# Problem with turbulent (?) flow prediction

#### brunopeniche123

Hi!!

I'm a student of pharmaceutical sciences which means that my physics knowledge isnt that great ahah, if I say a big mistake in the following text, a thousand "sorrys" ahah:

So, I am now writing my master thesis and currently I am finishing writing a part where fluid mechanics are involved:

I have a pressure driven flow system, where the flow rate is controlled by a capillary, which is much smaller in terms of diameter than the rest of the tubbing, thus, having more hydraulic resistance than the rest of the system, leading to flow control. So, the flow rate is dictated by the capillary.

During my experiments measuring the flow rates, we used the Haggen-Pouiselle law to predict and change flow rate (Q= (dP. pi. r^4)/(8. viscosity.lenght of the capillary)). In the smallest capillary diameters (0,127mm with lenghts of 150 and 60mm) flows predicted were very similar to the pratical ones (flow rates were predicted using https://www.dolomite-microfluidics.com/support/microfluidic-calculator/), which made sense because the flow rate calculator uses the haggen-poiseulle law to predict flows and the flow was laminar (Low reynolds number according to it)

The problem is when I increase capillary diameter, to the 0,180 and 0,250mm. The predicted values are bigger than the real ones. Which is logical because the online calculator tells me that the flow is turbulent (although I dont really understand that part because the value of the reynolds number given is smaller than is smaller than 2000).

So, first I was intrigued because if flow is turbulent, I should have real flows greater than the predicted, and the opposite was happening. But I forgot that part

So I though of the Darcy-Weisbach equation which allow us to calculate flow velocity of turbulent and laminar flows (dP = friction factor. (L/diameter). ((density.v^2)/2)). So I searched for the friction factor of my capillary which was a PEEK capillary used in HPLC. So in order to find the friction factor, I had to calculate the relative roughness of the capillary. I searched for the absolute roughness of PVC and other plastics and assumed a value of 0,0015 mm. I divided by the diameter of my capillary which in this calculation was the one with 0,250mm of diameter which gave me 0,006. I then used the Moody's diagram to find the friction factor. I used the reynolds number gave by the dolomite online calculator. Making the calculations (pressure drop was 10 psi which I converted to Pa), I had a flow value of 5ml/min of flow rate, much higher than the real one of 2,3ml/min and the predicted of 2,9ml/min.

The first problem that I find here is that for knowing the reynold number, I have to know the velocity of the fluid, something that I wanted to predict... So, this approach doesnt make much sense to me.

The second, is that I am getting real flow rates slower than the predictions of Haggen-Pouiselle law and it should be the opposite.

Can you help me please with this issue?
Is there any alternative equation to predict flow rate?
Can you see what is wrong here?

Thank you so much for your attention!!!