1.  Introduction

Flow rate is one of the most important process parameters in many sectors, such as the chemical, pharmaceutical, petroleum, energy, and power engineering industries, etc. [¹–⁵]. Currently, with the development of new technologies, the industrial devices and equipment present a trend of miniaturization [⁶–⁹]. More and more millimeter-scale devices/equipment are appearing, such as tube reactors, tube heat exchangers, etc. [⁶–⁸]. There is a practical need of effective techniques to measure the flow rate in millimeter-scale pipes.

Some commercial flowmeters may be used in millimeter-scale pipes, such as the constriction flowmeter, rotameter, turbine flowmeter, electromagnetic flowmeter, coriollis flowmeter, etc. [³–⁵,¹⁰,¹¹]. The constriction flowmeter, rotameter and turbine flowmeter have constriction elements or moving parts in the pipe that disturb the fluid flow and cause a pressure drop. The Coriollis flowmeter also disturbs the fluid flow. Furthermore, this kind of flowmeter has the disadvantage of high cost. Although the electromagnetic flowmeter do not disturb the fluid flow, electromagnetic flowmeters have relatively high cost and their practical application in millimeter-scale pipes is also limited [³–⁵,¹⁰,¹¹]. Therefore, it is necessary to seek more effective methods to solve the problem of flow rate measurement in millimeter-scale pipes.

Electrical conductivity is one of the basic physical parameters. Conductivity detection techniques have been adopted to implement flow rate measurements for many years and some achievements/progress have been obtained [¹²,¹³]. However, the conventional conductivity detection technique for flow rate measurement is based on contact conductivity detection. The detection electrodes are directly in contact with the fluid. That will cause polarization effects and electrochemical erosion effects. Meanwhile, if the electrode has been contaminated, unpredictable measurement errors will arise. Besides, the conventional conductivity detection techniques for flow rate measurement are used in normal scale pipes and few research works concerning the application of conductivity detection technique to the flow rate measurement in millimeter-scale pipes have been reported.

Capacitively Coupled Contactless Conductivity Detection (C⁴D) is a new conductivity detection technique, which was independently proposed by Zemann et al., da Silva and do Lago, in 1998 [¹⁴,¹⁵], on the basis of Gas' research work [¹⁶]. Compared with the conventional contact conductivity detection technique, C⁴D is a contactless detection technique and the electrodes of a C⁴D sensor are not in contact with the measured fluid. C⁴D can avoid the electrochemical reactions on the electrode surfaces and the polarization of the electrodes. Meanwhile, C⁴D sensors are low cost and more robust [¹⁷–²⁰]. Unfortunately, C⁴D is still a developing technique. To date, C⁴D has mainly been studied and applied in the research field of analytical chemistry for ion concentration/conductivity detection in capillaries (the inner diameter is usually less than 0.2 mm) [¹⁴–²⁰]. Little research concerning the application of C⁴D to the flow rate measurement in millimeter-scale pipes has been reported [¹⁷–¹⁹,²¹–²⁴].

The aim of this work was to propose a new method for flow rate measurement in millimeter-scale pipes, combining the C⁴D technique and the principle of cross correlation flow measurement. The research work included two steps: first, a new five-electrode C⁴D sensor, which was suitable for the flow rate measurement in millimeter-scale pipes, was developed. Second, with two conductivity signals obtained by the developed sensor, the flow rate measurement using the principle of cross correlation flow measurement was implemented. Meanwhile, experiments were carried out in five millimeter-scale pipes with different inner diameters of 0.5, 0.8, 1.8, 3.0 and 3.9 mm, respectively, to verify the feasibility and effectiveness of the proposed flow rate measurement method.

2.  Five-Electrode C⁴D Sensor

Figure 1 inllustrates the principle of the C⁴D technique. As shown in Figure 1(a), a conventional C⁴D sensor includes an insulating pipe, two metal electrodes (an excitation electrode and a pick-up electrode) placed cylindrically around the outside of the insulating pipe, an AC source and a current pick-up unit. Figure 1(b) shows the simplified equivalent circuit of conventional C⁴D sensor. C₁ and C₂ are the coupling capacitances formed by the two metal electrodes, the insulating pipe and the conductive fluid. R is the equivalent resistor of the fluid between the two electrodes. Thus, an alternating current path is formed. The application of an AC voltage V_i on the excitation electrode will lead to an AC current flowing through the AC path. From the AC current obtained by the AC current pick-up unit, the conductivity detection can be implemented [¹⁴–¹⁹].

For the conventional C⁴D sensor, from Figure 1 it can be seen that only the impedance of the resistor (R) is the useful signal. The impedances of the two coupling capacitances (C₁ and C₂) are the unfavorable background signals. The existence of the background signals has a negative influence on the conductivity detection and hence limits the resolution and the detection range [¹⁸–²⁷], so it is necessary to find an effective method to solve this problem [¹⁸–²⁰]. Laugere et al. have proposed a new detection method and have developed a four-electrode C⁴D sensor [²⁵–²⁸]. In the four-electrode C⁴D sensor, four electrodes are placed cylindrically around the pipe. The outer two are excitation electrodes and the inner two are pick-up electrodes. A fixed AC current source is connected between the outer electrodes and a resulting differential voltage between two inner electrodes can be obtained by a high input impedance voltmeter. Thus, from the measured differential voltage, the measurement value of the fluid conductivity can be obtained. Compared with the conventional C⁴D sensor, the way of implementing the conductivity detection is different. The conventional current measurement way has been changed to a voltage measurement way, thus the existence of coupling capacitances (C₁ and C₂) has no influence on the conductivity measurement. Unfortunately, the detection method proposed by Laugere et al. is also a developmental technique. According to the latest technique reports, the four-electrode C⁴D sensor has been used in a channel with dimensions of 106 μm × 170 μm [²⁴–²⁷]. However, there is still a lack of the knowledge and experience of the four-electrode C⁴D sensor in millimeter-scale pipes.

Although more research work should be undertaken, the detection method proposed by Laugere et al. provided a very useful reference for our research work. In this work, on the basis of Laugere et al.'s work, we developed a new five-electrode C⁴D sensor in millimeter-scale pipes. The construction of the new five-electrode C⁴D sensor was illustrated in Figure 2(a), along with a simplified equivalent circuit diagram given in Figure 2(b). The aim of this work was to implement the flow rate measurement using the principle of cross correlation flow measurement. According to the principle of cross correlation flow measurement, two independent conductivity signals were needed. A four-electrode C⁴D sensor could only obtain one conductivity signal, so an extra electrode was added. The new sensor became a five-electrode C⁴D sensor.

The new five-electrode C⁴D sensor consists of an insulating pipe, five cylindrical metal electrodes (one excitation electrode, one ground electrode and three pick-up electrodes), an AC source and a data acquisition unit. C_x1∼C_x5 are the coupling capacitances formed by the five metal electrodes, the insulating pipe and the conductive fluid. R_x1∼R_x4 are the equivalent resistors of the fluid between the two adjacent electrodes. C_x1, R_x1∼R_x4 and C_x5 form an alternating current path. When an AC voltage V_in is applied on the excitation electrode, an AC current I flowing through the AC path is generated. The AC current I can be determined by:

where V_in is the input voltage (AC source), and Z is the overall impedance of the AC path. At the frequency f of the input voltage signal, the overall impedance Z is:

The differential voltage U_x1 reflects the conductivity information of the fluid between electrode 2 and electrode 3 (up-stream sensor) and the differential voltage U_x2 reflects the conductivity information of the fluid between electrode 3 and electrode 4 (down-stream sensor). U_x1 and U_x2 can be expressed as:

Thus, two conductivity signals can be obtained by measuring the differential voltages U_x1 and U_x2.

Comparing Figures 1 and 2, it can be seen that the difference between the conventional C⁴D sensor and the new five-electrode C⁴D sensor is the way conductivity detection is implemented. The new five-electrode C⁴D sensor is based on the detection method proposed by Laugere et al., in which the conductivity measurement is mainly implemented by the voltage measurement technique and the existence of coupling capacitances has no influence on the conductivity measurement. Thus, in the new five-electrode C⁴D sensor, the unfavorable influence of the coupling capacitances on conductivity detection can be avoided and its linearity, accuracy, and sensitivity of the measurement can be improved [¹⁸,²⁵–²⁷]. However, with the increasing dimension of the pipes, the impedances of the fluid resistors R_x2 and R_x3 decrease. That causes more difficulties for the low voltage measurement and electrical circuit design in millimeter-scale pipes. The electrical circuit and the signal processing circuit need to be specially designed in our research work.

The data acquisition unit can be roughly divided into two parts: the signal processing circuit and the data processing computer system. The function of the signal processing circuit is to demodulate and to amplify the conductivity signals obtained from the AC path. The function of the data processing computer system is to acquire the data and display the detection results (in this work, the data processing computer system was based on the data acquisition module of National Instruments (NI cDAQ-9172) and a microcomputer).

As shown in Figure 3, the signal processing circuit includes three parts: (1) Bootstrapping high input impedance amplifier buffer, which is used to obtain the voltage signals V₁, V₂, V₃. (2) Differential amplifier, which is used to obtain two differential voltage signals V_1-2, V_2-3. (3) Demodulation circuit, which consists of Demodulation and Filter amplifier. Finally, two independent conductivity signals of fluid U₁, U₂ are obtained. Using these two independent conductivity signals, the flow rate measurement can be implemented according to the principle of cross-correlation measurement [⁴,⁵,²⁹–³¹].

3.  Flow Rate Measurement Implementation

The new five-electrode C⁴D sensor can obtain two independent conductivity signals (the differential voltages U₁ and U₂). According to the principle of cross correlation flow measurement, if we cross-correlate the two conductivity signals, the time delay τ between the two signals can be determined by searching for the time position of the maximum cross-correlation coefficient. The cross-correlation function R_U1U2(τ) can be defined as:

The cross-correlation coefficient ρ_U1U2(τ) can be defined as:

Then the average flow velocity V can be calculated by the following equation [⁴,⁵,²⁹–³¹]:

where L is the distance between the centre of the combination of electrode 2 and electrode 3 and the centre of the combination of electrode 3 and electrode 4, as shown in Figure 2 (in the developed five-electrode C⁴D sensor, the combination of electrode 2 and electrode 3 and the combination of electrode 3 and electrode 4 can be regarded as two independent sensors of conductivity detection, the up-stream sensor and the down-stream sensor).

Thus, the flow rate Q₀ can be calculated by the following equation:

where A is the sectional area of the pipe. Meanwhile, for practical cross correlation flow rate measurement, calibration is needed. The practical flow rate Q should be: where K is the calibration coefficient.

4.  Experimental Results and Discussion

The experiments included two parts: first, the conductivity measurement experiments of the new five-electrode C⁴D sensor in millimeter-scale pipes were carried out to test the conductivity measurement performance of the new five-electrode C⁴D sensor. Second, the flow rate measurement experiments were carried out to verify the effectiveness of the proposed flow rate measurement method in millimeter-scale pipes.

5.  Conclusions

In this research, a new method, which combined the C⁴D technique and the principle of cross correlation flow measurement, was proposed for flow rate measurement in millimeter-scale pipes. A new five-electrode C⁴D sensor was developed. Meanwhile, experiments were carried out in five millimeter-scale pipes with different inner diameters. The research results showed that the flow rate measurement of this method had a relative difference lower than 5%, which was considered accurate and satisfactory for the flow rate measurement application in millimeter-scale pipes. As a preliminary study, this research work verified the feasibility of the application of C⁴D technique to the flow rate measurement in millimeter-scale pipes. However, more research works (improving the measurement accuracy, extending the application scope, etc.) should be undertaken in this area. Optimization of the length of the electrodes and the distance between two adjacent electrodes will be topics for our further research works. Meanwhile, to improve the signal-to-noise rate of the five-electrode C⁴D sensor and then to implement the flow rate measurement without the help of the tracer is also a challenge in future.

This work was supported by National Natural Science Foundation of China (No. 51076141, No. 61074173 and No.11132008).