A venturi tube is measuring the flow of water; it has a main diameter of 4.1 cm tapering down to a throat diameter of 1.8 cm. If the pressure difference is measured to be 67 mm-Hg, what is the velocity of the water? Please help step by step!!!
To calculate the velocity of the water using the venturi tube, we can apply Bernoulli's principle and the continuity equation.
Step 1: Convert the pressure difference to the SI unit of pressure (Pascal).
The pressure difference is given in mm-Hg. We need to convert it to Pascal (Pa) by using the conversion factor:
1 mm-Hg = 133.3224 Pa.
Pressure difference = 67 mm-Hg
Pressure difference = 67 * 133.3224 Pa
Pressure difference = 8939.7918 Pa
Step 2: Apply Bernoulli's principle.
Bernoulli's principle states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume of a fluid remains constant throughout the flow streamline.
According to Bernoulli's principle:
P1 + (0.5 * ρ * v1^2) + (ρ * g * h1) = P2 + (0.5 * ρ * v2^2) + (ρ * g * h2)
For simplicity, we can assume that the fluid is incompressible (ρ = constant) and that the tube is horizontal and at the same height (h1 = h2). The equation then simplifies to:
P1 + (0.5 * ρ * v1^2) = P2 + (0.5 * ρ * v2^2)
Step 3: Apply the continuity equation.
The continuity equation states that the mass flow rate at any point in an incompressible fluid flow is constant.
According to the continuity equation:
A1 * v1 = A2 * v2
Where A1 and A2 are the cross-sectional areas at points 1 and 2, respectively.
Step 4: Calculate the cross-sectional areas.
The cross-sectional area of a circle is given by the formula A = π * r^2, where r is the radius of the circle.
A1 = π * (d1/2)^2
A2 = π * (d2/2)^2
Given:
d1 = 4.1 cm = 0.041 m (main diameter)
d2 = 1.8 cm = 0.018 m (throat diameter)
A1 = π * (0.041/2)^2
A2 = π * (0.018/2)^2
Step 5: Substitute the values into the equations.
Using the values we have calculated, let's substitute them into the equations we derived earlier.
For Bernoulli's principle:
P1 + (0.5 * ρ * v1^2) = P2 + (0.5 * ρ * v2^2)
For the continuity equation:
A1 * v1 = A2 * v2
Step 6: Solve the equations.
We have two equations and two unknowns (v1 and v2). We can rewrite the continuity equation as v2 = (A1 * v1) / A2 and substitute it into Bernoulli's equation:
P1 + (0.5 * ρ * v1^2) = P2 + (0.5 * ρ * ((A1 * v1) / A2)^2)
Simplifying the equation:
8 * (P1 - P2) = ρ * v1^2 * (A1^2 - A2^2)
Now, rearrange the equation to solve for v1:
v1 = sqrt((8 * (P1 - P2)) / (ρ * (A1^2 - A2^2)))
Step 7: Calculate the velocity.
Substitute the known values into the equation to calculate the velocity (v1):
ρ = density of water (assumed to be 1000 kg/m^3 for water at standard conditions)
A1 = π * (0.041/2)^2
A2 = π * (0.018/2)^2
P1 - P2 = 8939.7918 Pa
v1 = sqrt((8 * (8939.7918)) / (1000 * ((π * (0.041/2)^2)^2 - (π * (0.018/2)^2)^2)))
Now, simply calculate the value of v1.