Sizing a packed absorber

gas absorptionpacked tower heightHTU and NTUmass transfer coefficientscrubber sizing

Sizing a packed absorber: diffusivity and Schmidt number, the film coefficients, the absorption factor, and the packed height from HTU and NTU.

Wilke–Chang Liquid Diffusivity

D_{AB} = 7.4 \times 10^{-8} \, \frac{(\phi M_B)^{1/2} \, T}{\mu_B V_A^{0.6}}

An estimate of how fast a solute diffuses through a liquid, from the solvent's molar mass, viscosity and association behaviour plus the solute's molar volume. Wilke and Chang's 1955 correlation, still the default when nobody has measured the real thing.

Schmidt Number

\mathrm{Sc} = \frac{\mu}{\rho D}

The ratio of momentum diffusivity to mass diffusivity — how readily a fluid spreads motion compared with how readily it spreads molecules. The mass-transfer twin of the Prandtl number, and the fluid property every mass-transfer correlation is written around.

Chilton–Colburn Analogy for Mass Transfer

\mathrm{Sh} = \frac{f}{2} \, \mathrm{Re} \, \mathrm{Sc}^{1/3}

Mass transfer inferred from pressure drop. Chilton and Colburn found in 1934 that the j-factor for mass transfer equals half the Fanning friction factor, which lets a Sherwood number be predicted from nothing but a Reynolds number and a Schmidt number.

Sherwood Number

\mathrm{Sh} = \frac{k_c L}{D}

The convective mass transfer coefficient made dimensionless: transfer with the flow divided by transfer by diffusion alone. Named for Thomas Sherwood, whose 1934 wetted-wall work with Gilliland set the pattern every correlation since has followed.

Two-Film Overall Mass Transfer Coefficient

\frac{1}{K_{OG}} = \frac{1}{k_G} + \frac{m}{k_L}

Two films in series, and the resistances add. Lewis and Whitman's 1924 picture of a gas film and a liquid film meeting at an interface in equilibrium is still the working model for every absorber and stripper built.

Absorption Factor

A = \frac{L}{m V}

The single number that decides whether an absorber can do its job: the liquid's capacity to carry the solute away divided by the gas's capacity to deliver it. Below one, no number of stages will finish the separation.

Transfer Units for Dilute Absorption

N_{OG} = \frac{\ln \left[ \dfrac{y_1}{y_2} \left( 1 - \dfrac{1}{A} \right) + \dfrac{1}{A} \right]}{1 - \dfrac{1}{A}}

The number of transfer units a dilute absorber demands — the packed-column counterpart of a stage count. Colburn's 1939 integration of the design equation for a straight equilibrium line, and the number that multiplies HTU to give a height.

Packed Column Height from HTU and NTU

Z = H_{OG} \, N_{OG}

A packed column's height split into the two things that set it: how hard the separation is, and how good the packing is at it. Chilton and Colburn's transfer-unit idea, and the reason packing is specified in metres of HTU rather than in stages.

How they fit together

This chain runs from a molecular property to a tower height, and it is worth seeing as two halves that meet in the middle. The first half asks how fast the solute can move. Wilke-Chang estimates liquid diffusivity when you cannot measure it — an empirical correlation good to perhaps ±10 to 20%, and it is worth using a measured value whenever one exists. The Schmidt number puts that diffusivity in context against momentum transport, and its magnitude tells you immediately which phase you are dealing with: around 1 for gases, where momentum and mass diffuse at similar rates, and 1000 or more for liquids, where mass diffusion is far slower. That single contrast is the reason the liquid film so often controls.

The Chilton-Colburn analogy then gets a Sherwood number from friction factor, Reynolds and Schmidt, which is the practical trick of the field: pressure drop is easy to measure and mass transfer is not, so the analogy borrows one to predict the other. Sherwood is the dimensionless coefficient and unpacking it gives kc, the film coefficient in real units — Sherwood is to mass transfer exactly what Nusselt is to heat transfer, and a reader who knows one already knows the shape of the other.

The second half sizes the tower. The two-film overall coefficient adds the gas and liquid resistances in series, and the m/kl term in it is where the design decision lives. A very soluble gas has a small m, the liquid resistance collapses, and the tower is gas-film controlled — spend your money on gas distribution. A sparingly soluble gas has a large m and is liquid-film controlled, and the answer is liquid rate or a reactive solvent, while improving gas-side turbulence does nothing at all. Diagnosing which one you have before sizing anything is the most valuable thing in this set. The absorption factor A = L/mV is the same idea as a ratio, and its rule of thumb is worth carrying: design for A between 1.2 and 2.0. Below 1 the separation pinches no matter how tall the column, and much above 2 you are pumping solvent you will pay to regenerate. Number of transfer units follows from A and the inlet and outlet gas compositions — note this dilute form assumes a linear equilibrium and small concentration changes, so a concentrated absorber needs the general integral instead. Packed height closes it as Z = HTU × NTU, and the split between the two factors is the useful part: NTU is the difficulty of the separation you asked for and no packing choice changes it, while HTU is the performance of the hardware, typically 0.3 to 1 m. Demanding a tighter outlet raises NTU; buying better packing lowers HTU. Knowing which of the two your height problem lives in tells you whether to renegotiate the spec or reorder the internals.