Engineering Thermodynamics 2015

7/23/2019 Engineering Thermodynamics 2015

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Engineering Thermodynamics - A Graphical Approach

byIsrael Urieli(latest update: 6/22/2!"#

https://$$$%ohio%edu/mechanical/thermo/http://$$$%thermo&uids%net/

http://$$$%ou%edu/theob'ectlab/boo%html

http://$$$%academia%edu/!!)6)")*/T+E,.01A

I34A14E1GI1EE,I1G4A55,.A+4*th4EITI.142

!"This web resource is intended to be a totally self-contained learning resource in

Engineering Thermodynamics, independent of any textbook. It is designed to be

suitable for a two course sequence for Mechanical Engineering maors. It may,

howe!er, be used in any format and for any purpose, including self-study. The !arious

unique pedagogical features of this web resource are discussed inPaper AC 2010-47,

which was presented at the "#$#ASEE%nnual &onference. It is licensed under a

&reati!e &ommonsAttribution-Noncommercial-Share Alike 3.0 Unite

State!license and as such is freely a!ailable. &omments and constructi!e criticism arewelcomed by the author.

In 'art $ we introduce the"ir!t an Secon #a$! o% &hermo'namic!. (ather than

applying these laws in terms of components and processes we ha!e chosen a more

interesting approach of applying them to complete cycles or systems. The ideal

)tirling cycle machine is de!eloped as a prime example of both *aws +refer to a

paperA (eetin) bet$een *obert Stirlin) an Sai Carnot in 1+24presented at

the2014 ,SEC., and complete ideal heat engines, steam power plants and

refrigeration systems are e!aluated in &hapters and /. 0here appropriate, we

introduce graphical two-dimensional plots to e!aluate the performance of these

systems rather than relying on equations and tables. This enables intuiti!e

!isuali1ation of the solutions to a high degree of accuracy.
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Part 1 - ,ntrouction to the "ir!t an Secon #a$! o%

&hermo'namic!

Chapter 1,ntrouctor' Concept! Unit! an /e%inition!

Chapter 2Propertie! o% Pure Sub!tance!

a Pha!e Chan)e Propert' &able! an /ia)ram!

b &he ,eal a! Euation o% State

&hermo'namic Propertie! &able! an Chart!

Chapter 3 &he "ir!t #a$ o% &hermo'namic! %or Clo!e S'!tem!

a &he Ener)' Euation %or Clo!e S'!tem!

b ,eal Stirlin) C'cle (achine! En)ine! Cooler!

c &he Air Stanar /ie!el C'cle Compre!!ion-,)nition En)ine

&he Air Stanar 5tto C'cle Spark-,)nition En)ine

Chapter 4 &he "ir!t #a$ o% &hermo'namic! %or Control 6olume!

a &he Ener)' Euation %or Control 6olume!

b Steam Po$er Plant!

c *e%ri)eration S'!tem!

Chapter &he Secon #a$ o% &hermo'namic!

Chapter 8Entrop' - A Ne$ Propert'

a /e%inin) an E9aluatin) Entrop'

0e present an Entrop' Summar' Sheet, ,!entropic Proce!!e! Summar' Sheet,
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and an Aiabatic E%%icienc' Summar' Sheetof all the rele!ant equations relatingto this )ection.

b Aircra%t a! &urbine En)ine!

In 'art " we introduce the concept of Exergy to determine theoretical limits ofperformance of !arious thermodynamic components and systems, followed byad!anced application of steam power plants. The chapter on &arbon 2ioxide as arefrigerant does not appear in any textbook that I am aware of. 3ecause of the 4lobal0arming crisis, the currently used refrigerant, ($/a, will be banned from usage inautomobile air conditioning systems in Europe within a few years. %mong thealternati!es being de!eloped we prefer to return to &arbon 2ioxide as the refrigerantof choice. 5inally we introduce mixtures of water !apor and air and their application in

air-conditioning and cooling tower systems, and conclude with an introduction tocombustion processes.

Part 2 - Applie En)ineerin) &hermo'namic!

Chapter 7 E:er)' - (a:imum A9ailable ;ork Potential

a *e9er!ible ;ork ,rre9er!ibilit' Secon #a$ E%%icienc'

b E:ample! o% Aiabatic Control 6olume!

c Carbon /io:ie *744 &he Ne$ *e%ri)erant
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Chapter 10 Air - ;ater 6apor (i:ture!

a


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exclusi!ely )I units and then, when you reach maturity and are ready to take the'rofessional Engineering +'E exam, you find that the English system of units +7)&)is acceptable, and in some cases used exclusi!ely. This confusion reached a newclimax when in the 5all of $;;;, 9%)%


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Fne attempt to sol!e this paradox has been the introduction of a new unit of mass, theGslugG, thus

$ slug ? "." lbm

howe!er I challenge anyone to go to the grocery store and request a slug of potatos.

0e now consider the work done +0, the energy in transit requiring both the appliedforce +5 and mo!ement +x. If the force +5 is constant o!er the distance mo!ed +xthen the work done is gi!en by

:owe!er, in general the force +5 is not constant o!er the distance x, thus we need tosum all the incremental work processes taking into consideration the !ariation of theforce +5. This leads to the equi!alent integral form for determining work done +0 asfollows

A Unit! Sur9i9al ?it %or US Stuent!

F!er the years we ha!e de!eloped a basic 7nits )ur!i!al 6it +for the )I challenged inorder to help con!ert between the 7)&) +English system and the )I +Internationalsystem of units, as well as to de!elop a feel for the magnitudes of the !arious units.


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uick uiB- we all know +from reading our speedometers that ># mph is equi!alentto @# kmChr.$. 0hat is the accuracy of this con!ersion

". 7se this information to show that ; mph is equi!alent to / mCs.

0e find that with the abo!e sur!i!al kit we can determine many unit con!ersionsbetween )I H English units, typically as demonstrated in the following block

%s we progress and learn new concepts we will add to this )ur!i!al 6it.


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"orm! o% Ener)'

0e introduce the !arious forms of energy of interest to us in terms of a solid bodyha!ing a mass m BkgD. These include potential, kinetic and internal energy. 'otentialenergy +'E is associated with the ele!ation of the body, and can be e!aluated in terms

of the work done to lift the body from one datum le!el to another under a constantacceleration due to gra!ity g BmCs2D, as follows

6inetic energy +6E of a body is associated with its !elocity BmCsD and can bee!aluated in terms of the work required to change the !elocity of the body, as follows

Internal energy +7 of a body is that associated with the molecular acti!ity of the bodyas indicated by its temperature T B&D, and can be e!aluated in terms of the heat

required to change the temperature of the body ha!ing a specific heat capacity &BJCkg.&D, as follows

Cookin) $ith Potential Ener)'

In order to gain an intuiti!e appreciation for the relati!e magnitudes of the differentforms of energy we consider the +tongue-in-cheek example of an attempt to cook aturkey by potential energy. The turkey is brought to the top of a $## m building +about

# stories and then dropped from the ledge. The potential energy is thus con!ertedinto kinetic energy, and finally on impact the kinetic energy is con!erted into internalenergy. The increase in internal energy is represented by an increase in temperature,and hopefully, if this experiment is repeated enough times the temperature increasewill allow the turkey to cook. This remarkable experiment was first reported by(.&.4immi and 4loria J 3rowne - Cookin) $ith Potential Ener)', published inthe =ournal o% ,rreproucible *e!ult!+Kol , $;@A, pp "$-"".
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/energy/CookingPE.pdfhttp://www.jir.com/https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/energy/CookingPE.pdfhttp://www.jir.com/


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0hat a disappointmentL %t #.& per fall it will require repeating the experiment ##times ust to reach the cooking temperature of "##&.

&hermo'namic S'!tem!

5or purposes of analysis we consider two types of Thermodynamic )ystems

Clo!e S'!tem- usually referred to as a S'!temor a Control (a!!. This type

of system is separated from its surroundings by a physical boundary. Energy in

transit in the form of 0ork or :eat can flow across the system boundary,howe!er there can be no mass flow across the boundary. Fne typical example ofa system is a piston C cylinder de!ice in which the system is defined as the fixed


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mass of fluid contained within the cylinder.

5pen S'!tem- usually referred to as a Control 6olume. In this case, in

addition to work or heat, we ha!e mass flow of the working fluid across thesystem boundaries through inlet and outlet ports. In this course we will beexclusi!ely concerned with steady flow control !olumes, in that the net mass of

working fluid within the system boundaries remains constant +ie mass flow inBkgCsD ? mass flow out BkgCsD. The following sections refer mainly to systems -we will consider control !olumes in more detail starting with Chapter 4a.

Propertie! o% a S'!tem

The closed system shown abo!e can be defined by its !arious Propertie!, such as itspressure +', temperature +T, !olume +K and mass +m. 0e will introduce and definethe !arious properties of thermodynamic interest as needed in context. 5urthermore the

properties can be either E:ten!i9eor ,nten!i9e+or Speci%ic. %n extensi!e property isone whose !alue depends on the mass of the system, as opposed to an intensi!e

property +such as pressure or temperature which is independent of the system mass. %specific property is an intensi!e property which has been obtained by di!iding theextensi!e property by the mass of the system. Two examples follow - notice that

specific properties will always ha!e kilograms +kg in the units denominater.
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Fne often used exception to the abo!e definitions is the concept of )pecific 0eight,defined as the weight per unit !olume. 0e will not be using this concept throughoutthis text.

State an Euilibrium

The Stateof a system is defined by the !alues of the !arious intensi!e properties ofthe system. The State Po!tulatestates that if two independent intensi!e property!alues are defined, then all the other intensi!e property !alues +and thus the state ofthe system are also defined. This can significantly simplify the graphicalrepresentation of a system, since only two-dimensional plots are required. 9ote that

pressure and temperature are not necessarily independent properties, thus a boilingliquid will change its state from liquid to !apor at a constant temperature and pressure.

0e assume that throughout the system Euilibriumconditions pre!ail, thus there are

no temperature or pressure gradients or transient effects. %t any instant the entiresystem is under chemical and phase equilibrium.

Proce!! an C'cle

% Proce!!is a change of state of a system from an initial to a final state due to anenergy interaction +work or heat with its surroundings. 5or example in the followingdiagram the system has undergone a compression process in the piston-cylinderde!ice.

The Proce!! Pathdefines the type of process undergone. Typical process paths are

,!othermal+constant temperature process


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,!ochoricor ,!ometric+constant !olume process

,!obaric+constant pressure process

Aiabatic+no heat flow to or from the system during the process

0e assume that all processes are ua!i-Staticin that equilibrium is attained after eachincremental step of the process.

% system undergoes a C'clewhen it goes through a sequence of processes that leadsthe system back to its original state.

Pre!!ure

The basic unit of pressure is the 'ascal B'aD, howe!er practical units are kilo'ascalBk'aD, bar B$## k'aD or atm +atmosphere B$#$." k'aD. The a)e+or 6acuumpressure is related to the Ab!olutepressure as shown in the diagram below

The basic method of measuring pressure is by means of a (anometer, as shownbelow


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The atmospheric pressure is measured by means of a (ercur' Darometeras follows

Sol9e Problem 1.1 - U!in) a Darometer to etermine the


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Temperature is a measure of molecular acti!ity, and a temperature difference betweentwo bodies in contact +for example the immediate surroundings and the system is thedri!ing force leading to heat transfer between them.

3oth the "ahrenheitand the Cel!iu!scales are in common usage in the 7), hence it

is important to be able to con!ert between them. 5urthermore we will find that in somecases we require the Ab!olute+*ankineand ?el9in temperature scales +for examplewhen using the Ideal 4as Equation of )tate, thus we find it con!enient to plot all fourscales as follows

9otice from the plot that -/#& equals -/#5, leading to con!enient formulas forcon!erting between the two scales as follows

uick uiB- The temperature in &hicago in winter can be as low as $/5. 0hat is thetemperature in &, 6, and (. B-$#&, " 6, /A/(D 9ote that by con!ention " 6 isread G" 6el!in,G and not G" degrees 6el!inG.

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN

Engineering Thermodynamics by Israel 7rieliis licensed under a &reati!e &ommons%ttribution-9oncommercial-)hare %like .# 7nited )tates *icense
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44444444444444444444444444444444444444444444444444444-

Sol9e Problem 1.1 - U!in) a Darometer to etermine the

$

mm :g, and at the bottom of the building is A# mm :g. %ssume the density of

mercury O:g? $,## kgCm3, and that the a!erage density of the column of air Oair?

$.$@ kgCm3.

The approach to solution is illustrated in the following diagram. % free-body force

diagram on the column of air allows us to determine the height as a function of the

pressure difference P' from top to bottom.


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0ell, exactly $##m - ob!iously this is a contri!ed example. 0hen we first e!aluated

the height we came up with the result

In Engineering Thermodynamics we normally present results to within or /

significant digits. The question that one really should ask is GIs this a reasonable

method to measure the height of a buildingG and the answer is a resounding 9FL In


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the following we do an uncertainty analysis and find that unless we also measure the

air temperature during this experiment +why temperature doesn


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water is $### kgCm3, the density of mercury is $,## kgCm3, and that the

atmospheric pressure is $## k'a.

4444444444444444444444444444444444444444444444444444444444444

Chapter 2 Pure Sub!tance!

a Pha!e Chan)e Propert' &able! an /ia)ram!

In this chapter we consider the property !alues and relationships of a pure substance+such as water which can exist in three phases - solid, liquid and gas. 0e will notconsider the solid phase in this course.

In order to introduce the rather complex phase change interactions that occur in puresubstances we consider an experiment in which we ha!e liquid water in a piston-cylinder de!ice at "#& and $##k'a pressure.. :eat is added to the cylinder while the

pressure is maintained constant until the temperature reaches ##&, as shown in thefollowing T-vdiagram +temperature !s specific !olume


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5rom )tate +$ to )tate +" the water maintains its liquid phase and the specific !olumeincreases !ery slightly until the temperature reaches close to $##& +)tate +"

- Saturate #iui. %s more heat is added the water progressi!ely changes phasefrom liquid to water !apor +steam while maintaining the temperature at $##&+Saturation &emperature- Tsat until there is no liquid remaining in the cylinder+)tate +/ - Saturate 6apor. If heating continues then the water !apor temperatureincreases +T R Tsat and is said to be in the Superheate+)tate +>.

9otice that during this entire process the specific !olume of the water increased bymore than three orders of magnitude, which made it necessary to use a logarithmicscale for the specific !olume axis.

0e now consider repeating this experiment at !arious pressures, as shown in thefollowing T-vdiagram


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9otice that as we increase the applied pressure, the region between the saturated liquidand saturated !apor decreases until we reach the Critical Point, abo!e which there is

no clear distinction between the liquid and !apor states.

It is common practice to oin the loci of saturated liquid and saturated !apor points asshown in the T-vdiagram below.


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The saturation lines define the regions of interest as shown in the diagram, beingthe Compre!!e #iui region, the ualit' region enclosed by the saturation lines,

and the Superheat region +which also includes the &ran!criticalregion to the rightof the saturated !apor line and abo!e the critical point. 0e will use Propert'&able!associated with the regions in order to e!aluate the !arious properties. 9oticethat we ha!e pro!ided property tables of steam, (efrigerant ($/a, and &arbon2ioxide, which we belie!e is destined to become the future refrigerant of commonusage.

&he ualit' *e)ion

The ualit' *e)ion+also referred to as the Saturate #iui-6apor (i:ture*e)ion is enclosed between the saturated liquid line and the saturated !apor line, andat any point within this region the quality of the mixture +also referred to as thedryness factor is defined as the mass of !apor di!ided by the total mass of the fluid, asshown in the following diagram
https://www.ohio.edu/mechanical/thermo/property_tables/index.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/index.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/pure_fluid/quality.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/index.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/index.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/pure_fluid/quality.html


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9otice that properties relating to the saturated liquid ha!e the subscript f, and thoserelating to the saturated !apor ha!e the subscript g. In order to e!aluate the qualityconsider a !olume K containing a mass m of a saturated liquid-!apor mixture.

9otice from the !team propert' table!that we ha!e also included three newproperties internal energy u BkJCkgD, enthalpy h BkJCkgD, and entropy s BkJCkg.6D all ofwhich will be defined as needed in future sections. %t this stage we note that the
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equations relating quality and specific !olume can also be e!aluated in terms of thesethree additional properties.

&heP-v/ia)ram %or ;ater

The abo!e discussion was done in terms of the T-vdiagram, howe!er recall from&hapter $ when we defined the )tate 'ostulate that any two independent intensi!e

properties can be used to completely define all other intensi!e state properties. It isoften ad!antageous to use theP-vdiagram with temperature as the parameter as in thefollowing diagram

9otice that because of the extremely large range of pressure and specific !olume

!alues of interest, this can only be done on a log-log plot. This is extremelyincon!enient, so both the T-vand theP-vdiagrams are normally not drawn to scale,howe!er are sketched only in order to help define the problem, which is then sol!ed interms of the steam tables. This approach is illustrated in the following sol!ed

problems.


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Sol9e Problem 2.1 - Two kilograms of water at ">& are placed in a pistoncylinder de!ice under $## k'a pressure as shown in the diagram +)tate +$. :eat isadded to the water at constant pressure until the piston reaches the stops at a total!olume of #./ m3+)tate +". More heat is then added at constant !olume until the

temperature of the water reaches ##& +)tate +. 2etermine +a the quality of thefluid and the mass of the !apor at state +", and +b the pressure of the fluid at state +.

Step 1Al$a'!draw a complete diagram of the states and processes of the problemand include all the rele!ant information on the diagram. In this case there are threestates and two processes +constant pressure and constant !olume.

Step 2In the case of a closed system with a phase change fluid, al$a'!sketcha T_vorP_vdiagram indicating all the rele!ant states and processes on the diagram.%s mentioned abo!e this diagram will not be drawn to scale, howe!er it will help to

define the problem and the approach to solution. In the case of steam, as we determine!arious !alues from the !team table!we add these !alues to the diagram, typically asshown below


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9otice that the T_vdiagram is based exclusi!ely on intensi!e properties, hence mass isnot indicated on the diagram. Thus we indicate on the diagram that in order todetermine the quality at state +" we need to first e!aluate the specific !olume !",which can then be compared to the saturation !alues !fand !gat the pressure of $##k'a.

Thus !"? K C m ? #./ Bm3

D C " BkgD ? #." Bm3

C kgD

&oncerning state +, the problem statement did not specify that it is in the superheat

region. 0e needed to first determine the saturated !apor specific !olume !gat ##&.This !alue is #.#"$ m3C kg, which is much less than the specific !olume !of #."

m3C kg, thus placing state + well into the superheated region. Thus the two intensi!e

properties which we use to determine the pressure at state + are T? ##&, and !?#." m3C kg. Fn scanning the !uperheat table!we find that the closest !alues lie

somewhere between $." M'a and $./ M'a, thus we use linear interpolation techniqesto determine the actual pressure 'as shown below
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Sol9e Problem 2.2 - Two kilograms of water at ">& are placed in a pistoncylinder de!ice under ." M'a pressure as shown in the diagram +)tate +$. :eat isadded to the water at constant pressure until the temperature of the fluid reaches >#&+)tate +". 2etermine the final !olume of the fluid at state +".

In this example since the pressure is known +." M'a and remains constantthroughout the process, we find it con!enient to draw aP-vdiagram indicating the

process +$ - +" as follows.


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%s in the pre!ious example, on scanning the !uperheat table!we find that we need tointerpolate between pressure ' ? .# M'a and ' ? .> M'a in order to determine thespecific !olume at the required pressure of ." M'a as follows

Problem 2.3 - % piston-cylinder de!ice contains a saturated mixture of steam andwater ha!ing a total mass of #.> kg at a pressure of $# k'a and an initial !olume of

$## liters. :eat is then added and the fluid expands at constant pressure until it reachesa saturated !apor state.

a 2raw a diagram representing the process showing the initial and final states

of the system.

b )ketch this process on aP-vdiagram with respect to the saturation lines,

critical point, and rele!ant constant temperature lines, clearly indicating theinitial and final states.

c 2etermine the initial quality and temperature of the fluid mixture prior toheating. Bquality x$? #.$@", T$? $$.&D

d 2etermine the final !olume of the steam after heating. B#.>/ m3+>/

litersD

9ote $### liters - $ m3.
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Problem 2.4 - % pressure cooker allows much faster +and more tender cookingby maintaining a higher boiling temperature of the water inside. It is well sealed, andsteam can only escape through an opening on the lid, on which sits a metal petcock.0hen the pressure o!ercomes the weight of the petcock, the steam escapes,

maintaining a constant high pressure while the water boils.

%ssuming that the opening under the petcock has an area of @ mm2, determine

a the mass of the petcock required in order to maintain an operating pressure of

;; k'a gage. B@#.AgmD

b the corresponding temperature of the boiling water. B$"#."&D

9ote %ssume that the atmospheric pressure is $#$ k'a. 2raw a free body diagram ofthe petcock.

5n to Chapter 2b o% Pure Sub!tance!


https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter2b.htmlhttp://creativecommons.org/licenses/by-nc-sa/3.0/us/http://www.ent.ohiou.edu/~thermohttp://creativecommons.org/licenses/by-nc-sa/3.0/us/http://creativecommons.org/licenses/by-nc-sa/3.0/us/https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter2b.htmlhttp://creativecommons.org/licenses/by-nc-sa/3.0/us/http://www.ent.ohiou.edu/~thermohttp://creativecommons.org/licenses/by-nc-sa/3.0/us/http://creativecommons.org/licenses/by-nc-sa/3.0/us/


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Chapter 2 Pure Sub!tance!

b &he ,eal a! Euation o% State

0e continue with our discussion on 'ure )ubstances.

0e find that for a pure substance in the superheated region, at specific !olumes muchhigher than that at the critical point, the '-!-T relation can be con!eniently expressed

by the ,eal a! Euation o% Stateto a high degree of accuracy, as follows

' ! ? ( T

where ( is constant for a particular substance and is called the a! Con!tant

9ote that for the ideal gas equation both the pressure ' and the temperature T must beexpressed in absolute quantities.

&onsider for example the T-vduagram for water as shown below


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The shaded 1one in the diagram indicates the region that can be represented by theIdeal 4as equation to an error of less than $S. 9ote that at the critical point the erroris #S.

The gas constant ( can be expressed as follows

The three commonly used formats to express the Ideal 4as Equation of )tate are


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Sol9e Problem 2.- % piston-cylinder de!ice contains #.> kg saturated liquidwater at a pressure of "## k'a. :eat is added and the steam expands at constant

pressure until it reaches ##&.


of the system.

b )ketch this process on a T-v+temperature-specific !olume diagram with

respect to the saturation lines, critical point, and rele!ant constant pressurelines, clearly indicating the initial and final states.

c 7sing steam tables determine the initial temperature of the steam prior to

heating.

d 7sing steam tables determine the final !olume of the steam after heating

e 7sing the ideal gas equation of state determine the final !olume of the steam

after heating. 2etermine the percentage error of using this method compared to

that of using the steam tables.

9ote The critical point data and the ideal gas constant for steam can be found on thefirst page of the !team table!.

Solution Approach

E!en if questions a and b were not required, this should always be the first priorityitem in sol!ing a thermodynamic problem.
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c )ince state +$ is specified as saturated liquid at "## k'a, we use the !aturatepre!!ure !team table!to determine that T$? Tsat "##k'a? $"#."&.

d 5rom the T-! diagram we determine that state +" is in the superheated region, thuswe use the !uperheate !team table!to determine that !"? !"##k'a,##&? $.$"

m3Ckg. Thus K"? m,!"? +#,>kg.+$.$" m3Ckg ? #.>@ m3+>@ liters
https://www.ohio.edu/mechanical/thermo/property_tables/H2O/H2O_PresSat1.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/H2O/H2O_PresSat1.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/H2O/H2O_Super1.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/H2O/H2O_PresSat1.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/H2O/H2O_PresSat1.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/H2O/H2O_Super1.html


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9ote that in doing a units check we find that the following con!ersion appears so oftenthat we feel it should be added to our 7nits &on!ersion )ur!i!al 6it +recall Chapter1

5inally we determine the percentage error of using the ideal gas equation at state +"

Problem 2.8- &onsider a rigid container ha!ing a !olume of $## liters, filled withsteam at an initial state of /## k'a and ##&. The steam is then cooled until it reaches

a temperature of ;#&.


of the system.

b 7sing steam tables determine the mass of steam in the container. B#.$> kgD

c 7sing the ideal gas equation of state determine the mass of steam in the

container. B#.$>$ kgD2etermine the percentage error of using this method compared to that of using

the steam tables. B$SD

d )ketch this process on a T-v+temperature-specific !olume diagram with

respect to the saturation lines, critical point, and rele!ant constant pressurelines, clearly indicating the initial and final states.
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter1.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter1.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter1.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter1.html


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e 7sing steam tables determine the final pressure and quality of the fluid

mixture after cooling. BA#." k'a, U ? #."AAD

9ote The critical point data and ideal gas constant for steam can be found on the firstpage of the !team table!.

Sol9e Problem 2.7- %n automobile tire with a !olume of $## liters is inflatedto a gage pressure of "$# k'a. 2etermine a the mass of air in the tire if thetemperature is "#&, and b the increase in gage pressure if the temperature in the tirereaches >#&. %ssume that atmospheric pressure is $## k'a.

Solution Approach

0e always begin a thermodynamic problem with a sketch, indicating all the rele!antinformation on the sketch, thus

5or part b the temperature in the tire increases to >#& +"6, howe!er the !olumeand mass of air in the tire remains constant, thus


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+9ote for the )I challenged - initially the pressure was # psig, and then rose to >psig. Try to !alidate these !alues

Sol9e Problem 2.+- In aircraft design it is common practice to specify astandard temperature distribution in the atmosphere near the surface of the earth +up toan ele!ation 1 of $####m as T+1 ? +T#V a.1&, where T#at the earth&, and a is the Temperature *apse (ate +? -#.##>$& C m. 7sing the Ideal 4as

Equation of )tate, determine the pressure at an ele!ation of ###m if at 1 ? #, ' ? $#$k'a.

Solution Approach

0e first de!elop the solution in terms of the :ydrostatic Equation on an elementalheight of the column of air, the Ideal 4as Equation of )tate, and the Temperature*apse (ate equation, as follows


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Sol9e Problem 2.>- Wou may wonder why we would be interested inknowing the !alue of air pressure at ###m altitude. In the following example wecontinue with the abo!e de!elopment in order to e!aluate the payload that can be liftedto an altitude of ###m using a small hot air balloon +Kolume ?$### m3 ha!ing an

empty mass of $##kg. %ssume that the temperature of the air in the balloon is $##&.

Solution Approach

In this case we de!elop the solution in terms of a force balance between the bouyancyforce +weight of the displaced air and the gra!ity force including the weight of the hotair, the balloon empty mass, and the payload mass. The maximum altitude occurswhen those two forces are equal, as follows


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5inally - with $>/ kg payload at least " persons can share and enoy this wonderfulexperience. 7nfortunately they will not be able to enoy a decent cup of English tea.%t a pressure of ;.; k'a water will boil at +hea!ens forbid X ;#&L +)aturation


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temperature Tsatfrom the Steam &able!. uick uiBdetermine the temperatureof a cup of tea in 2en!er, &olorado +ele!ation >### ft, or on the peaks of the (ockyMountains +ele!ation $/### ft.Hint use the 7nits )ur!i!al 6it presentedin Chapter1to first con!ert from feet to meters

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN

Non-,eal a! Deha9ior

0e noticed in the abo!e T-vdiagram for water that the gasses can de!iate significantlyfrom the ideal gas equation of state in regions nearby the critical point and there ha!e

been many equations of state recommended for use to account for this non-idealbeha!iour. :owe!er, this non-ideal beha!iour can be accounted for by a correctionfactor called the Compre!!ibilit' "actorY defined as follows

thus when the compressibility factor Y approaches $ the gas beha!es as an ideal gas.9ote that under the same conditions of temperature and pressure, the compressibilityfactor can be expressed as

2ifferent fluids ha!e different !alues of critical point pressure and temperature data'&(and T&(, and these can be determined from the &able o% Critical Point /ata o%

6ariou! Sub!tance!.5ortunately the Principle o% Corre!ponin) State!states thatwe can normali1e the pressure and temperature !alues with the critical !alues asfollows

%ll fluids normali1ed in this manner exhibit similar non-ideal gas beha!iour within afew percent, thus they can all be plotted on a 4eneralised &ompressibility &hart. %number of these charts are a!ailable, howe!er we prefer to use the #ee-?e!lerlo)arithmic Compre!!ibilt' Chart, The use of the compressibility chart is shown inthe following example.
https://www.ohio.edu/mechanical/thermo/property_tables/H2O/H2O_PresSat1.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter1.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Critical.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Critical.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Zfactor.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Zfactor.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/H2O/H2O_PresSat1.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter1.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Critical.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Critical.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Zfactor.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Zfactor.html


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Sol9e Problem 2.10-&arbon 2ioxide gas is stored in a $## liter tank at M'a and #&. 2etermine the mass of &F"in the tank based on +a !alues obtainedfrom the &F"tables of data, +b the ideal gas equation of state, and +c the generali1edcompressibility chart.&ompare +b and +c to +a and determine the percentage error in

each case.

Solution Approach

0e first determine the &ritical 'oint data for &F"from the &able o% Critical Point/ata o% 6ariou! Sub!tance!

%fter e!aluating the (educed 'ressure and (educed Temperature we plot them onthe eneraliBe Compre!!ibilit' Chartin order to determine the &ompressibility5actor, as shown below
https://www.ohio.edu/mechanical/thermo/property_tables/gas/Critical.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Critical.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Zfactor.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Critical.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Critical.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/Zfactor.html


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The actual !alue of specific !olume !ais obtained from the C52Superheat &able!
https://www.ohio.edu/mechanical/thermo/property_tables/CO2/C02_Superheat.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/CO2/C02_Superheat.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/CO2/C02_Superheat.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/CO2/C02_Superheat.html


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The general rule is that if ' XX '&(or if T RR T&(then you are probably dealing withan ideal gas. If in doubt alwayscheck the &ompressibility 5actor Y on the&ompressibility &hart.


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Chapter 3: The First Law of

Thermodynamics for Closed Systems

Chapter 3 &he "ir!t #a$ o% &hermo'namic! %or Clo!e

S'!tem!

a &he Ener)' Euation %or Clo!e S'!tem!

0e consider the 5irst *aw of Thermodynamics applied to stationary closed systems asa conser!ation of energy principle. Thus energy is transferred between the system andthe surroundings in the form of heat and work, resulting in a change of internal energyof the system. Internal energy change can be considered as a measure of molecularacti!ity associated with change of phase or temperature of the system and the energyequation is represented as follows


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In this course we consider three modes of work transfer across the boundary of asystem, as shown in the following diagram

In this course we are primarily concerned with Dounar' ;orkdue to compressionor expansion of a system in a piston-cylinder de!ice as shown abo!e. In all cases we

assume a perfect seal +no mass flow in or out of the system, no loss due to friction,and quasi-equilibrium processes in that for each incremental mo!ement of the pistonequilibrium conditions are maintained. 3y con!ention positi!e work is that done bythe system on the surroundings, and negati!e work is that done by the surroundings onthe system, Thus since negati!e work results in an increase in internal energy of thesystem, this explains the negati!e sign in the abo!e energy equation.

3oundary work is e!aluated by integrating the force 5 multiplied by the incrementaldistance mo!ed x between an initial state +$ to a final state +". 0e normally dealwith a piston-cylinder de!ice, thus the force can be replaced by the piston area %

multiplied by the pressure ', allowing us to replace %.x by the change in !olume K,as follows

This is shown in the following schematic diagram, where we recall that integration canbe represented by the area under the cur!e.


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9ote that work done is a Path "unctionand not a property, thus it is dependent on the

process path between the initial and final states. (ecall in Chapter 1that weintroduced some typical process paths of interest

,!othermal+constant temperature process

,!ochoricor ,!ometric+constant !olume process

,!obaric+constant pressure process

Aiabatic+no heat flow to or from the system during the process

It is sometimes con!enient to e!aluate the specific work done which can berepresented by aP-vdiagram thus if the mass of the system is m BkgD we ha!e finally

0e note that work done by the system on the surroundings +expansion process ispositi!e, and that done on the system by the surroundings +compression process isnegati!e.
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5inally for a closed system Sha%t ;ork+due to a paddle wheel and Electrical;ork+due to a !oltage applied to an electrical resistor or motor dri!ing a paddlewheel will always be negati!e +work done on the system. 'ositi!e forms of shaftwork, such as that due to a turbine, will be considered in &hapter / when we discussopen systems.

,nternal Ener)' u

The third component of our &losed )ystem Energy Equation is the change of internalenergy resulting from the transfer of heat or work. )ince specific internal energy is a

property of the system, it is usually presented in the 'roperty Tables such as inthe Steam &able!. &onsider for example the following sol!ed problem.

Sol9e Problem 3.1 -(ecall the )ol!ed 'roblem "." in Chapter 2ain which

we presented a constant pressure process. 0e wish to extend the problem to includethe energy interactions of the process, hence we restate it as follows

Two kilograms of water at ">& are placed in a piston cylinder de!ice under ." M'apressure as shown in the diagram +)tate +$. :eat is added to the water at constantpressure until the temperature of the steam reaches >#& +)tate +". 2etermine thework done by the fluid +0 and heat transferred to the fluid +8 during this process.

Solution Approach

0e first draw the diagram of the process including all the rele!ant data as follows

9otice the four questions to the right of the diagram, which we should always askbefore attempting to sol!e any thermodynamic problem. 0hat are we dealing with -liquid pure fluid, such as steam or refrigerant ideal gas In this case it is steam, thuswe will use the steam tables to determine the !arious properties at the !arious states. Isthe mass or !olume gi!en If so we will specify and e!aluate the energy equation in
https://www.ohio.edu/mechanical/thermo/property_tables/H2O/index.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter2a.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/H2O/index.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter2a.html


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kiloJoules rather than specific quantities +kJCkg. 0hat about entropy 9ot so fast - weha!e not yet considered enthalpy +below - wait patiently until Chapter 8.

)ince work in!ol!es the integral of '.! we find it con!enient to sketch theP-vdiagram of the problem as follows

9otice on theP-vdiagram how we determine the specific work done as the area under

the process cur!e. 0e also notice that in the &ompressed *iquid region the constanttemperature line is essentially !ertical. Thus all the property !alues at )tate +$+compressed liquid at ">& can be determined from the saturated liquid table !aluesat ">&.
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Enthalp' h - a Ne$ Propert'

In the case studies that follow we find that one of the maor applications of the closed

system energy equation is in heat engine processes in which the system isapproximated by an ideal gas, thus we will de!elop relations to determine the internalenergy for an ideal gas. 0e will find also that a new property called Enthalp'will beuseful both for &losed )ystems and in particular for Fpen )ystems, such as thecomponents of steam power plants or refrigeration systems. Enthalpy is not afundamental property, howe!er is a combination of properties and is defined asfollows

%s an example of its usage in closed systems, consider the following constant pressureprocess


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%pplying the energy equation we obtain

:owe!er, since the pressure is constant throughout the process

)ubstituting in the energy equation and simplifying

Kalues for specific internal energy +u and specific enthalpy +h are a!ailable fromthe Steam &able!, howe!er for ideal gasses it is necessary to de!elop equations for Puand Ph in terms of )pecific :eat &apacities. 0e de!elop these equations in terms ofthe differential form of the energy equation in the following web page

Speci%ic


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444444444444444444444444444444444444444

3pecic +eat apacities o7 an Ideal Gas

5or a simple system, internal energy +u is a function of two independant !ariables,

thus we assume it to be a function of temperature T and specific !olume !, hence

)ubstituting equation +" in the energy equation +$ and simplifying, we obtain

9ow for a constant !olume process +! ? #

That is, the specific constant !olume heat capacity of a system is a function only of its

internal energy and temperature. 9ow in his classic experiment of $@/ Joule showed
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that the internal energy of an ideal gas is a function of temperature only, and not of

pressure or specific !olume. Thus for an ideal gas the partial deri!ati!es can be

replaced by ordinary deri!ati!es, and the change in internal energy can be expressed

as

&onsider now the enthalpy. 3y definition h ? u V ' !, thus differentiating we obtain

%gain for a simple system, enthalpy +h is a function of two independant !ariables,

thus we assume it to be a function of temperature T and pressure ', hence

)ubstituting equation + in the energy equation +>, and simplifying

:ence for a constant pressure process, since ' ? #

That is, the specific constant pressure heat capacity of a system is a function only of its

enthalpy and temperature. 9ow by definition

9ow since for an ideal gas Joule showed that internal energy is a function of

temperature only, it follows from the abo!e equation that enthalpy is a function of

temperature only. Thus for an ideal gas the partial deri!ati!es can be replaced by

ordinary deri!ati!es, and the differential changes in enthalpy can be expressed as


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5inally, from the definition of enthalpy for an ideal gas we ha!e

Kalues of (, &', &!and k for ideal gases are presented +at ##6 in the table

on Propertie! o% 6ariou! ,eal a!e!. 9ote that the !alues of &', &!and k are

constant with temperature only for mon-atomic gases such as helium and argon. 5or

all other gases their temperature dependence can be considerable and needs to beconsidered. 0e find it con!enient to express this dependence in tabular form and ha!e

pro!ided a table of Speci%ic


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In its original single cylinder form the working gas +typically airor helium is sealed within its cylinders by the piston and shuttled

between the hot and cold spaces by a displacer. The linkagedri!ing the piston and displacer will mo!e them such that the gaswill compress while it is mainly in the cool compression spaceand expand while in the hot expansion space. This is clearlyillustrated in the adacent animation which was produced by(ichard 0heeler +eph'ri! of ;ikipeia.

(efer also to the animation produced by (att ?e9ene'inhis Stirlin) en)ine animationwebsite. )ince the gas is at ahigher temperature, and therefore pressure, during its expansionthan during its compression, more power is produced duringexpansion than is reabsorbed during compression, and this netexcess power is the useful output of the engine. 9ote that there areno !al!es or intermittent combustion, which is the maor source ofnoise in an internal combustion engine. The same working gas isused o!er and o!er again, making the )tirling engine a sealed,closed cycle system. %ll that is added to the system is steady hightemperature heat, and all that is remo!ed from the system is lowtemperature +waste heat and mechanical power.

%thens, Fhio, is a hotbed of )tirling cycle machine acti!ity, both engines and coolers,and includes (H2 and manufacturing companies as well as internationally recogni1edconsultants in the area of )tirling cycle computer analysis. The parent company of thisacti!ity is Sunpo$er ,nc. It was formed by 0illiam 3eale in the early $;A# k0 )T-> %ir engine. This large single cylinder engine burns

biomass fuel +such as sawdust pellets or rice husks and can function as a cogenerationunit in rural areas. It is not a free-piston engine, and uses a bell crank mechanism toobtain the correct displacer phasing. %nother important early )tirling engine is*ehmann


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)tirling engines in $@A$. %ndy (oss of &olumbus, Fhio built a small working replicaof the #ehmann machine, as well as a moel air en)ine.

Solar


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'rocess "- is a constant !olume displacement process in which the gas is

displaced from the cold space to the hot expansion space. 9o work is done,howe!er as we shall see below, a significant amount of heat 8(is absorbed bythe gas from the regenerator matrix.

'rocess -/ is the isothermal expansion process. 0ork 0-/is done by thesystem and is shown as the area under theP-Vdiagram, while heat 8-/is addedto the system from the heat source, maintaining the gas at a constanttemperature T:.


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5inally, process /-$ is a constant !olume displacement process which completes

the cycle. Fnce again we will see below that heat 8(is reected by the workinggas to the regenerator matrix.

The net work 0netdone o!er the cycle is gi!en by 0net? +0-/V 0$-", where thecompression work 0$-"is negati!e +work done onthe system.

0e now consider the heat transferred during all four processes, which will allow us toe!aluate the thermal efficiency of the ideal )tirling engine. (ecall from the pre!ioussection that in order to do a 5irst *aw analysis of an ideal gas to determine the heattransferred we needed to de!elop equations to determine the internal energy change Puin terms of the Speci%ic


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0e will find in Chapter that this is the maximum theoretical efficiency that isachie!able from a heat engine, and usually referred to as the Carnotefficiency. 5ormore information on this subect, refer to a paper A (eetin) bet$een *obertStirlin) an Sai Carnot in 1+24presented at the 2014 ,SEC.
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter5.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.htmlhttp://www.centrostirling.com/isec2014/index-isec2014.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter5.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.htmlhttp://www.centrostirling.com/isec2014/index-isec2014.html


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9ote that if no regenerator is present the heat 8(must be supplied by the heater. Thusthe efficiency will be significantly reduced to Z th? 0netC +8inV 8(. 5urthermore thecooler will then ha!e to reect the heat that is normally absorbed by the regenerator,thus the cooling load will be increased to 8outV 8(. (ecall that 8"-? 8(? -8/-$.

9ote that the practical )tirling cycle has many losses associated with it and does notreally in!ol!e isothermal processes, nor ideal regeneration. 5urthermore since the5ree-'iston )tirling cycle machines in!ol!e sinusoidal motion, theP-Vdiagram has ano!al shape, rather than the sharp edges defined in the abo!e diagrams. 9e!erthelesswe use the ideal )tirling cycle to get an initial understanding and appreciation of thecycle performance.


Problem 3.2 - &he Sunpo$er E-1000 Stirlin) En)ineenerator

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

FFFFF

2. &he Stirlin) C'cle Cooler

Fne important aspect of )tirling cycle machines that we need to consider is that thecycle can be re!ersed - if we put net work into the cycle then it can be used to pump

heat from a low temperature source to a high temperature sink. Sunpo$er ,nc. hasbeen acti!ely in!ol!ed in the de!eplopment of )tirling cycle refrigeration systems andproduces )tirling cycle croygenic coolers for liquifying oxygen. In $;@/ )unpowerde!eloped a free piston /uple: Stirlin) (achineha!ing only three mo!ing partsincluding one piston and two displacers, in which a gas fired )tirling cycle engine

powered a )tirling cycle cooler. lobal Coolin) ,ncwas established in $;;> as aspinoff of )unpower, and was formed mainly in order to de!elop free-piston )tirlingcycle coolers for home refrigerator applications. These systems, apart from beingsignificantly more efficient than regular !apor-compression refrigerators, ha!e theadded ad!antage of being compact, portable units using helium as the working fluid

+and not the :5& refrigerants such as ($/a, ha!ing a 4lobal 0arming 'otential of$,##. More recently 4lobal &ooling decided to concentrate their de!elopment effortson systems in which there are !irtually no competiti!e systems - cooling between-/#& and -@#&, and they established a new company name Stirlin) Ultracol.

0e are fortunate to ha!e obtained two original M$##3 coolers from 4lobal &ooling.The one is used as a demonstrator unit, and is shown in operation in the following
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCogen/StirlCogen.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCogen/StirlCogen.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCogen/StirlCogen.htmlhttp://us.sunpowerinc.com/https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCooler/Duplex_Stirling.pdfhttp://stirlingultracold.com/who_we_arehttp://stirlingultracold.com/https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCogen/StirlCogen.htmlhttp://us.sunpowerinc.com/https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCooler/Duplex_Stirling.pdfhttp://stirlingultracold.com/who_we_arehttp://stirlingultracold.com/


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photograph. The second unit is set up as a ME Senior #ab proGectin which wee!aluate the actual performance of the machine under !arious specified loads andtemperatures.

% schematic diagram followed by an animated schematic of the cooler +both courtesyof lobal Coolin) are shown below
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCooler/CoolerLab.jpghttp://stirlingultracold.com/who_we_arehttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCooler/CoolerLab.jpghttp://stirlingultracold.com/who_we_are


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&onceptually the cooler is an extremely simple de!ice, consisting essentially of onlytwo mo!ing parts - a piston and a displacer. The displacer shuttles the working gas+helium between the compression and expansion spaces. The phasing between the

piston and displacer is such that when the most of the gas is in the ambientcompression space then the piston compresses the gas while reecting heat to theambient. The displacer then displaces the gas through the regenerator to the coldexpansion space, and then both displacer and piston allow the gas to expand in thisspace while absorbing heat at a low temperature.


Problem 3.3 -Stirlin) C'cle Cooler (100D - ,eal Anal'!i!

7nfortunately the analysis of actual )tirling cycle machines is extremely complex andrequires sophisticated computer analysis. 0e consider the idealised model of thiscooler defined in terms of theP-Vdiagram shown below in order to determine theideal performance of the M$##3 under typical operating conditions as described

below. +Note that the values presented here are not actual values of the M100B,however were devised y your instructor for purposes of this e!ercise only.


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'rocess +$-+" is the isothermal compression process at temperature T&? #&,

during which heat 8&is reected to the ambient. 'rocess +"-+ is the constant !olumedisplacement process during which heat 8(is reected to the regenerator matrix.'rocess +-+/ is the isothermal expansion process at temperature TE? -"#&, during

which heat 8Eis absorbed from the free1er, and finally process +/-+$ is the constant!olume displacement process during which heat 8(is absorbed from the regeneratormatrix. Thus the ideal )tirling cycle consists of four distinct processes, each one ofwhich can be separately analysed. )tate +$ is defined at a maximum !olume of >cmand a pressure of $.; M'a, and )tate +" is defined at a minimum !olume of #cm. The energy transferred during both the compression and expansion processes isindicated onP-Vdiagrams as follows

)ince the working fluid is helium which is an ideal gas, we use the ideal gas equationof state throughout. Thus ' K ? m ( T, where ( ? ".#AA kJCkg 6, and Pu ? &! PT,where &! ? .$$ kJCkg 6. +refer ,eal a! Propertie!
https://www.ohio.edu/mechanical/thermo/property_tables/gas/idealGas.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/idealGas.html


62/210

$. 2etermine the heat absorbed in the expansion space 8Eduring the expansionprocess + - +/ +Joules. 2etermine also the heat power absorbed +0atts. 9ote thatthe cycle frequency is the line frequency +f ? # :1. B8E? @.>J +power ? >$.0D

". 2etermine the net work done per cycle +Joules 0net? 0EV 0&+9ote that the

compression work 0&is always negati!e. 2etermine also the power supplied to thelinear electric motor +0atts. B0net? -$.;J +power ? -$#$0D

. E!aluate the &oefficient of 'erformance of the refrigerator defined as &F'(? 8EC0net. +heat absorbed in the expansion space di!ided by the net work done. B&F'(?>.#AD

/. 2etermine the amount of heat reected by the working fluid 8(as it passes throughthe regenerator matrix during process +" - +. B8(? -$./J +power ? -;@@ 0DIf there were no regenerator present then this heat would need to be remo!ed from the

gas by the expansion process in order to reduce the temperature to the coldtemperature of the free1er. :ow would this affect the performance of the cooler2iscuss the importance of an effecti!e regenerator in the )tirling cycle cooler.

Chapter 3 &he "ir!t #a$ o% &hermo'namic! %or Clo!eS'!tem!

b ,eal Stirlin) C'cle (achine! En)ine! Cooler!

1. &he Stirlin) C'cle En)ine

&onceptually the )tirling engine is the simplest of all heat engines. It has no !al!es,and includes an externally heated space and an externally cooled space. It was

in!ented by (obert )tirling, and an interesting website by Dob Sierincludes aphotograph of (obert )tirling, his original patent drawing of $@$, and an animatedmodel of )tirling


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In its original single cylinder form the working gas +typically airor helium is sealed within its cylinders by the piston and shuttled

between the hot and cold spaces by a displacer. The linkagedri!ing the piston and displacer will mo!e them such that the gaswill compress while it is mainly in the cool compression spaceand expand while in the hot expansion space. This is clearlyillustrated in the adacent animation which was produced by(ichard 0heeler +eph'ri! of ;ikipeia.

(efer also to the animation produced by (att ?e9ene'inhis Stirlin) en)ine animationwebsite. )ince the gas is at ahigher temperature, and therefore pressure, during its expansionthan during its compression, more power is produced duringexpansion than is reabsorbed during compression, and this netexcess power is the useful output of the engine. 9ote that there areno !al!es or intermittent combustion, which is the maor source ofnoise in an internal combustion engine. The same working gas isused o!er and o!er again, making the )tirling engine a sealed,closed cycle system. %ll that is added to the system is steady hightemperature heat, and all that is remo!ed from the system is lowtemperature +waste heat and mechanical power.

%thens, Fhio, is a hotbed of )tirling cycle machine acti!ity, both engines and coolers,and includes (H2 and manufacturing companies as well as internationally recogni1edconsultants in the area of )tirling cycle computer analysis. The parent company of thisacti!ity is Sunpo$er ,nc. It was formed by 0illiam 3eale in the early $;A# k0 )T-> %ir engine. This large single cylinder engine burns

biomass fuel +such as sawdust pellets or rice husks and can function as a cogenerationunit in rural areas. It is not a free-piston engine, and uses a bell crank mechanism toobtain the correct displacer phasing. %nother important early )tirling engine is*ehmann


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)tirling engines in $@A$. %ndy (oss of &olumbus, Fhio built a small working replicaof the #ehmann machine, as well as a moel air en)ine.

Solar


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'rocess "- is a constant !olume displacement process in which the gas is

displaced from the cold space to the hot expansion space. 9o work is done,howe!er as we shall see below, a significant amount of heat 8(is absorbed bythe gas from the regenerator matrix.

'rocess -/ is the isothermal expansion process. 0ork 0-/is done by thesystem and is shown as the area under theP-Vdiagram, while heat 8-/is addedto the system from the heat source, maintaining the gas at a constanttemperature T:.


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5inally, process /-$ is a constant !olume displacement process which completes

the cycle. Fnce again we will see below that heat 8(is reected by the workinggas to the regenerator matrix.

The net work 0netdone o!er the cycle is gi!en by 0net? +0-/V 0$-", where thecompression work 0$-"is negati!e +work done onthe system.

0e now consider the heat transferred during all four processes, which will allow us toe!aluate the thermal efficiency of the ideal )tirling engine. (ecall from the pre!ioussection that in order to do a 5irst *aw analysis of an ideal gas to determine the heattransferred we needed to de!elop equations to determine the internal energy change Puin terms of the Speci%ic


67/210

0e will find in Chapter that this is the maximum theoretical efficiency that isachie!able from a heat engine, and usually referred to as the Carnotefficiency. 5ormore information on this subect, refer to a paper A (eetin) bet$een *obertStirlin) an Sai Carnot in 1+24presented at the 2014 ,SEC.
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter5.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.htmlhttp://www.centrostirling.com/isec2014/index-isec2014.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter5.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.htmlhttp://www.centrostirling.com/isec2014/index-isec2014.html


68/210

9ote that if no regenerator is present the heat 8(must be supplied by the heater. Thusthe efficiency will be significantly reduced to Z th? 0netC +8inV 8(. 5urthermore thecooler will then ha!e to reect the heat that is normally absorbed by the regenerator,thus the cooling load will be increased to 8outV 8(. (ecall that 8"-? 8(? -8/-$.

9ote that the practical )tirling cycle has many losses associated with it and does notreally in!ol!e isothermal processes, nor ideal regeneration. 5urthermore since the5ree-'iston )tirling cycle machines in!ol!e sinusoidal motion, theP-Vdiagram has ano!al shape, rather than the sharp edges defined in the abo!e diagrams. 9e!erthelesswe use the ideal )tirling cycle to get an initial understanding and appreciation of thecycle performance.


Problem 3.2 - &he Sunpo$er E-1000 Stirlin) En)ineenerator

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

FFFFF

2. &he Stirlin) C'cle Cooler

Fne important aspect of )tirling cycle machines that we need to consider is that thecycle can be re!ersed - if we put net work into the cycle then it can be used to pump

heat from a low temperature source to a high temperature sink. Sunpo$er ,nc. hasbeen acti!ely in!ol!ed in the de!eplopment of )tirling cycle refrigeration systems andproduces )tirling cycle croygenic coolers for liquifying oxygen. In $;@/ )unpowerde!eloped a free piston /uple: Stirlin) (achineha!ing only three mo!ing partsincluding one piston and two displacers, in which a gas fired )tirling cycle engine

powered a )tirling cycle cooler. lobal Coolin) ,ncwas established in $;;> as aspinoff of )unpower, and was formed mainly in order to de!elop free-piston )tirlingcycle coolers for home refrigerator applications. These systems, apart from beingsignificantly more efficient than regular !apor-compression refrigerators, ha!e theadded ad!antage of being compact, portable units using helium as the working fluid

+and not the :5& refrigerants such as ($/a, ha!ing a 4lobal 0arming 'otential of$,##. More recently 4lobal &ooling decided to concentrate their de!elopment effortson systems in which there are !irtually no competiti!e systems - cooling between-/#& and -@#&, and they established a new company name Stirlin) Ultracol.

0e are fortunate to ha!e obtained two original M$##3 coolers from 4lobal &ooling.The one is used as a demonstrator unit, and is shown in operation in the following
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCogen/StirlCogen.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCogen/StirlCogen.htmlhttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCogen/StirlCogen.htmlhttp://us.sunpowerinc.com/https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCooler/Duplex_Stirling.pdfhttp://stirlingultracold.com/who_we_arehttp://stirlingultracold.com/https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCogen/StirlCogen.htmlhttp://us.sunpowerinc.com/https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCooler/Duplex_Stirling.pdfhttp://stirlingultracold.com/who_we_arehttp://stirlingultracold.com/


69/210

photograph. The second unit is set up as a ME Senior #ab proGectin which wee!aluate the actual performance of the machine under !arious specified loads andtemperatures.

% schematic diagram followed by an animated schematic of the cooler +both courtesyof lobal Coolin) are shown below
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCooler/CoolerLab.jpghttp://stirlingultracold.com/who_we_arehttps://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/StirlCooler/CoolerLab.jpghttp://stirlingultracold.com/who_we_are


70/210

&onceptually the cooler is an extremely simple de!ice, consisting essentially of onlytwo mo!ing parts - a piston and a displacer. The displacer shuttles the working gas+helium between the compression and expansion spaces. The phasing between the

piston and displacer is such that when the most of the gas is in the ambientcompression space then the piston compresses the gas while reecting heat to theambient. The displacer then displaces the gas through the regenerator to the coldexpansion space, and then both displacer and piston allow the gas to expand in thisspace while absorbing heat at a low temperature.


Problem 3.3 -Stirlin) C'cle Cooler (100D - ,eal Anal'!i!

7nfortunately the analysis of actual )tirling cycle machines is extremely complex andrequires sophisticated computer analysis. 0e consider the idealised model of thiscooler defined in terms of theP-Vdiagram shown below in order to determine theideal performance of the M$##3 under typical operating conditions as described

below. +Note that the values presented here are not actual values of the M100B,however were devised y your instructor for purposes of this e!ercise only.


71/210

'rocess +$-+" is the isothermal compression process at temperature T&? #&,

during which heat 8&is reected to the ambient. 'rocess +"-+ is the constant !olumedisplacement process during which heat 8(is reected to the regenerator matrix.'rocess +-+/ is the isothermal expansion process at temperature TE? -"#&, during

which heat 8Eis absorbed from the free1er, and finally process +/-+$ is the constant!olume displacement process during which heat 8(is absorbed from the regeneratormatrix. Thus the ideal )tirling cycle consists of four distinct processes, each one ofwhich can be separately analysed. )tate +$ is defined at a maximum !olume of >cmand a pressure of $.; M'a, and )tate +" is defined at a minimum !olume of #cm. The energy transferred during both the compression and expansion processes isindicated onP-Vdiagrams as follows

)ince the working fluid is helium which is an ideal gas, we use the ideal gas equationof state throughout. Thus ' K ? m ( T, where ( ? ".#AA kJCkg 6, and Pu ? &! PT,where &! ? .$$ kJCkg 6. +refer ,eal a! Propertie!
https://www.ohio.edu/mechanical/thermo/property_tables/gas/idealGas.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/idealGas.html


72/210

$. 2etermine the heat absorbed in the expansion space 8Eduring the expansionprocess + - +/ +Joules. 2etermine also the heat power absorbed +0atts. 9ote thatthe cycle frequency is the line frequency +f ? # :1. B8E? @.>J +power ? >$.0D

". 2etermine the net work done per cycle +Joules 0net? 0EV 0&+9ote that the

compression work 0&is always negati!e. 2etermine also the power supplied to thelinear electric motor +0atts. B0net? -$.;J +power ? -$#$0D

. E!aluate the &oefficient of 'erformance of the refrigerator defined as &F'(? 8EC0net. +heat absorbed in the expansion space di!ided by the net work done. B&F'(?>.#AD

/. 2etermine the amount of heat reected by the working fluid 8(as it passes throughthe regenerator matrix during process +" - +. B8(? -$./J +power ? -;@@ 0DIf there were no regenerator present then this heat would need to be remo!ed from the

gas by the expansion process in order to reduce the temperature to the coldtemperature of the free1er. :ow would this affect the performance of the cooler2iscuss the importance of an effecti!e regenerator in the )tirling cycle cooler.

Chapter 3 &he "ir!t #a$ o% &hermo'namic! %or Clo!e

S'!tem!

c &he Air-Stanar /ie!el C'cle Compre!!ion-,)nition

En)ine

The Air Stanar /ie!el c'cleis the ideal cycle for Compre!!ion-,)nition+&Ireciprocating engines, first proposed by (udolph 2iesel o!er $## years ago. The

following link by the ?ru!e &echnolo)' Partner!hipdescribes the %our-!troke ie!elc'cleoperation including a short history of (udolf 2iesel. The four-stroke dieselengine is usually used in motor !ehicle systems, whereas larger marine systemsusually use the t$o-!troke ie!el c'cle.Fnce again we ha!e an excellent animation

produced by (att ?e9ene'presenting the operation of the %our-!troke ie!el c'cle.
http://www.kruse-ltc.com/introduction.phphttp://www.kruse-ltc.com/Diesel/diesel_cycle.phphttp://www.kruse-ltc.com/Diesel/diesel_cycle.phphttp://auto.howstuffworks.com/diesel-two-stroke1.htmhttp://auto.howstuffworks.com/diesel-two-stroke1.htmhttp://www.animatedengines.com/http://www.animatedengines.com/diesel.shtmlhttp://www.kruse-ltc.com/introduction.phphttp://www.kruse-ltc.com/Diesel/diesel_cycle.phphttp://www.kruse-ltc.com/Diesel/diesel_cycle.phphttp://auto.howstuffworks.com/diesel-two-stroke1.htmhttp://www.animatedengines.com/http://www.animatedengines.com/diesel.shtml


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The actual &I cycle is extremely complex, thus in initial analysis we use an ideal Gair-standardG assumption, in which the working fluid is a fixed mass of air undergoing thecomplete cycle which is treated throughout as an ideal gas. %ll processes are ideal,combustion is replaced by heat addition to the air, and exhaust is replaced by a heatreection process which restores the air to the initial state.

The ideal air-standard diesel engine undergoes / distinct processes, each one of whichcan be separately analysed, as shown in theP-Vdiagrams below. Two of the four

processes of the cycle are aiabaticprocesses +adiabatic ? no transfer of heat, thusbefore we can continue we need to de!elop equations for an ideal gas adiabaticprocess as follows

&he Aiabatic Proce!! o% an ,eal a! H 0

The analysis results in the following three general forms representing an adiabatic

process

where k is the ratio of heat capacities and has a nominal !alue of $./ at ##6 for air.

'rocess $-" is the adiabatic compression process. Thus the temperature of the airincreases during the compression process, and with a large compression ratio +usuallyR $$ it will reach the ignition temperature of the inected fuel. Thus gi!en theconditions at state $ and the compression ratio of the engine, in order to determine the

pressure and temperature at state " +at the end of the adiabatic compression processwe ha!e

0ork 0$-"required to compress the gas is shown as the area under theP-Vcur!e, andis e!aluated as follows.
https://www.ohio.edu/mechanical/thermo/property_tables/gas/adiabatic/adiabatic.htmlhttps://www.ohio.edu/mechanical/thermo/property_tables/gas/adiabatic/adiabatic.html


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%n alternati!e approach using the energy equation takes ad!antage of the adiabaticprocess +8$-"? # results in a much simpler process

+thanks to student 9ichole 3lackmore for making me aware of this alternati!eapproach

2uring process "- the fuel is inected and combusted and this is represented by aconstant pressure expansion process. %t state +Gfuel cutoffG the expansion processcontinues adiabatically with the temperature decreasing until the expansion iscomplete.

'rocess -/ is thus the adiabatic expansion process. The total expansion work is 0 exp?+0"-V 0-/ and is shown as the area under theP-Vdiagram and is analysed asfollows


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5inally, process /-$ represents the constant !olume heat reection process. In an actual2iesel engine the gas is simply exhausted from the cylinder and a fresh charge of air is

introduced.

The net work 0netdone o!er the cycle is gi!en by 0net? +0expV 0$-", where asbefore the compression work 0$-"is negati!e +work done onthe system.

In the %ir-)tandard 2iesel cycle engine the heat input 8inoccurs by combusting thefuel which is inected in a controlled manner, ideally resulting in a constant pressureexpansion process "- as shown below. %t maximum !olume +bottom dead center the

burnt gasses are simply exhausted and replaced by a fresh charge of air. This isrepresented by the equi!alent constant !olume heat reection process 8 out? -8/-$. 3oth

processes are analy1ed as follows


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%t this stage we can con!eniently determine the engine efficiency in terms of the heatflow as follows

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN


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The following problems summari1e this section

Problem 3.4- % frictionless piston-cylinder de!ice contains #." kg of air at $##k'a and "A&. The air is now compressed slowly according to the relationP Vk?

constant, where k ? $./, until it reaches a final temperature of AA&.

a )ketch theP-Vdiagram of the process with respect to the rele!ant constant

temperature lines, and indicate the work done on this diagram.

b 7sing the basic definition of boundary work done determine the

boundary $ork oneduring the process B-A.$@ kJD.

c 7sing the energy equation determine the heat tran!%erreduring the

process B# kJD, and !erify that the process is in fact adiabatic.

"eriveall equations used starting with the basic energy equation for a non-flowsystem, the equation for internal energy change for an ideal gas +Pu, the basicequation for boundary work done, and the ideal gas equation of state BP#V $ %#TD.7se !alues of specific heat capacity defined at ##6 for the entire process.

Problem 3.- &onsider the expansion stroke only of a typical %ir )tandard 2ieselcycle engine which has a compression ratio of "# and a cutoff ratio of ". %t the

beginning of the process +fuel inection the initial temperature is "A&, and the airexpands at a constant pressure of ." M'a until cutoff +!olume ratio "$.

)ubsequently the air expands adiabatically +no heat transfer until it reaches themaximum !olume.

a )ketch this process on aP-vdiagram showing clearly all three states. Indicate

on the diagram the total work done during the entire expansion process.

b 2etermine the temperatures reached at the end of the constant pressure +fuel

inection process B$@##6D, as well as at the end of the expansionprocess B@#6D, and draw the three rele!ant constant temperature lines ontheP-vdiagram.

c 2etermine the total work done during the expansion stroke B$#@A kJCkgD.

d 2etermine the total heat supplied to the air during the expansion stroke B$#"@

kJCkgD.


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"eriveall equations used starting from the ideal gas equation of state and adiabaticprocess relations, the basic energy equation for a closed system, the internal energyand enthalpy change relations for an ideal gas, and the basic definition of boundarywork done by a system +if required. 7se the specific heat !alues defined at $###6 forthe entire expansion process, obtained from the table of Speci%ic


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temperature of ;##6 throughout the cycle to define the specific heat capacity !alues aspresented in the table of Speci%ic


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9otice that e!en though the problem requests Gnet work output per cycleG we ha!eonly calculated the heat in and heat out. In the case of a 2iesel engine it is muchsimpler to e!aluate the heat !alues, and we can easily obtain the net work from the

energy balance o!er a complete cycle, as follows


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Wou may wonder at the unrealistically high thermal efficiency obtained. In thisideali1ed analysis we ha!e ignored many loss effects that exist in practical heatengines. 0e will begin to understand some of these loss mechanisms when we studythe )econd *aw in Chapter .

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN

5n to Part o% &he "ir!t #a$ - 5tto C'cle En)ine!


Engineering Thermodynamics by Israel 7rieliis licensed under a &reati!e &ommons%ttribution-9oncommercial-)hare %like .# 7nited

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