Modern Engineering Thermodynamics

172 CHAPTER 6: First Law Open System Applications
where proper input and output signs are to be used in the summations. Also, the kinetic and potential energy terms
on the right side of Eq. (6.4) are of the center of gravity of the entire system, whereas the kinetic and potential energy
terms in the flow stream summation terms on the left side of this equation apply only to the point of entry or exit of
the flow stream from the system (see Figure 6.1 for an illustration of this notation).
As a working equation, Eq. (6.4) is really too complex to remember or write down conveniently during the solution
of each thermodynamic problem we face. Since most of our problems involve systems operating at steady
state with a single inlet and a single outlet flow stream, we simplify Eq. (6.4) to fit this case. For a steady state
process, the entire right side of Eq. (6.4) vanishes:
Note that this does not necessarily mean that _U,
Steady state
_E G = d dt ðU + mV2 /2g c + mgZ/g c Þ system = 0 (6.6)
KE, _ and PE _ are all zero but only that their sum vanishes.
At this point, we introduce the conservation of mass law for open systems. This law can easily be cast into the
form of a rate balance by using the general form of Eq. (2.14) as _m G = _m T , where the mass transport rate is
simply given by
_m T = ∑
inlet
_m − ∑
outlet
_m (6.7)
Thus, the general mass rate balance for the rate of gain of mass _m G for an open system is simply
_m G =
dm
= ∑ _m − ∑ _m (6.8)
dt system
inlet outlet
Now, if a system is operating at steady state, then, by definition,
dE
dt system
so that Eq. (6.8) gives the steady state mass rate balance as
= dm
= _E G = _m G = 0 (6.9)
dt system
∑
inlet
_m = ∑
outlet
The condition of equal mass inflows and outflows is called a steady flow:
∑
inlet
_m = ∑
oulet
_m
_m (6.10)
It should be clear from this development that any steady state open system is also (by definition) a steady flow
system. To keep this clearly in mind, we often write both statements, steady state and steady flow, explicitly,
even though it is not really necessary to do so.
If the system has only one inlet and one outlet flow stream, then the summation signs can be dropped in Eqs. (6.4),
(6.7), (6.8), and (6.10). The steady flow condition for a system with a single inlet and a single outlet flow stream
then becomes
_m inlet = _m outlet = _m (6.11)
Note that the inlet-outlet direction subscripts on the mass flow rate term can now be dropped because they are
superfluous.
Substituting Eqs. (6.6) and (6.11) into Eq. (6.4), and abbreviating the terms inlet and outlet as simply in and
out gives a simplified energy rate balance. We call the resulting formula the modified energy rate balance (MERB).
Thus, the open system modified energy rate balance applies only to systems that are
1. Steady state: ð _E G = 0Þ:
2. Steady flow: ð _m G = 0Þ:
3. Single inlet and single outlet: ð _m inlet = _m outlet = _mÞ:
and has the following form:
The open system modified energy rate balance (MERB)
_Q − _W + _m½h in − h out + ðVin 2 − V2 out Þ/ð2g cÞ + ðZ in − Z out Þðg/g c ÞŠ = 0 (6.12)