CHE 318 Lecture 18

Dimensionless Numbers In Mass Transfer

Author

Dr. Tian Tian

Published

February 23, 2026

Note

Recitation: What Have Learned So Far?

Note

Please use the bullet points in this recitation to review the previous topics.

Topic 1: steady-state mass transfer

Governing equation

Geometry

Applications

Topic 2: unsteady-state mass transfer

Governing equation

Geometry

Applications

Topic 3: convective mass transfer (mass-transfer coefficient)

Governing equation

Geometry

Applications

Learning outcomes

After this lecture, you will be able to:

  • Recall key governing equations from steady, unsteady, and convective mass transfer.
  • Describe the roles of Reynolds, Schmidt, and Sherwood numbers in coefficient correlations.
  • Identify how \(k_c'\), concentration, and flux are linked in convective mass transfer problems.

Dimensionless number 1: \(N_{Re}\)

  • Reynolds number measures ratio between kinetic vs viscous forces of fluid flow
  • \(L_D\): characteristic length of system
\[\begin{align} N_{\text{Re}} = \frac{L_D v \rho}{\mu} \end{align}\]

Meaning of \(N_{Re}\)

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

\(N_{Re}\) for exhaust pipe

Dimensionless number 2: \(N_{Sc}\)

  • Schmidt number: ratio between momentum diffusivity and molecular diffusivity
  • Related to ratio of hydrodynamic layer and mass transfer layer thickness
\[\begin{align} N_{\text{Sc}} = \frac{\mu}{\rho D_{AB}} \end{align}\]

Meaning of \(N_{Sc}\)

  • \(N_{Sc}\): fluid boundary layer thicker or mass transfer thicker?
  • Similar to Prandt number in heat transfer
  • \(N_{Sc}^{1/3} = \dfrac{\delta}{\delta_c}\)

Analog of Prandt number

Dimensionless number 3: \(N_{Sh}\)

  • Sherwood number: ratio between convective mass transfer and molecular mass transfer
  • Has \(k_c'\) inside! โ€“> Usually a back-calculated number
\[\begin{align} N_{\text{Sh}} = \frac{k_c' L}{D_{AB}} \end{align}\]

How are \(k_c'\) correlated by dimensionless numbers?

  • The combinations of these properties โ€“> dimensionless number groups
  • The Chilton-Colburn \(j_D\)-factor: link to \(N_{\text{Sc}}\), \(N_{\text{Sh}}\), \(N_{\text{Re}}\)
\[\begin{align} j_D = f/2 &= \frac{k_c'}{v_{av}} (N_{\text{Sc}})^{2/3} \\ &= \frac{N_{\text{Sh}}}{N_{\text{Re}} N_{\text{Sc}}^{1/3}} \end{align}\]

General procedure to calculate \(k_c'\)

  • Dimensionless numbers solely from geometry and property: \(N_{Re}\), \(N_{Sc}\)
  • Dimensionless number having \(k_c'\): \(N_{Sh}\)
  • Link between them: \(j_D\)
  • How to obtain \(j_D\)?
    • Expression for different geometry / fluid flow
    • Use Table / Chart

Summary

  • Overview of dimensionless numbers to correlate mass transfer coefficients
  • Dimensionless numbers: grouping different regimes
  • Use table / charts to correct \(k_c\) (will discuss in Lecture 19)
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