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Check valve with control port to enable flow in reverse direction

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

The Pilot-Operated Check Valve (TL) block models a check valve with an override mechanism to allow for backflow when activated. (A check valve in turn is an orifice with a unidirectional opening mechanism installed to prevent just that backflow.)

The override mechanism adds a third port—the *pilot*—to the valve.
During normal operation, the pilot port is inactive and the valve behaves as any other
check valve. Its orifice is then open only when the pressure gradient across it drops
from inlet to outlet. Backflow, which requires the reverse pressure gradient, cannot
occur. This mode protects components upstream of the valve against pressure surges,
temperature spikes, and (in real systems) chemical contamination arising from points
downstream.

When backflow is desired, the pilot port is pressurized and the control element of the valve—often a ball or piston—is forced off its seat. The valve is then open to flow in both directions, with a reverse pressure drop (aimed from outlet to inlet) sufficing to drive the flow upstream. (The seat, which lies in the path of the flow, determines if the valve is open. When it is covered—by a ball, piston, or other control element—the flow is cut off and the valve is closed.)

The valve opens by degrees, beginning at its cracking pressure, and continuing to the
end of its pressure regulation range. The cracking pressure gives the initial
resistance, due to friction or spring forces, that the valve must overcome to open by a
sliver (or to *crack* open). Below this threshold, the valve is
closed and only leakage flow can pass. Past the end of the pressure regulation range,
the valve is fully open and the flow at a maximum (determined by the instantaneous
pressure conditions).

The cracking pressure assumes an important role in check valves installed upside down. There, the weight of the opening element—such as a ball or piston—and the elevation head of the fluid can act to open the valve. (The elevation head can arise in a model from a pipe upstream of the inlet when it is vertical or given a slant.) A sufficient cracking pressure keeps the valve from opening inadvertently even if placed at a disadvantageous angle.

The opening of the valve depends both on the pilot pressure and on the pressure drop from inlet to outlet:

$${p}_{\text{Ctl}}={k}_{\text{X}}{p}_{\text{X}}+{p}_{\text{A}}-{p}_{\text{B}},$$

where *p* is gauge pressure and
*k* is the pilot ratio—the proportion of the pilot opening area
(*S*_{X}) to the valve opening area
(*S*_{R}). The subscript
`X`

denotes a pilot value and the subscripts
`A`

and `B`

the inlet and outlet values,
respectively. The port pressures are variables determined (against absolute zero)
during simulation.

The pilot pressure can be a differential value relative to the inlet (port
**A**) or a gauge value (relative to the environment). You can
select an appropriate setting—**Pressure differential (pX - pA)**
or **Pressure at port X**—using the **Pressure control
specification** dropdown list. If **Pressure at port
X** is selected:

$${p}_{\text{X}}={p}_{\text{X,Abs}}-{p}_{\text{Atm}},$$

where the subscript `Atm`

denotes the
atmospheric value (obtained from the Thermal Liquid Settings (TL)
or Thermal Liquid
Properties (TL) block of the model). The subscript
`X,Abs`

denotes the absolute value at the pilot port. If
**Pressure differential (pX - pA)** is selected:

$${p}_{\text{X}}={p}_{\text{X,Abs}}-{p}_{\text{A,Abs}}$$

where the subscript `A,Abs`

similarly denotes
the absolute value at the inlet of the valve (port **A**). The
pilot pressure differential is constrained to be greater than or equal to zero—if
its calculated value should be negative, zero is assumed in the control pressure
calculation.

The degree to which the control pressure exceeds the cracking pressure determines how much the valve will open. The pressure overshoot is expressed here as a fraction of the (width of the) pressure regulation range:

$$\widehat{p}=\frac{{p}_{\text{Ctl}}-{P}_{\text{Crk}}}{{P}_{\text{Max}}-{P}_{\text{Crk}}}.$$

The control pressure (*p*_{Ctl}), cracking
pressure (*p*_{Set}), and maximum opening
pressure (*P*_{Max}) correspond to the control
pressure specification chosen (`Pressure differential`

or
`Pressure at port A`

).

The fraction—technically, the overshoot *normalized*—is valued
at `0`

in the fully closed valve and `1`

in the
fully open valve. If the calculation should return a value outside of these bounds,
the nearest of the two is used instead. (In other words, the fraction
*saturates* at `0`

and
`1`

.)

The normalized control pressure, *p*, spans three pressure
regions. Below the cracking pressure of the valve, its value is a constant zero.
Above the maximum pressure of the same, it is `1`

. In between,
it varies, as a linear function of the (effective) control pressure,
*p*_{Ctl}.

The transitions between the regions are sharp and their slopes discontinuous.
These pose a challenge to variable-step solvers (the sort commonly used with
Simscape models). To precisely capture discontinuities, referred to in some
contexts as *zero crossing events*, the solver must reduce
its time step, pausing briefly at the time of the crossing in order to recompute
its Jacobian matrix (a representation of the dependencies between the state
variables of the model and their time derivatives).

This solver strategy is efficient and robust when discontinuities are present. It makes the solver less prone to convergence errors—but it can considerably extend the time needed to finish the simulation run, perhaps excessively so for practical use in real-time simulation. An alternative approach, used here, is to remove the discontinuities altogether.

**Normalized temperature overshoot with sharp transitions**

The block removes the discontinuities by smoothing them over a specified time scale. The smoothing, which adds a slight distortion to the normalized inlet pressure, ensures that the valve eases into its limiting positions rather than snap (abruptly) into them. The smoothing is optional: you can disable it by setting its time scale to zero. The shape and scale of the smoothing, when applied, derives in part from the cubic polynomials:

$${\lambda}_{\text{L}}=3{\overline{p}}_{\text{L}}^{2}-2{\overline{p}}_{\text{L}}^{3}$$

and

$${\lambda}_{\text{R}}=3{\overline{p}}_{\text{R}}^{2}-2{\overline{p}}_{\text{R}}^{3},$$

where

$${\overline{p}}_{\text{L}}=\frac{\widehat{p}}{\Delta {p}^{*}}$$

and

$${\overline{p}}_{\text{R}}=\frac{\widehat{p}-\left(1-\Delta {p}^{*}\right)}{\Delta {p}^{*}}.$$

In the equations:

*ƛ*_{L}is the smoothing expression for the transition from the maximally closed position.*ƛ*_{R}is the smoothing expression for the transition from the fully open position.*Δp*^{*}is the (unitless) characteristic width of the pressure smoothing region:$$\Delta {p}^{*}={f}^{*}\frac{1}{2},$$

where

*f*^{*}is a smoothing factor valued between`0`

and`1`

and obtained from the block parameter of the same name.When the smoothing factor is

`0`

, the normalized inlet pressure stays in its original form—no smoothing applied—and its transitions remain abrupt. When it is`1`

, the smoothing spans the whole of the pressure regulation range (with the normalized inlet pressure taking the shape of an*S*-curve).At intermediate values, the smoothing is limited to a fraction of that range. A value of

`0.5`

, for example, will smooth the transitions over a quarter of the pressure regulation range on each side (for a total smooth region of half the regulation range).

The smoothing adds two new regions to the normalized pressure overshoot—one for the smooth transition on the left, another for that on the right, giving a total of five regions. These are expressed in the piecewise function:

$${\widehat{p}}^{*}=\{\begin{array}{ll}0,\hfill & \widehat{p}\le 0\hfill \\ \widehat{p}{\lambda}_{\text{L}},\hfill & \widehat{p}<\Delta {P}^{*}\hfill \\ \widehat{p},\hfill & \widehat{p}\le 1-\Delta {P}^{*}\hfill \\ \widehat{p}\left(1-{\lambda}_{\text{R}}\right)+{\lambda}_{\text{R}},\hfill & \widehat{p}<1\hfill \\ 1\hfill & \widehat{p}\ge 1\hfill \end{array},$$

where the asterisk denotes a smoothed variable (the normalized control pressure overshoot). The figure shows the effect of smoothing on the sharpness of the transitions.

The valve is assumed to open linearly with the smoothed control pressure overshoot:

$$S=\left({S}_{\text{Max}}-{S}_{\text{Min}}\right)\widehat{p}+{S}_{\text{Min}},$$

where *S* is sonic conductance and the
subscripts `Max`

and `Min`

denote its values in
the fully open and fully closed valve. In terms of the smoothed control pressure
overshoot, the opening area becomes:

$${S}^{*}=\left({S}_{\text{Max}}-{S}_{\text{Min}}\right){\widehat{p}}^{*}+{S}_{\text{Min}}.$$

(Both expressions are used in the calculation of the pressure drop across the valve.)

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:

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

where *ρ* is density,
*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 valve is assumed to transition from laminar to turbulent:

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

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

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

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 port **A** or **B**.

The valve is modeled as an adiabatic component. No heat exchange can occur between
the fluid and the wall that surrounds it. No work is done on or by the fluid as it
traverses from inlet to outlet. With these assumptions, energy can flow by advection
only, through ports **A** and **B**. By the
principle of conservation of energy, the sum of the port energy flows must then
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) | Check Valve (TL) | Variable Area Orifice (TL)