CHE 318 Lecture 14

Mass Transfer Coefficients (II)

Author

Dr. Tian Tian

Published

February 4, 2026

Note

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Recap

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

Learning outcomes

After this lecture, you will be able to:

Recall flux equations written with mass transfer coefficients in different phases.
Describe why coefficient-based descriptions are useful for interfacial transport.
Identify the units and meanings of common mass transfer coefficients.
Apply conversions between coefficient forms used in gas- and liquid-phase analyses.

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!

Equilibrium concentration distribution

\[\begin{align} K = \frac{c_{g,i}}{c_{L,i}} \end{align}\]

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

Interfacial concentration profiles depend on equilibrium constraints and the relative resistance of each phase.
Matching fluxes across the interface allows us to determine interfacial compositions.
Mass transfer coefficients can be expressed in multiple forms, but their units and meanings must be tracked carefully.

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

--- title: "CHE 318 Lecture 14" subtitle: "Mass Transfer Coefficients (II)" author: "Dr. Tian Tian" date: "2026-02-04" format: html: {} revealjs: output-file: slides.html pdf: output-file: L14.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L14.pdf) - Handwritten notes 👉 [Open in pdf](./public/L14_annotated.pdf) ::: ::: ## Recap {.center} - Introduction to mass transfer coefficients - Link between mass transfer coefficient and diffusivity - Introduction to boundary concentration problem ## Learning outcomes {.center} After this lecture, you will be able to: - **Recall** flux equations written with mass transfer coefficients in different phases. - **Describe** why coefficient-based descriptions are useful for interfacial transport. - **Identify** the units and meanings of common mass transfer coefficients. - **Apply** conversions between coefficient forms used in gas- and liquid-phase analyses. ## 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: ```{=tex} \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_](./public/distribution-coeff-profile.png){width="95%"} ## Match of interfacial fluxes **At equilibrium**, if there is no resistance when transferring A between the interface, what is the mass balance equation? ```{=tex} \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**. ```{=tex} \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 ```{=tex} \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 ```{=tex} \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 ```{=tex} \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 ```{=tex} \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 ```{=tex} \begin{align} K = \frac{c_{g,i}}{c_{L,i}} \end{align} ``` 2. Flux matching ```{=tex} \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 ```{=html} <iframe width="100%" height="900" src="../../scripts/L14_boundary.html" title="Webpage example"></iframe> ``` ## 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 ```{=tex} \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 ```{=tex} \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<br/>(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 :::{.columns} :::{.column width="50%"} **Gas phase** ```{=tex} \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 $B$ - $c_T = p_T/(RT)$ ::: :::{.column width="50%"} **Liquid phase** ```{=tex} \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 - Interfacial concentration profiles depend on equilibrium constraints and the relative resistance of each phase. - Matching fluxes across the interface allows us to determine interfacial compositions. - Mass transfer coefficients can be expressed in multiple forms, but their units and meanings must be tracked carefully. ## 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

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 \(k\)	\(\text{m}\cdot\text{s}^{-1}\)	\(\dfrac{\text{kg mol}}{\text{s} \cdot \text{m}^2 \cdot \text{Pa}}\)	\(\dfrac{\text{kg mol}}{\text{s} \cdot \text{m}^2 \cdot \text{mol frac}}\)