A Flow Element
Flow Work with Imaginary Piston


Work
is needed to push the fluid into or out of the boundaries of a control
volume if mass flow is involved. This work is called the flow work
(flow energy). Flow work is necessary for maintaining
a continuous flow through a control volume.
Consider a fluid element of volume V, pressure P, and crosssectional
area A as shown left. The flow immediately upstream will force this fluid
element to enter the control volume, and it can be regarded as an imaginary
piston. The force applied on the fluid element by the imaginary piston is:
F = PA
The work done due to pushing the entire fluid element across the boundary
into the control volume is
W_{flow }= FL = PAL = PV
For unit mass,
w_{flow }= Pv
The work done due to pushing the fluid element out of the control volume
is the same as the work needed to push the fluid element into the control
volume.

Total Energy of a Flowing Fluid


The total energy of a simple compressible system consists of three parts:
internal, kinetic, and potential energy.
E = U + KE + PE
For unit mass,
e = u + ke + pe = u + v^{2}/2 + gz
where
e =
total energy
u = internal energy
v = velocity of the system
z = the elevation of the fluid
The fluid entering or leaving a control volume possess an additional
energy, the flow work (Pv). Hence, the total energy of a flowing fluid becomes
θ = Pv + u + v^{2}/2 + gz
where
θ = methalpy, the total energy of a flowing fluid
The definition of enthalpy gives
h = Pv + u
Replacing Pv + u by h yields
θ = h + v^{2}/2 + gz
By using the enthalpy instead of internal energy, flow work is not a concern.



The conservation of mass principle, which has been previously introduced,
in rate format, is:
During a steadyflow process, the total amount of mass contained within
a control volume does not change with time. That is,
dm_{system}/dt = 0
Hence the conservation of mass principle gives the total amount of
mass entering a control volume equal to the total amount of mass leaving
it. In an equation format, it is 
Mass and Energy balance for
Steadyflow Process 

(Total mass entering the control volume per unit time)
=
(Total mass leaving the control volume per unit time)
or,
where
i = inlet
e = exit
Also, the energy balance for a process, which has been previously
introduced, in rate format, is:
For a steadyflow process, the total energy content of a control volume
remains constant. That is,
dE_{system}/dt = 0
Thus, the amount of energy entering a control volume in all forms
(heat, work, mass transfer) must be equal to the amount of energy leaving
it for a steadyflow process. In an equation format, it is
(Rate of net energy transfer in by heat, work and mass)
=
(Rate of net energy transfer out by heat, work and mass)
or,
For a general steadyflow process, the energy balance
can be written as
If the sign introduced previously for heat and
work is used, the energy balance for a general steadyflow process can
be rewritten as:
