A 6.2 cm diameter pipe gradually narrows to 4.0 cm. When water flows through this pipe at a certain rate, the gauge pressure in these two sections is 30.0 kPa and 24.0 kPa, respectively. What is the volume rate of flow?
I think quastion shoud be rong because when the area will decrease due to cross section the pressure shoud increase not a decrease. can you got my point
To calculate the volume rate of flow, we can use the principle of continuity, which states that the volume flow rate of an incompressible fluid is constant at any point in a pipe.
The volume flow rate, Q, is given by the equation Q = A * v, where A is the cross-sectional area of the pipe and v is the velocity of the fluid.
First, let's find the cross-sectional area at each section of the pipe.
The area of a circular cross-section is given by the equation A = π * r^2, where r is the radius of the cross-section.
In the first section where the diameter is 6.2 cm, the radius (r1) is half of the diameter, so r1 = 6.2 cm / 2 = 3.1 cm = 0.031 m.
The area (A1) of the first section is A1 = π * (r1)^2 = π * (0.031 m)^2.
In the second section where the diameter is 4.0 cm, the radius (r2) is half of the diameter, so r2 = 4.0 cm / 2 = 2.0 cm = 0.02 m.
The area (A2) of the second section is A2 = π * (r2)^2 = π * (0.02 m)^2.
Now, let's find the velocity of the fluid in each section using Bernoulli's equation. Bernoulli's equation states that the sum of pressure energy, potential energy, and kinetic energy per unit volume is constant along streamline flow of an incompressible, nonviscous fluid.
Using Bernoulli's equation, we have:
P1 + 1/2 ρ v1^2 = P2 + 1/2 ρ v2^2
Where P1 and P2 are the gauge pressures at each section, ρ is the density of the fluid, v1 and v2 are the velocities of the fluid at each section.
Since the fluid is water, we can assume a constant density of ρ = 1000 kg/m^3.
Rearranging the equation to solve for the velocity at each section, we get:
v1^2 = (2(P2 - P1) / ρ) + v2^2
v1 = sqrt((2(P2 - P1) / ρ) + v2^2)
Now, we can calculate the velocity at the second section:
v2 is not given directly, but it is the same as the velocity at the first section (v1). Therefore, we can substitute v1 into the equation to find v2:
v2 = sqrt((2(P1 - P2) / ρ) + v1^2)
With the velocities at each section obtained, we can now calculate the volume flow rate:
Q = A1 * v1 = A2 * v2
Substituting the values we calculated earlier for A1, A2, v1, and v2, we can solve for Q, the volume flow rate.