Water fills and flows through an expanding pipeline that is inclined
upwards. On the centreline of the pipe, Point 1 is 0.3 m below point 2.
The velocities are V1 = 3.4 m/s and V2 = 1.7 m/s. The pressure at
point 1 is 9300 N/m
2
. What is the pressure at point 2 assuming no
energy loss?
To determine the pressure at point 2, we can use Bernoulli's equation, which states that the total pressure at any point in a fluid flow system is the sum of the static pressure, dynamic pressure, and gravitational potential energy per unit volume.
Bernoulli's equation can be written as:
P1 + 0.5ρV1² + ρgh1 = P2 + 0.5ρV2² + ρgh2
Where:
P1 and P2 are the pressures at points 1 and 2, respectively.
ρ is the density of the fluid (assumed constant).
V1 and V2 are the velocities at points 1 and 2, respectively.
g is the acceleration due to gravity.
h1 and h2 are the heights of points 1 and 2, respectively.
In this case, we are assuming no energy loss, so the equation can be simplified to:
P1 + 0.5ρV1² + ρgh1 = P2 + 0.5ρV2² + ρgh2
Given the information provided:
P1 = 9300 N/m^2
V1 = 3.4 m/s
V2 = 1.7 m/s
h1 = 0
h2 = 0.3 m
Since h1 = 0, the term ρgh1 becomes zero, simplifying the equation further:
P1 + 0.5ρV1² = P2 + 0.5ρV2² + ρgh2
Rearranging the equation, we can solve for P2:
P2 = P1 + 0.5ρV1² - 0.5ρV2² - ρgh2
Now, we need the density of water (ρ) to calculate the pressure at point 2. The density of water at room temperature is approximately 1000 kg/m^3.
Plugging in the values:
ρ = 1000 kg/m^3
P1 = 9300 N/m^2
V1 = 3.4 m/s
V2 = 1.7 m/s
h2 = 0.3 m
P2 = P1 + 0.5ρV1² - 0.5ρV2² - ρgh2
P2 = 9300 + 0.5*1000*3.4² - 0.5*1000*1.7² - 1000*9.8*0.3
Simplifying the equation will give us the pressure at point 2.