Given:
Discharge, Q = 260 L/s = 0.26 m³/s
Impeller outer diameter, D2 = 80 cm = 0.8 m
Head developed, H = 15.3 m

To find: Minimum starting speed, N1

Formula used:
Q = π/4 x D² x v
H = v²/2g

where,
v = velocity of water
D = diameter of the impeller
g = acceleration due to gravity
π = 3.14

Calculation:
- Convert the given discharge from litres to cubic meters per second
Q = 260 L/s = 0.26 m³/s

- Calculate the velocity of water through the impeller
Q = π/4 x D² x v
v = Q / (π/4 x D²)
v = 0.26 / (3.14/4 x 0.8²)
v = 7.8 m/s (approx)

- Calculate the starting head of the pump
H = v²/2g
H = (7.8)² / (2 x 9.81)
H = 30.3 m (approx)

- Calculate the starting speed of the pump
Using the affinity laws, we can write:
N1/N2 = (Q1/Q2) x (D2/D1) x (H1/H2)^(1/2)
where N1 and N2 are the starting and running speeds respectively,
D1 is the impeller diameter at the starting speed (unknown),
Q1 is the discharge at the starting speed (unknown), and
H1 is the head at the starting speed (30.3 m).

Assuming the impeller diameter remains the same, we can write:
N1/N2 = (Q1/0.26) x (0.8/D1) x (30.3/15.3)^(1/2)

To find D1, we can use the Euler's equation:
H = (v²/2g) - (u²/2g)
where u is the velocity of the fluid at the impeller exit.

Assuming the velocity of the fluid at the impeller exit is negligible compared to the velocity at the impeller inlet, we can write:
H = (v²/2g) - (v²/2g)
H = 0

At the shut-off condition, the head developed is zero. Therefore, using the given head and the head coefficient, we can find the velocity at the impeller inlet:
H = v²/2g x (1 - (D1/D2)²)
15.3 = v²/2g x (1 - (D1/0.8)²)
v² = 30.6 x (1 - (D1/0.8)²)

Substituting the value of v² in the affinity laws equation, we get:
N1/N2 = (Q1/0.26) x (0.8/D1) x (30.3/15.3)^(1/2)
N1/600 = (Q1/0.26) x (0.8/D1) x (2)^(1/2)

Assuming Q1 is proportional to N1 (as the pump follows the affinity laws),