I know I am 9 years late, but Seeing that your problem is the replacement of the carbon rods, why not instead using tungsten rods and a carbon enriched water (as sewage) to create the gas? Tungsten erodes much slower. 

this doesn't make any sense to me... you will still be making CO2 when you burn this fuel.

We currently use natural gas produced by fracking, which contains carbon from prehistoric decomposed vegetation. So that re-introduces prehistoric carbon into the atmosphere. This process produces natural gas without fracking, and the carbon rods could potentially be produced from carbon extracted from seasonal vegetation, so the carbon is actually sourced from the atmosphere. 

This method has been used in sewage treatment, and the resulting hydrocarbon derives it's carbon from the sewage while the UV sterilizes the treated water. 

So carbon responsibility depends on where the carbon comes from

I don't think I'd consider it renewable if you are using graphite arc-welding rods, as that is also prehistoric carbon.  I doubt there is an energy efficient way to produce carbon in arc-weldable form from vegetation.  Even if you could do that, whenever you deal with methane, some will inevitably leak into the atmosphere, and it is itself a potent greenhouse gas.  Finally, the CO2 produced from burning methane doesn't get recycled out of the atmosphere by plants in a trivially short time - that's a pretty complicated analysis and I don't know if there is a consensus on that yet.  If you already have electricity to use for arc welding, why not use that in an electric motor instead of burning a fuel in an inefficient combustion engine?

I was thinking of using it for heat actually, & the batteries needed to store enough to even out the ups and downs of solar or wind aren't exactly environmentally friendly either.  They require energy to produce, need to be disposed of eventually, contain chemicals that will also leak , etc.  Mostly this is an energy STORAGE solution, that would be usable with your currently installed natural gas appliances.  I spend more on heat than driving (especially this year)

Anyway, I am presently more focused on my silicon based life form in another project, so unless I have some epiphany to solve the automated rod feeding this will probably not go too far. I put this up here to see if anyone had  idea for that. I suppose I could take the time to asses the impact against other solutions once I have  working prototype, but the parts are scattered around the garage at the moment. 

Actually, I do have a question... Complete newb to the concept...

How do you *capture* the gas produced?

First just a 2 liter bottle with the top cut off, lit the proceeds with a match.

Then a 5 gal plastic water bottle with the bottom cut off, had a vacuum pump connected to the top, with the output to a refrigerant tank. I made a "trap" to catch water so I wouldn't suck water thru the vacuum pump, with some wires connected to a circuit that interrupted the pump automatically.  I was able to get a sustained flame from the pressurized refrigerant tank. ( I evacuated it first with the pump)

A better idea that I have is to use this corrugated plastic plenum tube, 2' in diameter & 8' long when extended , but 1" collapsed. I would need to seal the end with some plexiglass & connect the pump there, & the plexiglass top would be guided by 4 upright tubes so it rises vertically when filled with gas. (square plexiglass & holes in the corners for the guides) Then I could use some limit switch at the bottom for safety, & the pump would kick on at the top with another. Without doing the math I would guess some 20 cu' of gas as opposed to the 5gal that was a pain. 

I would then connect to a natural gas tankless boiler with enough gas for testing.  I have most of the parts laying around, I am however more focused on my other project. The holdup on this one is the underwater arc welding & self feeding mechanism.  I need to get an autodarkening welding mask before anything. 

I hope I explained this clearly..

I think I get it. Does the pump's intake go near the welding process (underwater)?

The welding would take place in like a 40 gal tote, filled with water. The corrugated plenum tube has one end minimally submerged, with the top capped off with plexiglass ( or something). the gas is generated below the corrugated tube, which would expand to it's full length of 10' as it filled with the gas.  The corrugated tube's travel would be guided, and maybe even assisted with counterweights, so that its weight would not force gas out from under itself. The hose to the vacuum pump would connect most likely at the top, but now that you have questioned me, I suppose the top could just be sealed and a hose could be inserted from underneath. This would avoid allowing for the movement of the expandable tube with all the plumbing. 

I have worked on this intermittently for 5-6 years, I kind of threw it up here since it's a green topic., and maybe more eyes on it will prompt some potentially useful suggestions.

I googled a bit, but didn't find anything exactly like that. I thought of forcing carbon powder out of a diminishing orifice with an auger, so that powdered carbon is all that is fed into the device. The 2 orifice's would nearly meet, and the compression of the powder would create a rod. The continuous feed would match the "use" and provide a continuous arc. 

Is there some device that would take the slurry and provide a continuous plasma arc? That's what I was hoping to find googling.

Thanks for asking! The reason this is up here is I'm kind of stuck at this part, and all feedback is welcome.  I made this clockwork contraption that I could kind of manually control the feed, but it was Gilligans Island quality (I machined the parts from a plastic cutting board), and I really need a self darkening helmet, or a camera that would allow monitoring.

Have you thought about using a carbon-water slurry in a plasma-torch type of arrangement?

I'm not convinced that this could be a green or efficient method, but it interests me as a possible simple, robust, inexpensive way to generate lots of methane from a very cheap and easily-transported fuel.

Coal-water slurry is a good fuel in diesel engines, which makes it interesting for peaking generators.

Germany has got a lot of coal and surplus electricity problems with their unpredictable solar and wind output, and Europe has an uncertain natural gas supply from Russia, so a cheap way to make natural gas from surplus electricity and coal could be valuable there.