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Poppet valve in an isothermal liquid network

**Library:**Simscape / Fluids / Isothermal Liquid / Valves & Orifices / Flow Control Valves

The Poppet Valve (IL) block models a flow-control valve in an isothermal liquid
system. The poppet can either have a cylindrical or ball control member. For ball
valves, you can choose between sharp-edged and conical seats. The poppet opens or closes
according to the displacement signal at port **S**. A positive signal
retracts the poppet and opens the valve.

**Poppet Valve Schematic**

**Poppet Valve Top View**

The opening area of the valve is calculated as:

$${A}_{open}=\pi h\mathrm{sin}\left(\frac{\theta}{2}\right)\left[{d}_{s}+\frac{h}{2}\mathrm{sin}\left(\theta \right)\right]+{A}_{leak},$$

where:

*h*is the vertical distance between the outer edge of the cylinder and the seat, indicated in the schematic above.*θ*is the**Seat cone angle**.*d*_{s}is the**Stem diameter**.*A*_{leak}is the**Leakage area**.

The opening area is bounded by the maximum displacement
*h*_{max}:

$${h}_{\mathrm{max}}=\frac{{d}_{s}\left[\sqrt{1+\mathrm{cos}\left(\frac{\theta}{2}\right)}-1\right]}{\mathrm{sin}\left(\theta \right)}.$$

For any stem displacement larger than
*h*_{max},
*A*_{open} is the sum of the maximum
orifice area and the **Leakage area**:

$${A}_{open}=\frac{\pi}{4}{d}_{s}^{2}+{A}_{leak}.$$

For any combination of the signal at port **S**
and the cylinder offset less than 0, the minimum valve area is the **Leakage
area**.

The opening area of the valve is calculated as:

$${A}_{open,sharp-edged}=\pi {r}_{O}\sqrt{{\left({G}_{sharp}+h\right)}^{2}+{r}_{O}^{2}}\left[1-\frac{{r}_{B}^{2}}{{\left({G}_{sharp}+h\right)}^{2}+{r}_{O}^{2}}\right]+{A}_{leak},$$

where:

*h*is the vertical distance between the outer edge of the cylinder and the seat, indicated in the schematic above.*r*_{O}is the seat orifice radius, calculated from the**Seat orifice diameter**.*r*_{B}is the radius of the ball, calculated from the**Ball diameter**.*G*_{sharp}is the geometric parameter: $${G}_{sharp}=\sqrt{{r}_{B}^{2}-{r}_{O}^{2}}.$$*A*_{leak}is the**Leakage area**.

The opening area is bounded by the maximum displacement
*h*_{max}:

$${h}_{\mathrm{max}}=\sqrt{\frac{2{r}_{B}^{2}-{r}_{O}^{2}+{r}_{O}\sqrt{{r}_{O}^{2}+4{r}_{B}^{2}}}{2}}-{G}_{sharp}.$$

For any ball displacement larger than
*h*_{max},
*A*_{open} is the sum of the maximum
orifice area and the **Leakage area**:

$${A}_{open}=\frac{\pi}{4}{d}_{O}^{2}+{A}_{leak}.$$

For any combination of the signal at port
**S** and the ball offset that is less than 0, the minimum
valve area is the **Leakage area**.

The opening area of the valve is calculated as:

$${A}_{open,conical}={G}_{conical}h+\frac{\pi}{2}\mathrm{sin}\left(\theta \right)\mathrm{sin}\left(\frac{\theta}{2}\right){h}^{2}+{A}_{leak},$$

where:

*h*is the vertical distance between the outer edge of the cylinder and the seat, indicated in the schematic above.*θ*is the**Seat cone angle**.*G*_{conical}is the geometric parameter: $${G}_{conical}=\pi {r}_{B}\mathrm{sin}\left(\theta \right),$$ where*r*_{B}is the ball radius.*A*_{leak}is the**Leakage area**.

The opening area is bounded by the maximum displacement
*h*_{max}:

$${h}_{\mathrm{max}}=\frac{\sqrt{{r}_{B}^{2}+\frac{{r}_{O}^{2}}{\mathrm{cos}\left(\frac{\theta}{2}\right)}}-{r}_{B}}{\mathrm{sin}\left(\frac{\theta}{2}\right)}.$$

For any ball displacement larger than
*h*_{max},
*A*_{open} is the sum of the maximum
orifice area and the **Leakage area**:

$${A}_{open}=\frac{\pi}{4}{d}_{O}^{2}+{A}_{leak}.$$

For any combination of the signal at port
**S** and the ball offset that is less than 0, the minimum
valve area is the **Leakage area**.

At the extremes of the orifice opening range, you can maintain numerical robustness in your
simulation by adjusting the block **Smoothing factor**. When the
smoothing factor is nonzero, a smoothing function is applied to every calculated
displacement, but primarily influences the simulation at the extremes of this
range.

The normalized orifice opening is:

$$\widehat{h}=\frac{h}{{h}_{\mathrm{max}}}.$$

The **Smoothing factor**, *s*, is applied to the
normalized opening:

$${\widehat{h}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\widehat{h}}_{}^{2}+{\left(\frac{s}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\widehat{h}-1\right)}^{2}+{\left(\frac{s}{4}\right)}^{2}}.$$

The smoothed opening is:

$${h}_{smoothed}={\widehat{h}}_{smoothed}{h}_{\mathrm{max}}.$$

This smoothed opening is used in the valve opening area, either
*A _{open,conical}* or

Mass is conserved through the valve:

$${\dot{m}}_{A}+{\dot{m}}_{B}=0.$$

The mass flow rate through the valve is calculated as:

$$\dot{m}=\frac{{C}_{d}{A}_{valve}\sqrt{2\overline{\rho}}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\right)}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}},$$

where:

*C*_{d}is the**Discharge coefficient**.*A*_{valve}is the current valve open area.*A*_{port}is the**Cross-sectional area at ports A and B**.$$\overline{\rho}$$ is the average fluid density.

*Δp*is the valve pressure difference*p*_{A}–*p*_{B}.

The critical pressure difference,
*Δp*_{crit}, is the pressure differential
associated with the **Critical Reynolds number**,
*Re*_{crit}, the flow regime transition
point between laminar and turbulent flow:

$$\Delta {p}_{crit}=\frac{\pi \overline{\rho}}{8{A}_{valve}}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{{C}_{d}}\right)}^{2}.$$

*Pressure loss* describes the reduction of pressure in the
valve due to a decrease in area. *PR*_{loss} is
calculated as:

$$P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}{\sqrt{1-{\left(\frac{{A}_{valve}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{{A}_{valve}}{{A}_{port}}}.$$

*Pressure recovery* describes the positive pressure change in
the valve due to an increase in area. If you do not wish to capture this increase in
pressure, set the **Pressure recovery** to
`Off`

. In this case,
*PR*_{loss} is 1.

Ball Valve (G) | Ball Valve (TL) | Gate Valve (IL) | Needle Valve (IL)