Objectives and Value of Dimensional Analysis

The phenomena involved in matter transformation can be described by fundamental momentum, mass and energy transport equations, coupled with equations of chemical or biological kinetics and the constitutive and rheological equations. Boundary conditions of the flow domain and initial conditions are associated with this system of equations. Unfortunately, it is usually impossible to resolve this system of equations because:

1) the form of the differential equations is too complex to be integrated over the entire flow domain, especially given the complex geometry of the industrial equipment;

2) the numerical models used in order to reach an approximate resolution are often incomplete and/or insufficiently accurate to exhaustively describe the physics of phenomena (particularly turbulence, as well as coupled transfers, interactions between phases, rheological laws, etc.) and/or often too expensive to develop in terms of both and resource needed time.

For these reasons, scientists are still ill-equipped when facing the question of predicting the evolution of matter contained in equipment or quantifying the causal relationship which exists between the operating conditions imposed and the system's output variables (e.g. properties of the products or chemical conversion). In order to overcome such difficulties, experimental studies are, therefore, essential. They are either carried out on real industrial equipment (1/1 scale) or on a laboratory or pilot-scale (equipment representing the industrial equipment, generally reduced in size but geometrically similar). Carrying out experiments on laboratory-scale equipment is justified given the difficulty of experimenting on the real system (investment costs of a 1/1 scale installation, high experimentation costs given the volume of products to test and resources to mobilize, coupled with the considerations on hazards, etc.).

These experimental studies provide a first analysis of the empirical relationships linking the variable being studied (target variable) to a set of dimensional physical quantities. Such an analysis is not fully satisfying in terms of understanding the conditions to adopt in order to reproduce the transformation on a different scale (scale-up or scale-down). Indeed, the mechanisms governing the transformation process are dependent from the experiment scale on which the transformation has been carried out, as explained below (see section 1.2). Therefore, a given transformation mechanism is the product of a collection of operating conditions, notably including the scale of the experiment, and depending on the size of the system studied, the nature of the phenomena which occur may change. The operator is also ill-equipped to understand whether a different experiment scale will conserve all the phenomena between the laboratory-scale and the 1/1 scale.

To do this, dimensional analysis must be used to group the dimensional variables which influence the operation in the form of a set of dimensionless numbers characteristic of the state of the system, generally along with a precise physical meaning (see section 1.1 on the configuration of the system). It then becomes possible to determine which operating conditions should be chosen to obtain a phenomenological identity between the laboratory-scale and industrial equipments. To do so, the equality of the numerical value of each dimensionless number must be verified on both scales (conservation of an identical operating point at both scales). It is essential to understand that complying with this principle (known as the theory of similarity) is the only way to generalize the results obtained from one scale to another. This fact explains why the notions of model (laboratory-scale equipment) and similarity are closely linked and inseparable. This aspect is addressed in Chapter 5.

The key challenge for the researcher, therefore, lies in the ability to construct dimensionless numbers characteristic of the process being studied, and in the calculation of the set of numerical values of these dimensionless numbers, which define the operating points of the system. In this context, dimensional analysis provides a powerful theoretical framework required to generate an unbiased set of dimensionless numbers.

The construction of dimensionless numbers is traditionally carried out using dimensionless differential equations which:

– govern transport phenomena (momentum, mass and energy);

– define the boundary conditions of the flow domain and the initial conditions.

When the problem is too complex (geometry, formulation of the boundary conditions or geometry of the system, choice of the macroparameters used as target variables, etc.), another alternative consists of obtaining these numbers using the list of dimensional physical quantities influencing the target variable. In chemical engineering, this makes up the majority of cases. In this approach, traditionally called “blind” dimensional analysis, this list can be established without a complete level of physical knowledge on the process studied. In some cases, this may be very weak, and very advanced in other cases, depending on the scientific maturity of the process or the product formulated. Subsequently, the dimensionless numbers formed will have a more or less explicit physical sense.

In addition to the advantage of being able to be carried out without in-depth knowledge of the process, the “blind” dimensional analysis, coupled with the theory of similarity and experiments in laboratory-scale equipment, provides a global modeling of the process in the form of semi-empirical correlations (linking the target dimensionless number to a set of dimensionless numbers and translating the causes of the target variable's variation), which can be extrapolated to another scale or another product. In this book, these will be called process relationships.

Regardless of the approach used, the procedure for determining these dimensionless numbers characteristic of the process being studied is accompanied by a reduction in the number of variables by which the physical phenomenon1 is described. This consequently minimizes the experiments to be undertaken to establish the process relationship linking the various dimensionless numbers (see section 1.3).

Furthermore, the process relationship established using the dimensional analysis coupled with the theory of similarity and experiments on laboratory-scale equipment helps to identify the predominant physical phenomena. These are the phenomena which control the target variable insofar as it is based on dimensionless numbers which have a precise physical meaning.

Finally, this physical description coupled with the space reduction of the quantities describing the phenomenon can provide a simplified graphic representation. This, therefore, constitutes:

– an aid for understanding and carrying out the process;

– a way to carry out a reverse engineering approach (process for defining the set of operating conditions providing a given value of the target variable (see sections 1.3 and 1.4)).

This chapter illustrate the objectives and power of dimensional analysis.

1.1 Grouping dimensional variables in the form of a set of dimensionless numbers with a precise physical sense

The work of British physicist Osborne Reynolds (1842–1912) deals notably with the study and the characterization of liquid flows in cylindrical ducts. In 1883, by adding a dye, he illustrated the existence of various flow regimes for liquid flows in transparent pipes with a cylindrical section imposing various experimental conditions (average fluid velocity v, diameter of the pipe D, dynamic viscosity and density of the fluid μ and ρ, respectively). He noted that:

– the end of the laminar regime, shown by the appearance of...