Page 80 - Fluid Power Engineering

Hydraulic Oils and Theor etical Backgr ound 57

By integrating Eq. (2B.6), the following expression for the value of
the velocity at a point at distance r from the center is obtained;

1 dp dp
u =− ∫ rdr =− r + C (2B.7)
2
2μ dx 4μ dx
At the pipe wall, r = R, u = 0. The following expression for the
integration constant C is obtained by substitution in Eq. (2B.7):

2
R dp
C = (2B.8)
4μ dx
The expression for velocity at a point r from the pipe becomes:
R − r dp
2
2
u = (2B.9)
4μ dx
The maximum velocity is at the pipe axis, r = 0;
2
2
u = R dp = D dp (2B.10)
max 4μ dx 16μ dx
Then, the velocity distribution is a parabolic profile of the form
2
y = ax + b (see Fig. 2B.2). The flow rate in the pipe is calculated as
follows:
( π R − r ) dp
2
2
dQ = u(2π rdr =) rdr (2B.11)
2μ dx
For a pipe of length L, the pressure gradient can be written as

dp = Δ p (2B.12)
dx L

⎛
πΔ p R πΔ p R 4 R ⎞
4
2
2
Then Q = ∫ ( R − r ) rdr = ⎜ − ⎟ (2B.13)
2 μ L 0 2 μ L ⎝ 2 4 ⎠
π R 4 π D 4
Q = Δ p = Δ p (2B.14)
8 μ L 128 μ L
128μ L
or ΔP = Q = RQ (2B.15)
π D 4
The term R expresses the resistance of the hydraulic transmission
line.
Equation (2B.15) is the Hagen-Poiseuille equation for laminar flow
in a pipe. It shows that the pressure loss is directly proportional to