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On The Importance of Calculations
Adam Kenvarg
September 17, 2012 - 5:24pm

By Adam Kenvarg, Autodesk Sustainability Education Fellow

When doing design work, some people (engineers included) gloss over initial, basic calculations. Some see it as needless, others are intimidated. With such powerful software tools, why do we even need to do initial calculations?  The main reason is ensuring that you can actually build what you propose.

Escher Waterfall

Figure 1: Just because you can design something doesn't necessarily mean you can actually build it

 

I have often come across designs or concepts that hover between tremendously impractical and patently impossible.  These designs are unfortunately not quick drawings on the back of a napkin, though; they are often long projects, either as part of a competition, school project, or research project.  A great deal of time, effort, and in some cases embarrassment, could have been saved with some basic back-of-the-envelope calculations.

Pause and Ask: “Hey, Wait a Second”

Whenever you see a number that seems either too big or too small or too good to be true, you should pause.  These are the places where basic calculations, and revisiting your assumptions and estimations, can be a great help.  

For instance, in one funded research project for Engineers for a Sustainable World, a group of students were tasked with developing a solar energy storage system for developing countries.  They settled on storing solar energy as potential energy by raising a large amount of water to a height and letting it drain down over time, generating electricity.  This concept of pumped-storage hydropower is a very real and reasonable one.  However, the scale needed to make it a viable solution to our power needs is truly immense.  The students determined that they would need to raise 40,000 gallons of water to a height of 40 feet.  This is where you can quickly use estimation to check if this design can reasonably be built.  

Quick Calculation

Figure 2: A quick calculation of the weight of the water required

 

40,000 gallons of water would weigh approximately 160,000 kilograms (the actual number is 151,416 kilograms).  To give a sense of scale, I looked up how much an elephant weighs.  According to Wikipedia, the largest elephant ever recorded weighed 10,900 kilograms.  This means that for this storage system to work you would need to raise the weight of about 15 elephants to a height of 40 feet. 

This is obviously a lot of weight, and it would cost a lot of money to build a structure strong enough to support that, not to mention just buying a tank that large.  I also wasn’t sure how big a tank that would be, so I looked up how big a tanker truck is.  The largest in the US are about 9000 gallons.  This means that you would need a tank as big as 4.5 very large tanker trucks.  Here, this easy estimation shows that perhaps a better solution would involve using some cheap and easy to find car batteries to store the solar energy, instead of an expensive and complex water storage system.  And, taking a deeper look at the user needs might help too.  Perhaps people don’t even need electricity 24 hours a day in developing nations, but only enough to charge portable devices, such as cell phones, during the day.  There are now many more ways to solve this problem, and they are more likely to be a path to success.

Elephant Tanker Truck

Figure 3: How much water is feasible to store in an elevated tank in the developing world? The equivalent of 15 elephants seems like a lot of weight, and 4.5 tanker trucks seems like a very large tank.

 

A different team, who was analyzing the feasibility of using solar panels to help power cargo ships, did a good job of basic estimation that yielded useful results early in the design process.  Their analysis found that the solar panels required to power a cargo ship would greatly outsize the ship requiring them, making the plan impossible using current or near future technology.

Ship and Panels

Figure 4: The area of required solar panel arrays, with the ship in the bottom left for scale

 

The blue area in this picture represents the area of solar panels required given 24 hours a day of clear skies and full sunlight (which is clearly impossible).  The red represents the required solar panels given 6 hours a day of the same conditions.  Even if their numbers are somewhat off we can still reach the same conclusions:  From these simple representations we can see that we will need to look for other ways to power ocean-going freighters.

The Impossible Light

Gravia Lamp Render

Figure 5: A pretty design; but will it work?

 

Another example of forgetting to do some basic calculations is the Gravia Lamp, the runner-up in the Core77 / Greener Gadgets Design Competition in 2008.  The Gravia was designed to slowly lower a 50 pound weight over 4 hours, powering a light that could illuminate a room.  While this sounds like a great idea, the basic physics were impossible.  

Many people did analyses of what was wrong with the design, but this one is perhaps the best (please note that it does contain some potentially offensive language).  A simple calculation of the potential energy stored by lifting a weight (mass * the acceleration of gravity * height) shows that the even using a theoretically perfect light with no losses in a 100% efficient system would only produce a tiny fraction of what the design promised.  This means that no matter how efficient LEDs get no light source could ever make this design work.  The laws of physics simply prevent it, regardless of technological improvement.

 

Update 12/21/2012:

Another "gravity powered" lamp has been making the rounds lately, this time designed for the developing world.  The design has already raised over $200,000 on a crowdfunding site, and is on pace to finish with significantly more than that.  It is claimed that by simply letting a bag full of dirt fall it will create enough energy to cast useful light for up to 30 minutes.  Doing some basic calculations shows that this is a very difficult claim to believe (but we are ready to be proven wrong!).

Assuming the bag weighs 10 kg (this is actually higher than the claim of 9 kg) lifted to a height of 2 m (the height a standard person can reach) we can calculate that the potential energy stored is:

mass * gravitational constant * height = 10 kg * 9.8 m/s2 * 2 m = 196 Joules

Let's round to 200 Joules to make our calculations easier.  To find the potential work we can get out of this system we simply divide energy by the time that it will be extracted over.  In this case it is 200 Joules over 30 minutes (or, in SI units, 1800 seconds).  From this we get:

Work = 200 Joules / 1800 Seconds = 1/9 Watts

If we use the claimed weight of 9 kg we instead get almost exactly one tenth of a Watt (of course, this is assuming 100% efficiency).  Knowing that even small LED flashlights use 1 Watt bulbs should give us pause, but let's keep going with the calculations.  Of course, we care about the actual amount of light cast, so we should look at the luminous efficiency of LEDs.  To make the calculations easier we will be very generous and use an efficiency of 100 lumens / watt (which is higher than any readily available bulb).  Using the above estimate of one tenth of a watt we get a luminous flux of:

Luminous Flux = 1/10 Watt * 100 Lumens/Watt = 10 Lumens

But how much is 10 lumens, really?  Well, it depends on the size of the area you are trying to illuminate.  For this, we use the unit of a Lux.  One lux is equal to one lumen per square meter.  So if the lamp was illuminating an area restricted to one square meter we would get an illuminance of 10 lux.  Using a table of standard required illuminance levels we find that reading and writing high quaility books requires at least 500 lux.  The 10 lux supplied by this lamp is therefore far short of even enough lighting for basic tasks.

While gravity lights might sound like a good idea, by doing some basic calculations we can find that they simply don't store enough energy to be useful in real-world scenarios.

Finally, let's look at a comment from the creators themselves: "With hand-cranked devices, it might require three minutes of turning a handle for half-an-hour's return," says Reeves. "With this amount of effort required from the consumer, it's often not seen as a particularly attractive trade-off. The GravityLight just needs three seconds of lifting for 30 minutes' return." It is important to remember that, for equally efficient systems, there's no shortcut for the amount of energy needed as input. If you input the energy faster, it just requires more power (shorter, more intense effort).  If it takes 1/60th of the time, you will need 60 times the power.

Further Update 1/15/2013:

According to an article on Ars Technica today the creators of the Gravity Light claim that "the gearing took the kinetic energy from a weighted bag descending at a rate of a millimeter per second."  Given their 18 minute higher power setting we can quickly calculate that the bag will only decend 1.08 meters over that time.  This gives an initial height of approximately half of our above calculations.  Using these numbers, we can again calculate a theoretical maximum output of 1/10th of a Watt for their highest power setting.

(10 kg * 9.8 m/s2 * 1.08 m) Joules / 1080 seconds = .1 Watts

On the lower setting, we can expect to get a theoretical maximum of .06 Watts.  Just a little more information from the creators.

If It Won’t Work, It Won’t Get Built

Skipping basic background estimation and calculation may save you a small amount of time and headache early in the design process, but it can cost you greatly in the long run.  Using good estimation techniques and basic calculations can ensure that your work ends up being useful, rather than something doomed to sit on paper or in a computer forever.

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