Effective emissivity of a blackbody cavity formed by two coaxial tubes Guohui Mei,* Jiu Zhang, Shumao Zhao, and Zhi Xie College of Information Science and Engineering, Northeastern University, Shenyang 110819, China *Corresponding author: [email protected]. Received 21 January 2014; revised 25 February 2014; accepted 12 March 2014; posted 13 March 2014 (Doc. ID 204812); published 10 April 2014

A blackbody cavity is developed for continuously measuring the temperature of molten steel, which consists of a cylindrical outer tube with a flat bottom, a coaxial inner tube, and an aperture diaphragm. The ray-tracing approach based on the Monte Carlo method was applied to calculate the effective emissivity for the isothermal cavity with the diffuse walls. And the dependences of the effective emissivity on the inner tube relative length were calculated for various inner tube radii, outer tube lengths, and wall emissivities. Results indicate that the effective emissivity usually has a maximum corresponding to the inner tube relative length, which can be explained by the impact of the inner tube relative length on the probability of the rays absorbed after two reflections. Thus, these results are helpful to the optimal design of the blackbody cavity. © 2014 Optical Society of America OCIS codes: (120.5630) Radiometry; (120.6780) Temperature; (130.6010) Sensors; (230.6080) Sources. http://dx.doi.org/10.1364/AO.53.002507

1. Introduction

Blackbody cavities are widely applied as standard sources in radiometry and radiation thermometry. It also can be used for measuring the inner average temperature of an object when the temperature difference is small. A sensor for measuring the temperature of molten steel [1] is schematically shown in Fig. 1. The cavity of the sensor is immersed into the molten steel, from which the detector registers the approximate blackbody radiation. Taking into account the molten steel erosion, the sensor is composed of the inner and outer layers. The inner is used for forming a blackbody cavity and the outer for resisting the erosion of the molten steel and slag layer. Molten steel temperature is usually above 1500°C, and the corundum tube is used as the inner cavity to keep the light path clean. In practical application, the length and diameter of the corundum tube is about 1 m and 25 mm, respectively. Because of the 1559-128X/14/112507-08$15.00/0 © 2014 Optical Society of America

poor thermal shock and high temperature creep deformation, fracture and bending of the corundum tubes are often observed, which result in an error of temperature measurement. Therefore, a new kind of sensor structure is proposed in Fig. 1(c); the inner cavity bottom is removed, which can enhance the sensor’s reliability and improve the speed of response. However, since the outer cavity is made of alumina/magnesia carbon refractory [2,3], its eutectic impurities will volatilize transparent gas under a high temperature region and condense under a low temperature region, which obscures the light path of temperature measurement. By N 2 (or inert gas) with a certain pressure as shown in Fig. 1(c), the volatile gas will be blocked and the cleanness of the inner cavity light path can be kept as well. This sensor has been successfully applied to continuously measuring the temperature of molten steel in the continuous casting tundish. In order to measure the temperature, the effective emissivity for the cavity of the sensor is first considered. Since experimental determination of the effective emissivity is difficult and limited in some areas 10 April 2014 / Vol. 53, No. 11 / APPLIED OPTICS

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Fig. 1. Schematic view of continuous temperature measurement system for molten steel (a), cylindrical cavity (b), and cavity formed by two coaxial tubes (c).

of applicability, the method of model calculation usually is adopted. Among all the methods, the Monte Carlo method (MCM) [4–10] is the most powerful and flexible, due to its applicability to arbitrarily shaped cavities. By the MCM, researchers [11–22] had studied the effective emissivity of various cavity shapes such as cylindrical, conical, square, cylindrical and cylindro-conical, cylindrical-spherical, cylindrical, and cylindro-conical with grooved walls, as well as cylindrical with an inclined bottom. However, the cavity formed by two coaxial tubes [Fig. 1(c)] has not been investigated. In this paper, numerical simulation based on MCM is carried out to study the radiation characteristics of the isothermal cavities. The numerical calculated method is presented. The results are analyzed, which are helpful to optimize the parameters of the cavity. 2. Statement of the Problem A.

Cavity Model

The cavity is composed of an outer tube with a flat bottom, a coaxial inner tube, and an aperture diaphragm (see Fig. 2 for details), where Ro , Ri , and

Ra represent the radii of the outer tube, inner tube, and aperture. Lo and Li are the length of the outer and inner tubes, respectively. In order to uniform the scale, we use dimensionless geometrical parameters of the cavity and set Ro  1 in all simulations. For simplification, the following assumptions are applied: (1) The optical characteristics of the cavity walls are independent of temperature and wavelength. (2) The walls of the cavity are diffuse (according to Lambert’s law) and independent of the position. The walls of the outer and inner tubes are opaque, and their emissivities are εo and εi , respectively. (3) The thickness of the inner tube is ignored. B. Definition of the Effective Emissivity

Since the walls of the cavity are all diffuse and opaque, the radiance is the same for all directions and does not depend on the angular distribution of the incident radiation. So the effective emissivity is also independent of the direction, which can be defined as the ratio of the wall effective radiation (the sum of its intrinsic radiation and reflected radiation from surrounding objects) to that of pure blackbody at the reference temperature. Here, the reference temperature, T ref , is the temperature at the central point of the cavity bottom. The spectral local effective emissivity can be defined in a similar way as ελ;e λ; T ref ; T; ρ 

Fig. 2. Schematic drawing of a cavity formed by two coaxial tubes. 2508

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Lλ λ; T; ρ ; Lλ;bb λ; T ref 

(1)

where Lλ is the spectral radiance of a cavity at the temperature T, wavelength λ, and point ρ; Lλ;bb is

the spectral radiance of a perfect blackbody at the reference temperature T ref and wavelength λ. Using the Planck law, the spectral local effective emissivity can be written by ελ;e λ; T ref ; T; ρ 

Lλ λ; T; ρ ; c1 · fπ · λ expc2 ∕λT ref  − 1g−1 (2) 5

where c1  3.743 × 10−16 W · m2 and c2  1.4388 × 10−2 m · K are the first and second radiation constants, respectively. According to the Stefan–Bolzmann law, the total local effective emissivity can be written by πLT; ρ ; εt;e T ref ; T; ρ  σT 4ref

ελ;a

(3)

Z Sa

ελ;e λ; T ref ; T; ρdSa ;

for the zenith angle, ψ, and the azimuth angle, ϕ, of the local spherical coordinate system: ψ  arcsin

where σ  5.67 × 10−8 W · m−2 · K −4 is the Stefan– Boltzmann constant. For an isothermal cavity, ελ;e λ; T ref ; T; ρ  εt;e T ref ; T; ρ, in order to make a distinction with the nonisothermal cavity, the local effective emissivity is expressed as ε0e . The average spectral effective emissivity can be defined as 1  Sa

Fig. 3. Schematic drawing of the ray tracing. The solid and dotted lines represent the direction of an incident ray and the reflected rays, respectively.

(4)

where Sa is the effective region of the cavity bottom. For an isothermal cavity, the average effective emissivity can be expressed as ε0e;a . In fact, only the radiation of the cavity bottom central region can reach the detector in the optical system for the temperature sensor. The average effective emissivity corresponds to the integrated effective emissivity of the cavity where the detector is located at an infinite distance from the cavity bottom. This research focuses on the local effective emissivity distribution and average effective emissivity for the isothermal cavity formed by two coaxial tubes. 3. Numerical Calculated Method

In this paper, the backward ray tracing method of the MCM is employed for calculating the effective emissivity. The incident radiation is considered to consist of a rather large number of rays, and each ray is assigned the unit energy. For calculating the local effective emissivity, all the incident rays hit the same point of the cavity bottom. For calculating the average effective emissivity, the incident points are uniformly distributed over the cavity bottom. When a ray hits the cavity wall, the energy being absorbed or reflected is chosen by the pseudorandom number uε . If uε < ε (the emissivity of the wall), the energy is absorbed; otherwise it is reflected. For the diffuse cavity, the direction of the reflection is random. The following relationships [6] are usually employed

p uψ ;

ϕ  2πuϕ ;

(5)

where uψ and uϕ are a pair of pseudo-random numbers. The reflected rays propagate according to ψ and ϕ (shown in Fig. 3), which hits a cavity wall again or escapes through the opening aperture. The paths of the ray are pursued until the energy of the ray is absorbed or the ray escapes through the opening aperture after reflections in the cavity. The effective absorptivity of the cavity can be derived from the ratio of the rays absorbed by the cavity to the total incident rays. According to the Kirchhoff’s law, the effective emissivity of the opaque cavity can be written as ε0e  α0e 

Nα ; N

(6)

where ε0e is the effective emissivity of the isothermal cavity, α0e is the effective absorptivity, N is the total number of the incident rays, and N α is the number of the rays absorbed by the cavity wall, given by N α  N α0  N α1  N α2      N αi     ;

(7)

where N αi is the number of the rays absorbed by the cavity wall after the ith reflection. In order to confirm the validity and accuracy of the algorithms, the calculation results are compared with those in [11] for the cylindrical cavity with an aperture diaphragm under the isothermal condition. The characteristic parameters of the cavity are listed in Table 1. The 109 rays are traced to obtain the effective emissivity, and the standard deviation of the result is less than 7 × 10−7 near the bottom and 5 × 10−6 at other regions. The results are in very good agreement with those in [11], and the discrepancies do not Table 1.

Characteristic Parameters of the Cylindrical Cavity with Diffuse Walls

Length of the Radius of the Radius of the Emissivity of the Cavity (Ro ) Aperture (Ra ) Wall(ε) Cavity (Lo ) 500 mm

30 mm

25 mm

0.885

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Table 2.

Radial Distance to Bottom Center (r∕mm) 3 9 15 21 27 Axial distance to bottom (z∕mm) 60 140 220 300 380 415 435 455 475 495

Effective Emissivity Distribution Calculated for the Cylindrical Cavity

Ref. [11]

Discrepancies

0.999705 0.999705 0.999705 0.999706 0.999708

−7

5.8996 × 10 4.08075 × 10−7 6.27258 × 10−7 2.68323 × 10−7 4.65223 × 10−7

0.999706 0.999706 0.999707 0.999707 0.999708

−1 × 10−6 −1 × 10−6 −2 × 10−6 −1 × 10−6 0.7 lie between inner and outer tubes; the space near these points is more closed as

the inner tube length increases. When Li ∕Lo > 0.6, the intersection of curves is observed. It indicates that there is an inversion of the effective emissivity with increasing Li ∕Lo. The results are consistent

Fig. 9. Local effective emissivity as a function of the distance from center to edge of the cavity bottom for Li ∕Lo  0, 0.2, 0.4, 0.6, 0.8, εo  εi  0.85, Ro  1, Ri  0.7, Lo  4, where (a) without a diaphragm Ra  Ri . (b) With a diaphragm, Ra  0.5Ro . 10 April 2014 / Vol. 53, No. 11 / APPLIED OPTICS

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with that of the average effective emissivity as above. In comparison, the local effective emissivity of a cavity with a diaphragm is higher than that without a diaphragm, and their distributions are similar except that one of the bottom edge points increases more gently with Li ∕Lo . 5. Conclusions

A blackbody cavity is developed for continuously measuring the temperature of molten steel, which consists of a cylindrical outer tube with a flat bottom, a coaxial inner tube, and an aperture diaphragm. The effective emissivity of the cavity has been calculated by the MCM. For εi > εo, each average effective emissivity curve has a maximum corresponding to the inner tube relative length Li ∕Lo , whose location is dependent on the inner tube radius, Ri , and outer tube length, Lo . On the other hand, εi < εo , the average effective emissivity monotonously decreases with increasing Li ∕Lo. It can be explained by the impact of the inner tube relative length on the probability of the rays absorbed after two reflections. Thus, the results are helpful to the optimal design of the cavity. This research is limited to the isothermal cavity with the diffuse wall, but the algorithms can also be applicable to the diffuse and specular wall. Moreover, the numerical results and application of the nonisothmal cavity formed by two coaxial tubes will be discussed in the subsequent paper. We would like to thank the Fundamental Research Funds for the Central Universities (No. N110404029) and National High Technology Research and Development Program of China (No. 2006AA040309) for financial support. References 1. Z. Xie, Y. Ci, H. J. Meng, and H. Zhang, “Development of continuous temperature measuring sensor for liquid steel based on blackbody cavity,” Chin. J. Scientific Instrum. 26, 446–448, 456 (2005). 2. S. M. Zhao, G. H. Mei, J. Zhang, and Z. Xie, “Finite element analysis of composite structure continuous temperaturemeasuring sensor for liquid steel,” J. Northeast. Univ. Nat. Sci. 133, 926–929 (2012). 3. S. M. Zhao, G. H. Mei, J. Sun, W. Yang, and Z. Xie, “Estimation of effective diffusion coefficient of gaseous species in MgO-C

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Effective emissivity of a blackbody cavity formed by two coaxial tubes.

A blackbody cavity is developed for continuously measuring the temperature of molten steel, which consists of a cylindrical outer tube with a flat bot...
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