# Effects of Tube Radius and Surface Tension on Capillary Rise Dynamics of Water/Butanol Mixtures
## 3 Results and Discussion
### Effect of Tube Inner Radius on Capillary Dynamics
The length of Stage III, which is a purely viscous region, was reduced from 23.1% to 9.3% with the increase in the capillary tube radius from 0.4 to 0.85 mm. Additionally, the time expressed as \(t_{2}\) corresponding to \(h_{2}\) was reduces from 0.23 to 0.1 s with the capillary tube radius increasing from 0.4 to 0.85 mm, as shown in Figure 7a. Based on these results, the capillary rise dynamics can be controlled by the tube diameter, regardless of whether the flow is inertial dominant or viscous dominant in all the capillary rise applications. The effect of the tube radius on the capillary rise height is shown in Figure 7a. At the end of Stage III, gravitational forces become significant owing to the balance of gravity and capillarity. The meniscus stops at the equilibrium capillary rise height (\(h_{\rm e}\)) expressed as follows [15; 36]:  
\[h_{\rm e}=\frac{2\sigma\cos\theta_{\rm e}}{\rho g\tau}, \tag{6}\]  
where \(\sigma\) is the surface tension (mN/m), \(g\) is the standard gravity (m/s\({}^{2}\)), \(\rho\) is the density of the liquid (kg/m\({}^{3}\)), \(r\) is the capillary radius (mm), and \(\theta_{\rm e}\) is the contact angle. From Figure 7a and Table 1, \(h_{\rm e}\) was 35.9, 24.1, and 16.1 mm at tube radii 0.4, 0.6, and 0.85 mm, respectively. The theoretical prediction values were 36.8 at \(r=0.4\) mm, 24.5 at \(r=0.6\) mm, and 17.3 at \(r=0.85\) mm. In this theoretical prediction for Equation (6), we used cos\(\theta_{\rm e}\) = 1 in all the test conditions because the contact angle for water at \(h_{\rm e}\) was measured to be very small by [17]. In Figure 7a, it is expected that the contact angle \(\theta\) varies in time, and the value of cos\(\theta\) extracted from Equation (3) increases with increasing the time (or rise height). At the equilibrium state, the value of cos\(\theta\) reaches unity (cos\(\theta_{\rm e}\) = 1) and its value in this study was 0.98, 0.98, and 0.93 at tube radii 0.4, 0.6, and 0.85 mm, respectively. In addition, the effect of capillary tube radii on the rise height depends on the capillary pressure (\(P_{\rm C}\)). The isothermal pressures in gas (\(P_{\rm G}\)) and liquid (\(P_{\rm L}\)) phases are defined as the \(P_{\rm C}\) that is:  
\[P_{\rm C}=P_{\rm G}-P_{\rm L}. \tag{7}\]  
The equilibrium relation between the pressures on the two sides of an interface can be expressed by well-known Young-Laplace equation as given below for a spherical meniscus in a cylindrical tube [37; 38]:  
\[P_{\rm C}=\frac{2\sigma}{r}\cos\theta, \tag{8}\]  
where \(r\) is the radius of capillary tube, \(\sigma\) is the surface tension between the gas and liquid phases, and \(\theta\) is the contact angle of the gas-liquid interface with respect to the solid surface. Figure 7: Effect of tube diameter (\(2r=0.8\), 1.2, and 1.7 mm) on capillary rise dynamics of DW. (**a**) height and (**b**) velocity. In Equation (8), the surface tension of and the tube radius are the parameters that affect capillary pressure at fixed value of cos\(\theta\) = 1. At \(r\) = 0.4, 0.6, and 0.85 mm in radii, the \(P_{\text{C}}\) values were 360, 240, and 169 N/m\({}^{2}\), respectively. The trend of capillary rise height with a variation in tube radii was consistent with the results of capillary pressures. As shown in Figure 7b, the larger the radius of the capillary tube, the lower is the maximum instantaneous rise velocity. At \(r\) = 0.4, 0.6, and 0.85 mm in radii, the peak velocity values were 0.46, 0.36, and 0.27 m/s, respectively. The velocity decreases after reaching the peak value and tends to converge to zero near he. These trends are in qualitative agreement with a previous publication [20; 23; 39; 40]. Regarding the effects of changes in the radius, we demonstrated that the capillary rise is sensitive to the radius; thus, the capillary radius is an important control parameter in the capillary rise dynamics.