According to penetration theory, the mass transfer coefficient k_c depends on the molecular diffusivity D_AB as

k_c proportional to the square root of D_AB

k_c proportional to D_AB squared

The Chilton-Colburn analogy, f/2 = j_H = j_D, is practically useful chiefly because it

lets a measured friction factor estimate an unmeasured heat or mass transfer coefficient

eliminates the need to measure friction factor at all

lets a measured friction factor estimate an unmeasured heat or mass transfer coefficient

replaces the need for an equilibrium stage calculation

Murphree stage efficiency for a real distillation tray is defined as

the ratio of vapour to liquid flow on the tray

the number of real trays divided by the column diameter

the reflux ratio divided by the minimum reflux ratio

Which of the following are true about minimum and total reflux in a distillation column?

Minimum reflux corresponds to a pinch point where the operating and equilibrium lines touch.

Minimum reflux requires an infinite number of theoretical stages.

Total reflux gives zero net product draw but the fewest theoretical stages.

Minimum reflux corresponds to a pinch point where the operating and equilibrium lines touch.

Minimum reflux requires an infinite number of theoretical stages.

Total reflux gives zero net product draw but the fewest theoretical stages.

A real column is designed to operate exactly at minimum reflux for maximum efficiency.

The key difference between leaching and liquid-liquid extraction is that

leaching dissolves a soluble solute out of a SOLID with a liquid solvent, while extraction separates two miscible LIQUIDS using a third, immiscible solvent

leaching separates two immiscible liquids, extraction separates a solid and a liquid

leaching dissolves a soluble solute out of a SOLID with a liquid solvent, while extraction separates two miscible LIQUIDS using a third, immiscible solvent

leaching requires no solvent at all

extraction is only used for gases

In batch drying of a wet solid, the constant-rate period is limited by

external heat and mass transfer conditions at the surface, since surface moisture is still freely available

the rate of moisture diffusion from inside the solid to its surface

external heat and mass transfer conditions at the surface, since surface moisture is still freely available

the adsorption isotherm of the solid

Reverse osmosis for desalination requires operating pressure to exceed

the feed water's own osmotic pressure

the feed water's vapour pressure

the feed water's own osmotic pressure

According to the Chilton-Colburn analogy, if a system's friction factor f is doubled while flow conditions otherwise stay comparable, the Colburn j-factors for heat and mass transfer would be expected to

also roughly double, since f/2 = j_H = j_D

stay exactly the same regardless of f

fall to half their original value

become undefined once f changes

Diffusion, Mass Transfer Coefficients and Separation Operations

Section 6 of the GATE Chemical Engineering (CH) paper, the discipline that separates what reaction and mixing put together. Start with Fick's laws and molecular diffusion, the mass-transfer analogue of Fourier's law; then the three theories (film, penetration, surface-renewal) that turn a diffusion coefficient into a usable mass transfer coefficient, and the analogies that relate it back to momentum and heat transfer; then stage-wise and continuous contacting, stage efficiency and the HTU/NTU framework that ties the two together; and finally the separation operations themselves — distillation, absorption, leaching, extraction, drying, humidification, adsorption and membranes — each a different application of the same underlying driving force.

1. Fick's laws, diffusion, mass transfer coefficients and analogies

Fick's first law, N_A = -D_AB(dC_A/dz) (plus a bulk-flow term when net molar flux is not zero), is the mass-transfer counterpart of Fourier's law, with diffusivity D_AB playing the role thermal conductivity plays for heat. Molecular diffusion in a stagnant or laminar fluid is slow and purely concentration-gradient driven; in a real process stream it is almost always augmented by convection, and the practical quantity engineers actually use is a mass transfer coefficient k_c (or k_y, k_x depending on the driving-force units chosen), defined by N_A = k_c(C_A,interface - C_A,bulk), exactly the way h was defined for convective heat transfer.

Three theories of the mass transfer coefficient
Theory	Physical picture	k_c depends on D_AB as
Film theory	A stagnant film of fixed thickness carries all the resistance	k_c proportional to D_AB (first power)
Penetration theory	Fluid elements are exposed to the interface for a short, fixed contact time then replaced	k_c proportional to square root of D_AB
Surface-renewal theory	Contact time itself has a distribution (a renewal rate), not a single fixed value	k_c also proportional to square root of D_AB, generalising penetration theory

🎯 The Chilton-Colburn analogy links three transfer coefficients through the same friction factor

Momentum, heat and mass transfer share the same underlying eddy-transport mechanism in turbulent flow, which is why the Chilton-Colburn analogy relates the friction factor f to the Colburn j-factors for heat (j_H) and mass (j_D) transfer as f/2 = j_H = j_D. This lets a measured pressure drop (which gives f) estimate an UNMEASURED heat or mass transfer coefficient directly, without a separate correlation — the single most practically useful shortcut in this section.

2. Stage-wise and continuous contacting, stage efficiency, HTU and NTU

An IDEAL (equilibrium, or theoretical) stage is one where the two leaving streams are in complete equilibrium with each other — the definition used to count the number of ideal stages a separation needs. A real stage never reaches this, so Murphree stage efficiency (actual composition change achieved, divided by the change an ideal stage would have achieved) corrects the ideal-stage count up to the actual number of real trays or stages a column needs; overall column efficiency (ideal stages needed divided by actual stages provided) is the single number most often quoted for a whole column.

A packed column has no discrete stages at all — mass transfer happens continuously along its height — so it is instead characterised by the height of a transfer unit (HTU, the packing height that achieves one 'unit' of concentration change) and the number of transfer units (NTU, essentially the total concentration change divided by the average driving force across the column); total packed height = HTU x NTU. This directly parallels the ideal-stage count for a tray column: NTU plays the role stage count plays, and HTU plays the role of 'how much column height one ideal stage is worth'.

3. Distillation and absorption

Distillation separates a liquid mixture by exploiting differences in volatility between components, using vapour-liquid equilibrium data (relative volatility) rather than a solvent; the McCabe-Thiele graphical method steps ideal stages between the equilibrium curve and two operating lines (rectifying and stripping, meeting at the q-line set by feed condition) to find the number of theoretical stages a given reflux ratio and separation demand. Absorption instead uses a liquid SOLVENT to selectively dissolve a component out of a gas stream, so it is analysed on an operating line (from an overall mass balance across the column) against an equilibrium curve, with the minimum liquid-to-gas ratio set by where the two lines would touch — exactly parallel to distillation's minimum reflux.

⚠️ Minimum reflux gives infinite stages, not zero stages

It is a common slip to think 'minimum' reflux ratio means the column needs the FEWEST stages — it is the opposite. At minimum reflux, the operating and equilibrium lines touch (a pinch point), the driving force there goes to zero, and an infinite number of stages would be needed to actually achieve the separation; total reflux (all overhead vapour condensed and returned) is the other extreme, needing the MINIMUM number of stages but doing zero net separation of product. Every real column operates between these two limits.

4. Leaching, extraction, drying, humidification, adsorption and membranes

Leaching dissolves a soluble solute out of a solid using a liquid solvent (sugar from cane, oil from seeds); liquid-liquid extraction instead separates two miscible liquids by adding a third, immiscible solvent that preferentially dissolves the solute — both are governed by a distribution coefficient (how the solute splits between the two phases) and analysed with the same triangular or right-angled phase-diagram tools. Drying removes moisture from a solid by evaporation into a gas, and its own rate curve has two distinct periods: a constant-rate period (surface moisture evaporates as fast as heat arrives, and the rate depends only on external conditions) followed by a falling-rate period (moisture must diffuse from inside the solid to the surface, and the rate depends on the solid's own internal structure).

Humidification and dehumidification transfer both mass AND heat simultaneously between a gas (usually air) and a liquid (usually water), which is why a cooling tower and an adiabatic humidifier both approach the WET-BULB temperature line on a psychrometric chart rather than a simple mass or energy balance alone.
Adsorption removes a solute from a gas or liquid onto the surface of a solid adsorbent, described by an equilibrium isotherm (Langmuir, Freundlich or BET) at fixed temperature, and a fixed-bed adsorber runs until its outlet concentration rises past a set limit — the breakthrough point — after which the bed must be regenerated or replaced.
Membrane separation drives a species across a semi-permeable barrier by a concentration, pressure or electrical-potential difference (reverse osmosis for desalination is pressure-driven, needing pressure above the feed's own osmotic pressure) rather than by phase change or a second solvent, which is exactly why it can be far less energy-intensive than distillation for a dilute separation.

Key takeaways

Fick's first law is the diffusion analogue of Fourier's law; the practical mass transfer coefficient k_c, defined the way h was for heat, is what film, penetration and surface-renewal theories each try to predict from D_AB.
The Chilton-Colburn analogy, f/2 = j_H = j_D, lets a measured friction factor estimate an unmeasured heat or mass transfer coefficient directly.
Murphree stage efficiency corrects an ideal-stage count to real trays; HTU x NTU plays the same role for a packed column, which has no discrete stages at all.
Minimum reflux in distillation needs infinite stages (a pinch point of zero driving force); total reflux needs the fewest stages but zero net product separation — every real column sits between the two.
Drying has a constant-rate period limited by external heat/mass transfer and a falling-rate period limited by internal moisture diffusion; membrane separation drives species by a concentration, pressure or potential gradient rather than phase change.

Practice questions (11)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Fick's first law, N_A = -D_AB*(dC_A/dz), is the mass-transfer analogue of
1. Newton's law of cooling
2. Fourier's law of conduction
3. Stefan-Boltzmann's law
4. Bernoulli's equation
Show answer
Answer: B — Fourier's law of conduction
Both laws state that a flux is proportional to a negative concentration (or temperature) gradient, with diffusivity D_AB playing the role thermal conductivity k plays in Fourier's law — the same mathematical form describing a different transported quantity.
According to penetration theory, the mass transfer coefficient k_c depends on the molecular diffusivity D_AB as
1. k_c proportional to D_AB
2. k_c proportional to the square root of D_AB
3. k_c proportional to D_AB squared
4. k_c independent of D_AB
Show answer
Answer: B — k_c proportional to the square root of D_AB
Penetration theory models each fluid element as being exposed to the interface for a short, fixed contact time before being replaced by fresh fluid, which mathematically gives k_c proportional to sqrt(D_AB/contact time) — a square-root dependence, unlike film theory's direct proportionality.
The Chilton-Colburn analogy, f/2 = j_H = j_D, is practically useful chiefly because it
1. eliminates the need to measure friction factor at all
2. lets a measured friction factor estimate an unmeasured heat or mass transfer coefficient
3. applies only to laminar flow
4. replaces the need for an equilibrium stage calculation
Show answer
Answer: B — lets a measured friction factor estimate an unmeasured heat or mass transfer coefficient
Because momentum, heat and mass transfer share the same turbulent eddy-transport mechanism, a friction factor obtained from a pressure-drop measurement can, through this analogy, directly estimate the Colburn j-factors and hence the heat or mass transfer coefficients, without a separate correlation for each.
Murphree stage efficiency for a real distillation tray is defined as
1. the ratio of vapour to liquid flow on the tray
2. actual composition change achieved, divided by the change an ideal stage would achieve
3. the number of real trays divided by the column diameter
4. the reflux ratio divided by the minimum reflux ratio
Show answer
Answer: B — actual composition change achieved, divided by the change an ideal stage would achieve
Murphree efficiency directly compares what a real tray actually achieves to the equilibrium (ideal-stage) benchmark, and is exactly the correction factor used to go from a theoretical-stage count to the actual number of real trays a column must be built with.
A packed column's total packing height is computed as
1. HTU + NTU
2. HTU x NTU
3. NTU / HTU
4. HTU alone, independent of NTU
Show answer
Answer: B — HTU x NTU
HTU is the packing height equivalent to one transfer unit and NTU is how many transfer units the separation actually needs, so total packed height is their product — exactly parallel to (ideal stages) x (height per stage) for a tray column.
Which of the following are true about minimum and total reflux in a distillation column?
1. Minimum reflux corresponds to a pinch point where the operating and equilibrium lines touch.
2. Minimum reflux requires an infinite number of theoretical stages.
3. Total reflux gives zero net product draw but the fewest theoretical stages.
4. A real column is designed to operate exactly at minimum reflux for maximum efficiency.
Show answer
Answer: A — Minimum reflux corresponds to a pinch point where the operating and equilibrium lines touch.; B — Minimum reflux requires an infinite number of theoretical stages.; C — Total reflux gives zero net product draw but the fewest theoretical stages.
The fourth statement is false: operating exactly at minimum reflux would need an infinite column, so real columns instead run at some multiple (commonly 1.2 to 1.5 times) of minimum reflux, trading a finite, economical stage count for a manageable reflux ratio. The first three describe the two limiting cases correctly.
The key difference between leaching and liquid-liquid extraction is that
1. leaching separates two immiscible liquids, extraction separates a solid and a liquid
2. leaching dissolves a soluble solute out of a SOLID with a liquid solvent, while extraction separates two miscible LIQUIDS using a third, immiscible solvent
3. leaching requires no solvent at all
4. extraction is only used for gases
Show answer
Answer: B — leaching dissolves a soluble solute out of a SOLID with a liquid solvent, while extraction separates two miscible LIQUIDS using a third, immiscible solvent
Leaching's feed is a solid containing a soluble component that a liquid solvent dissolves out (sugar from cane, oil from seed); extraction's feed is already a liquid mixture, and a third immiscible solvent is added to preferentially pull one component into a separate liquid phase — different feed phases entirely.
In batch drying of a wet solid, the constant-rate period is limited by
1. the rate of moisture diffusion from inside the solid to its surface
2. external heat and mass transfer conditions at the surface, since surface moisture is still freely available
3. the solid's melting point
4. the adsorption isotherm of the solid
Show answer
Answer: B — external heat and mass transfer conditions at the surface, since surface moisture is still freely available
As long as free surface moisture is available, evaporation proceeds as fast as heat can arrive and vapour can leave, so the rate is set purely by external conditions (air temperature, humidity, velocity); once surface moisture is depleted, the rate falls and is instead limited by internal diffusion of moisture to the surface — the falling-rate period.
Reverse osmosis for desalination requires operating pressure to exceed
1. the feed water's vapour pressure
2. the feed water's own osmotic pressure
3. atmospheric pressure only
4. the membrane's melting point
Show answer
Answer: B — the feed water's own osmotic pressure
Natural osmosis drives water from lower to higher solute concentration; to reverse that flow and push pure water OUT of the concentrated (saline) side, the applied pressure on the feed side must exceed the feed's own osmotic pressure, forcing water against its natural direction across the membrane.
A packed absorption column needs 6 theoretical transfer units (NTU = 6) and the packing's height equivalent to one transfer unit (HTU) is 0.8 m. What total packing height, in metres, is required (height = HTU x NTU)?
Numerical answer — type the value.
Show answer
Answer: 4.8
Total packing height = HTU x NTU = 0.8 m x 6 = 4.8 m. This is the packed-column analogue of multiplying an ideal-stage count by the height per stage in a tray column.
According to the Chilton-Colburn analogy, if a system's friction factor f is doubled while flow conditions otherwise stay comparable, the Colburn j-factors for heat and mass transfer would be expected to
1. also roughly double, since f/2 = j_H = j_D
2. stay exactly the same regardless of f
3. fall to half their original value
4. become undefined once f changes
Show answer
Answer: A — also roughly double, since f/2 = j_H = j_D
The analogy states f/2 = j_H = j_D as a direct proportionality, so doubling f (holding the analogy's assumptions valid) implies both j_H and j_D roughly double as well — this is exactly what lets a measured pressure-drop change be used to estimate how an unmeasured heat or mass transfer coefficient would shift.