A fluidized-bed, immobilized-cell bioreactor is used for the conversion of glucose to ethanol by Z mobilis cells immobilized in \(\kappa\)-carrageenan gel beads. The dimensions of the bed are \(10 \mathrm{~cm}\) (diameter) by \(200 \mathrm{~cm}\). Since the reactor is fed from the bottom of the column and because of \(\mathrm{CO}_{2}\) gas evolution, substrate and cell concentrations decrease with the height of the column. The average cell concentration at the bottom of the column is \(X_{0}=45 \mathrm{~g} / \mathrm{l}\), and the average cell concentration decreases with the column height according to the following equation: $$ X=X_{0}(1-0.005 Z) $$ where \(Z\) is the column height \((\mathrm{cm})\). The specific rate of substrate consumption is \(q_{S}=2 \mathrm{~g} S / \mathrm{g}\) cells \(\mathrm{h}\). The feed flow rate and glucose concentration in the feed are \(5 \mathrm{l} / \mathrm{h}\) and \(160 \mathrm{~g}\) glucose/l, respectively. a. Determine the substrate (glucose) concentration in the effluent. b. Determine the ethanol concentration in the effluent and ethanol productivity \((\mathrm{g} / \mathrm{l} \cdot \mathrm{h})\) if \(Y_{P / S}=0.48 \mathrm{~g}\) ethanol/g glucose.

Step by step solution

01

Calculate the volume of the reactor

Determine the volume of the reactor using the formula for the volume of a cylinder: \[ V = \frac{\text{Area of base} \times \text{Height}}{1000} \text{ l} = \frac{\frac{\text{π}}{4}(d^2) h }{1000} \] where \[ d = 10 \text{ cm (converted to meters: 0.1 m)} \] \[ h = 200 \text{ cm (converted to meters: 2 m)} \]After calculations, we find: \[ A = \frac{\text{π}}{4}(0.1 m)^2 \approx 0.00785 m^2 \] \[ V = A \times 2 m \approx 0.0157 m^3 (or 15.7 l) \]

02

Calculate the total biomass in the reactor

Using the provided average cell concentration formula: \[ X = X_0 (1 - 0.005 Z) \]Integrate over the height of the column (0 to 2 m). The average cell concentration, \(\bar{X}\), can be found using integration: \[ \bar{X} = \frac{1}{h} \times \bigg( \text{integral of } X_0 (1 - 0.005 Z) \text{ over 0 to h} \bigg) \] \[ \bar{X} = \frac{1}{2} \times \bigg( X_0 \times Z - \frac{X_0 \times 0.005}{2} Z^2 \bigg)\bigg|_0^2 \] Substitute the values to get the average biomass concentration.

03

Calculate the rate of substrate consumption

Using the specific rate of substrate consumption defined: \[ q_S = 2 \text{ g S}/\text{g cells.h} \] \[ Total \text{ substrate consumption rate} = q_S \times \bar{X} \times \text{Volume} \]

04

Determine substrate concentration in the effluent

Utilize the mass balance equation for substrate consumption: \[ \text{Inflow rate} ( S_{in} - S_{out}) = \text{Volume} \times \text{Consumption rate} \times \text{Residence time} \] Solve for \( S_{out} \) i.e., effluent glucose concentration: \[ S_{in} = 160 \text{ g/l} \] Total substrate consumption = [calculation from above step]

05

Determine the ethanol concentration in effluent

Convert the substrate (glucose) concentration change to ethanol concentration using yield factor \( Y_{P/S} = 0.48 \text{ g ethanol/g glucose} \). \[ P = (\text{Initial concentration - effluent concentration}) \times Y_{P/S} \]

06

Determine the ethanol productivity

Calculate productivity using \[ \text{Productivity} = \frac{\text{Ethanol concentration}}{\text{Residence time}} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

glucose to ethanol conversion

The conversion of glucose to ethanol is a vital process in biotechnology, especially for producing biofuels and beverages. In this particular exercise, Zymomonas mobilis cells are used because they efficiently convert glucose to ethanol. This microorganism is immobilized in \(\kappa\)-carrageenan gel beads within a fluidized-bed bioreactor. The glucose enters the bioreactor and is metabolized by the cells, producing ethanol and carbon dioxide as byproducts.
The reaction is represented as:
\[C_6H_{12}O_6 \rightarrow 2C_2H_5OH + 2CO_2\].
This means that for every mole of glucose that is consumed, two moles of ethanol and two moles of carbon dioxide are produced. Understanding this conversion is crucial for optimizing the process, maximizing ethanol yield, and ensuring efficient substrate utilization.

immobilized-cell bioreactor

In an immobilized-cell bioreactor, cells are confined within a solid support material, such as \(\kappa\)-carrageenan gel. This setup allows continuous operation and high cell densities, preventing cell washout. The immobilized cells stay active over longer periods, thus improving process stability and efficiency.
The fluidized-bed design involves suspending the immobilized cells in a flowing liquid or gas. This enhances mass transfer and ensures that all cells have sufficient contact with the substrate (glucose). The bioreactor in this exercise measures 10 cm in diameter and 200 cm in height. As a result, substrate and cell concentrations vary along the column height, which must be considered for accurate calculations.
This design leads to a more efficient process and higher ethanol productivity.

specific rate of substrate consumption

The specific rate of substrate consumption \(\q_S\) is a measure of how effectively the cells utilize the glucose. It is defined as the amount of substrate consumed per unit mass of cells per hour. In this problem, \(\q_S\) is given as 2 g S/g cells.h. This means that each gram of Z. mobilis cells consumes 2 grams of glucose every hour.
To calculate the total substrate consumption rate, multiply \(\q_S\) by the average cell concentration \(\overline{X}\) and the bioreactor's volume (calculated as 15.7 liters in step 1). The mass balance for the substrate helps to determine how much glucose is converted and how much remains in the effluent.

ethanol productivity

Ethanol productivity is a key performance indicator for any bioreactor system. It measures the amount of ethanol produced per unit volume of bioreactor per hour. To determine this, the ethanol concentration in the effluent needs to be known. Use the yield factor \(\Y_{P/S} = 0.48\), which indicates that 0.48 grams of ethanol are produced per gram of glucose consumed.
Calculate the ethanol concentration by multiplying the difference between the initial glucose concentration and the effluent concentration by \(\Y_{P/S}\). Once you have the ethanol concentration, determine productivity through:
\[ \text{Productivity} = \frac{\text{Ethanol concentration}}{\text{Residence time}}\].
The residence time is the time the substrate spends in the reactor, derived from the feed flow rate. High ethanol productivity indicates efficient conversion and effective operational conditions in the bioreactor.