Del currículo al aula. Actividad como ejemplo de implantación

6.2.1 The dimensional model

The model equations for a n reactor cascade with recycle around the whole cascade are given by,

Equations in the first reactor

V dS₁

dt = F (S₀− S1)− V X1

µ(S₁, X₁)

α + F R(S_n− S1), (6.2.1) V dX₁

dt =−F X1+ RF (CX_n− X1) + V X₁µ(S₁, X₁)− KdV X₁. (6.2.2)

Equations in the ith reactor, (1 < i≤ n)

V dS_i

dt = F (1 + R)(Si−1− Si)− V Xi

µ(S_i, X_i)

α , (6.2.3)

V dX_i

dt = F (1 + R)(X_i−1− Xi) + V X_iµ(S_i, X_i)− KdV X_i. (6.2.4) Specific growth rate (Contois model).

µ(Si, Xi) = µmax

( S_i K_sX_i+ S_i

)

. (6.2.5)

In the following i (1 < i≤ n) denotes the ith reactor in a cascade containing n re-actors. All parameters in the model are strictly non-negative. For details of the assumptions and definition of model parameters, we refer readers to chapter (4) and appendix (D) respectively.

For a specific wastewater, a given biological community and a particular set of envi-ronmental conditions the parameters K_s, K_d, α and µ_maxare fixed. The parameters that can be varied are C, R , S₀, X₀ and τ_t. In our numerical simulations we use parameter values for the anaerobic digestion of ice-cream wastewater proposed by Hu et al. [39]. These are: α= 0.2116 (gVSS)(g COD)⁻¹, µmax= 0.9297 (day⁻¹), K_d =0.0131 (day⁻¹) and K_s=0.4818 (COD)(g VSS)⁻¹.

The operation of the settling unit is characterized by two parameters: a concentrat-ing factor (C) and a recycle parameter (R). The maximum value of the concentratconcentrat-ing factor that be achieved in a specific settling unit is related to the value of the recycle parameter.

The value of the maximum concentrating factor Cmax is given by

C_max= (1 + 1

R). (6.2.6)

Alternatively the value of the maximum recycle R_max is given by

R_max= 1

C− 1, C ≥ 1. (6.2.7)

The cases (R = R_max) and (C = C_max) represent perfect recycle. The constraint on the maximum concentration factor and maximum recycle ratio in (6.2.6 and 6.2.7) is derived by considering a mass balance for the case when all the microorganism concentration leaving the effluent stream of reactor cascade is captured by the set-ting unit. This means that there are no microorganisms flowing out of the reactor cascade.

The value of the recycle ratio depends upon the type of the reactor. Typically, the designed value can vary between 0.1 to 1 [80][pp.117, 446, 622-664]. Furthermore, it is recommend that the recycle pumps should be capable of delivering flows between one-half and twice the design value [80]. The concentration factor (C) typically varies between one to three [80][pp.117, 446-447, 664].

6.2.2 The dimensionless model

By using the transformations introduced in chapter (5), the equations, (6.2.1-6.2.4) can be written into following dimensionless form,

Equations in the first reactor.

All dimensionless parameters in the model are strictly non-negative. For more de-tails of the assumptions and definition of model parameters, we refer readers to chapter (4) and appendix (D) respectively.

A feature of our dimensionless scheme is that there is a one to one relationship between our dimensionless variable and their dimensional counterparts. Hence, we write often residence time, rather than dimensionless residence time.

6.3 Results

The steady-state solution of equations (6.2.8-6.2.11) and their stability are deter-mined numerically. Steady state diagrams showing the variation of the effluent concentration (S_n^∗) as a function of the total residence (τ_t^∗) time are plotted. It should be noted that only the stable physical meaningful solutions are presented.

In section (6.3.1), the positive and boundness of the solution are determined and in section (6.3.2) the steady state solutions branches are discussed. In section (6.3.3), the stability of the washout solution for an n-reactor cascade (n = 2, ..., 5) is de-termined. In section (6.3.4) we compares the performance of a two-reactor cascade with recycle around the whole cascade against a two reactor cascade without recycle.

We discuss the effect of changing the concentration factor (C) for a given recycle ratio for a two-reactor cascade in section (6.3.5). In section (6.3.6) we compare the performance of a two-reactor cascade with that of a three-reactor cascade and finally in section (6.3.7) we provide asymptotic solutions for the substrate and cell-mass concentrations for an n-reactor cascade (n = 1, ..., 5) at large total residence times.

We show that the washout solution is globally stable when K_d^∗ ≥ 1. Hence, we

assume in the following that 0 < K_d^∗ < 1.

6.3.1 The positive and bounded of the solution

In this section the aim is to establish the basic properties of invariance for the region Ω ={(S₁^∗, ..., S_n^∗, X₁^∗, ..., X_n^∗)∈ R²ⁿ₊, 0≤ S_i^∗, X_i^∗ ≥ 0, i = 1, ..., n.}.

On the boundary of Ω, if X₁^∗ = 0 then ^dX_dt∗^1∗ = ^nRC_τ∗

t X_n^∗, if X_n^∗ > 0 then the vector field of X₁^∗ point inside Ω and if X_n^∗ = 0 then the plane X₁^∗ = 0 is invariant. Likely, if S₁^∗ = 0 then ^dS_dt∗^1∗ = _τⁿ_∗

t + ^nR_τ∗

t (S_n^∗), which is strictly positive, then the vector field of S₁^∗ point inside Ω. By the same way, if X_i^∗ = 0 then ^dX_dt∗^i∗ = ^n(1+R)X_τ∗ ^i−1∗

t ., if X_i^∗₋₁ > 0 then the vector field of X_i^∗ point inside Ω and if X_i^∗₋₁ = 0 then the plane X_i^∗ = 0 is invariant. On the other hand, if S_i^∗ = 0 then ^dS_dt∗^i∗ = ^n(1+R)_τ∗

t S_i^∗₋₁, if S_i^∗₋₁ > 0 then the vector field of S_i^∗ point inside Ω and if S_i^∗₋₁ = 0 then the plane S_i^∗ = 0 is invariant.

We conclude that Ω is positively invariant.

For the boundness of S_i^∗ and X_i^∗, suppose that S_n^∗(t)→t→+∞ +∞, so we can extract a subsequence (t_j) such that t_j →j→+∞ +∞ and Sn^∗(t_j)↗j→+∞ +∞ (is increasing to +∞).

Hence,

S˙_n^∗(t_j)≥ 0

It follows that

0≤ ˙Sn^∗(t_j)≤ n(1 + R)

τ_t^∗ (S_n^∗₋₁(t_j)− Sn^∗(t_j)) so

S_n^∗(t_j)≤ Sn^∗−1(t_j)

By the way,

S_n^∗₋₁(t_j)→j→+∞ +∞

By the same way we can extract a subsequence, noted also (t_j) to simplify the redaction, such that t_j →j→+∞ +∞ and Sn^∗−1(t_j)↗j→+∞ +∞ Using the same reasoning, we prove that

S_i^∗(t_j)↗j→+∞ +∞ ∀ i = 1, ..., n

which is contradiction. So, S_n^∗ is bounded. The boundness of S_i^∗,, i = 1, ..., n− 1 follows immediately from this fact. Note that by convention ˙S^∗ = ^dS_dt∗^∗.

In order to prove X_i^∗ (i=1,2,...,n) is bounded, we introduce a new variable that is Z =∑n

i=1X_i^∗. Hence, equations (6.2.9-6.2.11) become, dZ

Obliviously this fact implies that Z is bounded and hence that X_i^∗ is bounded for all i = 1, .., n.

6.3.2 Steady state solutions

Under washout conditions, the concentration of the effluent is equal to the concen-tration of the influent. This state of operation must be avoided. The washout,

steady state solution is given by:

(S_j^∗ = 1, X_j^∗ = 0), (1≤ j ≤ 5). (6.3.15)

It is not possible to find the no-washout steady state of the system (6.2.8-6.2.11) analytically. However, some interest points can be derived from the system (6.2.8-6.2.11). After some algebra, we find the following properties.

S_n^∗ = 1 + X_n^∗

α^∗(RC − (1 + R)) − τ_t^∗ α^∗n

∑n i=1

X_i^∗K_d^∗, (6.3.16)

Equation (6.3.16) shows when the recycle around the whole a cascade is perfect (RC− (1 + R) = 0) with no death rate of the microorganism then only value of the steady state solution is S_n^∗ = 1, i.e, washout solution. Since, the washout steady state solution is unstable when the recycle around whole a cascade is perfect , see equa-tion (6.3.20), then the soluequa-tion of the microorganism concentraequa-tion should move out from the washout of steady state solution to infinity. Once should noted that equation (6.2.10) has washout solution when the substrate concentration leaving the i th reactor is less/or equal to that leaving the (i+1) th reactor. Thus, along the no-washout solution, the substrate concentration leaving the i th reactor is always higher than that leaving the (i+1) th reactor.

6.3.3 Stability of the steady state solutions

6.3.3.1 Global stability of washout solution when K_d^∗ ≥ 1

Theorem 6.3.1 If K_d^∗ > 1 then the washout solution is globally asymptotically stable on Ω.

Proof. To prove global stability of the globally washout solution whenever K_d^∗ > 1, we consider the argument from [170]. We introduce a new variable Z = ∑_n

i=1X_i^∗. 1, 2, ..., n are positive and bounded (see section (6.3.1)). Thus as

Z(t^∗)≤ Ae^{(1−Kd∗)t∗}

where A is a constant.

Hence, the variable Z is a decreasing function of t^∗, reaching zero when t^∗ increases to infinity. It follows that X₁^∗, ..., X_n^∗ are decreasing functions of t^∗ reaching zero when t^∗ increases to infinity. Thus, the washout solution is globally stable when

K_d^∗ ≥ 1. 2

6.3.3.2 Stability along the washout branch

The stability of the washout branch for an n-reactor cascade (n = 2, ..., 5) is deter-mined by evaluating the Jacobian matrix at the washout steady state. The eigen-values are determined for an n-reactor cascade (n = 2, ..., 5) and it is presented in

appendix (A.2). It is found that the real parts of all the eigenvalues are negative, i.e., the washout branch is stable, when the following formula hold,

τ_t^∗ < τ_w^∗ = It is conjectured that this formula (6.3.18) holds for all n. This condition can never be satisfied for a reactor employing perfect recycle as τ_w^∗(C = C_max) = 0.

In appendix (A.0.1) we show that the critical value of the residence time (τ_w^∗) is a decreasing function of the recycle ratio (R). It follows that its maximum value is,

τ_w^∗(R = 0) = n

1− K_d^∗. (6.3.19)

and its minimum value is,

τ_w^∗(R = R_max) = 0. (6.3.20)

6.3.3.3 Stability along the no-washout branch

The Jacobian matrix for the system of equations (6.2.8-6.2.11) along the no-washout branch is,

k₁ = ^2(1+R)_τ∗ facts that along the no-washout branch,

1) ^{τt∗Kd∗+2(R+1)}_τ∗

The polynomial characteristic associated with matrix (J ) has been obtained by using the maple package [173] and given by,

z(λ) =λ⁴+ b₁λ³+ b₂λ²+ b₃λ + b₄, (6.3.21)

The Routh-Hurwitz Criterion is general technique that can be used to determine if the eigenvalue of characteristic polynomial have negative real part. The explicitly Routh-Hurwitz stability conditions for fourth degree of polynomial equation (6.3.21) are given by [180],

b₁ > 0, b₃ > 0, b₄ > 0 and b₁b₂b₃− [b²3+ b²₁b₄] > 0 (6.3.22)

We consider each coefficient of the Characteristic Polynomial. For first coefficient b₁, It is clear that b₁ > 0.

For second coefficient b2, it includes a two negative terms,−a10a6and−a7a3. Adding the two negative terms to the following two positive terms from b₂, a₁₂a₅ and a₈a₁, we have,

a₁₂a₅− a10a₆ = k₉k₄k₅+ k₆[k₁k₈+ k₉] > 0, a₈a₁− a7a₃ = k₁[2

τ^∗ + k₇+ k₂] + k₂k₇ > 0.

Thus, the second coefficient b2 is positive.

The third coefficient b₃ has four negative terms, −a10a₆a₈, −a10a₆a₁, −a12a₇a₃ and

−a7a₃a₅. Adding these four negative terms to the following four positive terms from the expression b3, a12a5a8, a12a5a1, a12a8a1 and a5a8a1, we obtain,

[a₈+ a₁][a₁₂a₅− a10a₆] + [a₁₂+ a₅][a₈a₁− a7a₃] = [a₈+ a₁][k₉k₄k₅+ k₆(k₁k₈+ k₉)] + [a₁₂+ a₅][k₁(_τ²_∗ + k₇+ k₂) + k₂k₇] > 0.

Thus, the third coefficient b3 is also positive.

The fourth coefficient b₄has two negative terms,−a10a₆a₈a₁and−a12a₇a₃a₅. Adding these two negative terms to the following positive terms from the expression b₄, a12a5a8a1 and a10a6a7a3 we obtain,

[a₁₂a₅− a10a₆][a₈a₁− a7a₃] = [k₉k₄k₅+ k₆(k₁k₈+ k₉)][k₁(_τ²_∗ + k₇+ k₂) + k₂k₇] > 0, Thus, the fourth coefficient b₄ is positive. Since all coefficients of equation (6.3.21) are a positive then the roots of this equation must be a negative if they are real.

However, this result is not true if the eigenvalue is complex. In this case, we need to satisfy the last condition of Routh-Hurwitz Criterion (b₁b₂b₃− [b²3+ b²₁b₄] > 0) to conclude that all eigenvalues of the characteristic equationz(λ) have negative real

parts. We have been unable to establish this analytically. However, it appears that this condition is satisfied numerically using our parameters in section (6.2). Thus, we believe that the no-washout branch is stable whenever it is physically meaningful.

The complexity of using the Routh-Hurwitz Criterion prevent us from attempting stability analysis for a cascade containing more than two reactors. This complexity has been reported by May [180]. Thus, the stability of the no-washout branch for of the three, four and five reactor cascades, was studied numerically. We observe that, for our parameters, all eigenvalues along the no-washout solution are negative. Thus, the no-washout branch, if it exist, is stable using our parameter in section (6.2).

6.3.4 Reactor Cascade With Recycle Around the whole Cas-cade

In this section, we particularly consider an analysis of a two reactor cascade and compare the performance of a two-reactor cascade with recycle, configuration (1), against that without recycle for three different values (minimum, moderate and maximum) of the concentration factor (C): C = C_min = 1, C = 2 and C = C_max. In the previous section, we established that the critical value of the residence time (τ_w^∗) for the two reactor cascade without recycle is greater than the critical value of the residence time for the two reactor cascade with recycle. That is, only the washout branch is stable below these critical values, and therefore process failure occurs.

6.3.4.1 Case C = C_min=1

Figure (6.2) shows the effluent concentration leaving a two reactor cascade, as a function of the total residence time when the concentration factor is at its mini-mum (C = C_min = 1) whereby it is seen from equation (6.2.7) that in this case the maximum value of the recycle parameter is infinity (R_max =∞).

It can be seen from figure (6.2) that there are three distinct regions for each value of R (R > 0). In the first region, washout occurs in the two reactor cascade. As the value of R increases from 0 to 1, the value of the washout point (τ_w^∗) decreases from 2/(1− Kd^∗) to zero. In this region the performance of the cascade with recycle is identical to that of a cascade without recycle. In the second region, the performance of the two reactor cascade with recycle is superior to the performance of the two reactor cascade without recycle. Thus in this region the effluent concentration can be minimised, at a specified total residence time, by selecting an optimal value of the recycle ratio, denoted by R_Opt. The dependence of the effluent concentration upon the recycle ratio for a residence time in this region is illustrated in figure (6.3(a)).

For a specified value of R the second region ends when the corresponding effluent concentration curve with recycle intersects the effluent concentration curve without recycle. Using the data in table (6.1) we see that the second region corresponding to the curves R = 0.5 and R = 1 in figure (6.2) ends at τ_t^∗ = 2.5 and τ_t^∗ = 2.4 respectively. The detail of intersection points for the cases R = 0.5 and R = 1 are given in table (6.1). For total residence times beyond the intersection points, the effect of recycle ratio (R) on the reactor performance is negative and vice-versa.

In the third region, the performance of a two reactor cascade with recycle is inferior

to the performance without recycle, i.e., recycle has a negative effect on the effluent concentration. Figure (6.3(b)) illustrates that in the third region where R_Opt = 0.

0.0001 0.001 0.01 0.1 1

0 1 2 3 4 5 6 7 8 9

Dimensionless substrate concentration ( S* e)

Dimensionless total residence time (τ^∗_τ) No recycle

R=0.5 R=1.0 R=R_Opt

Figure 6.2: Effluent concentration in a cascade of a two reactors with recycle around the whole cascade. Parameter value: C = C_min = 1.

6.3.4.2 Case C =2

Figure (6.4) shows steady-state diagrams of a two reactor cascade when C = 2.

When C > 1, equation (6.2.7) shows that there is a restriction on the maximum value of the recycle parameter.

When 0 < R < R_max = 1, we observe the same three distinct regions that were described in section (6.3.4.1). (When R = R_max, the first of these regions does not exist.) It can be seen that in the second region the performance of the two reactor cascade with recycle increases as the recycle ratio (R) increases and the optimal value of the recycle is R = R_max = 1. The intersection points for the cases R = 0.5

0.01 0.1 1

0 2 4 6 8 10

Dimensionless substrate concentration ( S* 2

Recycle Ratio (R)

(a) τ^∗= 2 (Region 2)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

0 20 40 60 80 100

Dimensionless substrate concentration ( S* 2)

Recycle Ratio(R)

(b) τ^∗= 8 (Region 3)

Figure 6.3: Diagram showing the effect of the recycle ratio on the effluent concen-tration (S₂^∗). In figure (6.3(a)), the filled-in circles is the optimum value of the recycle ratio (S₂^∗ = 0.0638, R_Opt = 1.08). In figure (6.3(b)), the optimum value of the recycle ratio is R_Opt= 0. Parameter value: C = C_min = 1.

and R = R_max= 1 are shown in table (6.1). In the third region the optimum value of the recycle ratio is zero, i.e. the use of a settling unit in this region increases the effluent concentration.

Figures (6.5(a)) demonstrates the effect of a settling unit upon the effluent concen-tration (S₂^∗) leaving a two reactor cascade for fixed total residence time in the second region. For sufficiently small values of R (0 ≤ R ≤ Rcr = 0.21) washout occurs as τ^∗ < τ_w^∗. For sufficiently large values of the recycle parameter (R > R_cr= 0.21) the effluent concentration decreases as the recycle ratio increases; the minimum effluent concentration is obtained when ROpt = 1.

Figure (6.5(b)) demonstrates the effect of a settling unit in the third region. In this region the effluent concentration is an increasing function of the recycle ratio, i.e ROpt = 0 and the effluent concentration is minimised by turning the settling unit off.

6.3.4.3 Case C = C_max = 1 + 1/R

Figure (6.6) shows steady-state diagrams of a two reactor cascade for the special case in which the concentration factor is maximized (C = Cmax= 1 + 1/R) for two values of the recycle ratio (R = 0.5 and R = 1). There are two distinct regions.

In the first region the effluent concentration in a cascade with recycle is lower than that without recycle. The effluent concentration leaving the cascade with R = 0.5 is lower than that leaving the cascade with R = 1.0. The intersection points for the cases R = 0.5 and R = 1 are shown in table (6.1). In the second region, for sufficiently large values of the total residence time, the effluent concentration leaving the reactor cascade with recycle is greater than that without recycle. In this region

0.0001 0.001 0.01 0.1 1

0 1 2 3 4 5 6 7 8 9

Dimensionless substrate concentration ( S Dimensionless total residence time (τ^∗_τ)

R=R_Opt=1

R=R_Opt=0 No recycle

R=0.5 R=1.0 R=R_Opt

Figure 6.4: Effluent concentration in a cascade of a two reactors with recycles around the whole cascade. Parameter value: C = 2, R_max= 1.

the performance of the cascade is maximised by removing the settling unit.

6.3.4.4 Summary

For each of the three cases considered there are two regions of practical interest. In the first region, the use of a settling unit improves the performance of the reactor cascade. In the second region the operation of a settling unit worsens the perfor-mance of the reactor cascade.

Extensive numerical investigations (not shown) an n-reactor cascade for (n = 3, 4 and 5) showed the presence of the same two regions of practical interest that arise for specific case of the two reactor cascade. The same conclusions were also reached by using Monod model as explained in appendix (A.1). This latter investigation was suggested by a referee of our paper [179].

0.01 0.1 1

0 0.2 0.4 0.6 0.8 1

Dimensionless substrate concentration ( S* 2)

Recycle Ratio(R)

(a) τ^∗= 1 (Region 2)

0.0008 0.00085 0.0009 0.00095 0.001 0.00105 0.0011 0.00115 0.0012 0.00125

0 0.2 0.4 0.6 0.8 1

Dimensionless substrate concentration ( S* 2)

Recycle Ratio (R)

(b) τ^∗= 8 (Region 3)

Figure 6.5: The effect of the recycle ratio upon the effluent concentration (S₂^∗).

Parameter value: C = 2, R_max = 1. In figures (6.5(a) and 6.5(b)), the optimum value of the recycle ratio is ROpt = 1 and ROpt= 0 respectively.

0.0001 0.001 0.01 0.1 1

0 1 2 3 4 5 6 7 8 9

Dimensionless substrate concentration ( S Dimensionless total residence time (τ^∗_τ)

No recycle R=0.5 R=1.0

Figure 6.6: Effluent concentration in a cascade of a two reactors with recycles around the whole cascade. Parameter value: C = C_max= 1 + 1/R.

6.3.5 The effect of changing the concentration factor (C)

Figure (6.7) shows the effect of varying the concentration factor on the effluent con-centration leaving a two reactor cascade for a fixed value of the recycle ratio (R = 0.5). For a value of R = 0.5, we have from equation (6.2.6) that Cmax= 3. At suffi-ciently low values of total residence time, it is apparent that the performance of the reactor cascade improves continually as the concentration factor increases. However, for each value of the concentration factor, there is a critical value for the residence time at which the performances of the two reactor cascade with and without recycle are identical. These values are given in table (6.1). This critical value increases as the concentration factor increases. If the residence time is greater (lower) than the critical value, the performance of the two reactor cascade with a settling unit is worse (superior) than its performance without recycle.

Furthermore, the effect of the concentration factor (C) on the performance of the two-reactor cascade decreases for larger values of total residence time (τ_t^∗ ≥ 9 ) as shown in figure (6.7). This agrees with our results presented in section (6.3.7) showed that in the asymptotic limit of large residence times the effluent concen-tration leaving the final reactor is independent of the concenconcen-tration factor (C) (a second-order effect). This behavior was observed for an n-reactor cascade (n = 3, 4 and 5) as well.

0.0001 0.001 0.01 0.1 1

0 1 2 3 4 5 6 7 8 9

Dimensionless substrate concentration ( S* e)

Dimensionless total residence time (τ^∗_τ) No recycle

C=1 C=2 C=3

Figure 6.7: Effluent concentration in a cascade of a two reactors with recycle around the whole cascade. The parameter value: R = 0.5.

6.3.6 Comparison of a two-reactor cascade and a three-reactor cascade

Figure (6.8) compares the performance of two-reactor and three-reactor cascades with and without recycle. For the cascades with recycle, the performance of the

Table 6.1: Intersection points in figure i of the recycle curve with the no recycle curve.

i R = 0.5 R = 1 C

6.2

S₂^∗

0.02 0.03 1

6.4 0.003 0.003 2

6.6 0.001 0.003 1+1/R

?? 0.00022 0.00016 2

6.2

τ_t^∗

2.5 2.4 1

6.4 4.3 4.5 2

6.6 6.3 4.4 1+1/R

A.1 46.60 68.11 2

i C = 1 C = 2 C = 3

6.7 S₂^∗ 0.028 0.003 0.0014

6.7 τ_t^∗ 2.5 4.4 6.4

three-reactor cascade is superior to the performance of the two-reactor cascade.

However, comparing the effluent concentration leaving a two-reactor cascade with recycle (Line 2A) against that leaving the three-reactor cascade without recycle (Line 3B) shows that there is a critical value of the total residence time in which the performances of these two configurations are identical. This value is (τ_t^∗ = 3.4, S_e^∗ = 0.004). If the desired effluent concentration is higher (lower) than this value, then the performance of the two-reactor cascade with recycle is superior (worse) to the performance of the three-reactor cascade without recycle.

1e-005 0.0001 0.001 0.01 0.1 1

0 1 2 3 4 5 6 7 8 9

Dimensionless substrate concentration ( S* e)

Dimensionless total residence time (τ^∗_τ) Two reactors.2A

Three reactors.3A Two reactors.2B Three reactors.3B

Figure 6.8: Comparison of performance in a two reactor cascade against a three reactor cascade. The parameter value: (A) R = 0.5, C = C_max and (B) No recycle .

6.3.7 Large residence time approximations

In this section we provide approximations at large total residence times for the concentrations of the substrate and microorganisms. These explain the interesting

behavior that was observed in sections (6.3.4-6.3.6) regarding the effect of recycle on the reactor cascade. We obtain the following approximations by using Taylor series,

S_i^∗ = A_i−1

Equations (6.3.23–6.3.24) give the concentrations of the substrate and microor-ganisms in the first five reactors of a n cascade. These expressions hold for any value

Equations (6.3.23–6.3.24) give the concentrations of the substrate and microor-ganisms in the first five reactors of a n cascade. These expressions hold for any value

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