Water flows horizontally in a tapered pipe at a rate of 500 m^3/h. The area at one section of the pipe is 6500 mm² and 4000 mm² at another section. Calculate the gauge pressure at the larger section if the pressure at the smaller section is atmospheric.
To calculate the gauge pressure at the larger section of the pipe, we need to consider the continuity equation and use Bernoulli's equation.
1. Let's start by converting the flow rate from cubic meters per hour to cubic meters per second:
Flow rate = 500 m^3/h = (500/3600) m^3/s = 0.139 m^3/s
2. Next, we need to calculate the velocity at the smaller section of the pipe using the equation:
Velocity = Flow rate / Area
Velocity at smaller section = 0.139 m^3/s / (6500 mm^2 * 1 m^2/1,000,000 mm^2)
Velocity at smaller section = 0.139 m^3/s / 0.0065 m^2 = 21.385 m/s (approximately)
3. Now, using Bernoulli's equation, we can relate the velocities and pressures at the two sections of the pipe. Bernoulli's equation states that the sum of the pressure energy, kinetic energy, and potential energy along a streamline remains constant.
At the smaller section:
Pressure + 0.5 * density * velocity^2 = atmospheric pressure + 0.5 * density * velocity^2
Pressure at smaller section = atmospheric pressure
At the larger section:
Pressure + 0.5 * density * velocity^2 = Pressure at smaller section + 0.5 * density * velocity^2
4. Since the pressure at the smaller section is atmospheric (which is usually taken as 0 gauge pressure), we can simplify the equation:
Pressure + 0.5 * density * velocity^2 = 0 + 0.5 * density * velocity^2
This simplifies to:
Pressure = -0.5 * density * velocity^2
5. To calculate the pressure at the larger section, we need the density of water. The density of water is approximately 1000 kg/m^3.
Pressure = -0.5 * 1000 kg/m^3 * (21.385 m/s)^2
Pressure = -0.5 * 1000 kg/m^3 * 458.421 m^2/s^2
6. Simplifying the equation, we find that the gauge pressure at the larger section is approximately 232012.5 Pa (or 232.0125 kPa). Note that gauge pressure measures the pressure relative to atmospheric pressure.
Therefore, the gauge pressure at the larger section of the pipe is approximately 232.0125 kPa.
To calculate the gauge pressure at the larger section of the pipe, we need to use the principle of continuity and Bernoulli's equation.
1. Start by converting the flow rate from m^3/h to m^3/s. Since there are 3600 seconds in an hour, divide the flow rate by 3600:
500 m^3/h ÷ 3600 s/h = 0.1389 m^3/s
2. Convert the areas of the two sections from mm² to m². Since there are 1,000,000 mm² in a m², divide each area by 1,000,000:
6500 mm² ÷ 1,000,000 m²/mm² = 0.0065 m²
4000 mm² ÷ 1,000,000 m²/mm² = 0.0040 m²
3. Apply the principle of continuity, which states that the mass flow rate of an incompressible fluid remains constant along a flow path. The mass flow rate is given by the formula:
m = ρAv
where m is the mass flow rate, ρ is the density of water (which we can assume to be 1000 kg/m³), A is the cross-sectional area, and v is the fluid velocity.
Since the density and flow rate are constant, we can write the equation as:
A₁v₁ = A₂v₂
where A₁ and v₁ are the area and velocity at the smaller section, and A₂ and v₂ are the area and velocity at the larger section.
4. Rearrange the equation to solve for velocity at the larger section:
v₂ = (A₁v₁) / A₂
v₂ = (0.0065 m² * 0.1389 m^3/s) / 0.0040 m²
v₂ ≈ 0.224 m/s
5. Using Bernoulli's equation, which states that the total energy of a fluid remains constant along a streamline, we can calculate the gauge pressure difference between the two sections. Neglecting height differences and assuming no energy losses, Bernoulli's equation can be simplified to:
P₁ + 0.5ρv₁² = P₂ + 0.5ρv₂²
Since the pressure at the smaller section is atmospheric (which we can assume to be 101,325 Pa), the equation becomes:
101,325 Pa + 0.5 * 1000 kg/m³ * v₁² = P₂ + 0.5 * 1000 kg/m³ * v₂²
6. Rearrange the equation to solve for the gauge pressure at the larger section:
P₂ = 101,325 Pa + 0.5 * 1000 kg/m³ * v₁² - 0.5 * 1000 kg/m³ * v₂²
P₂ ≈ 101,325 Pa + 0.5 * 1000 kg/m³ * (0.1389 m^3/s)² - 0.5 * 1000 kg/m³ * (0.224 m/s)²
P₂ ≈ 101,325 Pa + 9.176 Pa - 20.096 Pa
P₂ ≈ 90,404 Pa
Therefore, the gauge pressure at the larger section of the pipe is approximately 90,404 Pa.