APPLICATION OF THE Z AND LAPLACE TRANSFORMS TO PANIFICATION YEAST PRODUCTION USING AN AIRLIFT BIOREACTOR

area Nanotechnology and Thermodynamics, mainly on the following themes: nanostructures in packaging for preserving post-harvest fruits, nanostructures of whey proteins and protein conjugates to obtain new products, thermodynamics of nanotube interaction carbon, clays and oxides; polymer-surfactant interaction; two-phase aqueous system. NanoPack - Technologies in packaging. modeling of modeling of equipment animation computational process computer Abstract . The Z- and Laplace transforms are mathematical techniques applied to solve difference equations and differential equations, respectively. Mathematical models used to describe cell growth, substrate consumption and product formation in bioprocesses can be represented by these types of equations. Thus, in this work, the fermentation process of the yeast Saccharomyces cerevisiae was modeled using different models from the literature, and the Z- and Laplace transforms were applied to solve the equations. Once the equations were solved, the models were represented in state space and simulated in Octave® software. Finally, the models were compared to experimental data from previous studies and to each other. Verhulst was the model that best described the process, with an average error of 4.74% for cell growth and 13.9% for substrate consumption. This work is unprecedented since no works that use the Z transform and discrete models for the representation of fermentation of this yeast were found in the literature. Even more importantly, this work proved that discrete-time models can be applied to bioprocesses with the same precision as continuous-time models.


Introduction
According to a report by Mcwillians [1] in BCC Research, the market for the yeast This scenario stimulates the development of studies aimed at optimizing processes.
Many tools in the literature can be used to improve the productivity of a process, including mathematical modeling.
According to Finlayson [2], this technique is able to determine, based on experimental results, the best operating conditions for a process and, consequently, to predict the best operating conditions when some process parameters change. Many types of models can be applied to describe a process: linear models, nonlinear models, network models, multiobjective optimization and discrete models. After a model is established, it is necessary to simulate it. However, the mathematical resolution of these models is not always trivial, and in most cases, the use of mathematical techniques is necessary.
The Laplace transform method is very useful for solving ordinary differential equations. The Laplace transform allows the conversion of more complex functions into algebraic equations of a complex variable, "s", including the conversion of differential equations into algebraic equations. Once differential equations are transformed into algebraic equations, the necessary operations are performed on the domain "s" and then returned to the time domain by the inverse Laplace transform [3].
The Z-transform is widely used for discrete-time system analysis. It can be applied in linear differece equations, which, when converted to the Z domain, are transformed into algebraic equations that are easier to solve. After solving the algebraic equation in the Z domain, the inverse Z-transform is applied, thus providing the final system response [4].
It can be said that the Z-transform has the same function as the Laplace transform, except for discrete-time systems.
Although many studies have already been performed in the area of bioprocessing modeling, no work that has applied the Z-transform to construct discrete models for fermentation under the conditions studied in this work has been found in the literature. In this work, we applied different kinetic models found in the literature, in addition to the Z-and Laplace transforms, to the fermentation process of the yeast Saccharomyces cerevisiae and compared the results with experimental results and with each other.

Methodology
Yeasts play the major role in various processes in the food industry and in bioethanol production. The most commonly used yeast is Saccharomyces cerevisiae. In addition, other yeasts known as nonconventional yeasts (Scheffersomyces stipitis, Yarrowia lipolytica, Kluyveromyces lactis and Dekkera bruxellensis) are already being used in some processes [5]. Currently, there are many studies looking for alternative yeasts for the baking process; however, Saccharomyces cerevisiae still dominates the market.
Yeasts can metabolize glucose in two different ways: by oxidation or by fermentation.
The oxidation mechanism provides cell growth, while fermentation provides the formation of ethanol. The metabolism of yeasts is influenced by oxygen and glucose concentrations; for example, a high concentration of glucose combined with a low concentration of oxygen may result in the Crabtree effect, which inhibits cell growth and favors the formation of ethanol [6]. Saccharomyces cerevisiae yeast shows cellular growth at both high and low oxygen concentrations, which is why it is so widely used [7].
Carbon and nitrogen are important sources of energy for Saccharomyces cerevisiae cells, and the availability and concentration of nitrogen can be associated with biosynthesis and ethanol; there are studies relating these effects [8].
Variations in temperature and pH impact the ethanol production in the fermentation process of Saccharomyces cerevisiae yeast [9]. A decrease in pH indicates a reduction in the formation of the product, whereas the inverse effect occurs with the temperature: the higher the temperature is, the smaller the amount of product formed.
With regard to the operation regime, the fermentation process can be batch, semicontinuous or continuous [13]. According to Cinar [14], the batch process continues to be the one most commonly applied in bioprocesses because it is less susceptible to contamination. The batch process consists of placing the inoculum and the culture medium in the bioreactor at the beginning of the process and removing the products only after the fermentation is completed; subsequently, the bioreactor is sterilized, and a new cycle begins.
Bioreactors can be divided into two main types: conventional bioreactors (stirred tank type and aerated mechanical agitation) and pneumatic bioreactors. Each bioreactor is better adapted to the culture of a certain species of microorganism [15]. Among the different types of pneumatic bioreactors is the concentric cylinder airlift type. In this type of bioreactor, the air is injected by the base through the central cylinder (riser) and is returned the base by the external cylinder (downcomer). The air promotes both the agitation of the culture medium and the aeration, with the advantage of causing less trauma to the cells and lower energy consumption than mechanical agitation bioreactors.
A model is nothing more than a dynamic relation between variables. There are many ways to represent the same mathematical model, but a given representation may apply better to one system than to another [16]. According to Aguirre [17], the most commonly used types of representation of mathematical models are transfer functions and statespace representations. As mentioned by Jacob and Zwart [18], representation in state space is able to describe the relationship among several variables (inputs and outputs), and its difference from the transfer function is its ability to establish general control strategies.
The simulation of continuous mathematical models involves the resolution of differential equations, while the simulation of discrete models involves the resolution of difference equations. To solve analytically linear differential equations, the Laplace transform is used, and to solve linear difference equations, the Z-transform is used [17].

Modeling of the Saccharomyces cerevisiae fermentation process
Experimental data: The results from Ferreira et al. [19], who carried out four experiments in an airlift bioreactor, were used to calculate the parameters and compare the results of the models developed in this paper. The experiments were performed in a 6 L bioreactor at 32 ºC. Table 1 shows the experimental conditions of each experiment. Table 1 Experimental conditions of the airlift bioreactor [19].
The curves for cell growth and substrate consumption of experiment 4, used to calculate the parameters and adjust all the models, are shown in Figure 2. Microorganism: A fresh commercial yeast was utilized in this work. This yeast is found in baker's yeast or supermarkets that sells baker yeast. This kind of yeast (Saccharomyces cerevisiae) has 30% dry mass and 70% humidity [19].
Culture Media: The culture media were prepared in an appropriated recipient with distilled water, and the pH was regulated to 4.6 using acid. This media was sterilized in an autoclave with all necessary equipment and tools [8]. Table 2 shows the composition of the culture media.

Results and discussion
The first step was to apply the mass balance on the Saccharomyces cerevisiae To solve the continuous Verhulst model, the differential Equations (9 and 10) and the ode 45 technique were applied. This model was not presented in state space representation because it is nonlinear.
The Z transform cannot be applied to a nonlinear model.
After obtaining the mathematical representations for the Monod, Andrews and Verhulst models, the next step was to calculate the parameters. Table 3 shows the values obtained for the parameters.   Compared with the points of experiment 4, the same trend is observed for the discrete Monod model (Figure 4). Conceptually, the discrete model is more suitable for describing fermentation processes. These processes can describe population variations because individuals (in this case, cells) are always integer numbers, not fractions (for example, a population cannot contain 1000.5 cells). Analyzing the Andrews models ( Figure 5), the models (discrete and continuous) show a slower cell growth rate for 2 h of fermentation. The substrate consumption also decreases more slowly than do the experimental points until it reaches 0 g.L -1 . The next step was to compare all continuous models and all discrete models. Figure   7 shows that even without calculating the accuracy of each model, the Verhulst model showed the best adjustment to the points of experiment 4 among the continuous and discrete models. To confirm the information observed in Figure 7, the medium error for each model was calculated. The results are shown in Table 4.

Table 4
Error comparison for the continuous and discrete Monod, Andrews and Verhulst models.
After the models were compared, the discrete and continuous Verhulst models were simulated for another initial condition, namely, a cell concentration of 1.  Finally, the precisions of the continuous and discrete Verhulst models were calculated, and the results are shown in Figure 9.

Conclusions
Although discrete representations are not largely applied to bioprocesses, this work shows that the discrete representation has the same accuracy as the continuous models do. This finding indicates that the Z transform can be applied to fermentation processes.
The parameters obtained for the Saccharomyces cerevisiae culture on a batch process in an Airlift bioreactor were µmáx = 0.81 h -1 and / = 0.35 gcel.gsac -1 .
The linear models (continuous or discrete) describe the cell growth for two hours of fermentation and the substrate consumption until it is completely depleted. The Andrews model does not accurately describe this process because it does not account for substrate inhibition.
The model that best represents this process is the nonlinear Verhulst model, as observed in a similar process by Costa [20], with a medium error of 0.166 g.L -1 for cell growth and 0.346 g.L -1 for substrate consumption. When the initial conditions were changed, the precision of the Verhulst model did change, but it still satisfactorily represented the process (Figure 9).