Question

Calculate the productivity (i.e., DP) of a chemostat under the following conditions: 1\. Assume Monod kinetics apply. Assume that negligible amounts of biomass must be converted to product \((<1 \%)\). 2\. Assume the Luedeking-Piert equation for product formation (eq. 6.18) applies. 3\. Assume steady state: $$ \begin{array}{ll} D=0.8 \mu_{m} & Y_{X J s}^{M}=0.5 \mathrm{~g} X / \mathrm{g} S \\ \mu_{m}=1.0 \mathrm{~h}^{-1} & S_{0}=1000 \mathrm{mg} / \mathrm{l} \\ K_{s}=10 \mathrm{mg} / \mathrm{l} & \beta=0.5 \mathrm{~h}^{-1} \mathrm{mg} P / \mathrm{g} X \\ \alpha=0.4 \mathrm{mg} P / \mathrm{g} X & \end{array} $$

Short answer

443.52 mg P/l-h.

Step-by-step solution

- Understand given parameters

Identify all given constants and initial values: \(D = 0.8 \, \mu_m \). \( Y_{X/S}^{M} = 0.5 \, \text{g X/g S} \). \( \mu_m = 1.0 \, \text{h}^{-1} \). \( S_0 = 1000 \, \text{mg/l} \). \( K_s = 10 \, \text{mg/l} \). \( \beta = 0.5 \, \text{h}^{-1} \, \text{mg P/g X} \). \( \alpha = 0.4 \, \text{mg P/g X} \).

- Calculate specific growth rate

Use the formula for the specific growth rate \( \mu \) under Monod kinetics: \[ \mu = \mu_m \frac{S}{K_s + S} \] Plug in the values: \[ \mu = 1.0 \frac{1000}{10 + 1000} = \frac{1000}{1010} = 0.99 \, \text{h}^{-1} \]

- Check Chemostat dilution rate

It is given that \(D = 0.8 \mu_m\), so, \[D = 0.8 \times 1.0 = 0.8 \, \text{h}^{-1} \] Since direct calculations conform with given steady-state assumption, the steady state in the chemostat is valid.

- Calculate biomass concentration (X)

Biomass concentration is determined from the dilution rate and yield: Use the mass balance on biomass: \[ X = Y_{X/S}^{M} (S_0 - S) \] Given the cells' growth exceeds substrate limit conversion, using balance: \[ X \approx Y_{X/S}^{M} S_0\] Plug in the values: \[X = 0.5 \times (1000 - 10) = 495 \, \text{mg/l} \]

- Apply Luedeking-Piert equation

The product formation rate \(q_p\) is: \[ q_p = \alpha \mu + \beta \] Plug in the values: \[q_p = 0.4 \times 0.99 + 0.5 = 0.396 + 0.5 = 0.896 \, \text{mg P/g X-h} \]

- Calculate productivity

Productivity (DP) is: \[ DP = q_p \times X \] Plug in the values: \[DP = 0.896 \times 495 = 443.52 \, \text{mg P/l-h} \]

Key concepts

Monod kinetics

Monod kinetics help to describe the relationship between microbial growth rate and substrate concentration. In a chemostat, the specific growth rate \( \mu \) can be described using the formula: \[ \mu = \mu_m \frac{S}{K_s + S} \] \( \mu_m \) is the maximum specific growth rate, \( S \) is the substrate concentration, and \( K_s \) is the half-saturation constant, where the substrate concentration gives half the maximum growth rate.
This formula is crucial because it simplifies the complexity of microbial growth into predictable terms. It allows us to quantify how quickly microorganisms convert substrates into biomass under specific conditions.
By using Monod kinetics, we can predict how changes in substrate levels will impact the microbial growth rate, which helps to optimize the chemostat operation.
In practice, by setting up a steady feed of nutrients (substrate), we can control the growth rate of microorganisms and maintain them in a balanced 'steady state.' This is important for consistent production in industrial applications, such as biofuel or bioproduct production.

Luedeking-Piert equation

The Luedeking-Piert equation is a model that describes the rate of product formation in microbial systems. According to this equation, the product formation rate \( q_p \) is a function of two terms: one proportional to the growth rate \( \mu \) and another constant term independent of growth: \[ q_p = \alpha \mu + \beta \] In this equation:

• \( \alpha \) represents the growth-associated product formation coefficient, meaning how much product is formed per unit of biomass growth.
• \( \beta \) is the non-growth-associated product formation coefficient, representing product formation regardless of biomass growth.

Practically, this means fermentation processes can be designed to maximize product yield either by enhancing microbial growth (increasing \( \mu \) to influence \( q_p \)) or by optimizing conditions to increase the non-growth-associated term \( \beta \). This dual dependency allows for flexibility in controlling production rates. When applied in chemostat systems, the Luedeking-Piert equation helps to differentiate between the parts of productivity that result from microbial growth and those that occur independently, aiding in the thorough understanding and optimization of the production process.

steady state in chemostat

A steady state in a chemostat is a condition where all variables (biomass concentration, substrate concentration, and product concentration) remain constant over time. This is achieved when the input and output rates of the system are balanced.
At steady state, the dilution rate \( D \) equals the specific growth rate \( \mu \): \[ D = \mu \] This balance ensures that the microorganism population remains constant because the rate at which cells are washed out is equal to the rate at which they grow. Maintaining a steady state is crucial for consistent and predictable production in industrial microbiology.
All calculations, including productivity determinations, depend on the assumption of steady state. This assumes that biomass formation, substrate uptake, and product formation are all occurring at constant rates. In the context of the given exercise, this allowed us to use the Monod kinetics and Luedeking-Piert equation effectively without additional variability factors. The steady state simplifies process control and enhances reliability, making it an essential aspect of industrial bioprocesses.