MARGINAL GAINS: ON PRESSURE AND FLOW.
If you've read any of my previous posts, you'll know two things, 1) I love an analogy and 2) pressure and flow in espresso is something that's been on my mind for a while. Whilst i'm still working on how and why they influence espresso extraction and uniformity (more on this to come soon) It seems like a good to to look at how pressure and flow are linked.
To do this we need to consider the relationship in two phases, firstly without restriction (known as steady flow) and then with restriction. After that we can think about the flow through espresso.
Flow and Pressure in Fluids.
The flow of fluids in a steady state is governed by the laws of conservation of energy, put simply energy cannot be created or destroyed, the sum of energy in any system must remain constant. Mathematician Daniel Bernoulli used this to describe the relationship between flow and pressure, postulating that at any point in a flow the sum of the energies must remain constant. Thus a change in the flow rate of the fluid must lead to a change in the pressure. The most familiar representation of this is the Aeroplane (or Aircar as my 2 year old calls them with crushingly direct logic), the shape of the wing forces the air to take different length paths across the top and bottom, this means that the air has to travel at different speeds creating a pressure difference with the result being lift (and a quick way to go on holiday).
The speed of the flow is linked to the kinetic energy of the system, the faster the flow, the more kinetic energy. With this rise there is a corresponding drop in the others energies, potential and internal. To visualise this, think about your toilet(!). The water in the cistern is stationary, it has zero kinetic energy. What it has is potential energy, the potential to do work, due to gravity. When we flush, we open a valve and the water drops into the cistern, losing its potential energy but gaining kinetic as it accelerates from a static position. The same follows for pressure, with an increase in kinetic energy, we see a corresponding increase in the dynamic pressure, the pressure due to the speed of the fluid flow, and a decrease in static pressure, the pressure due to its physical state. As water is in-compressible ( it cannot be compressed, which why belly flops hurt so much) static pressure can be measured at any point in the fluid flow, and it’s this that our pressure gauges measure.
On a practical level we use this to set our "pump pressure" by using a valve to change the flow on the exit of the pump. If you’ve ever done this you’ll know that to increase the pressure we see on our gauge we reduce the volume in the valve and so speed up the flow, contrary to everything we’ve just covered.
So, what’s changed? We’ve added a restriction to the water flow, our coffee puck. Suddenly the water flow hits this restriction stopping the flow. Now the dynamic pressure acts against the restriction, the mass of the water travelling at speed acts as a force ( Issac Newton’s second law of motion, Force = Mass x Acceleration), and this force works on the restriction in the form of pressure ( Hat Tip to Mr Blaise Pascal for this one, Pressure = Force / Area. btw, for those of you keeping score in scientist bingo, that’s Bernoulli, Newton and Pascal so far).
We mentioned before that water is in-compressible, as we lose dynamic pressure we see a rise in static pressure upstream from the restriction. To visualise this imagine that it’s summer, it’s sunny and warm (i.e. you’re not in the UK) and you’re watering the garden so you’ve got a running hose. Chances are that at some point you’ll get curious and place your thumb over the end of the hose, creating a restriction. What you’ll notice if you look at hose is that it’s less flexible, it’s stiffer as the pressure has backed up from your thumb throughout the hose, this is what we mean by upstream. Okay, so you all know what happens next, you release your thumb a tad, and the water sprays out, you soak your friend/sibling/partner, they get annoyed, grab the hose and turn it on you and much fun was had by all. Anyway, the water converges through the small gap you’ve left and speeds up dramatically with a subsequent drop in pressure.
Flow Through Espresso
Trying to fully explain the way water flows through espresso is somewhat of a fool’s errand, it’s interaction with the coffee is sufficiently complex that the mathematics are certainly beyond me but I’m confident that we can look at the broad principles. I’ve written about what I believe happens to extraction as you manipulate water flow before, as has Michael Cameron, read thisand the links it contains. But let’s consider just the physical water flow.
As we’ve just discussed, the pressure at the puck has a relationship to the dynamic pressure of the water flow. If the flow is slower ( i.e the pump spins slower) it exerts less force on the puck, hence less pressure on our gauge, and vice versa. The puck is our restriction, and it seems that it's ability to slow flow rises as it saturates, and then decreases as we lose mass and so in turn the pressure follows the same path. Pressure rises as the puck saturates and then drops as we extract solids.
Now we need to consider some of the other actors that will affect fluid flow through what is a granular bed (our puck of coffee grinds). What we are interested in here is the flow rate of the water through the bed, and we can think of this as the velocity a volume of water has passing a certain point. Seeing that we know the cross sectional area that the water flows through, we can consider the velocity as our indication of flow rate.
It has been shown that the velocity is directly proportional to the pressure drop across the bed and inversely proportional to the bed depth, this is known as Darcy's Law. What this means is that if we reduce the pressure drop or increase the bed depth, we also reduce the flow rate. (Darcy,Newton,Bernoulli, Pascal, BINGO!!)
What this suggests is that the the flow rate across the puck is directly related to the flow rate of the water as this creates a pressure difference, but this pressure difference is dependent on the restriction provided by the puck. So pressure and flow relationship is a constant feedback relationship.The speed of the water flow will influence the maximum pressure, which in turn will influence the flow rate across the puck, and both are influenced by the level of restriction the puck provides.
Okay, let’s put it all together to see what happens when we pull a shot:
When you start the shot the water quickly fills the puck and starts to saturate the grinds, you hear the pump kick in and often the machine strain under the pressure that builds up. Flow that was initially quick slows during this phase as we start to extract solids increasing the restriction. This increase in restriction slows the water flow and creates a pressure increase upstream from the puck (remember the hose!). This in turn sets up a pressure differential across the puck, high pressure at the face to atmospheric pressure at the bottom of the basket. Pressure and flow rate across the puck peaks as we start to remove mass and reduce the restriction.
As we reduce the restriction, we reduce the pressure at the puck face, which in turn influences the flow through the puck. However, remember what happened in our hose analogy, as we open the restriction slightly the water converges on the opening and flow rate increases. I believe this is what happens as we continue to extract solids from the puck. We’ve all seen the flow speed up at the end of the shot, I believe that this is akin to water squirting out of a hose, it’s faster than the water flow upstream of the puck.
All of this leads me to believe one thing, we should stop thinking about pressure for espresso, we need to think about flow rate. It’s the flow rate that sets up the conditions for flow across the puck, and flow rate that we should manipulate to influence the compounds we extract.