A valve junction has a pressure loss across the valve. When the valve closes this loss becomes infinite (Wylie, et al, 1993Wylie, E.B., V.L. Streeter & L. Suo, Fluid Transients in Systems, Prentice Hall, Englewood Hills, New Jersey, 1993., pp. 57). With two connecting pipes, the upstream and downstream pipe pressures are obtained from Compatibility Equation 1 and Equation 2:

From a mass balance:

The pressure drop is related to the flow rate according to the following:

where

Rvalve = Valve Resistance

Substituting the compatibility equations (Equations 1, 2) in and eliminating the pressures

Rearranging terms,

Using the quadratic equation, the solution for positive flow through the valve is

For negative flow, the solution is

After solving for flow rate, ṁ_valve, the upstream and downstream pressures can be obtained from the compatibility equations.

If the valve closes then 1/R_valve = 0 and ṁ_valve=0.

Exit Valves

If the valve is an exit valve, then the downstream pressure is known and

Valve Vapor Cavitation Theory

When the calculated pressure at the upstream or downstream side of a valve junction drops below vapor pressure, a vapor cavity forms on that side of the junction.

Vapor cavities can form on either side of the valve or on both sides.

As an example, when a vapor cavity forms downstream of a valve, the downstream pressure become fixed at the vapor pressure. The calculation then is similar to an exit valve, which also flows to a fixed pressure.

When a vapor cavity occurs on both sides of the valve, the pressure is at vapor pressure on both sides and thus there is no pressure drop across the valve. Accordingly, the flow rate goes to zero.

Similar to a pipe interior node, when the vapor volume is negative, the cavity collapses and the fluid pressure then rises above the vapor pressure.

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