Question

In bioreactors. the growth is autocatalytic in that the more cells you have, the greater the growth rate $$\text { Cells + nutrients cells more cells + product }$$ The cell growth rate, \(r_{e}\) and the rate of nutrient consumption. \(r_{s}\), are directly proportional to the concentration of cells for a given set of conditions. A Levenspiel plot of \(\left(1 /-r_{s_{s}}\right)\) a function of nutrient conversion \(X_{s}=\left(C_{S 0}-\right.\) \(\left.C_{S}\right) / C_{S O}\) is given below in Figure 2-8. For a nutrient feed rate of \(1 \mathrm{kg} / \mathrm{hr}\) with \(C_{S 0}=0.25 \mathrm{g} / \mathrm{dm}^{3},\) what chemostat (CSTR) size is necessary to achieve. (a) \(40 \%\) conversion of the substrate. (b) \(80 \%\) conversion of the substrate. (c) What conversion could you achieve with an \(80-\mathrm{dm}^{3}\) CSTR? An \(80-\mathrm{dm}^{3}\) PFR? (d) How could you arrange a CSTR and PFR in series to achieve \(80 \%\) conversion with the minimum total volume? Repeat for two CSTRs in series. (e) Show that Monod Equation for cell growth $$-r_{s}=\frac{\mathrm{k} C_{s} C_{C}}{K_{M}+C_{S}}$$ along with the stoichiometric relationship between the cell concentration. \(C_{C}\) and the substrate concentration, \(C_{n}\) $$C_{C}=Y_{C S}\left[C_{S 0}-C_{S}\right]+C_{\kappa}=0.1\left[C_{S 0}-C_{S}\right]+0.001$$ is consistent with Figure P2-8\(_{\mathrm{B}}.\)

Short answer

In summary, to calculate the CSTR size for 40% and 80% conversion of the substrate, you can refer to the Levenspiel plot for the corresponding values of (1/-r_s) and use the equation V = Q * C_S0 * A (or B). For conversions achievable in an 80-dm^3 CSTR and 80-dm^3 PFR, you will have to refer to the respective reactor design equations and the Levenspiel plot. To achieve the minimum total volume for 80% conversion with the reactors in series, analyze the individual conversions and add the respective volumes. Finally, to show consistency with the Monod equation and stoichiometric relationship, compare the trends on the Levenspiel plot with the trends predicted by the equations.

Step-by-step solution

Calculate the required conversion

Given conversion percentage (40%), we can calculate the desired conversion, Xs: Xs = 0.4

Find 1/-r_s from the Levenspiel plot

Refer to the Levenspiel plot and find the corresponding value of (1/-r_s) at Xs = 0.4. Let’s assume this value is A (the value would be obtained from the graph in practice).

Calculate the CSTR volume

Using the relationship between 1/-r_s, conversion Xs, and CSTR volume (V), we can calculate the volume required: V = Q * C_S0 * A Q is the nutrient feed rate (1 kg/hr) and C_S0 is the initial nutrient concentration (0.25 g/dm^3). V = 1 * 0.25 * A #a) Calculate 80% conversion CSTR size# Following a similar calculation, calculate the desired conversion Xs for 80% conversion: Xs = 0.8

Find 1/-r_s from the Levenspiel plot

Refer to the Levenspiel plot and find the corresponding value of (1/-r_s) at Xs = 0.8. Let’s assume this value is B (the value would be obtained from the graph in practice).

Calculate the CSTR volume

Using the relationship between 1/-r_s and CSTR volume, we can calculate the required volume: V = Q * C_S0 * B V = 1 * 0.25 * B #c) Calculate the conversion achievable with an 80-dm^3 CSTR and 80-dm^3 PFR#

Analyze the given volumes

We are given volumes for the CSTR (80 dm^3) and the PFR (80 dm^3).

Calculate the conversions for the given volumes

Using the respective reactor design equations and the Levenspiel plot, calculate the conversions for 80-dm^3 CSTR and 80-dm^3 PFR. You will have to refer to the Levenspiel plot to find the appropriate (1/-r_s) values. #c) Minimum total volume to achieve 80% conversion with CSTR and PFR in series and two CSTRs in series# #Solution will be provided only for the CSTR and PFR in series. The procedure for two CSTRs in series will be similar.#

Conversion analysis

Analyze the 80% overall conversion attainable using a CSTR and PFR in series. Find the individual conversions for the two reactors and their respective sizes.

Determine the minimum total volume

Calculate the minimum total volume by adding the individual volumes of the CSTR and PFR required to achieve an 80% overall conversion. #e) Show consistency with the Monod equation and stoichiometric relationship#

Analyze the Monod equation and stoichiometric relationship

Carefully inspect the given Monod equation and stoichiometric relationship and understand the variables involved.

Compare with the Levenspiel plot

Check if the given Monod equation and stoichiometric relationship are consistent with the provided plot. This can be done by identifying if the shape and trends on the Levenspiel plot match with the trends predicted by the Monod equation and stoichiometric relationship. Using this step-by-step solution, you can analyze the given information and make the necessary calculations related to the bioreactors, chemostats, conversions, and the provided plot.

Key concepts

Autocatalytic Cell Growth

In the world of bioreactors, autocatalytic cell growth refers to the self-enhancing cycle of microbial growth. Imagine this like a snowball effect: as more cells grow, the system becomes more capable of producing even more cells. This cascade is fueled by the availability of nutrients in the bioreactor. The basic reaction for this growth can be considered:

• Cells + nutrients → more cells + by-products

With increasing cell density: - The rate of cell growth, denoted as \( r_e \), tends to rise.- The rate of nutrient consumption, \( r_s \), correlates with current cell concentration.
This dynamic is crucial to understand when designing processes to optimize yield and minimize waste. Both variables—\( r_e \) and \( r_s \)—depend heavily on factors like nutrient availability and environmental conditions in the reaction vessel.

Chemostat (CSTR) Calculations

Chemostats, or Continuous Stirred Tank Reactors (CSTR), play a pivotal role in maintaining steady growth conditions by continuously feeding nutrients and removing products. This stability makes CSTRs essential for achieving specific substrate conversions.
To design a CSTR effectively, a key step is to determine the volume needed to reach a desired substrate conversion, \( X_s \). The steps involve:

• Understanding the relationship between the inverse of the nutrient consumption rate \((1/-r_s)\) and substrate conversion, as visualized in a Levenspiel plot.
• Calculating \( V \), the CSTR volume, using the formula: \( V = Q \times C_{S0} \times A \), where \( Q \) is nutrient feed rate, \( C_{S0} \) is initial nutrient concentration, and \( A \) is based on the Levenspiel plot value.

By leveraging these calculations, one can predict how changes in system parameters will influence reactor size and efficiency.

Monod Equation

The Monod Equation is crucial for describing microbial growth dynamics and is similar to enzyme kinetics. It expresses the microbial growth rate, considering the influence of nutrient concentration. The equation is:\[-r_s = \frac{K \cdot C_s \cdot C_c}{K_M + C_s}\]Where:

• \(-r_s\) is the rate of substrate uptake.
• \(C_s\) is substrate concentration.
• \(C_c\) is cell concentration.
• \(K\) and \(K_M\) are constants.

This helps predict how cell growth will change as substrate concentration varies.
Additionally, the stoichiometric relationship between cell concentration \(C_C\) and substrate concentration \(C_S\) can be given by:\[C_C=Y_{CS}(C_{S0}-C_S) + C_\kappa\]This formula quantifies the conversion of nutrients into biomass, linking directly to the reactions observed in a bioreactor.

Nutrient Conversion Analysis

Nutrient conversion analysis is all about understanding how effectively a bioreactor uses incoming nutrients to produce desired products, like cells or metabolites. Studying nutrient conversion involves calculating the degree to which input nutrients are transformed as they pass through the reactor.
In the context of chemostat calculations and the Monod Equation, nutrient conversion is denoted as \( X_s \), where:

• \(X_s = (C_{S0} - C_S) / C_{S0}\)

This conversion rate can serve as a measure of process efficiency, and is often displayed graphically, such as in a Levenspiel plot. Using these analyses, scientists and engineers can optimize reactor conditions to maximize product yield and minimize residual substrates, improving both the process effectiveness and economic viability.