Ideal mixing
Complete separation
Partial separation
Dilute limit
At 25 °C
Second-law efficiency
The real cost feed to handle linear in dilution, and usually binding
Three tasks concentration · fractionation · purification
Five mechanisms phase creation · phase addition · barrier · solid agent · force field
Two agents ESA (heat or work) · MSA (solvent, adsorbent, gas) — an MSA must itself be recovered
Conditions
General VLE
Phase rule
Raoult
Antoine
Tie line at
K value
Relative volatility
Constant-α curve
No separation when
Balances
Lever arm
Equilibrium
K and α
positive deviation · minimum -boiling azeotrope
negative deviation · maximum -boiling azeotrope
Limits
Margules (2-constant)
van Laar
What the constants mean
Definition
Existence test
At an azeotrope
Liquid splits when
Balances
Equilibrium
Energy
Degrees of freedom 2 — usually
With
With
Always through
Compositions
Rachford–Rice
Two-phase test and
Mass
Energy
Condenser duty
Signs
Rectifying, general
Rectifying, CMO
Stripping, CMO
Ratios
CMO needs equal molar latent heats · negligible sensible heat · no heat of mixing · adiabatic
Feed quality
Feed line
Conditions
Concurrency rectifying, stripping and feed lines meet at one point
Horizontal move equilibrium on a stage —
Vertical move balance between stages —
Optimum feed the first stage whose liquid passes the operating-line intersection
Partial reboiler counts as one equilibrium stage
Total reflux
Fenske
Minimum reflux
Practice
Vapour rate
General operating line
Δ bookkeeping start at add subtract
Across a feed
Across a liquid draw
Total condenser reflux and distillate share not a stage
Partial condenser counts as stage 1
Either way the operating line is unchanged
Overall
Trays to order
Murphree (vapour)
Graphically step against a pseudo-curve a fraction
Warning
Extractive high-boiling solvent alters
Heterogeneous azeotropic entrainer makes a two-liquid-phase azeotrope; a decanter does the separation
Pressure swing two columns, two pressures; the azeotrope moves, nothing added
Why a shortcut external balances do not close for
Specification fractional recoveries
Minimum stages
Non-key split
Consequence LNK → all distillate, HNK → all bottoms; only components near a key distribute
First equation
Root selection
Second equation
Coordinates
Molokanov fit
Stages
Feed stage
Standing assumptions constant
Which is which absorption: gas → liquid, operating line above equilibrium. Stripping: liquid → gas, below
Henry's law
Assumptions negligible heat of absorption · isothermal · non-volatile solvent · insoluble carrier
Constant flows
Mole ratios may exceed 1
Back again
Why the balance is linear in
Operating line
Minimum solvent interior tangent
Practice
Factors
Kremser
the lines converge — a maximum recovery no number of stages can beat
Economic optimum
Mass ratios
Operating line
Extraction factor
Cascades co-current = 1 stage whatever you build; countercurrent beats cross-flow at equal
Right triangle
Binodal inside it two phases; the ends of a tie line are the two layers; they merge at the plait point
Conjugate curve locus of
Lever rule
External balance and on
Difference point
Locating Δ intersection of
Stepping tie line → a stage; line through Δ → between stages. Count the tie lines.
Minimum solvent an operating line coincides with a tie line — search all of them; the pinch is usually interior
Practice
Linear Langmuir
Langmuir
Freundlich no Henry limit
Extended Langmuir
Selectivity
Isosteric heat
Velocity of a concentration
Shock velocity chord , not the tangent
Which happens favourable isotherm → shock on adsorption, spreading wave on regeneration; unfavourable swaps them
Retention
Capacity factor
Resolution
Working capacity the difference in loading between the two ends of the cycle — not
PSA / TSA PSA: pressure, seconds–minutes, bulk. TSA: temperature, hours, trace removal and drying
Flux partial pressures
Barrer
GPU
Selectivity
Balance
Flux ratio
Recovery
Pressure-ratio ceiling
Which limits?
Best one contact can do
Area
Units
Check
Watch out a permeate spec is cheap; a retentate spec is expensive, and
Profile local
Area
Collected permeate
Ranking well mixed < cross-flow < countercurrent; well mixed is conservative both ways
Permeate cascade polish the permeate, recycle stage-2 retentate — buys purity
Retentate cascade strip the retentate, recycle stage-2 permeate — buys recovery
Check the overall balance
Flooding velocity
Capacity factor
Flow parameter
Fair fit
Diameter
Total
Dry tray inches, ft/s
Holdup
Surface tension
Four failures jet flooding · downcomer flooding · weeping · dumping
Entrainment
Turndown sieve ≈ 2 · valve ≈ 4 · bubble cap ≈ 5
Packed columns far lower pressure drop; GPDC replaces Fair, HETP replaces efficiency
McCabe–Thiele, in order Draw the equilibrium curve and the Mark Draw the feed line from Draw the rectifying line from Join its intersection with the feed line to Step off: horizontal to the curve = a stage, vertical to the operating line = between stages. Switch lines at the first stage past the intersection. Count stages; the reboiler is one. Fenske–Underwood–Gilliland Pick the keys and write the two fractional recoveries. Fenske → Underwood 1 → Underwood 2 → Choose Kirkbride or the Fenske ratio → the feed stage. Absorber design Convert every stream to mole ratios Plot the equilibrium curve point by point from Mark Rotate the operating line until it first touches the curve → Take If the system is dilute, check with Kremser. Countercurrent extraction Plot From Stop when the raffinate is past Count the tie lines. Membrane sizing ladder Choose Balance + flux ratio → Flux law at those compositions → For a spec instead of a cut, bisect on Column diameter Fair fit at the chosen tray spacing → Correct for Operate at Net area = volumetric vapour / Envelope and lever rule x, y T x y z vapour liquid L/V = (y−z)/(z−x) McCabe–Thiele x y x D z F x B rectifying stripping feed The q-line fan x y q > 1 q = 1 0 < q < 1 q = 0 q < 0 z F Murphree efficiency x y equilibrium pseudo, E MV operating Absorber operating line X Y minimum L/G operating equilibrium top bottom Ternary construction plait Δ carrier x solute y tie line = 1 stage Isotherm shapes C q* favourable linear unfav. Shock vs spreading wave t C/C₀ shock spreading One membrane stage F, x F R, x r P, y p θ = P/F x F = θy p + (1−θ)x r purity ↓, recovery ↑ as θ rises Tray anatomy froth weir down- comer active area vapour h t = h d + h l + h σ Gas constant
At 25 °C
Pressure 1 atm = 760 mmHg = 101.325 kPa = 1.01325 bar; 1 bar = 750.06 mmHg = 75.006 cmHg
Molar volume 1 mol ideal gas = 22.414 L at STP (0 °C, 1 atm)
Barrer
GPU
Membrane flux
Surface tension 1 dyn/cm = 1 mN/m
Typical α benzene–toluene ≈ 2.4 · heptane–toluene ≈ 1.4 · p-/m-xylene ≈ 1.02
Tray efficiency
Reflux
Solvent rates absorber
Absorption optimum
Henry, water, 20 °C CO₂ 1420 · H₂S 483 · O₂ 40100 · N₂ 80400 atm
Module packing spiral-wound ≈ 1000 m²/m³ · hollow fibre ≈ 10 000 m²/m³
Separation energy distillation ≈ 50 % of US industrial energy; 10–15 % of national use