Problem Statement
The capillary rise in the glass tube used for measuring water level is not to exceed 0.5 mm. Determine its minimum size, given that the surface tension for water in contact with air is 0.07112 N/m.
Given Data
- Maximum Capillary Rise, \(h = 0.5 \, \text{mm}\)
- Surface Tension, \(\sigma = 0.07112 \, \text{N/m}\)
- Fluid: Water in contact with a glass tube
- Density of water, \(\rho = 1000 \, \text{kg/m}^3\)
- Angle of contact for water-glass, \(\theta = 0^\circ\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
1. Convert Units to SI
The capillary rise must be converted from millimeters to meters.
2. Apply and Rearrange the Capillary Rise Formula
The formula for capillary rise \(h\) is given below. For water in a clean glass tube, the contact angle \(\theta\) is \(0^\circ\), so \(\cos\theta = 1\).
We rearrange the formula to solve for the minimum diameter \(d\).
3. Substitute Values and Calculate
Now we substitute the known values into the rearranged formula.
4. Convert Diameter to a More Convenient Unit
The calculated diameter can be expressed in centimeters for practicality.
The minimum size (diameter) of the tube should be \( d = 5.8 \, \text{cm} \).
Explanation of Capillary Action
Capillary Action is the tendency of a liquid to rise or fall in a narrow tube. This effect is caused by the interplay of two forces: cohesion (the attraction between liquid molecules) and adhesion (the attraction between the liquid molecules and the tube’s surface). When adhesion is stronger than cohesion, as with water and glass, the liquid “climbs” the tube walls, and surface tension pulls the column of liquid up.
Physical Meaning
The result demonstrates the inverse relationship between the tube’s diameter and the height of the capillary rise. To restrict the rise to a very small amount (0.5 mm), the tube must have a relatively large minimum diameter of 5.8 cm.
This tells us that:
- To minimize the error caused by capillary action when measuring a water level, one must use a tube with a large diameter. A tube smaller than 5.8 cm would cause the water to rise more than the 0.5 mm limit, introducing a larger measurement error.
- Conversely, if the goal is to maximize the capillary effect (as in a wick or a medical test strip), a material with very small pores or tubes is needed.
