9.1 Mass Transfer and Diffusion Fundamentals

9.1 Mass Transfer and Diffusion Fundamentals

Quick Answer: Species move by diffusion down a gradient (Fick: (J_A = -c D_{AB}\nabla y_A) or similar) and by convection with the bulk flow. Engineering rates use mass-transfer coefficients and driving forces (\Delta y), (\Delta c), or (\Delta p). Film and two-film models explain gas–liquid contact. Flux is zero at equilibrium.

Domain C (transport phenomena: fluid, heat, and mass) carries about 20% of this guide's planning allocation. Chapters 7–8 covered momentum and heat. This chapter treats mass transfer—how components move between phases and through mixtures—and the major separation operations used in Qatar’s gas processing, refining, LNG, water treatment, and petrochemical plants: distillation, absorption/stripping, and extraction.

Equilibrium (Chapter 6) tells where a system wants to go. Mass transfer tells how fast it gets there and how large equipment must be. On exam items, confuse the two and you pick equilibrium answers for rate questions (or the reverse).

Molecular Diffusion and Fick’s Law

Molecular diffusion is the random molecular motion that produces a net flux of species from regions of higher concentration (more precisely, higher chemical potential) to lower. For a binary pair A–B under isothermal, isobaric conditions, Fick’s first law in one dimension is often written:

[ J_A = -D_{AB},\frac{dc_A}{dz} ]

or, with mole fraction and total molar concentration (c):

[ J_A = -c D_{AB},\frac{dy_A}{dz} ]

where:

• (J_A) = molar flux of A relative to the molar-average velocity (mol/(m²·s))
• (D_{AB}) = binary diffusivity (m²/s)
• (c_A) = molar concentration of A; (y_A) = mole fraction
• (z) = distance in the direction of the gradient

The minus sign means flux is down the gradient. Diffusivity depends on temperature, pressure (gases: roughly (D \propto T^{1.5}/P) conceptually), and species pair. Liquids have much smaller (D_{AB}) than gases—liquid-side mass transfer is often slower and more film-resistant.

Stefan flow / bulk contribution: when there is net molar flow (e.g., pure vapor A evaporating into stagnant B), total flux of A is diffusion plus convection of A with the bulk molar velocity. Exam qualitative point: equimolar counterdiffusion (classic binary distillation idealization) differs from unimolecular diffusion through stagnant gas (classic gas absorption of a dilute solute into a nonvolatile solvent with inert gas B stagnant)—the logarithmic mean driving force appears in the stagnant-gas case.

Worked intuition: Fick across a film

Gas film thickness (\delta = 0.5,\mathrm{mm} = 5\times 10^{-4},\mathrm{m}), (D_{AB} = 1.0\times 10^{-5},\mathrm{m^2/s}), total (c = 40,\mathrm{mol/m^3}), mole fractions (y_{A,1} = 0.20) at the bulk and (y_{A,i} = 0.05) at the interface (steady linear profile).

Diffusive flux magnitude:

[ J_A \approx c D_{AB}\frac{(y_{A,1}-y_{A,i})}{\delta} = 40\times 10^{-5}\times\frac{0.15}{5\times 10^{-4}} = 0.12,\mathrm{mol/(m^2\cdot s)} ]

If (\delta) doubles, flux halves—film thickness is a direct lever in film theory.

Convective Mass Transfer and Driving Forces

In equipment, turbulence and flow thin boundary layers and renew surface. Engineers rarely integrate Fick’s law through a complex velocity field; they define a mass-transfer coefficient (k) so that:

[ N_A = k,(\text{driving force}) ]

Common driving-force pairings:

Driving force is the difference between bulk composition and interface composition. The interface is assumed at local equilibrium with the other phase (two-film theory). If bulk and equilibrium compositions coincide, driving force is zero and net mass transfer stops—even if molecules still exchange both ways.

Analogy to heat transfer (Chapter 8):

Dimensionless groups (for recognition, not heavy correlation work on this exam):

• Sherwood (\mathrm{Sh} = kL/D_{AB}) — like Nusselt for mass
• Schmidt (\mathrm{Sc} = \nu/D_{AB}) — like Prandtl
• Reynolds — flow regime, as in fluids and heat

Higher Re generally raises (k); higher Sc means momentum diffusivity exceeds mass diffusivity (typical of liquids).

Film Theory

Film theory (Whitman) idealizes the resistance to mass transfer as a thin stagnant film of thickness (\delta) next to the interface, with linear concentration profile and pure diffusion across the film:

[ k_c \approx \frac{D_{AB}}{\delta} ]

Implications you can use on MCQs:

1. Anything that thins the film (higher velocity, better packing wetting, agitation) increases (k).
2. Higher diffusivity increases (k) (at fixed (\delta)).
3. Film theory is a model, not a measured geometric film; other models (penetration, surface renewal) change the exponent on (D_{AB}) (e.g., (k \propto \sqrt{D})), but the qualitative “resistance near the interface” story remains.

Equipment rate is often written per unit volume as (k_c a \Delta c), where (a) is interfacial area per volume. Column design balances height (enough transfer units) against diameter (capacity, flooding—linked to fluid mechanics).

Two-Film Concept for Gas–Liquid Contact

Gas absorption, stripping, and many distillation contacts involve a gas–liquid interface. Two-film theory places:

• A gas film from bulk gas to interface
• A liquid film from interface to bulk liquid

At the interface, phases are in equilibrium: e.g., (p_{A,i} = H x_{A,i}) (Henry) or a VLE relation (y_{A,i} = K x_{A,i}).

Overall gas-side coefficient (K_y) (or (K_G)) folds both resistances:

[ \frac{1}{K_y} = \frac{1}{k_y} + \frac{m}{k_x} ]

(schematic form; (m) is the local slope of the equilibrium curve (y^*=m x) in consistent units).

Exam reading: “The solute is very soluble in the solvent” often means equilibrium favors the liquid and liquid-side resistance may be less critical than for a sparingly soluble gas—but always check what the stem actually asks (rate vs equilibrium loading).

Linking to separators

Worked Conceptual: Overall Driving Force

Bulk gas: (y_A = 0.10). Equilibrium with bulk liquid would require (y_A^* = 0.02) (from measured liquid x and equilibrium curve). Overall gas-phase driving force is (y_A - y_A^* = 0.08). If a process change raises liquid solute level so (y_A^*) becomes 0.09, driving force collapses to 0.01 and transfer rate drops sharply even if hydrodynamics ((k)) are unchanged—rich solvent or high loading kills rate. That is why absorption uses lean solvent and stripping regenerates solvent (Section 9.3).

UPDA Exam Checklist for Section 9.1

1. Diffusion direction: high → low concentration/chemical potential; Fick has a minus sign.
2. Rate vs equilibrium: equilibrium sets interface compositions and zero driving force; kinetics/coefficients set flux.
3. Film theory: (k \sim D/\delta); thinner film or higher D raises k.
4. Two-film: series resistances; overall K combines gas and liquid films with equilibrium slope m.
5. Driving forces: know (\Delta y), (\Delta c), (\Delta p) pairings; LMTD-like log means appear in tower design language.
6. Do not invent numerical D values—use orders of magnitude and proportional reasoning.

Master diffusion, coefficients, and two-film logic first. Section 9.2 applies VLE and staging to distillation; Section 9.3 covers absorption, stripping, and extraction as rate-and-equilibrium hybrid unit operations.