Gradual Area Change (IL)
Constriction or expansion of pipework in isothermal liquid systems.
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Description
In the Area Change (IL) block, a sudden or gradual change in cross-sectional area is modelled for pipe systems of constant cross-section and variable flow direction. In the direction from port A to port B the channel narrows. To simulate channel expansion, the unit should be set up so that the fluid moves in the direction from port B to port A. The inlet and outlet areas can be the same.
Semi-empirical and tabular methods are available for calculating pressure losses to determine the dependence of losses on flow characteristics.
Semi-empirical method
In the analytical semi-empirical method, the dependence of pressure losses on flow velocities is determined by the hydraulic loss coefficient , which in turn is determined through user-defined parameters, Contraction correction factor and Expansion correction factor , more details in the author’s book [1]. The hydraulic loss factor is calculated from the expansion and contraction loss factors and the mass flow rates through the block.
In case of gradual contraction with cone angle in the range from 0° to 45°, the contraction loss factor is determined as follows:
,
where is the channel narrowing coefficient ; is the cone angle, the value of the Cone angle parameter.
And in the case of gradual narrowing with a cone angle between 45° and 180°, the loss coefficient will be determined by: :
.
In case of sudden narrowing of the channel, the cone angle is 180°. Then the loss factor is calculated as:
.
In case of gradual channel expansion with cone angle in the range from 0° to 45°, the loss factor will be determined as follows:
,
and in the case of gradual expansion with a cone angle between 45° and 180° :
.
Based on the data obtained, the hydraulic loss coefficient for a pipe segment with narrowing or widening of the channel will be determined:
where:
-
- is the mass flow rate through port A. The mass is stored in the block:
;
-
- the value of the mass flow rate at which a breakaway recirculation zone occurs is determined by the value of the Critical Reynolds number parameter, :
;
where:
-
- is the smallest cross-sectional area of the channel (either the values of the parameters Cross-sectional area at port A or Cross-sectional area at port B);
-
- kinematic viscosity of the fluid;
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- average density of liquid;
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- hydraulic diameter in section :
.
-
Tabular parameterization method
The hydraulic loss coefficient can also be determined using user-supplied interpolated data obtained at the minimum cross-section for different Reynolds numbers, i.e. data that is a function of the critical Reynolds number (Critical Reynolds number parameter value):
Intermediate values between neighbouring points are determined by linear interpolation, and beyond the table boundaries, the nearest neighbour method is used.
Parameters
Cone angle - angle of the cone forming the inner walls of the flow narrowing/expansion channel
30° (by default) | `positive scalar `
Angle (solution) of the cone forming the inner walls of the narrowing channel and located with its base to the port .
Dependencies
To use this parameter, set the Local loss parameterization parameter to Semi-empirical correlation - gradual area change.
Local loss parameterization - hydraulic loss model
Semi-empirical correlation - sudden area change (by default) | Semi-empirical correlation - gradual area change | Tabulated data - loss coefficient vs. Reynolds number.
Hydraulic loss model for channel contraction/expansion. You can choose between two analytical semi-empirical methods (sudden or conical contraction/expansion) or you can substitute your own data by selecting Tabulated data - loss coefficient vs. Reynolds number.
Cross-sectional area at port A - cross-sectional area at port A
0.02 m² (by default).
Cross-sectional area at inlet.
Cross-sectional area at port B - cross-sectional area at port B
0.01 m² (by default)
Cross-sectional area at outlet.
Reynolds number vector - vector of Reynolds number values in case of tabular parameterization method
[10, 20, 30, 40, 50, 100, 200, 500, 1000, 2000] (by default) | vector 1 to n.
Vector of Reynolds number values in the case of the tabular method of channel contraction/expansion parameterization. The elements of the vector shall correspond to the elements of the Contraction loss coefficient vector and Expansion loss coefficient vector. The values of the vector elements shall be listed in ascending order.
Dependencies
To use this parameter, set the Local loss parameterization parameter to Tabulated data - loss coefficient vs. Reynolds number.
Contraction loss coefficient vector - vector of contraction loss coefficients
[5, 2.7, 1.8, 1.46, 1.3, .9, .65, .42, .3, .2] (by default) | `vector of 1 by n'.
The vector of loss coefficients in the case of flow constriction, corresponding to the Reynolds number vector parameter, where n is the length of the Reynolds number vector. The elements must be listed in descending order and must be greater than 0.
Dependencies
To use this parameter, set the Local loss parameterization parameter to `Tabulated data - loss coefficient vs. Reynolds number'.
Expansion loss coefficient vector - vector of expansion loss coefficients
[3.1, 2.3, 1.65, 1.35, 1.15, .9, .75, .75, .65, .9, .65] (by default) | `vector of 1 by n'.
The vector of loss coefficients in the case of flow expansion, corresponding to the Reynolds number vector parameter, where n is the length of the Reynolds number vector. The elements must be listed in descending order and must be greater than 0.
Dependencies
To use this parameter, set the Local loss parameterization parameter to `Tabulated data - loss coefficient vs. Reynolds number'.
Expansion correction factor is the correction factor in the expansion pressure loss equation
1 (By default) | `positive scalar'.
A factor used in the semi-empirical method to calculate the loss factor in the case of flow expansion.
Dependencies
To use this parameter, set the Local loss parameterization parameter to `Semi-empirical correlation - sudden area change' or `Semi-empirical correlation - gradual area change'.
Contraction correction factor - correction factor in the constriction pressure loss equation
1 (By default) | `positive scalar'.
A factor used in the semi-empirical method to calculate the loss factor in the case of flow constriction.
Dependencies
To use this parameter, set the Local loss parameterization parameter to `Semi-empirical correlation - sudden area change' or `Semi-empirical correlation - gradual area change'.
Critical Reynolds number - upper limit of Reynolds numbers for laminar flow in the channel
150 (By default) | `positive scalar'.
Upper limit of Reynolds numbers characterising the laminar regime of fluid flow in the minimum cross-section.