# Gradual Area Change

Gradual enlargement or contraction

## Library

Local Hydraulic Resistances

## Description

The Gradual Area Change block represents a local hydraulic resistance, such as a gradual cross-sectional area change. The resistance represents a gradual enlargement (diffuser) if fluid flows from inlet to outlet, or a gradual contraction if fluid flows from outlet to inlet. The block is based on the Local Resistance block. It determines the pressure loss coefficient and passes its value to the underlying Local Resistance block. The block offers two methods of parameterization: by applying semi-empirical formulas (with a constant value of the pressure loss coefficient) or by table lookup for the pressure loss coefficient based on the Reynolds number.

If you choose to apply the semi-empirical formulas, you provide geometric parameters of the resistance, and the pressure loss coefficient is determined according to the A.H. Gibson equations (see [1] and [2]):

$${K}_{GE}=\{\begin{array}{ll}{K}_{cor}{\left(1-\frac{{A}_{s}}{{A}_{L}}\right)}^{2}\xb72.6\mathrm{sin}\frac{\alpha}{2}\hfill & \text{for0}\alpha \text{=}{45}^{o}\hfill \\ {K}_{cor}{\left(1-\frac{{A}_{s}}{{A}_{L}}\right)}^{2}\hfill & \text{for}{45}^{o}\text{}\alpha \text{}{180}^{o}\hfill \end{array}$$

$${K}_{GC}=\{\begin{array}{ll}{K}_{cor}\xb70.5{\left(1-\frac{{A}_{s}}{{A}_{L}}\right)}^{0.75}\xb71.6\mathrm{sin}\frac{\alpha}{2}\hfill & \text{for0}\alpha \text{=}{45}^{o}\hfill \\ {K}_{cor}\xb70.5{\left(1-\frac{{A}_{s}}{{A}_{L}}\right)}^{0.75}\xb7\sqrt{\mathrm{sin}\frac{\alpha}{2}}\hfill & \text{for}{45}^{o}\text{}\alpha \text{}{180}^{o}\text{}\hfill \end{array}$$

where

`K` | Pressure loss coefficient for the gradual enlargement, which takes place if fluid flows from inlet to outlet |

`K` | Pressure loss coefficient for the gradual contraction, which takes place if fluid flows from outlet to inlet |

`K` | Correction factor |

`A` | Small area |

`A` | Large area |

α | Enclosed angle |

If you choose to specify the pressure loss coefficient by a table, you have to provide a tabulated relationship between the loss coefficient and the Reynolds number. In this case, the loss coefficient is determined by one-dimensional table lookup. You have a choice of two interpolation methods and two extrapolation methods.

The pressure loss coefficient, determined by either of the two methods, is then passed to the underlying Local Resistance block, which computes the pressure loss according to the formulas explained in the reference documentation for that block. The flow regime is checked in the underlying Local Resistance block by comparing the Reynolds number to the specified critical Reynolds number value, and depending on the result, the appropriate formula for pressure loss computation is used.

The Gradual Area Change block is bidirectional and computes pressure loss for both the direct flow (gradual enlargement) and return flow (gradual contraction). If the loss coefficient is specified by a table, the table must cover both the positive and the negative flow regions.

Connections A and B are conserving hydraulic ports associated with the block inlet and outlet, respectively.

The block positive direction is from port A to port B. This means that the flow rate is positive if fluid flows from A to B, and the pressure loss is determined as $$p={p}_{A}-{p}_{B}$$.

## Basic Assumptions and Limitations

Fluid inertia is not taken into account.

If you select parameterization by the table-specified relationship

, the flow is assumed to be turbulent.`K=f(Re)`

## Parameters

**Model parameterization**Select one of the following methods for block parameterization:

`By semi-empirical formulas`

— Provide geometrical parameters of the resistance. This is the default method.`Tabulated data — loss coefficient vs. Reynolds number`

— Provide tabulated relationship between the loss coefficient and the Reynolds number. The loss coefficient is determined by one-dimensional table lookup. You have a choice of two interpolation methods and two extrapolation methods. The table must cover both the positive and the negative flow regions.

**Small diameter**Internal diameter of the small port,

**A**. The default value is`0.01`

m.**Large diameter**Internal diameter of the large port,

**B**. The default value is`0.02`

m. This parameter is used if**Model parameterization**is set to`By semi-empirical formulas`

.**Cone angle**The enclosed angle. The default value is

`30`

deg. This parameter is used if**Model parameterization**is set to`By semi-empirical formulas`

.**Correction coefficient**Correction factor used in the formula for computation of the loss coefficient. The default value is

`1`

. This parameter is used if**Model parameterization**is set to`By semi-empirical formulas`

.**Laminar transition specification**If

**Model parameterization**is set to`By semi-empirical formulas`

, select how the block transitions between the laminar and turbulent regimes:`Pressure ratio`

— The transition from laminar to turbulent regime is smooth and depends on the value of the**Laminar flow pressure ratio**parameter. This method provides better simulation robustness.`Reynolds number`

— The transition from laminar to turbulent regime is assumed to take place when the Reynolds number reaches the value specified by the**Critical Reynolds number**parameter.

**Laminar flow pressure ratio**Pressure ratio at which the flow transitions between laminar and turbulent regimes. The default value is

`0.999`

. This parameter is visible only if the**Laminar transition specification**parameter is set to`Pressure ratio`

.**Critical Reynolds number**The maximum Reynolds number for laminar flow. The value of the parameter depends on the orifice geometrical profile. You can find recommendations on the parameter value in hydraulics textbooks. The default value is

`350`

. This parameter is visible only if the**Laminar transition specification**parameter is set to`Reynolds number`

.**Reynolds number vector**Specify the vector of input values for Reynolds numbers as a one-dimensional array. The input values vector must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for smooth interpolation. The default values are

`[-4000, -3000, -2000, -1000, -500, -200, -100, -50, -40, -30, -20, -15, -10, 10, 20, 30, 40, 50, 100, 200, 500, 1000, 2000, 4000, 5000, 10000]`

. This parameter is used if**Model parameterization**is set to`By loss coefficient vs. Re table`

.**Loss coefficient vector**Specify the vector of the loss coefficient values as a one-dimensional array. The loss coefficient vector must be of the same size as the Reynolds numbers vector. The default values are

`[0.25, 0.3, 0.65, 0.9, 0.65, 0.75, 0.90, 1.15, 1.35, 1.65, 2.3, 2.8, 3.10, 5, 2.7, 1.8, 1.46, 1.3, 0.9, 0.65, 0.42, 0.3, 0.20, 0.40, 0.42, 0.25]`

. This parameter is used if**Model parameterization**is set to`By loss coefficient vs. Re table`

.**Interpolation method**Select one of the following interpolation methods for approximating the output value when the input value is between two consecutive grid points:

`Linear`

— Select this option to get the best performance.`Smooth`

— Select this option to produce a continuous curve with continuous first-order derivatives.

For more information on interpolation algorithms, see the PS Lookup Table (1D) block reference page. This parameter is used if

**Model parameterization**is set to`By loss coefficient vs. Re table`

.**Extrapolation method**Select one of the following extrapolation methods for determining the output value when the input value is outside the range specified in the argument list:

`Linear`

— Select this option to produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.`Nearest`

— Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.

For more information on extrapolation algorithms, see the PS Lookup Table (1D) block reference page. This parameter is used if

**Model parameterization**is set to`By loss coefficient vs. Re table`

.

## Restricted Parameters

When your model is in Restricted editing mode, you cannot modify the following parameters:

**Model parameterization****Interpolation method****Extrapolation method**

All other block parameters are available for modification. The actual set of modifiable
block parameters depends on the value of the **Model parameterization**
parameter at the time the model entered Restricted mode.

## Global Parameters

Parameters determined by the type of working fluid:

**Fluid density****Fluid kinematic viscosity**

Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

## Ports

The block has the following ports:

`A`

Hydraulic conserving port associated with the resistance inlet.

`B`

Hydraulic conserving port associated with the resistance outlet.

## References

[1] *Flow of Fluids Through Valves, Fittings, and Pipe*, Crane Valves
North America, Technical Paper No. 410M

[2] Idelchik, I.E., *Handbook of Hydraulic Resistance*, CRC Begell
House, 1994

## Extended Capabilities

## See Also

Elbow | Local Resistance | Pipe Bend | Sudden Area Change | T-junction

**Introduced in R2006b**