Study on the measurement of temperature field using laser holographic interferometry

Abstract

The temperature field of an axisymmetric ethylene diffusion flame is measured using laser holographic interferometry. Temperature field inversion is completed with the aid of components distribution divided from numerical simulation of combustion and air components assumption. Error analysis of key steps is conducted using the theoretical formula of interference temperature measurement and characteristic structure of fringes obtained from optical simulation. Based on the calculation and analysis, air components assumption will not cause significant error in the low temperature region but will result in high error in the high temperature region. Moreover, the small error in environmental temperature measurement transfer to a high temperature range will expand more than tenfold. Results of temperature measurement using air components assumption relative to combustion simulation require the greatest amendment amounting to seven percent.

Keywords

temperature field / flame / error analysis / holographic interferometry

Introduction

The measurement of temperature field is of great significance to scientific research and national economic development. This is because the measurement of thermal physical quantities is widely used in the implementation and control of industrial processes such as power, aerospace, chemical industry, and oil refining, among others [1]. Temperatures are usually measured by inserting thermocouples into tested fields and entire temperature fields are reconstructed using readings from the thermocouples. However, this method has drawbacks. Temperatures at different points are not obtained simultaneously and tested fields are disturbed by the thermocouples themselves. In addition, obtained readings need to be compensated against the effect brought about by radiation. Various optical methods that are full-field, sensitive, and non-contact, as well as include holographic interferometry [2,3], speckle shearing [4-6], Talbot interferometry [6-8], and shearing interferometry [6,7,9-11] have been widely used in measuring temperature fields.

The measurement of temperature fields using laser holographic interferometry has focused mainly on the natural convection of heated air. Luo and Huang [12] studied the natural convection of an elliptical tube and obtained clear fringes. Xiao et al. [13,14] used CT multi-projection reconstruction algorithm in the measurement of temperature fields of natural convection. Processing data from temperature field measurement using holographic interferometry is more difficult than low temperature hot air flow field measurement because the flame characteristic temperature is much higher than the normal hot air flow field. The phase gradient of fringes obtained through holographic reproduction is large and full-field components distribution is complex. However, with the increase in computer performance and the emergence of combustion numerical simulation tools and high-resolution camera equipment, a marked improvement in fringe phase extraction and concentration field processing has been seen and has contributed to the rapid development of the technology for interference measurement temperature [15].

Laser holographic interferometry method is based on the measurement of changes in the refractive index. As temperature increases, the refractive index of a tested flow field decreases accordingly. The relationship between temperature and refractive index can be expressed using Gladstone-Dale equation. Although the laser interferometry technique belongs to the field of optical imaging technology, Gladstone-Dale coefficients distribution of tested fields is associated with local components distribution. At the same time, with the inversion of phase difference distribution extracted from interference fringes, it can only obtain the distribution of space refractive index field. Full-field distribution of gas constant R_g has to be known, and temperatures are linked with the refractive index. However, the full-field distribution of R_g is also related to local components distribution. The effect of various compositions is usually neglected in holographic interferometry of flames [16,17]. Tieng et al. [18] studied the effect of composition in the measurement accuracy of two laminar propane-air flames. Xiao et al. [19,20] studied the temperature distribution of two-dimensional, steady-state, non-premixed, and premixed methane flames using laser holographic interferometry and provided the distribution of interference fringes. The previous studies discussed the role of components concentration in the process of temperature field inversion using refractive index. These studies concluded that the use of air components assumption is still appropriate for part premixed flames and non-premixed flames but that it induces large errors for non-premixed flames. The impact of ambient temperature was not considered. In this paper, the concentration field derived from combustion numerical simulation is obtained by referring to Ref. [21]. The error caused by interference measuring temperature using air components assumption is evaluated using the data of combustion numerical simulation. The inaccurate record of ambient temperature may lead to inexplicably high values of temperature inversion.

Experimental setup

The two-dimensional axisymmetric ethylene diffusion flame is emitted from a double-tube burner. The burner is a typical axisymmetric convergence burner filled with silver beads and steel wire balls to ensure a stable laminar diffusion flame and avoid flashing back of the flame (Fig. 1). The ethylene comes from a high-pressure cylinder with a purity of 99.99% and flows from the inner brass tube. The ethylene is adjusted to the desired pressure using the pressure regulating valve, flows into the mass flow controller, finally passing through the needle into the burner fuel tube. Air passes through the outer steel tube and is supplied by the air compressor to ensure sufficient supply of pressure. Ethylene and air flow are 129 and 189 L/min, respectively.

The layout of the experimental optical system is a typical off-axis holographic interferometer (Fig. 2). The double-exposure holographic interferometer is employed, and consists of a 40 mW He-Ne laser with a wavelength of 632.8 nm, a beam splitter, five flat mirrors, two expanders, two collimators, and a holographic plate. Laser light coming from the He-Ne laser is split by the splitter into two different light beams: one beam travels along the reverse direction and is parallel to the long side of the anti-vibration platform after two total reflections.

The beam becomes a large-diameter parallel beam, specifically, an object beam, after passing through the expander and the collimator. The object beam then passes through the tested object field above the burner to the vertical exposure holographic plate. After three total reflections, the other beam passes through the expander and the collimator to the exposure holographic plate with a large inclined angle, constituting the reference beam. Optical path difference of the object beam and the reference beam is maintained at a constant to form interference fringes. Figure 3 is the infinitely wide interference fringe of the flame produced by the double-exposure holographic interferometer.

Principle of temperature measurement

The relationship between the refractive index and temperature can be deduced by applying the ideal gas Eq. (1) and Gladstone-Dale relation (2).

(1)p=\rho R_{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{x}}T,

(2)n=\mathrm{1}+\rho K_{\mathrm{m}\mathrm{i}\mathrm{x}},

(3)T={\displaystyle \frac{p{K}_{\mathrm{m}\mathrm{i}\mathrm{x}}}{(n-\mathrm{1})R_{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{x}}}}={\displaystyle \frac{p{\displaystyle \sum {\phi }_i{k}_i}}{(n-\mathrm{1}){\displaystyle \sum {\phi }_i{k}_i/M_i}}},

where T is the absolute temperature; p is the pressure; n is the refractive index along the light path through the flame; R is universal gas constant; and {\phi }_i, M_i, k_i represent mass fraction, molecular weight, and G-D constant of the i-th species, respectively.

When X=\mathrm{2}\mathrm{\pi }(n-\mathrm{1})/\lambda and \mathrm{\Delta }X=X-X_{\mathrm{0}}, Eq. (3) can be expressed as

(4)T={\displaystyle \frac{\mathrm{2}\mathrm{\pi }p{K}_{\mathrm{m}\mathrm{i}\mathrm{x}}}{\lambda (\mathrm{\Delta }X+X_{\mathrm{0}})R_{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{x}}}},

where X is the phase function, \mathrm{\Delta }X is the phase difference, and X_{\mathrm{0}} is the phase function of the environment. K_{\mathrm{mix}} and R_{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{x}} become constant using the air components assumption for concentration field, that is,

(5)T={\displaystyle \frac{C(p,\lambda )}{\mathrm{\Delta }X+X_{\mathrm{0}}}},

where C(p,\lambda )=\mathrm{2}\mathrm{\pi }p{K}_{\mathrm{a}\mathrm{i}\mathrm{r}}/\lambda R_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{r}}. It is a constant under the influence of atmospheric pressure of the measured field and laser wavelength and known as the temperature measurement constant of air components.

The phase distribution of fringes can be determined by

(6)\theta (x,h)=\mathrm{2}{\displaystyle {\int }_{\mathrm{0}}^{\sqrt{R^{\mathrm{2}}-x^{\mathrm{2}}}}\mathrm{\Delta }X(\sqrt{x^{\mathrm{2}}+s^{\mathrm{2}}},h)}\mathrm{d}s,

where R indicates the impact radius of the flame.

Temperature field reconstruction can be concluded from Eqs. (1)-(6). Phase distribution \theta (x,h) is extracted from interference fringes, \mathrm{\Delta }X is obtained through projection inversion, and temperature field distribution T is obtained by combining the values of X_{\mathrm{0}} and the concentration field distribution.

Temperature field inversion based on air components assumption and concentration field of combustion numerical simulation

Experimental environment pressure is p = 1.013 × 10⁵ Pa. Laser wavelength \lambda is 632.8 nm and C(p,\lambda )=792.25\times {10}^3\mathrm{K}·\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{m}. The K_air is 2.26 × 10^-4 m³/kg, M_air is 28.97 × 10^-3 kg/mol, and R is 8.3143 J/(K·mol). Thus, when ambient temperature T₀ = 285 K, X₀ is 2.780 rad/mm. Temperature inversion is completed based on the air components assumption and the concentration field distribution of 11 kinds of main components, including C₂H₄, O₂, CO₂, H₂O, CO, N₂, O, H, H₂, C₂H₂, and CH₄ derived from combustion numerical simulation. Temperature inversion results at a given height of 3 cm based on simulative temperature, air components assumption, and simulative concentration field are demonstrated in Fig. 4.

The temperature field obtained from interference measurement temperature has the same partial peak distribution with the simulative result (Fig. 4). The result of the interference measurement temperature drops more quickly than the simulative result near the axis center. Taking into account that the error of axis center in the process of projection inversion is difficult to control, the result of interference temperature measurement is only for reference. The overall gradient at the right of the peak value of interference is clearly less than the simulative result because the valley point coordinate near 9 mm of the real fringes object is greater than the simulative fringes near 8 mm. This results in the increase of temperature from the environment region to the axis. A comparison of the interference measurement temperature based on air components assumption and combustion numerical simulation shows that the highest error (about 100 K) is obtained using air components assumption to simplify the concentration field if the complex components characteristics in the high temperature flame are neglected. However, using the air components assumption does not cause significant error in the low temperature region.

Error analysis

Error due to phase function

The curve of interference temperature measurement using air components assumption under standard atmospheric pressure is presented in Fig. 5. From air components assumption, Eq. (7) can be obtained.

(7)T={\displaystyle \frac{\mathrm{792.25}}{\mathrm{\Delta }X+X_{\mathrm{0}}}},

Obviously, Eq. (7) is a reciprocal distribution. The higher the temperature, the closer its denominator is to zero. Specifically, the small error of \mathrm{\Delta }X or X₀ can lead to a dramatic change in temperature measurement results in the high temperature region.

The \mathrm{\Delta }X is obtained from image processing of the double-exposure holographic interferometer and is significantly lower in the near-axis region than the normal value because of the effect of fringe image noise. This contributes to \mathrm{\Delta }X+X_{\mathrm{0}}\to \mathrm{0} and becomes the main reason for the inexplicably high temperature. On the other hand, even if there is no error in the process of image processing in an ideal situation, the value of temperature inversion remains inexplicably high due to inaccurate environmental temperature recording. If environmental temperature is given at random at a later image processing stage instead of taking note of it in a timely and accurate manner, it will also lead to a high value of temperature inversion.

Error due to concentration field

Figure 6 represents the distribution of correction factors of the flame at a height of 3 cm. The distribution changes simultaneously with the change of simulative temperature. The correction factors are all positive and the maximum does not exceed 6.5 percent. The correction factor is defined as

(8)\eta ={\displaystyle \frac{T}{T_{\mathrm{a}\mathrm{i}\mathrm{r}}}}={\displaystyle \frac{K_{\mathrm{m}\mathrm{i}\mathrm{x}}/K_{\mathrm{a}\mathrm{i}\mathrm{r}}}{R{g}_{\mathrm{m}\mathrm{i}\mathrm{x}}/R{g}_{\mathrm{a}\mathrm{i}\mathrm{r}}}}={\displaystyle \frac{R{g}_{\mathrm{a}\mathrm{i}\mathrm{r}}{\displaystyle \sum {\phi }_i{k}_i}}{K_{\mathrm{a}\mathrm{i}\mathrm{r}}{\displaystyle \sum {\phi }_{i}R/M_i}}}.

The disposal of concentration field is difficult in the interference measurement temperature of the flame. The simulative concentration field is different from the actual concentration field, but is a valid factor in assessing the required distribution of correction factors for the temperature inversion value using air components assumption. The correction factors are reasonable only above a certain height of flame. This is because the main gas component of the ethylene laminar diffusion flame near the exit plane of the inner tube is ethylene. The G-D constant and molecular weight of the ethylene differ greatly from air components. This leads to the demand for substantial amendment of the value of temperature inversion using air components assumption, thereby making interference measurement temperature meaningless.

Conclusion

Air components assumption and concentration field data of combustion numerical simulation were used to determine the temperature field distribution of axisymmetric, laminar, and ethylene diffused flame at a given height using a laser holographic interferometer. Air components assumption will not cause significant error in the low temperature region but may do so in the high temperature region. Error analysis due to the phase function and concentration field was conducted. In the experiment, the recording of environmental temperature was crucial because a small recording error of transfer to the high temperature region could expand more than tenfold. In the high temperature region, the distribution of temperature field using air components assumption relative to combustion numerical simulation required the greatest amendment amounting to seven percent.

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