Chapter03 Balance

8/13/2019 Chapter03 Balance

1/44

3 FUNDAMENTAL LAWSAND BALANCE

EQUATIONS FOR MASS,ENERGY, ENTROPY,EXERGY, ANDMOMENTUM

The purpose of studying thermodynamics is to predict the behavior

ofsystemsin terms of theirstatesas they respond to interactions

with theirsurroundings. Classical thermodynamics is an axiomaticscience; that is, the behaviors of systems can be predicted by

deduction from a few basic axiomsor laws, which are assumed to bealways true. A law is an abstraction of myriads of observations

summarized into concise statements that are self-evident and

certainly without any contradiction. e have already come across

the !eroth "aw of thermodynamics, which introduced temperature, athermodynamic property, as an arbiter of thermal e#uilibrium

between two ob$ects.

%n this chapter we will introduce the conservation of mass and

momentum principle, the &irst "aw and 'econd "aw ofthermodynamics and the concept of exergy. A uniform framewor( in

terms of balance e#uations will be developed. )ach fundamental

principle will be translated into a balance e#uation of a particularproperty. *ust as e#uations of state are the starting point for a state

evaluation, analysis of engineering systems and processes in the

future chapters will begin with the balance e#uations. hile thebalance e#uations are derived in this chapter, their applications to

closed and open systems are delegated to Chapter + and

respectively. To gain a comprehensive insight into these e#uations

Chapters , + and , therefore, should be iteratively studied.

Chapter


2/44

&undamental "aws and /alance )#uations &or 0ass, )nergy,

)ntropy, )xergy, and 0omentum.......................................................

. /alance )#uation.......................................................................1 2eynolds Transport )#uation 32T)4.......................................

. Classification of 'ystems .........................................................5

.. 6pen vs. Closed 'ystems ..................................................7..1 'teady vs. 8nsteady 'ystems ............................................7

.. 8nsteady 9rocess ..............................................................:

..+ 'ystem Tree .....................................................................+ 0ass )#uation ........................................................................

.+. &orms of 0ass /alance )#uation ...................................

. )nergy )#uation .....................................................................

.. &orms of )nergy /alance )#uation ................................+.5 )ntropy /alance )#uation .....................................................

.5. &orms of )ntropy /alance )#uation ...............................:.+?1.1:7 (*?(g

B and vc > .- .1:7 > .7: (*?(g B. The variable temperature

can be expressed as 1T c c x= + with c = B and 1 c = B?m.

8sing the ideal gas e#uation, pv RT= , )#. can be simplified asfollows.

1 1

; ;+ +

L Lb bp p bdx

B bdm dV dx dxv RT R T

= = = =

&or the evaluation of mass of the system, B m= ; therefore, b= .'ubstituting this and the linear temperature relation, )#. can beintegrated.

( )1 1

1 ;

1 1;1

1

1

ln+ +

l .


5/44

!iscussionThe evaluation of other properties such as energy is more

complicated since b as a function of x will complicate the integrand

of )#. . &or more complex systems, where variation can be in all

three directions and are not (nown in functional terms, integration of)#. may be impossible. &ortunately, the global properties of non-

uniform systems are seldom necessary to evaluate. )xamples ofproperty evaluation for uniform systems, which are more common,

can be found in )x. 1.< and 1.1.

3.2 Reynolds Transport Equation (RTE)

The fundamental laws are usually described with closed systems in

mind. &or instance, ewtonEs 'econd "aw which states that the net

external force on a particle e#uals its rate of change of momentum,implicitly assumes the particle, the system in our case, to be closed.

'imilarly, the conservation of mass principle, and the &irst and

'econd "aw are also easier to state as applied to closed systems.

)ach of these laws expresses the rate of change of a particularextensive property with respect to time in terms of other variables. %n

other words, a generic format for these laws can be written as

unit of(nown 3from the fundamental "aw4

s

cdB B

dt

=

The superscript c reminds us that this e#uation cannot be applied to

open system as is. The right hand side 32F'4 is prescribed by thespecific laws to be introduced shortly. ith the help of 2T) the

fundamental laws, which are (nown in the closed system format of

)#. , are expanded into balance e#uations applicable to any (ind of

system, open or closed.

e begin the development of the 2T) by considering a very

general open system at time tand t t+ as s(etched in &ig. 1.+. Theminor restriction of a single inlet and exit will be lifted as the laststep of this derivation. The system, defined by the dotted blac(

boundary, is allowed to have all possible interactions @ mass, heat

and wor( @ with its surroundings. As shown in the s(etch, even theshape of the system is allowed to change. As the wor(ing substance

passes through the system, we identify a closed systemmar(ed by

the red boundary at time t, which occupies the entire open system

plus a little region % near the inlet. The closed system becomesdeformed as it flows through the open system. After a small periodt , it still occupies the entire open system; however, the region-%

completely disappears and a new region, region-%%%, not necessarily

e#ual in size to region-%, appears near the exit. This is not acoincidence since for any given t , the region-% is carefully chosenso that the entire fluid inside that region flows into the system during

that interval. 6f course, t has to be sufficiently small so as not to


6/44

allow the closed system to loose its identity through disintegration,

and regions % and %% can be considered uniform so that

% %

t t t

iB m b= , and %%% %%%t t t t t t

eB m b+ + +=

/ecause B is an extensive property, an inventory of B for a system

can be obtained by combining contributions from different sub-regions comprising the system. 2eferring to &ig. ., the change in

B for the closed system as it passes through the open system 3region

%%4 can be written as

( ) ( ), , %%% %c t t c t t t t t t t B B B B B B+ + + = + +

o special superscript is necessary for the open system because it is

the system by default. 2earranging and substituting )#.

( ), , %%% %c t t c t t t t t t t t t t

e iB B B B m b m b+ + + +

= +

ividing both side by t and ta(ing a limit, ,

%%% %

; ; ; ;lim lim lim lim

t t t t t t t t tc t t c t

e i

t t t t

m b m bB BB B

t t t t

+ +++

= +

The "F' and the first term on the 2F' of this e#uation are clearlyderivatives of the extensive property B with respect to time for the

closed and open system respectively. Also, as t , t t te eb b+

,

and the last two terms approach e em b& and i im b& , where the

superscript tis not necessary anymore since each term in this

instantaneous expression refers to time t. The above e#uation, thus,

reduces toc

e e i i

dB dBm b m b

dt dt = + & &

=eneralizing for multiple inlets and exits, the "enol#s T$anspo$t

Equation32T)4 or the %ene$al balance equationcan be written as

i e

2ate of increase of 2ate of increaAet flow rate of Aet flow rate offor an open system into the system out of the system

c

i i e e

B B B

dB dBm b m b

dt dt = + & &

1 4 2 4 3 1 4 2 4 3 1 4 2 4 3se of

for a closed system.B

1 4 2 4 3

%t relates the rate of change of an extensive property B of an opensystem at a given instant to that of a closed system which happens topass through with the boundaries of the two systems aligning on top

of each other at that particular instant.

3.3 Classification of Systems

%n practical applications, thermodynamic systems or their behavior

are restricted in certain ways. Therefore the general template of the

5

&ig. .. A very general

system at two

neighboringmacroscopic instants.


7/44


8/44

6bviously this simplification is applicable to both open and closed

systems giving rise to four types of systems already. A closed system

passing through a steady open system need not be steady. %f youfollow a control mass of steam as a closed system entering the

turbine, it will surely undergo changes. That is why the last term in

)#. , which trac(s the changes in the closed system flowing through,cannot be set to zero.

%n the classification process, the second #uestion to as( is,

Goes the image of the system ta(en with a state camera change with

timeHI Although the answer is a simple yes or no, sometimes itdepends on the resolution or precision with which one answers the

#uestion. %nside a turbine 3ta(e a virtual tour of turbine in the T)'T

web site4 the rotors spins at a very high 290. Therefore,instantaneous snapshots at two different times cannot be identical.

Fowever, if the thermodynamic instant 3see 'ection ..14 is

stretched by increasing the camera exposure to a few milliseconds,

the pictures at two different times will be almost identical as all thefluctuations would average out in those few milliseconds. %n a

similar way, a car engine can be considered steady, as long as the

time resolution is large enough for the piston to execute severalcycles of stro(es. 6n the other hand if we are interested in a single

stro(e of the piston, the picture obviously changes and the system

must be considered unsteady.

3.3.3 Uns$e'"# P(!)ess

The time derivative of B is non-zero for an unsteady system. The"F' of the balance e#uations cannot be simplified any further if

instantaneous rate of change of B is important. &or example, if we

are interested in the rate of change of temperature of a cup of coffeeat a specific instant as it cools down, we have an instantaneous!

unsteady! closedproblem. The general balance e#uations, by default,

apply to instantaneous, unsteady, open systems.

6ften, in unsteady systems, the change of system propertiesover a finite interval is of greater interest than an instantaneous rate

of change. &or instance, in the compression stro(e of an automobile

engine cycle, we are interested in the state of the gas mixture at thebeginning and end of the stro(e rather than at any intermediate state.

'imilarly, in the charging of a propane tan(, another unsteadyphenomenon, the instantaneous rates maybe of less significance than

the overall changes during the entire process. The balance e#uationsunder such situations can be simplified by integrating with respect to

time.

An unsteady system is said to execute a p$ocess if it

undergoes changes from a beginning global state, called the b-stateor begin-state, to a final global state, called the &-stateor final-state.

:

&ig. .7. As water flows

through the constriction,

its pressure changes.

Fowever the open

system is a steady one ifthe global picture does

not change.

&ig. .:. 'ystem

classificationD 9rocess

vs. %nstantaneous rate.

&ig. .


9/44

The begin and finish states are also (nown as the anc'o$ statesof a

process. The anchor states must be in e#uilibrium for a process;

however, as the system moves from the b-state to f-state it does nothave to pass through a succession of e#uilibrium for the balance

e#uations to be simplified. &or system which is uniform at the

beginning and end of the process, the anchor states can be spotted onthe familiar p v diagram as s(etched for a compression process in&ig. .


10/44

3.3.* S#s$e% T(ee

The classification of systems introduced until now can be organized

in a tree structure as shown in &ig. ., called the sstem t$ee. Thenext two chapters will be devoted exclusively to the discussion of

closed and open systems respectively. &urther classification of

closed processand open steadysystems will be deferred until then.

%n T)'T start at the daemons page, by using the aemonslin( on the Tas( /ar, to classify a system. A simplification table

provides lin(s to all possible branches one can follow depending on

the answer to the #uestion posed at the table header. At any stage ofsimplification, a system schematic and the customized set of balance

e#uations appear below the simplification table. 6nce you gain

expertise in this step-by-step procedure, you can use the 0ap,

arranged li(e the tree of &ig. . and lin(ed from the Tas(-/ar inT)'T, to $ump to a specific category of systems by clic(ing on its

node.e now begin the development of fundamental laws into

balance e#uations and customize these e#uations for different classesof systems.

3.4 ass Equation

The conse$)ation o& mass p$inciplecan be stated through the

following simple postulate.

Mass cannot be created or destroyed#

&or a closed system the total mass cm must remain constant;

therefore, the time derivative of cm must be zero, i.e.,

cdm

dt=

'ubstitute )#. into the 2T), )#. , with B m= and .b = , toformulate the mass balance equationfor an open unsteady system.

2ate of increase of mass Aet mass flow rate Aet mass flow rate of for an open system. into the system. out of the system.

(g

si e

i e

dmm m

dt

= & &

1 4 2 4 3 1 4 2 43 1 4 2 43

The meaning of the three terms is explained with the help of a &lo(

#ia%$amin &ig. .1. The difference between the inflow andoutflow is accumulated in the balloon. 'imilar flow diagrams will be

constructed for other balance e#uations.

&ig. . The systemclassification tree. The

0ap in T)'T displays a

similar clic(able tree.

&ig. .1 &low diagram

for the mass balance

e#uation..


11/44

3.*.1 F!(%s !+ M'ss B''n)e E-'$!n

The general form of the mass balance e#uation can be simplified for

different categories of systems classified in &ig. ..

*lose# Sstem Simpli&ication&or a closed system the mass transfer

terms drop out. &or both steady and unsteady closed systems,therefore,

dm

dt= or, constantm=

This is almost a trivial result; therefore, a constant mass can beimplicitly assumed for a closed system without having to refer to this

e#uation.

Open Stea# Simpli&icationAs explained in section ..1, at steady

state the total mass, li(e all other global properties, remains constant.

dmdt

;, steady state

i e

i e

m m= & & ; or, (g si ei em m = & &

This form of mass conservation is often referred as Gwhat goes incomes outI. %f there is a single flow, i.e., only one inlet and one exit,

the e#uation can be further simplified using )#. .

i em m m= =& & & ; or, i i i e e em AV A V = =& , ori i e e

i e

AV AVm

v v= =&

Open P$ocess Simpli&ication&or a process involving an opensystem )#. can be integrated or, alternatively, )#. can be used to

produce

finish finish

begin begin

where, , and

" b i e

i e

i i e e

m m m m m

m m dt m m dt

= =

= =

& &

This form is further simplified if there is only a single inlet or asingle exit as in the case of charging a propane tan( or a whistling

pressure coo(er. iscussion of such specific cases, however, is

postponed until Chapter .

3.! Ener"y Equation

The conse$)ation o& ene$% p$inciplealso (nown as the +i$st La(of thermodynamics can be stated through the following postulates.

i4 The internal energy u o" a system is a thermodynamic property#

&ig. . &low diagram


e#uation, open steady

system.

&ig. .+ &low diagram


e#uation, open process.


12/44

ii4EnergyE $ %E &E= + + cannot be created or destroyed! onlytrans"erred through heat or 'or(# )n a rate basis this can be

expressed as

(* >(

s

cc cdE * +

dt

=

& &

'here!c*& is the net rate o" heat trans"er into the system and c+& is

the net rate o" 'or( or po'er trans"er out o" the system#

'ubstituting E, e and cE for B , b and cB respectively in the 2T)

and using the second postulate

Aet 2ate of2ate of increase heaAet energy flow rate Aet energy flowof for an into the system. rate out of the system.open system.

i i e ei e

E

dEm e m e *

dt= + && &

14 2 43 1 4 2 4 3 1 4 2 4 3

[ ]Aet 2ate ofheat transfert transferinto the system.into the system.

(+ &14 2 4314 2 43

where, *& and +& , evaluated based on the open system boundary, are

substituted forc*& and c+& respectively since the boundaries of the

closed and open systems become coincident as ;t . The energyflow rates at the inlet and exit can be also be expressed through the

symbol E me=& & , which is used in the flow diagram of &ig. .5.)#uation is now completely decoupled from the original closed

system and will be labeled the conse$)ati)e &o$mof the energye#uation.

ifferent modes of heat and wor( transfer, shown in the flow

diagram of &ig. .5, will be #uantitatively discussed in the next

chapter. As explained in 'ection .1.1.1, the transfer of heat throughthe ports can be neglected compared to the transfer through the rest

of the boundary. The same, however, is not true about wor( transfer

through the system ports, called the"lo' 'or(. As explained in

'ection .1.1.+ different types of wor( transfer can be classified intotwo ma$or categories, flow and external wor(, to distinguish open

and closed systems.

, , sh el

Aet 2ate of 'haft or( )lectricalor( transfer 6ut or( 6Aet &low Aet &lowout of the system.

or( 6ut or( %n

&low or(, ,

, e , i

i i

+

+ + + + + = + +

&

& & & & &14 2 43 1 2 3

14 2 43 14 2 43

1 4 44 2 4 4 43

ext

/oundaryor( 6utut

6ther or(,

ext

)xternal or(,

)

B

+

, ) B ,

+

+

+ + + + +

+

= + + = +

&

&

&1 2 31 2 3

1 4 4 2 4 4 3

& & & & &1 4 2 4 3

1

&ig. .5 &low diagram

explaining variousmodes of heat and wor(

transfer.

&ig. . &low diagram

for the conservative

form of the energybalance e#uation, open

unsteady system.


13/44

&or a closed system ,+ =& and there is no distinction between +&

and ext+& .

To evaluate the flow wor(, consider the small fluid element

of length ex in the simplified system of &ig..7 that is pushed outof the system by the pressure force from the left against the pressurefrom the right. The pressure force e e e, p A= does a wor( of e e, x 3see 'ection .1.1.4 in t . According to the sign convention, theexit wor( must be positive since wor( is done by the system. %n a

similar manner, as a fluid element is pushed into the system against

the resistance of the inlet pressure, a negative wor( transfer with a

magnitude of i i, x ta(es place in time t at the inlet. As t ,the net flow wor( rate or flow power can be written with the help of

)#. as

, ,

e e i i e e e i i i

, , e , i

e e i ie e e i i i e e i i e e e i i i

e i

, x , x p A x p A x

+ + + t t t t

AV AVp A V p AV p v p v m p v m p v

v v

= = =

= = =

& & &

& &

A port with a very small area still can have very large pv and, thus,

transfer a relatively significant amount of flow wor(.

)#uation can be generalized for multiple inlets and exits.

, e e e i i i

e i

+ m p v m p v= & & &

)ach term on the 2F' resembles flow rate of properties discussed in'ection 1.:. The flow wor( too, therefore, can be regarded as a flowproperty. 'ubstituting the above expression for flow wor( after

separating it from all other wor( terms, the conservative form of the

energy e#uation, )#. , can be rewritten as

( ) ( ) exti i i i e e e ei e

dEm e p v m e p v * +

dt= + + + & && &

%n this modified form the mass flow can be seen to carry a

combination property consisting of energy e and a term thatrepresents the flow wor( performed per unit mass of the flow. e

call this combination property the speci&ic &lo( ene$%and representit with the symbol - in the absence of any universally accepted

symbol for this important convenience property.

- e pv u (e pe pv h (e pe= + = + + + = + +

'ubstituting the symbol - for the specific flow energy, we obtain

the balance equation &o$ ene$%in its most general form.

&ig. .: The flow of

flow energy .& ise#uivalent to the flow of

energy E& and thetransfer of flow wor(

,+& across a control

surface.

&ig. .7 A fluid

element at the exit being

expelled by the system

against an external

pressure.


14/44

2ate of increase Aet flow rate of flow Aet flow rate of flowof for an energy into the system. energy out of the system.open system.

i i e ei e

E

dEm - m - *

dt= + && &

14 2 43 1 4 2 4 3 1 4 2 4 3

[ ]extAet 2ate of Aet 2ate ofheat transfer wor( transferinto the system. into the system.

1

ext

(

where, , and1 B )

+

V gz

- h (e pe h + + +

= + + = + + = +

&14 2 43 14 2 43

& & &

The energycarried by the flow E me=& & in the conservative form,)#. , is replaced in this e#uation by the"lo' energycarried by the

flow, . m-=& & . The advantage of this form is that only the readilyrecognizable external wor( appears in this e#uation and the hidden

wor( of flow can be completely ignored since it is already accounted

for in the use of the property - . %t may seem that this form of energy

e#uation is meant only for open systems. To the contrary, if we

substitute ,+ =&

and ,+ +=& &;

extext+ ++ =& & into )#. , the secondpostulate of the &irst "aw is immediately recovered ma(ing )#. themost general form from which all other forms should be derived.

The meaning of various terms in this e#uation is explained through

the flow diagram of &ig. .:.

3./.1 F!(%s !+ Ene(0# B''n)e E-'$!n

As we did with the mass balance e#uation, the energy e#uation iscustomized for the particular classes of systems introduced in the

system tree of &ig. ..

*lose# Sstem Simpli&ication&or a closed systemthe mass transferterms drop out and ext+ +=& & as there is no possibility of any flowwor(. The energy balance e#uation, )#. , reduces to the secondpostulate of the &irst "aw.

dE* +

dt= & &

6bviously, this forms suits any instantaneous unsteady closed

system. There is no need for the superscript canymore because weare deriving a restricted form from a more general form applicable to

both open and closed systems.

*lose# P$ocess Simpli&ication&or an unsteady closed systemgoing

through a process, )#. can be integrated from the b-state to the f-state as outlined in section .. producing

+

&ig. .< /y using

specific flow energy - instead of specific

energy e , thecumbersome flow wor(

can be forgotten.

&ig. .1 &or a closed

system there is no flow

wor(; therefore,

ext+ +=& &

.


15/44

finish finish finish

begin begin begin

where, , and

" b

B ) B )

E E E * +

* *dt + + dt + dt + +

= =

= = + = + & & &

This is an algebraic e#uation that relates two anchor states through

two process variables * and + .

*lose# Stea# Simpli&ication&or a steady system, open or closed,

the time derivative of any global property must be zero. The energy

e#uation, thus, simplifies to

* +=& &

The net rate of heat transfer to asteady closed systemmust be

exactly e#ual to the net rate of wor( delivered by the system.

Open Stea# Simpli&icationThe time derivative of all global

properties of the system must be zero at steady state as the globalpicture remains frozen at steady state. The energy e#uation

simplifies to what is commonly called the stea# &lo( ene$%equation3'&))4.

ext i i e ei e

m - m - * + = + & && &

/y rearranging the e#uation, it can be shown that the sum total of the

rate of flow of"lo' energyand heat into asteady open systemmustbe e#ual to the rate at which energy leaves the system through flow

energy and external wor(. "i(e the steady state mass balance

e#uation, it expresses 'hat goes in! comes outin terms of energy.

Open P$ocess Simpli&ication&or a process involving an opensystem )#. can be integrated from the begin to the finish state as

outlined in section .. for a generic property. 8sing the uniform

flow uniform state assumption, the energy e#uation reduces to

ext

finish finish

ext

begin begin

where, , and

" b i i e e

i e

ext

E E E m - m - * +

* *dt + + dt

= = +

= =

& &

The mass transfers in such a process has already been examined in

section .+..

3.# Entropy Balance Equation

The Secon# La( of thermodynamics can be stated through the

following postulates.

&ig. .1 )nergy flow

diagram for a closed

process.

&ig. .11 )nergy flow

diagram for a closed

steady system.

&ig. .1 )nergy flow

diagram for an open

steady system.

&ig. .1+ )nergy flow

diagram for an open

process.


16/44

i4Entropy Sis an extensive property that measures the degree o"

disorder in a system# The speci"ic entropy s is a thermodynamic

property#

ii4Entropy can be trans"erred across a boundary through heat but

not through 'or(# The rate o" entropy trans"er by *& crossing a

boundary at a temperature BT is given as ? B* T& #

iii4Entropy cannot be destroyed# /t can be generated by natural

processes,i#e#! gen S & .

iv4An isolated system achieves thermodynamic e0uilibrium 'henthe entropy o" the system reaches a maxima#

"et us go over these statements one at a time. &rom our

experience of chaos, we would tend to agree with the first postulatethat entropy, being a measure of total amount of chaos or disorder in

a system, is an extensive property; that is, doubling the size of a

uniform system will double its entropy.

Feat transfer to a system can be expected to increase themolecular disorder and, hence, entropy. %f a uniform system is at a

high temperature and, therefore, pretty chaotic to start with, addition

of heat cannot be expected to add as much entropy to the system as

would be the case for a cooler, less chaotic system. This provides$ustification as to why the boundary temperature, which is same as

the system temperature for a local system, occurs in the denominator

of the entropy transfer term in postulate-%%. 6bserve that transfer ofwor( does not seem to affect entropy of a system. or( involves

organized motion such as the rotation of a shaft, motion of aboundary, and, in the case of electricity, directed movement ofelectrons, etc. The chaotic motion of the system, therefore, remains

unaffected by the transfer of organized motion.

The third postulate states that every system has a natural

tendency towards generating entropy. /ecause entropy cannot bedestroyed, the generated entropy is a permanent signature of the

process. hen heat radiates from the 'un to earth, the coffee in the

stirred cup gradually comes to rest, electrons flow across a voltage

difference, a drop of in( dissipates in a buc(et of water, rubbing onehand against another ma(e them warm, natural gas burns in air

forming hot flames, a volcano erupts @ there is one thing that iscommon in all these apparently unrelated phenomena; they all tendto destroy a gradient of some (ind while generating entropy as

dictated by postulate-%%. %n the next chapter we will devote an entire

section going after these sources of spontaneous entropy generation.&or the time being, we will refer to all these gradient destroying

natural phenomena as %ene$ali,e# &$iction.

5

&ig. .1 CA2T66

Are you saying that the'econd "aw left those

footprintsH


17/44

=eneralized friction leave an indelible footprint in the form

of entropy generation. Any process involving generalized friction,

therefore, cannot be completely reversed and are called i$$e)e$sible,the degree of i$$e)e$sibilitbeing proportional to the entropy

generation. =eneralized friction due to system surroundings

interactions sometimes extends beyond the system into theimmediate surroundings. epending on the location where the

entropy is generated with respect to the system boundary, the

associated irreversibilities are called inte$nal if within the systemandexte$nal if outside or at the boundary. &or instance, entropy is

generated inside and in the immediate surroundings of a turbine

operating in a steady state. The sstems uni)e$seenclosed by the

outer boundary of &ig. .15 includes both the internal and externalgeneration of entropy. %n the limiting situation of no entropy

generated in the systemEs universe as a result of a particular process,

the system can be completely restored bac( to its original state

without leaving any clue that the original process ever too( place.The system or process is said to be $e)e$sibleunder that ideal

situation. The concept of entropy generation will be lin(ed in thenext chapter with the design of more efficient engines, refrigerators

and various other thermal devices.

The third postulate 3not to be confused with the Third "aw of

thermodynamics to be introduced in Chapter-:4 has tremendousimplications in predicting e#uilibrium, which will be discussed in

more details in Chapter : and . &or the time being, consider two

closed insulated systems, initially at two different temperatures,brought in diathermalcontact by removing insulations from two

walls and pressing the two bloc(s against each other on their un-insulated faces. The entropy of the combined system will start to

increase as entropy is generated due to heat transfer from the hotterbloc( to the colder one. e (now from our experience that at

e#uilibrium temperatures of the two bloc(s will become e#ual, at

which point entropy will cease to increase any further, all thetemperature gradient having been completely destroyed. Thus

entropy has been maximized as the isolated system, consisting of the

two bloc(s, comes to e#uilibrium. As a matter of fact, we will showin Chapter-:, that starting from the second law, the e#uality of

temperature at e#uilibrium can be predicted. Although this may seem

li(e a trivial exercise, the same principle will help us deduce inChapter @, the emissions from combustion, something far fromtrivial.

=etting bac( to our tas( of translating the fundamental laws

into balance e#uations, the second postulate can be written as.

gen gen

(;

B

c cc c

B

dS *S S

dt T

= +

&& &

7

&ig. .15 The

interactions between thesystem and its

surroundings causes

entropy generation

inside and in the

immediate surroundings

of a system.

&ig. .15 )ntropy is

generated in the shadedarea which extends

beyond the system

boundary.


18/44

where, genS& is the rate of entropy generation within the closed

system boundary andc*& is the rate of heat transfer into the closed

system of &ig. .+. 'ubstituting Sand s for B and b respectively

in the 2T), we obtain the %ene$alent$op balance equation.

2ate of increase Aet flow rate of Aet flow rate of Aet 2ate ofof for an entropy into entropy out ofopen system. the system. the system.

i i e ei e B

S

dS *m s m sdt T

= + && &14 2 43 1 4 2 43 1 4 2 43

gen

Aet 2ate ofgeneration ofentropy insideentropy transferthe system.through heat.

gen

(B

where, ;

S

S

+

&14 2 43

1 4 2 43

&

As mentioned before, the boundary of the closed system passingthrough the open system of &ig. .+ is almost identical to that of the

open system as t goes to zero. Therefore, c* *=& & and gen gencS S=& & .

The comments under each term are (eyed to the open system of &ig.

.+ as this general entropy e#uation completely stands on its ownwithout any reference to the closed system to which it owes its

origin. The flow diagram of &ig. .17 also explains the various termsof the entropy e#uation. An arrow with dots inside is used to signify

the generation of entropy.

&or most systems on earth, the heat interaction ta(es place

with the surrounding atmosphere. %f the system boundary is carefullydrawn to pass through the surrounding air, atmospheric temperature

can be used for BT . 6bviously the precise location of the boundary

does not affect *& or +& , which are flow rates of energy; however,

being a cumulative #uantity, genS& depends entirely on the selection

of boundary. The total rate of entropy generation in the turbine of

&ig. .15, for instance, can be expressed as the sum of the entropy

generation inside the system and in the immediate surroundingsexternal to the system.

gen,univ gen,int gen,ext

(

BS S S

= + & & &

where the subscript univis used to signify thesystem1s universe.

%f a system exchanges heat with different segments of the

surroundings at different temperatures as shown in &ig. .1:, theboundary of the extended system can be made to pass through (

segments each at a uniform temperature (T . The entropy balance

e#uation for the systemEs universe modifies as follows

gen,univ

(

B

(i i e e

i e ( (

*dSm s m s S

dt T

= + +

&&& &

:

&ig. .17 )ntropy isaccumulated due to

generation and transfer

through mass and

energy.

&ig. .1:

gen,univ gen gen,extS S S= +& & &

includes all sources ofentropy generation

inside and outside the

system.

&ig. .1< &low diagramof entropy for an

extended system with

surroundings at twodifferent temperatures.


19/44

The total entropy S , the mass flow rates im& or em& and the heat

transfer rate (*& are assumed not affected significantly by extending

the system to include the thin layer of immediate surroundings. The

entropy generation, however, can be huge outside the system, evenin a very thin layer. This will be discussed with examples in the next

chapter.

3..1 F!(%s !+ En$(!p# B''n)e E-'$!n

As we did with the mass and energy balance e#uations, we will

customize the entropy e#uation in a similar manner for differentclasses of systems. Although the following e#uations are written for

a system with a fixed boundary temperature BT , they can be modified

for an extended system by replacing genS& with gen,univS

& andB

*

T

&

with

(

( (

*T &

.

*lose# Sstem Simpli&ication&or a closed systemthe mass transfer

terms drop out and )#. reduces to

gen

B

dS *S

dt T= +

&&

6bviously, this form suits any instantaneous unsteady closed system.

*lose# P$ocess Simpli&ication&or an unsteady closed systemgoing

through a process, )#. can be integrated from the b-state to the f-

state as outlined in section .. producing

gen

finish finish

gen gen

begin begin

where, , and

" b

B

*S S S S

T

* *dt S S dt

= = +

= = & &

This is an algebraic e#uation that relates two anchor states through

two process variables * and genS . 6bviously gen S since gen S &

*lose# Stea# Simpli&ication&or a steady system, the timederivative of any global property must be zero. )#. simplifies to

genB

*S

T= +

&&

A number of 'econd "aw statements can be deduced from thise#uation in the next two chapter.

carries a fraction, ; ? (T T , of itself as exergy.ote that for reservoir (= , i.e., the ambient atmosphere, thisfraction reduces to zero. That is, there is no exergy is transferredthrough heat transfer between the system and the ambient

atmosphere. The implication of a cold reservoir, i.e., ;(T T< , will bediscussed in 'ection +...+.

The exergy delivered by the system as useful wor(, u+& ,

appears with a negative sign as it drains the system of its storedexergy.

&inally, the entropy generation can be seen to produce a term

that must be always non-positive since gen,univ S & 3'econd "aw4. %tis called the $ate o& exe$% #est$uctionor the $ate o& i$$e)e$sibilit

and is represented by the symbol /& .

1


of exergy for an

extended system.


26/44

)ach term of this e#uation s(etched in the flow diagram of

&ig. .+ will be explained with plenty of examples in the next two

chapters. A comparison of the flow diagrams for energy 3&ig. .4,entropy 3&ig. .174 and exergy 3.+4 can be helpful in understanding

the similarities and differences in the inventory of the three

properties in terms of a common framewor(.

3.2.1 F!(%s !+ Ee(0# B''n)e E-'$!n

As we did with the mass, energy and entropy balance e#uations, we

will customize the exergy e#uation in a similar manner for different

classes of systems.

*lose# Sstem Simpli&ication&or a closed systemthe mass transferterms drop out and )#. reduces to

;( u( (

Td* + /

dt T = & & &6bviously, this form suits any instantaneous unsteady closed system.

*lose# P$ocess Simpli&ication&or an unsteady closed systemgoingthrough a process, )#. can be integrated from the b-state to the f-

state as outlined in 'ection .. producing

;

finish finish

; gen,univ

begin begin

where, , and,

" b ( u

( (

( ( u u

T* + /

T

* * dt + + dt / T S

= =

= = =

& &

The simplified form of the exergy e#uation for a closed process can

be used to explore the physical meaning of some of its terms. &orinstance, when a closed system, say, a warm cup of coffee cools

down from a temperature bT to the room temperature ;T by re$ecting

loss* amount of heat, no useful wor( is produced. Fowever, the

exergy e#uation can be used to see if it is possible to construct a

clever device to extract useful wor( out of this cooling process. ith

;(T T= , )#. simplifies as

u b "+ /=

Clearly it is possible to convert some of the exergy in a coffee muginto useful wor(. %f the final state is the dead state, i.e., the coffee in

the mug reaches e#uilibrium with the environment, " = . /eing anon-negative #uantity, the irreversibility /can be seen to reduce the

useful wor( output. %n fact for a regular coffee cup, the exergy is

15

&ig. .+ )nergy flow

diagram for )#. . Thedirection of the heat

arrow is reversed since

loss* *= 3 loss* is apositive #uantity4.

&ig. .+ A smart

coffee mug thatproduces electricity as

the coffee cools down toroom temperature.

&ig. .+1 The exergy of

a warm coffee mug isthe maximum possible

useful wor( that can be

extracted as the coffee

comes to e#uilibrium

with the surrounding air.


27/44

completely destroyed by /. %f the irreversibility can be eliminated -

and the 'econd "aw does permit gen,univ S = as a limiting ideal case- the wor( produced is maximized.

,maxu b "+ = ;

/;

The exergy of a system, therefore, has the simple interpretation of

the maximum possible useful wor( that can be extracted out of it by

transferring heat with only the atmospheric T)2.

6ne may naturally as(, why cannot we use an energyanalysis instead to predict the maximum wor( transferH The next

chapter will be devoted to analysis such as this for closed system. As

a preview let us see what the energy and entropy e#uation predictabout the system at hand. 8sing the solid?li#uid model for the coffee,

the energy e#uation, )#. , can be simplified as

( ) B)" bE $ $ = + ;

9)+ ( )( ) ( )

;

loss

loss ; loss b " v b

* + * +

+ $ $ * mc T T *

= = = =

/y eliminating loss* completely it seems that the change in internal

energy can be completely converted into wor(, i.e.,

( )max ;v b+ mc T T = . The 'econd "aw however has been completelydisregarded in arriving at this conclusion. %n fact, an entropye#uation for the process, )#. , yields

( )lossgen gen,univ

;

; lossgen,univ

;

ln

" b

B

v

b

**S S S S S

T TT *

S mcT T

= = + = +

= +

The first term on the 2F' being negative, an elimination of loss*

would result in a negative gen,univS , which is a direct violation of the

'econd "aw. Any conclusions from the energy e#uation, therefore,must be tested for compliance with the 'econd "aw. Conclusions

derived from the exergy balance e#uation, on the other hand, do not

run into these types of difficulty as the exergy e#uation is firmly

rooted in the combination of mass, energy and entropy e#uations.*lose# Stea# Simpli&ication&or a steady system, the time

derivative of , a global property, is set to zero and )#. simplifiesto

; ( u( (

T* + /

T

=

& & &

17

&ig. .++ The change in

$ and S according tothe solid?li#uid model as

the temperature goes

from bT to ;T . 0ass of

the cup is neglected in

these expressions.


28/44

Open Stea# Simpli&icationThe steady state exergy e#uation,

similarly, can be expressed in an algebraic form as the time

derivative drops out.

; i i e e ( ui e ( (

Tm m * + /

T

= +

& & && &

The destruction of exergy term ma(es it impossible to express thise#uation in the what-comes-in-must-go-out format.

To explore the physical meaning of flow exergy, consider a

steady stream of fluid flowing through a system which has heat

interactions with only the atmospheric reservoir. The powerdelivered by this device can be obtained from )#. as

( )u i i e e i i e+ m m / m / = = & & && & &

The useful wor( is maximized when the exergy destruction is

eliminated and the flow exits at its dead state.

,maxu i i e e+ m m = & & &;

/ &;

i i im= = &&

The flow exergy, therefore, can be interpreted as the maximum

possible useful wor( delivered per unit mass of the flow if the flow

is brought to dead state by exchanging heat with the atmosphericT)2. Complete analysis of open systems will be carried out in

Chapter at which point this will be a simple exercise to show that a

&irst "aw analysis alone cannot be used for predicting the maximum

wor( transfer since the 'econd "aw may be violated.

Open P$ocess Simpli&ication&or a process involving an opensystem )#. can be integrated from the begin to the finish state.

8sing the uni"orm "lo' uni"orm state assumption, the exergy

e#uation reduces to

;" b i i e e (i i ( (

Tm m * /

T

= = +

where many of the symbols have been explained in connection withthe corresponding form of the energy and entropy e#uations.

3.& omentum Balance Equation

The momentum e#uation will not be used until chapter 7, where we

will discuss modern $et engines. Fowever, this is the appropriate

place to cast ewtonEs law into our common framewor( of a balancee#uation that applies to all systems, open or closed.

1:


of exergy simplified for

an open steady system.


29/44

Ne(tonsSecon# La( o& Motion for a closed system can be stated

as

The rate o" change o" momentum o" a closed system is e0ual to the

net external "orce applied on the system#

/ecause momentum and force are vectors, the momentum e#uationcan be split into three independent e#uations along x , y , and z

directions in the Cartesian coordinates. Along the x direction,ewtonEs 'econd "aw can be written as

[ ] ( (g.m

( ; where, (.s> s s

ccx x

x x

dM mV , M

dt

= =

6bserve that in this e#uation the unit of force is ( to be consistentwith all other balance e#uations and the unit of pressure, a deviation

from the standard use of in mechanics.

'ubstituting xM and ?xV for B and b respectively in

the 2T), )#. , we obtain the %ene$almomentum balance equation.

, ,

2ate of increase Aet -momentum flow rate Aet -momentum flow rateof -momentum into the system. out of the system.of an open system.

? ? Lx i x i e x e xi i

x xx

dMmV m V ,

dt= & &

1 4 2 43 1 4 4 2 4 4 3 1 4 4 2 4 4 3

[ ]Aet 2ate ofgenerationof -momentum.

(

x

1 2 3

As in the energy and entropy e#uation, the superposition of the

closed and open system is exploited to substitute c

x x, ,= . "i(e anyother extensive property, momentum can be transported in and out of

the system with mass. "i(e the entropy generation term in the

entropy e#uation, the net external force acts as a source ofmomentum.

&or closed systems, ewtonEs law of motion is recovered.

( )? or, or,

xx xx x x

d mVdM ma, , ,

dt dt = = =

where, xa is the acceleration in the x direction.

&or an open steadysystem)#. reduces to

, ,

i x i e x e x

i i

mV m V , = + & &

These are the only forms of the momentum e#uation that will beused in Chapter 7 and , although other forms can be derived as

easily.

1 s

s

xi x i e x e x

i i

xx

dMmV m V ,

dt

mVM

= +

=

& &

3.4.& C!se" S#s$e%s

Considerable simplification results as the mass transfer terms aredropped from the balance e#uations for closed systems. 0oreover,

flow wor( being completely absent, ext+ +=& & .

Mass 3)#. 4

[ ]

(g

; constant (gs

dm

mdt

= =

Ene$% 3)#. 4

( ) [ ] ( ;B )dE

* + * + + dt

= = +& && & &

Ent$op 3)#. 4

gen

(

BB

dS *S

dt T

= +

&&

Exe$% 3)#. 4

[ ]; (( u( (

Td* + /

dt T

=

& & &

Momentum3)#. 4

[ ] ( ; constantx x xdM

, Mdt

= =

3.4.3 C!se" P(!)ess

hen an unsteady closed system undergoes a change of state from a

begin-state to a finish-state, it is said to have executed a closed

process.

Mass 3)#. 4

&ig. .+5.1 'ystem

schematic to accompany

'ection .


32/44

constant M(gNm=

Ene$% 3)#.4

( )" b B )E E E * + * + + = = = +

Ent$op 3)#. 4 gen" bB

*S S S S T

= = +

Exe$%3)#. 4

;" b ( u( (

T* + /

T

= =

3.4.* C!se" S$e'"#

hen the image of a closed systemta(en with astate cameradoes

not change with time, the time derivative of all global properties

becomes zero and the system is said to be a closed steady system.

3losed cycles, as will be shown in the next chapter, can be treated as

a special case of a closed steady system.

Mass 3)#. 4 constantm=

Ene$% 3)#. 4 * += & &

Ent$op 3)#. 4 gen

B

*S

T= +

&&

Exe$%3)#. 4; ( u

( (

T* + /

T

=

& & &

3.4./ Open S$e'"#

hen the image of an open systemta(en with astate cameradoes

not change with time, the time derivative of all global properties

becomes zero and the system is said to be an open steady system.Mass 3)#. 4

(g

s

i e

i e

m m =

& &

Ene$% 3)#. 4

1

&ig. .+5. 'ystemschematic to accompany

'ection .


33/44

ext M(Ni i e ei e

m - m - * + = + & && &

Ent$op3)#. 4

gen

(

Bi i e e

i e B

*m s m s S

T

= + +

&&& &

Exe$% 3)#. 4

[ ]; (i i e e ( ui e ( (

Tm m * + /

T

= +

& & && &

Momentum 3)#. 4

[ ], ,

(

i x i e x e x

i i

mV m V , = + & &

3.4. Open P(!)ess

hen an unsteady open systemundergoes a change of state from a

begin-state to a finish-state, it is said to have executed an openprocess. The inlet and exit states are carefully chosen so that their

properties can be assumed to remain unchanged over time and over

the cross-sectional areas. This is (nown as the uni"orm stateuni"orm

"lo' assumption.

Mass3)#. 4

;" b i ei e

m m m m m = =

Ene$% 3)#. 4

ext" b i i e e

i e

E E E m - m - * + = = +

Ent$op3)#. 4

gen" b i i e e

i i B

*S S S m s m s S

T

= = + +

Exe$% 3)#. 4

;" b i i e e (i i ( (

Tm m * /

T

= = +

&ig. .+5.5 'ystem

schematic to accompany'ection .


34/44

EXAMPLE 3-/0))) )#uations for a Closed 9rocess.

evelop the appropriate form of 0))) 3mass, energy, entropy and

exergy4 e#uations for the following problem.

etermine the amount of heat necessary to raise the temperature of

(g of aluminum from o C to o C H

SOLUTIONThe customized form of balance e#uations for various

classes of systems have been already identified in this chapter.

Therefore, the tas( at hand is to simplify the problem with suitableassumptions and choose the appropriate bloc( of e#uations from

'ection .


35/44

applicable regardless of the model chosen. %ndividual terms of the

balance e#uations will be discussed in the next two chapters. otice

that the e#uations are derived here for the extended system. Alsoobserve that the balance e#uations in their current form are

independent of the material model.

EXAMPLE 3-30))) )#uations for a Closed 9rocess.

evelop the appropriate form of 0))) 3mass, energy, entropy andexergy4 e#uations for the following problem.

A piston-cylinder device initially contains 1 g of saturated water

vapor at (9a. A resistance heater is operated within the cylinder

with a current of .+ A from a 1+ O source until the volumedoubles. At the same time a heat loss of + (* occurs. etermine the

final temperature and the duration of the process.

SOLUTIONTo develop a customized set of 0))) e#uations.

Simpli&icationThe simplification carried out in )x. -1 applies tothis problem as well. %n addition to heat transfer, there are two

modes of wor( transfer, electrical and boundary wor(. The closed

process e#uations of 'ection .


36/44

evelop the appropriate form of 0))) e#uations for the following

problem.

A + (g aluminum bloc( at o C is dropped into an insulated tan(

that contains . mof li#uid water at 1 o C . etermine the entropy

generated in this process.

SOLUTIONTo simplify the problem so that the balance e#uationscan be reduced to one of the customized forms discussed in this

chapter.

Simpli&icationater and the bloc( constitute a non-uni&o$m close#

sstem going through a process in this problem. Two states, one forthe bloc( and one for water, can be used to describe the composite

begin state. At the end of the process, even though the temperature is

uniform, the finish-state still re#uires a composite description as thedensity is different for the two sub-systems. esignating the two

subsystems as A and /, and neglecting any changes in B) and 9) ,

the closed process e#uations can be simplified as follows.

Mass constant; constant;A Bm m= =

Ene$% E $ %E = + ;

&E+ ;

*=;

+;

( ) ( ), , , ,6r, " b A " A B " B A b A B b B$ $ $ m u m u m u m u = = + + =

Ent$op (" b

(

*S S S

T = =

;

gen,univ

(

S+

( ) ( ), , , , gen6r, " b A " A B " B A b A B b BS S S m s m s m s m s S = = + + =

Exe$% " b (* = =;

; u( (

T+

T

; ; genT S

( ) ( ), , , , ; gen,univ6r, " b A " A B " B A b A B b Bm m m m T S = + + =

Simpli&ication Usin% TESTavigate through the 'ystems, Closed,

9rocess, =eneric, on-8niform, on-0ixing, pages to display theprogressively simplified system schematic and balance e#uations.

!iscussionThe subsystems are closed themselves since there is no

mass transfer between them. %n T)'T such systems are called non-

mixin% non-uni&o$msystems. %n the following example, on theother hand, the subsystems of a non-uniform system can be seen to

be mixin%. As in the previous problem, the balance e#uations in their

current form are independent of the material model.

EXAMPLE 3-0))) )#uations for a 0ixing Closed 9rocess.

5

&ig. .+< The composite

system goes through a

non-mixing closed

process.

&ig. . The

composite closed system

goes through a mixing

process.


37/44


problem.

A . mrigid tan( containing hydrogen at + o C , 1 (9a is

connected to another mrigid tan( containing hydrogen at 1 o C ,

5 (9a. The valve is opened and the system is allowed to reach

thermal e#uilibrium with the surroundings at o C . etermine the

irreversibility in this process. Assume variable pc .

SOLUTIONTo simplify the problem so that the balance e#uations

can be reduced to one of the customized forms discussed in this

chapter.

Simpli&ication/y drawing the system boundary as shown in theaccompanying figure, gases in the two tan(s, each of which acts as

an open system during the process, behave li(e a closed system. %n

the resulting non-uniform system, two states, one for tan( A and one

for tan( /, must be used to describe the composite begin state. At theend of the mixing process, the finish state is uniform and can be

represented by a single state. eglecting any changes in %Eand

&E, the closed process e#uations can be simplified as follows.

Mass constant;A Bm m+ =

Ene$% E $ %E = + ;

&E+ ;

* += ;

( ) ( ), ,6r, " b A B " A b A B b B$ $ $ m m u m u m u * = = + + =

Ent$op gen(

" b

( (

*S S S S

T

= = +

( ) ( ), , gen,univ;

" b A B " A b A B b B*

S S S m m s m s m s S T

= = + + = +

Exe$% ;" b ((

T*

T

= =

;

u

(

+ ; ; gen,univT S

( ) ( ), , ; gen,univ " b A B " A b A B b Bm m m m T S = + + =


9rocess, =eneric, on-8niform, 0ixing, pages to display the

progressively simplified system schematic and balance e#uations.

!iscussionAn interpretation of different terms of the balance

e#uation is postponed until the next chapter. %f the valve is closed

before mixing is complete, the finish state must be expressed through

a composite state $ust li(e the begin state. The balance e#uations, itshould be noted, are independent of the material model.

7


38/44

EXAMPLE 3-20))) )#uations for a Closed 'teady 'ystem.


problem.

A m1bric( wall separates two chambers at B and B

respectively. %f the rate of heat transfer is . (?m

1

, determine theentropy generation rate and the rate of exergy destruction in the wall.

Assume the wall surface temperatures to be the same as the ad$acent

chamber temperatures. Also assume steady state.



chapter.

Simpli&icationThe bric( wall in this problem, obviously, constitutesa closed system at steady state. /ecause the area of the wall at the

edges are negligible compared to the two main faces, heat transfer

through the end faces can be neglected. Also the time derivatives of

B) and 9) can be assumed zero.

Mass constant;m=

Ene$% 2 3* * += +& & &;

; 2 3* * =& &

Ent$op gen,univ (

( (

*S

T= +

&& gen,univ

2

3 2

S *T T

=

& &

Exe$%; ( u

( (

T* +

T

=

& &

;

; gen,univT S &

; ;

; gen,univ 2 32 3

T T* * T S T T

=

& & &


'teady pages to display the progressively simplified system

schematic and balance e#uations .

!iscussion6nce again we will defer interpretation of various terms

until the next chapter. ith 2 3* *=& & , the exergy e#uation can beshown to reduce to entropy e#uation for this particular system.otice that the e#uations are derived here for the extended system.

EXAMPLE 3-0))) )#uations for an 6pen 'teady 'ystem.

evelop the appropriate form of 0))) e#uations for the followingproblem.

Carbon dioxide enters steadily a nozzle at psia, + o &, and 1

ft?s and exits at 1 psia and 1 o &. Assuming the nozzle to be

adiabatic and the surroundings to be at +.7 psia, 5 o &, determine

:

&ig. . A closed

system at steady state.

&ig. .1 A nozzle

operating at steady state.


39/44

3a4 the exit velocity, and 3b4 the entropy generation rate by the device

and the surroundings.


chapter.

Simpli&icationThe image of the nozzle ta(en with a state camera

remains frozen even though the state of the fluid flowing through thenozzle changes. Fence, a nozzle is an open steady device. Although

change in 9) can be neglected, the purpose of a nozzle is to

accelerate a flow and, therefore, the change in B) must be

considered significant. /ecause there is a sin%le &lo( through thenozzle, the summation over inlets and exits of the open, steady

e#uations of section .


40/44

the mixing chamber with a resistance heater with a power rating of

(. "i#uid water enters the chamber at (g?s, and the chamber

looses heat at a rate of (*?min with the ambient at 1 o C . %f the

mixture leaves at (9a and o C , determine 3a4 the mass flow

rate of steam, and 3b4 the entropy generation rate during mixing.



chapter.

Simpli&icationThe mixing chamber can be assumed to operate atsteady state. Although heat is transferred from the electrical heating

elements to the wor(ing fluid, it is electrical power el+& that crosses

the boundary and, therefore, must appear in the energy and exergy

e#uations as ext+& and ext,u+

& respectively. Two inlet states, i-'tate

and i1-'tate, and one exit state, e-state, are re#uired in this multi

&lo(mixin% configuration. The open, steady e#uations of section.


41/44

heater at 1. 09a and o C and leaves 1 o C below the exit

temperature of steam. eglecting any heat losses, determine 3a4 the

mass flow rate ratio and 3b4 the entropy generation rate of the device

and its surroundings. Assume surroundings to be at 1 o C .


chapter.

Simpli&icationThe closed feed water heater shown in the

accompanying figure is a heat exchanger, where the flow of water isheated by the flow of steam. &or this non-mixin% multi-&lo(configuration, two inlet states, i- and i1-states, and two exit states,

e- and e1-states, describe the two flows, flow-A from i to e andflow / from i1 to e1. Clearly there is no external wor( transfer for

this passive device. The open, steady e#uations of section .


42/44

reaches (9a when the valve is closed. etermine the final

temperature of the air in the tan(. Assume variable specific heats.


chapter.

Simpli&icationThe tan(, an open system, goes from a vacuum b-

state to a filled f-state as air from the supply line rushes in. %f the i-state is located above the position of the valve, its thermodynamicstateat all times can be considered identical to that in the supply

line. %n this open-p$ocess, there is no external wor( or heat

transfer. The open, process e#uations of section 1.


43/44

3.1 Summary

The fundamental governing e#uations for the interactions between a

system and its surroundings are derived in a common format calledthe balance e#uation in this chapter. The goal is to express the

governing e#uations in a customized format for a given system. The

2eynolds transport e#uation or the 2T) relates the rate of change ofany total extensive property of an open system at a given instant withthat of a closed system passing through, which happens to occupy

the entire open system at that time. ith the help of 2T) the

fundamental laws of thermodynamics, postulated for a closedsystem, are converted into balance e#uation for a very general

system.

%n 'ection . systems are classified into a tree structure with

different branches representing groups of systems that show somesimilar patterns. 0ass balance e#uation is derived and expressed in

different formats in 'ection .+. 'imilarly, energy, entropy, exergy,and momentum e#uations are derived in 'ections . through .:.&inally, in 'ection .< the complete set of e#uations, called the

0))) e#uations are summarized for important classes of systems

that are often encountered in the practice of thermodynamics.

The next two chapters are devoted to understanding thevarious e#uations derived in this chapter through comprehensive

analysis of various closed and open systems.

3.11 nde%

anchor states..........................

Chapter03 Balance

Documents