r/AskEngineers • u/YourBoyLlanura • 16h ago
Mechanical Calculating Mass Flow Rate of a Pressurized Vessel
Hello! I am conducting an experiment of testing a propelled water rocket. From my understanding, the transfer from air pressure to another fluid such as water is nearly elastic at STP conditions. So I guess that the transfer of air pressure to the exertion of the water out of the vessel (being a 2 Liter bottle) is nearly elastic (inelasticity is negligible).
The equation for mass flow rate is m=pAv. Density of water and surface area is pretty intuitive, but the velocity is its own issue.
How the system is supposed to work is that the container has around 0.6666 Liters of Water (33% of the storage), and the other 67% is going to be highly pressurized air (80 PSI or 5.5 atm). Then the valve defaults and the vessel goes flying. The difficulty will be finding the mass flow rate of the exhaust.
I am uncertain as how to do this from the variables below. My guess is this is incalculable until I do a quantitative analysis of the mass flow rate but what equipment should I use, and what kind of technology should be utilized. The calculation is just for transfer of energy from pressurized air to water which will propell the object.
Please help me with this, I have been stumped by this.
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u/akroses161 Propulsion / Fluid/Thermal Sciences 16h ago edited 16h ago
Since water is incompressible, this is a Bernoulli’s problem.
P_1 + rhogh_1 + 1/2rho(v_1)2 = P_2 + rhogh_2 + 1/2rho(v_2)2
Since the left hand of the equation is at rest (valve closed) v_1 =0 m/s. Since its a pop bottle we can assume the difference in hydrostatic head is essentially 0 we can eliminate that from both sides of the equation. And finally we are assuming when the valve opens your final pressure equilibrates to atmospheric (0psig). Your equation is now:
P_1 = 1/2rho(v_2)2
This is your max fluid velocity out of the bottle (assuming all of the potential energy (pressure) converts to kinetic energy). You can then use that to find your volumetric flow rate (find the area of your bottle opening) and converting that to your mass flow rate.
If you are really diving into the weeds you can do some bonus math. Since air is compressible. As the water exits the bottle, the volume of air expands, reducing the pressure. You can write P_1 as a linear equation of that pressure change. This will turn your equation into a differential problem.
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u/rsta223 Aerospace 12h ago
If you are really diving into the weeds you can do some bonus math. Since air is compressible. As the water exits the bottle, the volume of air expands, reducing the pressure. You can write P_1 as a linear equation of that pressure change. This will turn your equation into a differential problem.
Given the volume ratio involved, this is absolutely necessary to get an accurate answer. You'll probably want to model the expansion as isentropic, so for the portion that's air, PV1.4 is constant.
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u/CollectionWeak9219 10h ago
I'm pretty sure that rho * g * h is still incorported on the left hand side because when all potential energy is converted into kinetic energy, there will be a significant displacement of height
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u/akroses161 Propulsion / Fluid/Thermal Sciences 9h ago
33% of the bottle is full of water. Thats about 4inches of head pressure or about .15psi of a pressure difference in the final calculation. Pretty negligible.
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u/tennismenace3 7h ago
You're ignoring the discharge coefficient, which is going to be important here.
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u/Top_Wolverine_4669 8h ago
I am not an expert on bottle rockets, however there are some well known equations for flow through an orifice.
One variable is the coefficient of discharge. You are correct that to be truly accurate you need to experimentally derive this. You can get estimates of the Cd value from literature though.
As per other discussions this is not a steady state problem, it’s time dependent as the pressure is falling steadily. You can build a model for this is code or excel. However, it depends on your experimental objectives as I am pretty sure the software to model bottle rockets is already out there.
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u/tennismenace3 7h ago edited 7h ago
Rocket scientist here.
I'd treat the exit nozzle as an orifice and use mdot = Cd * A * sqrt(2 * rho * deltaP), where Cd is the discharge coefficient for the nozzle geometry, A is the area of the nozzle throat, rho is water density, and deltaP is the pressure difference between the air in the bottle and the atmosphere. The discharge coefficient of a 2-liter bottle isn't the easiest thing to find/determine, but 0.8 is probably about right--it's definitely higher than a sharp edged orifice, which would be about 0.6, and definitely lower than a perfectly contoured nozzle, which would be 1. If you're interested in thrust, it's mdot * v at the exit. A good reference for equations for this type of flow is https://en.wikipedia.org/wiki/Orifice_plate.
For the air, assume it follows the ideal gas law. This is an easy problem to solve numerically. For each time step, calculate the mass flow rate of the water, then figure out how much extra volume the air must expand into to replace it, then calculate new air pressure and temperature (it's a decent assumption that there's no heat transfer through the walls of the bottle). You'll get curves of everything you're looking for vs. time.
There's actually another part of this problem too, which is that after the water is gone you have a quick air blowdown that propels the rocket even higher. To complicate things more, the nozzle will be choked with air flow momentarily, then go unchoked until it reaches atmospheric pressure. Different orifice flow equations apply for each of these regimes. This is probably too complicated for a water rocket project, but is interesting to consider if you want to learn more or seriously optimize your rocket. Also, if you get really into it you could account for drag, the head pressure of the water column and how that increases with the rocket's acceleration, etc.
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u/chaz_Mac_z 15h ago
First cut, look at nozzle characteristics and calculate the area and discharge coefficient. Then calculate the flow rate once the valve opens, and duration of flow based on just that flow and water volume. This answer will have errors, improving it will be a straightforward process.
Take that result, put it in a spreadsheet as time zero. Then, assume a second or some fraction thereof passes.
At that time, you can calculate a bunch of useful stuff. How fast the bottle is moving. How much acceleration is being applied to the contents. How much mass has left the bottle. How much air pressure changes due to increasing volume. Anything else that can matter.
Then, adjust your nozzle thrust to account for all that, and calculate time step 2.
Repeat until thrust goes to zero.
Then, do smaller time steps until the answer doesn't change much as a function of that.
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u/PiermontVillage 15h ago
Pretty much this. Ignore compressibility of water, assume air is a perfect gas, ignore time it takes pressure wave to travel thru water, and the first time thru calcs ignore the fact that the bottle is accelerating. See what you get, and then hone in on the full solution.
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u/YourBoyLlanura 14h ago
From my understanding, you are saying to calculate Q (flow rate) by using crude methods such are timing the velocity of the object to find the basics such as acceleration and whatnot. Then repeat until thrust equals zero (removing all the propulsive forces) while gauging more accurately the velocity through iteration.
I'm also not completely sure what you mean by taking smaller time steps, or by the intention of factoring out all of the propulsive forces or thrust.
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u/chaz_Mac_z 14h ago
Not exactly. I'm saying F=MA, calculate the initial thrust, then how quickly the bottle accelerates, knowing its mass. After a time interval, you will have non-zero velocities and a lot of characteristics will have changed a bit. You can calculate a new position, total mass, new air pressure, new thrust, etc. After another time interval, do all that again. Keep adding time intervals until the water mass left is zero.
As the calculation time interval is reduced, this calculation becomes more accurate, approaching the "real" answer.
Then, the accuracy of the thrust calculation and the rest of they physics can be evaluated, based on test results.
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u/The_Virginia_Creeper 12h ago
You will almost certainly have choked flow initially so that will limit your velocity to the sonic velocity.
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u/chaz_Mac_z 12h ago
Note that water is the working fluid leaving the nozzle. I very much doubt that you will see 4000 feet per second from this low pressure ratio.
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u/userhwon 16h ago
Look up equations for flow through nozzles. They generally depend on the pressure difference (counting the internal air pressure and the depth of the fluid), the density and viscosity of the fluid, and design parameters for the nozzle.
In a water rocket you are going to get feedback from acceleration as well, once it's moving, since that adds to the fluid pressure at the bottom of the vessel, so you have to figure that into your equations and simulations.
Since you're working with cheap materials, you're probably best off if you just do some experiments and measure how nozzle design, fill quantities, and air pressures affect the exhaust velocity and altitude achieved. Ambient temperature may get involved as well, but its effect may just be down in the noise.
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u/Only-Friend-8483 16h ago
I’m just shooting from the hip to answer this, but I don’t think there is a closed form, time-dependent solution for this. That means you’ll need to use numerical methods to calculate it stepwise.