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Local flow restriction with a variable cross-sectional area

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

The Variable Area Orifice (TL) block models the flow through a local restriction with variable opening area. The orifice contains a control member—such as a ball, spool, or diaphragm—which determines by its displacement the instantaneous opening area. Elements such as this are characteristic of valves and are, in the Thermal Liquid library, the foundation upon which all directional valve blocks are based. See, for example, the 2-Way Directional Valve (TL) block. Use this block to create a custom component with variable orifices if such is not provided in the Thermal Liquid library.

The orifice is assumed to consist of a contraction followed by a sudden expansion in
flow area. The contraction causes the flow rate to rise and the pressure to drop. The
expansion allows the pressure to recover, though only in part: past the vena contracta,
where the flow is at its narrowest, the flow generally separates from the wall, causing
it to lose some energy. The extent of the pressure recovery depends on the discharge
coefficient of the orifice and on the ratio of the orifice and port areas. Set the
**Pressure recovery** to `Off`

to ignore
this effect if necessary.

The effect that the motion of the control member has on the opening area of the
orifice depends on the setting of the **Opening orientation** block
parameter. In the default setting of `Positive`

, the orifice
(if within its opening range) opens when the control member moves in the positive
direction. In the alternate setting of `Negative`

, the orifice
opens with motion in the negative direction.

The orifice is continuously variable. It shifts smoothly between positions, of
which it has two. One—the *normal* position—is that to which the
orifice reverts when its control signal falls to zero. Unless a control member
offset has been specified, the
**A**–**B** orifice is always
fully closed in this position. Another—the *working* position—is
that to which the orifice moves when its control signal rises to a maximum. The
orifice is generally fully open in this position. Note that whether the orifice is
in fact open and how open it is both depend on the value of the control member
offset.

Which position the orifice is in depends on the control member coordinate—a length
that, in the valve blocks based on this orifice model, is often referred to as the
*orifice opening*. This variable is calculated during
simulation from the control member offset, specified via the block parameter of the
same name, and from the control member displacement, a variable obtained from the
physical signal specified at port **S**:

$$h={h}_{\text{0}}+\delta x,$$

where:

*h*is the**A**–**B**orifice opening.*h*_{0}is the**A**–**B**opening offset.*δ*is the orifice orientation,`+1`

if`Positive`

,`-1`

if`Negative`

.*x*is the control member displacement.

A control member displacement of zero corresponds to a valve that is in its normal
position. The orifice begins to open when the orifice opening (*h*)
rises above zero and it continues to open until the orifice opening is at a maximum
value. This maximum is obtained from the **Maximum control
displacement** block parameter, in the linear orifice parameterization,
or from the specified data vectors, in the tabulated orifice
parameterizations.

The orifice is by default configured so that it is fully closed when the control
member displacement is zero. Such an orifice, when it represents a valve, is often
described as being zero-lapped. It is possible, by applying an offset to the control
member, to model an orifice that is *underlapped*—that is,
partially open when in the normal position. The orifice can also be
*overlapped*—fully closed over a range of control member
displacements extending past the normal closed position.

The figure graphs the orifice opening—*h(x)*—in the cases of
zero-lapped (**I**), underlapped (**II**), and overlapped (**III**) orifices.
The opening offset—*h*s_{0}—is zero in the
first case, greater than zero in the second, and smaller than zero in the third. The
control member must move right of its normal position (in the positive direction
along the *x*-axis) for the overlapped orifice to crack open; it
must move left of its normal position for the underlapped orifice to shut
tight.

The orifice opening serves during simulation to calculate the mass flow rate
through the orifice. The calculation can be a direct mapping from opening to flow
rate or an indirect conversion, first from opening to orifice area and then from
orifice area to mass flow rate. The calculation, and the data required for it,
depend on the setting of the **Valve parameterization** block
parameter:

`Linear area-opening relationship`

— Calculate the valve opening area from the control member position and from it obtain the mass flow rate through the valve. The opening area is assumed to vary linearly with the control member position. The slope of the linear expression is determined from the**Maximum valve opening**and**Maximum opening area**block parameters:$${S}_{\text{Lin}}=\frac{{S}_{\text{Max}}}{{h}_{\text{Max}}}h,$$

where

*S*_{Lin}is the linear form of the opening area,*S*_{Max}is the value of the**Maximum orifice area**block parameter,*h*_{Max}is the value of the**Maximum control displacement**block parameter. This expression is reformulated as a piecewise conditional expression so as to saturate the opening area at a small leakage value and ensure that transitions to the normal and working positions are smooth.`Tabulated data - Area vs. opening`

— Calculate the valve opening area from the control member position and from it obtain the mass flow rate through the valve. The opening area can vary nonlinearly with the control member position. The relationship between the two is given by the tabulated data in the**Valve opening vector**and**Opening area vector**block parameters:$${S}_{\text{Tab}}=S(h),$$

where

*S*_{Tab}is the tabulated form of the opening area, a function of the orifice opening,*h*.`Tabulated data - Mass flow rate vs. opening and pressure drop`

— Calculate the mass flow rate directly from the control member position and the pressure drop across the valve. The relationship between the three variables can be nonlinear and it is given by the tabulated data in the**Valve opening vector**,**Pressure drop vector**, and**Mass flow rate table**block parameters:$${\dot{m}}_{\text{Tab}}=\frac{{\rho}_{\text{Ref}}}{{\rho}_{\text{Avg}}}\dot{m}(h,\Delta p),$$

where $$\dot{m}$$ is the tabulated form of the mass flow rate, a function of the orifice opening,

*h*, and of the pressure drop across the orifice,*Δp*. The mass flow rate is adjusted for temperature and pressure by the ratio*ρ*_{Ref}/*ρ*_{Avg}, where*ρ*is the fluid density at some reference temperature and pressure (subscript`Ref`

) or at the averages of those variables within the orifice.

To ensure adequate simulation performance, the orifice opening area is smoothed over two small regions of the orifice opening, one near the fully closed state, the other near the fully open state. The smoothing is accomplished by means of polynomial expressions (to be incorporated into the final form of the opening area expression):

$${\lambda}_{\text{Min}}=3\Delta {h}_{\text{Min}}^{*}-2\Delta {h}_{\text{Min}}^{*3}\text{and}{\lambda}_{\text{Max}}=3\Delta {h}_{\text{Max}}^{*}-2\Delta {h}_{\text{Max}}^{*3},$$

where *ƛ* is the smoothing factor applied at the
minimum (subscript `Min`

) and maximum (subscript
`Max`

) portions of the opening area expression. The smoothing
factors are calculated as:

$$\Delta {h}_{\text{Min}}=\frac{h-{h}_{\text{Min}}}{\Delta {h}_{\text{Smooth}}}\text{and}\Delta {h}_{\text{Max}}=\frac{h-({h}_{\text{Max}}-\Delta {h}_{\text{Smooth}})}{\Delta {h}_{\text{Smooth}}},$$

where *h*_{Min} is the
minimum orifice opening and *Δh*_{Smooth} is
the range of orifice openings over which to smooth the linear form of the opening
area. The value of *S*_{Min} is calculated as:

$${h}_{\text{Min}}={h}_{\text{Max}}\frac{{S}_{\text{Leak}}}{{S}_{\text{Max}}},$$

where *S*_{Leak} is the
value of the **Leakage area** block parameter. The value of
*S*_{Smooth} is calculated as:

$$\Delta {h}_{\text{Smooth}}={f}_{\text{Smooth}}\frac{{h}_{\text{Max}}-{h}_{\text{Min}}}{2},$$

where *f*_{Smooth} is the
value of the **Smoothing factor** block parameter—a fraction
between `0`

and `1`

, with `0`

indicating zero smoothing and `1`

maximum smoothing. The final,
smoothed, orifice opening area is given by the piecewise expression:

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

**Orifice Area Smoothing**

The primary purpose of the leakage flow rate of a closed orifice is to ensure that
at no time a portion of the thermal liquid network becomes 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 flow is
generally present in real orifices but in a model its exact value is less important
than its being a small number greater than zero. The leakage flow rate is determined
from the **Leakage area** block parameter.

The volume of fluid inside the orifice, 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 orifice through one port must therefore equal that out of the orifice through the other port:

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

where $$\dot{m}$$ is defined as the mass flow rate *into* the
orifice through the port indicated by the subscript (**A** or **B**).

The causes of the pressure losses incurred in the orifice 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 orifice relative to the theoretical value that it would have in an ideal orifice. Expressing the momentum balance in the orifice 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 effect impressed by the pressure recovery that
in real orifices occurs between the vena contracta (the point at which the flow is
at its narrowest) and the outlet, assumed to be a small distance away. 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 value
of the pressure ratio depends on the setting of the **Pressure
recovery** block parameter. In the default setting of
`Off`

:

$${\xi}_{\text{p}}=1.$$

If `On`

is selected instead:

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

The orifice 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 always equal zero:

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

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

[1] *Measurement of fluid flow by means of pressure differential devices
inserted in circular cross-section conduits running full — Part 2: Orifice
plates (ISO 5167–2:2003)*. 2003.

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