As I mentioned earlier, the equation I have posted is not completely accurate in its present form. Therefore, I will expand it to account for several of the major factors that influence filtration capacity. Namely, the expanded equation will deal with temperature, chemical purification IFUs, and residual flow IFUs. Again, I'm going to copy and paste part of my thesis so that I won't have to type as much:

In chapter two, I presented my basic theory on stocking an aquarium with respect to filtration by aerobic bacteria and a basic equation was derived. However, as I have already mentioned, that equation does not account for all the major factors that affect filtration capacity in a given aquarium and is therefore incomplete. To rectify this situation, this chapter will examine those additional factors and build on to the f(MAT).

Please bear in mind that filtration capacity is only one aspect of the MOA and later chapters will deal with material that, in some cases, supersedes filtration capacity as being the dominant item determining how many fish can be kept in a given freshwater aquarium. That being said, it is impossible to keep fish alive for a prolonged period of time without some sort of filter and, subsequently, filtration capacity is often the foremost concern when setting up a new aquarium. Because filtration capacity is vital to the lives of captive fish, I have taken a great amount of care in studying the subject and developing useful means of extending the f(MAT). Specifically, this chapter addresses the effects of temperature, chemical filtration, residual flow, [and cleaning frequency and quantity].

Starting where we left off in chapter two, an approximation for the number of IFUs an aquarium can support is equal to the product of the M, A, and T multiplied by .0005 multiplied by the gallon size of the aquarium:

This approximation, though well-justified, seems contradictory to what we already know when one considers that the more accurate general guidelines in chapter one all agreed that the stocking requirements of temperate fish differed from those of tropical fish. That is, it seems that fish from cold water require more space and filtration capacity than their warm-water counterparts. This being the case, the problem with the above equation is that it makes no distinctions based on temperature and thus does not reflect reality, or does it?

To answer the question of whether or not our equation is accurate, we need to examine the major properties of aquarium systems with respect to temperature. These major properties that are contingent upon temperature are oxygen solubility and bacterial activity. Oxygen solubility is the relative capacity of water at different temperatures to absorb oxygen. Since aerobic bacteria require dissolved oxygen to convert wastes into nitrate, it is reasonable to assume that their ability to biologically filter the water will be dependent upon the oxygen solubility of the water. Bacterial activity, on the other hand, is a relative comparison of the metabolic rate of bacteria at various temperatures. The net result of bacterial activity is that a greater metabolic rate produces a greater filtration capacity as the bacteria will be processing more wastes per unit of time. It is these two properties of oxygen solubility and bacterial activity that are the ones related to temperature that have a measurable effect and it will be interesting to see which of them, if either, dominate filtration capacity.

Before we can examine temperature in detail, a relative standard must be decided upon that coincides with the standard conditions used to create the f(MAT). If you recall section 2.6, you might remember that the inch per gallon rule and the tropical conditional statements of the graduated area rule were used to determine the standard conditions of the f(MAT). Thus, as both rules dealt with tropical aquarium systems, it is reasonable to assume that a tropical temperature should be used as the standard temperature in our comparisons. As to the specific measure of that temperature, you can consult almost any reference book on aquarium husbandry to discover that a temperature of 78° Fahrenheit is recommended for any given tropical aquarium system. Since 78° Fahrenheit is our standard temperature, all measurements and equations will be adjusted so that the standard temperature returns a value of one. This simply means that the f(MAT) is accurate at 78° Fahrenheit, but will be inaccurate at any other temperature by a factor below or above the number one in value.

Oxygen solubility is inversely dependent upon temperature in such a way that as temperature increases, oxygen solubility decreases. This means that warm water (tropical water) will contain less dissolved oxygen than water of a lower temperature (temperate water). This relationship means that if a temperate fish is housed under tropical conditions, it will be forced to make due with less oxygen than it is used to. Nevertheless, the inverse relationship of oxygen solubility and temperature does not fully explain why temperate fish require more space and filtration capacity than tropical fish. In fact, based on oxygen solubility alone, there should be little difference between the requirements of these two groups of fishes provided they are kept in aquariums representative of their respective temperature ranges.

To be more precise in the description of oxygen solubility, I will produce an equation from some basic facts so that you can see how oxygen solubility varies with temperature:

Basic Facts: 78° (all temperatures will be in Fahrenheit) returns a value of 1; water at a temperature of 32° contains nearly twice as much oxygen as water at 86° .

Applicable Inverse Formula: O.S. = K / (T + h)

Respective Unsolved Formulas:

Equation 1: O.S. = 1 = k / (78° + h)

Equation 2: 2k / (86° + h) = k / (32° + h)

Proof:

2k / (86° + h) = k / (32° + h) Þ 64k +2kh = 86k +kh Þ kh = 22k Þ h = 22

Substituting the number 22 into Equation 1: 1 = k / (78 + 22) Þ 100 = k

Therefore: O.S. = 100 / (T + 22)

When the above formula is graphed, you can see how oxygen solubility changes with temperature. In the following graph, the red line represents the solubility factors of all temperatures from 40° to 140° Fahrenheit while the green boxes represent the solubility factors of the temperatures commonly seen in home aquaria (64° to 86° Fahrenheit). Note the curving, downward slope of the graph that coincides with the general trend that as temperature increases, oxygen solubility decreases.

[No graph available in this format]

Bacterial activity, unlike oxygen solubility, rises and then decreases as temperature increases. Thus, bacterial activity peaks at a specific temperature and is not appreciable past certain values. With regard to differences between temperate and tropical environments, this means that their will be a range of temperatures that produce higher filtration capacities than others.

To determine the formula for bacterial activity, I have found that the most useful information can be gathered from the culinary arts. In the food industry, the unsafe range of temperatures to store food is from 40° Fahrenheit to 140° Fahrenheit. This is the range of appreciable bacterial activity and is thus an appropriate base for a formula that approximates bacterial activity. Furthermore, the equation I will produce assumes that bacterial activity peaks at a point directly between the extremes dictated by the culinary arts--90° Fahrenheit. Based on information that I have read and observed, this assumption that bacterial activity peaks at 90° Fahrenheit is accurate enough for our purposes, which is to simply assign relative bacterial activity values to a given range of temperatures.

Basic Facts: 78° returns a value of 1; temperatures of 40° and 140° return values of 0.

Applicable Formula: B.A. (bacterial activity) = a (some constant) x (T (temperature) - 90 (shift to center of established range))² + k (constant of vertical shift)

Respective Unsolved Formulas:

Equation 1: 1 = a (78 - 90)² + k Equation 2: 0 = a (40 - 90)² + k

Proof:

-1 = -144a - k

-1 = 2356a

a = -1/2356

Substituting “a” into Equation 1:

1 = -144/2356 + k

k = 625/589

Therefore: B.A. = - (T - 90)² / 2356 + 625 / 589

[No graph available in this format]

The above graph of bacterial activity relative to temperature reveals that filtration capacity falls off as the temperature departs from 90° Fahrenheit. However, you will notice that the difference in values for bacterial activity is relatively larger than the difference in values for oxygen solubility over the range of common home aquarium temperatures. As in the previous graph, the red line represents the factors of all appreciable temperatures while the green boxes represent the factors associated with typical aquarium temperatures.

Now that the specific effects of oxygen solubility and bacterial activity have been examined, we can use them to manipulate the f(MAT) and determine an appropriate biological filtration capacity for all temperatures ranges. To do this, one might suppose that oxygen solubility and bacterial activity compromise to form a limiting factor. That is, the factor that affects the f(MAT) might be somewhere between the values of oxygen solubility and bacterial activity. However, this is not the case. In reality, one factor limits the other and therefore dominates depending on the specific conditions. For example, the oxygen solubility factor at low temperatures is fairly high in value, but the bacterial activity value is comparatively low. As such, the surplus of oxygen at low temperatures doesn’t mean anything since the bacteria’s metabolic rate can only be so high. Similarly, the bacterial activity factor at warmer temperatures is high, but the oxygen solubility factor at such temperatures is comparatively low. Consequently, oxygen solubility dominates warmer temperatures. Taking this relationship of limiting factors into account, the graph of the factors at common aquarium temperatures is as follows:

[No graph available in this format]

In the above graph, you can see that biological filtration by aerobic bacteria is most efficient at exactly 78° Fahrenheit. This is part of the reason why experts recommend a temperature of 78° Fahrenheit for tropical aquariums. Another thing to note about the graph is that it indicates that temperate conditions are indeed less efficient that tropical conditions. However, you will notice that the lowest factor on the graph is a little above 0.77 and this factor does not agree with the general rules discussed in chapter one. According to the general rules, temperate fish should require about double what tropical fish do and thus the appropriate factor should be around 0.5. Because of this, we still have not explained the general rules adequately.

To discover why the general rules use a factor of 0.5 for temperate aquarium systems, we must also consider their definition of a temperate fish. Most of the general rules define temperate fish as goldfish and other similar species, which are longer than your average tropical fish by about 2 inches (a difference of about 180 IFUs). Accordingly, the average temperate fish is going to require more space just because of its size. In the net, the general rules use a factor of 0.5 for temperate aquariums (indicating that only half as many fish should be housed in such an environment relative to a tropical environment) because the biological filtration complex is less efficient at those temperatures and temperate fish are generally larger than their tropical counterparts.

With regard to the MOA, though, we can be much more precise than the general guidelines because the MOA accounts for fish biomass with IFUs and it can assign specific factors to specific temperatures. That is, the general rules yield a ballpark estimate while the MOA can tell you how close you are within a few percentage points. Taking this into consideration, the f(MAT) can be expanded to the following form:

This equation now accounts for six factors that affect your aquarium: the biomass of the fish, the gallon size of the aquarium, the medium that the filter bacteria grow on, the air-water interface of the aquarium system, the turnover of the filtration system, and the temperature of the water. With regard to knowing what factor to use for a specific temperature, I have made the following table. To use the table, find the specific water temperature you plan on keeping your fish in and insert the corresponding factor into the above equation. Also, you will note that I have included two types of factors into the table--a specific factor that represents only the selected temperature and a factor that represents the selected temperature with three degrees tolerance. If your aquarium temperature is very stable, it is alright to use the specific factor. On the other hand, if your aquarium temperature fluctuates a bit or the thermometer that you use to measure the water temperature is not very accurate, the factor with three degrees tolerance should be used. This means that the factor will compensate for minor fluctuations in temperature so that you do not have to re-assess your aquarium every time the temperature changes.

Water Temperature (F)	Specific Factor	Factor with 3 Degrees Tollerance
64	0.7742	0.7042
65	0.7958	0.7284
66	0.8166	0.7517
67	0.8366	0.7742
68	0.8557	0.7958
69	0.8739	0.8166
70	0.8913	0.8366
71	0.9079	0.8557
72	0.9236	0.8739
73	0.9385	0.8913
74	0.9525	0.9079
75	0.9656	0.9236
76	0.9779	0.9385
77	0.9894	0.9525
78	1.0000	0.9656
79	0.9901	0.9615
80	0.9804	0.9524
81	0.9709	0.9434
82	0.9615	0.9346
83	0.9524	0.9259
84	0.9434	0.9174
85	0.9346	0.9091
86	0.9259	0.9009

Since I have built on to the f(MAT), this might be a good place to give an example of how the temperature factor changes the results of our previous equation. Consider a 29 gallon that has a MAT of 22,000 units. By the f(MAT) alone, such an aquarium could support 319 IFUs (29 x .0005 x 22,000 = 319 IFUs), but when temperature is taken into consideration the IFU value may change. If, for instance, the aquarium is maintained at a temperature within three degrees of 73° Fahrenheit, then the IFU value would be 284.3247 IFUs (319IFUs (as calculated by f(MAT)) x 0.8913 (3 degrees tolerance factor) = 284.3247 IFUs). As a general rule, real world conditions dictate that an actual biological system will be less efficient than what the f(MAT) predicts. Due to this fact, one should always consider temperature before stocking an aquarium with fish. 

While it is true that practically every filtration system seen in freshwater aquaria makes use of aerobic bacteria, that does not mean that biological filtration is the only type of filtration that can occur in a given aquarium. In fact, most aquarists prefer to use multiple filtration types to ensure that there is no possibility of a build-up of dangerous ammonia in the water. One of the categories that non-aerobic filtration can be divided into is chemical filtration--a process by which a medium chemically alters or absorbs a substance to make it inert.

In the modern aquarium world there are many, many, many chemical filtration systems out there, but I have only found two that give consistent, reliable results. These are activated carbon systems and zeolite systems. Both of them work by absorbing substances and therefore removing them from the water flow. In order for either to work properly, they must be placed in the water flow in such a way that they cannot be covered by debris. Thus, in a multi-stage filtration system, chemical purifiers occur after the biological medium. This keeps the surface of the chemical purification medium exposed so that it can absorb passing chemicals.

Specifically, activated carbon is charcoal that has been superheated. This heating changes its chemical properties and allows it to absorb certain types of chemical compounds (including many nitrogen compounds). Because it is the heating process that enables activated carbon to behave in such a manner, you must understand that not all packages that are labeled “Charcoal” or even “Aquarium Charcoal” will work. If you intend to use an activated carbon system, look explicitly for activated carbon or, better yet, GAC. GAC is a fairly common acronym for Granulated Activated Carbon and represents a very high quality product that is sure to produce results.

The performance of activated carbon is unquestionable but it has two major drawbacks. The first is that is that it does not distinguish between substance types very well. As such, it will remove dangerous substances and harmless substances alike. This means that some of its abilities will be wasted on the harmless substances and it will not perform ideally for removing the dangerous substances. Second, activated carbon does not last forever and thus has to be replaced at the expense of the aquarist. Most freshwater stocking levels dictate that the activated carbon will only last for about two weeks before it is spent, a month if you only keep a few fish or use double the normal amount of activated carbon.

Zeolite, on the other hand, absorbs substances in a manner similar to activated carbon but is much more discriminatory with regard to the specific types of substances it removes than activated carbon is. Zeolite is very good at removing ammonia from the water. In fact, zeolite will not remove any common, significant substance except for ammonia. This condition, even though it seems like an advantage, is the greatest shortcoming of zeolite since an aquarium that uses any appreciable degree of biological filtration will not contain much ammonia. Thus, zeolite does not serve a vital function in most properly maintained aquarium systems. That said, zeolite is a nice thing to have on hand should the primary filtration system fail for one reason or another. Zeolite is best thought of as added security in that it acts as a fail-safe should something go wrong.

Because chemical filtration removes or converts wastes, it has a direct impact on the number of IFUs an aquarium can support. These additional IFUs are called chemical purification IFUs and are added to the result of the equation in section 3.1. Consequently, the f(MAT) can be extended to account for both aerobic filtration and chemical filtration. To simplify things a bit, chemical purification IFUs will be abbreviated as the letter C. The value of C can be determined by either of the following equations:

If the activated carbon is changed monthly………………C = (ounces of activated carbon) x 30

Or, if the activated carbon is changed every two weeks….C = (ounces of activated carbon) x 64

Note that the above equations only use activated carbon in the computation of C. This is due to the fact that zeolite does not have an appreciable effect in a properly operating aquarium. Justification for the above equations can be found in several reference books that say that one-third of an ounce of activated carbon should be used per gallon for systems that rely primarily upon chemical filtration with the understanding that the activated carbon is changed monthly. Since we already determined in section 2.6 that the average tropical aquarium has about 10 IFUs per gallon, the number of IFUs that a whole ounce of activated carbon would yield per month is 30 IFUs (1/3 oz. x 3 = 1 oz.; 10 IFUs/(1/3 oz.) x 3 = 30 IFUs/(1 oz.)). Similarly, if the frequency that the activated carbon is changed is shifted to every two weeks, then the number of IFU would increase by a corresponding factor--one ounce would yield 64 IFUs per gallon (30 (number of days in a month) / 14 (number of days in two weeks) = 2.1429; 30 (IFUs/gal./month) x 2.1429 » 64 IFUs /gal./two weeks).

The new expansion of the f(MAT) is as follows:

As an example of adding chemical purification IFUs, consider a 10 gallon aquarium with a MAT of 40,000, constant temperature of 75° Fahrenheit, and 1.5 ounces of activated carbon in its filtration system. By the first part of the above equation, the aquarium can support 193.12 IFUs (40,000 (MAT) x .0005 (factor of MAT equation) x .9656 (specific temperature factor) = 193.12 IFUs) due to filtration by aerobic bacteria. However, additional IFUs can be added to this number since the aquarium system uses activated carbon. The number of IFUs added will depend on how often the activated carbon is changed. If it is changed every month, the new total filtration capacity will be 238.12 IFUs (193.12 IFUs + 1.5 (oz. of activated carbon) x 30 (factor for monthly changing) = 238.12 IFUs). Alternatively, if the frequency the activated carbon is changed is shifted to every two weeks, then the total IFUs the aquarium could support would be 289.12 IFUs (193.12 IFUs + 1.5 (oz. of activated carbon) x 64 (factor for bi-weekly changes) = 289.12). As you can see, adding 1.5 ounces of activated carbon to such a system does not make a huge impact on total IFUs, but the effect is significant enough to enable the addition of several reasonably sized fish to the aquarium system.

Residual flow is a phenomenon that happens whenever a strong flow of water is near a porous object or substance. When such a situation occurs, micro-vortexes form on the surface of the object or substance and a small percentage of water will flow through the item. This creates even more surface area that bacteria can grow on, but most of these bacteria will be anaerobic, denitrifying bacteria because of the extremely minuscule flow rate.

As mentioned before, anaerobic bacteria break nitrate down into inert substances that can not harm any of the aquarium inhabitants. The downside of the presence of these specific bacteria is that if the flow of water is not strong enough or their location is too far away from the flow of water, then they will produce dangerous byproducts. Additionally, many pathogens like areas that do not have a strong flow rate. This means that residual flow areas could produce disease organisms that can make your fish sick.

To maximize the potential of residual flow areas while nullifying their fatal potential, there are three basic things that should be done:

To be perfectly clear, objects or substances that can be influenced by residual flow include lava rock, pumice, lace rock, deep gravel (over 5 inches deep), depths of sand exceeding two inches in thickness, ceramic pots, clay ornaments, and sponge medium not inside a filter. If your aquarium does not coincide with the three rules for maximizing residual flow potential while minimizing its risks, then you want to avoid putting these things in your aquarium. On the other hand, if your aquarium conditions are suitable, then placing these items in your aquarium will be advantageous. On average, every pound of an object or substance that encourages residual flow adds 3 IFUs to the aquarium’s biological load potential. This is not much, but it does add.

As with chemical purification IFUs, residual flow IFUs will be abbreviate into a representative letter--R. To calculate the value of R, simply add up the number of pounds that the items in a given aquarium weigh that encourage residual flow and multiply by 3. For example, if you use a layer of sand in your aquarium that is greater than two inches in depth and has a total mass of 20 pounds, then you can add 60 IFUs to the total biomass that your filtration system can support (20 lbs. x 3 = 60 IFUs). This process is as follows:

Subsequently, we have further expanded the f(MAT) to account for the actions of anaerobic bacteria as well as aerobic bacteria, temperature, and chemical purification:

All right, it's your turn to telll me what you think!

"Tears aren't a sign of weakness, they're a sign of poor plumbing."

--Dead Men's Lies

Thinking about the exception that princessotfu (sp?) found made me think about some of the other exceptions regarding M,A, &T. To help you see what I'm talking about, I made a graph that roughly models the data gather in my original testing (temperature, C, R, and other such factors were held constant; tests were done on a fifty-gallon aquarium and two ten-gallon aquariums):

In the above graph, you can see that the MAT product and IFUs/gal. share a linear relationship from 2,000 - 80,000 MAT, but behave asymptotically on either side of those points. As was previously discussed, the graph indicates that an aquarium without a filter (0 MAT product) does have some biological filter potential due to the bacteria that colonize the remainder of the tank. However, the limit of the actions of these non-filter bacteria is about 0.7 IFUs/gal., which is almost ridiculously small. In fact, if you had a ten gallon aquarium with no filter, you could only keep one fish that displaces 7 IFUs, or about a 2.25 inch fish (.7 x 10 gal. = 7 IFUs total; 2.25 x .85 = 1.9125; 1.9125 x 1.9125 x 1.9125 = 6.9953 IFUs = about 7 IFUs).

With regard to the upper limit indicated by the graph, 43 IFUs/gal. is the maximum an aquarium can support because the water can only absorb so much oxygen--regardless of how much you agitate it. Thus, the equation that I gave does not account for MAT products below 2,000 or above 80,000. Realistically, though, these values are pretty rare.

For the MAT equation to be correct, the values of the variable must fall within the following boundaries:

MAT Product	2,000 – 80,000 MAT
M	31 – 1,300 sq. in. / gal.
A	1.3 – 51.3 sq. in. / gal.
T	0.5 – 20 cycles / hr.

Additionally, no aquarium, regardless of how much filtration capacity it has, can hold more than 43 IFUs / gallon and every normal aquarium can hold at least 0.7 IFus / gallon.

Sorry about not posting this earlier. Hopefully it will be helpful. Please tell me if I missed something or if you have a question.

"Tears aren't a sign of weakness, they're a sign of poor plumbing."

--Dead Men's Lies

I might as well provide you all with a table that lists a fish's overall length and its corresponding average IFU size so that you know what I'm talking about when I relate filtration capacity to biomass:

Overall Length (Inches)	IFU Size if Fish is of Average Proportions	Overall Length (Inches)	IFU Size if Fish is of Average Proportions
1.00	0.61	10.25	661.35
1.25	1.20	10.50	710.93
1.50	2.07	10.75	762.93
1.75	3.29	11.00	817.40
2.00	4.91	11.25	874.41
2.25	7.00	11.50	934.01
2.50	9.60	11.75	996.25
2.75	12.77	12.00	1061.21
3.00	16.58	12.25	1128.92
3.25	21.08	12.50	1199.46
3.50	26.33	12.75	1272.88
3.75	32.39	13.00	1349.23
4.00	39.30	13.25	1428.58
4.25	47.14	13.50	1510.98
4.50	55.96	13.75	1596.49
4.75	65.82	14.00	1685.16
5.00	76.77	14.25	1777.06
5.25	88.87	14.50	1872.24
5.50	102.18	14.75	1970.76
5.75	116.75	15.00	2072.67
6.00	132.65	15.25	2178.04
6.25	149.93	15.50	2286.92
6.50	168.65	15.75	2399.38