Where Does Sherwood
Hide His Unit?

DOFPro Team

Introduction

  • This video, Where Does Sherwood Hide His Unit? Principles, explains the basics of dimensionless numbers.
  • The companion video, What the Schmidt? Examples, presents examples of dimensionless numbers used in chemical and thermal principles.

This video covers three concepts

1 – Dimensionless numbers or dimensionless groups

2 – Dimensional homogeneity

3 – Arguments of transcendental functions are
      dimensionless.

Dimensionless Numbers

Reduce data complexity

Found using Buckingham \(\pi\) theorem

Example is Reynolds number, \(Re\).

\(Re = \frac{\rho u D}{\mu} = \frac{u D}{\nu}\)

\(\nu = \frac{\mu}{\rho}\)

\(Re\) is ratio of momentum to viscous drag.

\(\mathrm{Momentum} \propto \rho u\)

\(\mathrm{Viscous\ force} \propto \mu/D\)

\(Sh = \frac{k l}{\mathscr{D}}\) is ratio of convective to diffusive transport.

Dimensional Homogeneity

All terms in an equation must have the same dimensions.

\(\ \ \ \ x - x_0 = v_0 t + \frac{1}{2} a t^2\)

\(x\) and \(x_0\) have units of length.

\(v_0 t\) must have units of length. If \(t\) has units of time, then \(v_0\) must have units of length per time or velocity.

\(\frac{1}{2}a t^2\) must have units of length. If \(t^2\) has units of time squared, then \(a\) or \(\frac{1}{2}a\) must have units of length per time squared, or acceleration.

More examples in next video.

Transcendental Functions

  • Arguments of transcendental functions are dimensionless.

  • Transcendental functions are dimensionless as well.

  • What are the dimensions of \(\omega\) in \(y = \sin \omega t\)?

    • Since \(t\) is in [time] then \(\omega\) must be in \([\mathrm{time}^{-1}]\).

  • What are the dimensions of \(a\) and \(\omega\) in \(x = x_0 e^{-a t} \cos \omega t\)?

    • Since \(t\) is in [time] then both \(a\) and \(\omega\) must be in \([\mathrm{time}^{-1}]\).

  • What are the dimensions of \(\tau\) in \(T = T_0 e^{-t / \tau }\)

    • Since \(t\) is in [time] then \(\tau\) must be in [time].

Implied Units

If the argument of a transcendental function appears to have units, then there must be an implied unit, such as 1 milliwatt inside the function.

\(\ \ \ \ x\ \mathrm{dB} = 20 \log_{10} \frac{A}{A_0}\) or \(x\ \mathrm{dB} = 10 \log_{10} \frac{P}{P_0}\)

If \(A_0\) or \(P_0\) are not explicitly given, they are implied, e.g.,

    For \(\mathrm{dBm}\), \(P_0 = 1\ \mathrm{mW}\).

    For \(\mathrm{dBu}\), \(A_0 = 0.7746\ \mathrm{V}\).

The Takeaways

  1. There are ways to combine variables or measured quantities to create dimensionless numbers, which help reduce the measurements you need to make to characterize your system.
  2. All of the terms in an equation must have the same dimensions. In other words, equations must have dimensional homogeneity.
  3. Arguments of transcendental functions must be dimensionless.
  4. Decibels often have an implicit reference or denominator used in their calculation.






Thanks for watching!
The Full Story companion video is in the link in the upper left. The companion video in the series, What the Schmidt? Examples, is in the upper right. To learn more about Chemical and Thermal Processes, visit the website linked in the description.