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Time-varying flow resistance

Two-Phase Fluid/Elements

The Variable Local Restriction (2P) block models the pressure drop due to a time-varying flow resistance such as a valve. Ports A and B represent the restriction inlet and outlet. Port AR sets the time-varying restriction area, specified as a physical signal.

The restriction consists of a contraction followed by a sudden expansion in flow area. The contraction causes the fluid to accelerate and its pressure to drop. The expansion recovers the lost pressure though only in part, as the flow separates from the wall, losing momentum in the process.

**Local Restriction Schematic**

The mass balance equation is

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

$${\dot{m}}_{A}$$ and $${\dot{m}}_{B}$$ are the mass flow rates into the restriction through port A and port B.

The energy balance equation is

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

*ϕ*_{A}and*ϕ*_{B}are the energy flow rates into the restriction through port A and port B.

The local restriction is assumed to be adiabatic and the change in specific total enthalpy is therefore zero. At port A,

$${u}_{A}+{p}_{A}{\nu}_{A}+\frac{{w}_{A}^{2}}{2}={u}_{R}+{p}_{R}{\nu}_{R}+\frac{{w}_{R}^{2}}{2},$$

$${u}_{B}+{p}_{B}{\nu}_{B}+\frac{{w}_{B}^{2}}{2}={u}_{R}+{p}_{R}{\nu}_{R}+\frac{{w}_{R}^{2}}{2},$$

*u*_{A},*u*_{B}, and*u*_{R}are the specific internal energies at port A, at port B, and the restriction aperture.*p*_{A},*p*_{B}, and*p*_{R}are the pressures at port A, port B, and the restriction aperture.*ν*_{A},*ν*_{B}, and*ν*_{R}are the specific volumes at port A, port B, and the restriction aperture.*w*_{A},*w*_{B}, and*w*_{R}are the ideal flow velocities at port A, port B, and the restriction aperture.

The ideal flow velocity is computed as

$${w}_{A}=\frac{{\dot{m}}_{ideal}{\nu}_{A}}{S}$$

$${w}_{B}=\frac{{\dot{m}}_{ideal}{\nu}_{B}}{S}$$

$${w}_{R}=\frac{{\dot{m}}_{ideal}{\nu}_{R}}{{S}_{R}},$$

$${\dot{m}}_{ideal}$$ is the ideal mass flow rate through the restriction.

*S*is the flow area at port A and port B.*S*_{R}is the flow area of the restriction aperture.

The ideal mass flow rate through the restriction is computed as:

$${\dot{m}}_{ideal}=\frac{{\dot{m}}_{A}}{{C}_{D}},$$

*C*_{D}is the flow discharge coefficient for the local restriction.

**Local Restriction Variables**

The pressure difference between the ports is derived from the momentum balances in the contraction zone (the region between the inlet and the restriction aperture) and expansion zone (the region between the restriction aperture and the outlet). In the turbulent flow regime, with the flow directed from port A to port B:

$$\Delta {p}_{AB}=\frac{{w}_{R}\left|{w}_{R}\right|}{{\nu}_{R}}\left[\left(\frac{1+r}{2}\right)\left(1-r\frac{{\nu}_{A}}{{\nu}_{R}}\right)-r\left(1-r\frac{{\nu}_{B}}{{\nu}_{R}}\right)\right],$$

$$\frac{{S}_{R}}{S}.$$

$$\Delta {p}_{BA}=\frac{{w}_{R}\left|{w}_{R}\right|}{{\nu}_{R}}\left[\left(\frac{1+r}{2}\right)\left(1-r\frac{{\nu}_{B}}{{\nu}_{R}}\right)-r\left(1-r\frac{{\nu}_{A}}{{\nu}_{R}}\right)\right],$$

The equations indicate that the pressure difference between the ports varies with the square of the flow rate through the restriction. This relationship is characteristic of turbulent flows only. In the laminar regime, where the relationship becomes linear, the pressure difference is approximated as:

$$\Delta {p}_{lam}={w}_{R}\sqrt{\frac{\Delta {p}_{transition}}{2{\nu}_{R}}\left(1-r\right),}$$

$$\Delta {p}_{transition}={p}_{avg}\left(1-{B}_{lam}\right),$$

*p*_{avg}as the average of the pressures at port A and port B:$${p}_{avg}=\frac{{p}_{A}+{p}_{B}}{2}$$

*B*_{lam}as the**Laminar flow pressure ratio**parameter.

The laminar pressure difference equation is the same for both flow directions—from port A to port B or from port B to port A.

The pressure at the restriction aperture is computed from the momentum balance in the flow contraction zone. In the turbulent flow regime, with the flow directed from port A to port B:

$${p}_{R,AB}={p}_{A}-\frac{{w}_{R}\left|{w}_{R}\right|}{{\nu}_{R}}\left(\frac{1+r}{2}\right)\left(1-r\frac{{\nu}_{A}}{{\nu}_{R}}\right)$$

$${p}_{R,BA}={p}_{B}+\frac{{w}_{R}\left|{w}_{R}\right|}{{\nu}_{R}}\left(\frac{1+r}{2}\right)\left(1-r\frac{{\nu}_{B}}{{\nu}_{R}}\right)$$

$${p}_{R,lam}={p}_{avg}-\frac{{w}_{R}^{2}}{{\nu}_{R}}\left(\frac{1-{r}^{2}}{2}\right)$$

A cubic polynomial function is used to smooth the transition between laminar and turbulent flow regimes. The smoothing function blends the pressure difference between the ports as well as the pressure at the restriction aperture between the two flow regimes:

When $$\Delta {p}_{transition}\le {p}_{A}-{p}_{B},$$ then $${p}_{A}-{p}_{B}=\Delta {p}_{AB}$$ and $${p}_{R}={p}_{R,AB}.$$

When $$0\le {p}_{A}-{p}_{B}\le \Delta {p}_{transition},$$ then $${p}_{A}-{p}_{B}=\Delta {p}_{BA}$$ and $${p}_{R}={p}_{R,BA}.$$

When $$0\le {p}_{A}-{p}_{B}\le \Delta {p}_{transition},$$ then $${p}_{A}-{p}_{B}$$ is smoothly blended between

*Δp*_{AB}and*Δp*_{lam}and*p*_{R}is smoothly blended between*p*_{R,AB}and*p*_{R,lam}.When $$-\Delta {p}_{transition}\le {p}_{A}-{p}_{B}\le 0,$$ then $${p}_{A}-{p}_{B}$$ is smoothly blended between

*Δp*_{BA}and*Δp*_{lam}and*p*_{R}is smoothly blended between*p*_{R,BA}and*p*_{R,lam}.

To set the priority and initial target values for the block variables prior to simulation, use
the **Variables** tab in the block dialog box (or the
**Variables** section in the block Property Inspector). For more
information, see Set Priority and Initial Target for Block Variables.

**Minimum restriction area**Area normal to the flow path at the restriction aperture when the restriction is in the fully closed state. The area obtained from physical signal AR saturates at this value. Input values smaller than the minimum restriction area are ignored and replaced by the value specified here. The default value of

`1e-10`

m^2.**Maximum restriction area**Area normal to the flow path at the restriction aperture when the restriction is in the fully open state. The area obtained from physical signal AR saturates at this value. Input values greater than the maximum restriction area are ignored and replaced by the value specified here. The default value is

`0.005`

m^2.**Cross-sectional area at ports A and B**Area normal to the flow path at the restriction ports. The ports are assumed to be identical in cross-section. The default value,

`0.01`

m^2, is the same as the restriction aperture area.**Flow discharge coefficient**Ratio of the actual to the theoretical mass flow rate through the restriction. The discharge coefficient is an empirical parameter used to account for non-ideal effects such as those due to restriction geometry. The default value is

`0.64`

.**Laminar flow pressure ratio**Ratio of the outlet to the inlet port pressure at which the flow regime is assumed to switch from laminar to turbulent. The prevailing flow regime determines the equations used in simulation. The pressure drop across the restriction is linear with respect to the mass flow rate if the flow is laminar and quadratic (with respect to the mass flow rate) if the flow is turbulent. The default value is

`0.999`

.

A pair of two-phase fluid conserving ports labeled A and B represent the restriction inlet and outlet. A physical signal input port labeled AR controls the cross-sectional area of the restriction aperture, located between the restriction inlet and outlet.

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