in which n is the period of usefulness, or life of the plant, expressed in years, no allowance being made for scrap value.
A comparison of the annual expense of three different plants is shown in Table 32. It is evident from this comparison that the machinery with the least first cost is not always the least expensive when all items are considered.
A sinking fund is a sum of money to which additions are made annually for the purpose of renewing a plant at the expiration of its period of usefulness. The annual payment into the sinking fund is equivalent to the term Rr in the expression for annual cost, or in terms of C, r, and n, the annual payment is
Cr
(1 + r)n − 1.
It is the same as the capitalized depreciation multiplied by the rate of interest. The expression r
(1 + r)n − 1 is sometimes called the rate of depreciation.
The present worth of a machine is the difference between its first cost and the present value of the sinking fund. If m represents the present age of a plant in years, then the present worth is
P = C(1 – (1 + r)n − 1
(1 + r)m − 1).
| TABLE 32 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Comparison of Costs of Three Different Pumping Stations. Nominal Capacity Thirty Million Gallons per Day Raised Thirty Feet | ||||||||||||
| Equipment | Plant A | Plant B | Plant C | |||||||||
| One Acre of Land. Brick Building, Steel Trussed Roof, Slate Covered. Cross Compound Condensing Horizontal Pumping Engine | One Acre of Land. Brick Building. Steel Trussed Roof, Slate Covered. Compound Condensing Low Duty Horizontal Pumping Engine | One Acre of Land. Frame Building, Shingle Roof. Compound Duplex Non-Condensing Pumping Engine. | ||||||||||
| Annual Payment on First Cost | Years of Usefulness | Sinking Fund Payment | Total | Annual Payment on First Cost | Years of Usefulness | Sinking Fund Payment | Total | Annual Payment on First Cost | Years of Usefulness | Sinking Fund Payment | Total | |
| Land | 100 | 0 | 100 | 100 | 0 | 100 | 100 | 0 | 100 | |||
| Permanent Structures[[49]] | 1188 | 50 | 1080 | 2,260 | 1180 | 50 | 1080 | 2,260 | 810 | 50 | 775 | 1,585 |
| Pumps and Machinery | 440 | 15 | 435 | 875 | 390 | 15 | 395 | 785 | 360 | 15 | 352 | 712 |
| Boilers | 280 | 10 | 446 | 726 | 252 | 10 | 400 | 652 | 308 | 10 | 490 | 798 |
| Labor | 14,000 | 14,000 | 14,000 | |||||||||
| Fuel | 5,500 | 7,200 | 8,200 | |||||||||
| Repairs, etc. | 480 | 400 | 550 | |||||||||
| Total | 23,941 | 25,497 | 25,945 | |||||||||
Where straight-line depreciation is spoken of it is assumed that the worth of a machine depreciates an equal part of its first cost each year. For example, if the life of a plant is assumed to be 20 years, straight-line depreciation will assume that the plant loses 1
20 of its original value annually. The present worth of a plant under this assumption would be the product of its first cost and the ratio between its remaining life and its total life. This method of estimating depreciation and worth is frequently used, particularly for short-lived plants and for simplicity in bookkeeping, but it is less logical than the method given above.
89. Number and Capacity of Pumping Units.—In order to select the number and capacity of pumping units for the best economy, a comparison of the costs of different combinations of units should be made and the most economical combination determined by trial. The principles outlined in the preceding articles should be observed in making these comparisons. In a steam pumping station, when the number of units operating is less than the average daily maximum for the period, steam must nevertheless be kept on a sufficient number of boilers to operate the maximum number of pumps. This, and corresponding standby losses must not be overlooked, as they may show that a smaller number of larger units is ultimately more economical.