Finding the average water mass flow rate in a water rocket

[kelv_01]

Homework Statement

Hey, I'm writing a research paper on water rocket and the effect of volume on the maximum height achieved I am trying to also come up with a mathematical model however I am struggling to understand an equation I found to calculate the average mass flow rate of water which is
ṁ = A * Cd * √(2ρΔP)
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.

Relevant Equations

ṁ = A * Cd * √(2ρΔP)

Could anyone kindly kind me as to where this formula is derived from and how, because I searched and can't find it anywhere

 

[haruspex]

Problem Statement: Hey, I'm writing a research paper on water rocket and the effect of volume on the maximum height achieved I am trying to also come up with a mathematical model however I am struggling to understand an equation I found to calculate the average mass flow rate of water which is
ṁ = A * Cd * √(2ρΔP)
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.
Relevant Equations: ṁ = A * Cd * √(2ρΔP)

Could anyone kindly kind me as to where this formula is derived from and how, because I searched and can't find it anywhere

This is effectively a repost of what you posted on an old thread:
https://www.physicsforums.com/threads/average-water-mass-flow-rate.953904/#post-6192964As noted there, it is derived from Bernoulli’s equation applied to the water stream as it goes from a point just inside the rocket to a point just outside.
Referencing the equations in a post in that thread:
The first term on each side of the Bernoulli 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 rocket question, what is the change in height between the two points?
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)?

 

[kelv_01]

This is effectively a repost of what you posted on an old thread:
https://www.physicsforums.com/threads/average-water-mass-flow-rate.953904/#post-6192964As noted there, it is derived from Bernoulli’s equation applied to the water stream as it goes from a point just inside the rocket to a point just outside.
Referencing the equations in a post in that thread:
The first term on each side of the Bernoulli 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 rocket question, what is the change in height between the two points?
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)?

I'm unsure of the answer to the last 2 questions, but I know that ΔP is the average pressure in the rocket

 

[haruspex]

I'm unsure of the answer to the last 2 questions, but I know that ΔP is the average pressure in the rocket

We are considering the flow through the nozzle, so that's from a point just inside to a point just outside.

The Δ means a difference between two values, in this case, ΔP, a difference between two pressures. So yes, it is pressure difference between inside and outside.
In the Bernoulli equation there is a separate term for each pressure, so rearrange those to be on the same side of the equation as a difference and call that ΔP.

The z variables in Bernoulli are the heights of the two points. Again, rearrange to form the difference, z₁-z₂. Since the points we are considering in the flow are very close we can drop this term.

This leaves the two velocities. While the water is in the rocket, what is its velocity relative to the rocket?

 

[kelv_01]

Whilst in the rocket, water should have a higher velocity compared to the rocket

 

[haruspex]

Whilst in the rocket, water should have a higher velocity compared to the rocket

Not sure how you think that. Maybe you have the wrong model in your mind.
The idea is that we think of the water accelerating from rest (with respect to to the rocket) to exhaust velocity over a short distance through the nozzle. This is driven by the pressure difference. So for the purposes of Bernoulli's equation the water velocity inside the rocket is zero and that outside the rocket is the exhaust velocity.

 

[kelv_01]

Not sure how you think that. Maybe you have the wrong model in your mind.
The idea is that we think of the water accelerating from rest (with respect to to the rocket) to exhaust velocity over a short distance through the nozzle. This is driven by the pressure difference. So for the purposes of Bernoulli's equation the water velocity inside the rocket is zero and that outside the rocket is the exhaust velocity.

Ohh alright, I understand that bit now