2. Justificación de las propuestas
2.1. Proyecto de Innovación Docente
2.1.2. Metodología utilizada
We analysis the generalized reactor models by consider the cases 0 ≤ β ≤ 1 and 0≤ R∗ ≤ 1. In section (4.3.1), an invariant region is determined. In section (4.3.2), the steady state solutions branches are found and the conditions of no-washout
solution branch (The no-washout solution refers to when the microorganism con-centration is positive) to be physically meaningful are stated. In section (4.3.3) and (4.3.4) the stability of the steady state solutions are determined and, the washout solution (The washout solution refers to when the microorganism concentration is equal to zero) is shown to be globally stable if it is locally stable. In section (4.3.5) the no-washout solution (4.3.14) is globally asymptotically stable if it is locally sta-ble. In section (4.3.6) the transcritical bifurcation points are presented. Asymptotic solutions for large residence times are determined in section (4.3.7) while, in sec-tion (4.3.8), these solusec-tions are presented for residence times just higher than the washout point. In section (4.3.9), we discuss steady-state diagrams for the efflu-ent concefflu-entration and microorganism concefflu-entration leaving the reactor. In section (4.3.4) we show that the washout solution is globally stable when Kd∗ ≥ 1. Hence, we assume in the following that 0 < Kd∗ < 1.
4.3.1 Invariant region
In this section our aim is to establish the basic properties of invariance for the region Ω ={(S∗, X∗)∈ R2+, 0≤ S∗ ≤ 1, X∗ ≥ 0}.
On the boundary of Ω, if X∗ = 0 with S∗ > 0 then dXdt∗∗ = 0. Thus, the plane X∗ = 0 is invariant. If S∗ = 0 with X∗ > 0 then dSdt∗∗ = τ1∗ > 0. Thus, the vector field of S∗ is pointed to inside the region Ω. However, the point (S∗, X∗) = (1, 0) is a steady-state solution and solution trajectories can not leave through it.
The model (4.2.6, 4.2.7) is not defined at the point (S∗, X∗) = (0, 0). We show that no solution can enter this point. If X = 0, then dSdt∗∗ = τ1∗ > 0, thus S∗(t∗) is
an increasing function. Now suppose X∗ > 0. Then
Hence the region Ω is (positively) invariant.
For the boundness of S∗ and X∗, we have that,
Where C is an arbitrary constant. i.e. the solution trajectory is attracted into the invariant region. Hence, the substrate concentration is bounded. We now show that the concentration of microorganisms is bounded.
Let Z∗ = α∗S∗+ X∗, Adding equations (4.2.6) and (4.2.7) we have,
It follows that
Z∗ ≤ α∗
β(1− R∗)+ C1exp [
−β(1− R∗)t∗ τ∗
]
(4.3.12)
Where C1 is an arbitrary constant. Hence the function Z∗ = α∗S∗+ X∗ is bounded and as S∗ is bounded so is X∗.
4.3.2 Steady State Solutions Branches
Steady state solutions of the system for the equation (4.2.6, 4.2.7) are found by setting the derivatives terms equal to 0 and given by,
Washout branch:
(S∗, X∗) = (1, 0). (4.3.13)
No-washout branch:
(S∗, X∗) = (α∗
a ,α∗(β(R∗ − 1) + τ∗(1− Kd∗))
a(β(1− R∗) + τ∗Kd∗) ) (4.3.14) where
a = β(R∗− 1) + τ∗(1− Kd∗) + α∗.
The no-washout branch in equation (4.3.14) is physically meaningful only when the substrate and microorganism concentrations are positive (ie. where S∗ > 0 and X∗ > 0). Upon analysing both components of (4.3.14) we find that
τ∗ >β(1− R∗)
1− Kd∗ (4.3.15)
is required for S∗ > 0 and X∗ > 0.
By differentiating the S∗ component of equation (4.3.14), the effect of recycle on the
substrate concentration is determined. We obtain that dS∗
dR∗ =− α∗β
[−α∗+ β(1− R∗) + τ∗(Kd∗− 1)]2 < 0. (4.3.16) Now the denominator in equation (4.3.16) is equal to zero when
τ∗ = β(1− R∗)
(1− Kd∗) − α∗
1− Kd∗ < β(1− R∗) (1− Kd∗).
It is shown latter that this condition is the washout point. Hence, for physically meaningful solution dS∗
dR∗ < 0 which means that substrate concentration is a decreas-ing function of the effective recycle parameter (R∗). Thus, at fixed residence time, the performance of reactor reaches the highest level of efficiency at the maximum value of the effective recycle parameter (i.e, R∗ = 1).
From equation (4.3.14), it can be shown that, dS∗
dτ∗ =− α(1− Kd∗)
[−α∗+ β(1− R∗) + (Kd∗− 1)τ∗]2 < 0. (4.3.17) Since 0 < Kd∗ < 1 then the substrate concentration is a decreasing function of residence time along the no-washout branch.
4.3.3 Stability of the steady state solutions
The Jacobian matrix for the system (4.2.6) and (4.2.7) is given by
J (S∗, X∗) =
Evaluating the Jacobian matrix (4.3.18) at the washout steady state solution produces
The eigenvalues of this matrix are given by
λ1 = 1
τ∗ < 0 and λ2 = β(R∗− 1) − τ∗Kd∗
τ∗ + 1.
Hence, the washout branch is stable when λ2 < 0 which is when
β(1− R∗) > τ∗(1− Kd∗).
It can be seen that when
- Kd∗ > 1 then λ2 < 0 for all τ∗,
The Jacobian matrix along the no-washout branch can be written as
J (S∗, X∗) =
For physically meaningful solutions to the system of equations (4.2.6, 4.2.7), we require that X∗ > 0 and S∗ > 0. From the Jacobian matrix (4.3.20), the trace and determinant are:
Since the trace is negative and the determinant is positive for the Jacobian ma-trix then the no-washout branch is stable for all physically meaningful solutions of equations (4.2.6, 4.2.7).
Theorem 4.3.1 The washout solution (1,0) is globally asymptotically stable pro-vided τ∗ < τcr∗ on Ω.
Proof. To prove global stability of the washout solution (1,0) whenever τ∗ < τcr∗, we consider the following Lyapunov function V (S∗, X∗) = X∗,
V = X˙ ∗[ S∗
(S∗+ X∗) +β(R∗− 1)
τ∗ − Kd∗],
≤ X∗[1 + β(R∗ − 1)
τ∗ − Kd∗] = X∗[τ∗− τcr∗].
By hypothesis we have τ∗ < τcr∗, this leads to ˙V < 0. Since the region Ω is positively invariant, the washout solution (1,0) is globally asymptotically stable in Ω. The analysis shows that if Kd∗ > 1 then the washout solution is globally stable. Thus no practical biochemical plane can be operated when Kd∗ > 1. 2
4.3.4 Global stability for washout solution
In this section, we show that when the washout solution is locally stable, it is also globally stable. We do this by using the argument from [170].
There are three cases to consider:
1. Kd∗ > 1,
2. Kd∗ = 1 provided that R∗ ̸= 1 and
3. 0 < Kd∗ < 1 provided that τ∗ < β(1− R∗) 1− Kd∗ .
Proof for the first case can be done by assuming that X∗ > 0, then from equation (4.2.7) we have that
dX∗
dt∗ = S∗X∗
(S∗ + X∗) +β(R∗− 1)
τ∗ X∗− Kd∗X∗
≤ β(R∗− 1)
τ∗ X∗ − Kd∗X∗+ X∗ as S∗
(S∗+ X∗) ≤ 1
= Xτ∗∗(β(R∗− 1) + τ∗(1− Kd∗))
≤ X∗(1− Kd∗), since R∗max= 1
(4.3.23)
It follows that X∗ is decreasing function of t∗. We know that the region X∗ ≥ 0 is a positively invariant region. Hence, the microorganism concentration X∗ approaches zero as t∗ → ∞ and therefore, the washout solution is globally stable.
The cases ((2) and (3)) use a similar argument to the above. Furthermore, the case when linearized stability is not defined (ie. when λ2 = 0) can be also be considered.
Two more cases need to be consider for this circumstance. These are:
4. R∗ = 1 provided that Kd∗ = 1 and
5. τ∗ = β(1− R∗) 1− Kd∗ .
By using a similar argument as above it can be shown that the washout solution is globally stable for these specific cases.
4.3.5 Global stability of no-washout solution
Theorem 4.3.2 The no-washout solution (4.3.14) is globally asymptotically stable provided τ∗ > τcr∗ on Ω.
Proof. Since the system (4.2.6 and 4.2.7) is a plane dynamical system defined on Ω invariant domain and since the trajectories are bounded, we can apply the theorem of Poincare-Bendixson [171] assuring that there are three cases, for a trajectory which are,
1. It is a periodic solution,
2. It converges to a periodic solution,
3. It converges to an equilibrium point.
On the other hand, since the bounded domain containing trajectories is connected, we can apply Dulac-Bendixson criteria to prove that there is no periodic orbits.
Using Dulac’s test [172] it is shown that there are no periodic solution for the system of equations (4.2.6 and 4.2.7).
Let ρ = 1
X∗ then the Dulac’s test yields that d
dS∗[ρdS∗
dt ] + d
dX∗[ρdX∗ dt ] =−
[α∗(X∗ + S∗)2+ X∗τ∗(α∗S∗+ X∗) X∗τ∗α∗(X∗+ S∗)2
]
< 0.(4.3.24)
Equation (4.3.24) is negative when the components of (S∗, X∗, τ∗, α∗) are all positive.
This is true for physical meaningful solutions therefore, by Dulac’s test, the system of equation (4.2.6, 4.2.7) has no periodic solutions. So since there is no periodic solution and since τ∗ > τcr∗ then the washout solution is not stable then the only possibility that a trajectory has, is to converge into the no-washout solution. Thus, the no-washout solution (4.3.14) is globally asymptotically stable provided τ∗ > τcr∗
on Ω. 2
4.3.6 Transcritical bifurcation
A transcritical bifurcation occurs at the residence time value when
τcr∗ = β(1− R∗) 1− Kd∗
. (4.3.25)
The value of τcr∗ represents the maximum residence time at which treatment process fails. At lower residence times microorganisms are removed from the reactor at a rate greater than their maximum growth rate, resulting in process failure. At the residence times lower (higher) than the transcritical value, the washout (no-washout) solution is the only stable solution. The critical value of the residence time is zero when either R∗ = 1 or β = 0. Thus, increasing the recycle parameter R∗ or decreas-ing the reactor parameter β allows the reactor to operate at lower residence times.
Equation (4.3.25) is governed by Contois growth kinetics. It shows that the washout condition does not depend upon the effluent pollutant concentration (S0∗) for wastew-ater treatment processes. In vast contrast in which the treatment process is governed by Monod kinetics, the washout condition depends upon the effluent pollutant con-centration (S0∗) [132]. The practical implication of this is that, for processes governed by Monod kinetics, the wastewater that enters the reactor needs to be concentrated.
4.3.7 Residence time approximations near the washout point
When residence times is slightly higher than the washout point, the approximate values of the microorganism concentration and the substrate concentration are given by
X∗ = (1 − Kd∗)2ϵ
β (1− R∗) + O(ϵ2), (4.3.26)
S∗ = 1− (1− Kd∗) ϵ
α∗ + O(ϵ2). (4.3.27)
where ϵ = τ∗−β(1− R∗)
1− Kd∗ << 1.
From equations (4.3.26and4.3.27), increasing ϵ leads to, respectively, an increase and decrease in the microorganism and substrate concentrations. It is observe that for fixed ϵ, the increasing the value of expression β(1− R∗) leads to a decrease in the microorganism concentration.
4.3.8 Steady-State diagrams
Steady-state diagrams showing the physical meaningful solutions for the variation of substrate concentration (S∗) and microorganism concentration (X∗) as a function of the residence time (τ∗) time are demonstrated in figures (4.2), (4.3) and (4.4).
The Maple package [173] was used to obtain the steady-state diagrams.
In section (4.3.2), it was shown that the no washout branch is only physically mean-ingful when τ∗ > τcr∗. These physical meaningful solutions (i.e 0 < S∗ ≤ 1 and X∗ > 0) are plotted in figure (4.2). From this figure, it can be seen that for suffi-ciently low values of the residence time (τ∗ < τcr∗) the stable solution is the washout solution (given by the solid lines S∗ = 1 and X∗ = 0) and the unstable solution is the no-washout solution. For τ∗ > τcr∗, only the no-washout solution is stable (pre-sented by the solid curve for both S∗ and X∗) and the washout solution is unstable (presented by a dotted line for both S∗ and X∗).
Using these same parameter values as for figure (4.2), a general pattern of substrate S∗ and micro-organism X∗ behaviour with respect to low value residence time is shown figures (4.3 and 4.4). From these figures, it can be seen as the value of
β(1− R∗) decreases to 0, the critical residence time (τcr∗) also decreases. This means that the length of the residence time for where the washout solution is stable dimin-ishes too.
In the following discussions we assume that the residence time is sufficiently high (ie τcr∗ << τ∗) so that the no-washout solution branch is the stable solution. Figure (4.3) shows that for both fixed residence time (τ∗) and effective recycle parameter (R∗), the performance of the reactor increases as the value of the reactor parameter β decreases (from curve (a) to (e)). It also shows that if the values of β and R∗ are fixed, the performance of the reactor improves as τ∗ increases. In general, as the value of the expression β(1−R∗) decreases, the performance of the reactor improves.
That is there is effective removal of pollutant or substrate from the bioreactor.
Figure (4.4) shows that the microorganism concentration (X∗) is an increasing func-tion of the residence time (τ∗), except for the case where β(1− R∗) = 0. It also shows that if the residence time is fixed, then the microorganisms concentration (X∗) decreases as the value of β(1− R∗) increases (from curve (a) to (e)).
In the following section, we show that the steady state of microorganism concentra-tion (X∗) decreases towards zero when the residence time increases to infinity (i.e.
τ∗ → ∞). Thus, the microorganism concentration must increase to a maximum before it decreases towards zero.
0 0.02 0.04 0.06 0.08 0.1
0 2 4 6 8 10
Dimensionless microorganism concentration ( X* )
Dimensionless residence time (τ∗)
(a)
0.01 0.1 1
0 2 4 6 8 10
Dimensionless substrate concentration ( S* )
Dimensionless residence time (τ∗)
(b)
Figure 4.2: Dimensionless microorganism (a) substrate concentrations (b) as a func-tion of dimensionless residence time. Parameter value: dimensionless decay rate Kd∗ = 0.014091, dimensionless yield coefficient α∗ = 0.10194 and β(1− R∗)= 1 .
0.01 0.1 1
0 2 4 6 8 10
Dimensionless substrate concentration ( S* )
Dimensionless residence time (τ∗) a b c d e f
Figure 4.3: Dimensionless substrate concentration as a function of dimensionless residence time. Parameter value: dimensionless decay rate Kd∗ = 0.014091, dimen-sionless yield coefficient α∗ = 0.10194 and β(1− R∗)= 0 (a), 0.25 (b), 0.5( c), 0.75 (d) , 1 (e) and washout solution (f)
0 0.2 0.4 0.6 0.8 1
0 2 4 6 8 10
Dimensionless microorganism concentration ( X* )
Dimensionless residence time (τ∗) a
b c d e f
Figure 4.4: Dimensionless microorganism concentration as a function of dimension-less residence time. Parameter value: dimensiondimension-less decay rate Kd∗ = 0.014091, dimensionless yield coefficient α∗ = 0.10194 and β(1− R∗)= 0 (a), 0.25 (b), 0.5( c), 0.75 (d) , 1 (e) and washout solution (f)
4.3.9 Large residence time approximations
At large residence times, the approximate value of the substrate concentration is obtained by using Taylor series,
S∗ ≈ α∗
(1− Kd∗)τ∗ + O( 1
τ∗2), 0 < Kd∗ < 1. (4.3.28)
Equation (4.3.28) demonstrates that the substrate concentration decreases to zero at large value of the residence times. Interestingly, this equation also shows that the substrate concentration (S∗) is independent of the recycle parameter R∗ and the reactor parameter β. This means that equation (4.3.28) shows that, for processes controlled by Contois kinetics, the substrate concentration can be decreased to any desired level by operating the reactor at a sufficiently large the residence time. This is not the case for processes controlled by Monod kinetics whereby the substrate concentration has a limiting value, Kd∗
1 + Kd∗, [132].
At large residence times, the approximate value of the microorganism concentration is given by,
X∗ ≈ α∗
Kd∗τ∗ + O( 1
τ∗2), 0 < Kd∗ < 1. (4.3.29) Equation (4.3.29) shows that, for large values of the residence time, the microor-ganism concentration (X∗) is a decreasing function the residence time τ∗. Thus, in the limiting case, that is where τ∗ → ∞, the microorganism concentration tends to 0. The physical meaning of this is that the microorganism concentration becomes depleted due to the limited availability of substrate concentration.
4.3.10 Maximizing the microorganism concentration
In some bioreactor processes it is important to maximise the microorganism concen-tration, for instance if the bioreactor is being used to grow the microorganism. The microorganism concentration is zero (i.e. X∗ = 0) along the washout solution and in the limit that the residence time increase to infinity. Thus, there is a maximum value of microorganism concentration (X∗). Differentiating the expression of the microorganism concentration from equation (4.3.14), with respect to the resident time, we find that the maximum value occur when,
τmax∗ =β(1− R∗) 1− Kd∗ +
√Kd∗α∗β(1− R∗) Kd∗(1− Kd∗)
4.3.11 Discussion
The performance of a bioreactor processing industrial wastewaters can be charac-terized by using a number of criterions such as the specific utilization, the efficiency and the rate of waste treatment. Here, we will consider the process efficiency of bioreactor characteristic along the no-washout branch, that is when
τ∗ > β(1− R∗)
1− Kd∗ . (4.3.30)
The treatment efficiency is the percentage of substrate that has been removed by the reactor. It is defined by
ε = 100
(S0− S S0
)
. (4.3.31)
and therefore, the efficiency is given by
ε∗ = 100(1− S∗). (4.3.32)
From equation (4.3.32), it is clear that when S∗ < 1 along the no-washout branch, the efficiency is positive (ε∗ > 0). Substituting for S∗ from equation (4.3.14) into equation (4.3.32), we obtain the efficiency along the no-washout branch as
ε∗ = 100
The efficiency (equation (4.3.33)) as a function of the residence time (τ∗) are shown in figure (4.5). This figure shows that at fixed values for both the residence time and the parameter reactor β, the efficiency of the reactor increases as the value of the recycle parameter R∗ increases. The figure also shows that when the value of the parameters β and R∗ are given, then the efficiency characteristic of the bioreactor improves as the residence time increases. Furthermore, as the value of the expression (β(1− R∗)) decreases, the efficiency of the bioreactor improves.
When 0 < τ∗ − τcr∗ < 1. the values of the reactor parameter β and the recycle pa-rameter R∗ play a significant role in the efficiency of waste reduction. By expanding τ∗ about τcr∗ in equation (4.3.25), we find that the efficiency along the no-washout branch becomes
ε∗ = 100(1− Kd∗)
α (τ∗− τcr∗) + O((τ∗− τcr∗)2), (4.3.34)
At large residence times, the efficiency becomes
ε∗ = 100(1− α∗
τ∗(1− Kd∗)) + O( 1
τ∗2), (4.3.35)
As the residence time approaches infinity, the efficiency of the process approaches 100 % and is independent of the influent pollutant concentration. In contrast for processes controlled by Monod kinetics, the maximum efficiency is bounded below 100 and depends upon the influent pollutant concentration (S0) [132].
Finally, at high values of the residence time processes efficiency is independent of the reactor parameter (β) and the recycle parameter (R∗). To study the efficiency as a function of the parameter β(1− R∗), table (4.1) shows the residence time required to achieve efficiencies of 99%, 99.9% and 99.99% for varies value of β(1− R∗). It shows that the residence time to achieve a given efficiency increases as the value of β(1−R∗) increases. However, the dependence upon the parameter β(1−R∗) is very week.
Table 4.1: The residence time in single reactor to achieve an efficiency of 99.9%, 99.99% and 99.999% for varies value of β(1− R∗) = 0, 0.25, 0.5, 0.75 and 1.
Efficiency β(1− R∗)
ε∗ 0 0.25 0.5 0.75 1 percentage increase.
99% 11.01 11.28 11.55 11.82 12.10 9.9%
99.9% 111.104 111.37 111.64 111.92 112.19 0.977%
99.99% 1112.04 1112.31 1112.58 1112.86 1113.13 0.098%
0 20 40 60 80 100
0 1 2 3 4 5 6 7 8 9
Dimensionless Efficiency
Dimensionless residence time
a b c d e
Figure 4.5: The dimensionless efficiency (ε∗) in an ideal reactor as a function of dimensionless residence time. The parameter values: dimensionless decay rate Kd∗ = 0.014091, dimensionless yield coefficient α∗ = 0.10194 and β(1− R∗)= 0 (e), 0.25 (d), 0.5( c), 0.75 (b) and 1 (a)