CHE 318 Lecture 17

Dimensionless Numbers In Mass Transfer

Author

Dr. Tian Tian

Published

February 13, 2026

Note

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Note

This lecture contains the overview of dimensionless number relation to mass transfer coefficients. Detailed discussions will be included in Lecture 18, Lecture 19, Lecture 20 and Lecture 21

Learning outcomes

After this lecture, you will be able to:

Recall the physical meaning of key dimensionless numbers in mass transfer.
Describe how Reynolds, Schmidt, and Sherwood numbers correlate mass transfer behavior.
Identify suitable dimensionless correlations for different flow geometries.

Recap: boundary layer theory

The global mass transfer coefficient in a tube:

\[\begin{align} k_c' = \frac{0.664 D_{AB}}{L} N_{Re}^{0.5} N_{Sc}^{1/3} \end{align}\]

$N_{Re}$: Reynolds number
$N_{Sc}$: Schmidt number

Why do we need these dimensionless numbers?

Expressing fluxes using $k$ coefficients are easy
But do we need to measure $k$ for each system specifically?
- Of course NO!
We can correlate the values of $k$ measured in different geometries, velocities using dimensionless numbers
Similar treatment exists in heat and momentum (fluid) transfer

Dimensionless numbers in mass transfer

General form: $N_{\text{name}} = \dfrac{\text{Scale of effect 1}}{\text{Scale of Effect 2}}$
Schmidt number (ratio between momentum diffusivity and molecular diffusivity)

\[\begin{align} N_{\text{Sc}} = \frac{\mu}{\rho D_{AB}} \end{align}\]

Sherwood number (ratio between convective mass transfer and molecular mass transfer)

\[\begin{align} N_{\text{Sh}} = \frac{k_c' L}{D_{AB}} \end{align}\]

Reynolds number (ratio between kinetic vs viscous forces of fluid flow)

\[\begin{align} N_{\text{Re}} = \frac{L v \rho}{\mu} \end{align}\]

$L$: characteristic length of system
Location specific $N_{\text{Re}, x}$ also used

Meaning of dimensionless numbers – $N_{Re}$

$N_{Re}$: laminar flow vs turbulent flow
Varies with characteristic length $L_D$ (diameter for a pipe)

How are $k_c'$ correlated by dimensionless numbers?

Idea: mass transfer in flowing fluid described by $v$, $\rho$, $\mu$, $c$, $D_{AB}$ and geometry (characteristic length $L$)
The combinations of these properties –> dimensionless number groups ($N_{\text{Sc}}$, $N_{\text{Sh}}$, $N_{\text{Re}}$)
The Chilton-Colburn $j$-factor analog has most successful use in mass transfer

\[\begin{align} j_D = f/2 \\ &= \frac{k_c'}{v_{av}} (N_{\text{Sc}})^{2/3} \\ &= \boxed{\frac{k_c' L}{D_{AB}}} \boxed{\frac{\rho D_{AB}}{\mu}} \boxed{\frac{\mu}{L v_{av} \rho}} (N_\text{sc})^{2/3} \\ &= N_{\text{Sh}} N_{\text{Sc}}^{-1} N_{\text{Re}}^{-1} (N_\text{sc})^{2/3} \\ &= \frac{N_{\text{Sh}}}{N_{\text{Re}} N_{\text{Sc}}^{1/3}} \end{align}\]

$f$ is the Fanning Friction Factor (can be found in table)

General procedure to calculate $k_c'$

Calculate Reynolds number $N_{Re}$ from fluid properties + geometry
Determine flow regime (liquid)

$N_{Re} < 2100$ → laminar flow
$N_{Re} \ge 2100$ → turbulent flow

Evaluate friction factor $f$

Laminar flow:
\[ f = \frac{16}{N_{Re}} \]
Turbulent flow:
\[ f = \frac{\tau_s}{\tfrac12 \rho v^2},\qquad \tau_s = \frac{\Delta P_f\,\pi R^2}{2\pi R\,\Delta L} \]

Compute mass-transfer $j$-factor
\[ j_D = \frac{f}{2} \]
Obtain mass-transfer coefficient
\[ k_c' = j_D\,v_{av}\,N_{Sc}^{-2/3} \]

Use of empirical mass transfer laws

In many systems, flux and / or concentration profiles become hard to have simple form
Luckily we can simplify typical mass transfer problems as different geometries
- Cyliner / Pipe
- Parallel plates
- Flow around sphere
- Packed bed
We will show a few case studies for different geometries
Dimensionless numbers ($N_{Re}$, $N_{Sc}$, $N_{Sh}$) help determine governing equations

Case 1: mass transfer for flow inside pipes

Usually use the Linton & Sherwood chart
Valid for gas / liquid in both laminar & turbulent regimes

Flow inside pipe: chart

Flow inside pipes: solution procedure

Governing dimensionless quantity:

\[\begin{align} \frac{W}{D_{AB}\rho L} &= N_{Re}N_{Sc} \frac{D}{L} \frac{\pi}{4} \\ &= \frac{\text{[Total Forced Flow]} \text{(kg/s)}}{\text{[Total Diffusive Flow]} \text{(kg/s)}} \end{align}\]

If gas 👉 use the “rodlike flow” line
If liquid, distinguish 2 cases
- parabolic flow ($N_{Re} < 2100;\ \frac{W}{D_{AB}\rho L} > 400$)
- turbulent flow ($N_{Re} > 2100;\ 0.6 < N_{Sc} < 3000$)

Flow inside pipes: solution for liquid

Parabolic flow

\[\begin{align} \frac{c_A - c_{A,s}}{c_{A,i} - c_{A,s}} &= 5.5 \left[ \frac{W}{D_{AB}\,\rho\,L} \right]^{-\tfrac{2}{3}} \end{align}\]

$c_A$: exit concentration
$c_{A,i}, c_{A,s}$: inlet & surface concentration
$W$: flow rate in (kg/s)
$k_c'$ can be calculated by $j_D$

Turbulent flow

\[\begin{align} N_{Sh} &= k_c'\left(\frac{D}{D_{AB}}\right) \\ &= \frac{k_c\,p_{BM}}{P} \left(\frac{D}{D_{AB}}\right) \\ &= 0.023 \left(\frac{\rho D v}{\mu}\right)^{0.83} \left(\frac{\mu}{\rho D_{AB}}\right)^{0.33} \\ &= 0.023\, N_{Re}^{0.83}\, N_{Sc}^{0.33} \end{align}\]

Similar to the $j_D$ analog
Just need $N_{Re}$ and $N_{Sc}$ to determine $k_c'$
Characteristic length $D$ is pipe diameter!

Case 2: flow past parallel plates

Can be used for gases or evaporation of liquid
Distinguished between laminar & turbulent flow
$N_{Re}$ regime cutoff different in gas & liquid!
Characteristic length $L$: length of plate in flow direction

Flow past parallel plates: results

Laminar flow ($N_{Re} < 15,000$)

\[\begin{align} j_D &= 0.664 N_{Re, L}^{-0.5} \\ \frac{k_c' L}{D_{AB}} &= 0.664 N_{Re, L}^{0.5} N_{Sc}^{1/3} \end{align}\]

This follows our derivation of boundary layer theory

Turbulent flow

Gas: $15,000 < N_{Re}< 300,000$

\[\begin{align} j_D = 0.036 N_{Re, L}^{-0.2} \end{align}\]

Liquid: $600 < N_{Re}< 50,000$

\[\begin{align} j_D = 0.99 N_{Re, L}^{-0.5} \end{align}\]

Case 3: flow past single sphere

Frequent geometry in particle solutions
Low Reynolds regime 👉 solution for stagnant diffusion on spherical surface
High Reynolds regime 👉 correct $N_{Sh}$ and back calculate $k_c'$

Flow past single sphere: results

Low Reynolds ($N_{Re} < 2$)

\[\begin{align} N_A &= \boxed{\frac{2 D_{AB}}{D_p}} (c_{A1} - c_{A2}) \\ &= k_c (c_{A1} - c_{A2}) \\ &= \frac{k_c'}{x_{Bm}}(c_{A1} - c_{A2}) \\ \end{align}\]

For $x_{Bm} \approx 1$, we have:

\[ k_c' = \frac{2D_{AB}}{D_p} \]

Sherwood number: $N_{Sh} = 2$

High Reynolds ($N_{Re} > 2$)

Gas:

\[\begin{align} &N_{Sh} = 2 + 0.552 N_{Re}^{0.53} N_{Sc}^{1/3} \\ &0.6 < N_{Sc} < 2.7\;\ N_{Re} < 48000 \end{align}\]

Liquid:

\[\begin{align} N_{Sh} &= 2 + 0.95 N_{Re}^{0.5} N_{Sc}^{1/3};\ N_{Re} < 2000\\ N_{Sh} &= 0.347 N_{Re}^{0.62} N_{Sc}^{1/3};\ 2000 < N_{Re} < 17000 \end{align}\]

Back calculate $k_c' = N_{Sh} \frac{D_{AB}}{D_p}$

Case 4: mass transfer for packed beds

Very common geometry for chemical engineering
- Adsorption and desorption through solid particles (gases and liquids)
- Catalytic processes with very large surface area
Geometry characteristics: void fraction $\varepsilon$: \[ \varepsilon = \frac{\text{void space}}{\text{total space}} = \frac{\text{void space}}{\text{void space} + \text{solid space}} \]
- Typically $0.3 < \varepsilon < 0.5$
- Void fraction is difficult to measure experimentally

Correlation equations in packed bed

Correlation 1, applicable to:

gase with $10 < N_{Re} <10,000$
liquid with $10 < N_{Re} < 1500$

\[\begin{align} j_D &= j_H = \frac{0.4548}{\varepsilon} N_{Re}^{-0.4069} \\ N_{Re} &= \frac{D_p v' \rho}{\mu} \end{align}\]

$D_p$: (average) particle diameter
$v’$: superficial velocity in the tube without packing

Correlation equations in packed bed (II)

Correlation 2, applicable to:

liquid with $0.0016 < N_{Re} < 55$, $165 < N_{Sc} < 70000$

\[\begin{align} j_D = \frac{1.09}{\varepsilon} N_{Re}^{-2/3} \end{align}\]

liquid with $55 < N_{Re} < 1500$, $165 < N_{Sc} < 10690$

\[\begin{align} j_D = \frac{0.250}{\varepsilon} N_{Re}^{-0.31} \end{align}\]

Correlation equations in packed bed (III)

Correlation 3, applicable to fluidized beds

$10 < N_{Re} < 4000$ (gas & liquid)

\[\begin{align} j_D = \frac{0.4548}{\varepsilon} N_{Re}^{-0.4069} \end{align}\]

$1 < N_{Re} < 10$ (liquid only)

\[\begin{align} j_D = \frac{1.1068}{\varepsilon} N_{Re}^{-0.72} \end{align}\]

Packed bed calculation steps

Known value from operational column: $\varepsilon$, $V_b$ (total volume), $D_p$, $D_{AB}$, $\mu$, $\rho$, etc.
Depend on the operational range, calculate $N_{Re}$, $N_Sc$ 👉 choose the equation for $j_D$
Obtain $k_c$ from $j_D$ value
Calculate flux $N_A$
Estimate effective area $A_{\text{eff}}$ inside the columne 👉 $\overline{N}_A = A_{\text{eff}} N_A$

Caveats in packed bed problems (1)

Estimate the effective area?
- First calculate the effective surface area per volume $a$ then $A_{\text{eff}}$
\[\begin{align} a &= \frac{6 (1 - \varepsilon)}{D_p} \\ A_{eff} &= a V_b \end{align}\]

Caveats in packed bed problems (2)

Use log-mean driving force (see Lecture 21)

\[\begin{align} \overline{N}_A &= A_{\text{eff}} N_A \\ &= A_{\text{eff}} k_c \frac{ (c_{A, i} - c_{A1}) - (c_{A, i} - c_{A2})} { \ln\!\frac{(c_{A, i} - c_{A1})}{(c_{A, i} - c_{A2})} } \end{align}\]

where

$c_{A, i}$: surface concentration
$c_{A1}, c_{A2}$: in- and outlet concentrations

Caveats in packed bed problems (3)

Mass-flow balance

\[\begin{align} \overline{N}_A &= A_{eff} N_A \\ &= Q (c_{A2} - c_{A1}) \end{align}\]

where $Q$ is the volumetric flow rate (unit m$^3$/s).

These equations will give rise to solving the flow in packed bed problem.

Summary

Dimensionless numbers can be used to correlate mass transfer problems in different flow rate, dimension etc
Typically, start with a known geometry (pipe? parallel plate? sphere? packed bed?)
Find the correlation with dimensionless numbers $N_{Re}$, $N_{Sc}$
Calculate the final mass transfer rate

--- title: "CHE 318 Lecture 17" subtitle: "Dimensionless Numbers In Mass Transfer" author: "Dr. Tian Tian" date: "2026-02-13" format: html: {} revealjs: output-file: slides.html pdf: output-file: L17.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L17.pdf) - Handwritten notes 👉 [Open in pdf](./public/L17_annotated.pdf) ::: ::: ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} This lecture contains the overview of dimensionless number relation to mass transfer coefficients. Detailed discussions will be included in [Lecture 18](../L18), [Lecture 19](../L19), [Lecture 20](../L20) and [Lecture 21](../L21) ::: ::: --- ## Learning outcomes {.center} After this lecture, you will be able to: - **Recall** the physical meaning of key dimensionless numbers in mass transfer. - **Describe** how Reynolds, Schmidt, and Sherwood numbers correlate mass transfer behavior. - **Identify** suitable dimensionless correlations for different flow geometries. ## Recap: boundary layer theory The global mass transfer coefficient in a tube: ```{=tex} \begin{align} k_c' = \frac{0.664 D_{AB}}{L} N_{Re}^{0.5} N_{Sc}^{1/3} \end{align} ``` - $N_{Re}$: Reynolds number - $N_{Sc}$: Schmidt number ## Why do we need these dimensionless numbers? - Expressing fluxes using $k$ coefficients are easy - But do we need to measure $k$ for each system specifically? - Of course **NO**! - We can correlate the values of $k$ measured in different geometries, velocities using **dimensionless numbers** - Similar treatment exists in heat and momentum (fluid) transfer ## Dimensionless numbers in mass transfer {.smaller} - **General form**: $N_{\text{name}} = \dfrac{\text{Scale of effect 1}}{\text{Scale of Effect 2}}$ - **Schmidt number** (ratio between momentum diffusivity and molecular diffusivity) ```{=tex} \begin{align} N_{\text{Sc}} = \frac{\mu}{\rho D_{AB}} \end{align} ``` - **Sherwood number** (ratio between convective mass transfer and molecular mass transfer) ```{=tex} \begin{align} N_{\text{Sh}} = \frac{k_c' L}{D_{AB}} \end{align} ``` - **Reynolds number** (ratio between kinetic vs viscous forces of fluid flow) ```{=tex} \begin{align} N_{\text{Re}} = \frac{L v \rho}{\mu} \end{align} ``` - $L$: characteristic length of system - Location specific $N_{\text{Re}, x}$ also used ## Meaning of dimensionless numbers -- $N_{Re}$ - $N_{Re}$: laminar flow vs turbulent flow - Varies with characteristic length $L_D$ (diameter for a pipe) ![$N_{Re}$ for exhaust pipe](./public/exhause-pipe-reynolds.jpeg) ## How are $k_c'$ correlated by dimensionless numbers? - Idea: mass transfer in flowing fluid described by $v$, $\rho$, $\mu$, $c$, $D_{AB}$ and geometry (characteristic length $L$) - The combinations of these properties --> dimensionless number groups ($N_{\text{Sc}}$, $N_{\text{Sh}}$, $N_{\text{Re}}$) - The Chilton-Colburn $j$-factor analog has most successful use in mass transfer ```{=tex} \begin{align} j_D = f/2 \\ &= \frac{k_c'}{v_{av}} (N_{\text{Sc}})^{2/3} \\ &= \boxed{\frac{k_c' L}{D_{AB}}} \boxed{\frac{\rho D_{AB}}{\mu}} \boxed{\frac{\mu}{L v_{av} \rho}} (N_\text{sc})^{2/3} \\ &= N_{\text{Sh}} N_{\text{Sc}}^{-1} N_{\text{Re}}^{-1} (N_\text{sc})^{2/3} \\ &= \frac{N_{\text{Sh}}}{N_{\text{Re}} N_{\text{Sc}}^{1/3}} \end{align} ``` - $f$ is the Fanning Friction Factor (can be found in table) ## General procedure to calculate $k_c'$ :::{.columns} :::{.column width="50%"} 1. Calculate Reynolds number $N_{Re}$ from fluid properties + geometry 2. Determine flow regime (liquid) - $N_{Re} < 2100$ → laminar flow - $N_{Re} \ge 2100$ → turbulent flow 3. Evaluate friction factor $f$ - Laminar flow: $$ f = \frac{16}{N_{Re}} $$ - Turbulent flow: $$ f = \frac{\tau_s}{\tfrac12 \rho v^2},\qquad \tau_s = \frac{\Delta P_f\,\pi R^2}{2\pi R\,\Delta L} $$ ::: :::{.column width="50%"} 4. Compute mass-transfer $j$-factor $$ j_D = \frac{f}{2} $$ 5. Obtain mass-transfer coefficient $$ k_c' = j_D\,v_{av}\,N_{Sc}^{-2/3} $$ ::: ::: ## Use of empirical mass transfer laws - In many systems, flux and / or concentration profiles become hard to have simple form - Luckily we can simplify typical mass transfer problems as different geometries - Cyliner / Pipe - Parallel plates - Flow around sphere - Packed bed - We will show a few case studies for different geometries - Dimensionless numbers ($N_{Re}$, $N_{Sc}$, $N_{Sh}$) help determine governing equations ## Case 1: mass transfer for flow inside pipes - Usually use the Linton & Sherwood chart - Valid for gas / liquid in both laminar & turbulent regimes ## Flow inside pipe: chart ![](./public/fluid-pipe-chart.png) ## Flow inside pipes: solution procedure - Governing dimensionless quantity: ```{=tex} \begin{align} \frac{W}{D_{AB}\rho L} &= N_{Re}N_{Sc} \frac{D}{L} \frac{\pi}{4} \\ &= \frac{\text{[Total Forced Flow]} \text{(kg/s)}}{\text{[Total Diffusive Flow]} \text{(kg/s)}} \end{align} ``` - If gas 👉 use the "rodlike flow" line - If liquid, distinguish 2 cases - parabolic flow ($N_{Re} < 2100;\ \frac{W}{D_{AB}\rho L} > 400$) - turbulent flow ($N_{Re} > 2100;\ 0.6 < N_{Sc} < 3000$) ## Flow inside pipes: solution for liquid :::{.columns} :::{.column width="50%"} **Parabolic flow** ```{=tex} \begin{align} \frac{c_A - c_{A,s}}{c_{A,i} - c_{A,s}} &= 5.5 \left[ \frac{W}{D_{AB}\,\rho\,L} \right]^{-\tfrac{2}{3}} \end{align} ``` - $c_A$: exit concentration - $c_{A,i}, c_{A,s}$: inlet & surface concentration - $W$: flow rate in (kg/s) - $k_c'$ can be calculated by $j_D$ ::: :::{.column width="50%"} **Turbulent flow** ```{=tex} \begin{align} N_{Sh} &= k_c'\left(\frac{D}{D_{AB}}\right) \\ &= \frac{k_c\,p_{BM}}{P} \left(\frac{D}{D_{AB}}\right) \\ &= 0.023 \left(\frac{\rho D v}{\mu}\right)^{0.83} \left(\frac{\mu}{\rho D_{AB}}\right)^{0.33} \\ &= 0.023\, N_{Re}^{0.83}\, N_{Sc}^{0.33} \end{align} ``` - Similar to the $j_D$ analog - Just need $N_{Re}$ and $N_{Sc}$ to determine $k_c'$ - Characteristic length $D$ is pipe diameter! ::: ::: ## Case 2: flow past parallel plates - Can be used for gases or evaporation of liquid - Distinguished between laminar & turbulent flow - $N_{Re}$ regime cutoff different in gas & liquid! - Characteristic length $L$: length of plate in flow direction ## Flow past parallel plates: results :::{.columns} :::{.column width="50%"} **Laminar flow** ($N_{Re} < 15,000$) ```{=tex} \begin{align} j_D &= 0.664 N_{Re, L}^{-0.5} \\ \frac{k_c' L}{D_{AB}} &= 0.664 N_{Re, L}^{0.5} N_{Sc}^{1/3} \end{align} ``` - This follows our derivation of boundary layer theory ::: :::{.column width="50%"} **Turbulent flow** - Gas: $15,000 < N_{Re}< 300,000$ ```{=tex} \begin{align} j_D = 0.036 N_{Re, L}^{-0.2} \end{align} ``` - Liquid: $600 < N_{Re}< 50,000$ ```{=tex} \begin{align} j_D = 0.99 N_{Re, L}^{-0.5} \end{align} ``` ::: ::: ## Case 3: flow past single sphere - Frequent geometry in particle solutions - Low Reynolds regime 👉 solution for stagnant diffusion on spherical surface - High Reynolds regime 👉 correct $N_{Sh}$ and back calculate $k_c'$ ## Flow past single sphere: results :::{.columns} :::{.column width="35%"} **Low Reynolds** ($N_{Re} < 2$) ```{=tex} \begin{align} N_A &= \boxed{\frac{2 D_{AB}}{D_p}} (c_{A1} - c_{A2}) \\ &= k_c (c_{A1} - c_{A2}) \\ &= \frac{k_c'}{x_{Bm}}(c_{A1} - c_{A2}) \\ \end{align} ``` - For $x_{Bm} \approx 1$, we have: $$ k_c' = \frac{2D_{AB}}{D_p} $$ - Sherwood number: $N_{Sh} = 2$ ::: :::{.column width="65%"} **High Reynolds** ($N_{Re} > 2$) - Gas: ```{=tex} \begin{align} &N_{Sh} = 2 + 0.552 N_{Re}^{0.53} N_{Sc}^{1/3} \\ &0.6 < N_{Sc} < 2.7\;\ N_{Re} < 48000 \end{align} ``` - Liquid: ```{=tex} \begin{align} N_{Sh} &= 2 + 0.95 N_{Re}^{0.5} N_{Sc}^{1/3};\ N_{Re} < 2000\\ N_{Sh} &= 0.347 N_{Re}^{0.62} N_{Sc}^{1/3};\ 2000 < N_{Re} < 17000 \end{align} ``` - Back calculate $k_c' = N_{Sh} \frac{D_{AB}}{D_p}$ ::: ::: ## Case 4: mass transfer for packed beds - Very common geometry for chemical engineering - Adsorption and desorption through solid particles (gases and liquids) - Catalytic processes with very large surface area - Geometry characteristics: void fraction $\varepsilon$: $$ \varepsilon = \frac{\text{void space}}{\text{total space}} = \frac{\text{void space}}{\text{void space} + \text{solid space}} $$ - Typically $0.3 < \varepsilon < 0.5$ - Void fraction is difficult to measure experimentally ## Correlation equations in packed bed Correlation 1, applicable to: - gase with $10 < N_{Re} <10,000$ - liquid with $10 < N_{Re} < 1500$ ```{=tex} \begin{align} j_D &= j_H = \frac{0.4548}{\varepsilon} N_{Re}^{-0.4069} \\ N_{Re} &= \frac{D_p v' \rho}{\mu} \end{align} ``` - $D_p$: (average) particle diameter - $v’$: superficial velocity in the tube without packing ## Correlation equations in packed bed (II) Correlation 2, applicable to: - liquid with $0.0016 < N_{Re} < 55$, $165 < N_{Sc} < 70000$ ```{=tex} \begin{align} j_D = \frac{1.09}{\varepsilon} N_{Re}^{-2/3} \end{align} ``` - liquid with $55 < N_{Re} < 1500$, $165 < N_{Sc} < 10690$ ```{=tex} \begin{align} j_D = \frac{0.250}{\varepsilon} N_{Re}^{-0.31} \end{align} ``` ## Correlation equations in packed bed (III) Correlation 3, applicable to fluidized beds - $10 < N_{Re} < 4000$ (gas & liquid) ```{=tex} \begin{align} j_D = \frac{0.4548}{\varepsilon} N_{Re}^{-0.4069} \end{align} ``` - $1 < N_{Re} < 10$ (liquid only) ```{=tex} \begin{align} j_D = \frac{1.1068}{\varepsilon} N_{Re}^{-0.72} \end{align} ``` ## Packed bed calculation steps 1. Known value from operational column: $\varepsilon$, $V_b$ (total volume), $D_p$, $D_{AB}$, $\mu$, $\rho$, etc. 2. Depend on the operational range, calculate $N_{Re}$, $N_Sc$ 👉 choose the equation for $j_D$ 3. Obtain $k_c$ from $j_D$ value 4. Calculate flux $N_A$ 5. Estimate effective area $A_{\text{eff}}$ inside the columne 👉 $\overline{N}_A = A_{\text{eff}} N_A$ ## Caveats in packed bed problems (1) - Estimate the effective area? - First calculate the effective surface area per volume $a$ then $A_{\text{eff}}$ ```{=tex} \begin{align} a &= \frac{6 (1 - \varepsilon)}{D_p} \\ A_{eff} &= a V_b \end{align} ``` ## Caveats in packed bed problems (2) - Use log-mean driving force (see [Lecture 21](../L21)) ```{=tex} \begin{align} \overline{N}_A &= A_{\text{eff}} N_A \\ &= A_{\text{eff}} k_c \frac{ (c_{A, i} - c_{A1}) - (c_{A, i} - c_{A2})} { \ln\!\frac{(c_{A, i} - c_{A1})}{(c_{A, i} - c_{A2})} } \end{align} ``` where - $c_{A, i}$: surface concentration - $c_{A1}, c_{A2}$: in- and outlet concentrations ## Caveats in packed bed problems (3) - Mass-flow balance ```{=tex} \begin{align} \overline{N}_A &= A_{eff} N_A \\ &= Q (c_{A2} - c_{A1}) \end{align} ``` where $Q$ is the volumetric flow rate (unit m$^3$/s). These equations will give rise to solving the flow in packed bed problem. ## Summary - Dimensionless numbers can be used to correlate mass transfer problems in different flow rate, dimension etc - Typically, start with a known geometry (pipe? parallel plate? sphere? packed bed?) - Find the correlation with dimensionless numbers $N_{Re}$, $N_{Sc}$ - Calculate the final mass transfer rate