Water is pumped at a steady uniform flow rate of 0.01 m3/s through a h...
Given Information:
- Flow rate: 0.01 m3/s
- Pipe diameter: 100 mm
- Reynolds number: 800
- Acceleration due to gravity: 9.81 m/s2
Calculating Velocity:
To calculate the velocity of water flowing through the pipe, we can use the formula:
Velocity (V) = Flow rate (Q) / Cross-sectional area (A)
Given that the pipe is circular, we can calculate the cross-sectional area using the formula for the area of a circle:
A = π * (d/2)2
Substituting the values, we get:
A = π * (0.1/2)2 = 0.00785 m2
Now, substituting the values of flow rate and cross-sectional area, we can calculate the velocity:
V = 0.01 m3/s / 0.00785 m2 ≈ 1.27 m/s
Calculating Reynolds Number:
The Reynolds number is a dimensionless quantity that represents the flow characteristics of a fluid. It can be calculated using the formula:
Re = (ρ * V * D) / μ
Where:
- ρ is the density of the fluid
- V is the velocity of the fluid
- D is the diameter of the pipe
- μ is the dynamic viscosity of the fluid
Given that the Reynolds number is 800 and the diameter of the pipe is 100 mm, we can calculate the dynamic viscosity:
800 = (ρ * 1.27 * 0.1) / μ
Calculating Head Loss:
The head loss due to friction in a pipe can be calculated using the Darcy-Weisbach equation:
hL = (f * (L/D) * (V2/2g))
Where:
- hL is the head loss
- f is the Darcy friction factor
- L is the length of the pipe
- D is the diameter of the pipe
- V is the velocity of the fluid
- g is the acceleration due to gravity
To calculate the head loss, we need to determine the Darcy friction factor. For laminar flow, the Darcy friction factor can be calculated using the formula:
f = 16 / Re
Substituting the values, we get:
f = 16 / 800 = 0.02
Given that the length of the pipe is 1 km (1000 m), we can now calculate the head loss:
hL = (0.02 * (1000/0.1) * (1.27^2/2 * 9.81)) ≈ 66.10 m
Therefore, the head loss per km length due to friction in this pipe is approximately 66.10 meters.