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Chapter 6

MASS AND ENERGY

ANALYSIS OF CONTROLVOLUMES

Fundamentals of Thermal-Fluid Sciences, 3rd EditionYunus A. Cengel, Robert H. Turner, John M. Cimbala

McGraw-Hill, 2008

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Objectives Develop the conservation of mass principle.

Apply the conservation of mass principle to various systems

including steady- and unsteady-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, mixers,

heaters, and heat exchangers.

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

Mass is conserved even during chemical reactions.

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 aprocess.

Control volumes: Mass can cross the boundaries, and so we mustkeep track of the amount of mass entering and leaving the controlvolume.

Mass m and energy Ecan be converted to each other according to

where cis the speed of light in a vacuum, which is c = 2.9979 108 m/s.

The mass change due to energy change is absolutely negligible.

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Mass and Volume Flow Rates

The average velocity Vavg is

defined as the average speed

through a cross section.

The volume flow rate is the

volume of fluid flowing through a

cross section per unit time.

Definition of

average velocity

Mass flow

rate

Volume flow rate

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Conservation of Mass Principle

Conservation of mass principle

for an ordinary bathtub.

The conservation of mass principle for a control volume: The net mass transfer

to or from a control volume during a time interval Dtis equal to the net change

(increase or decrease) in the total mass within the control volume duringD

t.

General conservation of mass

General conservation of mass in rate form

or

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Mass Balance for Steady-Flow Processes

Conservation of mass principle for a two-

inletone-outlet steady-flow system.

During a steady-flow process, the total amount of mass contained within a

control volume does not change with time (mCV = constant).

Then the conservation of mass principle requires that the total amount of mass

entering a control volume equal the total amount of mass leaving it.

For steady-flow processes, we are

interested in the amount of mass flowing per

unit time, that is, the mass flow rate.

Multiple inletsand exits

Single

stream

Many engineering devices such as nozzles,

diffusers, turbines, compressors, and

pumps involve a single stream (only oneinlet and one outlet).

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Special Case: Incompressible Flow

During a steady-flow process, volume

flow rates are not necessarily conserved

although mass flow rates are.

The conservation of mass relations can be simplified even further when

the fluid is incompressible, which is usually the case for liquids.

Steady,

incompressible

Steady,

incompressible

flow (single stream)There is no such thing as a conservation of volume

principle.

However, for steady flow of liquids, the volume flow

rates, as well as the mass flow rates, remain

constant since liquids are essentially incompressible

substances.

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FLOW WORK AND THE

ENERGY OF A FLOWING FLUID

Schematic for flow work.

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

required to push the mass into or out of the controlvolume. This work is necessary for maintaining a

continuous flow through a control volume.

In the absence of acceleration, the force

applied on a fluid by a piston is equal to the

force applied on the piston by the fluid.

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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

The product is the energy

transported into control volume by

mass per unit time.

iimq&

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

Many engineering systems such

as power plants operate under

steady conditions.

Under steady-flow conditions, the mass

and energy contents of a control volumeremain constant.

Under steady-flow conditions,

the fluid properties at an inlet

or exit remain constant (do not

change with time).

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Mass and Energy balances

for a steady-flow process

A water

heater in

steady

operation.

Mass

balance

Energy

balance

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Under steady operation,shaft work and electrical

work are the only forms of

work a simple compressible

system may involve.

Energy balance relations with sign conventions

(i.e., heat input and work output are positive)

when kinetic and potential energy

changes are negligible

Some energy unit equivalents

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SOME STEADY-FLOW ENGINEERING 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.

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.

At very high velocities,

even small changes invelocities can cause

significant changes in

the kinetic energy of the

fluid.

Total electric power generation of Korea: 55,000 MW

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Nozzles and Diffusers

Nozzles and diffusers areshaped so that they cause large

changes in fluid velocities and

thus kinetic energies.

Nozzles and diffusers are commonly

utilized in jet engines, rockets,

spacecraft, and even garden hoses.

A nozzle is a device that increasesthe velocity of a fluidat the expense

of pressure.

A diffuseris a device that increases

the pressure of a fluidby 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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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 sourcethrough a rotating shaft.

A fan increases the pressure of a gas

slightly and is mainly used to mobilize a

gas.

A compressoris capable of compressing

the gas to very high pressures. Pumps work very much like compressors

except that they handle liquids instead of

gases.

Energy balance for the

compressor in this figure:

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Throttling valvesThrottling valves are any kind of flow-restrictingdevices

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 alarge drop in temperature, and for that reason throttling

devices are commonly used in refrigeration and air-

conditioning applications.

The temperature of an ideal gas does

not change during a throttling

(h = constant) process since h = h(T).

During a throttling process, the enthalpy

of a fluid remains constant. But internal

and flow energies may be converted to

each other.

Energy

balance

t

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[Example 6-8] Expansion(throttling) of R-134a [p.221]R-134a enters the capillary tube.Inlet (State 1): 0.8 MPa, sat. liquidExit (State 2): 0.12 MPaFind (1) exit quality, (2) exit temperatureAssume steady flow, KPE=0Then first law of thermodynamics becomes 21 hh =

kJ/kg47.951 =hable A-12 (p.1008)kJ/kg47.952 =h

( )fgf hhxhh -+= 22fg

f

hh

hhx

-

-=

2

2

3403.0

49.2297.236

49.2247.952 =

-

-=x

C32.22 o2 -=t

Table A-12 (p.1008)

Table A-12 (p.1008)

C31.31 o1 =t

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Mixing chambers

In engineering applications, the section

where the mixing process takes place is

commonly referred to as a mixingchamber.

The T-elbow of an ordinary shower

serves as the mixing chamber for the

hot- and the cold-water streams.

Energy balance for the

adiabatic mixing chamber in

the figure is:

10C

60C

43C

140 kPa

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Heat exchangers

Heat exchangers are

devices where two moving

fluid streams exchange heatwithout mixing. Heat

exchangers are widely used

in various industries, and

they come in various

designs.

A heat exchanger

can be as simple as

two concentric pipes.

Mass and energy

balances for the adiabatic

heat exchanger in the

figure is:

The heat transfer associated with a heat

exchanger may be zero or nonzero depending on

how the control volume is selected.

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[Example 6-10] Heat exchanger, cooling R-134a by water [p.224]Water (State 1): 15, 300 kPa comp. liquidWater (State 2): 25, 300 kPa comp. liquidR-134a (State 3): 6 kg/min, 1 MPa, 70 sup. vaporR-134a (State 4): 35, 1 MPa comp. liquidFind (1) flow rate of water, (2) heat transfer fromrefrigerant(R-134a) to water

Assume steady flow, insulated (Q=0), KPE=0, W=0Then first law of thermodynamics becomes43213311 hmhmhmhm &&&& +=+

+=+outin

hmWhmQ &&&&st law

kJ/kg982.621 =hable A-4 (p.994) kJ/kg83.1042 =hTable A-13 (p.1010) kJ/kg85.3033 =hTable A-11 (p.1007) kJ/kg865.1004 =h

minkg1.29

minkg6

982.6283.104865.10085.3033

12

431 =

--=

--= mhhhhm &&

2111 hmhmQ waterref &&& =+ ( )121 hhmQ waterref -= &&

( ) kJ/min1218982.6283.1041.29 =-=waterrefQ&

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Pipe and duct flow

The transport of liquids or gases in

pipes and ducts is of great importance

in many engineering applications. Flowthrough a pipe or a duct usually satisfies

the steady-flow conditions.

Heat losses from

a hot fluid

flowing through

an uninsulated

pipe or duct tothe cooler

environment

may be very

significant.

Pipe or duct flow may involve more than

one form of work at the same time.

Energy balance

for the pipe flow

shown in the

figure is

for liquid ( )1212 TTChh p --

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[Example 6-11] Electric heating of air [p.227]Duct with electric heater (15 kW)Air in (State 1): 100 kPa, 17, 150 m3/minHeat loss 0.2 kWFind the exit temperature of airAssume air as an ideal gas with constant Cp,KPE=0

2211 hmWhmQ &&&& +=+st law

( ) ( )1212 ttCmhhmWQ p -=-=- &&&&

( )kg

m8327.0

100

15.27317287.03

1

11 =

+==

P

TRu

s

kg00.3

min

kg14.180

kg

m

8327.0

min

m150

3

3

1

1 ====u

Vm

&

&

C9.2117005.100.3

)15(2.0 o12 =+

---=+

-= t

Cm

WQt

p&

&&

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[Example 6-12] Charging of a rigidinsulated tank by steam [p.230]Tank: initially evacuatedSupply steam: 1 MPa, 300Steam enters the tankFinal state of tank inside: 1 MPaDetermine the final temperatureAssume KPE=0, Q=0, W=01st law WhmumumhmQ eeii ++-=+ 1122

22umhm ii =

kJ/kg6.3051=ihable A-6 (p.999)2mmi = 2uhi =

Final state (state 2) kJ/kg6.30512 =uMPa,12 =P

T () u (kJ/kg)400 2957.9T2 = ? u2 = 3051.6500 3125.0

Linear interpolationC1.456 o2 =T

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Summary

Conservation of mass

Mass and volume flow rates

Mass balance for a steady-flow process

Mass balance for incompressible flow

Flow work and the energy of a flowing fluid

Energy transport by mass

Energy analysis of steady-flow systems Some steady-flow engineering devices

Nozzles and Diffusers

Turbines and Compressors

Throttling valves

Mixing chambers and Heat exchangers

Pipe and Duct flow