Tile size (2 cm/day) and spacing (3 cm/day)

(i) Size of tiles for 2 cm/day

Each lateral drains 200 m × 50 m = 1 ha. The discharge is

$$Q = 0.02\times\frac{10000}{86400} = 0.00231\ \text{m}^3/\text{s} = 2.31\ \text{l/s}$$

For a pipe flowing full ($A=\pi D^2/4$, $R=D/4$, $n$ = 0.011, $S$ = 0.003):

$$D=\left(\frac{4^{5/3}\,nQ}{\pi\sqrt{S}}\right)^{3/8}=\left(\frac{4^{5/3}\times0.011\times0.00231}{\pi\times\sqrt{0.003}}\right)^{3/8} = 87\ \text{mm}$$

Adopt the next standard size, 100 mm. Full-flow capacity of the 100 mm tile:

$$Q_{full}=\frac{1}{0.011}\times\frac{\pi(0.1)^2}{4}\times\left(\frac{0.1}{4}\right)^{2/3}\times\sqrt{0.003} = 0.00334\ \text{m}^3/\text{s} = 3.34\ \text{l/s}$$

(ii) Spacing at 3 cm/day

The same 100 mm tile at the same grade can carry only 3.34 l/s. The area it can serve at a drainage coefficient of 0.03 m/day is

$$A = \frac{Q_{full}}{q} = \frac{0.00334}{0.03/86400} = 9630\ \text{m}^2$$

For a lateral of length 200 m:

$$\text{Spacing}=\frac{9630}{200} = 48.1\ \text{m}$$

So the laterals must be placed about 48 m apart (slightly closer than 50 m). If the 50 m spacing is to be kept, the tile size must be increased: at 3 cm/day the discharge is 3.47 l/s, which needs the next size, 125 mm.

Answer: (i) 100 mm tiles; (ii) spacing ≈ 48 m for 100 mm tiles.