Welcome to ES.
First off, a 72V, 40Ah pack is going to be big and heavy. If the range you're after is around 15 miles then you probably don't need such a big pack, 10Ah would probably be OK.
You need to start off by estimating the power that the bike will draw on the route you want to take. You've already settled on a 5000 series motor (which is quite heavy) an powerful controller and a high pack voltage (which implies you're after some speed) and you said that you have to deal with hills, so I'm going to guess that you might want to plan on using around 500 to 600 watts average power at an average speed of around 25mph. This works out to between 20 and 24 watt hours per mile. The calculation to get these numbers is:
Average speed = 25mph
Distance = 15 miles
Average journey time = 15miles/25mph = 0.6hr
If average power = 500 watts, then Wh per journey = 500 x 0.6 = 300 watt hours
If average power = 600 watts, then Wh per journey = 600 x 0.6 = 360 watt hours
Watt hours per mile is between 300 / 15 = 20Wh/mile and 360 / 15 = 24Wh/mile
Next you need to translate these power requirements into the battery capacity you need. The battery needs to be around 20 to 30% greater in capacity than the basic numbers indicate, because you don't want to have to discharge it too deeply in normal use. This extends its life. Increasing the figures above to give 30% spare amp hour capacity gives:
Amp hour capacity required = 1.3 x 300 = 390 watt hours and 1.3 x 360 = 468 watt hours
So you need a battery pack that's somewhere between 390 and 468 watt hours to get the range you're looking for, with a small reserve, at an average power consumption of around 500 to 600 watts. You've already settled on 72V as the supply voltage, so the battery pack amp hour capacity needed is:
Ah capacity required is between 390 / 72 = 5.41 Ah and 468 / 72 = 6.5Ah, both pretty low figures. The smallest practical pack is around 10Ah, which will give you a bit of extra reserve capacity and will mean that the cells will have an easier time delivering the up to 40 amps you need for the controller peak power. A 72V 10Ah pack has a capacity of 720 watt hours, so in theory you could draw 720 watts (around 6 times more than an average cyclist can manage) for an hour from it.
Next you need to look at the cells you need to make up the pack. The rule here is that connecting cells in series increases the voltage, connecting cells in parallel increases the capacity in Ah. For around 10 Ah you need three of the 3.3Ah cells you've linked to, connected in parallel. This will give you 3.3 x 3 = 9.9Ah. These cells are 3.2V, so to get around 72 volts you need to connect 72 / 3.2 = 22.5 cells in series. In practice you would either connect 22 or 23 cells in series. 24 cells in series would give you 24 x 3.2 = 76.8V nominal, but hot off charge these cells will be at about 3.6V per cell, so the voltage could rise to 24 x 3.6 = 86.4V or a little more, which may be more than the 72V controller can take (best check first).
If you decide to use a pack with 24 cells in series, then you need to buy 24 x 3 = 72 cells and connect them as 24 groups of 3 paralleled cells. This will give you a pack with a nominal capacity of about 76.8V x 9.9Ah = 760 watt hours, comfortably in excess of your needs and around 1/4 of the pack size you were looking at.
Next, you need to check the battery will safely deliver the current you need. This is given by the cell C rate, which is a multiple of the cell capacity, given in amps. For example, a 10Ah 2C battery could deliver 10 x 2 = 20 amps. The cells you've linked to have a maximum C rate of 2C continuous (they state 6.6A, 6.6 / 3.3 = 2) and 5C maximum (they state 16.5A, 16.5 / 3.3 = 5), so a 9.9Ah pack could deliver 9.9 x 2 = 19.8A continuous and 9.9 x 5 = 49.5A maximum. This equates to a power rating for this battery of 19.8A x 76.8V = 1520 watts continuous and 3801 watts maximum, which is just about OK for your needs. Most of the time you will be running the battery at a discharge rate of below 1C, which is a reasonable operating point.
To connect the batteries, controller etc up you need fairly heavy gauge wire, to avoid power losses in the wiring. I'd use 12g, or maybe even 10g wire, on a set up like this.
There are many other battery choices you could make. For example you could look at using RC LiPo packs for a smaller, lighter battery with the same performance. Hopefully the above might help you work out just what you need.
Jeremy
never mind like i said dumb q?
