Local Restriction (TL)

Restriction in flow area in thermal liquid network

Libraries:
Simscape / Foundation Library / Thermal Liquid / Elements

Description

The Local Restriction (TL) block models the pressure drop due to a localized reduction in flow area, such as a valve or an orifice, in a thermal liquid network.

Ports A and B represent the restriction inlet and outlet. The input physical signal at port AR specifies the restriction area. Alternatively, you can specify a fixed restriction area as a block parameter.

The block icon changes depending on the value of the Restriction type parameter.

Restriction Type Block Icon

Restriction Type	Block Icon
`Variable`
`Fixed`

Variable

Fixed

The restriction is adiabatic. It does not exchange heat with the environment.

The restriction consists of a contraction followed by a sudden expansion in flow area. The fluid accelerates during the contraction, causing the pressure to drop. In the expansion zone, if the Pressure recovery check box is cleared, the momentum of the accelerated fluid is lost. If the Pressure recovery check box is selected, the sudden expansion recovers some of the momentum and allows the pressure to rise slightly after the restriction.

Local Restriction Schematic

Mass Balance

The mass balance in the restriction is

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

where:

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

$\begin{array}{l} Δ p = \frac{ρ}{2} (1 - \frac{S_{R}^{2}}{S^{2}}) v_{R} \sqrt{v_{R}^{2} + v_{R c}^{2}} \\ v_{R} = \frac{{\dot{m}}_{A}}{C_{d} ρ S_{R}} \\ v_{R c} = \frac{{Re}_{c} μ}{C_{d} ρ} \sqrt{\frac{π}{4 S_{R}}} \end{array}$

where:

Δp is the pressure differential.
ρ is the liquid density.
μ is the liquid dynamic viscosity.
S is the cross-sectional area at ports A and B.
S_R is the cross-sectional area at the restriction.
v_R is fluid velocity at the restriction.
v_Rc is the critical fluid velocity.
Re_c is the critical Reynolds number.
C_d is the discharge coefficient.

If pressure recovery is off, then

$p_{A} - p_{B} = Δ p,$

where:

p_A is the pressure at port A.
p_B is the pressure at port B.

If pressure recovery is on, then

$p_{A} - p_{B} = Δ p \frac{\sqrt{1 - {(\frac{S_{R}}{S})}^{2} (1 - C_{d}^{2})} - C_{d} \frac{S_{R}}{S}}{\sqrt{1 - {(\frac{S_{R}}{S})}^{2} (1 - C_{d}^{2})} + C_{d} \frac{S_{R}}{S}} .$

Energy Balance

The energy balance in the restriction is

$ϕ_{A} + ϕ_{B} = 0,$

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.

Assumptions and Limitations

The restriction is adiabatic. It does not exchange heat with its surroundings.
The dynamic compressibility and thermal capacity of the liquid in the restriction are negligible.

Examples

Engine Cooling System

Model a basic engine cooling system using custom thermal liquid blocks. A fixed-displacement pump drives water through the cooling circuit. Heat from the engine is absorbed by the water coolant and dissipated through the radiator. The system temperature is regulated by the thermostat, which diverts flow to the radiator only when the temperature is above a threshold.

Open Model

Water Hammer Effect

Model shows how the Thermal Liquid foundation library can be used to model water hammer in a long pipe. After slowly establishing a steady flow within the pipe by opening a valve accordingly, the same valve is shut within a few milliseconds. This triggers a water hammer effect. The valve is modeled using a Variable Local Restriction (TL) block and the pipe is divided into four segments using the Pipe (TL) block. Breaking the pipe into more segments increases fidelity at the expense of simulation performance.

Open Model

Ports

Input

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AR — Restriction area control signal, m^2
physical signal

Input physical signal that controls the flow restriction area. The signal saturates when its value is outside the minimum and maximum restriction area limits, specified by the block parameters.

Dependencies

This port is visible only if you set the Restriction type parameter to Variable.

Conserving

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A — Inlet or outlet
thermal liquid

Thermal liquid conserving port associated with the inlet or outlet of the local restriction. This block has no intrinsic directionality.

B — Inlet or outlet
thermal liquid

Thermal liquid conserving port associated with the inlet or outlet of the local restriction. This block has no intrinsic directionality.

Parameters

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Restriction type — Specify whether restriction area can change during simulation
`Variable` (default) | `Fixed`

Select whether the restriction area can change during simulation:

Variable — The input physical signal at port AR specifies the restriction area, which can vary during simulation. The Minimum restriction area and Maximum restriction area parameters specify the lower and upper bounds for the restriction area.
Fixed — The restriction area, specified by the Restriction area block parameter value, remains constant during simulation. Port AR is hidden.

Minimum restriction area — Lower bound for the restriction cross-sectional area
`1e-10 m^2` (default)

The lower bound for the restriction cross-sectional area. You can use this parameter to represent the leakage area. The input signal AR saturates at this value to prevent the restriction area from decreasing any further.

Dependencies

Enabled when the Restriction type parameter is set to Variable.

Maximum restriction area — Upper bound for the restriction cross-sectional area
`0.005 m^2` (default)

The upper bound for the restriction cross-sectional area. The input signal AR saturates at this value to prevent the restriction area from increasing any further.

Dependencies

Enabled when the Restriction type parameter is set to Variable.

Restriction area — Area normal to flow path at the restriction
`1e-3 m^2` (default)

Area normal to flow path at the restriction.

Dependencies

Enabled when the Restriction type parameter is set to Fixed.

Cross-sectional area at ports A and B — Area normal to flow path at the ports
`0.01 m^2` (default)

Area normal to flow path at ports A and B. This area is assumed to be the same for the two ports.

Discharge coefficient — Ratio of actual mass flow rate to theoretical mass flow rate through restriction
`0.64` (default)

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.

Pressure recovery — Specify whether to account for pressure recovery
`On` (default) | `Off`

Select the check box to account for pressure recovery at the local restriction outlet.

Critical Reynolds number — Reynolds number for transition between laminar and turbulent regimes
`12` (default)

The Reynolds number for the transition between laminar and turbulent regimes. The default value corresponds to a sharp-edged orifice.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2013b

Local Restriction (TL)