The First Law ofThermodynamics:

Chapter 4

Thermodynamics:Control Volumes

Objectives• Develop the conservation of mass principle.

• Apply the conservation of mass principle to various systemsincluding steady- and unsteady-flow control volumes.

• Apply the first law of thermodynamics as the statement of theconservation of energy principle to control volumes.

• Identify the energy carried by a fluid stream crossing a controlsurface as the sum of internal energy, flow work, kinetic energy,surface as the sum of internal energy, flow work, kinetic energy,and potential energy of the fluid and to relate the combination ofthe internal energy and the flow work to the property enthalpy.

• Solve energy balance problems for common steady-flow devicessuch as nozzles, compressors, turbines, throttling valves, mixingchamber and heat exchangers.

• Apply the energy balance to general unsteady-flow processes withparticular emphasis on the uniform-flow process as the model forcommonly encountered charging and discharging processes.

Control VolumesControl Volumes

•A control volume ( open system) differsfrom a closed system where it involvesmass transfer. Mass carries energy with it,and thus the mass and energy in theand thus the mass and energy in thesystem change when mass enters orleaves.

Mass flow through a cross section per unittime is called the mass flow rate and isdenoted m. It is expressed as

where = density, kg/m3 (= 1/v)

= average fluid velocity normal to A, m/s

Mass Flow RateMass Flow Rate

.

= average fluid velocity normal to A, m/s

A = cross-sectional area, m2

The average velocity Vavg isdefined as the average speedthrough a cross section.

Volume Flow RateVolume Flow Rate

The fluid volume flowing through a crosssection per unit time is called the volumeflow rate V. It is given by

.

The volume flow rate is thevolume of fluid flowing through across section per unit time.

Mass & Volume Flow RateMass & Volume Flow Rate

The mass and volume flow rates arerelated by

where AVV av.

Mass & Energy BalanceMass & Energy Balance

The mass and energy balances for anysystem undergoing any process can beexpressed as

Conservation of mass principlefor an ordinary bathtub.

The mass and energy balances for anysystem undergoing any process can beexpressed in the rate form as

Rate of Mass & EnergyBalance

Rate of Mass & EnergyBalance

(kJ / s)

FLOW WORK AND THEENERGY OF A FLOWING FLUID

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 acontinuous flow through a control volume.

Schematic for flow work.

In the absence of acceleration, the forceapplied on a fluid by a piston is equal to theforce applied on the piston by the fluid.

Total Energy of a flowing fluid

gzV

upekeue

2

2

Total energy of a simple compressible system

The fluid possesses an additional form ofenergy –the flow energy (flow work)

pekeuPvePv

pekeh Total energy of a flowing fluid

Total Energy of 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.

The total energy consists of three parts for a non-flowing fluid and four parts for aflowing fluid.

Energy Transport by Mass

When the kinetic and potential energiesof a fluid stream are negligible

The product is the energytransported into control volume bymass per unit time.

iim

of a fluid stream are negligible

When the properties of the mass ateach inlet or exit change with timeas well as over the cross section(very seldom)

VV

22

We can rewrite the energy balance

TheThe equationequation inin thethe raterate formform cancan bebe writtenwritten asasfollowingfollowing ::

cvee

eenetii

iinetEgz

Vhmgz

Vhm WQ

22

22

This represent workThe flow workis included inthe enthalpyterm

This representheat transfer

Thermodynamic processes involvingcontrol volumes can be considered in twogroups:

1. Steady-flow processes2. Unsteady-flow processes.

Control Volume

2. Unsteady-flow processes.

During a steady-flow process, the fluidflows through the control volume steadily,experiencing no change with time at a fixedposition. The volume, mass and totalenergy content of the control volumeremain constant during a steady-flowprocess.

The Steady flow process– A process during which a fluid flows through a

control volume steadily

– Steady means no change with time

0

0

cv

cv

E

m

Under steady-flow conditions, the massand energy contents of a control volumeremain constant.

outin mm

022

2

,

2

,

ee

eeoutnetii

iiinnetgz

Vhmgz

Vhm WQ

ii

iiee

eegz

Vhmgz

Vhmoutnetinnet WQ

22,,

22

Taking heat transfer to the system and work done by thesystem to be positive quantities, the conservation of massand energy equations for steady-flow processes areexpressed as

Steady-flowSteady-flow

where the subscript i stands for inlet and e for exit. Theseare the most general forms of the equations for steady-flowprocesses.

for each exit for each inlet

Energy balance relations with sign conventions(i.e., heat input and work output are positive)

when kinetic and potential energychanges are negligible

Some energy unit equivalents

If there is no kinetic or potential energy

• For single-stream (one-inlet--one-exit) systems such asnozzles, diffusers, turbines, compressors, and pumps, thesteady flow equations simplify to

Steady-flow for one entrance and one exitSteady-flow for one entrance and one exit

In the above relations, subscripts 1 and 2 denote the inletand exit states, respectively.

Some common steady flowdevices

Only one in and one out

Single Stream Steady Flow System

• Nozzles

• Diffusers

• Turbines

• Compressors

ieie

ie zzgVV

hhmWQ2

22

• Compressors

• Throttling Valve

2

Often the change in kinetic energyof the fluid is small, and thechange in potential energy of thefluid is small

SOME STEADY-FLOWENGINEERING DEVICES

Many engineering devices operate essentially under the same conditionsfor long periods of time. The components of a steam power plant (turbines,compressors, heat exchangers, and pumps), for example, operate nonstop formonths before the system is shut down for maintenance. Therefore, thesedevices can be conveniently analyzed as steady-flow devices.

Nozzles and Diffusers

A nozzle is a device thatincreases the velocity of afluid

A diffuser is a device thatslows a fluid down

At very high velocities,even small changes invelocities can causesignificant changes inthe kinetic energy of thefluid.

Nozzles and Diffusers

2

022

ie

ie

VVhh

Is there work in this system? NO

Is there heat transfer? Usually it can be ignored

Does the fluid changeelevation? NO

2

Does the kinetic energy change?Yes!

20

22ie

ie

VVhh

In a nozzle, enthalpy is converted intokinetic energy

How can you find the mass flow rate in a nozzle?

e

ee

i

ii

v

AV

v

AVm

kinetic energy

Example 5-4 Deceleration of air in a diffuser

Air at 10°C and 80 kPa enters the diffuser of a jetengine steadily with a velocity of 200 m/s. The inlet areaof the diffuser is 0.4 m2. The air leaves the diffuser witha velocity that is very small compared with the inletvelocity. Determinevelocity. Determine

(a) the mass flow rate of the air

(b) the temperature of the air leaving the diffuser

Turbines

In steam, gas, or hydroelectric power plants, thedevice that drives the electric generator is theturbine.

As the fluid passes through the turbine, work is doneagainst the blades, which are attached to the shaft.

As a result, the shaft rotates and the turbineproduces work. The work done in a turbine ispositive since it is done by the fluid.

A turbine is a device that produces work at theexpense of temperature and pressure.

Compressors

Compressors, as well as pumps and fans, are devicesused to increase the pressure of a fluid.

Work is supplied to these devices from an externalsource through a rotating shaft. Therefore, compressorsinvolve work inputs (negative work).involve work inputs (negative work).

A compressor is a device that increases the pressure ofa fluid by adding work to the system

Turbines and Compressors

ie hhmW ie hhw

Is there work in this system? Yes!

Is there heat transfer? Usually it can be ignored

Does the fluid change elevation?

Does the kinetic energy change? Sometimes it can be ignored

ieW

Usually it can be ignored

Example 5-6 Compressing Air by a Compressor

Air at 100 kPa and 280 K is compressed steadily to 600 kPa and400 K. The mass flow rate of the air is 0.02 kg/s, and a heat lossof 16 kJ/kg occurs during the process. Assuming the changes inkinetic and potential energies are negligible, determine thenecessary power input to the compressor.

Example 5-7 Power Generation by a Steam Turbine

The power output of an adiabatic steam turbine is 5MW, and the inlet and the exit conditions of the steamare as indicated in figure below;

(a) Compare the magnitudes of

h, ke and pe.

(b) Determine the work done per(b) Determine the work done per

unit mass of the steam flowing

through the turbine.

(c) Calculate the mass flow rate

of the steam.

Throttling Valve

A throttling valve reducesthe fluid pressure

For example, the waterthat comes into your housegoes through a throttlingvalve, so it doesn’t haveexcessive pressure in yourhome.

Throttling Valve

ie hh 0

Is there work in this system? NO

Is there heat transfer? Usually it can be ignored

Does the fluid change elevation? NO

Does the fluid change velocity? Usually it can be ignored

Throttling Valves

• hin = hout

• Pin > Pout

• For gases that are not ideal, the temperaturegoes down in a throttling valvegoes down in a throttling valve

• The pressure drop in the fluid is oftenaccompanied by a large drop in temperature,and for that reason throttling devices arecommonly used in refrigeration and air-conditioning applications.

Throttling Valves

• What happens if the gas is ideal?

• For ideal gases

Dh = Cp DT

– But Dh = 0– But Dh = 0

– So… DT = 0

– The inlet and outlet temperatures are thesame!!!

Throttling Valve

For an ideal gas, the temperature does notchange in a throttling valve!!!change in a throttling valve!!!

During a throttling process, the enthalpy of a fluid remainsconstant for gases that are not ideal. But internal and flowenergies may be converted to each other.

Example 5-8

Refrigerant-134a enters the capillary tube of arefrigerator as saturated liquid at 0.8 MPa and isthrottled to a pressure of 0.12 MPa. Determine thequality of the refrigerant at the final state and thetemperature drop during this process.

Mixing Chamber

Mixing two or more fluidsis a common engineeringprocess

MixingChamber

Mixing Chamber

iieehmhm0

Is there any work done? No

Is there any heat transferred? No

Is there a velocity change?

Is there an elevation change?

Usually it can be ignored

Usually it can be ignored

Mixing Chamber

mm

Mass Balance

iemm

321 mmm

Mixing Chamber

Energy Balance

Example 5-9

Consider an ordinary shower where hot water at 600C ismixed with cold water at 100C. If it is desired that asteady stream of warm water at 450C be supplied,determine the ratio of the mass flow rates of the hot tocold water. Assume the heat losses from the mixingchamber to be negligible and the mixing to take place atchamber to be negligible and the mixing to take place ata pressure of 150 kPa.

Heat Exchanger

A heat exchanger is adevice where two movingfluids exchange heatwithout mixing.

Longer pipe coil in theheat exchanger, more heatwill be absorbed ortransfer to the fluid in thepipe.

Evaporator

Condenser

Heat Exchangers

Your analysis approach will depend on how youdefine your system

Heat Exchangers

• Energy balance is the same as a mixingchamber, but…

– Two inlets

– Two outlets– Two outlets

• Material Balance

– Divide into two separate streams with equalinlet and outlet flow rates

Example 5-10

Refrigerant 134a is to be cooled by water in a condenser. Therefrigerant enters the condenser with a mass flow rate of 6 kg/minat 1 MPa and 700C and leaves at 350C. The cooling water enters at300 kPa and 150C and leaves at 250C. Neglecting any pressuredrops, determine

a) The mass flow rate of the cooling water required

b) The heat transfer rate from the refrigerant to water

R-134a

R-134a

Water

Water

Pipe Flow

Q W

The transport of liquids or gases inpipes and ducts is of great importancein many engineering applications. Flowthrough a pipe or a duct usually satisfiesthe steady-flow conditions.

Pipe Flow

.

There’s work going intothe pump

There’s an elevationchange

ieie

ie zzgVV

hhmWQ2

22

Pipe Flow

2

Is there work in this system? Sometimes

Is there heat transfer? Usually

Does the fluid change elevation? Sometimes

Does the kinetic energy change? Not usually

Example

In a simple steam power plant, steam enters a boiler at

3 MPa, 600°C, and leaves a turbine at 2 MPa, 500°C.

Determine the in-line heat transfer from the steam per

kilogram mass flowing in the pipe between the boilerand the turbine.

Unsteady Flow

• During unsteady-flow (transient–flow) process the energycontent of a control volume changes with time. Themagnitude of change depends on the amount of energytransfer across the system boundaries as heat and work aswell as on the amount of energy transport into and out ofthe control volume by mass during the process.

• When analyzing an unsteady-flow process, we must keeptrack of the energy content of the control volume as well asthe energies of the incoming and outgoing flow stream.

• The general unsteady-flow process, is difficult to analyzebecause the properties of the mass at the inlets and exitsmay change during a process.

Unsteady FlowCharging of a rigid tank from asupply line is an unsteady-flowprocess since it involves changeswithin the control volume.

The shape and size of a controlvolume may change during anunsteady-flow process.

We need to look at ourbalances equations

cvsystemexitin mmmmm )(12

cvee

eeii

ii EgzV

hmWgzV

hmQ

22

22

We aren’t using the rate form of thebalances here. Why?

Unsteady-flowUnsteady-flow

At any instant during the process, the state of thecontrol volume is uniform ( it is the same throughout).Consequently the state of the mass leaving the controlvolume at any instant is the same as the state of themass in the control volume.

Most unsteady-flow processes, can be representedreasonable well by the uniform-flow process, whichinvolves the following idealization:1.

mass in the control volume.The fluid flow at any inlet or exit is uniform and steady,and thus the fluid properties do not change with time orposition over the cross section of an inlet or exit. If theydo, they are averaged and treated as constant for theentire process.

2.

The conservation of energy equation for a uniform-flowprocess can be expressed as:

cvee

eeii

ii EgzV

hmWgzV

hmQ

22

22

PEKEUE

PEKEUE PEKEUE

This is the kinetic energy ofThis is the kinetic energy ofthe systemthe system

This is the potentialThis is the potentialenergy of the systemenergy of the system

Usually, bothequal 0

cvee

eeii

ii EgzV

hmWgzV

hmQ

22

22

Usually both the kinetic energy and potential energy of

the fluid are zero too

1122 umumhmhmWQ eeii

Time 2 and time 1Inlet and Exit conditions,assuming only a singleinlet and a single exitstream

Example 5-12

A rigid, insulated tank that is initially evacuated isconnected through a valve to a supply line that carriessteam at 1 MPa and 3000C. Now the valve is opened andsteam is allowed to flow slowly into the tank until thepressure reaches 1 MPa, at which point the valve isclosed. Determine the final temperature of the steam inthe tank.

END OF CHAPTERFOUR.

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