The whole mobile phone idea is just a really bad joke. Let's say something like an iPhone 6 uses about 10 Wh of power to charge it (0.01 kWh):
http://www.telegraph.co.uk/technology/apple/iphone/11128043/How-much-power-is-used-to-charge-an-iPhone-6.html.
If we assume that 21 can make a 50 GH/s ASIC that runs at 0.1 J/GH (way way better than anything available right now) then that 10 Wh of power will last less than 2 hours while the ASIC is active (much less in fact because the Wi-Fi or LTE modems and either the main CPU or an NPU have to be active too to connect to anything).
Given that the single biggest complaint almost every phone user has about their phone is battery life does anyone really thing that people are going to be happy with phones that need charging every 90 minutes? Let's say they throttle back to a duty-cycle that's only 25% (6 hours when the phone is charging overnight) then the phone has a whopping 12.5 GH/s of hashing; that's 1 28 millionth of the current hash rate so to supply 50% of the network hashrate will require 28M mobile phones (at the current hash rate).
Of course this is a zero-sum problem so those 28M phone users will between them mine half of the 25 BTC per block (going down to 12.5 BTC next year). That's 44.6 satoshis each per block (call it 45 with the tx fee). If the indications about reward sharing are correct then 21.co get 75%, or 33.75 of those 45 satoshis so our phone owner has 11.25 satoshis per block or 1620 satoshis per day. After a year they've got 591k satoshis, so that's 5.91 mBTC ($1.37 at current prices).
At an average of 12.5 GH/s and 0.1 J/GH then they used 24 * 12.5 * 0.1 = 30 Wh of power per day, or 10.95 kWh per year (assuming a perfect charger; not that they exist - the numbers will be higher). If the video's quoted price of $0.12 per kWh is correct (retail electricity pricing) then mining that $1.37 costs at best $1.31 but also costs the battery on the phone draining, the phone getting hotter, etc. The heat thing is not to be dismissed because phone enclosures aren't designed to dump sustained heat this way.
Basically they'll have to persuade 28M phone owners that they should spend their electricity mining for 21.co, getting a worse phone experience and having to pay more for the phone in the process (to cover the extra ASIC and the necessary heat dissipation). Now if they persuade 56M phone owners then 21.co's mining operation is huge; it now nets them 2/3 of all of the bitcoins but our phone users now only get 3.94 mBTC ($0.91) but their costs stay the same.
When the block reward halves then the numbers get much worse from the consumer's perspective. Now they have the same costs but only half of their previous reward.
Oh and here's the irony: the claim is that this is to enable "decentralisation" right? Well guess what, our 28M phone miners are delivering all of the blocks to 21.co's mining pool; that's 50% of the network. Can we please see the necessary freak out along the lines of when GHash.io didn't actually quite get to 50% of the network last year? Our 56M phone miners would give 21.co 66% of the network.
Remind me again how all of this is a good idea?
