# Variable Local Restriction (TL)

Time-varying reduction in flow area

## Library

Thermal Liquid/Elements

## Description

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

### Mass Balance

The mass balance in the restriction is

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

where:

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

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

### Momentum Balance

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

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

where:

• pA is the pressure at port A.

• pB is the pressure at port B.

• Cd is the discharge coefficient of the restriction aperture.

• SR is the cross-sectional area of the restriction aperture.

• ρu is the liquid density upstream of the restriction aperture.

• ${\stackrel{˙}{m}}_{\text{Ac}}$ is the critical mass flow rate at port A.

The critical mass flow rate at port A is calculated as

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

where:

• Rec is the critical Reynolds number,

${\text{Re}}_{c}=\frac{|{\stackrel{˙}{m}}_{\text{Ac}}|D}{{S}_{\text{R}}{\mu }_{\text{u}}},$

D is the hydraulic diameter of the restriction aperture.

• μu is the liquid dynamic viscosity upstream of the restriction aperture.

### Energy Balance

The energy balance in the restriction is

$0={\varphi }_{\text{A}}+{\varphi }_{\text{B}}+\frac{{\stackrel{˙}{m}}_{\text{A}}}{{\rho }_{\text{u}}}\left({p}_{\text{A}}-{p}_{\text{B}}\right)\text{\hspace{0.17em}}+{p}_{R}S\left({v}_{\text{A}}+{v}_{\text{B}}\right),$

where:

• ϕA is the heat flux into the restriction through port A.

• ϕB is the heat flux into the restriction through port B.

• pR is the average of the pressures at ports A and B.

• `S` is the cross-sectional area of the restriction inlets.

• `v`A and `v`B are the liquid velocities into the local restriction through inlets A and B.

The flow velocity at port A is

${v}_{\text{A}}=\frac{{\stackrel{˙}{m}}_{\text{A}}}{S{\rho }_{\text{A,u}}},$

while that at port B is

${v}_{\text{B}}=\frac{{\stackrel{˙}{m}}_{\text{B}}}{S{\rho }_{\text{B,u}}},$

where:

• vA is the flow velocity at port A.

• vB is the flow velocity at port B.

• ρA,u is the liquid density at port A.

• ρA,u is the liquid density at port B.

## Assumptions and Limitations

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

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

## Dialog Box and Parameters

### Parameters

Minimum restriction area

Enter the smallest cross-sectional area for the local restriction. The restriction area physical signal saturates at this value. The default value is `1e-10` m^2 .

Pipe cross-sectional area

Enter the cross-sectional area of the adjoining pipes. The default value is `1e-2` m^2 .

Characteristic longitudinal length

Enter the restriction length along the flow direction. The default value is `1e-1` m.

Flow discharge coefficient

Enter the discharge coefficient associated with the minor loss of the restriction. The default value is `0.7`.

Critical Reynolds number

Enter the Reynolds number at which flow transitions from laminar to turbulent. The default value is `12`.

### Variables

Use the Variables tab to set the priority and initial target values for the block variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.

## Ports

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