For many, many years my filtration formula has been linear. This means that my spreadsheets assumed that a filter's ability to process was was directly proportional to its turnover. This system seemed to work for most cases, but it fell extremely short in two areas: 1) it proved inaccurate for canister filters on several occasions and 2) it did not work for large aquariums in general. Now, I must admit that my spreadsheets were never meant for specialized setups (which are often associated with large tank systems and canister filters) and that there are always going to be data points that do not fit the trend-line nicely--the problem was that these filter issues came up too often to simply be flukes. This being the case, I have done more research on filtration systems and have come up with a new set of equations that can match most of data I have available. These equations will be included in the final production run of FishsheetA7 (not the prototype).
These equations are inverse relations (or forms of them) and allow me to compensate for a phenomenon I call "volumetric decay." More specifically, volumetric decay is the process by which an aquarium environment tends to become more effective with regard to filtration as the volume is increased. The net result of volumetric decay is that filters used on larger aquariums do not require as large a turnover as those used on smaller aquariums. This fact can be easily verified by taking a walk through your local pet shop; as you compare the flow rates of one brand/type of filter with its rating (the volume of aquarium it was made for), you find that the ratio of flow rate to tank size decreases as the tank size increases. This changing ratio means that my old linear model was not sufficient.
Before I go much farther I should clarify what I mean by turnover. Turnover, in a basic sense, is literally how often a filter can "turn over" the volume of the tank it is attached to. The particular unit that I often use is cycles per hour despite that most instances that involve cycles are measured in Hertz (cycles per second). The reason for a non-standard unit is that using Hertz would produce very small numbers that are hard to evaluate (1 cycle/hour is equal to one-three-thousand-six-hundredth (1/3,600th) of a Hertz). Moreover, cycles per hour (which I tend to use interchangeably with turnover) is typically much easier to calculate in that most filters are advertised in units based on hours rather than seconds (gallons per hour or liters per hour). In many situations it is relatively easy to assess a filter's efficiency by using the turnover of a particular system. For example, a filter with a GPH of 100 placed on a 20-gallon aquarium would have a turnover of 5 (100 (GPH) / 20 (Gal.) = 5 Cycles per Hr.). This is much more efficient than placing the filter on a 50-gallon aquarium, in which case the turnover would drop to 2 (implying that the same filter on the bigger aquarium could only process 2/5ths of the waste load of the bigger aquarium).
Using turnover as my comparative, I visited several manufacturer's sites and organized the data by filter type (see attachment). Initially, I had a very hard time finding a formula that fit the data, until I stumbled upon the idea of using a moderated inverse relation. A moderated inverse relation is one in which it is assumed that some value (turnover, in our case) is inversely proportional to some other value (tank volume, in our case) with the provision that the relation can never get so low that it approaches a value of zero (it is "moderated" to stay in the realm of real possibilities). The general form of this equation is as follows:
In the above formula, k and c are constants for each family of filters such that internal filters, powerheads, power filter, canister filters, etc. all have different values of k and c.
My first couple of attempts to apply this formula to filter families were quite successful. Below are two charts that reveal how the new formulas compare to what the manufacturers recommended for their products. You will note that, in every case, the turnover values of the specific data points and the new formulas are not only fairly close, but also slope downward and thus follow the rule of volumetric decay.
For all filters in the two above categories, I found no appreciable differences between brand or specific filtration technique. However, for power filters (HOB's, Hang-On-Back filters), I noticed a significant difference between models that either used extra media or had a Bio-Wheel and those that did not have such additions. There were no appreciable differences in performance between models with extra media and those with Bio-Wheels. These conditions led me to create two classes of HOB's: 1) those with added filtration mechanics and 2) those without any added material or devices. The two charts for the different HOB classifications are as follows:
The most significant difference in performance, nonetheless, occurred when I examined canister filters. Most canister filters on the market obeyed the rule set out by the inverse relation, yet Eheim(TM) filters did not. Instead, the only way to make Eheim filters fall in place was to include another variable in the formula: the volume of the canister itself. It would have been more accurate to use only the volume available for media, but using the external measure proved to be enough of an accurate portrayal to make a new formula fit the data. The formula for Eheim(TM) filters is still an inverse relation, but what has changed is that it is now inversely proportional to canister volume rather than tank volume (though the tank volume was included to maintain the validity of the ratio):
In the above equation, E is a coefficient that is specific to different types of Eheim(TM) filters. The coefficients k and c maintain their original significance.
Since some canister filters behave so oddly, FishsheetA7 (not the prototype) has several different formulas to account for the disparities. Below are two of the formulas:
The careful observer will note that media was not a significant factor in many of the above formulas (excluding the canister filters), but this should not be taken to mean that media was not evaluated for each filter type (the attachment below shows the media considerations that went into each data set). Also, the above formulas do not mean that an aquarist does not have to use media. Quite contradictorily, the above formulas indicate that if all of the available space in a filter set aside for media is filled with media, then the filter will perform similarly to most other filters of its same general type. Many persons who advocate one filter medium over another may disagree with the scope of this statement even though there is no significant data to the contrary. Even in the case of the Eheim(TM) filters, the specific media used made no difference on the outcome so long as all the space available for media was, indeed, used for an appropriate media. In this discussion, activated carbon, charcoal, zeolite, and other chemical purifiers are not considered media; since performance in units that relied on these materials was far below par, they were stricken from the data set.