Modern Engineering Thermodynamics

6.2 Mass Flow Energy Transport 169
theslugmovesadistancej V ! jdt = Vdt and sweeps out an incremental volume dV = AV dt, which has an
associated mass of dm = ρ dV = ρAV dt: Dividing by dt gives
dm
= ð _mÞ
dt
flow stream = ðρAVÞ flow stream (6.1)
flow stream
This equation is a very convenient way to calculate the flow stream mass flow rate from the easily measured
variables of density (ρ), cross-sectional area (A), and average fluid velocity ðj V ! j = VÞ:
The incremental energy required to push the mass slug across the system boundary and into the system is the
product of the force acting on it, p × A, and the distance moved, dL. Consequently, the flow work energy increment
is
dW mass
flow
= − pA dL = − pdV = − p dm = − pv dm
ρ
where v =1/ρ is the specific volume of the slug. The flow work is negative here because adding dm to the system
represents work done on the system.
The total mass flow energy entering the system with this incremental mass is then
dE mass
= dE m − dW mass
flow
flow
= ðu + ke + peÞ dm + ðpvÞ dm
= ðu + pv + ke + peÞ dm
Because the sum of the terms u and pv always appears in this equation, it is convenient to combine them into a
single term (as explained in Chapter 3) called specific enthalpy, h = u + pv.
In general, we have more than one mass flow stream in any given open system. To accurately account for all the
mass flow energies, we sum them in two groups. One group accounts for all inlet flow streams and the other
for all exiting flow streams. Therefore, we write
dE mass
flow
= ∑
inlet
ðh + ke + peÞ dm − ∑ ðh + ke + peÞ dm
outlet
On dividing this equation through by dt, we obtain
_E mass
= ∑
flow inlet
_mðh + ke + peÞ − ∑
outlet
_mðh + ke + peÞ (6.2)
and on integration of this equation, we obtain
1
E mass
flow
!
2
Z 2
Z 2
= ∑ _mðh + ke + peÞ dt − ∑ _mðh + ke + peÞ dt (6.3)
inlet
outlet
1
1
where ke = V 2 /2g c and pe = gZ/g c are the specific kinetic and potential energies of the flow streams at the point
where they cross the system boundary. Note that these equations already contain the proper thermodynamic
signs for input (+) and output (−) mass flow energy transport.
Each flow stream has its own average velocity V and height Z; in addition, the center of gravity of the entire
system has unique and usually different V and Z values (see Figure 6.1). The student must be careful not to get
these velocities and heights confused.