Mechanical seal failure flow rate calculation

you cannot calculate this. rather strange set up, you have a fixed and a floating bush? normal seal leakage is approx 5-10cc/hr. are you injecting N2 where the green arrows are? normally there is only 1 bush and this appears to be for safety because the product is not environmentally safe when it leaks. if thisi s the case then you have the wrong seal. one needs to know what the produst is, if you have a high Vapour Pressure then you will have a gas leak, if low VP then liquid leakage. so you question is not clear, do you want to know what the max leakage is in the event of a seal failure or normal leakage?
send some more information about product and what you are trying to achieve.

I would have liquid ammonia at the red arrow if the seal failed. I would like to know the max leakage at the event of a failure.

Define fail. Your valve is 100% functional to 100% failed. You could calculate the flow rate without seals to determine max leak rate. Then you will know you will be between this number and zero.

Humaidq
Max leakage in the event of failure needs to define the mode of failure. Looking at the drawing there are no pressures shown but if we assume flow through a 1/2" pipe with 1/8" orifice and 20PSI Delta P then leakage is around 1.5Gal/min. orifice area represents the bush clearance but the leakage depends on the level of damage. i think that you need to have a dual pressurised seal anyway for safety. I remember years ago getting a whiff of ammonia and it was not pleasant. if this were mine i would convert to double pressurised seals.

I now understnad why you have a double bush configeration which is rather unusual in fact this is the first time i have ever seen it. I think it is used like this to minimise any leakage to atmosphere in the event of a seal failure. not the best way to seal this if you are after safety.