Question

Which one of the following is valid for a laminar flow of filtrate through a cake deposited on septum?

• A. Kozney-Carman equation
• B. Leva's equation
• C. Blake-Plummer equation
• D. Hagen – Poiseuille equation

Hint:

Packed Bed Flow Formulations: - Laminar Flow Regime (\(Re < 1\)) \(\implies\) Kozeny-Carman Equation (\(\Delta P \propto v_0\)). - Turbulent Flow Regime (\(Re > 1000\)) \(\implies\) Blake-Plummer Equation (\(\Delta P \propto v_0^2\)). - Complete Flow Range \(\implies\) Ergun Equation (Sum of laminar and turbulent terms).

Answer 1

The Correct Option is A

Concept: A filter cake consists of an accumulation of solid particles trapped on a porous filter medium (septum). The voids between these packed particles form a network of tiny, tortuous channels through which the filtrate must flow. Because these channels are extremely narrow and the fluid velocity is low, the flow of filtrate through a packed bed or filter cake is almost always laminar (\(Re < 1\)). To model fluid flow through a porous bed of particles under laminar conditions, we use the Kozeny-Carman equation. This equation modifies the classical Hagen-Poiseuille pipe flow model to account for the specific surface area of the particles, the void fraction (porosity), and the winding path (tortuosity) of the channels.

Step 1: Analyzing the mathematical structure of the Kozeny-Carman relation.
The Kozeny-Carman equation expresses the pressure drop (\(\Delta P\)) across a packed bed of height \(L\) under laminar flow conditions as: \[ \frac{\Delta P}{L} = \frac{150 \cdot \mu \cdot v_0}{D_p^2} \cdot \frac{(1 - \epsilon)^2}{\epsilon^3} \] Where:
• \(\mu\) is the dynamic viscosity of the filtrate fluid.
• \(v_0\) is the superficial fluid velocity entering the cake face.
• \(\epsilon\) is the porosity or void fraction of the deposited cake.
• \(D_p\) is the effective mean diameter of the solid particles.

Step 2: Differentiating from alternative fluid flow models.
Let's look at why the other options are incorrect for this system:
• Blake-Plummer Equation: Used to model fluid flow through packed beds under highly turbulent conditions, where inertial losses dominate over viscous forces (\(\Delta P \propto v_0^2\)).
• Hagen-Poiseuille Equation: Valid only for laminar flow through straight, uniform circular pipes, not for tortuous networks of irregular particle voids.
• Leva's Equation: Primarily used to analyze pressure drops in fluidized beds rather than stationary packed filter cakes. Therefore, the Kozeny-Carman equation is the correct model for laminar filtrate flow through a filter cake.