Understanding the behaviour of fluids and their interaction with mechanical systems is fundamental for Mechanical Production Technicians in Kenya. Applying dimensional and model analysis allows technicians to predict fluid behaviours in scaled models, saving costs and ensuring safety in design and testing. This chapter covers the essential principles of dimensional homogeneity, methods for solving dimensional analysis problems, and the concept of similitude, which permits reliable model testing before full-scale implementation in industries such as manufacturing, water supply systems, and agro-processing plants.

9.1 Definition of Terms

Dimensional analysis is the study of the dimensions of the physical quantities in a problem. It checks that an equation is sound, helps to work out how one quantity depends on others when no theory is available, and reduces a long list of variables to a short list of dimensionless groups. Engineers use it to plan experiments on pumps, pipes and models, because a few dimensionless groups need far fewer tests than every separate variable.

The terms used in the topic are:

• A dimension is the kind of physical quantity that is measured, such as length, mass and time. The dimension does not depend on the unit. A length is a length whether it is measured in metres or in feet.
• A unit is the standard in which a dimension is measured. The SI unit of length is the metre, of mass the kilogram and of time the second.
• Fundamental (primary) dimensions are those that cannot be built from others. In mechanics they are mass M, length L and time T. Temperature θ is added when heat is involved.
• Derived (secondary) dimensions are built from the fundamental ones, such as velocity (L T⁻¹) and force (M L T⁻²).
• A dimensionless quantity has no dimensions. An angle, a ratio of two lengths and the Reynolds number are examples.

The dimensions of the quantities often used in fluid mechanics:

9.1.1 Dimensional Homogeneity

An equation is dimensionally homogeneous when every term in it has the same dimensions, so that the dimensions on the left-hand side equal those on the right-hand side. Quantities can only be added, subtracted or equated when they have the same dimensions: a length cannot be added to a time.

The checks follow from the rule:

• Every term in a sum has the same dimensions.
• The two sides of the equation have the same dimensions.
• The arguments of a logarithm, an exponential or a trigonometric function are dimensionless.

Worked examples:

1. The equation v = u + a × t. The dimension of v and of u is L T⁻¹. The term a × t is (L T⁻²) × T = L T⁻¹. All terms agree, so the equation is homogeneous.
2. The discharge through a small orifice, Q = C_d × A × √(2 g h), where C_d has no dimensions. The right-hand side is L² × √[(L T⁻²) × L] = L² × L T⁻¹ = L³ T⁻¹, which is the dimension of Q. The equation is homogeneous.
3. The Hagen-Poiseuille equation for laminar pipe flow, Q = π × R⁴ × G ÷ (8μ), where G is the pressure gradient. The dimensions are L⁴ × (M L⁻² T⁻²) ÷ (M L⁻¹ T⁻¹) = L³ T⁻¹, so the equation is homogeneous.
4. A faulty equation: a student writes the power of a pump as P = ρ × g × Q × H², where H is the head. The right-hand side is (M L⁻³)(L T⁻²)(L³ T⁻¹)(L²) = M L³ T⁻³, whereas power is M L² T⁻³. The two do not agree, so the equation contains an error, and the true equation has H, not H², as P = ρ × g × Q × H. Dimension checks like this find errors before any numbers are put in.

Some empirical formulas, such as the Manning formula for open channels, are not homogeneous because the constants in them carry hidden dimensions. They hold only in the units for which they were made, and the units are stated with them.

A homogeneous equation is correct in any consistent system of units. This is what lets an equation found by dimensional analysis be used in tests on models.

9.1.2 Methods of Solving Dimensional Analysis

Two methods find how a quantity depends on others: Rayleigh's method, which gives the result as a product of powers, and the Buckingham π method, which gives the result as a relation between dimensionless groups. Both start from a list of the variables that affect the quantity.

9.1.2.1 Rayleigh's theorem

Rayleigh's method writes the dependent variable as a constant multiplied by the independent variables, each raised to an unknown power. The dimensions on both sides are matched, and the equations for the powers are solved. There are only three equations (M, L and T), so the method fixes the powers directly when there are at most three unknown powers, that is, at most four variables in all. With more variables, some powers are left in terms of the others, and the π method is easier.

Steps:

1. List the variables the quantity depends on.
2. Write the quantity as K × (variable 1)^a × (variable 2)^b × (variable 3)^c, where K is a dimensionless constant.
3. Write the dimensions of each variable in M, L and T.
4. Equate the powers of M, L and T on the two sides and solve.
5. Substitute the powers. The constant K is found by experiment.

Example 1. The time of swing t_p of a simple pendulum depends on its length l, its mass m and the acceleration due to gravity g. Write t_p = K × l^a × m^b × g^c.

• Dimensions: t_p = T, l = L, m = M and g = L T⁻².
• M: 0 = b, so b = 0
• L: 0 = a + c
• T: 1 = −2c, so c = −½, and then a = ½

So t_p = K × √(l ÷ g). The mass does not matter. Experiment and theory give K = 2π.

Example 2. In laminar flow in a pipe the pressure drop per unit length, Δp ÷ L, depends on the viscosity μ, the mean velocity V and the diameter D. Write Δp ÷ L = K × μ^a × V^b × D^c.

• Dimensions: Δp ÷ L = M L⁻² T⁻², μ = M L⁻¹ T⁻¹, V = L T⁻¹ and D = L.
• M: 1 = a
• T: −2 = −a − b, so b = 1
• L: −2 = −a + b + c, so c = −2

So Δp ÷ L = K × μ × V ÷ D². This is the form of the Hagen-Poiseuille result, in which K = 32. Dimensional analysis gives the way the pressure drop depends on each variable, and a test or a theory gives K.

9.1.2.2 Buckingham π theorem

The Buckingham π theorem states that if a problem has n variables and these involve m fundamental dimensions, the relation between them can be written as a relation between n − m independent dimensionless groups, called π terms. In mechanics m is usually 3 (M, L and T).

Steps:

1. List the n variables, including the dependent one, and write their dimensions.
2. Find m, the number of fundamental dimensions, and so the number of π terms, n − m.
3. Choose m repeating variables. They must together contain all of M, L and T, and must not form a dimensionless group among themselves. A usual choice is one geometric variable (a length D), one kinematic variable (a velocity V) and one dynamic variable (the density ρ).
4. Form each π term from the repeating variables and one of the remaining variables, with an exponent that makes the group dimensionless.
5. Write the result as π₁ = φ(π₂, π₃, ...), and name the groups, using products or reciprocals if needed.

Example. The pressure drop Δp in a pipe flow depends on the pipe length L, the diameter D, the mean velocity V, the density ρ and the viscosity μ. These are n = 6 variables, with m = 3 fundamental dimensions, so there are 6 − 3 = 3 π terms. Take ρ, V and D as the repeating variables.

• π₁ from Δp: Δp × ρ^a × V^b × D^c with dimensions (M L⁻¹ T⁻²)(M L⁻³)^a (L T⁻¹)^b L^c. For M: 1 + a = 0, so a = −1. For T: −2 − b = 0, so b = −2. For L: −1 − 3a + b + c = 0, so −1 + 3 − 2 + c = 0 and c = 0. So π₁ = Δp ÷ (ρ × V²).
• π₂ from L: it has the dimension of length, and dividing by D gives π₂ = L ÷ D.
• π₃ from μ: μ × ρ^a × V^b × D^c gives M: 1 + a = 0, so a = −1. T: −1 − b = 0, so b = −1. L: −1 − 3a + b + c = 0, so −1 + 3 − 1 + c = 0 and c = −1. So π₃ = μ ÷ (ρ × V × D), which is 1 ÷ Re, the reciprocal of the Reynolds number.

The result is Δp ÷ (ρ × V²) = φ(L ÷ D, Re). Experiments show that Δp is proportional to L ÷ D, so Δp = f × (L ÷ D) × (ρ × V² ÷ 2), where the friction factor f depends only on Re. This is the Darcy-Weisbach equation, which has been found from three groups and not from six separate variables.

The advantage of the π method over Rayleigh's method is that it works for any number of variables, and that the groups it gives are the dimensionless numbers (Reynolds, Froude, Euler, Weber, Mach) that are used in model testing.

Chapter Summary

This chapter explored key concepts in dimensional and model analysis as applied to fluids within thermodynamics and fluid mechanics. It began by defining essential terms including dimensional homogeneity, which ensures equations are dimensionally consistent, and presented methods for solving dimensional analysis such as Rayleigh's theorem and the Buckingham Pi theorem. The discussion then progressed to dimensional analysis similitude, emphasizing the importance of geometric, kinematic, and dynamic similarity in creating effective fluid models. Various dimensionless numbers critical to fluid flow characterization were examined, including Reynolds, Froude, Euler, Weber, and Mach numbers, each describing different physical phenomena in fluid systems. The chapter concluded with model test analysis and calculations, detailing the classification of models into undistorted and distorted types and explaining the use of scale ratios for distorted models to accurately represent fluid behavior in experimental settings. This comprehensive treatment equips students with the theoretical and practical tools needed to analyze and model fluid mechanics problems effectively.