I have a 4" hose attached to a 2" nozzle, me resistance coeffient of the nozzle is .12, and the jet of water exiting the nozzle is 24.4 m/sec. The water is flowing from the 4" to the 2". What I am having trouble finding is Pressure 1 and the the velocity 1.
What I know is that P2=0, areas of the hose is 4" = 12.56 in^2 converts to .008m^2, 2" = 3.14 in^2 converts to .002m^2, I think Q=V2*A2 = 24.4*.002= .049 m^3/sec
then v1= 6.125 m/s?
then would I plug this into the bernoulli's equ.??
Thank You
Yes, plug it into Bernoulli's equation.
On the continuity equation, you didn't need to do all that converting...The area of the larger is four times the smaller, so
V1A1=V2A2
V1*4=24.4 m/sec
To find the velocity (V1) of the water in the 4" hose, you can rearrange the equation from the continuity equation:
V1 * A1 = V2 * A2
Since the area of the 4" hose (A1) is four times the area of the 2" nozzle (A2), you can substitute 4 * V1 for V2:
V1 * A1 = (4 * V1) * A2
Simplifying the equation further:
V1 * A1 = 4 * V1 * A2
Now divide both sides of the equation by V1:
A1 = 4 * A2
Since you know that A2 is 3.14 in^2 (or approximately 0.002 m^2), you can plug it into the equation:
4 * A2 = 4 * 0.002 = 0.008 m^2
Now you can solve for V1:
V1 * 0.008 = 24.4
Divide both sides of the equation by 0.008:
V1 = 24.4 / 0.008 = 3050 m/s
So the velocity (V1) of the water in the 4" hose is approximately 3050 m/s.
Now you can move on to Bernoulli's equation:
P1 + (0.5 * ρ * V1^2) + (ρ * g * h1) = P2 + (0.5 * ρ * V2^2) + (ρ * g * h2)
In this equation, P1 is the pressure in the 4" hose, V1 is the velocity of the water in the 4" hose (3050 m/s), P2 is the pressure at the nozzle (0, since P2 = 0), V2 is the velocity of the water at the nozzle (24.4 m/s), ρ is the density of the water, g is the acceleration due to gravity, h1 is the height of the water in the 4" hose, and h2 is the height of the water at the nozzle.
Since you want to find P1, rearrange the equation:
P1 = - (0.5 * ρ * V1^2) - (ρ * g * h1) + P2 + (0.5 * ρ * V2^2) + (ρ * g * h2)
Substitute the known values into the equation:
P1 = - (0.5 * ρ * (3050)^2) - (ρ * g * h1) + 0 + (0.5 * ρ * (24.4)^2) + (ρ * g * h2)
Now you can calculate P1 by plugging in the values for density (ρ), acceleration due to gravity (g), height of the water in the 4" hose (h1), and height of the water at the nozzle (h2).