Mixing of solids and pastes

Why it matters

Pharmaceutical tablets must contain the stated dose in every tablet, fertiliser granules must carry the right nutrient ratio, and doughs, rubbers and ceramic pastes must be uniform before they are shaped. Solids and pastes cannot be mixed like liquids: there is no turbulence or molecular diffusion to help, free-flowing powders tend to un-mix (segregate), and pastes demand large power. Engineers need the right mixer and a way to measure how well mixed a batch is.

Key ideas

Why solids are different. In a liquid, once bulk flow has spread the components, diffusion finishes the job. In a powder, particles move only when the mixer moves them, and a perfectly ordered arrangement is never reached; the best possible state is a random mixture, in which the probability of finding a particle of a component is the same everywhere.

Mechanisms of solids mixing.

• Convective mixing: groups of particles are moved from one place to another by blades, ribbons or screws (dominant in ribbon and screw mixers).
• Diffusive mixing: individual particles redistribute over a freshly exposed surface, as in tumbling mixers; slow but fine-scale.
• Shear mixing: slip planes form within the bed and smear layers into each other; dominant for cohesive powders and pastes.

Segregation. Free-flowing particles that differ in size (most important), density or shape separate when the bed is moved: fines percolate down through gaps between coarse particles, and coarse particles roll to the outside of heaps. Over-mixing can therefore make a free-flowing blend worse. Cohesive powders segregate little but need shear to break agglomerates. Remedies include matching particle sizes, adding a little liquid, and minimising handling after mixing.

Equipment for dry solids.

• Tumbling mixers (drum, double-cone, V-blender): gentle, easy to clean, good for free-flowing solids of similar size; prone to segregation for dissimilar particles.
• Ribbon blenders: a horizontal trough with inner and outer helical ribbons moving material in opposite directions; mainly convective; handles light pastes too.
• Screw mixers (orbiting-screw conical mixers), paddle mixers and high-speed plough-share mixers for cohesive powders.

Equipment for pastes and plastic masses. Pastes are too viscous to flow to the mixing element, so the mixer must bring the element to all the material and shear it hard.

• Change-can and planetary mixers: blades orbit as they rotate so they sweep the whole can (doughs, creams).
• Kneaders and dispersers (twin sigma-blade or Z-blade mixers): two blades counter-rotate at different speeds, folding, stretching and shearing heavy pastes and rubbers.
• Muller mixers: heavy wheels roll over the mass (foundry sand, clays).
• Pug mills and continuous extruders: screw or bladed shafts in a trough or barrel for continuous mixing. Power consumption is high, and much of it appears as heat, so jacket cooling is often needed.

Measuring mixedness. Take N spot samples, each containing n particles, and measure the fraction x of the key (tracer) component in each. If the overall fraction is p:

• before mixing (completely segregated), the variance is σ₀² = p(1 − p);
• for a perfectly random mixture of samples of n particles, the variance is σ_R² = p(1 − p)/n;
• the measured sample variance s² lies between them. The Lacey mixing index M compares s² with these limits: M = 0 when unmixed, M = 1 when random. Because σ_R² depends on n, sample size must be stated. Many solids mixers approximately follow first-order kinetics in time, so 1 − M ≈ exp(−k·t) and test data at one time can estimate the time for a target M.

Formulas

σ₀² = p·(1 − p) σ_R² = p·(1 − p) / n s² = Σ(xᵢ − p)² / N (p known); if p is estimated from the samples use Σ(xᵢ − x̄)² / (N − 1) Lacey index: M = (σ₀² − s²) / (σ₀² − σ_R²) First-order mixing: 1 − M = exp(−k·t) Shear stress in a Newtonian paste: τ = μ·γ̇

• p: overall fraction of tracer (−); n: particles per sample (−); xᵢ: tracer fraction in sample i; N: number of samples; s²: sample variance; M: mixing index (−); k: mixing rate constant (s⁻¹ or min⁻¹); t: mixing time; τ: shear stress (Pa); μ: (apparent) viscosity (Pa·s); γ̇: shear rate (s⁻¹).

Worked examples

Example 1 (standard): Lacey index. A blend contains 30% tracer by number (p = 0.3). Samples of n = 50 particles taken after 5 min give a sample variance s² = 0.0100. Find M.

1. σ₀² = 0.3 × 0.7 = 0.21.
2. σ_R² = 0.21/50 = 0.0042.
3. M = (0.21 − 0.0100)/(0.21 − 0.0042) = 0.200/0.2058 = 0.972.

Example 2 (GATE level): from samples to mixing time. Ten samples of 100 particles each from a blend with p = 0.30 contain tracer fractions 0.20, 0.38, 0.31, 0.22, 0.41, 0.27, 0.35, 0.24, 0.33 and 0.29. (a) Find M. (b) In a separate test the same mixer gave M = 0.6 after 2 min; assuming first-order behaviour, how long is needed for M = 0.95?

1. Squared deviations from p: 0.0100, 0.0064, 0.0001, 0.0064, 0.0121, 0.0009, 0.0025, 0.0036, 0.0009, 0.0001; sum = 0.0430.
2. s² = 0.0430/10 = 0.0043.
3. σ₀² = 0.21; σ_R² = 0.21/100 = 0.0021.
4. M = (0.21 − 0.0043)/(0.21 − 0.0021) = 0.2057/0.2079 = 0.989.
5. Kinetics: k = −ln(1 − 0.6)/2 = −ln 0.4/2 = 0.458 min⁻¹.
6. t = −ln(1 − 0.95)/k = −ln 0.05/0.458 = 2.996/0.458 = 6.5 min.

Common mistakes

• Assuming longer mixing always improves a free-flowing blend; segregation can make it worse.
• Computing σ_R² without dividing by the number of particles per sample.
• Using spot samples so large that they average out non-uniformity (or so small that σ_R² dominates).
• Choosing a tumbling mixer for powders of very different size, or for cohesive pastes that need shear.
• Treating a paste as Newtonian without checking; most are shear-thinning or have a yield stress, so viscosity values are "apparent" at a stated shear rate.

For GATE CH

Expect: Lacey mixing index from given variance or sample data; time for a target mixing index from first-order kinetics; mechanisms of solids mixing and segregation; matching mixers (tumbler, ribbon, sigma-blade kneader, muller, planetary) to materials. Practise computing σ₀², σ_R² and M quickly.

Quick check

1. What is the variance of a completely segregated mixture with 40% tracer?
2. What value does the Lacey index take for a perfectly random mixture?
3. Which mixer suits a stiff, heavy rubber-like paste?
4. What is the main cause of segregation in free-flowing powders? Answers: 1. 0.4 × 0.6 = 0.24. 2. 1. 3. A twin sigma-blade (Z-blade) kneader. 4. Differences in particle size.