Mechanics of Fluids

Compressible flow in pipes of constant cross-section 531
To account for the changes of fluid properties we must consider initially an
infinitesimal length of the pipe. For adiabatic flow the relation involving the
friction force may be integrated over a finite length of the pipe by introducing
the continuity equation and the energy equation. For isothermal conditions
integration is made possible by inserting the condition T = constant.
Applying the steady-flow momentum equation to an element with crosssectional
area A, perimeter P and length δl (Fig. 11.26), we have
pA − (p + δp)A − τ 0Pδl − ρgAδl sin α = ρAuδu
Substituting τ 0 = 1
2 f ρu2 (eqn 7.3), δl sin α = δz and dividing by ρgA we
obtain
−δz = δp uδu f δl
+ +
ρg g A/P
u 2
2g
(11.62)
The final term corresponds to Darcy’s formula for head lost to friction.
Equation 11.62 applies to any steady one-dimensional flow where A is constant,
but, for integration, the relation between density and pressure must
be known and this depends on the degree of heat transfer. We consider first
the case of zero heat transfer.
11.10.1 Adiabatic flow in a pipe: Fanno flow
Before embarking on the integration of eqn 11.62 for adiabatic conditions it
will be instructive to look at the problem in general thermodynamic terms.
When no external work is done and changes of elevation may be neglected,
the steady-flow energy equation (11.12) and the continuity equation m =
ρAu together give
Constant h0 = h + 1
2 u2 = h + 1
� �2 m
(11.63)
2 ρA
For given values of m, A and the stagnation enthalpy h0, curves of h against
ρ could be plotted from eqn 11.63. A more significant relation, however, is
that between h and specific entropy s. For any pure substance, s, like h and
ρ, is a function of state and so may be determined from values of h and ρ.
In particular, for a perfect gas s − s1 = cv ln(h/h1)(ρ1/ρ) γ −1 . Starting from
a specified state (point 1 on Fig. 11.27) the curve of h against s traces the
Fig. 11.26