Velocity of a flowing liquid

Question

Isopropanol with a density of ρ = 781 kgm3 flows uniformly through a tube that increases from a cross-sectional area A1 = 3.8 · 10−3 m2 to a cross-sectional area A2 = 3 A1. The static pressure difference between the narrow and the wide section of the pipe is 6500 Pa, the wide section is 31 cm below the narrow. At what speed does the isopropanol flow at the inlet?

Answer 1

We need to use Bernoulli's equation assuming the fluid is moving steadily and incompressible.

P₁ + (1/2)ρv₁² + ρgh₁ = P₂ + (1/2)ρv₂² + ρgh₂

Continuity equation helps us considering a steady flow:

A₁ x V₁ = A₂ x V₂

A₂ = 3A₁

Let's substutute these into Bernoulli's equation:

First we rearrange the equation as follows:

(1/2)ρv₁² - (1/2)ρv₂² = P₂ - P₁ + ρg(h₂ - h₁)

Substitute the numbers

0.5 x 781 x V₁² - 0.5 x 781 x (A₂/A₁) x V1² = 6500 + 781 x 9.81 x 0.31 where (A₂/A₁) = 3

Solve for V₁:

V₁ =~ 1.68 m/s