End Impedance
Use the End Impedance node to model conditions at the end of a pipe. The condition can either model an infinite pipe and thus represent the characteristic impedance of the pipe system at that point. This results in a zero reflection condition. Alternatively, the condition can represent the radiation impedance of an open pipe in either a flanged (in an infinite baffle) or unflanged (a pipe ending in free open space). The end impedance can also be user defined and could then represent modeled or experimental values for a specific pipe configuration.
The wave speed c in the pipe can be different from the speed of sound cs in an open space. It then depends on the elastic properties of the pipe structure. It is defined in Equation 3-17 in the Governing Equations section.
The wave speed can be evaluated as sqrt(1/patd.invc2) or sqrt(1/pafd.invc2) during the analysis and results stage.
End Impedance
Select an Impedance model.
For The Pipe Acoustics, Transient Interface select Infinite pipe (low Mach number limit) (the default) or User defined. The Infinite pipe (low Mach number limit) models and infinite pipe by specifying the characteristic impedance at that point. This condition creates a non-reflecting boundary. The expression is valid for small values of the Mach number Ma = u0/c. For User defined enter an End impedance Zend (SI unit: Pa⋅s/m). The default is patd.rho*(sqrt(1/patd.invc2)), which is ρ·c.
For The Pipe Acoustics, Frequency Domain Interface select from the following — Infinite pipe (low Mach number limit) (the default), Infinite pipe, Flanged pipe, circular, Flanged pipe, rectangular, Unflanged pipe, circular (low ka limit), Unflanged pipe, circular, or User defined.
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For Infinite pipe enter a Wave number k (SI unit: rad/m). The default expression is pafd.omega*(sqrt(pafd.invc2)). This end impedance models the infinite pipe using the full (nonlinear) dispersion relation. It is valid for all Mach numbers but requires the additional input of the wave number k.
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For Flanged pipe, circular enter an Inner radius a (SI unit: m). The default expression is pafd.dh/2. This end impedance models the radiation impedance of a circular pipe terminated in an infinite baffle. It is an exact analytical result valid for all frequencies and pipe radii. In the low frequency limit it reduces to the classical results:
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For Flanged pipe, rectangular enter an Inner width wi (SI unit: m). The default is 5 cm (0.005 m). Also enter an Inner height hi (SI unit: m). The default is 10 cm (0.01 m). This end impedance models the radiation impedance of a pipe of rectangular cross section terminated in an infinite baffle. The model is only valid in the low frequency range where and .
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For Unflanged pipe, circular (low ka limit) or Unflanged pipe, circular enter an Inner radius a (SI unit: m). The default expression is pafd.dh/2. These two end impedance models prescribe the radiation impedance of an unflanged circular pipe (a pipe ending in free open space). The first model is the classical low-frequency approximation valid for . While the second model extends the frequency range to ka < 3.83.
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For User defined enter an End impedance Zend (SI unit: Pa⋅s/m). The default expression is pafd.rho*(sqrt(1/pafd.invc2)), which is ρ·c.
For a detailed review of the end impedance models see: Theory for the Pipe Acoustics Boundary Conditions.