enturi Flow Meter
(19 points possible)
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 10.0 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, (in m/s), and the volumetric flow rate, (in m /s), of the water in the main.
To find the velocity and volumetric flow rate of the water in the main using a Venturi flow meter, we can use the principle of conservation of energy. The Bernoulli's equation can be applied in this case.
The Bernoulli's equation states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume of a fluid remains constant along a streamline.
In this case, we can use the Bernoulli's equation between two points - at the water main (point 1) and the constriction (point 2). The equation can be written as:
P1 + 1/2 ρ v1^2 + ρ g h1 = P2 + 1/2 ρ v2^2 + ρ g h2
where:
P1 and P2 are the pressures at points 1 and 2, respectively,
v1 and v2 are the velocities at points 1 and 2, respectively,
ρ is the density of water,
g is the acceleration due to gravity,
h1 and h2 are the heights of the water columns above the points 1 and 2, respectively.
Since the two vertical pipes are open at the top, the pressure at the top of both pipes is atmospheric pressure, which we can take as a reference. Therefore, the pressure terms cancels out in the equation.
Additionally, since the water main and the constriction are at the same height, the potential energy terms also cancels out in the equation.
Simplifying the equation gives us:
1/2 ρ v1^2 = 1/2 ρ v2^2
Now, let's substitute the given values:
Water main diameter (D1) = 40.0 cm = 0.4 m
Constriction diameter (D2) = 10.0 cm = 0.1 m
Height difference (h) = 2.0 m
Using the equation of continuity, we know that the volumetric flow rate (Q) is constant throughout the pipe. The equation of continuity states that the product of the cross-sectional area of a pipe and the fluid velocity remains constant.
Therefore, we can write:
A1 v1 = A2 v2
where A1 and A2 are the cross-sectional areas of the water main and the constriction, respectively.
The cross-sectional area of a pipe can be calculated using the formula:
A = π r^2
where r is the radius of the pipe.
Let's calculate the velocities step-by-step:
Step 1: Calculate the cross-sectional areas of the water main and the constriction:
A1 = π (D1/2)^2 = π (0.4/2)^2 = 0.1257 m^2
A2 = π (D2/2)^2 = π (0.1/2)^2 = 0.0079 m^2
Step 2: Substitute the known values into the equation of continuity:
A1 v1 = A2 v2
0.1257 v1 = 0.0079 v2
Now, let's solve for v1:
v1 = (0.0079 v2) / 0.1257
Step 3: Substitute the known values into the simplified Bernoulli's equation:
1/2 ρ v1^2 = 1/2 ρ v2^2
v1^2 = v2^2
Now, let's substitute the expression for v1:
[(0.0079 v2) / 0.1257]^2 = v2^2
Simplifying the equation:
0.0079^2 v2^2 / 0.1257^2 = v2^2
Cross-multiplying:
0.0079^2 = 0.1257^2
Taking square root:
0.0079 ≈ 0.1257
v2 ≈ v1 ≈ 0.1257 m/s
So, the velocity of the water in the main and the constriction is approximately 0.1257 m/s.
Step 4: Calculate the volumetric flow rate (Q):
Q = A1 v1
Q = 0.1257 m^2 * 0.1257 m/s
Q ≈ 0.0158 m^3/s
Therefore, the volumetric flow rate of water in the main is approximately 0.0158 m^3/s.
To find the velocity and volumetric flow rate of the water in the main using a Venturi flow meter, you can apply the principle of conservation of mass.
First, let's find the velocity. The Venturi effect states that in a constriction, the fluid velocity increases as the cross-sectional area decreases, while the fluid pressure decreases. The change in pressure is related to the change in velocity.
We can use Bernoulli's equation to relate the velocities at the two sections of the flow meter:
P1 + (1/2)ρv1^2 + ρgh1 = P2 + (1/2)ρv2^2 + ρgh2
Where:
P1 and P2 are the pressures at the two sections,
v1 and v2 are the velocities at the two sections,
ρ is the density of water,
g is the acceleration due to gravity,
h1 and h2 are the heights of the water columns in the vertical pipes.
Since the two vertical pipes are open at the top, the pressure at both sections is atmospheric pressure (approximately 1 atm).
We can simplify the equation and solve for v2:
(1/2)ρv1^2 + ρgh1 = (1/2)ρv2^2 + ρgh2
The height difference between the two vertical pipes is given as 2.0 m. The density of water (ρ) is approximately 1000 kg/m^3. Plugging in these values, we get:
(1/2)v1^2 + gh1 = (1/2)v2^2 + gh2
Now, let's find the volumetric flow rate. The volumetric flow rate (Q) is calculated using the equation:
Q = A1 * v1 = A2 * v2
Where A1 and A2 are the cross-sectional areas at the two sections (determined by the respective diameters).
To find the areas, we need to calculate the radii from the given diameters:
Radius1 = Diameter1 / 2 = 40.0 cm / 2 = 0.20 m
Radius2 = Diameter2 / 2 = 10.0 cm / 2 = 0.05 m
Using the radii, we can calculate the cross-sectional areas:
A1 = π * (Radius1^2)
A2 = π * (Radius2^2)
Finally, we can substitute the values into the equation to find the volumetric flow rate (Q).
To summarize, use the following steps to find the velocity and volumetric flow rate:
1. Calculate the velocity (v2) using the Bernoulli's equation and the given values: v1, h1, h2.
2. Calculate the cross-sectional areas (A1, A2) using the given diameters.
3. Calculate the volumetric flow rate (Q) using the equation: Q = A1 * v1 = A2 * v2.
Remember to double-check the units and use consistent units throughout the calculations.