If d is 10% smaller than D, how do the pressure drops experienced by the fluid across region I and region II compare?

   ΔP = - 0.262 ρ v₁²

Explanation:

This is a fluid mechanics problem, let's use the subscript for region I and the subscript for region II. Let's write Bernoulli's equation

         P₁ + ½ ρ v₁² + ρ g y₁ = P₂ + ½ ρ v₂² + ρ g y₂

suppose the pipe is horizontal, therefore

           y₁ = y₂

         P₁ -P₂ = ½ ρ (v₁² -v₂²)             1

Now let's use the continuity equation

         v₁ A₁ = v₂ A₂                              2

The area of a circle is

         A = π r²

also the radius is half the diameter r = d /2

         A₁ = π d₁² / 4

let's substitute in equation 2

         v1 π d₁² / 4 = v2 π d₂² / 4

          v₁ d₁² = v₂ d₂²

          v₂ = v₁ d₁² / d₂²

let's substitute in equation 1

          P₁ - P₂ = ½ ρ (v₁² - v₁² (d₁² / d₂²)² )

Now let's use that the diameter d₂ = d is 10% smaller than the larger diameter d₁ = D, we assume that it is in region 1

           d₂ = D -0.1D = 0.9 D

we substitute in the previous equation

          P₁ - P₂ = ½ ρ v₁² (1 - (D / 0,9D)⁴)

          P₁- P₂ = ½ ρ v₁² (1 - 1 / 0,9⁴ )

          P1 - P2 = ½ ρ v12 (- 0.524)

          ΔP = - 0.262 ρ v₁²

in this solution we assume that the data in zone I is known

answer: he got 23

explanation:

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explanation: