Formula sheet

Separation Processes and Distillation Formula Sheet

Distillation and separation formulas for VLE, relative volatility, balances, reflux, stages, duties, flooding, pressure drop, absorption, extraction, membranes, and validation.

Branch: Chemical Engineering
Content: Formula sheet
Updated: Jun 23, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass equations used in separation processes and distillation engineering. It is intended for screening, design review, troubleshooting, operating-window checks, and preparation before more detailed simulation or vendor design.

Use these equations with a stated basis. Distillation and separation calculations are sensitive to composition units, thermodynamic model, feed condition, pressure, temperature, tray or packing efficiency, fouling, hydraulics, utility limits, control response, and validation data. A formula that closes a mass balance does not prove that a column, absorber, extractor, membrane, filter, or dryer can operate safely at the proposed condition.

Symbols and Conventions

Symbol	Meaning	Common units
$F$	feed molar flow rate	$\text{kmol/h}$
$D$	distillate molar flow rate	$\text{kmol/h}$
$B$	bottoms molar flow rate	$\text{kmol/h}$
$z_i$	feed mole fraction of component $i$	dimensionless
$x_i$	liquid mole fraction of component $i$	dimensionless
$y_i$	vapor mole fraction of component $i$	dimensionless
$K_i$	vapor-liquid equilibrium ratio	dimensionless
$\alpha_{LK/HK}$	relative volatility of light key to heavy key	dimensionless
$R$	reflux ratio, $L/D$	dimensionless
$L$	internal liquid reflux flow	$\text{kmol/h}$
$V$	internal vapor flow	$\text{kmol/h}$
$N$	number of ideal stages	dimensionless
$E$	tray, stage, or overall efficiency	dimensionless
$\lambda$	latent heat of vaporization or condensation	$\text{MJ/kmol}$
$\Delta P$	pressure drop	$\text{kPa}$
$J$	membrane flux	volume per area per time

State whether compositions are mole fractions, mass fractions, volume fractions, ppm by volume, ppm by mass, or concentrations. Most distillation-stage formulas assume mole fractions and vapor-liquid equilibrium.

Total and Component Balances

For a steady separator without reaction:

\displaystyle F=\sum O_j

where $O_j$ are outlet stream flow rates.

For component $i$ :

\displaystyle Fz_i=\sum O_jx_{i,j}

For a binary distillation column:

F=D+B

Fz_F=Dx_D+Bx_B

Solving for distillate:

\displaystyle D=\frac{F(z_F-x_B)}{x_D-x_B}

Bottoms:

B=F-D

Light-key recovery to distillate:

\displaystyle \eta_{LK,D}=\frac{Dx_D}{Fz_F}

Worked Check

For:

F=120\ \text{kmol/h},\quad z_F=0.42,\quad x_D=0.94,\quad x_B=0.06

the distillate rate is:

\displaystyle D=\frac{120(0.42-0.06)}{0.94-0.06}=49.1\ \text{kmol/h}

Bottoms:

B=120-49.1=70.9\ \text{kmol/h}

Light-key recovery:

\displaystyle \eta_{LK,D}=\frac{49.1(0.94)}{120(0.42)}=0.916=91.6\%

The balance is physically consistent, but it says nothing yet about equilibrium stages, reflux, duty, pressure drop, or hydraulic capacity.

Vapor-Liquid Equilibrium Ratio

The equilibrium ratio or K-value is:

\displaystyle K_i=\frac{y_i}{x_i}

so:

y_i=K_ix_i

For an ideal vapor phase and ideal liquid solution at low to moderate pressure:

\displaystyle K_i\approx\frac{P_i^{sat}}{P}

For a nonideal liquid with activity coefficient $\gamma_i$ :

\displaystyle K_i\approx\frac{\gamma_iP_i^{sat}}{P}

These expressions are screening relationships. Use an appropriate thermodynamic model for azeotropic, polar, associating, electrolyte, high-pressure, reactive, or strongly nonideal systems.

Relative Volatility

Relative volatility of light key $LK$ to heavy key $HK$ is:

\displaystyle \alpha_{LK/HK}=\frac{K_{LK}}{K_{HK}}

Equivalently:

\displaystyle \alpha_{LK/HK}=\frac{y_{LK}/x_{LK}}{y_{HK}/x_{HK}}

For a binary with constant relative volatility:

\displaystyle y=\frac{\alpha x}{1+(\alpha-1)x}

where $x$ and $y$ refer to the light component.

Worked Check

If:

\alpha=2.4,\quad x=0.30

then:

\displaystyle y=\frac{2.4(0.30)}{1+(2.4-1)(0.30)}

\displaystyle y=\frac{0.72}{1.42}=0.507

The vapor is enriched in the light component. If $\alpha$ approaches $1$ , distillation becomes difficult because vapor and liquid compositions become similar.

Reflux Ratio and Internal Traffic

Reflux ratio:

\displaystyle R=\frac{L}{D}

Reflux flow:

L=RD

For a first-pass total-condenser, constant-molar-overflow estimate above the feed:

V\approx L+D=(R+1)D

Worked Check

Using the distillate rate from the balance example:

D=49.1\ \text{kmol/h}

and:

R=1.8

the reflux flow is:

L=1.8(49.1)=88.4\ \text{kmol/h}

Overhead vapor condensed:

V=(1.8+1)(49.1)=137.5\ \text{kmol/h}

Higher reflux can improve separation, but it also increases vapor traffic, liquid traffic, condenser duty, reboiler duty, pressure drop, flooding risk, and utility load.

Minimum Stages: Fenske Equation

For a binary or key-component split at total reflux:

\displaystyle N_{min}=\frac{\ln\left[\left(\frac{x_{D,LK}}{x_{D,HK}}\right)\left(\frac{x_{B,HK}}{x_{B,LK}}\right)\right]}{\ln(\alpha_{LK/HK})}

This gives the minimum number of ideal stages at total reflux. It does not include finite reflux, tray efficiency, pressure drop, feed condition, nonideal thermodynamics, or hydraulic constraints.

Worked Check

For:

x_{D,LK}=0.94,\quad x_{D,HK}=0.06,\quad x_{B,LK}=0.06,\quad x_{B,HK}=0.94

and:

\alpha_{LK/HK}=2.4

then:

\displaystyle N_{min}=\frac{\ln\left[\left(\frac{0.94}{0.06}\right)\left(\frac{0.94}{0.06}\right)\right]}{\ln(2.4)}

\displaystyle N_{min}=\frac{\ln(245.4)}{0.875}=6.29

At least about seven ideal stages are implied at total reflux. Operating at finite reflux requires more stages.

Minimum Reflux: Underwood Screening

For multicomponent distillation with constant relative volatilities, Underwood screening uses a root $\theta$ satisfying:

\displaystyle \sum_i \frac{qz_i}{\alpha_i-\theta}=1

Then:

\displaystyle R_{min}+1=\sum_i \frac{\alpha_ix_{D,i}}{\alpha_i-\theta}

where $q$ is the feed thermal-condition parameter. The root must lie between the relative volatilities of the key components for the relevant split.

Engineering Use

Underwood calculations are useful for screening, but they are easy to misuse. They assume a simplified thermodynamic basis and sharp split structure. For nonideal mixtures, side draws, azeotropes, close boiling systems, heat-integrated columns, or reactive systems, use rigorous simulation or verified design methods.

Operating Stage Count and Efficiency

If $N_{theoretical}$ ideal stages are required and the overall efficiency is $E_o$ :

\displaystyle N_{actual}\approx\frac{N_{theoretical}}{E_o}

For tray-by-tray review, efficiency may vary with location, composition, loading, foaming, weeping, entrainment, fouling, and mass-transfer rate. Do not apply a single efficiency value without checking its basis.

Worked Check

If a simulation estimates:

N_{theoretical}=12

and the overall tray efficiency is:

E_o=0.55

then:

\displaystyle N_{actual}\approx\frac{12}{0.55}=21.8

Use about $22$ actual trays before adding design allowances, feed-stage review, maintenance considerations, and vendor-specific internals checks.

Packing Height

For packed columns, a common structure is:

H=N_{OG}H_{OG}

where $N_{OG}$ is the number of overall gas-phase transfer units and $H_{OG}$ is the height of an overall gas-phase transfer unit.

For absorption with dilute gas and equilibrium relationship $y^*=mx$ :

\displaystyle N_{OG}=\int_{y_2}^{y_1}\frac{dy}{y-y^*}

This integral depends on the operating line, equilibrium line, solvent rate, temperature, pressure, and mass-transfer basis.

Reboiler and Condenser Duty

For a first-pass vaporization or condensation estimate:

\dot{Q}\approx V\lambda

where $V$ is molar vapor flow and $\lambda$ is latent heat per mole.

Include sensible heat when feed, reflux, bottoms, distillate, or utility streams change temperature:

\dot{Q}_{sens}=\dot{m}C_p(T_{out}-T_{in})

Heat-exchanger area screening:

\dot{Q}=UA\Delta T_{lm}

Worked Check

Using:

V=137.5\ \text{kmol/h},\quad \lambda=31\ \text{MJ/kmol}

the vaporization duty is:

\dot{Q}=137.5(31)=4263\ \text{MJ/h}

Convert to megawatts:

\displaystyle \dot{Q}=\frac{4263}{3600}=1.18\ \text{MW}

This duty should be checked against reboiler area, steam pressure, condensate removal, condenser area, cooling-water flow, fouling, turndown, relief cases, and temperature-control stability.

Cooling-Water Requirement

Cooling-water mass flow for condenser duty is:

\displaystyle \dot{m}_{cw}=\frac{\dot{Q}}{C_p\Delta T}

For water:

C_p\approx4180\ \text{J/(kg K)}

Worked Check

If:

\dot{Q}=1.18\ \text{MW},\quad \Delta T=10\ \text{K}

then:

\displaystyle \dot{m}_{cw}=\frac{1.18\times10^6}{4180(10)}=28.2\ \text{kg/s}

With density near $1000\ \text{kg/m}^3$ :

Q_{cw}=28.2\ \text{L/s}=101.5\ \text{m}^3/\text{h}

The result should be compared with cooling-water header pressure, return-temperature limit, cooling tower capacity, fouling, seasonal water temperature, and plant utility sharing.

Hydraulic Loading and Flooding Screen

Fraction of validated flooding vapor traffic:

\displaystyle f_{flood}=\frac{V_{op}}{V_{flood}}

For sustained operation, many screening reviews target operation below a selected fraction of flooding, such as:

f_{flood}\leq0.80\text{ to }0.85

depending on service, internals, uncertainty, foaming, pressure stability, control response, and consequence of entrainment.

For a velocity-based screen, the Souders-Brown flooding velocity is often written:

\displaystyle u_f=C\sqrt{\frac{\rho_L-\rho_V}{\rho_V}}

where $C$ depends on internals, spacing, surface tension, service and allowable entrainment.

Operating velocity fraction:

\displaystyle f_u=\frac{u_{op}}{u_f}

Worked Check

Use:

C=0.11\ \text{m/s},\quad \rho_L=720\ \text{kg/m}^3,\quad \rho_V=2.1\ \text{kg/m}^3

Then:

\displaystyle u_f=0.11\sqrt{\frac{720-2.1}{2.1}}

u_f=0.11(18.49)=2.03\ \text{m/s}

If operating vapor velocity is:

u_{op}=1.55\ \text{m/s}

then:

\displaystyle f_u=\frac{1.55}{2.03}=0.76

The velocity screen is below flooding, but vendor data and measured pressure drop are still required before approving operation.

Pressure Drop

Column pressure drop per tray:

\displaystyle \Delta P_{tray}=\frac{\Delta P_{column}}{N_{active}}

Pressure-drop increase factor:

\displaystyle I_{\Delta P}=\frac{\Delta P_{new}}{\Delta P_{baseline}}

Worked Check

If a $24$ tray column rises from:

\Delta P_{baseline}=18\ \text{kPa}

to:

\Delta P_{new}=27\ \text{kPa}

then:

\displaystyle I_{\Delta P}=\frac{27}{18}=1.50

A $50\%$ pressure-drop increase during a rate or reflux change is operational evidence. It should be reviewed with vapor traffic, foaming, entrainment, flooding indicators, instrument health, and product quality.

Absorption and Stripping Removal

Removal efficiency:

\displaystyle \eta=\frac{C_{in}-C_{out}}{C_{in}}

Contaminant load removed from a gas stream:

\dot{n}_{removed}=Q_G(C_{in}-C_{out})

when $Q_G$ and concentrations use compatible molar or normal-volume units.

For linear equilibrium:

y^*=mx

Absorption becomes difficult as the operating line approaches the equilibrium line because the mass-transfer driving force decreases.

Worked Check

For:

C_{in}=1200\ \text{ppm},\quad C_{out}=80\ \text{ppm}

removal efficiency is:

\displaystyle \eta=\frac{1200-80}{1200}=0.933=93.3\%

If:

Q_G=5000\ \text{Nm}^3/\text{h}

then contaminant removed on a normal-volume basis is:

Q_{removed}=5000(1200-80)\times10^{-6}=5.6\ \text{Nm}^3/\text{h}

The solvent, regeneration system, emissions control, corrosion allowance, foaming margin, and waste stream must be checked separately.

Liquid-Liquid Extraction

Distribution coefficient:

\displaystyle K_D=\frac{C_{solute,extract}}{C_{solute,raffinate}}

For a simplified single-stage extraction with immiscible carrier phases and dilute solute, the fraction extracted can be screened as:

\displaystyle E=\frac{K_DS}{F+K_DS}

where $S$ is solvent flow and $F$ is feed carrier flow on a compatible basis.

Worked Check

If:

K_D=3.0,\quad S=50\ \text{kg/h},\quad F=100\ \text{kg/h}

then:

\displaystyle E=\frac{3.0(50)}{100+3.0(50)}=\frac{150}{250}=0.60

The single-stage screen predicts $60\%$ extraction. The real design must check selectivity, phase disengagement, solvent recovery, emulsion tendency, density difference, viscosity, toxicity, flammability, residual solvent limits, and waste handling.

Membrane Separation Checks

Membrane flux:

\displaystyle J=\frac{Q_p}{A}

Recovery:

\displaystyle R=\frac{Q_p}{Q_f}

Rejection:

\displaystyle \mathcal{R}=1-\frac{C_p}{C_f}

Transmembrane pressure for a simple pressure-driven membrane:

\displaystyle TMP=\frac{P_{feed}+P_{concentrate}}{2}-P_{permeate}

Worked Check

For:

Q_f=10\ \text{m}^3/\text{h},\quad Q_p=7\ \text{m}^3/\text{h},\quad A=140\ \text{m}^2

recovery is:

\displaystyle R=\frac{7}{10}=70\%

Flux:

\displaystyle J=\frac{7}{140}=0.050\ \text{m}^3/(\text{m}^2\text{h})

If:

C_f=500\ \text{mg/L},\quad C_p=25\ \text{mg/L}

rejection is:

\displaystyle \mathcal{R}=1-\frac{25}{500}=0.95=95\%

These checks do not prove long-term operation. Fouling, scaling, cleaning, concentration polarization, temperature, pressure drop, membrane aging, and feed pretreatment often govern performance.

Filtration Rate Screen

Average filtrate flux:

\displaystyle J_f=\frac{V_f}{A t}

Cake-filtration resistance often causes flux to decline with time. A constant-flux or clean-water test is not enough when real solids are compressible, sticky, broad in particle size, or sensitive to chemistry.

Dryer Solvent Removal

Solvent mass removed:

m_{removed}=m_{initial}-m_{final}

Drying rate over a time interval:

\displaystyle \dot{m}_{dry}=\frac{m_{removed}}{\Delta t}

Solvent emissions or recovery load should close with condenser, adsorber, scrubber, ventilation, or waste records. Drying is a separation and a safety problem when flammable solvent, dust, high temperature, oxygen, or static electricity are credible.

Energy Intensity

Energy per distillate product:

\displaystyle e_D=\frac{\dot{Q}_R}{D}

Energy per feed:

\displaystyle e_F=\frac{\dot{Q}_R}{F}

These simple metrics are useful for comparing operating points, but they should not be separated from purity, recovery, utility temperature, pressure drop, and equipment limits.

Worked Check

If:

\dot{Q}_R=1.18\ \text{MW}=4248\ \text{MJ/h}

and:

D=49.1\ \text{kmol/h}

then:

\displaystyle e_D=\frac{4248}{49.1}=86.5\ \text{MJ/kmol distillate}

If reflux is increased, $e_D$ usually rises even if purity improves. The right operating point balances product quality, recovery, energy, capacity, and reliability.

Mass-Balance Closure

Mass-balance closure error:

\displaystyle \epsilon_m=\frac{\sum \dot{m}_{out}-\sum \dot{m}_{in}}{\sum \dot{m}_{in}}

Component closure error:

\displaystyle \epsilon_i=\frac{\sum \dot{n}_{i,out}-\sum \dot{n}_{i,in}}{\sum \dot{n}_{i,in}}

Worked Check

If a separator feed is:

\dot{m}_{in}=10{,}000\ \text{kg/h}

and measured outlets sum to:

\displaystyle \sum\dot{m}_{out}=9{,}930\ \text{kg/h}

then:

\displaystyle \epsilon_m=\frac{9930-10000}{10000}=-0.007=-0.7\%

The closure may be acceptable for an operating survey, but not necessarily for custody transfer, emissions reporting, or tight product-yield accounting. State the acceptance band.

Guarded Release Margin

For a measured margin $M_{nom}$ and combined uncertainty $u_c$ :

M_{guard}=M_{nom}-ku_c

Release condition:

M_{guard}\geq M_{required}

Worked Check

Suppose a column operates at:

f_{flood}=0.79

and the maximum allowed sustained fraction is:

f_{limit}=0.85

Nominal hydraulic margin:

M_{nom}=0.85-0.79=0.06

If combined uncertainty is:

u_c=0.02

and:

k=2

then:

M_{guard}=0.06-2(0.02)=0.02

The guarded margin is positive but small. The release should include monitoring, staged rate changes, and stop criteria.

Validation Data to Preserve

A separation calculation should leave an evidence trail:

feed, product, recycle, purge, waste, and utility flow bases;
composition method, sample location, analyzer calibration, and lab turnaround;
thermodynamic model and property data source;
pressure, temperature, reflux, duty, and control-loop data at the same timestamp basis;
tray, packing, membrane, filter, or exchanger basis;
pressure-drop, flooding, fouling, and cleaning evidence;
off-spec disposition and environmental stream records;
uncertainty assumptions and release margins.

Common Mistakes

Common mistakes include:

mixing mass fractions and mole fractions;
reporting purity without recovery;
applying constant relative volatility across a strongly nonideal composition range;
using Fenske minimum stages as an operating-stage count;
increasing reflux without checking flooding, pressure drop, reboiler duty, and condenser duty;
ignoring feed thermal condition and pressure effects;
accepting a membrane flux from clean-water data for a fouling feed;
treating a proxy temperature as composition evidence without validation;
closing the product balance while losing solvent, impurity, or waste mass elsewhere;
approving a new operating point without guarded margin and monitoring thresholds.

The strongest separation calculation is not the most detailed equation. It is the one that makes its basis visible, connects purity to recovery and capacity, checks utilities and hydraulics, and states which field evidence would prove that the operating window is real.

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