Question

A value of \(k_{L} a=30 \mathrm{~h}^{-1}\) has been determined for a fermenter at its maximum practical agitator rotational speed and with air being sparged at \(0.51 \mathrm{gas} / \mathrm{l}\) reactor volume-min. E. coli with a \(q_{\mathrm{O}_{2}}\) of \(10 \mathrm{mmol} \mathrm{O}_{2} / \mathrm{g}\)-dry wt-h are to be cultured. The critical dissolved oxygen concentration is \(0.2 \mathrm{mg} / \mathrm{l}\). The solubility of oxygen from air in the fermentation broth is \(7.3 \mathrm{mg} / \mathrm{l}\) at \(30^{\circ} \mathrm{C}\). a. What maximum concentration of \(E\). coli can be sustained in this fermenter under aerobic conditions? b. What concentration could be maintained if pure oxygen was used to sparge the reactor?

Short answer

The maximum concentration of E. coli that can be sustained with air sparging is 0.656 g/L. With pure oxygen, it is 3.42 g/L.

Step-by-step solution

Determine the Oxygen Transfer Rate (OTR)

The oxygen transfer rate (OTR) can be determined using the formula: \[OTR = k_L a (C^* - C_0)\] Where: - \(k_L a\) is the mass transfer coefficient. - \(C^*\) is the oxygen solubility in the broth. - \(C_0\) is the critical dissolved oxygen concentration. Given the values: \(k_L a = 30 \mathrm{~h}^{-1}\)\(C^* = 7.3 \mathrm{mg / l}\)\( C_0 = 0.2 \mathrm{mg / l}\)Substitute the values into the formula to get the OTR: \[OTR = 30 (7.3 - 0.2) \mathrm{~mg / l / h} = 210 \mathrm{~mg / l / h}\]

Convert OTR to mmol/L/h

Given that the molecular weight of O2 is approximately 32, convert \(210 \mathrm{mg/L/h}\) to mmol/L/h: \[210 \mathrm{mg/L/h} \times \frac{1 \mathrm{mmol}}{32 \mathrm{mg}} = 6.56 \mathrm{mmol/L/h}\]

Define and Use the Specific Oxygen Uptake Rate (qO2)

Using the specific oxygen uptake rate \( q_{O_2} \), we can find the maximum biomass concentration that the OTR can sustain. The given \( q_{O_2} \) for E. coli is \( 10 \mathrm{mmol} \mathrm{O_2} / \mathrm{g} - \mathrm{dry} \mathrm{wt}-h\). From step 2, we know the OTR is \( 6.56 \mathrm{mmol/L/h}\). Now use the formula: \[C_{X, max} = \frac{OTR}{q_{O_2}} = \frac{6.56 \mathrm{mmol/L/h}}{10 \mathrm{mmol} \mathrm{O_2} / \mathrm{g} - \mathrm{dry} \mathrm{wt}-h} = 0.656 \mathrm{g/L}\]

Determine the Maximum Concentration with Pure Oxygen

When pure oxygen is used, the solubility of oxygen in the broth can be assumed to increase by a factor of approximately 5, as pure oxygen is 100% O2, compared to air which is 21% O2. Thus, \(C^*_{pure} = 7.3 \times 5 = 36.5 \mathrm{mg / l}\). Using the values: \(k_L a = 30 \mathrm{~h}^{-1}\)Now recalculate the OTR: \[OTR_{pure} = 30 (36.5 - 0.2) \mathrm{~mg / l / h} = 1095 \mathrm{~mg / l / h}\]Convert to mmol/L/h: \[1095 \mathrm{mg/L/h} \times \frac{1 \mathrm{mmol}}{32 \mathrm{mg}} = 34.22 \mathrm{mmol/L/h}\]Using the same \(q_{O_2}\) value: \[C_{X, max_{pure}} = \frac{34.22}{10} = 3.42 \mathrm{g/L}\]

Key concepts

mass transfer coefficient

The mass transfer coefficient, often denoted as \(k_L a\), is a critical parameter in bioprocess engineering. It measures how efficiently oxygen transfers from gas to liquid phase in a fermenter. This coefficient depends on several factors like agitator speed, air sparging rate, and reactor design.

For instance, in the given problem, \(k_L a = 30 \text{ h}^{-1}\). This indicates how rapidly oxygen is being transferred at the conditions of maximum practical agitator speed and specified air sparging rate. Higher \(k_L a\) values mean better oxygen transfer, which is crucial for sustaining aerobic microorganisms like E. coli.

Understanding and optimizing \(k_L a\) can lead to more efficient fermentation processes. Engineers often adjust the mixing and aeration strategies to get the best \(k_L a\) values.

Ensuring a high mass transfer coefficient ensures that microorganisms receive the necessary oxygen, keeping the process efficient and productive.

specific oxygen uptake rate

The Specific Oxygen Uptake Rate \(q_{O_2}\) represents the amount of oxygen consumed per unit of microbial biomass per hour. It is a measure of how much oxygen each cell uses to survive and grow.

In the given problem, E. coli has a \(q_{O_2}\) of \(10 \text{ mmol } O_2 / \text{ g-dry wt-h}\). This rate is integral to deciding how much biomass can be sustained in a reactor. By interfacing \(q_{O_2}\) with the oxygen transfer rate (OTR), we can determine the maximum biomass concentration.

For example, if the OTR calculated is \(6.56 \text{ mmol/L/h}\), the biomass that can be sustained is \(C_X = \frac{OTR}{q_{O_2}} = 0.656 \text{ g/L}\). This means that the oxygen availability limits the biomass concentration to 0.656 g/L under the given conditions.

Specific oxygen uptake rates are vital for system design and scale-up in bioprocess engineering. They influence how we manage and maintain effective growth conditions in bioreactors.

aerobic fermentation

Aerobic fermentation is a process where microorganisms grow and ferment in the presence of oxygen. Unlike anaerobic conditions, where oxygen is not present, aerobic fermentation uses oxygen as the final electron acceptor in the electron transport chain.

In the context of the given problem, E. coli requires specific oxygen levels to thrive. Aerobic fermentation enables these cells to efficiently convert substrates to biomass, generating energy.

Aerobic conditions are managed by sparging air or pure oxygen into the fermenter. The amount of oxygen supplied must match or exceed the microorganisms' demand, quantified by the Specific Oxygen Uptake Rate (qO2).

Optimal aerobic fermentation conditions result in maximum microbial productivity. But suboptimal conditions, such as low oxygen levels, can lead to insufficient growth or even microbial death.

This problem illustrated how aerobic fermentation parameters are used to compute the maximum concentration of E. coli. Using pure oxygen increases the oxygen solubility, potentially boosting biomass formation, showcasing one approach to optimization in aerobic fermentation systems.