---
title: "CHE 318 Lecture 08"
subtitle: "Steady State Mass Transfer: More Examples"
author: "Dr. Tian Tian"
date: "2026-01-21"
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output-file: slides.html
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output-file: L08.pdf
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::: {.callout-note}
- Slides 👉 [Open presentation🗒️](./slides.html)
- PDF version of course note 👉 [Open in pdf](./L08.pdf)
- Handwritten notes 👉 [Open in pdf](./public/L08_annotated.pdf)
:::
:::
## Recap
- Mass transfer equations in other geometries
- Use M.B. for $\overline{N}_A$, not $N_A$
- Be careful with the coordinate system (sphere? cylinder?)
- Solving examples with pseudo steady state assumption
## Learning Outcomes {.center}
(Continue from Lecture 07) After today's lecture, you will be able to:
- **Solve** P.S.S problems in spherical coordinates
- **Recall** difference between steady and unsteady state solutions
- **Analyze** time-dependent concentration profiles for U.S.S situations.
# P.S.S Examples (continued)
## Example 1: Diffusion Through Stagnant B with Changing Path Length
_Adapted from Geankoplis 6.2-3_
Water vapor diffuses through a stagnant gas in a narrow vertical tube, dry air is constantly blown at the top of tube.
At time $t$, the liquid level is a distance $z$ from the tube top (i.e., the diffusion path length is $z$).
As diffusion proceeds, the liquid level drops slowly, so $z$ increases with time. The liquid has density $\rho_A$, and molecular weight $M_A$
1) Derive an expression for the time $t_F$ required for the level to drop such that the diffusion path length changes from $z=z_0$ at $t=0$ to $z=z_F$ at $t=t_F$.
## Example 1: solutions
:::{.callout-tip}
1. Use pseudo steady-state assumption
2. $N_A$ change over time!
:::
Answer
$$
t_F = \frac{\rho_A (z_F^2 - z_0^2) RT p_{Bm}}{2 D_{AB} M_A p_{T}} \frac{1}{(p_{A1} - p_{A2})}
$$
## Example 2: Determine $D_{AB}$ Through Evaporation
_Adapted from Griskey 10-2_
Sample setup as example 4, a vertical tube of diameter $D=0.01128$ m
contains a liquid volatile species $A$ (chloropicrin, $CCl_3NO_2$)
evaporating into stagnant air ($B$) at 1 atm. The gas-phase
diffusion of $A$ occurs through the air column above the liquid
surface.
At $t=0$, the distance from the tube top to the liquid surface is $z_0 = 0.0388$ m, after $t=1$ day, the distance is $z_1 = 0.0412$ m.
- Vapor pressure at the interface: $p_{A1} = 3178.3$ Pa
- Liquid density: $\rho_A = 1650$ kg/m$^3$
- Molecular weight: $M_A = 164.39$ kg/kmol
1) Use your expression from example 4, determine the binary diffusivity $D_{AB}$ of $A$ in air.
## Example 2: solutions
:::{.callout-tip}
Pseudo steady state solution and assuming $N_A=\text{const}$ solution differ very little. Why?
:::
Answer:
1. Pseudo-steady state: $D_{AB} = 8.56\times{}10^{-6}\ \text{m}^2/\text{s}$
1. $N_A$ constant: $D_{AB} = 8.75\times{}10^{-6}\ \text{m}^2/\text{s}$ ($+2.2\%$ error)
## Example 3: P.S.S For Diffusion Through Sphere
_Adapted from Geankoplis Ex 6.2-4_
A sphere of naphthalene having a radius of 2.0 mm is suspended in a large volume of still air at 318 K and
$1.01325 \times{}10^5$ Pa (1 atm). The surface temperature of the naphthalene can be assumed to be at 318 K and
its vapor pressure at 318 K is 0.555 mm Hg. The $D_AB$ of naphthalene in air at 318 K is $6.92\times{}10^{−6}\ \text{m}^2/\text{s}$.
1. Calculate the rate of evaporation of naphthalene from the surface.
2. Write the expression for the time $t_F$ to evaporate a sphere from radius $r_0$ to $r_F$. The solid density for naphthalene is $\rho$ and molecular weight is $M_A$.
3. What is the $t_F$ value when the sphere is completely evaporated?
## Example 3: solutions
:::{.callout-tip}
Similar setup as example 5. $N_A$ is time-dependent
:::
Answer:
1) $N_A = 9.68\times{}10^{-8}\ \text{kg mol}/\text{m}^2/\text{s}$
2) Expression for $t_F{r=r_F}$:
$$
t_F(r=r_F) = \frac{\rho RT p_{Bm}}{2 M_A D_{AB} p_T} \frac{1}{(p_{A1} - p_{A2})} (r_0^2 - r_F^2)
$$
3) Expression for $t_F(r=0)$:
$$
t_F(r=0) = \frac{\rho RT p_{Bm} r_0^2}{2 M_A D_{AB} p_T} \frac{1}{(p_{A1} - p_{A2})}
$$
Compare the solutions with Example 4. We can also measure the
diffusivity of volatile organic molecules using the sphere evaporation
method!
<!-- # Introduction to Unsteady State Mass Transfer -->
<!-- ## What Is Unsteady-State Mass Transfer? -->
<!-- {width="65%" .center} -->
<!-- ```{=tex} -->
<!-- \text{[In]} - \text{[Out]} + \text{[Gen]} = \text{[Acc]} -->
<!-- ``` -->
<!-- - Concentration varies with time $\partial c/\partial t \neq 0$ -->
<!-- - Accumulation term is non-zero $\text{[Acc]} \neq 0$ -->
<!-- - Requires time-dependent mass balances -->
<!-- - Common in transient diffusion, start-up, and response problems -->
<!-- - **More general** than S.S. -->
<!-- ## Governing Equation for U.S.S M.T. -->
<!-- Consider a control volume in 1D transport, the mass balance equation becomes -->
<!-- ```{=tex} -->
<!-- \begin{align*} -->
<!-- \text{[Acc]} &= \text{[In]} - \text{[Out]} + \text{[Gen]} \\ -->
<!-- S \Delta z \frac{\partial c_A}{\partial t} -->
<!-- &= S (N_A \vert_{z=z} - N_A \vert_{z=z+\Delta z}) + \text{[Gen]} S \Delta z \\ -->
<!-- \frac{\partial c_A}{\partial t} -->
<!-- &= -->
<!-- -\frac{\partial N_A(z)}{\partial z}\vert_z + r_{A} -->
<!-- \end{align*} -->
<!-- ``` -->
<!-- where $r_{A}$ is the generation rate for A (e.g. local reaction). -->
<!-- This is the governing equation for all time-dependent mass transfer! -->
<!-- ## Comparison between Flux equation and Mass Balance -->
<!-- :::{.columns} -->
<!-- :::{.column width="50%"} -->
<!-- #### Flux equation -->
<!-- ```{=tex} -->
<!-- \begin{align*} -->
<!-- N_A = J_{Az}^* + x_A(N_A + N_B) -->
<!-- \end{align*} -->
<!-- ``` -->
<!-- - Amount of material moved **in** and **out** of controled volume -->
<!-- - $J_{Az}^*$: Fick's first law of diffusion -->
<!-- - Can be used for S.S ($d N_A/dz = 0$) and U.S.S -->
<!-- ::: -->
<!-- :::{.column width="50%"} -->
<!-- #### Mass balance eq -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- \frac{\partial c_A}{\partial t} = - \frac{\partial N_A}{\partial x} + r_A -->
<!-- \end{align} -->
<!-- ``` -->
<!-- - Change of local $c_A$ over time -->
<!-- - Need flux equation solution first -->
<!-- - Can be used for S.S ($\frac{\partial c_A}{\partial t} = 0$) and U.S.S. -->
<!-- ::: -->
<!-- ::: -->
<!-- ## Mass Balance: Extension to 3D -->
<!-- {width="75%" .center} -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- \frac{\partial c_A}{\partial t} -->
<!-- &= r_A -->
<!-- - \frac{\partial N_{Ax}}{\partial x} -->
<!-- - \frac{\partial N_{Ay}}{\partial y} -->
<!-- - \frac{\partial N_{Az}}{\partial z} \\ -->
<!-- &= r_A - \nabla \cdot \vec{N}_A -->
<!-- \end{align} -->
<!-- ``` -->
<!-- - $\nabla \cdot$ is the **divergence** operator (not $\Delta$, not gradient!) -->
<!-- - $\vec{N}_A$ is generally a 3D vector field -->
<!-- --- -->
<!-- ## Dissecting the General Equation for Mass Balance -->
<!-- ```{=tex} -->
<!-- \begin{align*} -->
<!-- r_A - \frac{\partial c_A}{\partial t} -->
<!-- &= \nabla \cdot \left[\vec{J}_{A}^{*} + x_A(\vec{N}_A + \vec{N}_B) \right] \\ -->
<!-- &= \nabla \cdot \left[\vec{J}_{A}^{*} + c_A \vec{v}_m \right] \\ -->
<!-- &= \nabla \cdot \left[ -D_{AB} \nabla c_A + c_A \vec{v}_m \right] -->
<!-- \end{align*} -->
<!-- ``` -->
<!-- - We have fluid velocity $\vec{v}_m$ on the R.H.S -->
<!-- - Do we know $N_A$ and $N_B$ relation? -->
<!-- - In general this is hard to solve (coupling fluid with mass transfer) -->
<!-- - -->
<!-- - Often interested in several limiting cases -->
<!-- ## Special Cases of Unsteady-State Mass Transfer -->
<!-- - Case 1: Constant $D_{AB}$ -->
<!-- ```{=tex} -->
<!-- \begin{align*} -->
<!-- frac{\partial c_A}{\partial t} -->
<!-- &= -->
<!-- D_{AB}\nabla^2 c_A -->
<!-- - c_A \nabla \cdot \vec{v}_m -->
<!-- - \vec{v}_m \cdot \nabla c_A -->
<!-- + r_A -->
<!-- \end{align*} -->
<!-- ``` -->
<!-- - Case 2: EMCD for gases at constant $p_T$, $r_A = 0$ -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- \frac{\partial c_A}{\partial t} -->
<!-- &= -->
<!-- D_{AB}\nabla^2 c_A -->
<!-- \qquad \text{(Fick’s second law)} -->
<!-- \end{align} -->
<!-- ``` -->
<!-- - Case 3: Constant $\rho$ and $D_{AB}$ (e.g. imcompressible liquids) -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- \frac{\partial c_A}{\partial t} -->
<!-- &= -->
<!-- D_{AB}\nabla^2 c_A -->
<!-- - \vec{v}_m \cdot \nabla c_A -->
<!-- + r_A -->
<!-- \end{align} -->
<!-- ``` -->
<!-- where $\nabla \cdot \vec{v}_m = 0$ -->
<!-- ## What exactly do we solve? -->
<!-- For U.S.S M.T, we typically need -->
<!-- 1. Governing equation (PDE) from any limiting case -->
<!-- 2. Initial conditions $c_A(z, t=0)$ -->
<!-- 3. Boundary conditions (B.C.) -->
<!-- - Dirichlet B.C. (e.g. $c_A(z=0) = c_0$) -->
<!-- - Neumann B.D. (e.g. $N_A(z=0) = N_{A0}$, constant flux) -->
<!-- 4. Solving analytically or numerically -->
<!-- 5. Get $c_A(z, t)$, $x_A(z, t)$, $N_A(x, t)$ -->
<!-- 6. Steady-state solutions often means $c_A(z, t\to \infty)$ -->
<!-- ## B.C. Case A. Concentration at surfaces -->
<!-- {width="75%"} -->
<!-- - Interface can be gas|liquid, liquid|solid, gas|solid -->
<!-- - Often assuming equilibrium -->
<!-- $$ -->
<!-- c_{A}\vert_{\text{surf}} = c_{As}\qquad\text{eq. solubility} -->
<!-- $$ -->
<!-- ## B.C. Case 2: Chemical Reactions -->
<!-- {width="75%"} -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- N_A\big|_{\text{surf}} = \nu_A\, r_A -->
<!-- \end{align} -->
<!-- ``` -->
<!-- - Surface reaction couples mass transfer and kinetics -->
<!-- - Molar flux at surface determined by reaction rate -->
<!-- - Generally Neumann boundary -->
<!-- - $\nu_A$: stoichiometric ratio -->
<!-- ## U.S.S Example: Diffusion Through Stagnant B -->
<!-- We have seen in previous examples how to solve the molar flux of -->
<!-- liquid evaporating into stagnant air. Let's see the same system but in unsteady state. -->
<!-- **Question**: liquid methanol (A) evaporates inside stagnant air (B) -->
<!-- inside a vertical tube at constant temperature $T$ and pressure -->
<!-- $p_T$. At the vent of the system dry air is continuous blown. Plot the -->
<!-- molar fraction $x_A$ as a function of $z$ and time $t$. Assume the liquid level is $z_0$ away from the vent and does not change during the evaporation process. -->
<!-- ## Step 1: Species Mass Balance (Unsteady, 1D) -->
<!-- For a differential slice $A\,dz$, write the mass balance -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- \text{[IN]} - \text{[OUT]} &= \text{[ACC]} \\ -->
<!-- A N_A|vert_{z=z} - A N_A|vert_{z=z+\Delta z} -->
<!-- &= \frac{\partial}{\partial t}\left(A\,dz\,c_A\right) -->
<!-- -\frac{\partial N_A}{\partial z} = \frac{\partial c_A}{\partial t} -->
<!-- \end{align} -->
<!-- ``` -->
<!-- We would have $\frac{\partial c_B}{\partial t} = -\frac{\partial N_B}{\partial z}$ -->
<!-- ## Step 2: Couple With Flux Equation -->
<!-- This is a diffusion through stagnant B case, we can directly write -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- N_A(z, t) = -c_T D_{AB}\frac{\partial x_A(z, t)}{\partial z} + x_A \left[N_A(z, t) + N_B(z, t)\right] -->
<!-- \end{align} -->
<!-- ``` -->
<!-- - Can we use $N_B=0$ in this case? -->
<!-- - **No**, $N_B$ changes by $z, t$! -->
<!-- - $N_B=0$ only at $z=0$ (liquid interface) -->
<!-- ## Step 3: Conservation Equations -->
<!-- Generally, we still need to know the relation between $N_A$ and $N_B$ -->
<!-- to solve the mass-balance-flux equations. -->
<!-- The total concentration $c_T=c_A + c_B$ is conserved, therefore we have constrains -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- \text{[In]}_{T} - \text{[Out]}_{T} &= 0 \\ -->
<!-- \text{[In]}_A - \text{[Out]}_A &= -\text{[In]}_B + \text{[Out]}_B \\ -->
<!-- \frac{\partial N_A(z, t)}{\partial z} &= -\frac{\partial N_B(z, t)}{\partial z} -->
<!-- \end{align} -->
<!-- ``` -->
<!-- ## Step 4: Boundary Conditions -->
<!-- Boundary conditions (Left, Right, any time) -->
<!-- - $x_A(0, t) = x_{A0}$ (equilibrium vapor fraction) -->
<!-- - $x_A(L, t) = 0$ (dry air) -->
<!-- - $N_B(0, t) = 0$ (No-flux boundary for B) -->
<!-- The last B.C for $N_B(0, t)$ gives: -->
<!-- ```{=tex} -->
<!-- \begin{align} -->
<!-- N_A(0, t) = -\frac{c_T D_{AB}}{1 - x_{A0}}\frac{\partial x_A(0, t)}{\partial z} -->
<!-- \end{align} -->
<!-- ``` -->
## Summary
- Pseudo-steady state solutions to diffusion-evaporation problems
- We will discuss about unsteady state mass transfer next lecture!