The rate of water gushing out of a pipe of radius 5 cm is `100 L m i n^(-1)`. The Reynolds number for the flow is of the orderof [density of water = `1000 kg m^(-3)` ,coefficient of viscosity of water = 1m Pa s ]

Options

A. `10^(6)`
B. `10^(4)`
C. `10^(3)`
D. `10^(2)`

Answer

To calculate the Reynolds number for the flow of water through a pipe, we can follow these steps:

### Step 1: Convert the given flow rate to SI units
The flow rate is given as \(100 \, \text{L/min}\). We need to convert this to cubic meters per second (m³/s).

1. **Convert liters to cubic meters**:
\[ 100 \, \text{L} = \frac{100}{1000} \, \text{m}^3 = 0.1 \, \text{m}^3 \]

2. **Convert minutes to seconds**:
\[ 1 \, \text{min} = 60 \, \text{s} \]

3. **Combine the conversions**:
\[ Q = \frac{0.1 \, \text{m}^3}{60 \, \text{s}} = \frac{1}{600} \, \text{m}^3/\text{s} \]

### Step 2: Calculate the cross-sectional area of the pipe
The radius of the pipe is given as \(5 \, \text{cm}\). We need to convert this to meters and then calculate the area.

1. **Convert radius to meters**:
\[ r = 5 \, \text{cm} = \frac{5}{100} \, \text{m} = 0.05 \, \text{m} \]

2. **Calculate the area**:
\[ A = \pi r^2 = \pi (0.05)^2 = \pi (0.0025) \approx 0.00785 \, \text{m}^2 \]

### Step 3: Calculate the velocity of the flow
Using the flow rate and the area, we can find the velocity of the water.

\[ v = \frac{Q}{A} = \frac{\frac{1}{600}}{0.00785} \approx 0.0253 \, \text{m/s} \]

### Step 4: Calculate the Reynolds number
The Reynolds number \(Re\) is given by the formula:

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

Where:
- \(\rho\) = density of water = \(1000 \, \text{kg/m}^3\)
- \(v\) = velocity = \(0.0253 \, \text{m/s}\)
- \(D\) = diameter of the pipe = \(2r = 2 \times 0.05 = 0.1 \, \text{m}\)
- \(\mu\) = coefficient of viscosity = \(1 \, \text{m Pa s} = 1 \, \text{kg/(m s)}\)

Substituting the values:

\[ Re = \frac{1000 \times 0.0253 \times 0.1}{1} = 2.53 \times 10^2 = 253 \]

### Step 5: Determine the order of the Reynolds number
The Reynolds number we calculated is approximately \(253\), which is of the order of \(10^2\).

### Final Answer
The Reynolds number for the flow is of the order of \(10^2\).

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