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A concentrated solution of sugar dissolved in water is flowing through a capillary tube with an inside diameter of 2 mm. The capillary tube will be used to find the viscosity of this solution. The length of the capillary tube is 10 cm. The density of the solution is 1200 kg m⁻³. For a flow rate of 60 cm³ min⁻¹, the pressure drop per length of the tube was found to be 1.0 mmHg cm⁻¹.

What is the viscosity (Pa·s) of the fluid?

Answer :

To determine the viscosity of a sugar solution flowing through a capillary tube, we use Poiseuille's law and convert all units to the SI system. By substituting the given values into the Poiseuille law equation, we can solve for the viscosity of the fluid in Pascal seconds (Pa s).

To calculate the viscosity of a sugar solution flowing through a capillary tube, we can apply Poiseuille's law. This law relates the flow rate (Q), viscosity (η), length of the tube (L), pressure drop (ΔP), and radius of the tube (r) in the following equation for a cylindrical tube:

Q = (πr⁴ΔP) / (8ηL)

First, we need to consistently convert all units to the SI system. The flow rate Q is 60 cm³/min which is equivalent to 60 × 10⁻⁶ m³/min or 10⁻⁶ m³/s (since 1 min = 60 s). The pressure drop per length ΔP/L is 1.0 mmHg cm⁻¹. We convert this to Pa/m by recalling that 1 mmHg is approximately 133.3 Pa and 1 cm is 0.01 m, giving us ΔP/L = 133.3 Pa/0.01 m = 13330 Pa/m. The radius r of the tube is half its diameter, so r = 1 mm = 1 × 10⁻³ m. Lastly, the length L of the capillary tube is 10 cm = 0.1 m.

Substituting the values into the equation, we have:

10⁻⁶ = (π × (1 × 10⁻³)⁴ × 13330) / (8η × 0.1)

After rearranging the equation to solve for η, we obtain:

η = (π × (1 × 10⁻³)⁴ × 13330) / (8 × 10⁻⁶ × 0.1)

Finally, we calculate the viscosity to find out the value for η in Pa s.

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