Question

Consider a 1000-l CSTR in which biomass is being produced with glucose as the substrate. The microbial system follows a Monod relationship with \(\mu_{m}=0.4 \mathrm{~h}^{-1}, K_{S}=1.5 \mathrm{~g} / \mathrm{l}\) (an unusu-ally high value), and the yield factor \(Y_{X / S}=0.5 \mathrm{~g}\) biomass/g substrate consumed. If normal operation is with a sterile feed containing \(10 \mathrm{~g} / \mathrm{l}\) glucose at a rate of \(100 \mathrm{l} / \mathrm{h}\) : a. What is the specific biomass production rate \((\mathrm{g} / \mathrm{l}-\mathrm{h})\) at steady state? b. If recycle is used with a recycle stream of \(10 \mathrm{l} / \mathrm{h}\) and a recycle biomass concentration five times as large as that in the reactor exit, what would be the new specific biomass production rate? c. Explain any difference between the values found in parts \(a\) and \(b\).

Short answer

a. The specific biomass production rate is calculated by first determining the dilution rate, followed by applying the Monod equation and using the substrate and yield factor relationships to find biomass production. b. After considering the new effective dilution rate due to recycle, the specific biomass production rate is recalculated. c. The difference arises due to changes in the retention time and substrate availability caused by recycle.

Step-by-step solution

Determine the Dilution Rate (D)

The dilution rate is defined as the ratio of the feed flow rate to the reactor volume. Hence, \( D = \frac{Q}{V} \), where \( Q \) is the feed flow rate (100 l/h) and \( V \) is the reactor volume (1000 l). Calculate \( D \).

Specific Growth Rate (\( \mu \)

The specific growth rate follows the Monod equation: \( \mu = \mu_m \frac{S}{K_S + S} \), where \( \mu_m = 0.4 \mathrm{~h}^{-1}, K_S = 1.5 \mathrm{~g} / \mathrm{l} \), and \( S \) is the substrate concentration in the reactor. At steady state, \( D = \mu \). Set up the equation \( D = 0.4 \frac{S}{1.5 + S} \) and solve for \( S \).

Specific Biomass Production Rate for Part (a)

The specific biomass production rate is given by \( r_X = \mu X \). Using the yield coefficient \( Y_{X / S} = 0.5 \mathrm{~g}/\mathrm{g} \), relate substrate consumption to biomass production: \( r_S = D(S_{in} - S) \); hence, \( r_X = D Y_{X / S}(S_{in} - S) \), where \( S_{in} = 10 \mathrm{~g}/\mathrm{l} \). Calculate \( r_X \) using the steady-state \( S \) found in Step 2.

Calculate New Dilution Rate and Effective Growth Rate with Recycle (Part b)

With recycle, the effective flow rate increases. The new dilution rate, \( D_{eff} \), is given by \( D_{eff} = \frac{Q_r + Q}{V} \), where \( Q_r \) is the recycle stream flow rate (10 l/h). Hence, \( D = \frac{110}{1000} \). Calculate \( D_{eff} \), then find the new steady-state substrate concentration \( S \) using the Monod equation as in Step 2.

Specific Biomass Production Rate with Recycle

Calculate the new specific biomass production rate using the effective dilution rate and the yield factor as in Step 3. Use the new steady-state substrate concentration found in Step 4.

Explain Differences

Compare the specific biomass production rates from part (a) and (b) and explain the difference based on how recycle affects the system's retention time, substrate concentration, and microbial growth dynamics.

Key concepts

Monod equation

The Monod equation describes the relationship between the microbial growth rate and substrate concentration. It is a key aspect in understanding how bacteria or other microorganisms grow in bioreactors. The formula is given by: \[ \mu = \mu_{m} \frac{S}{K_{S} + S} \], where:

• \( \mu \) is the specific growth rate
• \( \mu_{m} \) is the maximum specific growth rate
• \( S \) is the substrate concentration
• \( K_{S} \) is the half-saturation constant

This equation helps us predict the growth rate based on the available substrate. When the substrate concentration is low, the growth rate increases significantly with a small increase in substrate. When the substrate concentration is high, the growth rate approaches its maximum value, \( \mu_{m} \), and doesn’t change much with further increases in substrate. Knowing this relationship is crucial for optimizing the conditions in the reactor to maximize biomass production.

Dilution rate

The dilution rate (\( D \)) is the rate at which fresh medium is fed to a bioreactor relative to the reactor volume. It’s calculated using the formula \( D = \frac{Q}{V} \), where:

• \( Q \) is the feed flow rate (in liters per hour)
• \( V \) is the reactor volume (in liters)

A high dilution rate means fresh medium enters the reactor quickly, while a low dilution rate means the medium enters slowly. In continuous stirred-tank reactors (CSTRs), the dilution rate also determines the growth rate of the biomass since \( \mu = D \) at steady state. This balance ensures that microorganisms are continuously supplied with nutrients while waste products are removed, maintaining a stable environment for microbial growth.

Specific biomass production rate

The specific biomass production rate (\( \dot{r}_{X} \)) indicates how quickly biomass is produced in the reactor. It’s derived from the specific growth rate using the formula \( \dot{r}_{X} = \mu X \), where:

• \( \mu \) is the specific growth rate
• \( X \) is the biomass concentration

This value tells us how effectively the microorganisms are converting substrates into biomass over time. In the exercise, this was further calculated using the yield coefficient (\( Y_{X/S} \)), which relates the amount of substrate consumed to the amount of biomass produced. The specific biomass production rate is crucial for understanding the efficiency of the bioreactor and ensuring optimal production.

Recycle process in bioreactors

The recycle process in bioreactors involves taking a portion of the biomass that exits the reactor and returning it to the reactor. This increases the biomass concentration in the reactor and can improve productivity. In the given exercise, a recycle stream with a flow rate of 10 liters per hour and a biomass concentration five times higher than the reactor exit stream was used. By employing this technique, the effective dilution rate is adjusted, and the system's retention time is increased. The effective dilution rate can be calculated as \( D_{eff} = \frac{Q + Q_r}{V} \), where:

• \( Q \) is the original feed flow rate
• \( Q_r \) is the recycle stream flow rate
• \( V \) is the reactor volume

Recycling enhances the contact time between the microorganisms and the substrate, potentially leading to higher biomass production efficiency. This method allows for better control over the growth environment and can be used to improve the quality and quantity of the biomass produced.