11.1 Particle Properties and Solid Processing

11.1 Particle Properties and Solid Processing

1. Particle Shape and Sphericity

Solid particles are widely processed in chemical engineering, including pharmaceuticals, catalysts, and minerals. Unlike fluids, solid materials are discrete and heterogeneous. Sphericity, denoted by $\phi_s$, is a dimensionless parameter characterizing particle shape, defined as the ratio of the surface area of a sphere of equal volume ($V_p$) to the actual surface area of the particle ($S_p$): $\phi_s = \frac{\pi^{1/3} (6 V_p)^{2/3}}{S_p} = \frac{6 V_p}{D_p S_p}$ where $D_p$ is the equivalent volume-based diameter. Sphericity ($\phi_s$) ranges from 0 to 1, with a perfect sphere having $\phi_s = 1.0$. Sphericity is critical in packed bed pressure drop calculations (e.g., Ergun equation), where the effective particle size is taken as $\phi_s D_p$ to account for the increased drag of non-spherical shapes.

2. Sieve Analysis and Size Distributions

Sieve analysis determines particle size distributions by vibrating a solid sample through a stack of screens of decreasing aperture size (mesh numbers). A higher mesh number indicates more openings per linear inch, hence a smaller opening size. Sieve analysis data are typically represented as:

• Differential Analysis: Shows the mass fraction ($x_i$) retained in each size interval. The mean size ($d_{pi}$) is typically the arithmetic average of the screen openings of the two bounding sieves.
• Cumulative Analysis: Tracks the cumulative mass fraction of particles smaller than (or larger than) a given size. The $D_{p,80}$ size represents the sieve opening through which 80% of the material passes, which is a key metric in size reduction.

3. Bulk Mean Diameters

When analyzing mixtures of particles of varying sizes, several statistical mean diameters are defined. The two most common are:

• Arithmetic Mean Diameter ($d_N$): Simple number-average diameter: $d_N = \frac{\sum N_i d_{pi}}{\sum N_i}$ where $N_i$ is the particle count in size interval $i$ of mean diameter $d_{pi}$.
• Volume-Surface (Sauter) Mean Diameter ($d_{vs}$): Preserves the specific surface-area-to-volume ratio of the bulk mixture: $d_{vs} = \frac{1}{\sum_{i=1}^n \frac{x_i}{d_{pi}}}$ where $x_i$ is the mass fraction of particles in size interval $i$. This is widely used in packed-bed pressure drop and fluidization calculations since fluid drag is a surface phenomenon.

4. Particle Forces and Angle of Repose

Solid particles are influenced by gravitational forces (dominant for particles $>100\ \mu\text{m}$), fluid drag, buoyancy, electrostatic forces, and cohesive van der Waals forces. For fine particles ($D_p < 10\ \mu\text{m}$), cohesive forces dominate, causing agglomeration. The angle of repose ($\theta$) is the steepest angle relative to the horizontal to which bulk solids can be piled without slumping: $\tan(\theta) = \frac{H}{R}$ where $H$ is the conical pile height and $R$ is the base radius. Free-flowing materials have low angles ($20^\circ - 30^\circ$), whereas cohesive materials have high angles ($>45^\circ$).

5. Size Reduction (Comminution)

Comminution is the mechanical process of reducing particle size through crushing or grinding. Three empirical laws relate specific energy ($E$, $ ext{kWh/short ton}$) to feed size ($D_1$) and product size ($D_2$):

• Kick's Law: Assumes energy is proportional to the log reduction ratio. Used for coarse crushing ($D > 50\text{ mm}$): $E = K_K \ln\left(\frac{D_1}{D_2}\right)$
• Rittinger's Law: Assumes energy is proportional to the new surface area created. Used for fine grinding ($D < 0.1\text{ mm}$): $E = K_R \left(\frac{1}{D_2} - \frac{1}{D_1}\right)$
• Bond's Law: Assumes energy is proportional to crack length. Widely used for industrial mill sizing: $E = 10 W_i \left( \frac{1}{\sqrt{D_2}} - \frac{1}{\sqrt{D_1}} \right)$ where $W_i$ is the Bond Work Index, representing the energy required to reduce the material from infinite size to 80% passing $100\ \mu\text{m}$. Crucially, in Bond's equation, $D_1$ and $D_2$ must be in micrometers ($\mu\text{m}$).

6. Crystallization

Crystallization is a separation process that produces high-purity solids from a liquid solution. The driving force is supersaturation, defined as: $\Delta C = C - C^*$ Crystallization proceeds via nucleation and crystal growth. For a cooling crystallizer without solvent evaporation, the yield of hydrated crystals ($C_y$) is calculated by a mass balance: $C_y = R \cdot \frac{M_s (x_1 - x_2)}{1 - x_2(R-1)}$ where $M_s$ is the mass of solvent, $x_1$ and $x_2$ are the initial and final solubilities (kg anhydrous solute/kg solvent), and $R$ is the hydrate-to-anhydrous molecular weight ratio: $R = \frac{M_{\text{hydrate}}}{M_{\text{anhydrous}}}$ If the product is anhydrous, $R = 1$ and the equation simplifies to $C_y = M_s (x_1 - x_2)$.

7. Worked Examples

Example 1: Sauter Mean Diameter Calculation A mixture of catalyst particles is analyzed by sieving, yielding the following size fractions:

• $30%$ by weight has a mean size of $1.5\text{ mm}$
• $50%$ by weight has a mean size of $1.0\text{ mm}$
• $20%$ by weight has a mean size of $0.5\text{ mm}$ Calculate the Sauter mean diameter ($d_{vs}$) of the mixture in mm. Solution: $\sum_{i=1}^3 \frac{x_i}{d_{pi}} = \frac{0.30}{1.5\text{ mm}} + \frac{0.50}{1.0\text{ mm}} + \frac{0.20}{0.5\text{ mm}} = 1.10\text{ mm}^{-1}$ $d_{vs} = \frac{1}{1.10\text{ mm}^{-1}} \approx 0.909\text{ mm}$

Example 2: Size Reduction Energy Requirement A mineral ore has a Bond Work Index ($W_i$) of $14.2\text{ kWh/short ton}$. Calculate the power in kW required to crush $80\text{ short tons/hour}$ of this ore from an 80% passing feed size of $16\text{ mm}$ ($16,000\ \mu\text{m}$) to an 80% passing product size of $2.5\text{ mm}$ ($2,500\ \mu\text{m}$). Solution: Calculate the specific energy required per short ton: $E = 10 \cdot 14.2 \cdot \left( \frac{1}{\sqrt{2,500}} - \frac{1}{\sqrt{16,000}} \right)$ $E = 142 \cdot \left( \frac{1}{50} - \frac{1}{126.49} \right) = 1.718\text{ kWh/short ton}$ Calculate the total power required: $P = 1.718\text{ kWh/short ton} \cdot 80\text{ short tons/hour} = 137.4\text{ kW}$