How to model a water plant valve

I'm looking for a mathematical model of resistance in valve. Some relation between drop of pressure $\Delta p$ and volumetric flow rate $Q$.

I got this tip from some colleague to use formula for resistance of tube written below:

\begin{equation} R =\frac{8\mu l}{\pi r^4} \end{equation} where $\mu$ is viscosity of water ($\mu = 10^{-3}\,\mathrm{N\,s\,m^{-2}}$), $l$ is length of the tube and $r$ is its radius. So the relation would be $Q = \frac{\Delta p}{R}$. The problem with this approach is, that it would mean linear dependence going from 0 to some value, as showen below. However, the data I'm getting from the real system doesn't look like this is what's going on. See the graph below. At time $\mathrm{143\,s}$ the valve is completely opened. It's a level of a water in tank, which has a tube coming from the bottom. There is the valve I want to model. So the drop of pressure is actually pressure at the bottom oh the tank, to be clear. The way I see it, the level is steadily decreasing with the time for all pressures. The speed of level falling does not decrease with the lower pressure in the tank. This graph below is data from real system. There is something unrelated going on until the time about 400 s (filling the tank). Around that time the ventil opens. The green line is what it would look like, if I use the constant resistance approach. Doesn't look right.

In this full experiment there is one other valve letting the water out from the tank. That one is already modeled well and cause exponential look of the line. I wanted to show here, that the constant resistance approach doesn't fit the data. I have gathered some data from sensor that measures flow through the tube. Nobody has any idea what units the sensor gives, but it should be in some linear relation to the $\mathrm{m^3s^{-1}}$. Something like $\mathrm{m^3s^{-1}} = 2\cdot10^{-5} \cdot[sensor data]$. The valve is proportional, so the data is given for few setups of valve (opened on 25 % to 100 %) It seems the relation will be more complicated. So how can I express the relation between pressure and flow rate in valve?

What is physically happening there? I had an idea, that there is some limit to the flow rate through the ventil. Maybe no metter how much pressure there is, the flow cannot go higher. But others told me that was wrong idea and that doesn't happen in reality. Teacher told me it would ruin the valve, if the pressure was too high. I don't have an experience with that. Is that true?

Any idea how to obtain the information needed for successful model of the valve?

• You might want to take a look at this earlier answer relating to a vessel draining through a nozzle. – Floris Jan 13 '16 at 23:43
• Except for the bottom emptying out faster because it is most likely not flat and the pipe above the valve adding extra pressure your real data curve looks kind of exponential... your simulation time constant is wrong, did you fit it to the data? How did you arrive at that? You have to add the resistances of the valve and all the pipes, did you do that? – CuriousOne Jan 13 '16 at 23:43
• @CuriousOne I don't know what "simulation time constant" means, but the simulation fits for everythink but this tank emptying. It's more complicated system than one single tank but everything else fits well enough. When I add the resistance of a pipe above, it just make this R constant bigger. Shape of the simulated data will be the same. Actually the full experiment is not just this one valve letting water out of the tank. There is one other opened valve. So maybe that's not the best example, I'm sorry. However the data from sensor are gathered while only that one pump is open. – user50222 Jan 16 '16 at 12:44
• I can't know about "everything else" that you didn't mention, I am afraid, that would take supernatural powers, which I don't have. Looking at the data it's very doubtful that you have fitted the correct coefficients here, that much I can say. – CuriousOne Jan 17 '16 at 1:24

My experience with the type of system you've described is that it does not follow the Poiseuille equation your friend gave you. That equation better fits low flow rates of viscous fluids through tubes of small diameter compared to their length. The pressure -flow behavior is best fit with an empirical based quadratic model: $$\Delta P = K_2Q^2 + K_1Q$$ As David White said the vendor of the valve may have a graph of data that would allow you to fit to that equation or you may need to run your own experiments to obtain data.