBL is given by

𝐽ions=−2𝐷

𝜕 𝑐

𝜕 𝑥 (7.48)

a result we can also have arrived at from adding up𝐽₊and𝐽₋, based on the NP-equation without convection, Eq. (7.2). We can then integrate Eq. (7.48) across the film layer, thus from bulk to surface, and obtain

𝐽ions=2𝑘

L (𝑐_∞−𝑐^∗) (7.49)

where as before𝑘

L=𝐷/𝛿, and𝛿is the thickness of the film layer (DBL), and𝑐^∗is the salt concentration at the surface.

We can also derive an expression for current density from Eq. (7.2) based on𝐽

ch=𝐼/𝐹= 𝐽₊−𝐽₋, resulting in

𝐽ch=−2𝐷 𝑐

𝜕 𝜙

𝜕 𝑥 (7.50)

which can be combined with Eq. (7.47) which then results in 𝜆= 𝜕ln𝑐

𝜕 𝜙 (7.51)

and if𝜆is the same at all positions in the film layer (which is the case for the simple film model of §7.1.1), then Eq. (7.51) can be integrated over the DBL to^ix

𝑐^∗=𝑐_∞·exp(𝜆 𝜙

dbl) (7.52)

where𝜙

dblis the potential across the film layer (potential at surface minus that in bulk).

These equations define transport in the film layer, or DBL, when diffusion and electromigration are both important driving forces. In combination with knowledge of ion reactions at the surface, or transport through it, for instance because of a membrane, the factor𝜆is established at that position, and for a given current𝐽_chwe then know𝐽_ions, and the above equations provide information on the surface concentration𝑐^∗, and potential drop over the DBL.

We can integrate Eq. (7.47) across the film and obtain 𝐽ch= 2𝑘

L

𝜆

(𝑐_∞−𝑐^∗) → 𝑐^∗ =𝑐_∞− 𝐽

ch𝛿 2𝐷 𝜆

. (7.53)

ixThis equation is an extension of Eq. (2.118b) in Vetter (1967). Many of the equations in the present and former sections can be found there.

182 Solute Transport If both𝜆and current𝐽

chare positive, then Eq. (7.53) predicts that the surface concentration is lower than the bulk concentration, thus the concentration of both ions goes down towards the membrane. For a current𝐽

chthat is positive in the direction to the surface, the potential over the DBL,𝜙

DBLis negative which inspection of Eq. (7.50) also shows, and then Eq. (7.52) shows the same, that for𝜆 >0, that𝑐^∗< 𝑐_∞. This decrease in salt concentration towards the surface is indeed the general observation in (models of) electrodialysis (ED) for the concentration profile of ions near a membrane in the diluate channel. The case of positive 𝐽_chand positive𝜆relates to the surface being a cation exchange membrane. For an anion- exchange membrane, both𝐽

chand𝜆are negative and again we have the concentration going down towards the membrane in the diluate channel. In the concentrate channels the situation is reversed, and at the membranes the salt concentration is higher than in the center of the flow channels.

If we consider the case of𝜆=±1, we have a perfectly selective interface that only allows one type of ion to go through. Then𝐽

ions only has a contribution from the ion that reacts at, or moves through, the surface. This ion we can call the counterion. The other ion is the coion and this ion cannot cross the interface, and thus it is at equilibrium throughout the DBL, and thus its concentration profile must follow Boltzmann’s law. With𝜆=1 and 𝐽_ch>0, then anions are the coions, thus Eq. (7.52) should be the Boltzmann equation for anions, which indeed it is. If the membrane only allows anions through, then𝜆=−1 and now Eq. (7.52) results in the Boltzmann distribution for cations. Thus for a perfectly selective surface, the Boltzmann relation is established for the coions over the entire film layer. Note that this is only the case in the classical DBL model of a fixed thickness, that was explained in §7.1.1.

Next we continue with models in which a dispersion effect is included by the refreshment concept of adding a term(𝑐−𝑐_∞) /𝜏in the salt mass balance that was explained in §7.1.3.

For neutral solutes this led to the same relationship between𝐽_𝑖and surface concentration as for the standard film model, just with a different definition of𝑘

L. That will then also be the case for a 1:1 salt solution, thus Eq. (7.53) is again valid. But the concentration and voltage profiles are now different. Compared to the basic film model of §7.1.1 where transport is across a layer of thickness𝛿, the result we arrive at is more elegant because it integrates simultaneously over the entire bulk phase (such as a channel) and the DBL, without an ad hoc dividing plane.

To find the potential across a channel including the DBL, we use Eq. (7.10) in which we implement Eq. (7.47), resulting in

𝑐(𝑥˜)=𝑐_∞−𝐽

ch𝜆 2𝑘_L

·exp(−𝑥˜/𝛿) (7.54)

Simplified solutions for a symmetric 1:1 salt 183 where𝑘

L=√︁

𝐷/𝜏and𝛿=√

𝐷 𝜏. Eq. (7.54) can be implemented in Eq. (7.50)^xto obtain an expression for the potential,𝜙

ch, across a layer or channel with thickness𝐿

ch, as function of the current density (directed across the channel) and salt concentration𝑐_∞

𝜙ch=− 𝐽

ch

2𝑘

L𝑐_∞ 𝐿

ch

𝛿

−ln

1− 𝜆 𝐽

ch

2𝑘

L𝑐_∞

. (7.55)

This𝜙

ch is the total voltage drop across the channel, all the way up to the surface; thus it includes the DBL. For low currents, Eq. (7.55) simplifies to

𝜙ch|

Ohmic=− 𝐽

ch

2𝑘_L𝑐_∞ 𝐿ch

𝛿

=−𝐽

ch𝐿

ch

2𝐷 𝑐_∞ (7.56)

which describes the Ohmic potential drop over a channel with concentration𝑐_∞, independent of refreshment parameters such as𝜏and𝛿. In the other limit, Eq. (7.55) leads to a limiting current (LC). For 𝜆 =±1 we arrive at 𝐽

ch,LC = ±2𝑘

L ·𝑐_∞. When the current density approaches this limit, the potential drop over the entire layer,𝜙_ch, diverges.^xi

In Fig. 7.3 we provide calculation results for voltage versus current across a channel of thickness 𝐿

ch for a 1:1 salt that is next to a perfectly selectivity interface (𝜆=1). [These results also apply to a ‘half-channel’ with a symmetry plane, with a surface on both sides, as in the diluate channel in the ED-process.] We use the refreshment model, Eq. (7.55), scaled to the Ohmic voltage drop given by Eq. (7.56). We use a value of𝛿that is 10% of the channel thickness𝐿_ch, and we scale𝐽_chto𝐽_ch,LC. We also plot the result for the standard DBL model given by Eq. (7.52) for a DBL of thickness𝛿, to which we must add the Ohmic voltage drop across the remaining channel of thickness 𝐿

ch-𝛿. The standard model with fixed film layer thickness predicts a lower voltage across the channel than Eq. (7.55). This can be understood because in the standard film model 90% of the channel is at the bulk salt concentration, and only 10% is perturbed, while in the refreshment model, the decrease in salt concentration penetrates further into the channel, i.e., the average salt concentration will be somewhat less, implying a higher resistance across the full channel thus a higher voltage.

For low currents, the slope of𝜙_ch/𝜙_ch|

Ohmicvs.𝐽_ch/𝐽_ch,LCin the refreshment model is given by𝛿/𝐿

ch, which in this calculation then has a value of 0.1. The standard film model has a slope in this limit that is exactly half that value. This difference has the consequence that if a transfer coefficient,𝑘

L, is derived on the basis of (the deviation from linearity of) a curve of

xThis equation is also valid with refreshment included, see Eq. (7.36) for the case considered of a 1:1 symmetric salt.

xiFor the refreshment model with variable𝜏of §7.1.4, the𝑐(𝑧)-function is given by Eq. (7.17) which can be implemented in Eq. (7.50) and then integrated to obtain an expression for𝜙

ch. This is possible but the result is a very lengthy expression.

184 Solute Transport

0.9 1 1.1 1.2 1.3 1.4

0 0.2 0.4 0.6 0.8 1

ϕch/ϕch|Ohmic

J_ch/J_ch,LC fixed refreshment model

standard film model

Fig. 7.3:The voltage across a channel that has a selective interface on one side, as function of current density, according to the standard film model, and based on the fixed-𝜏 refreshment theory. For parameter settings see main text.

voltage versus current in the limit of low currents, that we obtain a a difference by a factor of two in the predicted value of𝑘

L, depending on the chosen model. The standard film model with fixed thickness would lead to the prediction of a twice lower 𝑘

Lthan when the same data of voltage vs. current are analysed using the constant-𝜏refreshment model.