What pressure difference is required between the ends of a 2.0m long, 1.0mm diameter horizontal tube for 40 degrees Celsius water to flow through it at an average speed of 4.0 m/s ?
In order to determine the pressure difference required for water to flow through a tube at a specific speed, we need to consider the principles of fluid dynamics, specifically Bernoulli's equation. Bernoulli's equation relates the pressure, velocity, and height of a fluid within a system.
The formula for Bernoulli's equation is as follows:
P₁ + (1/2)ρv₁² + ρgh₁ = P₂ + (1/2)ρv₂² + ρgh₂
Where:
P₁ and P₂ are the pressures at points 1 and 2,
ρ is the density of the fluid (in this case, water),
v₁ and v₂ are the velocities at points 1 and 2,
g is the acceleration due to gravity, and
h₁ and h₂ are the heights at points 1 and 2 (in this horizontal scenario, the heights are the same).
Since the tube is horizontal, the height difference (γ₁ - γ₂) is zero, which means we can ignore the terms ρgh₁ and ρgh₂ in Bernoulli's equation.
The equation, with the modifications for the horizontal tube, simplifies to:
P₁ + (1/2)ρv₁² = P₂ + (1/2)ρv₂²
Given the information provided in the question, we need to solve for the pressure difference, which is (P₂ - P₁). Let's rearrange the equation to isolate the pressure difference:
(P₂ - P₁) = (1/2)ρv₁² - (1/2)ρv₂²
Now, let's substitute the known values into the equation:
P₂ - P₁ = (1/2)ρv₁² - (1/2)ρv₂²
P₂ - P₁ = (1/2)ρ(v₁² - v₂²)
We have the necessary information to calculate the pressure difference. The missing values are the density of water (ρ) and the velocities (v₁ and v₂).
The density of water is approximately 1000 kg/m³.
The diameter of the tube is given as 1.0 mm, which means the radius (r) is 0.5 mm or 0.0005 m (since radius = diameter / 2).
From the average speed of 4.0 m/s, we can determine the velocities as follows:
v₁ = 4.0 m/s (since it represents the average speed at point 1),
v₂ = 0 m/s (since the water is not flowing at point 2).
Now, let's calculate the pressure difference:
P₂ - P₁ = (1/2)ρ(v₁² - v₂²)
P₂ - P₁ = (1/2)(1000 kg/m³)(4.0 m/s)² - (1/2)(1000 kg/m³)(0 m/s)²
P₂ - P₁ = 2000 Pa
Therefore, a pressure difference of 2000 Pascal (Pa) is required between the ends of the tube for 40 degrees Celsius water to flow through it at an average speed of 4.0 m/s.