Download Report - Academy of Motion Picture Arts and Sciences

Since there is a steady diminution in the efficiency
o{ the Tungsten lamp, it follows that there is some
point in its life at which the use of any lamp should
te ended, regardless of its actual life. In order to
determine the truth of this theory and to find the
length of the most economical time of use, two
*"iitod. of computation are used; one by graphics;
the other, by ordinary analytical algebraic calculation.
The graphical I shall roughly sketch and then
the anal]'tiial equation for the time of use, which
is most ..otto*icul. The lamp is assumed to absorb
3,000 watts and deliver 66,000 lum-ens,. new' Figure
( 1) shows the curves' Curve No' 1 represents
cost of lamp, per hour, and the ordinate for any
value of time on the X axis, is equal to the original
cost of the lamp, in cents, divided by the time for
each value of X. For instance, if the lamp cost is
1,500 cents ($15) and the time o{ use 300 hours,
the cost, per'hour, is 5 cents. For a use o{ 500
hours, tfie cost per hour of use is 3 cents' In this
way the ordinates for curve No' 1 are computed and
the curve plotted.
Curves 2 and 3 are, respectively' the costs of
current {or the lamp. Curve 2 is lor current at 1'45
cents per K. W. hour, and curve 3 is {or the rate
of 2 cents per K. W. hour. Obviously, these are
straight lines at the elevations 4.35 and and 6 respectfvely,
the lamp being assumed as a 3 K' W'
lamp, so that the cost' per lamp hour for-,current at
1.4j cents is 4.35 cents per hour, while at 2'00
cents per K. W. hour the cost is 6 cents per hour'
The siraight line L--L, extending diagonally across
the sheet, represents the lumens delivered by the
lamp after any time of use as indicated by the
ubscissae. The diminution in the lumens is assumed
to be 40 per cent for 1000 hours of use. The
curves C, and C" represents the total cost of operation
of the lamp, per hour, at the current rates of
4.35 cents and 6 cents per lamp hour, respectively'
These curves are plotted from ordinates obtained
by adding to the ordinates of curve No. 1, the values
4,35, or 6 as the case may be. Curve C1 represents
the total cost per hour for a lamp receiving current
at 4.35 cents per hour, while curve C, shows the
cost per hour for operating a lamp receiving current
at 6 cents per hour' the cost of the lamp, per
lamp hour, being included, as before indicated. The
final curves, 4 and 5 show, respectively, the cost
per lumen in cents per hour and the ordinates are
Iound by dividing the total cost, in cents per hour,
by the number of lumens given by the lamp at any
plriod in its time of use. By plotting these final
values it is seen that the curves, 4 and 5 droop
downward for a certain time, then turn back upwards,
showing a definite minimum value, which
is marked on the figure'
Of course, the results obtained graphically can
be computed analytically.
If K be the cost of the lamP in cents.
If E be the cost of electricity per K' W. hour.
If A be the energy consumption in K. W.
ts0l
If L be the lumens delivered by the lamp when
new.
If Q be the factor of reduction in lumens with
use, or reduction of efrciency per hour of usage.
If X be the time of use, or number of hours of
use at which the point of maximum economy is
attained:
Then QX will be the reduction in the output
of iumens at the end oi the period o{ use, X.
The lamp cost, distributed over the total time of
K
use, X, is -
gsn15 per hour.
X
The current cost is EA cents per hour.
Lumens output after X hours will be L-LQX
which is equal to L(l-QX).
K
EA+X
Then the total cost per lumen i5 ---
gsn65.
L( 1_QX)
EAX+K
L( l-QX)
.....-..( 1 )
Differentiating, and equating to zero to determine
the time period, X, which gives the minimum cost
per lumen.
OV
- - LEAQX' + 2KLQX - KL :O...-.-.(2)
dx
(2KLQ) KL
Whence, X'?+X-
which re-
(LEAQ) LEAQ
duces to,
dy2KK
dx EA EAQ
*t/ffir'K
Whence,X - I V - -Fl -- -
-(4)
EAQ \EA7 EA
Obviously, only the positive value before the radical
can be used, because there is no negative value of
elapsed time.
By substituting thc proper numerical values the
economic length of time for use of the lamp can be
computed.
Note that the value of X is independent of the
actual quantity of lumens which the lamp delivers,
the term "L" having cancelled out in the derivation
of the formula.
Example :
K, or lamp cost, $15, or 1,500 cents.
E, the current cost, 2 cents per K. W. hour.
A, the energy consumption, 3 K. W.
Q, the reduction in efficiency with time of use
taken at 40 per cent for 1,000 hours, and
therefore 0.4 of the total illumination within
that time. This is equal to a diminution
of 0.0004 per cent of the lumens new per
hour,