http://www.diva-portal.org This is the published version of a paper published in Experimental Thermal and Fluid Science. Citation for the original published paper (version of record): Lundström, H. (2021) Investigation of heat transfer from thin wires in air and a new method for temperature correction of hot-wire anemometers Experimental Thermal and Fluid Science, 128: 110403 https://doi.org/10.1016/j.expthermflusci.2021.110403 Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Permanent link to this version: http://urn.kb.se/resolve?urn=urn:nbn:se:hig:diva-35964
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http://www.diva-portal.org
This is the published version of a paper published in Experimental Thermal and FluidScience.
Citation for the original published paper (version of record):
Lundström, H. (2021)Investigation of heat transfer from thin wires in air and a new method for temperaturecorrection of hot-wire anemometersExperimental Thermal and Fluid Science, 128: 110403https://doi.org/10.1016/j.expthermflusci.2021.110403
Access to the published version may require subscription.
N.B. When citing this work, cite the original published paper.
Permanent link to this version:http://urn.kb.se/resolve?urn=urn:nbn:se:hig:diva-35964
Experimental Thermal and Fluid Science 128 (2021) 110403
Investigation of heat transfer from thin wires in air and a new method for temperature correction of hot-wire anemometers
Hans Lundstrom Department of Building Engineering, Energy Systems and Sustainability Science, University of Gavle, SE-801 76 Gavle, Sweden
A R T I C L E I N F O
Keywords: Hot-wire anemometer Fluid temperature variations Temperature correction
A B S T R A C T
Variations in fluid temperature are the most important error source in hot-wire anemometer measurements and must sometimes be compensated for. Although many temperature compensation schemes have been proposed over the years, no consensus seems to be reached regarding correction methods. In this paper new precision measurements on heat transfer from heated wires have confirmed that, provided the fluid properties are eval-uated at the mean temperature between the air and sensor temperatures, the Nusselt number is a consistent function of the Reynolds number without any further corrections. Based on this knowledge a new calibration function which accounts for temperature dependencies in fluid properties is proposed. The aim has been to come up with a calibration function that is useful in practical anemometry. The only parameters in the function are air- and sensor temperatures. Velocity calibration is only necessary at one air temperature because temperature dependencies on conductivity and viscosity are incorporated in the calibration function.
1. Introduction
1.1. Background
A hot-wire anemometer is equipped with a sensor consisting of a thin metal wire usually made of platinum or tungsten, typically 5 μm in diameter and a few millimeters long. The sensor is heated by means of Joule heating. It is usually operated in constant temperature mode, which means that electric current through the sensor is controlled by means of electronic control circuitry, so that the temperature of the sensor wire is kept at a constant value irrespective of the surrounding velocity- and temperature- fields. Ignoring heat losses through radiation and end conduction, the electrical power delivered to the sensor to maintain its temperature is a measure of the heat transfer from the sensor to the air and can be related to velocity. This is achieved through calibration to known air velocities.
Besides cooling velocity, a number of parameters affect the heat transmission, but variations in fluid temperature are the most important error source in hot-wire anemometry, and to measure velocity accu-rately the temperature dependence must be eliminated. A typical situ-ation is when measurements are performed at an air temperature other than the calibration temperature. Air flows with temperature fluctua-tions may also be measured if the fluctuations are slow enough to be
captured by a temperature sensor near the hot wire. Due to the lack of an exact theory for convective heat transfer from a wire, temperature compensation techniques must be investigated empirically. A number of compensation schemes have been proposed over the years. Some are just variations of earlier proposals and some are contradictory. This study attempts to clarify the possible effects from variations in air temperature on a hot-wire anemometer and to provide a practical usable correction method for anemometry.
1.2. The constant temperature anemometer
The output from a constant temperature anemometer (CTA), is a voltage proportional to the sensor wire voltage. As the sensor resistance is constant, the output voltage squared represents the heat transferred from the sensor and the output voltage may be correlated to velocity by means of calibration to a known air flow. This correlation is only valid for measurements at the same air temperature as at calibration. For a CTA the sensor temperature TS is constant and any variations in the air temperature TA will also introduce variations in ΔT (the temperature difference between the wire and the air) and consequently also in the anemometer output voltage. In addition, errors also arise through the temperature dependence of the fluid properties.
Table 1 shows the measured anemometer output for an air temper-ature change of 5 K for three different velocities and three different
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sensor operating temperatures. The equipment used was a Dantec1
56C01 anemometer system with a 56C17 bridge and 55P11 wire probe. Sensor temperatures are the mean temperatures adjusted by means of the anemometer bridge. The probe was calibrated at an ambient tem-perature of 20 ◦C, and measurements were taken at 25 ◦C. Errors given in the table are in percent per degree temperature change.
We note that significant errors in measurements arise from a rather small variation in fluid temperature with an uncompensated CTA. Larger errors for lower sensor temperatures are caused by the variations in ΔT but we see that errors also increase with decreasing velocity. This ve-locity dependence is obviously a complication when searching for a temperature correction scheme. We will examine this issue below.
1.3. Convection heat transfer from a thin wire
Convection is the combined effect of conduction and heat transport due to motion of the fluid. The main thermal resistance in convection heat transfer between a surface and a viscous flow is exerted by the boundary layer which develops on the surface. The lower the thickness of the boundary layer and the higher the thermal conductivity of the fluid, the higher the heat transfer rate will be. To obtain a quantitative description of this process the energy transport through the boundary
layer is characterized by the heat transfer coefficient h, defined as:
Q = h⋅a⋅ΔT (1)
where Q denotes the amount of thermal energy transferred to the air per unit time, h is the average heat transfer coefficient over the wire area, a is the area of the wire surface and ΔT is the difference between the wire temperature TS and ambient air temperature TA. Thus ΔT = TS − TA. The heat transfer coefficient is unique for a given system. In the idealized case of an infinitely long heated cylinder in air crossflow, h depends on the dimensional variables: fluid velocity, fluid properties and geometry. If the wire is heated by Joule heating and provided that all heat trans-ferred from the wire to the environment is due to forced convection, Q is equal to the electric power P delivered to the wire. Consequently, we may express the heat transfer coefficient as:
h =P
a⋅ΔT=
Pa(TS − TA)
=E2
Rs⋅a(TS − TA)(2)
where P denotes the electric power for the Joule heating, E is the voltage over the sensor wire and RS is the sensor resistance. For a given sensor wire and with constant fluid properties, h is a function of fluid velocity alone. Thus, by knowing P, TS and TA we can calculate h, and by cali-bration to known velocities establish the relationship:
h = f (U) (3)
where U is air velocity and f denotes a functional dependence. For gases, where the Prandtl number is almost invariant, this may also be expressed in nondimensional form by means of the Nusselt number Nu and the Reynolds number Re as:
Nu = f (Re) (4)
The Nusselt and Reynolds numbers are defined as:
Nu =h⋅dk
Re =U⋅d
υ (5)
where d is a characteristic length (for a cylindrical shape, d equals the diameter), and k and ν are thermal conductivity and kinematic viscosity for air respectively. Eq. (4) conforms to the grouping predicted by dimensional analysis but is only of principal character. As the underly-ing equations for heat transfer from a cylinder cannot be solved, there is still insufficient knowledge to formulate a more detailed relation. Nevertheless Eqs. (4) and (5) reveal the principal influence from the fluid properties k and ν on the heat transfer process.
We may also note that for cylinders with diameter in the µm range in low velocity air flow the boundary layer thickness is of the same order as the diameter. Furthermore, the mean free path for air at atmospheric pressure is approximately one-tenth of the wire diameter, so the crite-rion for applying the boundary layer approximation is barely fulfilled. However, as long as density changes are small, continuum flow as-sumptions are still adequate for anemometer sensors and any molecular effects appear as small corrections to the continuum heat transfer and are absorbed in the calibration constants [1,2].
1.4. Prior work on temperature correction of anemometers and proposed temperature correction schemes
Air properties k and ν vary fairly linearly with temperature within the actual range for anemometric use but with different temperature derivatives. The air properties will vary with the temperature gradient in the boundary layer which may lead to non-similarity effects when temperature changes. Consequently, Eq. (4) is not uniquely defined in a non-isothermal system as it is not clear which temperatures the air properties should be referred to. For appropriate measurements of ve-locity in flows with varying air temperatures we may use either a cali-bration function that accounts for the variations in air properties or a
Nomenclature
a wire surface area d wire diameter h heat transfer coefficient k thermal conductivity of air kTm thermal conductivity evaluated at the film temperature Nu Nusselt number NuTm Nu with properties evaluated at film temperature P electrical power supplied to wire Q heat loss from wire due to convection Re Reynolds number ReTm Re with properties evaluated at film temperature TA air temperature TAcal air temperature at calibration occurrence Tm film temperature TS mean wire temperature ΔT TS – TA U air velocity ν kinematic viscosity νTm kinematic viscosity evaluated at film temperature
Table 1 Measured errors for a typical CTA due to a change in fluid temperature.
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correction function that corrects the measured values. In either case the influence from temperature- induced variations in air properties on the heat transfer process must be known and this is a long-standing issue. Attempts have been made to include the temperature variation of physical properties in theoretical analyses. Analytical investigations of the laminar two-dimensional flow around circular cylinders at low Reynolds numbers go back to the early work of Stokes who, however, found that the accompanying boundary layer problem had no solution. Oseen modified Stokes equations and found an approximate solution to the problem and analytical solutions based on the Oseen approximation have been obtained by for instance Cole and Roshko [3] and Wood [4]. Hieber and Gebhart [5] presented a similar solution based on the method of matching two asymptotic expansions of the temperature as described by Proudman and Pearson [6]. However, analytical solutions are rather complicated and involve some compromises; in most situa-tions empirical investigations give more reliable results [7]. The vast majority of the empirical investigations on the subject consists of modifications to the classical relation Nu = A + BRen. Under isothermal conditions A, B and n are constants but become variables depending on both temperature and velocity if the air temperature varies.
In the early twentieth century Nusselt introduced the idea that there is a single temperature, between the wire temperature and the free stream temperature, at which all the fluid properties involved can be evaluated to give a correct correlation over a range of velocities and air temperatures. This has been known as the “reference temperature method”. Different ways for finding a suitable reference temperature have been proposed, see e.g. Douglas and Churchill [8] for a review. For engineering calculations in heat transfer the mean of the surface- and free stream temperatures, Tm, also called “film temperature”, is often chosen as the reference temperature. The use of the film temperature as reference temperature has also been considered for correlations in anemometry by many workers but the results from these investigations are diverging. Some investigators had to introduce a correction factor, often called “temperature loading factor” based on different tempera-ture relations to make their data fit the equations. Here we can mention Hilphert [9] and the widely recognized work by Collis and Williams [1]. Later Koch and Gartshore [10] reported that they had to modify the temperature loading factor in [1] to make the equation fit their data, and in 1990 Kostka and Vasanta Ram [11] argue, after meticulous study, that the film temperature concept is not suitable for accurate instrument work such as hot-wire anemometry. However, Andrews et al. [12], Douglas and Churchill [8] and later Hultmark and Smits [13] reported good results with the expression Nu = f(Re) without correction factors and with properties evaluated at Tm. There is no doubt that the reference temperature method is promising but more validation is necessary along with adequate knowledge about its applicability.
For the sake of completeness, we also mention a few other proposals on the subject: Davies and Fisher [14] used a different approach and developed a relationship for the heat transfer by evaluating the thermal conductivity at the wire temperature and the kinematic viscosity at the fluid temperature. Bearman [15] proposed evaluation of both properties at the sensor temperature. For low overheats and low velocity ranges Lundstrom et al. [16] evaluated the properties at the air temperature, thereby eliminating the need to know the sensor temperature. Manshadi and Esfeh [17] proposed another method; they kept the air properties constant and chose to combine all temperature dependencies into a single temperature loading factor.
1.5. Present paper
Looking back, it remains clear that there is great divergence in the results from prior investigations. In 1975 Morgan [18] concluded, after reviewing various correlations, that the dependence on physical prop-erties was still undefined and declared the need for additional experi-mental work, and this has not lost its validity despite the decades that have passed Notwithstanding the lack of an analytical description of the
convective heat transfer from wires, insufficient accuracy in the mea-surements must at least be a partial explanation for the discrepancies encountered. A great deal of the experimental work described above was performed decades ago, and measuring equipment and techniques (as well as knowledge) have evolved over the years. In reviewing the existing literature, it remains clear that no general rules for temperature dependence of heat transfer from wires have been found. This state of affairs and the fact that no theoretical solution exists has highlighted the necessity of a new close experimental investigation of the problem. In order to enable a better understanding of the underlying mechanisms for the temperature sensitivity of a hot-wire anemometer new empirical investigations on heated wires have been performed (with hopefully at least somewhat higher accuracy than in earlier experiments). Exami-nation of the measured data with application to constant temperature anemometry revealed that the Nusselt number is a consistent function of the Reynolds number provided air properties are evaluated at the mean temperature. A new temperature correction method based on these data is presented. In this method a function, which includes the temperature dependencies for heat conductivity and viscosity, is used for correlating calibration data for the anemometer.
The paper is organized as follows: In the next section the new in-vestigations which include measurements of heat transfer from two wires of diameter 2.5 and 5 µm, with length-to-diameter ratios 300 and 10,000 are presented. In Section 3 the measured data are analyzed and it is concluded that using the reference temperature method is the most promising way to go and in the following Section 4, the reference tem-perature method is investigated. Continuing from this a new formula for correlation of anemometer data is presented in Section 5, which is the main results part of the paper. Finally, discussion and conclusions are given in Sections 6 and 7.
2. Measurements of forced convection heat transfer from thin heated wires in air
The objective of this section is to examine the nature of convective heat transfer from fine wires in air, as well as to develop the means to perform corrections for temperature dependencies. We particularly want to clarify if the mean temperature is suitable for evaluation of heat conduction and viscosity. Measurements were made on two platinum- plated tungsten wires of diameters 2.5 and 5 µm. Table 2 shows the different wires and investigated temperature and velocity ranges.
2.1. Method and measurement techniques
The wire under test was mounted between prongs and connected to a variable power supply through a series resistor and a current shunt resistor. The voltages over the wire and the shunt resistor were measured using a data acquisition unit Keysight 34970A2, and the wire resistance as well as the heat dissipation rate in the wire were calculated. The wire temperature was calculated from the resistance by means of calibrated data.
Measurements for both wires in the velocity range 0.3–3 m/s were performed in a small closed loop wind tunnel where air velocity and temperature could be varied independently in the ranges 0.3–3 m/s and 20–50 ◦C respectively. For wire 1 complementary measurements in the
Table 2 Investigated wires.
Wire No
d [µm] l/ d ratio
Wire temp [◦C]
Air temp range [◦C]
Velocity range [m/s]
1 2.5 10,000 70 20 and 40 0.3–30 2 5 300 70 20 and 50 0.3–3
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5–30 m/s range were made in an open circuit calibration wind tunnel. For selected values of the wind tunnel temperature and velocity, the current through the tested wire was adjusted manually to achieve a desired ΔT, and steady state measurements were then performed as mean values over a one-minute period. By this procedure a map of data for a number of air velocities, air temperatures and wire temperatures was achieved. From that map, data for different operating conditions of the sensor wire could be assembled. For the present investigation measuring points with equal wire temperature were gathered to reveal the behavior of the sensor wire as a constant temperature anemometer.
Before the heat transfer measurements, each wire was temperature calibrated in a calibration chamber with circulating air in the range 15–75 ◦C and the resistance–temperature data were fit to a second-order polynomial. To maintain high accuracy in the heat transfer measure-ments, it is important that the wire temperature is confirmed by cali-bration instead of relying on calculations from a linear approximation of resistance–temperature relationship, which is the usual procedure in hot-wire anemometry measurements. As the maximum attainable tem-perature in the calibration chamber was 75 ◦C it was decided to limit the wire temperature in this investigation to no more than 70 ◦C. This is lower than what is common in hot-wire anemometry measurements but the heat transfer mechanisms should be the same. The use of calibrated values for both air and wire temperatures, as well as the simplicity of the electronic circuitry and the absence of complete anemometer equipment (which might manipulate the signals in unpredicted ways) provides for high accuracy in these measurements.
2.2. Uncertainties
An uncertainty budget for the measured heat transfer will comprise estimated errors in temperature sensors and associated electronics, er-rors in the measurements of the electrical power delivered to the wires and errors in air velocity measurements. Uncertainties are given as expanded uncertainties, estimated at approximately 95% confidence level. For temperature calibrations a S12243 precision thermometer with calibrated uncertainty of ±0.005 K was used. However, due to stability and resolution of the wire sensors and instrumentation used, maximum uncertainty for (TS − TA) is estimated at ±0.05 K corre-sponding to no more than ± 0.2%. The electrical power is measured with uncertainty <±0.2% and the uncertainty for the air velocity is estimated to ±(0.02 m/s + 0.5%) for velocities up to 3 m/s (i.e. in the smaller precision wind tunnel) and ± (0.05 m/s + 0.5%) for higher velocities. The turbulence intensity was less than 0.7% and 0.5% respectively.
No influence from end effects could be observed and it is widely accepted that, for thin wires, radiation influence is negligible and nat-ural convection can be ignored for velocities above 0.2 m/s (e.g. Orlü [19]).
3. Heat transfer investigations based on the measured data
3.1. Wire 1
Fig. 1 shows the measured data plotted as U = f(P) for Wire 1 at two air temperatures 20 and 40 ◦C respectively and with the wire tempera-ture constant at 70 ◦C. Thus, these data are representative for a constant temperature anemometer in air flow of varying temperature.
The power consumption of the wire is affected in two ways by air temperature variations: changes in ΔT and temperature-induced varia-tions in the fluid parameters. For a CTA the sensor temperature TS is constant and any variations in the air temperature TA will also introduce variations in ΔT. In fact, the essential part of the temperature-induced error for a CTA is due to the change in ΔT. For the case of constant properties, the CTA is characterized by the equation:
U = f(
PTS − TA
)
(6)
which is plotted in Fig. 2. The curves show measurements corrected for the change in ΔT and thus the remaining differences are due to varia-tions in air properties. In the lower diagram, which shows the difference between the curves, we note a positive temperature dependence of the heat transfer for low velocities up to approximately 6 m/s (Re = 1) and negative dependence for higher velocities. Similar behavior was also observed by Kanevce and Oka [20] and and Bruun [21].
Apparently, at low velocities, the temperature dependence caused by variations in air properties increases rapidly with decreasing velocity (cf. data in Table 1). In order to reveal how the fluid parameters exercise influence on the heat transmission we use the well-known correlation function Nu = A+BU0.5 where A and B are constants. Converting to measurable quantities by inserting Eq. (5) gives:
h⋅dk
= A + B(
U⋅dν
)0.5
(7)
where d is a characteristic length, in this case the wire diameter. Rear-ranging and absorbing d in A and B results in:
h = A⋅k + Bk
ν0.5U0.5 (8)
from which we immediately can infer the impact of k and ν. Remembering that the temperature dependence for the properties
may be expressed as:
kkref
=
(T
Tref
)0.84 ννref
=
(T
Tref
)1.78
(9)
where kref , νref and Tref are reference values. We note that k varies about 0.3%/K in the room temperature range while the temperature depen-dence for the group k/ν0.5 is − 0.02%//K.
We can see that at low velocities the heat transfer is dominated by conduction/diffusion in the boundary layer resulting in a positive tem-perature coefficient but as velocity increases the temperature depen-dence will decrease due to influence from the viscosity. Thus, the positive temperature dependence at low velocities (diffusion dominates) gets lower with increasing velocity and eventually turns negative (advection dominates). Apparently, the temperature dependence of the heat transfer moves towards the value of heat conduction for air when the velocity approaches zero.
3.2. Wire 2
In Fig. 3 measured data for Wire 2 are plotted. The physical di-mensions of Wire 2 conform well to sensor wires used in anemometry. The curves show measured values corrected for the change in ΔT and
Fig. 1. U vs. P for Wire 1 at wire temperature 70 ◦C and air temperatures 20 and 40 ◦C in the velocity range 0.3–30 m/s.
3 SYSTEMTEKNIK AB, Sweden.
H. Lundstrom
Experimental Thermal and Fluid Science 128 (2021) 110403
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with air properties constant. As velocities are low, heat transfer shows a positive temperature
dependence over the whole investigated velocity range. We note a temperature sensitivity of up to 0.6%/K and a clear velocity dependence.
4. Investigating the reference temperature method
The investigations above reveal how the interaction between temperature-induced variations in the heat conduction and viscosity imposes a velocity dependency in the temperature sensitivity of the heat transfer process. We conclude that it may not be appropriate or possible to achieve accurate temperature corrections only by multiplying measured heat transfer data by a temperature-dependent factor. It fol-lows that it is deemed necessary that k and ν be incorporated in the correlation function so that their respective temperature dependencies can act upon the heat transfer function as in the real mechanisms of action. This might possibly be achieved by means of the “reference temperature method”. To move forward we undertake an investigation of this method for temperature correction of an anemometer.
4.1. Identifying the reference temperature
The reference temperature method assumes that if both k and ν are calculated at a certain reference temperature, between the air and sensor temperatures, the heat transfer will be invariable to changes in air temperature. The reference temperature may be expressed as Tref = TA +c(TS − TA) where c is a constant in the range 0–1. The mean temperature between TA and TS, denoted Tm (c = 0.5), is often recom-mended in the literature for calculation of heat transfer from flat and curved surfaces in laminar flow. However, the diverging experience
Fig. 2. U vs. P/ΔT for Wire 1 at wire temperature 70 ◦C and air temperatures 20 and 40 ◦C in the velocity range 0.3–30 m/s. Insert shows data for the very low velocities. The lower diagram shows the difference between the curves calculated as: UTA=40 − UTA=20
UTA=20100 .
Fig. 3. U vs. P/ΔT for Wire 2 at wire temperature 70 ◦C and air temperatures 20 and 50 ◦C in the velocity range 0.3–3 m/s.
H. Lundstrom
Experimental Thermal and Fluid Science 128 (2021) 110403
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from use of Tm in anemometry reported in earlier papers calls for a renewed investigation.
For this purpose, the measured data for the two investigated air temperatures are combined into one data set. For some selected values of reference temperatures, a fourth-order polynomial regression is per-formed on the combined data set and the R2-factor is noted. This was performed for both Wire 1 and Wire 2, see Figs. 4 and 5.
We note a pronounced maximum for c-values near 0.5 for both wires. This confirms the choice of Tm as a suitable temperature for evaluation of fluid properties. This has been shown to be valid for 0.05 < Re < 4.7, TS = 70◦ with TA in the range 20–50 ◦C and for l/d of 10,000 as well as 300. The latter is a common size in hot wire anemometry.
In Figs. 6 and 7 the measured data from the two investigated wires are plotted in nondimensional variables ReTm and NuTm, and with TS constant (ReTm and NuTm denotes Re and Nu evaluated at Tm). We note the good correlation between the curves for both wires4.
5. A new calibration function, built on the reference temperature method, that corrects for temperature induced variations in fluid properties as well as in ΔT
The finding that NuTm = f(ReTm) really is a temperature-independent correlation function for a constant temperature anemometer and no additional corrections are necessary opens the possibility to establish a calibration function which includes air properties and their respective temperature dependencies. Such a function would be most beneficial in
practical anemometry. Combining Eq. (2), (4) and (5) and absorbing constants in the
function f in Eq. (10) we may formulate an appropriate relation for a practical hot-wire anemometer as:
UνTm
= f(
PkTm (TS − TA)
)
(10)
Here νTm and kTm denote kinematic viscosity and thermal conduc-tivity respectively evaluated at the film temperature and f is any func-tion that correlates the measured data. As only the temperature dependencies of the fluid properties are of importance here, there is no need to calculate numeric values for νTm and kTm.. Using Eq. (9) we can write Eq. (10) in the following form:
UTm
1.78 = f(
PTm
0.84(TS − TA)
)
(11)
and by inserting: Tm = TS+TA2 we arrive at:
U(TS + TA)
1.78 = f(
P(TS + TA)
0.84(TS − TA)
)
Account for air properties Accounts for ΔT
(12)
Here temperatures are in absolute temperatures and f indicates a functional dependence, not the same in all expressions. The factors (TS +
TA)− 1.78 and (TS + TA)− 0.84 account for temperature dependencies in the
Fig. 4. R2-factors for regression of datafor Wire 1 at different reference temperatures.
Fig. 5. R2-factors for regression of data for Wire 2 at different reference temperatures.
Fig. 6. ReTm vs. NuTm for Wire 1 U = 0.3–30 m/s.
Fig. 7. ReTm vs. NuTm U = 0.3–3 m/sfor Wire 2.
4 Note that for calculation of nondimensional variables in this study the nominal values for length and diameter of the wires are used. These are only approximate and will not allow for comparisons between the numerical Nu- values in the curves for the two wires. This is no inconvenience for the present study, as for investigating temperature influence on the heat transfer it is enough to study each wire individually.
H. Lundstrom
Experimental Thermal and Fluid Science 128 (2021) 110403
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air properties. In the derivation above constant factors are absorbed in the function f and we assume that constants in f take dimensions to make the equation dimensionally consistent. The variables in Eq. (12) are numerically very small and it can be convenient to multiply both vari-ables by a factor (e.g. 106) for easier handling.
5.1. Application to Wire 1
The diagram in Fig. 8 shows the calibrated data from Wire 1 applied to Eq. (12) and we can see that the data collapse nicely to a single curve. The data in the lower curve shows the difference between the 20- and 40-degree curves and reveals that the remaining errors after tempera-ture correction using Eq. (12) are less than 0.15%/K (i.e. per degree air temperature change).
5.2. Application to Wire 2
Fig. 9 shows the data from Wire 2 applied to Eq. (12). We note a maximum remaining error of 0.05%/K after correction.
5.3. Application to practical anemometer measurements
The operating conditions in the above investigation, with the low wire temperature of 70 ◦C and the large change in air temperature of 20 and 30 K are unusual for hot-wire anemometer measurements (but put high demands on the temperature correction procedure and are thus suitable for evaluation purposes). In order to demonstrate the applica-bility of the derived calibration function to conditions more realistic for anemometric use, measurements by means of a commercial anemometer are performed. The equipment used is a Dantec5 56C01 anemometer system with a 56C17 bridge and 55P11 wire probe. A sensor operating temperature of 240 ◦C is adjusted by means of the anemometer bridge. Calibrations are performed in the velocity range 1–30 m/s at ambient temperatures 23 and 29 ◦C in an open circuit calibration wind tunnel where temperature and velocity can be adjusted separately.
In practical anemometry the output from the anemometer is a voltage proportional to the sensor voltage. Accordingly, we use E2
instead of P and for a CTA Eq. (12) then converts to:
U(TS + TA)
1.78 = f
(E2
(TS + TA)0.84
(TS − TA)
)
(13)
where E is the output voltage from the anemometer. The measured data is shown in curve B in Fig. 10. We see the excellent collapse of the data for the two air temperatures.
It was noted that the proposed method shows weak dependence on static errors in sensor temperature, in the sense that, for moderate var-iations in air temperature, a difference between the actual sensor operating temperature and the value given for TS in the equation has low
Fig. 8. U(TS + TA)− 1.78 vs P(TS + TA)
− 0.84(TS − TA)
− 1 for Wire 1 at wire tem-perature 70 ◦C and air temperatures 20 and 40 ◦C in the velocity range 0.3–30 m/s.
Fig. 9. U(TS + TA)− 1.78 vs P(TS + TA)
− 0.84(TS − TA)
− 1 for Wire 2 at wire tem-perature 70 ◦C and air temperatures 20 and 50 ◦C in the velocity range 0.3–3 m/s.
Fig. 10. Curve B) U(TS + TA)− 1.78 vs E2(TS + TA)
− 0.84(TS − TA)
− 1at sensor temperature 240 ◦C and air temperatures 23 and 29 ◦C in the velocity range 1–30 m/s. Curve A) same data but using TS = 250 ◦C in equation. Curve C) Using TS = 230 ◦C in equation. 5 https://www.dantecdynamics.com.
Experimental Thermal and Fluid Science 128 (2021) 110403
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impact on the measurement precision. This is demonstrated in the curves A and C in Fig. 10 which show the calibration functions generated when the value for TS in Eq. (13) is varied in the span 230–250 ◦C, while the actual sensor temperature is maintained at 240 ◦C. Varying the value for TS affects ΔT and thus different calibration curves are generated. This has no effect on the measurements because the same calibration curve is used for calibration and measurements. However, no significant degrading in the correction for air temperature changes can be noticed. The R2-factors for the curves in Fig. 10 are 0.99985, 0.99988 and 0.99991 respectively.
Thus, the method tolerates comparatively large mismatch between the value given for TS and the actual sensor temperature. It must of course be pointed out that the grade of intolerance is dependent on the relation between the actual ΔT and the span of the temperature varia-tions we intend to compensate for. Nevertheless, it is an important feature because the sensor wire resistance is set by means of the anemometer equipment and the corresponding temperature is calcu-lated by means of the linear temperature coefficient of resistance for the wire and the value for the temperature coefficient given by wire man-ufacturers may differ from the actual value.
Although a certain deviation between the actual resistance value and the value used in the calibration formula is allowed, it is important that the resistance to temperature relation is stable throughout calibration and measurements. To minimize drift, it is highly recommended to heat treat an anemometer sensor before use. Experience of how this should be done differ between workers but there seems to be some consensus that operating the sensor at high temperature (preferably higher than during measurements) for a few hours should suffice [19,22]. Be aware that etched tungsten and drawn platinum wires behave quite differently during heat treatment [22].
6. Discussion
It appears there are very few temperature correction methods in earlier proposals, that claim to operate with calibration at only one air temperature. Here we mention Refs. [1] and [9], but although highly recognized in their time these methods have been shown not to be very accurate and may be considered as outdated today. Refs. [8], [12] and [13] have contributed by confirming the use of the mean temperature Tm but did not present a practical usable formula.
The method for temperature correction used the most nowadays seems to be the method first proposed by Kanevce and Oka in 1973 [20], which modifies the anemometer output in accordance with the actual air temperature. If the anemometer is calibrated at the air temperature TAcal and a correlation is made between E2 and U, then measurements may be performed at the air temperature TA, different from TAcal, if the output is modified using Eq. (14).
E2corr = E2TS − TAcal
TS − TA(14)
where E2corr is the modified output. By this method the factor
(TS − TAcal)− 1 which is inherent in E2, is replaced by (TS − TA)
− 1. This is
equal to using the calibration function in Eq. (6), and the remaining errors after correction are the ones presented in Fig. 2. The method only corrects for changes in ΔT but not for variations in air properties.
It is common nowadays to use calibration functions that correlate E to U and then Eq. (14) converts to:
Ecorr = E(
TS − TAcal
TS − TA
)0.5
(15)
Eq. (15) has more or less become a standard method, described in literature and recommended by anemometer manufacturers. In order to also correct for changes in air properties it has been proposed to experimentally adjust the exponent until a proper fit is achieved [23,24], but for this, calibration at a minimum of two air temperatures is
needed. There is little doubt that the present calibration function (Eq. (13)), with implicit correction for air property variations, will be more convenient to use compared to existing methods for temperature correction of hot-wire anemometers in the investigated ranges.
A remaining question is, however, under what conditions we can expect Tm to be a valid reference temperature. The present investigation has showed that the use of Tm is valid for Re ≤ 4.7. In this regime the flow completely wraps around the wire and the heat transfer is char-acterized by the boundary layer, but for higher Re the onset of separa-tion and circulation flow in the wake will inevitably change the characteristic of the heat transfer. It is left for future investigations to establish an upper limit for the use of Tm as reference temperature for correction of anemometer data.
7. Conclusion
Changes in air temperature affect the constant temperature anemometer in two ways: variations in ΔT but also in the air properties involved in the heat transfer mechanism. Variations in ΔT are easy to compensate for but the air properties impose a velocity dependence in the temperature sensitivity of the heat transfer process, which compli-cates the temperature correction. For proper temperature correction the temperature variations of heat conduction and viscosity must both be addressed.
In this paper careful experimental investigations have confirmed that, provided that the heat conduction and kinematic viscosity for air are evaluated at the mean temperature between wire temperature and ambient air temperature, the heat transfer from heated wires in air crossflow may be described by the relation Nu = f(Re) with no further corrections. The actual values for the air properties need not be deter-mined, only their respective sensitivity to temperature. As it was found that the governing temperature is the mean temperature, these sensi-tivities may be expressed as simple functions of air temperature and sensor temperature and included in the calibration function.
A new calibration function for constant temperature anemometers in varying air temperatures is proposed (Eqs. (12) and (13)). As the tem-perature dependencies to heat conduction and kinematic viscosity for air are incorporated in the function it will compensate for changes in air temperature during measurements. Velocity calibration is only needed at one air temperature and it is not necessary to calculate the air prop-erties for different temperatures.
It has been demonstrated that the proposed method works well in the velocity range 0.3–30 m/s (with a 2.5 µm sensor wire) and for temper-atures changes of up to 30 ◦C. The proposed function is convenient to use as the only parameters involved are the air and sensor temperatures.
Declaration of Competing Interest
The author declare no known competing financial interests or per-sonal relationships that could have appeared to influence the work re-ported in this paper.
Acknowledgments
Many thanks to Professor Mats Sandberg, University of Gavle, for helpful discussions. The support from Professor Ivo Martinac, KTH Royal Institute of Technology is gratefully acknowledged.
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