Modern Engineering Thermodynamics

670 CHAPTER 16: Compressible Fluid Flow
where _X = dX=dt is the rate of change of X within the system (either closed or open) we are analyzing, X G is the
net gain in X by the system, X T is the net amount of X transported into the system, and X P is the net amount of
X produced inside the system boundaries by some internal process. We note in Chapter 2 that, if X is conserved,
then X P = _X P = 0.
This balance equation is conceptually accurate. The only problem that arises in its use is that the form of the derivative
for the net gain rate of X by the system depends on whether the system is open or closed. We never had to
consider this in the past, because these two derivative forms happen to be identical for the simple one-dimensional
analysis discussed in this book. However, many times (especially in the study of fluid mechanics), a onedimensional
analysis is not sufficient to solve the problem; and we have to expand to a multidimensional
approach, which is often supported by a computer-based numerical technique. Because of the basic power of the
Reynolds transport theorem in being able to transform differentiation operations back and forth between multidimensional
closed and open systems, because our general rate balance equations have such differentials, and
because we deal with both closed and open systems in thermodynamics, the Reynolds transport theorem is being
introduced at this point in its full three-dimensional form. We use it only to transform the left side of Eq. (2.12),
the system rate of gain term. The remaining transport and production rate terms are handled differently.
The Reynolds transport theorem is named after the British engineer and physicist Osborne Reynolds (1842–1912).
Its derivation is quite complex and is not presented here, but the interested reader may wish to consult fluid
mechanics texts for more information. Because there is a difference between the differentiation operation for a
closed system and that for an open system, we must create a new notation that acknowledges this difference. 2
Therefore, let
_X G = DX
Dt
= the rate of gain of X by a closed ðLagrangianÞsystem
where we use the operator symbol D/Dt to denote a fixed mass or closed system time derivative. The same time
derivative measured in an open system is given by the Reynolds transport theorem as
DX
Dt
⎵
Closed system
ðLagrangianÞ rate
Z
=
V
Z
∂ðρxÞ
dV +
∂t
⎵
A
Open system
ðEulerianÞ rate
ρxðV . dAÞ
(16.24)
where x = X/m is the intensive (specific) version of X, andV is the velocity vector of x as it crosses the surface
area element dA, as shown in Figure 16.19. The transport and production rate terms on the right side of
Eq. (2.12) can be generalized for a closed system to be of the form
Z
_X T = − J x
.dA (16.25)
and
A
Z
_X P = σ x dV (16.26)
V
where J x is the flux (flow per unit area per unit time) of x through the area element dA, and σ x is the local production
rate per unit volume of x inside the volume element dV. Note that the closed and open systems described by
the Reynolds transport theorem do not have the same system boundary and in fact are not the same physical system.
A closed system that is to be considered equivalenttoagivenopensystemmustbemuchlargerthanthe
open system. It must be large enough to include all the mass that crosses the boundary of the open system during
the analysis period, and it is therefore always much larger than the open system it emulates. Consequently, equivalent
closed systems are generally quite awkward and difficult to define, and it was this characteristic that ultimately
provided the motivation for developing the open system (or Eulerian) analysis technique.
Combining Eqs. (16.24), (16.25), and (16.26) into the general rate balance Eq. (2.12) and rearranging it slightly
gives the generalized open system rate balance equation:
Z
v
Z
∂ðρxÞ
dV = −
∂t
A
Z
J x
.dA −
A
Z
ρxðV .dAÞ + σ x
.dV (16.27)
V
2 In fluid mechanics texts, this derivative is often given a special name, such as the material derivative or the substantial derivative.