Question

In a fluidized-bed biofilm reactor, cells are attached on spherical plastic particles to form biofilms of average thickness \(L=0.5 \mathrm{~mm}\). The bed is used to remove carbon compounds from a waste-water stream. The feed flow rate and concentration of total fermentable carbon compounds in the feed are \(F=21 / \mathrm{h}\) and \(S=2000 \mathrm{mg} / \mathrm{l}\). The diameter of the column is \(10 \mathrm{~cm}\). The kinetic constants of the microbial population are \(r_{m}=50 \mathrm{mg} S / \mathrm{cm}^{3} \cdot \mathrm{h}\) and \(K_{S}=25 \mathrm{mg}\) \(S / \mathrm{cm}^{3}\). The specific surface area of the biofilm in the reactor is \(2.5 \mathrm{~cm}^{2} / \mathrm{cm}^{3}\). Assuming first- order reaction kinetics and an average effectiveness factor of \(\eta=0.7\) throughout the column, determine the required height of the column for effluent total carbon concentration of \(S_{0 i}=100 \mathrm{mg} / 1\).

Short answer

The required column height is approximately 5.72 meters.

Step-by-step solution

Identify known variables

Given values from the problem are: - Feed flow rate, \(F\): 21 l/hr - Inlet carbon concentration, \(S_0\): 2000 mg/l - Effluent carbon concentration, \(S_{0i}\): 100 mg/l - Column diameter, \(D\): 10 cm - Average biofilm thickness, \(L\): 0.5 mm = 0.05 cm - Maximum reaction rate, \(r_m\): 50 mg S/cm³·h - Substrate constant, \(K_S\): 25 mg S/cm³ - Specific surface area, \(a_s\): 2.5 cm²/cm³ - Effectiveness factor, \(eta\): 0.7

Calculate the volume flow rate

Convert the volumetric flow rate from liters per hour to cubic centimeters per hour:$$ F = 21 \text{ l/hr} \times 1000 \text{ cm}^3/\text{l} = 21000 \text{ cm}^3/\text{hr} $$

Calculate the first-order reaction rate constant

Using the given values:First-order reaction rate constant $$ k_1 = \frac{r_m}{K_S} = \frac{50 \text{ mg S/cm}^3\text{·h}}{25 \text{ mg S/cm}^3} = 2 \text{ hr}^{-1} $$

Calculate the substrate removal rate

Using the given effectiveness factor, the modified reaction rate constant is:$$ k = eta k_1 = 0.7 \times 2 \text{ hr}^{-1} = 1.4 \text{ hr}^{-1} $$

Calculate the cross-sectional area of the column

Using the diameter of the column, calculate the cross-sectional area:$$ A = \frac{\pi D^2}{4} = \frac{\u03C0 \times (10 \text{ cm})^2}{4} = 78.54 \text{ cm}^2 $$

Calculate volume of the reactor

Using the mass balance for the reactor, the reactor volume (\(V\)) is:$$ V = \frac{F}{k} \times \text{ln}\bigg(\frac{S_0}{S_{0i}}\bigg) = \frac{21000 \text{ cm}³/\text{hr}}{1.4 \text{ hr}^{-1}} \times \text{ln}\bigg(\frac{2000 \text{ mg/l}}{100 \text{ mg/l}}\bigg) $$Simplify the logarithm:$$ \text{ln}\bigg(\frac{2000 \text{ mg/l}}{100 \text{ mg/l}}\bigg) = \text{ln}(20) \thickapprox 2.9957 $$Then calculate:$$ V \thickapprox \bigg(\frac{21000}{1.4}\bigg) \times 2.9957 \thickapprox 44956.5 \text{ cm}^3 $$

Calculate column height

Finally, using the total volume of the reactor and the cross-sectional area to solve for height (\(H\)):$$ V = A \times H $$$$ 44956.5 \text{ cm}^3 = 78.54 \text{ cm}^2 \times H $$$$ H \thickapprox \frac{44956.5 \text{ cm}^3}{78.54 \text{ cm}^2} \thickapprox 572.45 \text{ cm} $$Convert height to meters:$$ H = 5.72 \text{ m} $$

Key concepts

Bioprocess Engineering

Bioprocess engineering is a field that combines principles from biology and engineering. It focuses on the design and development of systems and processes that involve biological organisms or molecules. One of the key applications of bioprocess engineering is in the development of reactors for industrial-scale production, like the fluidized-bed biofilm reactor. This reactor is particularly useful for processes requiring high surface areas for biological reactions, such as wastewater treatment.
By using a fluidized-bed biofilm reactor, the engineering process ensures efficient contact between the biofilm and the wastewater stream, optimizing degradation of pollutants. The designed biofilm's thickness, surface area, and flow rates are critical factors that must be engineered carefully for the optimal balance of microbial activity and mass transfer.
This ensures maximum efficiency in processes such as removing carbon compounds from wastewater, showcasing the integral role bioprocess engineering plays in environmental and industrial biotechnology applications.

Reaction Kinetics

Reaction kinetics in bioprocess engineering describes the rate at which biochemical reactions occur. In the context of a fluidized-bed biofilm reactor, understanding these kinetics is key to designing efficient systems.
The exercise assumes first-order reaction kinetics, meaning the rate of reaction is directly proportional to the concentration of the substrate. The formula used is:
\[ k_1 = \frac{r_m}{K_S} \]
Where r_m is the maximum reaction rate and K_S is the substrate constant.
Additionally, because the reactor operates under realistic conditions, the effectiveness factor (η) is applied to the rate constant to reflect actual operational efficiency:
\[ k = η \times k_1 \]
This modified reaction rate is vital for accurately predicting the substrate removal rate and designing the reactor to meet the effluent concentration requirements. By calculating the effective reaction rate, engineers can determine the required parameters for optimal reactor performance.

Wastewater Treatment

Fluidized-bed biofilm reactors are widely used in wastewater treatment to remove organic pollutants. These reactors are highly efficient due to the increased surface area provided by biofilms attached to small carrier particles.
The process involves streamlining wastewater over a bed of these particles, where microorganisms in the biofilm degrade organic compounds into less harmful substances. Maintaining an effective biofilm thickness and flow rate is critical to ensure that the microbial population has enough surface area and nutrient contact to function optimally.
One must also control factors like concentration gradients, biofilm sloughing, and the hydraulic retention time of wastewater in the reactor. By optimizing these parameters, fluidized-bed biofilm reactors ensure the effective removal of pollutants, thereby producing cleaner effluent suitable for environmental discharge or reuse.

Biofilm Thickness

Biofilm thickness in a fluidized-bed biofilm reactor is a crucial factor that influences the efficiency of substrate degradation. The given exercise describes an average biofilm thickness of 0.5 mm.
This thickness affects diffusion rates of nutrients and oxygen into the biofilm and waste products out of it. A thinner biofilm might not provide enough biomass for efficient substrate degradation, while a thicker biofilm could lead to diffusion limitations, reducing the reactor's performance.
Control of biofilm thickness is typically managed through operational parameters such as shear forces within the reactor and periodic biofilm detachment processes. Ensuring the right balance in thickness helps in maximizing the reactor's effectiveness and efficiency in treating wastewater.

Column Height Calculation

Calculating the column height of a fluidized-bed biofilm reactor involves several steps to ensure the reactor meets the required effluent quality. The height calculation integrates the volume flow rate, reaction kinetics, and substrate concentration changes.
First, convert the flow rate to an appropriate unit if necessary. Then, calculate the reaction rate constant and apply the effectiveness factor to get the modified rate. Using a mass balance approach, the volume of the reactor can be determined:
\[ V = \frac{F}{k} \times \text{ln}\bigg(\frac{S_0}{S_{0i}}\bigg) \]
Finally, obtain the column height by dividing the reactor volume by the cross-sectional area:
\[ H = \frac{V}{A} \]
The cross-sectional area is derived from the column diameter. This calculated height ensures that the reactor has sufficient capacity to reduce the carbon concentration in the wastewater to the desired effluent level.