The effect of initial diameter on rainbow positions and temperature distributions of burning single-component n-Alkane droplets

Journal of Quantitative Spectroscopy & Radiative Transfer ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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The effect of initial diameter on rainbow positions and temperature distributions of burning single-component n-Alkane droplets Haipeng Li, Christopher D. Rosebrock, Thomas Wriedt, Lutz Mädler n Foundation Institute of Material Science (IWT), Department of Production Engineering, University of Bremen, Bremen, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 15 September 2016 Received in revised form 3 January 2017 Accepted 3 January 2017

The effect of initial diameter on rainbow positions of burning single-component n-Alkane droplets has been investigated experimentally for the first time. The droplet diameters are determined with interferometric laser imaging for droplet sizing, and the temperature distributions inside burning droplets are assessed by rainbow refractometry together with a droplet combustion model developed in our previous work. Temperature gradients inside burning droplets influence rainbow positions, which first make the experimental scattering angles of the rainbow maxima increase and then decrease. The variations of initial diameter lead to variations of both experimental rainbow maxima and simulated temperature of n-Alkane burning droplets. & 2017 Elsevier Ltd. All rights reserved.

Keywords: Rainbow refractometry Single-component droplet combustion Droplet diameter Temperature gradients Rainbow positions

1. Introduction Spray combustion has been widely used in many industrial processes, such as furnaces, boilers, gas turbines, diesel engines, liquid rocket engines [1,2], and even nanoparticle production [3,4]. During spray combustion, droplets vaporize and burn in a gas environment. The understanding and modeling of droplet vaporization and combustion are fundamental for the design of energy efficient process. Non-intrusive laser light scattering techniques have been widely applied to determine droplet size, velocity, composition, and temperature. Interferometric laser imaging for droplet sizing (ILIDS) takes advantage of the interference pattern of light scattered in the forward direction to measure droplet size [5–10] and can reach an accuracy of 2% [6,9]. Rainbow refractometry (RRF) can provide information on refractive index, temperature and size of droplets [11–15]. Roth et al. [11] first described RRF and used the angular position of the rainbow (rainbow angle) to determine the droplet temperature. Most previous studies of RRF are focused on homogeneous droplets with constant size and constant temperature. However, droplet sizes and droplet temperatures change with time in spray combustion. Temperature variations result in refractive index gradients inside vaporizing or burning droplets, n Correspondence to: Department of Production Engineering, Foundation Institute of Material Science (IWT), University of Bremen, Badgasteiner Str. 3, Bremen 28359, Germany. E-mail address: [email protected] (L. Mädler).

which change the rainbow scattering pattern. Therefore, temperature or refractive index gradients have to be considered in the application of RRF to vaporizing or burning droplets. Theoretical and experimental studies of rainbow positions or RRF on droplets with temperature or refractive index gradients have been reported earlier. Kai et al. [16] used a finely stratified sphere model to calculate scattering electromagnetic fields from radially inhomogeneous spherical droplets. They found that radial gradients of the refractive index change the ray paths within large spherical droplets into curved lines, resulting in time variant rainbow angles. Anders et al. [17] investigated the effect of refractive index gradients within droplets on rainbow positions, and pointed out that refractive index gradients lead to a shift in rainbow angles compared with homogeneous droplets with constant refractive index. van Beeck et al. [18] observed the unusual temperature evolution inside a burning droplet, and attributed it to the strong temperature gradients during its transient heating phase. Massoli [14] calculated the light scattering of a radially inhomogeneous vaporizing droplet with reducing diameter and varying temperature, and indicated that the internal temperature gradients could produce intrinsic uncertainty to temperature measurement of inhomogeneous droplets. Vetrano et al. [19,20] generalized the rainbow Airy theory to a single droplet exhibiting internal refractive index gradients, and assessed refractive index gradients inside a burning n-Octane droplet with RRF. Saengkaew et al. [21] applied RRF to particles with radial refractive index gradients, and quantified the effect of radial gradients on rainbow measurements with a high accuracy. These investigations indicate

http://dx.doi.org/10.1016/j.jqsrt.2017.01.004 0022-4073/& 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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that temperature gradients and refractive index gradients inside droplets influence rainbow positions. For single-component droplet combustion, refractive index gradients are caused by temperature gradients inside burning droplets. Therefore, measurement of rainbow positions can be used to assess temperature distributions inside burning singlecomponent droplets [17,19,20]. Our previous work has confirmed that varying temperature gradients inside burning micro-size n-Alkane droplets can be detected with RRF together with a droplet combustion model [22]. The burning droplets investigated in our previous work have all similar initial droplet diameters of 100 mm. Droplets with different initial diameter have different specific surface area or volume for heat absorption during combustion, which will lead to different evolutions of temperature gradients inside burning droplets and then affect the rainbow scattering pattern. In principle, a small droplet will heat up faster than a large droplet due to its large specific surface area or small volume. As a result, this kind of influence should be reflected on rainbow positions and temperature distributions inside burning droplets with different initial diameter. However, to the best of our knowledge, experimental evidences is still missing in order to clarify this effect. The purpose of this paper is to investigate the effect of initial diameter on rainbow positions and temperature distributions of burning single-component n-Alkane droplets. We will first describe the experimental and analytical procedure, then present experimental results of droplet diameter and rainbow maxima, finally we will compare between experiment and simulation and discuss the simulated temperature distributions.

2. Experimental setup and procedure 2.1. Experimental setup and materials A sketch of experimental setup for the detection of refractive index and diameter of single isolated droplet during combustion is shown in Fig. 1 (left). The experimental setup includes a dropleton-demand generator (Piezodropper [23]), a green laser (Opus 3 W – 532 nm TEM00), ignition electrodes, gas supply, a solvent reservoir, a mass flow controller (Bronkhorst - EL Flow) and two linear array CCD cameras (DALSA - Spyder 3). The laser beam with an initial diameter of 2 mm is expanded to a laser light sheet by

lens for the purpose of covering the whole combustion process of the droplets. As shown in Fig. 1 (right), the cross section of the laser light sheet is in a shape of a vesical piscis, which is 20 mm in length and 2 mm in width. Pure n-Alkane is stored in a solvent reservoir, and fed to the droplet generator with the help of a plastic tube. The droplet generator reproducibly ejects single isolated droplets at a frequency of 4 Hz. The initial droplet velocities are in the range from 0.5 to 1 m/s. After ignition by the spark electrodes, the burning droplets move upwards through a laser light sheet. Co-flowing oxygen (purity 99.95%) is delivered to the plastic square frustum at a flow rate of 1.4 L/min. The oxygen sheath around the upward droplets provides an oxidizing atmosphere, keeping the flame spherical and concentric around the droplet. Thus, buoyancy effects on the burning droplets can be neglected in our experiments [24,25]. It is necessary to mention that rainbow refractometry is very sensitive to the shape of the droplet, and deformation of spherical droplet will cause caustics [26] and change the rainbow scattering pattern [27]. The scattering light from the droplets is recorded with two linear array CCD cameras (Spyder3 CL, S3-24-01K40-00-R) at a frequency of 67 kHz. To focus objects from infinity, one lens is fixed in the front of each CCD camera with a focal length of 6.56 70.25 mm. Both CCD cameras have a center of image area of 30 mm in length, and a scattering range of 10.2° for recording the scattering light. The linear array CCD camera uses a single line of sensor pixels to build up a two dimensional out-of-focus image, which consists 1024
1024 pixels and a resolution of 96 pixels/inch. Both CCD cameras are synchronized with the droplet-on-demand generator in order to obtain time-resolved images. While the backward CCD camera captures the rainbow scattering pattern, the forward CCD camera records the interference pattern. The two CCD cameras and the burning droplet are located at the same vertical level. N-Octane (SIGMA Aldrich, 98% assay), n-Nonane (SIGMA Aldrich, 99% assay), and n-Hexadecane (SIGMA Aldrich, 99% assay) are used as combustion liquids. The properties of n-Alkane investigated in this study are shown in Table 1 [28]. Their refractive indices in the temperature range from 20 to 90°C are measured with an Abbe refractometer (Carl Zeiss [29]) (See Fig. 2). 2.2. Calibration and diameter measurement For the purpose of calibration, the rainbow scattering pattern (see Fig. 3(a)) and the interference pattern (see Fig. 3(b)) were

Fig. 1. Sketch of experimental setup for the detection of refractive index and diameter of single isolated droplet during combustion (left), and the laser light sheet before it is interacting with the droplets (right). θILIDS (acute angle) is the scattering angle of the forward camera with respect to the laser sheet, and θRRF (obtuse angle) the scattering angle of the backward camera with respect to the laser sheet.

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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Table 1 Properties of n-Alkane investigated in this study [28]. n-Octane n-Nonane n-Hexadecane Molecular formula Boiling point [K] Surface tension (at 25 °C) [mN/m] Viscosity (at 25 °C) [mPa s] Heat of vaporization (at 25 °C) [kJ/mol] Heat of vaporization (at boiling point) [kJ/mol] Thermal conductivity (at 25 °C) [W/ m K]

C8H18 399 21.14 0.508 41.49 34.41

C9H20 424 22.38 0.665 46.55 37.18

C16H34 560 27.05 3.032 81.35 51.84

0.128

0.131

0.140

Fig. 4. Calibration of light scattering patterns of a 103 mm non-burning n-Nonane droplet: (a) rainbow scattering pattern; (b) interference pattern. The unit of intensity, a.u., represents arbitrary unit. The intensity of scattering light shown here are obtained by normalizing the real intensity first and then multiplied by 255. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 2. Refractive indices of n-Octane, n-Nonane and n-Hexadecane in the temperature range from 20 to 90 °C. Points with mark are measured, and dotted line are linearly fitted.

recorded for non-burning droplets. In Fig. 3, the horizontal axis represents the scattering angle (totally 10.2°), and the vertical axis represents the time (totally 15.2 ms). The scattering intensity versus rainbow angle at different times are captured at different vertical positions of the upwards moving droplet. The rainbow scattering pattern and the interference pattern can be extracted for a specific time, as shown in Fig. 4 (blue curves). Lorenz Mie theory is applied to simulate the light scattering patterns in order to calibrate the scattering angle ranges of the two CCD cameras (see Fig. 4).

In principle, the RRF technique makes use of the angular distance between the first rainbow and the second rainbow to measure droplet size. However, previous studies have indicated that the angular distance between the first rainbow and the second rainbow are sensitive to both droplet size and refractive index [12,13,15] and it is strongly dependent on droplet size and weakly dependent on the refractive index [15]. Thus, the measurement of droplet diameter with RRF can be influenced by refractive index variations (temperature variations) inside the burning droplet. Therefore, ILIDS is selected to determine droplet diameter because the angular distance between successive interference maxima is almost independent of temperature [13]. The equation to determine the droplet diameter in ILIDS can be expressed as [12,30]:

⎛ ⎛ θ⎞ 2λ ⎜ dp≅ ⎜ cos ⎝⎜ ⎠⎟ + 2 Δθ ⎜ ⎝

n sin

() θ 2

1+n2 −2n cos

() θ 2

⎞−1 ⎟ ⎟ ⎟ ⎠

(1)

where dp , ∆θ , and n are droplet diameter, the angular distance between successive interference maxima, and refractive index of the droplet, respectively. Moreover, λ is the wavelength of laser light, and θ is the mid-point of the covered scattering angle range, where values of both are constant.

Fig. 3. Light scattering patterns of a 103 mm non-burning n-Nonane droplet: (a) rainbow scattering pattern; (b) interference pattern.

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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Fig. 5. Rainbow scattering patterns of a non-burning (a) and a burning (b) n-Nonane droplet with the same initial diameter of 138 mm.

2.3. Temperature gradients and droplet combustion model For a droplet of uniform temperature, the refractive index inside the droplet is constant and hence the scattering angle of the rainbow does not change with time (see Fig. 5(a)). For a burning droplet, however, temperature increases considerably due to heat transfer from the flame. Due to high vaporization rate in combustion, temperature gradients inside the droplet exist throughout the entire droplet life time [31]. The temperature gradients result in varying refractive index gradients inside the burning droplet, which will influence the rainbow scattering pattern (see Fig. 5(b)). The detailed explanation of the effect of temperature gradients on the rainbow pattern will be shown later in the part of experimental rainbow maxima. Since the rainbow scattering pattern is very sensitive to the refractive index and the refractive index is a function of temperature for pure liquids (see Fig. 2), interior temperature distributions can be obtained by measuring the transient evolution of the rainbow scattering pattern. A fast Fourier transformation (FFT) proposed by Hom et al. [15] is applied to filter the high frequency oscillations of the rainbow scattering pattern and extract the angular position of the rainbow. The scattering angle of the rainbow maxima, defined as the angle position of the maximum intensities in the rainbow region, is extracted to evaluate rainbow positions. The droplet combustion in this work is modeled according to [22]. A time-varying heat conduction equation of a sphere is applied to describe the heat conduction process:

ρ ( T ) cp ( T )

∂T (r , t ) ∂ ⎛ ∂T (r , t ) ⎞ 1 = ⎜ r (t )2λ (T ) ⎟ 2 ⎝ ∂t r (t ) ∂r (t ) ∂r (t ) ⎠

(2)

where ρ, cp, λ, r, t, and T represent density, heat capacity, thermal conductivity, radius, time and temperature, respectively. The density, heat capacity, and thermal conductivity of liquids are temperature dependent, whose values are taken from polynomial equations [32] as follows:

ρ

(T ) =

A

(

D

)

(3)

cp (T ) = A+BT +C (T )2 +D (T )3+E (T )−2

(4)

1+ 1− T

B

C

where A, B, C, D, and E are coefficients in polynomial equations. Their values can be obtained from Table 2. The moving boundary is front fixed by the Crank-Nicolson finite difference scheme, which shows the decreasing droplet diameter with the instantaneous droplet radius [33,34]. The boundary conditions are:

⎛ ∂T ⎞ ⎜ ⎟ =0 ⎝ ∂r ⎠r = 0

(6)

( T )r = rs ( t)=TS

(7)

where rs ( t ) is the instantaneous droplet radius. The exponential varying surface temperature is given as follows:

(

Ts ( t )=T0+( ΔT ) ∙ 1 − exp ( −βt )

)

(8)

where ΔT is the temperature difference between the boiling point and the ambient temperature, and β represents pre-factor of the rate of the surface temperature increase. The larger the pre-factor, the faster the increase of the droplet surface and core temperature, and also the faster evolution of the scattering angle of the rainbow maximum [22]. The pre-factor is an adjustable parameter to match the experimental and simulated results. In [22], experimental and simulated rainbow maxima of droplets with similar diameters show good agreement for β ¼1000 s  1. Therefore, the same value is applied in this study. The simulated temperature distributions can be converted to refractive index distributions using the linear relation between refractive index and temperature (See Fig. 2). In order to clarify the droplet combustion model, the input and output parameters are listed in Table 3. The simulated scattering angles of the rainbow maxima are determined with an improved recursive algorithm for light scattering using the multilayered sphere [35]. For the same liquid component, according to Eq. (2), the different initial droplet diameters will cause different temperature gradients dynamics. These different temperature gradients will lead to different refractive index distributions, resulting in different scattering angles of the rainbow maxima. Therefore, experimental and simulated results are required to investigate the effect of the droplet initial diameters in order to verify the droplet combustion model. 3. Results and discussion

λ (T ) =

A+BT +C (T )2 +D (T )3+E (T )4

(5)

Three n-Octane droplets with initial droplet diameters of 142,

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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Fig. 6. Diameters and normalized diameters of n-Alkane droplets burning in pure oxygen. The ignition point of the droplet corresponds to a time or normalized time equal to zero.

96, and 88 mm, three n-Nonane droplets with initial droplet diameters of 138, 103, and 87 mm, and three n-Hexadecane droplets with initial droplet diameters of 103, 87, and 85 mm are investigated. 3.1. Droplet diameter and experimental rainbow maxima Fig. 6 shows the diameters evolutions of n-Octane droplets,

n-Nonane droplets, and n-Hexadecane droplets burning in pure oxygen. It can be seen from Fig. 6(b), (d) and (f) that the combustion of n-Alkane droplets follow the classical d2-law, i.e., the square of the droplet diameter decreases linearly with time. Moreover, the evolutions of normalized droplet diameters for the same component are very similar. These data will be used in the droplet combustion model to assess temperature distributions inside the burning droplets.

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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Table 2 The coefficients used in Eqs. (2)–(4) for n-Octane, n-Nonane and n-Hexadecane [32]. Density ρ [kg/m3]

A B C D E

Heat capacity cp [J/g K]

Thermal conductivity cp [W/mK]

n-Octane

n-Nonane

n-Hexadecane

n-Octane

n-Nonane

n-Hexadecane

n-Octane

n-Nonane

n-Hexadecane

0.1456282 0.02156 599.713 0.09275 0

0.1966136 0.01488 620.483 0.08106 0

2.1751004 0.04846 751.551 0.12742 0

1952.1  1.482 0.007936 0.000045 0

1452.2 2.58 0 0 0

1124.6 3.559 0 0 0

0.2150  0.02839  0.00072 0.00203  0.00206

0.2096 -0.02727 0.00046 -0.00109 0.00095

0.2106  0.0243 0.0006  0.00059 0.00017

Table 3 The input and output parameters used in the combustion model. Input parameters

Sources of input parameters

Output parameters

Density ρ Heat capacity cp Thermal conductivity λ Radius evolution r (t)

polynomial Eq. (1) polynomial Eq. (2) polynomial Eq. (3) ILIDS measurement (Fig. 6) ILIDS measurement (Fig. 6) Table 1. Table 4.

Temperature distributions T(r, t)

Droplet lifetime t Boiling point Tb Droplet temperature before ignition T0 Pre-factor of the rate of the surface temperature increase β

Refractive index distributions n(r, t)

1000 s  1 (Optimum parameter obtained from ref. [22])

Table 4 Temperature before ignition of n-Alkane droplets in this study. n-Octane Initial diameter [mm] Temperature before ignition [K]

n-Nonane

n-Hexadecane

88 96 142 87 103 138 85 295 297 291 299 293 293 297

87 295

103 296

Fig. 7 shows the scattering angles of the first and second rainbow maxima of n-Octane droplets, n-Nonane droplets, and n-Hexadecane droplets burning in pure oxygen. Before droplet ignition, the scattering angles of the rainbow maxima of n-Hexadecane droplets are larger than that of n-Nonane droplets, and the scattering angles of the rainbow maxima of n-Nonane droplets are larger than n-Octane droplets. This responds to the refractive index evolutions in Fig. 2, i.e., n-Alkane with more carbon atoms will have a higher refractive index at the same temperature. Since the scattering angle of the rainbow maxima is nearly a linear function of the refractive index for uniform liquid droplets [22], the droplet with the higher refractive index will have a larger scattering angle of the rainbow maxima at the same temperature. It can also be seen from Fig. 7 that the smaller initial droplet diameter show a faster reduction of the scattering angle of the rainbow maxima. After ignition, the scattering angles of the rainbow maxima of n-Alkane droplets first increase, and then decrease with time. This characteristic suggests that temperature gradients exist within the burning n-Alkane droplets. Kai et al. [16] calculated rainbow positions of 10 mm droplets, and found that the first rainbow angle increases and then decreases during heating process. Anders et al. [17] investigated the influence of refractive index gradients on the angular positions of the rainbow, and attributed the curved evolution of the rainbow angle to the presence of a temperature gradient.

Fig. 7. Scattering angles of the first and second rainbow maxima of n-Alkane droplets burning in pure oxygen.

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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Fig. 8. Rainbow scattering patterns of a non-burning and a burning n-Nonane droplet with the same initial diameter of 138 mm (Left) and the sketches of the geometric optics approximation for rainbow rays (Right). The path of the rainbow rays of the non-burning droplet in the first sketch appears in the last two sketches as a baseline for comparison. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

During the combustion of n-Alkane droplets, the surface temperature is assumed to increase rapidly close to the boiling point due to heat transfer from the flame, while the core temperature increases slowly due to the slow liquid-phase thermal diffusion. The temperature gradient inside the burning droplet causes an increasing refractive index gradient from the surface to the core. Thus, the rainbow rays will bend towards the droplet core according to the geometric optics approximation [19,20,36]. As a result, the scattering angles of the rainbow maxima increase first, as shown in the sketches of the geometric optics approximation in Fig. 8 (red curves). As the combustion proceeds, the core temperature of the droplet increases and temperature gradients inside droplet become weaker. The bent rainbow rays resume to be straight, resulting in a decrease of the scattering angles of the rainbow maxima, as seen in the sketches of the geometric optics approximation in Fig. 8 (blue curves). Due to the increased droplet temperature from core to surface, the scattering angles of the rainbow maxima become lower in comparison with the initial non-burning droplet. 3.2. Rainbow maxima and simulated temperature distributions In order to verify the simulated temperature distributions obtained from the droplet combustion model, the simulated and experimental scattering angles of the rainbow maxima are compared. In Fig. 9(a), (c), (e), Fig. 10(a), (c), (e), and Fig. 11(a), (c), (e), the agreement between the simulated and experimental scattering angles of the rainbow maxima of n-Octane droplets, n-Nonane

droplets, and n-Octane droplets is very high. This demonstrates that the droplet combustion model used in this work is reliable to assess temperature distributions inside burning n-Alkane droplets. The rainbow maxima comparison is an indirect confirmation of the simulated temperature distributions because the direct inversion of temperature from the experimental rainbow maxima is limited by both the algorithm and refractive index/temperature gradients. There are mainly two methods to directly obtain droplet temperatures from the experimental results. The conventional method is to use the function relation between the angular positions of the maxima and refractive index/temperature, but this method is restricted to uniform droplet and it will result in misleading results for droplets with refractive index/ temperature gradients [12,22]. The other method modifies the inversion algorithms to account for non-uniform droplets. A simulated annealing algorithm was investigated in our previous work where we could prove the applicability of the approach, but also highlighted existing deficits [22]. It is necessary to mention that discrepancies exist in the comparison between the simulated and experimental scattering angles of the rainbow maxima of n-Alkane droplets. These discrepancies can be a result of the spark-induced deformations of the droplet, which cause oscillations of experimental rainbow maxima (a short period of time after ignition), as shown in Fig. 11 (a) and (c). Moreover, these discrepancies can be also attributed to the droplet combustion model, since it does not consider soot formation between the flame and the droplet surface. The simulated temperature distributions inside the burning

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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Fig. 9. Comparison of the experimental and simulated scattering angles of the rainbow maxima: (a), (c), (e); and simulated temperature distributions: (b), (d), (f) of n-Octane droplets. In images (a), (c), and (e), red curves represent the simulated scattering angles of the rainbow maxima, and black curves are the experimental scattering angles of the rainbow maxima. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

droplet are shown in Fig. 9(b), (d), (f), Fig. 10(b), (d), (f), and Fig. 11 (b), (d), (f). For droplets of the same component, the one with largest initial diameter will have the strongest temperature gradient at the same time. For example, at t¼4 ms in Fig. 10 (b), (d), and (f), the n-Nonane droplet with an initial diameter of 138 mm has the strongest temperature gradient in comparison with the

other two burning n-Nonane droplets. The simulated surface and core temperatures, and temperature differences between surface and core of n-Octane droplets, n-Nonane droplets, and n-Hexadecane droplets are shown in Fig. 12. It can be seen from Fig. 12(a) that the initial core temperature of the 96 mm droplet is the highest, which is caused by

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Fig. 10. Comparison of the experimental and simulated scattering angles of the rainbow maxima: (a), (c), (e); and simulated temperature distributions: (b), (d), (f) of n-Nonane droplets. In images (a), (c), and (e), red curves represent the simulated scattering angles of the rainbow maxima, and black curves are the experimental scattering angles of the rainbow maxima. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

the high room temperature before combustion. The droplet temperature before ignition is uniform and is equal to room temperature, which can be measure by thermocouple. The evolution of the surface temperature is almost the same for these three n-Octane droplets. The core temperature of the small droplet rises fastest. Due the large volume, the large droplet needs to absorb

more heat to increase its core temperature, resulting in a slower increase of its core temperature compared with the small droplet. Similar results are also obtained for n-Nonane droplets and n-Hexadecane droplets (see Fig. 12(c) and (e)). The effect of initial droplet diameter on simulated core temperature demonstrates the slow liquid-phase heat transfer inside the burning droplets.

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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Fig. 11. Comparison of the experimental and simulated scattering angles of the rainbow maxima: (a), (c), (e); and simulated temperature distributions: (b), (d), (f) of n-Hexadecane droplets. In images (a), (c), and (e), red curves represent the simulated scattering angles of the rainbow maxima, and black curves are the experimental scattering angles of the rainbow maxima. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In Fig. 12(b), (d) and (f), the temperature differences between droplet surface and core first increase, attain a peak and then decrease, which is very similar to the evolution of the experimental rainbow maxima. Temperature differences between droplet surface and core attain a lower peak and reduce faster for small droplets, which corresponds to the faster reduction of the

scattering angles of the rainbow maxima of droplets with the smaller initial diameters. 4. Summary and conclusions Rainbow refractometry combined with a droplet combustion

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Fig. 12. Surface and core temperature: (a), (c), (e); and temperature difference between surface and core: (b), (d), (f) of n-Octane droplets, n-Nonane droplets, and n-Hexadecane droplets, respectively.

model have been employed to study the effect of initial diameter on temperature distributions within burning single-component n-Alkane droplets. The effect of initial diameter on rainbow positions of burning droplets is investigated experimentally for the first time, to the best of our knowledge. The evolutions of droplet diameter used in the droplet combustion model are determined with interferometric laser imaging for droplet sizing. Conclusions

could be drawn as follows: Temperature gradients exist inside burning single-component n-Alkane droplets, which influence the rainbow scattering pattern of droplets via changing the refraction paths of rainbow rays. The scattering angles of the rainbow maxima first increase, reach a peak, and then decrease with time. The effect of initial droplet diameter on both experimental rainbow maxima and simulated

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i

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temperature can be seen, even for small diameter variations, which supports the slow liquid-phase heat transfer inside the burning droplets. This study indicates that the pre-factor β ¼ 1000 s  1 is suitable for modeling the burning n-Alkane droplets with the initial droplet diameter from 85 mm to 142 mm. Discrepancies present in the comparison between simulated and experimental rainbow maxima, which can be attributed to deformations of the burning droplets as well as not considering the flame and the soot formation. This demands for further improvements of the inversion algorithms and the droplet combustion model in the further investigations. Due to the effect of non-sphericity on rainbow refractometry, it is very important to keep the droplet spherical when applying rainbow refractometry to single burning droplet in this study.

Acknowledgements The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for funding this project under grants of MA 3333/4-1. The authors also thank Horst Woyczechowski from the Foundation Institute of Material Science (University of Bremen) for providing technical supports for the droplet-on-demand generators.

References [1] Faeth GM. Current status of droplet and liquid combustion. Prog Energy Combust Sci 1977;3:191–224. [2] Aggarwal SK. A review of spray ignition phenomena: present status and future research. Prog Energy Combust Sci 1998;24:565–600. [3] Teoh WY, Amal R, Mädler L. Flame spray pyrolysis: an enabling technology for nanoparticles design and fabrication. Nanoscale 2010;2:1324–1347. [4] Mädler L, Kammler HK, Mueller R, Pratsinis SE. Controlled synthesis of nanostructured particles by flame spray pyrolysis. J Aerosol Sci 2002;33:369–389. [5] Tropea C. Optical particle characterization in flows. Annu Rev Fluid Mech 2011;43:399–426. [6] König G, Anders K, Frohn A. A new light-scattering technique to measure the diameter of periodically generated moving droplets. J Aerosol Sci 1986;17:157– 167. [7] Glover AR, Skippon SM, Boyle RD. Interferometric laser imaging for droplet sizing: a method for droplet-size measurement in sparse spray systems. Appl Opt 1995;34:8409–8421. [8] Maeda M, Kawaguchi T, Hishida K. Novel interferometric measurement of size and velocity distributions of spherical particles in fluid flows. Meas Sci Technol 2000;11:L13. [9] Dehaeck S, Van Beeck JPAJ. Designing a maximum precision interferometric particle imaging set-up. Exp Fluids 2007;42:767–781. [10] Bilsky AV, Lozhkin YA, Markovich DM. Interferometric technique for measurement of droplet diameter. Meas Sci Technol 2011;18:1–12. [11] Roth N, Anders K, Frohn A. Simultaneous measurement of temperature and size of droplets in the micrometer range. J Laser Appl 1990;2:37–42.

[12] Lemoine F, Castanet G. Temperature and chemical composition of droplets by optical measurement techniques: a state-of-the-art review. Exp Fluids 2013;54:1–34. [13] Van Beeck JPAJ, Riethmuller ML. Nonintrusive measurements of temperature and size of single falling raindrops. Appl Opt 1995;34:1633–1639. [14] Massoli P. Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets. Appl Opt 1998;37:3227–3235. [15] Hom J, Chigier N. Rainbow refractometry: simultaneous measurement of temperature, refractive index, and size of droplets. Appl Opt 2002;41:1899– 1907. [16] Kai L, Massoli P, D'Alessio A. Some far‐field scattering characteristics of radially inhomogeneous particles. Part Part Syst Charact 1994;11:385–390. [17] Anders K, Roth N, Frohn A. Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry. Part Part Syst Charact 1996;13:125–129. [18] van Beeck JP, Riethmuller ML, Lavergne G, Biscos Y, Atthasit A. Processing droplet temperature measurement data obtained with rainbow thermometry. In: Proceedings of the international symposium on optical science and technology. International Society for Optics and Photonics; 2001; p. 251–64. [19] Vetrano MR, van Beeck JPAJ, Riethmuller ML. Generalization of the rainbow Airy theory to nonuniform spheres. Opt Lett 2005;30(6):658–660. [20] Vetrano MR, van Beeck JPAJ, Riethmuller ML. Assessment of refractive index gradients by standard rainbow thermometry. Appl Opt 2005;44:7275–7281. [21] Saengkaew S, Charinpanitkul T, Vanisri H, Tanthapanichakoon W, Biscos Y, et al. Rainbow refractrometry on particles with radial refractive index gradients. Exp Fluids 2007;43:595–601. [22] Rosebrock CD, Shirinzadeh S, Soeken M, Riefler N, Wriedt T, et al. Time-resolved detection of diffusion limited temperature gradients inside single isolated burning droplets using Rainbow Refractometry. Combust Flame 2016;168:255–269. [23] Ulmke H, Wriedt T, Bauckhage K. Piezoelectric droplet generator for the calibration of particle-sizing instruments. Chem Eng Technol 2001;24:265–268. [24] Law CK, Chung SH, Srinivasan N. Gas-phase quasi-steadiness and fuel vapor accumulation effects in droplet burning. Combust Flame 1980;38:173–198. [25] Miyasaka K, Lae CK. Combustion of strongly-interacting linear droplet arrays. Symp (Int) Combust 1981;18:283–292. [26] Nye JF. Rainbow scattering from spheroidal drops - an explanation of the hyperbolic umbilic foci. Nature 1984:531–532. [27] Onofri FR, Ren KF, Sentis M, Gaubert Q, Pelcé C. Experimental validation of the vectorial complex ray model on the inter-caustics scattering of oblate droplets. Opt express 2015;23:15768–15773. [28] Haynes WM, Lide DR. Handbook of chemistry and physics, 6. Boca-Raton: CRC Press; 2007. p. 1–191. [29] Rheims J, Wriedt T. Refractive-index measurements in the near-IR using an Abbe refractometer. Meas Sci Technol 1997;8:601. [30] Glantschnig WJ, Chen SH. Light scattering from water droplets in the geometrical optics approximation. Appl Opt 1981;20:2499–2509. [31] Law CK, Sirignano WA. Unsteady droplet combustion with droplet heating—II: conduction limit. Combust Flame 1977;28:175–186. [32] VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen. In: VDI Heat Atlas (German version). Springer-Verlag Berlin Heidelberg; 2006, p. Dca1Dca38. [33] Crank J, Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. In: Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press; 1947, p. 50-67. [34] Crank J. Two methods for the numerical solution of moving-boundary problems in diffusion and heat flow. Q J Mech Appl Math 1957;10:220–231. [35] Yang W. Improved recursive algorithm for light scattering by a multilayered sphere. Appl Opt 2003;42:1710–1720. [36] Schneider M, Hirleman ED. Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry. Appl Opt 1994;33:2379–2388.

Please cite this article as: Li H, et al. The effect of initial diameter on rainbow positions and temperature distributions of burning singlecomponent n-Alkane droplets. J Quant Spectrosc Radiat Transfer (2017), http://dx.doi.org/10.1016/j.jqsrt.2017.01.004i