FINALES CAPÍTULO I

In this section we provide a brief historical discussion of the literature relating to the mathematical analysis of cascade reactors. Cascade reactors using Contois kinetic

are discussed separately in chapter (3).

The optimal design of a reactor cascade has been receiving attention since the early 1960s. A reactor cascade is optimized by adjusting the physical properties of the reactor cascade, such as volume, interconnection structure of different tanks, recircu-lation rates, tanks shapes, with respect to the design objective such as the residence time.

Harmand et al. [10] analyzed the optimal design of a two-reactor cascade including two feed streams (multi-stream flow) and/or a recirculation loop in which a single reaction occurs. The model assumed that the microorganism does not decay. The objective of this study is to minimize the total volume of the cascade given a spec-ified conversion. The steady-state design problem was analyzed for a generalized growth rate law that included Monod kinetics subject to substrate inhibition.

The obtained result was that the appropriate allocation of the influent into the re-actor cascade in order to minimize the total volume depend on the ratio rate. The ratio rate depends upon the effluent concentration leaving the cascade. They showed that the distribution of the influent and introducing a recycle into the reactor cas-cade is only required to minimize the total volume when the ratio rate is less than one. Harmand et al. [9] revisited the optimal design of a two-reactor cascade whilst investigating enzymatic biological reactions. Latter Rapaport et al. [102] extended the study on optimal design of two-reactor cascade to includes several microbial species, instead of a single specie. The important finding of their study was that the optimal design does not allow the coexistence of several species.

Powell and Lowe [103] studied the behaviour of a cascade of N reactors of equal

vol-ume with recycle and with no death rate. Recycle was considered to occur between the last reactor of the cascade and the first reactor. As the number of the reactors increased, they found that the behaviour of the system approached that of an ideal tubular fermenter with plug flow.

Erickson and Fan [104] studied the optimum hydraulic regime for several activated sludge systems composed of N reactors (N = 2 or 3) with recycle and a non-zero death rate. They considered two points of view: the minimum total reactor volume of the cascade required to produce the desired effluent concentration; and an esti-mate of the cost of the organic waste being discharged and the total cost for the volume of the total reactor cascade. They compared the results from the reactor cascades with these obtained using a single tank.

Erickson et al. [105] examined the optimisation of a multi-stream cascade with N reactor (N = 2, 3, 4, and 5), recycle, and a zero death rate. They considered two stages in each reactor, a mixing stage and an aeration stage. The influent entered the reactor cascade in the mixing stage, where no reactions occurred. The objective function was to minimize the total volume of the reactor cascade by determining the appropriate allocation of the influent into the mixing tanks and the distribution of reactor volumes for specific concentrations of the organisms and substrate in the recycle stream. The parameter value of the concentration factor (C) was 2 which is also commonly used in practice.

The optimal cascade designed can be divided to two groups which are static [106]

or dynamical [107]. Several investigators have sought to determine the optimum design of two-reactor cascade in series [107–110]. Yang and Su [110] conducted their

experiment in each reactor whilst residence times in each were equivalent. In their findings, they observed a superior performance for the two reactor cascade as com-pared to a single reactor of the same dilution rate. Basing their arguments on the later observation, they inferred that enormous output will be experienced when two reactor cascade are connected in series.

Chen et al. [109] examined the possibility of increasing performance in a single re-actor and two-rere-actor cascade in series with a variable yield coefficient as a result of natural oscillations generation. For a single reactor using Monod, Tessier and Moser growth models, they found out that the improvement of the reactor performance can be achieved by operating at the optimal steady-state residence time which always above the oscillatory region. Basing their study on the later observations, they com-pared the cascade reactor which has equal residence time against the single reactor that has substrate inhibition in its growth kinetics. While analyzing two more com-plex microbial systems, Balakrishnan and Yang [108] visited again Monod-growth model. In general, the output of a single chemostat system is poor as compared to a two-chemostat-in-series system with the same total residence time.

Nelson and Sidhu [107] re-investigated the biological system of a two-reactor cascade in series with no death rate that has been originally considered by Balakrishnan and Yang [108], Chen et al. [109] and Yang and Su [110] in which the growth rate is given by a Monod growth kinetics with a variable yield coefficient. The criteria they used to compare between the two configurations were different from that proposed in [108–110] wherein the performance of optimal single reactor compared with a cascade performance when the residence time of former is the same, or smaller than

that of the latter.

They found that the improvement of performance is obtained by using two-reactor cascade than a single reactor for the choice of specific parameter. The finding of the comparison with respect of the reactor productivity, is the single reactor surprisely is superior than the cascade. It also showed that an insignificant improvement of the cell mass efficiency of the cascade which is not as the first finding that reported in [108] and [110].

Grady and Lim [80] studied biological wastewater treatment using reactor cascades with and without recycle. Their study included the use of a single feed stream and multiple feed streams. They investigated the optimisation of the reactor design from two points of view, i.e., 1) the minimum effluent concentration that can be obtained for a specified total volume and 2) the minimum reactor volume that is required to deliver a specified effluent concentration. They found that the recycle ratio had no significant effect on either the substrate concentration or the microorganism con-centration in each reactor. They concluded that recycle is not a significant tool to reduce the substrate concentration.

Using several and differentiated equipments, Mantzaris, et al. [111] explained how the growth process in multiple bioreactors can be modelled and also addressed the following issues: (1) the chemical environment is likely to vary in diverse reactors and (2) the biomass arises as distinct cells which are in reality separate and remain separated from each other and thus a constituent of the biomass that go into the reactors will not mix up with the biomass that was previously there. They devel-oped different numerical algorithms for different models to solve the steady-state

and transient problems and concluded that the heterogeneity in the biomass can be ignored for the purposes of calculating the concentrations of biomass, nutrients, and products in the abiotic environment if the structured model is linear in the state vector.

Literature review of Contois growth kinetics

3.1 Contois growth kinetics

In many biological processes, the rate at which a population of microorganisms (X) grows and the rate at which the substrate (S) is consumed are essential part of the kinetic models. Often researchers have assumed that the specific growth rate µ is not dependent on the concentration of the microorganisms [30,48,112–115]. However, it has been found experimentally that in some systems, the specific growth rate model should include the concentrations of both the substrate and the microorganisms.

Such a growth kinetics model was introduced by Contois [13], who presented evidence to show that the specific growth rate can also depend upon the microorganism concentration from studies of batch cultures growing under conditions of nutrient

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limitation. The Contois growth model is given by,

where the specific growth rate µ(S, X) is a function of microorganism concentra-tion (X) and the concentraconcentra-tion of the substrate (S). The parameters µmax and Ks

are the maximum specific growth rate and the saturation constant respectively.

These are constants under defined conditions.

Contois’s model shows that as the microorganism concentration increases, the growth rate (µ(S, X)) decreases. The growth rate is given by,

G = µ_max

There are two extreme cases for the Contois model as the population density of biomass increases. In these cases, the Contois model reduces to first-order kinetics for either biomass or substrate concentration. These approximations are [116],

1. First-order kinetics for biomass growth.

S

X ≫ Ks=⇒ G ∼= µmaxX. (3.1.3) 2. First-order kinetics for substrate consumption.

S

The Contois growth rate has been used as a surface limiting model to explain the mass transfer limitations due to the limited surface area when the biomass con-centration is high [46,117–120]. In this case, the specific growth rate is expressed as,

Thus, if the population density of biomass increases, leading to increased obstruction to substrate uptake and growth of any particular microbe, then the Contois rate law reduces to,

G≈ µmax

( S K_s

)

. (3.1.6)