In the 2-tiered pipe shown, the area of the lower pipe is 50 cm2, the pressure in the lower pipe is 400 kPa, and the velocity of the water flowing through the lower pipe is 0.5 m/s. If the height difference between the lower and upper pipe is 12 m and the area of the upper pipe is 10 cm2, what is the water pressure in the upper pipe? Ignore the viscosity of water and assume the water always fills the pipe. (The density of water is 1000 kg/m3.)
To find the water pressure in the upper pipe, we can use the principle of conservation of energy in fluid dynamics, known as Bernoulli's equation, which states:
P1 + 0.5 * density * v1^2 + density * g * h1 = P2 + 0.5 * density * v2^2 + density * g * h2
Where:
P1 = Pressure in the lower pipe
v1 = Velocity of water flowing through the lower pipe
h1 = Height of the lower pipe
P2 = Pressure in the upper pipe (which we want to find)
v2 = Velocity of water flowing through the upper pipe (we assume it to be 0 m/s)
h2 = Height of the upper pipe
density = Density of water
g = Acceleration due to gravity
First, let's calculate the velocity of water flowing through the upper pipe:
A1 * v1 = A2 * v2
50 cm^2 * 0.5 m/s = 10 cm^2 * v2
v2 = (50 cm^2 * 0.5 m/s) / 10 cm^2
v2 = 2.5 m/s
Now, let's plug in the values into Bernoulli's equation:
400 kPa + 0.5 * 1000 kg/m^3 * (0.5 m/s)^2 + 1000 kg/m^3 * 9.81 m/s^2 * 12 m = P2 + 0.5 * 1000 kg/m^3 * (2.5 m/s)^2 + 1000 kg/m^3 * 9.81 m/s^2 * 0 m
400 kPa + 0.25 kPa + 1177.2 kPa = P2 + 3.125 kPa + 0 kPa
1577.45 kPa = P2
Therefore, the water pressure in the upper pipe is 1577.45 kPa.