Discussion Forum

Hi Campbell,

Sorry but I do not understand your model. Where does your 1D line go? From a fluid container into a porous electrode? Along a single pore in the electrode? Radially in a single pore? What are your boundaries? Is the deposition uniform through the electrode? Is there a certain distribution in pore sizes?

Regards,

Pieter

The 1D "stack" is the structure of a very simple battery - anode, liquid electrolyte, porous cathode terminating in a cathode current collector. The (ionic) current flows through that stack, with the anode grounded (phis = 0)
and a constant current extracted from the collector. One of the output variables is then the potential at the collector, i.e. the battery voltage.

The porosity is initially uniform (P0), but in principle can change as a function of position (x) in the porous cathode due to the deposition
porosity = P0 - Aa*Ldep(x)
where Aa is the active area and Ldep is the local thickness of deposited product.

I hope that is clearer.

So in this porous region your liquid is partially transformed into a solid deposit, and now you want to know to what extent this results in a convective flow of electrolyte through the boundary between the 'free' region and the porous region? I would say you have to impose a flow at the boundary equal to

(1/{density of fluid} - 1/{density of deposit}) * {total mass of fluid reacting to solid per unit time}

If you want to know the convective flow in the complete porous region this would come down to imposing such boundaries throughout the porous region, so if
- x=0 at the free-porous interface
- x=x_el at the porous-charge collector interface
- a=1/{density of fluid} - 1/{density of deposit}
- r(x) mass of fluid reacting to solid per unit time at position x
- phi(x) the vollume flow at position x
you would get

phi(x)=a * integral[x to x_el] r dx

differentiating with respect to x:

d phi/dx = a * r

I hope this addresses your question.

Pieter:
Thanks for the help. Your suggestion led me to examine the velocity field more closely.
I believe that the Darcy velocity is solving correctly. If I impose BCs of no flow at the anode, atmospheric pressure and outflow at the (open) end of the porous cathode, the outflow velocity corresponds exactly to the deposition rate.
However, the ion concentrations in the electrolyte are rising. This is unphysical because the anion is not involved in any reaction, so its integrated concentration must be constant, and therefore the cation also by neutrality. (These are dilute species in the tertiary current node.)
I find that the increase in concentration corresponds exactly to the loss of volume in the cathode, so the ions are not "flowing" with the electrolyte, although I am coupling Darcy flow (dl.u) to the convection field in the electrolyte. Clearly, I do not fully understand the equations, and must somehow compensate for the loss in electrolyte volume.
After a short time the solver stops converging and reports a possible singularity that I have not yet identified.
I will keep working on it, but would welcome any suggestions.
Campbell

Hi Campbell,

So what is reacting? Only your cations, or also the solvent? Have you coupled both the flow and the ion concentrations at the interface between the free and the porous region? Are you actually removing the reactants from the solution when you are depositing (maybe a wrong sign in the reaction rate?)?

Hi Pieter:
A dilute neutral species in the electrolyte is reduced at the cathode, and combines with the cations to form the insoluble deposit.
When I model this without reduction of porosity, the results are sensible: the thickness of the deposit grows smoothly with time, the concentrations of cation and neutral species are depleted in the cathode, the concentrations of cation and anion grow in the free region close to the anode such that the total amounts (integrated concentration) remain constant.
I can use the (local) thickness of deposit to slow the reaction rate (actually increase the over-potential at constant current) by reducing the active area in proportion to Ldep, and/or by decreasing the exchange current.
So I believe that I have the physics correct up to the point of inducing flow due to the lost volume in the porous region. And as I said last time, the flow velocity and pressure appear to make sense. The ternary current node and Darcy's Law node both apply in the free and porous regions. Does that not make the concentrations and (Darcy) velocity continuous at the interface?

Thanks, Campbell

Campbell:

As far as I can infer, your 1-D model has a point (A) for the anode (which is grounded) then a "line (A-B)" representing the electrolyte region and then another "line (B-C)" where C is the current collector side of the porous cathode.

1) Could you explain how you have set up the boundary conditions for the Darcy-flow equation at each of these points?

2) From what I understand, point A is grounded, point C is where a fixed current density BC is used and the line (B-C) is where the reaction occurs. You have species P+, Q- and R in your solution. In the porous electrode, P+ and R electrochemically form an insoluble species. The deposit starts at C and then slowly increases in thickness going towards B. Is the deposit conductive? Does it change the overall electronic conductivity of the porous electrode?

From what I can infer, when you reduce the porosity of the electrode, you are somehow effectively decreasing the volume available but the number of moles of your anion are not correspondingly reducing causing an increase in anion concentration. Have you checked to see if your cation and neutral species concentration are also increasing? If they are decreasing, are they decreasing by an amount that you expect?

Sri Puranam

Sri:
1. Point A has no flow, point B has no BC (i.e. I am assuming Darcy continuity), point C has constant pressure (1 atm) and an outflow given by the rate of volume decrease.

2. Not quite. The deposition is occurring everywhere that there is active area in the cathode, so the local thickness is just nanometers, where the total cathode thickness (length BC) is 1 mm. The deposit is non-conductive and the local exchange current and/or active area is reduced accordingly. (This is a macroscopic representation of microscopic heterogeneity.) The porous electrode retain its initial conductivity.

(One thought here - the sum of the relative volumes of porous conductor and electrolyte is now less than one, but I would not think this matters since the node has variables for each.)

Yes cation and anion concentrations have identical profiles (neutrality is preserved) and the neutral species also has unphysical increasing concentration.

So the only difference between the case which gives results as expected and the case where you include the decreased porosity is that you have included an outflow at boundary C?

The working models had no Darcy node and there was no convective flow in the electrolyte.
I added Darcy and set the outflow to the volume loss, coupling the velocity field back to convection in the tertiary current.

Question - does the tertiary current node account for the Darcy velocity - i.e. a velocity field that accounts for porosity and reduced cross-sectional area?