Question

Apple juice (viscosity = 1.6 cP) is being filtered through a 2 m\(^2\) filter under a constant pressure. The juice has solid concentration of 0.04 g.mL\(^{-1}\) in filtrate. The total pressure drop is 325.33 kPa. The values of cake and filter medium resistance are \(1.85 \times 10^{11}\) m.kg\(^{-1}\) and \(3.50 \times 10^{11}\) m\(^{-1}\), respectively. The time required to filter 800 litres of juice is ________ hour. (Answer in integer)

Hint:

When calculating filtration time, remember to account for both the filter medium and cake resistance, and ensure all units are properly converted.

Answer 1

We are given:
Viscosity of the juice \( \mu = 1.6 \, {cP} = 1.6 \times 10^{-3} \, {Pa.s} \),
Filter area \( A = 2 \, {m}^2 \),
Solid concentration in filtrate \( C_s = 0.04 \, {g/mL} = 40 \, {g/L} \),
Total pressure drop \( \Delta P = 325.33 \, {kPa} = 325.33 \times 10^3 \, {Pa} \),
Cake resistance \( R_c = 1.85 \times 10^{11} \, {m.kg}^{-1} \),
Filter medium resistance \( R_f = 3.50 \times 10^{11} \, {m}^{-1} \),
Volume to be filtered = 800 L = 0.8 m\(^3\).
The filtration rate is governed by the Darcy’s law for filtration: \[ \frac{1}{Q} = \frac{1}{K} \cdot \frac{1}{\Delta P}, \] where \(Q\) is the flow rate and \(K\) is the filtration constant. 
Step 1: Calculate the total resistance \(R_{{total}}\):
The total resistance consists of the resistance due to the cake and filter medium: \[ R_{{total}} = R_c \cdot C_s + R_f. \] Substitute the values: \[ R_{{total}} = (1.85 \times 10^{11} \cdot 40) + 3.50 \times 10^{11} = 7.4 \times 10^{12} + 3.50 \times 10^{11} = 7.75 \times 10^{12} \, {m}^{-1}. \] Step 2: Calculate the flow rate \(Q\):
Now, we can calculate the filtration rate using the Darcy’s law equation. First, we find the filtration constant \(K\) using the following relation: \[ K = \frac{\mu A}{R_{{total}}}. \] Substitute the values: \[ K = \frac{1.6 \times 10^{-3} \times 2}{7.75 \times 10^{12}} = \frac{3.2 \times 10^{-3}}{7.75 \times 10^{12}} = 4.13 \times 10^{-16} \, {m}^3 {Pa}^{-1} {s}^{-1}. \] Next, calculate the flow rate: \[ Q = \frac{K \Delta P}{1} = 4.13 \times 10^{-16} \times 325.33 \times 10^3 = 1.34 \times 10^{-10} \, {m}^3/{s}. \] Step 3: Calculate the time required to filter 800 liters of juice:
The time required to filter the desired volume \(V = 0.8 \, {m}^3\) is: \[ t = \frac{V}{Q} = \frac{0.8}{1.34 \times 10^{-10}} = 5.97 \times 10^9 \, {s}. \] Convert seconds to hours: \[ t = \frac{5.97 \times 10^9}{3600} \approx 1 \, {hour}. \] Thus, the time required to filter 800 liters of juice is 1 hour.