Available via license: CC BY 4.0

Content may be subject to copyright.

sensors

Communication

A Semi-Empirical Approach to Gas Flow Velocity Measurement

by Means of the Thermal Time-of-Flight Method

Jacek Sobczyk * , Andrzej Rachalski and Waldemar Wodziak

Citation: Sobczyk, J.; Rachalski, A.;

Wodziak, W. A Semi-Empirical

Approach to Gas Flow Velocity

Measurement by Means of the

Thermal Time-of-Flight Method.

Sensors 2021,21, 5679. https://

doi.org/10.3390/s21175679

Academic Editor: Vincenzo Spagnolo

Received: 9 July 2021

Accepted: 20 August 2021

Published: 24 August 2021

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional afﬁl-

iations.

Copyright: © 2021 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

Strata Mechanics Research Institute, Polish Academy of Sciences, Reymonta 27, 30-059 Krakow, Poland;

rachalski@imgpan.pl (A.R.); wodziak@imgpan.pl (W.W.)

*Correspondence: sobczyk@imgpan.pl

Abstract:

This paper presents a method of measuring gas ﬂow velocity based on the thermal time-of-

ﬂight method. The essence of the solution is an analysis of the time shift and the shape of voltage

signals at the transmitter and at a temperature wave detector. The measurements used a probe

composed of a wave transmitter and a detector, both in the form of thin tungsten wires. A rectangular

signal was used at the wave transmitter. The time-of-ﬂight of the wave was determined on the basis

of the time shift of two selected characteristic points of the voltage waveform at the transmitter and

the wave detector. To obtain the correct velocity indication, a correction in the form of a simple power

function was applied. From the measurements performed, the relative uncertainty of the method

was obtained, from approx. 4% of the measured value at an inﬂow velocity of 6.5 cm/s to 1% for an

inﬂow velocity of 50 cm/s and higher.

Keywords: thermal anemometer; thermal time-of-ﬂight (TTOF); thermal wave; low ﬂow velocity

1. Introduction

1.1. Introduction to the Thermal Time-of-Flight Method

Measurements of the gas ﬂow velocity constitute an important branch of metrology.

They are widely used in industrial and laboratory measurements. The need to measure gas

ﬂow velocity appears in the chemical, aviation, and automotive industries, in the control

of ventilation systems. Another issue is the study of air ﬂows in closed spaces (such as

production halls, ofﬁces, storage rooms, etc.). There is a great variety in the methods of

measuring the gas ﬂow velocity.

In simpliﬁed terms, the following types of methods of measuring the gas ﬂow velocity

can be distinguished (with exemplary devices):

•Based on pressure measurement (damming tubes, oriﬁces, nozzles);

•Mechanical (vane anemometers);

•Thermal (hot-wire anemometers);

•Marker (LDA, PIV);

•Ultrasonic.

The thermal time-of-ﬂight method uses a heated volume of ﬂowing ﬂuid as a marker.

Its advantages and disadvantages are best presented against the background of other meth-

ods. The most important advantage is the ability to measure very low ﬂow velocities, even

in the order of mm/s, i.e., in the range where pressure-based and mechanical anemometers

cannot be used. Compared to other marker methods, it does not require the introduction

of a foreign phase (solid or liquid) into the ﬂow, so it is minimally invasive. Another ad-

vantage of this method is its low sensitivity to changes in temperature or the composition

of the ﬂowing gas, i.e., in conditions where the use of hot-wire anemometers is difﬁcult or

even impossible. It requires neither complicated nor expensive apparatus such as LDA and

PIV methods. The main drawback of thermal time-of-ﬂight method is the low bandwidth

determined by the time of the marker’s ﬂight, which distinguishes it negatively from other

Sensors 2021,21, 5679. https://doi.org/10.3390/s21175679 https://www.mdpi.com/journal/sensors

Sensors 2021,21, 5679 2 of 16

methods. The spatial resolution of the velocity measurement, although better than that of

mechanical anemometers, is inferior to that of hot-wire anemometers due to the complexity

of the probe, which must include a wave transmitter and detector. Due to the fact that the

speed is determined on the path between the transmitter and the wave detector, the correct

orientation of the probe in the measured ﬂow is important, therefore it is difﬁcult to use it

in conditions where the velocity direction changes during the measurement.

1.2. Motivation of the Study

Interest in the thermal time-of-ﬂight method and work on its development stem from

the need to measure very low gas ﬂow velocities, in conditions where the gas composition

and temperature are unknown or change during the measurement. The second reason is

the need to conduct such measurements in real conditions (outside the laboratory), where

ﬂows are turbulent.

This paper describes an attempt to modify the thermal wave method so that it is

enough to use only one wave detector to determine the gas ﬂow velocity. The idea is

to determine the ﬂow velocity on the basis of an analysis of the time intervals between

the characteristic points of the voltage waveforms at the temperature transmitter and the

detector. This simple approach, although more empirical than physical by nature, is in fact

a step forward in meeting the needs mentioned above.

2. Materials and Methods

2.1. Basics of the Thermal Time-of-Flight Method

The basis for determining the velocity of a ﬂowing gas stream using the thermal wave

method is the measurement of the temperature wave propagation time in the tested ﬂow

over a known distance. The test probe comprises a wave transmitter, usually in the form of

a ﬁne wire, and one or two wave detectors (resistance thermometers), also in the form of

thin wires, positioned downstream of the transmitter. An important issue is the method of

determining the time-of-ﬂight of the wave. Two interconnected phenomena are involved

in the process of propagation of a temperature wave in ﬂowing gas: Wave drift with the

ﬂow velocity, and thermal diffusion. For a sinusoidal wave, the dependence of the wave

phase velocity on the ﬂow velocity is described by the relationship [1,2]:

vT=vr1+4κ2ω2

v4, (1)

where v

T

is the temperature wave phase velocity (m/s), vis the gas velocity (m/s),

ω

is the

wave frequency (rad/s), and

κ

is the gas temperature diffusivity (m

2

/s). From

Equation (1)

,

it follows that the velocity v

T

of the temperature wave is always greater than the drift

velocity vand depends on the frequency of the wave. If in Formula (1) the fraction is

negligibly small, which can be expressed in terms of the ratio of the Strouhal and Peclet

numbers: Sr

Pe =

κω

v21, (2)

then, the phenomenon of temperature diffusion can be ignored and the phase velocity of

the thermal wave can be assumed to be equal to the ﬂow velocity, i.e., v=v

T

. Additionally,

if the thermal diffusion is negligible, the signal does not change its shape. In this case, it is

enough to measure the time interval between any chosen characteristic point of the signal

waveform registered by the detectors. The above analysis can be applied to a waveform of

any shape—Equation (1) then describes the propagation of the i-th harmonic components

of the signal with frequencies ωi.

2.2. Main Approaches to the Thermal Time-of-Flight Method

In order to generate a thermal wave in the ﬂowing gas, the temperature of the transmit-

ter must change over time. In the thermal wave method, various types of electric current

waveforms heating the wave transmitter are used: It may be a pulse signal [

3

–

6

], a sinu-

Sensors 2021,21, 5679 3 of 16

soidal signal [

1

,

2

,

7

–

10

] or—easiest to implement and most commonly used—a rectangular

signal [

11

,

12

]. Other signals that have been used include a pseudo-stochastic signal [

13

], a

signal composed of the sum of sinusoidal waveforms with various appropriately selected

frequencies and amplitudes [

14

], and a rectangular multifrequency binary sequence (MBS)

signal [

15

]. MBS signals have the property that the major part of the signal power is

concentrated in several harmonic components [16].

To determine the time-of-ﬂight of the wave, the following solutions are used: A

direct method, determination of mutual correlation of signals on the detectors, and cal-

culation of the phase shift of the signals. In the direct method, the time of ﬂight of the

wave is determined on the basis of selected characteristic points in the time waveforms

of the signals recorded by the detectors. The most frequently selected points are the

beginning of the signal rise [

17

] or the maximum signal [

5

,

10

]. The correlation method

consists of determining the time interval between the signals using the cross-correlation

function [

13

,

18

]. The phase shift of the signals is determined using the spectral analy-

sis [1,12,15].

The correct measurement, especially at very low ﬂow velocities, is possible provided

that the inﬂuence of temperature diffusivity and the phenomenon of wave dispersion

on the signal shape is taken into account. A method based on the harmonic analysis of

temperature waveforms exists followed by the determination of phase shifts of individual

harmonics [

12

,

13

,

15

]. In this method, the ﬂow velocity is determined by matching the

measured phase shifts of harmonic components to the theoretically calculated relationship.

This method is insensitive to changes in the temperature diffusivity of the gas, and as a

consequence, enables the measurement of the ﬂow velocity in non-isothermal ﬂows and

ﬂows with a variable composition of the ﬂowing gas. A certain drawback of this method is

the need to use two wave detectors. This leads to complications in the measuring probe

and increases its sensitivity to aerodynamic ﬂow disturbances.

Even if we determine the transit time of the wave correctly and take into account the

temperature diffusion and wave dispersion, a problem arises related to the phenomenon

of the so-called aerodynamic shadow formed behind the transmitter and wave detectors

placed in the ﬂow [

19

,

20

]. This consists of a reduction in the ﬂow velocity behind the

obstacle. In fact, we measure the velocity of the ﬂowing gas vbetween the transmitter

and the wave detector or between the two detectors, while what interests us is the inﬂow

velocity v

∞

. Unfortunately, since the aerodynamic shadow practically coincides with the

temperature trace (in which the detector must be placed), it is impossible to completely

eliminate the inﬂuence of the former on the measurement result by changing the probe

geometry [20].

2.3. Measurement Stand

Measurements were carried out in a closed-circuit TANPOZ wind tunnel (Strata

Mechanics Research Institute of the Polish Academy of Sciences, Krakow, Poland). Its main

features are as follows [21]:

•Measurement chamber dimensions: 0.5 ×0.5 ×1.5 m (W ×H×L);

•Velocity range: 0.01–62.0 m/s;

•Turbulence level: <0.4%;

•Temperature and relative humidity: Controlled.

Therefore, measurements were carried out under controlled conditions at low and

very low inﬂow velocities. The inﬂow velocity was controlled using a Schmidt thermal

anemometer (model SS 20.500), whose measurement range is 0.07–2.50 m/s and measure-

ment uncertainty is 1.5% of the indicated value, but not less than 0.07 m/s. Velocities below

0.07 m/s were estimated using the frequency of the wind tunnel fan inverter with similar

uncertainty.

The thermal wave anemometer probe was placed close to the center of the measure-

ment chamber, at a sufﬁcient distance from the Schmidt anemometer to avoid the mutual

inﬂuence of the two devices.

Sensors 2021,21, 5679 4 of 16

For the generation and detection of temperature waves, a computer-controlled digital

anemometer-thermometer (CCC2002) was used [

22

]. This enables the imposition of various

types of voltage signal on the transmitter and the measurement of voltage on wave detec-

tors, which are resistance thermometers. The transmitter and the two wave detectors were

made of tungsten wire; 8

µ

m in diameter and 6 mm in length (transmitter) and 3

µ

m in

diameter and 3 mm in length (detectors). Thermal signal propagation measurements were

made for inﬂow velocities ranging up to 2.5 m/s. The transmitter operated in a constant

temperature-anemometer (CTA) system with a rectangular input. The overheating ratio

of transmitter wire alternated between 1.0 and 1.8. The wave frequency was 0.25, 0.5 or

1.0 Hz. Each measurement lasted no less than 10 periods of the thermal wave.

Data acquisition from both anemometers was performed with the use of an analog-

to-digital converter (ADC) card (16-bit) from the NI and NI Signal Express software. The

voltage range was set to 0–10 V. The resulting resolution was 0.15 mV. The sampling

rate of the card was set to 10 kHz per channel. In order to keep the level of signal-to-

noise ratio (S/N) as high as possible the measuring system was connected to an “on-line”

uninterruptible power supply (UPS). This type of UPS isolates connected devices from the

mains and its disturbances.

The geometric conﬁguration of the probe is shown in Figure 1b. The distance of the

detector T1 from the transmitter N (denoted as dxNT1) was 3.7 mm.

Sensors 2021, 21, x FOR PEER REVIEW 4 of 16

0.07 m/s were estimated using the frequency of the wind tunnel fan inverter with similar

uncertainty.

The thermal wave anemometer probe was placed close to the center of the measure-

ment chamber, at a sufficient distance from the Schmidt anemometer to avoid the mutual

influence of the two devices.

For the generation and detection of temperature waves, a computer-controlled digi-

tal anemometer-thermometer (CCC2002) was used [22]. This enables the imposition of

various types of voltage signal on the transmitter and the measurement of voltage on

wave detectors, which are resistance thermometers. The transmitter and the two wave

detectors were made of tungsten wire; 8 μm in diameter and 6 mm in length (transmitter)

and 3 μm in diameter and 3 mm in length (detectors). Thermal signal propagation meas-

urements were made for inflow velocities ranging up to 2.5 m/s. The transmitter operated

in a constant temperature-anemometer (CTA) system with a rectangular input. The over-

heating ratio of transmitter wire alternated between 1.0 and 1.8. The wave frequency was

0.25, 0.5 or 1.0 Hz. Each measurement lasted no less than 10 periods of the thermal wave.

Data acquisition from both anemometers was performed with the use of an analog-

to-digital converter (ADC) card (16-bit) from the NI and NI Signal Express software. The

voltage range was set to 0–10 V. The resulting resolution was 0.15 mV. The sampling rate

of the card was set to 10 kHz per channel. In order to keep the level of signal-to-noise ratio

(S/N) as high as possible the measuring system was connected to an “on-line” uninter-

ruptible power supply (UPS). This type of UPS isolates connected devices from the mains

and its disturbances.

The geometric configuration of the probe is shown in Figure 1b. The distance of the

detector T1 from the transmitter N (denoted as dx

NT1

) was 3.7 mm.

(a) (b)

Figure 1. Photograph of the thermal wave anemometer probe (a) and a diagram of its geometric

configuration (b). The wires (invisible on the photograph and shown on the diagram as black lines)

are welded at the tips of the supports.

2.4. Data Analysis and Visualisation

To analyse and visualize data, the OriginLab OriginPro software was used. All the

presented analyses were done by hand in order to track all the phenomena that can be

distinguished in the signals. Localizations of the characteristic points (vide Section 3.2.)

were estimated with the uncertainty ranging from ±0.0012 s for the lowest inflow veloci-

ties to ±0.0001 s for inflow velocities equal to 0.4 m/s and higher. These values resulted

from the S/N ratio of the detector T1 voltage signal and the sampling rate of the ADC card.

Figure 1.

Photograph of the thermal wave anemometer probe (

a

) and a diagram of its geometric

conﬁguration (b). The wires (invisible on the photograph and shown on the diagram as black lines)

are welded at the tips of the supports.

2.4. Data Analysis and Visualisation

To analyse and visualize data, the OriginLab OriginPro software was used. All the

presented analyses were done by hand in order to track all the phenomena that can be

distinguished in the signals. Localizations of the characteristic points (vide Section 3.2)

were estimated with the uncertainty ranging from

±

0.0012 s for the lowest inﬂow velocities

to ±0.0001 s for inﬂow velocities equal to 0.4 m/s and higher. These values resulted from

the S/N ratio of the detector T1 voltage signal and the sampling rate of the ADC card.

2.5. Shapes of Recorded Voltage Waveforms

The voltage signal supplied to the transmitter of the thermal wave N had a shape

similar to a rectangle. Figure 2a shows the course of 11 consecutive pulses, while Figure 2b

shows one selected pulse.

Sensors 2021,21, 5679 5 of 16

Sensors 2021, 21, x FOR PEER REVIEW 5 of 16

2.5. Shapes of Recorded Voltage Waveforms

The voltage signal supplied to the transmitter of the thermal wave N had a shape

similar to a rectangle. Figure 2a shows the course of 11 consecutive pulses, while Figure

2b shows one selected pulse.

(a) (b)

(c)

Figure 2. Voltage signal supplied to the transmitter by the CCC2002 device: course of 11 consecu-

tive pulses (a), one selected pulse (b), and structure of the overdrive (c).

The overdrive visible at the beginning of each pulse is to obtain the steepest possible

edge of the signal heating the transmitter. The structure of this overdrive is shown in Fig-

ure 2c.

Detectors T1 and T2 working in thermometer mode react to temperature changes in

their surroundings. The voltage signal recorded on the resistance bridge of each of the

detectors is proportional to this temperature. The thermal wave propagating from the

transmitter to the detectors quickly weakens over time. Therefore, the signal recorded by

detector T2 is usually much weaker than that recorded by detector T1.

Figure 3 shows the shapes of voltage signals recorded by detector T1 in conditions of

no flow (v∞ = 0 m/s,) for three wave frequencies: f = 0.25 Hz, f = 0.50 Hz, and f = 1.00 Hz.

Figure 2.

Voltage signal supplied to the transmitter by the CCC2002 device: course of 11 consecutive

pulses (a), one selected pulse (b), and structure of the overdrive (c).

The overdrive visible at the beginning of each pulse is to obtain the steepest possible

edge of the signal heating the transmitter. The structure of this overdrive is shown in

Figure 2c.

Detectors T1 and T2 working in thermometer mode react to temperature changes in

their surroundings. The voltage signal recorded on the resistance bridge of each of the

detectors is proportional to this temperature. The thermal wave propagating from the

transmitter to the detectors quickly weakens over time. Therefore, the signal recorded by

detector T2 is usually much weaker than that recorded by detector T1.

Figure 3shows the shapes of voltage signals recorded by detector T1 in conditions of

no ﬂow (v

∞

= 0 m/s,) for three wave frequencies: f= 0.25 Hz, f= 0.50 Hz, and f= 1.00 Hz.

In all three signals presented in Figure 3, one can distinguish the rising edge of the

signal, the peak related to the voltage signal overdrive at the transmitter, the beginning of

a further slow increase of the signal, and ﬁnally fall of the signal. The differences in the

shapes of these three pulses are due to the different lengths of the thermal wave period.

The change in the shape of the pulses towards sawtooth with the increase in the frequency

of the wave is caused by the shorter heating time of the transmitter in the successive

periods of the thermal wave. As a result, the fragment of pulses related to the heating of the

transmitter supports is shortened. Further increasing the frequency of the thermal wave

would shorten this fragment further until the waveform was very close to the sawtooth. At

the same time, the amplitude of this waveform would be further reduced.

Sensors 2021,21, 5679 6 of 16

Sensors 2021, 21, x FOR PEER REVIEW 6 of 16

(a) (b)

(c)

Figure 3. Shapes of voltage signals recorded by detector T1 in conditions of no flow for three differ-

ent frequencies of the transmitter wave: f = 0.25 Hz (a), f = 0.50 Hz (b), and f = 1.00 Hz (c).

In all three signals presented in Figure 3, one can distinguish the rising edge of the

signal, the peak related to the voltage signal overdrive at the transmitter, the beginning of

a further slow increase of the signal, and finally fall of the signal. The differences in the

shapes of these three pulses are due to the different lengths of the thermal wave period.

The change in the shape of the pulses towards sawtooth with the increase in the frequency

of the wave is caused by the shorter heating time of the transmitter in the successive peri-

ods of the thermal wave. As a result, the fragment of pulses related to the heating of the

transmitter supports is shortened. Further increasing the frequency of the thermal wave

would shorten this fragment further until the waveform was very close to the sawtooth.

At the same time, the amplitude of this waveform would be further reduced.

A very narrow peak (pin) visible just before the beginning of the signal growth rec-

orded by detector T1 is formed when the transmitter is overdriven. It is the result of cross-

talk between the channels of the ADC card’s multiplexer or is caused by the transmission

of an electromagnetic wave between the transmitter and detector.

In the case when the inflow velocity is non-zero, the amplitude of the signal on de-

tector T1 increases and its shape changes. This is illustrated in Figure 4. The change of the

signal shape is related to two phenomena. The first is the transport of the heated medium

towards the detectors. This transport shortens the time between the generation of the ther-

mal wave and the moment of its detection, which reduces the level of thermal energy

dissipation and increases the temperature recorded by the detectors. The second phenom-

enon is the increased transfer of thermal energy from the transmitter to the medium due

to cooling. The flow that cools the transmitter also changes the nature of the voltage–cur-

rent relationship, although its actual temperature does not change significantly, since the

transmitter operates in a constant temperature system (CTA).

Figure 3.

Shapes of voltage signals recorded by detector T1 in conditions of no ﬂow for three different

frequencies of the transmitter wave: f= 0.25 Hz (a), f= 0.50 Hz (b), and f= 1.00 Hz (c).

A very narrow peak (pin) visible just before the beginning of the signal growth

recorded by detector T1 is formed when the transmitter is overdriven. It is the result

of crosstalk between the channels of the ADC card’s multiplexer or is caused by the

transmission of an electromagnetic wave between the transmitter and detector.

In the case when the inﬂow velocity is non-zero, the amplitude of the signal on detector

T1 increases and its shape changes. This is illustrated in Figure 4. The change of the signal

shape is related to two phenomena. The ﬁrst is the transport of the heated medium towards

the detectors. This transport shortens the time between the generation of the thermal wave

and the moment of its detection, which reduces the level of thermal energy dissipation

and increases the temperature recorded by the detectors. The second phenomenon is the

increased transfer of thermal energy from the transmitter to the medium due to cooling. The

ﬂow that cools the transmitter also changes the nature of the voltage–current relationship,

although its actual temperature does not change signiﬁcantly, since the transmitter operates

in a constant temperature system (CTA).

Sensors 2021,21, 5679 7 of 16

Sensors 2021, 21, x FOR PEER REVIEW 7 of 16

(a) (b)

(c)

Figure 4. Pulse shapes recorded by detector T1 for three thermal wave frequencies: f = 0.25 Hz (a), f

= 0.50 Hz (b), and f = 1.00 Hz (c), at the inflow velocity v∞ = 0.065 m/s.

The graphs of pulses from the T1 detector presented in Figure 4 were recorded at an

inflow velocity of v∞ = 0.065 m/s. They differ from those shown in Figure 3 in having more

than five times greater amplitude and no initial peak. A further increase in the inflow

velocity causes the shape of the pulses recorded by detector T1 to approach a rectangular

shape.

Figure 5 shows the signals recorded by detector T2 at v∞ = 0.0 m/s. The thermal signal

reaching detector T2 is greatly weakened in the absence of forced convection. This is

mainly due to the distance of detector T2 from the transmitter, but also due to the fact that

the signal encounters an obstacle in its path in the form of detector T1. In Figure 5, the

thermal signal is marked in dark grey. The nature of the graphs indicates that they were

recorded at the limit of ADC card resolution. The amplitude of these signals does not

exceed a few mV at the measuring range of 10 V. The curves created by smoothing the

measured waveforms are marked in green. Signals recorded by detector T2 resemble the

corresponding signals recorded by detector T1, but have a much lower amplitude. For the

thermal wave frequency f = 1 Hz, the signal has a sawtooth shape.

Figure 4.

Pulse shapes recorded by detector T1 for three thermal wave frequencies: f= 0.25 Hz (

a

),

f= 0.50 Hz (b), and f= 1.00 Hz (c), at the inﬂow velocity v∞= 0.065 m/s.

The graphs of pulses from the T1 detector presented in Figure 4were recorded at an

inﬂow velocity of v

∞

= 0.065 m/s. They differ from those shown in Figure 3in having

more than ﬁve times greater amplitude and no initial peak. A further increase in the inﬂow

velocity causes the shape of the pulses recorded by detector T1 to approach a rectangular

shape.

Figure 5shows the signals recorded by detector T2 at v

∞

= 0.0 m/s. The thermal

signal reaching detector T2 is greatly weakened in the absence of forced convection. This

is mainly due to the distance of detector T2 from the transmitter, but also due to the fact

that the signal encounters an obstacle in its path in the form of detector T1. In Figure 5,

the thermal signal is marked in dark grey. The nature of the graphs indicates that they

were recorded at the limit of ADC card resolution. The amplitude of these signals does

not exceed a few mV at the measuring range of 10 V. The curves created by smoothing the

measured waveforms are marked in green. Signals recorded by detector T2 resemble the

corresponding signals recorded by detector T1, but have a much lower amplitude. For the

thermal wave frequency f= 1 Hz, the signal has a sawtooth shape.

Sensors 2021,21, 5679 8 of 16

Sensors 2021, 21, x FOR PEER REVIEW 8 of 16

(a) (b)

(c)

Figure 5. Shapes of voltage signals recorded by detector T2 in conditions of no flow and for three

thermal wave frequencies: f = 0.25 Hz (a), f = 0.50 Hz (b), and f = 1.00 Hz (c).

At an inflow velocity of v∞ = 0.065 m/s, as in the case of detector T1, the amplitude of

the signals recorded by detector T2 increases significantly (Figure 6). In this case, however,

it increases by more than an order of magnitude. The shape of the pulses also changes in

a similar way.

(a) (b)

Figure 5.

Shapes of voltage signals recorded by detector T2 in conditions of no ﬂow and for three

thermal wave frequencies: f= 0.25 Hz (a), f= 0.50 Hz (b), and f= 1.00 Hz (c).

At an inﬂow velocity of v

∞

= 0.065 m/s, as in the case of detector T1, the amplitude of

the signals recorded by detector T2 increases signiﬁcantly (Figure 6). In this case, however,

it increases by more than an order of magnitude. The shape of the pulses also changes in a

similar way.

Sensors 2021, 21, x FOR PEER REVIEW 8 of 16

(a) (b)

(c)

Figure 5. Shapes of voltage signals recorded by detector T2 in conditions of no flow and for three

thermal wave frequencies: f = 0.25 Hz (a), f = 0.50 Hz (b), and f = 1.00 Hz (c).

At an inflow velocity of v∞ = 0.065 m/s, as in the case of detector T1, the amplitude of

the signals recorded by detector T2 increases significantly (Figure 6). In this case, however,

it increases by more than an order of magnitude. The shape of the pulses also changes in

a similar way.

(a) (b)

Figure 6. Cont.

Sensors 2021,21, 5679 9 of 16

Sensors 2021, 21, x FOR PEER REVIEW 9 of 16

(c)

Figure 6. Pulse shapes recorded by detector T2 for three thermal wave frequencies: f = 0.25 Hz (a), f

= 0.50 Hz (b), and f = 1.00 Hz (c), at the inflow velocity v∞ = 0.065 m/s.

An increase in the inflow velocity to v∞ = 1.97 m/s results in a change in the voltage

waveform recorded by detector T2 (Figure 7). It shows a further, though relatively small,

increase in the signal amplitude, a significant change in shape towards rectangular, and a

disturbance occurring in the area of maximum voltage values.

The pulse shapes are adversely affected by aerodynamic disturbances caused by the

presence of the transmitter and detector T1. The influence of aerodynamic disturbances

from the transmitter was visible in Figure 4, where the waveform in the area of maximum

voltage values showed a slight undulation. In the case of detector T2, standing behind two

obstacles in the form of pairs of supports with stretched resistance wires, this disturbance

is more visible [23,24].

Figure 7. The shape of the pulse recorded by detector T2 for the thermal wave frequency f = 1 Hz

and the inflow velocity v∞ = 1.97 m/s.

3. Results and Discussion

3.1. Characteristic Points of Recorded Voltage Signals

The proposed method of velocity measurement consists of determining the time in-

terval between the characteristic points of the voltage waveform on the wave transmitter

and detector. Figure 8 shows the signal from the transmitter. It essentially has only two

characteristic points—marked Na and Nc. The third (Nb) results only from the initial

overdrive.

Figure 6.

Pulse shapes recorded by detector T2 for three thermal wave frequencies: f= 0.25 Hz (

a

),

f= 0.50 Hz (b), and f= 1.00 Hz (c), at the inﬂow velocity v∞= 0.065 m/s.

An increase in the inﬂow velocity to v

∞

= 1.97 m/s results in a change in the voltage

waveform recorded by detector T2 (Figure 7). It shows a further, though relatively small,

increase in the signal amplitude, a signiﬁcant change in shape towards rectangular, and a

disturbance occurring in the area of maximum voltage values.

Sensors 2021, 21, x FOR PEER REVIEW 9 of 16

(c)

Figure 6. Pulse shapes recorded by detector T2 for three thermal wave frequencies: f = 0.25 Hz (a), f

= 0.50 Hz (b), and f = 1.00 Hz (c), at the inflow velocity v∞ = 0.065 m/s.

An increase in the inflow velocity to v∞ = 1.97 m/s results in a change in the voltage

waveform recorded by detector T2 (Figure 7). It shows a further, though relatively small,

increase in the signal amplitude, a significant change in shape towards rectangular, and a

disturbance occurring in the area of maximum voltage values.

The pulse shapes are adversely affected by aerodynamic disturbances caused by the

presence of the transmitter and detector T1. The influence of aerodynamic disturbances

from the transmitter was visible in Figure 4, where the waveform in the area of maximum

voltage values showed a slight undulation. In the case of detector T2, standing behind two

obstacles in the form of pairs of supports with stretched resistance wires, this disturbance

is more visible [23,24].

Figure 7. The shape of the pulse recorded by detector T2 for the thermal wave frequency f = 1 Hz

and the inflow velocity v∞ = 1.97 m/s.

3. Results and Discussion

3.1. Characteristic Points of Recorded Voltage Signals

The proposed method of velocity measurement consists of determining the time in-

terval between the characteristic points of the voltage waveform on the wave transmitter

and detector. Figure 8 shows the signal from the transmitter. It essentially has only two

characteristic points—marked Na and Nc. The third (Nb) results only from the initial

overdrive.

Figure 7.

The shape of the pulse recorded by detector T2 for the thermal wave frequency f= 1 Hz

and the inﬂow velocity v∞= 1.97 m/s.

The pulse shapes are adversely affected by aerodynamic disturbances caused by the

presence of the transmitter and detector T1. The inﬂuence of aerodynamic disturbances

from the transmitter was visible in Figure 4, where the waveform in the area of maximum

voltage values showed a slight undulation. In the case of detector T2, standing behind two

obstacles in the form of pairs of supports with stretched resistance wires, this disturbance

is more visible [23,24].

3. Results and Discussion

3.1. Characteristic Points of Recorded Voltage Signals

The proposed method of velocity measurement consists of determining the time

interval between the characteristic points of the voltage waveform on the wave transmitter

and detector. Figure 8shows the signal from the transmitter. It essentially has only two

characteristic points—marked Na and Nc. The third (Nb) results only from the initial

overdrive.

Sensors 2021,21, 5679 10 of 16

Sensors 2021, 21, x FOR PEER REVIEW 10 of 16

(a) (b)

Figure 8. Selected characteristic points of the voltage signal provided at the transmitter (a). Precise

position of point Nb (b).

In the case of signals recorded by detectors, due to their complex shape, several char-

acteristic points can be distinguished. Due to the change in the shape of the signal as a

function of the inflow velocity, not all points can be determined for each inflow velocity.

Four points (from T1a to T1d), the easiest to determine with the use of automatic methods,

were selected for further analysis. They are shown in Figure 9, taking as an example the

signal from detector T1 for the inflow velocity v∞ = 0.065 m/s and the wave frequency f =

1.0 Hz. The points T1a and T1c correspond to points Na and Nc, respectively, on the trans-

mitter, and points T1b and T1d are the inflection points of the rising and falling signal

portions, respectively.

Figure 9. Selected characteristic points of signals from the detectors, using the example of the signal

from detector T1 for the inflow velocity v∞ = 0.065 m/s and the wave frequency f = 1.0 Hz.

Determination of the moment of the characteristic point occurrence can be done in

many ways. For the binary transmitter signal, simple thresholding may be just enough.

However, detectors signals require more sophisticated handling. One of the simplest ap-

proaches starts with the calculation of derivative of the voltage signal together with the

appropriately chosen smoothing. The derivative amplifies changes in the original wave-

form while “ignoring” the constant values. This enables a much easier determination of

all chosen characteristic points in the detectors signals by hand. Unfortunately, precise

automatic detection of points T1a, T2a, T1c, and T2c may be difficult and in certain cases

even impossible.

On the other hand, the four remaining points are easy to be determined by simply

finding the locations of peaks of the derivatives (Figure 10).

Figure 8.

Selected characteristic points of the voltage signal provided at the transmitter (

a

). Precise

position of point Nb (b).

In the case of signals recorded by detectors, due to their complex shape, several

characteristic points can be distinguished. Due to the change in the shape of the signal as a

function of the inﬂow velocity, not all points can be determined for each inﬂow velocity.

Four points (from T1a to T1d), the easiest to determine with the use of automatic methods,

were selected for further analysis. They are shown in Figure 9, taking as an example the

signal from detector T1 for the inﬂow velocity v

∞

= 0.065 m/s and the wave frequency

f= 1.0 Hz. The points T1a and T1c correspond to points Na and Nc, respectively, on the

transmitter, and points T1b and T1d are the inﬂection points of the rising and falling signal

portions, respectively.

Sensors 2021, 21, x FOR PEER REVIEW 10 of 16

(a) (b)

Figure 8. Selected characteristic points of the voltage signal provided at the transmitter (a). Precise

position of point Nb (b).

In the case of signals recorded by detectors, due to their complex shape, several char-

acteristic points can be distinguished. Due to the change in the shape of the signal as a

function of the inflow velocity, not all points can be determined for each inflow velocity.

Four points (from T1a to T1d), the easiest to determine with the use of automatic methods,

were selected for further analysis. They are shown in Figure 9, taking as an example the

signal from detector T1 for the inflow velocity v∞ = 0.065 m/s and the wave frequency f =

1.0 Hz. The points T1a and T1c correspond to points Na and Nc, respectively, on the trans-

mitter, and points T1b and T1d are the inflection points of the rising and falling signal

portions, respectively.

Figure 9. Selected characteristic points of signals from the detectors, using the example of the signal

from detector T1 for the inflow velocity v∞ = 0.065 m/s and the wave frequency f = 1.0 Hz.

Determination of the moment of the characteristic point occurrence can be done in

many ways. For the binary transmitter signal, simple thresholding may be just enough.

However, detectors signals require more sophisticated handling. One of the simplest ap-

proaches starts with the calculation of derivative of the voltage signal together with the

appropriately chosen smoothing. The derivative amplifies changes in the original wave-

form while “ignoring” the constant values. This enables a much easier determination of

all chosen characteristic points in the detectors signals by hand. Unfortunately, precise

automatic detection of points T1a, T2a, T1c, and T2c may be difficult and in certain cases

even impossible.

On the other hand, the four remaining points are easy to be determined by simply

finding the locations of peaks of the derivatives (Figure 10).

Figure 9.

Selected characteristic points of signals from the detectors, using the example of the signal

from detector T1 for the inﬂow velocity v∞= 0.065 m/s and the wave frequency f= 1.0 Hz.

Determination of the moment of the characteristic point occurrence can be done in

many ways. For the binary transmitter signal, simple thresholding may be just enough.

However, detectors signals require more sophisticated handling. One of the simplest

approaches starts with the calculation of derivative of the voltage signal together with

the appropriately chosen smoothing. The derivative ampliﬁes changes in the original

waveform while “ignoring” the constant values. This enables a much easier determination

of all chosen characteristic points in the detectors signals by hand. Unfortunately, precise

automatic detection of points T1a, T2a, T1c, and T2c may be difﬁcult and in certain cases

even impossible.

On the other hand, the four remaining points are easy to be determined by simply

ﬁnding the locations of peaks of the derivatives (Figure 10).

Sensors 2021,21, 5679 11 of 16

Sensors 2021, 21, x FOR PEER REVIEW 11 of 16

Figure 10. Pulses recorded by detector T1 (a) and T2 (b) at the thermal wave frequency f = 1.00 Hz,

and the inflow velocity v∞ = 0.065 m/s and their derivatives, (c,d), respectively. Dashed lines connect

characteristic points from the corresponding graphs.

Derivatives presented in Figure 10c,d despite the preliminary smoothing are still

“jagged”—exhibit some noise. Further smoothing would make them less noisy, but the

stronger the smoothing, the more the peak positions will be shifted. Therefore, in order to

find positions of peaks accurately it is indispensable to apply approximation with a func-

tion. In this study, the best results (the lowest χ2 values) were obtained using the Gumbel

probability density function (3) [25–27], which was chosen only due to its shape:

𝑈=𝑈+

𝐴

𝑒𝑥𝑝 −𝑒𝑥𝑝− 𝑡−𝑡

𝑤−𝑡−𝑡

𝑤+1, (3)

where U is the voltage (V), U0 is the voltage offset (V), A is the amplitude (V), t is the time

(s), tc is the centre of the peak (s), and w is the peak’s width (s). Fits were made with the

use of nonlinear estimation (Levenberg–Marquardt algorithm [28,29]).

The procedure described here is very precise—the expected accuracy for signals with

sufficient S/N value is better than the time resolution resulting from the ADC sampling

rate.

3.2. Selection of Characteristic Points

Analysis of the time intervals between the characteristic points of the corresponding

voltage pulses from the transmitter and both detectors, with knowledge of the actual dis-

tances between these three elements, enables the preparation of a velocity map (Figure

11). The values determined are the velocities resulting from the reference of time to dis-

tance. The time intervals were determined by pairing the characteristic points on the volt-

age waveforms of the transmitter N and the detectors T1 and T2 within the same period

of the thermal wave. The analysis was performed for two distant periods of the thermal

wave in order to check the repeatability of the results.

Figure 10.

Pulses recorded by detector T1 (

a

) and T2 (

b

) at the thermal wave frequency f= 1.00 Hz,

and the inﬂow velocity v

∞

= 0.065 m/s and their derivatives, (

c

,

d

), respectively. Dashed lines connect

characteristic points from the corresponding graphs.

Derivatives presented in Figure 10c,d despite the preliminary smoothing are still

“jagged”—exhibit some noise. Further smoothing would make them less noisy, but the

stronger the smoothing, the more the peak positions will be shifted. Therefore, in order

to ﬁnd positions of peaks accurately it is indispensable to apply approximation with a

function. In this study, the best results (the lowest

χ2

values) were obtained using the

Gumbel probability density function (3) [25–27], which was chosen only due to its shape:

U=U0+A exp−exp−t−tc

w−t−tc

w+1, (3)

where Uis the voltage (V), U

0

is the voltage offset (V), Ais the amplitude (V), tis the time

(s), t

c

is the centre of the peak (s), and wis the peak’s width (s). Fits were made with the

use of nonlinear estimation (Levenberg–Marquardt algorithm [28,29]).

The procedure described here is very precise—the expected accuracy for signals with

sufﬁcient S/N value is better than the time resolution resulting from the ADC sampling

rate.

3.2. Selection of Characteristic Points

Analysis of the time intervals between the characteristic points of the corresponding

voltage pulses from the transmitter and both detectors, with knowledge of the actual dis-

tances between these three elements, enables the preparation of a velocity map

(Figure 11)

.

The values determined are the velocities resulting from the reference of time to distance.

The time intervals were determined by pairing the characteristic points on the voltage

waveforms of the transmitter N and the detectors T1 and T2 within the same period of the

thermal wave. The analysis was performed for two distant periods of the thermal wave in

order to check the repeatability of the results.

Sensors 2021,21, 5679 12 of 16

Sensors 2021, 21, x FOR PEER REVIEW 12 of 16

(a) (b)

Figure 11. Velocity maps made with the use of selected characteristic points related to the voltage

signals rising (a) and falling (b) edges. For clarity, velocities resulting from pairs of points not cor-

responding to each other were omitted. The same applies to pairs containing point Nb. The numbers

along the horizontal axes indicate whether the data columns correspond to the first or second ther-

mal wave period.

In Figure 11, the horizontal red line marks the inflow velocity v∞ = 0.352 m/s. The

velocity values determined by pairing characteristic points obtained from voltage wave-

forms from the detectors (the pairs T1a-T2a and T1c-T2c) are closest to the set inflow ve-

locity. However, the main limitation of the velocity determination method based on infor-

mation from both detectors is the requirement of steady flow. Even slight changes in the

flow direction lead to a deterioration or even loss of the signal at detector T2.

If all points related to detector T2 are removed from the graphs in Figure 11, only

four series of points will remain. The first two series, referring to the characteristic points

Na-T1a and Nc-T1c, give velocity values that are around twice as high as the correct val-

ues, and fails to ensure repeatability of the results. In the other two series, two points relate

to the pair of characteristic points Na-T1b (in Figure 11 they correspond to values closer

to the inflow velocity), and two relate to the pair of characteristic points Nc-T1d (in Figure

11 they correspond to values further from the inflow velocity). Therefore, the characteris-

tic points Na-T1b were selected for further considerations.

3.3. The Method of Correcting the Velocity Measurement Results

An analysis was made of measurement data obtained in a cycle of 13 measurements

for a velocity range from 0.0 to 2.5 m/s. Only one thermal wave frequency, f = 1.0 Hz, was

taken into account. The calculations of every single velocity value used three adjacent pe-

riods of the thermal wave occurring not earlier than five periods from the moment of ac-

tivation of the transmitter. The obtained results, relating to the characteristic points Na-

T1b, were averaged to obtain one velocity value. The results are shown in Figure 12a and

are marked with green squares. Figure 12b shows a subset of these results limited to ve-

locities below 1 m/s, to better visualise their nature in the range of lowest velocities.

Figure 11.

Velocity maps made with the use of selected characteristic points related to the voltage

signals rising (

a

) and falling (

b

) edges. For clarity, velocities resulting from pairs of points not

corresponding to each other were omitted. The same applies to pairs containing point Nb. The

numbers along the horizontal axes indicate whether the data columns correspond to the ﬁrst or

second thermal wave period.

In Figure 11, the horizontal red line marks the inﬂow velocity v

∞

= 0.352 m/s. The ve-

locity values determined by pairing characteristic points obtained from voltage waveforms

from the detectors (the pairs T1a-T2a and T1c-T2c) are closest to the set inﬂow velocity.

However, the main limitation of the velocity determination method based on information

from both detectors is the requirement of steady ﬂow. Even slight changes in the ﬂow

direction lead to a deterioration or even loss of the signal at detector T2.

If all points related to detector T2 are removed from the graphs in Figure 11, only

four series of points will remain. The ﬁrst two series, referring to the characteristic points

Na-T1a and Nc-T1c, give velocity values that are around twice as high as the correct values,

and fails to ensure repeatability of the results. In the other two series, two points relate to

the pair of characteristic points Na-T1b (in Figure 11 they correspond to values closer to

the inﬂow velocity), and two relate to the pair of characteristic points Nc-T1d (in Figure 11

they correspond to values further from the inﬂow velocity). Therefore, the characteristic

points Na-T1b were selected for further considerations.

3.3. The Method of Correcting the Velocity Measurement Results

An analysis was made of measurement data obtained in a cycle of 13 measurements

for a velocity range from 0.0 to 2.5 m/s. Only one thermal wave frequency, f= 1.0 Hz,

was taken into account. The calculations of every single velocity value used three adjacent

periods of the thermal wave occurring not earlier than ﬁve periods from the moment of

activation of the transmitter. The obtained results, relating to the characteristic points

Na-T1b, were averaged to obtain one velocity value. The results are shown in Figure 12a

and are marked with green squares. Figure 12b shows a subset of these results limited to

velocities below 1 m/s, to better visualise their nature in the range of lowest velocities.

Figure 12 compares the inﬂow velocity measured with the Schmidt anemometer

(horizontal axis) with the velocity determined with the thermal wave anemometer (vertical

axis) using the method described above. The ﬁgure also includes a dashed line that shows

the ideal relationship of the two velocities (ratio 1:1). If the velocity values determined with

the use of the thermal wave anemometer were close to the actual values, the measurement

points (squares) should appear on this straight line.

Sensors 2021,21, 5679 13 of 16

Sensors 2021, 21, x FOR PEER REVIEW 13 of 16

(a) (b)

Figure 12. Velocity measurement results without correction (squares), ideal relationship (dashed

line), measurement results after applying the correction (circles): all the results (a), and a subset of

the results limited to velocities below 1 m/s (b).

Figure 12 compares the inflow velocity measured with the Schmidt anemometer

(horizontal axis) with the velocity determined with the thermal wave anemometer (verti-

cal axis) using the method described above. The figure also includes a dashed line that

shows the ideal relationship of the two velocities (ratio 1:1). If the velocity values deter-

mined with the use of the thermal wave anemometer were close to the actual values, the

measurement points (squares) should appear on this straight line.

Figure 12 shows that the higher the inflow velocity, the greater the difference be-

tween the results of the measurements using the thermal wave anemometer and the actual

values. The presentation of these deviations as a function of the velocity in the wind tunnel

reveals their exponential character (Figure 13a). The imposition of a correction in the form

of a power function (4) of Belehradek type [30] on the measurement results leads to a

significant improvement in the indications of the thermal wave anemometer. Adjustment

of the values of parameters a and n require only a few iterations. Measurement points with

the applied correction are marked in Figures 12 and 13 with circles.

𝑣 =𝑣+𝑎∙𝑣−𝑣 (4)

Designations used:

• vNT1—velocity with applied correction;

• vNT1′—measured velocity;

• vp—velocity of thermal wave propagation in the current medium;

• a, n—parameters of the Belehradek function, which are determined during fitting.

Determined values of the parameters of Equation (4):

• a = 0.409;

• n = 1.650;

• vp = 0.055 m/s.

Figure 12.

Velocity measurement results without correction (squares), ideal relationship (dashed

line), measurement results after applying the correction (circles): all the results (

a

), and a subset of

the results limited to velocities below 1 m/s (b).

Figure 12 shows that the higher the inﬂow velocity, the greater the difference between

the results of the measurements using the thermal wave anemometer and the actual values.

The presentation of these deviations as a function of the velocity in the wind tunnel reveals

their exponential character (Figure 13a). The imposition of a correction in the form of a

power function (4) of Belehradek type [

30

] on the measurement results leads to a signiﬁcant

improvement in the indications of the thermal wave anemometer. Adjustment of the values

of parameters aand nrequire only a few iterations. Measurement points with the applied

correction are marked in Figures 12 and 13 with circles.

vNT1=vNT10+a·vNT10−vpn(4)

Designations used:

•vNT1—velocity with applied correction;

•vNT10—measured velocity;

•vp—velocity of thermal wave propagation in the current medium;

•a,n—parameters of the Belehradek function, which are determined during ﬁtting.

Determined values of the parameters of Equation (4):

•a= 0.409;

•n= 1.650;

•vp= 0.055 m/s.

Figure 13b shows a graph of the relative differences (expressed as a percentage value)

between the velocity values corrected using the power function and the set values in

the wind tunnel. Deviations of the corrected velocity values from the set values amount

to about 1% for velocities above 0.5 m/s, and increase to about 4% with a decrease in

velocity. The increase is systematic, hence it will be possible to apply a simple second-order

correction to reduce the value of the deviations in the range of lowest velocities. The higher

deviations in the range of lowest velocities may also result from the metrological properties

of the instrument used as a reference, the Schmidt anemometer.

Sensors 2021,21, 5679 14 of 16

Sensors 2021, 21, x FOR PEER REVIEW 14 of 16

(a) (b)

Figure 13. Differences in velocity values between measurements and inflow velocity (a) and resid-

uals of measurement with correction (b). The measurement results are marked with squares, and

the measurement results after applying the correction are marked with circles.

Figure 13b shows a graph of the relative differences (expressed as a percentage value)

between the velocity values corrected using the power function and the set values in the

wind tunnel. Deviations of the corrected velocity values from the set values amount to

about 1% for velocities above 0.5 m/s, and increase to about 4% with a decrease in velocity.

The increase is systematic, hence it will be possible to apply a simple second-order correc-

tion to reduce the value of the deviations in the range of lowest velocities. The higher

deviations in the range of lowest velocities may also result from the metrological proper-

ties of the instrument used as a reference, the Schmidt anemometer.

In the graphs in Figures 12 and 13a, one point is visible with behaviour different from

the others. This point was determined in the absence of inflow (v∞ = 0 m/s). It was placed

on these graphs since the related velocity value appears in Equation (4) as the parameter

vp. In practice, the value of this parameter should be determined at the beginning of each

measurement (since it is the velocity of thermal wave propagation in the current medium),

and then the actual measurement should be carried out with discrimination against sig-

nals with such low amplitudes (see Figure 3). Otherwise, measurements at velocities close

to the thermal wave propagation velocity vp and lower will carry greater measurement

uncertainty.

4. Conclusions

In light of the presented results, one can draw the following conclusions:

• The estimated thermal time-of-flight value strongly depends on the chosen charac-

teristic points;

• for the single detector probe the most accurate flow velocity estimations can be

achieved with the use of the following points: The beginning of the transmitter signal

rise (point Na) and the inflection point of the detector signal rising slope (point T1b);

• flow velocity values resulting purely from the thermal time-of-flight estimations

(with the use of Na-T1b pair of characteristic points) vary from the inflow velocity,

and the difference increases with the increasing velocity;

• application of a simple numerical correction transfers the achieved results into an

acceptable region.

The main features of the described measurement method are as follows:

• It does not require the use of information provided by detector T2;

• The algorithm calculating flow velocity values directly from the thermal time-of-

flight estimations requires only two arguments—the positions of two points, Na and

T1b. Moreover, only the position of T1b depends on flow;

Figure 13.

Differences in velocity values between measurements and inﬂow velocity (

a

) and residuals

of measurement with correction (

b

). The measurement results are marked with squares, and the

measurement results after applying the correction are marked with circles.

In the graphs in Figures 12 and 13a, one point is visible with behaviour different from

the others. This point was determined in the absence of inﬂow (v

∞

= 0 m/s). It was placed

on these graphs since the related velocity value appears in Equation (4) as the parameter

v

p

. In practice, the value of this parameter should be determined at the beginning of each

measurement (since it is the velocity of thermal wave propagation in the current medium),

and then the actual measurement should be carried out with discrimination against signals

with such low amplitudes (see Figure 3). Otherwise, measurements at velocities close

to the thermal wave propagation velocity v

p

and lower will carry greater measurement

uncertainty.

4. Conclusions

In light of the presented results, one can draw the following conclusions:

•

The estimated thermal time-of-ﬂight value strongly depends on the chosen character-

istic points;

•

for the single detector probe the most accurate ﬂow velocity estimations can be

achieved with the use of the following points: The beginning of the transmitter signal

rise (point Na) and the inﬂection point of the detector signal rising slope (point T1b);

•

ﬂow velocity values resulting purely from the thermal time-of-ﬂight estimations (with

the use of Na-T1b pair of characteristic points) vary from the inﬂow velocity, and the

difference increases with the increasing velocity;

•

application of a simple numerical correction transfers the achieved results into an

acceptable region.

The main features of the described measurement method are as follows:

•It does not require the use of information provided by detector T2;

•

The algorithm calculating ﬂow velocity values directly from the thermal time-of-ﬂight

estimations requires only two arguments—the positions of two points, Na and T1b.

Moreover, only the position of T1b depends on ﬂow;

•

Of all the characteristic points of the signal from detector T1, T1b is the easiest to

determine with the use of automatic methods. Moreover, its determination is the most

unambiguous and accurate.

Further research will be carried out for probes with a different spatial arrangement,

i.e., with various mutual positions of the transmitter and detector. Work will also aim to

improve the accuracy of the method.

Author Contributions:

Conceptualization, J.S.; methodology, J.S. and A.R.; validation, J.S., A.R. and

W.W.; formal analysis, J.S.; investigation, J.S. and W.W.; writing—original draft preparation, J.S. and

Sensors 2021,21, 5679 15 of 16

A.R.; writing—review and editing, J.S., A.R. and W.W.; visualization, J.S.; supervision, J.S. All authors

have read and agreed to the published version of the manuscript.

Funding:

This research was completed in 2020, as part of a statutory work carried out at the Strata

Mechanics Research Institute of the Polish Academy of Sciences in Krakow (Poland), funded by the

Ministry of Science and Higher Education.

Data Availability Statement:

The data presented in this study are available on request from the

corresponding author.

Acknowledgments:

The authors express their gratitude to Marek Gawor for the valuable guidance

and help in preparatory work.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1.

Kiełbasa, J.; Piwowarczyk, J.; Rysz, J.; Smolarski, A.Z.; Stasicki, B. Heat Waves in Flow Metrology. In Proceedings of the

Conference on FLOMEKO 1978 IMEKO Flow Measurement of Fluid, Groningen, The Netherlands, 11–15 September 1978;

pp. 403–407.

2. Rachalski, A. High-Precision Anemometer with Thermal Wave. Rev. Sci. Instrum. 2006,77, 095107. [CrossRef]

3.

Bradbury, L.J.S.; Castro, I.P. A Pulsed-Wire Technique for Velocity Measurements in Highly Turbulent Flows. J. Fluid Mech.

1971

,

49, 657–691. [CrossRef]

4.

Tombach, I.H. An Evaluation of the Heat Pulse Anemometer for Velocity Measurement in Inhomogeneous Turbulent Flow. Rev.

Sci. Instrum. 1973,44, 141–148. [CrossRef]

5.

Avirav, Y.; Guterman, H.; Ben-Yaakov, S. Implementation of Digital Signal Processing Techniques in the Design of Thermal Pulse

Flowmeters. IEEE Trans. Instrum. Meas. 1990,39, 761–766. [CrossRef]

6.

Mathioulakis, E.; Poloniecki, J.G. A Pulsed-Wire Technique for Velocity and Temperature Measurements in Natural Convection

Flows. Exp. Fluids 1994,18, 82–86. [CrossRef]

7.

Kovasznay, L.S.G. Hot-Wire Investigation of the Wake behind Cylinders at Low Reynolds Numbers. Proc. R. Soc. Lond. Ser. A

Math. Phys. Sci. 1949,198, 174–190. [CrossRef]

8. Walker, R.E.; Westenberg, A.A. Absolute Low Speed Anemometer. Rev. Sci. Instrum. 1956,27, 844–848. [CrossRef]

9. Kiełbasa, J. Measurments of Steady Flow Velocity Using the Termal Wave Method. Arch. Min. Sci. 2005,50, 191–208.

10.

Byon, C. Numerical and Analytic Study on the Time-of-Flight Thermal Flow Sensor. Int. J. Heat Mass Transf.

2015

,89, 454–459.

[CrossRef]

11.

Biernacki, Z. A System of Wave Thermoanemometer with a Thermoresistive Sensor. Sens. Actuators A Phys.

1998

,70, 219–224.

[CrossRef]

12.

Rachalski, A.; Poleszczyk, E.; Zi˛eba, M. Use of the Thermal Wave Method for Measuring the Flow Velocity of Air and Carbon

Dioxide Mixture. Measurement 2017,95, 210–215. [CrossRef]

13.

Berthet, H.; Jundt, J.; Durivault, J.; Mercier, B.; Angelescu, D. Time-of-Flight Thermal Flowrate Sensor for Lab-on-Chip Applica-

tions. Lab Chip 2011,11, 215–223. [CrossRef] [PubMed]

14.

Rachalski, A. Absolute Measurement of Low Gas Flow by Means of the Spectral Analysis of the Thermal Wave. Rev. Sci. Instrum.

2013,84, 025105. [CrossRef]

15.

Bujalski, M.; Rachalski, A.; Lig˛eza, P.; Poleszczyk, E. The Use of Multifrequency Binary Sequences MBS Signal in the Anemometer

with Thermal Wave. In Proceedings of the 10th International Conference on Measurement, Smolenice, Slovakia, 10–11 February

2015; pp. 297–300.

16.

Henderson, I.A.; Mcghee, J. Compact Symmetrical Binary Codes for System Identiﬁcation. Math. Comput. Model. Int. J.

1990

,14,

213–218. [CrossRef]

17.

Mosse, C.A.; Roberts, S.P. Microprocessor-Based Time-of-Flight Respirometer. Med. Biol. Eng. Comput.

1987

,25, 34–40. [CrossRef]

18.

Engelien, E.; Ecin, O.; Viga, R.; Hosticka, B.J.; Grabmaier, A. Calibration-Free Volume Flow Measurement Principle Based on

Thermal Time-of-Flight (TToF). Procedia Eng. 2011,25, 765–768. [CrossRef]

19.

Ong, L.; Wallace, J. The Velocity Field of the Turbulent Very near Wake of a Circular Cylinder. Exp. Fluids

1996

,20, 441–453.

[CrossRef]

20. Kiełbasa, J. Measurement of Aerodynamic and Thermal Footprints. Arch. Min. Sci. 1999,44, 71–84. (In Polish)

21.

Bujalski, M.; Gawor, M.; Sobczyk, J. Closed-Circuit Wind Tunnel with Air Temperature and Humidity Stabilization, Adapted for

Measurements by Means of Optical Methods; Prace Instytutu Mechaniki Górotworu PAN: Krakow, Poland, 2013. (In Polish)

22. Lige¸za, P. Four-Point Non-Bridge Constant-Temperature Anemometer Circuit. Exp. Fluids 2000,29, 505–507. [CrossRef]

23.

Sobczyk, J. Experimental Study of the Flow Field Disturbance in the Vicinity of Single Sensor Hot-Wire Anemometer. In EPJ Web

of Conferences; EDP Sciences: Les Ulis, France, 2018. [CrossRef]

24.

Gawor, M.; Sobczyk, J.; Wodziak, W.; Lig˛eza, P.; Rachalski, A.; Jamróz, P.; Socha, K.; Palacz, J. Distribution of Flow Velocity in the

Vicinity of the Thermal Wave Anemometer Probe; Prace Instytutu Mechaniki Górotworu PAN: Krakow, Poland, 2019. (In Polish)

25. Gumbel, E.J. The Return Period of Flood Flows. Ann. Math. Stat. 1941,12, 163–190. [CrossRef]

Sensors 2021,21, 5679 16 of 16

26. Gumbel, E.J. Les Valeurs Extrêmes Des Distributions Statistiques. Ann. Inst. Henri Poincaré1935,5, 115–158.

27.

Help Online—Origin Help—Extreme. Available online: https://www.originlab.com/doc/Origin-Help/Extreme-FitFunc (ac-

cessed on 14 August 2021).

28.

Levenberg, K. A Method for the Solution of Certain Non-Linear Problems in Least Squares. Q. Appl. Math.

1944

,2, 164–168.

[CrossRef]

29.

Marquardt, D.W. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math.

1963

,11, 431–441.

[CrossRef]

30.

Help Online—Origin Help—Belehradek. Available online: https://www.originlab.com/doc/Origin-Help/Belehradek-FitFunc

(accessed on 28 June 2021).