Page 270 - Plant design and economics for chemical engineers

238 PLANT DESIGN AND ECONOMICS FOR CHEMICAL ENGINEERS

period involved, the total amount put in each year is $l/T, and the factor,
based on Eqs. (23) and (12), is
1 erT - 1
-e-m = j7
Factor = r r c (33)

For example, if the time period involved is the second five years (i.e., the
6th through the 10th years) and r represents 20 percent, the appropriate
factor, as shown in Table 3, is

Factor = i( e’“‘2~~~- ‘)(A) = $( 2*71tg:- I)& = 0.232

Cd) Discount factors to give present worths for cash flows declining to zero at a
constant rate over a period of years starting with the reference point. For this
case, the assumption is made that the continuous cash flow declines linearly
with time from the initial flow at time zero to zero flow at time r+ A
situation similar to this exists when the sum-of-the years-digits method is
used for calculating depreciation in that depreciation allowances decline
linearly with time from a set value in the first year to zero at the end of the
life.t$ For the case of continuous cash flow declining to zero at a constant
rate over a time period of n,, the linear equation for R is

R=a-gn (34)
where g = the constant declining rate or the gradient
R = instantaneous value of the cash flow
a = a constant
The discount factor is based on a total amount of one dollar of cash flow
over the time period nT and converts this total of one dollar to the present
worth at time zero. Under these conditions, g equals 2/(nT)*,§ and the

tSee Chap. 9 (Depreciation) for information on the sum-of-the-years-digits method for calculating
depreciation.
SEquation (35) does not represent a true sum-of-the-years-digits factor. Normally, the constant
declining rate or gradient for the sum-of-the-years-digits method. of depreciation is l/C?Tn =
2/n,(n, + 1). For the true case of continuous cash flow declining to zero at a constant rate, nr is
replaced by nrm as m -+ m, and the constant gradient becomes 2/(nr)‘.
IBv _ definition of terms and conditions. R is zero when n = nr and R is n when n = 0. Also, if a
-
total of one dollar is the cash flow during nr
pn = anT - !y = $10
0
Because E is zero when n = nr, a = gn,. Therefore,
gw* dnT)* 2
$1.0 = g(Q)2 - 2 = - and g = (“r)i
2