Lubricating oil, with a relative density of 0.79, flows around a 90¨¬ bend. The pipe diameter is 0.45m, and the oil has a pressure head of 7m and the flow is 1.7m3/s. Find the force exerted by the oil on the bend.
Oil density = 0.79 X 103
Volumetric mass flow rate = 1.7 m3/s
Pipe diameter = 0.45 m
Pressure head (¥÷) = 7m
Use the momentum change law.
The force exerted on the fluid equals the rate of change of the fluid's momentum, which is a vector. For a 90 degree turn, the momentum change vector bisects the angle betweeb the pipes and equals sqrt2 times the momentum flow rate.
To find the force exerted by the oil on the bend, we can use Bernoulli's equation, the continuity equation, and the concept of pressure difference.
1. Bernoulli's equation: Bernoulli's principle relates the pressure, velocity, and elevation of a fluid in a streamline flow. The equation is given by:
P1 + (1/2)ρv1^2 + ρgh1 = P2 + (1/2)ρv2^2 + ρgh2
Where:
P1 and P2 are the pressures at two points along the streamline (in this case, before and after the bend)
ρ is the density of the fluid (given as 0.79 x 10^3 kg/m^3)
v1 and v2 are the velocities of the fluid at the two points
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h1 and h2 are the elevations of the two points (in this case, they can be considered the same as the pipe is horizontal)
2. Continuity equation: The continuity equation relates the cross-sectional area and velocity of a fluid in a pipe. The equation is given by:
A1v1 = A2v2
Where:
A1 and A2 are the cross-sectional areas of the pipe before and after the bend, respectively.
3. Pressure difference: The force exerted by the oil on the bend can be calculated by finding the pressure difference before and after the bend.
Force = Pressure difference x Area
Now, let's plug in the given values:
Pipe diameter = 0.45 m
Radius (r) = 0.45/2 = 0.225 m
Cross-sectional area (A) = πr^2 = 3.14 x (0.225)^2 = 0.159 m^2
Volumetric mass flow rate = 1.7 m3/s
Density (ρ) = 0.79 x 10^3 kg/m^3
Velocity (v1 and v2) = Volumetric mass flow rate / cross-sectional area = 1.7 / 0.159 = 10.69 m/s
Pressure head (h) = 7 m
Using Bernoulli's equation, let's evaluate the pressure difference:
P1 + (1/2)ρv1^2 + ρgh = P2 + (1/2)ρv2^2 + ρgh
P1 + (1/2)(0.79 x 10^3)(10.69)^2 + (0.79 x 10^3)(9.8)(7) - P2 - (1/2)(0.79 x 10^3)(10.69)^2 - (0.79 x 10^3)(9.8)(0)
Simplifying this equation, we can cancel out similar terms and solve for the pressure difference.
Now, plug in the values into the equation to calculate the pressure difference.
Once you have the pressure difference, use the formula:
Force = Pressure difference x Area
Multiply the pressure difference calculated in the previous step by the cross-sectional area of the pipe to find the force exerted by the oil on the bend.