1.1. Fig 1.1

1.2

Fig1.5

1.3 Fig1.6

1.4

Fig1.8

1.6 Fig1.11

1.7 Fig1.12

We might well nsk whether all the equipment in the power plant, such as the steam generator, the turbine, the condenser, and the pump, is necessary. Is it possible to produce electrical energy from the fuel in a more direct manner?

The fuel cell accomplishes this objective. Figure 1.5 shows a schematic arrange-

At the present time tbe fuel used in fuel cells is usually either hydrogen or a mixture of aseous b drocarbons and b dro en. The oxidizer is \lsuall o en. However, current

asked about the steam power plant-is it possible to accomplish our objective iD a more direct manner? Is it possible, in the case of a refrigerator, to use the electrical energy (which goes to the electric motor that drives the compressor) to produce cooling in a more direct manner and thereby to avoid the cost of the compressor, condenser, evaporator, and all the related oioim:t?

will give a higher tluusl per unit mass race of now e>f reaccancs. Liquid oxygen is fre. queotly used as cbc oxidizer in liquid-propellnot rockets, nod liquid hydrogen Is ftequcncty used as the fuel.

Much work has also been de>ne (lll solid-propellant rockets. They have been very successfullv used fe>r iec-assisled takeoffs of airolanes militarv missiles and Sllace vehi·

2.1 Fig2.1 , fig2.2

2.2

2.3

2.4

Fig2.5

Fig2.6

2.7

process. A quasi-e ui 1 IUD p.m~~~-!~.2!1~ m w •c e vtalton om rmo ynallllc C<!l!i!illcium.!L inj(esimal, and all thc; .. ~tat_es t!ii.~Y.§~m paS,es tliiough (hitmg a ·qllnsi· ~l!ilill;ium pr~£~.!-'l!~YJe~onsldc.i'd t;qu_ilj![ium_sta!c·s: Many ociliaiproccsscs ciosoly

: (ft), which at present i:

~ 1ft he foot

~ ' ·· ·- t11 ......

Consider as a system a certain gas at a iven ressure and temperarure within a tank or pressure vessel. en considered from the molec ar v~ three general fomts of energy:

@J.nrennolecular potential energy, which is associated with the force molecules.

~olecular kinetic energy, which is associated with the trunslational veloc vtdual molecules. -

&rramolecular energy (that with' the individual molecules), whiclt i8 with the molecular and atomic st ture and related forces.

I, /lV u- tm -

6V-.lY /lnt

where /lV' is the smalle~t volume for which the mass can be considered a continuum:

2.8

Fig2.11

2.10

2.11

3.1

m trus text, toe ttn¢ vorume ana a¢nSJ wtU be ven e1ther on a mass or on a mole basis. bar over the S)mbol (lowercase) will be used to designate the propeny on a mole basis. us ii will desi te molal dfic volume and- will desi ate the molal den i . In SI units, those for specific volwne are m11kg and m1/mol (or m1ikmol); for density the corre-

The zeroth law of thermodynamics states that when two b 1es ave equa ty o tempernture with a third body, they in tum have equality of temperature with each other.

these scales is

~

A pure substance is one that has a homogeneous and invariable chemical composition. lt may eKist in more thao one phase, but the chemical compos ilion is the same in all phases. Thuu 1tnu1t1 \t.M~ I tlr A m•vrttrP n t u ruun w:.rPr :.nn w;ur.r ~nnr t!\leam 1. ant• a ·m iXture o r ICC

3.2

Fig3.3

Fig3.4

Fig3.5

:( If a substance exists as liquid at the saturation temperature and pressur~. it is called <Simrated liqtilll:> If the temperature of the li uid · ver than the saturation temperature for the existing pressure, it is called eithe subcool~d li · implying that th ra­rure is lower than the saturation temperature for the given pressure) or a mpressed li -uid {implying that the pressure is greater than the saturation pressure for t e given temperature). Either tenn may be used, but the Iauer term will be used in this text.

When a substance exists as art li uid and an va or at the saturation temperature, it uali is defined as the ratio of the mass ofva or to the total mas Tbus, in Fig. 3. lb, if the mass of the vapor is 0.2 kg and the mass of the liquid is 0.8 kg, the quality is 0.2 01'

20%. The quality may be considered an ~nsive prO!?,!?r)y and bas the symbol x. Qualiry bas meaning only when the substance is ina saturated state, that is, at saturation pressure and temperature. ~

If a substance exists as va saturation temperature, it is called turated vapor. (Sometimes the te · ry saturated vaPQ · used to empbasi2e that the qua tty is 100%.) When the vapor is at a empera re greater than the saturation temperature, it is said to exist ~The pressure and temperature of superbe.ated vapor are

a cy m cr ass own tg. . , t ere WI never e two p es present e state own in I'ig. 3. Jb will never ex.ist.lnstead, there will be a continuous change in density and at all times there will be ouly one pbase present. The question then is when do we have a l!q­ltid and when do we have a vapor? The answer is tbat this is not a valid question at Sllper­critical pressures. We simply ienn the substance a fhud. However, rather arbitrarily, at temperatures below tile critical tcmperehtre we usually refer to it as a compressed liquid and at tern cratures nbO\'C the critical tem crature as A su erhented vo r It h ulcl

.s9 .. l jJUUJl u u; u,:c f U C H ,:,O

l C$:iturated solla:>F

UClUlVJUl . U.U\T"CYCl l llC: llti iJ~IH.IHU t t:'W I.'CU:UUlC d.liU ClJU '-d l l CUltJCUI\UJC ViU Y l!'lt'dUV

from one substance to another. For example, the critical temperature of helium, as given in Table A.2, is 5.3 K. Therefore, the absolute temperature of helium at ambient condi­tions is over 50 lime$ greater than the critical temperillure. In contrast, water has a critical temperature of374.14°C (647.29 K), and at ambient conditions the temperatlll'e of water is less than half the critical tempemtme. Most metals have a much higher critical tempera· turc than water. When we consider the behavior of a 8Ubstanco in a 2ivcn state, it is often

Fig3.6

Fig3.7

3.3

3.4 Fig3.8

Fig3.10

3.5 Fig3.18

Fig3.19

3.6

Finally, it should be pointed out thar a pure substance can exist in a number of dif-1 1 ferent solid phases. A transition from one solid pbase to another is called an allotropic ~.~~.-~.-.~~~~---~ .. ~--.~~~.-~ .. ~.~7~.~.-... ~o~.~--.. ~~-~-~.~-~.n.~.~.~-~-~.~.-.-,.~-~ .. -.~A-~--,-~.~ .. ~.~ .. ~.~.~.~

E nRT, _ ____... ·-·

Fig3.21

Fig3.22

Fig3.23

lion remon ll .e. hH!h.,. or lOW 1'1, thc.ttas Denavtor oecomes closer to me taeat-eas moaet. A more quantitative study of the question of the ideal-gas approximalion can be

conducted by introducing the compressibility factor Z, defined as 1 -

Z=~ I

.. Is there a way in wbicb we can put all of these subslances on a couunoo p, we "reduce" !he propertiC8 with respect to tbe values at the critical point

~ op0rtics nre defined as

reduced pressure = P, = : , c

P, ~ critical pressure

reduced temperature ,. T, ::. f. T< = cri1ical temperature (3.8 y "'-"' ' J 1 . r -,~~.. .. , n1 v 1 a t pvun.

If lines ofconsta11t T, arc plotted on a Z versus P, diagram, a plot such as that in Fig. @)s obtained. Tile striking filet is that when StiCh Z versus P, diagrams are prepared for a number of different substances, all of them verv neatlv coincide. eSQeci~llv when the wttcthcr, on a giVen arcumstance, it is rensonnble to assume ideal-1>1\S beiiOVJOr as n model. For example, we note from the cb.:lrt that if tltc pressure is very low (that is, 4.P,), Ute ideal gns model con be assumed with good occlln!cy, regardless of the temperature. Furthcnllorc at high temperatures (th~t is, greater than about twice TJ, tile ideal-gas model can be as sumed with good accuracy to pressures as high as four or five times P •. \Vhen rhe tempera rure is less than about twice tbe critical temperature and the pressure is not extremely lo"' we are in a region, commonly termed superheated vapor, in wbich the deviation from ideal !!as behavior mav be consick>rable. ln this rel!ion it is oreferable tn .,., t•hl .. < ... r ·• .J.

4

4.1

4.2

4.3

JJ=lNm

Poweris the time rate of doing work and is designated by the symbol II':

f JP= oW I dt

The unit for power is a rate of work of one joule per second, which is a watt 1

~ A famJtiar unit for power in English units is the horsepower (bp~ where

~ ,_,_, _ iL . o .. 1.. ····-'· -- ---:-- •L• L-··- -'· -· . r...a. . ···-• · .. : . r: . "t :. ~~ .. .._ ...

at the moving boundary of a simple compressible system during a quasi-equilibrium roces.s .

' J .... ., ... o• ••••• •l> ..... __,.

11Y2 ~ J: llll' = {' P dV

4.4

4.5

..>U\ol ... vuu uuu "'J\..U.I..U.J-11""' V1 Ull-' """"-VUU ~.1 1J.._ Vl l UUl.IJUUll l 11.>1dUUU:'Ulll' J:\.

' polytropic process, one in which

PV" = constant

10ut the process. The exponent II may possibly be any value from -co to +• the particular process. For this type of process, we can integrate Eq. 4.4 as f.

PV" = constam = P 1 Vj ~ P2V';

p = constant= P, V~ = P11'; · v• Y" v•

f.' p dY - COMlant r '!!. .. constant ( _v:·:\ )I: J' p v•r•-· p ~v· -· p dV =constant (VJ-· _ v:-·) = > 2 2 - 1 1 1

1 1 - 11 1 - n

P~v, - P1V1 = l- n

1e resulting Eq. 4.5 is v.ilid for any exponent n, except ,

PV = constant = P1 1'1 = P1 V2

1g~A v:. tu\1": ~in .. n f whlrb r an

1W1 = - J,' ~ dA

, ..... ~. . . ..... ,, .. . ·--61V = - 'ti dt

IJV2"' - f.''8idt

'II as a rate equation for wo

• I)JY • W = - =-'eSt

dt

4.6

4.7

Simple compressible system

Stretched wire

Surface film

System in which the work is completely ele·ctrical

SW = P dY - '!J dL - !! dA - ~ dZ + · · ·

1W2 = fPdV , wl~ - r~dL

,w, "' - J,2

~ dA

I w, = - I.'~ dZ (4 .15)

re<:~uu:ed to r01se I " o water rom f fi t4 s•c 1s.s•c to Heat transferred ro a system is considered positive, and heat transferred from

o system is negative. Thus, positive heat represents cocrgy transferred to a system, and n~o•tiv~ heat renrcsents encriTV transferred !Tom a SV'ltem. The svmbol 0 reoreSents beat. A in 'uhir h ih orr io no-> ""' " ·••·· rn - m ; • .-ollp .f •n -

From a mathematical perspective, heat, li.ke work, is a path function and is recog-nized as an inexact differential. 'That is, tbe amount of heat tmnsferred when a system un-

>erahlre difference nd t to in Fourier's law of conduction

Q= -kA dT (W) tlr

h~ 0\tr; nfhta t tntnsfP.r :.c:; nmnnrtinn:ll tn thl» N'lnrlnr""

maucm or numenca1 so1unon JS n01 avauame. Values of the conductivity, lc, range from the order of I 00 Wlm K for metals, 1 to

10 for ooometallic solids as glass, ice and rock, from 0.1 to 10 for liquids, around 0.1 for insulation materials, and from 0.1 down to less than 0.01 for cases.

A A;rr ........... ..... ,..,l . .a ,..r h•"' ...... . ,.. ... r ..... ............ ~ ........... ... 1 . .......... - .... J: .•. - : • .!'!-···=-- __ u • .J

UV\YUt ll( :tiU'-' un .• r.~.nuUl\.1& l I Lit!

New1on's law of cooling as

Q "' All b.T

5.1

5.2

neat tra.nStt:r CUCIIICittu JUt a. ):.IVCll SJlUH.UUJI,

Typical values for the convechon coelfictent (alltn W/mK'J are

NQtural convection Fon:<:d convection Boiling phase change

h = ~,-"2S, gas h = 25-250, gas

h = 25()0-100 000

II= S(}-1000, liquid h = 5(}-20 000, liquid

(4.20) .... • • r-

form is written

'w"'~•:...---------I Btu ~ 778.17 ft lbf

dE = /lQ -llW

Because E is a property, its derivative is written dE. When ~ q tntttal state l to a final state 2. we bnve

E "' lntemal eoergy + kinetic energy + potential energy E = U+ KB+ PB

d(mV'} dE = dU + -

2- + d(mgZ) ~ 6Q - 6JY

Assuming g is a constant. Jn the integrated fonn of this equation,

m(Vl - IP,) U2 - U1 + 2 + mg(Zl - Z1) "' ,Q1 - 1W1

5.5.

5.6

: Cl l.lt;W V ... lCill!»JVC }Jl

I Ho:U+PV I

or,~oss, h"' u ~

As for internal energy, we could speak oKSjjecilic enlhalpy, ]i>: . . . . ..

... L. L • • ~·· · · 6L. • .••• ..1 . "'-- - .1 •.• ...:- - 6.L . _ . ............ : ... ,._ . . .. t -. .. 1.-\. .. .l:t£'.,..,.,.._..,.,. : .., .11 .. .. n tt - -..1,..,.1

u= h-Pu

Students often become confused about the validity of this calculation when analyz­ing system processes that do not occur at constant pressure, for which enthalpy has no physical significance. We must keep in mi.od that enthalpy, being a property, is a state or point function, and it$ usc in calculating intemal energy at the same state is not related to, or dependent on any process that may be taking place.

• I 1 I • , '" • I 1.·. ~ t

""' "nmrl >VOIM "'"' · h • lnHn" ~~~ Tnhl• U I A ,;~ .... h.

stances for which compressed-liquid tables arc not available, the enthalpy is takell as tlmt of saturated liquid at lhe same temperature.

In this section we will conside>;-=-.::;::=;<f===-T~=~==~;;.r..;.;;;~;;;;;~~rn'i--' tion. Thls phase may be a solid, a liquid, or a gas, ut no chao c o se w occur will then define a variable tenned the specific beat, the amount of heat required per unit nutss to raise the tomperuture by one degree. Since it would be of interest to exa!lline tbe

5.7

1. ConStant volume, or which the work tenn (P dV) is zero, so th

-~t-%~mM1'11Tiiiwne) isi---------.,-----,

c.~~(~) ... ~(~). = (~). 2. onsmnt presrure for which the work tenn can be integrated a•

e wlial and fma l states can be associated with the in~ os in Section 5.5 , thereby leading to tho conclusion that the heat pressed in tenns of the enthalpy change. The corresponding sp slant pressure) is

_ I (8Q) _ I (iJH) _ (ah) C, - iii I;T , - iii iJT P - iJT P

Solids and Liquids

As a special case, consider either a solid or a liquid. Since both of these phases are nearly incompressible,

dlr = drr + d(Pu) ~ drr + 11 dP (5.16)

Also, for both of these phases, tbe specific volume is very small, such thm in many cases

dil "' dtt "' C dT (5.17)

the two

. That is, for an ideal gas,

Pu = RT and 11 = f(T) only I

Because the internal energy of an ideal gas Is not a Jiwction of specillc volume, for an ideal gas we can write

C ., drr .o dT

dtt = C<lldT

lwbcre tbe subscript 0 deuotes the specific heal of an ideal gas.IFor a given mass m,

(5.20)

Therefore,

c? = (:~), ; a function of the tempt

c - dh 'til dT

dh = CJIJdT

C,.o = f(T)

On a :note basis this equation is written

C,g - C.o = R

Let us consider the specific beat C . 'fbere "re three sslbilities to examine. The situa6on is simplest if we assume constant spect c teat that is, no temperature depen­dence. Then it is possible to integrate Eq. 5.24 directly to

l 1zz - h, = Crf;(T2 - T,) I (5.29)

.. ... <J•··· . .. ·- ~ el" ' ... - ............... .

The second posMbility for the speeific heat is to ysc an analyrica! equation for c;.. as a function of temperature. Because the results of specific-heat calculations from statistical tll.ermodyoamics do not lend themselves to convenient mathematical forms, these results have been approximated empirically. The equations for C,g as a function of ternpemture nre listed in Table A6 for a number of gases.

Th.c third possibility is to imcgrnte the results of the calculations of statistical ther­modynatnics from an arbitrary reference tcmperatwe to any other temperature T and to define a function

hr = Jr CJIJ dT r,

This function can then be tabulat:cd~in;=a;:;su;;::;;;.;;?;(te-::,•mperature) table. Then, between any two sUites I and 2,

fr, J'' hl - It, = T, C,g dT - r, cp'l dT ~ hr, - hT, (5.30)

5.8

6.1

6.2

6.3

2.

Therefore, the f.!te equation form of the :first law is

We could also write this in the fonn

: is wrinen as

e = 11+~V2 +gZ

• ""'_..._,.express t11e rate or now wu111 ••

J PV dA ~ PV = Ptni• . ·• . ·---'· : .... \., ,..: ....... "'~" ..

d ension oft he Jirst law ofthermodynamics from Eq. 6.4 becomes

dEc.v. · · · . . . . ~ = Qc.v.- IYc.v. + mser- m,tt + H'ncwt. - nrtbYJ>O"~

:tih• tinn nrFn h .c; oivP.q

1es, so a summanon over tnose terms rs onen neeaea. 1 ne Mat rom1 or me nrst mw or :ermodynamics then becomes

dEc.v. · · . ( 1 ) . ( ) --;j/ = Qc.v. - Wc.v. + L Ill; h, + 2 V! + gZ, - L Ill, "·+ t v; + gZ, (6.7)

1. The assumption that the control volume does not move refauve to the coord>Dale . frame means !hat all veloeities measured relntivc to tbe coordinate frame are o.lso velocities relative to the control surface, nnd tbere is no work associated witb the

· . nf th• rnntrnl vol•tme.

6.4 Heat exchanger

Nozzle

Throttle

Turbine

Compressor and pumps

6.5

Example 6.11

Table 6.1

7

7.1

Fig7.3

Fig7.6

·-~· ~ ~· . , , These two examples lead us to

tor w c IS a so re erre

'"" tuw"tctnJ.l"l••mc uuuy. tue uutumu "' L.De ncar transrerwm oe evident 1t0m the context . ..,.. __ A:..::.,t .:.:th:!:lis point, it is appropriate to introduce the concept of henna! efficiency of a llteat engine! In general. we say that efficiency is lhe ratio of output, t e energy soug t, to input, the energy that costs, but the output and input must be clearly defined. At the risk of oversimplification, we may say that in a heat engine t.he energy sought is the work, and the energy that costs money is the beat from the high-temperature source (indire-ctly, the cost of the fuel). Thennal efficiency is defined as

II"( energy sougbt) Q11 - QL QL ~..., = Q11(enecgy that costs) = Q11 = 1 - Q11

(7.1)

Heat engines vary greatly in size and shape, from large steam engines, gas turbines, or jet engines, to gasoline engines for cars and diesel engines for trucks or cars, to much smaller engines for lawn mowers or hand-held devices soch as chain saws or trimmers. Typical values for the !henna! efficiency of real engines arc about 35-50"/o for large power plants, 30-35% for gasoline engines, and 35-40% for diesel engines. Smaller utility-type engines may have only about 20% efficiency, owing to their simple carburetioo and con-

OIC' ~nt1 1"\ t h4 Mf't fhftt ~ .... ....... ,,..., .., .. ,.,.. ,....,. , .,. ri.Ot'ti ... ..-..,_J·I~•~·••:• S.. .,,:,_., ·-"' ••••oo• C~ •• ·• • • • • •

The "efficienc " of a. refri eralor is ex res sed in terms of the coefficient of perfor· mance, which we designate with the.symbol {3. For a refrigerator the objective, that is, 1e

energy sought, is Qt. the heat transferred from the refrigerated sp.,ce. The energy that costs is the work W. Thus, the coefficient of performance, /3,1 is

Qt(cnergy sought) QL 1 (3 = W(energy that costs) = Qu - (h ~ Qt!Qt - 1

(1.2)

A household refrigerator mny have a coefficient of performance (often referred tons @ of about 2.5, whereas a deep freeze unit 1vill be c_loser to 1.0. Lower oold tem~era­fllrl'. •n•rl'. nfh ivhr.r w1m11temneralllre snace will resultw lower values of COP. as will be

, QH(cncrgy sought) QH ~ IV (energy that costs) = Qn - QL =

It also follows tbat fo a given cycle. {3' - f3 ~ I

~- .:-,,...., .. --,.----::-~-;;_,.-;,,.;_-i---,...-rr .. :-.=--"'--"'-"·"'·"·-:!-• . . -· t'..S --- - ··-· A ..

duccd A thennal KServoir is a bod to whiclt and from wllich beat can be transferred in· 1te y without change in the temperature of the reservoir Thus, a lhennal reservoir

a wnvs remruns a const:mt temnr.rnhttt: .

7.2

Fig7.8

Fig7.9

7.3

7.4

Ull me oaSIS Ot lhe matter constderetl .n th e ·~rlinn \VA ... n <'UU rPo.!v • • <lol~

the second law of tltennodynamics There are two classical nls of the second law, ! I lu!own as the Kelvin-Plsuckstatet •entHna_ e_l,.tausnts smr~entJ

The Kelvin-Pia lick statement: It is impossible to conslntet a device that will operate in a cycle and produce no effect other than the raising of a weight and the exchange ofbe3t with a single rc~ervoir. See Fig. 7 .8. ,...-,.! . -~ .•

U l.ClL \,o<III!)1U"' ( U .IU 41~\.1 ILUIU U.& .... U'--Q l \•.U!;UlV l V U u .. &VlV• ...VUijJC:IdUJlC: VVUJ. J.l.U:. UUJJUI:b Ul4\

I it is impossible to build a beat engine that hru; a thennal efficiency of JOO"A.. I

Tire Clausius statement; lt is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a cooler body to a hotter body. See Fig. 7 .9.

engmc of 100% efficiency, what is the maximum efficiency one can bove? The first step in the aoswer to tltis question is to defJM au ideal process, which is called n re· vcrsible process. '

A reversibl.: process for a system is defined as a process that once having taken lace can be reversed and in so doin leave no chan e in i ·

1-\ lJUUIUCI UJ U U ICI LISClVl~ U..rd.P..,~ fJl V\.t; C\'CI;)IUU, .. , UUt IU""} \YU I IJVl u~,o '-VUOI""-L ... \-1 uo

detail here. steresis effects the R loss e countered in electrical circuits ore both fac-

7.5

7.6

7.7

is reversed, the heat en · e bec<>mes a n:fri erator. Jn the next section we will show that this is the most efficient cycle that can operate between two constant-temperature reservoirs 1 ts c 1 c Cantot cycle an IS n:une a era French engineer, Nicolas Leonard Sndi Camot (l796-1832), who expressed Ute foWJdations of the second Jaw

1. A reversible isothermal process in which heat is troosferred to or from the high· tempemwre reservoir.

2. A reversible adiabatic process in which the temperature of the worKing fluid de· creases from the high tcmpecull•re to the low tempcrnture.

3. A reversible isothem:till process in which heat is transferred to or from the low· temperature reservoir.

4. A reversible adiabatic process in which the temperature of tb.e working fluid J.n. creases from the low temperarure to the high temperature.

First Proposition

It is impossible to construct an engine that operates between two given reservoirs and is mo~ efficient tban a reversible en ine o eratin between the same wo reservoirs.

Second Proposition

All engines tllat operate on the Camot cycle between two given cons!ant-temperature reservoirs have the same efllcienc . The roof of this >ro ilion is similar to the roof ... ...

IJ'CS so defined, the efficiency of a Camot cycle 101

tem emtures. i::::" Q, T,

11""""' ~ l - -Q = I - -T II H

. . . .

7.9

8.1

lC~ ClllCJCIU, SUCil UUU

I U"' 1101 0 \ '-'11 I U V\U """'l NIY'VI \oi\IVU v • t.u v ,.,....,.,.,.,J •t v .,. .. ,, ..,, , ... .,..11"./ - ov .. .,. .... .., .... ..... •-• •••

equality of Clausius, which is

The ioC<IuaUtv of Clausius is a corollarv or a coMequcncc of the second law of lhennody­namlcs.Lil w ill be demonstrated to be valid for all oossiblc cvcles mclud inr- both re-I

I "'"'ihle and· I• h~al en2incs and refti2erntorsl Since any reversible cycle can be

FIGURE 8.2 Reversible rcfrige18lion eyole for defl)()l1Sjmuon of tl:c incqU31ity or Clim!:Cius.

TL and receive the same amount of beat Q,. as the reversible refrigerator ofFi~&-l!d<...!:Jl!!!!!, the second law, we ccnelude that the work input required will be greater or the irrc­'"'rsoble refrlgernlorj or

0

()

._l _____ _jl r,.

7.2

QH"' > QH "v carnot

.e irreversible rcfdgcmtor o the I c y c le 1

:ted by tbe reversible refri emto

f 8Q = -QuilT + QL <

8Q = _ QHilr + Q~ < Q T Tn TL

efficiency

.. . . . . ,... ,.,.,

~c~y~c~lew~no~t~a~tio~n----~t~~-------------should be re m oved "

Sincd the p 80fT is the same for aU reversible paths between states I and 2) • • • .1 • •• • • • ~ . -'· · · • • • .J._ ... .l'.L ... _ ., .\. .. _., : .. : ... ... A .... .., t O"'v. ,... r,,., • .,n;-~ et"

sl- s. "' I: e;)..v Tn • , .,.. • • . w~ must know the rciBilon between Tand 0, and illustrations ~viii be given subsequently. The important point is that since entropy is a property, the change in the entropy of a substance in going from one state to another is tbe same for aU

""'h ' · oo<l · · ~veen these two s tates. Eauatioo 8.3 enables·

Eauation 8.3 eoables us to calculate changes of entropy, but it tells us nothing about !absolute values of entropy. From the third Jaw of thennodynamtes, which tsoasea on oO­jServations of low-temperature chemical reactions, it is concluded that the entrop y of all !Pure substances (in the appropriate structural form) can be assigned tbe absolute value of zero at tb.e absolute zero of teroperature.lt also follows from tl1e subject of statistical ther­modyruunics that all pure substances ln the (bypotlteticnl) ideal-gas state at absolute zero temperature have uro entropy.

However, when tb.ere is no change of composition, as would occur in a chemical re· action, for example, it is qu.itc adequate to give values of entropy relative to some arbitrar· ilv selected reference state such as wos done earlier when tabulating values of internal

8.3

Fig8.5

Fig8.6

8.4 Fig 8.8

s .., (I - x)31 + xs.

s = sr+ .vsfl (8.4)

The entropy o a compresse tqut Ill ra u ate m 1e same manner as te o or properties. These properties arc primarily a function of the temperoturc and are not greatly di fl'erent from those for saturated li uid ot the s;~mc temperature. Table 4 of the stealll ta­bles, which is summarized in Table B.l.4, give the entropy of compressc iqtn water m the same manner as for other ro rties.

The thermodynamic propett ies of a substance are often shown on a temperamre­dia m and on an enthal -entro din ram which is also called a Mollier

diagram, nfler Richard Mollier ( 1863- 1935) of Germany. Figures 8.5 and 8.6 show th~

_ W.,, area 1- 2-3-4-1 Tim --=

Qn area 1-2-lHl-1

1 f2 (8Q~), 1 fl q lr S2 - S t = s .. = - - = - /5Q c .!...1 = ..f!. ,. 111

1 T ,.., mT 1 T T

ver&tote Drocess. The important conclusion to draw here is that for processes that are internally re­

versible, the area underneath tbe process line on a temperntnre-entropy diagram repre· ~eots the quantity of heat transferred. This is not true for irreversible processes, as wiU be . .

8.5

8.6

AI this poiJlt ve derive two importa.tt thennodynamic relations for a simple compressible substance. T esc relations are

T dS = dU + P dJI

T dS=dH - VdP

Tile fin;t of these relations can be derived by considering a simple compressible Sllbstance!m the absence of motton or gravitational effects. IThe first Jaw for a cbange of state under lbese conditions can be 1vrinen

~Q= dU+ llW

11tc equations we ore dcri •·ing here deal first with the changes of state in which rhe stnte of the substance can identified at nll tim • · ut-

cess or to use the tenn introduced in the last cha ter a reversible roccss For a reversible process of a simple compressible substance, we can write

\VIIl'-ll I~ \Jlt: ~VUU lC'UlUUU UUU W:Q_bC'l_VU~ l\J UtJJ. \'C:. U le-K '-\'YV ti'AlJlC::!>:)IVJI~ J.:.~:>. O . ..J UHU

8.6, are two forms of the lbermodyuamic property relation and are frequently called Gibbs equations.

These equations can nlso be wrinen for a unit mass,

Tds = du + Pdu

Tds = dh - vdP

' CYJl'l.til:ws..Wil:!'ll::II=~LII:Jtlli' ~· ly in certain subsequent sections of this ·e remove "simple ompositioo other tbnn a simple comp1 . ' compressible cr than those just given for a simp! ;u substance" that r a reversible process we can w e~

assumption -9'dA - dZ+ ...

!bat more general expression for the thennodynamic property relatio W·

TdS = dU+ PdV - '(fdf.. - 9'dA - '$ dZ + .. ·

(8.7)

(8.8\

8.7

8.8

oQI« = T liS - T liS;;

t for lite reversH e case for l e sam :rsibl~ process. the wor~s o !on the first law is

~ •alid, +

~ \ '

oW;,, = PdV- ToS,.,

nal entropy generation and ellber mcreased or decreased b heat transfer de ndJn on the direction of that iransfer ln this section, we examine the effect of l1eat tronsfer on th change of st~te in the surround in s, ns well as on the control mass itself.

Fig8.15

8.9

(8.17)

(8.19)

Now, as was mentioned In Section .S. , for many processes involving u solid or liquid, we may assume that the sptdfic heat rc m.in:s c;:onslant, in which case Eq. 8.19 cnn bo inte­grated. The re.suh is

r, 1 s, - •• - C In T, I (8.20)

If Isentropic (Reveslbie+Adla.) the n... onstant. then comrnooly Cis lcncnvn as a function ofT, in which

L.:..:..:.:..:.:..._ ________ ~---,...J•lso be iolegrated to find the entropy cbauge. Eqootion 8.20 illustrates what hapll!!if.n a reversible tds - 0) adiabatic (dq m 0) pra<ess, wbich therefore is isentrooicJfu this crocess the approx.imation of constant u leads to c .. •onstant temperature, I which t."<Dioius whv oumpmg 1q\ud Cloes not change tlle tempcrnture.l

8.10

Ex

Tds = du + Priu

and

ds = c.., ar +Rdu T u

fl d1' v s, - s1 =

1 C.o T + R In 11:

tfh = c,. dT and

ds - cdT _R dP - ""T y

0 """\ __ .J 0 ..... 04.4 ... . . ... , _ _ ---· ..... . . . .

narnics from reference temperature T0 to any other temperature T and define the stan­dard entropy

fr cp) S~· - - . rlT

r, 1 (8.27)

8.11

Fig 8.18

• ~tvunn:.) I..Ull~ldlJl.\PC\:IUC ll~cU . · Hl

.ce, the result is Eq. 8.25 with the left idt

0 r, Pl

- s1 = = Cpe In- - R In -T1 P1

(8.35)

This is a special case of a polytropic process in which the polytropic exponent n 1s cqua to the specific heat ratio k.

a reversible polytropic process can be derived from the relatior

1 W1 = J,2

P dV and P V' = constan

w, = f P dV"" const:mt J,1 V: P71'2 - P1V1 mR(Tz - T1)

= I - n ;:;; 1-n

llly value of n except n = I . The polytropic proces~ for various values of narc shown i

diagrams. The values of 11 for some familiar processes are

ric process: Jsotbennal process: Isentropic process: lsochoric process:

1l = o. n = I, T= constant n = k, s = con.~tant

n = oo, u = constant

8.12

'·