A horizontal segment of pipe tapers from a cross sectional area of 50 cm2 to 0.5 cm2. The pressure at the larger end of the pipe is 1.26 105 Pa and the speed is 0.029 m/s. What is the pressure at the narrow end of the segment? Pa
Please help!
Due to the flow contraction, the velocity at the exit (Vout) will be 100 times the velocity at the entrance (Vin). That results from the incompresible-flow continuity equation.
The pressure at the entrance, combined with the Bernoulli Equation, Vin and Vout, will tell you the pressure at the exit.
You should use a ^ before exponents, when typing equations and scientific-notation numbers. I am sure you mean 10^5 and not 105.
To find the pressure at the narrow end of the pipe segment, we can use the principle of continuity equation which states that the product of the cross-sectional area and the fluid velocity remains constant for an incompressible and steady flow of fluid.
First, let's determine the fluid velocity at the narrow end of the pipe segment. We can use the continuity equation:
A1 * v1 = A2 * v2
where:
A1 and A2 are the cross-sectional areas at the larger and narrow ends respectively, and
v1 and v2 are the velocities at the larger and narrow ends respectively.
Given:
A1 = 50 cm^2 = 50 * 10^(-4) m^2
A2 = 0.5 cm^2 = 0.5 * 10^(-4) m^2
v1 = 0.029 m/s
We can rearrange the equation to solve for v2:
v2 = (A1 * v1) / A2
v2 = (50 * 10^(-4) * 0.029) / (0.5 * 10^(-4))
v2 = 290 / 0.5
v2 = 580 m/s
Now, using Bernoulli's equation, we can relate the pressure and velocity of the fluid at different points along the pipe.
P1 + (1/2) * ρ * v1^2 = P2 + (1/2) * ρ * v2^2
where:
P1 and P2 are the pressures at the larger and narrow ends respectively,
ρ is the density of the fluid (which we will assume to be constant), and
v1 and v2 are the velocities at the larger and narrow ends respectively.
Given:
P1 = 1.26 * 10^5 Pa
v1 = 0.029 m/s
v2 = 580 m/s
We can rearrange the equation to solve for P2:
P2 = P1 + (1/2) * ρ * (v1^2 - v2^2)
Since we don't have the value for ρ, we can assume it to be the density of water, which is approximately 1000 kg/m^3.
P2 = 1.26 * 10^5 + (1/2) * 1000 * (0.029^2 - 580^2)
P2 = 1.26 * 10^5 + (1/2) * 1000 * (-336701)
P2 = 1.26 * 10^5 - 168350500
P2 ≈ -168238500
Since the obtained pressure is negative, it is likely that there was an error in the calculation or given values. Please double-check the values or the equation.