# Discharge of a Pressure Vessel Containing Liquid Nitrous Oxide

Alexander Fernandes

## Homework Statement

I want to design a closed tank with an orifice hole (that can be opened and closed remotely) at the bottom of the tank and model how the mass flow rate of liquid nitrous oxide changes with respect to time, as the liquid nitrous oxide leaves the tank due to a pressure difference across the orifice. I want to also try and stay away from using CFD software as complete accuracy is not really overly important here. I have seen a lot of problems online where they solve discharge problems but all seem to use pressurised gases and the Ideal Gas Law. My problem, however, involves a pressurised liquid or am I missing something.

Known Data:
Initial Tank Pressure (assumed Uniform)
Initial Tank Temperature (assumed Uniform)
External Pressure (let's assume atmospheric)
Orifice Diameter / Area
Tank Diameter / Area
Fluid Density
Tank length
Coefficient of Discharge

## Homework Equations

dm/dt = cd * Ao * rho * sqrt(2 * ((P(t)-Pa)/rho) - gh(t))
where:
dm/dt = mass flow rate
cd = coefficient of discharge
Ao = orifice cross-sectional area
rho = fluid density
P(t) = tank pressure as a function of time
Pa = atmospheric pressure
g = acceleration due to gravity
h(t) = head of fluid as a function of time

## The Attempt at a Solution

I have tried using Bernoulli's equation, shortly after realising that the problem might be unsteady-flow and that the fluid might have to be considered compressible making the aforementioned formula useless. I also used the formula above which I derived from Bernoulli's equation assuming Quasi-steady flow but I'm not sure that is correct either.

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Mentor

Mentor
model how the mass flow rate of liquid nitrous oxide changes with respect to time
Can you comment about any flammability issues with this setup? Are you familiar with Intrinsic Safety and Explosion Proof considerations? Thanks.

Alexander Fernandes
Well Actually I have already designed a similar tank and calculated the longitudinal stress and hoop stress that the nitrous oxide might impart on the vessel in question. I also carried out FEA on the tank to back up hand any calculations. All stresses are well below the youngs modulus of the tanks material (Aluminium Alloy 6061T6). I used a similar tank with the hybrid rocket motor I designed as part of my FYP(Final year project), however the discharge time of the tank was alot longer than predicted due to decrease in pressure over the course of the discharge period. Hence, why I would like a model of the mass flow rate. As for flamability, nitrous oxide is not flamable at room temperature (nor at its cryogenic temperature). Nitrous oxide needs to heated to the point at which it breaks it nitrogen-oxygen bonds (577 degrees c) before any combustion could take place.

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Mentor
Mechanistically, what would you assess the flow and phase behavior to be within a few orifice diameters of the exit orifice? What would be the state of the N2O right at the exit orifice? What do you think the streamlines in the region would look like?

Alexander Fernandes
The N20 would be in the gaseous phase and will want to expand at the exit plane of the orifice.

Mentor
The N20 would be in the gaseous phase and will want to expand at the exit plane of the orifice.
Please dope this out in more detail. What is happening as the N2O approaches the exit orifice?

Also, is the tank sealed at the top? Is there N2O vapor above the liquid, with no air in the head space? What is the fraction of liquid and vapor in the tank to start?

Alexander Fernandes
There is a relief valve at the top of the tank, so at the start the tank can be considered to be comprised of 100% liquid nitrous oxide pressurised at 50 bar with a temperature of 180K. A valve at the bottom of the tank with a diameter (Do) is opened. The exit plane of the orifice can be considered to have 1 bar of pressure and a temperature of 298K (initially). My guess it that the liquid starts to vapourise at the entrance of the orifice (is that correct). Ultimately I just want to model the pressure drop in the tank, mass flow rate and the time needed for the system to reach equilibrium.

Mentor
Gravity is a complicating factor, because there will be vapor formed close to the exit orifice, and the vapor will tend to rise. Would you be willing to temporarily assume that there is no gravity, and solve it that way first? What is the equilibrium vapor pressure of N2O at 180 K? (Antoine equation parameters are available on the NIST data sheet). Is the temperature going to somehow stay at 180 K, or does this have to be adiabatic?

Without gravity, I would envision a spherically converging flow toward the exit. At some radial distance from the exit, there would be a phase transition from liquid to vapor. The phase transition would occur at the equilibrium vapor pressure.

That's my thinking so far.

Alexander Fernandes
The vapour pressure for nitrous oxide at 180K is around 0.6152 bar. I don't mind the assumption that the system is adiabatic but I think the system should not be isothermal as the tank will cool as the vapoursied fluid leaves the tank. As for gravity I am not sure. I only really care about solving this problem as a one dimensional problem, so the stream line can be considered going straight in the axial direction of the orifice.(accuracy is not overally important here otherwise I would have just used CFD).

Mentor
So, if the pressure is 0.6 bars, what would cause the liquid to come out of the tank?

Alexander Fernandes
This graph was taken from the NIST webbook and shows a vapour pressure of 0.6 bar at 180K, ok so maybe the liquid does not vapourise in the tank but the liquid must come out because you have a pressure difference of 49 bar across the orifice.

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$$65000+\rho g h+0=100000+0+\frac{1}{2}\rho v^2$$where the left side of this equation applies to the top liquid surface, the right hand side of the equation applies to the exit orifice, ##\rho## is the fluid density, h is the liquid level in the tank, and v is the liquid velocity coming out of the exit orifice.