Mrs. **Reynolds** probably must have said that to her husband sometime in 1883. She meant that as a compliment, not a concern or a reproach, hence the exclamation point at the end of the sentence. In that year, Mr. Reynolds wrote a publication in which his, for process technologists, famous **dimensionless number** is first mentioned.

**Dimensionless, what are you talking about?**

Those of you who have read my e-book, "How Not Sexy?" will have seen that I had already paid some attention to the importance of proper dimensions or units in the formulas used in process technology to calculate all sorts of things.

With a dimension or unit, you give a meaning to a value. Or in other words, you give a category to which a certain measurable quantity belongs. It seems like an open door, but there are quite a few units that are used and used interchangeably. For example, for pressure, you use the following units:

1 atmosphere = 101.325 Newtons per square meter (^{N/m2}) = 101.325 Pascal = 1.013 bar = 760 millimeters of mercury pressure (mmHg) and more...

*Some commonly used quantities with corresponding standard units according to the SI unit system*

**The relationship must be consistent**

Obviously, if a wrong result emerges from a calculation, it can have major consequences. Since so many units are used interchangeably, a mistake is easily made. It is therefore important to look carefully at the units in a formula. This is called **dimension analysis**.

The premise is that for any equation between a number of variables (that is, the numbers in the formulas that vary, hence the name...), the relationship between the units must be the same. Take the simple mathematical equation:

In this, x and y are the variables. So if you successively enter for x: 0, 1, 2, 3... the result is y = 0, 2, 4, 6... Suppose y has the unit meter then x must also have the unit meter. For example, it cannot be the case that x has as its unit ˚C and y has meter. This is meant by consistency of units.

Ok this is obvious right? Although they are not directly formulas from process technology, they are the two most famous formulas of contemporary exact science and lend themselves well to an explanation of what dimension analysis means: Einstein's famous formula

and Newton's second law:

In words: Energy is equal to mass times speed of light squared and Force is mass times acceleration. Now for the dimension analysis:

The unit of energy is Joule, abbreviated J (pronounced *djoel*). This is equivalent to Newton meters (Newton times meters and not Newton per meter), abbreviated **Nm**. The unit of mass is kilograms**(kg**), the unit of acceleration is meters per second squared, abbreviated ** ^{m/s2}**, the unit of force is Newton, abbreviated

**N**and finally the unit of (light) speed is meters per second, abbreviated

**m/s**. Here goes... In the place of the variables in the above formulas, we now fill in the units:

**Einstein**: Nm = kg (m/s^{)2}, or Nm = kg ^{m2/s2}, or Nm = kg m/s2^{m }

**Newton**: N = kg ^{m/s2}

If we replace kg ^{m/s2} in Einstein's formula with Newton's formula , then we get Nm = N m. So it is correct! Just goes to show that these two gentlemen knew what they were doing.

**Numbers without dimension **

All right, now back to dimensionless numbers. The word says it all, they do not have a unit or dimension as in the formula above. It's just a number. Strange, yet very useful and widely used in process technology, in describing flows and heat transport. Mr. Reynolds was the first to define a dimensionless number, then more followed.

The **Reynolds number**, *Re*, is calculated by the formula:

Specific gravity is different for each substance: for example, 1 liter of water weighs 1 kg, 1 liter of oil weighs 0.8 kg. Viscosity is a measure of "viscosity." Water has a lower viscosity than peanut butter.

If you enter the units into this formula you will see that they cancel out against each other and thus the Reynolds number is dimensionless:

**Laminar and turbulent**

If a low Reynolds number is calculated, the flow is *laminar*. In laminar flow, molecules flow in the direction of flow in layers at the same velocity. At high Reynolds numbers, the flow is *turbulent*. Here the molecules in the flow direction move chaotically through each other. In the video below, the difference between the types of flows can be clearly seen.

*If the water is completely blue, then the ink is perfectly mixed and the flow is turbulent. And where the ink flows through the water in lines, the flow is laminar. You can also see it on the outside of the water where it comes out of the pipe.*

After Mr. Reynolds, there have been about twenty more gentlemen (I don't believe there are any ladies who have linked their names to a dimensionless number), who also defined dimensionless numbers to describe phenomena in heat transport (Nusselt, Prandtl) or fluid dynamics . We process technologists are still grateful to them for that today....

Do you have an opinion on this? If so, give the editors a tip and we can write a blog about it or respond to it and we can all learn something from it.