Average water mass flow rate

  • #1
Hey, I'm a student who has to complete a written assignment for physics. I chose to launch water rockets and investigate the effect of volume on the range. I am trying to also come up with a mathematical model however I am struggling to understand an equation

I found the following equation to calculate the average mass flow rate:

ṁ = A * Cd * √(2ρΔP)
(https://www.palmbeachschools.org/sc.../05/Bottle_Rocket_Calculations_Simplified.pdf)

Where:
ṁ is the mass flow rate
A is the area of the nozzle
Cd is the coefficient of drag
p is the density
and ΔP is the average Pressure acting on the water

Does anyone know where this equation is derived from cause I can't find it anywhere?

Thanks in advance for all replies.
 

Answers and Replies

  • #2
20,971
4,603
This equation starts out from ##\dot{m}=\rho v A##, where v is the exit velocity. Are you familiar with that equation, which is the same as the volumetric flow rate times the density? If the exit velocity were determined by the Bernoulli equation, then it would satisfy:
$$\frac{1}{2}\rho v^2=\Delta P$$##C_d## is a nozzle discharge factor to correct for nozzle shape and non-Bernoulli deviation. It is typically on the order of unity.
 
  • #3
This equation starts out from ##\dot{m}=\rho v A##, where v is the exit velocity. Are you familiar with that equation, which is the same as the volumetric flow rate times the density? If the exit velocity were determined by the Bernoulli equation, then it would satisfy:
$$\frac{1}{2}\rho v^2=\Delta P$$##C_d## is a nozzle discharge factor to correct for nozzle shape and non-Bernoulli deviation. It is typically on the order of unity.

Thanks so much, I now understand. I watched some videos and found this helpful diagram:
bernoul.gif

But I don't see how to get the average pressure(ΔP) from Bernoulli's???
 

Attachments

  • bernoul.gif
    bernoul.gif
    19 KB · Views: 897
  • #5

Attachments

  • upload_2018-8-24_1-56-26.png
    upload_2018-8-24_1-56-26.png
    22.6 KB · Views: 288
  • upload_2018-8-24_1-57-9.png
    upload_2018-8-24_1-57-9.png
    2.2 KB · Views: 810
  • upload_2018-8-24_1-57-25.png
    upload_2018-8-24_1-57-25.png
    24.3 KB · Views: 776
  • #7
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
35,867
6,588
ṁ = A * Cd * √(2ρΔP)
The equation makes sense except for the Cd term. In many rocketry equations there is a drag term for the air drag on the rocket, but that would not be relevant here since we are concerned only with the speed of the exhaust, not the acceleration of the rocket. So presumably it represents the nozzle drag on the water flow. But in that case the greater the drag the lower the flow rate, which is not what the equation says.
I would have expected ṁ = A * √(2ρΔP/Cd) (though not necessarily the factor 2).
 
  • #8
20,971
4,603
The equation makes sense except for the Cd term. In many rocketry equations there is a drag term for the air drag on the rocket, but that would not be relevant here since we are concerned only with the speed of the exhaust, not the acceleration of the rocket. So presumably it represents the nozzle drag on the water flow. But in that case the greater the drag the lower the flow rate, which is not what the equation says.
I would have expected ṁ = A * √(2ρΔP/Cd) (though not necessarily the factor 2).
Cd is the discharge coefficient, related to the nozzle shape (Google discharge coefficient). Typically, Cd is < 1, so it does result in reduced discharge mass flow rate.
 
  • #9
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
35,867
6,588
Cd is the discharge coefficient, related to the nozzle shape (Google discharge coefficient). Typically, Cd is < 1, so it does result in reduced discharge mass flow rate.
Ok, thanks.
 
  • #10
6
0
Hi guys, pleas can someone run me through exactly how the equation is derived? Up until you end up at
ṁ = A * Cd * √(2ρΔP)
 
  • #11
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
35,867
6,588
Hi guys, pleas can someone run me through exactly how the equation is derived? Up until you end up at
ṁ = A * Cd * √(2ρΔP)
Can you answer any of my questions in post #6?
 
  • #12
6
0
Can you answer any of my questions in post #6?
No, unfortunately I cannot. Can you answer mine?
 
  • #13
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
35,867
6,588
No, unfortunately I cannot.
Then I'll give you some hints.
We are applying Bernoulli’s equation to the water stream as it goes from a point just inside the rocket to a point just outside.
The first term on each side of the first equation refers to the pressure at a point in the stream flow. What does ##\Delta P## mean in the other equation?
The middle term on each side of the first equation refers to a height at a point in the stream flow. In the actual question, what is the change in height?
The last term each side refers to a velocity. What do we know about the velocities of the water flow at the two points (relative to the rocket)?
 

Related Threads on Average water mass flow rate

Replies
6
Views
624
  • Last Post
Replies
3
Views
5K
Replies
2
Views
4K
  • Last Post
Replies
3
Views
1K
Replies
8
Views
691
Replies
1
Views
4K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
1
Views
775
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
1
Views
6K
Top