Page 41 - Mechanical Engineers Reference Book

1/30 Mechanical engineering principles

1.5.6.2 Turbulent boundary layers The simplest is the rigid column theory, which assumes that
the fluid is incompressible, and that the valve is closed
For Re, > 500 000, the boundary is assumed to be turbulent. relatively slowly. This is often applied to water flow in pipes.
In a turbulent boundary layer the velocity distribution is often
written in a power form:
1.5.7.1 Slow valve closure
(1.83)
When a fluid flowing through a pipe with a velocity vo
undergoes a change in velocity there is an associated change in
The index n vanes between 6 and 9, depending on Re,. pressure. Equating the force due to the pressure change to the
Because of the presence of the laminar sublayer, the rate of change of momentum during closure gives the resulting
turbulent regime is not continuous down to the plate surface, pressure rise Ap over a length of pipe L:
and (dvldy),,o does not give a useful result.
The equation used for T~ is dv
Ap -pL - (1.90)
T, = 0.0225p*R6-0.25 (1.84) dt
based on work on smooth pipes by Blasius. The solution to this equation depends on a knowledge of the
Taking n = 7 and using the same techniques as for laminar relationship between v and t (the valve closure rate in terms of
boundary layers gives: the flow velocity).
Equation (1.90) is only applicable to relatively slow valve
6 closure rates in which the closure time should not be less than
- = 0.37 (1.85)
X 2LIC (where C is the speed of sound in the fluid).
and
CD = 0.072 (1.86) 1.5.7.2 Time to establish flow
This result assumes that the turbulent boundary layer obtains The rigid column theory is also often used to calculate the time
over the whole length of the plate to L. required to establish flow in a pipe on opening a valve. The
Prandtl suggested a more realistic expression which takes theory implies that the time required to fully establish the flow
into account the presence of a laminar boundary layer near the is infinite and so the time t to achieve 99% of the final velocity
leading edge: vo is usually accepted:
C, = 0.074 - 1700 ReL-' (1.87) t = 2.646- LVO (1.91)
This may be used for 5 X 10' < ReL < lo7. For lo7 < ReL gH
< lo9 Schlichting (1960) suggests a logarithmic velocity distri- where His the supply head to the pipe entrance. The time t,
bution and required to reach x% of the final velocity is given by
CD = 0.44(1og~0Re~)-~.~~ 3.91(lnReL)-2.58 (1.88)
=
(1.92)
Again, equation (1.82b) may be applied to find the drag force
on the whole plate.
1.5.7.3 Rapid valve closure
1.5.6.3 Laminar sublayers
When a fluid is brought to rest instantaneously from a velocity
The analyses in Sections 1.5.6.1 and 1.5.6.2 above assume that of vo by the closure of a valve at the exit of a pipe of diameter
the plate surface is smooth or at least hydraulically smooth. A D there will be a relatively high pressure rise at the valve. If
surface is regarded as hydraulically smooth if the average the valve closure time is less than 2(Llc) then the resultant
roughness height k is less than the laminar sublayer thickness pressure rise is as if it were instantaneous; c is the speed at
&,. For a turbulent layer with a velocity distribution power which the pressure wave travels through the fluid, which is the
index of n = 1/7, the laminar sublayer thickness at a point at a sonic velocity.
distance x along the plate from the leading edge is given by On such a rapid valve closure the kinetic energy of the flow
is converted into strain energy in both the pipe material and
(1.89) the fluid (even liquids are acknowledged as compressible in
this context). The resulting pressure wave is transmitted
through the fluid away from the valve as shown in Figure 1.45.
Thus 8, may be compared with the roughness height, k, if the The pressure rise produced is
boundary layer thickness, 6, is known.
Ap = PCVO (1.93)
1.5.7 Pressure transients (water hammer) For a fluid of bulk modulus G, in a pipe of wall thickness x,
Water hammer is the common name for the rattling which of a material with a Young's modulus E and Poisson's ratio CT,
occurs in water pipes as result of pressure transients. This the velocity of the pressure wave is
phenomenon is due either to the collapse of cavitation bubbles {k
or to rapid valve closure. The former is not uncommon in C= [ p-+- L( 1.25-u )]}O'j (1.94a)
domestic water pipe work as an irritating vibration or noise on
valve closure or opening (usually of the hot taps). It can be 01
much more serious on a larger scale, where high-pressure rises (t2
over short periods may cause severe damage. Similar effects c= [ p-+- 3l-O.j (1.94b)
due to valve closure can be analysed on different levels of
sophistication.
if longitudinal stress is small compared to hoop stress.