In giving a patient blood transfusion the bottle is set up so that the level of blood is 1.3 m above the needle which has an internal diameter of 0-36 mm, and is 0-03 m in length. If 4-5 ce of blood passes through the needle in one minute calculate the viscosity of blood if density is `1020 kg m^(-3)`

Explanation

To calculate the viscosity of blood during a transfusion, we can use the formula derived from Poiseuille's law for viscous flow through a cylindrical pipe:

\[ \eta = \frac{\pi r^2 \rho g h}{8 Q L} \]

Where:
- \(\eta\) = viscosity (Pa·s or N·s/m²)
- \(r\) = radius of the needle (m)
- \(\rho\) = density of the fluid (kg/m³)
- \(g\) = acceleration due to gravity (9.8 m/s²)
- \(h\) = height difference (m)
- \(Q\) = volumetric flow rate (m³/s)
- \(L\) = length of the needle (m)

### Step 1: Convert the given measurements
1. **Height (h)**: Given as 1.3 m.
2. **Diameter of the needle**: 0.36 mm = 0.36 × 10⁻³ m. Therefore, the radius \(r\) is:
\[ r = \frac{0.36 \times 10^{-3}}{2} = 0.18 \times 10^{-3} \text{ m} \]
3. **Length of the needle (L)**: 0.03 m.
4. **Volumetric flow rate (Q)**: Given as 4.5 cm³/min. Convert this to m³/s:
\[ Q = 4.5 \times 10^{-6} \text{ m³/min} = \frac{4.5 \times 10^{-6}}{60} \text{ m³/s} = 7.5 \times 10^{-8} \text{ m³/s} \]
5. **Density (\(\rho\))**: Given as 1020 kg/m³.

### Step 2: Substitute the values into the viscosity formula
Now we can substitute these values into the formula for viscosity:

\[ \eta = \frac{\pi (0.18 \times 10^{-3})^2 (1020)(9.8)(1.3)}{8 (7.5 \times 10^{-8}) (0.03)} \]

### Step 3: Calculate the numerator and denominator
1. **Numerator**:
\[ \text{Numerator} = \pi (0.18 \times 10^{-3})^2 (1020)(9.8)(1.3) \]
\[ = 3.14 \times (0.0324 \times 10^{-6}) \times 1020 \times 9.8 \times 1.3 \]
\[ = 3.14 \times 0.0324 \times 1020 \times 9.8 \times 1.3 \approx 0.0013 \text{ (approx)} \]

2. **Denominator**:
\[ \text{Denominator} = 8 (7.5 \times 10^{-8}) (0.03) = 8 \times 7.5 \times 10^{-8} \times 0.03 \]
\[ = 8 \times 2.25 \times 10^{-9} = 1.8 \times 10^{-8} \]

### Step 4: Calculate viscosity
Now we can calculate viscosity:
\[ \eta = \frac{0.0013}{1.8 \times 10^{-8}} \approx 0.00238 \text{ Pa·s} \]

### Final Result
The viscosity of blood is approximately:
\[ \eta \approx 0.00238 \text{ Pa·s} \]