## 1. Introduction

Organic acids can be produced from biomass-derived sugars through fermentation. One of these products, butyric acid, has many applications in the food and perfume industries. Pure butyric acid is used in food flavors, and esters of butyric acid are widely used as additives in the perfume industry [1,2]. Butyric acid is also used to produce the biodegradable polymer β-hydroxybutyrate and in the production of several drugs [3]. Butyric acid might also find application as an intermediate in the production of one of the next generation of biofuels, biobutanol, through two-step fermentation as described by Ramey [4]. Butyric acid is currently produced chemically, which starts from the oxosynthesis of propylene [5]. The fermentation process to produce butyric acid is difficult because it forms multiple products at low concentrations, which greatly increases downstream separation costs.

There are several Clostridial strains which can convert sugars to butyric acid [6–14]. Of these strains, Clostridium tyrobutyricum is attractive because it only requires a simple medium (yeast extract) for growth, and can produce butyric acid in high yield, selectivity and concentration [15]. C. tyrobutyricum is a rod-shaped, Gram-positive bacterium which grows under anaerobic conditions. It produces butyric acid and acetic acid as its main fermentation products from sugars, and also produces hydrogen and carbon dioxide as gaseous by-products. The production of acetic acid diminishes the feasibility of the fermentation process with C. tyrobutyricum by decreasing the selectivity and yield for butyric acid. Some research has been done to improve this fermentation by trying to inhibit or eliminate the production of acetic acid. Several fermentation modes and operation parameters were investigated [10–12,16]. During these studies, the selectivity varied from 71% to 89% in batch fermentation, depending on the glucose concentration. For continuous fermentation, the selectivity varied from 60% to 96%, depending on the dilution rate. It was also found that continuous fermentation with partial cell recycle gave high selectivity at low dilution rate and high cell recycle ratio [17]. A novel extractive fermentation, with 10% alamine 336 in oleyl alcohol as a solvent, selectively extracted butyric acid from the fermentation with a butyric acid selectivity of 91% [15]. Further, selective separation using electrodeionization has given high selectivity at relatively high productivities [18]. Gene manipulation methods have also been used to disrupt the gene associated with the acetic acid formation pathway. In this way, more butyric acid was produced, although acetic acid production was not eliminated [3,19,20].

The cell growth rate of C. tyrobutyricum, glucose concentration within the system and nutrient supply have significant influences on selectivity. A possible explanation for the influence of cell growth rate on selectivity is that the production of acetic acid produces more ATP, which is needed for rapid cell growth [3,10,13]. Additionally, the influence of glucose concentration and supply modes are dependent upon the cell growth rate. For example, in a continuous fermentation of C. tyrobutyricum, the selectivity for butyric acid decreases with increased dilution rate [11]; the relationship between dilution rate and cell growth rate exists because the dilution rate equals the specific cell growth rate for continuous fermentation [21]. The selectivity for butyric acid increases with increasing glucose concentration. This could be explained by product inhibition of cell growth at higher product concentrations, which then prefers the production of butyric acid. A continuous fermentation with total cell recycle showed a selectivity for butyric acid of 96.8%, which was much higher than batch fermentation results, because cell recycle inhibited cell growth [12]. Although experiments have shown the impact of cell growth rate, a comprehensive simulation explaining batch and continuous behavior has not been published for this fermentation.

#### 1.1. Theory

The objective of this paper was to develop a mathematical model to describe cell growth rate, product formation, and substrate consumption for the fermentation of glucose to butyric acid with C. tyrobutyricum. By solving these coupled differential equations with initial cell, glucose and products concentrations, a time course for the batch fermentation could be simulated. By introducing the mass balance for continuous fermentation, a steady state simulation for continuous fermentation could also be obtained.

The classical Luedeking-Piret equation shows how the rate of product formation changes with cell mass and cell growth rate [22,23], which illustrates that cell growth rate is important to describe most fermentation processes. The Monod equation expresses the change in cell growth rate [24]. These two equations, combined with an equation describing substrate consumption, are often used for fermentation simulation, mainly for the production of lactic acid, glutamic acid, and ethanol [22,23,25–28]. Although these fermentations produce a single product, the Luedeking-Piret equation is also applicable to multiple-product fermentations [23]. Therefore, it is possible to develop a mathematical model to predict the production of butyric acid and acetic acid, and then the selectivity for butyric acid in batch and continuous fermentations, by combining the Luedeking-Piret equation, the Monod equation, and an equation describing substrate consumption.

Cell growth over time during fermentation can be divided into the lag phase, the exponential growth phase, the stationary phase, and the death phase. The rate of cell growth in the exponential phase can be written as:

where X is the concentration of cells (dry cell mass) in the reactor in g L^{−1}; t is the time in hours; and μ is the specific cell growth rate in h^{−1}.A simplified equation to describe the specific cell growth rate is the Monod equation, assuming glucose is the limiting growth component of the fermentation [23]:

where S is the concentration of substrate in the reactor, in g L^{−1}; μ_{0} is the maximum specific cell growth rate in g L^{−1} h^{−1}; and K_{s} is the semi-empirical substrate saturation constant in g L^{−1}.The Monod equation allows for good prediction in a low inhibition environment. If there is a strong inhibition with a maximum cell concentration, it is necessary to introduce a cell inhibition term. X_{m} is the maximum cell concentration that can be reached and the constant m represents how strongly the inhibition affects the growth rate equation [21], shown as:

After describing the cell growth rate with a modified Monod equation, the production formations of butyric acid and acetic acid must be described. The Luedeking-Piret equation with empirical constants α_{i} and β_{i} correlates product formation, dP_{i}/dt with cell growth rate, dX/dt and cell concentration, X, respectively:

In cultivation with a composite medium (a medium specially designed for this organism), a 95%–100% carbon source is used for product formation, meaning that almost all the carbon is going to products and little is going to cell maintenance [8]. Thus, the rate of substrate utilization, dS/dt, may be shown as:

where Y_{B/S} is the butyric acid yield factor in g butyric acid/g glucose; Y_{A/S} is the acetic acid yield factor in g acetic acid/g glucose; and dP_{A}/dt and dP_{B}/dt are the rates of formation of acetic and butyric acid, respectively.In batch culture the cell growth, product formation and substrate consumption change with time and then terminate after a period of time. In continuous culture, steady state is reached, and cell growth rate can be obtained from the mass balance [23]:

where D is the dilution rate in h^{−1}; F is the feed stream flow rate in L h^{−1} and V_{R} is the volume of the reactor in L.#### 1.2. Definitions

Selectivity was defined as the ratio of the weight of butyric acid to the weight of all acids produced. Y_{P/S} was used to define how much glucose is converted to products, both desirable (organic acids) and undesirable (carbon dioxide). It should be noted that the stoichiometry of producing butyric acid dictates that 1 mole of carbon dioxide is made for every mole of butanol. A Y_{P/S} of 1 meant all of the glucose was converted to products. The yield of butyric acid was defined as the ratio of the weight of butyric acid produced to the weight of glucose consumed. The conversion of glucose was defined as the ratio of the consumed glucose to the initial glucose.

## 2. Results and Discussion

#### 2.3. Continuous Fermentation of C. tyrobutyricum with Total Cell Recycle

An extreme condition available for use to evaluate the influence of cell growth rate on continuous fermentation was total cell recycle by ultrafiltration or microfiltration. Total cell recycle greatly inhibited cell growth, and kept the cell growth rate close to zero. However, it could never be truly continuous as the fermentation vessel would eventually fill with dead cell mass. Theoretically, the cell growth rate for total cell recycle equals the cell death rate [21]. Continuous fermentation experimentation with cell recycle was conducted at a glucose concentration of 17 g L^{−1} and a dilution rate of 0.078 h^{−1}, yielding a butyric acid selectivity of 97.3%, as shown in Table 1. Normally, the cell concentration in a fermenter with cell recycle would be much higher than without cell recycle [12]. For this particular fermentation, however, the cell concentration was only slightly higher, which again showed that product concentration increased cell growth inhibition.

#### 2.4. Batch Fermentation Modeling

In order to accurately model the fermentation characteristics, the adjustable parameters from Equations (3)–(5) were determined. Thus, Equation (4) was rearranged to yield:

To obtain α_{i} and β_{i}, (1/X) × (dP_{i})/dt and (1/X) × dX/dt were obtained using the Matlab curve fit toolbox. As shown in Figures 2 and 3 which contains all the data in Figure 1, Luedeking-Piret equations for butyric acid and acetic acid formation were generated:

Both of these equations showed very good fits (R^{2} > 0.975), strongly indicating butyric acid was non-growth rate-related and the production of acetic acid was growth-rate-related. The strength of these correlations indicated that low growth conditions (such as late in batch fermentation or in high rates of cell recycling) were best for high selectivities of butyric acid.

To simplify the Monod equation, an assumption was made that cell inhibition was low. The following equation represented cell growth when initial glucose concentration was low:

which indicated that the maximum specific cell growth rate would not exceed 0.15 h^{−1}, which was estimated as the limit of the data in Table 1. When the extended Monod model was needed, at higher glucose concentrations, a K_{s} of 4 and m = 1/3 (based on literature) were used, and X_{m} was found to be 5 g/L; this was the maximum cell growth rate for this media. In this work we modified K_{s} at different concentrations for a better fit but one K_{s} could be used. These numbers would need to be modified if a different medium with different growth conditions was used.Equation (5), describing substrate consumption, includes the parameters Y_{B/S} and Y_{A/S}. These parameters could be obtained from reaction stoichiometry, showing that one mole of butyric acid was produced from one mole of glucose and two moles of acetic acid were produced from one mole of glucose [8]. Thus:

Matlab was used to solve these three coupled differential equations. These were solved simultaneously with the mass balance in order to assure carbon balance. Figure 4 shows the simulated time courses of batch fermentation against experimental data for the last batch fermentation, which was initiated with 39.6 g L^{−1} of glucose. The simulation predicted the kinetics of this batch fermentation very well. Y_{P/S} was calculated from experimental results to be close to 1. This proved the assumption that for this composite medium, 95%–100% of the glucose was converted to products instead of cell mass. It also shows that this model can be used as the basis for continuous fermentation modeling.

#### 2.5. Steady State Continuous Fermentation Modeling

The equation describing product formation during the batch fermentation was used to find the steady-state mass balance for the continuous culture to obtain predicted output concentrations of products and selectivity for butyric acid. A mass balance on acetic acid is shown below, in Equation (13). From batch fermentation kinetics:

This equation is empirically derived from data obtained by Figure 3. A transient simulation of the mass balance on acetic acid applied to the fermenter is:

where F is the feed flow rate in and out in L h^{−1} and C_{A}_{0} and C_{A} represent the starting and present concentration of acetic acid in the fermenter. When the continuous fermentation reached steady state, one would assume the concentration of acetic acid does not change with time. However, previous work in our group [18] has shown that acetic acid metabolizes to butyric acid and thus this assumption does not hold true. Thus:
and the production of acetate was assumed to use the following format:
where α_{a} is assumed to be unchanged from the batch fermentation model; and β_{a} is evaluated by the following equation, which is a recombination of the continuous culture mass balance:
A similar equation was derived for butyrate production:

although the parameter β_{b} would be unchanged from the batch estimation since butyric acid did not further metabolize to any other products. Thus, the substrate consumption was defined as:
Finally, the selectivity was defined as the amount of butyric acid produced over the total amount of acids produced. If acetic acid was not metabolized to butyric acid this selectivity was given as:

However, if acetic acid was metabolized to butyric acid then Equations (14), (16)–(18) were solved simultaneously using Matlab and the selectivity was calculated on a point by point basis where the term β_{a} is concentration dependent with a big effect occurring at high concentrations. The results of using this model with several dilution rates at several different glucose concentrations are given in Figure 5. First, if no acetic to butyric acid conversion was assumed, the model clearly under-predicts the selectivity: butyric acid selectivity was predicted to be about 0.90 at a low dilution rate of 0.02 h^{−1}, for example, while the actual values at various concentrations of glucose are closer to 0.95. However, using the adjusted model it was found that the selectivity prediction was closer to the actual values at several different glucose feed concentrations. This provided proof that not only was acetic acid to butyric acid conversion taking place, but also that it was possible to model the conversion accurately with few data points. Although Michel-Savin et al. [11] showed conversion of acetic acid to butyric acid, this phenomenon had not been accurately modeled until this work.

#### 2.6. Predicting Optimum Operating Conditions

Since the model was proven at several different glucose concentrations, it was then used to predict optimum conditions for glucose concentration and dilution rate. The prediction of butyric acid selectivity, productivity, and glucose conversion are given in Figures 6, 7 and 8, respectively. With all glucose concentrations, it was predicted that butyric acid selectivity was maximized at low dilution rates. This was because butyric acid was produced when cell growth rates were low. The best conditions to achieve these maxima are either a fed batch fermentation system, when cell density can remain high when feed conditions are low, or in a high cell recycle environment in a continuous system with cell recycling. To maximize productivity, glucose concentration must be kept high (>75 g L^{−1}) at all cell growth rates. However, a look at overall glucose conversion shows the opposite effect; that is, that glucose conversion is higher when glucose concentration is low. This opposite effect shows that with this fermentation, one always fights the battle of high selectivity and high conversion in optimizing these conditions. Thus, as has been proposed earlier, cell recycling [17] and product separation [18] can make it possible to get high selectivities, conversions, and productivities. These levels are >95% selectivity is certain cases.