fluid_mechanics

188 Chapter 5 ■ Finite Control Volume Analysis
Newton’s second law of motion leads to the conclusion that forces can result from or cause
changes in a flowing fluid’s velocity magnitude and/or direction. Moment of force 1torque2 can result
from or cause changes in a flowing fluid’s moment of velocity. These forces and torques can
be associated with work and power transfer.
The first law of thermodynamics is a statement of conservation of energy. The second law
of thermodynamics identifies the loss of energy associated with every actual process. The mechanical
energy equation based on these two laws can be used to analyze a large variety of steady,
incompressible flows in terms of changes in pressure, elevation, speed, and of shaft work and loss.
Good judgment is required in defining the finite region in space, the control volume, used
in solving a problem. What exactly to leave out of and what to leave in the control volume are important
considerations. The formulas resulting from applying the fundamental laws to the contents
of the control volume are easy to interpret physically and are not difficult to derive and use.
Because a finite region of space, a control volume, contains many fluid particles and even
more molecules that make up each particle, the fluid properties and characteristics are often average
values. In Chapter 6 an analysis of fluid flow based on what is happening to the contents of
an infinitesimally small region of space or control volume through which numerous molecules
simultaneously flow (what we might call a point in space) is considered.
5.1 Conservation of Mass—The Continuity Equation
The amount of
mass in a system is
constant.
5.1.1 Derivation of the Continuity Equation
A system is defined as a collection of unchanging contents, so the conservation of mass principle
for a system is simply stated as
or
time rate of change of the system mass 0
DM sys
where the system mass, M sys , is more generally expressed as
Dt
0
(5.1)
M sys sys
r dV
(5.2)
and the integration is over the volume of the system. In words, Eq. 5.2 states that the system mass
is equal to the sum of all the density-volume element products for the contents of the system.
For a system and a fixed, nondeforming control volume that are coincident at an instant of
time, as illustrated in Fig. 5.1, the Reynolds transport theorem 1Eq. 4.192 with B mass and b 1
allows us to state that
D
Dt sys
r dV 0 0t cv
r dV cs
rV nˆ dA
(5.3)
System
Control Volume
(a) (b) (c)
F I G U R E 5.1 System and control volume at three different
instances of time. (a) System and control volume at time t Dt.
(b) System and
control volume at time t, coincident condition. (c) System and control volume at
time t Dt.