Gas-Phase Mass Transfer Rate Calculator
Two-film theory: the gas-phase absorption rate is the product of the overall gas coefficient, the specific interfacial area, and the driving force y − y*. Solve for NA, Ky, a, y or y*, see the two-film resistance split, and read the full concentration profile across the gas film, interface, and liquid film.
Two films in series: gas and liquid resistances add
Whitman's two-film theory (1923) pictures the interface between gas and liquid as two stagnant films — one on each side — separated by a sharp interface. All the resistance to mass transfer is assumed to lie in these two films, where transport is by molecular diffusion. Inside the bulk phases, turbulent mixing is fast enough to keep concentrations uniform. The overall resistance is the sum of the two film resistances: 1/K_y = 1/k_y + m/k_x. This additive structure is the reason a poorly-soluble gas (large m, high Henry constant) is liquid-film controlled, and a highly-soluble gas (small m) is gas-film controlled.
The Henry slope m controls which film dominates
The factor m in m/k_x is what couples the two resistances. For NH3 in water, m ≈ 0.001 — the liquid-film term m/k_x is negligible, and K_y ≈ k_y (gas-film controlled). For O2 in water, m ≈ 40 — the liquid-film term dominates, and K_y ≈ k_x/m (liquid-film controlled). The transition happens when 1/k_y ≈ m/k_x. This is why physical absorption of CO2 or O2 into water needs liquid-side enhancement (chemical reaction, or a reactive solvent like amine) to be industrially viable — the liquid film resistance alone would otherwise dominate the column height.
The volumetric coefficient K_y·a is what you actually measure
You cannot measure K_y and a independently from column experiments — you measure their product, K_y·a. That is why packed column design correlations (Onda, Billet-Schultes, etc.) are formulated for K_G·a directly. The specific area a depends on packing geometry and liquid loading; the coefficient K_y depends on gas and liquid velocities and physical properties. Splitting them requires either separate correlations for each, or empirical estimates of wetted area versus total area. In practice, simulation packages like Aspen Plus or ProTreat include built-in correlations for K_G·a for hundreds of packings.
The driving force (y − y*) shrinks along the column
In an absorption column, the gas enters at the bottom with high solute concentration y_bottom and exits at the top with low y_top. The liquid enters at the top with low solute x_top and exits at the bottom with high x_bottom. So y − y* varies continuously from bottom to top — large at the bottom, small at the top. The simple form N_A = K_y·a·(y−y*) applies at a single point. To size a real column you integrate: Z = (G / (K_y·a·A_c)) ∫ dy/(y − y*), which is the packed-column height equation. This tool gives the point rate; the column height requires the integral.
How to Use This Calculator
The two-film mass transfer rate is a simple product of three factors — the overall gas coefficient, the specific interfacial area, and the driving force. Pick what to solve for, fill in the rest, and read the resistance split.
1. Choose what to solve for
Select NA, Ky, a, y, or y* from the toggle. All five rearrangements are algebraic, so results are exact for the given inputs.
2. Enter the overall coefficient Ky
If you have the gas-film and liquid-film coefficients (ky, kx) and the Henry slope m, the two-film panel computes Ky = 1/(1/ky + m/kx). Use the "Apply" button to feed it into the main field.
3. Set the specific interfacial area a
Use the packing manufacturer's data if available. As a rough guide: 50 mm Raschig rings ≈ 100 m²/m³, 50 mm Pall rings ≈ 200 m²/m³, structured packing 250Y ≈ 500 m²/m³.
4. Read the result and the resistance split
The result card shows NA and the four key inputs. The two-film panel below shows what fraction of the total resistance is in the gas film versus the liquid film, so you know which phase to modify if you want to speed up absorption.
5. Check the concentration profile
The diagram shows the actual concentration profile: y in the bulk gas, dropping through the gas film to yi at the interface, then continuing through the liquid film to y* in the bulk liquid. The steeper the film slope, the larger that film's resistance.
6. Switch to Advanced for depth
Advanced mode adds charts of NA vs Ky and vs a, a sensitivity panel with partial derivatives, and companion metrics including the absolute molar rate, the column height per transfer unit, and the classification of which film controls.
What This Tool Offers
Built around the two-film theory relation NA = Ky · a · (y − y*), the standard model of interphase mass transfer.
Five-way solving
Solve for NA, Ky, a, y, or y*. The calculator rearranges NA = Ky·a·(y−y*) algebraically for whichever you pick.
Live two-film schematic
A concentration profile showing y in the gas bulk, yi at the interface, and y* in the liquid bulk — with the physical film layers drawn beneath.
Film resistance split
The two-film panel decomposes Ky into gas-film and liquid-film contributions: 1/Ky = 1/ky + m/kx. The stacked bar shows which film controls.
Henry slope integration
The m input couples the two films. Small m (soluble gas like NH3) gives gas-film control; large m (sparingly-soluble gas like O2) gives liquid-film control — the tool shows the crossover.
Four coordinated charts
NA vs y, NA vs y*, NA vs Ky, and NA vs a — with three-curve families to show sensitivity to the operating point.
Reference tables
Typical KG·a values for common packings, Henry's law constants for gas-water systems, mass transfer correlations, and a complete variable reference.
Reference Tables
Typical volumetric mass transfer coefficients for packed columns, Henry's law slopes for gas-water systems, and the standard variable reference.
Typical Volumetric Coefficients KG·a for Packed Absorbers
Order-of-magnitude values at moderate gas and liquid loads. Real values depend strongly on flow rates and physical properties.
| Packing | Nominal size | Specific area a (m²/m³) | KG·a (kmol/(m³·s)) | Typical use |
|---|
Henry's Law Slope m = y*/x for Gas-Water Systems
Approximate dimensionless Henry slopes at 25 °C. Small m = highly soluble (gas-film controlled); large m = sparingly soluble (liquid-film controlled).
| Gas | m = y*/x at 25 °C | Solubility | Controlling film | Typical industrial solvent |
|---|
Standard Mass-Transfer Correlations
Commonly used correlations for individual-film coefficients in packed columns.
| Correlation | Phase | Form | Notes |
|---|---|---|---|
| Frossling | Gas (single sphere) | Sh = 2 + 1.1·Re0.6·Sc1/3 | Baseline for particle-gas transfer; older but well-validated |
| Sherwood–Holloway | Liquid (packed bed) | kL·(μL/(ρL·g))1/3 = f(ReL, ScL) | Classic liquid-film correlation for random packing |
| Onda et al. (1968) | Both films | Separate correlations for kG, kL, and wetted area aw | Widely used for random and structured packing |
| Billet–Schultes | Both films | Based on channel model, uses packing geometry directly | Modern; handles structured packing well |
| Rocha–Bravo–Fair | Structured packing | Separate correlations for kG, kL, and effective area | Standard for structured packing in the chemical industry |
| Higbie penetration | Liquid film | kL = 2·√(D/(π·texposure)) | Surface-renewal theory; used with interface age estimation |
| Danckwerts surface renewal | Liquid film | kL = √(D·s) | s = fractional rate of surface renewal; more realistic than Higbie for turbulent contactors |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| NA | Volumetric molar transfer rate of A | 10⁻⁴ – 5 | kmol/(m³·s) |
| Ky | Overall gas-phase coefficient | 0.0005 – 0.05 | kmol/(m²·s) |
| ky | Gas-film individual coefficient | 0.001 – 0.1 | kmol/(m²·s) |
| kx | Liquid-film individual coefficient | 0.001 – 0.05 | kmol/(m²·s) |
| a | Specific interfacial area | 50 – 800 | m²/m³ |
| Ky·a | Volumetric overall coefficient | 0.01 – 5 | kmol/(m³·s) |
| y | Bulk gas mole fraction of A | 0.001 – 0.5 | — |
| y* | Equilibrium gas mole fraction | 0 – y | — |
| yi | Interfacial gas mole fraction | y* < yi < y | — |
| m | Henry's law slope (y* = m·x) | 0.0001 – 100 | — |
| x | Bulk liquid mole fraction of A | 0 – 0.1 | — |
| HTU | Height of a transfer unit | 0.3 – 2 | m |
| NTU | Number of transfer units | 1 – 20 | — |
What Does NA = Ky · a · (y − y*) Measure?
The gas-phase mass transfer rate NA is the volumetric molar flux of a solute A moving from the gas bulk into the liquid phase, per unit volume of the contacting device. It is the product of three factors: the overall gas-phase mass transfer coefficient Ky (which captures the combined resistance of both films), the specific interfacial area a (the amount of gas-liquid contact area packed into each cubic metre of column), and the driving force (y − y*), which is the difference between the actual bulk gas composition and the composition that would be in equilibrium with the bulk liquid. Where any of these three factors is small, the transfer rate is small; the entire art of absorber design is choosing conditions and packings that maximise their product.
Gas-Phase Mass Transfer Formula
The defining relationship is a simple product:
where NA is in kmol/(m³·s), Ky in kmol/(m²·s), a in m²/m³, and y, y* are dimensionless mole fractions. Rearranged to solve for the other four variables:
a = NA / (Ky · (y − y*))
y = NA / (Ky · a) + y*
y* = y − NA / (Ky · a)
The two-film decomposition of the overall coefficient is:
And the interfacial composition follows from the flux continuity across both films:
xi = yi / m
For a packed column of height Z and cross-section Ac, the total molar rate of absorption is:
where the driving force is evaluated as a log-mean over the column height:
Mass Transfer Variables
| Symbol | Meaning | Notes |
|---|---|---|
| NA | Volumetric molar flux of A | Rate of solute transferred per unit column volume |
| Ky | Overall gas-phase coefficient | Combines both film resistances, based on gas mole fraction |
| ky | Gas-film coefficient | Resistance of the gas-side stagnant film |
| kx | Liquid-film coefficient | Resistance of the liquid-side stagnant film |
| a | Specific interfacial area | Wetted gas-liquid area per unit packed volume |
| y | Bulk gas mole fraction | Solute mole fraction in the bulk gas phase |
| y* | Equilibrium gas mole fraction | = m · x, the composition in equilibrium with bulk liquid |
| yi | Interfacial gas mole fraction | Actual composition at the interface, between y and y* |
| m | Henry's law slope | y* = m·x; small m = high solubility |
How Each Variable Affects the Transfer Rate
The formula is linear in Ky, a, and (y − y*). Doubling the overall coefficient doubles the rate; doubling the specific area doubles the rate; doubling the driving force doubles the rate. This makes the trade-offs in absorber design easy to see. If you want more absorption but the driving force is already fixed by process constraints (inlet gas and liquid compositions), you can either increase Ky (by increasing gas and liquid flow rates, which increases turbulent mixing in the films) or increase a (by using a packing with more surface area per unit volume, or by using more packing in the column). Both usually mean more pressure drop and more energy, so the design is an economic balance.
The rate is not linear in Ky when Ky itself is derived from the two-film model, because Ky depends on m in a non-trivial way. If m is large (sparingly soluble gas), 1/Ky ≈ m/kx, and Ky ≈ kx/m — small values of m have a huge effect on the overall rate. If m is small (highly soluble gas like NH3), 1/Ky ≈ 1/ky, and the liquid-side resistance doesn't matter. This is why selecting the right solvent (low m) is often the single most important absorber design decision — it can shift the rate-limiting film entirely.
Mass Transfer Rate Examples
NA = 0.005 × 200 × (0.10 − 0.02) = 0.005 × 200 × 0.08 = 0.080 kmol/(m³·s)
Ky = 0.080 / (200 × 0.08) = 0.005 kmol/(m²·s)
1/Ky = 1/0.008 + 0.5/0.020 = 125 + 25 = 150
Ky = 0.006667 kmol/(m²·s)
Gas-film resistance fraction = 125/150 = 83.3%
Liquid-film resistance fraction = 25/150 = 16.7%
1/Ky = 1/0.008 + 40/0.020 = 125 + 2000 = 2125
Ky = 0.000471 kmol/(m²·s)
Gas-film resistance fraction = 125/2125 = 5.9%
Liquid-film resistance fraction = 2000/2125 = 94.1%
yi = (0.008 × 0.5 × 0.10 + 0.020 × 0.02) / (0.020 + 0.008 × 0.5)
= (0.0004 + 0.0004) / 0.024 = 0.0008 / 0.024 = 0.0333
xi = 0.0333 / 0.5 = 0.0667
Frequently Asked Questions
Two-film theory (Whitman, 1923) is the simplest model of interphase mass transfer. It pictures the gas and liquid phases as separated by two stagnant films — one on each side of the interface — with all the mass transfer resistance concentrated in these films. Within each film, transport is by molecular diffusion; the bulk phases are assumed perfectly mixed. At the interface itself, the two phases are in equilibrium (y_i = m·x_i), so there is no resistance to crossing the interface. The model's key result is that the resistances of the two films add in series: 1/K_y = 1/k_y + m/k_x. Although real films are not perfectly stagnant, this additive-resistance structure has held up remarkably well and remains the basis for absorber design today.
Compare the two resistances: gas-film resistance is 1/k_y, liquid-film resistance is m/k_x. Whichever is larger controls the overall rate. For highly soluble gases (NH₃, HCl, SO₂ in water), m is very small — usually below 0.1 — so m/k_x is negligible and the gas film controls. For sparingly soluble gases (O₂, N₂, H₂, CO₂ in water), m is 1.5 or larger — often 30–80 for permanent gases — so m/k_x dominates and the liquid film controls. Between these extremes, the controlling film depends on the actual values. This calculator's two-film panel shows the split directly, with a stacked bar that makes the controlling film visually obvious.
The "specific area" a is the wetted interfacial area — the portion of the packing surface that is actually in contact with both gas and liquid. In a real column, part of the packing surface is dry (gas-side only), part is fully flooded (liquid-side only), and only part has a genuine gas-liquid interface. The wetted fraction depends on liquid flow rate, packing wettability, and surface tension. Correlations like Onda's estimate aw/at (wetted area / total area) as a function of liquid Reynolds number and liquid properties. Typical wetted fractions are 0.6–0.95 for aqueous systems on ceramic or metal packing, but can be much lower for organic liquids or for poorly-wetted plastics. When you multiply K_y by a, you are effectively multiplying by the "effective" area, which is what actually participates in mass transfer.
Yes — the same equation applies to stripping, just with the driving force reversed. In stripping, the solute moves from liquid to gas, so the driving force is (y* − y) rather than (y − y*), and the "overall liquid-phase coefficient" K_x (based on liquid mole fraction) is often more convenient. The two-film model is symmetric: 1/K_x = 1/k_x + 1/(m·k_y). The same physical coefficients k_y, k_x, and m apply. In practice, the choice of which coefficient to use (K_y or K_x) is a matter of convenience — use the one that puts the larger resistance in the denominator, so the driving force is in the phase that controls the rate.
This calculator gives the local volumetric rate at a single point in the column. To size a real column, you integrate the driving force over the height: Z = HTU × NTU, where HTU = G/(K_y·a·A_c) is the height of a transfer unit and NTU = ∫(dy/(y − y*)) is the number of transfer units. HTU has units of length (typically 0.3–2 m for industrial packings) and captures the mass transfer efficiency; NTU is dimensionless and captures how difficult the separation is (large NTU = high purity or large composition change). This decomposition — HTU·NTU — is the standard way to think about packed column design, and it follows directly from the rate equation you are computing here.
The two-film model is a simplification and can fail in several situations. (1) When the liquid film is not stagnant but has significant surface renewal — then penetration or surface-renewal models (Higbie, Danckwerts) are more accurate. (2) When chemical reaction occurs in the liquid film, enhancing the local mass transfer rate — the enhancement factor E must be applied, and the effective k_x becomes E·k_x. (3) When the interface is not flat but has ripples, waves, or droplets — the effective area and film thickness are no longer well-defined. (4) When surface-active contaminants build up at the interface and add an extra resistance that the model does not account for. In industrial practice, the simple additive form 1/K_y = 1/k_y + m/k_x is almost always used as the starting point, with empirical correlations for k_y and k_x that incorporate these effects implicitly.
Ky is based on mole-fraction driving force (Δy) and KG is based on partial-pressure driving force (Δp_A). Since p_A = y·P, we have Δp_A = P·Δy, so KG = Ky/P where P is the total pressure. In practice, values are quoted in kmol/(m²·s·atm) for KG and kmol/(m²·s) for Ky. The relationship is exact for a constant total pressure. If you are converting from one to the other, multiply (or divide) by the total pressure in the appropriate units. This calculator works entirely in mole fractions, so it uses Ky throughout, but if your data is in KG, just multiply by the total pressure (in atm) to convert to Ky.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| N_A | Molar mass transfer rate per volume | mol/m3.s |
| Ky | Overall gas-phase mass transfer coefficient | mol/m2.s |
| a | Interfacial area per unit volume | m2/m3 |
| y | Bulk gas-phase mole fraction | |
| y* | Gas-phase mole fraction in equilibrium with the bulk liquid |
What it means
The two‑film theory is a simple model for mass transfer across a phase interface (e.g., gas‑liquid). It assumes that the resistance to mass transfer is confined to two stagnant films (one on each side of the interface), and that the bulk fluids are well‑mixed. The overall mass transfer rate N_A (per unit volume) is given by N_A = K_y a (y − y*), where K_y is the overall gas‑phase mass transfer coefficient, a is the interfacial area per unit volume, y is the mole fraction of the transferring component in the bulk gas, and y* is the mole fraction in equilibrium with the bulk liquid. This equation is used in the design of packed columns, tray columns, and other gas‑liquid contactors. The overall coefficient K_y combines the individual film coefficients for gas and liquid phases. The driving force is the difference between actual and equilibrium composition. The two‑film theory is a cornerstone of absorption, stripping, and distillation design. Although simplified, it provides a practical framework for estimating mass transfer rates and is widely used in chemical engineering practice.
Worked example
Mass Transfer Rate – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Ky | 0.05 |
| a | 100 |
| y | 0.05 |
| y* | 0.02 |
| Parameter | Value |
|---|---|
| Ky | 0.03 |
| a | 150 |
| y | 0.1 |
| y* | 0.04 |
Common mistakes
- Mass transfer coefficient K_y: Based on gas‑phase driving force (y − y*). Units: mol/(m²·s) or kmol/(m²·s) – check.
- Interfacial area a: Specific interfacial area per unit volume (m²/m³). For packed columns, depends on packing.
- Driving force (y − y*): y is bulk gas mole fraction, y* is equilibrium mole fraction at the interface.
- Two‑film theory: Assumes equilibrium at the interface and steady state; for fast reactions, modify.
- Units: N_A in mol/(m³·s) if a is per volume; or in mol/(m²·s) if area is total.
Applications
The two‑film theory describes mass transfer between a gas and a liquid, with the transfer rate given by N_A = K_y·a·(y − y*). This equation is used to design gas absorption, stripping, and humidification equipment. The overall mass transfer coefficient K_y accounts for resistances in both gas and liquid films, while the interfacial area a is provided by the contacting device (packing, trays). Engineers use this formula to size absorption towers, to evaluate the performance of packed columns, and to optimise operating conditions. It is also applied in environmental engineering for air stripping of volatile organics. By understanding mass transfer, professionals can achieve efficient separation and pollution control.
- Design of gas absorption columns (e.g., CO₂, SO₂, NH₃ removal)
- Sizing of stripping columns for volatile contaminants
- Design of humidifiers, dehumidifiers, and cooling towers
- Selection of packing or tray types for gas‑liquid contactors
- Environmental control of industrial emissions
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