Temperature measurement of air convection using a Schlieren system

ARTICLE IN PRESS Optics & Laser Technology 41 (2009) 233–240

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Temperature measurement of air convection using a Schlieren system C. Alvarez-Herrera, D. Moreno-Herna´ndez , B. Barrientos-Garcı´a, J.A. Guerrero-Viramontes ´n, Guanajuato 37150, Me´xico Centro de Investigaciones en Optica A.C., Loma del Bosque 115, Lomas del Campestre, Leo

a r t i c l e in fo

abstract

Article history: Received 25 January 2008 Received in revised form 27 June 2008 Accepted 1 July 2008 Available online 29 August 2008

In this work, we have developed a procedure for full-field measurement of temperature of a fluid flow by using the schlieren technique. The basic idea is to relate the intensity level of each pixel in a schlieren image to the corresponding knife-edge position measured at the exit focal plane of the system. The method is applied in the measurement of temperature fields of the air convection caused by a heated rectangular metal plate (7.3 cm  12 cm). Our tests are carried out at plate temperatures of 50 1C and 80 1C. To validate the proposed method, the schlieren temperature results are compared to those obtained by a thermocouple. Thermocouple data are obtained along two mutually perpendicular directions (one direction along the optical axis, z-direction, and other direction along the x-axis, which is perpendicular to the optical axis) at points located on a 9  9 grid with a variable spacing. The thermocouple measurements were integrated along the z-axis in order to be compared with the measurements obtained by the schlieren technique. The results from the two methods show good agreement between them. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Schlieren Fluid flow Temperature

1. Introduction The measurement of temperature is an important task in several areas of science. Optical techniques are widely used for that purpose. Generally, full-field techniques are desired, such as laser-induced fluorescence, holographic interferometry, laser speckle photography, and schlieren or a combination of them. The success of the schlieren technique to measure temperature is due to its relatively easy implementation, low cost, use of conventional light sources, and because of its high and variable sensitivity. The schlieren technique was developed as a visualization technique in the eighteenth century by Toepler [1]. The interest of Toepler [2–14] was to identify inhomogeneities (‘‘schlieren’’ in German) in images from microscopes and telescopes. Since then, the technique has been improved to face different kinds of problems in different fields of knowledge. Despite the schlieren technique was primarily conceived as a visualization technique, it has also been used to obtain quantitative temperature data in flows [8–14]. For this, several approaches have been proposed, such as color schlieren [8–10], background schlieren [11,12] and calibration schlieren [13,14]. In color schlieren, the angular deflection of the light rays is related to both the color of the image and temperature. Background schlieren is similar to speckle photography, but instead of using speckle patterns generated by a double exposure of highly seeded

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E-mail address: [email protected] (D. Moreno-Herna´ndez). 0030-3992/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2008.07.004

flows, speckle patterns are generated by a ground glass in order to derive density gradient information. An alternate schlieren technique used in the measurement of density fields has been proposed in [14]. It is based on a calibration procedure that relates light variation at the observation plane to transverse knife position. In the schlieren measurements, the integral of flow properties along the optical path of a given ray contributes to the light intensity at the observation plane. Classical schlieren systems include a point-like light source, usually a Xenon flash lamp. The light from the source is collimated by a lens, which is at least as large as the object that is being analyzed. After the collimated beam passes through the test section, it is refocused by a second lens. A knife-edge is then placed at this focal point, which is the exit focal point of the system. When the knife-edge is placed horizontally, positive density gradients are visualized as light areas and negative gradients as dark ones. The reason for this is that light rays that are deflected upwards by the positive gradients miss the knife-edge, whereas those deflected downwards by the negative gradients are blocked out by the knife. An extensive description of the schlieren technique can be found in Merzkirch and Settles [15,16]. In this work, we present an extension of the calibration schlieren technique, but instead of using a photodetector that provides time-resolved measurements, but which gives just one spatial point measurement at a time, a digital camera is used to cover a full region of interest at once [14]. The proposed technique allows us to measure flow temperature by relating the gray-level values of a schlieren image recorded at the observation plane to

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the corresponding transverse knife position measured at the exit focal plane of the system. By this technique, we have been able to measure temperature fields of the natural convection of a heated horizontal rectangular metal plate. The schlieren results are compared to thermocouple measurements obtained along the zand x-directions on a grid of 9  9 points with a variable spacing. To do this comparison, the thermocouple data is integrated along the z-axis.

wavelength of 600 nm. Additionally, we used a white-light source rather than a monochromatic source; however, since K is only weakly dependent on wavelength, the resulting error is relatively small [15]. For an ideal gas at constant pressure, the temperature can be found from the Gladstone–Dale’s equation, which reduces to the following form [18]: T¼

2. Theoretical development It is well known that when a light ray passes through an inhomogeneous medium it suffers a deviation in its trajectory by a certain angle [15–17]. This angle depends on the refractive index and thickness of the medium under test. The equation of the ray path through an inhomogeneous medium is expressed as   d d~ r n ¼ rn, (1) ds ds where n ¼ n(x, y, z) is the refractive index of the medium, ds is arc length, and ~ r is a position vector. By considering the approximation of small angles of deviation, e, the measured deviation at the observation plane (the exit focal plane) is dx ¼ f2taneEf2e, where f2 is the focal length of the focusing mirror in a 2-mirror schlieren system (in this type of systems the first mirror serves to collimate the entering light). Then, Eq. (1) can be written as [15,16] Z z2 qn x ¼ dz; (2) z1

qx

where x can be x or y, depending on the direction in which the knife blocks out the light. In this work the analysis is done for the x-direction. By combining the Gladstone–Dale’s equation, (n1) ¼ Kr, and Eq. (2), we get the following equation:

rx ¼

qr dx ¼ , qx f 2 hK

(3)

where r is the gas density, h is the thickness of the inhomogeneous medium under test in the ray propagation direction and K is the Gladstone–Dale’s constant. The Gladstone–Dale’s constant is a function of both the wavelength of the light source and the physical properties of the gas. In the present study, its value was taken as 2.259  104 m3/kg, for an air temperature of 293 K and a

ro no  1 T . T ¼ r o n1 o

(4)

In Eq. (4), no and ro are the refractive index and density at reference temperature To, respectively, and T is the temperature of interest. 2.1. Method to measure temperature In a schlieren system, the blockage of light by a knife-edge placed at the exit focal plane is due to the deviation of light by an inhomogeneous medium. This blockage can be similarly obtained in a homogeneous medium by a knife-edge which is allowed to be translated laterally a quantity Dx [14]. By considering this, we can establish a relationship between the light levels at the exit focal plane to the corresponding transverse knife-edge position. This transverse position may cover the conditions from no cutoff of light (maximum intensity) to full cutoff of light (minimum intensity). An algorithm for measuring temperature that takes into account the above-mentioned relationship is described next. First, a calibration procedure that relates the knife position and intensity level is described. Let I(n,m)x be a schlieren image recorded at the observation plane with the condition of no-flow, where n ¼ 0,1,y,N and m ¼ 0,1,y,M, with N and M denoting the image pixel size; x indicates the x-knife position Dx at which the schlieren image was recorded, i.e. x can take on values from kDx0 to kDx0 with k ¼ 0,1,2,y,l, with Dx0 denoting the spacing between consecutive lateral displacements of the knife; x ¼ 0 represents the condition when the knife-edge is in its reference position, which in this analysis corresponds to a knife position such that the light intensity at the observation plane presents an intermediate intensity value between cutoff and no-cutoff, i.e. when the intensity value is about 40% of the light observed for the condition of no-cutoff. We used this value because experimentally it yielded the largest dynamic range for Dx and a quasi-linear relationship between Dx and the schlieren intensities are observed. In Fig. 1a, we show a curve representing light intensity

120

100 80

mean std

100

60 80

20

 I(n, m)

Δx I(200, 200)

40

0

60 40

-20 -40

20 -60 -80 -1200-1000 -800 -600 -400 -200 0 Δx (μm)

200 400 600 800

0 -1200 -1000 -800 -600 -400 -200 0 Δx (μm)

200

400

600

Fig. 1. Calibration curves. (a) Calibration curve for pixel (n, m) and (b) mean and standard deviation of the M  N calibration curves for schlieren images.

800

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levels for different knife-edge positions for a single pixel (n, m). Notice that the curve is presented in terms of (Ixðn;mÞ  I0ðn;mÞ ) vs. Dx i.e. the light intensity is expressed as a deviation from the value obtained when the knife is at its reference position. This is equivalent to subtract the intensity of a schlieren image in presence of flow from a schlieren image with no flow or with undisturbed conditions. Curves relating light intensity to transverse knife position may serve for calibration purposes as described below. Since a calibration curve can be constructed for each pixel of a schlieren image, then a total of N  M calibration curves may be necessary. Now, let I(n,m)p be a schlieren image recorded at the observation plane in presence of flow with the knife at the reference position. Subtracting both Ixðn;mÞ and Ipðn;mÞ from the image captured at x ¼ 0 with the condition of no-flow, i.e. I0ðn;mÞ , we can write the following expressions, x x 0 ID ðn;mÞ ¼ Iðn;mÞ  Iðn;mÞ ,

(5)

dx ¼ Ipðn;mÞ  I0ðn;mÞ . Iðn;mÞ

(6)

Dx

Here, Iðn;mÞ gives the relationship between image intensity variations and corresponding knife position Dx defined in the calibration procedure, for each pixel (n, m). On the other hand, the dx recorded intensity values in presence of flow, Iðn;mÞ , are used to find the corresponding transverse knife position via being compared with the closest intensity values produced by Eq. (5) and represented in Fig. 1a, for example; this process is done for each pixel. Thus, the corresponding calibration transverse knife position for pixel (n, m) of the schlieren image in the presence of flow can be obtained by using the following criterion:

dxðn;mÞ ¼ Dx, where Dx corresponds to the condition    x Dx  minIðdn;m Þ  Iðn;mÞ ,

(7)

where to find the minimum of this equation, we need to make reference to calibration curves of the type shown in Fig. 1a, for each pixel.

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When the match between intensity values with flow and those from the calibration curve is not perfect, interpolation is applied. Finally, once that transverse knife position Dx from the calibration curves has been assigned to its corresponding intensity value for all pixels of a schlieren image, in the presence of flow, the rx value can be determined via Eq. (3). The density value can be stated as [16] Z z2 1 rðxÞ ¼ r0 þ dx dx. (8) f 2 hK z1 The trapezoidal numerical integration algorithm is used to integrate Eq. (8).

3. Experiment This experiment is aimed at measuring temperature fields of the convection of air caused by a hot rectangular metal plate (7.3 cm  12 cm) that is set to two different temperature values, 50 1C and 80 1C. The metal plate corresponds to a programmable chilling/heating plate provided by Torrey Pines Scientific. The experiment also includes a K-type thermocouple probe provided by Fluke and a conventional schlieren device. The programmable plate allows us to induce plate temperatures from 10 1C to 100 1C. A Z-type configuration schlieren system [15,16] is used in this research, and it includes two similar spherical mirrors of diameter D ¼ 0.15 m and focal distance f ¼ 1.54 m, and a knife-edge that can be translated in the x- or y-direction by a 2D micrometer stage. The system also includes a 5 mm diameter white high-brightness LED with luminous intensity of 2500 lm and dominant wavelength of 0.31 nm. Fig. 2 shows a schematic representation of the experimental setup. The metal plate under study is positioned horizontally on an optical table. The stream-wise direction of the fluid flow is the ydirection, and the knife-edge is positioned parallel to this direction. For comparison reasons, the temperature is measured by the proposed schlieren technique and by the thermocouple probe.

Fig. 2. Z-type schlieren experimental arrangement.

Fig. 3. Thermocouple temperature measurements at plate temperatures of (a) 50 1C and (b) 80 1C. The measurements are at the center of the plate.

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A mechanical system is used to move transversally the probe so that the average local point temperature could be recorded at points located on a grid of 9  9 points. The grid is in the y–z plane, where the z-direction corresponds to the optic axis of the system, which is in turn the propagation direction. The schlieren images are acquired with a Lumenera digital camera, which can provide 15 frames per second at a spatial resolution of 640  480 pix. These images are stored in BMP format and digitized by an 8-bit gray-level frame grabber. The camera is driven by software that lets us take either a single frame or a set of them. The selected spatial and temporal resolutions of

the camera allow us to capture the main characteristics of the dynamics of the fluid flow under study. The first step of the experiment consists in the recording graylevel schlieren images with no flow for several transverse positions of the knife-edge, covering the conditions of no cutoff of light to full cutoff of light. These images are used for calibration purposes as it was explained in Section 2.1. By the thermocouple, probe measurements are taken over the surface of the plate at the center of it (x ¼ 0 cm) and on a grid of 9  9 points located along the y- and z-coordinate axes. These data are obtained in the stream-wise direction from y ¼ 0.1 cm (just above the plate

x 10-3

ϑ

Calibration curve

I(n,m)

Subtraction

1 0 -1 300 250 200 150 100 y (Pixel) 50 0

Density for temperature calculation

Temperature

250 200 150

0 50 100 150 200 250 300 350 400

No flow

y (Pixels)

2

Images for different knife position

∂ρ/∂χ (kg/m )

236

Δx

x (Pixel)

100 50 50 100 150 200 250 300 350 x (Pixels)

Searching for Δx Schlleren image With flow

Subtraction

Light variations after subtraction

Fig. 4. Flowchart for temperature fields calculation.

Fig. 5. Schlieren images for different knife-edge positions for calibration purposes.

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surface) to y ¼ 4.0 cm with a variable step size and along the optical axis at z ¼ 0.0, 1.0, 2.0, 3.0, 3.6, 4.0, 5.0, 6.0, and 7.0 cm. Fig. 3, shows a plot of the thermocouple measurements at

237

temperatures of 50 1C and 80 1C. We acquired a total of 200 schlieren images. All measurements were done at a room temperature of 20 1C.

Fig. 6. (a) An instantaneous schlieren image for a plate temperature of 50 1C and (b) density gradient field calculated from (a).

Fig. 7. Instantaneous schlieren images and contour plots of temperature fields for different plate temperatures, (a) 50 1C and (b) 80 1C. Magnified sections of the images are included.

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4. Data analysis In order to show the feasibility of the technique, we determined the temperature fields of a convective fluid flow from experimental data as outlined in Section 3. For each temperature under test, a total of 200 schlieren images were recorded.

4.1. Temperature data analysis The value of rx can be determined using Eqs. (3) and (7); integration of this quantity is required to obtain the value of r, as it is shown in Eq. (8). The resulting density values are substituted in Eq. (4) in order to obtain the temperature of interest. In Fig. 4, a flowchart of the procedure is depicted. To obtain calibrations curves for each pixel, the knife is laterally displaced from 1 to 1 mm with a step size of Dx ¼ 100 mm, and at each knife position a schlieren image is recorded. Fig. 5 shows calibration images for different knife-edge positions. All the schlieren images are subtracted from the image with the knifeedge in its reference position at x ¼ 0 (upper path in the chart flow). In the flowchart and Fig. 5, the schlieren image for the reference knife-edge position is indicated by a white square. In Fig. 1b, the average and standard deviation of the calibration curves for the M  N pixels of the schlieren images are shown. Deviations in calibration curves for different pixels may be due to lack of uniformity in the illumination. On the lower path of the flowchart, a particular schlieren image in presence of flow, with the knife-edge at its reference position, is shown. This image is subtracted from a schlieren image with no flow and knife-edge at its reference position. The light deviation intensity for each image pixel is directly related to its corresponding transverse knife position from the calibration curves. In order to increase the resolution in Dx, each calibration curve is interpolated from 20 points to 200 points by using a piecewise cubic Hermite interpolating polynomial provided by MatLab. This algorithm does not modify the shape of the curve. Fig. 6 shows an instantaneous schlieren image and density gradient field calculated for a plate temperature of 50 1C. In Fig. 7, we show an instantaneous schlieren image and their corresponding contour plots of temperature fields for two different plate temperatures, 50 1C and 80 1C. An averaging filter was applied to the temperature data to reduce high frequency noise. In the figure, the x- and y-coordinates have been normalized against the rectangular plate width, h ¼ 7.3 cm. We found that the temperature close to the surface of the plate is consistent with the value of temperature set to the programmable plate. To obtain a better insight of the flow at the plate–air interface, in Fig. 7, insets of magnified regions of the images have been included.

To be able to compare the schlieren temperature measurements over the surface of the plate with thermocouple measurements, first, to reduce the influence of fluid flow fluctuations, the average of the temperature values from 200 schlieren images is obtained. Note that in schlieren measurements, the light intensity in the image reflects an integral of flow properties along the entire light path of a given ray. Therefore, thermocouple measurements have to be integrated along the z-axis before being compared with the schlieren measurements. Fig. 8 shows the comparison between both results; the results show good agreement between them. The agreement is better for points close to the plate surface. This is because the convective fluid flow becomes more fluctuating for points further than y/h ¼ 0.5. Also, the averaged values show that the case for 50 1C plate temperature is more fluctuating than the 80 1C plate temperature case. 4.1.1. Error analysis in temperature measurement The uncertainty in the temperature measurement will be characterized in this section. Since the primary measurement is the temperature, Eq. (4) can be expressed in function form as T ¼ Tðr; r0 ; T 0 Þ,

(9)

where r is an integral that depends of Dx, f2, K and h; i.e. Eq. (3). In our analysis, the values of r0, T0, f2, K and h are considered as constants and r is the only independent variable. Then the uncertainty in T is given by   qT sT ¼ sr . (10)

qr

Eq. (10) can be stated as s  s  r T ¼ , T r

(11)

where sT and sr are the uncertainty of temperature and density measurements, respectively. Table 1 Uncertainty calculations for temperature measurements y/h

sT (50 1C)

sT (80 1C)

0.0137 0.0274 0.0411 0.0548 0.0685 0.1370 0.2740 0.4110 0.5479

0.3 0.9 1.5 2.1 2.7 3.4 4.1 7.5 8.2

0.3 0.8 1.2 1.5 1.7 1.6 1.2 1.5 1.5

Fig. 8. Comparison between average temperature measurements obtained by schlieren and by a thermocouple probe for metal plates at: (a) 50 1C and (b) 80 1C.

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Fig. 9. Estimated relative error between schlieren and thermocouple temperature measurements for plate temperatures of (a) 50 1C and (b) 80 1C.

Eq. (11) was used to study errors caused by measuring temperature. Since we obtained 200 density values from schlieren images, the density uncertainty has been determined for these experimental data. Therefore, the standard deviation sr has been calculated in order to estimate uncertainty values of the density. Table 1 shows the uncertainty values for each temperature under test, i.e. 50 1C and 80 1C. The first column corresponds to the normalized stream-wise direction position from y ¼ 0.1 to 4.00 cm; the second and third columns correspond to uncertainty values for 50 1C and 80 1C measurements, respectively. We obtain minimum and maximum uncertainty values of 0.3 1C and 8 1C, respectively, for 50 1C temperature measurements, and minimum and maximum uncertainty values of 0.2 1C and 1.4 1C, respectively, for 80 1C temperature measurements. Close to the surface, uncertainty values for both temperatures under test are very similar; however, farther from it, difference in the uncertainty values become more significant. This may be due to the difference in the level of fluctuations present in both cases. Now, the relative error estimated between schlieren and thermocouple measurements is shown in Fig. 9. Some deviation can be observed when comparing these measurements, mainly at points close to the plate surface. As the first five thermocouple measurements were done with a separation of one millimeter and the stage used to translate the thermocouple had a resolution of one millimeter, then the uncertainty arising from positioning the thermocouple is larger than any other uncertainty source.

5. Discussion and conclusion We presented a method to measure temperature fields of a convective fluid flow caused by a heated rectangular metal plate. A full-field schlieren technique was used for that purpose. Despite the presented schlieren technique integrates the optical effects along the optical axis it is attractive, because only one image is needed for retrieving the desired information, such as the instantaneous spatial structure of a fluid flow. The technique can be used where fluid flow fluctuations are negligible and where errors caused by integrating along the optical axis are small. However, if this technique is used in fluctuating fluid flows, the errors caused by integrating along the optical axis can be reduced by using statistical values such as the mean and the standard deviation (or rms) of the flow properties. The proposed method makes use of a calibration procedure that allows us to directly convert intensity level for each pixel in a schlieren image into a corresponding transverse knife position. For the measurement of different temperature fields, as long as the optical setup configuration remains unchanged, it is not

necessary to update the calibration procedure. However, when changes in the ambient illumination occur, a recalibration procedure becomes necessary. This issue is the aim of a further research. The selection of the knife-edge reference position is done such that the dynamic range for the measurements is set within the quasi-linear part of the relationship between knife position and intensity. For comparison reasons, schlieren and thermocouple measurements of temperature of the convective flow were carried out. The results from both experimental techniques showed good agreement between them.

Acknowledgments David Moreno–Herna´ndez gratefully acknowledges the support granted by CONACYT (Me´xico) and CONCYTEG (Me´xico) Grants 47795 and 04-K117-038, respectively. C. Alvarez-Herrera thanks the CONACYT for his doctoral grant. The reviewers are also acknowledged for their substantive comments which helped improve the manuscript. References [1] Toepler A. Observations by new optical method in Fluid. In: Meyer-Arendt JR, Thomson Brian J, editors. Selected Papers on Schlieren Optics. SPIE Milestone Series, USA, 1992. [2] Barnes NF, Bellinger SL. Schlieren and shadowgraph equipment for air flow analysis. J Opt Soc Am 1945;35:497–509. [3] Burton RA. A modified Schlieren apparatus for large areas of field. J Opt Soc Am 1949;39:907–8. [4] Decker G, Deutsch R, Kies W, Rybach J. Computer-simulated Schlieren optics. Appl Opt 1985;24:823–8. [5] Korpel A, Yu TT, Snyder HS, Chen YM. Diffraction-free nature of Schlieren sound-field images in isotropic media. J Opt Soc Am 1994;11:2657–63. [6] Peale RE, Summers PL. Zebra Schlieren optics for leak detection. Appl Opt 1996;35:4518–21. [7] Stanic S. Quantitative Schlieren visualization. Appl Opt 1978;17:837–42. [8] Agrawal AK, Butuk NK, Gollahalli SR, Griffin D. Three-dimensional rainbow Schlieren tomography of a temperature field in gas flows. Appl Opt 1998;37:479–85. [9] Tregub VP. A color Schlieren method. J Opt Technol 2004;71:785–90. [10] Wong T, Agrawal AK. Quantitative measurements in an unsteady flame using high-speed rainbow Schlieren deflectometry. Meas Sci Technol 2006;17:1503–10. [11] Popova EM. Processing Schlieren-background patterns by constructing the direction field. J Opt Technol 2004;71:572–4. [12] Raffel M, Richard H, Meier AGEA. On the applicability of background oriented optical tomography for large scale aerodynamic investigations. Exp Fluids 2000;28:477–81. [13] Teese RB, Waters MM. Inexpensive Schlieren video technique using sensor dead space as a grid. Opt Eng 2004;43(11):2501–2.

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