What is the pressure change in methyl alcohol (SG =0.791) flowing from 4.0 cm diameter pipe to a 1.5 cm diameter pipe if the velocity in the larger pipe is 0.40 m/s?
rho = density in kg/m^2 = 0.791* 1000 kg/m^3 = 791 kg/m^3
P1 + (1/2) rho V1^2 = P2 + (1/2) rho V2^2
V * area = Q the volume flow rate = constant
so
V1 * pi (2)^2 = V2 * pi * .75^2 (note using cm because only ratio important)
V2 = V1 * 4/.5625 = 0.40 m/s * 7.11 meters/second = 2.84 m/s
so
P1 + (1/2)(791) (.4^2) = P2 + (1/2)(791)(2.84)^2
P2 - P1 = (1/2)(791)(.4^2-2.84^2) Pascals or Newtons/m^2
note negative because it is a pressure drop
Thank you so much!
To calculate the pressure change in methyl alcohol flowing from a larger pipe to a smaller pipe, we can use the equation for Bernoulli's principle, which states that the total energy per unit mass of a fluid is conserved along a streamline.
The equation for Bernoulli's principle is:
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,
ρ is the density of the fluid (methyl alcohol),
V1 and V2 are the velocities at points 1 and 2 respectively,
g is the acceleration due to gravity, and
h1 and h2 are the heights at points 1 and 2 respectively.
In this case, the height difference (h1 - h2) is negligible since the fluid is flowing horizontally.
Since the velocity of the fluid is changing from V1 in the larger pipe to V2 in the smaller pipe, the equation can be simplified to:
P1 + 1/2 * ρ * V1^2 = P2 + 1/2 * ρ * V2^2
We are trying to find the pressure change, which is given by:
Pressure change = P2 - P1
To calculate the pressure change, we need to find the pressures at points 1 and 2.
Given that the velocity in the larger pipe (V1) is 0.40 m/s and the diameter of the larger pipe is 4.0 cm, we can find the velocity in the smaller pipe (V2) using the continuity equation, which states that mass flow rate is constant for an incompressible fluid.
The equation for the continuity equation is:
A1 * V1 = A2 * V2
Where A1 and A2 are the cross-sectional areas of the larger and smaller pipes respectively.
The cross-sectional area of a pipe can be calculated using the formula:
Area = π * (diameter / 2)^2
By substituting the values, we can solve for V2.
Once we have V2, we can use the simplified Bernoulli's equation to calculate the pressure change (P2 - P1).
Please provide the numerical values for the diameter of the smaller pipe to proceed with the calculations.