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## Brazilian Journal of Chemical Engineering

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*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol. 14 no. 2 São Paulo June 1997

#### http://dx.doi.org/10.1590/S0104-66321997000200009

**Start-up Strategy for Continuous Bioreactors**

**A.C. da Costa, E.L. Lima and T. L.M. Alves**

Programa de Engenharia Química, COPPE, Universidade Federal do Rio de Janeiro - Cx. Postal 68502

CEP 21945-970, Rio de Janeiro, RJ, Brazil - E-mail: aline@peq.coppe.ufrj.br

*(Received: October 26, 1996; Accepted: May 6, 1997)*

ABSTRACT -The start-up of continuous bioreactors is solved as an optimal control problem. The choice of the dilution rate as the control variable reduces the dimension of the system by making the use of the global balance equation unnecessary for the solution of the optimization problem. Therefore, for systems described by four or less mass balance equations, it is always possible to obtain an analytical expression for the singular arc as a function of only the state variables. The steady state conditions are shown to satisfy the singular arc expression and, based on this knowledge, a feeding strategy is proposed which leads the reactor from an initial state to the steady state of maximum productivity.

KEYWORDS: Bioreactors, optimization, start-up strategy.

**INTRODUCTION**

The continuous mode of operation has some advantages when compared to the batch and fed-batch modes. If a steady state is attained, the quality of the final product is constant, as the operational conditions are invariable. Besides, the volume of the reactor can usually be smaller, as shut-downs for harvesting and sterilization are less frequent. However, due to the high probability of contamination and/or culture degeneration caused by a long operational time, continuous processes are used only in a few cases in the fermentation industry, such as cell mass and ethanol production.

Before a continuous reactor reaches steady state operation, there is a transient period in which it is filled up to the working volume. This period is called the start-up of the reactor and refers to the sequence of operations that leads it from an initial state to a final state of continuous operation. The modes of operation usually used for the start-up are the fed-batch and batch modes, the latter being a particular case of the former (feed rate equal to zero).

Determining an efficient start-up strategy is important as it saves time. Biochemical reactions are usually slower than chemical reactions and an inappropriate start-up may cause the reactor to spend too much time to reach the steady state conditions.

In the last years there has been little work dealing with the start-up of continuous bioreactors. Yoshida and Yamané (1974) used the batch and fed-batch modes of operation for the start-up of a continuous reactor. They proposed switching from a constantly fed-batch culture to a continuous operation in the steady state. They assumed, however, that the volume of the culture was invariable. Later, Dunn and Mor (1975) extended the idea to consider the volume variation. Takamatsu *et al.* (1975) solved the problem of the start-up to a previously specified steady state as the time optimization of a fed-batch culture, by using the optimal control theory. They used Green's theorem and Pontryagin's maximum principle cooperatively to determine an optimal feeding strategy. Yamané *et al.* (1979) established a time optimal start-up strategy by applying Miele's extremization method based on Green's theorem. Later, Brooks (1988) discussed the use of the fed-batch mode for the start-up of a continuous chemical reactor. In his work the reactor was operated in fed-batch mode until it reached the point of maximum product concentration. Once this point was reached, the output rate was set equal to the feed rate to start continuous operation.

In this work the start-up strategy for a continuous bioreactor will be defined using results from the singular control theory applied to optimize a fed-batch fermentation.

In the literature, there are many works dealing with the optimization of fed-batch fermentations. One approach to solving this problem is to use Pontryagins maximum principle (Pontryagin *et al.*, 1962) and the substrate feed rate as the control variable. It leads to a singular control problem and in this case Pontryagin's maximum principle fails to yield a complete solution. Thus, the singular control theory is used. Modak *et al.* (1986) studied the general characteristics and the theoretical development of feed rate profiles for various fed-batch processes. Based on this work, Lim *et al.* (1986) developed computational algorithms to calculate the optimal substrate feed rates which were valid for processes described by four or less mass balance equations. Modak and Lim (1987) developed a closed-loop optimization scheme which attenuated uncertainties and was independent of the initial conditions. Palanki *et al.* (1993) derived optimal state feedback laws for the singular interval by using the Lie brackets concept. In 1994, the same authors derived optimal state feedback laws for the singular interval for free final time problems with an escalar inequality constraint. Costa (1996) proposed a methodology for the analytical solution of this optimization problem which was valid for processes described by four or less mass balance equations. In this work, a new control variable was proposed.

The problem of determining the optimal substrate feed rate can also be solved by using Mieles method, which is based on Greens theorem. Ohno and Nakanishi (1976) used this method to optimize a lysine fermentation. Later, Cazzador (1988) determined the optimal substrate feed rate profile in a process for biomass production.

When the analytical solution of the optimal control problem is not possible, the use of Pontryagins maximum principle and the singular control theory usually requires much computational effort. Other methods have been proposed in the literature to deal with the computational difficulties. Cuthrell and Biegler (1989) proposed a simultaneous optimization and solution strategy based on successive quadratic programming and orthogonal collocation on finite elements. Later, Luus (1993) used iterative dynamic programming to determine the optimal feed rate profile in fed-batch fermentations. In the same year, this author extended the iterative dynamic programming to provide piecewise linear continuous control policies.

The objective of this work is to develop a practical feeding strategy for the start-up of a continuous reactor by using the singular control theory. It is shown that an adequate choice of the control variable can make possible the analytical solution of the optimization problem.

**CHOICE OF THE CONTROL VARIABLE**

Considering a fed-batch fermentation in which a micro-organism x produces a product p utilizing a substrate s, the mass balance equations are

(1)

where X, S and P are concentration of cell mass, substrate and product, respectively; Sf is the feed substrate concentration; , and are the specific rates of cell growth, substrate consumption and product formation, respectively, that may be functions of X, S and P; F is the volumetric substrate feed rate and V is the volume of the reactor.

In the literature, the variable most usually chosen as the control variable is the volumetric substrate feed rate.

The equations, rewritten in terms of the dilution rate (D=F/V), are

(2)

If the dilution rate is used as the control variable, the mass balance equations for biomass, substrate and product are not explicit functions of the volume of the reactor. Thus, although the global balance equation is used to calculate the volume of the reactor, it is not used in the solution of the optimization problem. The dimension of the system is reduced and processes described by four or fewer mass balance equations can be solved analytically.

**OPTIMIZATION PROBLEM**

Writing equation 2 in a vectorial form results in

(3)

with

; ; and

u=D

The global balance equation is not used in the solution of the optimization problem.

Constraints are imposed on the dilution rate and the final fermenter volume.

(4)

(5)

The optimization problem is to find the optimal temporal profile of the dilution rate that leads the reactor from a given initial state to the final state that maximizes an objective function, S(**x**(tf)).

This problem can be solved by using Pontryagins maximum principle. According to this principle, the optimal solution must maximize the Hamiltonian, which is defined as

(6)

where** **(t) is the adjoint vector which satisfies

(7)

The Hamiltonian can be rewritten as

(8)

where

(9)

(10)

Since the Hamiltonian is linear in the control variable, it is easy to determine u(t) which maximizes this Hamiltonian by examining the sign of the function (t).

if (t) > 0 u=umax

(11)if (t) < 0 u=umin

(12)

However, if (t) is equal to zero over a finite time interval (t1,t2), Pontryagin´s maximum principle fails to give u(t) during this interval and the singular control theory has to be used. This is called the singular interval.

During the singular interval, (t) is equal to zero and hence its derivatives must also vanish. According to the singular control theory, the expression of the control variable in the singular interval can be determined by deriving (t) until the control variable appears explicitly. In the cases studied in this article, the singular dilution rate is determined by making the second derivative of (t) equal to zero.

The following equations can be written in the singular interval:

(13)

(14)

(15)

In addition, if the fermentation time (tf) is not specified a priori, the Hamiltonian on the optimal trajectory is equal to zero. Then, during the singular interval,

(16)

In this work, Lie brackets (Kravaris and Kantor, 1990) are used to obtain the singular arc and the singular dilution rate expressions. By using this differential geometric tool, the first derivative of can be written as

(17)

where [**f,g**] is the Lie bracket, which is a vector field defined by

(18)

Its components are given by

(19)

The second derivative of (t) can be written as

(20)

where **adf2g(x)** and **adg2f(x)** are the notations used for the iterated Lie brackets **[f,[f,g]]** and **[g,[f,g]]**, respectively.

The singular control is then given by

(21)

For systems described by four mass balance equations, as the global balance equation is not used, the effective state variables vector, and thus the adjoint variables vector, are three dimensional (the volume is not used as a state variable). In this case, it is possible to write two adjoint variables as a function of the third adjoint variable and the state variables. From equation 13 one can write

(22)

And from equation 16

(23)

The substitution of equations 22 and 23 into equation 17 leads to

(24)

Then, from equation 14, it follows that, in the singular interval

(25)

Also, substituting equations 22 and 33 into equation 20, the following equation is obtained:

(26)

From equation 15 it follows that

(27)

which is the equation of the control acting during the singular interval as a function only of the state variables.

In equations 22 to 27, A(**x**), B(**x**), G(**x**), a(**x**) and b(**x**) are functions of the state variables.

Once the expression of the control variable in the singular interval is determined, the next problem is the determination of the initial and final time of the singular interval. As the singular arc expression (equation 25) was obtained as a function only of the state variables, these switching times are easily determined: the first is the one in which equation 25 is satisfied and the second is the time at which the reactor reaches its maximum volume.

**THE START-UP STRATEGY**

In the problem described above, the objective was to maximize a function of the final state in a free final time, starting from known initial conditions. For different initial conditions, different optimum values would be obtained.

In the start-up of a continuous reactor, the known conditions for the state equations are the final conditions. They are given by the optimal steady state to be attained.

In the steady state

(28)

Using equations 3, 17 and 18, it can be shown that

(29)

This equation is equal to equation 14, which is valid in the singular interval. Thus, the steady state conditions satisfy the singular arc.

As, during the start-up, the reactor is operated in fed-batch mode, it should be said that what is called the "steady state" in this work is a state in which, although the volume of the reactor increases, the values of the state variables remain constant.

The proposed start-up strategy is derived from the knowledge that the steady state satisfies the singular arc conditions. The reactor is led from an initial state, determined *a priori,* to the singular arc, at the point which corresponds to the optimal steady state. In the singular arc, the state variables are kept constant, until the reactor reaches the maximum volume. At this point, the output rate is set equal to the feed rate and the continuous operation starts.

The initial conditions that lead the system to the optimal steady state should be determined by back-integrating the mass balance equations of the system using the maximum or minimum dilution rate.

**NUMERICAL EXAMPLES**

**Example 1. Cell Mass Production**

The objective in this example is to start-up a reactor for cell mass production from an initial state to the steady state of maximum productivity. The utilized model was studied by Weigand *et al.* (1979). The specific rates are given below.

(30)

;

(31)

In this case there are only two state equations and hence two adjoint variables. Equations 9 and 10 are given by

(32)

(33)

By using equations 32 and 33, one of the adjoint variables can be written as a function of the other and the state variables. It can be done in two different ways:

(34)

(35)

For this example, equation 17 is equal to

(36)

The substitution of equation 34 into equation 36 leads to

(37)

And the substitution of equation 35 into equation 36 to

(38)

The singular arc expression must satisfy equations 37 and 38.

For the kinetic model considered, it is easy to notice that equation 37 is always satisfied. Therefore, the singular arc can be described by two expressions, derived from equation 38:

(39)

(40)

Equation 39 describes the relationship between the concentrations of biomass and substrate for the steady state in a continuous fermentation. Equation 40 describes the maximum growth rate. As in this problem the goal is to reach the steady state, equation 40 will be chosen to describe the singular arc.

Equation 20 in this example is given by:

(41)

The substitution of equation 39 into equation 41 leads to

(42)

This is the expression of the dilution rate in the steady state in a continuous fermentation.

The final conditions of the optimization problem are the conditions of the steady state to be reached. In this work it is defined as the steady state of maximum productivity.

In this case, the productivity is defined as DX, where D is given by equation 42 and X by equation 39.

(43)

Making the derivative of equation 43 with respect to the substrate concentration equal to zero, the substrate concentration in the optimal steady state is determined. The biomass concentration is obtained from equation 39.

Figure 1 shows the productivity *versus* substrate concentration for a fermentation with the feed substrate concentration equal to 10 g/l. In this case, the substrate concentration that leads to the maximum productivity is 0.22 g/l, which corresponds to a concentration of biomass of 4.89 g/l and to a dilution rate of 0.80 h-1.

Before the start-up strategy can be applied for this example, it is also necessary to determine the initial conditions that lead the system to the desired steady state. Takamatsu *et al.* (1975) proved that, for a bioreaction described only by mass balance equations for biomass and substrate and whose equations are written in terms of the dilution rate, a steady state described by equation 39 will only be attained if the initial conditions satisfy this equation. Figure 2 shows equation 39, which describes both the steady state relationship between substrate and biomass and the singular arc expression. Point P in this figure shows the conditions of the optimal steady state.

Any initial conditions in the curve given in Figure 2 lead to the steady state described by point P. Initial conditions situated to the right of point P correspond to high initial concentrations of substrate and, to reach this point, the reactor must be initially operated in batch mode. On the other hand, initial conditions situated to the left of point P correspond to low initial concentrations of substrate and hence the initial dilution rate must be the maximum one. Once point P is reached, the dilution rate assumes the singular value, which is equal to the steady state value, until the reactor is full. In this case, the state variable values are kept constant and the singular arc is described by point P.

**Figure 1:** Productivity *versus* substrate concentration in the steady state (Sf =10 g/l).

**Figure 2: **Initial concentrations of biomass and substrate that lead to the steady state.(Sf=10 g/l).

The choice of a specific set of initial conditions depends on economical and operational factors and will not be discussed in this work. If the initial conditions are fixed and do not satisfy equation 39, some operation has to be done to bring the reactor to a point where this equation is satisfied before the start-up strategy begins.

Figure 3 shows the concentrations of biomass and substrate* versus* time for a fermentation with high initial substrate concentration. Figures 4a, 4b and 4c show the volume, dilution rate and substrate feed rate *versus* time for the same fermentation. The reaction starts with a batch period given in the figures by the time interval from zero to t1. In this interval, the substrate concentration is reduced from 1 g/l to 0.22 g/l, which corresponds to the optimal concentration in the steady state. In the singular interval, shown in the figures by the time interval from t1 to t2, the biomass and substrate concentrations are kept constant at the value corresponding to point P. This interval lasts until the reactor is full, when the output rate is set equal to the feed rate and the continuous operation starts. In the figures an interval of continuous operation is also shown.

In Figures 4a, 4b and 4c the difference between the fed-batch and continuous modes of operation can be noticed. In the fed-batch mode, although the dilution rate is constant, the volume of the reactor is increasing and then the volumetric substrate feed rate is variable. In the continuous operation, the volume, the dilution rate and the volumetric substrate feed rate are constant.

**Figure 3:** Biomass and substrate concentrations *versus* time.

Operational conditions: X(0)=4.5 g/l, S(0)=1 g/l; V(0)=1 L, Sf=10 g/l, Vmax= 5l.

**Figures 4a, 4b and 4c: **Volume, dilution rate and volumetric feed rate *versus* time.

Operational conditions given in Figure 3.

The proposed strategy to start up the continuous reactor led it to the steady state of maximum productivity in 2.1 h.

**Example 2.** **Alcohol Fermentation**

The fermentation process for ethanol production by *Sacharomyces cerevisae* was studied in 1968 by Aiba *et al.* The kinetic expressions are given by

(44);

(45)

(46)

The Hamiltonian is given by equation 8, with

(47)

(48)

Equation 17 in this case is equal to

(49)

s, p,, s, p, s and p are the derivatives of , and with respect to substrate and product concentrations.

The substitution of the steady state equations into equation 49 leads to

(50)

Thus, the steady state conditions satisfy the singular arc expression.

In this example, the optimal steady state corresponds to the maximum ethanol productivity (PD), where P is given by dP/dt=0. The resulting expression is

(51)

Figure 5 shows the ethanol productivity *versus* the dilution rate in the steady state for a fermentation with a feed substrate concentration equal to 100 g/l. It can be seen in this figure that the maximum productivity corresponds to a dilution rate of 0.161 h-1. The optimal biomass, substrate and product concentrations are 8.86 g/l, 11.4 g/l and 32.5 g/l, respectively.

The initial conditions that lead to the optimal steady state were determined by back-integrating the mass balance equations from the optimal steady state with the dilution rate value equal to zero.

**Figure 5: **Ethanol productivity *versus* dilution rate in the steady state (Sf =100 g/l).

Figure 6 shows the trajectory obtained and its projection in planes XS, XP and PS. Any initial conditions in the curve AB lead to the desired steady state if the reactor is initially operated in batch mode. The optimal steady state is given in the figure by point B.

Figure 7 shows biomass, substrate and product concentrations *versus* time for a fermentation whose initial conditions are situated in curve AB. In Figures 8a, 8b and 8c the variations of the volume, dilution rate and volumetric feed rate with time can be seen. In these figures, three distinct phases are noticed. The first is a batch phase, in which the biomass concentration is reduced and the substrate and product concentrations increase. This corresponds to a movement in trajectory AB in Figure 6 until point B is attained. In the second phase, the reactor is operated with the steady state dilution rate. In this phase the state variable values are kept constant in the optimal steady state. It is shown in the graphics as the time interval from t1 to t2. When the reactor reaches the maximum operational volume, the output rate is set equal to the feed rate and the continuous operation begins.

**Figure 6:** Initial conditions that lead to the optimal steady state with a batch mode of operation (Sf =100 g/l).

**Figure 7:** Biomass, substrate and product concentration *versus* time. Operational conditions: X(0)=2.1 g/l, S(0)=78.5 g/l, P(0)=10.9 g/l, V(0)=5 L, Sf=100 g/l, Vmax=20 L.

**Figures 8a, 8b and 8c: **Volume, dilution rate and volumetric substrate feed rate *versus* time.

Operational conditions in Figure 7.

**CONCLUSIONS**

The use of the dilution rate as the control variable reduced the dimension of the problem by making the use of the global balance equation unnecessary for the solution of the optimization problem. Thus, systems described by four or less mass balance equations can be solved analytically. This avoids the numerical solution of the optimization problem, which usually requires much computational effort.

The expression obtained for the singular arc, independent of the adjoint variables, facilitated the interpretation of the results and showed that the steady state conditions satisfied the singular arc expression.

A simple strategy for the start-up of continuous reactors was developed. The reactor was started from an initial state (with D=0 or D=Dmax). Once it reached the singular arc, the singular dilution rate was applied and the state variables were kept constant while the volume of the reactor increased. When the reactor was full, the output rate was set equal to the feed rate and continuous operation began.

The steady state attained was that of maximum productivity, as the initial conditions which led to this point of operation were determined during the solution of the problem.

**NOMENCLATURE**

A(**x**):** **Function of the state variables

B(**x**): Function of the state variables

a(x): Function of the state variables

b(x): Function of the state variables

D: Dilution rate, h^{-1 }**f**(**x**),**g**(**x**): Vectors in equation, 1

F: Volumetric feed rate, l/h

G(**x**): Function of the state variables

H: Hamiltonian

H_{0}: l ^{T}**f(x)**

'p: Derivative with respect to product concentration

P: Product concentration, g/l

's: Derivative with respect to substrate concentration

S: Substrate concentration, g/l

S_{f}: Feed substrate concentration

t: Time, h

T: Transpose of a vector

u: Control variable

V: Volume, l

X: Cell mass concentration, g/l

**x**: Vector of state variables

Y_{X/S}: Cell mass to substrate yield, g cell mass/g substrate

*Greek Letters*

f : l ^{T}**g(x) **

**l**

**: Vector of adjoint variables**

m : Specific growth rate, h

^{-1 }p : Specific product formation rate, h

^{-1 }s : Specific substrate consumption rate h

^{-1}

*Subscripts*

f: Final

max: Maximum

min: Minimum

sing Singular

0: Initial

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