Hence, depending on the outcome, the water flow is either laminar (where the water flows in a smooth, relatively orderly manner with all the particles flowing in the same direction) or turbulent (where the water moves in a disorderly fashion - chaotic eddies, vortices and other flow instabilities).

The origin of the Reynolds number dates to 1883 when the Irish scientist Osborne Reynolds developed a number that predicts fluid flow based on the balance of static and dynamic properties including the velocity, density, dynamic viscosity (the measure of a fluid's resistance to flow) and characteristics of a given fluid ².

Unlike the maintenance of the highest standards of cleanroom air (EU GMP Grade A), the objective of a pharmaceutical water system is for the water to flow in a turbulent manner since this is optimal for guarding against microbial adhesion. For non-microbial control applications, many engineers will be keen on laminar flow since it causes less wear and attrition to the pipes or open channel walls. This helps to improve the performance of pumps and requires less energy.

Calculating the Reynolds number

To verify the Reynolds number, data input and an equation are required. The output is a dimensionless (adimensional) value (with ‘Reynolds’ typically abbreviated to ‘Re’). The Reynolds number formula is:

Re = VDρ/μ 

or 

Re = VD/v

Here:

• V is the fluid velocity (user calculated)
• D is the characteristic distance (user calculated)
• ρ is the fluid density (user calculated)
• ν is the kinematic viscosity (acquired from data tables)
• μ is the dynamic viscosity (acquired from data tables)

For a pipe, the characteristic length is the hydraulic diameter. This is calculated from:

L = dh

Here:

dh = hydraulic diameter (metres)

The Reynolds number for the flow in a duct or pipe with the hydraulic diameter can be expressed as:

Re = ρ u dh / μ = u dh / ν        

Here: 

dh = hydraulic diameter (metres)

Alternatively, online calculators are available.

Transitional state

Given our two possible outcomes are laminar or turbulent flow, there will be a difference in the balance of forces and a cutoff point (in terms of the expressed Reynolds number) that signifies either ‘laminar’ or ‘turbulent’.

• Laminar flow: This outcome occurs when viscous forces are dominant. The Reynolds number for laminar flow is typically Re < 2100
• Turbulent flow: This outcome occurs when inertial forces are dominant. Turbulent flow definition is typically Re > 4000

This leaves an ‘absolute value’, the space between a Reynolds number of more than 2100 and a Reynolds number of less than 4000. What is occurring here? The answer is a state of uncertainty or rather fluctuating states. This state is described as intermittent or transitional flow, where the flow will change from laminar to turbulent and then back again ³.

This state poses a greater contamination risk than laminar flow as the increased fluid flow (towards or parallel to a substratum surface) results in faster adhesion of microorganisms due to higher mass transport. This is despite the presence of higher fluid shear stresses stimulating detachment. 

Yet when fluid flow exceeds a critical limit, the resulting wall shear stresses may become high enough to prevent adhesion, which is what is occurring under higher Reynolds numbers (>4000) ⁴. Once a true and consistent turbulent flow is achieved, it will remove significantly more bacterial cells than laminar flow ⁵.

Variables

The extent to which flowing water transitions from laminar to turbulent depends on the different factors expressed in the above equations, such as the shape and dimensions of the pipework ⁶.

Another important consideration is the material used for the pipework and its relative smoothness (which connects with the importance of correct material specifications). PVC, for example, has a low degree of roughness. The friction on a pipe surface relates to its roughness - if a pipe surface is too ‘rough’ the flow will be affected (the greater the friction, the lower the Reynolds number according to a logarithmic scale).

Account must also be taken of the water temperature when considering the calculations needed for the Reynolds number, since water viscosity varies with temperature. For example, the kinematic viscosity of water at 20oC 1.004·10-6 m2/s whereas, at 0oC it is 1.787·10-6 m2/s and at 100oC it is 0.29·10-6 m2/s.

Summary

For those interested in microbial control of pharmaceutical water systems, it is important to specify and to periodically verify Reynolds numbers of > 4000 ⁶.