A novel light fluctuation spectrum method for in-line particle sizing

Abstract

This paper discusses two problems in in-line particle sizing when using light fluctuation method. First, by retrieving the ratio of particle concentrations at different time, the intensity of incident light is obtained. There exists narrow error between the calculated and pre-detected value of the intensity of incident light. Secondly, by combining spectrum analysis with Gregory’s theory, a multi-sub-size zone model is proposed, with which the relationship between the distribution of turbidity and the particle size distribution (PSD) can be established, and an algorithm developed to determine the distribution of turbidity. Experiments conducted in the laboratory indicate that the measured size distribution of pulverized coal conforms well with the imaging result.

Keywords

in-line measurement / particle size distribution (PSD) / incident light intensity / particle concentration / light fluctuation

Introduction

Since the in-line measurement of particle concentration and size distribution in two-phase flow is very important in many industries and scientific research, many attempts and efforts have been made to seek reliable on-line or in-line measurement techniques, including R&D of mechanical vibration, electrical capacitance tomography (ECT), differential pressure, electric charge transfer, acoustic emission, microwave injection, etc [1]. Optical techniques become one of the major contenders for in-line measurement in recent years [2,3] because they are insensitive to the interference of most environment elements, such as temperature, humidity, electrostatics, etc, and highly sensitive to the changes of concentration or size of particles. Consequently optical techniques have been recognized more appropriate for in-line particle sizing. In addition, compared with other techniques, optical techniques are more cost-effective and robust, which is especially suitable for long-time working.

Of all the optical methods, the so-called light transmission fluctuation (LTF) method is perhaps the most feasible choice for in situ and in-line particle sizing proposed by Shifrin et al. [4,5] and Gregory [6]. In this method the intensity fluctuation of the transmitted light due to particle number fluctuation in the measurement volume was analyzed. The mean particle size and mean particle number density can, therefore, be calculated by getting the transmitted light. The LTF method has already been applied in on-line monitoring on account of its simplicity in both theory and the optical layout [7].

In LTF the intensity of incident light will be known in advance for calculating the particle size and concentration. However, many factors expose effect on this measurement, such as optical path polluted during the in-line monitoring, light source drifting or temperature variation for long time running and etc. In addition, in-line measurement means that the measurement should take place whenever the equipment is in operation; as a result, some impurities are left in the measurement volume making the measurement volume not exactly the original intensity of incident light. Unfortunately it is impossible for to turn off the equipment to measure the original intensity of incident light. This fatal defect disables the calculation. Another problem is that in-line measurement can only be used for calculating the mean size but not the size distribution of particles which is of utmost importance in many applications. In this paper, a novel method is proposed to solve these problems.

Theory of light fluctuation spectrum

Light extinction and fluctuation

Light extinction is one of the most widely used conventional optical methods. While a light beam is passing through a suspension, the light intensity is attenuated (Fig. 1(a)). Then the following expression applies:

(1){\displaystyle \frac{\overline{I}}{I_{\mathrm{0}}}}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{\mathrm{\pi }}{\mathrm{4}}}LN{D}^{\mathrm{2}}E(\lambda ,m,D)\right),

where \overline{I} is the intensity of transmitted light passing through an optical path length (L) of suspension, I₀ is the intensity of incident light, N is the number concentration of particles per unit volume, and E is the light scattering coefficient which is the function of wavelength \lambda, particle size D and refractive index m of the particles. E could be calculated with Mie’s light scattering theory. When D is much larger than \lambda, the value of E tends to be two according to the Mie’s light scattering theory [8]. Then, Eq. (1) may be rewritten as

(2){\displaystyle \frac{\overline{I}}{I_{\mathrm{0}}}}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}LN{D}^{\mathrm{2}}\right).

This indicates that the attenuation of the intensity of incident light depends only on the particle size and the concentration. The average number of particles n in the measurement volume can be defined as

(3)n=NAL,

where A is the cross-sectional area of the light beam. Substituting Eq. (3) into Eq. (2), the following equation can be obtained

(4){\displaystyle \frac{\overline{I}}{I_{\mathrm{0}}}}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{nC}{A}}\right),

where C=\mathrm{\pi }{D}^{\mathrm{2}}/\mathrm{2}.

During the suspension flowing through the measurement volume, particle number and size vary randomly. If the cross-sectional area of the light beam is much larger than the projective area of particles, the effect of variation of particles flowing into and out of the measurement volume caused by Brownian motion can be neglected. The intensity of transmitted light, therefore, remains constant (shown in Fig. 1(c)).

But in the case that the cross-sectional area of the light beam is comparable to the projective area of particles, the variation of particle number flowing into or out of the measurement volume is, therefore, noticeable, the fluctuation of the corresponding intensity of transmitted light is significant as illustrated in Fig. 2. This fluctuation signal contains the information of particle size and concentration.

Based on this phenomena, Gregory [6] proposed the theory of light fluctuation. The particle number in the finite measurement volume changes because the particles flow into or out of the measurement volume randomly, as do in the pneumatic conveying. On the other hand, the particles flowing out of the volume do not affect the particles flowing into the volume. Thus the particle number in the measurement volume can be described by the Poisson distribution function, and the probability P(n) of the particle number in a defined volume can be given by

(5)P\left(n\right)={\displaystyle \frac{\exp \left(-v\right)v^n}{n!}}.

The probability follows the Poisson distribution. Both the mean and variance are equal to v. The standard deviation is v^1/2. The number of particle is approximately higher or lower than the mean value with the standard deviation (i.e., v±v^1/2). By combining Eqs. (1), (4) and (5) for a monodisperse suspension, the deviation of the signals of intensity of transmitted light is

(6)I_{\mathrm{r}\mathrm{m}\mathrm{s}}=I_{\mathrm{0}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{vC}{A}}\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\displaystyle \frac{v^{\mathrm{1}/\mathrm{2}}C}{A}}\right)=\overline{I}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\displaystyle \frac{v^{\mathrm{1}/\mathrm{2}}C}{A}}\right).

Assuming that the measurement volume is small, there is not many particles in the volume, and comparing small C to A, lives , Eq. (6) can be simplified as Eq. (7) based on the properties of Hyperbolic function [9

(7)I_{\mathrm{r}\mathrm{m}\mathrm{s}}=I_{\mathrm{0}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{vC}{A}}\right)\left({\displaystyle \frac{v^{\mathrm{1}/\mathrm{2}}C}{A}}\right)=\overline{I}\left({\displaystyle \frac{v^{\mathrm{1}/\mathrm{2}}C}{A}}\right).

Combining Eqs. (2), (3) and (7), the following equation can be obtained [10]:

(8){\displaystyle \frac{\mathrm{l}\mathrm{n}\left(I_{\mathrm{0}}/\overline{I}\right)}{I_{\mathrm{r}\mathrm{m}\mathrm{s}}/\overline{I}}}={(NAL)}^{\mathrm{1}/\mathrm{2}}.

By Eqs. (8) and (2), once the intensity of incident light I₀ and the time series intensity of transmitted light are measured, the particle concentration N and the particle size D can be obtained.

Evaluation of intensity of incident light

With Eq. (6), the following two equations can be obtained

(9)\begin{array}{l}{I}_{\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{1}}={\overline{I}}_{\mathrm{1}}{\displaystyle \frac{n_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}C}{A}},\hfill \\ I_{\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{2}}={\overline{I}}_{\mathrm{2}}{\displaystyle \frac{n_{\mathrm{2}}^{\mathrm{1}/\mathrm{2}}C}{A}},\hfill \end{array}

where subscript 1 and 2 denote the first and second measurement, respectively.

Usually, one measurement with 1024 records needs a few milliseconds, and the time interval between two measurements is a few milliseconds, too. Assuming that the intensity of incident light and the average diameter of measured particles during such short time interval do not change or have a negligible change, the intensity of incident light I₀ and the scattering cross section C in two measurements would be identical. Then, with Eq. (9) the relation of relative change of particle number in two measurements is expressed as

(10){\displaystyle \frac{I_{\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{1}}/{\overline{I}}_{\mathrm{1}}}{I_{\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{2}}/{\overline{I}}_{\mathrm{2}}}}=\sqrt{{\displaystyle \frac{n_{\mathrm{1}}}{n_{\mathrm{2}}}}}.

With Eq. (3) similar relationship of average transmitted light intensities between two measurements may be obtained

(11){\displaystyle \frac{\mathrm{l}\mathrm{n}\left({\overline{I}}_{\mathrm{1}}/I_{\mathrm{0}}\right)}{\mathrm{l}\mathrm{n}\left({\overline{I}}_{\mathrm{2}}/I_{\mathrm{0}}\right)}}={\displaystyle \frac{n_{\mathrm{1}}}{n_{\mathrm{2}}}}.

Combine Eqs. (10) and (11), we will get

(12){\displaystyle \frac{\mathrm{l}\mathrm{n}\left({\overline{I}}_{\mathrm{1}}/I_{\mathrm{0}}\right)}{\mathrm{l}\mathrm{n}\left({\overline{I}}_{\mathrm{2}}/I_{\mathrm{0}}\right)}}={\left({\displaystyle \frac{I_{\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{1}}/{\overline{I}}_{\mathrm{1}}}{I_{\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{2}}/{\overline{I}}_{\mathrm{2}}}}\right)}^{\mathrm{2}}.

In Eq. (12), I_{\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{1}}, I_{\mathrm{r}\mathrm{m}\mathrm{s}\mathrm{2}}, {\overline{I}}_{\mathrm{1}} and {\overline{I}}_{\mathrm{2}} are measurable, and only I₀ is unknown. By solving this equation the intensity of incident light I₀ can be obtained.

Calculation of size distribution

Physical model for polydisperse particle system

As described in Sect. 2.1, the particle diameter for monodisperse system or the average diameter for polydisperse system can be derived by giving the intensities of incident light and transmitted light. However, the obtainment of particle size distribution (PSD) for polydisperse system is not mentioned. In order to match the actual industry application, a novel model to measure the PSD for the polydisperse system with the light fluctuation method was proposed.

The fluctuation of transmitted light in the polydisperse system is caused by the random variation of particle number and size in the measurement region. Assuming a polydisperse system with particle number concentration N(D) can be divided into n sub-size zones, as depicted in Fig. 3. Each sub-size zone can be characterized by a mean diameter D_i and the number concentration N_i. Thus, N_{\square }(D_{\square }) can be rewritten as

(13)N(D)={\displaystyle \sum _{i=\mathrm{1}}^n{N}_i}(D_i).

The total attenuation of transmitted light \mathrm{\Delta }I is the sum of the attenuations of each sub-size zone accordingly,

(14)\mathrm{\Delta }I=I_{\mathrm{0}}-I={\displaystyle \sum _{i=\mathrm{1}}^n\mathrm{\Delta }{I}_i}={\displaystyle \sum _{i=\mathrm{1}}^n\left(I_{\mathrm{0}}-{\overline{I}}_i\right),}

where \mathrm{\Delta }{I}_i is the attenuation of transmitted light by particles in the ith sub-size zone, and {\overline{I}}_i is the mean intensity of transmitted light for the ith sub-size zone. According to Eq. (1), for each sub-size zone, there is

(15)\begin{array}{l}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{{\overline{I}}_i}{I_{\mathrm{0}}}}\right)=-{\displaystyle \frac{\mathrm{\pi }}{\mathrm{4}}}L{N}_i{D}_i^{\mathrm{2}}E(\lambda ,m,D_i),\hfill \\ i=\mathrm{1},\mathrm{2},\cdots ,n.\hfill \end{array}

Substitute Eqs. (14) and (15) into Eq. (7), and consider the light extinction coefficient E as 2 when measuring large particles, a set of equations can be obtained as follows:

(16)\begin{array}{l}{\displaystyle \frac{\mathrm{l}\mathrm{n}\left(I_{\mathrm{0}}/\overline{I_i}\right)}{{(I_{rms})}_i/\overline{I_i}}}={(N_{i}LA)}^{\mathrm{1}/\mathrm{2}},\hfill \\ i=\mathrm{1},\mathrm{2},\cdots ,n.\hfill \end{array}

Now the determination of the values of {\overline{I}}_i and {(I_{\mathrm{r}\mathrm{m}\mathrm{s}})}_i in Eq. (16) enable us to get N_i and D_i from Eq. (15), and furthermore, to get the PSD and the total concentration of measured particles.

Evaluation of {\overline{I}}_i and {(I_{\mathrm{r}\mathrm{m}\mathrm{s}})}_i

Known by the theory of gas-solid two-phase flow [11], the final velocity of different particles is dissimilar in the fully developed straight pipe flow. Larger particles move more slowly through the sample volume than do smaller ones. Hence, the residence time of large particles in the measuring light beam is longer than that of small ones. Therefore, the transmitted light signal consists of a wide frequency band signals. Lower frequency relates to bigger particles and higher frequency to smaller ones [12,13]. Figure 4 demonstrates the raw signal measured in a pulverized coal pipe and 6 signals with different frequent bands that consist of the raw signal.

The measured transmitted light signals were processed with FFT to get the power spectrum [14] as displayed in Fig. 5. Actually it characterizes the power spectrum of \ln (I_i/I_{\mathrm{0}}). Therefore, the integral of the power spectrum at all frequencies is related to the integral of the corresponding total turbidity of all particles, and sum of the power spectrum of different frequency band corresponds to the turbidity of the appropriate particle size range. Via this process, one polydisperse system is divided into many monodisperse systems which are appropriate for Gregory’s theory. According to the above principles, there is the following relation between the total attenuation of transmitted light and the attenuations of transmitted light in each sub-size zone:

(17){\displaystyle \frac{\mathrm{l}\mathrm{n}\left({\overline{I}}_i/I_{\mathrm{0}}\right)}{\mathrm{l}\mathrm{n}\left(\overline{I}/I_{\mathrm{0}}\right)}}={\displaystyle \frac{{\tau }_i}{\tau }}={\displaystyle \frac{S_i}{S}},

where {\tau }_i and \tau are the turbidity of the ith frequency band and the total turbidity, respectively, and S_i and S are the corresponding integrations of measurable parameters of the power spectrum of the ith frequency band and all frequencies, respectively.

Therefore, the mean intensity of the transmitted light of the ith sub-size zone {\overline{I}}_i can be obtained from Eq. (17). And the standard deviations {(I_{\mathrm{r}\mathrm{m}\mathrm{s}})}_i of the signals of each frequency band can be determined from the filtered data corresponding to the frequency band. With {\overline{I}}_i and {(I_{\mathrm{r}\mathrm{m}\mathrm{s}})}_i the particle concentration N_i and size D_i can be obtained from Eqs. (16) and (15), respectively, and the PSD can be fixed by

(18)N^{*}(D_i)={\displaystyle \frac{N(D_i)}{{\displaystyle \sum _{i=\mathrm{1}}^{n}N(D_i)}}}\times \mathrm{100}\%.

Equation (18) is viable for in-line measurement because not any additional parameters are needed.

Experiments and discussion

Experimental setup

To verify the above method, experiments were conducted. The schematic diagram of the experimental setup is given in Fig. 6. During the experiments, particles flow through the measurement volume where laser beam is injected and transmitted light is collected by the receiving lens and detected by a photo detector. The measurement volume may be varied by changing the focus length of the convex lens, and the flow rate can be adjusted too.

Before particles were added into the flow the intensity of incident light I₀ were measured as background signal as shown in Fig. 7. Two kinds of particles, glass beads and pulverized coal were measured. Figure 8 is the image of microscopes of glass beads and pulverized coal. Several tens of records were performed for each concentration and more than 7000 data were recorded in every measurement.

Experimental results of glass beads

As shown in Fig. 8(a) the glass beads are nearly monodispersed, and its Rosin-Rommler (R-R) size distribution function parameters is \overline{D}=\mathrm{31.3}\mathrm{\mu }\mathrm{m} and k = 15.6. The mass of glass beads of 1, 2, 3 and 4 g were added into the water cycle system to form 4 different concentrations. Figure 9 presents the increase of fluctuation of transmitted light increases with the increase of the concentration.

Two data acquisition frequencies of 2.5 and 6.25 kHz were adopted during experiments. According to the algorithm described in Sect. 2.2, any pair of measured transmitted light intensities in different concentrations may be used to evaluate the intensity of incident light I_{\mathrm{0}}^{\prime } based on {\overline{I}}_i and {(I_{\mathrm{r}\mathrm{m}\mathrm{s}})}_i. The results are listed in Table 1. The pre-measured intensity of incident light I_{\mathrm{0}} is 3.243 V, and the avarege evaluated intensity of incident light I_{\mathrm{0}}^{\prime } is 3.265 V in 6.25 kHz and 3.272 V in 2.5 kHz, respectively. Both of the densities agree well with the pre-detected value and the avarege error is only 0.83%. The maximum error in different concentrations is 2.23%, which is accurate enough in most engineering applications. Table 1 summarizes the experimental results of glass beads. As can be seen from Table 1, the evaluated values with two sampling frequencys are almost identical, which means the sampling frequency is of no significance to the results.

The evaluated \overline{D} and k are 31.28 μm and 17.08, respectively. Figure 10 exhibits the measured size distributions by the two methods.

Experimental results of pulverized coal

Two groups of concentration of pulverized coal were measured with the experimental setup. The attenuation of transmitted light in the higher concentration is 73%–54%, and the maxium attenuation is 53.7%. In the lower concentration the attenuation is 97%–93%. The evalued intensities of incident light and the parameters of R-R function are listed in Table 2. The smapling frequency is 2.5 kHz in the experiments, and the pre-measured intensity of incident light is 2.904 V. The size distribution of pulverized coal is \overline{D}=\mathrm{27.14}\mathrm{\mu }\mathrm{m} and k = 2.1 measured with the imaging analyzing method (Fig. 11). The smaller k means wide size distribution that can be seen clearly from Fig. 8(b).

From Table 2, it can be seen that the errors between the pre-measured I₀ and the evaluated I_{\mathrm{0}}^{\prime } in the lower concentration are smaller than those in the higher concentration. On the contrary, the errors of parameters of R-R function in the lower concentration are bigger. But both are within the acceptable range in industry applications. This indicates that there is an optimal concentration range with this method. The measurement error is accepteble in this concentration range. If the concentration is higher or lower than this range the measurement errors would be too big and not be acceptable.

Conclusions

The correct measurement of the background signal correctly in the light fluctuation method poses a challenge for in-line measurement. Another problem concerning this method is that it may only measure the mean size of particles in stead of the size distribution which is important in many applications. To solve these two problems a novel method, the LFS method, was developed, by which the background signal or the intensity of incident light could be evaluated based on the data of transmitted light intensities. Besides, a multi-sub-size zone model was proposed to measure the PSD. Experimental results for measuring glass beads and pulverized coal in different concentrations prove that the background signal may be evaluated correctly with the developed algorithm, and the error is less than 2.5%, which is acceptable in industry applications. For the PSD measurement, measurement error in the higher concentration is smaller than that in the low concentration. There is an optimal concentration range for getting good results. Due to its simplicity in measurement principle and configuration, the LFS method is especially suitable for in-line measurement of particle two-phase flow.

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