Question

A waste-water stream of \(F=1 \mathrm{~m}^{3} / \mathrm{h}\) with substrate at \(2000 \mathrm{mg} / \mathrm{l}\) is treated in an upflow packed bed containing immobilized bacteria in form of biofilm on small ceramic particles. The effluent substrate level is desired to be \(30 \mathrm{mg} / \mathrm{l}\). The rate of substrate removal is given by the following equation: $$ r_{s}=\frac{k X S}{K_{s}+S} $$ By using the following information, determine the required height of the column \((H)\). $$ k=0.5 \mathrm{~h}^{-1}, X=10 \mathrm{~g} / \mathrm{l}, K_{s}=200 \mathrm{mg} / \mathrm{l}, L=0.2 \mathrm{~mm}, a=100 \mathrm{~m}^{2} / \mathrm{m}^{3}, A=4 \mathrm{~m}^{2}, \eta=0.8 $$

Short answer

The required height of the column is 75.95 meters.

Step-by-step solution

Identify the given parameters

Extract the given parameters from the problem:1. Flow rate, F = 1 m³/h2. Initial substrate concentration, S0 = 2000 mg/L3. Effluent substrate concentration, Se = 30 mg/L4. Rate constant, k = 0.5 h⁻¹5. Biomass concentration, X = 10 g/L (which is 10,000 mg/L)6. Half-saturation constant, Ks = 200 mg/L7. Length of column, L = 0.2 mm8. Specific surface area, a = 100 m²/m³9. Cross-sectional area of column, A = 4 m²10. Effectiveness factor, η = 0.8

Write the substrate removal rate equation

The substrate removal rate is given by the equation:\[ r_s = \frac{k X S}{K_s + S} \]

Calculate the overall substrate removal rate

Using the effectiveness factor, the overall substrate removal rate equation becomes:\[ r_s = \frac{\text{Effectiveness factor} \times k \times X \times (S0 - Se)}{K_s + \frac{(S0 + Se)}{2}} \]Substitute the given values:\[ r_s = \frac{0.8 \times 0.5 \times 10000 \times (2000 - 30)}{200 + \frac{(2000 + 30)}{2}} \]

Simplify the substrate removal rate equation

Calculate the intermediate substrate concentration term and then simplify:\[ \frac{S0 + Se}{2} = \frac{2000 + 30}{2} = 1015 \text{ mg/L} \]Now substitute back into the equation:\[ r_s = \frac{0.8 \times 0.5 \times 10000 \times 1970}{200 + 1015} \]\[ r_s = \frac{0.8 \times 0.5 \times 10000 \times 1970}{1215} \]\[ r_s = \frac{7,880,000}{1215} \text{ mg/h} \]\[ r_s \rightarrow 6484.57 \text{ mg/L} \text{ converted to g/L} \rightarrow 6.48457 \text{ g/L} \]

Use the mass balance equation and height

The mass balance equation for the substrate over the height of the reactor is:\[ F \times (S0 - Se) = r_s \times V_{\text{reactor}} \]Where the volume of the reactor, V = A \times H. Substitute and solve for H:\[ V = 4 \times H \text{ m}^3 \]\[ 1 \times (2000 - 30) = 6.48457 \times (4 \times H) \]Solve for H:\[ 1970 = 25.93828 \times H \]\[ H = \frac{1970}{25.93828} \]\[ H = 75.95 \text{ m} \]

Key concepts

Biofilm Reactors

Biofilm reactors play a critical role in wastewater treatment processes. These reactors use microorganisms that grow on surfaces in the form of biofilms, which consist of bacterial colonies immobilized on a support material. These biofilms degrade organic contaminants in the wastewater. A key advantage of biofilm reactors is their high efficiency in removing organic pollutants. They are often employed in packed bed configurations where wastewater flows over media populated with biofilms. This configuration allows for increased contact time between the bacteria and contaminants, enhancing treatment efficiency.

Substrate Removal Rates

The substrate removal rate in bioprocess engineering is a measure of how quickly organic contaminants are degraded in a reactor. In our exercise, the substrate removal rate is described using the Michaelis-Menten kinetics equation: \( r_s = \frac{k X S}{K_s + S} \). This equation aligns with the principles of reaction kinetics, where **k** is the rate constant, **X** is the biomass concentration, and **S** is the substrate concentration. Effective substrate removal ensures that the treated wastewater meets environmental discharge standards, minimizing the impact on aquatic ecosystems.

Mass Balance in Bioreactors

Mass balance in bioreactors is essential to determine the reactor size and efficiency. In a bioreactor treating wastewater, mass balance equations account for the input, output, and reaction of the substrate. For our problem, we utilize: \[F (S_0 - S_e) = r_s V_{reactor}\] Here, **F** represents the flow rate, **S_0** and **S_e** are the initial and effluent substrate concentrations, respectively, and **V_{reactor}** is the volume of the reactor. Solving this equation helps us determine the required height of the column reactor to achieve the desired effluent quality.

Immobilized Bacteria

Immobilized bacteria are key in biofilm reactors, where they attach to surfaces like ceramic particles. This immobilization allows the bacteria to effectively degrade substrates over long periods. Bacteria immobilization increases the stability and retention time of microbial communities, enhancing the overall efficiency of the wastewater treatment process. Plus, immobilized bacterial systems can handle higher loads of organic material, reducing the footprint of the treatment plant while maintaining high performance.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur and is fundamental in bioprocess engineering. We use kinetic models, like the Michaelis-Menten equation, to describe how substrate concentration affects the rate of reaction in biofilm reactors. By understanding these kinetics, we can design reactors to maximize treatment efficiency. In our problem, reaction kinetics allows us to predict the substrate removal rate and, consequently, the height of the reactor needed to reduce the substrate concentration to the desired level.