Tuesday, August 03, 2010

Can You Spot the Math Error?

Trawling around the interwebs, I encountered the following critique of vertical farm concepts at TreeHugger:
This led us to wonder, "What would be the consequences of a vertical-farming effort large enough to allow us to remove from the landscape, say, the United States' 53 million acres of wheat?"...Our calculations, based on the efficiency of converting sunlight to plant matter, show that just to meet a year's U.S. wheat production with vertical farming would, for lighting alone, require eight times as much electricity as all U.S. utilities generate in an entire year [see calculations here].
This evoked a raised eyebrow from me, given the widespread claim that American agriculture uses vastly more fossil fuel energy than it delivers in terms of total food calories. Something doesn't add up here--clearly, someone is very, very wrong about the amount of energy embodied in our foodstuffs.

Fortunately, the included "calculations" reveal the source of the problem:
The following is a very rough estimate of the amount of power needed just for lighting vertical farms to grow the U.S. wheat crop. Note this is under ideal conditions for nutrients, temperature, and other productivity factors. Under excellent conditions, wheat has radiation use efficiency of 2.8 grams of biomass produced per 106 joule of photosynthetically active radiation (PAR). So to produce one metric ton (106 g) of wheat biomass requires [106 g / (2.8 g/106 J)] = 3.6 × 1011 joules of PAR over a season under ideal conditions.
Oops. >BANGS HEAD ON DESK< (3.6 x 1011 joules is the amount of energy in 10,526 liters of gasoline. That's a lot of energy for just one metric ton of wheat... )

To be fair I've seen equivalent mathematical goofs from the proponents of these fanciful vertical farm concepts. A lot of people aren't dotting their Is and crossing their Ts, so to speak. I actually like the concept, but it needs ample cheap energy to work--basically, it would have to be coupled with nuclear reactors. I have no problem with that myself. But I don't think it will be a near-term development in any case.

9 comments:

Joffan said...

I'm assuming those "106" values scattered around are actually one million, "10^6", and my browser is just crap. I do see some superscripts though.

1kWh is 3 600 000 J. So this is a good working figure for the noon solar energy reaching the ground per hour per sq m. Assuming a growing season of five months, we might get 700 hours of sunlight (@20% "CF") for 2.6 x 10^9 J. Random internet referencing turns up 600g/sqm as a grain yield, so a tonne of grain requires 1600 sqm, and 4.3 x 10^12 J of sunlight. Total biomass is maybe 4 times grain weight -> 10^12 J/t biomass.

Close enough for jazz.

There's a lot of joules in sunlight, and only a small fraction of its energy gets captured into energy in the foodstuff.

Joffan said...

subscribing...

Sovietologist said...

Yes, googling around I found that RUE is "close to 3.0 g/MJ PARA when roots are included (Fischer, 1983)." That would be around 3.3x10^11 J/t. I thought the "106" was, well, 106, given that's what their own linked calculations say. It makes their figures bewildering, to say the least...

There are a number of things you could do to get around this--among others, grow something with a better RUE. But the main application I can see for these sorts of farms/oversize greenhouses is growing out-of-season fruits anyway--a premium-cost field with a VERY long history.

GRLCowan said...

I also figured a million rather than twice 53, and when that is done, I don't think there's any significant math error. Powering any kind of staple food crop with any kind of electric lamp is just silly. Doing it on every floor of a tower block, and having to buy more power to pump out the ~99 percent of the lamps' output that the plants convert to waste heat, would be foolishly silly and dumb.

Typical biodiesel yields are 2700 litres of oil per hectare-year. They are obtained in near-equatorial places where the annual average insolation is on the order of 2.75 MW/hectare, and 2700 L/(ha y) turns out to be, um, not such that one can say it ain't hay, except in the most literal sense. At 0.915 kg/L and 35 MJ/kg, it's 2740 watts per hectare, and the sunlight-to-oil efficiency is 0.1 percent.

That ain't not hay. Figuratively speaking.

Sovietologist said...

I wonder if it might be possible to engineer a unicellular organism with the goal of converting process heat to usable food energy with maximum efficiency. There are a wide variety of known bacteria with exotic metabolic processes--those that live in the deep ocean, inside of rocks, etc. If you had to create a way of producing large quantities of food in a closed environment without benefit of sunlight (on a deep-space flight, or in a bomb shelter after an atomic war, for example) such technology might prove quite useful... if not necessarily particularly appetizing.

GRLCowan said...

... engineer a unicellular organism with the goal of converting process heat to usable food energy with maximum efficiency. There are a wide variety of known bacteria with exotic metabolic processes ...

Yes, but they're always boringly thermodynamically orthodox. If they're going to be like that, how will they turn process heat into anything? If it's heat that is inside them?

For converting process heat, you want organisms that confine it in lifeless matter, within which it is much more concentrated than living tissue can bear, at scale such that it doesn't all leak away too quickly.

We are those organisms.

(How fire can be domesticated)

Sovietologist said...

I don't think I was quite specific enough. One doesn't violate thermodynamics. I was imagining that one could use process heat to synthesize a convenient chemical feedstock that would the basis of the organisms' metabolism. Then perhaps one could engineer a whole suite of organisms that could convert this biological feedstock into a variety of culinary building blocks, with abiological steps wherever convenient. Basically a whole artificial ecology for artificial environments deprived of sunlight, engineered with the express purpose of providing humans with food.

Ray Van De Walker said...

Sovietologist, the research has already been done. Try this: Pyrolyze organic garbage, sewage sludge or other biomass with steam to get synthesis gas, a mixture of CO and H2. Syngas is run through a water-gas shift to adjust the carbon/hydrogen ratio, and reacted with a copper catalyst to form methanol, C3H3OH. The methanol, ammonia and trace gases can be separated by fractionation. Ammonia is reacted with refined salts or metals to make fertilizer for the last step. The methanol can be reacted at 400C with a gamma alumina catalyst to form dimethyl ether, (CH3)2O. DME can be reacted with an acidic alumina-silicate zeolite (such as ZSM-5) to get ethylene. Ethylene can be reacted with an acid catalyst, usually Phosphoric acid, to get edible ethyl alcohol. Refined edible ethyl alcohol can be fed directly to Candida Utilis (torula yeast), with water, ammonia salts and mineral salts. Since the feed stock, ethyl alcohol, is also edible, produced from an immisicible gas already present in traces in natural food (ethylene), using a trace-edible catalyst (Phosphoric Acid) the yeast is food grade. Torula grown this way is more than 50% protein, and can be eaten directly in soups, tofu or bread, or spun into something very like meat, with a strong umami flavor. It can also be chemically processed to extract sugars and fats. In 2010, Ohlev, a Minnesota food-additive processor, operated a 30-ton/day plant that grew Torula on ethanol.

Dawn said...

I'm listening!!! (And following.)