Using some simple mathematics and dimensions we can do some prismatic cell electrode estimation. The real challenge was:

by giving following inputs (length, width & thickness of cathode, anode & separator, fixed mandrel length and dimensions of the prismatic cell case), can we get the thickness of the jelly roll? Is there any direct formula for that?

The basic inside dimensions of the prismatic case and the jelly roll shown fitting those dimensions with a gap above and below.

The tolerance of the jelly roll versus the dimensions of the case are an unknown for production at scale and we have here assumed a direct fit.

The above schematic is just one of many prismatic cell layouts and we will need to cover the other orientations and the cases where there is more than one jelly roll.

The simplest form of the mandrel is assumed to be two posts with a diameter of d_{mandrel} that are centre a distance apart of l_{mandrel}.

The resultant winding is then squashed in thickness to fit the inside dimension of the case.

In this process the length of the overall jelly roll increases.

For now let us layout the simple estimation using the PHEV2 size of prismatic cell.

Outer Width | 148 | mm |

Outer Height | 91 | mm |

Outer Thickness | 26 | mm |

Outer Volume | 0.350 | litre |

Case Material Thickness | 0.5 | mm |

Inner Width, L_{in} | 147 | mm |

Inner Height, H_{in} | 90 | mm |

Inner Thickness, W_{in} | 25 | mm |

Inner Volume | 0.331 | litre |

Basic dimensions of the case and calculated volumes are quite simple.

This post has been built based on the support and sponsorship from: **Thermo Fisher Scientific**, **Eatron Technologies**, **About:Energy** and **Quarto Technical Services**.

Top Jelly Roll to case, h_{top} | 2 | mm |

Bottom Jelly Roll to case, h_{bot} | 2 | mm |

Next we need to calculate the thickness of the active material. Firstly we need the dimensions of the active layers.

Cathode Active Thickness | 71 | µm |

Cathode Electrode Thickness | 12 | µm |

Separator Thickness | 12 | µm |

Anode Active Thickness | 88 | µm |

Anode Electrode Thickness | 15 | µm |

The stack is anode, separator, cathode and a further separator:

Stack Thickness, *d _{stack}* = 381 µm

The number of winds, *n _{wind}* =

*W*/ 2

_{in}*d*

_{stack}*n _{wind}* = 25 / (2 x 0.381) = 32.8

The easiest way to calculate the length of the electrodes is to break the problem down geometrically. I consider the straight sections and the curved end sections separately.

The distance between the mandrel centres, *l _{mandrel}* =

*L*

_{in}– W_{in }The length of electrode in these two straight sections = 2x *n _{stack}* x

*l*

_{mandrel}= 2 x 32.8 x (147 – 25) = 8,003 mm

Now we make a big assumption that the compressed form of the jelly roll has circular end forms, with a diameter of *W _{in}*. In this assumption

*d*= 0mm which again means this is compressed.

The length of electrode in the curved ends can be calculated using the cylindrical electrode formula for a spiral. Both ends of the prismatic cell when put together form an equivalent to a cylindrical equivalent.

= 3.141 x 25^{2} / (4 x 0.381) = 1,288 mm

The total electrode length is the summation of the straight and curved sections

= 8,003 + 1,288 = 9,291 mm = 9.291 m

This calculation makes a lot of assumptions and hence is a very rough estimate. Also, I currently think that the minimum radius is handled incorrectly and should be project out from the mandrel to the corners. However, that means we know the thickness. An iterative approach is required.

In the literature and patents we see that the mandrels are not the simple two posts form and take more of an elliptical shape.

However, based on these being patented there is little detail as to the actual shape. The these shapes will improve the tension and reduce the stress on the active layers when the jelly roll is compressed.

#### Cylindrical Cell Electrode Estimation

Knowing the outer and inner diameter of the spiral along with it’s thickness we can calculate the length of the material to create it.

D is the inner diameter of the cylindrical can.

The inner diameter is that of the mandrel around which we wind the spiral.