The small town of Groin, Mo., has an excellent source of water. The water board says its wells and distribution system are coasting along at about half the maximum capacity. Since demand is growing at less than 2 percent per year, it would be reasonable to assume the current system is more than adequate for many years.
The city council, however, has concerns. Groin is increasing in popularity with many families who are tired of the crowding in St. Louis, which is about 30 miles from Groin.. The council's survey data indicate that Groin's population could increase by 5 percent per year.
If we assume that water usage in the future will be directly proportional to the number of residents, how long will Groin's existing system last? But, what if our assumption of water usage is incorrect? After all, as population grows, more than just individual consumption must be satisfied. With a greater number of residents, there will be more swimming pools and hot tubs and irrigation will increase dramatically. If actual demand were to increase by 7 percent annually, how long would the current system last?
Certain types of change, such as population growth, can be difficult to fully comprehend. On the other hand, changes that occur in some straightforward proportion are much easier to grasp. For example, if I said that gas prices increased from $2 to $4 per gallon during 2008, you would immediately recognize that they doubled. This is not necessarily the case when change is continuous and steady during the long term. We refer to change that occurs at a steady rate as exponential. The affinity laws, for instance, reveal that the head developed by a centrifugal pump varies as the square of a change in speed, while horsepower varies as the cube. We think we understand the effect of these exponents, but do our brains really comprehend the magnitude? The following exercise will introduce their often unexpected influence.
Take a plain sheet of 8.5 x 11 in paper and fold it in half from top to bottom. Fold it in half again from side to side. Continue folding it in this fashion until you have completed 10 folds. Do this now before reading further.
Now, if you somehow knew that this exercise was an impossible task, you may already have a good understanding of the exponential function. If, however, you forged ahead on faith alone, you should definitely continue reading. What you probably noticed as you folded the paper was that it went pretty smoothly for the first four or five iterations. The sixth fold was more difficult and the seventh was virtually impossible. In attempting to fold the paper back on itself, you were witnessing the exponential function in action.
Each time you folded it, the number of layers doubled, so the overall thickness doubled. After one fold there were two layers of paper (21), after two folds there were four (22), after three folds-eight layers (23) and so on. Had you been able to fold it nine times (29) there would be 512 layers-about the thickness of a ream of copier paper (2 in). That tenth fold (210) would have produced the equivalent of two reams! All in all, your single sheet would result in a pile almost 4 in thick! If you could continue this process for another 15 folds (a total of 25, 225) the result would be a stack a little more than two miles high! If you could complete 50 folds (250), a 70 million mile-high monster would appear before you!
Do not be concerned if you attempted to complete this exercise. When confronted with such an apparently simple task, most of us will do the same thing. Let it be a lesson-numbers can fool us, especially when they are presented in a way that is not intuitively obvious. An important component of exponential change is something called doubling (or halving) time. Doubling time is the time it takes for something, growing at a steady rate, to double in size. The reason it is so important is because doubling (or halving) is a much easier concept for us to comprehend whereas the exponent itself may not be. The following simple equation allows us to calculate doubling time based on some steady rate of growth.
Doubling Time = 70 / Growth Rate Percent
Here is an example everyone can appreciate. The doubling time equation predicts that an investment returning 10 percent annually will double in value every seven years. This, of course, is known as compound interest and represents interest earned on both the principle and interest.
In our example, the town of Groin was experiencing an increase in demand of less than 2 percent per year and its system was operating around 50 percent capacity. If this continued, the existing system would be good for about 35 years, but the city council forecasts a population growth rate of 5 percent each year. At first glance, this does not seem to be an unusually large increase (just 3 percent), but based on the exponential function and the doubling time equation above, the population (and water usage) will double in just 14 years. An increase of just 3 percent reduces the system life by 60 percent. Suppose our estimate was wrong and the actual increase in usage is 7 percent per year. This would reduce the remaining life of the existing system to just 10 years!
It becomes easy to see the importance of doubling time. It takes some "fuzzy" numbers and puts them in a readily comprehensible perspective. Planning and building for growth is an ongoing process. In the case of Groin, 10 years is not a long time, especially when one considers the services that must be scaled up to meet the needs of a growing population. Doubling time is a tool that can help portray exponential change in a more understandable format.
Pumps and Systems, August 2009