There are 3 basic types of relief valves:

1. Conventional
2. Balanced Bellows
3. Pilot Operated Relief Valves ( PORV)

Conventional Relief Valve

The conventional relief valve is a differential pressure device. Back pressure on the outlet of the relief valve effects it. They are really only any use if they discharge to atmosphere. Discharging to a relief/flare header is not practical as the back pressure is always variable and will effect when they conventional relief valve pops.

Balanced Bellows Relief Valve

The balanced bellows relief valve addresses the back pressure problem with conventional relief valves but is not a complete solution. The balanced bellows relief valve flow performance starts getting impacted at back pressures in excess of 30% of relief valve set point. Also the bellows that eliminates the effect of back pressure on the ability of the relief to pop has mechanical integrity issues. Large balance bellows relief valves ( such as 6x8 ) cannot take more than 60 psig back pressure without it causing a failure of the bellows.

Piloted Operated Relief Valves

The pilot operated relief valve is not normally effected by back pressure (as to its ability to pop at set). The effect of back pressure on the performance of the relief valve is also minimal. They can have up to 80% back pressure with it only derating the capacity by 20%. There is one interesting effect: if the back pressure exceeds the upstream pressure of the relief valve the relief valve pops even if it is not at set. This can be prevented by purchasing a back flow preventor. PORVs also can handle high inlet losses by using a remote pilot. The sensing line for the remote pilot must use a special pickup to allow it to capture the dynamic pressure part of the gas.

Farris
w = area*C*Kd*P*Kb*SQRT[Mw/(T*Z)]
Kd = 0.953 Series 2600 valves
Kd = 0.975 Series 3000 PORV's

area used here is "effective area" (actual * 0.9)
Farris provides 11.11 percent extra area (1/.9)
e.g. a R orifice (API 16 square inches) is really 17.78 square inches

Anderson Greenwood

w = area*C*Kd*P*SQRT[Mw/(T*Z)]
Kd = 0.809 ( actual is 0.899 )
discharge coef is so low because the nozzle is not
particularly smooth entry

Actual area is the same as API orifice.


GPE Controls

w = area*C*Kd*P*SQRT[Mw/(T*Z)]
Kd = 0.860 ( actual is 0.9*0.956 )


Actual area is the same as API orifice.

Consolidated
w = area*C*Kd*1.041*P*SQRT[Mw/(T*Z)]
Kd = 0.95
to be compatible with standard formulae use
Kd = 0.98895 , making Consolidated the highest capacity of all
the manufacturers

note Consolidated has an extra factor in their
formulae
the 1.041 takes care of the difference between the
so called ASME area (actual) and the API area)
The API area for an R orifice is 16.0 square inches
actual area for Consolidated valve is 18.6 square inches

CRANE
w = area*C*Kd*P*SQRT[Mw/(T*Z)]
Kd = 0.97 ! based on old catalog Believe its still right
! Crane seems to be doing the effective area game

area = API orifice area (actual for Anderson Greenwood and GPE)
C = 520*Sqrt(k[(2./(k+1))**(k+1)/(k-1)])
P = Relieving pressure, psia
Mw = mole weight


Effect of backpressure on valve performance varies with valve type
and with manufacturer :

Conventional valve can not reasonable be used for
variable back pressures as it effects the pop pressure

Balanced bellows are good for variable back pressure
in as far as set point not being effected (up to 80 to 90%)
however flowrate is effected! Also it cannot be calculated
as just the effect of backpressure on a constant orifice
device because the backpressure can effect the lift of
the PSV. You need to check each manufacturer.

PORV's are best here. not only is set point uneffected,
the flowrate is not greatly effected and can be easily
calculated. As far as I know all manufactures are the same
here. So if back pressure starts hitting the 60 to 80 percent
region give some thought to PORV's.


Inlet losses

Calculate flow rate for inlet loss as flow rate at pop/0.9
You divide by 0.9 to take into account the derating done
as required by ASME. Note this will be similar as using
rated capacity, since rated capacity includes 10 % overpressure
The above method is a little better since it works for
multiple psv's which may have 116 % of set pressure and for
fire cases.

Note balanced bellows and conventional psv's may only attain
25 percent capacity at pop and require the full 10 percent
over pressure to attain full lift. This would tend to prevent
chattering. Still I think assuming full flow is best.

Note: the above method is especially important for pilot operated
psv's as they attain full lift immediately. Some people
are tempted to size inlet line based on required flow rate
This is not correct!

Inlet lines to PSVs

Inlet lines are sized based on the rated capacity of the relief valve at rated conditions. In the case of multiple relief valves, rated capacity and conditions will be at 10% accumulation for purposes of inlet line calculations. The allowable pressure drop is 3%. The pressure drop is based on the difference of stagnation pressures. The stagnation pressure is the static pressure plus the velocity head. The stagnation pressure is the total pressure. The 3% pressure loss is essentially the frictional losses not changes in pressure as a result of conversion of static pressure to velocity head.

Intial assumptions of inlet lines should be one line size larger than the relief valve inlet. The gate valve should be on the large line. All valves on relief valve inlets and outlets are required to be full port. Note that api gate valves 4 inch and larger are full port. 3 inch and smaller gate valves can be either reduced port of full port, so in the case of relief valve isolation specify full port.

Outlet lines from PSVs

The lines running from the relief valves to the common header are also sized based on rated capacity. The common relief header is sized based on required capacity of any relief valves that can conceivable relieving at the same time.