Glycerine flows steadily through a horizontal tube of length 1.5 m and radius 1.0 cm. If the amount of glycerine collected per second at one end is 4.0 × 10 –3 kg s –1 , what is the pressure difference between the two ends of the tube? (Density of glycerine = 1.3 × 10 3 kg m –3 and viscosity of glycerine = 0.83 Pa s). [You may also like to check if the assumption of laminar flow in the tube is correct].
Answers
Given data:-
Length of the horizontal tube, l = 1.5 m
Radius of the tube, r = 1 cm = 0.01 m
Diameter of the tube, d = 2r = 0.02 m
Glycerine is flowing at a rate of 4.0 × 10–3 kg s–1.
M = 4.0 × 10–3 kg s–1
Density of glycerine, ρ = 1.3 × 103 kg m–3
Viscosity of glycerine, η = 0.83 Pa s
Volume of glycerine flowing per sec:
V = M / ρ
= 4 × 10-3 / (1.3 × 103) = 3.08 × 10-6 m3s-1
According to Poiseville’s formula, we have the relation for the rate of flow:
V = πpr4 / 8ηl
Where, p is the pressure difference between the two ends of the tube
∴ p = V8ηl / πr4
= 3.08 × 10-6 × 8 × 0.83 × 1.5 / [ π × (0.01)4 ]
= 9.8 × 10^2 Pa
Reynolds’ number is given by the relation:
R = 4pV / πdη
= 4 × 1.3 × 103 × 3.08 × 10-6 / ( π × 0.02 × 0.83)
= 0.3
Reynolds’ number is about 0.3. Hence, the flow is laminar.