A Proposed Method for Estimating Nitrate Production in Aquarium Systems
Updated on: May 10, 2014 Among the many factors constraining the design of aquarium systems, the production of nitrogenous waste, as a result of proteinaceous decomposition and nitrification, tends to dictate much of the design and maintenance of most systems. After all, an aquatic system simply cannot thrive without adequate means to process, and ultimately remove, nitrogen that is introduced into the system. There are many rules of thumb, such as the "inch of fish rule", that are commonly employed in the aquarium trade. However, such rules of thumb convey very little useful information, and generally serve as heuristic guidelines, in the absence of technical data. In order to accurately design a proper habitat, for a particular grouping of organisms, it is essential to at least estimate the expected bioload of the system. This article aims to outline a very simple and useful method of estimating the production of nitrogenous waste, for use in the design of aquarium systems. Methodology In order to develop a reliable model of the production of nitrogenous waste in aquarium systems, some consideration must be given to the various sources of nitrogen, as well as the overall movement of nitrogen within the system. Various simplifying assumptions can be made, in order to produce an idealized model that lends itself to analysis. Such considerations are simplified by the fact that atmospheric nitrogen, present in the system as a dissolved diatomic gas, is essentially bound and chemically inert. When the atmospheric nitrogen present in the system is ignored, and only the fixed portion is considered, the system can be modelled as an essentially closed system, with respect to nitrogen. Conservation of matter dictates that, absent any addition or removal of fixed nitrogen, the mass of fixed nitrogen within the system remains constant. Although it may change form within the system, from protein to amino acid to ammonia to nitrite to nitrate, the total mass of fixed nitrogen within the system remains unchanged. Therefore, a useful model need only consider the following three factors: the addition of fixed nitrogen to the system, the movement of fixed nitrogen within the system, and the removal of fixed nitrogen from the system. It is worth noting at this point that, although atmospheric nitrogen can be excluded from consideration as potential nitrogenous waste, the movement of nitrogen between bound and fixated forms cannot. The processes of nitrogen fixation and denitrification, if present in the system, will create an interchange between the fixed nitrogen of the system, and the bound nitrogen of the atmosphere. By arbitrarily setting the boundary of the system at this interchange, the two processes can simply be modelled as addition and removal of fixed nitrogen, respectively. As such, use of the word "nitrogen", from this point forward, will be understood to mean fixed nitrogen only (i.e. excluding diatomic nitrogen). Addition of Nitrogen to the System There are at least four general sources of nitrogen in a typical aquarium system, 1. Food 2. Source Water 3. Miscellaneous Nitrogenous Material 4. Fixation Fixation: Starting from the bottom, fixation is negligible to nonexistent in most aquarium systems, and probably not very common at all, except under certain circumstances. There are, of course, cyanobacteria capable of fixating diatomic nitrogen, that may well occasionally colonize some systems. In such cases, allowances would have to be made for the additional nitrogen entering the system. Keep in mind that this particular source of nitrogen is relatively uncommon, and only receives consideration here for the sake of completeness. Source Water: The source water itself can potentially contribute a significant amount of nitrogen. In drinking water, safe levels of nitrogen (as nitrate) are typically set at 10 ppm or lower, in which case the addition of water to the system contributes to the addition of nitrogen as well. Miscellaneous Nitrogenous Material: Any organic material added to the tank necessarily carries nitrogen with it. As the material decomposes, nitrogen will be released as soluble waste, thereby contributing to the addition of nitrogen to the system. It should be noted that bacterial supplements are included as organic material, and must also be considered as a source of nitrogen into the system. Other sources might include leaves, wood, invertebrate shells containing decomposing tissue, etc. Inorganic sources of nitrogen include most plant fertilizers and supplements. Food: The primary and most significant source of nitrogen, that is also common to all aquarium systems, is the food provided to the various inhabitants. An artificial aquatic environment can thrive without nitrogen fixation, nitrogenated water, or miscellaneous organic debris, but it absolutely cannot function for long without a food source. As such, any reliable model of nitrate production must necessarily include food as the primary contributor of nitrogen into the system. Fortunately, the task of accounting for the nitrogen content of a food source is greatly simplified by the fact that it is often well known, and commonly printed on retail packaging. Proteins are generally the only significant source of nitrogen in a given food source (carbohydrates and fats contributing little to none). Most commercial food products provide an estimate of protein content in terms of the "Minimum Crude Protein". The methods used to determine crude protein content lend themselves well to estimating nitrogen content, because they are in fact based on the actual nitrogen content of the food, rather than the true protein content. Typically, the measured nitrogen content of a food source is multiplied by 6.25 to arrive at the crude protein content (based on the fact that the average nitrogen content of amino acids is roughly 16%). Therefore, the actual measured nitrogen content of the food source can be determined by simply taking 16% of the crude protein content. Movement of Nitrogen within the System Once a quantity of nitrogen has entered the system, it will be conserved as it is exchanged between various forms and structures within the system. There are generally two forms in which nitrogen entering the system will eventually be stored, and begin to accumulate. Eventually, it will either be incorporated into the tissues of the various organisms inhabiting the system, or it will accumulate in the water column as excess nitrate. The portion that is stored in the tissues of organisms (keep in mind that this list includes fish, invertebrates, plants, algae, fungi, bacteria, protozoa, etc.) can be thought of as temporarily removed from the system. At some later time, for example during an algae die off or a plant leaf rotting, much of the stored nitrogen will be metabolized and re-released into the system. Again, by arbitrarily setting the boundary of the system in this way, the various organisms inhabiting the system can be viewed as alternating sinks and sources of nitrogen. The particular case of nitrogen being completely metabolized, and stored as excess nitrate, is of particular interest, in that it represents the movement of nitrogen through a 100% metabolic system. In this way, it can be viewed as a sort of baseline rate, about which the actual rate will tend to either fluctuate or simply lag consistently. This baseline rate can be used for design purposes, as an estimate of the expected rate of nitrate production in the system. Removal of Nitrogen from the System There are at least four general ways in which nitrogen is permanently removed from aquarium systems, 1. Water Change 2. Skimming 3. Denitrification 4. Adsorption Essentially, nitrogen is either directly removed from the system, bound into diatomic nitrogen and allowed to escape, or adsorbed to some chemical media for removal. Ultimately, the goal of a well balanced system is to balance the rate of addition with an equal rate of removal. This can, of course, be accomplished quite simply through periodic water changes, or by more sophisticated means through the addition of various skimmers, denitrators, and/or chemical media. The means by which this is accomplished, however, are beyond the scope of this discussion. Each method, as well as each particular implementation, will require its own unique analysis. For the purposes of this model, it will be assumed that the various mechanisms of nitrogen removal can be lumped together into one composite rate of removal. Quantitative Analysis As stated previously, conservation of matter dictates that, absent any addition or removal, the mass of nitrogen within the system remains constant. It follows then, that the change in the mass of nitrogen within the system, over time, will simply be equal to the difference of the mass of nitrogen added and the mass of nitrogen removed. Quantitatively, this can be expressed as, $$\Delta M_{N,sys}(t)=M_{N,in}(t)-M_{N,out}(t)$$ Therefore, given some initial value, the mass of nitrogen within the system, as a function of time, can be represented by the following general equation, $$M_{N,sys}(t)=M_{N,initial}+M_{N,in}(t)-M_{N,out}(t)$$ Assuming that all terms contained within \(M_{N,in}(t)\) and \(M_{N,out}(t)\) can be linearized with respect to time, the concentration of nitrate within the system can be expressed as a simple linear function, $$N(t)=\frac{\dot{N}t}{V_{sys}}+N_0$$ where, \(N(t)\) is the concentration of nitrate in mg/L (ppm) \(\dot{N}\) is the overall rate of nitrate production in mg/day \(N_0\) is the initial concentration of nitrogen as nitrate in mg/L (ppm) \(V_{sys}\) is the total system volume in L Regarding linearity, it should be noted that some of the factors, discussed previously, exhibit the potential for nonlinear relations, the rate often depending on the instantaneous value of \(N(t)\). For example, a flow-through system, in which water is continuously being added and removed, will present a removal rate that is proportional to the instantaneous concentration. Such cases will typically result in a linear differential equation, that can generally be solved using standard methods. This scenario, and others, will be the topic of a future article. In the case of a linear function, as shown above, \(\dot{N}\) will be proportional to the difference of nitrogen entering the system versus nitrogen being incorporated into the tissues of organisms and/or being permanently removed from the system, $$\dot{N}=\dot{N}_{in}-\dot{N}_{tissue}-\dot{N}_{out}$$ \(\dot{N}_{in}\) will be the sum of nitrogen entering the system in food, released from decaying organic material, introduced as inorganic material, and produced by nitrogen fixation, $$\dot{N}=\dot{N}_{food}+\dot{N}_{organic}+\dot{N}_{inorganic}+\dot{N}_{fix}-\dot{N}_{tissue}-\dot{N}_{out}$$ \(\dot{N}_{food}\) can be determined from the feed rate as follows, $$\dot{F}=0.16(CP)(FR)$$ where, \(FR\) is the feed rate in mg/day \(CP\) is the Minimum Crude Protein of the feed, as a fraction \(\dot{F}\) is the total mass of atomic nitrogen from feed in mg/day Note that 0.16 represents the average mass percentage of atomic nitrogen in amino acids (as discussed previously). Multiplying by the molar mass ratio of nitrate to atomic nitrogen, $$\dot{F}*\frac{MM NO_3: 62.01 g/mol}{MM N: 14.01 g/mol}=0.16(CP)(FR)*4.43=0.71(CP)(FR)$$ gives \(\dot{N}_{food}\) in units of mg of nitrogen as nitrate per day, $$\dot{N}_{food}=0.71(CP)(FR)$$ All other terms contained in \(\dot{N}\) will depend on the particular system being analyzed, and must either be estimated or determined experimentally. Note that terms must be expressed in units of mg of nitrogen AS NITRATE per day, for consistency. \(N_0\) simply represents the concentration of nitrogen as nitrate in the source water, and can be calculated as, $$N_0=NO_{3,ini}+1.35*NO_{2,ini}+3.54*NH_{3/4,ini}$$ where, \(NO_{3,ini}\) is the initial concentration of nitrate in the source water in mg/L (ppm) \(NO_{2,ini}\) is the initial concentration of nitrite in the source water in mg/L (ppm) \(NH_{3/4,ini}\) is the initial total ammonia concentration (i.e. free and ion) in the source water in mg/L (ppm) Note that 1.35 and 3.54 represent the molar mass ratios of nitrate to nitrite and ammonia, respectively. Expanding terms in the original function gives the following generalized linear equation for nitrate concentration as a function of time, $$N(t)=\frac{[0.71(CP)(FR)+\dot{N}_{organic}+\dot{N}_{inorganic}+\dot{N}_{fix}-\dot{N}_{tissue}-\dot{N}_{out}]t}{V_{sys}}+N_0$$ $$N_0=NO_{3,ini}+1.35*NO_{2,ini}+3.54*NH_{3/4,ini}$$ where, \(N(t)\) is the nitrate concentration, as a function of time (days), in mg/L (ppm) \(CP\) is the Minimum Crude Protein of the feed, as a fraction \(FR\) is the feed rate in mg/day \(\dot{N}_{organic}\) is the rate of nitrogen, as nitrate, introduced from organic material, in mg/day \(\dot{N}_{inorganic}\) is the rate of nitrogen, as nitrate, introduced from inorganic material, in mg/day \(\dot{N}_{fix}\) is the rate of nitrogen fixation, as nitrate, in mg/day \(\dot{N}_{tissue}\) is the rate of incorporation into tissue of nitrogen, as nitrate, in mg/day \(\dot{N}_{out}\) is the rate of removal of nitrogen, as nitrate, in mg/day \(V_{sys}\) is the total system volume in L \(NO_{3,ini}\) is the initial concentration of nitrate, in the source water, in mg/L (ppm) \(NO_{2,ini}\) is the initial concentration of nitrite, in the source water, in mg/L (ppm) \(NH_{3/4,ini}\) is the initial total ammonia concentration (i.e. free and ion), in the source water, in mg/L (ppm) It is worth noting that there are two assumptions implicit in this idealized model, that deserve mention. The first is that decomposition of proteins, amino acids, and nitrogenous compounds occurs instantaneously. In the actual system, of course, decomposition will occur at some finite rate. However, it is the gradual accumulation of nitrate, rather than the instantaneous concentration, that this model seeks to estimate and predict. As such, the assumption is sufficient to study nitrate concentration in aggregate, over the course of days or weeks. The second is that biological filtration (i.e. nitrification) in the system is both complete and sufficient to maintain negligible levels of ammonia and/or nitrite. Were this not the case, nitrogen would begin to accumulate in other forms, and the system itself would have far greater problems than controlling nitrate levels. It is also worth noting that this model could be modified to estimate ammonia concentration, in an uncycled system, by simply converting all terms to represent nitrogen as ammonia, rather than as nitrate. Experimental Verification As a simple case study, consider the example of a mature female Betta splendens, kept in a 3 gallon tank (see Figure A below), and fed the equivalent of a standard betta diet (approximately 16 mg/day @ 42% minimum crude protein). The tank was fully cycled, prior to the introduction of the betta, and the water was replaced initially, to remove residual nitrogen content. No organic material was added to the tank at any time, other than the betta, the biofilter, and the food. No inorganic sources of nitrogen were added to the tank at any time. No algae growth (including cyanobacteria) was observed in the tank, for the duration of the study. Also, no chemical filtration, skimming, denitrification, or any other means of removing nitrogen were employed for the duration, other than a small quantity of activated charcoal in the filter. The parameters of the source water were as follows,
\(TDS\): 101 ppm \(LRpH\): 7.6 \(HRpH\): 7.4 \(GH\): 35.8-53.7 ppm \(KH\): 35.8-53.7 ppm \(NH_3/NH_4^+\): 0.25 ppm \(NO_2^-\): 0.0 ppm \(NO_3^-\): 0.0 ppm \(PO_4^{3-}\): 0.0 ppm \(Ca^{2+}\): < 20 ppm \(Cu^+\): 0.0 ppm and remained within safe levels for the duration. The tank was kept at a constant 80° F, and was aerated continuously. Evaporation losses were replaced periodically with distilled water. And finally, the tank was kept on a 50% light/dark cycle (i.e. 12 hours on, 12 hours off). Applying the proposed quantitative model to this scenario, and assuming a 100% metabolic system, yields the following relation for the expected nitrate concentration, $$N(t)=\frac{[0.71(0.42)(16)+0+0+0-0-0]t}{11.36}+0+1.35(0)+3.54(0.25)=0.42t+0.89$$ The chart below compares the predicted nitrate concentration (represented by the solid blue line) with the actual measured nitrate concentration (represented by colored dots), for the duration of the study (see Figure B below). All nitrate measurements were obtained using the API Freshwater Master Test Kit. The coloring of the dots represents the general range of the measurement, within the color range of the test kit. Vertical bars are also included, to indicate the relative uncertainty of the measurement. The final water parameters, at the end of the study, were as follows,
\(TDS\): 143 ppm \(LRpH\): 7.6 \(HRpH\): 7.8-8.0 \(GH\): 71.6-89.5 ppm \(KH\): 71.6-89.5 ppm \(NH_3/NH_4^+\): 0.0 ppm \(NO_2^-\): 0.0 ppm \(NO_3^-\): ~40 ppm Conclusion A quantitative model, for the estimation of nitrate production in aquarium systems, has been presented and shown to correlate well with the results of a simple case study. In addition to the simple study presented above, the author has found the proposed model to correlate well with a variety of cases he has observed. Future articles will expand on the use of this model, and explore its application to more sophisticated systems. |