Study notes

Qwestrum Engineering360 · Chemical Engineering · Mechanical Operations

Sedimentation and Centrifugation

Gravity settling is governed by Stokes’ terminal velocity (valid only for particle Reynolds number below one), and a centrifuge simply replaces g with the far larger centrifugal acceleration rω², dramatically speeding separation of fine particles.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Stokes law valid for particle Re < 1; use drag correlations otherwise
Hindered settling modifies effective density and viscosity at high solids
A centrifuge achieves a large g-equivalent via high ω

Topic details

Introduction

This McCabe-Smith topic covers solid-liquid separation by density difference. You compute the Stokes terminal velocity, verify the low-Reynolds-number assumption, size gravity thickeners from settling tests, and translate a required settling velocity into centrifuge speed using the g-multiplier or sigma-factor concept.

Key relations & formulas

v_{t} = g d_{p}^{2} \frac{(\rho_{p} - \rho_{f})}{(18 \mu)}

(Stokes terminal velocity, laminar)

t = \frac{h}{v_{t}}

(batch settling time for height h)

Σ = V \omega^{2} / (g ln(\frac{r_{2}}{r_{1}}))

(centrifuge sigma factor — equivalent settling area)

Notation and sign conventions

Relation 1 —

v_{t} = g d_{p}^{2}

v_{t} = g d_{p}^{2} \frac{(\rho_{p} - \rho_{f})}{(18 \mu)}

(Stokes terminal velocity, laminar)

Write this relation with symbols exactly as in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

t = \frac{h}{v_{t}}

t = \frac{h}{v_{t}}

(batch settling time for height h)

Relation 3 —

Σ = V \omega^{2} /

Σ = V \omega^{2} / (g ln(\frac{r_{2}}{r_{1}}))

(centrifuge sigma factor — equivalent settling area)

Concept in depth

A particle settling under gravity accelerates until drag balances its net weight, reaching a constant terminal velocity that Stokes’ law gives for small, slow particles in the viscous regime. Because that velocity scales with the square of diameter, fine particles settle painfully slowly — which is why centrifugation exists: spinning the fluid substitutes an acceleration of thousands of g for the 1 g of gravity, cutting separation time proportionally. At high solids concentration particles interfere, so hindered settling slows things further and requires corrected density and viscosity. Always check the Reynolds number, because Stokes’ law fails for coarse or dense particles.

Assumptions and validity limits

State assumptions explicitly before using any relation for sedimentation and centrifugation — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Mechanical Operations viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Mechanical Operations papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to sedimentation and centrifugation.
4. Use equation 1:

v_{t} = g d_{p}^{2}

.
5. Use equation 2:

t = \frac{h}{v_{t}}

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Sedimentation and Centrifugation appears in solids processing industries. In Indian chemical curricula this topic is tested because it connects theory to size reduction, filtration, and fluidization.

GATE and semester exams often combine sedimentation and centrifugation with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use sedimentation and centrifugation?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

The signature trap is applying Stokes’ law without checking that particle Re < 1, which fails for coarse particles. Students also forget the density difference (using particle density alone), mix up radius terms in the centrifuge equation, and ignore hindered settling at high solids loading.

Quick revision checklist

Before attempting sedimentation and centrifugation problems, confirm you can:

1. Stokes law valid for particle Re < 1; use drag correlations otherwise
2. Hindered settling modifies effective density and viscosity at high solids
3. A centrifuge achieves a large g-equivalent via high ω

Revise the solved examples in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Stokes terminal velocity

Problem

A 50 µm particle (ρ_p = 2500 kg/m³) settles in water (ρ = 1000, µ = 10⁻³ Pa·s). Find its terminal velocity.

Solution

v_t = g d²(ρ_p − ρ_f)/(18µ) = 9.81×(50e-6)²×1500/(18×10⁻³) = 9.81×2.5e-9×1500/0.018 = 2.04×10⁻³ m/s. Check Re ≈ 0.1 < 1, so Stokes is valid.

Conceptual check — Sedimentation and Centrifugation

Problem

In a Mechanical Operations semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of sedimentation and centrifugation." What should a complete answer include?

Exams & GATE

Check particle Re before applying Stokes — an exam trap for coarse particles.

📖 Standard books (India)

Unit Operations of Chemical Engineering — McCabe, Smith & Harriott
Read: Syllabus unit
Momentum, heat, and mass transfer operations

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📚Key formulas & points