Conservation of Mass Principle

The conservation of mass principlefor a control volume can be expressed

as:The net mass transfer to or from a control volume during a time interval

t is equal to the net change (increase or decrease) in the total mass within

the control volume duringt. That is,

or

(5–8)

where mCVmfinal
minitialis the change in the mass of the control volume

during the process (Fig. 5–5). It can also be expressed in rate formas

(5–9)

where m

.

inand m

.

outare the total rates of mass flow into and out of the control

volume, and dmCV/dtis the time rate of change of mass within the control vol-

ume boundaries. Equations 5–8 and 5–9 are often referred to as the mass bal-

anceand are applicable to any control volume undergoing any kind of process.

Consider a control volume of arbitrary shape, as shown in Fig. 5–6. The

mass of a differential volume dVwithin the control volume is dmrdV.

The total mass within the control volume at any instant in time tis deter-

mined by integration to be

Total mass within the CV: (5–10)

Then the time rate of change of the amount of mass within the control vol-

ume can be expressed as

Rate of change of mass within the CV: (5–11)

For the special case of no mass crossing the control surface (i.e., the control

volume resembles a closed system), the conservation of mass principle

reduces to that of a system that can be expressed as dmCV/dt0. This rela-

tion is valid whether the control volume is fixed, moving, or deforming.

Now consider mass flow into or out of the control volume through a differ-

ential area dA on the control surface of a fixed control volume. Let n→be

the outward unit vector of dAnormal to dAand V

→

be the flow velocity at dA

relative to a fixed coordinate system, as shown in Fig. 5–6. In general, the

velocity may cross dAat an angle uoff the normal of dA, and the mass flow

rate is proportional to the normal component of velocity V

→

nV

→

cos urang-

ing from a maximum outflow of V

→

for u0 (flow is normal to dA) to a min-

imum of zero for u90° (flow is tangent to dA) to a maximum inflowof V

→

for u180° (flow is normal to dAbut in the opposite direction). Making use

of the concept of dot product of two vectors, the magnitude of the normal

component of velocity can be expressed as

Normal component of velocity: (5–12)

The mass flow rate through dAis proportional to the fluid density r, normal

velocity Vn, and the flow area dA, and can be expressed as

Differential mass flow rate: dm#rVn dAr 1 V cos u 2 dAr 1 V (5–13)

S#

n

S

2 dA

VnV cos uV

S#

n

S

dmCV

dt



d

dt

(^)

CV
r dV
mCV

CV
r dV
m

in
m

outdmCV>dt¬¬^1 kg>s^2
min
mout¢mCV¬¬ 1 kg 2
a
Total mass entering
the CV during ¢t
b
a
Total mass leaving
the CV during ¢t
ba
Net change in mass
within the CV during ¢t
b
222 | Thermodynamics
Water
∆mbathtub
= min

• mout

= 20 kg

min = 50 kg

FIGURE 5–5

Conservation of mass principle for an
ordinary bathtub.

→

→

Control

volume (CV)

Control surface (CS)

dV

dm

dA

n

V

u

FIGURE 5–6

The differential control volume dVand
the differential control surface dAused
in the derivation of the conservation of
mass relation.