A horizontal pipe carries oil whose coefficient of viscosity is 0.00010 ns/ m^{2}}. The diameter of the pipe is 5.5 cm, and its length is 60 m.
What pressure difference is required between the ends of this pipe if the oil is to flow with an average speed of 1.2 m/s?
What is the volume flow rate in this case?
To determine the pressure difference required for the oil to flow with the given average speed, you can use the Poiseuille's law equation:
Q = (πΔPd^4)/128μL
Where:
Q = Volume flow rate
ΔP = Pressure difference
d = Diameter of the pipe
μ = Coefficient of viscosity
L = Length of the pipe
First, convert the diameter of the pipe from centimeters to meters:
d = 5.5 cm = 0.055 m
Substitute the values into the equation:
1.2 = (πΔP(0.055^4))/(128(0.00010)(60))
Simplify the equation:
1.2 = (πΔP(0.00001763))/(7.68x10^(-6))
Multiply both sides by 7.68x10^(-6) to isolate ΔP:
ΔP = (1.2 x 7.68x10^(-6))/(π(0.00001763))
ΔP ≈ 0.0258 Pa
Therefore, a pressure difference of approximately 0.0258 Pa is required for the oil to flow with an average speed of 1.2 m/s.
To calculate the volume flow rate, substitute the pressure difference into the Poiseuille's law equation:
Q = (π(0.0258)(0.055^4))/(128(0.00010)(60))
Calculate the volume flow rate:
Q ≈ 8.718x10^(-6) m^3/s
Therefore, the volume flow rate in this case is approximately 8.718x10^(-6) m^3/s.
To find the pressure difference required for the oil to flow with a given average speed, we can use the equation for the volume flow rate of a fluid through a pipe:
Q = (π * r^4 * ∆P) / (8 * η * L),
where:
Q is the volume flow rate,
π is the mathematical constant pi (approximately 3.14159),
r is the radius of the pipe,
∆P is the pressure difference across the pipe,
η is the coefficient of viscosity of the oil, and
L is the length of the pipe.
First, let's convert the diameter of the pipe to radius:
radius (r) = diameter / 2 = 5.5 cm / 2 = 2.75 cm = 0.0275 m.
The volume flow rate (Q) is given as 1.2 m/s.
Rearranging the above equation, we can solve for ∆P:
∆P = (Q * 8 * η * L) / (π * r^4).
Plugging in the values:
∆P = (1.2 m/s * 8 * 0.00010 ns/m^2 * 60 m) / (π * (0.0275 m)^4).
Now, we need to solve this equation using the appropriate order of operations.
The pressure difference (∆P) needed for the oil to flow with an average speed of 1.2 m/s can be calculated to find the answer.
To determine the volume flow rate in this case, we can use the formula:
Q = A * v,
where:
Q is the volume flow rate,
A is the cross-sectional area of the pipe, and
v is the average velocity of the fluid.
The cross-sectional area (A) can be calculated using the formula:
A = π * r^2,
where:
r is the radius of the pipe.
Plugging in the values:
A = π * (0.0275 m)^2.
Finally, we can calculate the volume flow rate by multiplying the cross-sectional area by the average velocity:
Q = (π * (0.0275 m)^2) * 1.2 m/s.