Calculating Nozzle Reaction Force for Pressurized Gas Cavity

[sappo14]

Lets say I have a tank of argon gas at high pressure, let's say around 17MPa. It is connected to a hose with a nozzle of a certain diameter d, and opened in ambient conditions, and the hose is used to pressurize a cavity, is there a simple way to calculate the nozzle reaction force, or the force required to keep the nozzle in place?

 

[russ_watters]

Welcome to PF...

Multiply the pressure by the cross sectional area of the hose for a rough approximation.

 

[dgOnPhys]

Hi Russ and sappo14,

Russ is basically providing a static approximation, as if we had a closed valve at the end of the hose.

I think that including dynamics the requirements become more stringent though: let me know if you spot anything wrong with this line of reasoning:
you can think of the hose as a variable mass system (it is like holding a rocket trying to take off)
F=v.dM/dt
in our case it becomes
F=v.rho.A.v=rho.A.v^2
v can be estimated assuming energy losses in the hose are negligible (unlikely but that should reduce velocity hence lessen the requirements) and the transformation adiabatic, then for a compressible fluid (e.g. see http://en.wikipedia.org/wiki/Bernoulli%27s_principle" )
v^2/2 + psi + gamma/(gamma-1) p/rho = const
if we now neglect the contribution of conservative forces, psi term (e.g. gravity) and use the ideal monotomic (argon) gas approximation for gamma=5/3, we have
rho.v^2= 5.p0
where we also assumed the pressure at the end of the hose is negligible compared to the pressure in the tank and also that the density in the gas stream does not change immediately after the hose opening.
That said we end up with an estimate for the force required to hold the hose of
F=5.A.p0
that is 5 times the static case.
Convincing at all?