# Estimating energy density

A couple of years ago I made an Excel spreadsheet to estimate what the gravimetric energy density (or, more accurately, *specific energy*) of a scaled-up lithium-sulfur (Li-S) cell would be based on various parameters and performance indicators from experimental tests. At the time, I was working in a project with a stated goal of producing a 400 Wh/kg cell at the end of the project (an ambitious target, and although the project was generally very successful and productive we were in the end quite a way off this target).

I’ve recently translated that spreadsheet into a Shiny app, and now it’s available here. I thought it would be interesting to pair this app with a page describing the various parameters, where the default values I’ve chosen come from, and what values seem realistic given current literature. I hope this app will be quite instructive, and maybe give some insight as to why it’s so hard to make such a high energy battery.

Open the app here

The app is fairly simple, and should look like the screenshot below. Just change the values however you like and see what happens.

Energy density is calculated very simply:

$$\text{energy density} = \frac{Q_d \cdot E_{\text{mean}}}{\sum m_x}$$

where $$Q_d$$ is the “surface capacity” (or, areal discharge capacity) of the electrode in mAh/cm2, $$E_\text{mean}$$ is the average discharge voltage, and $$\sum m_x$$ is the sum of all the masses of all of the components, in g/cm2. The product of Q and E gives an energy, and divided by the sum of all of the masses gives an energy density for the whole cell stack.

The different parameters that can be changed are as follows:

• Sulfur utilisation (mAh/g) – more recognisably, the discharge capacity expressed per unit mass of sulfur. The default is 900, which is roughly what we achieve in the typical test batteries we’ve been creating for the last 18 months. Theoretical is 1672. Literature values are all over the place, typically between 600 and 1000, but values of ~1200 and sometimes higher have been reported, but more rarely.
• Mean discharge voltage (V) - default is 2.13, which is also typical for our cells. Theoretical, according to the Gibbs free energy of formation, is 2.24.
• Sulfur loading (mg/cm2) - The amount of sulfur on the positive electrode. Default is 3, which is typical for our current research and a reasonable minimum value for a practical system. Literature values are most commonly between 0.5 and 2, but values >4 are relatively rarely reported.
• S fraction (%) - The fraction of the positive electrode composite which is elemental sulfur. The remaining fraction is typically conductive additives and binders. Default is 65, which is what we use. Literature values are most frequently between 40 and 60, with up to 80 being reported.
• Electrolyte/sulfur ratio (µL/mg) - The ratio of the electrolyte volume to the mass of sulfur in the electrode. Default is 5, which is the lowest integer value at which our cells cycle for more than a few cycles (we’ve used 6 in most of our recent publications). Literature values are most often not reported, but have varied between 6 and 100. A couple of recent papers have reported the “optimum” as being between 10 and 20. I’ve not seen values lower than 6 in any published papers so far, with the exception of a 2010 paper from Sion Power [here] – working backwards from a figure in this paper describing 2.8 Ah cells, one can estimate an E/S ratio of approx. 2.4 corresponded to batteries with a cycle life of ~50 cycles.
• Electrolyte density (g/cm3) - self explanatory. Default is 1.09, which is what I previously estimated the density of our standard electrolyte to be (1 M LiTFSI, 0.25 M LiNO3, 1:1 DME:DOL – a more accurate number may be out there somewhere).
• Separator mass (mg/cm2) - default is 0.894, which is the mass of the porous PE separators we have used in recent years.
• Li thickness (µm) - Thickness of the Li negative electrode. Default is 50, which is reasonable for a practical battery. We use both 125 and 30 µm Li foils. Literature values are rare. Li foil is sold in a wide range of thicknesses. The most commonly used foils in academic labs are probably in the range 100 – 1000 µm.
• Al thickness (µm) - Thickness of the Al (positive) current collector. Default is 14, reasonable for typical Li-ion batteries.
• Cu thickness (µm) – Thickness of the Cu (negative) current collector. Default is 9, reasonable for typical Li-ion batteries. It has been suggested that if the excess of the Li negative electrode is high enough, it could be its own current collector, and Cu would therefore not be needed; the density of Li metal is a fraction of that of Cu, and is (apparently) cheaper per unit area.

Try changing the parameters within the ranges I’ve suggested above to start with. There are plenty of things to think about: How high will the energy density be with values in these ranges? How much of the cell is active mass (i.e., Li and sulfur?) What’s the biggest contribution to the mass? What more do we need to improve to get up to, say, 400 or 500 Wh/kg?

Note! Cell packaging is not (currently) included in these calculations! The energy density of the cell will of course depend on the mass of the packaging. This partly depends on the format (size, type etc) of the cell. I would guess a reasonable estimate for the cell packaging would be about 10% of the total cell mass, but this is just an educated guess.

Also note: The excess on the negative electrode is also given in the app – it’s important! If it is negative, it means the capacity on the lithium electrode is limiting, and the energy density of the cell will be reduced. This is factored into the calculations.

Worthwhile literature on the topic: Hagen et al (2015), Urbonaite et al (2015), Pope et al (2015), Mikhaylik et al (2010), Berg et al (2015).

I also wrote a separate post providing some context to this page. You can find that here.