Page 258 - Plant design and economics for chemical engineers

228 PLANT DESIGN AND ECONOMICS FOR CHEMICAL ENGIN ERS
i
periods in n years is mn. With m annuity payments per year, let R represent
the total of all ordinary annuity payments occurring regularly and uniformly
throughout the year so that R/m is the uniform annuity payment at the end of
each period. Under these conditions, Eq. (21) becomes
s = i? [ 1 + ( ,/m)](m’rXm) - 1
- (22)
m r/m
For the case of continuous cash flow and interest compounding, m
approaches infinity, and Eq. (22), by use of Eq. (10, becomes?

(23)

Present Worth of an Annuity

The present worth of un annuity is defined as the principal which would have to
be invested at the present time at compound interest rate i to yield a total
amount at the end of the annuity term equal to the amount of the annuity. Let
P represent the present worth of an ordinary annuity. Combining Eq. (5) with
Eq. (21) gives, for the case of discrete interest compounding,
p = R (1 + 9” - 1 (3
i(1 + i)”
The expression [Cl + i)” - l]/[i(l + i)“] is referred to as the discrete
uniform-series present-worth factor or the series present-worth factor, while the
reciprocal [i(l + i)“]/[(l + i)” - l] is often called the capital-recovery factor.
For the case of continuous cash flow and interest compounding, combina-
tion of Eqs. (12) and (23) gives the following equation which is analogous to Eq.
(24):
m - 1
p,RL (25)
rem
Example 5 Application of annuities in determining amount of depreciation with
discrete interest compounding. A piece of equipment has an initial installed value
of $12,000. It is estimated that its useful life period will be 10 years and its scrap

tThe same result is obtained from calculus by noting that the definition of E is such that the
differential change in S with n is equal to 8, which is the constant gradient during the year, plus
the contribution due to interest, or dS/dn = R + rS. This expression can be integrated as follows to
give Eq. (23):

R + rs em - 1
In - =rn o r S=R - P)
R ( r 1