In this paper, we set up and completed five sets of flow rate calibration experiments at different pressure differentials: 2600 Pa, 3000 Pa, 4000 Pa, 6000 Pa, and 8000 Pa. These experiments were carried out simultaneously by the mass method and image processing technology. Additionally, each set included at least three droplet generation cycles. In this section, we will first analyze each set of experiments to ensure that the flow rates were in the nanoliter range and met the flow control design expectations. Then, we will use the results from the gravimetric method to verify the feasibility of using image processing technology to achieve the rapid calibration of the nanoliter per second flow rate.
3.1. Results of the Gravimetric Method
Using the gravimetric method, we recorded the weight of each droplet and the time it took to accumulate. The specific data are shown in
Table 3, indicating that the weight of each droplet at different pressure differentials is 1.5 mg. Based on the balance between surface tension and gravity, as expressed in Equation (10), the theoretical weight of the droplet under the specific pipeline design is 1.4961 mg when the fluid is stationary. The deviation between the ideal weight and the experimental value is 0.56%. Considering that the resolution of the electronic balance used in the experiment is 0.1 mg, this deviation can be disregarded. Hence, we can conclude that when the flow rate is stable and less than 1.44 nL/s, the equilibrium between surface tension and gravity is applicable, and the weight of the droplet can be calculated by surface tension.
After we obtained the data about the weight of droplets and the time it took them to accumulate, we used Equation (7) to calculate the experimental value of the average flow rate at different pressure differentials and Equation (5) to calculate the corresponding theoretical flow rate. As shown in
Figure 4, the red line is the theoretical value, and the black line is the experimental value, which is very close to the red line and almost linear. Additionally, the blue dotted line is the deviation, which is the experimental value minus the theoretical value. The maximum error is 0.037 nL/s, which is less than the 0.1 nL/s anticipated resolution of the flow supply system, meaning that the flow supply system achieves the design expectations.
Considering that the maximum deviation occurred in the set of the 4000 Pa pressure differential, we must take into account two additional sources of error that may affect the accuracy of our measurements. First, the electronic balance used in the experiment has a resolution of 0.1 mg, which limits the precision of our weight measurements. Second, some degree of leakage may occur in the vacuum chamber, which could affect the pressure differential and the flow rate. However, we believe that the impact of these factors is minimal, given that the deviation between the theoretical and experimental values is within an acceptable range. Based on the experimental results, the flow control system was able to adjust the flow rate with a resolution of 0.1 nL/s, as anticipated. Moreover, the theoretical value or the linear fit to the experimental value can be used to predict the flow rate of the flow control system at a given pressure differential. Hence, the result confirms the validity of the experimental setup and provides a reliable basis for subsequent analyses.
3.2. Results of Image Processing Technology
Section 3.1 of our study detailed the gravimetric method employed for droplet analysis, while
Section 3.2 introduced a novel approach using image processing technology for droplet measurement. High-speed imaging was employed to capture the droplet growth process, as demonstrated in
Figure 5. The images clearly revealed that during droplet growth, the liquid emerging from the emitter outlet accumulated at different locations, ranging from the emitter needle port to the emitter sidewall, before sliding down to the emitter needle port for accumulation until droplet detachment. Based on this observation, we divided the droplet growth process into three stages, corresponding to
Figure 5a–c: (i) formation of a liquid film at the emitter needle port; (ii) droplet attachment on the emitter sidewall; and (iii) droplet detachment and further accumulation at the emitter needle port. However, it is noteworthy that in an ideal scenario, liquid flow from the emitter needle port should not infiltrate upwards along the emitter outer wall and accumulate on one side. Considering the size effect of the emitter, we propose three assumptions: (i) the emitter is not fixed vertically along the direction of gravity, resulting in liquid outflow on the emitter side with a small wetting angle and upward infiltration of the emitter along the outer wall, which continues to pull subsequent outflow until the resistance along the emitter’s outer wall can no longer offset the accumulated liquid’s own gravity, leading to downward sliding towards the emitter needle port; (ii) the emitter needle port is not flat, and there exists a fine gap for liquid infiltration; (iii) prior to the formal experiment, the liquid flow rate in the pipeline was initially set at a relatively high rate (>5 nL/s) and then subsequently reduced to a lower flow rate (<1 nL/s). As a result, liquid backflow occurred at the emitter needle port, leading to the initial droplet accumulation at the needle mouth. This accumulation contaminated the outer wall of the emitter, facilitating liquid infiltration.
After obtaining high-speed camera images of the droplet growth process (as displayed in
Figure 5), image segmentation and threshold separation operations are performed to identify and extract droplet contours using the Canny–Sobel. The extracted contours enable precise droplet volume calculations, as shown in
Figure 5d–f. However, the extent of droplet infiltration on the emitter outer wall during the second stage of droplet growth is not entirely reflected by one-dimensional imaging and gray value processing. This is because assuming that the uninfiltrated side of the emitter outer wall always serves as the droplet boundary in axisymmetric droplet modeling can lead to a calculated volume larger than the actual volume. Despite yielding an experimental value greater than the actual value, this error gradually reduces as droplets continue to accumulate and increasingly conform to the axisymmetric droplet model.
In this experiment, the outer diameter of the emitter, with a diameter of 0.25 mm, was obtained by employing image processing techniques, yielding an outline of 41 pixels. Based on Equation (4), the scale factor D for this experiment was calculated as 0.0061 mm/pixel. With this factor, we were able to compute the volume of each droplet from the droplet image frames. To obtain the droplet volume for comparison with the volume obtained through the gravimetric method, we utilized the image processing method to obtain the droplet volume prior to the droplet falling and subsequently subtracted the droplet volume after falling. Due to the limitations of the image memory, every 1635 shots must be saved, and the image data is lost during the image storage period. If the droplet falls during this period, the mass data of the falling droplet under the image method cannot be obtained. Therefore, to increase the control sample between the gravimetric method and the image processing technology method in terms of droplet mass measurement, electronic balance mass records and images were retained after each pressure differential grouping experiment (without controlling for pressure differential changes due to air leakage), as shown in
Table 4.
Table 4 shows that the droplet masses acquired through image processing are in agreement with those obtained by the mass method. The maximum relative error is only 3.15%, which can be attributed to the image vibration noise and the measurement accuracy of the electronic balance. It should be noted that in the context of microflow measurement, the error introduced by mass measurement is substantially smaller than the required accuracy for flow measurement. Therefore, we can confidently assert that the droplet volume and mass data derived from high-resolution camera imaging, image processing techniques, and the axisymmetric droplet model are highly reliable.
Figure 6a displays the volume data obtained from image measurements at a pressure differential of 4000 Pa. Each straight line in the graph represents a distinct droplet growth period, encompassing nine relatively complete droplet growth periods. It is important to note that due to the attachment of liquid to the emitter sidewall during the droplet growth period, it was not possible to intercept the emitter completely in each image. Consequently, the volume of the emitter exposed in the image was calculated as the initial value for each line. The presence of gaps between the discrete data points represents the loss of data during the image saving period, as mentioned previously. Despite these gaps, a clear linear correlation between droplet volume and time is observed, as depicted in the scatter plot of the image volume data. For detailed analysis, a single cycle of droplet growth data was selected, as illustrated in
Figure 6b. A distinct break point, indicated by the red box, is observed on the graph. This break point corresponds to the accumulation of droplets slipping from the emitter sidewall to the emitter needle port, aligning with the three-stage droplet growth process described in
Section 3.2. As the first stage provides a limited number of samples at a shooting rate of 1 frame per second, resulting in approximately 10 data points at the set pressure differential of 4000 Pa, it is not suitable for individual data analysis. Therefore, the data from the first and second stages were combined, as shown in
Figure 6c, while the third stage data are plotted separately in
Figure 6d. A linear fit was performed on the data at both ends to obtain the corresponding slope as the average flow rate. The slopes obtained for both sets of data are 0.4818 and 0.4768, which are similar to the flow rate of 0.47 nL/s obtained by the gravimetric method. The fitted flow rate obtained in the second stage is greater, confirming the analysis of the source of error in the second stage calculation. Using the flow rate obtained by the gravimetric method as the true value, the error in the larger flow rate obtained by fitting the data obtained by the image method in the second stage is only 2.5%. This result demonstrates the feasibility of calibrating microflows using the image method. The high accuracy of the image processing technique for droplet images based on the axisymmetric droplet model, as described in
Section 2.3, further supports the reliability of the obtained droplet volume and mass data.
In order to expedite flow calibration, a cumulative average flow rate was computed from the droplet volume acquired through image recognition, leading to a cumulative flow curve as a function of time for four different pressure differentials, as depicted in
Figure 7. The blue line represents the cumulative average flow rate curve, the red line denotes the flow rate value obtained by the gravimetric method corresponding to the pressure differential, and the orange dotted line represents the gravimetric method flow rate value with an error margin of plus or minus 5%. For the experiment conducted at a pressure differential of 2600 Pa, the cumulative average flow rate calculated by image processing required only 26.3% of the droplet drop time to converge to the “true flow rate”, while the other three groups required 18.2%, 3.8%, and 9.5% of the droplet drop time, respectively, at their respective pressure differentials. In contrast to the gravimetric method, which calculates the stable flow by weighing the dripping liquid, the image processing method can save more than two-thirds of the time by accumulating the average flow. Therefore, assuming a stable flow, the cumulative flow rate obtained by the image processing technique can converge to within plus or minus 5% of the actual flow rate obtained by the mass method in significantly less time than the droplet generation time. This approach demonstrates a considerable improvement in the speed of nanoliter per second flow calibration and highlights the potential for using image processing techniques to obtain accurate flow measurements.
During the high hold state of the vacuum chamber, the operation of the pump set generates vibrations that can cause shaking of the droplet image. This, in turn, can result in errors in the extraction of boundary contours, typically at the single-digit pixel level. Particularly at low flow rates where the volume change rate of the droplets is extremely small, these errors can become comparable to the error induced by the vibration. To overcome this issue, the cumulative average flow is computed to characterize the flow at the outlet of the emitter. However, due to the one-dimensional imaging and vertical fixation of the emitter, an error arises between the actual droplet volume and the identified droplet volume in the first and second phases of droplet accumulation. Nevertheless, by computing the cumulative average flow rate, the error between the two is continuously reduced, resulting in a better characterization of the rate of change of the droplet volume (i.e., flow rate). As shown in
Figure 7, the accumulation of droplets sliding from the emitter side to the emitter exit causes significant step changes in the late cumulative average flow calculation.