Dear Adi,

The origin of th horizontal reaction force is somewhat like this ( this derivation was told to me by a person who I consider as my guru in stress analysis):


The second half of the equation – AP -- is obvious. The first part -- 129W√[kt/((k+1)M)] – represents the momentum part of the equation (i.e., F = Aρv2, where A = cross-sectional area, ρ = density, v = velocity). Writing this another way:



F = Aρv*v = W*v (where W = Aρv, mass flow in kg/sec).



So the question is, what is the maximum possible velocity when a vent is opened, releasing a hot gas into the atmosphere (or an open system)? To estimate that, we can turn to de Laval’s nozzle equation (from rocketry):







where:


Ve
= Exhaust velocity at nozzle exit, m/s

T
= absolute temperature of inlet gas, K

R
= Universal gas law constant = 8314.5 J/(kmol·K)

M
= the gas molecular mass, kg/kmol (also known as the molecular weight)

k
= cp / cv = isentropic expansion factor

cp
= specific heat of the gas at constant pressure

cv
= specific heat of the gas at constant volume

Pe
= absolute pressure of exhaust gas at nozzle exit, Pa

P
= absolute pressure of inlet gas, Pa




If we assume that the valve vents into the atmosphere, then Pe approaches 0, so this equation approaches:



V = √(TR/M)(2k)/(k-1)



So:



F = W*√(TR/M)(2k)/(k-1)



Since R = 8314.5, √2R = 129, reducing the equation to:



F = 129W*√(kT)/[(k-1)M]



This is almost exactly the API-520 equation (we have “k-1” in the denominator inside the radical, while API-520 has “k+1”. I don’t know if that is an error on their part, or if they are using a completely different basis for their estimate than I am, or if the Pe/P part of the equation that I dropped out makes a difference…


Regards