Sudden expansion or contraction in flow area

Thermal Liquid/Pipes & Fittings

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 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.

The momentum conservation equation in the sudden area change is

$${p}_{A}-{p}_{B}=\frac{{\dot{m}}^{2}}{2\rho}\left(\frac{1}{{S}_{B}^{2}}-\frac{1}{{S}_{A}^{2}}\right)+{\varphi}_{Loss},$$

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

$${\varphi}_{Loss}={K}_{Loss}\frac{{\dot{m}}^{2}}{2\rho {S}_{B}^{2}},$$

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

$${K}_{Loss}={K}_{e}{\left(1-\frac{{S}_{B}}{{S}_{A}}\right)}^{2},$$

while for a sudden contraction it is computed as

$${K}_{Loss}=\frac{{K}_{c}}{2}\left(1-\frac{{S}_{B}}{{S}_{A}}\right),$$

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:

$${K}_{Loss}={K}_{e}{\left(1-\frac{{S}_{B}}{{S}_{A}}\right)}^{2}+\lambda \left[\frac{{K}_{c}}{2}\left(1-\frac{{S}_{B}}{{S}_{A}}\right)-{K}_{e}{\left(1-\frac{{S}_{B}}{{S}_{A}}\right)}^{2}\right],$$

where

$$\lambda =3{\overline{\dot{m}}}^{2}-2{\overline{\dot{m}}}^{3},$$

and

$${\dot{m}}_{Cr}={\mathrm{Re}}_{Cr}\sqrt{\frac{\pi}{4}{S}_{B}\mu .}$$

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.

The energy conservation equation in the sudden area change is

$${\varphi}_{A}+{\varphi}_{B}=0,$$

where:

*Φ*_{A}and*Φ*_{B}are the energy flow rates into the sudden area change through ports A and B.

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

**Cross-sectional area at port A**Area normal to the direction of flow at inlet A. This value must be greater than the cross-sectional area at B. The default value is

`2e-2`

m^2.**Cross-sectional area at port B**Area normal to the direction of flow at inlet B. This value must be smaller than the cross-sectional area at A. The default value is

`1e-2`

.**Characteristic longitudinal length**Average distance traversed by the fluid from inlet A to inlet B. This value must be greater than zero. The default value is

`0.1`

m.

**Loss coefficient specification**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. The default setting is`Tabulated data — Loss coefficient vs. Reynolds number`

.**Contraction correction factor**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`

. The default value is`1`

.**Expansion correction factor**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`

. The default value is`1`

.**Critical Reynolds number**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`

. The default value is`10`

.**Reynolds number vector**Vector of Reynolds numbers with which to build a loss coefficient lookup table. 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`

. The default vector is a 10-element array ranging from`10.0`

to`2000.0`

.**Contraction loss coefficient 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`

. The default vector is a 10-element array ranging from`4.0`

to`0.2`

.**Expansion loss coefficient 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`

. The default vector is a 10-element array ranging from`4.0`

to`0.65`

.

**Mass flow rate into port A**Mass flow rate into the component through port

**A**at the start of simulation. The default value is`1 kg/s`

.

A — Thermal liquid port representing inlet A

B — Thermal liquid port representing inlet B