**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).

Nomenclature:

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.

Formulation:

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, *c*_{energy} (dollars per kilowatt hour)

-initial cost of the bulb, *c*_{bulb} (dollars)

-life of the bulb, *l* (hours)

-power consumption, *P* (watts)

With this information, a formula can be developed

where

and

Complications:

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.

Assumptions:

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.

Example:

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.

Calculator:

I compiled all of this simple math into one, easy-to-use calculator available here: bulb calculator

Results:

- 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.*

Wuddup Jason, Seth here… you have to consider the lumen output of each bulb in relation to you lighting needs as well… 210 lumens is not going to be enough to light a room. Maybe a desk. I will give you a real world example of cost effectiveness of CFL’s though. I replaced all of the commonly used lights in my house with CFL’s and I saw a 40% reduction in my power bill.

I went from a total 1800w usage via incandescent to 410w usage via CFL’s.

Also, go to Ikea… the CFL’s there are way cheaper than H-Deezy.

Right, but the idea is that this is scaleable. You simply figure out which bulb is the most efficient and then use as many bulbs as will produce the required lighting (lumens).

Very true… I’m doing similar calculations to figure out lighting for bicycles.

So far it looks like a dual setup is going to be the way to go… a 32 LED low beam (10W MR16 Halogen equiv) and a 20W overdriven high beam (12v 20W MR16 run at 13.2 V). Driving the 12v halogen at 13.2v gives a better lumen to watt ratio at the expense of a 50% reduction in bulb life. However, seeing as the bulb life to begin with is 5000 hours and the cost of the bulb is around 1.75, the reduction in life is not a big factor when you look at how much the bulb will be used (a few hours a week at most).

I’m moving this to FM.

All your GIF and JPG are missing.

oops. fixed.

I think you need to divide kWh by 1000 to get Wh, rather than multiply by 1000. If I am right you have over represented the running cost by a factor of 1000000. Any other views? Andy from the UK