## 833 Payback Time

The payback time Np is defined as the ratio of extra capital cost DCcap over first year savings:

F first year savings

The inverse of Np is sometimes called return on investment. If one neglects discounting, one can say that after Np years, the investment has paid for itself and any revenue thereafter is pure gain. The shorter the Np, the higher the profitability. As selection criterion, the payback time is simple, intuitive, and obviously wrong because it neglects some of the relevant variables. There has been no lack of attempts to correct for that by constructing variants such as a discounted payback time (in contrast to which Eq. 8.44 is sometimes called simple payback time), but the resulting expressions become so complicated that one might as well work directly with life cycle savings or internal rate of return.

The simplicity of the simple payback time is, however, irresistible. When investments are comparable to each other in terms of duration and function, the payback time can give an approximate ranking that is sometimes clear enough to discard certain alternatives right from the start, thus avoiding the effort of detailed evaluation.

To justify the use of the payback time, let us recall Eq. 8.42 for the internal rate of return and note that it can be written in the form

The rate of return is uniquely determined by the payback time Np and the system life N. This equation implies a simple graphical solution for finding the rate of return if one plots (P/A,rr,N) on the x-axis vs. rr on the y-axis as in Figure 8.5. Given N and Np, one simply looks for the intersection of the line x = Np (i.e., the vertical line through x = Np) with the curve labeled N; the ordinate (y-axis) of the intersection is the rate of return rr. This graphical method can be generalized to the case where the annual savings change at a constant rate re.

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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