CHAPTER 5

MASS AND ENERGY ANALYSIS OF CONTROL VOLUMES

Lecture slides by

Dr. Fawzi Elfghi

Thermodynamics: An Engineering Approach 8th Edition in SI Units

Yunus A. Çengel, Michael A. Boles

McGraw-Hill, 2015

2

Objectives

• Develop the conservation of mass principle.

• Apply the conservation of mass principle to various systems

including steady-flow control volumes.

• Apply the first law of thermodynamics as the statement of the

conservation of energy principle to control volumes.

• Identify the energy carried by a fluid stream crossing a control

surface as the sum of internal energy, flow work, kinetic energy,

and potential energy of the fluid and to relate the combination of

the internal energy and the flow work to the property enthalpy.

• Solve energy balance problems for common steady-flow devices

such as nozzles, compressors, turbines, throttling valves,

• Apply the energy balance to general unsteady-flow processes with

particular emphasis on the uniform-flow process as the model for

commonly encountered charging and discharging processes.

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CONSERVATION OF MASS Conservation of mass: Mass, like energy, is a conserved property, and it cannot be created or destroyed during a process.

Closed systems: The mass of the system remain constant during a process.

Control volumes: Mass can cross the boundaries, and so we must keep track of the amount of mass entering and leaving the control volume.

Mass m and energy E can be

converted to each other

according to

where c is the speed of light in a

vacuum, which is c = 2.9979 108

m/s.

The mass change due to energy

change is negligible.

4

Conservation of Energy for Control volumes

The conservation of mass and the conservation of energy principles for open systems

or control volumes apply to systems having mass crossing the system boundary or

control surface. In addition to the heat transfer and work crossing the system

boundaries, mass carries energy with it as it crosses the system boundaries. Thus,

the mass and energy content of the open system may change when mass enters or

leaves the control volume.

Thermodynamic processes involving control volumes can be considered in two

groups: steady-flow processes and unsteady-flow processes. During a steady-flow

process, the fluid flows through the control volume steadily, experiencing no change

with time at a fixed position.

5

A pressurized water nuclear reactor steam power plant has many

examples of open system operation. Some of these are the

pressure vessel, steam generator, turbine, condenser, and pumps.

Photo courtesy of Progress Energy Carolinas, Inc.

6

A heat exchanger, the heater core from a 1966 289 V8

Mustang, is another example of an open system. Cooling

water flows into and out of the tubes and air is forced

through the fin sturucture.

7

A hair drier is an example of a one entrance, one exit open

system that receives electrical work input to drive the fan

and power the resistance heater.

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Let’s review the concepts of mass flow rate and energy transport by mass.

One should study the development of the general conservation of mass presented in

the text. Here we present an overview of the concepts important to successful

problem solving techniques.

Mass Flow Rate

Mass flow through a cross-sectional area per unit time is called the mass flow rate .

Note the dot over the mass symbol indicates a time rate of change. It is expressed

as

m

where is the velocity normal to the cross-sectional flow area.

Vn

9

If the fluid density and velocity are constant over the flow cross-sectional area, the

mass flow rate is

aveave

V Am V A

v

where is the density, kg/m3 ( = 1/v), A is the cross-sectional area, m2; and is the

average fluid velocity normal to the area, m/s. aveV

Example 5-1

Refrigerant-134a at 200 kPa, 40% quality, flows through a 1.1-cm inside diameter, d,

tube with a velocity of 50 m/s. Find the mass flow rate of the refrigerant-134a.

At P = 200 kPa, x = 0.4 we determine the specific volume from table A12, Page 918

3

0.0007533 0.4(0.0999 0.0007533)

0.0404

f fgv v xv

m

kg

2

2

3

4

50 / (0.011 )

0.0404 / 4

0.117

ave aveV A V dm

v v

m s m

m kg

kg

s

10

The fluid volume flowing through a cross-section per unit time is called the volume

flow rate . The volume flow rate is given by integrating the product of the velocity

normal to the flow area and the differential flow area over the flow area. If the

velocity over the flow area is a constant, the volume flow rate is given by (note we are

dropping the “ave” subscript on the velocity)

V

( / )V VA m s

3

The mass and volume flow rate are related by

( / )m VV

vkg s

Example 5-2

Air at 100 kPa, 50oC, flows through a pipe with a volume flow rate of 40 m3/min. Find

the mass flow rate through the pipe, in kg/s.

Assume air to be an ideal gas, so From table A1, page 898 , we find the gas constant

(R)

vRT

P

kJ

kg K

K

kPa

m kPa

kJ

m

kg

0 287273

100

0 9270

3

3

.(50 )

.

11

/ min

. /

min

.

mV

v

m

m kg s

kg

s

40

0 9270

1

60

0 719

3

3

Conservation of Mass for General Control Volume

The conservation of mass principle for the open system or control volume is

expressed as

or

( / )m m m kg sin out system

Steady-State, Steady-Flow Processes

Most energy conversion devices operate steadily over long periods of time. The

rates of heat transfer and work crossing the control surface are constant with time.

The states of the mass streams crossing the control surface or boundary are constant

with time. Under these conditions the mass and energy content of the control volume

are constant with time.

12

0CVCV

dmm

dt

Steady-state, Steady-Flow Conservation of Mass:

Since the mass of the control volume is constant with time during the steady-state,

steady-flow process, the conservation of mass principle becomes

or

( / )m m kg sin out

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Flow work and the energy of a flowing fluid

Energy flows into and from the control volume with the mass. The energy required to

push the mass into or out of the control volume is known as the flow work or flow

energy.

The fluid up steam of the control surface acts as a piston to push a unit of mass into

or out of the control volume. Consider the unit of mass entering the control volume

shown below.

As the fluid upstream pushes mass across the control surface, work done on that unit

of mass is

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FLOW WORK AND THE ENERGY OF A FLOWING FLUID

Flow work, or flow energy: The work (or energy)

required to push the mass into or out of the control

volume. This work is necessary for maintaining a

continuous flow through a control volume.

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Total Energy of a Flowing Fluid

The total energy consists of three parts for a nonflowing fluid and

four parts for a flowing fluid.

h = u + Pv

The flow energy is

automatically taken

care of by enthalpy.

In fact, this is the

main reason for

defining the property

enthalpy.

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Energy Transport by Mass

When the kinetic and potential energies

of a fluid stream are negligible

When the properties of the mass at

each inlet or exit change with time

as well as over the cross section

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ENERGY ANALYSIS OF STEADY-FLOW SYSTEMS

Steady-flow process: A process

during which a fluid flows through

a control volume steadily.

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The term Pv is called the flow work done on the unit of mass as it crosses the control

surface.

The total energy of flowing fluid

The total energy carried by a unit of mass as it crosses the control surface is the sum

of the internal energy, flow work, potential energy, and kinetic energy.

u PvV

gz

hV

gz

2

2

2

2

Here we have used the definition of enthalpy, h = u + Pv.

Energy transport by mass

Amount of energy transport across a control surface:

2

(kJ)2

mass

VE m m h gz

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Rate of energy transport across a control surface:

2

( )2

mass

VE m m h gz kW

Conservation of Energy for General Control Volume

The conservation of energy principle for the control volume or open system has the

same word definition as the first law for the closed system. Expressing the energy

transfers on a rate basis, the control volume first law is

or

E E E kWin out system

Rate of net energy transfer by heat, work, and mass

Rate change in internal, kinetic, potential, etc., energies

( )

Considering that energy flows into and from the control volume with the mass, energy

enters because net heat is transferred to the control volume, and energy leaves

because the control volume does net work on its surroundings, the open system, or

control volume, the first law becomes

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where is the energy per unit mass flowing into or from the control volume. The

energy per unit mass, , flowing across the control surface that defines the control

volume is composed of four terms: the internal energy, the kinetic energy, the

potential energy, and the flow work.

The total energy carried by a unit of mass as it crosses the control surface is

u PvV

gz

hV

gz

2

2

2

2

Where the time rate change of the energy of the control volume has been written

as ECV

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Steady-State, Steady-Flow Processes

Most energy conversion devices operate steadily over long periods of time. The

rates of heat transfer and work crossing the control surface are constant with time.

The states of the mass streams crossing the control surface or boundary are constant

with time. Under these conditions the mass and energy content of the control volume

are constant with time.

dm

dtm

dE

dtE

CVCV

CVCV

0

0

Steady-state, Steady-Flow Conservation of Mass:

( / )m m kg sin out Steady-state, steady-flow conservation of energy

Since the energy of the control volume is constant with time during the steady-state,

steady-flow process, the conservation of energy principle becomes

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E E E kWin out system

Rate of net energy transfer by heat, work, and mass

Rate change in internal, kinetic, potential, etc., energies

( )

0 or

or

Considering that energy flows into and from the control volume with the mass, energy

enters because heat is transferred to the control volume, and energy leaves because

the control volume does work on its surroundings, the steady-state, steady-flow first

law becomes

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Often this result is written as

where

Q Q Q

W W W

net in out

net out in

Steady-state, steady-flow for one entrance and one exit

A number of thermodynamic devices such as pumps, fans, compressors, turbines,

nozzles, diffusers, and heaters operate with one entrance and one exit. The steady-

state, steady-flow conservation of mass and first law of thermodynamics for these

systems reduce to

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where the entrance to the control volume is state 1 and the exit is state 2 and is the

mass flow rate through the device.

When can we neglect the kinetic and potential energy terms in the first law?

Consider the kinetic and potential energies per unit mass.

m

keV

2

2

For V = 45m

s

V = 140m

s

kem s kJ kg

m s

kJ

kg

kem s kJ kg

m s

kJ

kg

( / ) /

/

( / ) /

/

45

2

1

10001

140

2

1

100010

2

2 2

2

2 2

pe gz

For z m pem

sm

kJ kg

m s

kJ

kg

z m pem

sm

kJ kg

m s

kJ

kg

100 9 8 1001

10000 98

1000 9 8 10001

10009 8

2 2 2

2 2 2

./

/.

./

/.

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When compared to the enthalpy of steam (h 2000 to 3000 kJ/kg) and the enthalpy

of air (h 200 to 6000 kJ/kg), the kinetic and potential energies are often neglected.

When the kinetic and potential energies can be neglected, the conservation of energy

equation becomes

( ) ( )Q W m h h kW 2 1

We often write this last result per unit mass flow as

q w h h kJ kg ( ) ( / )2 1

qQ

m

w

W

m

where and .

Some Steady-Flow Engineering Devices

Below are some engineering devices that operate essentially as steady-state, steady-

flow control volumes.

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We may be familiar with nozzles, diffusers, turbines,

compressors, compressors, heat exchangers, and mixing

devices. However, the throttle valve may be a new device to

many of us. The Throttle may be a simple as the expansion

tube used in automobile air conditioning systems to cause

the refrigerant pressure drop between the exit of the

condenser and the inlet to the evaporator.

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SOME STEADY-FLOW ENGINEERING DEVICES

Many engineering devices operate essentially under the same conditions

for long periods of time. The components of a steam power plant (turbines,

compressors, heat exchangers, and pumps), for example, operate nonstop for

months before the system is shut down for maintenance. Therefore, these devices

can be conveniently analyzed as steady-flow devices.

A modern land-based gas turbine used for electric power

production. This is a General Electric LM5000 turbine. It

has a length of 6.2 m, it weighs 12.5 tons, and produces

55.2 MW at 3600 rpm with steam injection.

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Nozzles and Diffusers Nozzles and diffusers are commonly

utilized in jet engines, rockets,

spacecraft, and even garden hoses.

A nozzle is a device that increases the

velocity of a fluid at the expense of

pressure.

A diffuser is a device that increases

the pressure of a fluid by slowing it

down.

The cross-sectional area of a nozzle

decreases in the flow direction for

subsonic flows and increases for

supersonic flows. The reverse is true

for diffusers.

Energy

balance for

a nozzle or

diffuser:

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Deceleration

of Air in a

Diffuser

30

Acceleration

of Steam in a

Nozzle

= 2.8 kJ/kg

Steam

5 kg/s

1 1.8 MPa, 400C

0.02 m2

2 1.4 MPa

275 m/s

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Turbines and

Compressors Turbine drives the electric generator In

steam, gas, or hydroelectric power plants.

As the fluid passes through the turbine,

work is done against the blades, which are

attached to the shaft. As a result, the shaft

rotates, and the turbine produces work.

Compressors, as well as pumps and

fans, are devices used to increase the

pressure of a fluid. Work is supplied to

these devices from an external source

through a rotating shaft.

A fan increases the pressure of a gas

slightly and is mainly used to mobilize a

gas.

A compressor is capable of compressing

the gas to very high pressures.

Pumps work very much like compressors

except that they handle liquids instead of

gases.

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Compressing Air

by a Compressor

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Power Generation

by a Steam

Turbine

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Throttling

valves

Throttling valves are any kind of flow-restricting devices that

cause a significant pressure drop in the fluid.

What is the difference between a turbine and a throttling

valve?

The pressure drop in the fluid is often accompanied by a large

drop in temperature, and for that reason throttling devices are

commonly used in refrigeration and air-conditioning applications.

Energy

balance

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Expansion of

Refrigerant-134a

in a Refrigerator