A straight horizontal pipe with a diameter of 1.0 cm and a length of 46 m carries oil with a coefficient of viscosity of 0.11 N · s/m2. At the output of the pipe, the flow rate is 8.4 10-5 m3/s and the pressure is 1.0 atmosphere. Find the gauge pressure at the pipe input.
Answer in atm
To find the gauge pressure at the pipe input, we need to use the equation for pressure difference in a fluid flowing through a pipe. The equation is:
ΔP = 4ηQ / πr^4
where:
ΔP is the pressure difference
η is the coefficient of viscosity
Q is the flow rate
r is the radius of the pipe, which is half of the diameter
First, let's convert the diameter into meters:
1.0 cm = 0.01 m
Next, calculate the radius:
r = 0.01 m / 2 = 0.005 m
Now, plug in the values into the equation:
ΔP = (4 * 0.11 N · s/m^2 * 8.4 * 10^-5 m^3/s) / (π * (0.005 m)^4)
Simplify the equation:
ΔP = (4 * 0.11 * 8.4 * 10^-5) / (π * 0.005^4) N/m^2
Now, convert the pressure into atmospheres:
1 N/m^2 = 1 Pa = 1.01325 × 10^-5 atm
So, the gauge pressure at the pipe input is:
ΔP = (4 * 0.11 * 8.4 * 10^-5) / (π * 0.005^4) * 1.01325 × 10^-5 atm
Calculating this value will give you the gauge pressure at the pipe input in atm.