CHE 318 Lecture 14

Mass Transfer Coefficients (II)

Dr. Tian Tian

2026-02-04

Recap

  • Introduction to mass transfer coefficients
  • Link between mass transfer coefficient and diffusivity
  • Introduction to boundary concentration problem

Learning Outcomes

After today’s lecture, you will be able to:

  • Remember the flux equations using mass transfer coefficient in different phases
  • Recall the motivation of using mass transfer coefficient
  • Analyze mass transfer coefficients’ units
  • Distinguish between mass transfer coefficients in different forms

Realistic Interfacial Concentration Profiles

At the boundary, it is often convenient to use the equilibrium concentration ratio between gas (\(c_i\)) and liquid (\(c_{Li}\)). This constant, often called the equilibrium distribution coefficient, is defined as:

\[\begin{align} K &= \frac{\text{[Conc. at gas side]}}{\text{[Conc. at liquid side]}} \\ &= \frac{c_{i}}{c_{Li}} \end{align}\]
  • Similarity: Henry’s law (\(H = \dfrac{p_{\text{gas}}}{c_{\text{aq}}}\)), remember in gas \(p_{\text{gas}} = c_{\text{gas}} RT\)
  • We have already seen similar concepts in solubility in liquid / solid diffusion equations!
  • \(K\) can be a value range from \(0\) to \(\infty\) 👉 what does that mean?

Interfacial Concentration Profile For Different \(K\)

  • The interfacial concentration values depend on the value of \(K\)!
  • What else determines the interfacial balance? 👉 matching of interfacial fluxes \(N_A\) at both sides!

Geankoplis Figure 7.1.3

Match of Interfacial Fluxes

At equilibrium, if there is no resistance when transferring A between the interface, what is the mass balance equation?

\[\begin{align} \text{[In]}\vert_{\text{left}} - \text{[Out]}\vert_{\text{right}} + \text{[Gen]}\vert_{\text{interface}} &= \text{[Acc]}\vert_{\text{interface}} \\ N_A\vert_{\text{left}} = N_A\vert_{\text{right}} \end{align}\]

How do we model \(N_A\) in each phase?

  • May be a combination of diffusion and convection
  • Fluid velocity at interface may be turbulent (hard to model)
  • In general we use the phenomenal relation \(\text{[Flux]} = \text{[Driving Force]} / \text{[Resistance]}\)

General Flux Equation For Convective Transport Regime

  • The driving force is the concentration difference “bulk concentration” and “interfacial concentration”
  • The resistance is lumped into one coefficient \(k_c'\), mass transfer coefficient.
\[\begin{align} N_A = k_c' (c_{L, b} - c_{Li}) \end{align}\]
  • \(k_c'\) is inversely related to the resistance.
  • \(k_c'\) means driving force is concentration & convection term is EMCD-like
  • The unit of \(k_c'\)? 👉 \(\text{m}\cdot\text{s}^{-1}\)
  • \(k_c'\) can be \(\infty\)! 👉 No transfer resistance inside the bulk phase

Where Does \(k_c'\) Come From (I)?

Simplified view: transport resistance of A from bulk to the interface occurs inside an interfacial film with thickness \(\delta\). We can write \(N_A\) at steady state using already known results

  • Case 1: EMCD / diffusion-controlled / dilute transport in liquid
\[\begin{align} N_A &= \boxed{\frac{D_{AB}}{\delta}} (c_{L, b} - c_{L, i}) \\ &= k_c' (c_{L, b} - c_{L, i}) \end{align}\]
  • Case 2: transport in stagnant film with non-negligible convection
\[\begin{align} N_A &= \boxed{\frac{D_{AB}}{\delta (1 - \frac{c}{c_T} )}} (c_{L, b} - c_{L, i}) \\ &= k_c' \frac{1}{x_{Bm}} (c_{L, b} - c_{L, i}) \\ &= k_c (c_{L, b} - c_{L, i}) \end{align}\]
  • Can be written both by \(k_c'\) or \(k_c\) terms
  • One can expect \(k_c\) contain the \(1/x_{Bm}\) term!

Where Does \(k_c\) Come From (II)?

The mass transfer coefficient \(k_c\) is even valid for systems with effective \(D_{AB}\)!

  • Case 3: mass transfer in porous solid materials
\[\begin{align} N_A &= \boxed{\frac{\epsilon}{\tau}\frac{D_{AB}}{\delta}} (c_{L, b} - c_{L, i}) \\ &= k_c' (c_{L, b} - c_{L, i}) \end{align}\]
  • Case 4: turbulent mass transfer

The turbulence in the fluid contributes to an additional term \(\epsilon_m\) in diffusion terms

\[\begin{align} N_A &= \boxed{\frac{D_{AB} + \epsilon_m}{\delta}} (c_{L, b} - c_{L, i}) \\ &= k_c' (c_{L, b} - c_{L, i}) \end{align}\]
  • \(\epsilon_m\) is the “Eddie diffusivity” (correction to \(D_{AB}\) due to turbulence)
  • General case for \(\epsilon_m\) is non-trivial to solve!

Implications of Mass Transfer Coefficient \(k_c'\)

  • Really convenient to use!
  • In reality, \(k_c'\) is not a physics-based quantity, it depends on system / condition
  • \(k_c' \propto D_{AB}^n\) in realistic systems
  • We will discuss about different theories that explains the relation between \(k_c'\) and \(D_{AB}\) (penetration theory, film theory, boundary theory) in coming weeks

Balance Equations At Interfaces

We now have 2 equations to determine the interfacial concentrations!

  1. Equilibrium concentration distribution
\[\begin{align} K = \frac{c_{g,i}}{c_{L,i}} \end{align}\]
  1. Flux matching
\[\begin{align} N_{A}\vert_{\text{liq}} &= N_{A}\vert_{\text{gas}} \\ k_{c, \text{liq}}' (c_{L, b} - c_{L, i}) &= k_{c, \text{gas}}' (c_{g, i} - c_{g, b}) \end{align}\]

We can solve \(c_{L, i}\) and \(c_{g, i}\) given information about:

  • Bulk concentrations \(c_{L, b}\), \(c_{g, b}\)
  • Equilibrium distribution coefficient \(K\)
  • Mass transfer coefficients \(k_c\) in each phase

Demonstration of Mixed Boundary Conditions

Mass Transfer Coefficient In Different Forms

When using \(k\) to express flux, we have the same form (Geankoplis Table 7.2.1)

  • Flux equations for EMCD
\[\begin{align} \text{Gases:}\quad N_A &= k_c'\,(c_{A1}-c_{A2}) = k_G'\,(p_{A1}-p_{A2}) = k_y'\,(y_{A1}-y_{A2}) \\ \text{Liquids:}\quad N_A &= k_c'\,(c_{A1}-c_{A2}) = k_L'\,(c_{A1}-c_{A2}) = k_x'\,(x_{A1}-x_{A2}) \end{align}\]
  • Flux equations for diffusion through stagnant B
\[\begin{align} \text{Gases:}\quad N_A &= k_c\,(c_{A1}-c_{A2}) = k_G\,(p_{A1}-p_{A2}) = k_y\,(y_{A1}-y_{A2}) \\[4pt] \text{Liquids:}\quad N_A &= k_c\,(c_{A1}-c_{A2}) = k_L\,(c_{A1}-c_{A2}) = k_x\,(x_{A1}-x_{A2}) \end{align}\]

But:

  • \(k_c'\) and \(k_c\) are two different coefficients
  • \(k_c'\), \(k_L'\), \(k_G'\) have different units

Naming Convention and Units

  • Superscript: EMCD \(k'_{\text{driving force}}\); Convective / stagnant B \(k_{\text{driving force}}\)
Phase / Driving force Concentration \(c_A\) Partial pressure \(p_A\) Mole fraction
(gas \(y_A\), liquid \(x_A\))
Gas phase \(k_c\), \(k_c'\) \(k_G\), \(k_G'\) \(k_y\), \(k_y'\)
Liquid phase \(k_c\), \(k_c'\) – \(k_x\), \(k_x'\)
Liquid (alt. form) \(k_L\), \(k_L'\) – –
Unit of \(k\) \(\text{m}\cdot\text{s}^{-1}\) \(\dfrac{\text{kg mol}}{\text{s} \cdot \text{m}^2 \cdot \text{Pa}}\) \(\dfrac{\text{kg mol}}{\text{s} \cdot \text{m}^2 \cdot \text{mol frac}}\)

Conversions Between Mass Transfer Coefficients

Gas phase

\[\begin{align} k_c' \, c_T &= k_c' \frac{p_T}{RT} = k_c \frac{p_{Bm}}{RT} \\ &= k_G' \, p_T = k_G \, p_{Bm} \\ &= k_y' = k_y \, y_{Bm} \\ &= k_c \, y_{Bm} \, c_T = k_G \, y_{Bm} \, p_T \end{align}\]
  • \(p\) : total pressure
  • \(p_{Bm}\) : log-mean partial pressure of inert \(B\)
  • \(y_{Bm}\) : log-mean mole fraction of \(B\)
  • \(c_T = p_T/(RT)\)

Liquid phase

\[\begin{align} k_c' \, c &= k_L' \, c = k_L \, x_{Bm} \, c \\ &= k_L' \, \frac{\rho}{M} = k_x' = k_x \, x_{Bm} \end{align}\]
  • \(\rho\) : liquid density
  • \(M\) : molecular weight
  • \(x_{Bm}\) : log-mean mole fraction of solvent \(B\)

Summary

In this lecture, we talked about

  • The difficulty of studying boundary problems in mass transfer
  • The rise of equilibrium distribution coefficient \(K\) and mass transfer coefficient \(k\)
  • Link between mass transfer coefficient \(k\) and transfer layer

What To Learn Next

The concept of mass transfer coefficient \(k\) is both beautiful and ugly.

  • We gain the simplicity of expressing flux equations using simple formula,
  • We lose physical understanding about its origin in many systems

In next lectures, we will see:

  • How to use mass transfer coefficients for different phases (gas, liquid, solid)
  • How convective fluid transport is expressed using coefficients
  • How to perform mass transfer analysis based on transfer coefficients