In my “Physics of Energy and the Environment” class [1], we try to construct estimates of how much energy is required by various aspects of modern civilization. Transportation is the main focus — it’s important, energetically costly, and the physics underlying it can be grasped by non-science-major undergrads in a few weeks [2]. Teaching this course this past Winter, I wanted to estimate the energetic cost of food. This, it turns out, is much more challenging. Here, I’ll write about the energy requirements of meat, and how this leads to a fun puzzle that combines estimates of energy with biophysical scaling relationships about the metabolism and lifespan of animals. Sadly, since it would be impossible (and bizarre) to combine my energy class and my general-education biophysics class into one super-course, I’ll never be able to assign this puzzle to students!

### Energy and meat

All aspects of food, from its growth to its processing to its transportation, take energy. Can we estimate the *minimal* amount of energy required, bounded by constraints of biology or physics? It’s well known that in general, meat requires more energy than plant-based foods, so let’s consider the raising of animals. (Being vegetarian, I find this somewhat weird to write about, and I think my writing sounds a bit ghoulish.) How much energy does meat need? Can we make sense of the empirical fact that beef requires considerably more power than pork, which requires more than chicken?

The only place I’ve seen this estimated is in David MacKay’s excellent *Sustainable Energy – without the hot air* (p. 77-78). The idea is to ask how much animal needs to be kept alive to supply some quantity of meat. Altering MacKay’s treatment a bit: Let’s consider a pig, as the “average” animal, and let’s use some rough numbers. (I’ll be more general in the next section.) The average American eats half a pound of meat per day. The average farm pig weighs about 200 pounds, and has a lifespan of about 200 days. If half of the pig is “edible,” this means one pig provides meat for one human through the course of its life — a nice one-to-one mapping. (As I told my class, you can imagine a pig that follows you around, and you take a 1-lb. bite every day of its life.) The mass of a pig is similar to the mass of a human, so we suppose that the body of a pig has roughly the same power requirements as the body of a human: about 100 Watts. (Or in more familiar units, about 2000 calories/day.) Just as the average American uses a lot more power than his or her 100 W body requires– for transportation, lights, manufacturing, etc. — we expect that keeping a pig alive in a modern industrial farm takes some multiple of 100W. For humans in the U.S.A. that factor is about 100. We’d expect the factor for pigs to be less, since their lives are less luxurious than ours; let’s say 5.

So, we expect eating meat to require, very roughly, around 5 x 100 = 500 W per person, or to contribute about 1/20 of the total 10,000 W U.S. per capita power consumption. This is equivalent to about 100 gallons of fuel per year, or about one 10,000 mile plane flight. (See my earlier post on travel.)

We made a rough guess of 5 for this “industrial factor.” From what I’ve read, the actual number is very variable; it can be ~2x or ~10x depending on farming methods. And, of course, it could be zero. As MacKay notes, a sheep left to forage on a rocky pastureland is an “automated self-replicating biofuel-harvesting machine” that has hardly any energy costs associated with its maintenance. However, most meat isn’t raised like this, but rather comes from animals that are fed foods farmed by energy-intensive agriculture, and that are housed in energy-intensive spaces.

Of course, plant-based foods also require energy to produce, and in some cases (especially for transportation), these costs can be substantial. This is hard to estimate, though. It is claimed that ) a vegetarian diet has about 1/2 the carbon footprint of a non-vegetarian one, but I haven’t looked at this carefully.

### A biophysical puzzle!

Conveniently, our hypothetical pig’s lifespan matched the amount of time it took for its body to be completely one person’s meat consumption. What happens for animals of different sizes and lifespans? (Before reading on, you might like to think about what happens to the above estimate if the pig lives longer.)

Let’s call the rate of meat consumption *c*, and fold the fraction of the animal that’s eaten into this. In other words, for the typical American, *c* = 1 lb./day, accounting for the 0.5 lb./day of meat and the fact that only half the animal is edible. We can think of *c* as the amount of flesh required per day.

Let’s call the average rate of flesh the animal can *supply* per day *s*. This is related to the animal’s mass (*M*) and lifespan (*T*) by *s = M /T*. For the pig, this was *s = 1 lb./day* — at the end of its 200 day lifespan, it gives 200 lbs.

The number of animals per person that need to be maintained is therefore *N = c / s*. For the pig, *N=1.*

Each animal’s body requires some amount of power input *P_0 = pM*, where *M* is its mass, and *p* is the power per unit mass of this organism’s metabolism (i.e. how many Watts per kg of flesh). The total power required is *P_0 = bpM*, where *b* is the factor that represents the “industrial” power needs of the animal’s housing, transport, etc.

The total amount of power needed is therefore *P = N P_0*, which, putting everything together, is *P = (c T / M) b p M*, or ** P = c T b p**.

Already, we see a few interesting things:

- There’s no explicit dependence on the animal’s mass (
*M*); it canceled out. - The longer the animal’s lifespan (
*T*), the greater the power requirement. Note that cows live longer than pigs which live longer than chickens.

But: Let’s think a bit more about mass. Bigger animals live longer. (Elephants live for decades; mice don’t.) In fact, there’s an empirical observation that lifespan scales as the one-quarter power of mass for a wide range of animals (*T ~ M^(1/4)*). Bigger animals also use less power per unit body mass, with a metabolic rate that scales inversely with the one-quarter power of mass. (Mice, for example, have very fast hearbeats.) Therefore, *p ~ M^(-1/4)*.

Combining these dependences on Mass:

*P ~ M^(1/4) x M^(-1/4), *so* *

*P ~ M^0.*

Te two factors cancel out, therefore the requisite power should be *independent* of the mass of the animal that’s raised for meat, and so should be the same for a chicken as for a cow. This is very wrong, severely contradicting the reality of energy costs of different livestock animals. (For example, beef has a much greater energy cost than chicken.)

**The puzzle:** Figure out why! (I.e. what’s wrong with the above reasoning.) I’ll put the answer in the comments in a day or two! *Hint:* this doesn’t require any special biophysical knowledge, or any knowledge about energy.

** Update:** the answer is posted; see the comments section!

### Today’s illustration…

A warthog, based on a photo in this book. It didn’t turn out too well, but as my wife points out: warthogs are ugly.

### Notes

[1] A few related posts here, here, and here; the syllabus.

[2] Raghuveer Parthasarathy, “Cars and Kinetic Energy – Some Simple Physics with Real-World Relevance,” The Physics Teacher 50: 395-397 (2012). [Link]

Here’s the answer to the puzzle: It’s the *natural* lifespan of animals that scales as M^(1/4), not the actual lifespan of farm animals, which is far shorter. The lifespan of beef cattle is about 2 years; the natural lifespan is about 20 years. “Fattening” Pigs: about 6-7 months versus 20 years. “Broiler” chickens: 40 days versus 8-15 years. A chicken is about 300x less massive than a cow, but its lifespan on a farm isn’t 300^(1/4) = 4.2x shorter, but rather about 20x shorter. (Lifespan numbers: http://www.four-paws.us/campaigns/farm-animals-/farm-animal-life-expectancy/)