termotehnica automotive engeneering

8/10/2019 Termotehnica Automotive Engeneering

1/189

Chapter 1. Basic concepts

1.1.Thermodynamics as a science

Thermodynamics is a science based on phenomena which studies the matter

from the point of thermal motion; it studies the body properties and energytransfer among bodies,which are produced by internal molecular motion.

The matter is made of microscopic particle which interacts and are in

perpetual motion, known as thermal motion.Thermodynamics studies the

phenomena produced at microscopic level to groups of particles

investigating their macroscopic, measurable effects and establishes relations

between measures which are observable and measurable, such as volume,

pressure, temperature, concentration of chemical solutions.

Thermodynamics stands on two relations from kinetic molecular theory :

Bernoulli relation determines the pressure of a as on a wall, a

macroscopic measure, as a function microscopic measures :

2

2 2vm

V

Np=

! no. of molecules in volume "

m mass of a molecule

2

2vm# mean kinetic energy of molecules in translation motion,

$a%well Bolt&mann relation determines the connection between

the temperature of a gas and mean speed of molecules.

22

2vmkT=

in which k is Bolt&mann constant kJk '().*,( 2=

+n other words pressure and temperature are measures which features thegroup, the assembly of molecules, not a given molecule.

The technical thermodynamics studies the processes of producing,

transmitting and use of energy in its form as heat and work, applying its own

laws to heat engines and installations.

Thermodynamics operates with measures and concepts which define thermal

phenomenon. %amples of thermal phenomena are as follows:


2/189

-eat transfer between two bodies having different temperatures

hase changes of substances

Transformation of heat in work and reverse in heat engines ;

$ain notions are thermodynamic system, state, process.

1.2 Thermodynamic systems

/enerally a physical system is an area of universe made of substance or

fields.

The thermodynamic system is made of a body or an assembly of bodies

having a finite number of particles,which is limited from surroundingsthrough a boundary surface and which interacts energetically. The system

changes substance 0mass1 or energy as heat or work with the surroundings.

n insulated system is not influenced in any way by the surroundings. This

means that no heat or work or mass crosses the boundary of the system.

n adiabatic or thermal insulated system does not change heat with the

surroundings. mechanical insulated system does not change work with the

surroundings.

3ome thermodynamic analysis involves a flow of mass into or out of adevice. 3o, we can discuss about open systems. +f there is no mass flow, the

system is closed. The terms closed system and open system are used as the

e4uivalent of the terms system 0fi%ed mass1 and control volume 0involving a

flow of mass1.The surface of a control volume is referred to as a control

surface. $ass, as well as heat and work, can flow across the control surface.

Thus, a system is defined when dealing with a fi%ed 4uantity of mass and a

control volume is specified when an analysis is to be made that involves a

flow of mass.

1.3. State. State parameters

The state may be identified or described by certain observable, macroscopic

properties, like temperature, pressure, density. ach of the properties of a

substance in a given state has only one definite value and these properties


3/189

always have the same values for a given state, regardless of how the

substance arrived at that state.

state property can be defined as any 4uantity that depends on the state of

the system and is independent of the path 0prior history1 by which the

system arrived at the given state.

5onversely, the state is specified or described by its properties.

Thermodynamic properties can be divided into two general classes:

intensive and e%tensive properties. n intensive property is independent of

mass; the value of an e%tensive property varies directly with the mass.

ressure, temperature and density are e%amples of intensive properties. $ass

and volume are e%amples of e%tensive properties. %tensive properties per

unit mass, such as specific volume, become intensive properties.

6e will refer not only to the properties of a substance, but also to the

properties of a system. Then we necessarily imply that the value of theproperty has significance for the entire system and this implies what is called

e4uilibrium. 7or e%ample, if a gas is in thermal e4uilibrium, the temperature

will be the same throughout the entire system and we may speak of the

temperature as a property of the system. 6e may also consider mechanical

e4uilibrium and this is related to the pressure. system is in mechanical

e4uilibrium if there is no tendency for the pressure at any point to change

with time as long as the system is isolated from the surroundings. +f the

chemical composition of a system does not change with time, that system is

in chemical e4uilibrium. +t means, no chemical reactions occur.6hen a system is in e4uilibrium as regards all possible changes of state, we

say that the system is in thermodynamic e4uilibrium.

7or fluids there are at least three macroscopic measures which are state

parameters, usually pressure, temperature and volume, called fundamental

state parameters. The values of parameters of state do not depend on

previous history or path of the system, depend only on instantaneous

coordinates of the system. 6hen a system passes from a state of

thermodynamic e4uilibrium to another one, the variation of a parameter of

state will cause the variation of other ones, showing a dependency between

parameters of state, called state e4uation.

1.3.1. Specific volume


4/189

The specific volume of a substance is defined a s the ratio between volume

" and mass m being noted with v 0m'kg1.

=

kg

m

m

Vv

The specific volume is inverse of density or specific mass'mkg .

=

kg

mv

(

"olume of a kilomole of substance is called molar volume "$

=== kmolm

Mv

M

m

V

n

V

VM

The molar volume is the volume reported to number of kilomol of

substances contained 0n#number of kilomol, $#molar mass1.

The kilomol has two definitions :

# molar mass of a substance e%pressed in kilograms 0 e%. ( kmol of o%igen

82 9 2 kg written as $82 92kg'kmol1 or

#a 4uantity of matter which contains in the same conditions a number of

molecules e4ual to vogadro number 0 ( kmol 9 !molecule 9 ,)2.()2

molecule1.

1.3.2.Temperature

lthough temperature is a property with which we are all familiar with , an

e%act definition of it is difficult.

7rom our e%perience we know that when a hot body and a cold body are

brought into contact, the hot body becomes cooler and the cold body

becomes warmer. +f these bodies remain in contact for some time, they

appear to have the same hotness or coldness.


5/189

Because of the difficulty in defining temperature, we define e4uality of

temperature.

5onsider two blocks of copper, one hot and the other cold, each one in

contact with a mercury in glass thermometer. +f these two blocks are

brought into thermal contact , the mercury column of the thermometer in the

hot block drops at first and in the cold block rises, but after a period of timeno further changes in height are observed. +t is said that both bodies have

the same temperature when they reached the thermal e4uilibrium.

The &eroth law of thermodynamics states that when two bodies have

e4uality of temperature with a third body, they in turn have e4uality of

temperature one with each other. +t is based on the transitivity of the thermal

e4uilibrium. Two systems found in thermal e4uilibrium with a third one are

in thermal e4uilibrium between them 0meaning that they have the same

temperature1.Based on this property, a thermometer 0or more general a

thermometric body 9 a body having a thermometric property1 can comparedifferent thermal e4uilibrium states of the bodies.

Temperature represents a parameter of state which describes the heating

state of a system, state dependable on the molecular energy of every

component.

Temperature is a physical fundamental measure and it can be determined

measuring the variation of a physical measure, generally called

thermometrical measure which sensitive, preferable as linear as possible, to

temperature variation. 7or e%ample when we measure the temperature of ourbody with a mercury thermometer we measure in fact the e%pasion of the

mercury produced at the contact with the body, e%pansion 4uantified by the

variation of height of mercury column in the thermometer.8ther

thermometric measures are variation of the length of a metal rod, variation

of the electric resistance 0thermometer is called thermo#resistance1, variation

of the thermo#electromotor voltage 0thermometer is called thermocouple1,

variation of a gas in a constant volume enclosure 0 gas thermometer1,

variation of light intensity of an incandescence source 0optic pyrometers1.

7or every type of thermometer it must be set a temperature measurement

scale in such a way that temperature measurement to be done through a

simple reading. 7i%ing the temperature scale is called calibration and it can

be done bringing thermometer in thermal contact with a body in a

reproducible state 0 e%. water in thermal e4uilibrium with vapours , water in

thermal e4uilibrium with ice, melting point of some metals1 To those perfect

reproducible states are associated accurate values of temperature.


6/189

Thermometers have a macroscopic property that changes considerably with

temperature 0for e%., the pressure of a gas, the electric resistance, e%pansion

etc1.

+f a thermometer is brought in thermal contact with a hot body 0-1, at

e4uilibrium, the variable property 0for e%. the length of fluid column of a

alcohol thermometer, 1 changes with - 0segment T-1. The origin of the

diagram corresponds to the initial e4uilibrium state of the thermometer.

+f the same thermometer is brought through thermal contact at the

e4uilibrium state of a cold body 051, the same property will change with 50segment T51.

The segment -5 corresponds to the change in the property of the

thermometer brought in thermal contact with the two bodies. This change

characterises the difference between the temperatures of the hot body andthe cold body.

To estimate this difference, it is necessary to adopt a scale for temperature

measurements.

Based on the assumption that the temperature and the property of the

thermometer are related linearly, results:

dt 9 ad sau or t 9 a


7/189

The temperature of the steam point is the temperature of water and steam,

which are in e4uilibrium at a pressure of ( atm.

line of e4ual ratios can be written which define different temperature

scales 5elsius, =eaumur, 7ahrenheit, >elvin, =ankine:

0 # 51'0 -# 51 9 0t# t51'0 t- # t51 9 t?5'()) 9 t?=e'*) 9 0t?7#21'(*) 9

0T>#2@,(A1'()) 9 0T=#C(,@1'(*)

or

t?5 9 At?=e' 9 A0t?7#21'C 9 T>#2@.(A 9 A0T=#C(,@1'C

> 5 7

@ , ( A > ( ) ) 5 2 ( 2 7) )

2 @ , ( A > ) 5 2 7) )

) > # 2 @ , ( A 5 # B : ) 7) )

8n 5elsius scale ))5 coresponds to a mi%ture water and ice at

e4uilibrium and ()))5 to boiling water. 5elsius grade e4uals a >elvin

grade, only origins of scales are different.

There are several temperature scales different through the origin of the scaleor the magnitude of the unit 0grade or degree1. 5elsius scale considers &ero

value to the temperature at which water solidifies and ()) value to the water

vaporisation temperature, at normal atmospheric pressure 7ahrenheit scale

considers 2 and respectively, 2(2?7 to the same water temperatures .

These two scales are defined based on well known and reproducible

temperatures.

+n thermodynamics it was considered to find a temperature scale


8/189

independent of the properties of a particular substance. 3uch a scale is called

a thermodynamic scale or an absolute temperature scale. This scale was

imagined starting from the properties of ideal gas when there is a variation

of its pressure with temperature. +f it was considered a gas thermometer

having a constant volume it was noticed that in the field of low pressures the

gas temperature is proportional with gas pressure 0at constant volume1. 8r it

can be written that gas temperature T varies linearly with gas pressure p or

T 9 a < bp,

in which a, b are constants of the gas thermometer. 3uch a scale is called

temperature scale of ideal gas and can be determined measuring gas pressure

at two reproducible temperatures such as water solidification and

vapori&ation at normal atmospheric pressure and the e4uation is determined

knowing a and b for 5elsius scale. +f the nature of the gas is changed from

0for e%ample o%ygen, B argon1 then similarly can be determined two

points and another proportional line like in fig.(.

7igure (. =elation between pressure and temperature for gas thermometer

+t can be noticed that no matter the nature of the gas the two lines intersects

in a point corresponding to &ero pressure and correspondent temperature to

this point is #2@,(A?5.This temperature is the lowest reachable temperature

attributing &ero value and the scale is called >elvin scale. t this

temperature the gas molecules are no longer in motion The value of a

p

0

Gas A

Gas B

-273,15 t (C )


9/189

constant is &ero and e4uation T 9 bp, meaning that it is enough one point to

define an absolute temperature scale. +t is considered that the absolute

temperature scale is identical with thermodynamic scale in the field of

linear variation of ideal gas, thus meaning all temperature range e%cepting

very low temperatures to which condensation appears and very high

temperatures to which dissociation and ionisation appear.

=elation between >elvin scale and 5elsius scale is

t59 T># 2@,(A .

degree 5elsius is e4ual to a >elvin degree , (?59( >, but the

origins of the scales are different. 7ahrenheit scale differs from >elvin scale

and 5elsius scale through the origin of the scale and magnitude of the

unit.The connection between 7ahrenheit and 5elsius scales is given by

relation :

t79 (,*t5 < 2 .

+n the British units system there is an absolute temperature scale

called =ankine in which absolute &ero is identic to absolute &ero in >elvin

scale , but the magnitude of the degree is different (=9(,* >.

=elation of transformation between =ankine and 5elsius degrees is

t=9 t7 < AC,@

degree =ankine e4uals a degree 7ahrenheit, (=9(?7, but the

origins are different.

In thermodynamic calculations it is used absolute temperature

expressed in Kelvin scale.

1.3.3 ressure

ressure is defined as normal force applied on unit of surface , for a static

fluid, pressure gas the same value on any direction. 7or fluids it used

hydrostatic pressure e%pressed as hgp = 0D # density of the fluid, g#

gravitational acceleration , h # the height of the fluid column1.

ressure is classified according to method of measurement in

- bsolute pressure pressure measured reported to absolute

vacuum 0p1


10/189

- =elative pressure 0pr1 pressure measured reported to atmospheric

pressure 0pa1

ar ppp =

The technical gauges measure relative pressure pr.

$anometers measure the increase of pressure from atmospheric pressure,

pressure called manometric pressure 0pmanom1 or supra pressure ,when app, and aman ppp =

"acuum#meters measure the decrease of pressure from atmospheric

pressure, pressure called vacuum#metric pressure or vacuum pressure pv or

0pvac1 when

app ,and

ppp avac =

6henapp in calculation is used also the EvacuumF e%pressed in percentage:

( ) ( ) ( )G())G())Ga

vac

a

r

p

p

p

pvacuum ==

+n other words the absolute pressure is measured reported 0 or having as

reference 1 the absolute vacuum :

p9paH pr

in which it is considered the plus sign when relative pressure is an over

pressure called manometric pressure pr9pmand the minus sign when

relative pressure is a depressure called vacuum pressure pr9pv.

Barometers measure absolute atmospheric pressure pbar.

In thermodynamic calculation it is used the absolute pressure.

=elations between pressure units


11/189

Init !'m2 bar kgf'm2 at

kgf'cm2atm torr

mm -g

mm -28

(

!'m29ascal

( ()#A ).()2 ).()2.()#

).C*@.()#

A

@A).()#A ).()2

( bar ()A ( ).()2.()A (.)2 ).C*@ @A) ).().()A

( kgf'm2 C.*( C.*(.()#A ( ()# C.*.()#A @A..()#

(

( at(kgf'cm2

C.*(.()

).C*( ()

( ).C* @A. ()

( atm (.)(.()A (.)( ().2 (.)( ( @) (.)(.()

( torr

(mm -g

(. (..()#

(. (..()# (.2.()# ( (.

(mm-28 C.*( C.*(.()#A ( ()# C.*.()#A @A..()#

(

# physical atmosphere ( atm 9 ()(2A !'m29 ().2.2@ kgf'm29 @)

torr.

#(bar 9 ()A!'m2

# technical atmosphere (at 9(kgf'cm2 9C.*(.()!'m2.

#4uivalent pressure of (mm column of mercury is called torr

( torr 9 (.22 !'m29 (.ACA( kgf'm29 (.(A@*C.()#atm.

# 4uivalent pressure of (mm column of water

( mm -289C.*( !'m2


12/189

+n order to compare the properties of gaseous substances they must be in the

same state of pressure and temperature. +t was defined a standardi&ed state

called normal physical state by:

barcmkgfmmHgpN )(.(').(@:)2 ===

CtsauKT NN)) )(A.2@ ==

+n thermodynamics is used a unit of volume measurement#normal cubic

meter

( Nm which is a unit of volume but also a unit of mass, representing

the mass of gas contained in a volume of ( min the conditions of normal

physical state.

There is also a tolerated unit called normal technical state defined by:

baricmkgfpn C*(.)'(2

==

CtKT nn)

2)(A.2C ==

1.3.!. "ensity #specific mass$

Jensity is the mass of unit of volume, being the reverse of specific volume.

=

m

kg

V

m

+n thermodynamics it is used also specific gravity K defined as

=

m

N

V

mg

or g= .

1.!. rocess measures. %or& and heat

The thermodynamic process or state transformation is a physical

phenomenon in which the bodies e%change energy in form of heat and

mechanic work. s a conse4uence of energy variation the thermodynamic

system modifies its state of energetic balance meaning the modification of

thermodynamic state. thermodynamic transformation means the passing of

a thermodynamic system from an initial e4uilibrium state to a final

e4uilibrium state, through continuous, successive, intermediate e4uilibrium


13/189


14/189

7ig.2 =epresentation of 4uasistatic 0a1 and non#static processes 0b1.

c. fter the procedure of passing from the initial state 0i1 into final state 0f1

and reverse, the thermodynamic processes can be divided in:

=eversibile process, in which the system passes from initial to final

state directly and reversely,e%actly through the same points, on the

same path.

+n order to perform such a process, the e%ternal conditions should modify

e%tremely slow so the system to adapt progresively to the new variationswhich gradually appears;

!on reversible process, in which the system passes from initial to

final state and reversed through different points,on other path.

=eal processes cannot be considered reversible . process can be considered

reversible if intermediate states when passing from initial to final state are

close enough to intemediate states when passing from final to initial state.

d. fter connection between initial and final state:

5yclic processes when initial state is the same with final state;

!on cyclic processes 0open1, when initial state differs from final state.

6ork and heat are macroscopic forms of energy transfer between bodies

6ork and heat do not feature the state of the system at a given moment 0they

are not state parameters1. 6ork and heat represent specific process measures.

6ork and heat are not forms of energy, but forms of energy transfer.

p

V

if

if

p

V(a) (b)


15/189

1.!.1.%or&

et us consider a gas in a cylinder of an internal combustion engine which

e%pands and actuates upon the piston. -itting the piston wall the moleculesmodify a%ial components of the speeds; the variation of molecular energy

will transmit to piston as work, which is a ordered form of energy transfer

because it affects only one direction components of the molecular speeds.

6ork sums at macroscopic level 0piston motion1 the effect of molecule

motions.

+n mathematical calculations there are used three formulas for work. +n

nglish literature the abbreviation of work is 6, in =omanian one is 0ucru mechanic#$echanic 6ork1.

a1 6ork produced by state transformation 0Boundary work1

+t is considered an enclosure with gas at pressure p. 8utside the enclosure

there is e%ternal pressure pe. +n time interval dthe volume of gas is

increasing with d". 7or an elementary surface d3 from initial surface with

the versor of normal direction and dn the motion of d3 on normal

direction. +ntegrating on the whole volume " it results the relation forelementary work by variation of the volume as result of pressure forces.

( )= ve dSdnpL

dVpL e=

ast formula e%presses the mechanical work produced modifying thevolume of the fluid as a conse4uence of pressure forces.

8bservations

(1Because work is not a state parameter, its elementary variation id not a

total differential # so M represent a infinitesimal 4uantity of work .


16/189

7inite work released or consumed in a thermodynamic process when

passing from an initial state ( 0 parameters p(,"(, T(1 to a final state 2

0 parameters p2,"2,T21 is noted with:

=2

(

2( LL , never =2

(

(2 LLL

21 +n relation of work appears pe and dVpL e=

+f e%ternal pressure is identical with internal pressure ppe = or dpppe = ,then pdVL= 0neglecting the infinitesimals of second order 1.

These conditions are met when the processes are reversible and 4uasi#static.

+n thermodynamic calculations all real processes are replaced with

e4uivalent 4uasi#static processes and elementary work is calculated with

formula pdVL= ,in which p pressure of the fluid .

The signs of the work are deducted for the elementary work formula.

pdVL=

s )p then the sign of work is given by the sign of variation of elementaryvolume: in e%pansion processes ( ))dV )L )

(2

L , the performed worktowards e%terior of the system is positive; similar, in compression processes,

( ))dV )L )(2 L ,and the work received by the system 0 performed bye%terior upon the system1 is negative.

/raphical representation of the processes


17/189

+t is considered the e%pansion process from figure from initial state i to final

state f .

7ig. 6ork of the state transformation

The work of the e%pansion process is pd", in which :

pdVL =

and M is elementary work given by a current value of the pressure

0pressure is considered constant for a infinitesimal variation of volume d"1

=2

(

2( pdVL

The work is e4ual to area under curve made with abscissa ". 7rom graphic it

is noticed that transformation i#f or (#2 can actuate also on other paths and

work of the transformation (#2 could have different values according to

specific path 0intermediate states 1 on which the system works. +n other

words in the transformation from state ( to state 2, the work depends on the

path of the transformation.

b$7low 6ork 0work consumed to actuate a fluid1

5onsidering a pipe through which a fluid is flowing, if we imagine three

&ones of the same length l at constant pressure p 9 constant. +t is called work

p

i

pi

fp

e

V

Vi Vf


18/189

consumed to actuate a volume of fluid " in an environment of constant

pressure p or flow work , the product

pVpSlLd ==

d work consumed to actuate a volume of fluid " at constant pressure p.

The fluid from area + actuates upon the fluid from area ++, the fluid fromarea ++ actuates upon the fluid from area +++ and so on resulting the motion of

the fluid. This type of work does not increase fluid energy, dcontributes

only to the increase of energy of the fluid accumulated in the reservoir at the

end of the flow pipe.

8ne of the forms of energy applied to a fluid is enthalpy, +, being the sum

between I, internal energy and product p".

( )JpV!enthalp" += .

c$3haft work is the total mechanical work performed upon or consumed by

a heat engine taking into account both the thermodynamic processes of the

working agent in the engine and intake and e%haust processes into and

outside engine. +t is considered the same source of working agent which

enters and leave the engine.

evadmt LLpdVL ++= 2

(

((==

==

releaseVp#orkflo#LL

receiveVp#orkflo#LL

dev

dadm

22

N

((

O

.

.

( ) ( ) ==++=2

(

2

(

2

(

((22

2

(

VdppVdpdVVpVppdVLt

=2

(

VdpLt # shaft work is the total work produced or consumed by a

working agent in a heat engine. The shaft work is e4uivalent to area between

the graphic of state transformation and coordinate a%is of pressure p.


19/189

7ig.. 3haft work of a state transformation

1.!.2. 'eat exchan(e

-eat is a form of macroscopic transfer of energy, generally produced

between two bodies with different temperatures without mechanical

interactions.

6hat is called e%change of energy as heat at macroscopic scale is ane%change of molecular kinetic energy at microscopic level.

6hen water is heated in a bowl by a flame, the amplitude of the molecule

motion is increased. The molecules of the fluid took the energy from the

bowl, the water temperature increases and there is an e%change of kinetic

energy from gas to water. -eat is a disordered form of energy transfer the

flame contains highly activated molecules.

The heat e%change in an elementary process is e%pressed

mcdt$=

+n which m mass of the body , c# real specific heat and dt# difference of

temperature.

7or a chemical process # ===2

(

2

(

2

(

2(

t

t

t

t

cdtmmcdt$$

p

i

pi

fp

e

V

Vi Vf


20/189

s heat is not a parameter of state 0it is not a form of energy, but a form of

transfer of energy1, $ is not a total e%act differential.

-eat is considered positive when the system receives energy from

environment and negative when the system releases energy to the

environment.

1.).Specific heats

+t was e%perimentally noticed that in order to heat different bodies with the

same number of degrees are re4uired different heat 4uantities. 3o in order to

describe substance from this point of view it was introduced the term caloric

capacity. The caloric capacity is the ratio between the heat $ in an

elementary process and the corresponding variation of its temperature dT,

dT

dL5 = .

5aloric capacity can be also defined as the physical 4uantity of heat

absorbed by a body in order to modify its temperature with ( unit 0( grade1.

Init of measure is P'>.

3pecific heat is a physical property of the substances which depends on the

nature, phase of the body, temperature and for gases, on the nature of

thermodynamic process in which the heat transfer is done 0 at constantpressure or at constant volume1. 3pecific heat or the caloric capacity of unit

of mass is the physical measure numerically e4ual to heat 4uantity

e%changed by unit of mass of a body with the surroundings in order to

modify its temperature with ( unit. Between specific heat c and caloric

capacity 5, there is the following relation: 5 9 mc

3pecific heat can be classified according to unit of substance reported as

follows:

a1 3pecific heat reported to ( kg of mass 0mass specific heat1

=

=

mcdt$

kgK

J

tm

$c


21/189

with m, the mass of the body e%pressed in kg, Qt is temperature variation of

the body, in degrees .

b13pecific heat reported to ( kmol of substance 0molar specific heat 1

=

=

dtnc$

kmolK

J

tn

$c

M

M

with n, number of kilomoles , Qt is temperature variation of the body, in

degrees.

c13pecific heat reported to ( Nm

=

=

dtCV$

Km

J

tV

$C

NN

NN

N

with "!, volume e%pressed in normal state, Qt is temperature variation of

the body .

gas can be heated 0 or cooled1 in several ways, keeping some parameters

constant. gas can be heated at constant volume or at constant pressure.

%periments showed that the heat at constant pressure of the same amount of

gas for the same difference of temperature is higher than the heat at constant

volume 0of the same amount of gas for the same difference of temperature1.

+n other words a gas can have two specific heats according to the nature of

the process:

- 3pecific heat at constant pressure 0marked with inde% p1

Km

JC

kmolK

Jc

kgK

Jc

N

NpMp p ;;

- 3pecific heat at constant volume 0marked with inde% v1

=

Km

JC

kmolK

Jc

kgK

Jc

N

NVMV V ;;


22/189


23/189

( )2

2(

)

ttttccc

m

t

tm

+===

or

(2

(2(

)

2

)

tt

ctctc

t

t

t

t

m

=

+n thermodynamic applications, to make the calculations easier specific

heats are appro%imated to average values between two temperatures.

nother type of specific heat #latent heat

There are thermal processes in which even the heat is transmitted to the

body, its temperature does not vary 0e%. melting or vapori&ation1. +n this case

the heat transmitted is used for the change of phase of the body and in this

situation a new caloric coefficient is defined, called latent heat for phase

transformation 0 latent heat of vapori&ation, latent heat of fusion1

m

L= ,

This coefficient is defined as heat 4uantity re4uired for changing the phase

of the unit of mass from a substance, at a constant temperature and pressure.

The unit is P'kg. 7or the same substance the latent heat of vapori&ation ise4ual to latent heat of condensation and the latent heat of fusion is e4ual to

latent heat of solidification.

*uestions

(. 6hat studies thermodynamics S /ive e%amples of thermal

phenomena.2. 6hat is a physic systemS 6hat is a thermodynamic system S

. 6hat is an isolated systemS

. 6hat is the difference between an open and a closed system S

A. -ow do you e%press the state of a system S

. 6hat is thermodynamic e4uilibrium S

@. 6hat are e%tensive parameters S 6hat are intensive parameters S

*. -ow is classified pressure according to measurement method S

C. 6hat type of pressure is measured with manometers and vacuum#


24/189

meters S

().6hat type of pressure is measured with barometer S

((.6hat property is described by temperatureS

(2.6hat is a thermal measure 0or 4uantity1 S Jo you have some

e%amples S

(.5an be directly measured the temperature of a body S 6hy S

(.-ow do you enounce the &eroth law of thermodynamicsS(A.6hat is a scale of temperature S

(.-ow do you classify the scales of temperature and which are theyS

(@.6hich are the relations between the origins and units of the scales S

(*.-ow was determined the lowest temperature and what is its meaning S

(C.-ow is defined the specific volumeS

2).-ow is defined the molar volume S -ow is defined the kilomoleS

2(.6hich are normal physical gas state S

22.-ow is defined the density S -ow is defined the specific gravity S

2.6hat are finite thermodynamic processes S2.6hat is the difference between 4uasi#static and non#static processesS

2A.6hen a process is reversible S 6hen a process is irreversible S

2.6hat is mechanic work S

[email protected] is classified work in thermodynamics S

2*.+n p#" representation of a transformation of state of a gas which is the

significance of the boundary work 0 work of the state transformation 1

But of the shaft work S

2C.6hich is the sign rule for workS

).6hat is heat S +s a 4uantity of state or a processS

(.6hich is the sign rule for heat S2.-ow is e%pressed the heat change which produces the heating of a

body S

.-ow is e%pressed the heat change which produces the change of phase

of a body S

.6hat is specific heat S -ow is reported to different units of mass S

A.+s specific heat constant with temperature S

Chapter . 2. +irst la, of thermodynamics

3ome calculations and e%periments performed in the + th century

demostrated that mechanical work and any other form of energy can be

transformed in heat and reversed and it was determined the e4uivalency

ratio of transformation. +n technical system heat is e%pressed in kilocalories 0

a calorie is the amount of heat 0energy1 re4uired to raise the temperature of

one gram of water by ( ?51 and work in kgf.m 0work produced moving a
http://en.wikipedia.org/wiki/Degree_Celsiushttp://en.wikipedia.org/wiki/Degree_Celsius


25/189

body of ( kg on a length of ( m1, those units being in that period considered

as independent.

+n (*2 =obert $ayer introduces the mechanical e4uivalent of heat unit and

determined its value by calculations, in the same year Poule determined the

caloric e4uivalent of work and -elmholt& demostrated the e4uivalence

between the thermal and mechanical energy.

Lecture--The mechanical equivalent of heat

Joule*s Heat %pparatus+ ),(-,Joule*s apparatus for measuring the

mechanical e.uivalent of heat

/urther e0periments and measurements b" Joule led him to estimate the

mechanical e.uivalent of heatas ,1,ft2lbfof #ork to raise the temperature

of a pound of #aterb" one degree/ahrenheit' He announced his results at a

meeting of the chemical section of the3ritish %ssociation for the

%dvancement of Sciencein Corkin ),(1 and #as met b" silence'

Joule #as undaunted and started to seek a purel" mechanical demonstration

of the conversion of #ork into heat' 3" forcing #ater through a perforated

c"linder+ he #as able to measure the slight viscousheating of the fluid' He

obtained a mechanical e.uivalent of 445 ft2lbf63tu7(')(J6cal8' The fact that

the values obtained both b" electrical and purel" mechanical means #ere in

agreement to at least one order of magnitude#as+ to Joule+ compelling

evidence of the realit" of the convertibilit" of #ork into heat'
http://en.wikipedia.org/wiki/Mechanical_equivalent_of_heathttp://en.wikipedia.org/wiki/Foot-pound_forcehttp://en.wikipedia.org/wiki/Waterhttp://en.wikipedia.org/wiki/Fahrenheithttp://en.wikipedia.org/wiki/British_Association_for_the_Advancement_of_Sciencehttp://en.wikipedia.org/wiki/British_Association_for_the_Advancement_of_Sciencehttp://en.wikipedia.org/wiki/Cork_(city)http://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/British_thermal_unithttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Caloriehttp://en.wikipedia.org/wiki/Order_of_magnitudehttp://en.wikipedia.org/wiki/Image:Joule's_heat_apparatus.JPGhttp://en.wikipedia.org/wiki/Image:Joule's_Apparatus_(Harper's_Scan).pnghttp://en.wikipedia.org/wiki/Image:Joule's_heat_apparatus.JPGhttp://en.wikipedia.org/wiki/Mechanical_equivalent_of_heathttp://en.wikipedia.org/wiki/Foot-pound_forcehttp://en.wikipedia.org/wiki/Waterhttp://en.wikipedia.org/wiki/Fahrenheithttp://en.wikipedia.org/wiki/British_Association_for_the_Advancement_of_Sciencehttp://en.wikipedia.org/wiki/British_Association_for_the_Advancement_of_Sciencehttp://en.wikipedia.org/wiki/Cork_(city)http://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/British_thermal_unithttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Caloriehttp://en.wikipedia.org/wiki/Order_of_magnitude


26/189

Joule no# tried a third route' He measured the heat generated against the

#ork done in compressing a gas' He obtained a mechanical e.uivalent of

,&1 ft2lbf63tu 7('(1 J6cal8' !n ),(-+ Joule read his paper 9n the mechanical

e.uivalent of heat to the 3ritish %ssociation meeting in Cambridge' !n this

#ork+ he reported his best:kno#n e0periment+ involving the use of a falling

#eight to spin a paddle:#heel in an insulated barrel of #ater+ #hose

increased temperature he measured' He no# estimated a mechanicale.uivalent of ,); ft2lbf63tu 7('() J6cal8'

+n (*A), Poule published a refined measurement of @@2.C2 ftUlbf'Btu 0.(AC

P'cal1, closer to twentieth century estimates.

n important contribution had 5.$iculescu, a =omanian physicist who

established the value of mechanical e4uivalent of the calory : (kcal9(*A,@

P, a value very close to the closest value (kcal9(*A,A P.

2.1. Internal -ner(y

The first law of thermodynamics represents the energy conservation and

transformation law applied to thermodynamics processes in which energy

change is done as heat and work variation. 7irst law is based on a state

measure called internal energy.

body, which in thermodynamics is called thermodynamic system, is madeof very high, but finite number of particles in continuous, disordered

motion, which interact amongst them. +t means that the particles have a

kinetic energy corresponding to thermal, disordered motion and a potential

energy due to forces of interaction between them 0intermoleculare forces1

and due to interaction with other e%ternal forces 0 e%. gravitational field1.

ll these energies form internal energy of the system. 3o internal energy of a

system is made of kinetic energies corespunding to particle macroscopic

motions as well as potential energy of interaction of particles.

+nternal energy represents the sum of kinetic and potential energies of theparticles within a body and of the energies within the molecules 0e%. energy

of chemical bonds, inter and intra atomic1.The last energy, although is

contained in internal energy,does not change during thermodynamic

processes because it is not changed the structure of the body.That is why it is

of interest only the variation of internal energy due to kinetic and potential

energy. +nternal energy is noted with I and for thermodynamic processes is

the sum of kinetic and potential energy of molecules.
http://en.wikipedia.org/wiki/Cambridgehttp://en.wikipedia.org/wiki/Cambridge


27/189

noscillatiorotationtrasition

cincincincin

potcin

++=

+=

$olecules of li4uid and gas may have translation and rotation motions; in

the molecules the groups of atoms have oscilation motions.

7or e%ample, internal energy of a gas enclosed in a vessel is composed of :

kynetic energy of translation and rotation of gas molecules; potential energy

of molecules depending on molecular interaction forces; kinetic and

potential energies corresponding to atom oscilation within molecules;

electron energy from atoms; motion and interaction energy of particles

which compose the nucleus of atoms.The last two forms of energy are

contained in intermolecular energy ).

+nternal energy [ ]kcalJ , is a state measure or 4uantity meaning that itdepends only by the state of the system. 6hen a system passes from a state

having I(internal energy to another state having I2internal energy, nomatter if the process is reversible or not, variation I9I2#I(of internal

energy does not depend on intermediate states throgh which the system

passed, it depends only on the initial and final states 0 their internal

energies1. +nternal energy is an additive measure meaning that the internal

energy of a system is e4ual to the sum of energies of the components.The

ratio of internal energy to the mass od the system is called specific internal

energy and is noted with u.

( )

( )

= kgJu

kgm

mu '

2.2.-nouncements of first la, of thermodynamics

8n the basis of law there was the e%perimental observation that mechanic

work can be in heat and reversed.Transformation of the work in heat are met

at most friction processes between bodies, at gas compression and e%pasion,

when work is transformed in electric energy and then into heat.

a1 V-eat could be obtained from work and it can be transformed into work

always in the same e4uivalence ratio.F


28/189

+f in thermodynamic relationships appear heat e%pressed in kcal and work

e%pressed in Pouli or kgfm, in order to have homogeneus formula it must

e%pressed the ratio beween heat and work as the caloric e4uivalent of unit

for work

L'9 , # caloric e4uivalent of unit for work.

mkgfkcal B2@*:,B2:( =

kgfm

kcal%

B2@

(=

+f [ ] [ ]kgfmLkcal$

%L$= in technical system

b1 V+t can not be produced a heat engine in continuos operation to produce

work , without consuming an e4uivalent 4uantity of heat L.F

c1 8swald:

Eerpetual motion machine of the first kind does not e%ist.F perpetual

motion machine of the first kind is a machine which produces more work

than e4uivalent heat L, thus meaning it produces energy from nothing; in

this way, it violates the law of conservation of energy.

+n a thermodynamic process the variation of internal energy of the system

e4uals the sum of mechanic e4uivalents of all energy changes between the

system and surroundings.ny form of energy can be e%pressed through

mechanical e4uivalent,P.

2.3. athematical formulation of first la, for open systems

n open system is a thermodynamic system changing energy and mass with

the surroundings.

7or a heat engine 0a device that converts heat energy into mechanical energy

or more e%actly a systemwhich operates continuously and only heatand

workmay pass across its boundaries1 is e%pressed the energy balance of the

system for period of time, meaning the balance energy transfer formsand mechanical and thermal energies.
http://www.taftan.com/thermodynamics/SYSTEM.HTMhttp://www.taftan.com/thermodynamics/HEAT.HTMhttp://www.taftan.com/thermodynamics/WORK.HTMhttp://www.taftan.com/thermodynamics/SYSTEM.HTMhttp://www.taftan.com/thermodynamics/HEAT.HTMhttp://www.taftan.com/thermodynamics/WORK.HTM


29/189

7ig..3cheme of energy changes in a heat engine

+t is consider a heat engine from fig. in which point ( represents the intake

of thermal agent and point 2 represents the e%haust of thermal agent .The

heat engine is supplied by fuel which is burned releasing heat L(#2; the heat

engine produced shaft work noted t(#2. The thermal agent in point ( has

pressure p(,temperature T(, specific volume v(, specific internal energy u(and specific enthalpy i( and it gets into the engine with w( velocity level

difference h(.

The thermal agent in point 2 has pressure p2, temperature T 2, specific

volume v2, specific internal energy u2 Wi specific enthalpy i2 and it gets out

engine with w2 velocity at a a level difference h2.

$asic balance e4uation written between points ( and 2 indicates mass

conservation m(9m29 m .

nergy balance e4uation written on the control area between points ( and 2

is:

releasedreceived


30/189

( )Jmumghm#


31/189

closed system is a system which do not change mass with the

surroundings, for e%ample the gas from the cylinder of a piston engine. 7or

e4uations 0(1 and 0(X1 from aforementioned chapter for open systems,

considering intake and e%haust velocities &ero and the same value of

reference levels.

=

=

2(

2( )

hh

##

tl.vpvpuu =+ 2(((22(2

+t is obtained

( ) += 2

(

2

(

2((2 pvdvdp.uu

1 ==2

(

2(2(2((2 l.pdV.uu

8r for m kilos of thermal agent

2(2((2 = L$

7rom (X02X1 tl.ii = 2((2

and ( )JL$!! t= 2((2 for m kilos of agent in which ( )=2

(

JVdpLt .

The mathematical e%pressions in differential form of the first law of

thermodynamics:

1 kgpentruvdp.dipdv.du (

+=

=

and kgmpentru

Vdp$d!pdV$d

+=

=

The mathematical e%pression of the first law of thermodynamics for closed

systems


32/189

( )JL$ 2(2((2 =

favori&es the following hydraulic interpretation and analogy.

7ig.A. -ydraulic analogy of first law for close systems

The analogy emphasi&es that internal energy of the thermal agent varies in

function of value and sign of heat and work agent changes with the

surroundings.

3pecial cases :

.7or adiabatic processes )2( =$ the first law becomes a relationbetween I and .6hen system does not receive energy from e%terior,

meaning that is adiabatically isolated, then it could perform work only on

the variation of internal energy, L9) resulting 9 #dI and in this

situation work does not depend on intermediate states, meaning that in this

particular situation work is a total diferential 0d9#dI1.

B. 7or isochoric processes 0" 9 ct.1 with variation of volume &ero,work

of the isochoric transformation is &ero )=i=L and first law becomes a

relation between I and L. +f the system does not perform work upon

e%terior and e%terior does not perform work upon the system, the heat

received by the system from e%terior determines an increase of its internal

energy and 9pd"9) and dL 9 dI or

L(#29 I2 # I(. +n the isochoric process the heat is a total differential.

U

L1-2

Q1-2


33/189

5. 6hen

==

2(2(

2(

2(

.0tan...)

...)

L$

processisothermaltcons#orkpeformsagenttheL

heatreceivesagentthe$

6hen agent receives heat and performs work, if these 4uantities are

e4ual, internal energy remains constant.

+n other words when internal energy does not change during its

interaction with the environment, then system cannot perform work unless

it receives energy from e%terior.7or dI9), it is obtained 9 L or L(#29(#2.

2.) Caloric e/uation of state

The 4uantities intenal energy I and enthalpy + are called caloric state

4uantities repre&enting thermal forms of energy.

=ule EThe state of thermal e4uilibrium of a system is completely determined

if are known two intensive state parameters and masses mY of components of

the systemF. +ntensive parameters are pressure, temperature, specific

volume.

+t is considered a monocomponent system 0( body 1, having mass of ( kg

for which any state 4uantity can be e%pressed in function of two intensive

parameters.

+t is e%pressed:

( )

( )pTfivTfu

,

,

(=

=.By differentiation

+

=

+

=

dpp

idT

T

iid

dvv

udT

T

udu

Tp

Tv


34/189

7or ( kilo of agent which suffers an elementar heating at constant volume:

( ) dTc. vv =

ccording to first law for closed systems:

pdV.du = for ( ) ( ) ( ) dTcduand.dutconsv vvvv === tan

( )vtf ,=

v

vT

c

=

3imilarly results :

p

pT

ic

=

=eplacing in du and di results

+=

+=

dpp

idTcid

dvv

udTcdu

T

p

T

v

analog with capital letters for m kg :

+=

+=

dpp

!dTmc!d

dVv

dTmcd

T

p

T

v

These e4uations are called caloric e4uations of state.

*uestions

(. +s the work performed by a system a form of energy e%change S6hat

about the heat change S

2. 5an be mechanic work converted into heat S 5an be heat converted

into workS

. 6hat are units for work and heat S


35/189

. 6hat is internal energy of a system S

A. 6hich is the enouncement of the first law of thermodynamics S

. 6hich is the enouncement of the first law of thermodynamics for

closed systems S

@. 6hich is the hydraulic analogy of the first law of thermodynamics for

closed systems S

*. 6hich are the caloric e4uations of state S

Chapter 3. The ideal (as

+deal gas is a hypotetic notion # it represents a gaseous body having the

following properties:

- molecules are perfectly spherical;

- molecules are perfectly elastic;

- molecules have no interaction;

- moleculesX own volume can be neglected;

The perfect gas is also called ideal gasand, according to the kinetic

molecular theory, it cannot be li4uefied.

=elations e%pressing the properties of a perfect gas are:

a8 the e%pression of a gasX pressure 0Bernoulli1

2

22m#

V

Np=

b8 the e%pression of kinetic energy


36/189

7or a constant mass of gas 0! 9 constant1 it results :

constant=T

pV# the ecuation of state for a hypotetical perfect gas.

+n certain pressure and temperature conditions, gases in nature almost obey

the rigurous relations for the hypotetical perfect gas: these generic

conditions consist of small and medium pressures and medium and hightemperatures so states that are far enough from the li4uefying point.

/ases in the nature that are in such conditions can be considered perfect

gases; the simple laws established in "++#+ centuries that are not

rigurously correct were determined by e%periments on gaseous bodies in

the nature, in pressure and temperature conditions far away of li4uid states,

thus obtaining the laws of perfect gases. The appro%imation of the simple

laws of gases is sometimes under the errors introduced by mathematical

models of the phenomena.

3.1. 0a,s of the perfect (as

7or a constant mass 0kg1 of perfect gas, these laws are as follows:

a1 Thermal state e4uation

m>TpVct

T

pV == or

- the variables are e%pressed in different measurement units:

====

kmolT>pV

nkmolTn>pV

mkgm>TpV

kg>TpV

M

M

(

(

= is called the constant of the gas. +ts value doesnXt depend on its status, but

only on its nature and thermal properties.

7or a kmol of gas, the state e4uation becomes:

T>pV

M>TpvM

MM =

=, where>Mis the the universal constant of the perfect gas, being

independent of the nature of the gas and having a value that can be

computed out of the state e4uation of the gas in normal physical conditions.


37/189


38/189

- general e%pressions of caloric state e4uations:

dVV

dTmcd

T

v

+=

dpp

!dTmcd!

T

p

+=

7or the perfect gases:

)=

V

according to Poule law

)=

p

according to Poule law.

6e obtain the following e%pressions of the caloric state e4uations for perfect

gases:

=

=dTmcd!dTmcd

p

v

7or ( kg

=

=

dTcdi

dTcdu

p

v

and, integrated,

( )( )( )JTTmc!!

TTmc

pm

vm

=

=

(2(2

(2(2

3.2. Specific heat of perfect (ases

The ratio of specific heats is called adiabatic e%ponent and is noted as:

kC

C

c

c

c

c

Nv

Np

Mv

Mp

v

p ===


39/189

MccM =

NM Cc B(B,22=

+t can also be written >Tupvui +=+= and, in diferential form,

>dTdudi += or >dTdTcdTc vp +=

= kgKJ

>cc vp # $ayer relation

M>McMc vp = or

=

kmolK

J>cc MMM vp >M>M =

=

=

=

=

kgK

J>

k

kc

kgK

J

k

>c

>cc

k

c

c

p

v

vp

v

p

(

(

=

=

kmolK

J>k

kc

kmolK

J

k

>c

MMp

MMv

(

(

8ne can notice that specific heats of perfect gas can be determined as a

function of adiabatic e%ponent kand of the constants of the gas.

7or the hypotetic perfect gas, specific heats are considered constant and k9

constant.

3pecific heats of gases considered as perfect can vary with temperature.

%periments show that, for these gases, kstays constant in large intervals of

temperature. 7or first appro%imation computation, the following values areadopted:

-

for monoatomic gases 0-e1, k9 (.

-

for biatomic gases 0!2, 82, -2, 581, k 9 (.

-

for polyatomic gases 0582, 382Z.1, k9 (.


40/189

The hypotetic perfect gas with punctiform molecules behave like a

monoatomic gas, with cconstant and kconstant. 7or gases of the nature with

two or more atoms in the molecule, the degrees of freedom in moleculesX

movement appear progressively with the increase of temperature.

-

for low temperatures only translation

-for medium temperatures 0e.g. atmospheric temperature1 translation T, becomes for the real

gas:

( ) >Tbvv

ap =

+

2 , 0"an de 6aals e4uation1, wher aand bare two

constants.

The real gas is different from the ideal gas 0whose internal energy depends

only on temperature1, as the internal energy of the real gas depends also on

volume, so

? 7T8# for the ideal gas

? 7T+ V8# for the real gas


41/189

+n ideal gases, molecules donXt interract and internal energy consists only of

the kinetic energy of the molecules, that depends only on the temperature of

the gas. The internal energy of the real gas consists of the kinetic energy and

potential energy of the molecules, due to their interraction the potential

energy depends on the distance between molecules, so, on the volume of the

gas.

3.!. ixtures of perfect (ases

+n most installations, the gaseous thermal agents are not pure gases, but

mi%tures of gases.

%amples:

-the air is a mi%ture of gases, mainly 22 9N +

-

e%haust gases consist of .....22222 C9N9S99HC9 vap +++++

mi%ture of perfect gases behave like a perfect gas:

m>TpV=

7or the study of gaseous mi%tures, they are two hypoteses:

a1 mi%ed gases # molecules of each gas spread in whole the volume.

7ig.. $i%ture of molecules in the whole volume

/iven three different gases, , B and 5, with molecules of different si&es, it

can be considered that they occupy all the available volume:

VVVV C3% ===

A B C A A B C A B C

B B C B A A B C

A B C C A C B

C A B C A C

C B C A B C

A C B B A B C


42/189

Temperature of the components can be considered as e4ual with that of the

mi%ture:

TTTT C3% ===

artial pressure is the pressure of a gas on the enclosure walls if it should be

alone. 7or the chosen three gases, partial pressures are not e4ual:

C3% ppp # partial pressuresThe pressure of the mixture is the sum of the partial pressures of

component (ases pppp C3% =++ # JaltonXs law

The sum of component masses is the mass of the mi%ture:

( )kgmmmm C3% ++=They are called mass fractions 0concentrations1 the ratios:

m

mg %%=

m

mg 33 =

m

mg C

C= (=

n

i

ig

b1 component gases are separated by ima(inarywalls, each having the

same pressurep and temperature Tas the mi%ture.

7ig.@. $i%ture of molecules in partial 0imaginary1 volumes

7or this hypothesis

pppp C3% ===

TTTT C3% ===

C3% VVV

The partial volumes of the components are different because the partial

enclosures are different.

The sum of the partial volumes of the components is e/ual to the volume

of the mixture.

VVVV C3% =++# magat law

A A A

A

A

A A A

B B

B

B B

B

B B

B B

B

B B

C C

C

C

C C

C

C C

C C

C


43/189

There are called volumic fractions the ratios

V

Vr %% =

V

Vr 33 =

V

Vr CC= (=

n

i

ir

roperties of ideal (as mixtures

1. Constant of the mixture .

+t is considered the mi%ture made of gases , B and 5 in conditions

described in paragraph a1. 7or every component it can be written the

e4uation of state:

T>mVp %%% =

T>mVp 333 =

T>mVp CCC =

3umming the e4uations, it results:

( )T>m>m>mpV CC33%% ++=

m>TpV = e4uation of state of the mi%ture

CC33%% >m>m>mm> ++= and dividing by m, it results

i

n

i

iCC33%%CC

33

%% >g>g>g>g>

m

m>

m

m>

m

m> =++=++=

= constant of the mi%ture is e4ual to he sum of the gas constants of the

components, weighted with the mass fractions.

2. Specific mass of the mixture

V

m= 0density1

5onsidering the hypothesis from paragraph b, it can be written

CC33%%C3% VVVmmmm ++=++= and dividing to volume "

=++=n

i

iiCC33%% rrrr


44/189

Jensity of a mi%ture is e4ual to the sum of gas densities weighted withthe volume fractions.

3. Specific volume of the mixture v

==== iiiii vgm

vm

m

V

m

Vv

in whichp

T>v ii = are specific volumes of the components.

!. olar mass #,ei(ht$ of the mixture

$olar mass 0weight1 of the mi%ture is called conventional mass because it

does not have a measurable value in real life. +f it is considered the densityof the mi%ture and components

=n

i

iir in which

===

===

MT>

p

>T

p

V

m

MT>

p

T>

p

V

m

M

i

Mii

ii

+t results the e4ualityT>

p

M ==

n

i

iii

M

i MrMMT>

prM

$olar mass of the mi%ture $ e4uals the sum of molar masses of the

components weighted to volume fractions.

).artial pressures of the (as components

7or component i it is e%pressed the e4uations of state which correspond to

both hypothesis of the mi%tures which are considered to be e4ual:

T>mVp iii =

T>mVp iii =

prpV

Vp i

ii ==


45/189

The partial pressure of a component is 4ual to the pressure of the mi%ture p

weighted with the volume fraction of the component ri.

7or e%ample, for air at atmospheric pressure considered close to ( bar:

bar2(,)

bar@C,)

2

2

9

N

p

p

. Specific heat of the mixture

+t is considered a mi%ture which suffers an elementary heating process at

constant pressure or volume;heat change is e%pressed

mcdt$= in which specific heat c is either cpor cv.

-eat received by the mi%ture e4uals the sum of the heats received by every

component.

== dtcm$i$ ii

==n

i

ii

n

i

ii cgc

m

mc

3imilarly, it can be demonstrated that

= iMiM crc

= iNiN CrC

4. Conversion of the fractions

The mi%ture composition is e%pressed through mass or volume fractions.

By definition : ==

i

iii

m

m

m

mg

-

+n hypothesis b1 TM

>mT>mpV

i

Miiii ==


46/189

VMV

T>

p

MVT>

p

g

ii

M

ii

Mi

(

=

=

ii

iii

Mr

Mrg

By definition : ==

i

iii

V

V

V

Vr

m

M

m

p

T>

M

m

p

T>

r

i

iM

i

iM

i

(

=

=

i

i

i

i

i

M

g

Mg

r

%ample for air:

==

==

G@@G@C

G2G2(

2

22

2 NN

99

gr

gr

2.)2*@C.)22(.)

22(,)

2222

22

2=

+=

+=

00

0

MrMr

Mrg

NNoo

oo

9

@@,)2*@C.)22(.)

2*@C.)

2222

22

2=

+=

+=

00

0

MrMr

Mrg

NNoo

NN

N

5. ean temperature of a mixture

There are considered several gases initially at different temperatures and it is

re4uired the final temperature of the mi%ture, after mi%ing process. 7om

e4uation of heat balance considering that a component releases heat and the

others absorb heat :


47/189

( ) ( ) ( )222(((

TTcmTTcmTTcm +=

The temperature of the mi%ture is T

=

++++

=ii

iii

cm

Tcm

cmcmcm

TcmTcmTcmT

22((

222(((

Chapter 3. The ideal (as #continued$

3.). Thermodynamic processes applied to the ideal (as

6hen a heat engine is designed the real processes of heating, cooling,compressing and e%panding are replaced with one or more simple

thermodynamic processes or transformations.

-

The thermodynamic process which takes place at constant volume is

called isochoric transformation 0" 9 constant1

-

The thermodynamic process which takes place at constant pressure is

called isobaric transformation 0p 9 constant1

-

The thermodynamic process which takes place at constant temperature is

called isothermal transformation 0T 9 constant1

-

The thermodynamic process which takes place without heat e%change

with the surroundings is called adiabatic transformation 0L 9)1

-

The thermodynamic process which takes place with variation of all

parameters of state, in the condition of a constant specific heat of the

process 0cn9constant1 is called polytropic transformation.

7or each type of the trnsformation it is necessary to know the e4uation of the

transformation, work, heat change and graphical representation in p#"

coordinates.

3.).1. Isochoric transformation


48/189

The isochoric transformation takes place at constant volume V9 constant ,

0 ttanconsm= 1, from e4uation of state tconsT

pVtan= it results if Vis

constat ttanconsT

p= 0e4uation of the transformation 1 or

(

2

(

2

T

T

p

p=

The mechanical work of the isochoric transformation is

====2

(

2

(

2( tan) tconsVaspdVLL

-eat transfer of the isochoric transformation is

( ) ===2

(

(22(

2

(

T

T

vv TTmcdtmc$$

m

5aloric e4uations of state :

( ) 2((2(2 == $TTmc mv( )(2(2 TTmc!! pm =

/raphical representation of the transformation inp:Vcoordinates is like in

fig. *, where the transition from state ( to state 2 is characterised by keeping

the volume constant: VVV == 2(

7ig. *. lot of the isochoric transformation inp:Vcoordinates

V

1

2

p

p

2

p

1

Heating

C%%!ing


49/189


50/189

( ) ( )JTTmc vm (2(2 =

( ) 2((2(2 == $TTmc!! pm


fig. C, where the transition from state ( to state 2 is characterised by keeping

the pressure constant:ppp ==

2(

7ig. C. lot of the isobaric transformation inp:Vcoordinates

s the V6T ratio stays constant with volume increase, e.g. passing from state

( to state 2 0V&@V) 1, the temperature would increase, 0 T&@T)1 resulting a a

heating of the gas. +f the process inverts, from state 2 to state (, as the

volume decreases, the gas is cooling.

The physical model of a thermodynamic system undergoing such a

transformation is that of a fluid container with a mobile e%terior wall pushed

by a constant force that gives a constant pressure. -eating by an e%ternal

source gives an increase of the temperature and of the volume occupied by

the gas.

3.).3. Isothermal transformation

The isothermal transformationtakes place at constant temperature T 9

constant, 0 ttanconsm= 1, from e4uation of state

p

p

Heating

1 2

VV

1V

2

C%%!ing


51/189

tconsT

pVtan= it results if Tis constat ttanconspV = 0e4uation of the

transformation 1 or 22(( VpVp =

The mechanical work of the isothermal transformation is:

==2

(

2

(

2( pdVLL where 22(( VpVpttanconspV ===

===2

(

2

( (

2((((2(

V

VlnVp

V

dVVp

V

dVctL

( )Jp

pm>T

V

Vm>T

p

pVp

V

VVpL

2

(

(

2

2

(((

(

2((2( lnlnlnln ====

-eat e%change of the isothermal transformation is :

mcdT$= )=dT )$

The isothermal process has a specific heat of . +n order to solve such anundetermination, the general formulas of the (st principle are used:

( )

( )

==+=

===

VdpLL$Vdp$d!

pdVLL$pdV$d

tt

and, using the caloric state e4uations for the perfect gas:

=

=

dTmcd!

dTmcd

p

v

it results, forttanconsT

=

=

=

L$

d )

or( )JL$

2(2( =

8nly in the isothermal transformation, $e%changed by the agent with the

environment is e4uivalent with the mechanical work done or consumed

by the thermal agent during the transformation.

( )Jp

pm>T

V

Vm>T

p

pVp

V

VVp$

2

(

(

2

2

(((

(

2((2( lnlnlnln ====


52/189


fig. (), where the transition from state ( to state 2 is characterised by

keeping the productpV constant.

7ig. (). lot of the isothermal transformation inp:Vcoordinates

The plot is an e4uilateral hyperbole arch.

+n the isothermal transformation:

( ) )(2(2 == TTmc vm( ) )(2(2 == TTmc!! pm


transformation is that of an engineXs cylinder that is intensely e%changing

heat with the environment. 6hen the piston moves, the decrease of volume

gives an increase of pressure so the temperature tends to increase as well.

ssuming that cylinderXs walls allow the release of a heat thatXs large#

enough, it can result a constant temperature of the gas in the cylinder.

p1

2

V2

V1

p1

p2

&'pansi%n

C%pressi%n


53/189

3.).!.6diabatic transformation

The adiabatic transformation is the thermodynamic transformation without

any heat e%change with the e%ternal environment )) 2( == $$

arameters p+ T+ Vare variable and the goal is to find their relationship

under these conditions

(st principle:

+=

=

Vdp$d!

pdV$d

5aloric e4uations of state:

=

=dTmcd!dTmcd

p

v

=eplacing d+ d!and computing the ratio, it gives:

pdV

Vdp

c

c

v

p = where kc

c

v

p =

k is the adiabatic e%ponent that represents the ratio of specific heats of a gas

[n the isobaric respectively in the isochoric transformation.

+ntegrating the )=+V

dVk

p

dp e4uation gives ( ) tconsVp k tanlnln =+ that

leads to the e4uation of the transformation inpand Vcoordinates:

tconspVk tan= orkk

VpVp 22(( =

ogarithmation and differentiation is applied to the state e4uation, in the

form ttanconsT

pV= .

+t results ( ) )()

=

+

=+

=+

V

dVk

T

dT

V

dVk

p

dp

T

dT

V

dV

p

dp

,


54/189

that, after integration, gives the e4uation of the adiabatic transform with

parameters Tand V:

tconsTVk tan( = (22(

((

= kk VTVT

=eplacing Vgivestcons

p

T

k

ktan

(= or

k

k

p

p

T

T(

(

2

(

2

= that represents the

e4uation of the adiabatic transform withpand T.

The mechanical work of the adiabatic transform is:

==2

(

2

(

2( pdVLL wherekkk VpVptconspV 22((tan ===

==

2

(

2

(

2( dVVctV

dV

ctL

k

k

( )(((22((

++ +

= kk VVk

ctL

the constant is replaced withkVp

22 and then withkVp (( .

(

22((2(

=

k

VpVpL or

( )

=

=(

2(2(2( (

(( T

T

k

m>T

k

TTm>L

=

k

k

p

p

k

VpL

(

(

2((2( (

(

The heat e%change of the adiabatic transform is )=$ )2( =$ , that meansthat for this transformation, the specific heat is &ero: )=adc .

"ariation of caloric state variables:

( )( ) =

=(2(2

(2(2

TTmc!!TTmc

pm

vm where [ ]JL 2(2( = 2( !!Lt =

tconspVk tan=


55/189


fig. ((, where the transition from state ( to state 2 is characterised by

keeping the constant product: VVV == 2(

7ig. ((. lot of the adiabatic transformation inp:Vcoordinates

The plot is a hyperbole arch with greater slope than the e4uilateralhyperbole, as the values kR (.


transformation is that of an engineXs cylinder with a perfectly thermal

isolation, that doesnXt allow any heat e%change with the environment.

3.).).The polytropic transform

The polytropic transform is a general transformation in which p,T," vary

and energy is e%changed in form of L and with environment .

The polytropic transform is a a process din which all parameters vary so in

order to define the transformation it is accepted that the specific heat is

given and constant ( )nc .

cn9 polytropic specific heat

p1

2

V2

V1

p1

p2

&'pansi%n

C%pressi%n


56/189

dTmc$ n=

To find out the e4uation of the trasformation:

7irst law

+=

=

Vdp$d!

pdV$dwith

=

=

dTmcd!

mcvdTd

p

dTmc$ n=

=

=

pdVdTmcdTmc

VdpdTmcdTmc

nv

np =atio

pdV

Vdp

cc

cc

nv

np =

is noted with

nv

np

cc

ccn

= and

is called polytropic e%ponent. +t is considered that ratio constant

0n9constant 1 during a polytropic transform.

)

=+ V

dVn

p

dp

ttanconspV n

= # e4uation of the transformation in p,"or nn VpVp 22(( =

=+T

dT

V

dV

p

dp; )=+

V

ndV

p

dp ( ) )( =

+

V

dVn

T

dT

ttanconsTVn =( # e4uation of the transformation in T," or(

22(

(( = nn VTVT

ttancons

p

T

n

n =

( # e4uation of the transformation in T,p orn

n

p

p

T

T(

(

2

(

2

=

The work of the polytropic transform is ==2

(

2

(

2( pdVLL

nnnVpVpctpV 22(( ===

( )((

((22((

(2

2

(

2

(

2( +

=+

=== ++ nVpVp

VVn

ctdVVct

V

dVctL nnn

n

(

22((2(

=

n

VpVpL

( )

=

=(

2(2(2( (

(( T

T

n

m>T

n

TTm>L

=

n

n

p

p

n

VpL

(

(

2((2( (

(


57/189

The heat e%change of polytropic transform is:

dTmc$ n=

( ) ===2

(

2

(

(22( TTmcdTmc$$ nmn

5onsidering n constant, as known, for o given polytropic transform, it can be

deduced cnfrom ratio (

=

=n

cncc

cc

ccn

pv

n

nv

np

cu kc

c

v

p =vn c

n

knc

(=

dTcn

knmdTmc$

vn(

==

( ) ( )(2(22(

(TTc

n

knmTTmc$ vmnm

==

+t is usual to e%press 2($ in function of mec

L$d =

kn

k

kn

n

dTmc

dTmc

$

d

$

dL

n

v

+

=

===((

(((

Lk

nk$

(= , iar 2(2(

(

= L

k

nk$

The caloric state 4uantities of polytropic transform are:

( )( )( )JTTmc!!TTmc

pm

vm

=

=(2(2

(2(2

/iving to n values in the interval 0 + 1 it can be obtained an infinity ofpolytropic transform; not all corespond to heat engine processes.

%perimentally it was determined that:

(1 The compression and e%pansion processes form heat engines are

accompanied by heat e%change of the agent to the environment; this heat

e%change is not so intense to reach an isothermal process.

21 Juring real compression and e%pansion processes the heat e%change

with the environment cannot be completely avoided and these processes do

not undertake perfectly adiabatic.


58/189

5onclusions : =eal compression and e%pansion processes can be considered

as polytropic processes situated between isothermal and adiabatic

transformation.

7rom the e4uation of real polytropic transform ttanconspVn =

isothermalctpVn == (

adiabatectpVkn k ==

3o in thermodynamic calculation it is of interest the study of polytropic

transformation having n coplying the ine4uation kn( , in whichk9(,...(,.The graphic of the transform versus n values is :

p 1

V

2a 2p%

!

2i*


59/189

7ig.(2. =epre&entation of polytropic e%pansion and compression incoordinate p#"

8bservations:

/iving particular values to VnF the simple transformations can be deduced

from polytropic transformation.

a1 (=n ctpVctpVn == =

=

=vn cic

n

knc

(

b1 kn= ctpVctpV kn == )(

=

= advn ccn

knc

c1 )=n i=obarActpctpVn == vpvn kcccn

knc =

=(

d1 =n i=ocorActVctVpctpV nn ===(

vn cc =

The specific heats for different polytropic transformation can be obtained

according to n values from diagram.

p

V

1

2i* 2p%! 2a


60/189

7ig.(.3pecific polytropic heat versus polytropic e%ponent

The polytropic curves in function of n if it isconsidered that all curves pass

through a given point.

a1 n 9 ) p 9 ct (#(

b1 n 9 ( T 9 ct 2#2

c1 n 9 k adiabatic #

d1 n 9 H \ " 9 ct #

e1 n 9 #( A#A

f1 #\]n]#( # [ntre 0#, A#A1

n

p

+

0 1


61/189

g1 #(]n]) @#@ [ntre 0A#A, (#(1

h1 (]n]k *#* [ntre 02#2, #1

7ig.(. olytropic curves in the hypothesis that pass through a given point.

7romthe point of view of the heat e%change the adiabatic splits the p#" area

in two &ones in any transformation which starts from a point of the

adiabatic and undertakes upwards the adiabatic , in &one +, the heat

e%change is positive.

p

1 1

2

2

3

3

5

5.

7

7

/

/

A

V

.


62/189

7ig.(A.The &ones of the heat e%change defined by adiabatic curve

+ )$ # e%ample: X (, 2, A

+n any transformation which starts from and undertakes under adiabatic

curve #X 0&one ++1 the heat e%chanege is negative.

++ )$ # e%ample: , #(X, 2X,AX.

*uestions

(. 6hich are the hypothesis of the ideal gas S

2. 6hich is the e4uation of state of ideal gas S 6hat is the

meaning of the 4uantities usedS

. 6hat are the laws of ideal gas S

. -ow is defined the adiabatic e%ponent k S

p

1 1

2 5 3

2

35

V

A


63/189

A. 6hich is $ayerXs e4uationsS

. 6hat is the e4uation of real gas S +n what conditions the real

gas is close to ideal gas S

@. 6hich are the hypothesis to gas mi%ture S

*. 6hat is Jalton law S 6hat is magat law S

C. 6hat is a simple transformation S 6hich are the simple

transformations of ideal gas S(). 6hich is the e4uation of isochoric transformation S 6hich is

its representation in p#" S

((. 6hich is the e4uation of isobaric transformation S 6hich is its

representation in p#" S

(2. 6hich is the e4uation of isothermal transformation S 6hich is


(. 6hich is the e4uation of adiabatic transformation S 6hich is


(. 6hich is the e4uation of polytropic transformation S 6hich isits representation in p#" S

(A. -ow is defined the polytropic e%ponent S

Course

Chapter !. Second la, of thermodynamics

!.1. Thermodynamic cycles

+t is called thermodynamic cycle 0or a cyclic thermodynamic process1 a

series of succesive thermodynamic processes 0or transformations1 which

undertake in such a way that at the end of last transformation the thermal

agent is brought in the initial state of the first transformation.

+f it is considered the cycle (2B(


64/189

7ig.(. The thermodynamic cycle in coordinates p#"

+t is called the work of the cycle the sum of all works 0c 9 mecL 1performed in the transformations which compose the cycle.

=== (

(

(

(

2(

LLLL3%

c

+=+=2

(

(22(

(

2%

3%

3

c LLLLL

# in transformation (##2 )dV and ))2( = %LpdVL

# in transformation 2#B#( )dVand )) (23LpdVL =

(O22(2( %areaL

% =

O2O((2(2

3areaL3 =

(2((22((22( 3%areaLLLLL 3%3%c ==+=

5onsidering the sign of work , the cycles are divided in:

a1 6ork producing cycles # )cL when cycle is perfomed clock wise,characteristic to energ" producingaggregates, such as internal combustionengines, gas turbines.

b1 6ork consuming cycles # )cL when cycle is perfomed anti clock wise0trigonometric1, characteristic to energ" consumingaggregates such as

compressors, refrigerating installations.

p

V

#1 2

A

B

2

1


65/189

7or an arbitrary cycle the points ( and 2 are in contact to 2 adiabates.

7ig.(@. The cycle placed between two adiabates

7or a heat engine working on this cycle, the first law of thermodynamics is

e%pressed:

L$pdV$d ==

+t is applied the formula for the cycle (2B(

(1 L$d =

21 )=d

1 = cLL

1 +=(

2

2

( 3%

$$$

+n which =2

(

2( )

%

%$$

p

V

#1 2

A

B

2

1

Aiabate

Aiabate


66/189

The transformation (2 begins on the first adiabate and undertakes upwards

0beyond1 the adiabate.

=

(

2

(2)

3

3$$

The transformation 2B( begins on the second adiabate and undertakes

downwards 0beneath1 the adiabate.

+t is noted heat amount received by the agent during the transformations of

the cycle L, )$+t is noted heat amount released by the agent during the transformations of

the cycle L), )) $

)$$$ += or )) $$L$$$ c ==

+n an work producing cycle performed by an ideal gas, only a part of the

heat received by the agent 0L1 is transformed in work c, the rest of the heat

being released to the environment during the rest of the cycle

transformations.

+t is called thermodynamic efficiency of the cycle the e%pression :

$

Lct=

$

$

$

$$t

))(=

=

The efficiency e%presses the thermodynamic 4uality of the cycle.

!.2.eversible and irreversible processes

s presented in chapter (., the thermodynamic processes can be divided in:

=eversibile process, in which the system passes from initial to final

state directly and reversely,e%actly through the same points, on the

same path.


67/189


68/189

only of thermal nature. The physicist notices that heat engines can not

transform totally absorbed heat in work. The second law asserts the

irreversibility of the natural processes e%plaining terms of conversion of

heat into work L$ and rounds the first law.

7irst law: 6ork turns into heat on the same e4uivalent ratio, meaning

$L and L$= ,but it does not say anything about the possibility ofreverse transformation, emphasi&ing Yust the e4uivalence.

3econd law says the possibility and the sense in which the processes are

undertaken.

6ork turns into heat $L spontaneously, integrally 0on its own # de la

sine1. -eat turns into work L$ by means of a heat engine in which

irreversible processes take place and L$ transformation is partial.

The second la,says that temperature differences between systems in

contact with each other tend to become e4ual and workcan be producedfrom non#e4uilibrium differen

termotehnica automotive engeneering

Documents