11.2 Solids Transportation and Storage

11.2 Solids Transportation and Storage

1. Solids Transportation Systems

Moving bulk solids is a critical operational task in chemical plants. Transportation systems are split into mechanical conveying and fluid-based piping systems.

Belt Conveyors: Used for horizontal or slightly inclined transport ($<20^\circ$) over long distances. The volumetric capacity ($Q$, $\text{m}^3/\text{h}$) is determined by the cross-sectional area of the solid load on the belt ($A$, $\text{m}^2$) and belt velocity ($v$, $\text{m/s}$):

$$Q = 3600 \cdot A \cdot v$$

The cross-sectional area $A$ depends on the belt width, the shape of the idlers (troughed vs. flat), and the material's dynamic surcharge angle.

Screw Conveyors: Consist of a rotating helical screw flight in a stationary trough. They are ideal for short distances ($<30\text{ m}$) and can operate at steep inclines. Capacity ($Q_s$, $\text{m}^3/\text{h}$) is calculated as:

$$Q_s = 60 \cdot \frac{\pi}{4} (D_s^2 - (D_s^c)^2) \cdot P \cdot n \cdot \eta_L$$

where $D_s$ is the screw diameter, $D_s^c$ is the shaft diameter, $P$ is the pitch, $n$ is rotational speed in rpm, and $\eta_L$ is the loading factor. The loading factor is typically 15% to 30% for abrasive solids to minimize wear, and up to 45% for non-abrasive, free-flowing solids.

Pneumatic Conveying: Uses a gas stream (usually air) to transport particles through pipelines.

• Dilute Phase: Particles are fully suspended in high-velocity air ($15 - 35\text{ m/s}$) at low pressure. The air velocity must exceed the saltation velocity (horizontal pipes) or choking velocity (vertical pipes) to prevent settling. High velocities cause pipe erosion and particle attrition.
• Dense Phase: Particles slide or move in plugs along the pipe at low velocities ($1 - 10\text{ m/s}$) under high pressure. This regime minimizes erosion and is ideal for fragile or abrasive materials.

Slurry Transport: Involves pumping solid-liquid mixtures through pipelines. To prevent settling, the slurry velocity must exceed the critical deposition velocity ($v_D$), which Durand's equation estimates:

$$v_D = F_L \sqrt{2 g D_p \left(\frac{\rho_s - \rho_f}{\rho_f}\right)}$$

where $F_L$ is an empirical factor, $g$ is gravity, $D_p$ is pipe diameter, $\rho_s$ is solid density, and $\rho_f$ is fluid density.

2. Solids Storage and Silo Design

Bulk solids are stored in vertical bins, silos, or hoppers. Unlike liquids, which exert hydrostatic pressure increasing linearly with depth ($P = \rho g h$), bulk solids transfer their weight to the walls through friction. Consequently, the vertical pressure in a silo asymptotes to a maximum limit at deep levels. This behavior is modeled by Janssen’s Equation:

$$\sigma_v = \frac{\rho_b g R_h}{\mu_w K} \left[ 1 - e^{-\frac{\mu_w K z}{R_h}} \right]$$

where $\sigma_v$ is vertical stress, $\rho_b$ is bulk density, $R_h$ is hydraulic radius of the silo cross-section, $\mu_w$ is the wall friction coefficient ($\tan\phi'$), and $K$ is the lateral-to-vertical pressure ratio. The lateral stress on the wall is $\sigma_h = K \sigma_v$.

3. Flow Patterns and Discharge Obstructions

When discharging solids from a hopper, two main flow patterns can occur:

• Mass Flow: The entire solid mass is in motion. This produces a "first-in, first-out" sequence, eliminating stagnant zones and segregation. It requires steep, smooth walls, which increases height and causes wall wear.
• Funnel Flow: Material flows only through a central channel. Solids near the walls remain stagnant, producing a "first-in, last-out" sequence. This occurs in shallow or rough hoppers and can lead to product degradation or silo instability.

Hoppers can fail to discharge due to two major obstructions:

• Arching (Bridging): A stable cohesive arch forms over the outlet, blocking flow.
• Ratholing (Piping): Solid drains from the center, leaving a stable empty core surrounded by stagnant material.

To prevent arching, the minimum hopper outlet width ($B$) is designed as:

$$B \ge \frac{H(\theta') \bar{\sigma}_1}{g \rho_b}$$

where $H(\theta')$ is a geometry factor and $\bar{\sigma}_1$ is the bulk strength of the solid.

4. Worked Examples

Example 1: Slurry Transport Critical Velocity A sand slurry ($\rho_s = 2650\text{ kg/m}^3$) in water ($\rho_f = 1000\text{ kg/m}^3$) is transported in a $0.20\text{ m}$ diameter pipe. If the empirical factor $F_L$ is $1.40$, calculate the critical deposition velocity ($v_D$) in m/s.

Solution: Durand's equation is:

$$v_D = F_L \sqrt{2 g D_p \left(\frac{\rho_s - \rho_f}{\rho_f}\right)}$$

1. Calculate the density ratio term:

   $$\frac{\rho_s - \rho_f}{\rho_f} = \frac{2650 - 1000}{1000} = 1.65$$

2. Substitute parameters:

   $$v_D = 1.40 \cdot \sqrt{2 \cdot 9.81\text{ m/s}^2 \cdot 0.20\text{ m} \cdot 1.65}$$ $$v_D = 1.40 \cdot \sqrt{6.4746} \approx 1.40 \cdot 2.5445 \approx 3.56\text{ m/s}$$

The critical deposition velocity is 3.56 m/s.

Example 2: Screw Conveyor Capacity Sizing Determine the volumetric capacity in $\text{m}^3/\text{h}$ of a screw conveyor with a screw diameter of $0.40\text{ m}$, a shaft diameter of $0.10\text{ m}$, and a pitch of $0.35\text{ m}$ rotating at $50\text{ rpm}$ with a trough loading factor of $25\%$.

Solution:

1. Calculate cross-sectional area:

   $$A = \frac{\pi}{4} (D_s^2 - (D_s^c)^2) = \frac{\pi}{4} (0.40^2 - 0.10^2) = \frac{\pi}{4} (0.16 - 0.01) = \frac{\pi}{4} (0.15) \approx 0.1178\text{ m}^2$$

2. Use capacity formula:

   $$Q_s = 60 \cdot A \cdot P \cdot n \cdot \eta_L$$ $$Q_s = 60 \cdot 0.1178\text{ m}^2 \cdot 0.35\text{ m} \cdot 50\text{ rpm} \cdot 0.25$$ $$Q_s = 60 \cdot 0.1178 \cdot 4.375 = 30.92\text{ m}^3/\text{h}$$

The capacity is approximately 30.9 $\text{m}^3/\text{h}$.