To find the Reynolds number for the flow of water through a pipe, we will follow these steps:
### Step 1: Convert the flow rate from liters per minute to cubic meters per second.
Given:
- Flow rate = 100 liters/minute
We know that:
- 1 liter = \(10^{-3}\) cubic meters
- 1 minute = 60 seconds
So, the flow rate in cubic meters per second is calculated as follows:
\[
\text{Flow rate} = 100 \, \text{liters/minute} \times \frac{10^{-3} \, \text{m}^3}{1 \, \text{liter}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = \frac{100 \times 10^{-3}}{60} \, \text{m}^3/\text{s} = \frac{1}{600} \, \text{m}^3/\text{s} \approx 0.00167 \, \text{m}^3/\text{s}
\]
### Step 2: Calculate the cross-sectional area of the pipe.
Given:
- Radius of the pipe, \(r = 5 \, \text{cm} = 0.05 \, \text{m}\)
The area \(A\) of the circular cross-section of the pipe is given by:
\[
A = \pi r^2 = \pi (0.05)^2 = \pi (0.0025) \approx 0.00785 \, \text{m}^2
\]
### Step 3: Calculate the velocity of the water flow.
The velocity \(V\) can be calculated using the formula:
\[
V = \frac{\text{Flow rate}}{A}
\]
Substituting the values we found:
\[
V = \frac{0.00167 \, \text{m}^3/\text{s}}{0.00785 \, \text{m}^2} \approx 0.212 \, \text{m/s}
\]
### Step 4: Calculate the diameter of the pipe.
The diameter \(d\) of the pipe is:
\[
d = 2r = 2 \times 0.05 \, \text{m} = 0.1 \, \text{m}
\]
### Step 5: Calculate the Reynolds number.
The Reynolds number \(Re\) is given by the formula:
\[
Re = \frac{\rho V d}{\eta}
\]
Where:
- \(\rho = 1000 \, \text{kg/m}^3\) (density of water)
- \(V \approx 0.212 \, \text{m/s}\) (velocity)
- \(d = 0.1 \, \text{m}\) (diameter)
- \(\eta = 1 \, \text{mPa s} = 1 \times 10^{-3} \, \text{Pa s}\)
Substituting the values:
\[
Re = \frac{1000 \times 0.212 \times 0.1}{1 \times 10^{-3}} = \frac{21.2}{1 \times 10^{-3}} = 21200
\]
### Step 6: Determine the order of the Reynolds number.
The Reynolds number is approximately \(21200\), which is of the order of \(2 \times 10^4\).
Thus, the Reynolds number for the flow is of the order of \(2 \times 10^4\).
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