Page 96 - Mechanical Engineers' Handbook (Volume 4)

11 Viscous Fluid Flow in Ducts 85
p p Vp L 2ln
p
2
2
2
2
ƒ
1
1
2
1
1
D p 1
or, in terms of the initial Mach number,
p p kM p L 2ln
p
2
2
2
2
2
ƒ
1
1
1
2
D p 1
The downstream pressure p at a distance L from section 1 may be obtained by trial by
2
neglecting the term 2 ln(p /p ) initially to get a p , then including it for an improved value.
1
2
2
The distance L is a section where the pressure is p is obtained from
2
2
p
1
L p 1
2
ƒ 1 2ln
D kM 2 1 p 1 p 2
A limiting condition (designated by an asterisk) at a length L* is obtained from an
expression dp/dx to get
dp pƒ/2D (ƒ/D)( V /2)
2

2
dx 1 p/ V 2 kM 1
For a low subsonic flow at an upstream section (as from a compressor discharge) the pressure
gradient increases in the flow direction with an infinite value when M* 1/ k 0.845
for k 1.4 (air, for example). For M approaching zero, this equation is the Darcy equation
for incompressible flow. The limiting pressure is p* p M k, and the limiting length is
1
1
given by
ƒL* 1 1
1 ln
D kM 2 1 kM 2 1
Since the gas at any two locations 1 and 2 in a long pipe has the same limiting condition,
the distance L between them is

ƒL ƒL* ƒL*
D D D
M 1 M 2
Conditions along a pipe for various initial Mach numbers are shown in Fig. 35.
For adiabatic flow the limiting Mach number is M* 1. This is from an expression
for dp/dx for adiabatic flow:
dp ƒkp M 1 (k 1)M 2 ƒ V 2 1 (k 1)M 2
2
dx 2D 1 M 2 D 2 1 M 2
The limiting pressure is
1
p* 2[1 ⁄2(k 1)M ]
2
1
M
p 1 1 k 1
and the limiting length is
¯
ƒL* 1 M 2 k 1 (k 1)M 2
1 ln 1
2
1
D kM 2 1 2k 2[1 ⁄2(k 1)M ]
1