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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 energy transfer 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 aas on a wall, a macroscopic measure, as a
function microscopic measures :
23
2 2vm
V
Np
N – no. of molecules in volume V
m – mass of a molecule
2
2vm - mean kinetic energy of molecules in translation motion,
Maxwell – Boltzmann relation determines the connection between the temperature of a
gas and mean speed of molecules.
22
3 2vmkT
In which k is Boltzmann constant kJk /10.38,1 23
In other words pressure and temperatureare measures which features the group, the assembly of
molecules, not a given molecule.
The technical thermodynamicsstudies 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.Examples of thermalphenomena are as follows:
Heat transfer between two bodies having different temperatures;
Phase changes of substances ;
Transformation of heat in work and reverse in heatengines.
Main notions are thermodynamic system, state, process.
1.2 Thermodynamic systems
Generally a physical system is an area of universe made of substance or fields.
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The thermodynamic system is made of a body or an assembly of bodies having a finite number
of particles,which is limited fromsurroundings through a boundary surface and which interacts
energetically.The system changes substance (mass) or energy as heat or work with the
surroundings.
An 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.
An adiabatic or thermal insulated system does not change heat with the surroundings. A
mechanical insulated system does not change work with the surroundings.
Some thermodynamic analysis involves a flow of mass into or out of a device. So, we can
discuss about open systems. If there is no mass flow, the system is closed. The terms closed
system and open system are used as the equivalent of the terms system (fixed mass) and control
volume (involving a flow of mass).
The surface of a control volume is referred to as a control surface. Mass, as well as heat and
work, can flow across the control surface.
Thus, a system is defined when dealing with a fixed quantity 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. Each of the properties of a substance in a given state has only
one definite value and these properties always have the same values for a given state, regardless
of how the substance arrived at that state.
A state property can be defined as any quantity that depends on the state of the system and is
independent of the path (prior history) by which the system arrived at the given state.
Conversely, the state is specified or described by its properties.
Thermodynamic properties can be divided into two general classes: intensive and extensive
properties. An intensive property is independent of mass; the value of an extensive property
varies directly with the mass. Pressure, temperature and density are examples of intensive
properties. Mass and volume are examples of extensive properties. Extensive properties per unit
mass, such as specific volume, become intensive properties.
We 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 the property has significance for the entire system
and this implies what is called equilibrium. For example, if a gas is in thermal equilibrium, the
temperature will be the same throughout the entire system and we may speak of the temperature
as a property of the system. We may also consider mechanical equilibrium and this is related to
the pressure. A system is in mechanical equilibrium 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. If the
chemical composition of a system does not change with time, that system is in chemical
equilibrium. It means, no chemical reactions occur.
When a system is in equilibrium as regards all possible changes of state, we may say that the
system is in thermodynamic equilibrium.
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For 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.When a system passes from a state of thermodynamic
equilibrium 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 equation.
1.3.1.Specific volume
The specific volume of a substance is defined a s the ratio between volume V and mass m
being noted with v (m3/kg).
kg
m
m
Vv
3
The specific volume is inverse of density or specific mass 3/ mkg .
kg
mv
31
Volume of a kilomole of substance is called molar volume VM
kmol
mMv
M
m
V
n
VVM
3
The molar volume is the volume reported to number of kilomol of substances contained (n-
number of kilomol, M-molar mass).
The kilomol has two definitions :
- molar mass of a substance expressed in kilograms ( ex. 1 kmol de O2 = 32 kg as MO2 =
32kg/kmol) or
-a quantity of matter which contains inthe same conditions a number of molecules equal to
Avogadro number( 1 kmol = NA molecule = 6,023.1026
molecule).
1.3.2.Temperature
Although temperature is a property familiar to us , an exact definition of it is difficult.
From our experience 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. If these bodies remain in
contact for some time, they appear to have the same hotness or coldness.
Because of the difficulty in defining temperature, we define equality of temperature.
Consider two blocks of copper, one hot and the other cold, each one in contact with a mercury –
in – glass thermometer. If 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
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period of time no further changes in height are observed. It is said that both bodies have the
same temperature when they reached the thermal equilibrium.
The zeroth law of thermodynamics states that when two bodies have equality of temperature
with a third body, they in turn have equality of temperature with each other. It is based on the
transitivity of the thermal equilibrium. Two systems found in thermal equilibrium with a third
one are in thermal equilibrium between them (meaning that they have the same
temperature).Based on this property, a thermometer (or more general a thermometric body = a
body having a thermometric property) can compare different thermal equilibrium states of the
bodies.
Temperature represents a parameter of statewhich 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 is sensitive,
preferable as linear as possible, to temperature variation. For example when we measure the
temperature of our body with a mercury thermometer we measure in fact the expasion of the
mercury produced at the contact with the body,expansion quantified by the variation of height
of mercury column in the thermometer.Otherthermometric measures are variation of the length
of a metal rod, variation of the electric resistivity (thermometer is called thermo-resistance),
variation of the thermo-electromotor voltage (thermometer is called thermocouple), variation of
a gas in a constant volume enclosure ( gas thermometer), variation of light intensity of an
incandescence source (optic pyrometers).
Bimetal strip thermometer Glass thermometer
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Thermocouple Thermoresistance
Gas thermometer Pyrometer
For 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. Fixing the temperature
scale is called calibration and it can be done bringing thermometer in thermal contact with a
body in a reproducible state ( ex. water in thermal equilibrium with vapours , water in thermal
equilibrium with ice, melting point of some metals). To those perfect reproducible states are
associated accurate values of temperature.
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S2.
1.3.2.Temperature (continued)
Thermometers have a macroscopic property that changes considerably with
temperature (for ex., the pressure of a gas, the electric resistance, expansion
etc).
If a thermometer is brought in thermal contact with a hot body (H), at
equilibrium, the variable property (for ex. the length L) changes with LH
If the same thermometer is brought through thermal contact at the
equilibrium state of a cold body (C), the same property will change with LC
The difference between LH and L
C 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
and the cold body.
To estimate this difference, it is necessary to adopt a scale for temperature
measurements.
Based on the assumption that the temperature t and the property of the
thermometer ( for example the heigth of the mercury termometer, L) are
related linearly, it results:
dt=adLor t=aL+B.
If the thermometerisbrought in contact with a cold body, at tC
tC =aLC +B
and if the thermometerisbrought in contact with a hot body, at tH
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tH=aLH+B.
For anyintermediatetemperaturebetweentC and tH ,called t, corresponds a
length L. Substituting a and B itisobtained for anyintermediatetemperature t
.
t=tC+ OB
OB/CE=OA/AE -Thales theorem
OB=(OAxCE)/AE=(L-Lc)(tH -tC) /(LH-LC)
t =tC+ (L-Lc)(tH -tC) /(LH-LC), based on interpolation formula and similarity
of triangles.
This is the equation of any empirical temperature scale.An empirical
temperature scale is a scale based on the properties of a given substance. In
order to compare the measurements of different thermal states, these must be
submitted to the same thermal states. For example for empirical scale
Celsius there were adopted two fixed, easily reproducible points, based on
water properties, the ice point point and the steam point.
The temperature of the ice point is defined as the temperature of a mixture of
ice and water, which is in equilibrium with saturated air at a pressure of 1
atm.
The temperature of the steam point is the temperature of water and steam,
which are in equilibrium at a pressure of 1 atm.
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A line of equal ratios can be written which define different temperature
scales Celsius, Reaumur, Fahrenheit, Kelvin, Rankine:
(L-LC)/(LH-LC) = (t-tC)/(tH-tC) = t°C/100 = t°Re/80 = (t°F-32)/180 = (TK-
273.15)/100 = (TR-491.67)/180
or
t°C = 5t°Re/4 = 5(t°F-32)/9 = TK-273.15 = 5(TR-491.67)/9
On Celsius scale00C coresponds to a mixture water and ice at
equilibrium and 1000C to boiling water. A Celsius grade equals a Kelvin
grade, onlyorigins of scales are different.
There are several temperature scales different through the origin of the scale
or the magnitude of the unit (grade or degree). Celsius scale considers zero
value to the temperature at which water solidifies and 100 value to the water
vaporisation temperature, at normal atmospheric pressure Fahrenheit scale
considers 32 and respectively, 212°F to the same water temperatures (Some
historical sources said that zero Fahrenheit is the temperature of brine
solidification ( -17.8 °C) and 100° F (37.7° C) is the max. temperature of
human body). These two scales are defined based on well known and
reproducible temperatures.
In thermodynamics it was considered to find a temperature scale
independent of the properties of a particular substance. Such 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.If 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 (at constant volume)Or it
can be written that gas temperature T varies linearly with gas pressure p or
T = a + bp,
in which a, b are constants of the gas thermometer. Such a scale is called
temperature scale of ideal gas and can be determined measuring gas pressure
K C F
373,15K 100 C 212 F0 0
273,15K 0 C 32 F0 0
0K -273,15 C -460 F0 0
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at two reproducible temperatures such as water solidification and
vaporization at normal atmospheric pressure and the equation is determined
knowing a and b for Celsius scale. If the nature of the gas is changed from
A to B then similarly can be determined two points and another proportional
line like in fig.1.
Figure 1. Relation between pressure and temperature for gas thermometer
It can be noticed that no matter the nature of the gas the two lines intersects
in a point corresponding to zero pressure and correspondent temperature to
this point is -273.15°C.This temperature is the lowest reachable temperature
attributing zero value and the scale is called Kelvin scale. At this
temperature the gas molecules are no longer in motion The value of a
constant is zero and equation T = bp, meaning that it is enough one point to
define an absolute temperature scale. It 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 excepting very
low temperatures to which condensation appears and very high temperatures
to which dissociation and ionisation appear.
Relation between Kelvin scale and Celsius scale is
tC = TK–273.15 .
A degree Celsius isequal to a Kelvindegree,1°C=1 K,but the
origins of the scales are different.Fahrenheit scalediffersfromKelvinscale
and Celsius scalethrough the origin of the scale and magnitude of the
unit.Theconnectionbetween Fahrenheit and Celsius scalesisgiven by relation
:
tF = 1.8tC + 32 .
In the British units system thereis an
p
0
Gas A
Gas B
-273,15 t (°C )
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absolutetemperaturescalecalled Rankine in whichabsolutezeroisidentic to
absolutezero in Kelvin scale , but the magnitude of the degreeisdifferent
1R=1,8 K.
Relation of transformation between Rankine and Celsius degreesis
tR = tF + 459.67
A degree Rankine equals a degreeFahrenheit,1R=1°F,but the
origins are different.
In thermodynamiccalculationsitisusedabsolutetemperatureexpressed in
Kelvin.
Chapter 1.Basic concepts
1.3.3 Pressure
Pressure is defined as normal force applied on unit of surface, for a static
fluid,pressure gas the same value on any direction. For fluids it used
hydrostatic pressure expressed as hgp (ρ - density of the fluid, g-
gravitational acceleration, h - the height of the fluid column).
Pressure is classified according to the method of measurement in:
- Absolute pressure (p)– pressure measured reported to absolute
vacuum
- Relative pressure (pr) – pressure measured reported to atmospheric
pressure (pa)
ar ppp
The technical gauges measure relative pressure pr.
Manometers measure the increase of pressure from atmospheric pressure,
the pressure calledmanometric pressure (pmanom) or supra pressure,when
app ,and aman ppp
Vacuum-metersmeasure the decrease of pressure from atmospheric pressure,
pressure called vacuum-metric pressure or vacuum pressure pvor(pvac) when
app ,and
ppp avac
When app in calculation is usedalso another indicator called “vacuum”
expressed in percentage:
%100%100%a
vac
a
r
p
p
p
pvacuum
In other words the absolute pressure is measured reported ( or having as
reference ) the absolute vacuum :
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p=pa ± pr
in which it is considered the plus sign when relative pressure is an over
pressure called manometric pressure pr=pmand the minus sign when relative
pressure is a loss of pressure called vacuum pressure pr=pv.
Barometers measure absolute atmospheric pressure pbar.
In thermodynamic calculation, it is used the absolute pressure.
Relations between pressure units
Unit N/m2 bar kgf/m
2 at
kgf/cm2
atm torr
mm Hg mm H2O
1
N/m2=Pascal
1 10-5 0.102 0.102.10
-
4 0.987.10
-
5 750.10
-5 0.102
1 bar 105 1 0.102.10
5 1.02 0.987 750 0.10.105
1 kgf/m2 9.81 9.81.10
-5 1 10-4 9.68.10
-5 735.6.10-4 1
1 at
1kgf/cm2
9.81.104 0.981 10
4 1 0.968 735.6 104
1 atm 1.013.105 1.013 10.332 1.013 1 760 1.013.10
4
1 torr
1mm Hg 133.3 1.333.10
-
3 13.6 13.6.10
-4 1.32.10-3 1 13.6
1mmH2O 9.81 9.81.10-5 1 10
-4 9.68.10-5 735.6.10
-4 1
-A physical atmosphere 1 atm = 101325 N/m2 = 10.332.276 kgf/m
2 = 760 torr.
-1bar = 105N/m
2
-A technical atmosphere 1at =1kgf/cm2=9.81.10
4N/m
2 .
-Equivalent pressure of 1mm column of mercury is called torr
1 torr = 133.3223 N/m2 = 13.5951 kgf/m
2 = 13.15789.10
-4 atm.
-Equivalent pressure of 1mm column of water
1 mm H2O=9.81N/m2
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In order to compare the properties of gaseous substances they must be in the
same state of pressure and temperature. It was defined a standardized state
called normal physical state by:
barcmkgfmmHgpN 013.1/033.1760 2
CtorKT NN
0015.273
In thermodynamics is used a unit of volume measurement-normal cubic
meter 31 Nm which is a unit of volume but also a unit of mass, representing
the mass of gas contained in a volume of 1 m3in the conditions of normal
physical state ( at pNand TN).
There is also a tolerated unit called normal technical state defined by:
barcmkgfpn 981.0/1 2
CtKT nn
02015.293
1.3.4. Density (specific mass)
Density is the mass of unit of volume,being the reverse of specific volume.
3m
kg
V
m
In thermodynamics it is used also specific gravity γ defined as
3m
N
V
mg
or g .
1.4.Process measures. Work and heat
The thermodynamic process or state transformation is a physical
phenomenon in which the bodies exchange energy in form of heat and
mechanic work.As a consequence of energy variation the thermodynamic
system modifies its state of energetic balance meaning the modification of
thermodynamic state. A thermodynamic transformation means the passing of
a thermodynamic system from an initial equilibrium state to a final
equilibrium state,through continuous,successive, intermediate equilibrium
states. Any thermodynamic process is featured by specific measures called
process measures which depend on the path the system passes that mean
they depend on intermediate states of the system.The intermediate state can
be equilibrium states or not. Classification of thermodynamic processes can
be done after several criteria.
a. After relative variation of state parameters:
Differential processes or infinitesimal – for which the relative
variation of state parametersis very small;
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Finite processes when at least one parameter suffer a relatively high
variation.
b. After the nature of intermediate states:
Quasistatic processes (at equilibrium ), in which intermediate states
can be considered close enough to equilibrium states in every moment
of the process;
Non–static processes, in which intermediate states of the system
cannot be completely described from a thermodynamic point of view.
When a thermodynamic system leaves the equilibrium state, after a period of
time called time of relaxation, it recovers its initial state. When the
thermodynamic processes performs with smaller speeds than relaxation
speed, in any step of the process, the state parameters have values
corresponding to equilibrium state and it is said that the process is
quasistatic. Real processes are non –static and quasistatic processes are only
approximations of real processes.Quasistatic processes can be represented in
diagrams, for example in p-V ( pressure –volume), by means of a continuous
line between initial and final state,(fig.2a) andnon-static processes cannot be
represented like a continuous line because in the intemediate states which
are not in equilibrium the state parameters have not a unique value for the
whole system (fig.2b).
Fig.2 Representation of quasistatic (a) and non-static processes (b).
c. After the procedure of passing from the initial state (i)into final state (f)
and reverse, the thermodynamic processes can be divided in:
p
V
i
f
i
f
p
V
(a) (b)
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Reversibile process, in which the system passes from initial to final
state directly and reversely,exactly through the same points, on the
same path.
In order to perform such a process, the external conditions should modify
extremely slow so the system to adapt progresively to the new variations
which gradually appears;
Non reversible process, in which the system passes from initial to
final stateand reversed through different points,on other path.
Real processes cannot be considered reversible .A 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. After connection between initial and final state:
Cyclic processes when initial state is the same with final state;
Non cyclic processes (open), when initial state differs from final
state.
Work and heat are macroscopic formsof energy transfer between bodies
Work and heat do not feature the state of the system at a given moment (they
are not state parameters).Work and heat represent specific process measures.
Work and heat are not forms of energy, but forms of energy transfer.
1.4.1.Work
Let us consider a gas in a cylinder of an internal combustion engine which
expands and actuates upon the piston. Hitting the piston wall the molecules
modify axial 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.
Work sums at macroscopic level (piston motion) the effect of molecule
motions.
In mathematical calculations there are used three formulas for work. In
English literature the abbreviation of work is W,in Romanian one is L
(Lucrumechanic-Mechanic Work).
a) Work produced by state transformation (Boundary work)
It is considered an enclosure with gas at pressure p.Outside the enclosure
there is external pressure pe. In time interval dthe volume of gas is
increasing with dV. For an elementary surface dSfrom initial surface with
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the versor of normal direction and dn – the motion of dSon normal
direction.Integrating on the whole volume V it results the relation for
elementarywork L by variation of the volume as result of pressure forces.
v
e dSdnpL
dVpL e
Last formula expresses the mechanical work produced modifying the
volume of the fluid as a consequence of pressure forces.
Observations
1) Because workis not a state parameter,its elementary variation id not a
total differential -so δLrepresent a infinitesimal quantity of work .
Finite work released or consumed in a thermodynamic process when
passing from an initial state 1 ( parameters p1,V1, T1) to a final state 2 (
parameters p2,V2,T2) is noted with:
2
1
21 LL , never
2
1
12 LLL
2) In relation of work appears pe and dVpL e
If external pressure is identical with internal pressure ppe or dpppe ,
then pdVL (neglecting the infinitesimals of second order ).
These conditions are met when the processes are reversible and quasi-static.
In thermodynamic calculations all real processes are replaced with
equivalent quasi-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
As 0p then the sign of work is given by the sign of variation of elementary
volume:in expansion processes 0dV 0L 012 L , the performed work
towards exterior of the system is positive; similar, in compression processes,
0dV 0L 012 L ,and the work received by the system ( performed by
exterior upon the system) is negative.
Graphical representation of the processes - It is considered the expansion
process from figure, from initial statei to final state
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f.
Fig.3 Work of the state transformation
The work of the expansion process ispdV, in which :
pdVL
and δL is elementary work given by a current value of the pressure (pressure
is considered constant for a infinitesimal variation of volume dV)
2
1
21 pdVL
The work is equal to area under curve made with abscissa V.From graphic it
is noticed that transformation i-f or 1-2 can actuate also on other paths and
work of the transformation 1-2 could have different values according to
specific path (intermediate states ) on which the system works. In other
words in the transformation from state 1 to state 2, the work depends on the
path of the transformation.
b)Flow Work (work consumed to actuate a fluid)
Considering a pipe through which a fluid is flowing,if we imagine three
zones of the same lengthl at constant pressure p = constant.It is called work
consumed to actuate a volume of fluid V in an environment of constant
pressure p or flow work ,the product pVpSlLd
Ld – work consumed to actuate a volume of fluid V at constant pressure p.
The fluid from area I actuates upon the fluid from area II,the fluid from
area II actuates upon the fluid from area III and so on resulting the motion of
the fluid. This type of work does not increase fluid energy, Ldcontributes
p
i
pi
f pe
V
Vi Vf
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only tothe increase of energy of the fluid accumulated in the reservoir at the
end of the flow pipe.
One of the forms of energy applied to a fluid is enthalpy, I, being the sum
between U, internal energyand product pV.
JpVUIenthalpy .
c)Shaft 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 exhaust processes into and
outside engine. It is considered the same source of working agent which
enters and leave the engine.
evadmt LLpdVL 2
1
releasedVpworkflowLL
receivedVpworkflowLL
dev
dadm
22
"
11
'
.
.
2
1
2
1
2
1
1122
2
1
VdppVdpdVVpVppdVLt
2
1
VdpLt - shaft work is the total work produced or consumed by a
working agent in a heat engine.The shaft work is equivalent to area between
the graphic of state transformation and coordinate axis of pressure p.
Fig.3. Shaft work of a state transformation
p
i
pi
f pe
V
Vi Vf
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1.4.2.Heat exchange
Heat is classified in sensible heat and latent heat. Sensible heat is related to changes
in temperature of a gas or object with no change in phase (I).Latent heat is related to
changes in phase between liquids, gases, and solids (II).
Heat is a form of macroscopic transfer of energy, generally produced between two
bodies with different temperatureswithout mechanical interactions.
What is called exchange of energy as heat at macroscopic scale is an exchange of
molecular kinetic energy at microscopic level.
When 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 (sensible heat) and there is an exchange of kinetic energy from
gas to water. Heat is a disordered form of energy transfer – the flame contains highly
activated molecules.
The heat exchange inan elementary process is expressed mcdtQ
in which m – mass of the body , c- real specific heat and dt- difference of
temperature.
For a chemical process
2
1
2
1
2
1
21
t
t
t
t
cdtmmcdtQQ
As heat is not a parameter of state (it is not a form of energy,but a form of transfer of
energy), Q is not a total exact differential.
Heat is considered positive when the system receives energy from environment and
negative when the system releases energy to the environment.
1.5.Specific heats
(I).It was experimentally noticed that in order to heat (or cool) different bodies with
the same number of degrees are required different heat quantities. So 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 Q in an elementary
process and the corresponding variation of its temperature dT,
.
Caloric capacity can be also defined as the physical quantity of heat absorbed by a
body in order to modify its temperature with 1 unit (1 grade). Unit of measure is J/K.
Specific 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 ( at constant pressure or at constant
volume). Specific heat or the caloric capacity of unit of mass is the physical measure
numerically equal to sensible heat quantity exchanged by unit of mass of a body with
the surroundings in order to modify its temperature with 1 unit. Between specific heat
c and caloric capacity C, there is the following relation: C = mc
dT
dQC
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Specific heat can be classified according to unit of substance reported as follows:
a) Specific heat reported to 1 kg of mass (mass specific heat)
mcdtQ
kgK
J
tm
Qc
with m, the mass of the body expressed in kg, ∆t –is temperature variation of the
body, in degrees .
b) Specific heat reported to 1 kmol of substance (molar specific heat )
dtncQ
kmolK
J
tn
Qc
M
M
with n, number of kilomoles , ∆t –is temperature variation of the body,in degrees.
c) Specific heatreported to 31 Nm
dtCVQ
Km
J
tV
QC
NN
NN
N
3
with VN, volume expressed in normal state, ∆t –is temperature variation of the body .
A gas can be heated ( or cooled) in several ways, keeping some parameters constant.
A gas can be heated at constant volume or at constant pressure. Experiments 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 (of the same amount of gas
for the same difference of temperature). In other words a gas can have two specific
heats according to the nature of the process:
- Specific heat at constant pressure (marked with index p)
Km
JC
kmolK
Jc
kgK
Jc
N
NpMp p 3;;
- Specific heat at constant volume (marked with index v)
Km
JC
kmolK
Jc
kgK
Jc
N
NVMV V 3;;
For gases , cp and cv have different values (cp>cv) but to solid and liquid substances
the difference between values is very small and is neglected.
There are the following equivalent relations between the three types of specific heats
AvogadromkmollCc
kgMkmolMcc
NNM
M
3414.22414.22
1
The expressions above can be explained like this:
Molar specific heat is equal to the product between molar mass of the gas (M) and
mass specific heat (c).
Molar specific heat is equal to the product between the constant 22.414 and specific
heatreported to 31 Nm (CN).
Page 20
Specific heat of bodies increases with the increase of temperature the variation c=c(t)
can be done graphically or analytically.
Analytically c is made of a sum of polynomials which follow the form of graphic:
tbac
linearforor
gtftetdtbtac
11
3322
..
.......
The specific heat values depend on the nature of substances and vary with
temperature;their values are measured and can be found in thermodynamic tables.
( http://www.engineeringtoolbox.com/specific-heat-solids-d_154.html,http://www.engineeringtoolbox.com/specific-
heat-fluids-d_151.html,http://www.engineeringtoolbox.com/spesific-heat-capacity-gases-d_159.html)
Example Air properties
Temperature
(oC)
Density
(kg/m3)
Specific heat -
cp -
(kJ/kg.K)
-100 1.980 1.009
0 1.293 1.005
40 1.127 1.005
80 1.000 1.009
140 0.854 1.013
180 0.779 1.022
250 0.675 1.034
300 0.616 1.047
400 0.524 1.068
a) In order to select c values according to temperature variation it is considered the
linear variation of VpMMVp CCcccc
Vp,,,,,
b) It can be considered also average (mean) values of c given between t0 and t.
tfcc t
tpmp 0
Considering the hypothesis of linear variation
Page 21
2
21
0
ttttccc m
t
tm
or12
121
0
2
0
tt
ctctc
t
t
t
t
m
In thermodynamic applications, to make the calculations easier specific heats are
approximated to average values between two temperatures.
II.Latent heats
The word latent comes from Latin latere, meaning to lie hidden. There are thermal
processes in which even the heat is transmitted to the body, its temperature does not
vary (ex. melting or vaporization)so the heat exchange is not sensitive, is latent
(hidden) and it cannot be measured with a thermometer. In 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 ( latent heat
of vaporization, latent heat of fusion)
,
This coefficient is defined as heat quantity required for changing the phase of the unit
of mass from a substance,at a constant temperature and pressure. The unit is
J/kg,being an intensive measure. For the same substance the latent heat of
vaporization is equal to latent heat of condensation and the latent heat of fusion is
equal to latent heat of solidification.
m
Q
Page 22
Questions
1. What studies thermodynamics ? Give examples of thermal phenomena.
2. What is a physic system?What is a thermodynamic system ?
3. What is an isolated system?
4. What is the difference between an open and a closed system ?
5. How do you express the state of a system ?
6. What is thermodynamic equilibrium ?
7. What are extensive parameters ? What are intensive parameters ?
8. How is classified pressure according to measurement method ?
9. What type of pressure is measured with manometers and vacuum-meters ?
10. What type of pressure is measured with barometer ?
11. What property is described by temperature?
12. What is a thermal measure (or quantity) ? Do you have some examples ?
13. Can be directly measured the temperature of a body ? Why ?
14. How do you enounce the zeroth law of thermodynamics?
15. What is a scale of temperature ?
16. How do you classify the scales of temperature and which are they?
17. Which are the relations between the origins and units of the scales ?
18. How was determined the lowest temperature and what is its meaning ?
19. How is defined the specific volume?
20. How is defined the molar volume ?How is defined the kilomole?
21. Which are normal physical gas state ?
22. How is defined the density ? How is defined the specific gravity ?
23. What are finite thermodynamic processes ?
24. What is the difference between quasi-static and non-static processes?
25. When a process is reversible ?When a process is irreversible ?
26. What is mechanic work ?
Page 23
27. How is classified work in thermodynamics ?
28. In p-V representation of a transformation of state of a gas which is the
significance of the boundary work ( work of the state transformation ) But of
the shaft work ?
29. Which is the sign rule for work?
30. What is heat ?Is a quantity of state or a process?
31. Which is the sign rule for heat ?
32. How is expressed the heat change which produces the heating of a body ?
33. How is expressed the heat change which produces the change of phase of a
body ?
34. What is specific heat ? How is reported to different units of mass ?
35. Is specific heat constant with temperature ?
Chapter . 2. First law of thermodynamics
Some calculations and experiments performed in the XIX th century demostrated that
mechanical work and any other form of energy can be transformed in heat and
reversed and it was determined the equivalency ratio of transformation.In technical
system heat is expressed in kilocalories ( a calorieis the amount of heat (energy)
required to raise the temperature of one gram of water by 1 °C)and work in
kgf.m(work produced moving a body of 1 kg on a length of 1 m),those units being in
that period considered as independent.
In 1842 Robert Mayer introduces the mechanical equivalent of heat unit and
determined its value by calculations, in the same year Joule determined the caloric
equivalent of work and Helmholtz demostratedthe equivalence between the thermal
and mechanical energy.
Lecture--The mechanical equivalent of heat
Joule's Heat Apparatus, 1845, Joule's apparatus for measuring the mechanical equivalent of heat
Page 24
Further experiments and measurements by Joule led him to estimate the mechanical equivalent of
heat as 838 ft·lbf of work to raise the temperature of a pound of water by one degree Fahrenheit.
He announced his results at a meeting of the chemical section of the British Association for the
Advancement of Science in Cork in 1843 and was met by silence.
Joule was undaunted and started to seek a purely mechanical demonstration of the conversion of
work into heat. By forcing water through a perforated cylinder, he was able to measure the slight
viscous heating of the fluid. He obtained a mechanical equivalent of 770 ft·lbf/Btu (4.14 J/cal). The
fact that the values obtained both by electrical and purely mechanical means were in agreement to
at least one order of magnitude was, to Joule, compelling evidence of the reality of the
convertibility of work into heat.
Joule now tried a third route. He measured the heat generated against the work done in
compressing a gas. He obtained a mechanical equivalent of 823 ft·lbf/Btu (4.43 J/cal). In 1845,
Joule read his paper On the mechanical equivalent of heat to the British Association meeting in
Cambridge. In this work, he reported his best-known experiment, involving the use of a falling
weight to spin a paddle-wheel in an insulated barrel of water, whose increased temperature he
measured. He now estimated a mechanical equivalent of 819 ft·lbf/Btu (4.41 J/cal).
In 1850, Joule published a refined measurement of 772.692 ft·lbf/Btu (4.159 J/cal), closer to
twentieth century estimates.
An important contribution had C.Miculescu, a Romanian physicist who established
the value of mechanical equivalent of the calory : 1kcal=4185,7 J, a value very close
to the closest value 1kcal=4185,5 J.
2.1.Internal Energy
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. First law is based on a state measure called internal
energy.
A body, which in thermodynamics is called thermodynamic system, is made of very
high, but finite number of particles in continuous, disordered motion, which interact
amongst them. It means that the particles have a kinetic energy corresponding to
thermal, disordered motion and a potential energy due to forces of interaction
between them (intermoleculare forces) and due to interaction with other external
forces ( ex. gravitational field). All these energies form internal energy of the system.
So internal energy of a system is made of kinetic energies corespunding to particle
macroscopic motions aswell as potential energy of interaction of particles.
Internal energy represents the sum of kinetic and potential energies of the particles
within a body and of the energies within the molecules (ex. energy of chemical
bonds, inter and intra atomic).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.Internal energy is noted with U and for
thermodynamic processes is the sum of kinetic and potential energy of molecules.
Page 25
noscillatioUrotationUtrasitionUU
UUU
cincincincin
potcin
Molecules of liquid and gas may have translation and rotation motions; in the
molecules the groups of atoms have oscilation motions.
For example, 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 E0.
Internal energy kcalJU , is a state measure or quantity meaning that it depends only
by the state of the system. When a system passes from a state having U1internal
energy to another state having U2internal energy, no matter if the process is reversible
or not, variation U=U2-U1of internal energy does not depend on intermediate states
throgh which the system passed,it depends only on the initial and final states ( their
internal energies).Internal energy is an additive measure meaning that the internal
energy of a system is equal 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
muU /
2.2.Enouncements of first law of thermodynamics
On the basis of law there was the experimental 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 expasion, when work is
transformed in electric energy and then into heat.
a) „Heat could be obtained from work and it can be transformed into work always in
the same equivalence ratio.”
If in thermodynamic relationships appear heat expressed in kcal and work expressed
in Jouli or kgfm,in order to have homogeneus formula it must expressed the ratio
beween heat and work as the caloric equivalent of unit for work A
Q/L=A , A-caloric equivalent of unit for work.
mkgfkcal 42786,4261
kgfm
kcalA
427
1
If kgfmLkcalQ
ALQ in technical system
b) „It can not be produced a heat engine in continuos operation to produce work L,
without consuming an equivalent quantity of heatQ.”
Page 26
c) Oswald:
“Perpetual motion machine of the first kind does not exist.”A perpetual motion
machine of the first kind is a machine which produces more work L than equivalent
heat Q,thus meaning it produces energy from nothing; in this way,it violates the law
of conservation of energy.
In a thermodynamic process the variation of internal energy of the system equals the
sum of mechanic equivalents of all energy changes between the system and
surroundings.Any form of energy can be expressed through mechanical equivalent,J.
2.3. Mathematical formulation of first law for open systems
An open system is a thermodynamic system changing energy and mass with the
surroundings.
For a heat engine (a device that converts heat energy into mechanical energy or more
exactly a system which operates continuously and only heat and work may pass
across its boundaries) is expressed the energy balance of the system for period of
time, meaning the balance energy transfer forms and mechanical and thermal
energies.
Fig.4.Scheme of energy changes in a heat engine
It is consider a heat engine from fig. 4 in which point 1 represents the intake of
thermal agent and point 2 represents the exhaust of thermal agent .The heat engine is
supplied by fuel which is burned releasing heat Q1-2; the heat engine produced shaft
work noted Lt1-2. The thermal agent in point 1 has pressure p1,temperature T1, specific
volume v1 , specific internal energy u1and specific enthalpy i1and it gets into the
engine withw1velocity level difference h1.
Heat engine
h1 h2
Reference plan
Q 1-2 L t1-2
1
2
w1
w2
Page 27
The thermal agent in point 2 has pressure p2, temperature T2, specific volume v2 ,
specific internal energy u2şi specific enthalpy i2and it gets out engine
withw2velocity at a a level difference h2.
Masic balance equation written between points 1 and 2 indicates mass conservation
m1=m2= m .
Energy balance equation written on the control area between points 1 and 2 is:
releasedreceivedEEEE
21
For a time interval
Jmumghmw
E 11
2
11
2
Jmumghmw
E 22
2
22
2
1121VpQE
received
3
11 mmvV
22VpLE
treleased 3
22 mmvV
22Vp - flow work
Replacing in energy balance:
2222
2
2112111
2
1
22VpLmumgh
mwVpQmumgh
mwt
kg
J
m
Ll
kg
J
m
Qq
tt
2121
Replacing in energy balancefor 1 kgof thermal agent passing through the engine
kg
Jlqvpvphhg
wwuu t21112212
2
1
2
212
2
1’) tlqhhgww
ii
2112
2
1
2
212
2
In which pvui
Diferentiating it is obtained
1)
kg
Jlqdhg
wddi t 21
2
2
In which 2
1
vdplt
Relation 1)has a general character and can be applied in any open system having L
and Q as forms of energy transfer.The second equation is true for mechanical energy
transfers (pumps, etc.).
2.4.Mathematical formulation of first law for closed systems
Page 28
A closed system is a system which do not change mass with the surroundings,for
example the gas from the cylinder of a piston engine.For equations (1) and (1’) from
aforementioned chapter for open systems,considering intake and exhaust velocities
zero and the same value of reference levels.
21
21 0
hh
ww
tlqvpvpuu 21112212
It is obtained
2
1
2
1
2112 pvdvdpquu
3)
2
1
21212112 lqpdVquu
Or for m kilos of thermal agent
212112 LQUU
From 1’ (2’) tlqii 2112
and JLQII t 2112 for m kilos of agent in which 2
1
JVdpLt .
The mathematical expressions in differential form of the first law of
thermodynamics:
4) kgpentruvdpqdi
pdvqdu1
and kgmpentru
VdpQdI
pdVQdU
The mathematical expression of the first law of thermodynamicsfor closed systems JLQUU 212112
favorizes the following hydraulic interpretation and analogy.
Fig.5. Hydraulic analogy of first law for close systems
The analogy emphasizes that internal energy of the thermal agent varies in function
of value and sign of heat and work agent changes with the surroundings.
Special cases :
U
L 1-2
Q 1-2
Page 29
A.For adiabatic processes 021 Q the first law becomes a relation between U and
L.When system does not receive energy from exterior, meaning that is adiabatically
isolated, then it could perform work only on the variation of internal energy, Q=0
resultingL = -dU and in this situation work does not depend on intermediate states,
meaning that in this particular situation work is a total diferential (dL=-dU).
B. For isochoric processes (V = ct.) with variation of volume zero,work of the
isochoric transformation is zero 0izL and first law becomes a relation between U
and Q.If the system does not perform work upon exterior and exterior does not
perform work upon the system, the heat received by the system from exterior
determines an increase of its internal energy and L=pdV=0 and dQ = dU or
Q1-2 = U2 - U1 . In the isochoric process the heat is a total differential.
C.When
2121
21
21
).(tan...0
...0
LQ
processisothermaltconsUworkpeformsagenttheL
heatreceivesagenttheQ
When agent receives heat and performs work, if these quantities are equal,internal
energy remains constant.
In other words when internal energy does not change during its interaction with
the environment, then system cannot perform work unless it receives energy from
exterior.For dU=0,it is obtainedL = Q or Q1-2 = L1-2.
2.5 Caloric equation of state
The quantities intenal energy U and enthalpyIare called caloric state quantities
reprezenting thermal forms of energy.
Rule “The state of thermal equilibrium of a system is completely determined if are
known two intensive state parameters and masses mj of components of the system”.
Intensive parameters are pressure, temperature,specific volume.
It is considered a monocomponent system (1 body ), having mass of 1 kgfor which
any state quantity can be expressed in function of two intensive parameters.
It is expressed: pTfi
vTfu
,
,
1
.By differentiation
dpp
idT
T
iid
dvv
udT
T
udu
Tp
Tv
For 1 kilo of agent which suffers an elementar heating at constant volume: dTcq vv
According to first law for closed systems:
pdVqdu for dTcduandqdutconsvvvvv tan
Page 30
vtfU ,
v
vT
Uc
Similarly results :
p
pT
ic
Replacing in duand di results
dpp
idTcid
dvv
udTcdu
T
p
T
v
analog with capital letters for m kg :
dpp
IdTmcId
dVv
UdTmcdU
T
p
T
v
These equations are called caloric equations of state.
Questions
1. Is the work performed by a system a form of energy exchange ?What about the
heat change ?
2. Can be mechanic work converted into heat ? Can be heat converted into work?
3. What are units for work and heat ?
4. What is internal energy of a system ?
5. Which is the enouncement of the first law of thermodynamics ?
6. Which is the enouncement of the first law of thermodynamics for closed
systems?
7. Which is the hydraulic analogy of the first law of thermodynamics for closed
systems ?
8. Which are the caloric equations of state?
Page 31
Chapter 3. The ideal gas
Ideal gas is a hypotetic notion - it represents a gaseous body having the following
properties:
- molecules are perfectly spherical;
- molecules are perfectly elastic;
- molecules have nointeraction;
- molecules’ own volume can be neglected;
The perfect gas is also called ideal gas and, according to the kinetic–moleculartheory,
it cannot be liquefied.
Relations expressing the properties of a perfectgas are:
a) the expression of a gas’ pressure (Bernoulli)
23
2 2mw
V
Np
b) the expression of kinetic energy Ecas a function of the velocity distributionw
22
3 2mwkT (Maxwell Boltzmann)
m –molecule mass
N – number of molecules in V
Out of the two relations, 2610.38,1 where, kkNTpV -the Boltzmann constant
For a constant mass of gas (N = constant)it results :
constantT
pV - the ecuation of state for a hypotetical perfect gas.
In certain pressure and temperature conditions, gases in nature almost obey the
rigurousrelations for the hypotetical perfect gas: these generic conditions consist of
small and medium pressures and medium and high temperatures – so states that are
far enough from the liquefying point.
Gases in the nature that are in such conditions can be considered perfect gases; the
simple laws established in XVII-XIXcenturies – that are not rigurously correct –were
determined by experiments on gaseous bodies in the nature, in pressure and
temperature conditions far away of liquid states, thus obtaining the laws of perfect
gases. The approximation of the simple laws of gases is sometimes under the errors
introduced by mathematical models of the phenomena.
3.1.Laws of the perfect gas
For a constant mass (kg) of perfect gas, these laws are as follows:
a) Thermal state equation
Page 32
mRTpVctT
pV or
- the variables are expressed in different measurement units:
kmolTRpV
nkmolTnRpV
mkgmRTpV
kgRTpV
M
M
1
1
R is called the constant of the gas. Its value doesn’t depend on its status, but only on
its nature and thermal properties.
For a kmol of gas, the state equation becomes:
TRpV
MRTpvM
MM
, where RM is the the universal constant of the perfect gas, being
independent of the natureof the gas and having a value that can be computed out of
the state equation of the gas in normal physical conditions.
kmolK
J
T
VpR
N
MNN
M 4.831415.273
414.22101325
and
Kkg
J
MM
RR M
.
4.8314
2m
Np ,
kg
mvmV
33 , KT ,
kmolnkgm ,
kgK
JR
b)Boyle Mariotte law
When constantT then constantpV
c) Gay – Lussac law
When constantp then constantT
V
d) Charles law
When constantV then constantT
p
e) Avogadro law(1811)
At the same pressure p and temperature T, in equalvolumesthey are the same numbers
of molecules.
Page 33
For gases 1 and 2 :
21 pp 21 TT 21 VV 21 NN - numbers of molecules
1 kmol contains NA= 6.023 · 1026
molecules
3414.221 Nmkmol
In normalstate, 1 kmol occupies the same volumeV.
f) Joule law
Internal energy of perfect gases depends only on temperature.
TUU and 0
p
U0
V
U (independence on p or V)
Entalpyof perfect gases depends only on T.
mRTUpVUI TfI 1
Consequences:
- general expressions of caloric state equations:
dVV
UdTmcdU
T
v
dpp
IdTmcdI
T
p
For the perfect gases:
0
V
Uaccording to Joule law
0
p
Uaccording to Joule law.
We obtain the following expressions of the caloric state equations for perfect gases:
dTmcdI
dTmcdU
p
v
For 1 kg
dTcdi
dTcdu
p
v
and, integrated,
J
TTmcII
TTmcUU
pm
vm
1212
1212
Page 34
3.2.Specific heat of perfect gases
The ratio of specific heats is calledadiabaticexponent and is noted as:
kC
C
c
c
c
c
Nv
Np
Mv
Mp
v
p
MccM
NM Cc 414.22
It can also be written RTupvui and, in diferential form,
RdTdudi or RdTdTcdTc vp
kgK
JRcc vp - Mayer relation
MRMcMc vp or
kmolK
JRcc MMM vp
RMRM
kgK
JR
k
kc
kgK
J
k
Rc
Rcc
kc
c
p
v
vp
v
p
1
1
kmolK
JR
k
kc
kmolK
J
k
Rc
MMp
MMv
1
1
One can notice that specific heats of perfectgas can be determined as a function of
adiabatic exponent k and of the constants of the gas.
For the hypotetic perfect gas, specific heats areconsidered constant and k = constant.
Specific heats of gases considered as perfectcan vary with temperature.Experiments
show that, for these gases, kstaysconstant in large intervals of temperature. For first
approximation computation, the following values are adopted:
- for monoatomic gases (He),k = 1.66
- for biatomic gases(N2, O2, H2, CO),k = 1.4
- for polyatomic gases(CO2, SO2….),k = 1.33
The hypotetic perfect gaswith punctiform moleculesbehave like a monoatomic gas,
with c constant and k constant. For gases of the nature with two or more atoms in the
molecule, the degrees of freedom in molecules’ movement appear progressively with
the increase of temperature.
- for low temperatures – only translation
- for medium temperatures (e.g. atmospheric temperature) –translation + rotation
- for high temperatures–translation + rotation+ oscillation.
Page 35
As a consequence of progressing in rotation and, then, in oscillationof molecule’s
atoms, specific heats increase with temperature. For termotechnical computation, in
the first approximation, for biatomic gas the adopted value isk = 1.4.
3.3 Real gases
The differences between ideal and real gas can be described as follows:
- a gas behaves like an ideal gas when it is very rarefied – in this case, the own
volume of molecules – versus the whole volume occupied by the gas – can be
neglected and molecular interraction forces are very low due to large distances
between molecules.
- a gas behaves like a real gas at high pressures and low temperatures, when neither
own molecules’ volume nor molecular interraction forces can be neglected.
The status equation for a kilomol ideal gas, pv = RT , becomes for the real gas:
RTbvv
ap
2, (Van de Waalsequation), wher a and b aretwo constants.
The real gas is different from the ideal gas (whose internal energy depends only on
temperature), as the internal energy of the real gas dependsalso on volume, so
U = U(T) - for the ideal gas
U = U(T, V) - for the real gas
In ideal gases, molecules don’t 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 gasconsists of the kinetic energy and potentialenergy of the
molecules, due to their interraction – the potentialenergy depends on the distance
between molecules, so, on the volume of the gas.
3.4. Mixtures of perfect gases
In most installations, the gaseous thermal agents are not pure gases, but mixtures of
gases.
Examples:
- the air is a mixture of gases, mainly 22 ON
- exhaust gases consist of .....22222 CONOSOOHCO vap
A mixture of perfect gases behave like a perfect gas:
mRTpV
Page 36
For the study of gaseous mixtures, they are two hypoteses:
a) mixed gases - molecules of each gas spread in whole the volume.
Fig.6. Mixture of molecules in the whole volume
Given three different gases, A, B and C, with molecules of different sizes, it can be
considered that they occupy all the available volume:
VVVV CBA
Temperature of the components can be considered as equal with that of the mixture:
TTTT CBA
Partial pressure is the pressure of a gas on the enclosure walls if it should be
alone.For the chosen three gases, partial pressures are not equal:
CBA ppp - partial pressures
The pressure of the mixture is the sum of the partial pressures of component
gases: pppp CBA - Dalton’s law
The sum of component masses is the mass of the mixture:
kgmmmm CBA
They are called mass fractions (concentrations) the ratios:
m
mg A
A m
mg B
B m
mg C
C 1n
i
ig
b) component gases are separated by imaginary walls, each having the same
pressure p and temperature Tas the mixture.
Fig.7.Mixture of molecules in partial (imaginary) volumes
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
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 A B B C A A
B B C A A C B B
C
Page 37
For this hypothesis pppp CBA
TTTT CBA
CBA 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 equal to the volume of the
mixture. VVVV CBA
- Amagat law
There are called volumic fractions the ratios
V
Vr AA
V
Vr BB
V
Vr CC 1
n
i
ir
Properties of ideal gas mixtures
1. Constant R of the mixture .
It is considered the mixture made of gases A,B and C in conditions described in
paragraph a). For every component it can be written the equation of state: TRmVp AAA
TRmVp BBB
TRmVp CCC
Summing the equations, it results:
TRmRmRmpV CCBBAA
mRTpV equation of state of the mixture
CCBBAA RmRmRmmR and dividing by m, it results
i
n
i
iCCBBAACC
BB
AA RgRgRgRgR
m
mR
m
mR
m
mR
R constant of the mixture is equal to he sum of the gas constants of the components,
weighted with the mass fractions.
2.Specific mass of the mixture
V
m (density)
Considering the hypothesis from paragraph b, it can be written
CCBBAACBA VVVmmmm and dividing to volume V
n
i
iiCCBBAA rrrr
Density of a mixture is equal to the sum of gas densities weighted with the volume
fractions.
Page 38
3.Specific volume of the mixture v
ii
iiivg
m
vm
m
V
m
Vv
in whichp
TRv i
i are specific volumes of the components.
4. Molar mass (weight) of the mixture M Molar mass (weight) of the mixture is called conventional mass because it does not
have a measurable value in real life. If it is considered the density of the mixture and
components
n
i
iir in which
MTR
p
RT
p
V
m
MTR
p
TR
p
V
m
M
i
Mii
ii
It results the equality TR
p
M
n
i
iii
M
i MrMMTR
prM
Molar mass of the mixture M equals the sum of molar masses of the components
weighted to volume fractions.
5.Partial pressures of the gas components
For component i it is expressed the equations of state which correspond to both
hypothesis of the mixtures which are considered to be equal: TRmVp iii
TRmVp iii
prpV
Vp i
ii
The partial pressure of a component is qual to the pressure of the mixture p weighted
with the volume fraction of the component ri.
For example, for air at atmospheric pressure considered close to 1 bar:
bar21.0
bar79.0
2
2
O
N
p
p
6.Specific heat of the mixture
It is considered a mixture which suffers an elementary heating process at constant
pressure or volume;heat change is expressed
mcdtQ in which specific heat c is either cp or cv.
Heat received by the mixture equals the sum of the heats received by every
component.
dtcmQiQ ii
n
i
ii
n
i
ii cgcm
mc
Page 39
Similarly, it can be demonstrated that
iMiM crc
iNiNCrC
7.Conversion of the fractions
The mixture composition is expressed through mass or volume fractions.
By definition :
i
iii
m
m
m
mg
- In hypothesis b) TM
RmTRmpV
i
Miiii
VMV
TR
p
MVTR
p
g
ii
M
ii
Mi
1
ii
iii
Mr
Mrg
By definition :
i
iii
V
V
V
Vr
m
M
m
p
TR
M
m
p
TR
r
i
iM
i
iM
i
1
i
i
i
i
i
M
g
M
g
r
Example for air:
%77%79
%23%21
2
22
2 NN
OO
gr
gr
23.02879.03221.0
3221.0
2222
22
2
xx
x
MrMr
Mrg
NNoo
ooO
77.02879.03221.0
2879.0
2222
22
2
xx
x
MrMr
Mrg
NNoo
NNN
Page 40
8. Mean temperature of a mixture
There are considered several gases initially at different temperatures and it is required
the final temperature of the mixture, after mixing process. Fom equation of heat
balance considering that a component releases heat and the others absorb heat : 333222111 TTcmTTcmTTcm
The temperature of the mixture is T
ii
iii
cm
Tcm
cmcmcm
TcmTcmTcmT
332211
333222111
Page 41
Chapter 3. The ideal gas (continued)
3.5.Thermodynamic processes applied to the ideal gas
When a heat engine is designed, the real processes of heating, cooling, compressing
and expanding are replaced with one or more simple thermodynamic processes or
transformations.
- The thermodynamic process which takes place at constant volume is called
isochoric transformation (V = constant)
- The thermodynamic process which takes place at constant pressure is called
isobaric transformation (p = constant)
- The thermodynamic process which takes place at constant temperature is called
isothermal transformation (T = constant)
- The thermodynamic process which takes place without heat exchange with the
surroundings is called adiabatic transformation (Q =0)
- The thermodynamic process which takes place with variation of all parameters of
state, in the condition of a constant specific heat of the process (cn =constant) is
called polytropic transformation.
For each type of transformation it is necessary to know the equation of the
transformation ( relation between the parameters of state in initial and final points) ,
mechanic work variation , heat exchange and graphical representation, typically in p-
V coordinates.
3.5.1. Isochoric transformation
The isochoric transformation takes place at constant volume V= constant ,
( ttanconsm ), from equation of state tconsT
pVtan it results if V is constant
ttanconsT
p (equation of the transformation ) or
1
2
1
2
T
T
p
p
The mechanical work of the isochoric transformation is
2
1
2
1
21 tan 0 tconsVaspdVLL
Heat transfer of the isochoric transformation is
2
1
1221
2
1
T
T
vv TTmcdtmcQQm
Caloric equations of state : 211212 QTTmcUU
mv
1212 TTmcII pm
Page 42
Graphical representation of the transformation in p-Vcoordinates is like in fig. 8,
where the transition from state 1 to state 2 is characterised by keeping the volume
constant: VVV 21
Fig. 8. Plot of the isochoric transformation in p-V coordinates
As the p/Tratio stays constant with pressure increase, e.g. passing from state 1 to state
2 (p2>p1), the temperature would increase, (T2>T1) resulting a heating of the gas. If
the process inverts, from state 2 to state 1, the gas would cool as the pressure
decreases.
The physical model of a thermodynamic system undergoing such a transformation is
that of a fluid container with fixed exterior walls. Heating by an external source gives
an increase of the temperature and of the pressure in the container.
3.5.2.The isobarictransform
The isobaric transformis characterized by a constant pressure, p= constant ,
( ttanconsm ), inthe state equation tconsT
pVtan . It results, for p constant,
tconsT
Vtan (the equation of the transformation) and
1
2
1
2
T
T
V
V
Mechanical work of the isobaric transformis:
2
1
2
1
2
1
21 dVppdVLL
JVVpL 1221
Heat exchange of the isobaric transformwith the external environment is
2
1
2
1
21 dTmcQQ p
V
1
2
p
p2
p1
Hea
ting
Coo
ling
Page 43
JTTmcQ pm 1221
For heating, 00 L,Q
For cooling, 00 L,Q
Modification of caloric state variables: JTTmcUU vm 1212
211212 QTTmcII pm
Graphical representation of the transformation in p-V coordinates is like in fig. 9,
where the transition from state 1 to state 2 is characterised by keeping the pressure
constant: ppp 21
Fig. 9. Plot of the isobaric transformation in p-V coordinates
As the V/T ratio stays constant with volume increase, e.g. passing from state 1 to
state 2 (V2>V1), the temperature would increase, ( T2>T1 ) resulting a heating of the
gas. If the process inverts, from state 2 to state 1, 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 exterior wall pushed by a constant force that
gives a constant pressure. Heating by an external source gives an increase of the
temperature and of the volume occupied by the gas.
3.5.3. Isothermal transformation
Theisothermal transformationtakes place at constant temperatureT= constant,
( ttanconsm ), from equation of state tconsT
pVtan it results if T is constant
ttanconspV (equation of the transformation ) or 2211 VpVp
The mechanical work of the isothermal transformation is:
p
p
Heating
1 2
V V1 V2
Cooling
Page 44
2
1
2
1
21 pdVLL where 2211 VpVpttanconspV
2
1
2
1 1
2111121V
VlnVp
V
dVVp
V
dVctL
Jp
pmRT
V
VmRT
p
pVp
V
VVpL
2
1
1
2
2
111
1
21121 lnlnlnln
Heat exchange of the isothermal transformation is : mcdTQ 0dT 0Q
The isothermalprocess has a specific heat of . In order to solve such an
undetermination, the general formulas of the 1st principle are used:
VdpLLQVdpQdI
pdVLLQpdVQdU
tt
and, using the caloric state equations for the perfect gas:
dTmcdI
dTmcdU
p
v
it results, for ttanconsT
LQ
dU 0or JLQ 2121
Only in the isothermal transformation, Qexchanged by the agent with the
environment is equivalent with the mechanicalwork done or consumed by the
thermal agent during the transformation.
Jp
pmRT
V
VmRT
p
pVp
V
VVpQ
2
1
1
2
2
111
1
21121 lnlnlnln
Graphical representation of the transformation in p-V coordinates is like in fig. 10,
where the transition from state 1 to state 2 is characterised by keeping the product
pVconstant.
Fig. 10. Plot of the isothermal transformation in p-V coordinates
p 1
2
V2 V1
p1
p2
Expansion
Compression
Page 45
The plot is an equilateral hyperbole arch.
In the isothermal transformation: 01212 TTmcUU vm
01212 TTmcII pm
The physical model of a thermodynamic system undergoing such a transformation is
that of an engine’s cylinder that is intensely exchanging heat with the environment.
When the piston moves, the decrease of volume gives an increase of pressure so the
temperature tends to increase as well. Assuming that cylinder’s walls allow the
release of a heat that’s large-enough, it can result a constant temperature of the gas in
the cylinder.
3.5.4.Adiabatic transformation
The adiabatic transformation is the thermodynamic transformation without any heat
exchange with the external environment 00 21 QQ
Parameters p, T, V are variable and the goal is to find their relationship under these
conditions
1st principle:
VdpQdI
pdVQdU
Caloric equations of state:
dTmcdI
dTmcdU
p
v
Replacing dU, dI and computing the ratio, it gives:
pdV
Vdp
c
c
v
p where k
c
c
v
p
k is the adiabaticexponent that represents the ratio of specific heats of a gas în the
isobaric respectively in the isochoric transformation.
Integrating the 0V
dVk
p
dp equation gives tconsVp
ktanlnln that leads to the
equation of the transformation in p and V coordinates:
tconspV k tan or kkVpVp 2211
Logarithmation and differentiation is applied to the state equation, in the form
ttanconsT
pV .
It results 01
0
V
dVk
T
dT
V
dVk
p
dp
T
dT
V
dV
p
dp
,
that, after integration, gives the equation of the adiabatic transform with parametersT
and V:
tconsTV k tan1 1
22
1
11
kk VTVT
Page 46
ReplacingVgives tcons
p
T
k
ktan
1
or
k
k
p
p
T
T1
1
2
1
2
that represents the equation of the
adiabatic transform with p and T.
The mechanical work of the adiabatic transform is:
2
1
2
1
21 pdVLL where kkk VpVptconspV 2211tan
2
1
2
1
21 dVVctV
dVctL k
k
1
1
1
2211
kk VVk
ctL
the constant is replaced with kVp 22 and then with kVp 11 .
1
221121
k
VpVpL or
1
212121 1
11 T
T
k
mRT
k
TTmRL
k
k
p
p
k
VpL
1
1
21121 1
1
The heat exchange of the adiabatic transform is 0Q 021 Q , that means that for
this transformation, the specific heat is zero: 0adc .
Variation of caloric state variables:
1212
1212
TTmcII
TTmcUU
pm
vm where JUUL 2121 21 IILt
tconspV k tan
Graphical representation of the transformation in p-V coordinates is like in fig. 11,
where the transition from state 1 to state 2 is characterised by keeping the constant
product: VVV 21
p 1
2
V2 V1
p1
p2
Expansion
Compression
Page 47
Fig. 11. Plot of the adiabatic transformation in p-V coordinates
The plot is a hyperbole arch with greater slope than the equilateral hyperbole, as the
values k> 1.
The physical model of a thermodynamic system undergoing such a transformation is
that of an engine’s cylinder with a perfectly thermal isolation, that doesn’t allow any
heat exchange with the environment.
3.5.5.The polytropic transform
The polytropic transform is a general transformation in whichp,T,V vary and energy
is exchanged in form of Q and L with environment .
The polytropic transform is a process in which all parameters vary, so in order to
define the transformation it is accepted that the specific heat is given and constant
nc .
cn = polytropic specific heat dTmcQ n
To find out the equation of the trasformation:
First law
VdpQdI
pdVQdUwith
dTmcdI
mcvdTdU
p
dTmcQ n
pdVdTmcdTmc
VdpdTmcdTmc
nv
npRatio
pdV
Vdp
cc
cc
nv
np
is noted with
nv
np
cc
ccn
and is called
polytropic exponent. It is considered that ratio constant (n=constant )during a
polytropic transform.
0V
dVn
p
dp ttanconspV n - equation of the transformation in p,V or
nnVpVp 2211
T
dT
V
dV
p
dp; 0
V
ndV
p
dp 01
V
dVn
T
dT
ttanconsTV n 1 -equation of the transformation in T,V or 122
111
nn VTVT
ttancons
p
T
n
n
1 -equation of the transformation in T,p or
n
n
p
p
T
T1
1
2
1
2
The work of the polytropic transform is
2
1
2
1
21 pdVLL
nnn VpVpctpV 2211
11
112211
12
2
1
2
1
21
n
VpVpVV
n
ctdVVct
V
dVctL nnn
n
Page 48
1
221121
n
VpVpL
1
212121 1
11 T
T
n
mRT
n
TTmRL
n
n
p
p
n
VpL
1
1
21121 1
1
The heat exchange of polytropic transformis: dTmcQ n
2
1
2
1
1221 TTmcdTmcQQ nmn
Considering n constant, as known, for o given polytropic transform, it can be deduced
cnfrom ratio1
n
cncc
cc
ccn
pv
n
nv
np cu kc
c
v
p vn c
n
knc
1
dTcn
knmdTmcQ vn
1
1212211
TTcn
knmTTmcQ vmnm
It is usual to express 21Q in function of Lmec
LQdU
kn
k
kn
n
dTmc
dTmc
Q
dU
Q
dL
n
v
11111
Lk
nkQ
1
, iar 2121
1
L
k
nkQ
The caloric state quantities of polytropic transform are:
J
TTmcII
TTmcUU
pm
vm
1212
1212
Giving to n values in the interval ( , )it can be obtained an infinity of polytropic
transform; not all corespond to heat engine processes.
Experimentally it was determined that:
1) The compression and expansion processes form heat engines are accompanied
by heat exchange of the agent to the environment; this heat exchange is not so intense
to reach an isothermal process.
2) During realcompression and expansion processes the heat exchange with the
environment cannot be completely avoided and these processes do not undertake
perfectly adiabatic.
Conclusions : Real compression and expansion processes can be considered as
polytropic processes situated between isothermal and adiabatic transformation.
From the equation of realpolytropic transform ttanconspV n
isothermalctpVn 1
adiabatectpVkn k
Page 49
So in thermodynamic calculation it is of interest the study of polytropic
transformation having n complying with the inequation kn1 ,in which
k=1.33...1.6.The graphic of the transform versus n values is :
Fig.12.Reprezentation of polytropic expansion and compression in coordinate p-V
Observations:
Giving particular values to „n” the simple transformations can be deduced from
polytropic transformation.
a) 1n ctpVctpV n
zvn cic
n
knc
1
b) kn ctpVctpV kn 01
advn cc
n
knc
c) 0n izobarăctpctpV n vpvn kcccn
knc
1
d) n izocorăctVctVpctpV nn
1
vn cc
The specific heats for different polytropic transformation can be obtained according
to n values from diagram.
p
V
1
2iz 2pol 2ad
p 1
V
2ad 2pol
2iz
Page 50
Fig.13.Specific polytropic heat versus polytropic exponent
The polytropic curves in function of n if it isconsidered that all curves pass through a
given point.
a) n = 0 p = ct 1-1
b) n = 1 T = ct 2-2
c) n = k adiabatic 3-3
d) n = ± ∞ V = ct 4-4
e) n = -1 5-5
f) -∞<n<-1 6-6 between (4-4, 5-5)
g) -1<n<0 7-7 between (5-5, 1-1)
h) 1<n<k 8-8between (2-2, 3-3)
Fig.14. Polytropic curves in the hypothesis that pass through a given point.
p
1 1
2
2
3
3
4
4 5
5 6
7
7
8
8
A
V
6
cn
cp
cv
0 n 1 k
Page 51
Fromthe point of view of the heat exchange the adiabatic splits the p-Varea in two
zones –in any transformation which starts from a point of the adiabatic and
undertakes upwards the adiabatic, in zone I, the heat exchange is positive.
Fig.15.The zones of the heat exchange defined by adiabatic curve
I 0Q - example: A4’ A –1, A2, A5
In any transformation which starts from A and undertakes under adiabatic curve 3-3’
(zone II) the heat exchange is negative.
II 0Q - example: A4, A-1’, A2’,A5’.
Questions
1. Which are the hypothesis of the ideal gas ?
2. Which is the equation of state of ideal gas?What is the meaning of the
quantities used?
3. What are the laws of ideal gas ?
4. How is defined the adiabatic exponent k ?
5. Which is Mayer’sequations?
6. What is the equation of real gas ? In what conditions the real gas is close to
ideal gas ?
7. Which are the hypothesis to gas mixture?
8. What is Dalton law? What is Amagatlaw?
9. What is a simple transformation?Which are the simple transformations of
ideal gas ?
10. Which is the equation of isochoric transformation ?Which is its
representation in p-V ?
11. Which is the equation of isobaric transformation ? Which is its
representation in p-V ?
p
1´ 1
2´ 5´
3´
2
3
5
I
II
V
4´
A
4
Page 52
12. Which is the equation of isothermal transformation ? Which is its
representation in p-V ?
13. Which is the equation of adiabatic transformation ? Which is its
representation in p-V ?
14. Which is the equation of polytropic transformation ? Which is its
representation in p-V ?
15. How is defined the polytropic exponent ?
Page 53
Course 6
Chapter 4. Second law of thermodynamics
4.1. Thermodynamic cycles
It is called thermodynamic cycle (or a cyclic thermodynamic process) aseries of
succesive thermodynamic processes (or transformations) 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.
If it is considered the cycle 1A2B1,
Fig.16. The thermodynamic cycle in coordinates p-V
It is called the work of the cycle the sum of all works (Lc = mecL )performed in the
transformations which compose the cycle.
1
1
1
1
21
LLLLBA
c
2
1
1221
1
2A
BA
B
c LLLLL
- in transformation 1-A-2 0dV and 00 21 ALpdVL
- in transformation 2-B-1 0dV and 00 12BLpdVL
1'22121 AareaL A
'2'11212 BareaL B
12112211221 BAareaLLLLL BABAc
Considering the sign of work , the cycles are divided in:
a) Work producing cycles- 0cL when cycle is perfomed clock wise, characteristic
to energy producing aggregates, such as internal combustion engines, gas turbines
b) Work consuming cycles - 0cL when cycle is perfomed anti clock wise
(trigonometric), characteristic to energy consuming aggregates such ascompressors,
refrigerating installations.
For an arbitrary cycle the points 1 and 2 are in contact to 2 adiabates.
p
V
Lc 1 2
A
B
2´
´´ 1´
Page 54
Fig.17. The cycle placed between two adiabates
For a heat engine working on this cycle,the first law of thermodynamics is expressed: LQpdVQdU
It is applied the formula for the cycle12B1
1) LQdU
2) 0 dU
3) cLL
4)
1
2
2
1 BA
QQQ
In which
2
1
21 0
A
AQQ
The transformation 1A2begins on the first adiabate and undertakes upwards (beyond)
the adiabate.
2
2
12 0
B
BQQ
The transformation 2B1 begins on the second adiabate and undertakes downwards
(beneath) the adiabate.
It is noted heat amount received by the agent during the transformations of the
cycleQ, 0Q
It is noted heat amount released by the agent during the transformations of the cycle
Q0, 00 Q
0QQQ or 00 QQLQQQ c
In an work producing cycle performed by an ideal gas, only a part of the heat
received by the agent (Q) is transformed in work Lc, the rest of the heat being
released to the environment during the rest of the cycle transformations.
It is called thermodynamic efficiency of the cycle the expression :
Q
Lct
Q
Q
Q
QQt
001
The efficiency expresses the thermodynamic quality of the cycle.
p
V
Lc 1 2
A
B
2´
´´ 1´
Adiabate I Adiabate II
Page 55
4.2.Reversible and irreversible processes
As presented in chapter 1.4, the thermodynamic processes can be divided in:
Reversibile process, in which the system passes from initial to final state
directly and reversely,exactly through the same points, on the same path.
In order to perform such a process, the external conditions should modify extremely
slow, so the system to adapt progresively to the new variations which gradually
appears;
Irreversible or non-reversible process, in which the system passes from initial
to final stateand reversed through different points, on other path.
Real processes cannot be considered reversible. A process can be considered
reversible if intermediate states when passing from initial to final state are close
enough to intermediate states when passing from final to initial state.
All thermal and mechanical processes in nature are irreversible,they undertake
naturaly in one way, to a state with higher probability of achievement;the causes of
ireversibility are:
-Processes have finite velocities (not very slow);
-friction during process;
-molecular difusion;
-heat exchange at finite temperature difference;
-finite variation of internal and external conditions.
In order to study real processes from heat engines, all real irreversible processes are
replaced with equivalent reversible processes.
Examples of irreversible processes:
Passing of gas ( on its own,naturally ) from an area of high pressure to an area of
lower pressure and never in reverse way.
At the contact of two bodies of different temperatures, after a period of time,the
bodies reach an intermediate temperature, never happened that hot body to
become hotter and the cold one to become colder, even the first law of
thermodynamics is obeyed.
If two gases ( or a water-sugar solution ) are introduced in the same enclosure the
tendency of the molecules will be to mix, never to separate, no matter how long we
will wait for.
Other irreversibile processes are the errosion of theEarth crust, metal corrosion,
aging of material and people.
4.3.Versions of second law
Page 56
First law represents the generalization of energy conservation law for thermal
processes and the second law was discovered during experimental research of heat
engines, and its content applies to other energy changes, not only of thermal
nature.The physicists noticed that heat engines cannot transform totally the absorbed
heat in work.The second law asserts the irreversibility of the natural processes
explaining terms of conversion of heat into work LQ and rounds the first law.
First law: Work turns into heat on the same equivalent ratio, meaning QL and
LQ ,but it does not say anything about the possibility of reverse
transformation,emphasizing just the equivalence.
Second law says the possibility and the sense in which the processes are undertaken.
Work turns into heat QL spontaneously, integrally (on its own-de la
sine).Heat turns into work LQ by means of a heat engine in which irreversible
processes take place and LQ transformation is partial.
The second law says that temperature differences between systems in contact with
each other tend to become equal and work can be produced from non-equilibrium
differences (temperature, pressure and density differences).For an isolated system all
parameters,particularly temperature will eventually have constant,uniform values. A
heat engine is a mechanical device that provides useful work from the difference in
temperature of two
bodies:
Since any thermodynamic engine requires such a temperature difference, it
follows that no useful work can be derived from an isolated system in equilibrium;
there must always be an external energy source (hot source or heat reservoir ) and a
cold source (sink).So,second law shows that work can be totally turned into heat, the
inverse transformation is not correct, heat cannot be totally turned into work.The
cause of the assymetry is the fact that work corresponds to a ordered motion of
particles and heat corresponds to a disordered one.Second law has many formulations
of the physicists who studied thermodynamics,which are equivalent.
Formulations
a.Clausius:“Heat generally cannot spontaneously flow from a material at lower
temperature to a material at higher temperature.”(irreversibility of spontaneous
phenomena).
Hot
source
Cold
sink Heat
engine Q Q0
L
Page 57
b. „It cannot be reversed on the same path ( through the same intermediate states) a
process in which friction generated heat.(irreversibility of processes accompanied by
friction)”.
c.Clausius: “A heat engine running continuously converting to work the heat
absorbed from a hot source without releasing heat to a cold source is impossible.”
d. Kelvin:”It is impossible to convert heat completely into work.”
“It is impossible to produce work in the surroundings using a cyclic process
connected to a single heat reservoir”.
d.Oswald : „Perpetual motion machine of the second kind is impossible.”
Perpetual motion machine of the second kind is a machine which runs only with a hot
source.
e. „The heat of a hot source cannot be converted into work without producing
changes to environment „(heat release to environment ).
f.A thermodynamic system will naturally evoluate from the state with lower
thermodynamic probability to the state with higher thermodynamic probability.
Explanation : To understand the thermodynamic probability it is considered that N
molecules occupy a given volumeand it is surveyed the distribution of the molecules
in two halves of the volume.It is considered that at a given moment, N1 moleculesare
situated in left side and N2in right side.
21 NNN
Any change in molecules distribution will lead to a new state regarding system
distribution. A state is defined if it is in equilibrium and there are known some
characteristic quantities: mean kinetic molecular energy,mean potential molecular
energy, molecular distribution in the volume.
Molecules are continuously in motion,changing position and velocity.The distribution
of molecular velocities and positions ( coordinates)in the enclosure at a given
moment determine a micro-state of the system.As a result of a perpetual molecular
motion, the micro-states vary continuously.
At the equilibrium, the macroscopic properties of the system do not vary in time
even at molecular level the microstates change; this is possible because the
thermodynamic properties appear as an mean effect of the processes produced at
molecular level.
23
2 2wm
V
Np , kT
wm
2
3
2
2
The number of microstates corresponding to an equilibrium state represents the
thermodynamic probability of the equilibrium state. The total number of microstates
determined by the variation (permutation) of velocities and positions, keeping
themean kinetic molecular energy constant represents the thermodynamic probability.
Coming back to the distribution of N moleculeswithin the enclosure,it can be noticed
N1
N2
Page 58
that, in theory, at molecular level,are possible any kind of distribution (for example
NNN 21 0 ); flowing of a molecule from right to left or the change of position of
two molecules determine a new micro-state; applying the thermodynamic
probability,it is noticed that it is possible any distribution between N1and N2, but
every distribution has a different thermodynamic probability.
For a system made of N = 100 molecules,thermodynamic probability P resulted from
N distribution in N1and N2:
!N!N
!NP
21
N1 0 10 20 30 40 45 50
N2 100 90 80 70 60 55 50
P 1 1,6.1013
5,25.1020
2,8.1025
1,31.1028
6,65.1028
1,12.1029
All the experiments showed that at macroscopic level the molecules will distribute
uniformely 21 NN meaning that they reached the state with the highest
thermodynamic probability.
If it is measured in the two halves of the enclosure the pressure and temperature of
gas, at equilibrium, the values are equal.
Although at microscopic level are possible all the distributions, at macroscopic level
is evident the state which corresponds to maximum probability.
Second law says that the thermodynamic processes undergoes in the sense of
reaching the maximum thermodynamic probability.The content of second law is
connected to the microscopic interpretation, being a statistic law.
4.4 Carnot cycle
The most efficient cycle of transformation LQ is the cycle in which the agent gets
in contact to two heat sources; a heat source (thermostat) is a body with constant
temperature having infinite caloric capacity.The cycle is formed of two adiabates and
two isothermes:
Fig.18. Representation of Carnot cycle (work producing)in p-V coordinates
1-2 isothermal expansion (T = ct), the agent receives heat sourcehotQQ 21
2-3 adiabatic expansion 023 Q
p
V
1
2
3
4
Lc
Q
Q0
Page 59
3-4 isothermal compression (T0 = ct), the agent released heat sourcecoldQQ 043
4 – 1 adiabatic compression 041 Q
The efficiency of Carnot cycle when the agent is an ideal gas is :
Q
Q
Q
QQ
Q
Lct
001
in which 0QQLc
For Carnot cycle when the agent is an ideal gas it can be written :
1
221
V
VlnmRTQQ
3
40430 ln
V
VmRTQQ in which 34 VV and 00Q .
4
300 ln
V
VmRTQ
1
2
4
30
0
ln
ln
11
V
VmRT
V
VmRT
Q
Qc
t
But 1
2
4
3
V
Vln
V
Vln ,explained as follows :
- for 2-3 133
122
kk VTVT
130
122
kk VTVT
- for 4-1 144
111
kk VTVT
140
11
kk VTTV
4
3
1
2
V
V
V
V
When the ratios of volumes are equal, also are their logarithms ratios so they can be
simplified. The expression of the efficiency is:
T
Tct
01
Postulate: The efficiency of Carnot cycle does not depend on the natureof the agent, it
depends only on the temperatures of heat sources.
It was established that T
T
Q
Q
Q
Lcc
t00
11 for Carnot cycle.
From two expressions of efficiency T
T
Q
Qc
t00
, 00 Q - the released heat to cold
source and 00
0 T
Q
T
QThis function is Carnot function.
No heat engine works on Carnot cycle.The thermodynamic efficiency t of the
Carnot cycle has maximum values if it is reported at the same temperatures T and T0.
Page 60
It is considered a Carnot cycle undergone in reverse sense ( anticlock wise or
trigonometric)
Fig.19. Representation of Carnot cycle (work consuming )inp-V coordinates
0cL - work consuming
- 1-4 adiabatic expansion
- 4-3 isothermal expansion(the agent receives heat )0340 QQ
- 3-2 adiabatic compression
- 2-1isothermal compression (the agent releases 012 QQ )
In an inversed Carnot cycle the agent receives Q0at T0 and releases Q at T.This cycle
is characteristic to refrigeration installations.
ccc LQQsauQQLLQQ 000 0,0,0
It is called coefficient of performance c
fL
Q0 . It is not of interest that heat is
released to environment.The value of the coefficient of performance can be higher
than 1.
p
V
1
2
3
4
Lc
Q
Q0
Page 61
Course 7.
Chapter 4. Second law of thermodynamics (continued)
4.5.Clausius integral . Entropy
It is considereda reversible thermodynamic cycle 1ab2cd1, which is intersected by
infinite number of adiabates in cycle area.
Fig.20. Reversible cycle as a sum of elementary Carnot cycles
It will result an infinite number of elementary cycles as abcd. Considering an
elementary cycle, reversible,abcd:
ab – isotherme at T constant
bc – adiabate
cd – isotherme at T0 constant
da – adiabate
It is noted δQ, the heat quantity received by the agent in elementary isothermal
transformation ab at Tand analogue, δQ0, the heat quantity released by the agent in
elementary isothermal transformation cd at T0.
For elementary Carnot cycle, it is expressed the Carnot function.
00
0 T
Q
T
Q ,then it is integrated , in which 00 Q .
0
0
0
T
Q
T
Q for all the cycles having the form of abcd and 0 T
Q
1
21
0BA
T
Q
T
Q
2
1
1
2
0
A BT
Q
T
Q as the transformation is reversible it
means that in can be performed,directly and reversely,through the same intermediate
states.
As 1
2
2
1B BT
Q
T
Q it results that
2
1
2
1A BT
Q
T
Q meaning that the integral of the ratio heat-
temperature does not depend on the path, having the same value for path A or B.
p
V
1 2
a b
d c
δQ
δQ0
Page 62
For the integral to be independent on the path it is needed as the ratioT
Qto be a total
diferential.
Clausius noted dST
Q
- being the mathematical expression of second law,in which
S quantity was called entropy.
Entropy S is a state quantity which can be written as msS in which
Kkg
Js is
specific entropy.
For a reversible cycle, variation of entropy is 0 and T
QdS
.
2
1
2
1
2
1
12T
QSS
T
QdS ,
δQ –elementary heat exchange between agent and surroundings at T.
For a reversible, adiabatic transformation 0 QT
QdS
dS = 0and S = constant,or 12 SS .
The reversible, adiabatic transformation is called isentropic because entropy is
constant ctS
Observation: In all transformations above,
T
Qis Clausius integral.
Ratio Q/T is called reduced heat.
In a reversible Carnot cycle,thealgaebrical sum of reduced heats is zero or Carnot
function is zero.
4.6.Entropy variation of ideal gas
The variation of entropy S is established for any process using general formulas, for
the case of ideal gas.
According to second law T
QdS
pdVQdU
VdpQdI
T
VdpdIdS
T
pdVdUdS
fundamental thermodynamics equations
Some particular relations for ideal gas are replaced in fundamental equations. dTmcdU v mRTpV
dTmcdI p
It will result :
Page 63
V
mRdV
T
dTmcdS v
V
mR
T
p
p
mRdp
T
dTmcdS
p
Integrating the relations
K
J
V
VmR
T
TmcSS
mv
1
2
1
212 lnln
K
J
p
pmR
T
TmcSS
mp
1
2
1
212 lnln
Replacing temperature from dS and integrating, it results:
1
2
1
212 lnln
V
Vmc
p
pmcSS pmvm
4.7.T-S diagram . Graphical representation of processes in T-S coordinates
The thermodynamic processes may be represented in temperature T-entropy S
coordinates.
Fig.21.Representation of a transformation in T-S coordinates
In T-S representation the elementary area Tdsabcdarea is equal to elementary heat:
TdsQ
2
1
2
1
21
2
1
21 '1'122areaQabcdareaTdSQQ
For any simple transformation of ideal gas for which there were made representation
in p-V coordinates, it must be ploted graphics in T-S coordinates as it follows:
Isochoric transformation V = constant
T
dTmc
T
QdS v
1
2
T
[K]
1´ 2´ dS
T
dT a b
d c S [J/K]
Page 64
1
212
T
TlnmcSS v
In isochoric transformation, entropy varies logharitmically with temperature.
Fig.21.Isochoric process in T-S coordinates
01
vctv
vc
T
mdS
dTtg
Tvmcvtg
BNBA
(βT-temperature scale)
The segment NB is proportional to cv.In T – S diagram,the representation of isochore
is an exponential curve with positive slope.
Isobaric transformationp = constant
K
J
T
TlnmcSS
T
dTmc
T
QdS p
p
1
212
In isobaric transformation entropy varies logarithmically with temperature.
01
pctp
pc
T
mdS
dTtg
pT mcAB (βT-temperature scale )
Fig.22.Isobaric process in T-S coordinates
1
2
T
[K]
αp
N
A B S[J/K]
1
2
T
[K]
αv
N
A B S [J/K]
Page 65
Observation: Comparing the formulas showing the slopes of V = constant and p =
constantprocesses, it can be written:
v
vc
T
mtg
1 and
p
pc
T
mtg
1
If are plotted two curves of V = constant and p = constant in the same T- S diagram
the curves intersecting in point N will be distincted as follows the exponential with
higher slope is the isochore and the exponential with smaller slope is the isobar.
Fig.23.Comparison between isochoric and isobaric curves in T-S
pv tgtg as
1kk
c
ccc
v
p
pv
Isothermal transformation(T = constant)
T
QdS
2
1
2
1
2112
1
K
J
T
QQ
TT
QSS for T = constant
2
1
1
2
2
111
1
2112121
p
plnmRT
V
VlnmRT
p
plnVp
V
VlnVpLQ
It can be written :
2
1
1
22112
p
plnmR
V
VlnmR
T
QSS
T
[K]
αp
N
A B S [J/K]
αv
p=constant
V=constant
Page 66
- Fig.24.Isothermal process in T-S coordinates
- If the process performs from point 1 to point 2 meaning that the difference of
entropy is positive, it means that the heat exchange is positive, the system performs
work and expands. 00 2112 QSS receiving heat.
Adiabatic transformation
Inadiabatic transformation the heat exchange and variation of entropy are zero. 0Q
021 Q 0dS ttanconsS
K
JSS 21
- The adiabatic transformation which is also reversible is called isentropic (of constant
entropy).
- - Fig.25.Isentropic process in T-S coordinates
-
Polytropic transformation
1
2
1
212
1
1
T
Tlnc
n
knm
T
TlnmcSS
dTcn
knmdTmcQ
T
QdS
medvmediun
vn
S
1
2
T
Co
mp
ress
ion
mar
ere
Ex
pan
sio
n
eree
T
Expansion
Destindere
1 2
S S1 S2
Compression
Page 67
Fig.26.Polytropic process in T-S coordinates
The graphic of the transformation is exponential with negative slope
vnpol
pc
T
kn
n
mc
T
mdS
dTtg
111
- for the tranformation having kn 1
0 ptg when p belongs to the second trigonometric quadrant, being obtuse.
Observations: 1) In T-S diagram a reversible Carnot cycle is represented as a
rectangle.
Fig.27. Carnot cycle in T-S coordinates
1-2 isothermal expansion – it receives heat
2-3 adiabatic expansion
3-4 isothermal compression –it releases heat
4-1 adiabatic compression
cicluAB
BALQQ
QAQQ
QAQQ
0
034430
1221
0
0
2)In T-S coordinates for ideal gas transformations the isochores ( isochoric curves)
are parralel among them,the isobars also, the polytropic curves with the same n are
parralel too.
4.8. Clausius integral and entropy variation in irreversible processes
There are considered two Carnot cycles working between two sourses of heat of
temperatures Tand T0.
T
T0
1 2
3 4
Q
Q0
S
T
S
M
αp 1
2
Expansion
B A
Page 68
The first cycle is reversible and the second is irreversible.
Assuming that : irevrev QQ
Fig.28. Reversible and irreversible cycles with the same sources
Taking into account the characteristics of irreversible processes it results that
revcirevc LL for the same amount of heat Q taken from the hot source.
irevccrev LL soirevrev
QQ 00
For reversible Carnot cycle
T
T
Q
Q
Q
L
rev
rev
rev
revct
rev00
11
irev
irev
irev
revict
irevQ
Q
Q
L0
1 resulting 00
000
T
Q
T
Q
T
T
Q
Q irevirev
irev
irev -expression of Carnot
function for irreversible Carnot cycle.
It is considered an irreversible cycle 1A2B1; the cycle intersects in p – V plane with
an infinite number of adiabates.
Fig.29.Decomposing of an irreversible cyle into an infinite number of Carnot cycles
There are formed an infinite number of elementary irreversible Carnot cycles having
the form abcd.
p
V
1 2
a b
d c
δQ
δQ0
Hot source , T
Cold source, T0
Heat
engine I
Heat
engine II
Qrev Qirev
Lcirev Lcrev
Q0rev Q0irev
Page 69
a-b – elementary isothermal transformation at constant T
b-c – adiabatic transformation
c-d – elementary isothermal transformation at constant T0
d-a – adiabatic transformation .
The Carnot function for elementary irreversible Carnot cycle abcd is :
00
0
T
Q
T
Q
Integrating all Carnot function for all elementary irreversible Carnot cycle having the
form abcd,
0,001210
0
BAT
Q
T
Q
T
Q
T
Q
Clausius integral for an irreversible cycle 1A2B1 is negative.
It is considered the cycle 1A2B1 irreversible
Fig.30. Cycle half reversible –half irreversible
Assuming that the cycle is made of 1A2 irreversible and 2B1 reversible.
0T
Q
1
21
0
BAT
Q
2
1
1
2
0
A B revirev T
Q
T
Q
In which for reversible transformation
1
2
2
1B B revrev T
Q
T
Q
2
1
2
1
2
1
12
B B B revrev
SST
QordS
T
Q resulting
2
1
12 0)(A
SST
Qor
2
1
12
A
SST
Q
The formula expresses the variation of entropy in an ordinary irreversible process.
Differentiating results leads to irevT
QdS
– mathematical formulation of the second
law of thermodynamics for irreversible processes.
4.9.Examples of thermodynamic processes,typically irreversible
a) Heat exchange between two bodies with finite temperature difference.
It is considered an isolated system made of two bodies of different temperatures.
1
A
2 B
p
V
Page 70
Fig.31.Adiabatic system
If the first body has a higher temperature than the second one it releases heat towards
the second Q 1-2 2121 QTT
In the system 0dS as th heat exchange at finite difference is an irreversible process.
For body I:the analysis of irreversible process is done replacing real proces with an
equivalent isothermal, reversible process.
01
QT
QS I - the body releases Q
Analog 02
QT
QS II - the body receives Q
For all the system, adiabatic and irreversible
021
ST
Q
T
QSSS III as T1>T2
b).Gas laminar flow
Definition: A laminar flow is a fluid flow through an orifice with a smaller section
than upstream which is produced by expanding without performing work to
surroundings.
Fig.32. Laminar flow of gas when passing through a diafragm
The minimum section is downstream orifice. Experimentally were noticed energy
loss (friction and vortices) which turn into heat which is absorbed by fluid, resulting
12 pp .
1 2
w1 w2
p1,T1,i1 p2 ,T2,i2
T1
T2
Q
I
II
Page 71
In order to find out the thermodynamic properties of laminar flow it is written the
first law for closed systems as energy balance.
kg
Jltqhhg
wwii 2112
21
22
122
In laminar flow 211212 qhhww negligible .
Gas has the exterior temperature and it is imposed the condition not to perform shaft
work , 0lt . The condition is obeyed if 211212 qhhww negligible, wich lead to
equality of the enthalpies of the two sections The laminary flow is an isenthalpic
process, resulting 12 ii .
The analysis of entropy variation: considering that gas has temperature equal to
environmental temperature TT 0 the laminar flow can be studied as a adiabatic
irreversible process.
Due to frictions heat is realeased and stored in gas and entropy increases.The
equivalent process do not consider temperature variation the process being replaced
by a reversible isothermal process in which the pressure decreases.
K
J
p
plnmR
V
VlnmRSS
2
1
1
212
2
1
1
212
p
plnR
V
VlnRss - for1 kilo of agent
Questions
1. What is a thermodynamic cycle?
2. What is the expression of work in a thermodynamic cycle ?
3. How are classified the thermodynamic cycles?
4. Which is the formula for thermodynamic efficiency ?
5. Enounce the second law of thermodynamics in several versions.
6. What is Carnotcycle ?Which is its thermodynamic efficiency ?
7. What is inversed Carnot cycle ?What is coefficent of performance of
refrigerating installations?
8. How is defined entropy and which is the connection to second law of
thermodynamics?
9. Which is the entropy variation in isobaric transformation and how is
represented in T-S coordinates ?
10. Which is the entropy variation in isochoric transformation and how is
represented in T-S coordinates ?
11. Which is the entropy variation in isothermal transformation and how is
represented in T-S coordinates ?
12. Which is the entropy variation in adiabatic transformation and how is
represented in T-S coordinates ?
13. Which is the entropy variation in polytropic transformation and how is
represented in T-S coordinates ?
14. What is the influence of irreversibility of processes upon entropy?
