2.6 Numerical Example
2.6.1 Bioreactor - A simulation case study
Mathematical modeling of bioreactor is of much interest due to its highly nonlinear behaviour. At the low concentration of input feed substrate, the growth rate of the cell-biomass dominates the reaction rate. Thus, increase in feed concentration results in an increase in productivity. However, beyond a certain value of the input feed concentration, the product and substrate inhibition prevails and thereupon productivity shows a negative gradient with respect to the input feed concentration.
Thus, there exists an optimum value of the bioreactor productivity. The necessary condition for optimality implies that the steady-state gain (product concentration vs. manipulated feed substrate concentration) changes its sign across the optimum point and the steady state gain is zero at the optimum. Thus, the system exhibits input multiplicity and there exists a singularity at the optimum operating point.
The mechanistic model of a bioreactor is given by the following differential equa- tions [77, 134],
dX
dt = −DX+µX (2.27)
dS
dt = D(Sf −S)− 1 YX/S
µX (2.28)
dP
dt = −DP + (αµ+β)X (2.29)
where X represents effluent cell-mass or biomass concentration, S represents sub- strate concentration and P denotes product concentration. Product concentration (P) and the cell-mass concentration (X) are measured process outputs, while dilu- tion rate (D) and the feed substrate concentration (Sf) are the process inputs. YX/S represents the cell-mass yield andα and β are the yield parameters for the product.
The specific growth rate, µ, is given by,
µ= µm(1−PPm) Km+S+KS2
i
(2.30) whereµm represents maximum specific growth rate, Pm,Km andKi are the product saturation constant, substrate saturation constant and substrate inhibition constant, respectively. The nominal model parameters are given in Table 2.1.
2.6. Numerical Example 46
Table 2.1: Nominal parameters for bioreactor Parameter Nominal Value
YX/S 0.4 g/l
α 2.2 g/l
β 0.2 h−1
µm 0.48 h−1
Pm 50 g/l
Km 1.2 g/l
Ki 22 g/l
For the present case the input-output data of SISO system of product concen- tration P against the feed substrate concentration Sf is taken for developing the empirical model. The operating point, around which the perturbation is given, has been chosen judiciously. In [134], the process optima has been chosen as the op- erating point and the perturbation had been given in the vicinity of the process optima. While the Laguerre Polynomial (LP) model, developed in [133], was able to capture the process nonlinearity around a narrow range of operating zone around the optima, it failed in the sub-optimal zone. Nevertheless, it can be argued that in real life scenario, the process data is generated as a random sequence and there is no guarantee that the data will be obtained around a specific point of one’s choice. An ideal model should be robust enough to capture the entire process behaviour with whatever data available at one’s disposal. With that argument, a sub-optimal point, obtained from [77], has been chosen in this work around which the perturbation is provided to collect the process data. The data required for model parameter esti- mation is obtained by giving a random signal inSf at its mean value of 20 and with variance of 0.1, perturbing the mechanistic model, eqn. (2.27)-(2.30), in open-loop.
Two sets of data are obtained, using one for parameter estimation and another for model validation. The Laguerre filter parameters are chosen such as, n= 3, p= 0.1
2.6. Numerical Example 47
from some a priori knowledge about the system,i.e., the open-loop step response and the sampling time of 0.1 hr. The wavelet parameters are estimated using backprop- agation algorithm. An error tolerance, ǫ = 0.001 is given as the stopping criterion for the network training.
0 100 200 300 400 500
10 15 20 25 30 35
Time (min)
Process Input
0 100 200 300 400 500
10 15 20 25 30 35
Time (min)
Process Output LWN Model Process LP Model
Figure 2.2: Dynamic response of the Wiener type Laguerre models for bioreactor process.
10 15 20 25 30 35
12 14 16 18 20 22 24 26 28
Feed Conc. (g/l)
Product Conc. (g/l)
LP Model Process LWN Model
Figure 2.3: Steady-state response of the Wiener type Laguerre models for bioreactor process.
2.6. Numerical Example 48
The wavelet parameter after training is shown in Appendix A (Table I). The resulting model is validated with another data set generated, and the result is shown in Figure 2.2. It could be noted that initially the process was run for 50 hours in the nominal steady state values as given in Table 2.1. Then the process was excited with a random signal about the steady state value. It could be observed from Figure 2.2 that the model is capable of capturing the process dynamics very well. Using the same Laguerre parameters, Laguerre polynomial proposed in [134] is developed with the maximum polynomial order as two. From Figure 2.2 it could be observed that Laguerre wavelet network model outperforms LP model, by closely following the response of the process.
The model is also validated for the steady state response against that of the process. Generally, model validation is performed using statistical methods such as Residual Analysis for finding out whether the developed model has missed out any necessary information. However, residual analysis cannot be used for black-box models, such as the Wiener type model developed in this thesis, as the system states and the process states are not equivalent for incorporating the replacement of the model’s one-step ahead predicted value with that of the process measurement. The feed substrate concentrationSf is varied from 10 to 35 and the corresponding steady state product concentration P is observed. From Figure 2.3, it could be found that the model is able to give better and very close steady state response matching with that of the process when compared against that of the Laguerre polynomial, used in [134]. Thus the model could efficiently capture the input multiplicity nonlinearity of the process over a wide range of operation. The prominent mismatches between process and LP model at 125 hr and 350 hr in Figure 2.2 are the indication that the LP model fails to capture the process steady state characteristics. This phenomenon is also supported by Figure 2.3. The Integral Squared Error (ISE) values of LP and LWN model are 120.9904 and 23.4225, respectively, which proves that the LWN model outperforms its polynomial counterpart. Although the training of the model with 10,000 input-output data points takes 54.098 seconds, once the model is trained it takes only 6.323×10−5 seconds to get the output from the model for a given input.
2.6. Numerical Example 49
This is one of the most welcomed results as it makes the model suitable even for fast processes in various applications.