# Trying to gain a deeper understanding of what a system curve (pressure vs flow) actually represents

I'm trying to make sense of what a system curve is, and I have three different conceptions in mind.

Note: I don't have an engineering background, and these are relatively new concepts to me, so I may be botching some of the ideas and terminology. I'm also purposely avoiding discussion of a pump or pump curve.

To keep this simple, let's consider a simple water filled syringe that is placed horizontally. Work can be done on this system by pressing the plunger.

Conception A:

The diameter of the syringe, the roughness of the material, and the size of the tip (by tip, I mean the small exit hole) are all going to determine the hydraulic resistance of this system. For example, a very large diameter tip will not restrict flow as much as a smaller tip. With a very small tip, if work is done on the water by depressing the plunger, then much of that energy will be experienced as an increase in the water pressure potential energy (pressure), and relatively little will be experienced as kinetic energy due to flow out of the tip. The system curve will thus reflect the relative partitioning of energy introduced into a system: a steeper curve means that more energy will be converted into pressure. In the extreme, if the tip is closed, then virtually all the energy will be converted into pressure.

Conception B:

This is more of a definition rather than a conception, as I don't really understand it. Here, the system curve describes the amount of pressure required to move water at a given flow rate. Here, the pressure has a causal relationship to the flow, whereas in conception A, pressure and flow are both effects that are caused by the work done to the system (by the plunger of the syringe). The hydraulic resistance of the system will determine the shape of this curve. The reason I have this definition in mind is simply due to online definitions I've come across. e.g. "A system curve, as shown in Figure 2, is a graphical representation of the pump head that is required to move fluid through a piping system at various flow rates." (from here).

Conception C:

Here, the system curve describes the loss of pressure (head loss), due to hydraulic resistance, as a function of flow. As flow increases, frictional losses increase, thus reducing pressure.

My intuition tells me that A and C may be two sides of the same coin, but I'm having trouble wrapping my head around it. In particular, in the extreme case, where the tip is closed, we get the steepest (i.e. vertical) system curve, as all work is converted into pressure. Yet because there is no flow, there are also no frictional losses.

A and B place pressure at different places in the causal hierarchy (A suggests pressure - and flow - are effects of the work done on the system, whereas B suggests pressure is a cause of flow). In my own understanding of fluid dynamics, pressure can be both a cause and an effect:

Pressure as a cause, where flow is the effect: When water pressure is high, then a tap that is opened along a pipe will have a high flow as the pressure energy is converted into kinetic energy. This is illustrated by the fact that the water can flow out of the pipe at a direction orthogonal to the direction of water flow within the pipe, which is consistent with the fact that water pressure potential energy is omnidirectional. In the case of the syringe, there is a bit of conflation, as the direction of the bulk flow within the syringe is the same as the direction of the exit out of the tip.

Pressure as an effect, where work done on system is a cause: When energy is introduced into a system, that energy can be converted into pressure potential energy. Similarly, if negative work is done on a system via friction, then pressure potential energy can be reduced.

Pressure as an effect, where change in flow is a cause: If water along a single streamline experiences an increase in flow without any additional energy being put into the system (i.e. the flow increases due to diameter of pipe becoming smaller), then the pressure in that smaller diameter pipe will decrease due to conservation of energy (Bernoulli's principle). I suppose that if you think about it deeply, then the change in flow really isn't the cause of the pressure drop - rather, both the change in pressure and the change in flow are effects of the change in the pipe configuration.

Conception B doesn't ring true to me, as I don't see pressure as being a cause of the bulk flow through the system. In an actual water network, while the pressure may be causally implicated in the rate of flow of water out of a tap, I don't think it's causally implicated in the bulk flow through the system, even though it's correlated with it. My guess is this definition is a sloppy one that confuses cause and effect.

I also suspect that I may be abusing the idea of hydraulic resistance being inextricably linked with friction. If you were to add more holes to the syringe, then the hydraulic resistance would decrease, yet frictional losses would presumably increase due to more flow. Perhaps it's better to conceptualize hydraulic resistance as simply a barrier to flow that is incidentally sometimes associated with frictional losses. In the case of a small hole, I'm guessing something like a high pressure barrier builds up near the hole, creating an action-reaction force pair with the force applied to the syringe, and this manifests as a compressive force upon the water, increasing its pressure. In the case of a completely closed tip, the barrier is the physical barrier of the syringe itself. So there is infinite hydraulic resistance that is associated with zero frictional losses. If this thinking is correct, then Conceptions A and C appear to be wholly distinct.

I'd greatly appreciate any guidance in helping me resolve these conceptions.

Update TLDR: I'm having trouble understanding why there are so many different definitions of a system curve, and whether they all mean the same thing or whether some definitions are incorrect.

Here are examples:

A:

"The system curve describes the increase in head resulting from increasing fluid flow through the pipework and other equipment in your plant." (from here)

B:

"A system curve, as shown in Figure 2, is a graphical representation of the pump head that is required to move fluid through a piping system at various flow rates." (from here)

C:

"A system curve is a graphical representation of the relationship between flow rate and the associated head losses. It is generated by calculating the total head losses at various flow rates and plotting them on a common set of axes." (from here)

In particular, I'm trying to wrap my head around how C relates to B or A.

Update 2: I think one of the sources of my confusion was that I was equating pressure head as the portion of the energy that was bound up in pressure potential energy. But after learning more about system curves, it appears that the head in this context is the total energy of the water (including velocity head). Once I understood this, things are starting to clear up. If I do end up answering my own question, I'll include it as an answer.

• TLDR; You may be overcomplicating things. Supposing a system strives to move fluid and some pressure resists that effort, the flow is the result the system achieves.
– Abel
Commented Jan 28 at 12:36

Inasmuch as the question is "pressure and flow rate: which way round are cause and effect?", the answer is "both ways round". All fluids are compressible to some extent, so there's an elastic equation that says the spatial variation (divergence) of the velocity can generate a rate of change of pressure; also, the Navier-Stokes momentum equation says the spatial variation (gradient) of the pressure can generate a rate of change of velocity.

• Thanks. Upon reflection, perhaps I am getting a bit hung up on the cause and effect. So perhaps conceptions A and B are somewhat equivalent. Commented Jan 28 at 15:36
• to clarify, AFAIK the "both ways round" is true even for an incompressible fluid Commented Jan 28 at 16:09
• @PeteW Yes, although for an incompressible fluid it takes the form "any divergence in the velocity would generate an infinite rate of change of pressure, therefore there can be no divergence in the velocity". Commented Jan 28 at 16:32

For a mathematical explanation of a simple example of pressure drop vs flow, which this won't be, one would usually start with laminar pipe flow and incompressible fluid. Laminar means the range of conditions where some complications due to momentum can be neglected. Other refinements can be added later but we should already have enough to demonstrate a pressure-drop-vs-flow.

The concept is "anchored" by the "no-slip" boundary condition at the walls - that is, the fluid velocity is zero at the walls. At macro scale this pretty much holds true.

The rest is a consequence of shear. In fluids unlike solids, the effect of viscosity means shear stress corresponds to rate of shear strain (i.e. in pipe flow, difference of axial velocities in the radial direction). You then get the parabolic velocity-vs-radius profile that's characteristic of laminar pipe flow. Very much glossing over the rest here, but to maintain the shear stress in the fluid as it flows, some energy must be consumed.

To the cause-and-effect portion of the question, it's both. If you have a pressure gradient in a section of a pipe, or just a pressure difference between the inlet and exit of some volume, fluid would move. If you force fluid to move, lets say its own momentum is carrying it through, then a pressure gradient would appear (and you could say that it's ultimately caused by the "sides").

• Thanks Pete. I've added a TLDR update to my original post. If I'm understanding your answer, you're saying that the no-slip condition results in shear strain, which is work done by the system that is not converted to flow or pressure (i.e. the energy is "consumed"). But how does this resolve definitions A and C? Commented Jan 28 at 16:49
• @spacediver - I was using laminar pipe flow as an example that's relatively simple to analyze, and to show that there is a relationship between pressure drop and flow (if some reasonable assumptions are met), and that the essential phenomenon is viscosity (iow, shear stress and strain). One can keep going deeper to get an honest physics explanation, which we often don't need to for engineering. Commented Jan 28 at 18:42
• But for practical purposes, given the geometry (and possibly operating conditions of machinery) and fluid properties, you'll generally get a 1-to-1 curve relating pressure drop to flow. Because it's 1-to-1, you can take either pressure or flow as the independent variable and you'll be fine either way. The definitions you have should be equivalent... You might be overthinking it a bit. Work thru some example problems and you'll see Commented Jan 28 at 18:42
• Thanks, that sounds like good advice. Commented Jan 28 at 19:00