Water is flowing in a pipe as depicted in the figure. As shown, p = 113 kPa, d = 4.57 cm, d' = 3.22 cm, v = 4.40 m/s, and h = 1.25 m. What pressure is indicated on the upper pressure gauge? Take y = 0 at the bottom of the left pipe so that the center of the left pipe is y1 = d/2.
To determine the pressure indicated on the upper pressure gauge, we can use Bernoulli's principle, which states that the total energy of a fluid flowing through a pipe is constant.
The formula for Bernoulli's principle is:
P1 + 0.5 * ρ * v1^2 + ρ * g * h1 = P2 + 0.5 * ρ * v2^2 + ρ * g * h2
Where:
P1 and P2 are the pressures at two different points in the fluid flow.
ρ is the density of the fluid.
v1 and v2 are the velocities at the two points.
g is the acceleration due to gravity.
h1 and h2 are the heights of the fluid above some reference point.
In this case, the known values are:
P1 = 113 kPa (given)
d = 4.57 cm
d' = 3.22 cm
v = 4.40 m/s
h = 1.25 m
y1 = d/2 = 4.57 cm/2 = 2.285 cm = 0.02285 m
First, let's calculate the velocity v2 at the upper point, using the principle of continuity, which states that the mass flow rate of a fluid remains constant between two points in a pipe of constant cross-sectional area.
According to the principle of continuity, the equation is:
A1 * v1 = A2 * v2
Where:
A1 and A2 are the cross-sectional areas at two different points in the fluid flow.
To find the cross-sectional areas, we'll use the formula for the area of a circle:
A = π * r^2
Where r is the radius of the pipe.
Given:
d = 4.57 cm
d' = 3.22 cm
We can calculate the radii:
r1 = d/2 = 4.57 cm/2 = 2.285 cm = 0.02285 m
r2 = d'/2 = 3.22 cm/2 = 1.61 cm = 0.0161 m
Now, we can find the cross-sectional areas:
A1 = π * r1^2
A2 = π * r2^2
Next, we'll calculate the velocity v2:
v2 = (A1 * v1) / A2
Now that we have the velocity v2, we can use Bernoulli's principle to find the pressure P2 at the upper point:
P2 = P1 + 0.5 * ρ * (v1^2 - v2^2) + ρ * g * (h1 - h2)
Substituting the given values and the calculated values, we can solve for P2.