What must be the pressure difference between the two ends of a 1.4 km section of pipe, 22 cm in diameter, if it is to transport oil (rho = 950 kg/m^3 ,eta = 0.20 Pa*s) at a rate of 780 cm^3/s? Please explain step-by-step.
Well, let's dive into the circus of fluid dynamics and see what we can find!
To calculate the pressure difference between the two ends of the pipe, we can use the Hagen-Poiseuille equation, which relates the pressure difference to the flow rate and the properties of the fluid and the pipe.
First, let's convert the given values into SI units:
- Pipe length, L = 1.4 km = 1400 m
- Pipe diameter, D = 22 cm = 0.22 m
- Oil density, ρ = 950 kg/m^3
- Oil viscosity, η = 0.20 Pa*s
- Flow rate, Q = 780 cm^3/s = 0.78*10^(-3) m^3/s
The Hagen-Poiseuille equation is given by:
ΔP = 8 * η * L * Q / (π * D^4)
Now, let's plug in the values and calculate the pressure difference:
ΔP = 8 * 0.20 Pa*s * 1400 m * 0.78*10^(-3) m^3/s / (π * (0.22 m)^4)
Simplifying this equation by evaluating the constants and doing some arithmetic, we get:
ΔP = 12.6726 Pa
So, the pressure difference between the two ends of the pipe must be approximately 12.6726 Pascal.
Now, I'm curious, do oil and clowns mix? I guess we'll never know unless we have a proper circus show with a clown slipping on some oil. Trust me, it's hilarious!
To find the pressure difference between the two ends of the pipe, we can use the Bernoulli's equation. This equation relates the pressure difference, density of fluid, velocity, and height difference across the pipe.
Step 1: Convert the given values to SI units.
- Distance (L) = 1.4 km = 1400 m
- Diameter (d) = 22 cm = 0.22 m
- Density of oil (rho) = 950 kg/m^3
- Viscosity of oil (eta) = 0.20 Pa*s
- Flow rate (Q) = 780 cm^3/s = 0.00078 m^3/s
Step 2: Calculate the cross-sectional area (A) of the pipe.
The formula for the area of a circle is A = π * r^2, where r is the radius of the pipe. Since the diameter (d) is given, we can find the radius (r) by dividing the diameter by 2.
r = d/2 = 0.22 m/2 = 0.11 m
Now, we can calculate the area using A = π * r^2.
A = π * (0.11 m)^2 = 0.0382 m^2
Step 3: Calculate the velocity (v) of the oil.
The velocity can be determined by dividing the flow rate (Q) by the cross-sectional area (A).
v = Q/A = 0.00078 m^3/s / 0.0382 m^2 = 0.0204 m/s
Step 4: Calculate the pressure difference (ΔP) using Bernoulli's equation.
The Bernoulli's equation states that the pressure difference (ΔP) is equal to the sum of the pressure difference due to elevation (ΔP_elev) and the pressure difference due to velocity (ΔP_vel).
ΔP = ΔP_elev + ΔP_vel
The ΔP_elev can be calculated as follows:
ΔP_elev = ρ * g * Δh
Where:
- ρ is the density of the fluid (950 kg/m^3)
- g is the acceleration due to gravity (9.8 m/s^2)
- Δh is the height difference across the pipe
The ΔP_vel can be calculated as follows:
ΔP_vel = 8 * η * L * Q / (π * d^4)
Where:
- η is the viscosity of the fluid (0.20 Pa*s)
- L is the length of the pipe (1400 m)
- Q is the flow rate (0.00078 m^3/s)
- d is the diameter of the pipe (0.22 m)
First, let's calculate ΔP_elev:
ΔP_elev = ρ * g * Δh
Since we have no information about the height difference (Δh), we assume it is negligible and set ΔP_elev = 0.
Now, let's calculate ΔP_vel:
ΔP_vel = 8 * η * L * Q / (π * d^4)
ΔP_vel = 8 * 0.20 Pa*s * 1400 m * 0.00078 m^3/s / (π * (0.22 m)^4)
Step 5: Calculate the pressure difference (ΔP).
ΔP = ΔP_elev + ΔP_vel
ΔP = 0 + ΔP_vel
Substitute the values and calculate:
ΔP = 8 * 0.20 Pa*s * 1400 m * 0.00078 m^3/s / (π * (0.22 m)^4)
Using a calculator, compute the value of ΔP.
The result will be the pressure difference (ΔP) required to transport oil through the 1.4 km section of pipe.
To find the pressure difference between the two ends of the pipe, we can use the Bernoulli's equation, which relates the pressure difference to the flow rate, pipe dimensions, fluid properties, and velocity.
1. Start by calculating the velocity of the fluid flowing through the pipe.
- The flow rate is given as 780 cm^3/s.
- Convert it to m^3/s: 780 cm^3/s = 780 * 10^(-6) m^3/s.
- The cross-sectional area of the pipe can be calculated using the formula for the area of a circle: π * (radius)^2. Since the diameter is 22 cm, the radius is 11 cm or 0.11 m. Thus, the area is π * (0.11 m)^2.
- Divide the flow rate by the area to find the velocity: velocity = flow rate / area.
2. Calculate the viscosity head loss.
- The viscosity (eta) of the oil is given as 0.20 Pa*s.
- The viscosity head loss is calculated using the equation: (8 * viscosity * length * velocity) / (π^2 * (radius)^4).
- In this case, the length is given as 1.4 km, so convert it to meters: 1.4 km = 1.4 * 1000 m.
3. Calculate the friction head loss.
- The friction head loss is calculated using the equation: (16 * viscosity * length * velocity) / (π^2 * (radius)^2 * density).
4. Add the viscosity head loss and the friction head loss to get the total head loss.
5. Apply Bernoulli's equation to find the pressure difference.
- Bernoulli's equation for incompressible flow is expressed as: pressure + 0.5 * density * velocity^2 + density * g * height = constant.
- In this case, we assume the height difference is negligible, and we are interested in the pressure difference.
- Rearrange the equation to solve for the pressure difference: pressure difference = 0.5 * density * velocity^2.
6. Calculate the pressure difference using the equation from step 5.
By following these steps, you should be able to find the pressure difference required to transport oil through the specified pipe.