Application to Relief Valves

Assumptions

The gas Equation of State is Eq. 3.1.1

The state 1 conditions are those in the pressure vessel; in particular this implies V1=0 as the vessel is much larger than the nozzle of a relief valve.

The flow from the vessel to the throat of the relief valve is isentropic (i.e. no friction and no heat exchange).

The law applies to the gas to characterize the isentropic expansion.

Eq. 3.1.2

Eq. 3.1.3

Eq. 3.1.4

Eq. 3.1.5

These are the assumptions made by all relief valve sizing standards.

Combining Eq. 3.1.3, 3.1.4 and 3.1.5, we get:

Eq. 3.1.6

Subsonic Flow Equations

Since V1=0, we can express equation 2.4.1 as follows:

Eq. 3.2.1

From equation 3.1.6 we have:

Eq. 3.2.2

From equation 3.1.1 and equation 3.1.2 we derive:

Eq. 3.2.3

And replacing in equation 3.2.2 we finally derive the change in internal energy of the gas during the isentropic expansion is:

Eq. 3.2.4

Similarly we have:

Eq. 3.2.5

Hence for the speed at the throat of the nozzle we get:

Eq. 3.2.6

The theoretical mass flow rate through the nozzle is:

and

So:

Eq. 3.2.7

Where A is the area of the nozzle and m is the mass flow rate.

In practice, the flow is always less than the theoretical flow rate and we introduce Kd, the coefficient of discharge. Many standards also add a safety factor asking that the flow be further de-rated by 10% so the sizing formula becomes:

Eq. 3.2.8

This is the formula used by the program.

Sonic Flow

As the pressure downstream is lowered, the velocity through the nozzle increases until it reaches the speed of sound. When this occurs, the flow becomes choked and lowering the pressure further on the downstream side does not produce an increase in flow rate. The pressure at which the flow becomes critical is called the critical pressure, Pcrit.

At this critical pressure, the speed through the nozzle is still given by equation 3.2.6 but it is also given by equation 2.2.2:

Eq. 3.3.1

and:

Eq. 3.3.2

and:

Eq. 3.3.3

Eliminating V2 we get the relationship between the critical pressure and the upstream pressure as a function of k.

Eq. 3.3.4

Because most of the time relief valves operate in choked mode (sonic flow), most standards insert this critical pressure ratio in Equation 3.2.8 Then, to cater for the cases when the flow is subsonic, they re-introduce formula 3.2.8. In the program we operate in a slightly different manner. The formula always remains the same (Eq. 3.2.8) but we calculate the critical pressure, PCRIT based on equation 3.3.4. If P2 is less than PCRIT we use PCRIT in the formula otherwise we use P2 and we have the advantage of having a single formula covering both cases.

Note: The various relief valve standards still require the calculation of the Critical Pressure Ratio.

Real Gases vs. Ideal Gases

Real gases do not quite behave according to equation 3.1.1 and a correction factor, Z, also called compressibility factor, is introduced:

Eq. 3.4.1

In practice nothing is changed except that T is replaced by ZT in all equations.

Z is best estimated by using either generalised compressibility charts or a 3rd order Equation of State generalized formula such as Redlich-Kwong or Peng-Robinson.

Purists will argue that at least two Z values should be used, one at the generating state and the other at the nozzle conditions, however the nozzle conditions at the critical pressure ratio also happen to be a maximum mass flow condition. Being a maximum, a first order deviation in pressure from the critical pressure induces a second order effect on the flow. We can therefore ignore the effect of a small error on Z at the critical pressure.