The Potential Importance of Tank Depth on Light Attenuation
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Old 02-26-2004, 02:46 AM   #1
Splash
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This message is a quasi-scientific analysis of light requirements for planted tanks, and I apologize beforehand if I lose anyone in presenting my analysis and for its length. I will be happy to answer any questions through this forum.

Most of us are familiar with the rule of thumb that 2-3 Watts per gallon (WPG) of light is required to support good growth in a planted tank. Many of us also recognize that light intensity attenuates with depth, so more light is required for a deeper tank than for a shallow one. What is generally not known is *how* much more light is required for specific tank dimensions of width (W), length (L), and depth (D). Below is my attempt to address this question.

From a scientific perspective, the use of a volume measure (i.e, gallons) in the rule of thumb is incorrect. Photosynthesis in a plant is driven by the uptake of light energy over the exposed surface area of all the leaves, so an areal measure is the appropriate starting place for developing a new rule of thumb.

"Watt" is a measure of radiant energy flux; that is, it is the energy radiating from a source (a bulb, in our case) per unit time (seconds, minutes, or whatever). "Intensity" is the areal density of that energy flux; that is, Watts per unit area (e.g., Watts/square inch). If we where to measure the light intensity on the surface of a leaf and multiply that value by the surface area of the leaf, we would have a measure of how much energy was available to the leaf per unit time. Photosynthesis is the uptake of light energy by the plant, so the photosynthetic rate (or the rate of nutrient uptake and carbon fixation into new plant tissue) is directly proportional to light intensity at the leaf surface.

Most hobbyists, including myself, do not have the means to measure light intensity at each leaf in our tanks, so we have to use a surrogate measure. The obvious measure is the wattage of the bulbs above our tanks divided by the horizontal area of the tank opening. So in our units of choice (Watts per square inch), we have a measure of the light energy per unit time striking the entire water surface in the tank.

At this point, I'm going to make a few assumptions and caveats to simplify and hopefully clarify my analysis. If these assumptions are not met for a specific tank, then correction factors will be required. Note that most of these same assumptions and caveats exist implicitly in the old rule of thumb, I'm just making them explicit.

------------------------------
Assumptions:

1) Bulbs are arranged over the tank such that the flux of radiant energy is distributed evenly over the water surface.

2) Bulbs of a certain format (e.g., compact fluorescents) are usually placed at about the same distance above the tank. (Small changes in this distance are not important, because light attenuates slowly in air.)

3) Differences in reflectors is a secondary factor, and won't greatly alter the solution.

4) Glass or acrylic tank tops are kept clean, so at least 90% of incident light energy will pass through to the water surface.

Caveats:

1) Some of the wattage of a bulb is radiated as heat or ultraviolet (UV) energy, not as visible-light energy. However, compact fluorescents run fairly cooly, so less power is lost to heat. Both the heat and UV losses will be "self correcting," as shown below.

2) Not all of the visible light from a bulb is useable by plants. That is, it is not "photosynthetically active radiation" or PAR. This caveat will also be "self-correcting."
------------------------------

Moving on now...

For the the hobbyist, the best surrogate measure for the areal flux of PAR at the water surface is the total bulb wattage divided by the water surface area (length of tank times its width). We know that the old rule of thumb provided adequate light for a planted tank, as long as the tank wasn't too deep. This rule of thumb was developed through trial and error by hobbyists with different tank sizes and shapes observing how well their plants grew given a certain wattage of appropriate bulbs. All of the secondary issues of energy loss to heat and UV, the difference between total energy flux and the flux of PAR, etc., are accounted for in the trial-and-error evolution of the old rule of thumb, so if I make use of the old rule in the development of a new (obstensibly, more correct rule), the issues or caveats will be self-correcting and accounted for.

The range of tank dimensions used by most hobbyists is fairly narrow, so I'm going to assert the existence of a "typical" or average tank, which I'm going to designate as the "reference tank." From my assessment of the planted-tank hobby, I would propose that a typical, average, reference tank is about 55 gallons in volume and has dimensions of about 36"L x 18"W x 20"D. If my assertion and assessment are valid (and I believe that they are!), I can be reasonably certain that the old rule of thumb of 2-3 WPG will provide good plant growth all the way to the bottom of the reference tank. 2-3 WPG in the reference tank translates to an areal energy flux at the water surface of about 0.17-0.25 Watts per square inch (W/in^2). This becomes the first part of the new rule of thumb. What I still require is an explicit correction for the effect of tank depth that ensures adequate light for good plant growth at the bottom of a tank of any given depth.

The attenuation of light intensity as it passes through water is generally described mathematically as a exponential decrease with depth. The equation looks like this:

I(d) = I(0) x exp(-k x d).

In words...the light intensity at depth (d) is equal to the intensity at the water surface (I at depth 0) times "e" (= 2.17183) raised to the power of (-k times depth d), where k is the "attenuation coefficient" and controls how rapidly light energy decreases with water depth. Note that for the 20"-deep reference tank, the required intensity at the water surface [I(0)] was 0.17-0.25 W/in^2. For a deeper tank, more light would be required at the surface. For a shallower tank, less light would be required at the surface.

Without going through the tedious algebra, its pretty simple to develop a correction factor that accounts for depth explicitly and ensures that the light intensity at the bottom of a tank of any depth is the same as the intensity at the bottom of the reference tank (which we know through the combined experience of planted-tank hobbyists has adequate light intensity for good plant growth).

The new rule of thumb for a tank of depth D now looks like this:

Required wattage = (0.17 to 0.25 Watts/in^2) x L x W x exp[k x (D - 20)]

where L, W, & D are tank length, width, & depth in inches. All that remains is to assign a value for the attenuation coefficient k.

The attenuation coefficient k is a function of the molecular structure of water, scattering and absorption of light energy by suspended particles, and scattering and absorption by dissolved substances (particularly colored substances like peat or blackwater extract.) If one had access to a light meter, it would be possible to measure light attentuation in a tank and determine k directly. Obviously, this isn't a pratical solution for most hobbyists. What I did is seek a value for k from the scientific literature on lakes. Most of us keep our water well filtered and clean, so I restricted my search to clean lakes. I found two values that might be representative. Crater Lake in Oregon is a very pristine lake with superb clarity and has a measured k of 0.10 per meter. Lake Tahoe is slightly less pristine but still very clear and has a measured k of 0.15 per meter. I took the average of these two values and converted meters to inches to get a "representative" k of about 0.0032 per inch. So the finished new rule of thumb is:

Required wattage = (0.17 to 0.25 Watts/in^2) x L x W x exp[0.0032 x (D - 20)]

where all length measurements are in inches.
----------------------------

So now we can examine explicity the potential importance of tank depth for the typical hobbyist...assuming of course that my analysis here is reasonably valid.

The depth range for hobbyist tanks is pretty narrow, say 12" deep at the low end to 30" deep at the high end. By my formula, the 12"-deep tank could get away with 2.5% less light intensity than the 20"-deep reference tank, and the 30"-deep tank would require about 3.3% more light intensity than the reference tank. So worrying about tank depth seems to be a waste of time for most hobbyists, which is really why the old rule of thumb of 2-3 WPG worked. Dimensionally it was incorrect to use a volume measure in the old rule of thumb, but since the depth ranges for most tanks is so narrow, it didn't matter. You could use a dimensionally correct areal rule of thumb like I have developed and either include or ignore the depth correction, or you could include the depth in the tank volume instead. All three methods would give the typical hobbyist very similar answers.

But what about some of the Amano tanks, pictures of which have been posted in this forum? One tank I saw was probably 4-5 feet deep. A 5-ft-deep tank would require about a 14% increase in bulb wattage to give the same light intensity at the bottom of the tank as that in the reference tank.
--------------------------

OK, that was a lengthy post, but I'm nearly done. Obviously, I've made a number of assumptions that could be in error. It would be great if someone would actually use a light meter to measure light attenuation in tanks of different depths. But I personally would not be surprised if it was found that tank depth was inconsequential to the planted-tank hobbysist. Peaty or blackwater tanks are another critter entirely, since they would require a much higher attentuation coefficient.

**The results you achieve at home might differ from those presented here!**
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Old 02-26-2004, 12:33 PM   #2
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Isn't a 55 gallon tank 48 x 13 x 20 ? A 30 gallon tank is 36 x 18 x 16.
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Old 02-26-2004, 01:30 PM   #3
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Splash,

I have been trying to tell people for a long time that the depth of the tank is basically irrelevant in the hobby. People get all bent out of shape because their tank is 24" deep and they JUST KNOW it's going to take a lot more light because of the depth. And they start talking about the inverse square law and such. But I point out to them that we are talking INCHES here and not feet.
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Old 02-26-2004, 01:55 PM   #4
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Great post, Splash. I'm sure that took a lot of time and careful effort. I've argued with many hobbyists that the correction factor for depth in most aquariums is relatively insignificant. One hobbyist took his expertise from a consultant for public aquarium displays, and obviously the depth of such displays required that they took aquarium depth into consideration.
Absolute intensity aside, I'm still concerned about the attentuation of various wavelengths in an aquarium, especially those above 650nm in the red portions of the visible spectrum, since they see attentuation in water at a rate many times that of wavelengths below that level. This is why we are seeing a prevalence of "pinkish" plant-grow bulbs recently in this hobby. While the increased red output might be unecessary at the water surface, a somewhat smaller portion of that red light is available two feet later.
The correction factor is still small, and as you mentioned, this is supported by the fact that the WPG rule has been supported by many years of empirical data.

Here's some more interesting reading on the subject at the APD archives:
http://fins.actwin.com/aquatic-plant.../msg00467.html
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Old 02-26-2004, 01:57 PM   #5
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It has not been directly said yet so: Great article. Way over my head in coming up with the calculations but informative, none the less.

I believe George Booth actually measured the lux of various brands of t-12 and metal halide lighting (aquaticconcepts dot thekrib dot com) . I'm not sure how long ago this was but it may be of some use if your interested. However, it would be nice to see how things compare with t-8's and PC lighting and maybe even the t-5's.
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Old 02-26-2004, 05:32 PM   #6
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Quote:
Originally Posted by SCMurphy
Isn't a 55 gallon tank 48 x 13 x 20 ? A 30 gallon tank is 36 x 18 x 16.
I wish there was 18" width in my 30!

standard formula (LxWxD)/231
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Old 02-27-2004, 01:59 AM   #7
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Thank you for your comments, GCA, you stimulated some additional neurons to fire!

The two lake attenuation coefficients that I cited are average values for all PAR wavelengths (300 nm to 720 nm). The main photosynthetic pigment for green plants, Chl a, has energy absorption peaks at wavelengths of 435 nm (violet) and 675 nm (red) and an absorption low at 550 nm (green). As you note in your post, light attentuation increases fairly dramatically with increasing wavelength, so it is harder to stimulate the 675-nm absorption peak in a deeper tank. If we knew the attenuation coefficient (k) for aquarium water at 675 nm, we could calculate a more conservative depth correction factor that would require an increase in bulb wattage until the amount of red light reaching the bottom of a deeper tank was the same as that in the shallower reference tank.

While poking around on the web, I came across the following table (see below) of wavelength-specific attentuation coefficients for pure (DI) water. I can't say how applicable these values are to lake or aquarium water, but I note that the PAR-averaged k that I used in my previous post (0.125/m) falls close to the middle of the PAR spectrum (~ 585 nm). So for our nice clean aquarium water, perhaps attentuation coefficients for DI water aren't too bad. If so, then the k for the 675 nm Chl a absorption peak is around 0.44/m, 3-1/2 times higher that the PAR-averaged k.

So a hobbyist who wanted red light (675 nm) intensities at the bottom of a 30"-deep tank to be the same as those at the bottom of the 20"-deep reference tank would need to increase bulb wattage about 12% beyond either the WPG rule of thumb or my (Watts per square inch tank opening) rule of thumb. This is still a fairly small depth correction factor, and I suspect that it is already built into both rules of thumb. 3 WPG is a 50% increase over 2 WPG, so rather than worrying about a calculated depth correction factor, the hobbyist with the 30"-deep tank should just go with the high-end value of the rule of thumb.

Your comments about the "pinkish" growth bulbs were also intriguing. I wonder whether that pink light is actually "violet" for stimulating the Chl a absorption peak at 435 nm. As you can see from the table below, the attenuation of violet light in water is very low, so all that spectral energy will be available right down to the bottom of even a 30"-deep tank. So, while the violet light might make colors in the tank look odd, it might be a very efficient growth bulb.
------------------------------

Table 1: Table of absorption coefficients for visible light (from Light and Photosynthesis in Aquatic Ecosystems, 2nd Edition, J.T.O. Kirk, 1994; based on Smith and Baker, 1981)

Wavelength Absorption coefficient
(nanometer) (per meter)
380...................0.023
390...................0.020
400...................0.018
410...................0.017
420...................0.016
430...................0.015
440...................0.015
450...................0.015
460...................0.016
470...................0.016
480...................0.018
490...................0.020
500...................0.026
510...................0.036
520...................0.048
530...................0.051
540...................0.056
550...................0.064
560...................0.071
570...................0.080
580...................0.108
590...................0.157
600...................0.245
610...................0.290
620...................0.310
630...................0.320
640...................0.330
650...................0.350
660...................0.410
670...................0.430
680...................0.450
690...................0.500
700...................0.650
710...................0.839
720...................1.169
730...................1.799
740...................2.38
750...................2.47
760...................2.55
770...................2.51
780...................2.36
790...................2.16
800...................2.07

Quote:
Originally Posted by GulfCoastAquarian
Great post, Splash. I'm sure that took a lot of time and careful effort. I've argued with many hobbyists that the correction factor for depth in most aquariums is relatively insignificant. One hobbyist took his expertise from a consultant for public aquarium displays, and obviously the depth of such displays required that they took aquarium depth into consideration.
Absolute intensity aside, I'm still concerned about the attentuation of various wavelengths in an aquarium, especially those above 650nm in the red portions of the visible spectrum, since they see attentuation in water at a rate many times that of wavelengths below that level. This is why we are seeing a prevalence of "pinkish" plant-grow bulbs recently in this hobby. While the increased red output might be unecessary at the water surface, a somewhat smaller portion of that red light is available two feet later.
The correction factor is still small, and as you mentioned, this is supported by the fact that the WPG rule has been supported by many years of empirical data.

Here's some more interesting reading on the subject at the APD archives:
http://fins.actwin.com/aquatic-plant.../msg00467.html
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Old 03-01-2004, 04:40 AM   #8
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My new tank will be 22-24 inches tall. Most people at LFS's kept saying I would have trouble with lighting my plants.

It would be nice (and more useful) to have this thread edited and uploaded in the article section.
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Old 03-01-2004, 02:02 PM   #9
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Hi Basilisk,

Even though my ideas might have a reasonable scientific basis, you should not accept them as "fact" until they are validated with actual measurements. We need someone with an accurate light meter to check the change in light intensity with depth in "natural" aquarium water to get a more useful estimate of the attenuation coefficient (k).

IMHO, the biggest problem with a deep tank is the strain on your back trying to work down inside of them.


Quote:
Originally Posted by Basilisk
My new tank will be 22-24 inches tall. Most people at LFS's kept saying I would have trouble with lighting my plants.

It would be nice (and more useful) to have this thread edited and uploaded in the article section.
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Old 03-01-2004, 03:10 PM   #10
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I agree with Splash, this is an interesting debate, but it does need some validation before it is engraved in stone. On the other hand, the WPG rule itself has seen a great deal of validation through experimentation with a myriad of hobbyists finding success in it.

I don't have time to run the compensation factors for the various wavelengths in the Photosynthetic Action Spectrum) PAS, but your 12% increase for 675nm seems about right. As a matter of fact, if you look at the spectral output of most "plant bulbs" like GE AquaRay 9325K, Sylvania Gro-Lux, etc. the balance of phosphors actually seems to have a built in "compensation factor" for the various wavelengths in the PAS, given the typical depth of 20"-24" in most aquaria.
This puts all the more stress on the use of Plant-Specific bulbs. Instead of increasing intensity across the board for all wavelengths, spectral output should be tailored to the plants needs, as well as attenuation of various wavelengths in water.
Sure, the bulbs might look pink to us, and some might not like the appearance of their fish in this type of light, but if we are putting so much effort into other areas of plant care (substrate, CO2, nutrients, etc.) then it is worthwhile to get used to the appearance of this type of light.
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Old 03-01-2004, 04:46 PM   #11
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Not to be contradictory , but having played around with these ideas and done a little literature search on pigment action spectra, I've become more of a fan for mixes of bulbs to get more intensity across a broader spectrum.

Consider the 6700K bulbs that I currently use over my tank. They have a fairly broad, but low intensity peak from about 400-500 nm, which is a good match for chlorophyll absorption at those wavelengths. The bulbs also have a sharper moderate-intensity peak at about 625 nm. That one seems a bit low, since the chlorophyll absortion peak is closer to 675 nm, but the cholorphyll will still catch some of it. The bulb's highest intensity peak is at 550 nm, which is where chlorophyll absorption has a low. This accounts for the yellow-green tone of the 6700K bulbs. They will provide plenty of green light to reflect off your plants and make them look pretty, but light at 550 nm might seem kind of wasted in terms of promoting plant growth. However...

...Carotenoids, which are the yellow-green-orange pigments, have a broad absorption peak in the range of 500-600 nm. So those of you who have nice red contrast plants in your tanks really ought to consider using at least one 6700K bulb. Your red plants will thank you for it. In fact, since I've starting using these bulbs, my Red Ludwiga has become much more red, and my crypts have shifted from very green to more of an olive brown....my plants are shifting their pigment composition to better utilize the bulb's high output at 550 nm.

The point of all this is that for a mix of plants in your tank, it seems like "something" is utilizing light across the entire PAR spectrum from 300-700 nm. Personally, I feel that two 6700K bulbs for my tank is overkill at 550 nm and weak at 435 nm and 675 nm. I need to change out one of my bulbs and add another that provides more light at the two chlorophyll absorption peaks. Hopefully, that would balance the growth of all of my plants and give me an nice balance of green and red pigments in the tank.

So let me leave a question for discussion on this forum. Rather than messing around with mixtures of bulbs, why shouldn't I just use two 10,000K bulbs that have a very broad, continuous spectrum from 400-700 nm? (Although they also appear to put out a bit of UV, which I'm not particularly keen on having in my tank.)




Quote:
Originally Posted by GulfCoastAquarian
Instead of increasing intensity across the board for all wavelengths, spectral output should be tailored to the plants needs, as well as attenuation of various wavelengths in water.
Sure, the bulbs might look pink to us, and some might not like the appearance of their fish in this type of light, but if we are putting so much effort into other areas of plant care (substrate, CO2, nutrients, etc.) then it is worthwhile to get used to the appearance of this type of light.
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Old 03-01-2004, 05:50 PM   #12
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Not contradictory at all, I agree. A mixture of bulbs is always best since no one combination of phosphors is going to provide all of the light a plant needs. I mix GE9325K bulbs with 6700K bulbs.

As for the question you posed about broad-spectrum bulbs, I believe the explanation is this: a bulb of a given wattage has the capacity of only so much excitation of the electrons in the tube. From this energy, the excited phosphors selected by the manufacturer provide light various parts of the visible spectrum. The balance and amount of the phosphors determines the "color" of the bulb.
So, while a good "daylight" bulb like the 10,000K bulb you mentioned with a broad, continuous spectrum from 400-700nm, more of that energy is put towards areas of the spectrum that plants are less able to utilize. If you overlapped a curve of a "plant bulb" with relative intensity, you would see higher output in areas of PAR-sensitivity.

Like you said, though, these elevated regions in the PAS might not necessarily "nail it" for the plants as far as their needs go. Especially red/dark plants. A mixture of bulbs ensures you span the entire necessary spectrum a plant needs.
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Old 03-01-2004, 11:04 PM   #13
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I think there's a misconception on my statement. You say the average height would be around 20-24 inches. I wasn't aware that was the avergae in the US. I live in Mexico, and most people kept saying that if I went over 18 inches tall I would have much trouble with lighting.

What I really thought when posting that was on the people around here that futilely concern about a couple more inches, when they already have over 2.5 wpg, or more when using PCs. In my case, the substrate will use three or four inches of the total depth, which leaves me with the initial 18-20 inches of actual height to be lit. I will send them a link to the thread.

I hope they allow me to conduct this experiment in the physics lab if there's a lightmeter.
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Old 03-02-2004, 12:52 PM   #14
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A lot of the glass tanks here run 18 inches deep, but some are 20 inches. The acrylic tanks seem to run deeper in general, 20-24 inches. And certain shapes run deeper still; for eaxmple, an acrylic bowfront tank generally runs 24-30 inches deep.

I think the deeper ones were originally designed for SW reef tanks. The aquarium industry hasn't really responded yet to the growing numbers of planted tank hobbyists.

I really like the bowfront shape, but I don't want a 24-30-inch deep tank. Not because of lighting, but because I don't want to have to hang upside down in the tank to reach the bottom for planting, trimming, and cleaning.




Quote:
Originally Posted by Basilisk
I think there's a misconception on my statement. You say the average height would be around 20-24 inches. I wasn't aware that was the avergae in the US.
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Old 03-02-2004, 01:44 PM   #15
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The true average height might be even closer to 12" since 90% of the fish tanks in the US must be 10 gallon tanks. Talk to an aquarium manufacturer and see how many 10g tanks they make vs the rest of their stock. But regardless, Basilisk is right to say that it is futile to worry over a few inches of height. But for this discussion's sake, red wavelengths are definitely attenuated quickly and the difference between 18" and 22" is significant.
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plantedtanker in limbo - all tanks currently in storage
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