Question

In a chemostat you know that if a culture obeys the Monod equation, the residual substrate is independent of the feed substrate concentration. You observe that in your chemostat an increase in \(S_{0}\) causes an increase in the residual substrate concentration. Your friend suggests that you consider whether the Contois equation may describe the situation better. The Contois equation (eq. \(6.36\) ) is: $$ \mu=\frac{\mu_{m} S}{K_{x} X+S} $$ a. Derive an expression for \(S\) in terms of \(D, \mu_{m}, K_{s x}\), and \(X\) for a steady-state CFSTR (chemostat). b. Derive an equation for \(S\) as a function of \(S_{0}, D, K_{s x}, Y_{X / S}^{M}\), and \(\mu_{m}\). c. If \(S_{0}\) increases twofold, by how much will \(S\) increase?

Short answer

S increases twofold when S_{0} increases twofold.

Step-by-step solution

Understanding the Contois equation

The Contois equation for microbial growth is given by \[ \mu = \frac{\mu_{m} S}{K_{x} X + S} \] where - \( \mu \) is the specific growth rate- \( \mu_{m} \) is the maximum specific growth rate- \( S \) is the substrate concentration- \( K_{x} \) is the half-saturation coefficient- \( X \) is the biomass concentration.

Derive expression for steady-state substrate concentration

In a continuous flow stirred-tank reactor (CFSTR) at steady state, the dilution rate \( D \) is equal to the specific growth rate \( \mu \). Therefore, we can write\[ D = \frac{\mu_{m} S}{K_{x} X + S} \]Rearranging this equation to solve for \( S \) in terms of \( D, \mu_{m}, K_{s x}, \) and \( X \), we get:\[ D(K_{x} X + S) = \mu_{m} S \]\[ D K_{x} X + D S = \mu_{m} S \]\[ D K_{x} X = S (\mu_{m} - D) \]\[ S = \frac{D K_{x} X}{\mu_{m} - D} \]

Derive expression for substrate concentration as a function of feed substrate concentration

For a CFSTR, the substrate balance at steady state is\[ D S_{0} = D S + \mu X \]Using the Contois equation for \( \mu \), substitute \( \mu \) into the substrate balance:\[ D S_{0} = D S + \frac{\mu_{m} S X}{K_{x} X + S} \]Rearranging the above equation to solve for \( S \) in terms of \( S_{0}, D, K_{s x}, Y_{X/S}^{M}, \) and \( \mu_{m} \), we get\[ D S_{0} K_{x} X + D S_{0} S = D S K_{x} X + S (\frac{\mu_{m} X}{Y_{X/S}^{M}}) \]Simplifying further,\[ S = \frac{D S_{0} (K_{x} X + S)}{\frac{\mu_{m} S}{Y_{X/S}^{M}}} \]\[ S = \frac{D S_{0}}{\frac{\mu_{m}}{Y_{X/S}^{M}} D} \]

Determine how much S increases if S_{0} increases twofold

From the derived equation for \( S \) as a function of \( S_{0}, D, K_{s x}, Y_{X/S}^{M},\) and \( \mu_{m} \), we see that if \( S_{0} \) increases twofold, the substrate concentration \( S \) will also see an increase. Specifically, we have\[ S_{new} = \frac{2 D S_{0}}{\frac{\mu_{m}}{Y_{X/S}^{M}} D} = 2S \]Thus, \( S \) will increase twofold if \( S_{0} \) increases twofold.

Key concepts

Chemostat

A chemostat is a bioreactor where a microbial culture is grown in a continuous and steady-state manner. Fresh nutrient medium is added, while culture liquid containing microorganisms, by-products, and unused substrate is removed at the same rate. This ensures that the culture volume remains constant.
In a chemostat:

• The dilution rate (D) is controlled, which is defined as the rate at which fresh medium is added per unit volume of culture.
• The system can maintain a steady state where the microbial growth rate equals the dilution rate.
• Preferable for studying microbial kinetics under controlled conditions.

Understanding the chemostat is crucial as it provides a consistent environment to analyze how microorganisms interact with their substrates and how their growth is influenced by various factors.

Contois Equation

The Contois equation is used to model the microbial growth rate in environments where the substrate concentration and biomass concentration both influence the growth rate. It is given by: \[ \mu = \frac{\mu_{m} S}{K_{x} X + S} \] where:

• \(\mu\) is the specific growth rate.
• \(\mu_{m}\) is the maximum specific growth rate.
• \(S\) is the substrate concentration.
• \(K_{x}\) is the half-saturation coefficient.
• \(X\) is the biomass concentration.

The Contois equation accounts for the impact of both substrate and biomass concentrations, which is particularly useful when the substrate is limiting or the biomass concentration is high. In contrast to the Monod equation, which considers only substrate concentration, the Contois equation provides a more comprehensive model for certain conditions.

Microbial Growth Rate

Microbial growth rate, denoted as \mu, is a measure of the rate at which microorganisms reproduce and increase in biomass. In the context of a chemostat, the growth rate depends on:

• Substrate concentration \(S\)
• Biomass concentration \(X\)

The growth rate can be influenced by the availability of nutrients, environmental conditions, and the inherent characteristics of the microorganisms. Using the Contois equation, we can express the microbial growth rate by accounting for both the substrate and biomass concentrations. This gives a more accurate depiction of microbial dynamics in systems where these factors are interdependent.

Steady-state CFSTR

In a Continuous Flow Stirred-Tank Reactor (CFSTR), often referred to simply as a chemostat, achieving steady state means that all variables (e.g., microbial population, substrate concentration) remain constant over time. Here:

• The dilution rate \(D\) equals the specific growth rate \(\mu\).
• Inflow of fresh medium and outflow of culture are balanced.

At steady state, we use the following equation from the Contois model: \[ D = \frac{\mu_{m} S}{K_{x} X + S} \] Solving for the substrate concentration \(S\), we get: \[ S = \frac{D K_{x} X}{\mu_{m} - D} \] which shows how substrate concentration at steady state is influenced by the dilution rate, maximum specific growth rate, and biomass concentration.

Substrate Concentration

Substrate concentration \(S\) in a chemostat is crucial as it serves as the main nutrient source for microbial growth. At steady state, the substrate concentration is determined by a balance of several factors:

• Feed substrate concentration \(S_{0}\)
• Biomass concentration \(X\)
• Specific growth rate \(\mu\)

From the steady-state substrate balance equation: \[ D S_{0} = D S + \frac{\mu_{m} S X}{K_{x} X + S} \] We can derive that substrate concentration at steady state can either increase or decrease depending on the changes in \(S_{0}\), dilution rate, and other parameters. For instance, if the feed substrate concentration \(S_{0}\) increases, so does the residual substrate concentration \(S\), as more substrate enters the system per unit time.