Why Does Hagen-Poiseuille Predict Higher Exit Velocity Than Bernoulli?

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Discussion Overview

The discussion revolves around the comparison of exit velocities predicted by the Hagen-Poiseuille equation and Bernoulli's principle in the context of fluid flow through a nozzle connected to a high-pressure syringe. Participants explore the implications of laminar versus turbulent flow, the role of viscosity, and the effects of pressure loss in a tube.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions why Hagen-Poiseuille predicts a higher exit velocity than Bernoulli, considering Hagen-Poiseuille accounts for viscosity.
  • Another participant notes that Hagen-Poiseuille applies to laminar flow and suggests checking the Reynolds number.
  • A participant points out that the Hagen-Poiseuille computation assumes constant velocity, while Bernoulli's computation considers acceleration from rest to a specified speed.
  • One participant reports extreme values from their Hagen-Poiseuille calculations and discusses the need for a combination of both equations, while also mentioning the importance of checking the Reynolds number.
  • Concerns are raised about the accuracy of calculations, with one participant admitting to a mistake in unit conversion.
  • Another participant discusses using the Darcy-Weisbach equation to estimate pressure loss and iterating this into their Bernoulli speed calculation, questioning the validity of their resulting exit velocity.
  • Participants clarify that Hagen-Poiseuille assumes laminar flow, while Bernoulli assumes inviscid flow, leading to different implications for the flow regime.
  • One participant suggests using a version of Bernoulli's equation that includes frictional energy dissipation, emphasizing the significance of pressure drop in the tube.
  • Another participant shares their methodology for estimating pressure drop and velocity, indicating a transition from theoretical predictions to practical calculations.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Hagen-Poiseuille and Bernoulli's equations under the conditions described, with no consensus reached on the correct approach or the implications of their calculations.

Contextual Notes

Participants highlight the importance of flow regime (laminar vs. turbulent) and the need to consider frictional losses in their calculations. There are indications of potential inaccuracies in initial assumptions and calculations, particularly regarding pressure loss and velocity estimations.

bjarke
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I am trying to calculate the exit velocity of a nozzle (0,25Ø) that is connected to a high pressure syringe (10 BAR), however I cannot understand why Hagen-Poiseuille will have a higher exit velocity then Bernoulli when HP take viscosity into account.

Screenshot-2021-04-12-155018.png
 
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Your H-P computation assumes constant velocity. Your Bernoulli computation assumes acceleration from 0 to 45m/s. So your Bernoulli computation also takes the acceleration of mass into account.
 
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Actually, I get much more dramatic values for H-P:
Q = 1.9e+4 m3/s, v = 390e3 m/s...

Ps: I said "Bernoulli *also* takes acceleration into account". Actually, Bernoulli has no friction of course so it *only* takes acceleration into account... What you would actually need is a combination of the two (and then you need to check Reynolds as @BvU suggested, otherwise check Moody's Diagram)
 
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Arjan82 said:
Q = 1.9e+4 m3/s,
seems like the math needs checking :wink:

##\ ##
 
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millimeters, not meters... 🙈... nvm, I cannot brain today...
 
BvU said:
Hello @bjarke ,
:welcome: !​

Hagen-Poiseuille is for laminar flow. Check the Reynolds number ...

##\ ##
Yeah, with these speeds the Reynolds number is turbulant for both scenarios, but it is my understanding that both HP and bernoulli assumes laminar flow.
But the "pipe" in HP is more an orifice. If I halve the L in HP I would get twice the speed, which doesn't seem right either.
Is there some kind of Diameter to length ratio where HP is no longer accurate?
 
This is the same problem, but expanded for my entire system.
I use the Darcy-Weisbach equation to figure out what my pressure loss is in my tube, and then I iterate this pressure loss into my bernoulli speed to approximate an exit velocity. v=sqrt((2P-static-2P-headloss)/rho)
Does this exit velocity of 94m/s make any sense?

Another problem is that the pressure loss (P-DW) is very close to, and almost ends up being larger than the initial static pressure. So that if I decrease the tube diamater (D2) the loss will be greater, and the resulting velocity will end up being an imaginary number

Screenshot 2021-04-13 094913.png


Screenshot 2021-04-13 094933.png
 
bjarke said:
Yeah, with these speeds the Reynolds number is turbulant for both scenarios, but it is my understanding that both HP and bernoulli assumes laminar flow.
No, H-P assumes laminar flow indeed. Bernoulli assumes inviscid flow (i.e. no friction at all)
 
  • #10
This problem calls for the use of the version of the Bernoulli equation which includes the frictional energy dissipation. Both the acceleration and the frictional drag are going to contribute to the overall pressure drop. The main frictional resistance is going to reside in the 1 mm tube rather than the nozzle. And the flow in the tube is going to be turbulent.

As best I can estimate, the pressure drop in the 1 mm tube is going to be about 0.4 Bars, so most of the pressure drop is going to be due to the acceleration in the nozzle.
 
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  • #11
Chestermiller said:
This problem calls for the use of the version of the Bernoulli equation which includes the frictional energy dissipation. Both the acceleration and the frictional drag are going to contribute to the overall pressure drop. The main frictional resistance is going to reside in the 1 mm tube rather than the nozzle. And the flow in the tube is going to be turbulent.

As best I can estimate, the pressure drop in the 1 mm tube is going to be about 0.4 Bars, so most of the pressure drop is going to be due to the acceleration in the nozzle.
That sounds much more reasonable than the 20 bar I got. What did you equation look like?
 
  • #12
bjarke said:
That sounds much more reasonable than the 20 bar I got. What did you equation look like?
Well, I started with the 45 m/s you got using Bernoulli without friction. From that, I estimate a velocity of 45/16 = 2.8 m/s in the 1 mm tube. From that, I got a Re of about 2800 in the 1 mm tube, which is just beyond the HP region. I then used Darcy Weissbach to get the pressure drop in the tube of about 0.47 Bars. So the actual acceleration pressure change would only be about 9.5 Bars, rather than 10 Bars. So then, one could continue to iterate until the sum of the pressure drops due to friction plus acceleration is 10 Bars.
 

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