Scaling Chemical Processes

Chapter 2

Dimensional Analysis

Abstract

This chapter provides the foundation for using a matrix-based procedure for identifying and defining dimensionless parameters.

Keywords

Dimension; fundamental dimension; derived dimension; unit systems; Dimension table; Dimension matrix; Total matrix

Introduction

To ensure that a prototype chemical process behaves similarly to its model chemical process, we must establish the relationships

$\begin{array}{l}{\mathrm{\Pi }}_{\mathrm{M}}^{\mathrm{Geometric}}={\mathrm{\Pi }}_{\mathrm{P}}^{\mathrm{Geometric}}\hfill \\ {\mathrm{\Pi }}_{\mathrm{M}}^{\mathrm{Static}}={\mathrm{\Pi }}_{\mathrm{P}}^{\mathrm{Static}}\hfill \\ {\mathrm{\Pi }}_{\mathrm{M}}^{\mathrm{Kinematic}}={\mathrm{\Pi }}_{\mathrm{P}}^{\mathrm{Kinematic}}\hfill \\ {\mathrm{\Pi }}_{\mathrm{M}}^{\mathrm{Dynamic}}={\mathrm{\Pi }}_{\mathrm{P}}^{\mathrm{Dynamic}}\hfill \\ {\mathrm{\Pi }}_{\mathrm{M}}^{\mathrm{Thermal}}={\mathrm{\Pi }}_{\mathrm{P}}^{\mathrm{Thermal}}\hfill \\ {\mathrm{\Pi }}_{\mathrm{M}}^{\mathrm{Chemical}}={\mathrm{\Pi }}_{\mathrm{P}}^{\mathrm{Chemical}}\hfill \end{array}$ (2.1)

(2.1)

where each ${\mathrm{\Pi }}_{\mathrm{M}}^{\mathrm{i}}$ and ${\mathrm{\Pi }}_{\mathrm{P}}^{\mathrm{i}}$ is a dimensionless parameter for the model M and the prototype P. The ...