I've always thought about making ice. What is the energy required to make ice? It's not such an easy question. The problem is that going from liquid water to solid ice, you DECREASE the total energy in the system. But look at the reverse question. How much energy does it take to melt ice? That's a much easier question. Taking solid ice to liquid water (at the same temperature) requires that you increase the energy of the system. Actually, this is exactly what I calculated in my estimation of the amount of ice you need to cool down your beer (or other preferred beverage).

So, how could you estimate the energy needed to make ice? The answer: look at ice-making machines. If I find some ice makers online, I can look up both the rate that they produce ice as well as their power consumption. From this, I can get a real-world estimate for the cost (both energy and money) to make ice.

I thought the best place to find this information would be on Amazon. Sadly, I was wrong. There are indeed ice makers listed but many of them do not also list the power requirements. Alas, after much google-fu I found the data that I wanted. There are two quantities for each ice maker that I need: the power and the rate that ice is created. For the ice rate (which I will call R), most sites listed this as the amount of ice that can be created in 1 day. I have converted this to units of kg/s.

Ok, here is the data. I have included both portable ice makers and commercial ice makers.

This looks linear enough that I can get a relationship between the rate of ice production and the power required. First, let's look at the y-intercept of this function. If I am making zero kilograms of ice per second, my ice maker would still draw 88 watts of power. I like that. Really, most machines still use electricity when they are not doing anything useful. Ok, but what about the slope? If I want to produce 1 kilogram of ice per second, it would require 4.55 x 10^5 ^Watts. Yes, that is a large power - but remember that is a whole kilogram of ice in just one second.

Just to be clear, let me write out the expression for ice maker power as a function of ice production rate.

But what about the cost to make ice? Suppose I want to make a quantity of m (in kg) of ice. If I look at the value in front of the ice rate in the linear fit above, I can re-write it as:

I can change the units since 1 Watt is equal to 1 Joule per second. But now this slope makes a bit more sense. This says that to make 1 kilogram of ice, it would require 4.55 x 105 Joules of energy. But how much does electric energy cost? Although different people have different electricity rates, I am going to go with a price of 10 cents per kilowatt hour. If you do some simple conversions, this is 2.78 x 10-8 dollars per Joule.

Now I can write an expression for the cost (C) as a function of mass of ice.

This says that to make 1 kilogram of ice, you would need to spend 1.26 cents for the electricity. That seems a bit lower than I would have guessed, but I'm going with that value. Oh, just let me point out that I ignored the 88 Watts to just have the machine on. That's probably a small enough factor to ignore.

There are some other things that might matter such as the starting temperature of the water. Also, I didn't include the cost of the water itself. Still, this seems like an ok estimate.

How Much Does the Ice Bucket Challenge Cost? ——————————————–

This will be my last mention of the Ice Bucket Challenge (IBC) - hopefully. But can I estimate the electricity cost that people have expended on this challenge? Well, of course I can estimate it - but it might be a terrible estimation. Clearly the limitation of accuracy has never stopped me before and it obviously won't stop me now.

We can start with some guesses:

How many people have completed the IBC? I'm going to guess that this is mostly a USA thing and that about 10% of the population have dumped ice water over their heads. With a USA population around 300 million, that puts the number of heads at about 30 million and 30 million buckets of ice.

How much ice is in a bucket? This is tough. Some people will go skimpy on the ice and some will load up on ice. I'm going to just go with 1 kg of ice per bucket.

This means that so far 30 million kilograms of ice have been used for the IBC (by my estimate). So, how much did it cost to make this ice? Since it is 1.26 cents per kilogram, this would be $378,000 in electricity bills. Ok, maybe I over estimated some things. I still feel pretty comfortable that it is over $200,000 in electricity to do the IBC. Does this mean it's silly? No, that really isn't that much money when we are talking about the WHOLE USA population.

Just for comparison, what if you estimated how much money is spent on electricity for ice used in drinks in the USA? Just as a rough estimate, I will guess that on average everyone (in the USA) uses 200 grams of ice per day. If I look at the total ice used over 2 months, that would be about 3.6 billion kilograms of ice. Over this two month period that would cost about 45 million dollars. So, in comparison the cost of ice for the IBC is just a drop in the bucket. Get it? Drop in the bucket?

Homepage image: jar ()/Flickr