Question

In cultivation of baker's yeast in a stirred and aerated tank, lethal agents are added to the fermentation medium to kill the organisms immediately. Increase in dissolved oxygen (DO) concentration upon addition of lethal agents is followed with the aid of a DO analyzer and a recorder. Using the following data, determine the oxygen transfer coefficient \(\left(k_{L} a\right)\) for the reactor. Saturation DO concentration is \(C^{*}=9 \mathrm{mg} / \mathrm{l}\). $$ \begin{array}{cc} \text { Time (min) } & \text { DO (mg/l) } \\ \hline 1 & 1 \\ 2 & 3 \\ 2.5 & 4 \\ 3 & 5 \\ 4 & 6.5 \\ 5 & 7.2 \\ \hline \end{array} $$

Short answer

Use the formula \(k_L a = \frac{ \frac{dC}{dt}}{(C^* - C)} \), calculate for each interval, and average the values.

Step-by-step solution

Understand the Concept

The oxygen transfer coefficient, denoted as \( k_L a \), represents the rate at which oxygen is transferred from the gas phase to the liquid phase. We can determine \( k_L a \) using the data provided for dissolved oxygen (DO) concentration over time and the given saturation DO concentration (C* = 9 mg/L).

Use the Formula

The rate of change of the dissolved oxygen concentration in the medium is given by: \ \frac{dC}{dt} = k_L a (C^* - C) \. Here, \(dC/dt\) is the rate of change of the oxygen concentration, \(C^*\) is the saturation concentration, and \(C\) is the concentration at time \(t\).

Calculate \(dC/dt\)

To find \(dC/dt\), calculate the differences in DO concentrations divided by the corresponding time intervals. For example, from 1 min to 2 min: \ \frac{3 - 1}{2 - 1} = 2 \; from 2 min to 2.5 min: \ \frac{4 - 3}{2.5 - 2} = 2 \; and so on. Calculate these velocities between each time interval.

Set Up and Solve for \(k_L a\)

Using the relation \(\frac{dC}{dt} = k_L a (C^* - C)\), rearrange to solve for \(k_L a\): \ k_L a = \frac{ \frac{dC}{dt}}{(C^* - C)} \. Substitute the values of \(\frac{dC}{dt}\) and \(C\) at each time point to solve for \(k_L a\).

Average \(k_L a\) Values

Compute the values of \(k_L a\) for each time interval, then take the average of these values to get a more accurate measurement of the oxygen transfer coefficient.

Key concepts

Dissolved Oxygen Concentration

Dissolved oxygen concentration (DO) refers to the amount of oxygen that is present in the liquid phase, typically measured in milligrams per liter (mg/L). In the context of fermentation processes, having the right DO is essential as it influences the health and productivity of the microorganisms involved.
During the cultivation of baker's yeast, the DO levels can provide insights into how well oxygen is being transferred from the air into the liquid medium.
It is important to monitor DO concentrations because they provide a direct indicator of the oxygen available for the yeast. Too low DO levels may lead to insufficient oxygen for the yeast to effectively perform aerobic metabolism.
Conversely, too high levels of DO can be wasteful and may lead to the formation of reactive oxygen species, which might harm the yeast cells.

Oxygen Transfer Rate

The oxygen transfer rate (OTR) is the rate at which oxygen moves from the gas phase into the liquid phase. This is a key factor in aerobic fermentation processes because the microorganisms, like baker's yeast, require a certain amount of oxygen to perform optimally.
The effectiveness of oxygen transfer is typically described by the oxygen transfer coefficient \( k_L a \), where \( k_L \) represents the liquid-side mass transfer coefficient and \( a \) stands for the specific gas-liquid interfacial area. The formula \( \frac{dC}{dt} = k_L a (C^{*} - C) \) captures how quickly dissolved oxygen accumulates in the liquid.
Understanding this transfer rate can help in optimizing conditions in a bioreactor to ensure that the yeast receives enough oxygen, which is crucial for efficient and productive fermentation.

Fermentation Process

The fermentation process involves the biochemical reactions where microorganisms, such as yeast, convert sugars into products like ethanol, carbon dioxide, and other metabolites. In baker's yeast cultivation, the primary goal is often to produce biomass or specific metabolites under controlled conditions.
During fermentation, the yeast cells metabolize oxygen to produce energy through aerobic respiration. This makes monitoring and controlling the dissolved oxygen concentration crucial for achieving high yields.
Factors like temperature, pH, and nutrient levels should be carefully balanced alongside oxygen levels to create an optimal environment for the yeast to thrive and perform the desired biochemical reactions efficiently.

Baker's Yeast Cultivation

Baker's yeast (Saccharomyces cerevisiae) is widely used in the baking industry for its ability to leaven dough by releasing carbon dioxide through fermentation. Cultivating baker's yeast involves growing these microorganisms under specific conditions that provide the necessary nutrients, temperature, and oxygen levels.
The process typically occurs in stirred and aerated tanks where conditions can be closely monitored and controlled. The introduction of lethal agents, as in the given exercise, allows for the immediate cessation of microbial activity, thus providing a clear picture of the oxygen transfer dynamics without the influence of ongoing biological processes.
This ensures the accurate determination of the oxygen transfer coefficient, which can then be used to optimize the conditions for yeast cultivation, ensuring high productivity and quality of the yeast.