Question

For a packed-bed comprising of uniform-sized spherical particles of diameter \(D_p\), the pressure drop across the bed is given by the Kozeny–Carman (laminar Ergun) equation when the particle Reynolds number \((Re_p)<1\). Under this condition, minimum fluidization velocity is proportional to \(D_p^{\,n}\). Which one of the following is the CORRECT value of exponent \(n\)?

• A. 2
• B. \(-1\)
• C. \(-2\)
• D. 1

Hint:

For \(Re_p<1\), only the viscous (Kozeny–Carman) term matters: \(\Delta P/L \propto \mu u / D_p^{2}\).
At minimum fluidization, \(\Delta P/L\) equals the (constant) bed weight per volume \(\Rightarrow u_{mf}\propto D_p^{2}\).
Remember: laminar \(u_{mf}\sim D_p^{2}\); if inertial effects dominated, the scaling would change.

Answer 1

The Correct Option is A

Step 1: For \(Re_p < 1\) (laminar regime), the Kozeny–Carman/laminar term of the Ergun equation gives the pressure drop per unit length: \[ \frac{\Delta P}{L} = \frac{150(1-\varepsilon)^2}{\varepsilon^3}\,\frac{\mu\,u}{D_p^{\,2}} , \] where \(\varepsilon\) is bed voidage, \(\mu\) is fluid viscosity, and \(u\) is the superficial velocity.

Step 2: At minimum fluidization, the pressure drop balances the apparent weight of solids per unit bed volume: \[ \left.\frac{\Delta P}{L}\right|_{u=u_{mf}} = (\rho_s-\rho_f)(1-\varepsilon_{mf})\,g , \] which is independent of particle diameter \(D_p\) (material and bed properties fixed).

Step 3: Equating the two expressions: \[ (\rho_s-\rho_f)(1-\varepsilon_{mf})\,g = \frac{150(1-\varepsilon_{mf})^2}{\varepsilon_{mf}^3}\, \frac{\mu\,u_{mf}}{D_p^{\,2}} . \] Solving for \(u_{mf}\): \[ u_{mf} \;\propto\; D_p^{\,2}. \]

Final Answer: \[ \boxed{n = 2} \]