If you want to find out about the basic operation of a Vanadium Redox Flow Battery – see this What is a Redox Flow Battery ? This article will carry on from that to look at the amount of energy that can be stored and the unit cost.

During discharge the reaction on the left (positive) side of the battery is

VO_{2}^{+} + 2H^{+} + e^{–} -> VO^{2+} + H_{2}O

Vanadium in a 5^{+} oxidation state (yellow) is reduced to Vanadium in a 4^{+} Oxidation state (blue).

On the other side of the battery V^{2+} (purple) gives up an electron to become V^{3+ }(green) in solution. The voltage generated by such a cell is 1.26V.

#### How much Vanadium is needed to store a particular amount of energy ?

For those who like to see their science:

- 1 electron moving through a potential of 1.26V releases energy e.V which is 1.6E-19 . 1.26V = 2.02E-19 Joules (J)
- 1 mole of electrons doing this produces 2.02E-19 x 6.022E23 = 121,600 J
- 1 Watt.hour (W.h) = 3600 J so the energy released by 1 mole of electrons is 33.8 W.h

For this to work you need 1 mole of Vanadium on the +ve side of the battery and another 1 mole of Vanadium on the -ve side of the battery – thus 2 moles of Vanadium for every 1 mole of electrons.

**1 mole of Vanadium can store 16.9 W.h of energy.**

1 mole of Vanadium weighs 50.94g, so 1Kg is 19.6 moles. Thus **1kg of Vanadium can store 331 W.h**.

#### How much is this in terms of the electrolyte ?

Well (the PNNL mixed acids paper) quotes the improved electrolyte formulation as 2.5M Vanadium + 2.5M SO_{4}^{2-} (Sulphuric Acid) + 6M Cl^{–} (Hydrochloric acid) where M is the Molar concentration per litre. These sound like pretty concentrated acids, however they are not that different to the 4.5-6.0 Molar Sulphuric acid found in a conventional Lead-acid battery.

So in 1 litre of Electrolyte we have 2.5M Vanadium (127g), 2.5M of H_{2}SO_{4} (245g) and 6M of HCl (216g) plus some water. How much water? Well it cannot be the full 1000g that you would normally have in 1 litre of water – as that would imply that chucking all that other stuff in would have not increased the volume of the mixture at all, which doesn’t seem a reasonable assumption.

Even adding 2.5+2.5M+6M =11 moles of a small species like water would displace 11/55ths of the original water you started with in your 1 litre (1 litre is 55 moles) – and as the SO_{4} and Vanadium complexes are bigger than the little old water molecule you might expect each of them to displace say 3 water molecules each. HCL appears more compact so we might guess that it displaces 1 water molecule. Imagine then putting everything that we have in the electrolyte into 1 litre (55 moles) of water – we can estimate that (2.5+ 2.5) * 3 + 6 * 1 = 21 moles of water molecules would be displaced from the original 55 moles that we had. This would leave 34 moles or 612g of water that might fit into 1 litre of electrolyte.

Thus 1 litre of electrolyte contains approximately:

- 127g Vanadium
- 245g of H
_{2}SO_{4} - 216g of HCL
- 612g of H
_{2}O (guessed)

and weighs in at 1200 g – so has a density of 1.2 – which seems not unreasonable.

Thus the vanadium is perhaps only 10.5% of the mass of the electrolyte. If 1 Kg of Vanadium gives 331 W.h then **1Kg of electrolyte can store 34.75 W.h** – this seems to agree with the oft-quoted figure of 35 W.h per Kg.

#### How much does this cost ?

V_{2}O_{5} would seem to be the correct starting material for making electrolyte with (as we don’t want Fe knocking around which might well interfere as an impurity) – this is currently trading at 6.4 USD/lb, which in proper units is 14.1 USD/Kg.

In V_{2}O_{5} only 56% is actually V by weight so it costs 14.1 USD to get 0.56 Kg of V.

Assuming that the other components of the electrolyte are negligible in comparison to Vanadium it would therefore cost 25 USD to get 1Kg of Vanadium, which allows you to store 331 W.h. To store 1 kW.h you would need 3.02 Kg of Vanadium which would cost 76 USD.

We are told that the Vanadium or the electrolyte comprises some 30% of the cost of a VRFB battery, if 30% = 76 USD then the overall cost would be **253 USD per kW.h**

Why the rest of the VRFB always has to cost two and half times the cost of the very large volumes of Vanadium that may be in the battery is not clear at present. One might expect this ratio to approach unity as the storage time of the battery capacity time gets very long, and VRFB per kW.h pricing to become Vanadium limited.

It would also not be unreasonable to expect a similar strong learning effect such as that termed Swanson’s Law in photovoltaics. VRFBs are only really just starting out on the commercial production path so have much price optimisation ahead of them, however they are already close to being very price-competitive with Lithium-Ion batteries once the longer life of VRFBs is taken into account, rather than disingenuously ignored.

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