Sources and References

These notes synthesize material from the standard canon of separation-process textbooks and equivalent courses at peer institutions. The primary reference for topic coverage and worked-example style is P. C. Wankat, Separation Process Engineering (5th ed., Pearson). Supporting references include J. D. Seader, E. J. Henley, and D. K. Roper, Separation Process Principles (Wiley), C. J. King, Separation Processes (McGraw-Hill), and McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering (McGraw-Hill). Thermodynamic background draws on Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics. Comparable undergraduate treatments appear in MIT OpenCourseWare 10.213 and 10.40, UC Berkeley CBE 140, and the Cambridge Chemical Engineering Tripos Part IIA separation modules.

Chapter 1: The Equilibrium Stage Concept

Chemical products almost always leave a reactor as mixtures. The economics of a chemical process are frequently dominated not by the reaction step but by the downstream separations that purify the target molecule. The central organizing idea of this course is the equilibrium stage: a control volume in which two immiscible (or partially miscible) phases are contacted, allowed to approach thermodynamic equilibrium, and then separated. If a real contactor approximates this idealization, we can chain many of them together and analyze the resulting cascade with nothing more than component balances and equilibrium relations.

1.1 Phases, Contacting, and Separation Factors

A separation exploits a difference in the affinity of species for two coexisting phases. Distillation uses vapor and liquid; absorption and stripping use gas and liquid; liquid-liquid extraction uses two immiscible liquid phases; leaching and washing use a liquid solvent in contact with a solid. Regardless of which phases are in play, the figure of merit is the relative volatility or, more generally, the separation factor

\[ \alpha_{ij} = \frac{K_i}{K_j}, \qquad K_i = \frac{y_i}{x_i} \]

where \( K_i \) is the distribution (or K-value) of species \( i \) between the two phases. The larger \( \alpha_{ij} \) is, the easier it is to separate \( i \) from \( j \) in a single stage.

1.2 The Ideal Stage

For a stage with two inlet streams and two outlet streams in equilibrium, a mass balance on each component plus the phase-equilibrium relation closes the system. This is the kernel of every method introduced later in the course.

Chapter 2: Washing and Simple Contactors

The simplest equilibrium stage problem is washing: a solid (or slurry) carrying solute is contacted with wash liquid, and the solute partitions between the adhering liquid in the cake and the bulk wash. Assuming the solute concentration in the entrained liquor equals the concentration in the bulk wash leaving the stage (the equilibrium assumption), each stage yields a geometric reduction of solute carried by the solid. The fraction of solute remaining after \( N \) ideal washing stages with a wash ratio \( R \) (mass wash solvent per mass retained liquor) is

\[ \frac{s_N}{s_0} = \frac{1}{1 + R + R^2 + \cdots + R^N} \cdot (1-\epsilon) + \cdots \]

For countercurrent washing the analysis mirrors countercurrent absorption and gives a Kremser-like expression. Washing is a useful warm-up because it introduces the countercurrent flow pattern, operating lines, and the logic of cascading stages without requiring complex thermodynamics.

Chapter 3: Gas Absorption and Liquid Stripping

3.1 Physical Picture

In absorption a solute is transferred from a gas stream into a liquid solvent; in stripping the direction is reversed. Industrially important examples include CO\(_2\) absorption into amines, SO\(_2\) scrubbing, and air stripping of volatile organics from groundwater. The solvent is usually chosen to be nonvolatile and the carrier gas to be insoluble, so the flows of carrier gas \( G' \) and solvent \( L' \) can be taken as constant up the column. It is convenient to work in solute-free mole ratios

\[ Y = \frac{y}{1-y}, \qquad X = \frac{x}{1-x} \]

so that operating lines are exactly straight.

3.2 Single-Stage Analysis

For a single equilibrium stage with inlet gas \( (G', Y_{\text{in}}) \) and inlet liquid \( (L', X_{\text{in}}) \), a component balance on the solute and the equilibrium relation \( Y_{\text{out}} = f(X_{\text{out}}) \) form two equations in two unknowns. For dilute systems where Henry’s law applies, \( Y = m X \), and the solution is algebraic. The single-stage recovery is limited; to achieve high purity one must cascade stages.

3.3 Countercurrent Multi-Stage Absorbers and the Kremser Equation

A countercurrent cascade of \( N \) ideal stages with dilute, linear equilibrium \( Y^* = m X \) admits a closed-form analysis. Define the absorption factor

\[ A = \frac{L'}{m G'} \]

which measures the capacity of the liquid to hold solute relative to the tendency of the gas to retain it. Material balances around the top \( n \) stages give a straight operating line of slope \( L'/G' \), and combining it with the equilibrium line yields the classical Kremser equation for the fraction of solute unabsorbed:

\[ \frac{Y_{N+1} - Y_1}{Y_{N+1} - m X_0} = \frac{A^{N+1} - A}{A^{N+1} - 1} \]

The complementary form for stripping uses the stripping factor \( S = m G'/L' = 1/A \). The Kremser equation is one of the most useful quick-estimation tools in separations practice: given inlet and outlet compositions and the equilibrium slope, the number of ideal stages follows from a single logarithm. It is also exact in the limit of straight operating and equilibrium lines, and provides an excellent approximation whenever both are gently curved.

3.4 Graphical Construction

For nonlinear equilibria the cascade is solved graphically on an \( X\text{-}Y \) diagram. The operating line is a material balance around one end of the column; stages are stepped off between the operating line and the equilibrium curve. The minimum solvent rate \( (L'/G')_{\min} \) corresponds to the operating line that just touches the equilibrium curve (a pinch), requiring an infinite number of stages. A practical design operates at roughly \( 1.2 \) to \( 1.5 \) times the minimum.

Chapter 4: Vapor-Liquid Equilibrium

Before attacking distillation, we need a working model of VLE. The foundation is equality of fugacity for each species in each phase. For low-to-moderate pressures and nonpolar mixtures, this reduces to the familiar forms below.

4.1 Raoult’s Law and Ideal Mixtures

Mathematically,

\[ y_i P = x_i P_i^{\text{sat}}(T) \]

so that \( K_i = P_i^{\text{sat}}/P \) and the relative volatility \( \alpha_{ij} = P_i^{\text{sat}}/P_j^{\text{sat}} \) depends only on temperature. Ideal behavior is a reasonable first approximation for mixtures of isomers or adjacent members of a homologous series (hexane-heptane, benzene-toluene).

4.2 Henry’s Law and Dilute Solutions

For species present only in trace amounts, linearity still holds but with a different slope: \( p_i = H_i x_i \). Henry’s constant \( H_i \) is specific to the solute-solvent pair and depends strongly on temperature. This is the form used throughout absorption and stripping.

4.3 Nonideal Mixtures

Real mixtures of polar species or molecules of very different sizes depart from Raoult’s law. The departure is captured by an activity coefficient \( \gamma_i \):

\[ y_i P = x_i \gamma_i P_i^{\text{sat}} \]

When \( \gamma_i > 1 \) the mixture shows positive deviation, which can lead to a minimum-boiling azeotrope; negative deviation \( (\gamma_i < 1) \) can produce a maximum-boiling azeotrope. Classical activity-coefficient models (Margules, Van Laar, Wilson, NRTL, UNIQUAC) fit \( \gamma_i(x,T) \) to binary data; UNIFAC predicts them from group contributions. At an azeotrope the vapor and liquid compositions are identical and ordinary distillation cannot cross the azeotropic composition.

4.4 Phase Diagrams

For a binary mixture, the three most useful plots are the \( T\text{-}x\text{-}y \) diagram (temperature against composition at fixed pressure, showing the bubble and dew curves), the \( P\text{-}x\text{-}y \) diagram (its isothermal counterpart), and the \( y\text{-}x \) diagram (equilibrium vapor mole fraction versus liquid mole fraction). The \( y\text{-}x \) diagram is the stage for all graphical distillation methods.

Chapter 5: Flash Distillation

A flash drum is a single equilibrium stage: feed at high pressure is let down through a valve into a vessel where vapor and liquid separate. Let \( F \), \( V \), and \( L \) be feed, vapor, and liquid molar flows, with compositions \( z_i \), \( y_i \), and \( x_i \). Defining the vapor fraction \( \psi = V/F \), the overall balance \( F z_i = V y_i + L x_i \) combined with \( y_i = K_i x_i \) yields the Rachford-Rice equation:

\[ \sum_i \frac{z_i (K_i - 1)}{1 + \psi (K_i - 1)} = 0 \]

For a binary mixture with constant relative volatility, this collapses to an algebraic expression, and the flash can be solved graphically on the \( y\text{-}x \) diagram by intersecting the equilibrium curve with the operating line

\[ y = -\frac{1-\psi}{\psi} x + \frac{z}{\psi} \]

which passes through \( (z,z) \) with slope \( -L/V \). Flash distillation achieves only modest separations and is mostly useful as a preconditioning step or as the conceptual building block for columns.

Chapter 6: Equilibrium Cascades and Binary Distillation

6.1 Why Cascade

A single stage gives at most one equilibrium split. To obtain near-pure products, stages are cascaded countercurrently with reflux: part of the overhead vapor is condensed and returned to the column top, and part of the bottom liquid is reboiled and returned to the column bottom. The result is a column with a rectifying section (above the feed) and a stripping section (below the feed), each behaving like a countercurrent cascade.

6.2 McCabe-Thiele Construction

Under the constant molar overflow (CMO) assumption - equal molar latent heats, negligible heat of mixing, adiabatic operation - the liquid and vapor molar flows are constant within each section. Material balances then give straight operating lines. The rectifying operating line is

\[ y_{n+1} = \frac{R}{R+1}\, x_n + \frac{x_D}{R+1} \]

where \( R = L/D \) is the external reflux ratio and \( x_D \) is the distillate composition. The stripping operating line is

\[ y_{m+1} = \frac{\bar L}{\bar V}\, x_m - \frac{B\, x_B}{\bar V} \]

Stages are stepped off as right-angle staircases between the operating line and the equilibrium curve on a \( y\text{-}x \) plot, beginning at \( (x_D, x_D) \) on the \( y=x \) diagonal. The number of triangles required to reach \( x_B \) is the number of ideal stages; the reboiler counts as one.

6.3 The Feed Line (q-line)

The feed condition is characterized by the quality

\[ q = \frac{\text{mol saturated liquid produced by feed per mol feed}}{1} \]

so that \( q=1 \) is a saturated-liquid feed, \( q=0 \) is a saturated vapor, \( 01 \) is subcooled liquid, and \( q<0 \) is superheated vapor. The locus of intersections of the two operating lines, obtained by simultaneous solution with the overall mass balance, is the q-line

\[ y = \frac{q}{q-1}\, x - \frac{z_F}{q-1} \]

passing through \( (z_F, z_F) \) with slope \( q/(q-1) \). The optimal feed stage is the one where the step crosses the q-line, minimizing the number of stages.

6.4 Minimum and Total Reflux

Two limiting operating conditions bracket real designs. At total reflux \( (R \to \infty) \) the operating lines coincide with the diagonal \( y=x \) and the number of stages is the minimum. For constant relative volatility this is given by the Fenske equation

\[ N_{\min} = \frac{\ln\!\left[\left(\dfrac{x_D}{1-x_D}\right)\!\left(\dfrac{1-x_B}{x_B}\right)\right]}{\ln \alpha} \]

At minimum reflux the rectifying operating line just touches the equilibrium curve (or the q-line/equilibrium intersection for a pinch at the feed), giving an infinite number of stages. For a saturated-liquid feed and a pinch at the feed point,

\[ R_{\min} = \frac{x_D - y'}{y' - x'} \]

where \( (x', y') \) is the pinch point. Economic optimum operation typically lies at \( R = 1.1 \) to \( 1.5 \, R_{\min} \), reflecting the tradeoff between capital cost (more stages at low \( R \)) and operating cost (larger reboiler and condenser duties at high \( R \)).

6.5 Stage Efficiency

Real trays do not reach equilibrium. The Murphree tray efficiency for the vapor phase is

\[ E_{MV} = \frac{y_n - y_{n+1}}{y_n^* - y_{n+1}} \]

which shortens each step on the \( y\text{-}x \) diagram. An overall column efficiency \( E_o = N_{\text{ideal}}/N_{\text{actual}} \) is a blunt but convenient aggregate measure. Typical values for well-designed sieve trays run 60-80%.

Chapter 7: Multicomponent Distillation

With more than two components, graphical methods fail and rigorous solutions require stage-by-stage numerical algorithms (Lewis-Matheson, Thiele-Geddes, inside-out methods, or equation-based MESH solvers). For preliminary design the Fenske-Underwood-Gilliland (FUG) shortcut is still indispensable.

7.1 Key Components and Splits

Two components are designated as the light key (LK) and heavy key (HK), the pair the column is nominally designed to separate. Components lighter than the LK are distributed almost entirely to the distillate; components heavier than the HK go to the bottoms. Non-keys intermediate in volatility may be distributed.

7.2 Fenske’s Equation for \(N_{\min}\)

Applied to the keys at total reflux,

\[ N_{\min} = \frac{\ln\!\left[\dfrac{(x_{\text{LK}}/x_{\text{HK}})_D}{(x_{\text{LK}}/x_{\text{HK}})_B}\right]}{\ln \alpha_{\text{LK,HK}}} \]

with \( \alpha \) taken as a geometric mean over the column. Fenske also gives the distribution of non-key components at total reflux, useful as an initial estimate of their split.

7.3 Underwood’s Equations for \(R_{\min}\)

Underwood’s derivation assumes constant relative volatilities and constant molar overflow across the pinch. Solve

\[ \sum_i \frac{\alpha_i\, z_{i,F}}{\alpha_i - \theta} = 1 - q \]

for the root \( \theta \) lying between \( \alpha_{\text{LK}} \) and \( \alpha_{\text{HK}} \); then

\[ R_{\min} + 1 = \sum_i \frac{\alpha_i\, x_{i,D}}{\alpha_i - \theta} \]

7.4 Gilliland’s Correlation

Gilliland correlated the ratio \( (N - N_{\min})/(N+1) \) against \( (R - R_{\min})/(R+1) \) from many rigorous solutions; Molokanov’s analytical fit is typical

\[ \frac{N - N_{\min}}{N+1} = 1 - \exp\!\left[\frac{1+54.4 X}{11+117.2 X} \cdot \frac{X-1}{\sqrt{X}}\right], \quad X = \frac{R - R_{\min}}{R+1} \]

The Kirkbride equation then estimates the ratio of rectifying to stripping stages from the feed composition and split. Together, FUG gives a defensible first-pass design that is refined in a process simulator.

Chapter 8: Batch Distillation

Small-scale, specialty, and pharmaceutical production often use batch distillation, where feed is charged to a still pot, heated, and a sequence of product cuts is taken overhead. The simplest case, differential (Rayleigh) distillation, treats the pot as a single stage with no reflux:

\[ \ln \frac{L_0}{L} = \int_{x}^{x_0} \frac{d x}{y^* - x} \]

relating the remaining charge \( L \) to the still composition \( x \). With reflux and trays, operation may be run at constant reflux ratio (distillate purity declines through the batch) or at variable reflux chosen to hold distillate purity constant, with \( R \) rising as the pot depletes. Both modes are analyzed by stepping McCabe-Thiele diagrams over successive snapshots in time.

Chapter 9: Liquid-Liquid Equilibria and Extraction

When distillation is uneconomic - because relative volatilities are low, because the mixture is heat-sensitive, or because an azeotrope blocks separation - liquid-liquid extraction (LLE) offers an alternative. A solvent partially miscible with the feed is contacted with it; the solute of interest partitions into the solvent.

9.1 Ternary Phase Diagrams

The thermodynamics of three-component LLE is displayed on an equilateral triangular diagram, with each vertex a pure component. The interior of the diagram is divided by a binodal curve into a single-phase region and a two-phase region; tie lines inside the two-phase region connect equilibrium raffinate and extract compositions. As the solute concentration rises toward the plait point, the tie lines shrink to zero length and the two phases become identical. Type I systems have one partially miscible pair; Type II systems have two.

9.2 Single-Stage Extraction

Given a feed of known composition and a chosen solvent-to-feed ratio, the mixing point of feed and solvent lies inside the two-phase region. The tie line through the mixing point identifies the equilibrium raffinate and extract compositions, and the lever rule gives their relative amounts. This is the direct analog of a single flash stage on a \( T\text{-}x\text{-}y \) diagram.

9.3 Multi-Stage Countercurrent Extraction

Countercurrent cascades are analyzed either on the triangular diagram (the Hunter-Nash method, using an operating point located by extending feed/solvent and extract/raffinate balances) or, when the solvent and diluent are nearly immiscible, on a solute-only \( Y\text{-}X \) diagram identical in form to the absorption plot. In the latter case the Kremser equation applies with an extraction factor \( E = K\, S/F \) in place of the absorption factor, and minimum solvent corresponds to a pinch against the equilibrium curve.

9.4 Solvent Selection

A good extraction solvent has high capacity (large \( K \) for the solute), high selectivity (large \( K_{\text{solute}}/K_{\text{diluent}} \)), low mutual solubility with the feed, favorable density contrast and interfacial tension for phase separation, low volatility (or convenient volatility for solvent recovery by distillation), chemical stability, low toxicity, and low cost. These criteria often conflict; solvent selection is a multi-objective decision.

Chapter 10: Unifying Themes

Across all the operations in this course, a common conceptual scaffold recurs. Each problem is built from four ingredients: (i) a thermodynamic equilibrium relation describing which compositions can coexist in two contacted phases; (ii) component and overall material balances written over chosen control volumes; (iii) an idealized stage that imposes equilibrium between its outlet streams; and (iv) a contacting pattern - single-stage, crosscurrent, or countercurrent - that dictates how stages are chained. The Kremser equation, the McCabe-Thiele construction, the FUG shortcut, the Hunter-Nash method, and flash calculations are all specializations of this scaffold to particular geometries and equilibrium forms.

Two lessons deserve emphasis. First, countercurrent contacting is almost always preferable to cocurrent or crosscurrent contacting because it permits the outlet of one phase to approach equilibrium with the inlet of the other, exploiting the full driving force along the cascade; this is why countercurrent columns achieve high recoveries that would be thermodynamically impossible in a single stage. Second, limiting conditions bracket the feasible design space: minimum reflux (or minimum solvent) sets a lower bound on energy/material consumption at the price of infinite stages, while minimum stages (at total reflux) set a lower bound on equipment size at the price of infinite energy. Real designs live between these extremes, with economics deciding the specific balance.

These ideas scale beyond the stagewise model. Continuous contactors like packed columns are analyzed with the analogous rate-based concepts of height of a transfer unit (HTU) and number of transfer units (NTU), where the latter plays the role of the number of equilibrium stages and is related to it by simple expressions when the operating and equilibrium lines are straight. Mastery of the equilibrium-stage picture is therefore a prerequisite for almost every more advanced topic in separations, from rigorous tray-by-tray simulators to reactive distillation, membrane cascades, and chromatography.