Question

At what velocity does water flow convert from laminar to transitional in a 25 mm diameter pipe at 25$^\circ$C? The density of water is 997 kg/m$^3$ and viscosity is 1 x 10$^{-3}$ kg/m.s}

• A. 0.063 m/s
• B. 0.084 m/s
• C. 0.84 m/s
• D. 0.042 m/s

Hint:

For fluid flow in pipes, always use $Re = 2100$ as the critical threshold for the transition from laminar flow. Ensure that the pipe diameter is converted to meters before calculating.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Fluid flow in a closed pipe is characterized as laminar, transitional, or turbulent based on the dimensionless Reynolds number ($Re$).
For flow in a circular pipe, the critical Reynolds number ($Re_c$) at which the flow transitions from stable laminar to transitional is approximately 2
The Reynolds number is a ratio of inertial forces to viscous forces.
Key Formula or Approach:
The Reynolds number formula is:
\[ Re = \frac{\rho \cdot v \cdot d}{\mu} \]
To find the transition velocity ($v$), we rearrange the equation:
\[ v = \frac{Re_c \cdot \mu}{\rho \cdot d} \]

Step 2: Detailed Explanation:
Let us identify the parameters from the problem:
The transition Reynolds number is:
\[ Re_c = 2100 \]
The dynamic viscosity of water is:
\[ \mu = 1 \times 10^{-3} \text{ kg/m}\cdot\text{s} \]
The density of water is:
\[ \rho = 997 \text{ kg/m}^3 \]
The pipe diameter is 25 mm, which we convert to meters (SI units):
\[ d = 0.025 \text{ m} \]
Substituting these values into the rearranged formula to solve for the velocity ($v$):
\[ v = \frac{2100 \cdot (1 \times 10^{-3})}{997 \cdot 0.025} \]
First, calculate the numerator:
\[ 2100 \cdot 10^{-3} = 2.1 \]
Next, calculate the denominator:
\[ 997 \cdot 0.025 = 925 \]
Now, solve for the velocity ($v$):
\[ v = \frac{2.1}{925} \]
\[ v \approx 0.08425 \text{ m/s} \]
This rounds to 0.084 m/s.
Therefore, water flowing through this pipe at a velocity above 0.084 m/s will begin to transition from laminar to transitional flow.

Step 3: Final Answer
The flow converts from laminar to transitional at a velocity of 0.084 m/s.