Page 81 - Fluid Power Engineering

58 Cha pte r T w o

the flow rate when the flow is laminar, (Re < 2300). It has been vali-
dated many times by experiments.

4 π
The mean velocity v = Q, or Q = D v (2B.16)
2
π D 2 4
ρ
32μ L 64 L v 2
ΔP = v = (2B.17)
D 2 ρ vD μ/ D 2
ρ
Then, ΔP =λ L v 2 (2B.18)
D 2
64
where, λ= = friction coefficient
Re
Re = Reynolds number; Re = vD = ρ vD
ν μ
The mean fluid velocity, v, in laminar flow is given by

Q Δ p 1
v = = D = u or v = 05. u (2B.19)
2
π D / 4 32 μ L 2 max max
2
Thus, in the case of laminar flow in pipe
• The shear stress is linearly proportional to the radius r.
• The velocity distribution is parabolic.
• The velocity is maximum at the pipe axis and zero in the
vicinity of the pipe wall.
• The average, mean, velocity is half of the maximum value.

where A = Area, m 2
D = Pipe inner diameter, m
dx = Length of the fluid element, m
P = Pressure, Pa
r = Radius of the cylindrical fluid element, m
R = Pipe radius, m
R = Hydraulic resistance, Pa s/m 5
u = Oil velocity, m/s
v = Mean oil velocity, m/s
τ= Shear stress, N/m 2
μ= Dynamic viscosity, Pa s