Calculate the average velocity of water in a tube of diameter 0.3 m so that the flow is (i) laminar (ii) turbulent. The viscosity of water is `10^(-3)" Nm"^(-2)s`.

Answer

To calculate the average velocity of water in a tube of diameter 0.3 m for both laminar and turbulent flow, we will use the Reynolds number formula:

\[ Re = \frac{\rho V D}{\mu} \]

Where:
- \(Re\) is the Reynolds number,
- \(\rho\) is the density of the fluid (for water, \(\rho = 1000 \, \text{kg/m}^3\)),
- \(V\) is the average velocity,
- \(D\) is the diameter of the tube,
- \(\mu\) is the dynamic viscosity of the fluid (for water, \(\mu = 10^{-3} \, \text{N m}^{-2} \text{s}\)).

### Step 1: Determine the Reynolds number for laminar flow
For laminar flow, the critical Reynolds number is typically taken as 2000.

### Step 2: Rearrange the Reynolds number formula to solve for average velocity
We can rearrange the formula to find the average velocity \(V\):

\[ V = \frac{Re \cdot \mu}{\rho \cdot D} \]

### Step 3: Substitute values for laminar flow
Now, substituting the known values for laminar flow:

- \(Re = 2000\)
- \(\mu = 10^{-3} \, \text{N m}^{-2} \text{s}\)
- \(\rho = 1000 \, \text{kg/m}^3\)
- \(D = 0.3 \, \text{m}\)

\[ V = \frac{2000 \cdot 10^{-3}}{1000 \cdot 0.3} \]

### Step 4: Calculate average velocity for laminar flow
Calculating this gives:

\[ V = \frac{2000 \cdot 10^{-3}}{300} = \frac{2}{3} \cdot 10^{-2} = 6.67 \times 10^{-3} \, \text{m/s} \]

### Step 5: Determine the Reynolds number for turbulent flow
For turbulent flow, the critical Reynolds number is typically taken as 3000.

### Step 6: Substitute values for turbulent flow
Now, substituting the known values for turbulent flow:

- \(Re = 3000\)

\[ V = \frac{3000 \cdot 10^{-3}}{1000 \cdot 0.3} \]

### Step 7: Calculate average velocity for turbulent flow
Calculating this gives:

\[ V = \frac{3000 \cdot 10^{-3}}{300} = 10 \cdot 10^{-2} = 0.01 \, \text{m/s} \]

### Final Results
- Average velocity for laminar flow: \(6.67 \times 10^{-3} \, \text{m/s}\)
- Average velocity for turbulent flow: \(0.01 \, \text{m/s}\)