Lesson 6 - Principles of Fluid Mechanics 3

Welcome back! So, what were your thoughts?

Let us start with the length of the pipe. We would expect the pressure loss to change depending on how short or long the pipe is. A longer pipe would possess a greater pressure drop and vice versa (think of a garden hose – the longer the hose, the more the water dribbles out at the outlet with a lower pressure, and hence greater pressure drop). So, ∆P would be proportional to L (∆P ∝ L).

Next, let us consider the flow rate of wort down the pipe. To make things simpler, we can think of the flow velocity rather than the flow rate for a given pipe diameter (we can look back at the continuity equation if we need a refresher of the validity of this). We would expect that a higher flow velocity would increase the pressure drop, (think of it as if the wort would need to be forced more through the pipe at higher velocities). So, ∆P would be proportional to u (∆P ∝ u).

A variant on the pipe velocity argument above would be consideration of the pipe diameter. For a given flow rate of fluid, if we decreased the pipe diameter, then we would increase the flow velocity (remind yourself of the continuity of mass if necessary). So ∆P would be inversely proportional to d (∆P ∝ 1/d).

What about the condition of the surface on the inside of the pipe wall? Well, we previously discussed that friction is responsible for pressure drop, so presumably, we would expect more friction to increase the pressure drop, and vice versa? So how might we define friction at the inside of the pipe wall?

We can think of it this way perhaps: an icy surface on the road does not offer much friction for our car’s tyres to grip on, whereas a roughened surface can offer more friction. Consequently, we would expect the surface roughness or other friction-related parameters of the inside of the pipe to increase pressure drop. So ∆P would be proportional to surface roughness (∆P ∝ surface roughness).

Now how about the fluid itself? Which of its parameters might be influential? Well, we would expect a denser and/or viscous wort to yield a higher pressure drop, yes? It is harder to have a ‘feel’ for such parameters and their likely influence. Do not worry we will come back to this in the lesson.

Summarising these points in a proportional relationship would give us the following:

If you got anywhere close to this, give yourself a pat on the back, or better still a beer!

The above exercise is how Henry Darcy, Julius Weisbach and others approached their study of frictional pressure loss in pipes. They wondered which parameters would be at play, considered their influence (proportional or inversely proportional), tested their ideas in experiments (in the laboratory or in nature) and then quantified the relationships between the parameters to arrive at a defining model, equation, or law. Let us quickly look at the two originators of the equation we are about to study further.

Henry Darcy (1803-1858) was a French engineer who unlike many of his contemporaries was both a practicing engineer and a leading researcher in his field of hydraulics. In addition to his studies on frictional pressure losses in pipes, he also developed a leading view on flow through porous beds. We utilise this view extensively in malting and brewing to quantify airflow through cereal beds during grain-drying and malting. We also use it for wort and beer flow during lautering and beer filtration.

Julius Weisbach (1806-1871) was a German mathematician and engineer who was prolific in many fields including hydraulics, mineralogy, mathematics, and mine surveying. He was instrumental in modifying earlier work of Henry Darcy to give us the Darcy-Weisbach equation, which we will now look at in-depth. The Darcy-Weisbach equation

The Darcy-Weisbach equation proposes that the pressure drop within a pipe due to frictional losses, ∆P _f between the moving fluid and the pipe wall, can be represented by:

Where φ (Greek symbol, phi, pronounced fi) is the friction factor (it is dimensionless – hence it has no units); L is the pipe length (m); d is the internal pipe diameter (m); ρ is the fluid density (kg/m³), and u is the average flow velocity (m/s).

All these parameters will be quite familiar by now, apart from the friction factor φ, so let us look at this in some more detail.