Forgotten Fundamentals of the Energy Crisis - Part 5

by Prof. Al Bartlett

V. Length of life of a finite resource when the rate of consumption is growing exponentially

Physicists would tend to agree that the world's mineral resources are finite. The extent of the resources is only incompletely known, although knowledge about the extent of the remaining resources is growing very rapidly. The consumption of resources is generally growing exponentially, and we would like to have an idea of how long resources will last. Let us plot a graph of the rate of consumption r(t) of a resource (in units such as tons / yr) as a function of time measured in years. The area under the curve in the interval between times t = 0 (the present, where the rate of consumption is r0 ) and t = T will be a measure of the total consumption C in tons of the resource in the time interval. We can find the time Te at which the total consumption C is equal to the size R of the resource and this time will be an estimate of the expiration time of the resource.

Imagine that the rate of consumption of a resource grows at a constant rate until the last of the resource is consumed, whereupon the rate of consumption falls abruptly to zero. It is appropriate to examine this model because this constant exponential growth is an accurate reflection of the goals and aspirations of our economic system. Unending growth of our rates of production and consumption and of our Gross National Product is the central theme of our economy and it is regarded as disastrous when actual rates of growth fall below the planned rates. Thus it is relevant to calculate the life expectancy of a resource under conditions of constant rates of growth. Under these conditions the period of time necessary to consume the known reserves of a resource may be called the exponential expiration time (EET) of the resource. The EET is a function of the known size R of the resource, of the current rate of use r0 of the resource, and of the fractional growth per unit time k of the rate of consumption of the resource. The expression for the EET is derived in the Appendix where it appears as Eq. (6). This equation is known to scholars who deal in resource problems5 but there is little evidence that it is known or understood by the political, industrial, business, or labor leaders who deal in energy resources, who speak and write on the energy crisis and who take pains to emphasize how essential it is to our society to have continued uninterrupted growth in all parts of our economy. The equation for the EET has been called the best-kept scientific secret of the century.6

Reprinted with permission from Bartlett, A., American Journal of Physics, 46(9), 876, 1978. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Association of Physics Teachers. Copyright 1978, the American Association of Physics Teachers.
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