Page 92 - Bird R.B. Transport phenomena

§3.1 The Equation of Continuity 77

(x + Ax, у + Ay, z + Az)

(pv x )\ x

Fig. 3.1-1. Fixed volume element Ax Ay
Az through which a fluid is flowing. The
arrows indicate the mass flux in and out
of the volume at the two shaded faces lo-
cated at x and x + Ax.

§3.1 THE EQUATION OF CONTINUITY
This equation is developed by writing a mass balance over a volume element Ax Ay Az,
fixed in space, through which a fluid is flowing (see Fig. 3.1-1):

rate of I Irate of [rate of
increase = { mass — < mass (3.1-1)
of mass 1 in 1 out
Now we have to translate this simple physical statement into mathematical language.
We begin by considering the two shaded faces, which are perpendicular to the
x-axis. The rate of mass entering the volume element through the shaded face at x is
(pv )\ Ay Az, and the rate of mass leaving through the shaded face at x + Ax is
x x
(pv )\ x+Ax Ay Az. Similar expressions can be written for the other two pairs of faces. The
x
rate of increase of mass within the volume element is Ax Ay Az(dp/dt). The mass balance
then becomes
Ax Ay Az -£ = Ay &z[(pv x)\ x - (pv x) \ X+Ax]

+ &z&x[(pv y)\ y- (pv XJ)\ lJ+b X]
+ Ax Ay[(pv z)\ z - (pv z)\ z+J (3.1-2)

By dividing the entire equation by Ax Ay Az and taking the limit as Ax, Ay, and Az go to
zero, and then using the definitions of the partial derivatives, we get

(3.1-3)

This is the equation of continuity, which describes the time rate of change of the fluid den-
sity at a fixed point in space. This equation can be written more concisely by using vector
notation as follows:

dp
= -(V-pv) (3.1-4)
dt
rate of net rate of mass
increase of addition per
mass per unit volume
unit volume by convection