You are asked to design a 180° reducing bend to attach to a pipe that conveys water at a water treatment plant (see figure). The water is supplied from a tank where the water surface elevation is kept constant at 100 m above the ground. The bend is horizontal and at ground level. Its exit opens out to the atmosphere (Patm 105 N/m²). The bend is to be attached to the supply pipe with flanges held in place by stainless steel bolts. The yield stress of steel is = 200 MPa, and a safety factor of 50% is to be applied. Your task is to calculate the smallest safe bolt diameter if 8 or 12 bolts are used to attach the bend to the pipe. =
Helpppp please ?
To calculate the smallest safe bolt diameter, we need to consider the stresses induced in the bolts due to the weight of the water and the pressure in the pipe. We also need to take into account the material properties of the bolts and apply a safety factor.
Assumptions:
Water is incompressible.
The bend is horizontal and at ground level, so we can ignore any potential energy changes.
The water flow is steady and the bend is fully filled with water.
The bend is symmetric, and the water exerts the same pressure on both sides of the bend.
Calculations:
The pressure at the inlet of the U-bend is the same as the pressure at the bottom of the water tank. Using the hydrostatic pressure formula, the pressure at the inlet can be calculated as follows:
P1 = ρgh
where ρ is the density of water, g is the acceleration due to gravity, and h is the height of the water surface above the inlet.
ρ = 1000 kg/m³ (density of water)
g = 9.81 m/s² (acceleration due to gravity)
h = 100 m (height of water surface above the inlet)
P1 = ρgh = 1000 x 9.81 x 100 = 981000 Pa
The pressure at the outlet of the U-bend is atmospheric pressure.
P2 = Paatm = 105 Pa
The pressure difference across the U-bend is therefore:
ΔP = P1 - P2 = 981000 - 105 = 881000 Pa
The stress induced in the bolts can be calculated using the following formula:
σ = F/A
where σ is the stress, F is the force, and A is the cross-sectional area of the bolt.
The force on the bolts can be calculated by summing the forces due to the pressure and the weight of the water in the U-bend. The weight of the water can be calculated using the following formula:
W = mg
where W is the weight, m is the mass, and g is the acceleration due to gravity.
The mass of the water in the U-bend can be calculated as follows:
m = ρV
where V is the volume of water in the U-bend.
The volume of water in the U-bend can be approximated as a frustum of a cone:
V = (π/3)h(R1² + R1R2 + R2²)
where h is the height of the frustum, R1 is the radius of the inlet, and R2 is the radius of the outlet.
h = 0.5 m (half the height of the U-bend)
R1 = 0.3/2 = 0.15 m (radius of the inlet)
R2 = 0.2/2 = 0.1 m (radius of the outlet)
V = (π/3)0.5(0.15² + 0.15 x 0.1 + 0.1²) = 0.024871 m³
m = ρV = 10000.024871 = 24.8709 kg
The weight of the water in the U-bend is:
W = mg =24.87099.81 = 243.9839 N
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