SOLID-STATE FERMENTATION MODEL FOR A PACKED-BED BIOREACTOR

Simulations can lead to many process advantages, like quality of the product, scaling-up, process control and optimization. However, to turn on the model used in the simulation reliable, it is necessary a good parameter set to fit the experimental data. In this manner, parameter estimation is very important to keep the model adjusted. The main goal of this study is to estimate the parameters of ordinary differential equations set in a solid-state fermentation process model. The process model is composed by a set of seven ordinary differential equations that represents respectively the cells profile, cells physiological state, temperature, substrate consumption and production of ethanol, carbon dioxide and oxygen by the cells metabolism. The model used in this work was statistically validated and the results have shown that it is able to predict the experimental data.


INTRODUCTION
The use of process simulations can lead to advantages, like optimization, process safety, product quality (Zhang et al., 2008), can be economically convenient and facilitate measuring some process variables (Yamada et al., 2005), or even in control developing (Salau et al., 2008). Besides a reliable model, good parameter estimation is required to keep the model adjusted to experimental data, usually through the minimization of an objective function (Schwaab et al., 2008).
In this work is presented a model for a solid-state fermentation (SSF), which is a fermentation process without any free water, i.e. only water that is attached to the substrate cells is available in amount enough to allow microorganisms grow (Pandey, 2003). SSF has lower energy requirements and wastewater production when compared to submerged fermentation, also it can use agroindustrial byproducts as substrate (Mazutti et al., 2010), and has easier products recovery (Rahardjo et al., 2006).

Solid-state fermentation modeling
The model presented in this work is based on the Verhulst Logistic Equation (Verhulst, 1838), on the work of Fanaei and Vaziri (2009) and Silveira et al. (2014). Also, it was included some yield coefficients to predict the substrate consumption, ethanol, CO2 and O2 production. Also, the substrate inhibition hypothesis was used to evaluate the specific growth variable µ, as it can be seen on Equation 1.
Further, the bed density and specific heat algebraic equations (Equations 2-3) were used to support the phenomenological heat transfer equation (Equation 4), which is strongly dependent on the cells growth profile through the metabolic heat term (YQ) and based on energy balance.

ρb = ɛ ρa + (1 -ɛ)ρs
(2) The cells growth profile is described by the use of the Verhulst logistic equation with an additional physiological factor state (Φ). Both equations can be seen on Equation 5 and 6, respectively.
The coefficients of the physiological state are described by Equations 7 and 8.
The substrate consumption, ethanol, CO2 and O2 production are described by Equation 9 to 12, respectively.

Parameter estimation problem
The parameters were estimated to fit the model to the experimental data obtained by Mazutti et al. (2010). In section 1.1 the model equations were shown and the parameters chosen to be estimated were: µmax, Ks, k1, , ρs, Cps, ρa, α, YS/X, YP/X, YCO2/X, YO2/X, YQ, Es, Ed, γs0 and γd0. The parameters values, units and descriptions can be found in Table 1. 17 parameters were estimated (NP=17), for 6 experiments (NE=6) measured at 25 different times (NY=25). Thus, according to Equation (13), 48 degrees of freedom (DF) were available for the statistical tests.

DF=NE.NY-NE.NP
The least squares equation was used to the parameter estimation as objective function, cf. Equation 14. , (14)

∑Fobj(X,T,S,P,CO2,O2) = min[f(x), f(T), f(S), f(P), f(CO2), f(O2)] = [║Xexp-Xmod║²
Where the subscripts exp denotes experimental data and mod denotes modeling data. As smaller the objective function is the better are the parameters estimated for the model, as the residuals will be smaller, i.e. the difference between experimental and simulated data will be more likely.
The software Matlab® function lsqnonlin was used in this work because it solves nonlinear least-squares problems. The chosen algorithm was Levenberg-Marquardt, which consists in an iterative damped least-squares method for minimization of nonlinear functions, it may be subjected to local minimum if a bad initial guess is given (Moré, 1977). Thus, the initial parameter guesses were always updated iteratively until the objective function value stops to change.

MATERIAL AND METHODS
The experimental data were obtained by Mauztti et al. (2010). The medium of the SFF process was sugarcane bagasse with cane molasses (10 wt%), corn steep liquor (30 wt%) and soybean bran (20 wt%). The yeast used was Kluyveromyces marxianus NRRL Y-7571. The bioreactor was a cylindrical stainless with air supplier with 95-100% of water. Inlet and outlet temperatures were monitored by a PT100 (NOVUS, Brazil). The microbial growth was calculated according to the measurements of oxygen uptake rate (Mazutti et al., 2010). Experiments were carried out by 24 hours with data acquisition of all state variables hour by hour. Also, temperature was measured at the bed inlet, 10 cm, 20 cm and 30 cm from the bed inlet and at the outlet.
The numerical integrator used was the Dormand-Prince pair, based on a Runge-Kutta of 4 th and 5 th order, which solves non-stiff differential equations (Dormand and Prince, 1980). The computer used for the procedures has an Intel® Core™ i7-3770 with 3.40 GHz processor and 12 Gb of RAM memory and it is running with the Windows 7 64 bits Operating System.

RESULTS AND DISCUSSION
As referred in item 1.2, Table 1 presents the estimated parameters obtained after several estimations with initial parameters guesses corrected after each estimation until the residuals become unalterable. The means and confidence intervals of the estimated parameters for each experiment dataset were computed according to Draper and Smith (1998) and Schwaab and Pinto (2007) works. For these parameters, the objective function value, cf. Equation 14, was found to be 11.2542. Also, Student's t-test and the Fisher's exact test were performed to verify if the means and the variances of the models correspond to the experimental data. The test results of the Student's ttest, as they can be seen in Table 2, have shown that the means for the model are correspondent to the experimental data, because their confidence intervals intercept each other. Fisher's exact test results, cf. Table 3, have shown that the limits corresponds to the variances ratio, denoting that, equally, they cannot be distinguished from the experimental variances.

Model simulations
Model simulations were performed with the estimated parameters in order to visually compare the simulated data to the experimental data. Figures 1-2 show the experimental data versus the model only for cells growth and temperature profiles for three experiments. These figures show how close the model is from the process, confirming what was seen in Tables 2 and  3.

CONCLUSIONS
According to Table 2, the state variables means cannot be distinguished between experimental and model data. It can be also seen on Table 3 that the variances of the model and the experiments are equivalent. Thus, the model with the parameters that were estimated, cf. Table 1, cannot be distinguished from the experimental data. In other words, the whole model is able to describe a solid-state fermentation using the Kluyveromyces marxianus NRRL Y-7571.
Moreover, the objective function is low, considering the amount of data analyzed. According to the results the model has presented a good agreement with the experimental data. The temperature profile seems to have a minor delay, however the behavior is much like the experimental data.