Castor oil, which has a density of 0.96×103kg/m3 at room temperature, is forced through
a pipe of circular cross section by pump that maintains a gauge pressure of 950 pa. The
pipe as a diameter of 2.6 cm and a length of 65 cm. The castor oil emerging from the free
end of the pipe and at atmospheric pressure is collected. After 90 s, a total of 1.23 kg has
been collected. What is the coefficient of viscosity of castor oil at this temperature?

To determine the coefficient of viscosity of castor oil under the given conditions, we can use the principles of fluid dynamics, specifically the Hagen-Poiseuille equation, which relates the flow rate of a viscous fluid through a pipe to the pressure difference, the viscosity, and the dimensions of the pipe.

Step-by-Step Calculation

1. Gather the Given Information

We have the following data:

• Density of castor oil, ρ = 0.96 × 10³ kg/m³
• Gauge pressure, ΔP = 950 Pa
• Diameter of the pipe, d = 2.6 cm = 0.026 m
• Length of the pipe, L = 65 cm = 0.65 m
• Mass of castor oil collected, m = 1.23 kg
• Time, t = 90 s

2. Calculate the Flow Rate (Q)

The flow rate (Q) can be calculated using the mass of the fluid collected over time:

Q = m / t = 1.23 kg / 90 s = 0.01367 kg/s

3. Convert Flow Rate to Volume Flow Rate (Q_v)

Using the density of castor oil, we can convert the mass flow rate to volume flow rate:

Q_v = Q / ρ = 0.01367 kg/s / (0.96 × 10³ kg/m³) = 1.42 × 10^-5 m³/s

4. Calculate the Radius of the Pipe

The radius (r) of the pipe is half of the diameter:

r = d / 2 = 0.026 m / 2 = 0.013 m

5. Apply the Hagen-Poiseuille Equation

The Hagen-Poiseuille equation for laminar flow is given by:

Q_v = (πr⁴ΔP) / (8ηL)

Where η is the dynamic viscosity. Rearranging this equation to solve for η gives:

η = (πr⁴ΔP) / (8Q_vL)

6. Substitute the Values

Now we can substitute the known values into the equation:

η = (π(0.013 m)⁴ × 950 Pa) / (8 × 1.42 × 10^-5 m³/s × 0.65 m)

7. Perform the Calculations

Calculating the numerator:

Numerator = π × (0.013⁴) × 950 ≈ 5.67 × 10^-6 Pa·m⁴

Calculating the denominator:

Denominator = 8 × 1.42 × 10^-5 × 0.65 ≈ 7.38 × 10^-5 m³·s

Now, substituting these values into the viscosity equation:

η = (5.67 × 10^-6) / (7.38 × 10^-5) ≈ 0.077 mPa·s

Final Result

The coefficient of viscosity of castor oil at this temperature is approximately 0.077 mPa·s. This value indicates how resistant the castor oil is to flow under the given conditions, which is essential for various applications in both industrial and culinary contexts.