Influence and Correction of Refraction Phenomenon in Liquid Contact Angle Measurement in Capillary Tube

Abstract

By using clear vapor–liquid interface line images of the liquid inside the capillary, the measurement coordinate points of the vapor–liquid interface line were measured. A new method for measuring liquid contact angle has been proposed, which was used to calculate the actual coordinate points and fit the actual vapor–liquid interface line of the liquid. Finally, an angle measurement tool is used to measure the angle of the actual vapor–liquid interface line and obtain the actual contact angle of the liquid. Effectively reducing the influence of refraction on the contact angle by correcting the errors caused by the refractive index of different materials, it can be used for the precise measurement of the static contact angle of liquids. By measuring the static contact angle of the upper and lower liquid surfaces of the liquid column, it was found that the presence of refraction caused a difference of [1.84°, 5.61°] between the actual and measured values of the static contact angle.

1. Introduction

The contact angle is an important parameter for characterizing the surface wettability of materials [1], and the interaction between the liquid and the pipe wall surface directly affects the efficiency of heat transfer and mass exchange. In 1805, Young’s equation [2] first proposed the concept of liquid contact angle, which refers to the static contact angle formed between the tangent of the solid–liquid interface and the tangent of the vapor–liquid interface of a droplet. When the contact angle is less than 90°, the wettability of the liquid on the material surface is good. And when the contact angle is greater than 90°, the liquid is less likely to wet the surface of the material. In addition to the static contact angle, the dynamic wetting behavior of solid surfaces should also be considered [3,4]. The advancing contact angle and receding contact angle are two important parameters that describe the dynamic wetting behavior of a liquid on solid surfaces. The measurement methods for contact angle include the sessile droplet method, the advancing/receding droplet method, the inclined plate method, the capillary rise method, etc. The sessile droplet method [5,6,7] places droplets directly on a solid surface and measures the contact angle between the droplet profile and the substrate through an optical system. The advancing/receding droplet method [8] measures the contact angles of the leading edge (advancing angle) and trailing edge (receding angle) of the droplet by increasing or decreasing the droplet volume or tilting the substrate, respectively, to characterize the surface wetting hysteresis. When measuring the contact angle using the inclined plate method [9], gradually tilt the base until the droplet begins to slide. At this point, the angles of the front and rear ends of the droplet correspond to the advancing and receding angles, respectively. The capillary rise method [10,11] measures the height of liquid rise in a vertical capillary and calculates the contact angle using the Young–Laplace equation. By comparing the limitations of these methods in practical measurement and application, it was found that the sessile droplet method was sensitive to surface roughness and chemical uniformity and difficult to measure dynamic contact angle. The advancing/receding droplet method requires precise control of the rate of droplet volume change, and the operation is complex. The inclined plate method can directly obtain dynamic wettability data, but it is greatly affected by gravity and is only suitable for larger droplets. The capillary rise method is suitable for powder or porous materials but requires known liquid surface tension and density and has extremely high requirements for tube wall cleanliness.

In order to obtain the contact angle value in a capillary tube, scholars at home and abroad have proposed measurement methods for contact angle under different influencing factors. Wang et al. [12] introduced a surface composition concept to scrutinize the wetting mechanism by considering the liquid–vapor density asymmetry and the fluid–solid van der Waals interaction. Al-Zaidi [13] directly measured the dynamic contact angle in a single glass capillary tube ranging from 100 to 250 µm as a function of saltwater concentration. Factors such as quartz glass aperture, surface tension, and liquid chemical structure were considered to reduce the influence of image distortion in contact angle analysis, and specific steps for contact angle measurement were proposed. Li [14] pointed out that the static contact angle in a capillary tube was different from that on flat glass surfaces and proposed a solution to overcome the problem of image distortion in a capillary tube. Gu [15] used a high-power optical microscope and software to measure the vapor–liquid and liquid–liquid contact angles in silica microtubes with inner diameters ranging from 5 to 1800 μm. It was found that the vapor–liquid and liquid–liquid contact angles in microtubes with sufficiently small inner diameters were significantly different from their contact angles on the plane. Cheong et al. [16] introduced Snell’s law using the reflection mechanism of light and proposed a new measurement technique and derivation formula. When the radius and height of the interface line are known, the contact angle can be determined. Lv et al. [17] used graphic processing software to identify and process the distribution of liquids without relying on data such as liquid density and surface tension when calculating contact angles.

In summary, the existing contact angle measurement methods directly measure the contact angle by selecting the local contour of the droplet. For the measurement of contact angle in a capillary tube, without considering the phenomenon of light refraction caused by the material and thickness of the tube wall, the vapor–liquid interface line undergoes two refractions of liquid–quartz glass and quartz glass–air. This will result in a deviation between the measured and actual values of the contact angle. A visual experimental system has been established to reduce the deviation between the measured contact angle values and the actual values. A pulsating heat pipe made of quartz glass with an inner diameter of 3 mm and an outer diameter of 6 mm. A pulsating heat pipe is used to study the contact angle of liquids in a capillary tube. During the measurement process, the effects of pipe wall thickness and light refraction were taken into account. The geometric relationship in the process of light refraction is used to draw the refracted ray diagram between the droplet and the recording position (digital camera). A liquid contact angle measurement method was proposed to correct the refraction phenomenon in a capillary tube. This measurement method was used to measure the static contact angle when the fluid was not heated and at the initial heating stage but with no movement of the vapor and liquid columns inside the pulsating heat pipe. During the measurement process, the temperature of the experimental system is around 20 °C, which is the indoor temperature. This method is particularly suitable for contact angle measurement in confined spaces, such as capillary and microchannel systems. By using this measurement method, it is possible to better control the shape of droplets and reduce external interference such as airflow or vibration. Meanwhile, based on the flow process of the working fluid, this method can measure the contact angle hysteresis.

2. Methodology

2.1. Introduction to the Experimental System

In order to measure the contact angle of liquid in a capillary tube, a visual pulsating heat pipe experiment system was built. The visual experiment system consists of a pulsating heat pipe specimen made of quartz glass, a vacuum pump, a filling device, a heating and cooling system, a data acquisition device, a digital camera, and an LED light, as shown in Figure 1.

Research [18,19] has shown that when the diameter of the capillary tube is within the following range, the capillary tube can operate normally, as shown in Equation (1). According to this Equation, the diameter of the capillary tube using distilled water as the working fluid is calculated to be 1.75 mm ≤ D ≤ 5.01 mm. The outer diameter of the pulsating heat pipe is 6 mm, and its inner diameter is 3 mm, as shown in Figure 2. The inner diameter matches the diameter of the pulsating heat pipe where the vapor–liquid column is alternately distributed when distilled water is used as the working fluid.

$$0.7\sqrt{{\displaystyle \frac{\sigma }{({\rho }_{\mathrm{l}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}}-{\rho }_{\mathrm{v}\mathrm{a}\mathrm{p}\mathrm{o}\mathrm{r}})g}}}\le D\le 2\sqrt{{\displaystyle \frac{\sigma }{({\rho }_{\mathrm{l}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}}-{\rho }_{\mathrm{v}\mathrm{a}\mathrm{p}\mathrm{o}\mathrm{r}})g}}}$$

(1)

The pulsating heat pipe is vertically placed, and its bottom is wound with a nickel–chromium resistance wire for heating. The heating power is adjusted by regulating the input voltage through a transformer. At the same time, a T-type thermocouple is arranged at the bottom to collect the instantaneous temperature. The top of the pulsating heat pipe specimen is cooled using a cooling water tank, and the cooling water is provided by a low-temperature constant temperature bath and maintained at 10 °C. A Canon 1DX-Mark II digital camera (Canon Inc., Tokyo, Japan) is used for shooting and recording. The total number of pixels of the camera is about 21.5 million, and it can take 50 photos per second continuously. The distance between the camera position and the center of the experimental system is 18 cm.

Arrange LED light with uniform illumination intensity around the pulsating heat pipe and use LED light to create a contrast between the liquid and the background. By using this method, the vapor–liquid interface of the liquid column is clearly displayed, as shown in Figure 3.

Distilled water is selected as the filling working fluid, and the physical parameters of distilled water at the saturation temperature of 101.325 kPa are shown in

Table 1. Properties of distilled water at 100 °C and 1 atm.

Working Fluid	tsat (°C)	ρl (kg/m3)	ρv (kg/m3)	cp (kJ/(kg·K))	Hlv (kJ/kg)	σ (N/m)
Distilled water	100	958	0.6	4.18	2256.7	0.0589

.

2.2. Measurement and Conversion Method of Liquid Contact Angle in Pulsating Heat Pipe

Under the action of surface tension, when the attraction between liquid molecules is greater than the attraction of the wall to the liquid, the interface line appears convex. But when the attraction between liquid molecules is less than the attraction of the wall to the liquid, the interface line appears concave [20]. The directly measured interface line is the tangent to the vapor–liquid interface line and then tangent b to the solid–liquid interface line. The angle formed by tangent a and tangent b is the contact angle $\mathrm{\angle }\theta$ between the liquid inside the tube and the tube wall, as shown in Figure 4. However, whether it is a convex or concave interface line, the vapor–liquid interface line of the liquid is refracted twice through liquid–quartz glass and quartz glass–air. Therefore, the directly measured vapor–liquid interface line is not the actual interface line of the liquid inside the capillary.

In order to reduce the errors caused by direct measurement of the liquid contact angle, a method for measuring and converting the liquid contact angle inside a capillary is proposed. Firstly, by using a digital camera, a clear image of the vapor–liquid interface line inside the capillary is captured, and the measurement coordinate points of the vapor–liquid interface line are calibrated using the software. Secondly, the proposed contact angle measurement method is used to calculate the actual coordinate points and fit the actual vapor–liquid interface line of the liquid. Finally, the angle measurement tool in the software is used to measure the actual liquid contact angle. This measurement method effectively reduces errors caused by different refractive indices of materials and can be used for precise measurement of the static contact angle of a liquid.

The measurement method for the liquid contact angle of capillary proposed in this article follows Snell’s law. This law calculates the refractive index of light rays based on the refractive index, incident angle, and refractive angle of two media, as shown in Formula (2). When light enters a medium with a higher refractive index from a medium with a lower refractive index, the light can cause objects in both media to appear to change in position and size.

$${\displaystyle \frac{\sin \left({\theta }_{\mathrm{i}\mathrm{n}}\right)}{n_2}}={\displaystyle \frac{\sin \left({\theta }_{\mathrm{r}\mathrm{e}}\right)}{n_1}}$$

(2)

The liquid contact angle inside the capillary undergoes two refractions: liquid–quartz glass and quartz glass–air. The refractive index of light in water is n_water = 1.333; the refractive index in quartz glass is n_si = 1.45847, and the refractive index in air is n_air = 1.00029. According to Snell’s law, light is deflected toward the normal direction at the interface between liquid and quartz glass, causing the liquid to appear larger than its actual size. Therefore, by directly making a tangent K₁ to the vapor–liquid interface curve, the contact angle between the liquid and the quartz glass wall can be obtained. The actual vapor–liquid interface line is tangent K₂. The actual contact angle between the liquid and the quartz glass wall obtained is smaller than the directly measured contact angle, as shown in Figure 5.

According to Snell’s law, when the refractive indices of two media are different, light rays will be deflected toward the normal direction at the interface between the two media. Using the geometric relationship during light refraction, a refracted light ray diagram between the liquid in the capillary and the recording position is drawn, as shown in Figure 6. Place the digital camera at point O and set the horizontal distance between the digital camera and the capillary to L. Select the lowest point A of the interface line as the coordinate origin, point B as the calibrated measurement coordinate point, and point C as the actual coordinate point calculated through functional relationships. The horizontal axis length of measuring coordinate point B is L_AB, and the actual length of coordinate point C on the horizontal axis is L_AC. The angles that have practical significance in the calculation process are ∠AOB and ∠β_in, where ∠AOB is the angle between the measurement coordinate point B and the digital camera placement point O, and ∠β_in is the angle between the actual coordinate point C and the digital camera position point O. The remaining angles assist in determining the distance L_AC between the actual coordinate point C and the coordinate origin point A.

L_AB and L are known. According to the Pythagorean theorem, the length L_BO between point B and the digital camera placement point O can be found.

$$L_{\mathrm{B}\mathrm{O}}=\sqrt{{L_{\mathrm{A}\mathrm{B}}}^2+L^2}$$

(3)

For a right-angled triangle ∆ORB, the definition of the sine function is

$$\sin \angle AOB={\displaystyle \frac{L_{\mathrm{A}\mathrm{B}}}{\sqrt{{L_{\mathrm{A}\mathrm{B}}}^2+L^2}}}$$

(4)

From the sine inverse function, the angle ∠AOB between the measurement coordinate point B and the digital camera placement point O can be calculated as follows:

$$\angle AOB=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{L_{\mathrm{A}\mathrm{B}}}{\sqrt{{L_{\mathrm{A}\mathrm{B}}}^2+L^2}}}$$

(5)

$$\angle AOB=\angle AOE$$

(6)

In the right-angled triangle ∆OAB, ∠BAO is a right angle. The ∠ABO can be obtained from the sum of the interior angles of a triangle theorem.

$$\mathrm{\angle }ABO={\displaystyle \frac{\pi }{2}}-\mathrm{\angle }AOB$$

(7)

$$\angle ABO=\angle ABE$$

(8)

According to the sine theorem, given the magnitude of ∠ABE, the side length of ∠ABE is L_AE; the side length of the opposite side of ∠BEA is L_AB, and ∠BEA is obtained as follows:

$$\mathrm{\angle }BEA=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{L_{\mathrm{A}\mathrm{B}}·\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }ABE}{L_{\mathrm{A}\mathrm{E}}}}$$

(9)

In the triangle ∆ABE, the sizes of ∠ABE and ∠BEA have been determined, and ∠EAB can be obtained from the sum of the interior angles of the triangle theorem.

$$\mathrm{\angle }EAB=\pi -\mathrm{\angle }ABE-\mathrm{\angle }BEA$$

(10)

In the triangle ∆ABE, since ∠BAO = π/2, ∠EAB has been obtained. So, ∠OAE can be found as follows:

$$\mathrm{\angle }OAE={\displaystyle \frac{\pi }{2}}-\mathrm{\angle }EAB$$

(11)

In the triangle, ∆AEO, ∠OAE, and ∠AOE have been obtained. The ∠AEO can be found from the sum of the triangle interior angles theorem:

$$\mathrm{\angle }AEO=\pi -\mathrm{\angle }OAE-\mathrm{\angle }AOE$$

(12)

Because ∠AEO and ∠α_re are supplementary angles to each other, ∠α_re can be found as follows:

$$\angle {\alpha }_{\mathrm{r}\mathrm{e}}=\pi -\angle AEO$$

(13)

Applying Snell’s law to the interface between quartz glass and air, when light enters the air from the glass, refraction can be found.

$$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }{\alpha }_{\mathrm{i}\mathrm{n}}·{\mathrm{n}}_{\mathrm{s}\mathrm{i}}=\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }{\alpha }_{\mathrm{r}\mathrm{e}}·{\mathrm{n}}_{\mathrm{a}\mathrm{i}\mathrm{r}}$$

(14)

$$\angle {\alpha }_{\mathrm{i}\mathrm{n}}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\angle {\alpha }_{\mathrm{r}\mathrm{e}}·{\mathrm{n}}_{\mathrm{a}\mathrm{i}\mathrm{r}}}{{\mathrm{n}}_{\mathrm{s}\mathrm{i}}}}$$

(15)

Using the triangle cosine theorem to solve L_DE, the L_DE range is [L_AE − L_AD, L_AE). When the equation solution is not satisfied, discard the unsatisfied solution.

$$L_{\mathrm{D}\mathrm{E}}^2={L_{\mathrm{A}\mathrm{D}}}^2-{L_{\mathrm{A}\mathrm{E}}}^2+2·L_{\mathrm{A}\mathrm{E}}·L_{\mathrm{D}\mathrm{E}}·\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\angle }{\alpha }_{\mathrm{i}\mathrm{n}}$$

(16)

Use the cosine theorem to obtain ∠ADE as follows:

$$\mathrm{\angle }ADE=\arccos {\displaystyle \frac{{L_{\mathrm{A}\mathrm{D}}}^2+L_{\mathrm{D}\mathrm{E}}^2-{L_{\mathrm{A}\mathrm{E}}}^2}{2·L_{\mathrm{A}\mathrm{D}}·L_{\mathrm{D}\mathrm{E}}}}$$

(17)

In the triangle ∆ADE, ∠α_in and ∠ADE have been obtained, and ∠EAD is obtained from the sum of interior angles of the triangle theorem.

$$\mathrm{\angle }EAD=\pi -\mathrm{\angle }{\alpha }_{\mathrm{i}\mathrm{n}}-\mathrm{\angle }ADE$$

(18)

Because ∠OAE and ∠EAD have been obtained, ∠DAC can be found.

$$\mathrm{\angle }DAC={\displaystyle \frac{\pi }{2}}-\mathrm{\angle }OAE-\mathrm{\angle }EAD$$

(19)

Because ∠ADE and ∠β_re supplement each other, ∠β_re can be found.

$$\mathrm{\angle }{\beta }_{\mathrm{r}\mathrm{e}}=\pi -\mathrm{\angle }ADE$$

(20)

Applying Snell’s law to the interface between liquid and quartz glass, light enters the glass from water and undergoes refraction.

$$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }{\beta }_{\mathrm{i}\mathrm{n}}·{\mathrm{n}}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}=\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }{\beta }_{\mathrm{r}\mathrm{e}}·{\mathrm{n}}_{\mathrm{s}\mathrm{i}}$$

(21)

$$\mathrm{\angle }{\beta }_{\mathrm{i}\mathrm{n}}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }{\beta }_{\mathrm{r}\mathrm{e}}·{\mathrm{n}}_{\mathrm{s}\mathrm{i}}}{{\mathrm{n}}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}}$$

(22)

In the triangle ∆ACD, ∠β_in and ∠DAC have been obtained, resulting in ∠ACD.

$$\mathrm{\angle }ACD=\pi -\mathrm{\angle }{\beta }_{\mathrm{i}\mathrm{n}}-\mathrm{\angle }DAC$$

(23)

According to the sine theorem, the edge length of the opposite side of $\mathrm{\angle }{\beta }_{\mathrm{i}\mathrm{n}}$ is L_AC; the edge length of $\mathrm{\angle }ACD$ is L_AD, and the distance L_AC between the actual coordinate point C and the coordinate origin point A is calculated. The range of L_AC is [0, L_AD].

$$L_{\mathrm{A}\mathrm{C}}={\displaystyle \frac{L_{\mathrm{A}\mathrm{D}}·\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }{\beta }_{\mathrm{i}\mathrm{n}}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }ACD}}$$

(24)

Thus, based on the actual length of coordinate point C on the horizontal axis, the actual coordinate points were determined. The actual vapor–liquid interface line of the liquid inside the pulsating heat pipe was fitted according to the actual coordinate points, and the true liquid contact angle was measured. When the liquid inside the tube is in a static state, the static contact angle of the liquid is measured.

2.3. Validate Constraints on All Measurement Angles

Use the clear vapor–liquid interface line inside the capillary as the marking object for measuring coordinates. In order to standardize the measurement scale, the image is first scaled before reading the coordinates. From the vapor–liquid interface line image captured by a digital camera, various marked points (X_i, Y_i) on the vapor–liquid interface line can be measured as the measurement coordinates for the liquid contact angle measurement method, as shown in Figure 7. When X₁ = 2.282, it can be seen from Figure 7 that the X values from X₁ to X₁₀ show an increasing trend from left to right, and the X values from X₂₁ to X₁₁ also show an increasing trend from left to right. Therefore, the size of the measured coordinates Xi increases from left to right. When Y₁ = 3.889, it can be seen from Figure 7 that the Y value from Y₁ to Y₁₀ shows a decreasing trend from bottom to top, and the X value from Y₂₁ to Y₁₁ also shows a decreasing trend from bottom to top. Therefore, the size of the measurement coordinate Y_i decreases from bottom to top. Finally, based on the contact angle measurement method proposed in this article, the actual coordinate points are calculated, and constraints are imposed on each angle and length during the calculation process. The constraint conditions are as follows: (1) The range for acute angles is [0, π/2); the range for obtuse angles is [π/2, π); the range for length L_CD is [0, L_AD); the range for length L_DE is [L_AE − L_AD, L_AE]; (2) Snell’s law verification: when n₁ > n₂, sin∠2 > sin∠1; (3) Verification of the sum of interior angles in a triangle: The magnitude of the sum of interior angles in a triangle is π. The final obtained contact angle is the average of the contact angles on both sides of the vapor–liquid interface line.

3. Results and Discussion

3.1. The Influence of Refraction Phenomenon on the Interface Line

Select a clear image of the liquid in the capillary and measure the static contact angles of the upper and lower liquid surfaces of the liquid column, as shown in Figure 8.

Adjust the power meter and heat the capillary. When there is no flow trend of the liquid inside the capillary during the initial heating stage, measure the static contact angle of the liquid inside the capillary, as shown in Figure 9. Based on the visual experiment (as shown by the black line), the proposed method for measuring the liquid contact angle of a capillary is used to obtain the actual interface line of distilled water (as shown by the red line). During the measurement process, it is necessary to measure the contact angles on the left and right sides of the interface line. Finally, the average of the contact angle values on both sides is taken to obtain the contact angle value representing the vapor–liquid interface line, as shown in Figure 9a. The left measurement value is 49.66°, the right measurement value is 46.98°, and the average measurement value is 48.32°; the actual value on the left is 46.01°; the actual value on the right is 43.39°, and the average of the actual values is 44.7°. The difference between the average of the measured values and the average of the actual values is 3.62°.

Measure the contact angle on both sides of the vapor–liquid interface line, and take the average of the contact angles on both sides to represent the contact angle value of the vapor–liquid interface line. Use standard deviation to measure the fluctuation of the contact angle deviation from the mean contact angle on both sides, as shown in Formula (25). Obtain the error bar chart for contact angle measurement, as shown in Figure 10.

$$SD=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{(x_i-\overline{x})}^2}$$

(25)

3.2. The Influence of Refraction Phenomenon on Static Contact Angle

When the liquid in the capillary is in a static state, use the above measurement method to measure the static contact angle of the liquid. As shown in Figure 11, the static contact angle of the liquid in a stationary state was measured to obtain the static contact angle value of distilled water in the capillary. From Figure 11, it can be seen that there is a dissymmetric phenomenon of the upper and lower liquid surfaces of the liquid column and a difference in the static contact angle between the upper and lower liquid surfaces of the liquid column. This is because the lower liquid surface of the liquid column is more tightly in contact with the tube wall due to gravity. At the same time, the lower liquid surface is subjected to greater hydrostatic pressure, resulting in an increase in curvature and a smaller static contact angle. The upper liquid surface of a liquid column is less affected by gravity and remains relatively flat. The curvature of the vapor–liquid interface decreases, resulting in a larger contact angle.

Select clear vapor–liquid interface lines in Figure 11 as the measurement objects for static contact angle. Measure the static contact angle of the upper and lower liquid surfaces of the liquid column separately are shown in

Table 2. Comparison between average measured and actual values of upper static contact angles without heating power.

Serial Number	Measurement Value (°)	Actual Value (°)	Difference (°)
1	54.32	49.22	5.10
2	54.17	51.16	3.01
3	54.53	52.69	1.84

and

Table 3. Comparison between average measured and actual values of lower static contact angles without heating power.

Serial Number	Measurement Value (°)	Actual Value (°)	Difference (°)
1	45.17	41.98	3.19
2	45.63	40.90	4.74
3	47.57	41.95	5.61

. The difference between the measured value and the actual value of the static contact angle is a minimum of 1.84° and a maximum of 5.61°.

When the working fluid is heated but in a stationary state, measure the contact angle of distilled water in the capillary tube, as shown in Figure 12. From Figure 12, it can be seen that due to the phenomenon of light refraction, the actual values of the upper and lower contact angles of the liquid column are both smaller than the measured values. There is a certain difference between the actual and measured static contact angles, with a minimum difference of 1.85° and a maximum difference of 4.67°.

4. Conclusions

This article proposes a method for measuring the contact angle of a liquid in a capillary. Considering that the vapor–liquid interface line of the liquid is refracted twice through liquid–quartz glass and quartz glass–air, which brings errors to the measurement results of the liquid contact angle in the capillary. Therefore, the actual coordinate points of the vapor–liquid interface line are calculated using the proposed liquid contact angle measurement method. Effectively reducing the errors caused by different refractive indices of materials can be used for the precise measurement of the static contact angle of a liquid.

When the liquid inside the capillary is in its initial distribution state, static contact angle measurements are taken on the upper and lower vapor–liquid interface lines of the liquid column. The difference between the measured and actual values of the static contact angle is [1.84°, 5.61°]. Adjust the power meter and heat the capillary. When there is no flow trend of the liquid, the static contact angle of the liquid is measured under different heating conditions. There is a certain difference between the actual value and the measured value of the static contact angle. The minimum difference in the measured static contact angle is 1.85°, and the maximum difference is 4.67°. The measurement results show that the proposed method for measuring the contact angle of liquid in a capillary effectively reduces the errors caused by different refractive indices of materials and can be used for precise measurement of the static contact angle of liquid.