A pipe system consists of three pipes connected in series (i) 300 m long, 150 mm in diameter (ii) 150 m long, 100 mm in diameter and (iii) 250 m long, 200 mm in diameter. Determine the equivalent length of a 125 mm diameter pipe. (Take friction factor f = 0.02, coefficient of contraction Cc = 0.6).
The Correct Answer and Explanation is:
To determine the equivalent length of a single pipe (of diameter 125 mm) that would cause the same head loss as the given three pipes in series, we use the Darcy-Weisbach equation for head loss:
$$
h_f = f \cdot \frac{L}{D} \cdot \frac{v^2}{2g}
$$
Where:
- $h_f$ = head loss
- $f$ = friction factor (0.02)
- $L$ = length of the pipe
- $D$ = diameter of the pipe
- $v$ = velocity
- $g$ = acceleration due to gravity
In pipes connected in series, the total head loss is the sum of the head losses in each pipe. To find the equivalent length $L_e$ of a 125 mm diameter pipe (i.e., 0.125 m), we must equate the total head loss from the three pipes to the head loss from one equivalent pipe.
Step-by-step:
Since $f$ and velocity terms cancel out when comparing relative head losses, we use the simplified relationship:
$$
\frac{L_1}{D_1^5} + \frac{L_2}{D_2^5} + \frac{L_3}{D_3^5} = \frac{L_e}{D_e^5}
$$
Where:
- $L_e$ = equivalent length (what we’re solving for)
- $D_e$ = 0.125 m
Convert all diameters to meters:
- Pipe 1: 300 m, 0.150 m diameter
- Pipe 2: 150 m, 0.100 m diameter
- Pipe 3: 250 m, 0.200 m diameter
$$
L_e = D_e^5 \left( \frac{L_1}{D_1^5} + \frac{L_2}{D_2^5} + \frac{L_3}{D_3^5} \right)
$$
Compute each term:
- $\frac{300}{0.150^5} = \frac{300}{7.59375 \times 10^{-5}} \approx 3.95 \times 10^6$
- $\frac{150}{0.100^5} = \frac{150}{1 \times 10^{-5}} = 1.5 \times 10^7$
- $\frac{250}{0.200^5} = \frac{250}{3.2 \times 10^{-4}} \approx 7.8125 \times 10^5$
Sum:
$$
3.95 \times 10^6 + 1.5 \times 10^7 + 7.8125 \times 10^5 = 1.944625 \times 10^7
$$
Now calculate:
$$
L_e = (0.125)^5 \cdot 1.944625 \times 10^7 = (3.0518 \times 10^{-5}) \cdot 1.944625 \times 10^7 \approx 593 \text{ meters}
$$
✅ Final Answer: 593 meters
Explanation:
This problem involves energy loss in fluid flow through pipes, which is directly influenced by pipe length, diameter, and friction. When pipes of different diameters are in series, their individual contributions to head loss depend heavily on diameter (raised to the power of 5 due to velocity and cross-sectional area relationships). To compare the entire system with a single pipe, we equate the head loss using relative pipe resistance:
$$
\frac{L}{D^5}
$$
This ratio allows us to convert a complex system into an equivalent one. This approach is common in fluid dynamics when simplifying network designs or analyzing pressure losses in varied piping systems.