Capillary Rise Between Closely Packed Vertical Rods: 04-BS-7, December 2013, Q1

Question 1: Capillary Rise Between Closely Packed Vertical Rods (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Rod diameter, d	1 mm = 0.001 m
Surface tension, σ	0.073 N/m
Wetting angle, θ	0° (cosθ=1)
Water density, ρ	1000 kg/m³

Square-packed rod array; the shaded curvilinear-square pore between four touching rods is the effective capillary passage.

Find. Height h above the free surface to which water rises in the pore between the rods.

Approach. Generalise the circular-tube capillary formula to an arbitrary cross-section, $h=(\sigma\cos\theta/\rho g)\times(\text{wetted perimeter}/\text{area})$, then evaluate that perimeter/area shape factor for the touching-rod unit cell.

Unit-cell pore area. Four rods of diameter d touch at the corners of a d×d square cell; each rod contributes one quarter-circle of area $\pi(d/2)^2/1$ into the cell (four quarters = one full circle), so the open pore area is $$A = d^{2}-\pi\left(\frac{d}{2}\right)^{2} = d^{2}\left(1-\frac{\pi}{4}\right) = (0.001)^{2}(1-0.7854) = 2.146\times10^{-7}\ \text{m}^{2}$$
Wetted perimeter. The pore is bounded by four quarter-circle arcs, one from each rod, whose arc lengths sum to exactly one full rod circumference: $$P = \pi d = \pi(0.001) = 3.1416\times10^{-3}\ \text{m}$$
Shape factor and rise. Substituting into the generalised capillary-rise relation, $$h = \frac{\sigma\cos\theta}{\rho g}\cdot\frac{P}{A} = \frac{0.073\times1}{1000\times9.81}\times\frac{3.1416\times10^{-3}}{2.146\times10^{-7}} = \boxed{0.1089\ \text{m} = 108.9\ \text{mm}}$$

Quantity	Value
Pore area	2.146×10⁻⁷ m²
Wetted perimeter	3.142×10⁻³ m
Capillary rise, h	108.9 mm

Topic: Capillarity in a non-circular passage.

Key relations: circular tube $h=2\sigma\cos\theta/(\rho g r)$; general passage $h=(\sigma\cos\theta/\rho g)(P/A)$, which reduces to the circular form since a circle's $P/A=2/r$.

Why this works: the capillary-rise balance equates the vertical component of the surface-tension force acting around the wetted perimeter, $\sigma\cos\theta\,P$, to the weight of the raised column, $\rho g A h$ — the perimeter/area ratio is exactly what survives when that force balance is solved for h, regardless of the passage's shape.

Common pitfall: treating the rod diameter itself as if it were a tube radius; the relevant length scale is the pore's own perimeter-to-area ratio, which is set by the packing geometry, not by the rod size directly.

Source: White, Fluid Mechanics, Ch. 1 (surface tension & capillarity).

Question 1 of 13: Capillary Rise Between Closely Packed Vertical Rods

Question 1: Capillary Rise Between Closely Packed Vertical Rods (5 marks)