Acoustics of a Pipe System with3D Bend and Junction

Acoustics of a Pipe System with
3D Bend and Junction

The propagation and response of acoustic signals in pipe systems is used to detect leaks and deformations in the pipes. This is, for example, relevant in the oil and gas industry or in water delivery piping systems. This model uses the pipe acoustics interfaces to model the propagation of acoustic waves in a pipe systems using a 1D model. The 1D model is coupled to a detailed 3D model of a junction and a bend. In the 3D domains, a pressure acoustics interface is used. The 3D domains could represent any complex geometry like valves, reservoirs, or pipe expansions where the 1D model is not valid. The tutorial sets up both a transient and a frequency domain versions of the model. A Pressure Acoustics, Boundary Mode interface is used to compute and analyze the propagating modes in the pipe cross section, as only plane propagating modes are supported in the 1D pipe formulation.

The model requires both the Acoustics Module and the Pipe Flow Module.

Model Definition

Pipe acoustics is used to model the propagation and response of acoustic signals in the long pipe segments using a 1D model. This interface is only valid for the analysis of plane wave modes. The 3D structures are modeled using pressure acoustics. To ensure correct coupling to the 1D model, the Acoustics-Pipe Acoustics Connection multiphysics coupling is used. The coupling requires a straight pipe segment connected to the 3D structure and it is only valid when the background fluid velocity speed is zero.

For a given pipe system in quiescent conditions (u0 = 0), constant background pressure, and completely rigid walls, the governing equations of the pipe system become:

(1)

(2)

where ρ is the fluid density, A is the cross-sectional area of the pipe, u is the area-averaged acoustic velocity, which is defined in the tangential direction u = uet, where et is the tangential vector, ∇t is the tangential derivative, τw is the tangential wall drag force, Z is the inner perimeter of the pipe, F is a volume force applied to the pipe, and c is the effective speed of sound in the pipe (in a rigid pipe this has the same value as the isentropic bulk speed of sound).

The pipe acoustics domain (pipe domain) is connected to the pressure acoustics domain (acoustic domain) through the Acoustics-Pipe Acoustics Connection multiphysics coupling. This coupling guarantees continuity of pressure and velocity between both domains through the equations:

(3)

(4)

Here, ppipe is the pipe pressure at the connection point, A is the total area of the connected acoustic boundary, pt is the total pressure in the acoustic domain and S is the connected acoustic boundary, n is the vector normal to the boundary, and upipe is the pipe velocity at the connection point. The possible dipole domain source, qd, is zero in this model.

The geometry of the pipe system, including both 1D and 3D domains, is shown in Figure 1.

Figure 1: Model geometry.

The model has two separated 3D domains, shown in Figure 2, which will be included in the model using the Pressure Acoustics interface.

Figure 2: T-junction and bend.

The pipe ends, not connected to the Pressure Acoustics domains, have an end impedance condition of an infinite pipe, allowing the acoustic waves to exit the system. In our case, this simply corresponds to the characteristic specific impedance.

(5)

The points where this impedance condition is imposed can be seen in Figure 3.

Figure 3: The pipe end impedance conditions are applied to the open ends of the pipe system.

The tutorial sets up both a transient and a frequency domain version of the model. Under the transient analysis, a wave train is created through the imposition of a volume force on one end of the pipe system. The force is defined through an analytical function created under the node. Figure 4 shows a plot of the analytical function used in the model to generate the wave train.

Figure 4: Imposed volume force as direct output of the plot functionality when defining an analytic function.

At last, a Pressure Acoustics, Boundary Mode interface is used to solve for the propagation modes on the pipe cross section at the driving frequency to check if higher order modes can propagate. This is easy and quick to check by solving an eigenvalue problem. The eigenvalue solver computes a specified number of solutions {pj, λj} to the equation

(6)

with

(7)

where pt is the total pressure, keq is the in-plane wave number, ω is the angular frequency, kn is the out-of-plane wave number, and λ = −ikn is the eigenvalue.

Results and Discussion

Figure 5 shows the transient pressure distribution in the pipe system at various times. The figure shows the continuity in the pressure at the Acoustics-Pipe Acoustics Connection points.

Figure 5: Evolution of the Pressure in the pipe system at various times.

The initial wave train travels through the pipe system producing reflections as the wave finds discontinuities at the T-junction and the bend. Figure 6 shows the pressure at different points of the pipe system. The reflected wave from the T-junction can be observed in Point A after 0.09 s and the reflected wave coming from the pipe bend can be seen after 0.13 s. By updating the parameter rbend, that controls the bending radius of the bend, to 160 cm this reflection can be greatly reduced.

Figure 6: Pressure at three points in the pipe system (Point B and Point C are offset on the y-axis for clarity).

This model can be updated to consider the compressibility of the pipe section based on the elastic modulus and thickness of the pipe wall. Analyzing the pipe system response in the time domain allows us to easily identify reflections or changes in the acoustic response caused by changes in pipe section or wall thickness and these changes can be linked to corrosion or leaks in the pipe system.

The problem can be solved in the frequency domain; the results will be equivalent to exciting the system in the time domain with a sine wave until you reach steady state. By solving the model in the frequency domain, a quick and inexpensive computation, insight into the pressure distribution for a given input can be obtained. Figure 7 shows the response of the pipe system under an excitation with a frequency of 120 Hz.

Figure 7: Pressure distribution in the frequency domain at 120 Hz.

The Pressure Acoustics, Boundary Mode interface solves for the propagating and nonpropagating modes on the pipe cross section at the driving frequency to check if higher order modes propagate at the given frequency or if it just the plane wave component. A mode with a real wave number will propagate, while a mode with imaginary wave number will not be able to propagate. Figure 8 shows the pressure distribution for the mode (2,0) which cannot propagate at 120 Hz.

Figure 8: Pressure distribution for mode (2,0), which cannot propagate at 120 Hz.

This tutorial presents an introduction to the modeling of pipe systems and a selection of the different analysis techniques that can be applied to them. Further details like the compressibility of the pipes or more complex 3D domains can be added to increase the fidelity of the model.

Acoustics_Module/Tutorials,_Pipe_Acoustics/acoustics_pipe_system

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

Click .

Geometry 1

With this, we import a fully parametric sequence that defines the geometry of the model. The parameters used to define the geometry can be modified and updated at any point of the tutorial. Detailed instructions for setting up the geometry are found at the appendix at the end of the instructions.

In the toolbar, click .

Browse to the model’s Application Libraries folder and double-click the file acoustics_pipe_system_geom_sequence.mph.

In the toolbar, click .

The image should look like that in Figure 1.

Global Definitions

Parameters 1

In the window, under click .

In the window for , type Geometry in the text field.

This group includes all the parameters that define the geometry of the model

Parameters 2

In the toolbar, click and choose .

In the window for , type Model Parameters in the text field.

Locate the section. Click .

Browse to the model’s Application Libraries folder and double-click the file acoustics_pipe_system_parameters.txt.

This group includes all the parameters that define the model like material properties, frequencies of interest, and the total time of the analysis.

Definitions

Rectangle 1 (rect1)

In the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 0.5*T0.

In the text field, type 3*T0.

Click to expand the section. In the text field, type T0.

Analytic 1 (an1)

In the toolbar, click and choose .

In the window for , type Volume force in the text field.

In the text field, type volforc.

Locate the section. In the text field, type rect1(t[1/s])*sin(2*pi*f0*t).

In the text field, type t.

Locate the section. In the text field, type s.

In the text field, type N/m^3.

Locate the section. In the table, enter the following settings:


Argument	Lower limit	Upper limit
t	0	3.5*T0

Click .

The image should look like that in Figure 4.

Explicit 1

In the toolbar, click .

In the window for , type 1D Pipes in the text field.

Locate the section. From the list, choose .

Select Edges 1, 2, 20, 37, and 60 only.

Explicit 2

In the toolbar, click .

In the window for , type Pipe Ends in the text field.

Locate the section. From the list, choose .

Select Points 1, 11, and 38 only.

Materials

In the toolbar, click and choose .

Add Material

Go to the window.

In the tree, select .

Click in the window toolbar.

Materials

Water, liquid 1 (mat2)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Water, liquid 2 (mat3)

In the window, click .

In the window for , locate the section.

From the list, choose .

Select Boundary 1 only.

Add Physics

In the toolbar, click to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

Pipe Acoustics, Transient (patd)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the fmax, sol text field, type f0.

Pipe Properties 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the di text field, type Di.

Pipe Properties 2

In the toolbar, click and choose .

Select Edge 20 only.

In the window for , locate the section.

From the list, choose .

In the di text field, type Di*2/3.

End Impedance 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Volume Force 1

In the toolbar, click and choose .

Select Edge 1 only.

In the window for , locate the section.

Specify the F vector as


volforc(t)	x
0	y
0	z

Pressure Acoustics, Transient (actd)

In the window, under click .

In the window for , locate the section.

In the fmax, sol text field, type f0.

Pressure Acoustics, Boundary Mode (acbm)

In the window, under click .

In the window for , locate the section.

Click .

Select Boundary 1 only.

Pipe Acoustics, Frequency Domain (pafd)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Pipe Properties 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the di text field, type Di.

Pipe Properties 2

In the toolbar, click and choose .

Select Edge 20 only.

In the window for , locate the section.

From the list, choose .

In the di text field, type Di*2/3.

End Impedance 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Volume Force 1

In the toolbar, click and choose .

Select Edge 1 only.

In the window for , locate the section.

Specify the F vector as


1	x
0	y
0	z

Multiphysics

Acoustic-Pipe Acoustic Connection 1 (apc1)

In the toolbar, click and choose .

Acoustic-Pipe Acoustic Connection 2 (apc2)

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Size

In the window, under click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type min(lam0/12,Di/2).

In the text field, type min(lam0/24,Di/6).

Click .

Edge 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Free Triangular 1

Right-click and choose .

Select Boundary 1 only.

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type Di/12.

Select the check box.

In the associated text field, type Di/12.

Click .

Root

In the toolbar, click and choose .

Add Study

Go to the window.

Find the subsection. In the tree, select .

Find the subsection. In the table, enter the following settings:


Physics	Solve
Pressure Acoustics, Boundary Mode (acbm)
Pipe Acoustics, Frequency Domain (pafd)
Pressure Acoustics, Frequency Domain (acpr)

Find the subsection. In the table, enter the following settings:


Multiphysics couplings	Solve
Acoustic-Pipe Acoustic Connection 2 (apc2)

Click in the window toolbar.

In the toolbar, click to close the window.

Study 1

Step 1: Time Dependent

In the window for , locate the section.

In the text field, type range(0,T0/24,Tend).

In the window, click .

In the window for , locate the section.

Clear the check box.

In the text field, type Study 1 - Pipe System Transient.

In the toolbar, click .

Add Study

In the toolbar, click to open the window.

Go to the window.

Find the subsection. In the table, enter the following settings:


Physics	Solve
Pipe Acoustics, Transient (patd)
Pressure Acoustics, Transient (actd)
Pipe Acoustics, Frequency Domain (pafd)
Pressure Acoustics, Frequency Domain (acpr)