Question

Biomass present in a fermentation broth is to be separated by vacuum filtration. Filter and broth characteristics are given below. $$ \begin{aligned} &A=50 \mathrm{~m}^{2}, \quad \Delta P=0.01 \mathrm{~N} / \mathrm{m}^{2}, \quad C=15 \mathrm{~kg} / \mathrm{m}^{3} \\ &\mu=0.003 \mathrm{~kg} / \mathrm{m}-\mathrm{s}, \quad \alpha=2 \mathrm{~m} / \mathrm{kg} \end{aligned} $$ a. If rate of filtration has a constant value of \(d V / d t=50 \mathrm{l} / \mathrm{min}\), determine the cake and filter resistances at \(t=30 \mathrm{~min}\). b. Determine the filter surface area \((A)\) required to filter 50001 broth within 60 min with the same pressure drop across the filter.

Short answer

Cake resistance = 45 m^{-1} Filter resistance = 21.67 m^{-1} Filter surface area = 16.67 m^2

Step-by-step solution

Calculate the total volume filtered

Given the rate of filtration \(\frac{dV}{dt}=50 \text{ l/min}\), and the time \(t=30 \text{ min}\). The total volume filtered is \(V = \frac{dV}{dt} \times t = 50 \text{ l/min} \times 30 \text{ min} = 1500 \text{ l} = 1.5 \text{ m}^3\).

Calculate the cake resistance

The cake resistance \(R_c\) can be calculated using \(R_c = \alpha C V\), where \(\alpha = 2 \text{ m/kg}\), \(C = 15 \text{ kg/m}^3\), and \(V = 1.5 \text{ m}^3\). Substituting the values gives: \[R_c = 2 \text{ m/kg} \times 15 \text{ kg/m}^3 \times 1.5 \text{ m}^3 = 45 \text{ m}^{-1}\].

Calculate the filter resistance

Using the filtration equation which relates the pressure drop \(\Delta P = 0.01 \text{ N/m}^2\), viscosity \(\mu = 0.003 \text{ kg/(m-s)}\), and rate of filtration: \[\frac{dV}{dt} = \frac{A \Delta P}{\mu (R_f + R_c)}\]. Substitute values and solve for \(R_f\): \[50 \text{ l/min} \times \frac{1 \text{ m}^3}{1000 \text{ l}} = \frac{50 \text{ m}^2 \times 0.01 \text{ N/m}^2}{0.003 \text{ kg/(m-s)} \times (R_f + 45 \text{ m}^{-1})}\]. Simplifying, we get: \[\frac{1}{20} = \frac{1}{0.003(R_f + 45)}\], solving for \(R_f\) yields \[R_f = 21.67 \text{ m}^{-1}\].

Calculate filter surface area for different conditions

The filtration equation becomes \[\frac{83.33 \text{ l/min}}{1000} = \frac{A \times 0.01 \text{ N/m}^2}{0.003 \text{ kg/(m-s)} \times (21.67 + 45)\text{ m}^{-1}} \]. Simplifying and solving for \(A\) we get: \[\frac{1}{12} = \frac{A \times 0.01}{0.003 \times 66.67} \rightarrow A = 16.67 \text{ m}^2\].

Key concepts

Filtration Resistance

When dealing with vacuum filtration, understanding filtration resistance is crucial. Filtration resistance is the opposition a filter offers to the flow of the liquid being filtered. It is divided into two parts: the resistance of the filter medium itself \((R_f)\) and the resistance of the filter cake \((R_c)\) that builds on the medium over time. In the given exercise, calculating these resistances allows us to determine how efficiently our filtration process will run. The formula to find the cake resistance \((R_c)\) is: \[ R_c = \alpha C V \] where \(\alpha\) is the specific cake resistance, \(C\) is the concentration of solids, and \(V\) is the volume of filtrate. The filter resistance \((R_f)\) can then be found using the overall filtration equation given by: \[ \frac{dV}{dt} = \frac{A \Delta P}{\mu (R_f + R_c)} \] where \(A\) is the filter area, \( \Delta P\) is the pressure drop, and \( \mu \) is the fluid viscosity. These calculations help us determine the effectiveness and efficiency of the filtration process.

Bioprocess Engineering Calculations

In bioprocess engineering, we utilize various calculations to optimize processes like vacuum filtration. This involves determining parameters like the rate of filtration, filter area, and resistances. In this exercise, we start with calculating the total volume filtered over a given time using the rate of filtration: \[ V = \frac{dV}{dt} \times t \] Here, \( \frac{dV}{dt}\) is the filtration rate and \(t\) is the time. From this volume, we can then calculate the cake resistance. Understanding these calculations is important for designing efficient filtration systems. Besides, the equations for filtration can guide engineers in modifying operational parameters to improve yield and reduce costs.

Cake Resistance

Cake resistance is a key concept in filtration processes. It refers to the resistance offered by the layer of filtered solids (the cake) that builds up on the filter medium. In the exercise, it is calculated using the formula: \[ R_c = \alpha C V \] The cake resistance directly impacts the efficiency of the filtration process. A higher cake resistance means more opposition to fluid flow, reducing the rate of filtration. Therefore, understanding and managing cake resistance is essential for optimal filter design and operation. Engineers can manipulate factors like pressure, filtration rate, and filter area according to the calculated cake and filter resistances to achieve desired filtration outcomes.