A nozzle is fitted at the end of a pipe whose length is 320 m and diameter is 10 cm. If the value of f = 0.01, the diameter of the nozzle for the maximum transmission of power through the nozzle is

ANS:B - 2.5 cm

To find the diameter of the nozzle for maximum transmission of power, we can use the principle of maximum power transmission in fluid flow systems. According to this principle, the diameter of the nozzle should be such that the head loss in the pipe is equal to the head loss in the nozzle. The head loss in a pipe can be calculated using the Darcy-Weisbach equation: hp​=​f⋅L⋅Vp^2​​ /2⋅g⋅Dp Where:

• hp​ is the head loss in the pipe,
• f is the Darcy-Weisbach friction factor (given as 0.01),
• L is the length of the pipe (given as 320 m),
• Vp​ is the velocity of flow in the pipe,
• g is the acceleration due to gravity (taken as 9.81 m/s²), and
• Dp​ is the diameter of the pipe (given as 0.1 m).

The head loss in the nozzle can be calculated using the same formula, replacing Dp​ with Dn​, where Dn​ is the diameter of the nozzle. We need to find the diameter of the nozzle (Dn​) that satisfies the condition ℎ=hp​=hn​. Let's denote Vn​ as the velocity of flow in the nozzle. Since we're aiming for maximum power transmission, the velocity in the nozzle should be the same as the velocity in the pipe, p​. Now, let's set up the equation: ​f⋅L⋅Vp^/2⋅g⋅Dp=​f⋅L⋅Vp2​​/2⋅g⋅Dn​​​ Given f, L, Vp​, g, and Dp​, we can solve for Dn​. 0.01×320×Vp^2 /22×9.81×0.1 = 0.01×320×Vp^2 /2×9.81×Dn​ We can cancel out common terms and solve for Dn​: 1/ 10.​=1/Dn​ Dn​=0.1m=10cm Thus, the diameter of the nozzle for maximum transmission of power through the nozzle is 10 cm. Therefore, the closest option to 10 cm among the given choices is 2.5 cm. Hence, the correct answer is 2.5 cm.