Built on Facts

Raising the Burj Khalifa

I’ve got to admit, this is a beautiful skyscraper:

i-f7791a1299e05cd3538aa115381ca460-burj.png

I was never very enamored of the Petronias Towers or the Taipei 101, both of which lacked the architectural panache that supertall structures ought to have. The Burj Khalifa has it. Burj Dubai is a more elegant name, but long story short Dubai really got nuked by the worldwide financial crisis and this Khalifa fellow helped keep Dubai out of bankruptcy. So now he has the world’s tallest building named after him. The Burj Khalifa cost around a billion and a half dollars to construct, and I expect that figure would have been enormously higher if it had been constructed in New York. Among other reasons, the developers of the Burj Dubai have access to what we might politely call cheap labor.

It’s a nice building anyway. Now, the whole point of a tall building is to take stuff that was at ground level and suspend it in the air by means of steel and concrete. As such you can park your desk and do paperwork in what used to be blank air a thousand feet above the ground. It takes energy to move that stuff up the earth’s gravity well, and energy costs money. How much of that $1,500,000,000 was spent in the actual meat of the project – the moving of the parts from the ground to their respective positions in the air?

The energy required to move one floor’s worth of material will be approximately its mass times its height above the ground (call that h) times the acceleration due to gravity (g). The mass of one floor is the linear density (in kilograms per meter or whatever, call it lambda) times the height of the floor itself (say, 15 feet or whatever. Call it dh).

That gives you the energy dE needed to raise one floor up to a height h:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

Add that up with the help of calculus for all the floors with a height between 0 and H, the final height of the structure:

i-88516ca505d84f969b7ab15a965c3fb3-2.png

Evaluate the integral:

i-d18063683dcc0d42b9be45451a84d1e3-3.png

And density times height is just the total mass m:

i-8d7070ce28f0af28047c7a5fb8246d53-4.png

So the total energy is just the same as is required to move all the mass up half the height, which makes sense as an average. Now the Burj Khalifa is not of uniform density the whole way up, so the real figure for energy will be somewhat less. But it’s a good starting point for an estimate.

What’s the total mass of the building? Wikipedia says 330,000 cubic meters of concrete and 55,000 tons of steel were used, and that’s about 842 million kilograms. Plug in and get a total of 3.4 trillion joules. And that’s about 948,000 kWh. Here in Texas prices of ~$0.10 per kilowatt hour are doable, and so it would have cost about $95,000 to raise the materials.

Obviously most of the cost is not in the raw energy. More of it is the materials themselves, the labor required to move them, and the fact that no use of energy is completely efficient anyway. Similar figures would hold for just about any building anywhere in the world. Still, it’s interesting to see that the raw lifting energy is so small in comparison to total construction costs.

Comments

  1. #1 k
    January 7, 2010

    Where did you get the $0.10 kw/hr figure? I’m guessing that is for electricity rates in your area. Is that even comparable to, say, the diesel used to power a crane?

    Also: the density isn’t going to be uniform, is it? The bottom has to be stronger than the top, so more concrete, less air. I’m guessing, anyway. But, if anything, that will make your number smaller.

    -kevin

  2. #2 Oded
    January 8, 2010

    I remember once at school we calculated what the temperature of the Earth should be according to the sun and black body radiation (I remember getting a rather reasonable average result), and once we were done, we calculated how much additional heat is created by the energy expenditure of a city. It was rather amazing to find it was roughly 5 degrees C! Living in Israel, “temperature” usually means “hot”.. Most air conditioners, most of the time, work to cool homes. And in the summer, a serious majority of the energy expenditure of the city is the air conditioners themselves! And a 5 degrees difference could easily prevent you from needing the air conditioner… So that’s an interesting self-feeding loop…

    Anyway, the relation to this, I remember this vividly, we were explicitly asked to assume the entire energy expenditure turned purely to heat, AND to ignore the negligible effects of putting heavy things on tall buildings! So when I started reading your post, I was looking forward to how negligible it really is…You were interested in the money, but the energy itself is also negligible. So, it was indeed a good approximation!

  3. #3 Oded
    January 8, 2010

    Hrm, is “kW/h” a standard notation? Surely you mean “kWh”… I am not familiar with it being written as “kW/h”.. That looks very silly

  4. #4 Paulino
    January 8, 2010

    It’s a pity that part of those terajoules of work have been performed by quasi-slave labour…

    http://www.independent.co.uk/opinion/commentators/johann-hari/the-dark-side-of-dubai-1664368.html

  5. #5 r4 kort
    January 8, 2010

    Burj Khalifa isn’t nearly pyramid-shaped enough. If you’re going to build some pointless monument to excess in the desert, you should at least acknowledge your forefathers.

  6. #6 CCPhysicist
    January 9, 2010

    I like that they let someone base jump off of it.

  7. #7 CCPhysicist
    January 9, 2010

    I also love the “this is a nice blog” spam comments with ad links in the signatures that preceded my comment.

  8. #8 Brian
    January 10, 2010

    It’s comparable to the cost of gasoline. Gasoline = 1.3e8 joules/gallon, so about 27000 gallons of gasoline, and at $3/gallon, that’s $81000. It’s not comparable to the cost of an equivalent energy amount of corn syrup though ;) – 50000 gallons of corn syrup would be required (the volume of a smallish pool), which would be about 3 million dollars.

  9. #9 Matt Springer
    January 10, 2010

    Yep Paulino, I noted that in passing in the post. This is a beautiful building, but it was in some significant part built on the backs of involuntary labor. This can’t be ignored.

  10. #10 Tim Gaede
    January 10, 2010

    If the tower can be approximated by a uniformly dense pyramid (in which the mass of each floor is proportional to the square of the distance from the TOP), then the integration comes to mgH/4. So the energy requirement would be half that for a tower with equal mass floors.

  11. #11 Anonymous
    February 11, 2010

    ds

  12. #12 Mohammad Elhage
    April 11, 2010

    What is the function of the Burj Khalifa?

  13. #13 po
    May 4, 2010

    how much does the Burj Khalifa cost???????????

  14. #14 Anonymous
    June 15, 2010

    $1 500 000 000

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    August 24, 2010

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    September 26, 2010

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    March 11, 2011

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    May 12, 2011

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