Engineering Optimization

19Scaled Variables and Dimensional Consistency

19.1 Introduction

The term “primitive variables” would refer to the variables we normally use when we measure or model a system, whether the system be a process, a procedure, a device, a product, or a combination. Primitive variables have units, and primitive variables can be either relative or absolute, or continuum or integer. The units may be very different for the same variable, which can be confounding for human understanding, for example, distance in miles or microns or light‐years, speed in rpm or kps, mass in kg or lbs, number of people or number of events, composition in mole fraction or ppm, energy in BTU or kJ, etc. Additionally, the mix of relative and absolute variables can also be confusing: temperature in °C or R, pressure in psig or Pa, and time in hours from the beginning of the local day or in years from the big bang.

By contrast, scaled variables represent a ratio of a primitive variable to its unit dimension, and common examples might be a fraction of a population as the number divided by total number, or distance as a portion of the total trip left. The scaling might be by a combination of variables, for instance, when activation energy in chemical kinetics is scaled by the product of temperature and the gas law constant or when fluid velocity is scaled by density, viscosity, and a characteristic dimension to form the Reynolds number. Scaled variables are dimensionless.

Although primitive variables directly ...