Dupuit's equation is expressed for series-connected pipes as: (where L1, L2, and L3 are lengths of pipe 1, 2 and 3; d1, d2, d3 are diameter of pipe 1, 2 and 3; L is equivalent length of pipe and d is equivalent diameter of pipe)
- AL / d⁵ = L₁ / d₁⁵ - L₂ / d₂⁵
- BL / d⁶ = L₁ / d₁⁶ + L₂ / d₂⁶ + L₃ / d₃⁶
- CL / d⁵ = L₁ / d₁⁵ - L₂ / d₂⁵ - L₃ / d₃⁵
- DL / d⁵ = L₁ / d₁⁵ + L₂ / d₂⁵ + L₃ / d₃⁵Correct
Solution
Concept
In series pipes, total head loss equals the sum of head losses in each pipe, and the discharge is the same in all pipes.
Formula
L/d5 = L1/d15 + L2/d25 + L3/d35
Step-by-step solution
1. Head loss in a pipe: hf = f L Q2 / (12.1 d5). 2. In series, Q is the same in every pipe and the total head loss is the sum of the pipe head losses. 3. The equivalent pipe has the same Q and the same total head loss. 4. Take f the same for all pipes: L/d5 = L1/d15 + L2/d25 + L3/d35. 5. So the answer is (D).
Explanation
Pipes in series carry the same discharge, so their head losses add up. Head loss is proportional to L/d5 for the same Q and f. This gives Dupuit's equation with a plus sign and power 5.
Why the other options are wrong
A: uses a minus sign and only two pipes. B: the power of d should be 5, not 6. C: head losses add in series, they are not subtracted.
Shortcut
Series pipes: add L/d5 terms (head loss is proportional to 1/d5).
Common mistake
Using d4 or d6, or subtracting the terms.
Correct answer: D) L / d⁵ = L₁ / d₁⁵ + L₂ / d₂⁵ + L₃ / d₃⁵