An air valve (sometimes called a vacuum breaker valve) allows air to flow into the pipe during low pressure events, and expels the air when the pressure rises again. Typically the cracking pressure is atmospheric. The theory is summarized in Wylie, et al., 1993, pp. 130-131Wylie, E.B., V.L. Streeter & L. Suo, Fluid Transients in Systems, Prentice Hall, Englewood Hills, New Jersey, 1993., and detailed here.

There are four cases that can occur – two for inflow and two for outflow. The inflow can be subsonic or sonic, as can the outflow. These make up the four cases. The flowrate equations are given as follows:

• Sonic Inflow -

• Subsonic Inflow -

• Sonic Outflow -

• Subsonic Outflow -

where:

• Pi        = pressure in the pipe at the junction node

• Po        = atmospheric pressure

• To        = atmospheric temperature

• TL        = liquid temperature

• Cd Ai        = flow area for inflow (discharge coefficient times inflow orifice area)

• Cd Aout        = flow area for outflow (discharge coefficient times outflow orifice area)

• R        = gas constant

• g        = specific heat ratio cp/cv

Equation of State

Besides the compatibility equation 1 and equation 2, we need another equation to form a complete set. The ideal gas equation of state is used:

Substituting terms one obtains

Eliminating the volumetric flowrate using the mass flowrate obtains

Using the Compatibility Equations to eliminate mass flowrate yields:

(1)

Now we can use the Flowrate equations for each of the four inflow and outflow cases to eliminate as follows:

Sonic Inflow

The pressure at the junction can be solved directly from Equation 1 as follows:

where is the sonic inflow which can be calculated from the Sonic Inflow Flowrate equation based completely on input data.

Subsonic Inflow

Equation 1 is modified using the Subsonic Inflow Flowrate equation and solved iteratively using Newton-Raphson. Defining Pr = Pi /Po obtains:

Defining terms for Newton-Raphson solution yields

where the function F is solved for Pr.

Subsonic Outflow

Equation 1 is modified using the Subsonic Inflow Flowrate equation and solved iteratively using Newton-Raphson. Defining Pr = Pi/Po obtains:

Defining terms for Newton-Raphson solution yields

where the function F is solved for Pr.

Sonic Outflow

Using the Sonic Outflow equation from above, the pressure at the junction can be solved directly from Equation 1 as follows:

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