Exercise set

Solid-Liquid Separation, Crystallization, Drying, and Cake Handling Exercises

Solved solid-liquid separation exercises for filtration, cake moisture, centrifuge recovery, crystallization yield, drying duty, solvent removal and release gates.

Branch: Chemical Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises focus on separations where solids, crystals, cakes or wet product must be recovered and released. They cover filtration rate, filter area, cake moisture, wash displacement, centrifuge recovery, crystallization yield, mother-liquor loss, dryer solvent removal, drying duty, residual solvent, solids handling capacity and release evidence.

Assume simplified screening calculations unless an exercise states otherwise. Real design must check particle size, crystal habit, compressible cake behavior, wash efficiency, centrifuge limits, dust hazard, solvent flammability, dryer oxygen control, product assay, containment and waste route.

Release Evidence Notes

Solid-liquid separation evidence should state slurry feed, solids concentration, filtrate quality, cake mass, cake moisture, wash basis, recovery, fines loss and equipment state.

Crystallization evidence should state solubility basis, yield, impurity rejection, seed or nucleation control, mother-liquor recycle and product assay.

Drying evidence should state wet-cake basis, solvent or water removed, heat duty, vapor handling, residual solvent, oxygen or inerting controls, dust controls and product release rule.

Engineering Boundary Notes

These calculations do not replace pilot filtration tests, centrifuge vendor sizing, crystallization kinetics, dryer hazard analysis, solvent LEL review, dust explosibility testing or product-quality validation. They are screening exercises for solids release.

Common Release Mistakes

reporting dry solids recovery while ignoring liquid retained in cake;
assuming wash displacement is perfect;
counting crystallized product while losing product in mother liquor;
drying to average moisture while local wet pockets remain;
releasing dried solids without residual solvent, dust and vapor-handling evidence.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Filtration and cake	1, 2, 3, 4, 5	slurry solids, area, cake moisture, wash and filtrate loss	Decide whether filter operation is releasable.
Crystallization and centrifuge	6, 7, 8, 9, 10	yield, mother liquor, centrifuge recovery, fines and purity	Decide whether product recovery is credible.
Drying and handling	11, 12, 13, 14, 15, 16, 17	solvent removal, heat duty, residual solvent, capacity and evidence	Decide whether dried product or residue can be released.
Release gate	18	all-of solids release	Decide whether the package can close.

Exercise 1: Slurry Solids Feed

A slurry feed is $12\ \text{m}^3/\text{h}$ at density $1150\ \text{kg/m}^3$ and $18\%$ solids by mass. Compute dry solids feed.

Solution

Slurry mass flow:

\dot{m}=12(1150)=13800\ \text{kg/h}

Dry solids:

\dot{m}_s=0.18(13800)=2484\ \text{kg/h}

Engineering Comment

Filter, centrifuge and dryer sizing should start from dry solids and liquid load, not only slurry volume.

Plausibility Check

Eighteen percent of about fourteen tonnes per hour is about two and a half tonnes per hour.

Exercise 2: Filter Area From Flux

Required filtrate flow is $9.5\ \text{m}^3/\text{h}$ . Sustainable filtrate flux is $0.38\ \text{m}^3/\text{m}^2\text{/h}$ . Compute filter area.

Solution

A=\dfrac{9.5}{0.38}=25.0\ \text{m}^2

Engineering Comment

Sustainable flux should come from process slurry tests, not clear-liquid data.

Plausibility Check

About ten cubic meters per hour at less than half a cubic meter per square meter-hour requires about twenty-five square meters.

Exercise 3: Cake Moisture

Wet cake is $3600\ \text{kg/h}$ and dry solids are $2484\ \text{kg/h}$ . Compute cake moisture by wet mass.

Solution

Liquid in cake:

\dot{m}_L=3600-2484=1116\ \text{kg/h}

Moisture:

M=\dfrac{1116}{3600}=0.31=31\%

Engineering Comment

Cake moisture sets dryer load, solvent loss and handling behavior.

Plausibility Check

About one third of the wet cake mass is liquid.

Exercise 4: Cake Wash Displacement

Cake contains $1116\ \text{kg/h}$ liquid with $2.5\%$ impurity. Washing displaces $70\%$ of retained liquid. Compute impurity remaining in cake liquid.

Solution

Initial impurity:

M_i=0.025(1116)=27.9\ \text{kg/h}

Remaining after wash:

M_r=27.9(1-0.70)=8.37\ \text{kg/h}

Engineering Comment

Wash efficiency should be validated with product assay, not assumed from wash volume alone.

Plausibility Check

Thirty percent of about twenty-eight kilograms per hour remains.

Exercise 5: Filtrate Product Loss

Filtrate flow is $9.5\ \text{m}^3/\text{h}$ and dissolved product concentration is $0.18\ \text{kg/m}^3$ . Compute product loss to filtrate.

Solution

M=9.5(0.18)=1.71\ \text{kg/h}

Engineering Comment

Filtrate loss may justify recycle, polishing or crystallization changes if product value is high.

Plausibility Check

Low concentration over about ten cubic meters per hour gives a few kilograms per hour or less.

Exercise 6: Crystallization Yield

A crystallizer feed contains $520\ \text{kg}$ dissolved product. Product crystals recovered are $410\ \text{kg}$ . Compute yield.

Solution

Y=\dfrac{410}{520}=0.788=78.8\%

Engineering Comment

Yield should be checked with mother-liquor assay and crystal purity.

Plausibility Check

Four hundred ten is a little under eighty percent of five hundred twenty.

Exercise 7: Mother-Liquor Product Loss

Mother liquor volume is $4.2\ \text{m}^3$ and dissolved product concentration is $18\ \text{kg/m}^3$ . Compute product remaining in mother liquor.

Solution

M=4.2(18)=75.6\ \text{kg}

Engineering Comment

Mother-liquor recycle can recover product but may build impurities.

Plausibility Check

Several cubic meters at tens of kilograms per cubic meter gives tens of kilograms.

Exercise 8: Crystal Purity

Dry crystal product is $410\ \text{kg}$ with $6.5\ \text{kg}$ impurity. Compute purity.

Solution

P=\dfrac{410-6.5}{410}=0.984=98.4\%

Engineering Comment

Purity alone does not prove release if yield, solvent, moisture or impurity trend fails.

Plausibility Check

Impurity is small relative to product mass, so purity is high.

Exercise 9: Centrifuge Solids Recovery

Centrifuge feed contains $2484\ \text{kg/h}$ solids. Solids lost in centrate are $86\ \text{kg/h}$ . Compute solids recovery.

Solution

R=\dfrac{2484-86}{2484}=0.965=96.5\%

Engineering Comment

Recovery must be balanced against cake moisture, fines loss and downstream drying load.

Plausibility Check

Losing less than one hundred kilograms out of about twenty-five hundred gives high recovery.

Exercise 10: Fines Loss Fraction

Total solids loss is $86\ \text{kg/h}$ . Fines below specification account for $52\ \text{kg/h}$ . Compute fines share of solids loss.

Solution

f=\dfrac{52}{86}=0.605=60.5\%

Engineering Comment

If fines dominate loss, particle-size control or flocculation may be more effective than more centrifuge speed.

Plausibility Check

Fifty-two is a little over half of eighty-six.

Exercise 11: Dryer Solvent Removal

Wet cake feed to a dryer is $3600\ \text{kg/h}$ at $31\%$ liquid. Product target is $5\%$ liquid by wet mass. Dry solids are $2484\ \text{kg/h}$ . Compute final wet product mass and liquid removed.

Solution

Final wet product mass:

m_f=\dfrac{2484}{1-0.05}=2615\ \text{kg/h}

Final liquid:

m_{L,f}=2615-2484=131\ \text{kg/h}

Initial liquid:

m_{L,i}=3600-2484=1116\ \text{kg/h}

Removed liquid:

m_{rem}=1116-131=985\ \text{kg/h}

Engineering Comment

Dryer vapor handling and condenser or abatement capacity should be sized on removed liquid.

Plausibility Check

The dryer removes most but not all retained liquid.

Exercise 12: Dryer Heat Duty

Liquid removal is $985\ \text{kg/h}$ and effective latent heat is $2300\ \text{kJ/kg}$ . Estimate heat duty.

Solution

\dot{Q}=985(2300)=2265500\ \text{kJ/h}

\dot{Q}=629\ \text{kW}

Engineering Comment

This excludes sensible heat, losses and inefficiency, so real utility demand will be higher.

Plausibility Check

About one tonne per hour of evaporation requires hundreds of kilowatts.

Exercise 13: Residual Solvent Limit

Dried product is $2615\ \text{kg/h}$ and residual solvent is $0.18\%$ by mass. Product release limit is $0.25\%$ . Compute margin.

Solution

\text{margin}=\dfrac{0.25-0.18}{0.25}=0.28=28\%

Engineering Comment

Residual solvent must be tied to sampling method and batch uniformity.

Plausibility Check

The result is below the limit with less than one third of the limit remaining as margin.

Exercise 14: Dryer Vapor Abatement Load

Solvent removed is $985\ \text{kg/h}$ . Condenser captures $92\%$ . Compute solvent sent to downstream abatement.

Solution

M=985(1-0.92)=78.8\ \text{kg/h}

Engineering Comment

Drying release needs vapor handling evidence, especially for flammable or toxic solvent.

Plausibility Check

Eight percent of about one thousand kilograms per hour is about eighty kilograms per hour.

Exercise 15: Solids Handling Capacity

Dry product conveyor capacity is $3000\ \text{kg/h}$ . Final wet product flow is $2615\ \text{kg/h}$ . Compute handling margin.

Solution

\text{margin}=\dfrac{3000-2615}{2615}=0.147=14.7\%

Engineering Comment

Handling margin should include bulk density, stickiness, dust and surge behavior.

Plausibility Check

The conveyor capacity is only a few hundred kilograms per hour above product flow.

Exercise 16: Batch Evidence Completion

A solids release package requires slurry feed, solids assay, filtrate assay, cake moisture, wash record, centrifuge loss, mother-liquor assay, dryer temperature, residual solvent, vapor abatement, product assay and packaging hold status. Nine of twelve records are complete. Compute completion.

Solution

C=\dfrac{9}{12}=75\%

Engineering Comment

The package is not release-ready if missing records include residual solvent, product assay or vapor abatement.

Plausibility Check

Nine of twelve is three quarters.

Exercise 17: Dryer Safety RPN

A high-solvent dryer discharge failure has severity $8$ , occurrence $4$ and detection $5$ . Compute RPN.

Solution

RPN=8(4)(5)=160

Engineering Comment

The score supports interlocks, oxygen control, validated sampling or hold-and-test release.

Plausibility Check

Moderate-high ratings multiply to a three-digit value.

Exercise 18: Solids Release Gate

A release gate requires crystal purity above $98\%$ , centrifuge recovery above $95\%$ , residual solvent below $0.25\%$ , handling margin above $10\%$ , and evidence completion above $90\%$ . Current values are purity $98.4\%$ , recovery $96.5\%$ , residual solvent $0.18\%$ , handling margin $14.7\%$ and completion $75\%$ . Decide release status.

Solution

Purity, recovery, residual solvent and handling margin pass. Evidence completion fails:

75\%<90\%

Release status:

\text{hold}

Engineering Comment

Product quality appears acceptable, but release should wait until batch and safety evidence is complete.

Plausibility Check

An all-of release gate fails when documentation and safety evidence are incomplete.

Validation Package Checklist

Slurry, filtrate, cake, crystals, mother liquor, dryer vapor and product streams close a mass balance.
Cake moisture, wash, centrifuge recovery and fines loss are measured on the same batch basis.
Drying calculations include removed liquid, duty, vapor abatement and residual solvent.
Product release separates purity, moisture, solvent, handling and evidence gates.

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