Overall Mass Transfer Coefficient (Two-Film Model)

Reference ID: MET-0CD2 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The overall mass‑transfer coefficient, often denoted as \(K_L\) for the liquid side, quantifies the rate at which a gaseous component (e.g., CO₂) transfers from a gas phase into a liquid phase (or vice‑versa) across a stagnant or flowing interface. In process engineering this parameter is essential for the design and scale‑up of:

Absorbers and strippers
Gas‑liquid reactors
Carbon capture units
Fermentation and bioreactor aeration systems

Accurate prediction of \(K_L\) enables engineers to size equipment, estimate energy consumption, and ensure that mass‑transfer limitations do not dominate the overall process performance.

Methodology & Formulas

1. Unit Conversions

All calculations are performed in SI units. The temperature and pressure supplied in practical units are converted as follows:

\[ T_{\text{K}} = T_{\;^\circ\text{C}} + 273.15 \] \[ P_{\text{Pa}} = P_{\text{bar}} \times 10^{5} \]

2. Solubility Factor

The Henry constant \(H\) relates the dissolved concentration to the partial pressure of the gas. Its reciprocal is used as a dimensionless solubility factor:

\[ s = \frac{1}{H} \]

3. Film‑side Mass‑Transfer Coefficients

Both the liquid‑side (\(k_L\)) and gas‑side (\(k_G\)) film coefficients are estimated with a Sherwood‑type correlation that accounts for diffusion and convective enhancement:

\[ k_L = 0.023 \; \frac{D_L^{\,2/3}\; u_L^{\,1/3}}{d_h^{\,1/3}} \] \[ k_G = 0.023 \; \frac{D_G^{\,2/3}\; u_G^{\,1/3}}{d_h^{\,1/3}} \]

where

\(D_L\) – liquid‑phase diffusivity (m² s⁻¹)
\(D_G\) – gas‑phase diffusivity (m² s⁻¹)
\(u_L\) – superficial liquid velocity (m s⁻¹)
\(u_G\) – superficial gas velocity (m s⁻¹)
\(d_h\) – hydraulic diameter of the conduit (m)

4. Overall Liquid‑Phase Mass‑Transfer Coefficient

The two‑film model treats the resistance of each side as series resistances. The overall coefficient expressed on the liquid side is:

\[ K_L = \left[ \frac{1}{k_L} + \frac{1}{k_G\,H} \right]^{-1} \]

5. Flow Regime Determination (Reynolds Number)

The Sherwood correlation above is valid for turbulent flow. The flow regime is assessed with the Reynolds number based on the hydraulic diameter:

\[ \text{Re}_L = \frac{\rho_L \, u_L \, d_h}{\mu_L}, \qquad \text{Re}_G = \frac{\rho_G \, u_G \, d_h}{\mu_G} \]

Typical regime limits are presented in the table below.

Phase	Reynolds Number	Regime	Applicable Correlation
Liquid	\(\text{Re}_L < 2000\)	Laminar	Not applicable – use laminar film theory
Liquid	\(\text{Re}_L \ge 2000\)	Turbulent	Sherwood correlation shown above
Gas	\(\text{Re}_G < 2000\)	Laminar	Not applicable – use laminar film theory
Gas	\(\text{Re}_G \ge 2000\)	Turbulent	Sherwood correlation shown above

6. Summary of Computational Steps

Convert temperature to Kelvin and pressure to Pascal.
Compute the solubility factor \(s = 1/H\).
Evaluate the liquid‑side film coefficient \(k_L\) using the Sherwood correlation.
Evaluate the gas‑side film coefficient \(k_G\) using the same form of the correlation.
Combine the two resistances to obtain the overall coefficient \(K_L\) via the two‑film expression.

7. Practical Use

Once \(K_L\) is known, the molar flux of the transferring component can be calculated from the driving force (difference between bulk gas partial pressure and equilibrium liquid concentration):

\[ N_A = K_L \, \bigl( C_{A,\; \text{bulk}} - C_{A}^{*} \bigr) \]

where \(C_{A}^{*}\) is the concentration in the liquid that would be in equilibrium with the bulk gas phase, obtained from Henry’s law \(C_{A}^{*}=H\,p_A\).

The overall mass transfer coefficient, usually denoted K_L (or K_G for gas‑phase driving force), quantifies the combined resistance of both the liquid and gas (or solid) films that surround the interface. In the two‑film model each phase is represented by a stagnant film of thickness δ, across which diffusion occurs. The overall coefficient relates the net flux N to the driving concentration difference between the bulk phases:

N = K_L (C_L,bulk – C_L,eq) for liquid‑side driving force, or
N = K_G (y_G,bulk – y_G,eq) for gas‑side driving force.

It is a single parameter that engineers use to design and size equipment such as absorbers, strippers, and membrane contactors.

The two‑film model combines the liquid‑side coefficient k_L and the gas‑side coefficient k_G (or their equivalents) as resistances in series. The relationship is:

If the driving force is expressed on the liquid side: 1/K_L = 1/k_L + (m/k_G)
If the driving force is expressed on the gas side: 1/K_G = 1/k_G + (1/(m·k_L))

where m is the equilibrium slope (C_L,eq / y_G,eq). To obtain K_L:

Determine or measure k_L (e.g., from a liquid‑phase diffusion experiment).
Determine or measure k_G (e.g., from a gas‑phase diffusion experiment).
Obtain the equilibrium relationship to calculate m.
Insert the values into the appropriate resistance equation and solve for K_L (or K_G).

The two‑film model is most useful for:

Steady‑state, isothermal gas‑liquid or liquid‑liquid systems where interfacial turbulence is low.
Processes where diffusion through thin stagnant layers dominates mass transfer (e.g., absorption, stripping, liquid‑liquid extraction).

Key limitations include:

Assumes each phase has a uniform, constant film thickness; real systems may have variable turbulence.
Neglects interfacial phenomena such as surface renewal, Marangoni effects, or chemical reactions occurring at the interface.
Accuracy decreases at high Reynolds numbers or when the interface is highly disturbed (e.g., in agitators or packed columns with intense mixing).
Requires a reliable equilibrium relationship (m); errors in m directly affect the calculated overall coefficient.

A typical laboratory procedure:

Set up a stirred tank or a bubble column with known geometry and operating conditions (temperature, pressure, agitation speed).
Introduce a gas stream of known composition and flow rate; measure inlet and outlet gas concentrations (e.g., using gas chromatography).
Allow the system to reach steady state; record the liquid bulk concentration at several times to confirm stability.
Calculate the molar flux N from the difference between inlet and outlet gas concentrations multiplied by the gas flow rate.
Determine the driving force: either the concentration difference between the liquid bulk and the equilibrium liquid concentration corresponding to the outlet gas composition, or the gas‑phase concentration difference using the equilibrium slope m.
Compute K_L using K_L = N / (C_L,bulk – C_L,eq) (or the gas‑side form if preferred).
Repeat at different agitation speeds or gas flow rates to assess the effect of hydrodynamics on K_L.

Worked Example – Estimating the Overall Liquid-Phase Mass-Transfer Coefficient for CO₂ Absorption in a Packed Column

A small packed absorption tower is being designed to remove CO₂ from a flue-gas stream using water at 30 °C. To size the column height, the process engineer needs the overall liquid-phase mass-transfer coefficient, \(K_L\), based on the two-film model. Laboratory hydrodynamic data and literature physico-chemical properties have been supplied for the operating conditions.

Knowns

Temperature: 30 °C (303.15 K)
System pressure: 1.2 bar (120 000 Pa)
Henry’s-law constant: 33 000 bar (3.30×10⁴ bar)
Diffusivity CO₂ in water, \(D_L\): 1.9×10⁻⁹ m² s⁻¹
Diffusivity CO₂ in air, \(D_G\): 1.6×10⁻⁵ m² s⁻¹
Liquid viscosity, \(\mu\): 8.0×10⁻⁴ Pa s
Packing hydraulic diameter, \(d_h\): 0.02 m
Superficial liquid velocity, \(u_L\): 0.02 m s⁻¹
Superficial gas velocity, \(u_G\): 0.1 m s⁻¹
Gas constant, \(R\): 8.314 J mol⁻¹ K⁻¹

Step-by-Step Calculation

Convert Henry’s constant to dimensionless form, \(H_{cc}\): \[ H_{cc}=\frac{H}{RT}=\frac{3.30\times10^{4}\ \text{bar}}{8.314\ \text{J mol}^{-1}\text{K}^{-1}\times303.15\ \text{K}}\times10^{-5}=3.03\times10^{-5} \]
Estimate the liquid-side film coefficient, \(k_L\), with a penetration-type correlation: \[ k_L=2\,\sqrt{\frac{D_L\,u_L}{\pi\,d_h}}=2\,\sqrt{\frac{1.9\times10^{-9}\times0.02}{\pi\times0.02}}=3.53\times10^{-8}\ \text{m s}^{-1} \]
Estimate the gas-side film coefficient, \(k_G\), using a Reynolds–Chilton–Colburn analogy for packed beds: \[ k_G=0.023\,\frac{D_G}{d_h}\,\left(\frac{\rho_G\,u_G\,d_h}{\mu_G}\right)^{0.8}=2.50\times10^{-5}\ \text{m s}^{-1} \]
Combine the resistances into the overall liquid-phase coefficient: \[ \frac{1}{K_L}=\frac{1}{k_L}+\frac{1}{H_{cc}\,k_G}=\frac{1}{3.53\times10^{-8}}+\frac{1}{3.03\times10^{-5}\times2.50\times10^{-5}} \] \[ \frac{1}{K_L}=2.83\times10^{7}+1.32\times10^{9}\approx1.35\times10^{9}\ \text{s m}^{-1} \] \[ K_L=3.53\times10^{-8}\ \text{m s}^{-1} \]

Final Answer

The overall liquid-phase mass-transfer coefficient for CO₂ absorption under the stated conditions is

\[ K_L=3.53\times10^{-8}\ \text{m s}^{-1} \]

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle