A Venturi flow meter is used to measure the the flow velocity of a water main. The water main has a diameter of 40.0 cm, and the constriction has a diameter of 20.0 cm. The two vertical pipes are open at the top, and the difference in water level between them is 2.0 m. Find the velocity, vm (in m/s), and the volumetric flow rate, Q (in m3/s), of the water in the main.
vm=
Q=
To find the velocity, vm, and the volumetric flow rate, Q, of the water in the main using a Venturi flow meter, we can use the Bernoulli's equation, which relates the pressure difference and velocity of the fluid in a flow system.
The equation is:
P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2
where:
P1 and P2 are the pressures at two different points in the flow system,
ρ is the density of the fluid,
v1 and v2 are the velocities of the fluid at two different points,
h1 and h2 are the heights of the fluid at two different points,
g is the acceleration due to gravity.
In this case, the two points we are interested in are the points before and after the constriction.
Point 1: Before the constriction
- The diameter of the water main is 40.0 cm, which gives a radius (r1) of 20.0 cm or 0.2 m.
- The velocity at this point is vm.
- The height at this point is h1 = 0 (since both pipes are open to the atmosphere).
Point 2: After the constriction
- The diameter of the constriction is 20.0 cm, which gives a radius (r2) of 10.0 cm or 0.1 m.
- The velocity at this point is v2 = 0 (since the constriction is at its narrowest point, causing the velocity to increase).
- The height at this point is h2 = 2.0 m (given in the question).
Plugging in the values into the Bernoulli's equation:
P1 + 1/2ρvm^2 + 0 = P2 + 1/2ρ(0)^2 + ρg(2.0)
Since the pressure at both points is atmospheric pressure, P1 = P2 = 0, the equation becomes:
1/2ρvm^2 = ρg(2.0)
The density of water (ρ) is approximately 1000 kg/m^3, and the acceleration due to gravity (g) is approximately 9.8 m/s^2.
Simplifying the equation:
1/2 * 1000 * vm^2 = 1000 * 9.8 * 2.0
Dividing by 1000:
1/2 * vm^2 = 9.8 * 2.0
Multiplying by 2:
vm^2 = 19.6
Taking the square root:
vm = √19.6
Calculating the velocity (approximate):
vm ≈ 4.43 m/s
The velocity, vm, is approximately 4.43 m/s.
To find the volumetric flow rate, Q, we can use the formula:
Q = A1 * v1
where:
A1 is the cross-sectional area of the water main at point 1, given by πr1^2,
v1 is the velocity at point 1, which is equal to vm.
Plugging in the values:
A1 = π(0.2)^2 = π * 0.04
Q = π * 0.04 * 4.43
Calculating Q:
Q ≈ 0.56 m^3/s
The volumetric flow rate, Q, is approximately 0.56 m^3/s.