A cylindrical pipe 30 cm in diameter carries water at a volumetric flow rate of 10-3 m3/s.
The pattern of the flow in the pipe ...
To determine the pattern of flow in the pipe, we need to calculate the Reynolds number. The Reynolds number is a dimensionless value that helps us understand the type of flow: laminar or turbulent.
The formula to calculate the Reynolds number is:
Re = (ρ * v * D) / μ
Where:
- Re is the Reynolds number
- ρ is the density of the fluid (water in this case)
- v is the velocity of the fluid flow
- D is the diameter of the pipe
- μ is the dynamic viscosity of the fluid (water in this case)
Let's calculate the Reynolds number step by step:
1. Density of water (ρ):
The density of water is approximately 1000 kg/m³.
2. Velocity (v):
The volumetric flow rate (Q) is given as 10^-3 m³/s. To find velocity, we divide the flow rate by the cross-sectional area of the pipe.
The cross-sectional area (A) of the pipe can be calculated using the formula:
A = (π * D²) / 4
Substituting the given diameter (30 cm = 0.3 m) into the formula:
A = (π * 0.3²) / 4
A = 0.0707 m²
The velocity (v) is then:
v = Q / A = (10^-3) / 0.0707
v ≈ 14.14 m/s
3. Dynamic viscosity of water (μ):
The dynamic viscosity of water at room temperature is approximately 0.001 kg/(m*s).
Now, we can calculate the Reynolds number (Re):
Re = (ρ * v * D) / μ = (1000 * 14.14 * 0.3) / 0.001
Re = 424,200
The flow pattern in the pipe can be determined based on the Reynolds number:
- If Re < 2,000, the flow is laminar.
- If Re > 4,000, the flow is turbulent.
- If 2,000 <= Re <= 4,000, the flow is in a transitional zone and could be either laminar or turbulent.
In this case, since the Reynolds number is quite large (424,200), the flow in the pipe is most likely turbulent.