Light at depth formula

This came up in another thread, but I thought it might be very useful and didn't want to hijiack the other thread, so here it is....

There is a way to calculate the amount of light reaching the bottom of your fishtank without a light meter. It's called Beer's Law, some of you may remember it from high school chemistry or physics. The equation looks like this:

Iz = Io*e(-kZ)

In English:

Light at the bottom = light at the surface^(-1*depth of aquarium)

K is the extinction coefficient, and can roughly be assumed to =1 in this case (trust me)

Light at the surface is your light fixture's wattage
depth of the aquarium should be from the water surface to the substrate, but the height of the aquarium will do. It needs to be in METERS for the calc to work. (Watts are metric units)

This won't calculate watts per gallon at the bottom of the tank, per se, what it will tell you is the amount of light reaching the bottom of the tank.

Here is an example calc on my 55 gal with 192 watts of light:

depth is 22" = 56 cm or 0.56m
light at the surface = 192 watts
light at bottom = 192watts^(-1per m*0.56m) = 110 watts reaching the bottom.

You'll need a scientific calculator or Excel (not the plant food!) to do the exponent.

What this can do is give you an idea of how the depth of your tank is affecting the amount of light reaching the bottom plants. So using the 110 watts and 55 gallons, you could say that at the bottom of this tank the lighting is 2 watts "per gallon", compared to the 3.5 watts per gallon at the top. So the light intensity at the bottom of this tank is a little more than 1/2 what it was at the top.

Please let me know if you need help with the calcs...It would be neat to have a bunch of folks do this and see what we all get. I don't know how to "post" a spreadsheet but if someone knows, let me know and I'll make one up so that people can just enter their numbers.

you forgot your *e in there

oh, ill be pasting a large spreadsheet in here in a moment

You could static the variables since most tanks > 20Gal are 21-24" deep.. So a simple light * static variable.. a simple graph could accomplish this. you are afterall dealing with known numbers (21, 22, 23 etc..) Most people won't have a problem multiplying two numbers..

Also. This could lead to someone trying to get 3WPG at the bottom of their tank (assuming WPG is accurate for their size tank) and needing 6WPG at the surface. (and crashing their tank with algae)

It makes a lot of assumptions. 100% reflectivity, 0% restrike (which is virtually impossible with a round tube) and all the light to hit the water surface between 45 and 135deg (the angle of entry for water).. It would also assume the water was crystal clear. The spectrum of the bulb is also important as reds are typically dissipated within the first 6" of water so only the blue percentage of the spectrum will penetrate the first 6" of water..

The value of K = 1 does not assume the water is crystal clear. That is a typical value for relatively clear freshwater lakes. K can be lower or higher than 1, it just can't be less than 0.

This is not trying to give an exact number, just as none of us know exactly how much light is hitting the surface of our tanks, either, because of all the variables Wizzard listed. All it does is give you an idea of what percentage of the light you have in the hood is reaching the bottom. All the wpg rules that everyone uses (very successfully!) still work, and should not be changed because of this. If you can grow a certain plant at 3 wpg surface, and you're really at 1.5 wpg bottom, then you don't need 3 wpg bottom. I just thought it would be interesting to see what values people get for the bottoms of their tanks, and then maybe use this as a guide when trying to light a weirdly shaped tank (like the 50 gal hex that started this). It also might be useful to explain why two people with the same wpg have different results with the same plant; if one plant is in a 10gal and the other in a 29 or 30 high, the loss of light due to depth might be the difference.

But please do heed Wizzard's point, newbies, you shouldn't use this to try and get high light at the bottom! You don't need it. I have 2 wpg at the bottom and lemme tell ya, it's plenty!

This is very interesting to me.

Making it very simple....it looks like if I have 2 wpg (simple math) in a 24" tall tank I'm going to have a very hard time growing dwarf hairgrass....... but if might grow in a 12" tank.

I also remember a 6 ft high tank somewhere.....that isn't growing anything on the bottom.

Thanks everyone for the work...it helped me!

Looks like you're on the right track, TwoHobbies.... For the "average" 20-24" high tank, you get about 1/2 the light at the bottom as you do at the top. Smaller tanks get more, and therefore might be able to get away with less light, or be able to grow higher light plants than their wpg might suggest. Reverse this for deep tanks.

I wasn't trying to be critical, just thorough.. The major flaw with lighting is the reflection of the light, if it's not at the correct angle it skips off the surface, and some of it isn't reflected to the water at all...

Linear, but interesting none-the-less.

Mine would be 265w surface (3.5WPG) and 151Watts bottom (2WPG) according to that chart..

75Gal 48x18x21.

I wasn't trying to be critical, just thorough..

Click to expand...

I knew you weren't being critical. Light is a super complex thing....in my field there are people who get PhDs on the properties of light in water. I figure, anything that we can use to better understand what's going on, is good. The linear assumption at least makes it easier to calculate, and for the shallow depths we're dealing with, is probably a good assumption. Most field studies of light take measurements every meter or so, which is deeper than 99% of the tanks most people have!

I would say much more then 99% of the tanks are < 1m.

and by linear I meant, wattage is not important. The Percentage lost at each given depth is the same..

"105 77.4" = 73.7%
"395 291.2" = 73.7%

Even looking at the formula.

"Iz = Io*e(-kZ)"

Light (depth) = Light(top) * .737 for a depth of 1 foot.

-kZ is constant for any given depth. so with that in mind you can simplify the formula with a graph of known percentages. It is also a good idea to include the source formula in any case.

And one of the more interesting experiments.. Take a mirror and place it near the bottom of your tank. (on the outside of the glass).. If you turn it to an angle that you can see your lighting it will show you the spectrum that reaches the bottom. the brightness of each band is it's intensity.. Most of us can't afford spectrometers, but it will at least show you the spectrum your lighting is putting out. (surprisingly, red does indeed make it to the bottom of the tank..)

Cool...I will try that....you sound like must be a fellow scientist, Wizzard

Nope. I work with electronics. I just find learning fun and try different things.. Figured the source of a rainbow was sunlight hitting water drops, so I tried to find a way to look up into a tank, used a mirror and observed the rainbow halo on each bulb. I also observed it was different for each of my 3 types of bulbs..

If you have a prism and a blank sheet of paper you can also find the bulbs spectrum at the source to compare the spectrum loss through water..

You could possibly use a prism at the bottom, but it's possible the distortion of the spectrum could have interesting effects.

When I was in college we did some light theory.

Being somewhat of a scientist (Engineer), I am a big fan of formulas and equations. But I have come to the conclusion that there are so many unknown variables when it comes to lighting, you are better off using the basics followed by experimentation (also know as trial and error).

JMO.

Very true, but the whole watts per gallon rule makes a ton of assumptions too. The reason I put the equation in there is because I remember reading several questions about a way to measure light at depth without a light meter, and I thought folks might like to try it.

Honestly I think a watts/surface area rule would be better than watts/gallon, because of the variability in tank shapes. But wpg is well known and does a good enough job, and people have a feel for how it behaves.