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 4.0 m. Find the velocity, vm (in m/s), and the volumetric flow rate, Q (in m3/s), of the water in the main.

p + (1/2) rho v^2 = constant

rho = water density = about 1000 kg/m^3

p = rho g h
p big pipe + (1/2) rho vbig^2 = plittle pipe + (1/2)rho vlittle^2

p big pipe - p little pipe =1000(9.81)(4)
= 39240 Newtons/m^2 or Pascals
so
39240 = (1/2)(1000)(vlittle^2-vbig^2)
now what can we say about vbig and vlittle?
Same amount of water per second through both the big pipe and the little pipe
so
Vbig(pi Dbig^2/4)=Vlittle(pi*Dlittle^2/4)
or
vlittle = Vbig (40^2/10^2) =Vbig(16)
so now
2*39.240 = ([16vbig]^2-vbig^2)=255 vbig^2
so
vbig = .555 m/s
Q = .555 * pipe area = .555(pi*.4^2/4)

To find the velocity and volumetric flow rate of the water in the main using a Venturi flow meter, we can use Bernoulli’s equation. Bernoulli's equation states that the pressure drop caused by the constriction in the Venturi flow meter is equal to the kinetic energy gained by the fluid.

Here's how to calculate the velocity and volumetric flow rate:

1. Calculate the pressure difference (ΔP) between the two vertical pipes. The pressure difference is given by the difference in water level between them, which is 4.0 m. Since the two pipes are open at the top, we can use the hydrostatic pressure equation:
ΔP = ρgh

Where:
- ΔP is the pressure difference
- ρ is the density of water (approximately 1000 kg/m^3)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- h is the height difference, which is 4.0 m

Plugging in the values:
ΔP = (1000 kg/m^3) * (9.8 m/s^2) * (4.0 m)
ΔP = 39,200 N/m^2 (or pascal, Pa)

2. Calculate the velocity of the water in the constriction (v_c) using Bernoulli's equation:
ΔP = (1/2) * ρ * v_c^2

Rearranging the equation to solve for v_c:
v_c = √((2ΔP) / ρ)

Plugging in the values:
v_c = √((2 * 39,200 N/m^2) / (1000 kg/m^3))
v_c ≈ √78.4 m^2/s^2
v_c ≈ 8.84 m/s

3. Calculate the velocity of the water in the main section (v_m) using the principle of continuity. Continuity states that the mass flow rate of an incompressible fluid is constant in a closed system. Since mass flow rate is equal to ρ * A * v, where ρ is density, A is the cross-sectional area, and v is the velocity, we can write:
ρ_m * A_m * v_m = ρ_c * A_c * v_c

Rearranging the equation to solve for v_m:
v_m = (ρ_c * A_c * v_c) / (ρ_m * A_m)

Where:
- ρ_m is the density of water (approximately 1000 kg/m^3)
- A_m is the cross-sectional area of the main pipe (π * r_m^2)
- v_m is the velocity in the main pipe
- ρ_c is the density of water (approximately 1000 kg/m^3)
- A_c is the cross-sectional area of the constriction (π * r_c^2)
- v_c is the velocity in the constriction

Plugging in the values:
v_m = (1000 kg/m^3 * π * (0.2 m)^2 * 8.84 m/s) / (1000 kg/m^3 * π * (0.05 m)^2)
v_m ≈ 35.36 m/s

4. Calculate the volumetric flow rate (Q) using the formula:
Q = A_m * v_m

Plugging in the values:
Q = π * (0.2 m)^2 * 35.36 m/s
Q ≈ 4.45 m^3/s

So, the velocity (v_m) of the water in the main pipe is approximately 35.36 m/s, and the volumetric flow rate (Q) is approximately 4.45 m^3/s.