Problem Statement
The capillary rise in a glass tube is not to exceed 0.2 mm of water. Determine its minimum size (diameter), given that the surface tension for water in contact with air is 0.0725 N/m.
Given Data
- Maximum Capillary Rise, \(h = 0.2 \, \text{mm}\)
- Surface Tension, \(\sigma = 0.0725 \, \text{N/m}\)
- Fluid: Water in contact with air
- 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 is given in millimeters, so we must convert it to meters to match the other SI units.
2. Apply the Capillary Rise Formula
The height \(h\) of capillary rise is given by the formula:
For water in contact with a clean glass tube, the angle of contact \(\theta\) is approximately \(0^\circ\), and \(\cos(0^\circ) = 1\). The formula simplifies to:
We need to find the minimum diameter \(d\), so we rearrange the formula:
3. Substitute Values and Calculate
Now we substitute the known values into the rearranged formula to find the diameter \(d\).
Rounding the result, we get:
4. Convert Diameter to a More Convenient Unit
The diameter is very small, so it's more practical to express it in millimeters or centimeters.
The minimum diameter of the tube should be \( d = 0.148 \, \text{mm} \).
Explanation of Capillary Action
Capillary Action is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity. It occurs because of the interplay between cohesive forces (attraction between liquid molecules) and adhesive forces (attraction between liquid molecules and the tube's surface).
In the case of water and glass, the adhesive forces are stronger than the cohesive forces. This causes the water to "climb" the walls of the glass tube, and the surface tension of the water pulls the rest of the column upwards until the weight of the risen water balances the adhesive and surface tension forces.
Physical Meaning
The result shows an inverse relationship between capillary rise and tube diameter (\(h \propto 1/d\)). To ensure the capillary rise does not exceed 0.2 mm, the tube's diameter must be at least 0.148 mm.
- Narrower Tube, Higher Rise: If the tube were any narrower than 0.148 mm, the capillary rise would be greater than 0.2 mm.
- Wider Tube, Lower Rise: A wider tube would result in a capillary rise of less than 0.2 mm.
This principle is crucial in many fields, including soil mechanics (water movement in soil), biology (transport of water in plants), and engineering (ink-jet printing and microfluidics).
