A pipe carrying 20°C water has a diameter of 3.1 cm. Estimate the maximum flow speed if the flow must be streamline.
To estimate the maximum flow speed of the water in a streamline flow, we can use the concept of the Reynolds number (Re). The Reynolds number helps determine whether the flow is laminar or turbulent.
The formula for calculating the Reynolds number is:
Re = (ρ * V * D) / μ
where:
- ρ is the density of the fluid (water in this case),
- V is the velocity of the flow,
- D is the diameter of the pipe, and
- μ is the dynamic viscosity of the fluid (water in this case).
To estimate the maximum flow speed, we need to determine the flow regime. Generally, when the Reynolds number is less than 2000, the flow is considered laminar or streamline.
Firstly, we need to find the properties of water at 20°C:
- Density of water (ρ) at 20°C is approximately 998 kg/m^3.
- Dynamic viscosity of water (μ) at 20°C is approximately 1.002 × 10^(-3) kg/(m·s).
Now, let's plug these values into the Reynolds number equation with an assumption of laminar flow:
Re = (998 * V * 0.031) / (1.002 × 10^(-3))
To estimate the maximum flow speed, we want to find the velocity (V), so let's solve for V:
V = (Re * 1.002 × 10^(-3)) / (998 * 0.031)
By substituting the values, we get:
V = (2000 * 1.002 × 10^(-3)) / (998 * 0.031)
Now, let's calculate the value of V:
V ≈ 2.03 m/s
Therefore, the estimated maximum flow speed for the streamline flow of 20°C water in a 3.1 cm diameter pipe is approximately 2.03 m/s.