Pressure of water coming out of a hose nozzle

[mr_sparxx]

Homework Statement

A fire engine is pumping water at a rate of 400l/min with a pressure of 7.5 atm into a hose with a diameter of 45 mm. The nozzle at the end of the hose has a diameter of 20 mm. The fireman holding the nozzle is 25 m above the fire engine. Calculate:
a) speed of water inside the hose
b) speed of water coming out of the nozzle
c) pressure of the water when entering the nozzle.
1 atm = 1.01 · 10^5 Pa ; rho = 1 g/cm^3 ; g=9.81 m/s²

Relevant Equations

Continuity equation: $$S_1 v_1 = S_2 v_2$$
Bernoulli's Theorem: $$p_1 + 1/2 \rho v_1^2 + \rho g h_1 = p_2 + 1/2 \rho v_2^2 + \rho g h_2 $$

I am using some solved excercises to put homework to my students but I cannot understand the proposed solution for c)
a)
$$ Q= 400 l/min = 6.67 ·10^3 m^3/s $$
applying continuity equation:
$$ Q = S_1 v_1 \Rightarrow v_1 = \frac Q S_1 = 4.19 m/s $$
This is exactly the same as the proposed solution

b) Considering how the section changes from the hose to the nozzle and using continuity equation:
$$ v_2 = \frac {S_1} {S_2} v_1= 21.24 m/s $$
This is exactly the same as the proposed solution

c) Using Bernouilli's Theorem and taking the hose section at the bottom (A) and the hose section just before the nozzle (B):
$$ p_A + 1/2 \rho v_A^2 + \rho g h_A = p_B + 1/2 \rho v_B^2 + \rho g h_B $$

Well, this means $p_A = 7.5 atm , v_A = 4.19 m/s, h_A = 0 m, v_B = 4.19 m/s$ (continuity) and $h_B = 25 m$ and it is just a matter of isolating $$ p_B = p_A + 1/2 \rho (v_A^2 - v_B^2) - \rho g h_B = p_A - \rho g h_B = 5.125 · 10^5 Pa $$

However the proposed solution takes the hose section just before the nozzle (A) and the water at the tip/exit of the nozzle (B)
$$ p_A + 1/2 \rho v_A^2 + \rho g h_A = p_B + 1/2 \rho v_B^2 + \rho g h_B $$
Stating that $p_B= 1 atm, h_A = h_B , v_A = 4.19 m/s, v_B = 21.24 m/s$ and then
$$ p_A = p_B + 1/2 \rho (v_B^2 - v_A^2) = 3.535 · 10^5 Pa $$

I would say that this is wrong and my intuition is that Bernouilli's theorem is applicable in the hose, but not so sure it is applicable when the water is exiting the nozzle... I am not sure that you can state that water pressure is 1 atm as soon as it gets out of the nozzle...

[Edit] I see that this must be the case... that water pressure is 1 atm at the exit, but I would like to understand why, and why can I get to a different (wrong) answer...

 

[haruspex]

Did you notice that the official solution nowhere uses the 7.5atm pressure?
Think about the number of degrees of freedom and the number of given values. Do you see an issue?

 

[erobz]

Homework Statement: A fire engine is pumping water at a rate of 400l/min with a pressure of 7.5 atm into a hose

[Edit] I see that this must be the case... that water pressure is 1 atm at the exit, but I would like to understand why, and why can I get to a different (wrong) answer...

The pressure change is communicated through the flow at the local speed of sound in the medium.

In our approximation for incompressible flow that implies that the local speed of sound $c \to \infty$. So the water jet "knows" the very instant it has left the end of the nozzle and the internal pressure of the flow matches the pressure of the new surroundings, atmospheric.

As for the first part of problem, they are either being careless, or getting the reader to think about viscous loss in the hose. A real fire hose and nozzle have losses, and the real pump that is generating the pressure adjusts volumetric output in response to those pressures its working against. If the nozzle is wide open, the losses are mainly in the hose and the significant pressure drop is across the hose. If however the nozzle is almost closed the significant pressure drop is across the nozzle...

For this problem you ignore the pump/hose/viscosity, and assume the pressure drop comes from accelerating the flow through the nozzle. So the pressure at the pressure at the hose inlet of $7.5 \text{atm}$ becomes extraneous. They are telling you about how much viscous loss is in the total system hose + nozzle on accident in my opinion. They likely didn't mean for the problem to be that real, unless they are going to ask you about the viscous loss across the hose in a subsequent problem (having assumed negligible losses across nozzle).

 

[erobz]

Oh, I didn’t read this well. The fireman is up in the air 25 m. That’s why they are saying the pump discharge is at 7.5 atm probably…whoops. It’s still extraneous to the asked questions.

 

[haruspex]

Oh, I didn’t read this well. The fireman is up in the air 25 m. That’s why they are saying the pump discharge is at 7.5 atm probably…whoops. It’s still extraneous to the asked questions.

Seems to me everything you wrote in post #3 remains valid.
If we ignore losses the pump pressure can be deduced from the rest of the information, and it's less than stated.

 

[erobz]

Yeah, ( I got lucky )I just was envisioning this without any elevation head, and when I read it again I thought I’d better say something about it since I had not calculated it out.

 

[mr_sparxx]

The pressure change is communicated through the flow at the local speed of sound in the medium.

In our approximation for incompressible flow that implies that the local speed of sound . So the water jet "knows" the very instant it has left the end of the nozzle and the internal pressure of the flow matches the pressure of the new surroundings, atmospheric.

Thank you: that perfectly explains one of my doubts.

As for the first part of problem, they are either being careless, or getting the reader to think about viscous loss in the hose.

I think that unfortunately it is the first case. This problem is for 17-18 years old who have just been introduced to the Bernouilli's principle. Although I have very basic notions of fluid mechanics I understand how the Benouilli's principle is based on Energy conservation and how viscosity introduces some energy loss. However, the intended audience of this problem are used to lossless situations when calculations are involved. So I agree with you.

Did you notice that the official solution nowhere uses the 7.5atm pressure?
Think about the number of degrees of freedom and the number of given values. Do you see an issue?

Yes, this is the key. I assume that when problems of this level state that the fire engine is pumping water at a rate of 400l/min with a pressure of 7.5 atm it is giving you actual values... So, I took this values for certain and discarded the 1atm pressure at the nozzle (as I am not so confident in fluid mechanics). I was pretty sure that something was wrong: either one of the pressures or the Bernouilli's principle application (losses) but could not see which.

By the way, any good introductory fluid mechanics book for a post graduate?

Thank you!

[Edit] BTW the 25 m elevation is extraneous as well...

 

[erobz]

By the way, any good introductory fluid mechanics book for a post graduate?

I don't know what level of analysis you yourself are looking for, but if you are looking to be introduced to fluid mechanics in a wide array of applications with plenty of worked examples from which you could probably re-teach some of it to enthusiastic high school students, I would stay in the "Engineering Fluid Mechanics" lane ( any book with the engineering approach will probably be fine). The text on my bookshelf is "Engineering Fluid Mechanics, Crowe, Elger, Williams, Roberson - 9th Edition"