Answer:
147.06 meters
Step-by-step explanation:
To determine the optimum thickness of cake for filtration pressure, we can use the Darcy's law which relates the flow rate through the cake to the pressure difference across the cake, the thickness of the cake, and the properties of the fluid and cake.
The Darcy's law is expressed as follows:
⇒ Q = (kA/μ)×(-DP/L)
where Q is the volumetric flow rate (m³/s), k is the permeability of the cake (m²), A is the cross-sectional area of the cake (m²), μ is the viscosity of the fluid (Pa·s), DP is the pressure difference (Pa), L is the thickness of the cake (m).
To apply this formula, we need to convert the given values to SI units:
- Whitting concentration = 100 kg/m³
- Density of Whitting = 2500 kg/m³
- Viscosity of water = 0.25 Pa·s
- Pressure difference (-DP) = 1000 kN/m² = 1e+9 Pa
- Flow rate = 0.03 cm³/s = 3e-8 m³/s
First, we need to calculate the permeability of the cake (k) from the flow rate and other parameters:
⇒ k = (QμL)/(A×(-DP))
The cross-sectional area of the cake can be calculated from the concentration of Whitting:
= 1000/100 = 10 m²/m³
Substituting the given values, we get:
⇒ k = (3e-8 × 0.25 × L)/(10 × 1e+9)
Simplifying, we get:
⇒ k = 7.5e-18 L
Next, we can use the given pressure difference (-DP) of 1000 kN/m² and the permeability of the cake to determine the optimum thickness of the cake:
⇒ L = (-DP)/(k×(1-ε))
where ε is the voidage of the cake.
Substituting the given values, we get:
⇒ L = (1e+9)/[7.5e-18 × (1-0.45)]
Simplifying, we get:
⇒ L = 147.06 m
Therefore, the optimum thickness of the filter cake for a filtration pressure difference of 1000 kN/m² is 147.06 meters.