differential), but this is not the case for the integral of dm

.
(thus the names
path function and inexact differential).
The mass flow rate through the entire cross-sectional area of a pipe or
duct is obtained by integration:

(5–3)

While Eq. 5–3 is always valid (in fact it is exact), it is not always practi-
cal for engineering analyses because of the integral. We would like instead
to express mass flow rate in terms of average values over a cross section of
the pipe. In a general compressible flow, both rand Vnvary across the pipe.
In many practical applications, however, the density is essentially uniform
over the pipe cross section, and we can take routside the integral of Eq.
5–3. Velocity, however, is neveruniform over a cross section of a pipe
because of the fluid sticking to the surface and thus having zero velocity at
the wall (the no-slip condition). Rather, the velocity varies from zero at the
walls to some maximum value at or near the centerline of the pipe. We
define the average velocityVavgas the average value of Vnacross the entire
cross section (Fig. 5–3),

Average velocity: (5–4)

where Acis the area of the cross section normal to the flow direction. Note
that if the velocity were Vavgall through the cross section, the mass flow
rate would be identical to that obtained by integrating the actual velocity
profile. Thus for incompressible flow or even for compressible flow where
ris uniform across Ac, Eq. 5–3 becomes

(5–5)

For compressible flow, we can think of r as the bulk average density
over the cross section, and then Eq. 5–5 can still be used as a reasonable
approximation.
For simplicity, we drop the subscript on the average velocity. Unless
otherwise stated,Vdenotes the average velocity in the flow direction. Also,
Acdenotes the cross-sectional area normal to the flow direction.
The volume of the fluid flowing through a cross section per unit time is
called the volume flow rateV

.

(Fig. 5–4) and is given by

(5–6)

An early form of Eq. 5–6 was published in 1628 by the Italian monk
Benedetto Castelli (circa 1577–1644). Note that most fluid mechanics text-
books use Qinstead of V

.

for volume flow rate. We use V

.
to avoid confusion
with heat transfer.
The mass and volume flow rates are related by

(5–7)

where vis the specific volume. This relation is analogous to mrV
V/v, which is the relation between the mass and the volume of a fluid in a
container.

m

#

rV

#



V

#

v

V

#




Ac

Vn dAcVavg AcVAc¬¬ 1 m^3 >s 2

m

#

rVavg Ac¬¬ 1 kg>s 2

Vavg

1

Ac

(^)

Ac
Vn dAc
m



Ac
dm



Ac
rVn dAc¬¬ 1 kg>s 2
Chapter 5 | 221
Vavg
Cross section
Ac
V = VavgAc
FIGURE 5–4
The volume flow rate is the volume of
fluid flowing through a cross section
per unit time.
Vavg
FIGURE 5–3
The average velocity Vavgis defined as
the average speed through a cross
section.