Morton
The pressure, density and/or temp in the first of the equations at the referenced site are misleading. Pressure and temp and /or density should be in terms of stagnation conditions. Use of static p, T or density will lead to some error depending upon the upstream Mach number.
I believe the original derivation of the equation was based on quasi steady flow from a large area reservoir thru a "much smaller area" orifice (negligible upstream velocity) such that upstream static and stagnation conditions are equivalent.
Regards