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Valve for the prevention of flow aimed counter to its intended direction

**Library:**Simscape / Fluids / Thermal Liquid / Valves & Orifices / Directional Control Valves

The Check Valve (TL) block models the opening
characteristics of a proportional valve that is forced shut when the pressure difference
between its ports is reversed (or, more precisely, dropped below a threshold known as
the *cracking pressure*). Check valves are common in backflow
prevention devices, such as those used in public water supply networks, where
contaminated water downstream of a water main must not be allowed to return upstream. No
specific valve shutoff mechanism—whether ball, disc, diaphragm, or other—is assumed in
the block. The allowed direction of flow is always from port **A** to
port **B**.

**Y-Shaped Check Valve with Piston-Type Control Member Partially
Retracted**

The valve cracks open when the pressure drop across it rises above the cracking pressure specified in the block. The opening area increases linearly with pressure, save for two small pressure intervals near the fully open and fully closed positions (over which nonlinear smoothing is applied in order to remove numerical discontinuities). When the pressure drop reaches the maximum value specified in the block, the valve is fully open and its opening area no longer increases with pressure. The flow rate through the valve is never truly zero as a small leakage area remains when the pressure falls below the cracking pressure.

The (smoothed) valve opening area is first computed as a linear function of
pressure, either that at the valve entrance (port **A**) or the
drop between those at the entrance and at the outlet (port **B**).
Which of these pressures—termed control pressures—features in the area calculations
depends on the setting of the **Pressure specification method**
block parameter:

$${p}_{\text{Control}}=\{\begin{array}{ll}{p}_{\text{A}},\hfill & \text{PressureatportAmethod}\hfill \\ {p}_{\text{A}}-{p}_{\text{B}},\hfill & \text{Pressuredifferentialmethod}\hfill \end{array},$$

where *p* is pressure; the subscript
`Control`

indicates the value to be used in determining the
opening area of the valve. Subscripts `A`

and `B`

denote the thermal liquid ports at which the pressures are obtained. The pressures
at the ports are always defined as absolute pressures. The cracking pressure, at
which the opening area is at a minimum, is similarly defined:

$${p}_{\text{Crack}}=\{\begin{array}{ll}{p}_{\text{Crack,G}}+{p}_{\text{Atm}},\hfill & \text{PressureatportAmethod}\hfill \\ \Delta {p}_{\text{Crack}},\hfill & \text{Pressuredifferentialmethod}\hfill \end{array},$$

where the subscript `Crack`

indicates a valve
just cracking open, subscript `G`

a gauge value, and subscript
`Atm`

the standard atmospheric value. The value of
*p*_{Crack,G} is obtained from the
**Cracking pressure (gauge)** block parameter; that of
*Δp*_{Crack} is obtained from the
**Cracking pressure differential** block parameter. The maximum
pressure, at which the opening area is at its largest, is:

$${p}_{\text{Max}}=\{\begin{array}{ll}{p}_{\text{Max,G}}+{p}_{\text{Atm}},\hfill & \text{PressureatportAmethod}\hfill \\ \Delta {p}_{\text{Max}},\hfill & \text{Pressuredifferentialmethod}\hfill \end{array},$$

where subscript `Max`

denotes a maximum pressure
(obtained as gauge). The value of *p*_{Max,G}
is obtained from the **Maximum opening pressure (gauge)** block
parameter; that of *Δp*_{Max} is obtained from
the **Maximum opening pressure differential** block parameter. The
control, cracking, and maximum pressures give for the linear form of the valve
opening area:

$${S}_{\text{Lin}}=\frac{{S}_{\text{Max}}-{S}_{\text{Crack}}}{{p}_{\text{Max}}-{p}_{\text{Crack}}}\left({p}_{\text{Ctl}}-{p}_{\text{Crack}}\right)+{S}_{\text{Crack}},$$

where *S* denotes the (linear) opening area. The
opening area at cracking is equal to the small value specified in the
**Leakage area** block parameter. The primary purpose of this
parameter is to ensure the numerical robustness of the model by ensuring that no
portion of a thermal liquid network becomes completely isolated during simulation.

**Valve opening area as a linear function of pressure**

The primary purpose of the leakage area of a closed valve is to ensure that at no time does a portion of the hydraulic network become isolated from the remainder of the model. Such isolated portions reduce the numerical robustness of the model and can slow down simulation or cause it to fail. Leakage is generally present in minuscule amounts in real valves but in a model its exact value is less important than it being a small number greater than zero. The leakage area is obtained from the block parameter of the same name.

To ensure adequate simulation performance, the valve opening area is smoothed over two small pressure intervals near the specified cracking and maximum pressures. The smoothing is accomplished by means of the polynomial expressions (to be incorporated into the final form of the opening area expression):

$${\lambda}_{\text{Crack}}=3{\gamma}_{\text{Crack}}^{2}-2{\gamma}_{\text{Crack}}^{3}\text{\hspace{1em}}\text{\hspace{0.22em}}\text{\hspace{0.22em}}\text{\hspace{0.22em}}\text{and}\text{\hspace{1em}}\text{\hspace{0.22em}}\text{\hspace{0.22em}}\text{\hspace{0.22em}}{\lambda}_{\text{Max}}=3{\gamma}_{\text{Max}}^{2}-2{\gamma}_{\text{Max}}^{3},$$

where *ƛ* is the smoothing factor applied at the
cracking (subscript `Crack`

) and maximum (subscript
`Max`

) portions of the surface area expression. The smoothing
factor is calculated from the normalized pressure differences *γ*:

$${\gamma}_{\text{Crack}}=\frac{{p}_{\text{Control}}-{p}_{\text{Crack}}}{\Delta {p}_{\text{Smooth}}}\text{\hspace{1em}}\text{\hspace{0.22em}}\text{\hspace{0.22em}}\text{\hspace{0.22em}}\text{and}\text{\hspace{1em}}\text{\hspace{0.22em}}\text{\hspace{0.22em}}\text{\hspace{0.22em}}{p}_{\text{Max}}=\frac{{p}_{\text{Crack}}-\left({p}_{\text{Max}}-\Delta {p}_{\text{Smooth}}\right)}{\Delta {p}_{\text{Smooth}}},$$

where *Δp*_{Smooth} is the
pressure smoothing region:

$$\Delta {p}_{\text{Smooth}}={f}_{\text{Smooth}}\frac{{p}_{\text{Max}}-{p}_{\text{Crack}}}{2}.$$

The parameter *f*_{Smooth}
is a value between `0`

and `1`

obtained from the
**Smoothing factor** block parameter. The final, smoothed,
valve opening area is given by the conditional expression:

$${S}_{\text{Smooth}}=\{\begin{array}{ll}{S}_{\text{Crack}},\hfill & \text{if}{p}_{\text{Control}}\le {p}_{\text{Crack}}\hfill \\ {S}_{\text{Crack}}\left(1-{\lambda}_{\text{Crack}}\right)+S{\lambda}_{\text{Crack}},\hfill & \text{if}{p}_{\text{Control}}\le {p}_{\text{Crack}}+\Delta {p}_{\text{Smooth}}\hfill \\ {S}_{\text{Lin}},\hfill & \text{if}{p}_{\text{Control}}\le {p}_{\text{Max}}-\Delta {p}_{\text{Smooth}}\hfill \\ {S}_{\text{Lin}}\left(1-{\lambda}_{\text{Max}}\right)+{S}_{\text{Max}}{\lambda}_{\text{Max}},\hfill & \text{if}{p}_{\text{Control}}\le {p}_{\text{Max}}\hfill \\ {S}_{\text{Max}},\hfill & \text{if}{p}_{\text{Control}}\ge {p}_{\text{Max}}\hfill \end{array}.$$

The volume of fluid inside the valve, and therefore the mass of the same, is assumed to be very small and it is, for modeling purposes, ignored. As a result, no amount of fluid can accumulate there. By the principle of conservation of mass, the mass flow rate into the valve through one port must therefore equal that out of the valve through the other port:

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

where $$\dot{m}$$ is defined as the mass flow rate *into* the
valve through one of the ports (**A** or **B**).

The causes of those pressure losses incurred in the passages of the valve are ignored in the block. Whatever their natures—sudden area changes, flow passage contortions—only their cumulative effect is considered during simulation. This effect is captured in the block by the discharge coefficient, a measure of the mass flow rate through the valve relative to the theoretical value that it would have in an ideal valve. Expressing the momentum balance in the valve in terms of the pressure drop induced in the flow:

$${p}_{\text{A}}-{p}_{\text{B}}=\frac{{\dot{m}}_{\text{Avg}}\sqrt{{\dot{m}}_{\text{Avg}}^{2}+{\dot{m}}_{\text{Crit}}^{2}}}{2{\rho}_{\text{Avg}}{C}_{\text{D}}{S}_{\text{Smooth}}^{2}}\left[1-{\left(\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}\right)}^{2}\right]{\xi}_{\text{p}},$$

where *C*_{D} is the
discharge coefficient, and *ξ*_{p} is the
pressure drop ratio—a measure of the extent to which the pressure recovery at the
outlet contributes to the total pressure drop of the valve. The subscript
`Avg`

denotes an average of the values at the thermal liquid
ports. The critical mass flow rate $${\dot{m}}_{\text{Crit}}$$ is calculated from the critical Reynolds number—that at which the
flow in the orifice is assumed to transition from laminar to turbulent:

$${\dot{m}}_{\text{Crit}}={\text{Re}}_{\text{Crit}}{\mu}_{\text{Avg}}\sqrt{\frac{\pi}{4}{S}_{\text{Lin}}},$$

where *μ* denotes dynamic viscosity. The
pressure drop ratio is calculated as:

$${\xi}_{\text{p}}=\frac{\sqrt{1-{\left(\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}\right)}^{2}\left(1-{C}_{\text{D}}^{2}\right)}-{C}_{\text{D}}\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}}{\sqrt{1-{\left(\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}\right)}^{2}\left(1-{C}_{\text{D}}^{2}\right)}+{C}_{\text{D}}\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}}.$$

The valve is modeled as an adiabatic component. No heat exchange can occur between
the fluid and the wall of the valve. No work is done on or by the fluid as it
traverses the valve. With these assumptions, energy can enter and exit the valve by
advection only, through ports **A** and **B**. By
the principle of conservation of energy then, the sum of the energy flows through
the ports must always equal zero:

$${\varphi}_{\text{A}}+{\varphi}_{\text{B}}=0,$$

where *ϕ* is defined as the energy flow rate
*into* the valve through one of the ports (**A** or **B**).

2-Way Directional Valve (TL) | 3-Way Directional Valve (TL) | 4-Way Directional Valve (TL) | Variable Area Orifice (TL)