Nutrient equation

The boundary conditions in (2.2) are fairly intuitive and appropriate for this type of equation. However, the boundary conditions are not defined in terms of the operating parameters of the simple chemostat. The problem will be considered in a heuristic way to see how the units compare between the simple chemostat and the chemostat without the assumption of well mixing. To keep matters simple, we work with the nutrient equation without consumption (equivalently, zero initial conditions for the microorganisms) the other cases will be clear by analogy. Under these circumstances, the simple chemostat takes the form... 

When considering the partial differential equation, the basic quantity S t, at) becomes a density, measured in units of mass per unit length. The nutrient equation, using a subscript to denote differentiation, is... 

A complex set of equations, proposed by Riley, Stommel, and Bumpus (1949) (5) first introduced the spatial variation of the phytoplankton with respect to depth into the conservation of mass equation. In addition, a conservation of mass equation for a nutrient (phosphate) was also introduced, as well as simplified equations for the herbivorous and carnivorous zooplankton concentrations. The phytoplankton and nutrient equations were applied to 20 volume elements which extended from the surface to well below the euphotic zone. In order to simplify the calculations, a temporal steady-state was assumed to exist in each volume element. Thus, the equations apply to those periods of the year during which the dependent variables are not changing significantly in time. Such conditions usually prevail during the summer months. The results of these calculations were compared with observed data, and again the results were encouraging. 

This is an old, familiar analysis that applies to any continuous culture with a single growth-limiting nutrient that meets the assumptions of perfect mixing and constant volume. The fundamental mass balance equations are used with the Monod equation, which has no time dependency and should be apphed with caution to transient states where there may be a time lag as [L responds to changing S. At steady state, the rates of change become zero, and [L = D. Substituting ... 

Assuming that a single nutrient is limiting, and eell growth is the only proeess eontributing to substrate utilization, substituting Equation 11-68 into Equation 11-66 and rearranging yields... 

The fed bateh reaetor (FBR) is a reaetor where fresh nutrients are added to replaee those already used. The rate of the feed flow U may be variable, and there is no outlet flowrate from the fermenter. As a eonsequenee of feeding, the reaetor volume ehanges with respeet to time. Figure 11-22 illustrates a simple fed-bateh reaetor. The balanee equations are ... 

Essentially all organic matter in the ocean is ultimately derived from inorganic starting materials (nutrients) converted by photosynthetic algae into biomass. A generalized model for the production of plankton biomass from nutrients in seawater was presented by Redfield, Ketchum and Richards (1963). The schematic "RKR" equation is given below ... 

Fig. 10-13. The links between the cycling of C, N, and O2 are indicated. Total primary production is composed of two parts. The production driven by new nutrient input to the euphotic zone is called new production (Dugdale and Goering, 1967). New production is mainly in the form of the upward flux of nitrate from below but river and atmospheric input and nitrogen fixation (Karl et al, 1997) are other possible sources. Other forms of nitrogen such as nitrite, ammonia, and urea may also be important under certain situations. The "new" nitrate is used to produce plankton protoplasm and oxygen according to the RKR equation. Some of the plant material produced is respired in the euphotic zone due to the combined efforts...

Oceanic surface waters are efficiently stripped of nutrients by phytoplankton. If phytoplankton biomass was not reconverted into simple dissolved nutrients, the entire marine water column would be depleted in nutrients and growth would stop. But as we saw from the carbon balance presented earlier, more than 90% of the primary productivity is released back to the water column as a reverse RKR equation. This reverse reaction is called remineralization and is due to respiration. An important point is that while production via photosynthesis can only occur in surface waters, the remineralization by heterotrophic organisms can occur over the entire water column and in the underlying sediments. 

The oxidative nutrients can be estimated from the RKR equation. From this model we might expect the four dissolved chemical species (O2,... 

A mathematical model for reservoir souring caused by the growth of sulfate-reducing bacteria is available. The model is a one-dimensional numerical transport model based on conservation equations and includes bacterial growth rates and the effect of nutrients, water mixing, transport, and adsorption of H2S in the reservoir formation. The adsorption of H2S by the roek was considered. 

D. Kirkham and W. V. Bartholomew, Equations for following nutrient transformation in soil, utilizing tracer data. Soil. Sci. Soc. Am. Proc. 1833 (1954). 

During the lifetime of a root, considerable depletion of the available mineral nutrients (MN) in the rhizosphere is to be expected. This, in turn, will affect the equilibrium between available and unavailable forms of MN. For example, dissolution of insoluble calcium or iron phosphates may occur, clay-fixed ammonium or potassium may be released, and nonlabile forms of P associated with clay and sesquioxide surfaces may enter soil solution (10). Any or all of these conversions to available forms will act to buffer the soil solution concentrations and reduce the intensity of the depletion curves around the root. However, because they occur relatively slowly (e.g., over hours, days, or weeks), they cannot be accounted for in the buffer capacity term and have to be included as separate source (dCldl) terms in Eq. (8). Such source terms are likely to be highly soil specific and difficult to measure (11). Many rhizosphere modelers have chosen to ignore them altogether, either by dealing with soils in which they are of limited importance or by growing plants for relatively short periods of time, where their contribution is small. Where such terms have been included, it is common to find first-order kinetic equations being used to describe the rate of interconversion (12). 

Traditionally, nutrient uptake from solution culture was taken to depend on the concentration of the external mineral nutrient, C , the amount of nutrientabsorbing surface, and the kinetics of uptake per unit surface area or unit length of root (22). The flux of nutrients into the roots, J, is described by one of two functionally equivalent equations. ... 

Several boundary conditions have been used to prescribe the outer limit of an individual rhizosphere, (/ = / /,). For low root densities, it has been assumed that each rhizosphere extends over an infinite volume of. soil in the model //, is. set sufficiently large that the soil concentration at r, is never altered by the activity in the rhizosphere. The majority of models assume that the outer limit is approximated by a fixed value that is calculated as a function of the maximum root density found in the simulation, under the assumption that the roots are uniformly distributed in the soil volume. Each root can then extract nutrients only from this finite. soil cylinder. Hoffland (31) recognized that the outer limit would vary as more roots were formed within the simulated soil volume and periodically recalculated / /, from the current root density. This recalculation thus resulted in existing roots having a reduced //,. New roots were assumed to be formed in soil with an initial solute concentration equal to the average concentration present in the cylindrical shells stripped away from the existing roots. The effective boundary equation for all such assumptions is the same ... 

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

In this case there is a continuous addition of nutrients to the culture by the feed stream. There is no effluent stream. The governing equations are ... 

In this case, there is a continuous supply of nutrients and a continuous withdrawal of the culture broth including the submerged free cells. The governing equations for continuous cultures are the same as the ones for fed-batch cultures (Equations 7.20-7.22). The only difference is that feed flowrate is normally equal to the effluent flowrate (Fm=Fout=F) and hence the volume. V, stays constant throughout the culture. 

When microorganisms use an organic compound as a sole carbon source, their specific growth rate is a function of chemical concentration and can be described by the Monod kinetic equation. This equation includes a number of empirical constants that depend on the characteristics of the microbes, pH, temperature, and nutrients.54 Depending on the relationship between substrate concentration and rate of bacterial growth, the Monod equation can be reduced to forms in which the rate of degradation is zero order with substrate concentration and first order with cell concentration, or second order with concentration and cell concentration.144... 

Control of nutrient transport dictates significant coupling between transported components in G1 epithelia. This complicates solute transport analysis by requiring a multicomponent description. Flux equations written for each component constitute a nonlinear system in which the coupling nonlinearities are embodied in the coefficients modifying individual transport contributions to flux. 

The main choice of these functions should be based on data from suitable laboratory experiments. This approach has been recently applied to the mathematical study of the allelopathic competition between Pseudokirchneriella subcapitata and Chlorella vulgaris in an open culture (a chemostat-like device), where the nutrient is a mixture of inorganic phosphates Pi. Here we will illustrate step by step the four mentioned features, observing that each equation of the mathematical model can be actually constructed as a balance equation. 

The two terms in the final expression of Equation 9.11 can compete against one another and with only a small membrane potential the concentration gradient chemical potential can be overcome. Electrochemical energy is thus harnessed to allow nutrients to flow into a cell and increase the local concentration. 

The values of nutrient nitrogen critical loads (CLnutrS) are calculated using the equation ... 

Microbes tend to form flocks as they grow, into which nutrients and dissolved oxygen must diffuse. The rate of growth thus depends on the diffusional effectiveness. This topic is developed by Atkinson (1974). Similarly enzymes immobilized in gel beads, for instance, have a reduced catalytic effectiveness analogous to that of porous granular catalysts that are studied in Chapter 7. For the M-M equation this topic is touched on in problems P8.04.15 and P8.04.16. 

For limiting nutrients, cellular concentrations are constant under conditions of steady-state growth. To ensure that the limiting nutrient is not diluted in the microbial population, kmt must be greater than the maximal growth rate, /imax. This limiting condition sets a minimum for the value of the Monod constant, Kmd = / max /[7]- Note that while Monod kinetics are more applicable than first-order kinetics for many ecological uptake processes, solutions of the above equations require considerably more a priori information [48]. 

The effect which nutrient discharges have on the dissolved oxygen of a stream is best demonstrated by reference to a simple organic molecule. If the total oxidation of glucose to carbon dioxide and water is considered (equation 10.1), one part by weight of glucose would require 1.06 parts of oxygen for complete oxidation. 

If an open system with renewal of substrate, nutrients, water, and electron acceptors can be supplied, the growth rate of the bacteria population is able to continue for an extended period of time until the remediation is complete. The Monod equation describes the type of bacterial growth that can be expected in an open system ... 

This equation describes the ratios with which inorganic nutrients dissolved in seawater are converted by photosynthesis into the biomass of "average marine plankton" and oxygen gas 02. The opposite of this reaction is respiration, or the remineralization process by which organic matter is enzymatically oxidized back to inorganic nutrients and water. The atomic ratios (stoichiometry) of this reaction were established by... 

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