Fluid Mechanics- inviscid fluid pressure in a pipe

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Discussion Overview

The discussion revolves around the application of Bernoulli's equation to a fluid mechanics problem involving water flow through a circular pipe with a horizontal elbow and a nozzle. Participants are attempting to determine the velocities at different sections of the pipe, the gauge pressure at one section, and the force exerted by the water on the elbow. The conversation includes theoretical considerations and calculations related to inviscid, incompressible flow.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose using Bernoulli's equation to find the gauge pressure, questioning whether to set the exit velocity to zero.
  • Others clarify that gauge pressure is defined as absolute pressure minus atmospheric pressure, suggesting that the exit pressure should be considered in the context of gauge pressure.
  • There is a discussion about whether to assume the exit pressure is zero, with some arguing that this assumption is incorrect if the exit is open to the atmosphere.
  • Participants express uncertainty about the proper application of Bernoulli's equation, particularly regarding the treatment of pressures as absolute or gauge.
  • Calculations are presented, with one participant calculating the gauge pressure at the inlet as 22308 Pa, but there is a request for clarity on the use of Bernoulli's equation in this context.
  • Some participants emphasize the importance of using absolute pressures in the Bernoulli equation, while others question the implications of this for the calculations being performed.

Areas of Agreement / Disagreement

Participants generally agree on the need to use Bernoulli's equation for the problem, but there is disagreement regarding the treatment of pressures (gauge vs. absolute) and the assumptions made about the exit pressure. The discussion remains unresolved on these points.

Contextual Notes

There are limitations regarding the assumptions made about the exit pressure and the definitions of gauge and absolute pressures, which are not fully resolved in the discussion.

Lee Harrington
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Homework Statement



Water flows through a circular pipe with a 180° horizontal elbow and exits to the atmosphere through a nozzle as shown in Fig. Q3. The diameter of the pipe is 300 mm and the diameter of the nozzle exit is 160 mm. The density of water is 999 kg/m3 . The mass flow rate of water is 140 kg/s. Incompressible, inviscid flow may be assumed. Determine:
(i) the velocity of the water at sections 1 and 2 respectively,
(ii) the gauge pressure at section 1, and
(iii) the magnitude and direction of the force Rx exerted by the water on the elbow

Homework Equations


Bernoulli's Equation?

The Attempt at a Solution


I have determined that Vin=1.98m/s and Vout=6.97m/s and i know how to determine Rx, but i have no idea how to get the gauge pressure

Do you use bernoulli's equation and let P2 equal atmospheric pressure? But does that mean you let the exit velocity equal zero?
 

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Lee Harrington said:

Homework Statement



Water flows through a circular pipe with a 180° horizontal elbow and exits to the atmosphere through a nozzle as shown in Fig. Q3. The diameter of the pipe is 300 mm and the diameter of the nozzle exit is 160 mm. The density of water is 999 kg/m3 . The mass flow rate of water is 140 kg/s. Incompressible, inviscid flow may be assumed. Determine:
(i) the velocity of the water at sections 1 and 2 respectively,
(ii) the gauge pressure at section 1, and
(iii) the magnitude and direction of the force Rx exerted by the water on the elbow

Homework Equations


Bernoulli's Equation?

The Attempt at a Solution


I have determined that Vin=1.98m/s and Vout=6.97m/s and i know how to determine Rx, but i have no idea how to get the gauge pressure

Do you use bernoulli's equation and let P2 equal atmospheric pressure? But does that mean you let the exit velocity equal zero?
I don't see any other way to analyze this pipe than to use the Bernoulli equation. I would not assume that if the gauge pressure is zero, the exit velocity is zero, which you have found is clearly not the case. If the velocity were zero, where would all the water go which enters the pipe at location 1?

Remember, gauge pressure = absolute pressure - atmospheric pressure, if that helps.
 
ok, so using bernoulli's equation:

pin = ρ((Vout2 - Vin2)/2) ?

And that's the gauge pressure?
 
Also for the next part, do you continue to assume that pout=0 ?
 
Lee Harrington said:
ok, so using bernoulli's equation:

pin = ρ((Vout2 - Vin2)/2) ?

And that's the gauge pressure?

It would be better if you showed your complete calculations.
 
SteamKing said:
It would be better if you showed your complete calculations.
converting mass flow rate to volume flow rate and using Q=VA:
Vin=1.98m/s ; Vout=6.97m/s

pin=999((6.972 - 1.982)/2)=22308 Pa
 
Lee Harrington said:
converting mass flow rate to volume flow rate and using Q=VA:
Vin=1.98m/s ; Vout=6.97m/s

pin=999((6.972 - 1.982)/2)=22308 Pa
Yeah, but where's your Bernoulli equation for this problem?
 
SteamKing said:
Yeah, but where's your Bernoulli equation for this problem?
sorry, being lazy

pin/ρ + Vin2/2 + gz = pout/ρ + Vout2/2 + gz

pout=0
pin/ρ + Vin2/2 = + Vout2/2

isolating pin
pin = ρ((Vout2 - Vin2)/2)
 
Lee Harrington said:
sorry, being lazy

pin/ρ + Vin2/2 + gz = pout/ρ + Vout2/2 + gz

pout=0
pin/ρ + Vin2/2 = + Vout2/2

isolating pin
pin = ρ((Vout2 - Vin2)/2)
Since you are not exhausting to a vacuum, Pout is clearly a gauge pressure.

In the Bernoulli equation, you should use absolute pressures in your calculations. Any results can be converted to gauge readings afterwards.
 
  • #10
SteamKing said:
Since you are not exhausting to a vacuum, Pout is clearly a gauge pressure.

In the Bernoulli equation, you should use absolute pressures in your calculations. Any results can be converted to gauge readings afterwards.

I'm not sure what you mean, so your saying let Pout equal to Patm as there is no gauge pressure on exit?

My understanding is that if there is a gauge pressure on exit you would need to have been told that in order to calculate Pin
 
  • #11
Lee Harrington said:
I'm not sure what you mean, so your saying let Pout equal to Patm as there is no gauge pressure on exit?

My understanding is that if there is a gauge pressure on exit you would need to have been told that in order to calculate Pin
In the absolute pressure scale, a perfect vacuum has P = 0 psia and atmospheric pressure is P = 14.7 psia.

P = 0 psig (psi gauge) is equivalent to P = 14.7 psia (psi absolute)

If you are dealing with pressure differentials, it makes no difference if both pressures are absolute or gauge, but Bernoulli's equation is written assuming the pressures are measured on the absolute scale.
 
  • #12
Your answers for the gauge pressures in and out of the section of pipe (22308 and 0) are correct. Now, what are your ideas on how to do part iv?

Chet
 

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