Calculate the mass of water flowing in 10 minutes throuth a tube 1 mm in diameter and 0.40 m long ,if there is a constant pressure heat of 20 cm of water . Coefficient of visosity of water is 0.00089 Pas.

Answer

To solve the problem, we will use the Hagen-Poiseuille equation, which describes the flow of a viscous fluid through a cylindrical pipe. The equation is given by:

\[ Q = \frac{\pi P r^4}{8 \eta L} \]

where:
- \( Q \) = volume flow rate (m³/s)
- \( P \) = pressure difference (Pa)
- \( r \) = radius of the tube (m)
- \( \eta \) = coefficient of viscosity (Pa·s)
- \( L \) = length of the tube (m)

### Step 1: Convert the given values

1. **Convert diameter to radius:**
- Diameter \( d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Radius \( r = \frac{d}{2} = \frac{1 \times 10^{-3}}{2} = 0.5 \times 10^{-3} \, \text{m} = 5 \times 10^{-4} \, \text{m} \)

2. **Convert pressure head to pressure:**
- Pressure head \( h = 20 \, \text{cm} = 0.2 \, \text{m} \)
- Pressure \( P = \rho g h \)
- Density of water \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity \( g = 9.81 \, \text{m/s}^2 \)
- Thus, \( P = 1000 \times 9.81 \times 0.2 = 1962 \, \text{Pa} \)

3. **Given values:**
- Length of the tube \( L = 0.4 \, \text{m} \)
- Coefficient of viscosity \( \eta = 0.00089 \, \text{Pa·s} \)

### Step 2: Calculate the volume flow rate \( Q \)

Using the Hagen-Poiseuille equation:

\[ Q = \frac{\pi P r^4}{8 \eta L} \]

Substituting the values:

\[ Q = \frac{\pi \times 1962 \times (5 \times 10^{-4})^4}{8 \times 0.00089 \times 0.4} \]

Calculating \( r^4 \):

\[ (5 \times 10^{-4})^4 = 6.25 \times 10^{-16} \, \text{m}^4 \]

Now substituting back:

\[ Q = \frac{\pi \times 1962 \times 6.25 \times 10^{-16}}{8 \times 0.00089 \times 0.4} \]

Calculating the denominator:

\[ 8 \times 0.00089 \times 0.4 = 0.002848 \]

Now calculating \( Q \):

\[ Q = \frac{3.14159 \times 1962 \times 6.25 \times 10^{-16}}{0.002848} \]

Calculating \( Q \):

\[ Q \approx 1.3513 \times 10^{-7} \, \text{m}^3/\text{s} \]

### Step 3: Calculate the total volume in 10 minutes

Convert 10 minutes to seconds:

\[ t = 10 \times 60 = 600 \, \text{s} \]

Now calculate the total volume:

\[ V = Q \times t = 1.3513 \times 10^{-7} \times 600 \approx 8.1078 \times 10^{-5} \, \text{m}^3 \]

### Step 4: Calculate the mass of water

Using the density of water to find mass:

\[ m = \rho V = 1000 \times 8.1078 \times 10^{-5} \approx 0.081078 \, \text{kg} \approx 8.1 \times 10^{-2} \, \text{kg} \]

### Final Answer

The mass of water flowing through the tube in 10 minutes is approximately **0.081 kg** or **81 g**.

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