Ultrasonic Flowmeter with Piezoelectric Transducers

Ultrasonic flowmeters are used to determine the velocity of the fluid flowing through a pipe. The principle is to send an ultrasonic signal across the flow at a skew angle. In case of no flow, the transmitting time between the transmitter and the receiver is the same for the signals sent in the upstream and the downstream directions. Otherwise, the downstream traveling wave moves faster than the one traveling upstream. In many cases piezoelectric transducers are used to send and receive the ultrasonic wave.

This tutorial shows how to simulate an ultrasonic flowmeter with piezoelectric transducers in the simplified no-flow case. The simulatoin approach is based on the discontinuous Galerkin (dG) method which is well suited for acoustically large transient problems. The model is a true multiphyiscs problem that involves acoustic-structure interaction and piezoelectric effect. The former is modeled with the Elastic Waves, Time Explicit and the Pressure Acoustics, Time Explicit physics interfaces coupled through the Pair Acoustic-Structure Boundary, Time Explicit multiphysics feature. The latter is handled with the Piezoelectric Effect, Time Explicit multiphysics feature that couples the Elastic Waves, Time Explicit and the Electrostatics physics interfaces. The model takes advantage of a geometry assembly and a nonconforming mesh.

Model Definition

The flowmeter consists of a main pipe and a signal pipe of a smaller diameter. The signal tube is tilted to the main pipe at the angle α = 45°. The dimensions of the pipes used in this tutorial are the same as the ones given in the model Ultrasound Flowmeter with Generic Time-of-Flight Configuration. The pipe walls are considered rigid. There are two transducers placed at either end of the signal pipe. They operate as a transmitter and a receiver. Both transducers are identical and consist of a piezoelectric unit, a matching layer, and a damping block. An input voltage signal applied to the transmitter results in the mechanical deformation of the piezoelectric transducer, which is due to the inverse piezoelectric effect. The mechanical deformation in its turn generates an acoustic wave in the fluid. When the acoustic wave reaches the receiver, the inverse process takes place: the mechanical load is being converted into an electric signal because of the direct piezoelectric effect.

As previously mentioned, this model studies the propagation of the acoustic wave is the simplified no-flow condition. Thus the main focus is the interaction between the fluid and the solid and the conversion input electric signal - acoustic wave - output electric signal. This model solves the full three-dimensional problem with the symmetry condition imposed on the sagittal plane of the structure.

The transmitter and the receiver can be used interchangeably; therefore, the principles of their construction are the same. The main part is a disk made of a piezoelectric material (here, PZT-5H) that is used for the conversion between the electric and the mechanical waves. Its thickness is taken to be 1/2 of the wavelength.

A direct propagation of the acoustic wave from the piezoelectric material to the fluid will result in significant reflections from the solid/fluid interface and, consequently, losses. This is due to the impedance mismatch: for water, the characteristic acoustic impedance

MRayl, and for PZT-5H,

MRayl. Therefore, a 1/4 wavelength thick matching layer is required to minimize the losses. Its impedance should be close to the geometric mean of those of the piezoelectric and the fluid materials, i.e.

MRayl.

The piezoelectric element is surrounded by a backing layer block (also called damping block) at the back. The damping block absorbs the waves radiated from the back face of the piezoelectric element. The properties of the used matching and damping materials are shown in Table 1 .

Table 1: Matching and damping material properties


Part	Material	Density, kg/m3	Longitudinal wave speed, m/s	Shear wave speed, m/s
Matching layer	Alumina/Epoxy	2280	3400	1920
Damping block	Tungsten/Epoxy	6580	1500	775

The input signal applied to the transmitter is a harmonic voltage pulse of the amplitude V0 = 50 V, the frequency f0 = 2.5 MHz, and duration of 2 μs. The voltage profile is depicted in Figure 1.

Figure 1: Input voltage applied to the transmitter.

Results and Discussion

The propagation of the pressure wave in solid and fluid domains is shown in Figure 2. The signal emitted by the transmitter propagates into the fluid at t = 4 μs (upper-left corner). The signal reaches the upper wall of the main pipe at t = 6 μs (upper-right corner) and propagates further in the signal pipe to the receiver at t = 8 μs (lower-left corner). The signal reaches the end of the signal tube and generates an elastic wave in the receiver at t = 10 μs (lower-right corner).

The elastic wave in the receiver piezoelectric element is converted into an electric signal. In Figure 3 you can see the profiles of the input voltage applied to the transducer and the output electric signal read on the receiver.

Figure 2: Propagation of the acoustic pressure signal at 4 time steps.

Figure 3: Input and output electric signals as functions of time.

Notes About the COMSOL Implementation

Geometry and Mesh

The model geometry is an assembly and therefore parts of the geometry are separated from one another and connected via Identity Boundary Pairs. The nodes of the generated mesh elements do not have to match on either side for the pairs thus making the mesh non-conformal. For wave propagation problems, feasible results are achieved when the mesh resolves the wavelengths of the propagating waves. In the solid domains, the minimal wavelength is given by the shear wave speed. Thus materials with lower speed of sound require finer mesh than those with higher speed of sound (compare the shear wave speeds for the matching and the damping material given in Table 1). The use of a non-conformal mesh in this tutorial makes it possible to reduce the number of DOFs solved for in the model.

Acoustics_Module/Ultrasound/flow_meter_piezoelectric_transducers

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

Click

Load the parameters that define the geometry and the physical properties of the system.

Global Definitions

Geometry Parameters

In the window, under click .

In the window for , locate the section.

Click

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

In the text field, type Geometry Parameters.

Model Parameters

In the toolbar, click

and choose .

In the window for , locate the section.

Click

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

In the text field, type Model Parameters.

Import the model geometry sequence from the geometry file. The instructions to the geometry are found in the appendix at the end of this document.

Geometry 1

In the toolbar, click and choose .

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

In the toolbar, click

Specify the driving voltage signal applied to the transmitter.

Global Definitions

Rectangle 1 (rect1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 0.5e-6.

In the text field, type 1.5e-6.

Click to expand the section. In the text field, type 1e-6.

Click

Analytic 1 (an1)

In the toolbar, click

and choose .

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

Locate the section. In the text field, type V0*sin(2*pi*f0*t)*rect1(t).

In the text field, type t.

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

In the table, enter the following settings:


Argument	Unit
t	s

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


Argument	Lower limit	Upper limit	Unit
t	0	10*T0	s

Click

The input electric signal should look like the one in Figure 1.

Create selections to simplify the model setup.

Definitions

Water

In the toolbar, click

Select Domains 1–3 only.

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

PZT

In the toolbar, click

Select Domains 5 and 9 only.

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

Matching

In the toolbar, click

Select Domains 6 and 7 only.

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

Backing

In the toolbar, click

Select Domains 4 and 8 only.

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

Symmetry

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Boundaries 2, 6, 16, 22, 30, 33, 40, 46, and 55 only.

In the text field, type Symmetry.

Solid

In the toolbar, click

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

Locate the section. Under , click

In the dialog box, in the list, choose , , and .

Click .

Define a coordinate system that corresponds to the piezoelectric material orientation: the Z-axis of the piezoelectric crystal points along the signal tube axis.

Transducer Coordinate System

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


	x	y	z
x1	cos(alpha)	0	-sin(alpha)
x3	sin(alpha)	0	cos(alpha)

Find the subsection. Select the check box.

In the text field, type Transducer Coordinate System.

Now, proceed to setting up the physics. Note that the model geometry is an assembly and therefore each physics interface automatically imposes the boundary condition on all boundary pairs.

Component 1 (comp1)

In the toolbar, click

and choose .

Add Physics

Go to the window.

In the tree, select .

Click in the window toolbar.

Pressure Acoustics, Time Explicit (pate)

In the window for , locate the section.

From the list, choose .

Add Physics

Go to the window.

In the tree, select .

Click in the window toolbar.

Elastic Waves, Time Explicit (elte)

In the window for , locate the section.

From the list, choose .

Add Physics

Go to the window.

In the tree, select .

Click in the window toolbar.

Electrostatics (es)

In the window for , locate the section.

From the list, choose .

Pressure Acoustics, Time Explicit (pate)

Impedance 1

In the window, under right-click and choose .

Select Boundaries 1 and 19 only.

Symmetry 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Elastic Waves, Time Explicit (elte)

In the window, under click .

Piezoelectric Material 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Low-Reflecting Boundary 1

In the toolbar, click

and choose .

Select Boundaries 21 and 52 only.

Symmetry 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Elastic Waves, Time Explicit Model 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Damping 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the f1 text field, type 0.99*f0.

In the ζ1 text field, type 0.01.

In the f2 text field, type 1.01*f0.

In the ζ2 text field, type 0.01.

Elastic Waves, Time Explicit Model 1

In the window, click .

Damping 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the f1 text field, type 0.99*f0.

In the ζ1 text field, type 0.025.

In the f2 text field, type 1.01*f0.

In the ζ2 text field, type 0.025.

Piezoelectric Material 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Mechanical Damping 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the f1 text field, type 0.99*f0.

In the ζ1 text field, type 0.005.

In the f2 text field, type 1.01*f0.

In the ζ2 text field, type 0.005.

Electrostatics (es)

In the window, under click .

Charge Conservation, Piezoelectric 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Ground 1

In the toolbar, click

and choose .

Select Boundaries 29 and 57 only.

Electric Potential 1

In the toolbar, click

and choose .

Select Boundary 32 only.

In the window for , locate the section.

In the V0 text field, type Vin(t).

Floating Potential 1

In the toolbar, click

and choose .

Select Boundary 54 only.

In the window for , locate the section.

Select the check box.

Symmetry Plane 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Add multiphysics coupling features.

Multiphysics

Piezoelectric Effect, Time Explicit 1 (pzete1)

In the toolbar, click

and choose .

Pair Acoustic-Structure Boundary, Time Explicit 1 (aspte1)

In the toolbar, click

and choose .

In the window for , locate the section.

Under , click

In the dialog box, in the list, choose and .

Click .

Now, set up the materials.

Materials

In the toolbar, click

and choose .

Add Material

Go to the window.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

Materials

Lead Zirconate Titanate (PZT-5H) (mat2)

In the window for , locate the section.

From the list, choose .

Matching Material

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type Matching Material.

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


Property	Variable	Value	Unit	Property group
Pressure-wave speed	cp	cp_match	m/s	Pressure-wave and shear-wave speeds
Shear-wave speed	cs	cs_match	m/s	Pressure-wave and shear-wave speeds
Density	rho	rho_match	kg/m³	Basic

Damping Material

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type Damping Material.

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


Property	Variable	Value	Unit	Property group
Pressure-wave speed	cp	cp_damp	m/s	Pressure-wave and shear-wave speeds
Shear-wave speed	cs	cs_damp	m/s	Pressure-wave and shear-wave speeds
Density	rho	rho_damp	kg/m³	Basic

Add absorbing layers (sponge layers) to truncate the computational domain.

Definitions

Absorbing Layer 1 (ab1)

In the toolbar, click

Select Domains 1 and 3 only.

Create a mesh. The mesh should be fine enough to resolve the shortest wavelength in each material. Note that the mesh element nodes on one side of the boundary pairs do not match with those on the other side. As a result, the mesh elements adjacent to the pairs have different size, which helps to reduce the number of DOFs in the model.

Mesh 1

Free Triangular 1

In the toolbar, click

and choose .

Select Boundaries 32, 35, 37, 42, 43, and 54 only.

Size 1

Right-click and choose .

Select Boundaries 32 and 54 only.

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type cs_pzt/f0/2.

Size 2

In the window, right-click and choose .

Select Boundaries 35, 37, 42, and 43 only.

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type cs_match/f0/2.

Swept 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Distribution 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 4.

Swept 2

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Distribution 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 2.

Free Tetrahedral 1

In the toolbar, click

When modeling using physics interfaces that are based on the DG method, it is important to avoid small mesh elements, since they control the time steps taken by the solver. Use the functionality available for the tetrahedral mesh to avoid this. This step is very important.

In the window for , click to expand the section.

From the list, choose .

Clear the check box.

Select the check box.

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type lam0/1.5.

Select the check box.

In the associated text field, type lam0/3.

Size 2

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type cs_damp/f0/1.5.

In the window, right-click and choose .

Root

In the toolbar, click

and choose .

Add Study

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

Study 1

Step 1: Time Dependent

In the window for , locate the section.

In the text field, type range(0, T0/5, 30*T0).

Click to expand the section. Find the subsection. From the list, choose .

Under , click

Store the results on the symmetry plane only.

In the dialog box, select in the list.

Click .

Step 1: Time Dependent

In the window, click .

In the toolbar, click

Plot the pressure in the fluid and solid domains and inspect the propagating acoustic signal at different times to get the results like the ones in Figure 2.

Results

Acoustic and Elastic Pressure

In the window for , type Acoustic and Elastic Pressure in the text field.

Click to expand the section. From the list, choose .

Selection 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Surface 2

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click to expand the section. From the list, choose .

In the toolbar, click

Plot the driving voltage applied to the transmitter and the voltage signal read on the receiver. The result should look like the one in Figure 3.

Sent and Received Signals

In the toolbar, click

and choose .

In the window for , type Sent and Received Signals in the text field.

Locate the section. From the list, choose .

In the text field, type range(0, T0/20, 30*T0).

Point Graph 1

Right-click and choose .

Select Point 39 only.

In the window for , locate the section.

In the text field, type V/V0.

Click to expand the section. Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
Relative driving voltage

Point Graph 2

Right-click and choose .

In the window for , locate the section.

Click

Select Point 73 only.

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


Legends
Relative received voltage

Sent and Received Signals

In the window, click .

In the window for , locate the section.

Select the check box.

In the table, select the check box for .

Locate the section. From the list, choose .

In the toolbar, click

Appendix — Geometry Modeling Instructions

From the menu, choose .

In the window, click

In the window, click

In the window, under click .

In the window for , locate the section.

Click

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

In the text field, type Geometry Parameters.

Model Parameters

In the toolbar, click

and choose .

In the window for , locate the section.

Click

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

In the text field, type Model Parameters.

Geometry 1

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type D/2.

In the text field, type L.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	0.5*D

Clear the check box.

Select the check box.

Cylinder 2 (cyl2)

In the toolbar, click

In the window for , locate the section.

In the text field, type D_transducer/2.

In the text field, type L_transducer.

Locate the section. In the text field, type L/2.

In the text field, type -L_transducer/2.

Cylinder 3 (cyl3)

In the toolbar, click

In the window for , locate the section.

In the text field, type D_transducer/2.

In the text field, type L_matching.

Locate the section. In the text field, type L/2.

In the text field, type L_transducer/2.

Cylinder 4 (cyl4)

In the toolbar, click

In the window for , locate the section.

In the text field, type D_transducer/4.

In the text field, type L_piezo.

Locate the section. In the text field, type L/2.

In the text field, type L_transducer/2+L_matching.

Cylinder 5 (cyl5)

In the toolbar, click

In the window for , locate the section.

In the text field, type D_transducer/2.

In the text field, type 2*L_piezo.

Locate the section. In the text field, type L/2.

In the text field, type L_transducer/2+L_matching.

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Click to select the

toggle button.

Select the object only.

Select the check box.

Copy 1 (copy1)

In the toolbar, click

and choose .

Select the objects , , and only.

In the window for , locate the section.

In the text field, type -L_transducer.

Mirror 1 (mir1)

In the toolbar, click

and choose .

Select the objects , , and only.

In the window for , locate the section.

In the text field, type -L_transducer/2.

Rotate 1 (rot1)

In the toolbar, click

and choose .

Select the objects , , , , , , and only.

In the window for , locate the section.

From the list, choose .

In the text field, type alpha.

Locate the section. In the text field, type L/2.

Work Plane 1 (wp1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Partition Objects 1 (par1)

In the toolbar, click

and choose .

Click the

button in the toolbar.

Click in the window and then press Ctrl+A to select all objects.

In the window for , locate the section.

From the list, choose .

Delete Entities 1 (del1)

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Domains 1, 3, and 5 only.

On the object , select Domain 1 only.

Union 1 (uni1)

In the toolbar, click

and choose .

Select the objects and only.

Delete Entities 2 (del2)

Right-click and choose .

On the object , select Boundaries 13, 14, 16, and 19 only.

Form Union (fin)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Make sure to select the check box to create an assembly with still matching surface pairs. This will eliminate the use of fallback and thus increase the performance.

Select the check box.

Click

Ignore Edges 1 (ige1)

In the toolbar, click

and choose .

On the object , select Edges 19, 20, 23, and 27 only.

In the window for , click