## Dimensional Analysis

Posted on August 22, 2023

Physical modeling is a well-recognized and eﬀective tool for the study of process ﬂows that are difficult or impractical to study directly in commercial environments, and the study of new processes under development prior to commercialization.

Some of the potential benefits provided by physical modeling include:

• reduced cost
• improved safety
• study of extended off-design operation,
• lack of impact on current operations
• easier access for observation and advanced measurement techniques.

To obtain valid results from a physical model, it is imperative that adequate similitude exists between the model and the commercial process. Dimensional analysis is a powerful technique which provides important insights into scaling relationships and ﬂow phenomena and is an essential first step in physical modeling studies.

Similitude between the industrial apparatus and the lab model is established and evaluated through the process of dimensional analysis, which is a method of expressing complex relationships in more compact form. Based on the principle of dimensional homogeneity – the requirement that additive terms in an equation must have the same dimensions.

In the SI system of units, there are seven base dimensions: mass, length, time, temperature, luminous intensity, electric current, and amount of substance. However, in ﬂuid mechanics and thermodynamics, problems are usually restricted to variables comprising units of mass, length, time, and temperature, and so the reduction in the number of parameters is usually not greater than four.

Dimensional analysis is one of the most important tools we use when it comes to building a physical model at reduced scale, which often operate at ambient temperature, i.e. “cold flow”. By applying the principles of the Buckingham-pi theorem, we can reduce the scale and complexity of large industrial equipment, be it a gasifier, gravity separator, heavy oil upgrader, or steam turbine for example, to a size that is manageable at lab scales, often using analogue fluids to study their behaviour.