Sudden Area Change (TL)

Sudden expansion or contraction in flow area

Libraries:
Simscape / Fluids / Thermal Liquid / Pipes & Fittings

Description

The Sudden Area Change (TL) block models the minor pressure losses due to a sudden change in flow cross-sectional area. The area change is a contraction from port A to port B and an expansion from port B to port A. This component is adiabatic. It does not exchange heat with its surroundings.

Sudden Area Change Schematic

The pressure drop across a sudden expansion is primarily due to turbulence mixing in the expansion zone. Across a sudden contraction, it is primarily due to flow detachment at the contraction zone entrance. The figure shows the expansion and contraction zones of the sudden area change.

Mass Balance

The mass conservation equation in the sudden area change is

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

where:

${\dot{m}}_{A}$ and ${\dot{m}}_{B}$ are the mass flow rates into the sudden area change through ports A and B.

Momentum Balance

The momentum conservation equation in the sudden area change is

$p_{A} - p_{B} = \frac{{\dot{m}}^{2}}{2 ρ} (\frac{1}{S_{B}^{2}} - \frac{1}{S_{A}^{2}}) + ϕ_{L o s s},$

where:

p_A and p_B are the pressures at ports A and B.
$\dot{m}$ is the average mass flow rate.
ρ is the average fluid density.
S_A and S_B are the flow cross-sectional areas at ports A and B.
Φ_Loss is the mechanical energy loss due to the sudden area change.

The mechanical energy loss is

$ϕ_{L o s s} = K_{L o s s} \frac{{\dot{m}}^{2}}{2 ρ S_{B}^{2}},$

where:

K_Loss is the loss coefficient.

If the Loss coefficient specification parameter is set to Semi-empirical formulation, the loss coefficient for a sudden expansion is computed as

$K_{L o s s} = K_{e} {(1 - \frac{S_{B}}{S_{A}})}^{2},$

while for a sudden contraction it is computed as

$K_{L o s s} = \frac{K_{c}}{2} (1 - \frac{S_{B}}{S_{A}}),$

where:

K_e is the correction factor in the expansion zone.
K_c is the correction factor in the contraction zone.

In the transition zone between sudden expansion and sudden contraction behavior, the loss coefficient is smoothed through a cubic polynomial function:

$K_{L o s s} = K_{e} {(1 - \frac{S_{B}}{S_{A}})}^{2} + λ [\frac{K_{c}}{2} (1 - \frac{S_{B}}{S_{A}}) - K_{e} {(1 - \frac{S_{B}}{S_{A}})}^{2}],$

where

$λ = 3 {\bar{\dot{m}}}^{2} - 2 {\bar{\dot{m}}}^{3},$

and

${\dot{m}}_{C r} = {Re}_{C r} \sqrt{\frac{π}{4} S_{B} μ .}$

If the Loss coefficient specification parameter is set to Tabulated data — Loss coefficient vs. Reynolds number, the block obtains the loss coefficient from tabular data provided as a function of the Reynolds number.

Energy Balance

The energy conservation equation in the sudden area change is

$ϕ_{A} + ϕ_{B} = 0,$

where:

Φ_A and Φ_B are the energy flow rates into the sudden area change through ports A and B.

Assumptions and Limitations

The flow is incompressible. The fluid density is assumed constant in the sudden area change.

Ports

Conserving

expand all

A — Section inlet
thermal liquid

Fluid entry or exit point to the area change section.

B — Section inlet
thermal liquid

Fluid entry or exit point to the area change section.

Parameters

expand all

Local loss parameterization — Method of flow loss calculating
`Semi-empirical` (default) | `Tabulated data - loss coefficient vs. Reynolds number`

Parameterization for calculating the loss coefficient due to the sudden area change. Select Semi-empirical formulation to automatically compute the loss coefficient from the cross-sectional areas at ports A and B. Select Tabulated data - Loss coefficient vs. Reynolds number to specify a 1-D lookup table for the loss coefficient with respect to the flow Reynolds number.

Cross-sectional area at port A — Inlet area at A
`0.02` `m^2` (default) | positive scalar

Area normal to the direction of flow at inlet A. This value must be greater than the cross-sectional area at B.

Cross-sectional area at port B — Inlet area at B
`0.01` `m^2` (default) | positive scalar

Area normal to the direction of flow at B. This value must be smaller than the cross-sectional area at A.

Critical Reynolds number — Transitional Reynolds number associated with smallest area in section
`150` (default) | positive scalar

Reynolds number at which flow transitions between laminar and turbulent regimes in the contraction portion of the sudden area change. This parameter is visible only when the Loss coefficient specification parameter is set to Semi-empirical formulation.

Contraction correction factor — Loss coefficient scaling factor
`1` (default) | positive scalar

Scaling factor for adjusting the loss coefficient value in the contraction portion of the sudden area change. The block multiplies the loss coefficient factor calculated from the semi-empirical expression by this factor. This parameter is visible only when the Loss coefficient specification parameter is set to Semi-empirical formulation.

Expansion correction factor — Loss coefficient scaling factor
`1` (default) | positive scalar

Scaling factor for adjusting the loss coefficient value in the expansion portion of the sudden area change. The block multiplies the loss coefficient factor calculated from the semi-empirical expression by this factor. This parameter is visible only when the Loss coefficient specification parameter is set to Semi-empirical formulation.

Reynolds number vector — Vector of Reynolds numbers for tabular parameterization
`[10, 20, 30, 40, 50, 100, 200, 500, 1000, 2000]` (default) | vector

Vector of Reynolds numbers for the tabular parameterization of loss coefficients. You specify the Contraction loss coefficient vector and Expansion loss coefficient vector parameters at these Reynolds numbers.

This parameter is visible only when the Loss coefficient specification parameter is set to Tabulated data - Loss coefficient vs. Reynolds number.

Contraction loss coefficient vector — Vector of loss coefficients for tabular parameterization
`[4, 2.7, 1.8, 1.46, 1.3, .9, .65, .42, .3, .2]` (default) | 1-by-n vector

Vector of loss coefficients for the contraction portion of the area change. Specify the loss coefficients at the Reynolds numbers in the Reynolds number vector parameter. The block uses the Reynolds number and loss coefficient vectors to construct a 1-D lookup table.

This parameter is visible only when the Loss coefficient specification parameter is set to Tabulated data — Loss coefficient vs. Reynolds number.

Expansion loss coefficient vector — Vector of loss coefficients for tabular parameterization
`[4, 2.3, 1.65, 1.35, 1.15, .9, .75, .65, .9, .65]` (default) | 1-by-n vector

Vector of loss coefficients for the expansion portion of the area change. Specify the loss coefficients at the Reynolds numbers in the Reynolds number vector parameter. The block uses the Reynolds number and loss coefficient vectors to construct a 1-D lookup table.

This parameter is visible only when the Loss coefficient specification parameter is set to Tabulated data — Loss coefficient vs. Reynolds number.

Extended Capabilities

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

Version History

Introduced in R2016a

Sudden Area Change (TL)

Description

Mass Balance

Momentum Balance

Energy Balance

Assumptions and Limitations

Ports

Conserving

A — Section inlet thermal liquid

B — Section inlet thermal liquid

Parameters

Local loss parameterization — Method of flow loss calculating Semi-empirical (default) | Tabulated data - loss coefficient vs. Reynolds number

Cross-sectional area at port A — Inlet area at A 0.02 m^2 (default) | positive scalar

Cross-sectional area at port B — Inlet area at B 0.01 m^2 (default) | positive scalar

Critical Reynolds number — Transitional Reynolds number associated with smallest area in section 150 (default) | positive scalar

Contraction correction factor — Loss coefficient scaling factor 1 (default) | positive scalar

Expansion correction factor — Loss coefficient scaling factor 1 (default) | positive scalar

Reynolds number vector — Vector of Reynolds numbers for tabular parameterization [10, 20, 30, 40, 50, 100, 200, 500, 1000, 2000] (default) | vector

Contraction loss coefficient vector — Vector of loss coefficients for tabular parameterization [4, 2.7, 1.8, 1.46, 1.3, .9, .65, .42, .3, .2] (default) | 1-by-n vector

Expansion loss coefficient vector — Vector of loss coefficients for tabular parameterization [4, 2.3, 1.65, 1.35, 1.15, .9, .75, .65, .9, .65] (default) | 1-by-n vector

Extended Capabilities

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

Version History

See Also

A — Section inlet
thermal liquid

B — Section inlet
thermal liquid

Local loss parameterization — Method of flow loss calculating
`Semi-empirical` (default) | `Tabulated data - loss coefficient vs. Reynolds number`

Cross-sectional area at port A — Inlet area at A
`0.02` `m^2` (default) | positive scalar

Cross-sectional area at port B — Inlet area at B
`0.01` `m^2` (default) | positive scalar

Critical Reynolds number — Transitional Reynolds number associated with smallest area in section
`150` (default) | positive scalar

Contraction correction factor — Loss coefficient scaling factor
`1` (default) | positive scalar

Expansion correction factor — Loss coefficient scaling factor
`1` (default) | positive scalar

Reynolds number vector — Vector of Reynolds numbers for tabular parameterization
`[10, 20, 30, 40, 50, 100, 200, 500, 1000, 2000]` (default) | vector

Contraction loss coefficient vector — Vector of loss coefficients for tabular parameterization
`[4, 2.7, 1.8, 1.46, 1.3, .9, .65, .42, .3, .2]` (default) | 1-by-n vector

Expansion loss coefficient vector — Vector of loss coefficients for tabular parameterization
`[4, 2.3, 1.65, 1.35, 1.15, .9, .75, .65, .9, .65]` (default) | 1-by-n vector

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