# Incandescent Electric Lighting/Efficiency

The Maximum Efficiency
of
Incandescent Lamps.
by
John W. Howell.
page
The Maximum Efficiency
of
Incandescent Lamps.
By John W. Howell.
The word efficiency, when applied to an incandescent lamp, is used to designate the amount of energy required by the lamp for the production of a given amount of light; thus we say that a given lamp has an efficiency of three watts per candle, at sixteen candles, meaning, that to produce an illumination of sixteen candles we must supply the lamp with forty-eight watts.
The word efficiency, when applied to a prime mover or to any piece of apparatus that changes energy from one form to another, or which transmits or utilizes energy, has a well-defined meaning, and is used to represent the ratio of the energy of the useful effect produced by the ​apparatus, to the energy necessarily supplied to the apparatus to enable it to produce that effect.
An incandescent lamp transforms electrical energy into heat and light, so the use of the word efficiency to denote the watts per candle required by the lamp is not a proper one. To denote properly the efficiency of an incandescent lamp we must be able to separate the energy of the light produced by it from the energy of the heat produced. Then the ratio of the energy of the light to the electrical energy required by the lamp will be a correct expression for the efficiency of the lamp, and we will have to find some other word to designate the watts per candle. In this paper the word efficiency is used in its ordinary improper sense to denote the watts per candle required by a lamp when producing a given amount of light.
The efficiency of a lamp varies with its candle-power. The curve. Fig. 1, shows the rate of this variation for a particular lamp. At 5 candles this lamp has an efficiency of 6.7 watts per candle, at 10 ​candles it is 4.2 watts per candle, and at 20 candles it is 2.66 watts per candle.
Any statement regarding the efficiency of a lamp must therefore, be accompanied by a statement of the candle-power at which it has the stated efficiency; without this it is meaningless. There is nothing in an incandescent lamp itself that fixes its proper efficiency or in any way indicates what it is. The lamp from which the curve. Fig. 1, was determined has, within the limits of the curve, any efficiency between 2 and 7 watts per candle. Thus, by simply changing the candle-power of the lump, we can operate it at any efficiency we choose, and get as much or as little light per watt as we choose.
In commercial practice the candle power of lamps is always marked on them and their efficiency at this candle-power is stated; but even this is not a proper index to the value of the lamp or to its proper efficiency. Experience has shown that lamps are almost universally run above their normal rating; lamps rated ​at 4.5 watts per Candle are usually run at
Fig. 1.
from 3.5 to 4 watts per candle, and in order to make lamps that will stand the ​strain of being run above their rated capacity, it is necessary to rate them considerably below the efficiency at which they will give the best results under ordinary circumstances.
Lamps have been made and sold in England which have a very high rated efficiency, but parties buying these lamps are told that they will get very much more satisfactory results if they run the lamp below their rated capacity. So we see that some lamps are rated above their capacity and some are rated below, and the rated efficiency of a lamp is not always the best efficiency at which to run it. How then, are we to determine the efficiency at which a lamp will give the best results? It is this question which I will attempt to answer.
The term "maximum efficiency" of a lamp, as used in the title of this paper, does not mean the highest efficiency at which a lamp can be operated, but the efficiency at which the best results are produced by the lamp: or more ​accurately, the efficiency at which the cost of operating the lamp is a minimum.
Taken in this latter sense, the maximum efficiency of a lamp is not its highest efficiency. As we increase the candle-power of a lamp its efficiency increases; consequently, by running the lamp high enough we can make its efficiency so high, that very little power is required to produce a given amount of light, and the cost of power to produce the light is very small. But, while the efficiency of the lamp increases, its life decreases, and if we run a lamp at too high an efficiency the saving in the cost of power is more than balanced by the increased cost of lamp renewals.
To determine the maximum efficiency for lamps under given conditions, we must determine the efficiency at which the sum of the costs of power and lamps is a minimum, and in order to do this w^e must know the rate of variation of the life of a lamp with its efficiency.
The curve. Fig. 2, shows this rate of variation. This curve is the result of ​very carefully conducted experiments
Curve showing lives of equally good lamps,
burned at different efficiencies.
Fig. 2.
made by the Edison company. These ​experiments extended over five years and consumed a very large number of lamps. Its accuracy when applied to Edison lamps is beyond question; but our experiments with lamps having an artificial surface on the carbon, or "flashed" lamps as they are called, show that their rate of variation of life and efficiency follows a different curve.
This curve does not apply to individual lamps. If we take two Edison lamps and burn them at different efficiencies, their lives for these efficiencies will probably not be such as indicated by the curve, nor will they be proportioned to these indicated lives. But if we take one hundred lamps and burn them at one efficiency, and another hundred equally good lamps and burn them at another efficiency, the average lives of the two sets will be proportional to the lives indicated by the curve for these two efficiencies.
In order to determine at what efficiency the cost of operating lamps of a given quality under given conditions is a ​minimum we must calculate what this cost is at different efficiencies. To do this we consider the total cost of operating the lamps to be made up of two parts, viz., the cost of the current and the cost of the lamps. The cost of the current is made up of every expense incurred in operating the lamps, including materials consumed, labor, taxes, insurance, rent and every other expense incurred in operating the plant, except the cost of lamps. The cost of the lamps is an item by itself, and is the amount which the lamp has cost when it is put in use. This is a natural division of the total cost of operating a plant, since to produce light by incandescence all that is necessary is a lamp and current to operate it.
If, in any case, we know the cost of the current required to operate the lamps,
the cost of the lamp, the quality of the lamps—that is, the life they will give when burned at a given efficiency—and the rate of variation of their life with efficiency, we can then calculate at what ​efficiency the cost of operating the lamps is a minimum, and this I call the maximiun efficiency of those lamps.
The following examples show what this maximum efficiency is, under varying conditions of the cost of lamps, the cost of current and the quality of the lamps. The coat of the lamps I have varied between 25 cents and $1.00 each. The cost of current varies between 2.5 cents and 10 cents per h.p, per hour. The quality of the lamps varies between 300 hours life at 3 watts per candle, and 2,400 hours life at 3 watts per candle.
In each of the following cases I have calculated the cost of operating 100 16 c, p, lamps 1,000 hours, at each of the efficiencies comprised in the curve of total cost. These curves do not show the cost of running the same lamps at different efficiencies, but the cost of running equally good 1 G candle lamps of the different efficiencies.
The first case we will consider is shown in the diagram Fig. 3. In this case the lamps are assumed to cost 85 cents each ​and to have a life of 600 hours at 3 watts per candle. The current is assumed to cost 10 cents per h.p. hour.
Lamps 85 cts. each—Life 600 hrs. at 3 Watts per Candle.
Current costs 10 cts. per ㏋. per hour.
The cost of the current is determined from the following formula:
Current cost=
w.p.c. × 16 × 100 × 1000 × cost of ct. per h. p. per hour ​And the cost of lamps from this formula:
Cost of lamps =
cost of one lamp × 100 × 1,000/Life at given efficiency
The curve marked total cost shows the total cost of running 100 16 c.p, lamps 1,0U0 hours, the efficiencies of the lamps varying between 2.5 and 4.25 watts per candle. The efficiencies are shown by the vertical lines, referring to the scale at bottom. The value of the total cost at any point of the curve is shown by the horizontal line through the point, referring to the scale at the left of the diagram.
The lowest point of the curve shows the point where the total cost is lowest. This is the minimum cost of operating these lamps under the given conditions. The mark at the lowest point of the curve shows this minimum cost to be $783, and a vertical line through this point to the scale at bottom of the diagram shows that this total cost is a ​minimum when lamps having an efficiency of 3.1 watts per candle are used.
Thus the maximum efficiency of these
Lamps $1.00 each-Life 600 hrs. at 3 Watts per Candle.
Current costs 10 cts. per ㏋. per hour.
Fig. 4.
lamps under the conditions assumed is 3.1 watts per candle.
The lamps considered in the case shown in Fig. 4 cost $1.00; all other conditions ​are the same as in the case shown in Fig. 3. This increases the total cost from $783 to $800, and necessitates using lamps of 3.18 watts per candle instead of 3.1,
Lamps $1.00 each-Life 600 hrs. at 3 Watts per Candle.
Current posts 5 cts. per ㏋. per hour.
Fig. 5.
to make the cost of operating a minimum.
In the case shown in the diagram. Fig. 5, the current costs 5 cents per h.p. per hour; the other conditions are the same ​as in the case assumed in Fig. 4. The minimum total cost in this case is $444, and to make this cost a minimum we must use lamps having an efficiency of 3.5 watts per candle.
Lamps $1.00 each—Life 300 hrs. at 3 Watts per candle.
Current costs 2.5 cts. per ㏋. per hour.
Fig. 6.
In this case shown in Fig. 6 the lamps cost the same as in the last, but are only half as good; the current costs just half as much as in the last case. The minimum total cost in this case is $273, and ​the maximum efficiency of the lamps is 4.33 watts per candle.
In the case shown in Fig. 7 the lamps cost 50 cents and have a life of 1,200
Lamps 50 cts. each—Life 1200 hrs. at 3 Watts per Candle.
Current costs 10 cts. per ㏋. per hour.
Fig. 7.
hours at 3 watts per candle. The current costs 10 cents per h. p. per hour. This is the cheapest and also the best lamp we have yet considered, but the current is expensive. In this case the ​minimum total cost is $654, and the maximum efficiency of the lamp is 2.62 watts per candle. In this case. Fig. 8, the current costs half as much as in the previous case, other conditions being the
Lamps 50 cts. each—Life 1200 hrs. at 3 Watts per Candle.
Current costs 5 cts. per ㏋. per hour.
Fig. 8.
same. The minimum total cost is reduced from $654 to $362. The maximum efficiency in this case is 2.88 watts per candle. In Fig. 9 the current costs twice as much as in the previous case and the ​lamps are only half as good. The minimum total cost is doubled, but the maximum
Lamps 50 cts. each—Life 600 hrs. at 3 Watts per Candle.
Current costs 10 cts. per ㏋. per hour.
Fig. 9.
efficiency is the same as in the previous case.
In this case, Fig. 10, the current costs one-half of that assumed in the previous ​case, other conditions being the same. The minimum total cost is $400, and the maximum efficiency 3.175 watts per candle.
The curve, Fig. 11, illustrates the case
Lamps 50 cts. each—Life 600 hrs. at 3 Watts per Candle.
Current costs 5 cts. per ㏋. per hour.
Fig. 10.
of very cheap and very good lamps, with moderate cost of current. The minimum total cost is low, $294, while the lamps are run at the high efficiency of 2.38 watts per candle.
​In the case shown in Fig. 12, the cost and quality of the lamps are the same as in the previous case, but the current costs twice as much. This increases the minimum
Lamps 25 cts. each—Life 2400 hrs. at 3 Watts per Candle.
Current costs 5 cts. per ㏋. per hour.
Fig. 11.
total cost from $294 to $535, and raises the maximum efficiency to 2.14 watts per candle.
In the case, Fig. 13, the cost of lamps and the cost of current are the same as in ​the case shown in Fig. 11, but the lamps are only half as good. The minimum total cost is increased from $294 to $327, and the maximum efficiency is reduced from 2.38 to 2.62 watts per candle.
Lamps 25 cts. each—Life 2400 hrs. at 3 Watts per Candle.
Current costs 10 cts. per ㏋. per hour.
Fig. 12.
The first and plainest inference drawn from these curves is that the maximum efficiency o£ any given lamp is not a fixed one, but varies with conditions outside ​the lamp itself. Identical lamps operated under different conditions of cost of current must be burned at different efficiencies to make the cost of operation a
Lamps 25 cts. each—Life 1200 hrs. at 3 Watts per Candle.
Current costs 5 cts. per ㏋. per hour.
Fig. 13.
minimum for the production of a given amount of light. In order to determine the maximum efficiency of lamps, therefore, we must know the quality of the lamps referred to some standard, the cost of the lamps to the consumer, and ​the cost of current under the actual conditions existing at the place where the lamps are to be used.
In the eleven cases shown in this paper the lamp of highest efficiency is obtained in the case shown in Fig. 12. This is a case where the lamps are very cheap and very good, and current is very expensive. The lamp having the lowest efficiency is obtained in the case shown in Fig. 6, in which the lamps are poor and highpriced and the current is very cheap.
There is a marked difference in the sharpness of these curves at the minimum points. An inspection will show that the sharpness of the bend in these curves depends upon the cost of the current, the curves in which the current costs 10 cents per h.p, per hour being the sharpest. Those, in which the current costs 5 cents, are next, and the one. Fig. 6, in which the current costs only 2½ cents per h.p, per hour, is very flat at the bottom or minimum point. In this comparison Fig. 7 is not considered, as it drawn on a different scale from the ​others, and is not as sharp as it should be.
This indicates that the more expensive the power is, the more carefully must the lamp efficiencies be chosen. In Fig. 12, which shows the sharpest curve, a very slight variation in the efficiencies of the lamps makes a very great change in the total cost of operation.
In the case shown in Fig. 6, in which the current is very cheap, we find that a very considerable change in the efficiency of the lamps used makes very little difference in the total cost of operation.
On each of the eleven diagrams two curves are drawn, one showing the total cost of operation, and the other showing the cost of lamps. On each of the curves showing the cost of lamps, a point is marked which indicates the cost of the lamps when the total cost is a minimum. The letter T marked on each of the figures denotes the minimum total cost of operating the lamps under the given conditions. The letter L denotes the cost of the lamps when the total cost is a ​minimum; and the expression L/T denotes the ratio of the cost of lamps when the total cost is a minimum, to the minimum total cost. An examination of all these curves shows that while the minimum total cost varies with each of the three quantities—price of lamps, quality of lamps, and cost of current—nevertheless, the total cost is always a minimum when the cost of lamps is about 14.5 or 15 per cent, of the total cost of operation.
This figure varies somewhat in the different examples considered, but the variation seems to follow no law. In Figs. 6 and 12, which show the highest and lowest efficiency lamps, the figures are 15.1 per cent, and 15.4 per cent, respectively.
The steepness of the curve showing the cost of lamps, and the difficulty of determining the exact minimum point of a curve which has been drawn by inaccurate methods, makes it difficult to get the cost of lamps accurately when total cost is a minimum. These curves and ​values are given just as they were determined, and no effort has been made to bring the results into closer agreement, as could readily have been done. I consider the variation shown by these carves to indicate closely enough that this ratio of cost of lamps to total cost at the minimum point is nearly, if not quite, constant and that its value is between .145 and .15.
This establishes a very simple law for determining whether or not lamps are being operated at their maximum efficiency; for if they are, the lamp bills will be about 15 per cent, of the total operating expenses of the plant. If the lamp bills are more than 15 per cent, of the total operating expenses the lamps are being burned above their maximum efficiency, and lower efficiency lamps should be obtained. If, on the other hand, the lamp bills are less than 15 per cent, of the total expenses, the lamps are being burned below their maximum efficiency, and higher efficiency lamps would reduce the cost of operating the plant. Where ​fuel is high priced, or where other caused operate to make the cost of generating current high, it is specially important to use lamps of the maximum efficiency, for we have seen from the above curves that where the cost of current is high, the use of lamps whose efficiency is a little above or below the maximum efficiency attainable under the conditions of operation, makes a very marked increase in the operating expenses of the plant.
If in any plant the lamp bills are only 10 per cent, of the total expenses, then increasing the efficiency of the lamps by increasing their candle-power does not diminish the total cost of operating the plant. In order to reduce the total cost, the lamps must he replaced by lamps of the same candle-power but of higher efficiency. If the efficiency of the lamps is increased by raising their candle-power, the cost of operating the plant per unit of light produced is reduced, but the total cost is increased. A plant which is paying less than 15 per cent, of its total expenses for lamps, and which brings ​the lamp bills up to 15 per cent, by increasing the candle-power of the lamps, does not decrease the cost per lamp of operating the plants but does decrease the cost per candle of light furnished. If they are paid for the increased light given by the lamps, the efficiency of the plant is made a maximum for the existing conditions; but if they are not paid for the additional light furnished, the efficiency of the plant is reduced.
This law enables any one operating an incandescent lamp plant to determine whether or not he is using the most suitable lamps for his plant. If the conditions of operation of a new plant are all known, and the quality of the lamps made by any lamp maker is known, we can determine before starting what is the most economical lamp to use.