The net flow rate into or out of the control volume through the entire con-
trol surface is obtained by integrating dm

.

over the entire control surface,

Net mass flow rate: (5–14)

Note that V

→

· n

→
Vcos uis positive for u90° (outflow) and negative for
u  90° (inflow). Therefore, the direction of flow is automatically
accounted for, and the surface integral in Eq. 5–14 directly gives the net
mass flow rate. A positive value for m

.
netindicates net outflow, and a nega-
tive value indicates a net inflow of mass.
Rearranging Eq. 5–9 as dmCV/dtm

.

out
m

.
in0, the conservation of
mass relation for a fixed control volume can then be expressed as

General conservation of mass: (5–15)

It states that the time rate of change of mass within the control volume plus
the net mass flow rate through the control surface is equal to zero.
Splitting the surface integral in Eq. 5–15 into two parts—one for the out-
going flow streams (positive) and one for the incoming streams (negative)—
the general conservation of mass relation can also be expressed as

(5–16)

where Arepresents the area for an inlet or outlet, and the summation signs
are used to emphasize that allthe inlets and outlets are to be considered.
Using the definition of mass flow rate, Eq. 5–16 can also be expressed as

(5–17)

Equations 5–15 and 5–16 are also valid for moving or deforming control vol-
umes provided that the absolute velocity V

→
is replaced by the relative velocity
V

→

r,which is the fluid velocity relative to the control surface.

Mass Balance for Steady-Flow Processes

During a steady-flow process, the total amount of mass contained within a
control volume does not change with time (mCVconstant). Then the con-
servation of mass principle requires that the total amount of mass entering a
control volume equal the total amount of mass leaving it. For a garden
hose nozzle in steady operation, for example, the amount of water entering
the nozzle per unit time is equal to the amount of water leaving it per
unit time.
When dealing with steady-flow processes, we are not interested in the
amount of mass that flows in or out of a device over time; instead, we are
interested in the amount of mass flowing per unit time, that is,the mass
flow rate m

.

. The conservation of mass principlefor a general steady-flow
system with multiple inlets and outlets can be expressed in rate form as
(Fig. 5–7)

Steady flow: a (5–18)
in

m

#

a

out

m

#

¬¬ 1 kg>s 2

d

dt

(^)

CV
r dVa
in
m#
a
out
m#¬or¬
dmCV
dt
a
in
m#
a
out
m#
d
dt
(^)

CV
r dVa
out


A
rVn dA
a
in


A
rVn dA 0
d
dt
(^)

CV
r dV

CS
r 1 V
S
#Sn 2 dA 0
m

net

CS
dm



CS
rVn dA

CS
r 1 V
S
#Sn 2 dA
Chapter 5 | 223
m
CV
̇ 1 = 2 kg/s m ̇ 2 = 3 kg/s
m ̇ ̇ ̇ 3 = m 1 + m 2 = 5 kg/s
FIGURE 5–7
Conservation of mass principle for a
two-inlet–one-outlet steady-flow
system.