Introduction & Context

In pneumatic conveying, maintaining a gas velocity above the saltation velocity (\(v_{salt}\)) is critical for system stability. Saltation occurs when the drag and lift forces provided by the conveying gas are insufficient to overcome gravitational forces, causing particles to drop out of suspension and form a settling bed in horizontal pipelines. This leads to increased pressure drop, unstable flow (slugging), and eventual pipe blockage.

The minimum conveying velocity (\(v_{min}\)) is the operational setpoint used to size blowers and compressors. Unlike terminal velocity, which describes a single particle in infinite fluid, saltation velocity is a system property heavily dependent on the pipe diameter (\(D\)) and the solids loading ratio (\(\mu\)). A velocity that works in a small pilot line will fail in a larger production line due to the Froude number relationship.

Methodology & Formulas

The Rizk & Marcus correlation is the industry standard for predicting saltation in dilute-phase horizontal conveying. It relates the solids loading ratio to the Froude number at the point of saltation.

Solids Loading Ratio (\(\mu\)) – The dimensionless mass flow ratio of solids to gas. \[ \mu = \frac{\dot{m}_s}{\dot{m}_g} = \frac{\dot{m}_s}{\rho_g \, v \, A} \]
Froude Number (\(Fr\)) – Characterizes the ratio of inertial forces to gravitational forces. \[ Fr = \frac{v}{\sqrt{g \, D}} \]
Rizk & Marcus Correlation – Defines the saltation condition. \[ \mu = \frac{1}{10^{\delta}} Fr_{salt}^{\chi} \quad \text{or} \quad Fr_{salt} = \left( 10^{\delta} \cdot \mu \right)^{1/\chi} \] Where the empirical parameters \(\chi\) and \(\delta\) are functions of the particle diameter (\(d_p\)) in meters:
- \(\chi = 1100 \, d_p + 2.5\)
- \(\delta = 1440 \, d_p + 1.96\)
Saltation Velocity (\(v_{salt}\)) – Rearranging the above to solve for velocity (noting that \(\mu\) also contains velocity): \[ v_{salt} = \left[ \sqrt{g \, D} \left( \frac{\dot{m}_s \cdot 10^{\delta}}{\rho_g \, A} \right)^{1/\chi} \right]^{\frac{\chi}{\chi+1}} \]
Design Minimum Velocity – Includes a safety margin (typically 20–30%) to account for pressure fluctuations and particle size distribution. \[ v_{min} = SF \cdot v_{salt} \]

Key Variables and Units
Symbol	Description	Unit
\(D\)	Pipe Internal Diameter	m
\(d_p\)	Mean Particle Diameter	m
\(\dot{m}_s\)	Solids Mass Flow Rate	kg s^-1
\(\rho_g\)	Gas Density (at operating P and T)	kg m^-3
\(A\)	Pipe Cross-sectional Area	m²

Saltation is governed by the Froude number (\(v/\sqrt{gD}\)). As the pipe diameter (\(D\)) increases, the velocity gradient near the bottom of the pipe changes, and the turbulent eddies required to keep particles suspended must be more energetic. Mathematically, to maintain the same Froude number (and thus the same suspension capability), the velocity must increase proportionally to \(\sqrt{D}\).

As \(\mu\) increases (more solids per kg of gas), the particles collide more frequently and steal momentum from the gas stream. This increases the saltation velocity. According to the Rizk & Marcus correlation, \(v_{salt}\) is proportional to \(\mu^{1/(\chi+1)}\). If you increase the production rate (increasing \(\mu\)) without increasing the blower speed, the system may drop below the new saltation threshold and block.

No. Horizontal sections are almost always the limiting factor because gravity acts perpendicular to the flow, forcing particles toward the wall. In vertical flow, the "choking velocity" is the equivalent limit, which is typically significantly lower than the horizontal saltation velocity. Always size your system based on the horizontal saltation requirements.

The fatal error is assuming saltation velocity is constant regardless of pipe size. If a 2-inch pipe conveys successfully at 15 m/s, a 10-inch pipe conveying the same material at 15 m/s will likely block. Based on Froude similarity, the 10-inch pipe would require approximately \(\sqrt{10/2} \approx 2.2\) times the velocity (roughly 33 m/s) to maintain the same suspension regime, assuming the same solids loading ratio.

Worked Example – Minimum Conveying Velocity for Plastic Pellets

A system must transport 10 t/h of plastic pellets through a 100 mm ID horizontal pipe. We must determine the minimum air velocity required to prevent saltation.

Input Data:

Solids mass flow, \(\dot{m}_s\) = 10,000 kg/h = 2.778 kg/s
Pipe diameter, \(D\) = 0.100 m
Particle diameter, \(d_p\) = 0.003 m (3 mm)
Air density, \(\rho_g\) = 1.20 kg/m³
Safety Factor, \(SF\) = 1.2

Calculation Steps:

Calculate empirical constants \(\chi\) and \(\delta\): \[ \chi = 1100(0.003) + 2.5 = 5.8 \] \[ \delta = 1440(0.003) + 1.96 = 6.28 \]
Calculate pipe area: \[ A = \frac{\pi \cdot 0.1^2}{4} = 0.007854 \, \text{m}^2 \]
Solve for Saltation Velocity (\(v_{salt}\)): Using the rearranged Rizk & Marcus equation: \[ v_{salt} = \left[ \sqrt{9.81 \cdot 0.1} \left( \frac{2.778 \cdot 10^{6.28}}{1.2 \cdot 0.007854} \right)^{1/5.8} \right]^{\frac{5.8}{6.8}} \] \[ v_{salt} = \left[ 0.9905 \cdot (5.62 \times 10^8)^{0.1724} \right]^{0.8529} \] \[ v_{salt} = \left[ 0.9905 \cdot 31.19 \right]^{0.8529} = 18.5 \, \text{m/s} \]
Determine Design Minimum Velocity: \[ v_{min} = 1.2 \cdot 18.5 = 22.2 \, \text{m/s} \]

Final Answer: The minimum conveying velocity required is 22.2 m/s. Note that using a terminal velocity approach (which might yield ~8 m/s) would result in immediate pipe blockage.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle