Variable Local Restriction (TL)

Narrow opening with variable cross-sectional area

Library

Thermal Liquid/Elements

Description

The Variable Local Restriction (TL) block represents a narrow opening with variable cross-sectional area. The restriction causes a pressure drop and temperature gain in the liquid flowing through it. Common restrictions include valves and orifices.

To compute the pressure drop across the restriction, the block uses a discharge coefficient. This coefficient relates the pressure drop to the kinetic energy of the upstream liquid. The restriction is adiabatic. It does not exchange heat with the environment.

The block provides physical signal port AR so that you can specify the restriction cross-sectional area. This signal saturates at the value set in the Restriction minimum area parameter of the dialog box.

The liquid volume in the local restriction is assumed small. As a result, the dynamic compressibility and thermal inertia of the liquid are negligible. The block ignores both of these effects.

The following equations govern the behavior of the local restriction:

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

${p}_{\text{A}}-{p}_{\text{B}}=\left\{\begin{array}{cc}\frac{{\text{Re}}_{\text{c}}{\mu }_{\text{u}}}{2{C}_{\text{d}}{}^{2}{\rho }_{\text{u}}A}{\stackrel{˙}{m}}_{\text{A}},& \text{if}\text{\hspace{0.17em}}\text{Re}\le {\text{Re}}_{\text{c}}\\ \frac{{\stackrel{˙}{m}}_{A}|{\stackrel{˙}{m}}_{A}|}{2{C}_{\text{d}}{}^{2}{\rho }_{\text{u}}{A}^{2}},& \text{else}\end{array},$

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

where:

• `A`p is the cross-sectional area of the pipes adjacent to the restriction.

• `A` is the cross-sectional area of the restriction.

• `C`d is the flow discharge coefficient.

• `p` is the liquid pressure inside the local restriction.

• ${\stackrel{˙}{m}}_{\text{A}}$ and ${\stackrel{˙}{m}}_{\text{B}}$ are the liquid mass flow rates into the local restriction at inlets A and B.

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

• Re is the Reynolds number.

• Rec is the critical Reynolds number.

• ρu is the upstream liquid density.

• μu is the upstream fluid dynamic viscosity.

• ϕA and ϕB are the thermal fluxes into the local restriction at inlets A and B.

The liquid velocities at inlets A and B follow from the mass flow rates at those inlets:

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

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

where ρA,u and ρB,u are the liquid mass densities at inlets A and B.

The Reynolds number in the restriction satisfies the expression:

$\text{Re}=\frac{|{\stackrel{˙}{m}}_{\text{A}}|D}{A{\mu }_{\text{u}}},$

The block smooths the transition between laminar and turbulent flow regimes (Re ≤ Rec and Re ≥ Rec, respectively). Smoothing occurs in a way that avoids zero-crossing events in both the flow regime transition and at zero flow.

Assumptions and Limitations

• Restriction is adiabatic. It exchanges no heat with the environment.

• Fluid dynamic compressibility and thermal inertia 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.