Page 35 - Mechanical Engineers Reference Book

1/24 Mechanical engineering principles

loss term, h~. Applied between two positions (1) and (2) in a where m is the ratio of the cross-sectional area of flow to the
pipe, the head equation gives: wetted perimeter known as the hydraulic mean diameter and C
is a coefficient which depends on the condition of the pipe
(1.41) wall.
(c) Pipe losses (changes in section) When a fluid flows
where the head loss term hL is the loss of energy per unit through a sharp (sudden) change in the cross section of a pipe,
weight of fluid flowing. energy is dissipated in the resulting turbulent eddies at the
Note that if a pump, say, is introduced between (1) and (2) edge of the flow stream, producing a loss of head (or energy
an energy gain per unit weight term h,, equivalent to the per unit weight). If the flow is from a smaller area to a larger
output of the pump written as a head, should be added to the one (sudden enlargement) the head loss is
left-hand side of the equation to give
(1.50)
(1.42)
When the flow is from a larger area to a smaller area (sudden
The relationship used to determine the head loss in a pipe contraction) the narrowed flow stream entering the smaller
depends on the flow regime in operation as well as the type pipe is known as a vena contractu. The loss of head is assumed
and surface finish of the pipe wall. to be that due to a sudden enlargement from the vena
A mathematical analysis of laminar flow may be used to contracta to the full area of the smaller pipe:
obtain an expression for the head loss along a pipe in terms of
('
the fluid properties, pipe dimensions and flow velocity. Relat- hL 3
ing the pressure change along a length, L, of pipe of diameter, 2g cc - 1)2 (1.51)
D, to the change in shear force across the flow produces
Poiseuille's equation: The contraction coefficient, C,, is the ratio of the vena
contracta area to that of the smaller pipe area. A typical value
(1.43) of C, is 0.6, which gives

If the flow regime is turbulent, then the relationships in the (1.52)
flow cannot be easily described mathematically, but the head
loss may be derived by equating the shear force at the pipe which is also the head loss at the sharp entry to a pipe from a
wall to the change in pressure force along the pipe. This gives reservoir. Energy dissipation at changes in section, and pipe
the D'Arcy equation: entry and exit, may be reduced by making the changes smooth
and gradual, though this may be relatively costly.
h 4fL v2 (1.44) Other pipe fittings, such as valves, orifice plates and bends,
L- D 2g produce varying values of head loss, usually quoted as a
fraction of the velocity head (v2/2g).
This relationship may also be established using dimensional
analysis. (d) Pipe networks A system of pipes may be joined together
Unfortunately, the friction coefficient, f is not a constant but either in series (one after the other) or parallel (all between
depends on the type of flow and the roughness of the pipe the same point). The friction head loss across a system of pipes
walls. There are general relationships between f and Re which in series is the sum of the losses along each pipe individually.
may be expressed as equations of varying complexity or as The flow rate through each pipe will be the same. Using
charts. For smooth pipes:
D'Arcy's head loss equation:
1
4 loglo (2ReV7) - 1.6 (1.45)
-= (1.53)
For rough pipes with a roughness size k this becomes: and
1 V = VIA1 = V~AZ . . . = v,A, (1.54)
=
v (1.46) If the system of pipes is connected in parallel the head loss
- 4 loglo (g) + 3.48
=
The Colebrook and White equation is a general or universal across the system is equal to the head loss along any one of the
friction equation: pipes, when the flow has settled down to steady. The flow rate
tkirough the system is the sum of the flow rates along each
(: R9$)
1
- 3.47 - 4 loglo - + - (1.47) pipe. Again using the D'Arcy equation:
v
=
It is, however, usually more useful to obtain values offfrom a
chart such as Figure 1.40. (Note: the value of f used in
American equations for head losses is four times that used in V = VIAl + vzAz + . . . v,A, (1.56)
the United Kingdom, so if values of f are obtained from In addition, the rate of flow into each junction of a network,
American texts they should be moderated accordingly or the either in series or parallel, is equal to the rate of flow out of it.
corresponding American equation used.) Pipe network problems are thus solved by setting up a
An empirical relationship widely used in water pipe work is number of such equations and solving them simultaneously.
the Hazen-Williams equation, usually written as: For a large number of pipes a computer program may be
0.54 needed to handle the number of variables and equations. An
v = 1.38 C rn0."(,) (1.48) example of a pipe network computer solution is given in
Douglas et al. (1986).