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Head Loss Calculations. Bernoulli and Pipe Flow. ○ The Bernoulli equation that we worked with was a bit simplistic in the way it looked at a fluid system.
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Head Loss Calculations Bernoulli and Pipe Flow ¢ The Bernoulli equation that we worked with was a bit simplistic in the way it looked at a fluid system ¢ All real systems that are in motion suffer from some type of loss due to friction ¢ It takes something to move over a rough surface 2 Pipe Flow
Bernoulli and Pipe Flow ¢ Consider flow in a constant-diameter pipe 3 Pipe Flow Bernoulli and Pipe Flow ¢ If we look at the energy at points 1 and 2. 4 Pipe Flow p 1 ρ
v 1
2
v 2
2
Bernoulli and Pipe Flow ¢ Since we are dealing with an uncompressible fluid, the pressures at points 1 and 2 should be the same. 7 Pipe Flow
ρ ⇒ p
= p
Bernoulli and Pipe Flow ¢ If we ran the system experimentally and measured the two pressures, they would not be the same. 8 Pipe Flow p
p
ρ ⇒ p
= p
Bernoulli and Pipe Flow ¢ The pressure at point 2 would be lower than the pressure at point 1. 9 Pipe Flow
ρ ⇒ p
= p
Bernoulli and Pipe Flow ¢ The pressure is being lost (actually the pressure energy) due to friction as the flow moves along the pipe. 10 Pipe Flow p
p
ρ ⇒ p
= p
Friction Losses ¢ The thickness of the quarter is dz and it has a perimeter length of P (πD) ¢ This makes the area in contact with the sides of the pipe equal to Pdz = πDdz ¢ At the wall there is a shearing stress, τw, which is the stress between the wall and the outer layer of the fluid 13 Pipe Flow Friction Losses ¢ Then the force from the wall on the section of fluid (opposing the fluid flow) will be equal to 14 Pipe Flow Fretarding = τ w Pdz = τ w π Ddz
Friction Losses ¢ The pressure on the upstream (left face) of the section will produce a force accelerating the section 15 Pipe Flow Fretarding = τ w Pdz = τ w π Ddz Faccelerating = pA = p π D
4 Friction Losses ¢ As we move from the left face to the right face (dz) the change in pressure will equal dp 16 Pipe Flow Fretarding = τ w Pdz = τ w π Ddz Faccelerating = pA = p π D
4
Friction Losses ¢ Therefore 19 Pipe Flow Fretarding = τ w π Ddz +( p + dp ) π D^2
π D^2
p π D^2
− τ w π Ddz −( p + dp ) π D^2
π D^2
Friction Losses ¢ Rearranging 20 Pipe Flow
Friction Losses ¢ So we have an expression for the rate at which the pressure changes as the flow moves downstream 21 Pipe Flow
2
2
Friction Losses ¢ In the simplest form 22 Pipe Flow − τ w π D π D
dp dz − 4 τ w D = dp dz
Friction Losses ¢ The friction factor, f , is the ratio of the friction forces to the inertia forces. 25 Pipe Flow − 4 τ w Dh = dp dz f = 4 τ w 1 2 ρ v
Friction Losses ¢ Combining the two expressions. 26 Pipe Flow − f 1 2 ρ v
D h = dp dz
27 Pipe Flow
h
28 Pipe Flow
h
L
Friction Losses ¢ So we have a modified form of the Bernoulli equation that takes into account the friction losses in the system 31 Pipe Flow p 1 ρ
v 12
ρ v^2
fL Dh ρ
p 2 ρ
v 22
Friction Losses ¢ Reducing 32 Pipe Flow p 1 ρ
v 12
fL Dh
p 2 ρ
v 22
Friction Losses ¢ And rewriting the expression in terms of head 33 Pipe Flow p 1 ρ g
v 12 2 g
p 2 ρ g
v 22 2 g
v 12 2 g
p 2 ρ g
v 22 2 g
Example
37 Pipe Flow
Example
38 Pipe Flow
Example
39 Pipe Flow
Example
40 Pipe Flow
