Control Volume Analysis Using Energy
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Transcript of Control Volume Analysis Using Energy
Introduction
To develop and illustrate the use of the control volume
form of the conservation of mass and conservation of energy
principles. Mass and energy balances are applied to control
volumes at steady state and for transient applications.
Devices such as turbines, pumps and compressors
through which mass flows can be analyzed in principle by
studying a particular quantity of matter (a closed system) as
it passes through the device, it is normally preferable to think
of a region of space through which mass flows ( CV).
As in the case of a closed system, energy transfer
across the boundary of a control volume can occur by means
of work and heat.
Conservation of Mass and the Control Volume
Control volume. A volume in space in which one has
interest for a particular study or analysis. The surface of this
control volume is referred to as a control surface and always
consist of a closed surface.
The size and shape of the control volume are
completely arbitrary.
The surface may be fixed, or it may move so that it
expands or contracts.
Mass as well as heat and work can cross the
control surface.
Mass in the control volume, as well as the properties
of this mass , can change with time.
The term control volume is used in open system.
A control volume differs from a closed system in that
it involves mass transfer. Mass carries energy with it, and
thus the mass and energy content of a system change
when mass enters or leaves.
The control volume approach will be used for many
engineering problems where a mass flow rate is present,
such as: Turbines, pumps and compressors, Heat
exchangers, Nozzles and diffusers, etc.
Conservation of Mass and the Control Volume
Control Mass. A system of fixed mass is called
a closed system, or control mass .
The closed system boundary does not have to
be fixed.
No mass can cross the closed system
boundary.
Energy in the form of heat and work can cross
the closed system boundary
The term control mass is some times used in
place of closed system.
Conservation of Mass and the Control Volume
Conservation of Mass
The physical law concerning mass, says that we
cannot create or destroy mass.
The rate of change of mass inside the control
volume equal to mass enters minus mass exit.
i.e Rate of change = + in out
Conservation of Mass and the Control Volume
Let us consider the conservation
of mass law as it relates to the control
volume. The law says that we cannot
create or destroy mass. We will express
this law in a mathematical statement
about the mass in the control volume.
To do this we must consider all the
mass flows into and out of the control
volume and the net increase of mass
within the control volume. As a control
volume we consider a tank with a
cylinder and piston and two pipes attaches as shown in Fig.
We know from the first law of thermodynamics for a control mass,
dECM / dt = QCM WCM …………………………………..(1) . .
Rate of energy change in Cm = (Energy that enters into the
Cm Energy that exit from the Cm)
Conservation of Mass and the Control Volume
For Control volume we can write
Rate of energy change in CV = Energy that enters into the
CV Energy that exit from the CV .………………….. (2)
The rate of change of mass inside the control volume equal to mass enters minus mass exit.
i.e Rate of change = + in out
Conservation of Mass and the Control Volume
............................... ……………..(3)
Equation 3 is the mass rate balance for control volumes with
several inlets and exits.
For several inlet and outlet
Conservation of Mass and the Control Volume
The First Law of Thermodynamics
for a Control Volume.
A control volume is shown in
Fig. that involves rate of heat
transfer, rates of work, and
mass flows. The fundamental
physical law states that we
cannot create or destroy
energy such that any rate of
change of energy must be
caused by rates of energy into or out of the control volume.
dEcv / dt = Energy in Energy out............................................... (3)
The fluid flowing across the control surface enters or leaves with an amount of energy per unit mass as
e i = u i + ½. V 2 + g Zi ( flow liquid energy)
and e e = u e + ½. V 2 + g Ze
Whenever a fluid mass enters a control volume at state i, or
exits at state e, there is a boundary movement work associated with that process. This is called flow work .
The First Law of Thermodynamics
for a Control Volume.
Accordingly, the conservation of energy principle applied to a
control volume states:
For the one-inlet and one-exit control volume with one-
dimensional flow shown in Fig. 4 the energy rate balance is:
Conservation of Energy for a
Control Volume
.…....(3)
where,
ECV = Energy of the control volume at time t.
Q and W = The net rate of energy transfer by heat and work
across the boundary of the control volume at t respectively.
Underlined Terms = the rates of transfer of internal, kinetic,
and potential energy of the entering and exiting streams.
If there is no mass flow in or out, the respective mass flow
rates vanish and the underlined terms of Eq. 3 drop out. The
equation then reduces to the rate form of the energy balance
for closed systems.
Conservation of Energy for a
Control Volume
Work is always done on or by a control volume where
matter flows across the boundary, it is convenient to
separate the work term into two contributions:
Wf = Work associated with the fluid pr as mass is
introduced at inlets and removed at exits i.e flow work.
= Includes all other work effects, such as those
associated with rotating shafts, displacement of the
boundary, and electrical effects.
Conservation of Energy for a
Control Volume
With these considerations, the work term W of the energy
rate equation, Eqn. 3, can be written as
.
Since A = m v / V ......................…......(4)
Substituting Eq. 4. in Eq. 3 and collecting all terms referring to
the inlet and the exit into separate expressions, the following
form of the control volume energy rate balance results
Conservation of Energy for a
Control Volume
or
Eqn 5 is the energy rate balance for single inlet and single
out let.
....(5)
Conservation of Energy for a
Control Volume
In practice there may be several locations on the boundary
through which mass enters or exits. This can be accounted
for by introducing summations as in the mass balance.
Accordingly, the energy rate balance is
Conservation of Energy for a
Control Volume
Application of First Law of Thermodynamics to flow process
Steady State. At the steady state of a system, any
thermodynamic property will have a fixed value at a particular
location, and will not alter with time. Thermodynamic
properties may vary along space coordinates, but do not vary
with time. `Steady state` means that the state is steady or
invariant with time.
Steady Flow Process. Steady flow` means that the
rates of flow of mass and energy across the control surface
are constant.
Assumptions.
The CV does not move relative to the co ordinate
frame i.e the CV is fixed.
The state of the mass at each point in the control
volume does not vary with time.
i.e dmCV / dt = 0 and dECV / dt = 0.
For the steady state process we can write
Continuity Eqn: = m
Application of First Law of Thermodynamics to flow process
The rates at which heat and work cross the control
surface remain constant. i.e heat transfer and work
transfer is fixed. If there is only one flow stream entering and
one leaving the CV.
For this type of process, we can write
Continuity Eqn: mi = me = m
and First Law :
= +
Application of First Law of Thermodynamics to flow process
Example of Steady Flow Process.
Nozzle and Diffuser. A
nozzle is a steady-state device
which increase the velocity or K.E
of a fluid at the expense of its pr
drop, whereas a diffuser increase
the pr of a fluid at the expense of
its K.E. Fig Shows a nozzle and a
Diffuser. The steady flow energy eqn of the control surface gives
The steady flow energy eqn of the control surface gives
QCv+ m [ hi +½ Vi2 +gZi] = Wcv+ m [ he+ ½ Ve
2 + gZe] ....... (1)
Assumption.
a. Nozzle and Diffuser are SSSF device.
b. Change in P.E is negligible i.e ∆ P.E =0.
c. Nozzle is insulated i.e no heat enters or leaves the
system (adiabatic). QCv = 0
d. Work transfer is negligible i.e Wcv= 0.
Example of Steady Flow Process
So energy eqn for nozzle and diffuser becomes
Example of Steady Flow Process
m ( hi + ½ Vi2 ) = m ( he + ½ Ve
2)
or, ( hi + ½ Vi
2 ) = ( he + ½ Ve 2)
or, Ve
2 = Vi2 + 2 ( hi he)
or Ve = Vi
2 + 2 ( hi he) [ Vi is very small as compare to Ve]
or Ve = 2 ( hi he)
Throttling Process. A throttling process occurs
when a fluid flows through a constricted passage, like a
partially opened valve, an orifice or a porous plug with a
significant drop in pr. This is a SSSF process with no heat
transfer to or from CV. There is no means for doing work and little or no change in PE.
Here, ∆PE = 0, ∆KE = 0, QCv = 0 and WCv = 0
Example of Steady Flow Process.
Steam enters a converging–diverging nozzle operating at steady state with p1 = 40 bar, T1 = 4000C, and a velocity of 10 m/s. The steam flows through the nozzle with negligible heat transfer and no significant change in potential energy. At the exit, p2 =15 bar, and the velocity is 665 m/s. The mass flow rate is 2 kg/s. Determine the exit area of the nozzle, in m2.
Problem-1
Steam enters a turbine operating at steady state with a mass flow rate of 4600 kg/h. The turbine develops a power output of 1000 kW. At the inlet, the pr is 60 bar, the temperature is 4000C, and the velocity is 10 m/s. At the exit, the pr is 0.1 bar, the quality is 0.9 (90%), and the velocity is 50 m/s. Calculate the rate of heat transfer between the turbine and surroundings, in kW.
Problem-2