Time-varying reduction in flow area

Thermal Liquid/Elements

The Local Restriction (TL) block models the pressure drop due to a time-varying reduction in flow area such as a valve. Ports A and B represent the restriction inlets. 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 pressure drop is assumed to persist in the expansion zone—an approximation suitable for narrow restrictions.

**Local Restriction Schematic**

The mass balance in the restriction is

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

where:

$${\dot{m}}_{\text{A}}$$ is the mass flow rate into the restriction through port A.

$${\dot{m}}_{\text{B}}$$ is the mass flow rate into the restriction through port B.

The pressure difference between ports A and B follows from the momentum balance in the restriction:

$${p}_{\text{A}}-{p}_{\text{B}}=\frac{{\dot{m}}_{A}{\left({\dot{m}}_{A}{}^{8}+{\dot{m}}_{Ac}{}^{8}\right)}^{1/8}}{2\text{\hspace{0.17em}}{C}_{\text{d}}^{2}{S}_{R}{\rho}_{\text{u}}},$$

where:

*p*_{A}is the pressure at port A.*p*_{B}is the pressure at port B.*C*_{d}is the discharge coefficient of the restriction aperture.*S*_{R}is the cross-sectional area of the restriction aperture.*ρ*_{u}is the liquid density upstream of the restriction aperture.$${\dot{m}}_{\text{Ac}}$$ is the critical mass flow rate at port A.

The critical mass flow rate at port A is

$${\dot{m}}_{\text{Ac}}={\mathrm{Re}}_{\text{c}}\sqrt{\pi {S}_{R}}\frac{{\mu}_{\text{u}}}{2},$$

where:

*Re*_{c}is the critical Reynolds number,$${\text{Re}}_{c}=\frac{\left|{\dot{m}}_{\text{Ac}}\right|D}{{S}_{\text{R}}{\mu}_{\text{u}}},$$

*D*is the hydraulic diameter of the restriction aperture.*μ*_{u}is the liquid dynamic viscosity upstream of the restriction aperture.

The discharge coefficient is the ratio of the actual mass flow rate through the local restriction to the ideal mass flow rate,

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

where:

$$\dot{m}$$ is the actual mass flow rate through the local restriction.

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

$${\dot{m}}_{ideal}={S}_{R}\sqrt{\frac{2{\rho}_{u}\text{\hspace{0.17em}}\left({p}_{A}-{p}_{B}\right)}{1-{\left({S}_{R}/S\right)}^{2}}}.$$

where

*S*is the inlet cross-sectional area.

The energy balance in the restriction is

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

where:

*ϕ*_{A}is the energy flow rate into the restriction through port A.*ϕ*_{B}is the energy flow rate into the restriction through port B.

The restriction is adiabatic. It does not exchange heat with its surroundings.

The dynamic compressibility and thermal capacity of the liquid are negligible.

**Minimum restriction area**Enter the lower bound for the restriction area range. This is the lowest allowed value for the restriction cross-sectional area. The input signal AR saturates at this value to prevent the restriction area from decreasing any further. The default value is

`1e-10`

m^2 .**Maximum restriction area**Enter the upper bound for the restriction area range. This is the highest allowed value for the restriction cross-sectional area. The input signal AR saturates at this value to prevent the restriction area from increasing any further. The default value is

`0.005`

m^2.**Cross-sectional area at ports A and B**Enter the flow cross-sectional area of the local restriction ports. This area is assumed the same for the two ports. The default value is

`0.01`

m^2 .**Characteristic longitudinal length**Enter the approximate longitudinal length of the local restriction. This length provides a measure of the longitudinal scale of the restriction. The default value is

`0.1`

m.**Discharge coefficient**Enter the discharge coefficient of the local restriction. The discharge coefficient is a semi-empirical parameter commonly used to characterize the flow capacity of an orifice. This parameter is defined as the ratio of the actual mass flow rate through the orifice to the ideal mass flow rate:

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

where

*C*_{d}is the discharge coefficient, $$\dot{m}$$ is the actual mass flow rate through the orifice, and $${\dot{m}}_{ideal}$$ is the ideal mass flow rate:$${\dot{m}}_{ideal}={S}_{r}\sqrt{\frac{2\rho \text{\hspace{0.17em}}\left({p}_{A}-{p}_{B}\right)}{1-{\left({S}_{r}/S\right)}^{2}}}.$$

The default value is

`0.7`

, corresponding to a sharp-edged orifice.**Pressure recovery**Specify whether to account for pressure recovery at the local restriction outlet. Options include

`On`

and`Off`

. The default setting is`On`

.**Critical Reynolds number**Enter the Reynolds number for the transition between laminar and turbulent flow regimes. The default value is

`12`

, corresponding to a sharp-edged orifice.

The block has two thermal liquid conserving ports, A and B, and one physical signal port, AR.

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