Maximum C-rate Calculation

[Johan]

Calculation model description
This thread focuses on the calculation of an approximate estimation of the maximum C-rate for a given battery and/or the cell temperature (increase) at full (dis)charge for a given C-rate.

It is conservatively assumed that heat cannot escape from the battery. Hence, all generated heat will contribute to a rise in the battery temperature. The calculation model only applies to a single full charge (or discharge), after which it is assumed that the battery is allowed to reach thermal equilibrium with the ambient temperature. Attached is an Excel-sheet (see also screenshot below) with the appropriate assumptions, in- and output fields, and equation derivations. The approach is extremely simple so you can easily check it yourself. You can enter your own battery specs.

Example
Given a CALB 3.2Vnom 180Ah LFP cell of 5.6kg with a 0.6mΩ internal resistance, 20°C ambient start temperature and 55°C maximum internal temperature, the maximum theoretical C-rate would be roughly 1.6 until the maximum internal temperature is reached. At a C-rate of 1, approximately 3.4% of the battery capacity would be lost as heat, roughly matching an earlier design statement that @electric made here.

Discussion
Note that factors other than the cell internal temperature may limit the maximum C-rate as well, but these are not identified nor discussed here. Hot spots and consecutive continuous cycling (etc) are neglected, suggesting an over-estimation of the max C-rate. On the other hand, in practice, heat can always escape because a perfect adiabatic condition is impossible. In this regard, the calculation model predicts an under-estimation of the max C-rate. These systematic errors may partially cancel each other out . Large uncertainties apply to the battery internal resistance, i.e. a factor 2 increase would reduce the modeled max C-rate by a factor 2. Note that cycle life degradation as a function of internal temperature is not addressed. Experimental validation of this model is lacking from my side. However, the model does suggest that (dis)charging at "fractional C-rates" (0 to 1C) would not lead to heating problems as @electric already suggested (again) here, provided that the ambient temperature is low enough. I am curious what improvements y'all propose for this calculation model.

I will try to process comments from below into the sheet and update it in this message.

Edit: When the cell specsheet only mentions an "impedance", then you could simply multiply that impedance by a factor 2 (or 4) to obtain a rough conservative estimation of the DC internal resistance for most (?)(not all) cells. I loosely base this statement on the following graph: http://liionbms.com/php/wp_resistance_vs_impedance.php

Screenshot (example of older version)

 

[offgriddle]

Good to know, I'm building an insulated, closed box for my lifepo's stored in a root cellar to keep from dropping below freezing. The current battery bank and charging system is small but some day it will not be and heat produced in an insulated box might become a factor then. Thank you.

 

[electric]

@Johan my hat is off to you for a lesson in thermodynamics calculations, I will certainly bookmark this post for future reference.
However, your max temp of 85C is way too high. Datasheets state max 55C for discharge and 45C for charge. In real life even at 1C and room temp ambient you can reach 45C limit for charging.

 

[Johan]

@grizzzman, @electric: Thanks for the replies.
@electric: Yes the datasheets show the (lower) 45-55°C, yet the reason that I picked a higher value is that my guess was that those lower values correspond to the temperature probe values as measured on the outside top or terminal of the cell. The internal maximum allowed temperature may be (I did not verify by calculation etc) 15°C higher simultaneously [1], and the 85°C randomly came from [1]. To stay on the safe side, I agree that a lower value would be better. Changing to 45°C, leaving all other values unchanged, would yield a max C-rate of 2.2.

In addition, in the real life case you mention (interesting to know, thanks), charging at 1C from 20°C to 45°C would mean an internal resistance of a factor ~2 higher (compared to the value I used), and/or a smaller specific heat value, and/or other neglected effects that cause heating, etc, so that is somewhere in the same ballpark.

References
[1] https://www.ev-power.eu/blog/LiFePO4/Operating-temperature-of-LFP-Cells.html

 

[electric]

@Johan I do not believe internal cell temp can ever be 15°C higher than surface temp. Maybe on those old plastic cells, but most newer cells nowadays come in aluminum casings. Internal heat spreads quickly into the casing on modern cells.
Internal resistance factor is a tricky one, it changes over the charge/discharge cycle, most prominent effect is at the end of charge, when temp rise is fastest. Also resistance of bus bars and terminal connections adds to thermal losses, but difficult to calculate as there is no good standard for bus bars materials and cross section.

 

[Johan]

@electric Interesting.

This made me wonder what exactly is inside an LFP cell, so an example of a prismatic LFP cell is shown in [1]: It seems to show a stack of copper sheets and aluminium sheets, wrapped between a zig-zagging plastic sheet. The thermal conductivity is roughly a factor 100 higher in the plane of the sheets compared to perpendicularly to that plane according to [2]: highly "anisotropic". Heat transfer time-scale analysis shows that sudden temperature differences across the stack plane vanish in the order of minutes (calculated in a separate tab in the Excel sheet), so indeed 15°C seems exaggerated and a more homogeneous temperature distribution seems more plausible. I will replace the default max temperature of 85°C by the more conservative 45°C.

Another correction: The mass specific heat ranges from 650 to 950 J/kg/K according to [2]. This is significantly lower (conservative) than my 1114 J/kg/K. Combined with the lower max temperature, this results in a theoretically lower allowable C-rate (1.6 instead of 6). I will use a value of 800 J/kg/K. I will update the sheet version with the date of today.

[1] Inside a Lithium Ion Electric Car Battery Cut Open by EV West
[2] https://www.politesi.polimi.it/bitstream/10589/501/1/Muratori_thesis.pdf

 

[Johan]

When the cell specsheet only mentions an "impedance", then you could simply multiply that impedance by a factor 2 to obtain a rough conservative estimation of the DC internal resistance for most (?)(not all) cells. I loosely base this statement on the following graph: http://liionbms.com/php/wp_resistance_vs_impedance.php

(this text was also added to the thread start post)