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Gardon gauge measurements of fast heat flux transients
Applied Thermal Engineering 100 (2016) 501–507
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Applied Thermal Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / a p t h e r m e n g
Research Paper
Gardon gauge measurements of fast heat flux transients Tairan Fu a,*, Anzhou Zong a, Jibin Tian a, Chengyun Xin b a Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Beijing Key Laboratory of CO Utilization and Reduction Technology, 2 Department of Thermal Engineering, Tsinghua University, Beijing 100084, China b School of Electric Power Engineering, China University of Mining and Technology, Xuzhou 221116, China
H I G H L I G H T S
• • • •
A fast heat flux transient measurement method using Gardon gauges is introduced. Simulations and experiments of heat flux transient measurements verify the method. Convert Gardon gauge with a limited response to fast-response heat-flux sensor. Gardon gauge measurement accuracy is improved for heat flux transients.
A R T I C L E
I N F O
Article history: Received 29 December 2015 Accepted 15 February 2016 Available online 23 February 2016 Keywords: Heat flux Transient radiative heat flux Gardon gauge Measurement method
A B S T R A C T
Heat flux measurements are widely used in thermal analyses. Gardon heat flux gauge are a widely used diffusion type heat flux gauge that can be used in harsh thermal environments such as in fires, combustors, and aerospace applications. However, Gardon gauges are usually regarded as quasi-steady-state sensors with a limited frequency response and are not often used for transient measurements. This paper introduces a fast radiative heat flux transient measurement method using Gardon gauges. The analysis assumes that the heat flux transient is a continuous function of time in every small/micro time interval. The transient heat flux density functions are determined from the transient output voltages of the gauge. Two cases with fast and slow heat flux transients were numerically investigated to verify the method. The analyses show that the gauge transient response must be corrected not only for fast heat flux transients, but also for slow heat flux transients. The gauges were then tested in experiments to measure radiative heat flux transients on a hot graphite plate in a vacuum chamber to evaluate the effects of the corrections. This method can use a limited frequency response Gardon gauge for fast heat flux measurements without other instrumentation adjustments. The Gardon gauges measurement accuracy is then greatly improved with transient measurements. The analysis shows how to accurately measure transient heat fluxes and improve the performance of Gardon gauges. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Heat flux measurements are often needed in thermal studies and have been widely used for engineering and research applications [1–8], for example, building and environmental studies, fire safety, electronic systems, material structures, combustion chambers and engines, and aerospace applications. Heat flux measurements are important for controlling heating processes, to assess device performance at high temperatures, to optimize systems for energy production and for designing the thermal protection systems for combustors in industrial and power engineering projects. The
* Corresponding author. Tel.: 86 10 62786006; fax: 86 10 62786006. E-mail address: [email protected] (T. Fu). http://dx.doi.org/10.1016/j.applthermaleng.2016.02.043 1359-4311/© 2016 Elsevier Ltd. All rights reserved.
interests in the measurements of unsteady heat transfer phenomena was increasing due to the need to understand the heat transfer process occurring in inherently unsteady environments. Transient heat flux measurements were widely needed in applications including high enthalpy plasmas, high power pulsed lasers, structural thermal tests and other fields. The well-established single-point heat flux gauges are widely available for various applications in thermodynamics and fluid. Measurement techniques of single-point heat flux, for example, null-point calorimeters, thin-film gauges, coaxial thermocouples, Schmidt–Boelter gauges, and Gardon gauges, have been developed to measure heat fluxes in various types of thermal environments. The transient null point calorimeter was first developed by Powars et al. [9] in 1972 and has been widely used to measure transient heat fluxes due to its simplicity, very wide flux range and favorable response time characteristics. Löhle et al. [10–12] made some
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significant improvements in the null-point calorimeter for high heat flux applications in a plasma wind tunnel facility. Thin-film gauges are classic fast-response sensors for very fast transient heat flux measurements with frequency responses up to 1 MHz [13–20]. They are usually fabricated by painting or sputtering a thin resistance layer onto an insulator substrate so they are usually fragile due to the weak layer-substrate bonding. Coaxial thermocouples [21–24] are more rugged than thin-film sensors and do not require frequent maintenance for transient heat flux measurements in harsh thermal environments. Irimpan et al. [23] designed a fast response coaxial thermocouple for shock tunnel applications, evaluated the coaxial thermocouple response against a platinum thin film gauge for heat flux measurements in a shock tunnel. However, the short working time characteristics of the thinfilm gauges and coaxial thermocouples greatly restrict their applications. Schmidt-Boelter gauges [21,25–28] and Gardon gauges [29–34] are available devices with excellent measurement characteristics with metal bodies, blackbody sensor foils and water cooled designs. Schmidt-Boelter and Gardon gauges are classified as diffusiontype or one-dimensional sensors that deduce the heat flux by measuring the temperature gradient in a material. The gauges are more suitable for high radiative or convective heat flux measurements with long working times in harsh environments such as in combustion flames, rocket motors, hyper sonic wind tunnels, etc. For example, Kidd et al. [21] developed a fast-response heat flux sensor, based on a modified Schmidt–Boelter gauge principle for heat transfer measurements in hypersonic wind tunnels. Gifford et al. [26] performed experiments to characterize the performance of Schmidt–Boelter heat flux gauges in stagnation and shear convective air flows. Nakos et al. [27] analyzed the heat transfer from an idealized Schmidt–Boelter heat flux gauge and showed that the theoretical sensitivity coefficients in radiative and convective environments differed. Sudheer et al. [28] measured the incident heat flux on a target for open gasoline pool fires with Schmidt– Boelter heat flux gauge measurements. The Gardon gauge (also called a circular foil heat flux gauge) was first introduced by Gardon in 1953 [29]. There have been many applications and calibrations of Gardon gauges [30–34]. The Gardon gauge is more useful for heat flux measurements over wide ranges than Schmidt–Boelter gauge so there is a great need for accurate Gardon gauges for measurements involving wide ranges of heat fluxes. Although Gardon gauges are widely used diffusion type heat flux gauges and have been extensively studied, the use of Gardon gauges in harsh environments is still restricted for transient applications. Gardon gauges as well as Schmidt–Boelter heat flux gauges are usually regarded as quasi-steady-state sensors with a limited frequency response and are rarely used in transient applications. It’s unable to obtain accurate heat fluxes using Gardon gauges in transient applications although the merit of high heat flux and long working times measurements is remarkable. Therefore, this paper describes an optimized measurement method of fast transient radiative heat fluxes based on a traditional Gardon gauge. The measurement accuracy of the Gardon gauges is also greatly improved for transient measurements. The merit of this work is to convert a traditional Gardon gauge with a limited frequency response to a fast-response heat-flux sensor which also has a wide range of heat fluxes and can be operated with long working times in thermal environments. The measurement principles of Gardon gauges are analyzed to improve the measurement method for transient heat fluxes. Numerical simulations and experimental measurements of typical heat fluxes verify the applicability and effectiveness of the measurement method. The analysis provides a useful reference for accurate measurement of transient heat fluxes and improves the measurement ability of Gardon gauges.
2. Principles 2.1. Traditional measurement method The Gardon heat flux gauge is a diffusion type heat flux. A sketch of a copper-constantan circular foil heat flux sensor is shown in Fig. 1. A circular constantan foil coated with a diffuse, highly absorbing coating is attached to a copper heat sink. The heat sink can be cooled to a low temperature so the gauge can be exposed to high heat flux environments. There is a large temperature gradient along the foil radial direction due to the effects of the heat flux on the foil surface. The metal leads are attached to the center and the edge (or the copper heat sink) of the foil to form a thermocouple joint. The voltage output of this thermocouple joint is proportional to the temperature difference across the foil. Therefore, the measured voltage gives a quantitative representation of the heat flux through the foil. The temperature gradient along the foil thickness is negligible and the heat losses from the back of the foil and from the center metal wire are also neglected. The foil surface of the gauge is assumed to be a highly absorbing gray diffuse surface. Then, the heat transfer equation and boundary conditions describing the radial temperature distribution in the foil are:
∂ T (r , τ ) ⎧ 1 ∂ ⎛ ∂T (r , τ ) ⎞ q (τ ) = ρc ⎜⎝ λr ⎟⎠ + ⎪ δ r r ∂ r ∂τ ∂ ⎨ ⎪T ∂ r =0 = T T = T ∂ T , , 0 0 ⎩ τ =0 r =R r =0
(1)
where T0 is the heat sink temperature, τ is the time, r is the foil radius axis, T (r, τ ) is the temperature at time τ and position r on the foil surface, q (τ ) is the heat flux reaching the foil surface due to convective or radiative heat transfer at time τ , δ is the thickness and R is the constantan foil radius. ρ is the foil density and c is the specific heat capacity of the foil which are both assumed to be constant. λ is the thermal conductivity of the constantan foil expressed as a linear function of temperature,
λ = λ0 (1 + b (T − T0 ))
(2)
where λ0 = 20.9 W/ (mK ) at T0 = 0°C and b = 0.00231 K −1 for a temperature range of 0~200 °C. The thermo-electromotive voltage output, E , of the copperconstantan thermocouple formed by the Gardon gauge is expressed as the function of the temperature difference, ΔT , between the center and the edge of the constantan foil as,
E = kΔT (1 + g ΔT ) where k = 0.0387 mV/K and g = 0.0012 K −1.
Fig. 1. Sketch of the Gardon heat flux gauge.
(3)
T. Fu et al./Applied Thermal Engineering 100 (2016) 501–507
The theoretical time constant τ c of the Gardon gauge is defined [29] to describe the transient response characteristics for heat flux,
τ c = ρcR 2 4λ0
(4)
The measurement sensitivity of the heat flux gauge, L , is defined as the ratio of the output voltage, E , to the measured heat flux, q . With the conventional measurement method, Gardon gauges are as quasi-steady-state sensors with q assumed to be a constant heat flux. Therefore, the Gardon gauge measurement sensitivity can be determined by assuming a constant heat flux and a constant thermal conductivity from Eqs. (1)–(3),
L=E q=
kR 2 4λ0δ
(5)
For the Gardon heat flux gauges, the measurement sensitivity is determined from experiment calibrations from the factory. The heat flux is then obtained from the measured output voltage and the known measurement sensitivity of the gauge using Eq. (5). However, for fast transient heat flux measurements, the measurement sensitivity of the Gardon gauge provided by the producer is not accurate because the heat flux is not constant.
503
determined when the C i is known. Equations (8) and (9) can then be combined as,
T (0, τ ) =
( (
a N∞ −ξa0βnτ J 2 ∑e λδ n=1 J12
) ∫ ∑C τ ′ β R)
βn R n
τ
τ ′=0
p
i
i −1 ξa0βnτ ′
e
dτ ′
(10)
i =1
The C i (i = 1, , p) are obtained from the measured center temperatures at m (m ≥ p) times using a least-squares algorithm or some other optimization algorithms. The center temperature, T (0, τ ) , can be deduced from the transient output voltage, E , of the Gardon gauge using Eq. (3). Therefore, the transient heat flux variation can be determined from the output voltage, E , using Eqs. (3), (9) and (10). The measurement frequency of output voltage and the choice of time interval in Eq. (9) determine the response characteristics of the transient heat flux measurements. For the traditional Gardon gauge measurement method in section 2.1, the constant heat flux is also obtained from the output voltage. Thus, this method enables a Gardon gauge with a limited frequency response to be a fast response heat flux sensor without any other instrument adjustments. The measurement accuracy of the Gardon gauges is also greatly improved in transient measurement applications.
2.2. Fast transient measurement method
3. Simulations
A measurement method for fast transient is introduced here for Gardon gauges. If the heat flux is constant, the foil surface temperature found by solving Eqs. (1) and (2) is,
The Gardon gauge transient heat flux measurement method was evaluated using simulations with fast and slow heat flux transients. The simulated Gardon gauge parameters were δ = 0.1 mm , R = 2.0 mm , T0 = 0°C , and a theoretical time constant τ c = 175 ms . The decrease of the gauge response time will reduce the measurement scale of heat flux. It’s difficult to fabricate a gauge with the time constant less than 50 ms. The standard measurement sensitivity of the gauge was obtained using Eqs. (1) –(5) as L = 0.0185 mV/ (kW/m2 ) . The simulated transient radiative heat flux, q , was applied to the foil surface of the Gardon gauge as the “true” heat flux, qtrue . The gauge output voltages at various times were calculated using Eqs. (3) and (8) and are reported as the “measured experimental values”. The heat flux calculated using the traditional measurement method was calculated using the standard measurement sensitivity from Eq. (5) and the “measured” output voltage and is defined as the “apparent” heat flux, qappa . The heat flux calculated using the transient measurement method using Eq. (10) based on the “measured” output voltage is defined as the “corrected” heat flux, qcal . The number of terms in the series, p , in Eq. (10) is assumed to be 4 ( p = 4 ) . The number of terms in the Bessel function was set to 15 ( N ∞ = 15) .
T = T0 +
R 2q 1 N∞ 1 − r 2 R 2 ) + ∑ Ane −ξa0βnτ J 0 ( 4ξλ0δ ξ n=1
(
βn r
)
(6)
qJ ( β R where ξ = 1 + b (T − T0 )and T = (T (r = 0) + T0 ) 2. An = − λδβ 2J 2 nβ ) R and n 1 ( n ) βn is the nth zeroth order Bessel function ( J 0 β R = 0 ). N ∞ is the number of terms in the Bessel function. If the heat flux is transient and a function of time, the foil surface temperature is obtained using the Duhamel theorem to solve Eqs. (1) and (6),
(
T (r , τ ) = ∫
τ
τ ′=0
=
)
q (τ ′ ) ∂T (r , τ − τ ′ ) dτ ′ δ ∂τ
( (
1 N∞ −ξa0βnτ J 2 ∑e ρcδ n=1 J12
)J( β R)
βn R
0
n
βn r
)∫
τ
τ ′=0
(7)
q (τ ′ ) eξa0βnτ ′ dτ ′
Then, the center temperature, T (0, τ ) , of the foil surface can be expressed as,
T (0, τ ) =
( (
1 N∞ −ξa0βnτ J 2 ∑e ρcδ n=1 J12
βn R
) )∫
βn R
τ
τ ′ =0
q (τ ′ ) eξa0βnτ ′ dτ ′
(8)
Equation (8) relates the center temperature, T (0, τ ) , to the transient radiative heat flux, q (τ ). When T (0, τ ) is known, q (τ )may be determined using an inverse analysis. However, unlike the equation for the constant heat flux used in the traditional measurement method with the Gardon gauge, an explicit expression for q (τ ) cannot be directly obtained due to the complex integration in Eq. (8). Therefore, assuming q (τ ) to be a continuous differentiable function of time in a small time interval (μs ~ ms), q (τ )can be expressed by a Taylor series expansion, p
q (τ ) = ∑ C iτ i−1
(9)
i =1
where p is the number of terms in the series and C i (i = 1, , p) are the series coefficients. The transient heat flux can then be
3.1. Case 1: fast heat flux transients For Case 1, the “true” fast heat flux transient was the sine function qtrue,1 = 500 sin (7.85τ ) kW/m2 . Thus, the maximum heat flux was 500 kW/m 2 with a period of 400 ms as shown in Fig. 2. The maximum of heat flux change frequency was up to 3.9 kW/ms. qappa,1 was calculated using the traditional method with qcal,1 calculated using the transient measurement method, as shown in Fig. 2. The absolute errors of qappa ,1 − qtrue ,1 and qcal ,1 − qtrue ,1 are shown in Fig. 3. The calculated “apparent” heat flux, qappa,1 , did not follow the rapid changes in the true transient heat flux, q1,true . The delay time of the peak response was up to 280 ms which is larger than the theoretical time constant τ c = 175 ms for the Gardon gauge. The peak qappa,1 is about 400 kW/m2 which is much less than the peak of the true heat flux. The absolute error, qappa ,1 − qtrue ,1 , in Fig. 3 has significant variations with a maximum error of 260 kW/m2. Therefore, qappa,1 is unable to accurately characterize the rapid changes in q1,true . The
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T. Fu et al./Applied Thermal Engineering 100 (2016) 501–507
500
Heat flux density, kW/m2
500
Heat flux, kW/m2
400
300
200
400
300
200
100
100
0
0
0
0.1
0.2
0.3
0.4
0
0.5
0.5
1
1.5
2
2.5
Time, s
Time, s
Fig. 4. Comparisons of ( qtrue,2 , qappa,2 , qcal,2 ) for a slow transient.
Fig. 2. Comparisons of ( qtrue,1 , qappa,1, qcal,1 ) for a fast transient.
limited Gardon gauge time constant makes limits the ability to accurately measure fast heat flux variations. However, the “corrected” heat flux, qcal,1, calculated using Eq. (10) agrees well with qtrue,1 . qcal,1 agrees well with q1,true at the peak response time point (0.2 s) and the discontinuity point (0.4 s). Thus, there is no obvious response time delay. The absolute error qcal ,1 − qtrue ,1 shown in Fig. 3 is very small over the entire time period. Therefore, this method to calculate qcal,1 greatly improves the traditional measurement response time of the Gardon gauge for fast heat flux transients. 3.2. Case 2: slow heat flux transients
where A and B are coefficients. Then, Eq. (8) is rewritten as,
T (0, τ ) =
∞
q (τ ) = Aτ + B
(11)
( (
⎡ N∞ J 2 ⎢∑ 2 ⎢⎣ n=1 J1
− ∑e
− ξa0βnτ
J2
βn R
) )
1⎛ A ⎞ Aτ + B − aξ βn ⎟⎠ βn R βn ⎜⎝
( (
J12
n=1
) 1 ⎛B − A aβ β R ) β ⎜⎝
βn R
n
n
ξ
(12)
⎞⎤ ⎟⎥ n⎠⎥ ⎦
The second term in Eq. (12) becomes smaller as time increases so it can be neglected. Then, Eq. (12) can be rewritten as,
T (0, τ ) =
The slow heat flux transient was given by the sine function qtrue,2 = 500 sin (1.57τ ) kW/m2 with a maximum heat flux of 500 kW/ m2 and a period of 2 s as shown in Fig. 4. The gauge time constant was much shorter than the signal period, 2 s, so this is a slow heat flux transient. qappa,2 was calculated using the traditional method with qcal,2 calculated using the transient method. As in Case 1, the “corrected” heat flux, qcal,2 , calculated using Eq. (10) agrees well with qtrue,2 . Although the theoretical time constant τ c = 175 ms of the Gardon gauge is much shorter than the signal period (2 s) of slow heat flux transient, there is still a response delay between qappa,2 and qtrue,2 . This again illustrates that the traditional Gardon gauge is more suitable for quasi-steady heat flux measurements. For slow heat flux transients, q (τ ) is assumed to be a linear function of time in a small time interval,
1 ξλ0δ
R2 (( Aτ + B) 4 − 0.0469 AR2 aξ ) ξλ0δ
(13)
Combining Eq. (13) with Eqs. (3) and (5) gives the heat flux,
q = A (τ − Z ) + B
(14)
where Z is a function of the gauge time constant and is equal to 0.7504τ c ξ . Comparing Eqs. (11) and (14) shows that Z reflects the heat flux measurement response delay time when using the traditional measurement method based on Eqs. (3) and (5). However, Eq. (14) provides another correction method for slow heat flux transient measurements. The actual heat flux at any moment can be obtained using the forward translation Z in the response time for the “apparent” heat flux. This is a simplification of the new measurement method in section 2.2. The translated curve of qappa,2 is defined as qappa ,2−translation which is shown in Fig. 5. The relative error
500 200
100 0 0
0.1
0.2
0.3
0.4
-100
0.5
Heat flux density, kW/m2
Absolute error in the heat flux, kW/m2
300
400
300
200
100
-200
0 -300
Time, s
Fig. 3. Absolute errors in the predicted heat fluxes for a fast transient.
0
0.5
1
Time, s
1.5
2
Fig. 5. Comparisons of ( qtrue,2 , qappa,2 , qappa ,2−translation ) for a slow transient.
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T. Fu et al./Applied Thermal Engineering 100 (2016) 501–507
505
3.5
Output voltage, mV
3.0 2.5 2.0 1.5 1.0 0.5 0.0
0
10
20
Time, s
30
40
50
Fig. 7. Output voltages as the gauge moved up and down. Fig. 6. Vacuum chamber heat flux measurement system.
between qappa,2 and qappa ,2−translation is less than 3% when the initial stage (0~0.2 s) and the discontinuity are neglected. Thus, this analysis illustrates that the transient response correction is still necessary for slow heat flux transients using the Gardon gauge. This simplified process method is effective for slow heat flux transients. The simulations in cases 1 and 2 provide evidence of sufficient improvement in transient measurement accuracy.
parent” heat flux density qappa * based on the traditional measurement mode of Gardon gauge was also calculated with the absolute error in the heat flux given as q = qappa * − qcal * . The results for qcal * and * , qappa q are shown in Fig. 8. Fig. 9 shows the distribution of qcal , * and * qappa q in the 31.2–32.2s (the fifth cycle). When the heat change frequency is below 0.02 kW/ms (28–29 s, 30–31.2 s and 32.2–34 s), the heat flux is quasi-steady so qcal * is equal to qappa * . There is no need to correct the transient character-
4. Experiments 400
100
50
200
0
100
-50
A
0 28
29
Absolute error
300
kw/m2
B
Heat flux, kw/m2
The heat fluxes on the high-temperature graphite plate in a vacuum chamber were measured using a Gardon gauge. The experimental system is sketched in Fig. 6. The graphite plate with a thickness of 3 mm, a width of 30 mm and a height of 50 mm was located in the vacuum chamber and directly heated to the desired steady-state high temperature by an alternating current power source. The vacuum chamber was connected to a vacuum pump to maintain the pressure at less than 1 × 10−2 Pa to avoid oxidation of the graphite at high temperatures. An optical pyrometer placed outside the vacuum chamber was used to monitor the graphite plate temperature through a viewing window. The Gardon gauge was then used to measure the radiative heat flux emitted by the hot graphite plate. The constantan foil thickness, δ , of the gauge was 0.1 mm, and the constantan foil diameter, D , was 3.0 mm. The foil surface emissivity was 0.92. The standard time constant of the gauge was 90 ms. The measurement sensitivity, L , was 0.008 mV/(kW/m2). The cooling water flow rate through the gauge was 15 ml/s and the inlet water temperature was 14 °C. The gauge was placed on an elevated platform in front of the graphite plate at a horizontal distance of 5 mm. The platform was moved up and down to adjust the radiative heat flux reaching the gauge due to the change in the view angle between the gauge and the graphite plate. The transient characteristics of the radiative heat flux are then related to the movement velocity of the elevated platform. The graphite temperature was set to 1450 °C monitored by an optical pyrometer [35–39], and the platform velocity supporting the gauge was 31.25 mm/s. The transient output voltage of the Gardon gauge shown in Fig. 7 was then measured as the gauge on the platform moved. Fig. 7 shows six heat flux cycles where the voltage rapidly increased and then decreased in each cycle. The heat flux change frequency is within 0~1.18 kW/ms that is not a very fast heat flux transient. The fifth cycle in the 28–34s was chosen to analyze the transient heat flux characteristics. The corrected heat flux, qcal * , was calculated using the fast transient measurement method. The “ap-
-100 30
31
32
33
34
Time, s Fig. 8. Distributions of qcal * , qappa * and q at various times in the fifth cycle (28–34 s).
Fig. 9. Distributions of qcal * , qappa * and q at various times (31.2–32.2 s).
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T. Fu et al./Applied Thermal Engineering 100 (2016) 501–507 400
20
10
200
0
100
-10
kW/m2
300
Absolute error
Heat flux, kW/m2
B
A 0 28
-20 29
30
31
32
33
34
Time, s Fig. 12. Distributions of qcal * , qappa * −translation and q at various times in the fifth cycle (28–34 s). Fig. 10. Sketch showing why the signal changes with time.
5. Conclusions istics. When the heat flux change frequency is from 0.02–1.18 kW/ ms (29–30 s and 31.2–32.2 s), the error, q , is obviously so the transient corrections of heat flux is necessary although it is a slow transient. As shown by the numerical analysis in section 3, the corrected heat flux, qcal * , significantly improves the response delay in the transient “apparent” heat flux calculated using the traditional * closely represents the true heat flux transient. method. Thus, qcal The absolute error, q , in the calculated heat flux was greatest at point A ( q A = −80 kW/m2) and point B ( q B = 75 kW/m2 ). The error is mainly due to the fast change in the radiative heat flux reaching the heat flux gauge at times A and B as shown in Fig. 10. The times for points A and B correspond to the positions where the gauge is moving into and out of the field view of the graphite plate. The radiative heat flux between the two surfaces is related to the changing view factor which is related to the change in the angle between the center of the gauge and the center of the graphite plate. Therefore, the changes in the heat flux at times A and B can be explained by the large change in the radiative view factor due to the Gardon gauge movement as shown in Fig. 11. The time axis translation method described in section 3.2 was also used to calculate the heat flux, qappa * −translation , shown in Fig. 12. The absolute error, q = qappa * −translation − qcal * , in the heat flux predicted by the translation method is much less than the error, q = qappa * − qcal *, shown in Fig. 8, but the results still has some little errors at times A and B which may be acceptable. The translation method is suitable for slow heat flux transients.
A
View factor
0.6
1.5
0.4
1.0
0.2
0.5
0.0
0.0
29
30
31
This work was supported by the National Natural Science Foundation of China (No. 51576110), the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No. 51321002), the Program for New Century Excellent Talents in University (NCET-13-0315) and the Beijing Higher Education Young Elite Teacher Project (YETP0091). We thank Prof. D.M. Christopher for editing the English. References
View factor
View factor rate of change
28
Acknowledgements
2.0
B
32
33
Time, s Fig. 11. Radiative view factor variations with times.
34
View facotr rate of change
0.8
Gardon gauges are widely used to measure heat fluxes in harsh environments due to their reliability and cooling water design. However, Gardon gauges are not good for fast heat flux transients. This study presents an accurate transient heat flux measurement method for Gardon gauges. A data analysis method is presented and verified using numerical simulations of fast and slow heat flux transients. The analysis shows that this method can accurately measure the transient heat flux characteristics. The transient response correction for the gauge is necessary not only for fast heat flux transients, but also for slow heat flux transients. A simplified method is presented for slow heat flux transients that are based on translation of the time axis that is also effective. Experiments using transient radiative heat flux measurements of the radiative heat flux from a hot graphite plate in a vacuum chamber using a Gardon gauge further illustrate the applicability of the heat flux transient analysis method. The analysis can be used to improve the instrument response of Gardon gauges for the measuring heat flux transients.
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Applied Thermal Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / a p t h e r m e n g
Research Paper
Gardon gauge measurements of fast heat flux transients Tairan Fu a,*, Anzhou Zong a, Jibin Tian a, Chengyun Xin b a Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Beijing Key Laboratory of CO Utilization and Reduction Technology, 2 Department of Thermal Engineering, Tsinghua University, Beijing 100084, China b School of Electric Power Engineering, China University of Mining and Technology, Xuzhou 221116, China
H I G H L I G H T S
• • • •
A fast heat flux transient measurement method using Gardon gauges is introduced. Simulations and experiments of heat flux transient measurements verify the method. Convert Gardon gauge with a limited response to fast-response heat-flux sensor. Gardon gauge measurement accuracy is improved for heat flux transients.
A R T I C L E
I N F O
Article history: Received 29 December 2015 Accepted 15 February 2016 Available online 23 February 2016 Keywords: Heat flux Transient radiative heat flux Gardon gauge Measurement method
A B S T R A C T
Heat flux measurements are widely used in thermal analyses. Gardon heat flux gauge are a widely used diffusion type heat flux gauge that can be used in harsh thermal environments such as in fires, combustors, and aerospace applications. However, Gardon gauges are usually regarded as quasi-steady-state sensors with a limited frequency response and are not often used for transient measurements. This paper introduces a fast radiative heat flux transient measurement method using Gardon gauges. The analysis assumes that the heat flux transient is a continuous function of time in every small/micro time interval. The transient heat flux density functions are determined from the transient output voltages of the gauge. Two cases with fast and slow heat flux transients were numerically investigated to verify the method. The analyses show that the gauge transient response must be corrected not only for fast heat flux transients, but also for slow heat flux transients. The gauges were then tested in experiments to measure radiative heat flux transients on a hot graphite plate in a vacuum chamber to evaluate the effects of the corrections. This method can use a limited frequency response Gardon gauge for fast heat flux measurements without other instrumentation adjustments. The Gardon gauges measurement accuracy is then greatly improved with transient measurements. The analysis shows how to accurately measure transient heat fluxes and improve the performance of Gardon gauges. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Heat flux measurements are often needed in thermal studies and have been widely used for engineering and research applications [1–8], for example, building and environmental studies, fire safety, electronic systems, material structures, combustion chambers and engines, and aerospace applications. Heat flux measurements are important for controlling heating processes, to assess device performance at high temperatures, to optimize systems for energy production and for designing the thermal protection systems for combustors in industrial and power engineering projects. The
* Corresponding author. Tel.: 86 10 62786006; fax: 86 10 62786006. E-mail address: [email protected] (T. Fu). http://dx.doi.org/10.1016/j.applthermaleng.2016.02.043 1359-4311/© 2016 Elsevier Ltd. All rights reserved.
interests in the measurements of unsteady heat transfer phenomena was increasing due to the need to understand the heat transfer process occurring in inherently unsteady environments. Transient heat flux measurements were widely needed in applications including high enthalpy plasmas, high power pulsed lasers, structural thermal tests and other fields. The well-established single-point heat flux gauges are widely available for various applications in thermodynamics and fluid. Measurement techniques of single-point heat flux, for example, null-point calorimeters, thin-film gauges, coaxial thermocouples, Schmidt–Boelter gauges, and Gardon gauges, have been developed to measure heat fluxes in various types of thermal environments. The transient null point calorimeter was first developed by Powars et al. [9] in 1972 and has been widely used to measure transient heat fluxes due to its simplicity, very wide flux range and favorable response time characteristics. Löhle et al. [10–12] made some
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significant improvements in the null-point calorimeter for high heat flux applications in a plasma wind tunnel facility. Thin-film gauges are classic fast-response sensors for very fast transient heat flux measurements with frequency responses up to 1 MHz [13–20]. They are usually fabricated by painting or sputtering a thin resistance layer onto an insulator substrate so they are usually fragile due to the weak layer-substrate bonding. Coaxial thermocouples [21–24] are more rugged than thin-film sensors and do not require frequent maintenance for transient heat flux measurements in harsh thermal environments. Irimpan et al. [23] designed a fast response coaxial thermocouple for shock tunnel applications, evaluated the coaxial thermocouple response against a platinum thin film gauge for heat flux measurements in a shock tunnel. However, the short working time characteristics of the thinfilm gauges and coaxial thermocouples greatly restrict their applications. Schmidt-Boelter gauges [21,25–28] and Gardon gauges [29–34] are available devices with excellent measurement characteristics with metal bodies, blackbody sensor foils and water cooled designs. Schmidt-Boelter and Gardon gauges are classified as diffusiontype or one-dimensional sensors that deduce the heat flux by measuring the temperature gradient in a material. The gauges are more suitable for high radiative or convective heat flux measurements with long working times in harsh environments such as in combustion flames, rocket motors, hyper sonic wind tunnels, etc. For example, Kidd et al. [21] developed a fast-response heat flux sensor, based on a modified Schmidt–Boelter gauge principle for heat transfer measurements in hypersonic wind tunnels. Gifford et al. [26] performed experiments to characterize the performance of Schmidt–Boelter heat flux gauges in stagnation and shear convective air flows. Nakos et al. [27] analyzed the heat transfer from an idealized Schmidt–Boelter heat flux gauge and showed that the theoretical sensitivity coefficients in radiative and convective environments differed. Sudheer et al. [28] measured the incident heat flux on a target for open gasoline pool fires with Schmidt– Boelter heat flux gauge measurements. The Gardon gauge (also called a circular foil heat flux gauge) was first introduced by Gardon in 1953 [29]. There have been many applications and calibrations of Gardon gauges [30–34]. The Gardon gauge is more useful for heat flux measurements over wide ranges than Schmidt–Boelter gauge so there is a great need for accurate Gardon gauges for measurements involving wide ranges of heat fluxes. Although Gardon gauges are widely used diffusion type heat flux gauges and have been extensively studied, the use of Gardon gauges in harsh environments is still restricted for transient applications. Gardon gauges as well as Schmidt–Boelter heat flux gauges are usually regarded as quasi-steady-state sensors with a limited frequency response and are rarely used in transient applications. It’s unable to obtain accurate heat fluxes using Gardon gauges in transient applications although the merit of high heat flux and long working times measurements is remarkable. Therefore, this paper describes an optimized measurement method of fast transient radiative heat fluxes based on a traditional Gardon gauge. The measurement accuracy of the Gardon gauges is also greatly improved for transient measurements. The merit of this work is to convert a traditional Gardon gauge with a limited frequency response to a fast-response heat-flux sensor which also has a wide range of heat fluxes and can be operated with long working times in thermal environments. The measurement principles of Gardon gauges are analyzed to improve the measurement method for transient heat fluxes. Numerical simulations and experimental measurements of typical heat fluxes verify the applicability and effectiveness of the measurement method. The analysis provides a useful reference for accurate measurement of transient heat fluxes and improves the measurement ability of Gardon gauges.
2. Principles 2.1. Traditional measurement method The Gardon heat flux gauge is a diffusion type heat flux. A sketch of a copper-constantan circular foil heat flux sensor is shown in Fig. 1. A circular constantan foil coated with a diffuse, highly absorbing coating is attached to a copper heat sink. The heat sink can be cooled to a low temperature so the gauge can be exposed to high heat flux environments. There is a large temperature gradient along the foil radial direction due to the effects of the heat flux on the foil surface. The metal leads are attached to the center and the edge (or the copper heat sink) of the foil to form a thermocouple joint. The voltage output of this thermocouple joint is proportional to the temperature difference across the foil. Therefore, the measured voltage gives a quantitative representation of the heat flux through the foil. The temperature gradient along the foil thickness is negligible and the heat losses from the back of the foil and from the center metal wire are also neglected. The foil surface of the gauge is assumed to be a highly absorbing gray diffuse surface. Then, the heat transfer equation and boundary conditions describing the radial temperature distribution in the foil are:
∂ T (r , τ ) ⎧ 1 ∂ ⎛ ∂T (r , τ ) ⎞ q (τ ) = ρc ⎜⎝ λr ⎟⎠ + ⎪ δ r r ∂ r ∂τ ∂ ⎨ ⎪T ∂ r =0 = T T = T ∂ T , , 0 0 ⎩ τ =0 r =R r =0
(1)
where T0 is the heat sink temperature, τ is the time, r is the foil radius axis, T (r, τ ) is the temperature at time τ and position r on the foil surface, q (τ ) is the heat flux reaching the foil surface due to convective or radiative heat transfer at time τ , δ is the thickness and R is the constantan foil radius. ρ is the foil density and c is the specific heat capacity of the foil which are both assumed to be constant. λ is the thermal conductivity of the constantan foil expressed as a linear function of temperature,
λ = λ0 (1 + b (T − T0 ))
(2)
where λ0 = 20.9 W/ (mK ) at T0 = 0°C and b = 0.00231 K −1 for a temperature range of 0~200 °C. The thermo-electromotive voltage output, E , of the copperconstantan thermocouple formed by the Gardon gauge is expressed as the function of the temperature difference, ΔT , between the center and the edge of the constantan foil as,
E = kΔT (1 + g ΔT ) where k = 0.0387 mV/K and g = 0.0012 K −1.
Fig. 1. Sketch of the Gardon heat flux gauge.
(3)
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The theoretical time constant τ c of the Gardon gauge is defined [29] to describe the transient response characteristics for heat flux,
τ c = ρcR 2 4λ0
(4)
The measurement sensitivity of the heat flux gauge, L , is defined as the ratio of the output voltage, E , to the measured heat flux, q . With the conventional measurement method, Gardon gauges are as quasi-steady-state sensors with q assumed to be a constant heat flux. Therefore, the Gardon gauge measurement sensitivity can be determined by assuming a constant heat flux and a constant thermal conductivity from Eqs. (1)–(3),
L=E q=
kR 2 4λ0δ
(5)
For the Gardon heat flux gauges, the measurement sensitivity is determined from experiment calibrations from the factory. The heat flux is then obtained from the measured output voltage and the known measurement sensitivity of the gauge using Eq. (5). However, for fast transient heat flux measurements, the measurement sensitivity of the Gardon gauge provided by the producer is not accurate because the heat flux is not constant.
503
determined when the C i is known. Equations (8) and (9) can then be combined as,
T (0, τ ) =
( (
a N∞ −ξa0βnτ J 2 ∑e λδ n=1 J12
) ∫ ∑C τ ′ β R)
βn R n
τ
τ ′=0
p
i
i −1 ξa0βnτ ′
e
dτ ′
(10)
i =1
The C i (i = 1, , p) are obtained from the measured center temperatures at m (m ≥ p) times using a least-squares algorithm or some other optimization algorithms. The center temperature, T (0, τ ) , can be deduced from the transient output voltage, E , of the Gardon gauge using Eq. (3). Therefore, the transient heat flux variation can be determined from the output voltage, E , using Eqs. (3), (9) and (10). The measurement frequency of output voltage and the choice of time interval in Eq. (9) determine the response characteristics of the transient heat flux measurements. For the traditional Gardon gauge measurement method in section 2.1, the constant heat flux is also obtained from the output voltage. Thus, this method enables a Gardon gauge with a limited frequency response to be a fast response heat flux sensor without any other instrument adjustments. The measurement accuracy of the Gardon gauges is also greatly improved in transient measurement applications.
2.2. Fast transient measurement method
3. Simulations
A measurement method for fast transient is introduced here for Gardon gauges. If the heat flux is constant, the foil surface temperature found by solving Eqs. (1) and (2) is,
The Gardon gauge transient heat flux measurement method was evaluated using simulations with fast and slow heat flux transients. The simulated Gardon gauge parameters were δ = 0.1 mm , R = 2.0 mm , T0 = 0°C , and a theoretical time constant τ c = 175 ms . The decrease of the gauge response time will reduce the measurement scale of heat flux. It’s difficult to fabricate a gauge with the time constant less than 50 ms. The standard measurement sensitivity of the gauge was obtained using Eqs. (1) –(5) as L = 0.0185 mV/ (kW/m2 ) . The simulated transient radiative heat flux, q , was applied to the foil surface of the Gardon gauge as the “true” heat flux, qtrue . The gauge output voltages at various times were calculated using Eqs. (3) and (8) and are reported as the “measured experimental values”. The heat flux calculated using the traditional measurement method was calculated using the standard measurement sensitivity from Eq. (5) and the “measured” output voltage and is defined as the “apparent” heat flux, qappa . The heat flux calculated using the transient measurement method using Eq. (10) based on the “measured” output voltage is defined as the “corrected” heat flux, qcal . The number of terms in the series, p , in Eq. (10) is assumed to be 4 ( p = 4 ) . The number of terms in the Bessel function was set to 15 ( N ∞ = 15) .
T = T0 +
R 2q 1 N∞ 1 − r 2 R 2 ) + ∑ Ane −ξa0βnτ J 0 ( 4ξλ0δ ξ n=1
(
βn r
)
(6)
qJ ( β R where ξ = 1 + b (T − T0 )and T = (T (r = 0) + T0 ) 2. An = − λδβ 2J 2 nβ ) R and n 1 ( n ) βn is the nth zeroth order Bessel function ( J 0 β R = 0 ). N ∞ is the number of terms in the Bessel function. If the heat flux is transient and a function of time, the foil surface temperature is obtained using the Duhamel theorem to solve Eqs. (1) and (6),
(
T (r , τ ) = ∫
τ
τ ′=0
=
)
q (τ ′ ) ∂T (r , τ − τ ′ ) dτ ′ δ ∂τ
( (
1 N∞ −ξa0βnτ J 2 ∑e ρcδ n=1 J12
)J( β R)
βn R
0
n
βn r
)∫
τ
τ ′=0
(7)
q (τ ′ ) eξa0βnτ ′ dτ ′
Then, the center temperature, T (0, τ ) , of the foil surface can be expressed as,
T (0, τ ) =
( (
1 N∞ −ξa0βnτ J 2 ∑e ρcδ n=1 J12
βn R
) )∫
βn R
τ
τ ′ =0
q (τ ′ ) eξa0βnτ ′ dτ ′
(8)
Equation (8) relates the center temperature, T (0, τ ) , to the transient radiative heat flux, q (τ ). When T (0, τ ) is known, q (τ )may be determined using an inverse analysis. However, unlike the equation for the constant heat flux used in the traditional measurement method with the Gardon gauge, an explicit expression for q (τ ) cannot be directly obtained due to the complex integration in Eq. (8). Therefore, assuming q (τ ) to be a continuous differentiable function of time in a small time interval (μs ~ ms), q (τ )can be expressed by a Taylor series expansion, p
q (τ ) = ∑ C iτ i−1
(9)
i =1
where p is the number of terms in the series and C i (i = 1, , p) are the series coefficients. The transient heat flux can then be
3.1. Case 1: fast heat flux transients For Case 1, the “true” fast heat flux transient was the sine function qtrue,1 = 500 sin (7.85τ ) kW/m2 . Thus, the maximum heat flux was 500 kW/m 2 with a period of 400 ms as shown in Fig. 2. The maximum of heat flux change frequency was up to 3.9 kW/ms. qappa,1 was calculated using the traditional method with qcal,1 calculated using the transient measurement method, as shown in Fig. 2. The absolute errors of qappa ,1 − qtrue ,1 and qcal ,1 − qtrue ,1 are shown in Fig. 3. The calculated “apparent” heat flux, qappa,1 , did not follow the rapid changes in the true transient heat flux, q1,true . The delay time of the peak response was up to 280 ms which is larger than the theoretical time constant τ c = 175 ms for the Gardon gauge. The peak qappa,1 is about 400 kW/m2 which is much less than the peak of the true heat flux. The absolute error, qappa ,1 − qtrue ,1 , in Fig. 3 has significant variations with a maximum error of 260 kW/m2. Therefore, qappa,1 is unable to accurately characterize the rapid changes in q1,true . The
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T. Fu et al./Applied Thermal Engineering 100 (2016) 501–507
500
Heat flux density, kW/m2
500
Heat flux, kW/m2
400
300
200
400
300
200
100
100
0
0
0
0.1
0.2
0.3
0.4
0
0.5
0.5
1
1.5
2
2.5
Time, s
Time, s
Fig. 4. Comparisons of ( qtrue,2 , qappa,2 , qcal,2 ) for a slow transient.
Fig. 2. Comparisons of ( qtrue,1 , qappa,1, qcal,1 ) for a fast transient.
limited Gardon gauge time constant makes limits the ability to accurately measure fast heat flux variations. However, the “corrected” heat flux, qcal,1, calculated using Eq. (10) agrees well with qtrue,1 . qcal,1 agrees well with q1,true at the peak response time point (0.2 s) and the discontinuity point (0.4 s). Thus, there is no obvious response time delay. The absolute error qcal ,1 − qtrue ,1 shown in Fig. 3 is very small over the entire time period. Therefore, this method to calculate qcal,1 greatly improves the traditional measurement response time of the Gardon gauge for fast heat flux transients. 3.2. Case 2: slow heat flux transients
where A and B are coefficients. Then, Eq. (8) is rewritten as,
T (0, τ ) =
∞
q (τ ) = Aτ + B
(11)
( (
⎡ N∞ J 2 ⎢∑ 2 ⎢⎣ n=1 J1
− ∑e
− ξa0βnτ
J2
βn R
) )
1⎛ A ⎞ Aτ + B − aξ βn ⎟⎠ βn R βn ⎜⎝
( (
J12
n=1
) 1 ⎛B − A aβ β R ) β ⎜⎝
βn R
n
n
ξ
(12)
⎞⎤ ⎟⎥ n⎠⎥ ⎦
The second term in Eq. (12) becomes smaller as time increases so it can be neglected. Then, Eq. (12) can be rewritten as,
T (0, τ ) =
The slow heat flux transient was given by the sine function qtrue,2 = 500 sin (1.57τ ) kW/m2 with a maximum heat flux of 500 kW/ m2 and a period of 2 s as shown in Fig. 4. The gauge time constant was much shorter than the signal period, 2 s, so this is a slow heat flux transient. qappa,2 was calculated using the traditional method with qcal,2 calculated using the transient method. As in Case 1, the “corrected” heat flux, qcal,2 , calculated using Eq. (10) agrees well with qtrue,2 . Although the theoretical time constant τ c = 175 ms of the Gardon gauge is much shorter than the signal period (2 s) of slow heat flux transient, there is still a response delay between qappa,2 and qtrue,2 . This again illustrates that the traditional Gardon gauge is more suitable for quasi-steady heat flux measurements. For slow heat flux transients, q (τ ) is assumed to be a linear function of time in a small time interval,
1 ξλ0δ
R2 (( Aτ + B) 4 − 0.0469 AR2 aξ ) ξλ0δ
(13)
Combining Eq. (13) with Eqs. (3) and (5) gives the heat flux,
q = A (τ − Z ) + B
(14)
where Z is a function of the gauge time constant and is equal to 0.7504τ c ξ . Comparing Eqs. (11) and (14) shows that Z reflects the heat flux measurement response delay time when using the traditional measurement method based on Eqs. (3) and (5). However, Eq. (14) provides another correction method for slow heat flux transient measurements. The actual heat flux at any moment can be obtained using the forward translation Z in the response time for the “apparent” heat flux. This is a simplification of the new measurement method in section 2.2. The translated curve of qappa,2 is defined as qappa ,2−translation which is shown in Fig. 5. The relative error
500 200
100 0 0
0.1
0.2
0.3
0.4
-100
0.5
Heat flux density, kW/m2
Absolute error in the heat flux, kW/m2
300
400
300
200
100
-200
0 -300
Time, s
Fig. 3. Absolute errors in the predicted heat fluxes for a fast transient.
0
0.5
1
Time, s
1.5
2
Fig. 5. Comparisons of ( qtrue,2 , qappa,2 , qappa ,2−translation ) for a slow transient.
2.5
T. Fu et al./Applied Thermal Engineering 100 (2016) 501–507
505
3.5
Output voltage, mV
3.0 2.5 2.0 1.5 1.0 0.5 0.0
0
10
20
Time, s
30
40
50
Fig. 7. Output voltages as the gauge moved up and down. Fig. 6. Vacuum chamber heat flux measurement system.
between qappa,2 and qappa ,2−translation is less than 3% when the initial stage (0~0.2 s) and the discontinuity are neglected. Thus, this analysis illustrates that the transient response correction is still necessary for slow heat flux transients using the Gardon gauge. This simplified process method is effective for slow heat flux transients. The simulations in cases 1 and 2 provide evidence of sufficient improvement in transient measurement accuracy.
parent” heat flux density qappa * based on the traditional measurement mode of Gardon gauge was also calculated with the absolute error in the heat flux given as q = qappa * − qcal * . The results for qcal * and * , qappa q are shown in Fig. 8. Fig. 9 shows the distribution of qcal , * and * qappa q in the 31.2–32.2s (the fifth cycle). When the heat change frequency is below 0.02 kW/ms (28–29 s, 30–31.2 s and 32.2–34 s), the heat flux is quasi-steady so qcal * is equal to qappa * . There is no need to correct the transient character-
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The heat fluxes on the high-temperature graphite plate in a vacuum chamber were measured using a Gardon gauge. The experimental system is sketched in Fig. 6. The graphite plate with a thickness of 3 mm, a width of 30 mm and a height of 50 mm was located in the vacuum chamber and directly heated to the desired steady-state high temperature by an alternating current power source. The vacuum chamber was connected to a vacuum pump to maintain the pressure at less than 1 × 10−2 Pa to avoid oxidation of the graphite at high temperatures. An optical pyrometer placed outside the vacuum chamber was used to monitor the graphite plate temperature through a viewing window. The Gardon gauge was then used to measure the radiative heat flux emitted by the hot graphite plate. The constantan foil thickness, δ , of the gauge was 0.1 mm, and the constantan foil diameter, D , was 3.0 mm. The foil surface emissivity was 0.92. The standard time constant of the gauge was 90 ms. The measurement sensitivity, L , was 0.008 mV/(kW/m2). The cooling water flow rate through the gauge was 15 ml/s and the inlet water temperature was 14 °C. The gauge was placed on an elevated platform in front of the graphite plate at a horizontal distance of 5 mm. The platform was moved up and down to adjust the radiative heat flux reaching the gauge due to the change in the view angle between the gauge and the graphite plate. The transient characteristics of the radiative heat flux are then related to the movement velocity of the elevated platform. The graphite temperature was set to 1450 °C monitored by an optical pyrometer [35–39], and the platform velocity supporting the gauge was 31.25 mm/s. The transient output voltage of the Gardon gauge shown in Fig. 7 was then measured as the gauge on the platform moved. Fig. 7 shows six heat flux cycles where the voltage rapidly increased and then decreased in each cycle. The heat flux change frequency is within 0~1.18 kW/ms that is not a very fast heat flux transient. The fifth cycle in the 28–34s was chosen to analyze the transient heat flux characteristics. The corrected heat flux, qcal * , was calculated using the fast transient measurement method. The “ap-
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Time, s Fig. 8. Distributions of qcal * , qappa * and q at various times in the fifth cycle (28–34 s).
Fig. 9. Distributions of qcal * , qappa * and q at various times (31.2–32.2 s).
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Time, s Fig. 12. Distributions of qcal * , qappa * −translation and q at various times in the fifth cycle (28–34 s). Fig. 10. Sketch showing why the signal changes with time.
5. Conclusions istics. When the heat flux change frequency is from 0.02–1.18 kW/ ms (29–30 s and 31.2–32.2 s), the error, q , is obviously so the transient corrections of heat flux is necessary although it is a slow transient. As shown by the numerical analysis in section 3, the corrected heat flux, qcal * , significantly improves the response delay in the transient “apparent” heat flux calculated using the traditional * closely represents the true heat flux transient. method. Thus, qcal The absolute error, q , in the calculated heat flux was greatest at point A ( q A = −80 kW/m2) and point B ( q B = 75 kW/m2 ). The error is mainly due to the fast change in the radiative heat flux reaching the heat flux gauge at times A and B as shown in Fig. 10. The times for points A and B correspond to the positions where the gauge is moving into and out of the field view of the graphite plate. The radiative heat flux between the two surfaces is related to the changing view factor which is related to the change in the angle between the center of the gauge and the center of the graphite plate. Therefore, the changes in the heat flux at times A and B can be explained by the large change in the radiative view factor due to the Gardon gauge movement as shown in Fig. 11. The time axis translation method described in section 3.2 was also used to calculate the heat flux, qappa * −translation , shown in Fig. 12. The absolute error, q = qappa * −translation − qcal * , in the heat flux predicted by the translation method is much less than the error, q = qappa * − qcal *, shown in Fig. 8, but the results still has some little errors at times A and B which may be acceptable. The translation method is suitable for slow heat flux transients.
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This work was supported by the National Natural Science Foundation of China (No. 51576110), the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No. 51321002), the Program for New Century Excellent Talents in University (NCET-13-0315) and the Beijing Higher Education Young Elite Teacher Project (YETP0091). We thank Prof. D.M. Christopher for editing the English. References
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Time, s Fig. 11. Radiative view factor variations with times.
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Gardon gauges are widely used to measure heat fluxes in harsh environments due to their reliability and cooling water design. However, Gardon gauges are not good for fast heat flux transients. This study presents an accurate transient heat flux measurement method for Gardon gauges. A data analysis method is presented and verified using numerical simulations of fast and slow heat flux transients. The analysis shows that this method can accurately measure the transient heat flux characteristics. The transient response correction for the gauge is necessary not only for fast heat flux transients, but also for slow heat flux transients. A simplified method is presented for slow heat flux transients that are based on translation of the time axis that is also effective. Experiments using transient radiative heat flux measurements of the radiative heat flux from a hot graphite plate in a vacuum chamber using a Gardon gauge further illustrate the applicability of the heat flux transient analysis method. The analysis can be used to improve the instrument response of Gardon gauges for the measuring heat flux transients.
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