Calculate the Reynolds number, Re for water flow in a circular pipe. The diameter of the pipe is 50 mm, the density of water is 998 kg/m³, the volumetric oil flowrate is 720 L/min, and the dynamic viscosity of water is 1.2 centipoise.
The correct answer and explanation is :
To calculate the Reynolds number (Re) for water flow in a circular pipe, we use the formula:
$$
Re = \frac{\rho \cdot V \cdot D}{\mu}
$$
Where:
- $\rho$ = density of fluid = 998 kg/m³
- $V$ = average velocity of fluid (m/s)
- $D$ = diameter of pipe = 50 mm = 0.05 m
- $\mu$ = dynamic viscosity of fluid = 1.2 cP = 1.2 × 10^{-3} Pa·s
First, convert the volumetric flow rate:
$$
Q = 720 \, \text{L/min} = \frac{720}{1000 \times 60} = 0.012 \, \text{m³/s}
$$
Next, calculate the cross-sectional area of the pipe:
$$
A = \frac{\pi D^2}{4} = \frac{\pi (0.05)^2}{4} = 1.9635 \times 10^{-3} \, \text{m}^2
$$
Then, calculate the average velocity $V$:
$$
V = \frac{Q}{A} = \frac{0.012}{1.9635 \times 10^{-3}} = 6.111 \, \text{m/s}
$$
Now compute the Reynolds number:
$$
Re = \frac{998 \cdot 6.111 \cdot 0.05}{1.2 \times 10^{-3}} = \frac{304.43}{1.2 \times 10^{-3}} = 253,692
$$
Final Answer:
$$
\boxed{Re = 253{,}692}
$$
Explanation (300 words):
The Reynolds number (Re) is a dimensionless parameter used in fluid mechanics to predict flow regimes in a pipe or over a surface. It provides insight into whether the flow is laminar, transitional, or turbulent. Laminar flow occurs at Re < 2000, transitional flow between 2000 and 4000, and turbulent flow at Re > 4000.
In this problem, water is flowing through a circular pipe with a diameter of 50 mm. The volumetric flow rate is 720 L/min, which we convert into cubic meters per second to maintain SI units. The dynamic viscosity is given in centipoise (cP), a common unit for fluids, especially in industry, and must be converted to Pascal-seconds (Pa·s) for consistency with SI units.
After calculating the cross-sectional area of the pipe using the diameter, we find the average velocity by dividing the volumetric flow rate by the area. The average velocity, density, pipe diameter, and dynamic viscosity are then substituted into the Reynolds number formula.
The final result, a Reynolds number of approximately 253,692, indicates highly turbulent flow. This is typical for many industrial systems where high flow rates are used. Turbulent flow affects heat transfer, pressure drop, and pipe design, so it’s crucial to know the flow regime when designing piping systems or choosing pumps.