Local Restriction (2P)
Restriction in flow area in twophase fluid network
 Library:
Simscape / Foundation Library / TwoPhase Fluid / Elements
Description
The Local Restriction (2P) block models the pressure drop due to a localized reduction in flow area, such as a valve or an orifice, in a twophase fluid 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 





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, which causes the pressure to drop. In the expansion
zone, if the Pressure recovery parameter is
off
, the momentum of the accelerated fluid is lost. If the
Pressure recovery parameter is on
, 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 equation is
$${\dot{m}}_{A}+{\dot{m}}_{B}=0,$$
where:
$${\dot{m}}_{A}$$ and $${\dot{m}}_{B}$$ are the mass flow rates into the restriction through port A and port B.
Energy Balance
The energy balance equation is
$${\varphi}_{A}+{\varphi}_{B}=0,$$
where 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 therefore, the change in specific total enthalpy is zero. At port A,
$${u}_{A}+{p}_{A}{\nu}_{A}+\frac{{w}_{A}^{2}}{2}={u}_{R}+{p}_{R}{\nu}_{R}+\frac{{w}_{R}^{2}}{2},$$
while at port B,
$${u}_{B}+{p}_{B}{\nu}_{B}+\frac{{w}_{B}^{2}}{2}={u}_{R}+{p}_{R}{\nu}_{R}+\frac{{w}_{R}^{2}}{2},$$
where:
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 block computes the ideal flow velocity as
$${w}_{A}=\frac{{\dot{m}}_{ideal}{\nu}_{A}}{S}$$
at port A, as
$${w}_{B}=\frac{{\dot{m}}_{ideal}{\nu}_{B}}{S}$$
at port B, and as
$${w}_{R}=\frac{{\dot{m}}_{ideal}{\nu}_{R}}{{S}_{R}},$$
inside the restriction, where:
$${\dot{m}}_{ideal}$$ 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 block computes the ideal mass flow rate through the restriction as:
$${\dot{m}}_{ideal}=\frac{{\dot{m}}_{A}}{{C}_{D}},$$
where C_{D} is the flow discharge coefficient for the local restriction.
Local Restriction Variables
Momentum Balance
The change in momentum between the ports reflects in the pressure loss across the restriction. That loss depends on the mass flow rate through the restriction, though the exact dependence varies with flow regime. When the flow is turbulent,
$$\dot{m}={S}_{\text{R}}\left({p}_{\text{A}}{p}_{\text{B}}\right)\sqrt{\frac{2}{\left{p}_{\text{A}}{p}_{\text{B}}\right{\nu}_{\text{R}}{K}_{\text{T}}}},$$
where K_{T} is defined as:
$${K}_{\text{T}}=\left(1+\frac{{S}_{\text{R}}}{S}\right)\left(1\frac{{\nu}_{\text{in}}}{{\nu}_{\text{R}}}\frac{{S}_{\text{R}}}{S}\right)2\frac{{S}_{\text{R}}}{S}\left(1\frac{{\nu}_{\text{out}}}{{\nu}_{\text{R}}}\frac{{S}_{\text{R}}}{S}\right),$$
in which the subscript in
denotes the inlet port and
the subscript out
the outlet port. Which port serves as the inlet and
which serves as the outlet depends on the pressure differential across the restriction. If
pressure is greater at port A than at port B, then
port A is the inlet; if pressure is greater at port
B, then port B is the inlet.
When the flow is laminar,
$$\dot{m}={S}_{\text{R}}\left({p}_{\text{A}}{p}_{\text{B}}\right)\sqrt{\frac{2}{\Delta {p}_{\text{Th}}{\nu}_{\text{R}}{\left(1\frac{{S}_{\text{R}}}{S}\right)}^{2}},}$$
where Δp_{Th} denotes the threshold pressure drop at which the flow begins to smoothly transition between laminar and turbulent,
$$\Delta {p}_{\text{Th}}=\left(\frac{{p}_{\text{A}}+{p}_{\text{B}}}{2}\right)\left(1{B}_{\text{L}}\right),$$
in which B_{Lam} is the Laminar flow pressure ratio parameter. The flow is laminar if the pressure drop from port A to port B is below the threshold value; otherwise, the flow is turbulent.
The pressure at the restriction area, p_{R} likewise depends on the flow regime. When the flow is turbulent:
$${p}_{\text{R,L}}={p}_{\text{in}}\frac{{\nu}_{\text{R}}}{2}{\left(\frac{\dot{m}}{{S}_{\text{R}}}\right)}^{2}\left(1+\frac{{S}_{\text{R}}}{S}\right)\left(1\frac{{\nu}_{\text{in}}}{{\nu}_{\text{R}}}\frac{{S}_{\text{R}}}{S}\right).$$
When the flow is laminar:
$${p}_{\text{R,L}}=\frac{{p}_{\text{A}}+{p}_{\text{B}}}{2}.$$
Assumptions and Limitations
The restriction is adiabatic. It does not exchange heat with its surroundings.