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EMRP ENG06 - D3.4: An ultrasound based temperature measurement method

Theory, calibration method, equipment, uncertainty

Version Date Author Description

1 17 October 2011 GJK Initial document

2 2 February 2012 GJK First complete draft version

3 21 February 2012 GJK Inclusion of several comments of PK and MvdB

4 29 February 2012 GJK Reordering of some sections

Gertjan Kok VSL Thijsseweg 11 2629 JA Delft the Netherlands

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1. Introduction ........................................................................................................................................ 4

2. Background ......................................................................................................................................... 4

3. Theory of ultrasound based temperature measurement ................................................................... 5

3.1. General idea ................................................................................................................................. 5

3.2. Ultrasound path average ............................................................................................................. 7

3.3. Representativeness of the path(s) ............................................................................................. 11

3.4. Enthalpic temperature/ energy flow ......................................................................................... 13

3.5. Effect of composition on the speed of sound ............................................................................ 18

3.6. Effect of flow speed ................................................................................................................... 19

3.7. Conclusions of theoretical background part .............................................................................. 20

4. Calibration method for ultrasound based temperature measurement ........................................... 21

4.1. Complete calibration by manufacturer ...................................................................................... 21

4.2. Calibration by user ..................................................................................................................... 22

4.2.1. Ultrasonic path lengths ....................................................................................................... 22

4.2.2. Temperature expansion coefficient .................................................................................... 23

4.2.3. Pressure expansion coefficient ........................................................................................... 23

4.2.4. Transmitter delays .............................................................................................................. 24

4.2.5. Speed of sound reference data ........................................................................................... 25

4.2.5.1. Off-line reference data................................................................................................. 25

4.2.5.2. Continuous on-line calibration reference data ............................................................ 28

4.2.6. Calibration of other auxiliary equipment ............................................................................ 29

4.3. Conclusion about calibration method........................................................................................ 30

5. Required equipment ......................................................................................................................... 31

5.1. Ultrasound sensor for temperature measurement ................................................................... 31

5.2. Auxiliary equipment ................................................................................................................... 32

5.2.1. Pressure sensor ................................................................................................................... 32

5.2.2. Composition determination ................................................................................................ 32

5.2.4. Equipment to be used for calibration ................................................................................. 32

5.2.5. Equipment to be used for validation of the method .......................................................... 34

5.2.6. Required equipment for speed of sound reference data ................................................... 38

5.2.6.1. Required equipment for off-line laboratory reference data ....................................... 38

5.2.6.2. Required equipment for continuous on-line calibration ............................................. 38

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5.3. Conclusion concerning required equipment ............................................................................. 38

6. Overall theoretical uncertainty budget ............................................................................................ 39

7. Experimental implementation of the different calibration methods ............................................... 40

7.0. Initial wet calibration ................................................................................................................. 40

7.1. Use of a real-time calibrator ...................................................................................................... 40

7.2. Use of external reference data .................................................................................................. 41

7.3. Creation of a traceable reference database .............................................................................. 41

7.4. Conclusion concerning experiments with calibration methods ................................................ 42

8. Overall conclusion ............................................................................................................................. 42

References ............................................................................................................................................ 42

Appendix A: Measured temperature per path ..................................................................................... 43

Appendix B: Full uncertainty budget .................................................................................................... 44

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1. Introduction This report has been made in the scope of the EU funded EMRP research project ENG06: ‘Metrology

for improved power plant efficiency’. This report is in fulfillment of the deliverable D3.4: ‘Report on

the development of a temperature calibration method and equipment for an ultrasound based

temperature method’.

This report gives theoretical background information concerning a method to measure temperature

with ultrasound in pipe flow. The equipment needed will be described together with a way how it

can be calibrated. In report D3.6 the quality of this measurement principle is described based on

experimental tests, whereas D3.5 deals with a multipoint reference sensor. In the scope of

deliverable D5.4 a webpage-handbook for performing ultrasound temperature measurements in

pipes will be made and in D5.5 a webpage linking to traceable speed of sound/ temperature data will

be created.

2. Background The background of this research topic is the question how the mean temperature of water flowing in

pipe lines where the water has a non-homogeneous temperature distribution can be best measured.

This problem is especially of interest in the hot leg of the primary circuit of nuclear power plants of

pressurized water reactors (PWR). In these pipe lines hot water with temperatures ranging from 310

to 330 °C flows at high speed with local temperature differences up to 20 °C (estimate based on

simulations performed by a nuclear power plant owner). The current temperature measurement

technique is estimated to have an uncertainty of at best 1 °C. In this research project it will be

investigated if the current measurement technique can be improved by means of using an ultrasonic

technique. Another advantage of the ultrasound based measurement would be the high speed with

which the measurement result is acquired and the fast reaction time to process fluid temperature

changes.

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3. Theory of ultrasound based temperature measurement In this chapter the general ideas behind the method are described. Only wetted ultrasonic sensors

are considered. For a clamp-on version the travel distance of the beam through the wall has to be

considered as well, adding some additional travel time and uncertainty. The trip of the beam

through the fluid itself follows the same logic as for wetted sensors. The application under

consideration will be pipe flows with water. All data in the report will be for water. However the

method can in principal be used for any fluid. The thermodynamic reference data in this report have

been obtained with the help of the NIST Refprop program (see reference [1]) and calculations have

been performed using Microsoft Excel.

3.1. General idea The speed of sound of a fluid changes with temperature. Therefore one can use speed of sound

measurement to derive the fluid temperature. As the speed of sound also depends on fluid pressure

and composition, these parameters need to be known as well. In case of homogeneous temperature

distribution in the pipe the temperature can accurately be measured by means of the speed of

sound (0.02 °C uncertainty may be possible). The idea now is to use this method also for

inhomogeneous temperature distributions. The ultrasound beam will travel through different parts

of the fluid having different temperatures. The beam will thereby consecutively travel at the speed

corresponding to the local fluid temperature. The total travel time is the sum of the travel times in

the different parts of the fluid and the resulting measured speed of sound over the whole traversed

path will be a somehow averaged speed of sound. The idea is to use this average speed of sound to

calculate an average pipe fluid temperature. In Figure 1 the temperature dependency and the

pressure dependency of the speed of sound of pure water are shown. The phase transition from

liquid to vapor is clearly visible. The temperature dependency is much stronger than the pressure

dependency. In Table 1 a few numerical values of the derivatives of c to T and P are shown. The last

column is a measure for the error in calculated temperature in °C by the ultrasound method when

the measured pressure is incorrect by 1 bar.

Figure 1: Speed of sound as function of temperature and as function of pressure for pure water. The phase transition from liquid to vapor is clearly visible.

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T [°C] P [barg] dc/ dT [m/s/°C] dc/ dP [m/s/bar] (dc/dP) · (dT/dc)

(=dT/dP) [°C /bar]

25 0 2.672 0.168 0.063

25 10 2.676 0.168 0.063

300 100 -6.373 0.919 -0.144

300 150 -5.702 0.807 -0.141

Table 1: Some numerical values of the derivatives of the speed of sound c of water to temperature T and pressure P.

The maximal allowable uncertainty for the measurement of speed of sound and pressure in order to

get a desired uncertainty in temperature (provided the fluid has homogeneous temperature) is

shown in Table 2. The maximal uncertainty is actually divided by 2 in order to allow for different

uncertainty contributions and not just one single maximal one.

Desired uncertainty in

T, u(T)

Nominal

temperature and

pressure

Maximal uncertainty in

c divided by 2, u(c)

Maximal uncertainty in

P divided by 2, u(P)

0.1 °C 20 °C/ 0 barg 0.13 m/s 0.8 bar

0.1 °C 300 °C/ 150 barg 0.3 m/s 0.3 bar

0.01 °C 20 °C/ 0 barg 0.013 m/s 0.08 bar

0.01 °C 300 °C/ 150 barg 0.03 m/s 0.03 bar

Table 2: Maximal uncertainty in speed of sound measurement and pressure to be able to attain a desired uncertainty in the temperature measurement.

The uncertainty aim of 0.1 °C should be possible at both low and high temperatures in view of the

required measurement uncertainty of speed of sound and pressure. At low temperatures 0.01 °C

should also be possible. At high temperatures it might be difficult to measure 150 barg with 0.03 bar

uncertainty (0.02%). Around the temperature of T0 = 70 °C the speed of sound attains a maximum

and the variation of the speed of sound with temperature is relatively small. Therefore it is more

difficult to get accurate values for the temperature in this region. Furthermore, in all cases it should

be known from other sources if the temperature is above or below T0.

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3.2. Ultrasound path average In this section it will be calculated what the ultrasonic temperature meter theoretically measures for

several inhomogeneous temperature profiles. For this purpose, consider the cross section of a pipe

with coordinate origin in the center, horizontal x-axis and vertical y-axis. Let a vertical ultrasound

path be parameterized by y ranging from R to -R with R the pipe radius and L = 2·R the pipe

diameter. So the beam travels from a point at the top of a horizontal pipe (inside of pipe) to a point

diametrically at the opposite at the bottom of the pipe (inside). The temperature at spot (x,y) equals

T(x,y) and the speed of sound will equal c = c(T(x,y), P) with pressure P. The ultrasound beam transit

time for the vertical path through the pipe center is now given by τ = ∫-RR 1/c(T(0,y), P) dy and the

measured average speed of sound will be cmeas = L/τ, which can be converted to a temperature when

the pressure is known. In Table 4 the difference between the true (arithmetic) average temperature

on a line (Tline,vol = (1/L)·∫-RR T(0,y) dy) and the ‘ultrasound temperature’ Tus (theoretical temperature

that will be measured by means of the ultrasound method) is shown for a few temperature profiles

described in Table 3. The ultrasound path under consideration goes vertically from top to bottom

through the center of the pipe.

Nr

. Temperature profile name Temperature profile mathematical description

Visualizati

on of

profile

1. Homogeneous, T = 25 °C T(x, y) = 25 °C, P = 2 barg

2. Homogeneous, T = 300 °C T(x, y) = 300 °C, P = 150 barg

3. Homogeneous with boundary layer

of 1% of L, Tin = 30 °C, Tout = 20 °C

T(x, y) = 30 °C for 0 < √(x2+y

2) < 0.98 R and T(x, y) = 20 °C

for 0.98 R < √(x2+y

2) < R. P = 2 barg.

4.

Homogeneous with boundary layer

of 0.1% of L, Tin = 300 °C, Tout = 250

°C

T(x, y) = 300 °C for 0 < √(x2+y

2) < 0.998 R and T(x, y) =

250 °C for 0.998 R < √(x2+y

2) < R. P = 150 barg.

5. Two homogeneous circular regions,

Tin = 25 °C, Tout = 24 °C

T(x, y) = 25 °C for 0 < x2 + y

2 < (R/2)

2, T(x, y) = 24 °C for

(R/2)2 < x

2 + y

2 < R

2. P = 2 barg.


Tin = 301 °C, Tout = 300 °C

T(x, y) = 301 °C for 0 < x2 + y

2 < (R/2)

2, T(x, y) = 300 °C for

(R/2)2 < x

2 + y

2 < R

2. P = 150 barg.


Tin = 28 °C, Tout = 22 °C

T(x, y) = 28 °C for 0 < x2 + y

2 < (R/2)

2, T(x, y) = 22 °C for

(R/2)2 < x

2 + y

2 < R

2. P = 2 barg.

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Nr

. Temperature profile name Temperature profile mathematical description

Visualizati

on of

profile


Tin = 303 °C, Tout = 297 °C

T(x, y) = 303 °C for 0 < x2 + y

2 < (R/2)

2, T(x, y) = 297 °C for

(R/2)2 < x

2 + y

2 < R

2. P = 150 barg.

9. Two equal homogeneous regions, Tup

= 25 °C, Tdown = 24 °C

T(x, y) = 24 °C for y < 0 and T(x, y) = 25 °C for y > 0. P = 2

barg.


= 301 °C, Tdown = 300 °C

T(x, y) = 300 °C for y < 0 and T(x, y) = 301 °C for y > 0. P =

150 barg.


= 28 °C, Tdown = 22 °C

T(x, y) = 22 °C for y < 0 and T(x, y) = 28 °C y > 0. P = 2

barg.


= 303 °C, Tdown = 297 °C

T(x) = 297 °C for y < 0 and T(x, y) = 303 °C for y > 0. P =

150 barg.


= 30 °C, Tdown = 5 °C

T(x, y) = 5 °C for y < 0 and T(x, y) = 30 °C for y > 0. P = 2

barg.


= 330 °C, Tdown = 300 °C

T(x, y) = 300 °C for y < 0 and T(x, y) = 330 °C for y > 0. P =

150 barg.

15. Linear profile from 22 °C to 28 °C T(x, y) = 25 + 3 y/R. P = 2 barg.

16. Linear profile from 297 °C to 303 °C T(x) = 300 + 3 y/R. P = 150 barg.

Table 3: Some theoretic temperature profiles. The temperature at the common boundary of two regions is assumed to be equal to the average of the temperatures of the two adjacent regions.

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Temperature profile True (arithmetic)

mean temperature

of path, Tline,vol [°C]

Ultrasound

measured path

temperature, Tus

[°C]

Difference Tus

- Tline,vol [°C]

Homogeneous, T = 25 °C 25.000 25.000 0.000

Homogeneous, T = 300 °C 300.000 300.000 0.000

Homogeneous with boundary layer of 1% of L, Tin = 30 °C, Tout

= 20 °C

29.800 29.765 -0.035

Homogeneous with boundary layer of 0.1% of L, Tin = 300 °C,

Tout = 250 °C

299.900 299.933 0.033

Two homogeneous circular regions, Tin = 25 °C, Tout = 24 °C 24.500 24.496 -0.004

Two homogeneous circular regions, Tin = 301 °C, Tout = 300 °C 300.500 300.503 0.003

Two homogeneous circular regions, Tin = 28 °C, Tout = 22 °C 25.000 24.856 -0.144

Two homogeneous circular regions, Tin = 303 °C, Tout = 297 °C 300.000 300.093 0.093

Two equal homogeneous regions, Tup = 25 °C, Tdown = 24 °C 24.500 24.496 -0.004

Two equal homogeneous regions, Tup = 301 °C, Tdown = 300 °C 300.500 300.503 0.003





Linear profile from 22 °C to 28 °C 25.000 24.952 -0.048

Linear profile from 297 °C to 303 °C 300.000 300.031 0.031

Table 4: Theoretically measured ultrasound path average compared with true path average for various temperature profiles. The homogeneous circular temperature distributions of Table 3 give the same results as the case of two homogeneous sections with the same temperature difference as the distance travelled by the beam in each temperature area is equal.

The last column of Table 4 shows that the difference between arithmetic mean temperature and

temperature measured by means of ultrasound can be considerable when big temperature

differences exist in the pipe. For temperature differences up to 6 °C the difference may be up to 0.14

°C. In case of differences of 25 °C the ‘ultrasound temperature’ may deviate more than 2 °C from the

arithmetic mean temperature.

To clarify Table 4 the graph in Figure 2 shows the measurement deviation of the ultrasound based

temperature measurement in case that the ultrasound beam travels half of the path in water of T1 =

20 °C and the other half of the path in water of T2 = 20 to 35 °C (pressure 2 barg). As the speed of

sound at lower temperatures is lower in this temperature domain, the beam will spend more time in

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the colder part and the deviation is negative. The same has been done for temperatures T1 = 300 °C

and T2 = 300 to 330 °C at a pressure of 150 barg, see Figure 3. As the speed of sound in this

temperature domain decreases with increasing temperature, the deviation is positive. The reference

temperature in both cases is the average of the temperatures of both sections, called Tvol

(‘volumetric temperature average’).

Figure 2: Deviation of ultrasound based temperature measurement in case of beam traveling half of the path length through water with T1 = 20 °C and the other half through water with temperature T2. The pressure is 2 barg.

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Figure 3: Deviation of ultrasound based temperature measurement in case of beam traveling half of the path length through water with T1 = 300 °C and the other half through water with temperature T2. The pressure is 150 barg.

It is seen that in the extreme case of a sharp 10 °C temperature difference in a pipe the ultrasound

based temperature method may be off by -0.4 °C at low temperatures and by 0.3 °C at high

temperatures. With increasing temperature differences this deviations increase rapidly. An

ultrasound temperature meter with multiple paths and smart software could possibly decrease this

deviation. It has to be noted that the studied profile is a very extreme case.

3.3. Representativeness of the path(s) The temperature that an ultrasound beam measures is an estimate for the average temperature

along the beam path. In general this is a better estimate of the average cross section temperature

than a spot temperature measured by a single Pt100 temperature sensor. However, it might still not

be completely representative for the entire cross section. In Figure 4 the difference between the

true section averaged temperature and the ‘ultrasound temperature’ is shown for different path

configurations (see Table 5) and temperature profiles (see Table 3). The numerical data used for this

figure are shown in Table 20 in appendix A. No (boundary) effects like reflection, refraction and

absorption at the interface of warm and cold water have been modeled. The refraction effect might

change the travel path and its length of the ultrasound beam, whereas absorption and reflection

would only cause an attenuation of the measurement signal.

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Path configuration name Path configuration description Visualization

One vertical path (yielding

temperature Tus,v1)

One path from the top of the

pipe to the bottom of the pipe,

perpendicular on pipe axis.

Three paths through center

(yielding temperature Tus,c3)

Three paths perpendicular on

pipe axis and through pipe

center. One from top to

bottom (0° to 180°), one from

50° to 230° and one from -50°

to 130°. All paths have weight 1

in the temperature calculation.

Three horizontal paths (yielding

temperature Tus,h3)

Three horizontal paths

perpendicular to pipe axis. One

at 28% of (vertical) pipe

diameter, one at 50% and one

at 72%. All paths have weight 1


Three vertical paths (yielding

temperature Tus,v3)

Three vertical paths

perpendicular to pipe axis. One

at 28% of (horizontal) pipe

diameter, one at 50% and one

at 72%. All paths have weight 1


Table 5: Different path configurations for ultrasonic temperature sensor. All paths are perpendicular on the pipe axis.

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Figure 4: Deviation of ultrasound based mean temperature for different path configurations and several temperature profiles. The reference temperature Tref is the volumetric mean section temperature Tsec,vol.

From Figure 4 it can be seen from the boundary layer and circular section temperature profiles (nr. 3

to 8) that all ultrasound path configurations miss the relative large section area that is in the outer

ring and therefore the temperature estimate is somewhat too high. For energy flow measurements

in practice this effect may however be compensated by a flow speed profile in which the speed close

to the wall is lower than in the center. Thus the larger boundary area travels at lower speed than the

smaller center area.

Furthermore, the points in the figure for profiles 13 and 14 seem to suggest that a design with three

horizontal layers gives the best results. However this is only true in this particular case, because of

the perfectly symmetrical temperature patterns and the three horizontal beams having the same

perfect symmetry. Further research with irregular flow patterns is needed. For this purpose a

software tool is planned which may be available on the Webpage-handbook of D5.4.

3.4. Enthalpic temperature/ energy flow Until now the focus was on the arithmetic path averaged and (arithmetic) section averaged

temperatures. However what really matters in the application in power plants is the energy or

enthalpy flow which is given by the enthalpy flow integral over the cross section A of the pipe

(section coordinates x and y, dA = dx·dy): H = ∫A v(x,y) · ρ(T(x,y), P) · h(T(x,y), P) · dA. Remark that the

local enthalpy is weighted with local (mass) density ρ(T(x,y), P) and local flow speed v(x,y)

(orthogonal to cross section dA). The enthalpic temperature Tflow,enth is now defined as the

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temperature that a fluid of homogeneous temperature of the same mass flow rate would have if it

transported the same amount of enthalpy. In formula form this yields:

Tflow,enth = h-1 P [ (∫A v(x,y) · ρ(T(x,y), P) · h(T(x,y), P) · dA) / (∫A v(x,y) · ρ(T(x,y), P) · dA) ],

where h-1 P[ . ] denotes the inverse enthalpy function as function of temperature at (constant)

pressure P. As hP(T) is almost linear at low temperatures, this function can for practical applications

at low temperatures be simplified to:

Tflow,enth ≈ Tflow,mass = (∫A v(x,y) · ρ(T(x,y), P) · T(x,y) · dA) / (∫A v(x,y) · ρ(T(x,y), P) · dA).

This is the mass flow averaged section temperature. The enthalpic temperature will be of interest in

the experimental testing of the method, which will be described in report D3.6.

In Table 7 the differences between the various definitions of average temperature are shown.

Hereby the notation and definitions of Table 6 are being used. The calculation of the enthalpic

temperature Tflow,enth also needs a definition of a flow pattern. When a homogeneous bulk flow

pattern is assumed (i.e. v(x,y) = constant) however, the flow speed v(x,y) drops out from the defining

equation for Tflow,enth. The symbol c-1P denotes the inverse function of speed of sound in order to

calculate temperature from a given speed of sound at a pressure P. To simplify notation the variable

P and subscript P for pressure will be dropped from ρ(T,P), h(T,P), c-1P and h-1

P.

Symbol Description Formula

Tus Temperature that a (wetted) ultrasound sensor

will measure on a path from top to bottom (or

bottom to top) of a pipe.

Tus = c-1 [L/(∫-RR 1/c(T(0,y)) dy)]

Tline,vol Arithmetic mean temperature on a line (e.g.

ultrasound path) from top to bottom of a pipe.

As there is no weighing with mass, it’s volume

based. The average of a big number of

temperature sensors on a row would tend

towards this value.

Tline,vol = (1/L)·∫-RR T(0, y) dy

Tline, mass Arithmetic mean temperature on a line (e.g.

ultrasound path) weighted with density on a

line from top to bottom of a pipe.

Tline, mass = (∫-RR ρ(T(0,y))·T(0,y) dy) /

∫-RR ρ(T(0, y)) dy

Tline, enth Arithmetic mean temperature on a line (e.g.

ultrasound path) using enthalpy on a line from

to bottom of a pipe. Physical true mean

temperature after mixing the fluid on the line.

Tline, enth =h-1[ (∫-RR ρ(T(0, y))· h(T(0, y))

dy) / ∫-RR ρ(T(0, y)) dy]

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Symbol Description Formula

Tsec,vol Arithmetic mean temperature over entire

circular cross section of a pipe. One could also

call it the volume mean temperature. The

average of a big number of temperature

sensors spread over the cross section would

tend towards this value.

Tsec,vol = (1/A)·∫A T(x, y) dA

Tsec, mass Arithmetic mean temperature over entire

circular cross section of a pipe weighted with

density.

Tsec, mass = (∫A ρ(T(x, y))·T(x,y) dA) /

∫A ρ(T(x, y)) dA

Tsec, enth Arithmetic mean temperature over entire

circular cross section of a pipe using enthalpy.

Physical true mean temperature after mixing

the fluid in the section.

Tsec, enth =h-1[ (∫A ρ(T(x, y))·h(T(x, y)) dA)

/ ∫A ρ(T(x, y)) dA]

Tflow,vol Arithmetic mean temperature over entire


flow speed.

Tflow,vol = (∫A v(x,y) · T(x,y) dA) / (∫A v(x,y)

dA)

Tflow,mass Arithmetic mean temperature over entire


flow speed and density.

Tflow,vol = (∫A v(x,y) · ρ(T(x,y)) · T(x,y) dA)

/ (∫A v(x,y) · ρ(T(x,y)) dA)

Tflow,enth Arithmetic mean temperature over entire


flow speed and using enthalpy. This is the true

energy or enthalpy flow mean temperature.

Tflow,enth = h-1 [ (∫A v(x,y) · ρ(T(x,y)) ·

h(T(x,y)) dA) / (∫A v(x,y) · ρ(T(x,y)) · dA)]

Table 6: Different ways of calculating an average line, section and bulk flow temperature.

The most relevant temperatures are Tus as it is the temperature the ultrasound sensor will give,

Tsec,vol as it is the temperature that a set of temperature sensors mounted in the pipe section (e.g.

Pt100s) would have as its mean temperature and Tsec, enth as it represents the true physical

temperature of the fluid when the cross section would be (adiabatically) mixed to a homogeneous

temperature. The other temperatures are mainly approximations of the true physical enthalpic

temperatures which are easier to calculate. It is instructive to see the size of the differences. The

flow speed weighted enthalpic temperatures Tflow,sec, Tflow,mass, Tflow,enth are not shown Table 7, as that

would need an assumption of the flow profile over the cross section of the pipe. This is not done

here. (If the flow profile is homogeneous bulk flow v(x,y) = constant, then Tflow,xxx = Tsec,xxx).

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Temperature profile Tus

[°C]

Tline,vol

[°C]

Tline, mass

[°C]

Tline, enth

[°C]

Tsec,vol

[°C]

Tsec, mass

[°C]

Tsec, enth

[°C]

Homogeneous, T = 25 °C 25.000 25.000 25.000 25.000 25.000 25.000 25.000

Homogeneous, T = 300 °C 300.000 300.000 300.000 300.000 300.000 300.000 300.000


of 1% of L, Tin = 30 °C, Tout = 20 °C

29.765 29.800 29.799 29.799 29.604 29.603 29.603


of 0.1% of L, Tin = 300 °C, Tout = 250

°C

299.933 299.900 299.890 299.890 299.800 299.780 299.781

Two homogeneous circular regions,

Tin = 25 °C, Tout = 24 °C

24.496 24.500 24.500 24.500 24.250 24.250 24.250


Tin = 301 °C, Tout = 300 °C

300.503 300.500 300.500 300.500 300.250 300.250 300.250


Tin = 28 °C, Tout = 22 °C

24.856 25.000 24.998 24.997 23.500 23.498 23.498


Tin = 303 °C, Tout = 297 °C

300.093 300.000 299.974 299.995 298.500 298.481 298.496

Two equal homogeneous regions,

Tup = 25 °C, Tdown = 24 °C

24.496 24.500 24.500 24.500 24.500 24.500 24.500


Tup = 301 °C, Tdown = 300 °C

300.503 300.500 300.500 300.500 300.500 300.500 300.500


Tup = 28 °C, Tdown = 22 °C

24.856 25.000 24.998 24.997 25.000 24.998 24.997


Tup = 303 °C, Tdown = 297 °C

300.093 300.000 299.974 299.995 300.000 299.974 299.995


Tup = 30 °C, Tdown = 5 °C

15.161 17.500 17.473 17.456 17.500 17.473 17.456


Tup = 330 °C, Tdown = 300 °C

318.149 315.000 314.174 314.969 315.000 314.174 314.969

Table 7: Differences between various ways of calculating a mean temperature. Tflow,vol , Tflow,mass and Tflow,enth are not calculated, as that would need an assumption about the flow profile.

In order to clarify the numbers of Table 7 the true temperature after mixing using enthalpy has been

compared with the approximations based on volume weighted averaging and mass weighted

averaging. Half of the volume has a fixed temperature T1 = 20 °C (c.q. 300 °C) and the other half has

temperature T2 ranging from 20 to 35 °C (c.q. 300 to 330 °C). See Figure 5 and Figure 6.

17

Figure 5: Error involved in approximate formulae to calculate temperature after mixing in case of half of the volume having temperature T1 = 20 °C and half of the volume having temperature T2 =20 to 35 °C. Pressure is 2 barg.

Figure 6: Error involved in approximate formulae to calculate temperature after mixing in case of half of the volume having temperature T1 = 300 °C and half of the volume having temperature T2 =300 to 330 °C. Pressure is 150 barg.

18

It is concluded that in case of low temperature the volume weighted approximate temperature has

an error less than 0.01 °C for temperature differences less than 10 °C. The mass weighted

approximate temperature is very good and has an error of 0.002 °C for a temperature difference of

15 °C.

In case of high temperatures the volume weighted approximate temperature has an error less than

0.05 °C for temperature differences up to 30 °C. The mass weighted approximate temperature has

an error less than 0.05 °C for temperature differences less than 8 °C. The fact that the mass weighted

temperature deviates is related to the fact that the water density is very sensitive to temperature

variations. The calorific value cp also changes with temperature. The final result is that the volume

weighted temperature is a very good approximation of the true temperature after mixing at high

temperatures.

3.5. Effect of composition on the speed of sound The speed of sound is directly dependent on the fluid composition. In this section it is studied what

the effect is of impurities in water on the ultrasound based temperature method. It is assumed that

the used reference data is valid for pure water, whereas the real fluid is water with a tiny bit of

ammonia dissolved or salt water with different salinities. In Table 8 the error induced in the

ultrasound based temperature measurement is shown for an ammonia solution at different

temperatures, for drinking water and for sea water. The ammonia concentration of 3.8 mg/l is the

highest concentration that was found to be used in power plants according to the survey that was

made.

The speed of sound in drinking and sea water is calculated by means of the equation of Coppens

(1981), see reference [4].

Real fluid Used data

base fluid

Nominal T and P Resulting error in

measured T

Ammonia solution (3.8 mg/l) Pure water 20 °C/ 2 barg 0.006 °C

Ammonia solution (3.8 mg/l) Pure water 300 °C/ 150 barg < 0.001°C

Drinking water (salinity 100 ppm) Pure water 20 °C/ 2 barg 0.036 °C

Limit drinking water (salinity 1000 ppm) Pure water 20 °C/ 2 barg 0.36 °C

Sea water (salinity 35 000 ppm) Pure water 10 °C/ 0 barg 13 °C

Table 8: The influence of water composition on measured temperature by means of ultrasound method.

19

According to the survey, the following chemicals are or were used in power plants (process water

and cooling water):

- Ammonia (0.2 – 4 mg/L) - Boric acid (0 - 1500 ppm) - Chlorine - Copper (0.02 - 0.2 mg/L) - Ferrous sulphate - Hydrazine (0.025 - 0.3 mg/L)

- HCL - H2 - Iron (0.01 - 10 mg/L) - Lithin - NaOH - Silicon (0.02-0.1 mg/L)

As concentrations are low, no significant influence on the speed of sound is expected. This was

verified for ammonia both based on RefProp reference data and experimentally, and for boric acid

experimentally.

3.6. Effect of flow speed In this section possible effects of the flow speed on the ultrasound based temperature measurement

method are studied.

An ultrasound transit time (flow) meter measures transit times t1 and t2 of two ultrasonic pulses

travelling in opposing directions between the transmitters. From the known path length L between

the two transducers and the angle α between the path and the pipe axis, flow speed in pipe axis

direction v and (approximate) speed of sound capp are calculated in the following way:

v = L/(2·cos(α)) · (t2 – t1)/(t1·t2)

capp = L/2 · (t1 + t2)/(t1·t2)

The speed of sound calculated in this way, is an approximation if the ultrasound transducers are

modeled as point sized and the water is homogeneous and flowing with speed v in the direction of

the pipe axis. The approximation lies in the fact that the effect of the ultrasound beam being

transported in axial direction by the fluid flow is neglected. At least theoretically this would imply

that the beam will not land on the point-sized transducer at the other side of the pipe, but

somewhat downstream. The correct formula for calculating the speed of sound in this setting is:

c = capp √[1 + (v/capp)2 · sin2(α)]

capp = c √[1 - (v/c)2 · sin2(α)]

and thus (slightly) depends on flow speed v. With transducers orthogonal to the pipe axis (α = 90°)

the inaccuracy is maximal and shown in Table 9.

This effect is not present if the point approximation of the transducers is invalid. For small pipes this

might indeed be the case: in a 0.1 m diameter pipe, with v = 10 m/s, the beam is displaced 0.7 mm

parallel to the pipe axis. As the transducer diameter will be much bigger than 0.7 mm, there won’t

20

probably any effect. For a pipe diameter of 1 m and a flow speed of v = 20 m/s, the beam will be 21

cm displaced parallel to the pipe axis. In this case, the effect may be present, as the transducers will

have a much smaller diameter than 21 cm.

Another effect is that by Bernoulli’s law the hydrostatic pressure is dependent on v, as the

hydrostatic pressure is diminished by the dynamic pressure ½ · ρ · v2. The fluid feels the static

pressure which is the total pressure diminished by the dynamic pressure. If the total pressure is

measured instead of the static pressure, then this will result in a final temperature measurement

error, as shown in Table 9.

This effect is not present if the pressure transducers measure static pressure and not total pressure.

This is probably the case. Care should be taken that they measure at a representative spot.

If the just described effects are present, in case of very high flow speeds and when additionally very

high accuracies are desired a flow meter for estimating the flow speed might be needed. In case of

using an ultrasonic flow meter for the ultrasound based temperature measurement this flow speed

information is readily available.

Nominal T

and P

Flow speed

v

Error in T due

to use of capp

Error in T due to

unmeasured reduced

hydrostatic pressure

Total resulting error in

measured T

20 °C/

2 barg

5 m/s -0.003 °C -0.007 °C -0.009 °C

20 °C/

2 barg

10 m/s -0.011 °C -0.027 °C -0.038 °C

300 °C/

150 barg

10 m/s 0.009 °C 0.051 °C 0.060 °C

300 °C/

150 barg

20 m/s 0.036 °C 0.206 °C 0.242 °C

Table 9: Induced temperature measurement error due to flow speed for orthogonal transducers (α = 90°). Both effects

probably don’t occur in practice for small size meters. For big size meters the first effect may occur, but the influence remains very small.

3.7. Conclusions of theoretical background part The method of measuring temperature with ultrasound is a priori a powerful idea. For homogeneous

temperatures measurement uncertainties down to 0.01 °C are theoretically possible for

temperatures around 20 °C. For high temperatures 0.1 °C seems a more realistic goal. In case of

temperature differences below 1 °C, the ultrasound method can still perform with the same

uncertainty. In case of higher temperature differences, the ultrasound temperature method path

mean will start to deviate from the path mean or true enthalpic mean temperature more and more.

For heterogeneous irregular temperature distributions with local temperature difference up to 10 °C

a measurement uncertainty of a few 0.1 °C is theoretically possible. At higher temperatures the

21

ultrasound method as presented here may start to deviate more. This can possibly be amended by

using multiple paths and some smart software.

Around the temperature of T0 ≈ 70 °C temperature measurement based on speed of sound is

difficult, as the speed of sound attains a maximum at this temperature. Furthermore, in all cases it

should be known from other sources if the temperature is above or below T0.

The flow speed has no significant effect on the method. Also the small amount of additives added to

the process water in power plants are not expected to influence the measurement in a noticeable

way.

In the next chapter the temperature calibration method will be described before moving on with a

more detailed description of the equipment.

4. Calibration method for ultrasound based temperature

measurement As is the case for most measurement devices, also an ultrasonic temperature measurement device

needs calibration. It depends however on the device, its manufacturer and the application of the

user how much work the user needs to perform for the calibration. Different cases will be

considered.

4.1. Complete calibration by manufacturer In this case the user can just install the ultrasound sensor in his pipe work and start working with it

without any calibration to be performed by the end user. The sensor should be equipped with its

own pressure measurement and the used fluid should be (sufficiently) equal to the fluid the

manufacturer had in mind (and possibly used in a factory calibration). Also the device should be very

robust such that ultrasonic path lengths don’t change when installing the device in the piping.

Therefore clamp on sensors delivered without any piping seems to be excluded for the ideal case, as

the determination of ultrasonic path length will need at least some calibration or configuration

effort.

In case the device doesn’t have its own pressure measurement, it should be possible to feed the

device with a representative pressure measurement by means of an analog or digital signal to the

temperature computing unit.

Finally the manufacturer should have implemented device software in which the temperature is

calculated from the measured ultrasonic transit times and pressure for the fluid in the piping.

One important issue to address is that the device should know if the temperature is above or below

T0 , the temperature at which the speed of sound attains a maximum (T0 ≈ 70 °C). This can either be

established by a conventional temperature measurement performed by the device, by an external

22

temperature measurement fed to the device or, most commonly, by prior knowledge about the

approximate fluid temperature. Furthermore, temperature measurement based on speed of sound

is problematic around T0.

4.2. Calibration by user In the basic case the end user only gets the ultrasonic transit times from the device. In this case he

has to determine some numerical factors himself. The user can either plug some theoretically

determined factors in the device or perform a real calibration (experimentally). For the calibration a

flow loop is needed in which homogeneous flow at different temperatures and pressures can be

created. Experimental calibration data are strongly preferred as they better reflect reality and can

also take into account or at least show not modeled effects, which would otherwise be overlooked.

These calibration factors will now be discussed as well as their approximate size and importance.

4.2.1. Ultrasonic path lengths

To calculate the speed of sound from transit times, the ultrasound path lengths should be divided by

the corresponding transit times. The path lengths could be measured geometrically before

installation of the device, taking into account the wall thickness for clamp on sensors. Another

possibility is to let flow a homogeneous fluid with known temperature and pressure and known

pressure-temperature-speed of sound relationship through the sensor. The homogeneous

temperature could be read from a reference temperature sensor close to the device and the same

for the pressure. The speed of sound can now be taken from the database and from this the path

lengths can be recalculated.

The contribution of the ultrasonic path length uncertainty to the total measurement uncertainty in

the temperature is shown in Table 10.

Parameter Nominal

value of

parameter

Uncertainty in

parameter

Nominal T

and P

Resulting

uncertainty in

measured T

Ultrasonic path

length

100 mm 0.001 mm 25 °C/ 0

barg

0.006 °C

Ultrasonic path

length

1000 mm 0.010 mm 300 °C/ 150

barg

0.002 °C

Table 10: Contribution of the ultrasonic path length uncertainty to the total measurement uncertainty of the temperature.

An inaccuracy in path length of 1 μm on a total path length of 100 mm gives an inaccuracy of 1/105

in the speed of sound which amounts to 1/105·1500 = 0.015 m/s or 0.006 °C. For comparison: a

human hair has a thickness of about 60 μm.

23

4.2.2. Temperature expansion coefficient

Depending on the pipe wall material the pipe diameter and thus also the ultrasonic path lengths will

change with temperature. Therefore the path lengths need correction with the best known wall

temperature estimate before calculating the speed of sound from path length and transit time. In

good approximation the temperature expansion correction can be taken in form of the linear

expansion coefficient α times the temperature increase from calibration reference temperature.

One can take a database value if the pipe material is known (e.g. 16 ppm/K for SS316). Another

possibility is to let flow a homogeneous fluid with known temperature and pressure and known

pressure-temperature-speed of sound relationship through the sensor and repeating this at different

temperatures. From the resulting data (transit times), the measured pressure and reference

temperature and the database data, the temperature expansion coefficient can be deduced.

The uncertainty contribution of the pipe temperature expansion to the total measurement

uncertainty of the measured temperature can be found in Table 11.

Parameter Nominal

value of

parameter

Uncertainty

in

parameter

Nominal T

and P

Resulting

uncertainty in

measured T

Temperature expansion

coefficient α (used for

extrapolation from 20 °C)

16 ppm/K 1 ppm/K 25 °C/ 0

barg

0.003 °C

Temperature expansion

coefficient α (used for

extrapolation from 20 °C)

16 ppm/K 1 ppm/K 300 °C/ 150

barg

0.05 °C

Wall temperature 25 °C 1 °C 25 °C/ 0

barg

0.009 °C

Wall temperature 300 °C 5 °C 300 °C/ 150

barg

0.01 °C

Table 11: Uncertainty contribution of the pipe temperature expansion to the total uncertainty in temperature measurement.

When the ultrasonic temperature measurement is corrected for pipe wall expansion, the remaining

uncertainty component is at most 0.05 °C, which is small compared to other effects. Nevertheless

the temperature expansion correction needs to be implemented somehow.

4.2.3. Pressure expansion coefficient

Depending on the pipe wall material and thickness the pipe diameter and thus also the ultrasonic

path lengths will change with pressure. However for most materials and low pressures, this effect

24

will be very minor. In the general case the path lengths need correction for the pressure. In good

approximation the pressure expansion correction can be taken of linear form. One can take a

theoretical value for the Young modulus E if the pipe material is known (e.g. E = 200 GPa for SS316),

measure or look up the pipe wall thickness b and use an expansion formula for thin or thick walled

pipes (e.g. dL/L = P L/ (2 b E) = β P) for thin walled pipes with diameter L and wall thickness b) to

calculate the pressure expansion coefficient. Another possibility is to let flow a homogeneous fluid

with known temperature and pressure and known pressure-temperature-speed of sound

relationship through the sensor and repeating this at different pressures. From the resulting data

(transit times), the measured pressure and reference temperature and the database data, the

pressure expansion coefficient can be deduced (which might also take into account some other

pressure effects).

The contribution of neglecting the pressure expansion for thin walled pipes to the total

measurement uncertainty (deviation/ error) in temperature is shown in Table 12.

Parameter Nominal

value of

parameter

Uncertainty in

parameter

Nominal T

and P

Resulting

uncertainty/

error in

measured T

Uncompensated

pressure expansion

(thin wall

coefficient: β= L/ (2

b E), L = 100 mm, b =

3 mm, E = 200 GPa)

β = 8.3 10-11

bar-1

8.3 10-11 bar-1 25 °C/ 2

barg

0.009 °C

Table 12: Error in ultrasound temperature measurement due to uncertainty in or negligence of pressure expansion of pipe for thin walled pipes.

At high pressures and temperatures, the pipe wall will be very thick, so that the thin wall

approximation is not valid anymore. If the pressure effect on the wall is bigger than desired, the pipe

wall thickness can be increased in the design (usual approach), or a correction factor can be applied

in the meter software. In any case the remaining uncertainty component is estimated to be relatively

small. Usually the pressure expansion correction doesn’t need to be determined.

4.2.4. Transmitter delays

The transit times measured by of the ultrasonic device τtransmitter might not be completely corrected

for delays τdelay in electronics and pipe walls. One can model this by the equation τtransit = τtransmitter –

τdelay for the true transit time τtransit. One can determine these delays by making a best fit through the

calibration data taken at a flow of homogeneous fluid with known temperature and pressure and

known pressure-temperature-speed of sound relationship through the sensor and repeating this at

different temperatures and pressures. From the resulting data (transit times), the measured

pressure and reference temperature and the database data, the temperature expansion coefficient,

25

the pressure expansion coefficient and the transmitter delays can be deduced. Normally this is not

needed however.

The uncertainty contribution of the transit time measurement is shown in Table 13.

Parameter Nominal

value of

parameter

Uncertainty in

parameter

Nominal T

and P

Resulting

uncertainty in

measured T

Transmitter delay

(D = 100 mm pipe,

τtransit = 67.446 μs )

0 μs 0.001 μs 25 °C/ 2

barg

0.008 °C

Transmitter delay

(D = 1000 mm pipe,

τtransit = 1034.531 μs )

0 μs 0.001 μs 300 °C/ 150

barg

0.0002 °C

Table 13: Uncertainty of ultrasound temperature measurement method due to uncertainty in transit time.

An inaccuracy in measured transit time of 1 ns on a total path length of 100 mm gives an inaccuracy

of 1.5/105 in the speed of sound which amounts to 1.5/105·1500 = 0.022 m/s or 0.008 °C at low

temperature and pressure. The uncertainty of this component is estimated to be relatively small and

normally this correction term doesn’t need to be determined.

4.2.5. Speed of sound reference data

The speed of sound reference data are not a calibration factor, but they are needed to convert

measured speed of sound (together with knowledge about pressure and composition) to a

temperature. This reference data can come under different forms: e.g. tables with values, an

equation of state, some polynomial functions, etc. All these forms will simply be called reference

data. In the following subsections several cases are considered for the type (not the form) of

reference data that can be used.

4.2.5.1. Off-line reference data

In this case an independent database with traceable speed of sound is being used. The database

gives the relationship between temperature, pressure and speed of sound for a given fluid. Once it is

established that the ultrasonic device is able to measure speed of sound correctly at different

pressures and temperatures, nothing else needs to be calibrated. Only the reference data have to be

looked up.

In deliverable D5.5 of this research project a webpage with references to existing traceable speed of

sound data will be created. If the end user wants to create high quality traceable reference data by

himself, considerable investments and work is needed. Description of the needed equipment

26

surpasses the scope of this report. Reference values can for example be taken from the equation of

state made by IAWPS for water (see Ref. [3]), or from the NIST RefProp program (see [1]).

The speed of sound reference data possess an uncertainty. The NIST RefProp program states an

uncertainty of 0.1 % (see reference [2]). The uncertainty for the speed of sound of water given by

IAWPS is shown in Figure 7. It has to be noted that here the speed of sound is calculated from a

general equation of state, which is in turn fitted based on a certain number of measurements.

Particular measurements of the speed of sound might have lower uncertainty. These data are

however harder to find and may only cover a small part of all interesting temperatures and

pressures. There also exist some particular equations which give speed of sound as function of

temperature and pressure with lower uncertainty in some small regions, especially at ambient

pressure and relatively close to ambient temperature.

In Table 14 the contribution of the uncertainty in speed of sound reference data to the total

measurement uncertainty is shown. This uncertainty can be reduced by a direct calibration of the

device with the correct fluid, temperature and pressure.

27

Figure 7: Uncertainty of speed of sound reference values for water given by IAPWS (see reference [3]).

Parameter Nominal

value of

parameter

Uncertainty in

parameter

Nominal T

and P

Resulting

uncertainty in

measured T

Speed of sound

reference value

1496.70

m/s

0.03 % / 0.45

m/s

25 °C/ 2

barg

0.17 °C

Speed of sound

reference value

966.6 m/s 0.2 % / 1.9 m/s 300 °C/ 150

barg

0.3 °C

Table 14: Uncertainty in ultrasound temperature measurement method due to uncertainty in database reference values.

From Table 14 it is seen that speed of sound reference data actually possesses a relatively high

uncertainty when converted to the induced uncertainty in ultrasonic temperature measurement.

28

This means that it is not possible to get low uncertainties in the ultrasound temperature

measurement without a wet temperature calibration of the device at approximately the working

temperature. The error in the speed of sound reference data will be systematic for small

temperature ranges, but might be random for larger temperature ranges. This means that

calibration data and speed of sound reference data cannot be extrapolated from e.g. 20 °C to 300 °C.

4.2.5.2. Continuous on-line calibration reference data

In this case the idea is that the needed speed of sound reference data is created by the end user

himself, using exactly the same fluid as in the process. Some liquid is taken out of the process, is

heated to the desired temperature and then pressure, temperature and speed of sound are

measured simultaneously in order to find their relationship. In Figure 8 this is schematically shown.

This method avoids the need for determining the fluid composition and the search for database

reference data for the pressure/ temperature/ speed of sound relationship for the particular fluid

composition. Another advantage is the quick response to changes in fluid composition. What is still

needed is an initial calibration of the main ultrasound device used in the process together with the

ultrasound device used for establishing the curve at different temperatures and preferably also

different pressures and compositions. This is needed to verify that both devices measure the same

(non-traceable) approximate speed of sound at different conditions. After this initial calibration the

on-line calibrator can be used for establishing the current reference curve for the current

composition.

29

Figure 8: Set-up for on-line calibrator of a speed of sound based temperature meter.

The disadvantages are the required additional investments for the measurement set-up and,

sometimes more important, the need of having a sampling point in the process. In the application in

the primary circuit of a nuclear power plant, this would mean taking out some radio-active water out

of the primary circuit. In practice this will never be done if it can be prevented. Furthermore, from

discussions with plant owners it followed that the fluid composition (concentration of chemicals

dissolved in the water) changes only very slowly. The interesting time scale is rather months than

hours. Therefore the off-line laboratory reference data approach seems to be more relevant for

practical applications together with monitoring of the water composition on a regular basis. Besides

that, the low concentration of chemicals used in power plants are expected to change the speed of

sound only in a very minor way, irrelevant in the current measurement uncertainty. This has been

verified for two chemicals (ammonia and boric acid) in the experimental tests and for ammonia also

using NIST RefProp reference data.

4.2.6. Calibration of other auxiliary equipment

The calibration of other auxiliary equipment (like pressure and temperature sensors, see next

chapter) is standard and will therefore not be described in this report.

30

4.3. Conclusion about calibration method In this chapter several issues related to the calibration needed for the ultrasonic temperature

measurement method were considered. Because of the limited accuracy of speed of sound

reference data and because it is difficult to measure speed of sound well without prior wet

calibration (e.g. only geometrical and theoretic calibration), a wet calibration of the device is needed

at the temperature at which the device will be used to get a good estimate of the path length.

Dependent on the construction of the device, the envisaged accuracy and on the range of working

temperatures, pressures and compositions, the temperature and pressure expansion coefficients

and transmitter delays may be calibrated as well. Often some theoretical values are good enough to

get a fair accuracy for the ultrasound based temperature measurement.

In case the process fluid only slowly changes composition and when reference data in literature

exist, it will be convenient to use the existing reference data instead of using a separate device to

determine the reference data concerning the temperature-pressure-speed-of-sound relationship

oneself.

In Table 15 a summary of the different calibration methods is given and when each of them is the

method of choice.

Calibration method Advantages Disadvantages Preferred when

Method 1:

Initial wet calibration,

use of real-time

calibrator

- Fast response to

changing composition

- No traceability

needed for speed of

sound

- Additional equipment

needed

- Additional calibration

and simultaneous

testing of devices

needed

- Fast changing

composition of process

fluid

- Unknown fluid

composition or use of

a fluid for which no

reference data exists

Method 2:


use of existing

(traceable) database

with reference data for

speed of sound

- No additional

equipment needed

- Database for all

relevant compositions,

pressures and

temperatures has to

be found

- Fluid composition

measurement might

be needed

- Fluid composition is

known and reference

data exists

31

Calibration method Advantages Disadvantages Preferred when

Method 3:


creation of traceable

database for speed of

sound

- Arbitrary

composition, pressures

and temperatures can

be used

- Database creation

and process can be

physically separated,

no simultaneous

calibration needed.

- Costly equipment is

needed for the

creation of a high

quality database and

additional work is

involved

- Composition

measurement of

process fluid might be

needed

- No reference data

exists for the fluid

- Establishment of

reference data and

main process should

be physically

separated

- Desire of universal

and traceable

reference data for

speed of sound

Table 15: Summary of different calibration concepts for an ultrasound based temperature method.

5. Required equipment This section will describe the needed equipment for an ultrasound based method for measuring

temperature in more detail. The main component is the ultrasound sensor, but also some other

equipment is needed.

5.1. Ultrasound sensor for temperature measurement The ultrasound sensor is the main part of the equipment. In order to measure the speed of sound, a

transit time based ultrasonic flow meter can be used. An alternative is to have a spool piece with

ultrasound transducers mounted orthogonally to the pipe axis on it. In both cases transit times of

short sound pulses at some ultrasonic frequency (e.g. 1 MHz) are being measured. These transit

times can either be a directly transferred to the user or converted to a speed of sound or

temperature by the (flow) meter transmitter itself.

The orthogonal design is more compact and the cross section measured by the ultrasound beams is

an orthogonal cross section and not a section on an angle, which may be advantageous when

dealing with irregular flow patterns. It may also allow for more accurate speed of sound

measurements. The advantage of an ultrasonic flow meter is obviously that it also measures the flow

rate.

Transit time ultrasonic flow meters are available from a number of commercial manufacturers.

However it depends on the manufacturer if measured transit times, speed of sound and/or

temperature are available as output from the device. Calibration of the device was discussed in the

last chapter.

32

5.2. Auxiliary equipment In order for the ultrasonic temperature method to work, some auxiliary measurement equipment is

needed. This consists of equipment directly used for the determination of the temperature (namely

pressure and possibly fluid composition measurement) and equipment used for calibration and

validation of the device.

5.2.1. Pressure sensor

As speed of sound depends on both pressure and temperature, it is not enough to measure only the

speed of sound for deducing the temperature. The hydrostatic pressure of the fluid at the position of

the ultrasound beam is needed as well. Thus a pressure sensor is needed. The required uncertainty

is determined by the sensitivity of the speed of sound to pressure variations (see Table 1 and Table

2). The influence of pipe wall expansion due to pressure is in general less important (see Table 12).

5.2.2. Composition determination

The speed of sound varies with the composition of the fluid. If one decides to use off-line speed of

sound reference data, the composition should be known with sufficient accuracy. The composition

should either be known beforehand and not change, or, when larger changes in composition are

expected to occur, should be measured on a regular basis with a suitable device. There is a number

of companies offering equipment for chemical analysis and monitoring the composition of fluids.

Depending on the estimated composition and the desired accuracy a device has to be selected.

If an on-line calibration reference data method is being used, the composition doesn’t need to be

determined. The on-line calibration device will automatically produce adapted calibration values in

case fluid composition changes.

In the case of power plants the concentration of solved additives into the process water is so low,

that no noticeable effect on the speed of sound is expected (deviating from the speed of sound in

pure water) and thus composition determination is not needed for the ultrasound based

temperature measurement method.

5.2.4. Equipment to be used for calibration

In case that ultrasonic path lengths are being measured geometrically a caliper or laser based

method is needed to determine these lengths. This is however unconventional and won’t include

electronic delays and other effects, which can integrally be determined in a direct calibration.

Therefore, this method is not advised.

A better idea is using a flow loop for determining the calibration settings. Besides the flow loop and

knowledge concerning pressure and composition, a reference temperature sensor is needed. Even

better are several reference sensors. With a so-called Pt100-Array, consisting of 9 Pt100

33

temperature sensors mounted at different positions in the cross section of a pipe, the homogeneity

of the fluid temperature can easily be verified, see Figure 9.

Figure 9: A Pt100-array: a spool-piece with 9 Pt100 temperature sensors mounted at different positions in the cross section.

The required uncertainty of a caliper or laser for a geometric calibration can be found in Table 10.

However a wet calibration is preferred. An overview of uncertainty sources in a wet calibration in a

flow loop is given in Table 16.

Uncertainty source Explanation Formula or

symbol

Numerical

estimate of error

or uncertainty

Reference

temperature

uncertainty

Depends a.o. on calibration of

reference, installation effects,

ambient temperature effects,

drift, etc.

δTcal + δTinstallation

+ δTamb + δTdrift

0.02 °C

Imperfect

homogeneous

temperature

Water temperature is non-

uniform over the pipe section.

Can be measured with a Pt100-

array.

(max(Ti) –

min(Ti))/2

0.01 °C

34

Uncertainty source Explanation Formula or

symbol

Numerical

estimate of error

or uncertainty

Heat leak between

ultrasound device

and reference

temperature sensor

Depends on insulation quality

(k), distance between

ultrasound device and reference

(L), mass flow rate (Qm), specific

heat of fluid (cp) and the

temperature difference

between ambient air and the

fluid (Tamb – T)

(Tamb – T) · k · L /

(Qm · cp)

0.001 °C

Time delay between

ultrasound device

and reference

temperature sensor

Effect depending on the size of

the temperature variation of

the water and the flow speed v

and the distance L

(L/ v) * max(

|dT/dt|)

0.01 °C

Fluid heating due to

viscous effects

Due to pipe wall friction and

viscous forces, fluid pressure

will be converted to heat

(pressure drop dP)

dP/(ρ·cp) 0.000 °C

Table 16: Uncertainties involved in calibrating an ultrasonic temperature sensor in a flow loop. The numerical values have to be estimated for each particular case. Example values are shown.

5.2.5. Equipment to be used for validation of the method

The validation of the ultrasound based temperature measurement method for homogeneous water

temperatures can be done with the same equipment as used in the experimental calibration, thus

comparing the ultrasound temperature with a reference temperature.

In order to validate the method for inhomogeneous water temperatures a method for creating

inhomogeneous temperature profiles in the fluid is needed. This can for example be done by

combining two flows of known flow rate and of known homogeneous temperature by using a T-

piece in the flow loop (see Figure 10 left) and two flow meters and two temperature sensors. One

can also design a special flow injector simulating special conditions, e.g. a stratifying flow injector

simulating stratified flow (see Figure 10 middle and right). The Pt100-Array described before can be

used to monitor the actual created temperature profile. The water temperature after perfect

(adiabatic) mixing is the enthalpy flow mean temperature. This temperature will be taken as

reference temperature. For further information see the chapter about the ‘Theory of ultrasound

based temperature measurement’.

35

Figure 10: Classical T-piece (left) and spool-piece for creating stratified flow pattern. There is a flow injection nozzle below the plate.

In Table 17 and Table 18 uncertainty budgets for the reference temperature in the case of mixing of

two flows with different temperature is given. Due to instable conditions, test set-up and delays,

some additional components may have to be taken into account in practice. See also Table 16. In the

last rows of Table 17 and Table 18 the approximate mixing temperature is calculated using mass

flows in order to see if volume flow and mass flow weighted temperatures form a good

approximation of the true temperature determined by enthalpy flows. The measurement equation is

given by

Tmix = h-1[(Q1·ρ1·h1 + Q2·ρ2·h2)/ (Q1·ρ1 + Q2·ρ2)]

and by

Tmix,vol = (Q1·T1 + Q2·T2)/ (Q1 + Q2)

and

Tmix,mass = (Q1·ρ1·T1 + Q2·ρ2·T2)/ (Q1·ρ1 + Q2·ρ2)

for the approximate temperatures using volume respectively mass flows.

Parameter Nominal

value of

parameter

Uncertainty in

parameter

Sensitivity

coefficient

Resulting

uncertainty in

measured T

Q1, flow 1 25 m3/h 0.05% (= 0.013

m3/h)

0.06 °C/ ( m3/h) 0.0007 °C

Q2, flow 2 5 m3/h 0.05 % (= 0.003

m3/h)

-0.28 °C/ (

m3/h)

0.00007 °C

T1, temperature of flow 1

30 °C 0.02 °C 0.8 °C/°C 0.017 °C

36

Parameter Nominal

value of

parameter

Uncertainty in

parameter

Sensitivity

coefficient

Resulting

uncertainty in

measured T

T2, temperature of

flow 2

20 °C 0.02 °C 0.17 °C/°C 0.003 °C

ρ1, water density in

flow 1

995.74 kg/m3 0.001 % (= 0.01

kg/m3)

0.0014 °C/

(kg/m3)

0.00001 °C

ρ2, water density in

flow 2

998.30 kg/m3 0.001 % (= 0.01

kg/m3)

-0.0014 °C/

(kg/m3)

0.00001 °C

Assuming constant

specific heat cp (*)

4.18 kJ/kg K 0.001 °C

Tmix, temperature after

mixing (using enthalpy

flow)

28.329 °C 0.017 °C

Tmix,mass, approximate

temperature after

mixing using mass

flow

28.330 °C 0.018 °C

Tmix,vol, approximate

temperature after

mixing using volume

flow

28.333 °C 0.021 °C

Table 17: Example uncertainty budget for reference (enthalpic) temperature at 2 barg with input flow temperatures 20 °C and 30 °C. The volume and mass flow weighted temperatures are shown for the purpose of comparison. In experiments some additional components may have to be taken into account. See also Table 16. It’s assumed that all uncertainty sources are independent and possess a normal distribution.

(*) The specific heat is not constant with changing temperature, but varies slightly with

temperature. The absolute value of the specific heat constant (or enthalpies) is not important for the

calculations, only its variation. Here, the assumption of constant cp is compared with using the best

known reference data values of h(T) in the temperature calculation. The difference is taken as very

small additional uncertainty which is linearly added to the final uncertainty. If one uses the

uncertainties stated for absolute enthalpy (1%) or of the (absolute) specific heat coefficient cp, one

will find higher values for the uncertainty (0.025 °C), but this neglects the fact that the uncertainty of

cp for temperatures close to each other is very correlated (‘the error is constant’).

37

Parameter Nominal value

of parameter

Uncertainty in

parameter

Sensitivity

coefficient

Resulting uncertainty

in measured T

Q1, flow 1 25 000 m3/h 0.5% (= 125 m

3/h) 0.0001 °C/ (

m3/h)

0.014 °C

Q2, flow 2 5 000 m3/h 0.5 % (= 25 m

3/h) -0.0006 °C/ (

m3/h)

0.014 °C

T1, temperature of flow 1 330 °C 0.02 °C 0.02 °C/°C 0.017 °C

T2, temperature of flow 2 310 °C 0.02 °C 0.003 °C/°C 0.003 °C

ρ1, water density in flow

1

650.0 kg/m3 0.1 % (= 0.7

kg/m3)

0.003 °C/ (kg/m3) 0.003 °C

ρ2, water density in flow

2

703.8 kg/m3 0.1 % (= 0.7

kg/m3)

-0.003 °C/

(kg/m3)

0.003 °C

h1, specific enthalpy in

flow 1 (**)

1518.5 kJ/kg 0.1 % (= 0.125

kJ/kg)

0.13 °C/ (kJ/kg) 0.19 °C

h2, specific enthalpy in

flow 2 (**)

1394.3 kJ/kg 0 % (uncertainty included in h1)

0.03 °C/ (kJ/kg) 0.0 °C

Induced error by

assuming constant

specific heat cp (**)

6.56 kJ/kg K 0.25 °C

Tmix, temperature after

mixing (using enthalpy

flow)

326.69 °C 0.19 °C

Tmix,mass, approximate

temperature after

mixing using mass flow

326.44 °C 0.44 °C

Tmix,vol, approximate

temperature after

mixing using volume

flow

326.67 °C 0.21 °C

Table 18: Example uncertainty budget for reference enthalpic temperature at 150 barg with input flow temperatures 310 °C and 330 °C. The volume and mass flow weighted temperatures are shown for the purpose of comparison. In experiments some additional components may have to be taken into account. See also Table 16.

(**) At high temperatures the assumption of constant specific heat capacity cp is not valid. It

would result in an error of 0.25 °C. Therefore the correct enthalpies should be used. In Table 18 the

full uncertainty of 0.1 % for cp has been used for the enthalpy 1, not taking into account correlations

between the error in cp at different temperatures, which would considerably reduce the total final

uncertainty.

38

5.2.6. Required equipment for speed of sound reference data

Two methods for getting the pressure-temperature-speed-of-sound relationship were described.

The first was the use of off-line laboratory reference data and the second was on line measurement

with the fluid being used. The equipment needed will now be described in brief.

5.2.6.1. Required equipment for off-line laboratory reference data

Reference data for the pressure/ temperature/ speed of sound relationship is available for a large

number of pure fluids and mixtures in various pressure and temperature ranges (e.g. the NIST

RefProp program, many alternatives exist). Thus in this case a computer suffices. If there is no data

available for the application under consideration, a facility can be build to measure speed of sound

accurately. However this requires mayor investments, considerable technical knowledge and is

beyond the scope of this document.

5.2.6.2. Required equipment for continuous on-line calibration

In this case the reference data for the pressure/ temperature/ speed of sound relationship is

established on-line. In this case a representative sampling point (valve) for the process liquid is

needed. This liquid has to be heated or cooled down to the desired reference temperature and may

be pressurized or depressurized if desired. When the liquid is of homogeneous temperature it can

flow through an ultrasonic meter for determination of the speed of sound. The liquid temperature

can be homogenized by using some pipe work or hoses going up and down in order to stimulate

mixing or by active mixing. Furthermore, the diameter of this sampling line will in general be smaller

than the process line, which enhances homogenization of the temperature. The relationship

between the values found by the reference device and the device measuring the temperature in the

process has to be established in an initial calibration effort. See Figure 8 for a schematic drawing of

an on-line calibrator set-up and the required parts.

The uncertainty components for the on-line calibrator are basically the same as for the calibration of

the ultrasound device (section 5.2.4.). The size of the uncertainty depends on the quality of the

components and the whole set-up.

5.3. Conclusion concerning required equipment The equipment needed for an ultrasound based temperature measurement method has been

described. In the ideal case a working calibrated device can be obtained from the manufacturer. It

has also been described which individual components are needed otherwise. Additional components

were presented for testing and validation of the device in case of heterogeneous fluid temperature

distribution. Finally a set-up for on-line calibration has been presented.

39

6. Overall theoretical uncertainty budget In Table 21 in appendix B all the relevant uncertainty components that have been discussed for an

ultrasound based temperature measurement method are presented. In Table 19 here below a

summary of the uncertainty budget for two cases is given only presenting the main components. In

principal each case will need its own uncertainty estimate (esp. different flow profiles).

Uncertainty

source

Explication

(Uncertainty due to:)

See table nr: Effect on final

temperature

uncertainty (20

°C/ 2 barg)

Effect on final

temperature

uncertainty

(300 °C/ 150

barg)

Calibration

uncertainty

The temperature reference

has some uncertainty

Table 16 0.02 °C 0.02 °C

Pressure

measurement

with 0.1 %

uncertainty

Conversion of speed of sound

to temperature depends on

pressure

Table 1 0.00 °C 0.02 °C

Transit time

measurement

Speed of sound is calculated

from transit time

measurement

Table 13 0.01 °C 0.00 °C

Pipe temperature

expansion

Path length increase caused

by the temperature expansion

of the pipe

Table 11 0.01 °C 0.05 °C

Composition 10 ppm-mol of NH3 added to

water changes speed of sound

Table 8 0.01 °C 0.00 °C

Temperature

profile vs. speed

of sound

averaging effect

The fact that averaging speed

of sound is not the same as

averaging temperature. Case

ΔT = 6 °C has been taken.

Table 4 0.14 °C 0.09 °C

Non-

representativity

effect

The fact that the ultrasound

beams don’t cover the entire

cross section.

Figure 4 0.15 °C 0.1 °C

Final uncertainty

in temperature

0.21 °C 0.15 °C

Table 19: Summary of uncertainty budget for measuring temperature with ultrasound.

For very homogeneous fluids, an uncertainty of a few 0.01 °C is theoretically possible at low

temperatures. At high temperatures the uncertainty is somewhat bigger. For heterogeneous fluids it

depends on the temperature gradient and profile. In general the uncertainty will be substantially

40

higher than for homogeneous fluids, namely in the order of 0.2 °C for a temperature difference of 6

°C (see also Figure 2 and Figure 3).

7. Experimental implementation of the different calibration methods Three different calibration methods were described in chapter 4. The needed equipment for the

ultrasound based temperature method and calibration was described in chapter 5. In this chapter

the experimental work considering the calibration is described. As far as possible with the limited

means and room for investments that were available, the three calibration methods were

implemented in an experimental setting. The experiments evaluating the measurement quality of

the ultrasound based temperature measurement method are described in a separate report, namely

D3.6.

7.0. Initial wet calibration All three calibration methods include a wet calibration of the device at homogeneous temperature.

This was done by mounting the ultrasound meter in a water flow loop with possibility of varying

temperature and pressure. The software of the transmitter was already quite advanced and gave a

temperature output. In the calibration run the path length setting of the three paths of the

ultrasound meter were adjusted such that the calculated temperature best matched the reference

temperature of the facility. The initial temperature and pressure coefficients of the manufacturer

were taken over. At high temperature also the conversion factors to convert speed of sound to

temperature were set.

7.1. Use of a real-time calibrator It was evaluated if the speed of sound of water significantly changed when a small amount of

different chemicals was added. This was not the case. The concept of a real-time calibrator was

partially tested by installing some hoses, an ultrasound sensor and a temperature array after each

other, see Figure 11. The hoses upstream of the ultrasound sensor create some distance to mix the

liquid to a homogenous temperature. The liquid could also be heated up to some desired

temperature. The ultrasound meter measures the speed of sound of the liquid. After the ultrasound

meter a Pt100-array with 9 Pt100 temperature sensors is installed to establish the reference

temperature and to check for temperature homogeneity. Pressure is also measured. In operation

the whole set-up is thermally insulated in order to stabilize the temperature as much as possible.

The measured composition-pressure-temperature-speed-of-sound relationship can now be used for

an ultrasound meter measuring temperature inside the main process as long as is guaranteed that

both ultrasound meters react in the same way on changing fluid composition, pressure and

temperature. The latter can be verified by an initial simultaneous calibration of both devices with

the same fluid, pressure and temperature. (It would also suffice that both devices measure speed of

sound in a traceable way, but this is a stronger requirement.)

41

Figure 11 Concept of a real-time calibrator: hoses to create some distance to mix (and possibly heat) the liquid, an ultrasound temperature meter and a reference temperature measurement, also checking for temperature homogeneity.

7.2. Use of external reference data In most of the experiments external reference data was used to convert measured speed of sound to

temperature. The initial wet calibration is to make sure that the device correctly measures the speed

of sound at different temperatures, pressures and possibly compositions. The reference data are

then used to convert speed of sound (in combination with pressure and composition) to a

temperature for cases not covered by the initial calibration. Preferably this is a temperature-

pressure-composition interpolation, although a limited extrapolation can be done as well. The

reference data were implemented in the transmitter of the ultrasound meter, and checked by an

off-line conversion of measured speed of sound to temperature as well. See deliverable D3.6 for

more details.

7.3. Creation of a traceable reference database A third way of being able to use the ultrasound based temperature meter is by creation of a

traceable reference database for the speed of sound oneself. Low quality reference data can be

made with the real-time calibrator set-up (see Figure 11) once the fluid composition is known. For

high quality traceable reference data a completely different set-up is needed. Inquiries were made

at some National Metrology Institutes which resulted in estimated costs of 100 to 200 k€. This was

far above the budget for investments for the current research project.

42

The analysis of the low quality reference data yielded that for small portions of added chemicals

(boric acid, ammonia) to the water, no measurable change in speed of sound was noticed. The used

small portions of additives are representative for the water composition in power plants.

For the reasons mentioned above it was decided not to create new traceable reference data, but to

gather the existing traceable speed of sound reference data and to present them on a website

(deliverable D5.5).

7.4. Conclusion concerning experiments with calibration methods The main method used in the experiments was the initial wet calibration with use of external

reference data. The concept of a real-time calibrator was partially implemented and tested. The

creation of new traceable reference data was not pursued as existing data already fitted the needs.

See deliverable D3.6 for more details regarding the experimental tests.

8. Overall conclusion In this report a general and practical frame work for temperature measurement using ultrasound for

flow in pipe lines has been presented. The necessary equipment has been described and different

calibration procedures have been proposed. Estimated uncertainties have been given. In the reports

of deliverables D3.5 and D3.6 the practical realization of these ideas will be presented together with

a performance evaluation.

For very homogeneous fluids, an uncertainty of a few 0.01 °C is theoretically possible at low

temperatures. At high temperatures the uncertainty is somewhat bigger. For heterogeneous fluids it

depends on the temperature gradient and profile. In general the uncertainty will be substantially

higher than for homogeneous fluids, namely in the order of 0.1 °C or more. Figure 2 and Figure 3

give an indication of the reachable uncertainty for extreme temperature profiles.

For (nuclear) power plants the ultrasound based temperature measurement method is very

promising from a theoretical point of view, as it may reduce the current uncertainty substantially.

References [1] NIST Reference Fluid Thermodynamic and Transport Properties Database (REFPROP), see

http://www.nist.gov/srd/nist23.cfmhttp://www.nist.gov/srd/nist23.cfm.

[2] NIST RefProp 8.0 help file on water.

[3] Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use, 2009, International Association for the Properties of Water and Steam.

[4] A.B. Coppens, Simple equations for the speed of sound in Neptunian waters (1981) J. Acoust. Soc. Am. 69(3), pp 862-863, see also http://resource.npl.co.uk/acoustics/techguides/soundseawater/content.html#COP

http://www.nist.gov/srd/nist23.cfmhttp:/www.nist.gov/srd/nist23.cfm

http://resource.npl.co.uk/acoustics/techguides/soundseawater/content.html#COP

43

Appendix A: Measured temperature per path In Table 20 the numerical values of the ultrasound based temperature measurement are shown per

path. The temperature at the line y = 0 for profiles 9 to 14 has been assumed equal to (Tup + Tdown)/2.

This has the consequence that the configuration with three horizontal paths gives the bests results.

This seems to be rather due to the studied symmetrical temperature profile than that it is true in

general. In deliverable D5.4 this will be studied in more detail using a software tool.

Temperature profile True volumetric

mean

temperature of

section, Tsec,vol

[°C]

Ultrasound

temperature,

one vertical

path, Tus,v1

[°C]

Ultrasound

temperature,

three center

paths, Tus,c3 [°C]

Ultrasound

temperature, 3

horizontal

paths, Tus,h3 [°C]

Ultrasound

temperature, 3

vertical paths,

Tus,v3 [°C]

Homogeneous, T = 25 °C 25.000 25.000 25.000

(m: 25.000,

b: 25.000,

t: 25.000)

25.000 - (m:

25.000, b:

25.000, t:

25.000)

25.000 - (m:

25.000, b:

25.000, t:

25.000)

Homogeneous, T = 300

°C

300.000 300.000 300.000 - (m:

300.000, b:

300.000, t:

300.000)

300.000 - (m:

300.000, b:

300.000, t:

300.000)

300.000 - (m:

300.000, b:

300.000, t:

300.000)

Boundary layer 1%, Tin =

30 °C, Tout = 20 °C

29.604 29.765 29.765 - (m:

29.765, b:

29.765, t:

29.765)

29.727 - (m:

29.765, b:

29.708, t:

29.708)

29.727 - (m:

29.765, b:

29.708, t:

29.708)

Boundary layer 0.1%, Tin

= 300 °C, Tout = 250 °C

299.800 299.933 299.933 - (m:

299.933, b:

299.933, t:

299.933)

299.922 - (m:

299.933, b:

299.917, t:

299.917)

299.922 - (m:

299.933, b:

299.917, t:

299.917)

Two homogeneous

circular regions, Tin = 25

°C, Tout = 24 °C

24.250 24.496 24.496 - (m:

24.496, b:

24.496, t:

24.496)

24.340 - (m:

24.496, b:

24.261, t:

24.261)

24.340 - (m:

24.496, b:

24.261, t:

24.261)

Two homogeneous


°C, Tout = 300 °C

300.250 300.503 300.503 - (m:

300.503, b:

300.503, t:

300.503)

300.345 - (m:

300.503, b:

300.266, t:

300.266)

300.345 - (m:

300.503, b:

300.266, t:

300.266)

Two homogeneous


°C, Tout = 22 °C

23.500 24.856 24.856 - (m:

24.856, b:

24.856, t:

24.856)

23.938 - (m:

24.856, b:

23.479, t:

23.479)

23.938 - (m:

24.856, b:

23.479, t:

23.479)

Two homogeneous


°C, Tout = 297 °C

298.500 300.093 300.093 - (m:

300.093, b:

300.093, t:

300.093)

299.138 - (m:

300.093, b:

298.66, t:

298.66)

299.138 - (m:

300.093, b:

298.66, t:

298.66)

Two equal homogeneous

sections, Tup = 25 °C,

Tdown = 24 °C

24.500 24.496 24.496 - (m:

24.496, b:

24.496, t:

24.496)

24.500 - (m:

24.500, b:

24.000, t:

25.000)

24.496 - (m:

24.496, b:

24.496, t:

24.496)

44

Temperature profile True volumetric

mean

temperature of

section, Tsec,vol

[°C]

Ultrasound

temperature,

one vertical

path, Tus,v1

[°C]

Ultrasound

temperature,

three center

paths, Tus,c3 [°C]

Ultrasound

temperature, 3

horizontal

paths, Tus,h3 [°C]

Ultrasound

temperature, 3

vertical paths,

Tus,v3 [°C]



Tdown = 300 °C

300.500 300.503 300.503 - (m:

300.503, b:

300.503, t:

300.503)

300.5 - (m:

300.500, b:

300.000, t:

301.000)

300.503 - (m:

300.503, b:

300.503, t:

300.503)



Tdown = 22 °C

25.000 24.856 24.856 - (m:

24.856, b:

24.856, t:

24.856)

25.000 - (m:

25.000, b:

22.000, t:

28.000)

24.856 - (m:

24.856, b:

24.856, t:

24.856)



Tdown = 297 °C

300.000 300.093 300.093 - (m:

300.093, b:

300.093, t:

300.093)

300.000 - (m:

300.000, b:

297.000, t:

303.000)

300.093 - (m:

300.093, b:

300.093, t:

300.093)



Tdown = 5 °C

17.500 15.161 15.161 - (m:

15.161, b:

15.161, t:

15.161)

17.5 - (m: 17.5,

b: 5, t: 30)

15.161 - (m:

15.161, b:

15.161, t:

15.161)



Tdown = 300 °C

315.000 318.149 318.149 - (m:

318.149, b:

318.149, t:

318.149)

315.000 - (m:

315.000, b:

300.000, t:

330.000)

318.149 - (m:

318.149, b:

318.149, t:

318.149)

Table 20: Error in ultrasound temperature due to non-representativity of paths. In the cell the mean temperature of the paths is shown, together with the individual paths (m = middle path, b = bottom, diagonal or off-center vertical path, t = top, diagonal or off center vertical path). In case of two homogeneous regions, the horizontal path at 50% is assumed to contain fluid of the average temperature. (Note: part of the temperature error is due to the ultrasonic averaging effect described in Table 4.)

Appendix B: Full uncertainty budget In this appendix an extensive uncertainty budget of the ultrasound temperature method is shown,

see Table 21. The numerical values are estimates and may need adjustment for a particular case.

Furthermore in experimental work the estimates may prove to be too conservative or too optimistic.

For a homogeneous temperature measurement at ambient temperature the uncertainty is

estimated to be 0.03 °C in this particular case, whereas in the case of an irregular inhomogeneous

temperature distribution with temperature differences up to 6 °C, the uncertainty is estimated to be

0.24 °C (both estimates at coverage factor k = 2). The measurement equations are as follows

(symbols are explained in Table 21):

45

t = tm – tdel

L = L0 + δLT + δLP

δLT = L·α·(Tw – T0)

δLP = β·(P – P0)

β = D/(2·b·E)

c = L/t

Tus = Tc-P + δTcal + δTcomp + δTc + δTP + δTcorr(c,P) + δTnon-lin + δTnon-repr

Transit time uncertainty

Variable xi Description Unit Value Expand

ed

uncerta

inty

U(xi)

Expand

ed

relative

uncerta

inty

U*(xi)

Probabi

lity

distribu

tion

Divis

or

Sensiti

vity

coeffici

ent ci

Standar

d

uncerta

inty

u(xi)

Standar

d

relative

uncerta

inty

u*(xi)

Expand

ed

uncerta

inty

U(xi)

Relative

expand

ed

uncerta

inty

U*(xi)

%

tm Average measured transit time s 6.9E-

05

1.0E-09 0.00% normal 2.00 1.00 5.0E-10

tdel Delay of electronics in US sensor s 2.0E-

06

5.0E-10 normal 2.00 -1.00 2.5E-10

t Average corrected transit time s 6.7E-

05

normal 2.00 5.6E-10 0.001% 1.1E-09 0.0017

%

Path length uncertainty


ed

uncerta

inty

U(xi)

Expand

ed

relative

uncerta

inty

U*(xi)

Probabi

lity

distribu

tion

Divis

or

Sensiti

vity

coeffici

ent ci

Standar

d

uncerta

inty

u(xi)

Standar

d

relative

uncerta

inty

u*(xi)

Expand

ed

uncerta

inty

U(xi)

Relative

expand

ed

uncerta

inty

U*(xi)

%

L0 Path length at T0 and P0 m 0.100 1.0E-06 0.00% rectang

ular

1.73 1.00 5.8E-07

46

δLT Temperature correction of L m 4.8E-

06

1.6E-06 34.08% rectang

ular

1.73 1.00 9.5E-07

- T0 Reference temperature °C 20 0.1 0.50%

- Tw Average wall temperature °C 23 1 4.35%

- α Linear thermal expansion coefficient 1/°C 1.6E-

05

1.0E-06 6.25%

δLP Pressure correction of L m 1.5E-

06

7.3E-08 5.00% rectang

ular

1.73 1.00 4.2E-08

- P0 Reference pressure barg 0 0.001

- P Actual process pressure barg 5 0.01 0.20%

- β Linear pressure expansion coefficient

for thin walled pipes

1/bar

g

2.9E-

06

1.5E-07 5.00%

- D Pipe diameter m 0.1 n.a.

- b Pipe wall thickness (thin wall) m 0.009 n.a.

- E Pipe Young modulus Pa 2.0E+1

1

n.a.

L Path length at T and P m 0.100 normal 2.00 1.1E-06 0.001% 2.2E-06 0.0022

%

Speed of sound uncertainty


ed

uncerta

inty

U(xi)

Expand

ed

relative

uncerta

inty

U*(xi)

Probabi

lity

distribu

tion

Divis

or

Sensiti

vity

coeffici

ent ci

Standar

d

uncerta

inty

u(xi)

Standar

d

relative

uncerta

inty

u*(xi)

Expand

ed

uncerta

inty

U(xi)

Relative

expand

ed

uncerta

inty

U*(xi)

%

L Path length (from above) m 0.100 2.2E-06 0.0022

%

normal 2.00 1.5E+0

4

1.1E-06

t Corrected transit time (from above) s 6.7E-

05

1.1E-09 0.0017

%

normal 2.00 -

2.2E+0

7

5.6E-10

c Speed of sound m/s 1492.

631

normal 2.00 0.021 0.001% 0.041 0.003%

47

Speed of sound based average cross section

temperature uncertainty


ed

uncerta

inty

U(xi)

Expand

ed

relative

uncerta

inty

U*(xi)

Probabi

lity

distribu

tion

Divis

or

Sensiti

vity

coeffici

ent ci

Standar

d

uncerta

inty

u(xi)

Standar

d

relative

uncerta

inty

u*(xi)

Expand

ed

uncerta

inty

U(xi)

Relative

expand

ed

uncerta

inty

U*(xi)

%

Tc-P =

f(c,P)

Temperature derived from c and P °C 23.21 0.027 0.12% rectang

ular

1.73 1 0.015

- c Speed of sound (from above) m/s 1492.

631

0.041

- P Process pressure (from above) barg 5 0.01

δTcal Corr./Unc. of calibration reference

temperature

°C 0 0.02

δTcomp Corr./Unc. due to impure water

composition

°C 0 0.01

δTc Corr./Unc. due to uncertainty in c °C 0 0.015

- df/dc Sensitivity coefficient df/dc °C/(m

/s)

0.355 n.a.

δTP Corr./Unc. due to uncertainty in P °C 0 -0.0006

- df/dP Sensitivity coefficient df/dP °C/ba

r

-0.06 n.a.

δTcorr(c,P) Correlation between c and P due to

pipe pressure expansion

°C negligi

ble for

final

result

n.a.

δTnon-lin Corr./Unc. of T due to averaging of 1/c

instead of averaging T inclusive non-

linearity of T=f(c,P) for dT = 6 °C in

pipe

°C 0 0.14 rectang

ular

1.73 1.00 0.08

δTnon-repr Corr./Unc. due to non-

representativeness of the

measurement paths due to

temperature profile

°C 0 0.15 rectang

ular

1.73 1.00 0.09

Tus Corrected cross section average

temperature

°C 23.21 normal 2.00 0.12 0.5% 0.24 1.0%

Table 21: Theoretical uncertainty budget for an ultrasound based temperature measurement method.

emrp eng06 - d3.4: an ultrasound based temperature ... · an ultrasound based temperature...

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