8/18/2012 UPDATE: From the comments, Andy pointed out that I had a mistake in my calculations. In the c_total equation, I should have divided by 1000, instead of multiplying by 1000. After he pointed this out, I checked the java script code as well. I had made an even more egregious mistake there. I had correctly divided by 1000 but I also had divided by power cost (dollars per kilowatt hour). I’ve now fixed the equation and the script. I also regenerated the results from the examples.
I am slowly upgrading the incandescent light bulbs in my house with more efficient compact fluorescent light bulbs. However the higher purchase price of the new bulbs has me wondering if they are actually “worth it.” I also thought whether an even more efficient type of bulb, perhaps an LED light bulb, would be better.
So, I set out to determine a value which would best quantify the “worth” of different bulbs and rank each based on this value.
I was frustrated by current use of vauge descriptors which rated fluorescent bulbs by power consumption (watts) without considering other factors. Even Howstuffworks.com fell short by comparing bulbs based on luminious efficacy (lumens per watt). I wanted to compare the total light energy output (lumen hours) to the total cost input (dollars).
I think the name “luminous cost efficacy” (symbol: η luminous cost) is an appropriate term for this quantity: “luminous” because the output concern is usable flux (human eye, see luminous vs. radiant flux), “cost” because the input concern is price, and “efficacy” because it is a non-unitless ratio.
Calculating the luminous cost efficacy requires several parameters of the bulb and the circumstances in which it is used:
-luminous flux, F (lumens)
-cost of energy, cenergy (dollars per kilowatt hour)
-initial cost of the bulb, cbulb (dollars)
-life of the bulb, l (hours)
-power consumption, P (watts)
With this information, a formula can be developed
As I feel it is always important to dig down to the most base level of understanding, the definition needs to be defined in terms of SI base units. This requires decomposing lumens into candelas and watts into m, kg, and s. This decomposition is only for completenes; in practicality, it will be difficult to find candela measurements for consumer light bulbs when moving back to real world problems.
1. Bulb parameters are constant over the entire life of the bulb. (this is not a practical assumption)
2. Bulb will last “average life of bulb” and not die out sooner, or later.
Using a standard compact fluorescent bulb found on Home Depot’s website: 27 watt, 1,750 lumens, and 10,000 average hour life, $7.99 cost, and my current home electricity bill of $0.1005 per kilowatt hour yields a luminous cost efficacy of 498,078.8 lumen hours per dollar.
I compiled all of this simple math into one, easy-to-use calculator available here: bulb calculator
- 40 watt incandescent from www.1000bulbs.com
(40 watts, 6,000hour, $1.20, 325 lumens)
77,627.4 lumen hours per dollar
60 watt incandescent from www.1000bulbs.com
(60 watts, 6,000hour, $1.20, 675 lumens)
108,929.5 lumen hours per dollar
27 watt compact flourescent from Home Depot
(27 watt, 10,000 hours, $7.99, 1,750 lumens)
498,078.8 lumen hours per dollar
120V LED Light Bulb from Home Depot
(1.3 watts, 60,000 hours, $39.99, “lighting equivalent of a 15 watt incandescent bulb” ~= 210 lumens)
273,734.5 lumen hours per dollar
From these simple calculations, it is evident that the CFL bulb is the best choice for my situation. Remember that these numbers are based on my energy cost. If my electricity were more expensive, say 20 cents per kilowatt hour, the results would be different.
This is a physics approach to the problem, not an economic, environmental, or social approach.
-No consideration has been given to present/future worth of the bulb or projected electricty prices.
-No consideration has been given to the enviromental impact of producing electricty versus producing light bulbs.