Cv/Kv and xT Loss Model
AFT xStream allows the user to enter valve loss coefficient data in terms of Kv or Cv. Conversions are handled seamlessly in the background. Kv is the metric equivalent to Cv.
The valve Cv loss model type is unique in that this loss model can be used as part of both the subsonic and sonic loss calculations across the valve.
Subsonic Loss
The relationship between valve Cv, xT, and flow is provided by ANSI/ISA-75.01.01-2012ANSI/ISA Standard 75.01.01-2012, Industrial-Process Control Valves - Part 2-1: Flow Capacity - Sizing equations for fluid flow under installed conditions, 2012, published by International Society of Automation, Research Triangle Park, North Carolina, USA standard as follows:
Where C is the flow coefficient (Cv or Kv) and N is a numerical constant found in Table 1 of the ANSI standard referenced above. The units for the flow, pressure and density will be based off of the numerical constant chosen. FP is a correction factor for piping geometry. This correction will be treated as 1 unless the Valve ID has been entered by the user.
The xsizing parameter is the ratio of the pressure drop across the valve to the inlet pressure of the valve for subsonic flow:
Y is the expansion factor, which accounts for the change in density as the gas passes through the vena contracta, and is calculated as follows:
where xT is the pressure ratio, x, when the valve has just begun to choke.
Sonic Loss
If choked flow occurs across the valve, then the mass flow equation above is still used. However, xsizing is now equivalent to xT, which is the pressure ratio when the valve first begins to choke.
Estimated Cv or Kv in Output
The Output window can display an estimated Cv or Kv when it was not explicitly defined by the user so long as the appropriate parameter is added in the Output Control. The estimated Cv or Kv value is calculated using the mass flow rate equation shown above, with xsizing estimated as follows:
Transient Theory
A valve junction has a pressure loss across the valve. When the valve closes this loss becomes infinite. With two connecting pipes, the upstream and downstream pipe parameters must be determined through iteration. The valve will be treated as a mass flow boundary. A guess for the velocity is made, which can be used to solve for the inflow conditions using the compatibility equations as shown for the Assigned Flow junction at the pipe inlet. This will provide values for the conditions at the valve outlet, including the mass flow at the valve.
The outlet static pressure can now be used to calculate the pressure ratio x and the expansion coefficient Y as are shown above. The calculated x and Y can then be used in the to determine the ANSI/ISA predicted mass flow as shown above.
The mass flow calculated from the assigned flow boundary solution and the mass flow calculated from the ANSI/ISA equation can now be compared. The difference between these two values will be driven to zero using iterations on velocity.
Exit Valves Transient Theory
If the valve is an exit valve, then the downstream pressure is known. The user specified exit properties are assumed to be static. The compatibility equations can be substituted into the ANSI/ISA mass flow equation shown above, which is then rearranged to solve for the inlet pressure at the valve.
If the flow reverses then a similar substitution can be done with the appropriate compatibility equation
Iterations are then performed by varying Pout (the pressure at the inlet of the connected pipe).
Sonic Choking Transient Theory
Restriction choking can occur at any time at the valve junction for either a regular or exit type valve. If the valve becomes choked, the valve Cv/Kv equation for choked flow must be used.
The mass flow term can be eliminated to find the equation in terms of pressure, density, and velocity only
The wave and particle path compatibility equations can then be substituted to eliminate density and velocity.
Though not shown here, this equation can be rearranged into a polynomial and solved for static pressure. Once the static pressure is known the choked valve can be solved using the typical transient sonic choking calculation.
At the end of the calculation x will be recalculated. If x is greater than xchoked then the solution is valid. Otherwise, the valve is not choked, and the non-choked valve calculation are repeated using the choked valve solution to bracket the velocity solution.
