Page 829 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

784 Appendix C: Miscellaneous Relations

P abs =P atm +P gage
Pressure of point A
P gage
P abs =P atm +P gage
Atmospheric pressure Atmospheric pressure
P gage
P abs
P atm Pressure of point A P atm

P abs
(a) Datum (b) Datum
FIGURE C.1 Graphical and algebraic pressure relations for conversions between absolute pressure and gage pressure. Pressure at level A is
the focus. (a) Positive gage pressure. (b) Negative gage pressure (vacuum).

C.3.3 NATURAL LOGARITHMS
TABLE C.1
In converting between common log and natural log values, the Amount after k Periods of Compounding
rationale must be reconstructed (unless this task is done
P n r k A
often). This section provides a review of the basics and can
also be used more expediently to glean only the immediate 1.00 1 1.00 1 2.00000000
needs of a conversion. 10 2.59374246
100 2.70481383
C.3.3.1 Compound Interest 1,000 2.71692393
We may look at say the way money is compounded as an 10,000 2.71814593
introduction as to how some things happen in nature, say 100,000 2.71826824
growth of a bacterial culture. Consider the money problem 1,000,000 2.71828047
first. The rate at which a sum of money grows depends on the
rate of interest, r. Suppose r ¼ 0.06=year (i.e., 6% per annum)
and the principal, P ¼ $1.00. The amount, A, accumulated at C.3.3.3 The Number e
k
the end of 1 year is $1.06, i.e., A ¼ P(1 þ r). After 2 years, i.e., The limit approached by the quantity, (1 þ 1=k) is denoted
by e:
n ¼ 2, A(2 year) ¼ A(1 year) (1 þ r) ¼ [P(1 þ r)] (1 þ r) ¼
2
P(1 þ r) . After n years
k
1
A(n) ¼ P(1 þ r) n (C:6) e ¼ L 1 þ (C:8)
k!1 k
Now suppose the interest is compounded every half year. ¼ 2:71828 (C:9)
For this case, r(0.5 year) ¼ 0.03, or r=2, and the formula is,
2n
A(n) ¼ P(þr=2) . To generalize, if we let the value be com- and
pounded k times per year, then
log e ¼ 0:43429426 (C:10)
kn
1 þ r
A(n) ¼ P (C:7)
k Therefore, $1 with 100% interest, compounded continuously
for 1 year would result in e dollars. The limiting value of A in
in which (C.6) is
A(n) is the compounded amount at the end of n years
P is the principal amount at t ¼ 0 A ¼ Pe rn (C:11)
r is the annual rate
n is the number of years which is the amount of any principal, P, after n years with
k is the number of times interest is compounded each year interest compounded continuously at any rate, r.
C.3.3.2 Continuous Growth C.3.3.4 Kinetics
Now suppose we compound the interest very often, or continu- For any physical quantity that increases or decreases at a rate
ously. The rate of growth is then proportional to the amount at proportional to the concentration, the expression is
any instant. To illustrate, let, P ¼ 1.00, n ¼ 1, r ¼ 1.00, and let
k ¼ 1,10, 100,1000. Table C.1 shows the amounts, A, for each k. dA ¼ rA (C:12)
The limiting value of A is seen to be 2.718. dt