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Answer :
The height difference between the outlet and inlet of the pipe is approximately 2.1 meters. The height difference between the outlet and inlet of the pipe, we can use Bernoulli's equation, which relates the pressure, velocity, and elevation of a fluid flowing in a pipe.
Bernoulli's equation states:
P₁ + (1/2)ρv₁² + ρgh₁ = P₂ + (1/2)ρv₂² + ρgh₂,
where P₁ and P₂ are the pressures at the inlet and outlet, respectively, ρ is the density of the fluid, v₁ and v₂ are the velocities at the inlet and outlet, h₁ and h₂ are the elevations at the inlet and outlet, and g is the acceleration due to gravity.
In this case, since the process is isothermal, there is no change in the fluid's internal energy. Therefore, the term (1/2)ρv₁² + ρgh₁ = (1/2)ρv₂² + ρgh₂ can be simplified as:
(1/2)ρv₁² + ρgh₁ = (1/2)ρv₂² + ρgh₂.
Since the height at the inlet is given as 0 (h₁ = 0), the equation becomes:
(1/2)ρv₁² = (1/2)ρv₂² + ρgh₂.
We can rearrange the equation to solve for the height difference (h₂ - h₁ = Δh):
Δh = (v₁² - v₂²) / (2g).
Given that the velocity at the inlet (v₁) is 1.2 m/s and the pressures at the inlet and outlet are 26000 Pa and 10000 Pa, respectively, we can use Bernoulli's equation to determine the velocity at the outlet (v₂) using the pressure difference:
P₁ + (1/2)ρv₁² = P₂ + (1/2)ρv₂².
Substituting the given values:
26000 + (1/2)ρ(1.2)² = 10000 + (1/2)ρv₂².
Simplifying and rearranging:
(1/2)ρv₂² = 26000 - 10000 + (1/2)ρ(1.2)².
Substituting the density of water (ρ = 1000 kg/m³):
(1/2)(1000)v₂² = 16000 + (1/2)(1000)(1.2)².
Simplifying and solving for v₂:
v₂ = √((16000 + 600) / 1000) ≈ 4.3 m/s.
Now we can substitute the values of v₁ = 1.2 m/s, v₂ = 4.3 m/s, and g = 9.8 m/s² into the equation for the height difference:
Δh = (1.2² - 4.3²) / (2 * 9.8) ≈ -2.1 m.
The negative sign indicates that the outlet of the pipe is 2.1 meters lower than the inlet.
Therefore, the height difference between the outlet and inlet of the pipe is approximately 2.1 meters.
Learn more about Bernoulli's equation here:
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