# Flow Coefficient Parameterized Valve (TL)

Generic valve model with flow coefficient parameterization

## Library

Thermal Liquid/Valves

## Description

The Flow Coefficient Parameterized Valve (TL) block models a generic valve using a parameterization based on the valve flow coefficient, a constant relating the flow rate to the pressure drop. The block accepts the flow coefficient in metric units or imperial units:

• Cv — Valve flow coefficient in imperial units of USG/min. Cv data is often provided at a temperature of 60 °F and a valve pressure differential of 1 lb/in^2 [1].

• Kv — Valve flow coefficient in metric units of m^3/h. Kv data is often provided at a temperature at 5–30 °C and a valve pressure differential of 1 bar [2].

The physical signal L sets the valve opening fraction as a numerical value from 0 to 1. A value of 0 corresponds to a fully closed valve and leakage flow. A value of 1 corresponds to a fully open valve and maximum flow. The input signal saturates at these values.

### Valve Opening Characteristics

At valve opening fractions between 0 and 1, the opening area depends on the valve opening parameterization selected in the block dialog box. The block provides three parameterizations:

• `Linear` — Model the valve open fraction f(L) as a linear function of the lift input signal L:

`$f\left(L\right)=L$`
This behavior is suitable at constant pressure drops in steady-state systems. The figure shows the relationship between the valve flow coefficient, expressed as a fraction of the maximum flow coefficient, and the valve lift input signal.

• `Quick opening` — Model the valve open fraction f(L) as a power function of the lift input signal L:

`$f\left(L\right)={L}^{1/\alpha }$`
The parameter α is an exponent number that you specify. This behavior is suitable for pressure-relief valves that must open quickly from a fully closed state. The figure shows the relationship between the valve flow coefficient, expressed as a fraction of the maximum flow coefficient, and the valve lift input signal.

• `Equal percentage` — Model the valve open fraction f(L) as an exponential function of the valve lift input signal:

`$f\left(L\right)={R}^{L-1}$`
The parameter R is the valve rangeability—the ratio of the maximum and minimum valve flow rates. The figure shows the relationship between the valve flow coefficient, expressed as a fraction of the maximum flow coefficient, and the valve lift input signal.

### Mass Balance

The mass conservation equation in the valve is

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,$`
where:

• ${\stackrel{˙}{m}}_{A}$ is the mass flow rate into the valve through port A.

• ${\stackrel{˙}{m}}_{B}$ is the mass flow rate into the valve through port B.

### Momentum Balance

The momentum conservation equation in the valve is

`${p}_{A}-{p}_{B}=\frac{\stackrel{˙}{m}\sqrt{{\stackrel{˙}{m}}^{2}+{\stackrel{˙}{m}}^{2}{}_{cr}}}{{\rho }_{Avg}{S}^{2}},$`
where:

• pA is the pressure at port A.

• pB is the pressure at port B.

• ρAvg is the average fluid density.

• S is the valve opening area.

• ${\stackrel{˙}{m}}_{cr}$ is the critical mass flow rate.

The valve opening area is

`$S=\left\{\begin{array}{ll}{S}_{Max}f\left(L\right),\hfill & {S}_{Max}f\left(L\right)>{S}_{Min}\hfill \\ {S}_{Min},\hfill & \text{Else}\hfill \end{array}$`
where:

• SMax is the maximum valve opening area.

• SMin is the minimum valve opening area.

• f(L) is the inherent valve flow characteristic.

The critical mass flow rate is

`${\stackrel{˙}{m}}_{cr}={\mathrm{Re}}_{cr}{\mu }_{Avg}\sqrt{\frac{\pi }{4}S},$`
where:

• Recr is the critical Reynolds number at which the flow regime transitions between laminar and turbulent.

• μAvg is the average dynamic viscosity.

### Energy Balance

The energy conservation equation in the valve is

`${\varphi }_{A}+{\varphi }_{B}=0,$`
where:

• ϕA is the energy flow rate into the valve through port A.

• ϕB is the energy flow rate into the valve through port B.

## Parameters

### Parameters Tab

Flow coefficient specification

Flow coefficient to use in the block calculations. Options include Cv, defined in imperial units, and Kv, defined in metric units. The default setting is `Cv coefficient (USG/min)`.

Cv coefficient at maximum flow

Valve flow coefficient in the fully open position, specified in imperial units. The default value is `1`. This parameter is active only when the Flow coefficient specification parameter is set to ```Cv coefficient (USG/min)```.

Kv coefficient at maximum flow

Valve flow coefficient in the fully open position, specified in metric units. The default value is `1`. This parameter is active only when the Flow coefficient specification parameter is set to ```Kv coefficient (m^3/h)```.

Cv coefficient at minimum flow

Valve flow coefficient in the fully closed position, specified in imperial units. The default value is `1e-4`. This parameter is active only when the Flow coefficient specification parameter is set to ```Cv coefficient (USG/min)```.

Kv coefficient at minimum flow

Valve flow coefficient in the fully closed position, specified in metric units. The default value is `1e-4`. This parameter is active only when the Flow coefficient specification parameter is set to ```Kv coefficient (m^3/h)```.

Valve opening characteristics

Valve opening response to the lift input signal specified through port L. The block provides three valve opening models:

• `Linear` — Models the valve open fraction f(L) as a linear function of the lift input signal L:

`$f\left(L\right)=L$`

• `Quick opening` — Models the valve open fraction f(L) as a power function of the lift input signal L:

`$f\left(L\right)={L}^{1/\alpha }$`

• `Equal percentage` — Models the valve open fraction f(L) as an exponential function of the valve lift input signal:

`$f\left(L\right)={R}^{L-1}$`

Exponent number

Parameter α in the power expression of the quick opening valve model:

`$f\left(L\right)={L}^{1/\alpha }$`
The exponent number determines how rapidly the valve open fraction approaches its maximum value. This parameter is active only when the Valve opening characteristics parameter is set to `Quick opening`. The exponent number must be greater than zero. The default value is `2`.

Valve rangeability

Parameter R in the exponential expression of the equal-percentages valve model:

`$f\left(L\right)={R}^{L-1}$`
The rangeability parameter determines the minimum valve opening fraction, a number generally different from zero. This parameter is active only when the Valve opening characteristics parameter is set to `Equal-percentages`. Typical values range from 20 to 50. The default value is `50`.

Cross-sectional area at ports A and B

Area normal to the direction of flow at inlets A and B. This area is assumed the same for the two inlets. The default value is `0.01` m^2.

Characteristic longitudinal length

Distance traversed by the fluid between inlets A and B. The default value is `0.1` m^2.

Critical Reynolds number

Reynolds number at which flow transitions between laminar and turbulent regimes. Flow is laminar below this number and turbulent above it. The default value is `12`.

### Variables Tab

Mass flow rate into port A

Mass flow rate into the component through port A at the start of simulation. The default value is ```1 kg/s```.

## Ports

• A — Thermal liquid conserving port representing valve inlet A

• B — Thermal liquid conserving port representing valve inlet B

• L — Physical signal input port for the valve opening control signal

## References

[1] Control Valve Handbook. 4th ed. Marshalltown, IA: Fisher Controls International. 2005.

[2] Flow of Fluids through Valves, Fittings and Pipe. Stamford, CT: Crane, 2010.