Dude, Where’s My Gas?
Problem:
Design a portable, pneumatic system to cycle air (or whatever your favorite gas is) with high precision over periodic intervals from atmosphere to an inflatable actuation reservoir, across a wide range of flow rates, and volumes, while being flexible and easily assembled without much in the way of tooling.
Initial Condition:
A good pump that could be calibrated across various flow targets at a range of altitudes that human beings could be reasonably expected to inhabit. A servomechanism with a baro sensor, along with the pump is defined. The servo mechanism integrates flow over the flow delivery time period to determine the gas volume delivered. However, just because the pump can deliver an accurate flow, doesn’t mean you will get accurate volumes at the reservoir. And just to be clear this isn’t a leak issue, I’m assuming that the design is sound mechanically and all leak points have been sufficiently sealed.
Tubing Selection:
Rigid tubing or piping will not do as part of this system, since they aren’t flexible, and most certainly can only be assembled with some amount of tooling hardware. I know, this may not necessarily be how you’d approach the project, but bear with me here because I am trying to make a point about volumetric flow targeting.
So allow me to suggest flexible tubing such as one manufactured from thin-walled ethyl vinyl acetate or low density polyethylene. These are flexible but durable, and the walls can be corrugated to prevent against tubing collapse or kinking when bent. And to prevent any funny flow profiles, the length of tubing is at least 10x the diameter (for practical purposes, this ensures the flow is fully developed when the system is in use).
Now, the pump is configured to know that whether you want to flow a volume of 10 litres in 30 seconds or 100 ml in 3 seconds, it will deliver whatever you ask for with negligible error. However, some of this volume won’t get to the desired destination because of a tubing property that leads to errors in delivered gas volumes:
Compliance Volume:
Compliance means that the system expands due to pressure as a fluid moves through it and traps some amount of fluid within said expanded volume, leading to a loss of delivered fluid. This is a function of:
Tubing material;
Tubing length and diameter;
Pressure Resistance in System;
Temperature;
Fluid density and viscosity also affect compliance volume, but these factors are generally part of the resistance pressure in the system. Compliance is non-linear across all its variants listed above (let alone when varying any 2 or more), making predicting exact errors somewhat difficult. Generally, the compliance volume increases logarithmically as the total volume delivered from the pump increases. If the tubing diameter and length remain the same, it becomes stiffer as more gas volume is pumped into the closed system. The rise in pressure in the system increases at a faster rate than the volume expansion resulting in the logarithmic function. On the other hand, increasing the diameter of a tube could exponentially increase its compliance, all other factors remaining constant. And we haven’t even begun considering the effects of temperature. As you can see, trying to model a one size fits all algorithm to predict compliance in a system could get pretty damn complicated.
Respiratory Ventilation:
However, in the medical respiratory industry, many ventilator patient circuit designers use something known as a tubing compliance factor, CT, which is defined in terms of volume per unit of pressure, across the entire range of breath ventilation. The goal here is to then multiply CT by the peak differential pressure reached between the patient circuit and patient lungs to determine the gas volume that is not delivered to the patient lungs and is trapped in the expanded patient circuit. The respirator could then be adjusted to include this lost volume if the clinician determines the volume error is too high to ignore for patient safety. CT is usually calculated by pressurizing the circuit tubing to a particular value, then determining how much volume it took to achieve that pressure. Conversely, sometimes it is determined by increasing the volume in the circuit by a particular amount, and then determining how much pressure was needed to do so.
For these respiratory circuits, CT is usually determined using the pressure required to expand a low breath volume. This is because compliance volume would be most dangerous to a patient at low breath volumes since the error could be a significant percentage of the total volume delivered by a respiratory ventilator. This CT is then used to adjust across all different volumes, even though CT would change logarithmically depending on what volume it is calculated at. However, in respiratory ventilation, standards dictate a maximum allowable compliance volume, therefore this change in CT as the delivered volume changes is small and the approximation is sufficient for clinical purposes.
Solution:
The simplest rule in nature: Conservation of Mass, this is the most reliable way to ensure that the flow delivers an accurate volume at the reservoir with each cycle.
Instead of calibrating volume based on targeting a particular flow across the system, the volumes would be targeted based on pressure at the flow target. Given that the gas in the system is air, it can be expected to display the properties of a Newtonian fluid. And even though the volume in the tubing expands because of compliance, the flow is still incompressible. Hence, by ensuring the pressure at the reservoir matches the pressure for a flow targeted for a particular volume at the pump, it can be ensured that the delivered volume is accurate, and compensates for any gas volume trapped in the expanded tubing.
Or simply put, in the diagram, P1 = P2!