Question

Air flows steadily along a streamline from point (1) to point (2) with negligible viscous effects. The following conditions are measured: At point (1) z₁ = 2 m and p₁ = 0 kPa; at point (2) z₂ = 10 m, p₂ = 20 N/m² and V₂ = 0. Determine the velocity at point (1). Also calculate the change in each form of energy in the Bernoulli equation and describe, in words, the energy gain or loss by each form as the air flows from point (1) to point (2).

Answer 1

The velocity at point (1) can be found by rearranging Bernoulli's equation to account for given pressure and elevation differences between the two points. Based on given values and Bernoulli's principle, the change in each form of energy along the streamline relates to the conservation of total mechanical energy.

Step-by-step explanation:

The student's question involves applying Bernoulli's equation to determine the velocity at point (1) where air flows steadily along a streamline, considering the different parameters given for two points. To find the velocity at point (1), we can rearrange Bernoulli's equation, which, when neglecting viscous effects and assuming an incompressible fluid, is given as: P1 + 1/2pv1² + pgh1 = P2 + 1/2pv2² + pgh2.

Because the question states that the velocity at point (2) is zero (V2 = 0), and assuming air density (ρ) is constant and atmospheric (standard air density can be used if not provided), the equation simplifies to:

P1 + pgh1 = P2 + pgh2

Plugging in the values:

(0 kPa + ρ * 9.81 m/s² * 2 m) = (20 N/m² + ρ * 9.81 m/s² * 10 m)

ρ cancels out since it's on both sides of the equation, and converting kPa to N/m², we can solve for P1 in N/m².

Now, in terms of energy changes, the elevation head increases, the pressure head increases at point (2), and the kinetic energy is zero at point (2) where the velocity is zero. All energy changes are interconnected by Bernoulli's principle - the total mechanical energy in a steady, incompressible, non-viscous flow is conserved.