Light Wattage vs. Tank Size

[VaughnH]

It has long been known that a small tank needs more watts of light for a given light intensity than a medium size tank needs, and a very large tank needs less watts of light for that intensity. This never did make sense to me. Last night, as I was trying to sleep I started working on this in my head, trying to work a calculus problem - WARNING: Don't try this, it leads to headaches!.

This morning I worked out the equation relating light intensity to tank size and bulb wattage. It isn't that difficult to do, if you can do integral calculus. The explanation is:
Tubular fluorescents usually produce a fixed amount of light per inch of length, consuming a fixed amount of watts per inch of length. Four foot long T12 bulbs are 40 watt bulbs, 10 watts per foot. 2 foot T12's are 20 watts, etc. So, a tubular fluorescent bulb is a length of light above the tank with about the same brightness per inch all along its length. Do a mind experiment: Take a four foot long tank, with a four foot long bulb above it. Now insert a pair of opaque dividers in the tank so that there is two feet between them and only two feet of the bulb lights the substrate between them. Same bulb, same tank, but the light intensity at the substrate drops considerably. That is because that section no longer gets light from the blocked off ends of the four foot long bulb. Move those opaque dividers close together, so they are an inch apart - the substrate in that section is almost dim, but it is still the same bulb up there.

The reason a ten gallon tank needs more wattage of light to get the same light intensity is that it doesn't have as long a bulb above it shining light from the whole length of the bulb on every spot on the substrate.

Now to the math: (only the solution, not the process)
The light intensity from a bulb producing a fixed amount of light per inch of length on a substrate at a distance "D" from the bulb, where the tank is "L" long is proportional to the natural log of the following ratio: [L/2 + the square root of D squared plus L/2 squared] divided by D.

Figuring all of that out for various tank sizes gives the following table:

This table shows that a ten gallon tank gets only about a quarter as much light intensity at the substrate as a 55 gallon tank, when both are lighted with T8 or T5, etc. bulbs. It also shows that a 125 gallon tank gets about 30% more light intensity at the substrate as a 55 gallon tank when both are lighted with T8, T5, etc. bulbs.

So, if a 55 gallon tank needs one 54 watt T5 bulb to get adequate light at the substrate, a 10 gallon tank would need about 40 watts of T5 light for the same light intensity.

Being a retired engineer, I find this fascinating.

EDIT: Another question that comes up is how much more light do we need for a deep tank, like a standard 90 gallon tank. This analysis gives that answer too. For a 90 gallon tank we only need about 15% more bulb wattage than for a 55 gallon tank to get the same light intensity at the substrate.

This analysis only works for a tubular light that extends the length of the tank. It doesn't work for any light fixture that has a bulb shorter than the tank, including HQI or Power Compact bulbs which are shorter than the tank is long.

 

[Calmer]

EDIT: Another question that comes up is how much more light do we need for a deep tank, like a standard 90 gallon tank. This analysis gives that answer too. For a 90 gallon tank we only need about 15% more bulb wattage than for a 55 gallon tank to get the same light intensity at the substrate.

So with my 90 gal.
I take 54 + 15% = 54 + 8.1 = 62.1w for a 1w/gal. reference.
I have 4-32w tubes 4' long which is 128w total.
Therefore 128/62.1 = 2.06w/gal using the reference.
Whereas 128/90 = 1.422w/gal without using the reference.
Does this seem right?

 

[VaughnH]

This isn't as clear to me as I would like it to be, so I'm still figuring out just what those numbers mean, too. But, the calculation is based on the length of the tank and bulb, and using the same kind of bulb, meaning the same wattage per inch of bulb length and same light output per inch of bulb length. A 90 gallon tank is just a tall 55 gallon tank for this calculation. So, the numbers mean you would get about 15% less light intensity at the substrate in the 90 gallon tank as you got with that same light on a 55 gallon tank. So, the 90 gallon tank would need 1.15 x 54 watts, or, 62 watts to get that intensity of light. In reality it would mean lowering the fixture closer to the top of the tank.

I'm still thinking about how the front to back distance of the tank affects all of this, since that wasn't part of the analysis.

 

[Tom Barr]

I just test the micromols and then I'm done with the guess work

I know what the plants need and how fast they will grow etc at that level of light.

Tank size does not matter in such cases where you can measure the light on equal terms.

Regards,
Tom Barr

 

[VaughnH]

Where this matters is when you don't have a light and need to get one. What do you need to buy? It is a little like a drop checker in that things like this are a way to get into the right ballpark, so you have a shot at getting the right amount of light. As data taken with PAR meters starts being available we should accumulate enough data to have a chart showing what each light fixture provides on each tank size. Then, we can forget the estimating and guessing part. I think shifting the standard location for a light fixture to a hanging one will help too, so, with a PAR meter you can adjust the light to get the intensity you want.

 

[Calmer]

Have a look here and see what you think.
New Lights?

 

[VaughnH]

Of course I disagree with much of that, and I doubt that being a surprise.

Let's start with tank depth: Light that is not from an exact point source located exactly at the focus of a perfect parabolic reflector that is very deep, compared to its focal length, drops off with the square of the distance from the bulb. Our bulbs are far from being point sources, and our reflectors are crude, even the best of them, compared to a perfect parabolic reflector. So, distance from the bulb does decrease the intensity of the light. That means tank depth does affect the intensity of the light at the bottom of the tank, where it is most critical. A linear source of light, one that is long, but doesn't have zero diameter, and isn't in a perfect 2 dimensional parabolic reflector drops off directly proportional to the distance from the bulb, if the bulb length is long compared to the distance at which we measure the intensity. That is close to what we have with T5 bulbs in their individual reflectors. And, again the intensity drops off with distance from the bulb, so tank depth does matter.

If I take a tank with a 55 gallon tanks length and height, but is 10 times the 55 gallon tanks depth from front to back, making it a 550 gallon tank, and put a 54 watt T5HO light on top of the tank in the middle, front to back, the only thing that would prevent having exactly the same light intensity at the substrate directly below the bulb is reflected light from the front and back glass. That alone would make a watts per gallon standard meaningless, except that that watts per gallon "rule" was never meant to apply to any tank except one with standard geometry.

At this point I'm sure about what effect the height of the tank has on light intensity directly under the bulb, and the effect the length of the tank and bulb have on that intensity in the middle of the tank end to end. But, I'm still unsure about the effect of the front to back dimension on the intensity. As far as I can see, if we are looking at single tube lights, the full length of the tank, it is only light reflected off the front and back glass that change the intensity as the front to back dimension changes. I will be thinking about this aspect and doing some research before I have any good idea about that aspect of light intensity.

I tend to forget that my exercise with calculus was never meant to do anything except help me understand why "2 watts per gallon" on a 10 gallon tank is far less light intensity than "2 watts per gallon" on a 55 gallon or 125 gallon tank. So, the analysis applies only to the intensity directly under the center of the bulb. I think that makes the analysis interesting and useful for understanding light a bit more, but certainly not good for deciding on what light to use for any tank. This is an academic exercise only.

 

[VaughnH]

Well is time to eat some crow - I do this often enough to have developed a taste for it. My calculations above were made with the assumption that light intensity drops off proportional to distance in the aquarium, which isn't true. For a calculus approach to this, the emitter of the light is infinitestimally small, so the light intensity has to drop with the square of the distance. Redoing those calculation with that assumption gives entirely different numbers. In fact that calculation shows a 10 gallon tank with the same light intensity per inch of bulb as on a 55 gallon tank actually has a higher intensity at the substrate level in the middle of the tank. Oops!

But, that triggered an all day effort to include reflected light from the front and back glass into the calculation. I finally was able to do that, assuming 100% reflection from the glass, which isn't correct, but it is close enough. Again, the 10 gallon tank has the highest intensity at the middle of the substrate level. Then studying the equation I derived I decided that the most significant factor is the depth of the tank, so I redid the calculations using the same 12 inch depth for each tank size (keeping the other dimensions accurate, so each tank became a smaller tank, except for the 10 gallon tank.) This time the 55 gallon tank has the highest intensity, but not by enough of a factor to be very significant.

Now I'm convinced that tank depth is the most significant factor in how much light is needed for the tank. But, the variation with depth is still small enough that I can't see a use for the knowledge. One exception is a 90 gallon vs a 55 gallon tank, where the deeper tank really does need more light.

The equation for the intensity at the substrate level in the center of the tank is: Intensity is proportional to 1 divided by the square root of [Depth squared + front to back squared] that multiplied by the arctan of [Tank length divided by {2xthe square root of (depth squared + front to back squared)}]

This has been fun! Now, why does a 10 gallon tank require so much more light than a 55 gallon tank to get a good light intensity for growing plants?

 

[SniperLk]

Sorry I don't have the time right now to read all the topic (but I will later )

Just to say that we "should" speak in W per square foot rather than W per gallons.. It explains a lot of things..

If a 100 gallons doesn't need 10 times the light wattage of a 10 gallons it's just because usually the area of a 100 gallons is not ten times the one of a 10 gallons.

This graphic compares the W per square meter (blue bar) Versus W per 1L (yellow bar) (actually per 1000L)

There must be like more than 50 tanks compared, from 21L to 2916L. Most of the tanks are from Amano as far as I remember.

What we can see is that the smallest tank is lighted with almost 2W/L whereas the biggest one with 0.2W/L. This is a 1:10 ratio. Ok but we all know that.

The interesting part is that the W per square meter values are way more homogeneous.
It goes from 500W/m² for the smallest tank to 125W/m² for the biggest tank. It's a 1:4 ratio..

So speaking in W per square meter (or square foot) instead of W/liters, gallons seems more accurate and appropriate

PS : 1 square foot = (approx) 0.1 m²

 

[Tom Barr]

Not really , because watts and PAR are also not part of the ratio either and PAR is what plants use, not watts.

Having tested PAR on ADA lights and tanks, there was a difference of 2-3x less PAr on ADA tanks vs the Coralife using the same bulbs.

So 1:4 is now about 1:10, you are back where you started.

W/meter is not really that much different than volume, gallons etc.

Not enough to make any difference as far as a successful aquarium when you also consider common sense and experience

I've used it for well over 20 years now and it's never failed once to date.

So while folks carry on about the "need to change the rule", as far as I can tell, not a single critic has yet to have bothered to test the relevant parameter that plants use to grow and sequester carbon. Adding to that, few have the experience and examples to show that the rule fails.

So.............it gets a bit old hearing about it. This light hogwash has been going on for no less than 20 years in this hobby, yep, before the internet. It's when you start focusing the question and asking the right kinds of questions do you get anywhere.

Now a PAR is not cheap, but they are not that costly either.
And then you know specifically for an individual system set up what the light is and can truly compare and know the light limits of plants.

You will never do that with W/gal or W/m^2.
Folks waste/spend a lot of time on this and I really have not bothered with good reason.

The large difference in Watt per unit PAR is a good example why with ADA.
You also cannot compare research studies well either, they all use PAR.

I've had many folks approach me about coming up with some super comparative thing, large graphs, long studies, correlations and what not, then there's the wind bag clowns that whine about the need to test and measure water quality that poo poo on me yet never test the most relevant parameters

Sorry to be crotchety about this, it's just after seeing and hearing the same things 100X over and over, and no one gets anywhere, seeing history repeat again and again.......it's not any one person. But once we see this, we can approach things from a different angle and hopefully answer the question.

This is not always the most popular road to take
Some things need tested so we can then know more about it and make better predictions. Then we no longer need to test/or much testing it all.


Regards,
Tom Barr

 

[aaronnorth]

wouldnt it be better to use lumens per square inch?

 

[SpeedEuphoria]

I dont think so, I think depth would be a better rating than surface area

 

[essabee]

I am not a mathematician but a simple hobbyist, and when I tried to find a answer as to why we need more lights in a smaller tank, this is what came into my mind:-

A 3' cubic tank holds 27 cft water and the area of its 4 sides total 36 sft a volume to area of sides ratio of 1 : 1.33

A 2' cubic tank holds 8 cft water and the area of its 4 sides total 16 sft a volume to area of sides ratio of 1 : 2

A 1' cubic tank holds 1 cft water and the area of its 4 sides total 4 sft a volume to area of sides ratio of 1 : 4

So I felt that it was the loss of light from the sides of the tank was the culprit and the reason for the need of extra illumination in smaller tanks.

 

[Tom Barr]

I get a similar PAR reading on a small 2 gallon tank with 13 W of Pc light as I might on a 20 Gallon with 55 W of PC light.

This has little to do with watts and a lot to do with the efficacy of those smaller cheap bulbs used on smaller tanks.

The light is the same (in terms of PAR), however, the watts used are far more relative to tank size. Then there's also the reduced distance between the light source, various other reflections etc in smaller tanks vs the larger ones. The watt to watt comparison is a bit like comparing normal FL's to nice T5's with good reflectors, often lacking and restrike issues as well with smaller tanks.

So watts vs watts are no good when you get into smaller lighting systems.
Good reason to make all comparison's using PAR.

Still, you can see why, or at least have a much better idea about what is going on, if you measure using a PAR meter. Rather than carrying on about how "bad" the watt/gal rule is and using different parameters to compare.

Just cut to the chase, use a PAR meter.

Regards,
Tom Barr