Stokes and Newtonian regimes in hydrocyclones

 Explain Stokes and Newtonian regimes in hydrocyclones

Stokes and Newtonian Regimes in Hydrocyclones

Hydrocyclones are devices that separate particles based on size and density by utilizing centrifugal forces in a swirling fluid. The behavior of particles in a hydrocyclone depends on the flow regime, which can be categorized into Stokes' regime (laminar flow) and Newtonian regime (turbulent flow). These regimes determine the settling behavior of particles and influence the separation efficiency of hydrocyclones.

---

1. Stokes' Regime

Description:

• Stokes' regime occurs when the flow around the particle is laminar, characterized by smooth and orderly fluid motion.
• This regime is valid for fine particles where the Reynolds number ($Re$) is very low ($Re < 1$).

Governing Equation:

The terminal settling velocity ($v_t$) of a particle in Stokes' regime is given by Stokes' law:

$$v_t = \frac{2 (\rho_p - \rho_f) g r^2}{9 \mu}$$

Where:

• $v_t$: Terminal settling velocity.
• $\rho_p$: Particle density.
• $\rho_f$: Fluid density.
• $g$: Acceleration due to gravity.
• $r$: Radius of the particle.
• $\mu$: Fluid viscosity.

Key Characteristics:

1. Dominant Forces:
   • Drag force and buoyant force dominate; inertial effects are negligible.
2. Particle Motion:
   • Particles settle smoothly with constant velocity, proportional to their size squared ($r^2$).
3. Particle Size:
   • Stokes' regime applies primarily to small particles (typically <10 µm).

Impact on Hydrocyclone Performance:

• Fine particles may not settle effectively due to their low terminal velocity, leading to reduced separation efficiency.
• This regime often results in higher recovery to the overflow stream (lighter fraction).

---

2. Newtonian Regime

Description:

• Newtonian regime occurs when the flow around the particle is turbulent, characterized by irregular and chaotic fluid motion.
• This regime is valid for coarse particles where the Reynolds number ($Re > 1000$) is high.

Governing Equation:

The terminal settling velocity ($v_t$) of a particle in the Newtonian regime is given by:

$$v_t = \sqrt{\frac{4 (\rho_p - \rho_f) g r}{3 C_d \rho_f}}$$

Where:

• $C_d$: Drag coefficient, which depends on the particle shape and flow conditions.

Key Characteristics:

1. Dominant Forces:
   • Inertial forces dominate; drag force is nonlinear and increases with velocity.
2. Particle Motion:
   • Coarse particles experience turbulence and high velocities, settling much faster than in Stokes' regime.
3. Particle Size:
   • Newtonian regime applies to larger particles (typically >50 µm).

Impact on Hydrocyclone Performance:

• Coarse particles are effectively directed to the underflow due to their high settling velocity.
• Separation is sharp, with minimal misplacement of coarse particles in the overflow.

---

3. Intermediate Regime

For particles in the size range between fine (Stokes' regime) and coarse (Newtonian regime), the flow behavior lies in the intermediate regime. This regime is characterized by a transition between laminar and turbulent flows. The drag coefficient ($C_d$) must be determined experimentally or through empirical correlations.

---

Comparison of Stokes and Newtonian Regimes

Aspect	Stokes' Regime	Newtonian Regime
Flow Type	Laminar	Turbulent
Reynolds Number	$Re < 1$	$Re > 1000$
Dominant Forces	Viscous drag, buoyant force	Inertial force
Applicable Particle Size	Fine particles (<10 µm)	Coarse particles (>50 µm)
Settling Velocity	Proportional to $r^2$	Proportional to $\sqrt{r}$
Efficiency in Hydrocyclones	Low efficiency for fine particle separation	High efficiency for coarse particles

---

Practical Implications in Hydrocyclones

1. Fine Particle Recovery:

   • In the Stokes' regime, fine particles tend to remain in the fluid and are more likely to report to the overflow due to their low settling velocities.
   • Hydrocyclones often face challenges in efficiently separating fine particles, especially when operating under laminar conditions.

2. Coarse Particle Recovery:

   • In the Newtonian regime, coarse particles are effectively separated to the underflow due to their high inertia and settling velocity.
   • This regime enhances separation efficiency and provides sharp cuts for larger particle sizes.

3. Design Considerations:

   • Adjustments to feed flow rate, cyclone diameter, and fluid viscosity are necessary to optimize the separation for particles in both regimes.
   • High centrifugal forces generated in the hydrocyclone help accelerate fine particle separation, mitigating the limitations of the Stokes' regime.

---

Conclusion

The Stokes' and Newtonian regimes describe the two extremes of particle settling behavior in hydrocyclones, influencing their separation efficiency. Understanding these regimes is crucial for optimizing hydrocyclone design and operation, ensuring efficient recovery of both fine and coarse particles.