Dimensionless anlysis

DIMENSIONAL ANALYSIS

Dimensional analysis depends upon the fundamental principle that any equation or relation between variables must be ‘dimensionally consistent', that is, each term in the relationship must have the same dimensions. Thus, in the simple application of the principle, an equation may consist of a number of terms, each representing, and therefore having, the dimensions of length. It is not permissible to add, say, lengths and velocities in an algebraic equation because they are quantities of different characters. The corollary of this principle is that if the whole equation is divided through by any one of the terms, each remaining term in the equation must be dimensionless. The use of these dimensionless groups, or dimensionless numbers as they are called, is of considerable value in developing relationships in chemical engineering.

The requirement of dimensional consistency places a number of constraints on the form of the functional relation between variables in a problem and forms the basis of the technique of dimensional analysis which enables the variables in a problem to be grouped into the form of dimensionless groups. Since the dimensions of the physical quantities may be expressed in terms of a number of fundamentals, usually mass, length, and time, and sometimes temperature and thermal energy, the requirement of dimensional consistency must be satisfied in respect of each of the fundamentals. Dimensional analysis gives no information about the form of the functions, nor does it provide any means of evaluating numerical proportionality constants.

The study of problems in fluid dynamics and in heat transfer is made difficult by the many parameters which appear to affect them. In most instances further study shows that the variables may be grouped together in dimensionless groups, thus reducing the effective number of variables. It is rarely possible, and certainly time consuming, to try to vary these many variables separately, and the method of dimensional analysis in providing a smaller number of independent groups is most helpful to the investigated.

The application of the principles of dimensional analysis may best be understood by considering an example. It is found, as a result of experiment, that the pressure difference (AP) between two ends of a pipe in which a fluid is flowing is a function of the pipe diameter d, the pipe length /, the fluid velocity u, the fluid density p, and the fluid viscosity /u,. The relationship between these variables may be written as:

ΔP = fi (d, l, u , p, µ) --------------------(1)

The form of the function is unknown, though since any function can be expanded as a power series, the function may be regarded as the sum of a number of terms each consisting of products of powers of the variables. The simplest form of relation will be where the function consists simply of a single term, or:

ΔP = constant dⁿ¹ lⁿ² uⁿ³  pⁿ⁴µⁿ⁵-------------------- (2)

The requirement of dimensional consistency is that the combined term on the right-hand side will have the same dimensions as that on the left; that is, it must have the dimensions of pressure.

Each of the variables in equation 2 may be expressed in terms of mass, length, and time.

The conditions of dimensional consistency must be met for each of the fundamentals of M, L, and T and the indices of each of these variables may be equated. Thus

Thus three equations and five unknowns result and the equations may be solved in terms of any two unknowns.

Substituting in the equation for L:

Thus, substituting into equation 2:

The group udp/µ, known as the Reynolds number, is one which frequently arises in the study of fluid flow and affords a criterion by which the type of flow in a given geometry may be characterised. Equation 1.8 involves the reciprocal of the Reynolds number, although this may be rewritten as: