Technical Thermodynamics

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Technical Thermodynamics

Egon P. Hassel

Prof. Dr.-Ing. habil. Dipl.-Phys.

Institute of Technical ThermodynamicsDepartment of Mechanical Engineering and Shipbuilding

University of Rostock, GermanyTel: +49 381 498 9400

email: egon.hassel (et) uni-rostock.dewebsite: www.LTT-Rostock.de

or http://web.me.com/egon.hassel

December 23, 2009


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CONTENTS 1

Contents

7 Pure materials and Clausius Rankine process 37.1 The complex behavior of the density of water with the temperature at constant pressure . . 37.2 Water cooking at 1 bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.3 The boiling procedure of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77.4 Three dimensional diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77.5 Vapor pressure diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7.6 Definition of vapor content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.6.1 Quantitative T-s-diagram and log(p)-1/T-diagram . . . . . . . . . . . . . . . . . . 13

7.7 Properties of state of real fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.8 Equations of state (EOS) with virial coefficients . . . . . . . . . . . . . . . . . . . . . . . 177.9 Realgasfaktor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.10 Liquid vapor region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.11 Sketch of the most important h-s-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 207.12 T-s-diagram with evaporation heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.13 Mathematics: linear interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.14 Clausius Rankine process, classical power plant station process . . . . . . . . . . . . . . . . 22

7.14.1 Exergy effi

ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29


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2LTT-Rostock, Prof. Dr. Egon Hassel

CONTENTS

2 LTT-Rostock, Prof. Dr. Egon Hassel

CONTENTS


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3

Chapter 7

Pure materials and Clausius Rankine

process

7.1 The complex behavior of the density of water with thetemperature at constant pressure

Pure materials are for example pure water or pure nitrogen. From such materials we know that they appearin three different phase states, as solids, as liquids or as gases. The most important substance of water weknow as ice, liquid water and water vapor. Since all technical processes work with substances, we shouldknow their behavior in order to design and understand these processes. As an impressive example of thecomplexity of the behavior of the properties of even seemingly simple substances, look at the density versustemperature for water at ambient pressure, figures 7.1 and is 7.2.We already notice here that the phase transition is an unsteadiness. At ambient pressure p and at t = 0Ctwo phases are present simultaneouly at the same time and with the same temperature T and the same

pressure p, and T and p remain constant although heat is added or subtracted. In figure 7.2 the densityof water at constant pressure, e.g. ambient pressure p = 1bar, between t = 100C and t = 200C isdrawn qualitatively. During the process heat is added. We start at t = 100C. The following specialcharacteristics of water properties behavior are to be seen:

1. The solid body Ice behaves like a normal solid body, it expands with rising temperature.

2. During the phase transition (ice - > liquid) the water contracts unlike most materials, volume changeis about 9 %.

3. Starting from t = 0C the specific volume of the water decreases with rising temperature until a

minimum with t = 4

C is reached, there the density of the liquid is largest. One recalls that this iscalled the anomaly of the water. This effect makes it possible for fish to survive in the winter if thelake above freezes over, but the water on the bottom t = 4C is still liquid.

4. Starting from t = 4C the density decreases with rising temperature, as with each other normal liquid.

5. At t = 100C at 1 bar there happens the phase change (liquid - > vapor), with a volume increase ofapproximately a factor of 800. This is also an unsteadiness, with two phases, liquid and vapor, existingsimultaneously with the same temperature and the same pressure in thermodynamic equilibrium. Duringthe phase change the temperature and the pressure stays constant.

6. Starting from t = 100

C the water vapor behaves similar to any other gas, expanding with increasingtemperature (at p = const). For very high temperatures the water vapor can be treated as an idealgas, because the molecules interaction (potential) energies due to the van der Waals hydrogen bindingforces are small in comparison to the kinetic energy due to the high temperature.


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Pure materials and Clausius Rankine process

+100C

quantitative water and ice, density versus temperature

temperature in C and F

water

ca. 9% volume increase

under cooled liquidice

!100C0C(= +32F)

!50C(= !58.0F) 50C(= 122.0F)

densitykg/m!

an under cooled liquid

is still a liquid!

Figure 7.1: water density quantitative versus temperature, ice -> liquid

density water qualitative p = const e.g. 1 bar:

ice, melting (0OC), anomaly (4OC), liquid, evaporation (100OC), vapor

temperature

C/F-100C 0C +100C +200C

-148F -32F +212F +392F

densityqualitative(kg/m^3)

anomaly at 4 OC

largest density

approx 9% volume

decrease

ice -> liquid

approx 800-fold

expansion with

evaporation

liquid -> vapor

approx 1 kg/m3

Figure 7.2: water density quanlitative versus temperature from -100 C to +200 C




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LTT-Rostock, Prof. Dr. Egon Hassel

Water cooking at 1 bar 5

Isobaric boiling of water, p=1.0133 bar

heating

20 OC

over

heating

150 OC

end of

boiling100 OC

boiling100 OC

start of

boiling100 OC

liquid vapor vapor! gas

= vapor bubbles = last liquid droplets

Figure 7.3: The isobaric evaporation of water at p=1,0133bar, scheme

Conclusion from this: even such a common substance like water, which we might consider as simple, exhibitsa very complex and strange behavior in its state variables dependence on temperature and pressure. Andin oder to design and understand technical processes we should understand this behavior. Many materialsbehave similarly, so that in this basic course of technical thermodynamics water serves as a model substancefor the other ones. The chapter is separated into two subsections A) The general behavior of pure materials,B) power station processes with water in the h s-diagram.

7.2 Water cooking at 1 bar

Before we start with this section, we want to observe the well known boiling procedure of water. This issimilar to our water boiling in the morning for breakfast eggs or coffee or tea. The air (oxygen gas andnitrogen gas) which is present can be neglected for these observations. From this experiment later on we arebetter able to understand the diagrams. See figure 7.3.

The water from the water tap, state (1), has a temperature of t = 20C perhaps, this state (1) is calledundercooled liquid. With the heat of the stove plate the liquid is warmed up first to t = 100C, state (2),where the liquid begins to evaporate. From state (1) to state (2) the liquid expands a little bit. With furtherheating more and more vapor evolves. The vapor has a much larger specific volume than the liquid, theexpansion is approximately 800-fold. Temperature and pressure remain constant during the entire boiling

procedure, i.e. liquid and vapor are present simultaneously. State (3) is a state in the so called two phaseregion with liquid and vapor simultaneously. We add heat until all liquid water is evaporated, state (4),which is called saturated vapor. Starting from state (4) we have only water vapor which is like a real gas,approximately like an ideal gas if the pressure is 1 bar, that is the vapor expands with increasing temperature.


Water cooking at 1 bar5


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T, p = const vapor

liquid

two phase region

1

2 3 4

5

Isobaric boiling of water, p=1.0133 bar, T-v-diagram

Figure 7.4: The isobaric evaporation of water at p=1,0133bar, T-v-Diagram

State (2) is called boiling liquid or saturated liquid, state (4) is called condensing vapor or saturated vapor.

This behavior is plottet in a T-v-diagram in figure 7.4 as an isobaric line.Conclusion: Even such a seemingly simple substance such as water already shows a very complicated behavior,which we want to be able to describe. The first and second law of thermodynamics describe natural processesin general without making statements about material properties. In order to be able to use these relationspractically, one needs additional data concerning the characteristics of the materials, which can be describedin the thermal and caloric equations of state (EOS). The theory of the classical thermodynamics makes nodeclaration about these EOS. The equations of state (EOS) are the result of measurements. Between thethermal equation of state pV = mRT and the caloric equation of state dU = mcvdT the Gibbs fundamentalrelation exists:

T ds = du +pdv = dh vdp. (7.1)

In general three phases are distinguished: gas, liquid and solid. The solid can occur further in differentphases, e.g. diamond, graphite, amorphous carbon. The characteristics of material mixtures are still muchmore complex than that of pure materials, which we (nearly) only regard. From our observations so far,we want to show and to develop some new diagrams and characteristics and define some new terms forpure substances. Fortunately the different diagrams look similar to each other and additionally they are alsosimilar for different substances. Mostly in the course of studies we see diagrams of water, because wateris the most important substance for us humans, naturally and technically, and thus it is also one of the

best-investigated substance. For the equilibria of each phase of a pure material there is a thermal equationof state: f(p, v, T) = 0. In a three-dimensional p v T-diagram, p as function of v and T, the functionf(p, v, T) = 0. represents gives a two dimensional surface p = p(v, T), which we will show later. The tripelpoint (TrP) of water is the point of state of water in which water exists in all three phases simultaneously.




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The boiling procedure of water 7

T-v-diagram of water

liquid

two phase region

vapor

Figure 7.5: T v diagramm of water with several isobaric lines.

as ice, as liquid and as vapor. Tripel point of water:

Ttrp = 273, 16K, ptrp = 0, 00611bar

The critical point (kP) is the point in the diagrams (p v, T s, hs) which marks the maximum pressureabove which no phase transition liquid to vapor can be detected. Critical point of water:

Tk = 647, 3K, pk = 22, 115MPa , vk = 0, 00315m3

kg

7.3 The boiling procedure of water

Now we want to study the boiling of water in more detail, see again figure 7.3, and above all we want toextend it into the higher pressure range. First we want to show for the process form figures 7.3 and 7.4 aT-v-diagram with several isobaric lines, figure 7.5, and then also a p-v-diagram with several isotherms, figure7.6.In figure 7.7 we see a p-v-diagram for a pure substance with isothermal lines and solid state area, it is notfor water because the substance expands on the phase transition from solid -> liquid.

7.4 Three dimensional diagrams

The two-dimensional diagrams are only projections on a two dimensional area of the the respective ther-modynamic functions, e.g. p = p(T, v). Remember: For a one component system we need exactly two

independent variables of state for the exact description, each third variable is function of those two. In thenext figure 7.8 an example of a three dimensional diagram for a pure substance (not water) is shown. Itis not for water because it expands on phase transition from solid to liquid. One can follow the paths, i.e.isobaric processes A->F, G->I and L->M.


The boiling procedure of water7


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liquid

two phase region

vapor

p-v-diagram of water

Figure 7.6: p v diagramm of water with several isothermal ines.

p-v-diagram of wateraccording to H.D. Baehr, Hannover

solid

two phase

region l-v

vapor

Isothermal lines

Figure 7.7: p v diagram of a pure substance (not water) with several isothermal lines.




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Vapor pressure diagram 9

solid

two

phase

melting

thawing

desublimation

sublimation gas

p,v,T-plane of a pure substance (not water), according to H.D. Baehr, Hannover

p

v

We can follow the path,

A-F, or L-M, or G-I

see book of Baehr(Thermodynamik in

German) for moreon this

This is not water,because the solid expands

when it thaws.

Figure 7.8: Three dimensional diagram of a pure substance (not water), the p-v-area is shown, according toH.D. Baehr, Thermodynamik, Springer Verlag.

7.5 Vapor pressure diagramA further projection from the three-dimensional diagram is those of the so-called vapor pressure diagram,see figure 7.9, in which the vapor pressure as function of the temperature for a material is represented. Thefigure 7.9 is only one part of the complete area. A larger area is to be seen in figure 7.10. In this diagramalso the boundary lines between solid and gas, and solid and liquid are drawn. One can see the tripel pointand also the critical point. Also the difference between the behavior of water and normal substances canbe seen in the boundary line solid to liquid. And if we draw the vapor pressure curves in the form of ln(p)over 1/p they are (nearly) straight lines, see figure 7.11 in which the vapor pressure curves for Ar, N2, CO2and H2O are shown. Among other things this is very favorable for the measurement procedure, because oneneeds only to measure very few points in order to draw the straight lines for sufficient accuracy.

7.6 Definition of vapor content

As with each new chapter, after having observed the strange and complex behavior of even pure substances,we need a nomenclatur to handle it. First we define some abbreviations:

msolid = ms

mliquid = ml

mvapor = mv

On the boiling-point line: mliquid,boiling = m

On the line of saturated vapor: mvapor,saturated = m


Vapor pressure diagram9


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p -T- diagram, vapor pressure curve for water, in principle

gas

vapor pressure curve

liquid

1.0133

bar

100 C

Figure 7.9: Vapor pressure diagram for water p over T..

phase equilibrium in p-T-diagram

trP

sublimimate

desublimate

normal

substance

water

krP

gassolid

liquidcondensingsolidifying

melting

evaporating

Figure 7.10: Vapor pressure diagram p over T with solid state area.




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Definition of vapor content 11

vapor pressure curves as ln(p) versus 1/T(according to H.D. Baehr, Hannover)

1 T

ln(p)

1

1000K

!

"#$

%&

Figure 7.11: Vapor pressure diagram for water.

definition of quality x = vapor contents

saturated

liquid 2phasesubcooled

liquid saturated

vapor

superhatedvapor

heating until

boiling pointevaporation overheating

1 2 34 5

x =mvapor

mliquid + mvapor=

m ''

m '+ m ''definition of quality:

Figure 7.12: To the definition of the vapor content or quality, definition of quality and boiling process.


Definition of vapor content11


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Definition of quality, example of isobaric evaporation of water

!x1= not defined, x

2= 0, 0 "x

3"1,

x4=1, x

5= not defined

heating

20 OC

overheating

150 OC

end ofboiling

100 OCboiling

100 OC

start of

boiling

100 OC

Definition of quality:

x1

0 < x3

!!!!!!< 1x

4= 1

x2=

0

x5

Figure 7.13: To the definition of the vapor content or quality, where is x defined.

Definition of vapor content or quality:

x = mvapor

mliquid + mvapor= m

m + m(7.2)

Only in the two-phase area the quality or vapor content is defined as x, see also figures 7.12, and 7.13. Infigure 7.13 we see x qualitatively as follows: x1 = not defined x2 = 0., x3 0, 0001, x4 = 1. and x5 = notdefined.In the figure 7.14 we see how an extensive state variable is the sum of the individual state variables of thedifferent phases. One can deduce thus:

V = V + V

= mv + mv(7.3)

v =V

m(7.4)

v = Vm

= Vm+m

= (1 x) v + xv

= v + x (v v)(7.5)

x =v v

v v(7.6)

x =v v

v v=

h h

h h=

s s

s s=

u u

u u(7.7)




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Properties of state of real fluids 13

Definition quality: fraction of mass of vapor to total mass

1 2 34 5

quality:

V1= mv

liq

U, H, S the same procedure

V3= m

3'v

3'+ v

3''v

3'' = m

3v

3

Figure 7.14: To the definition of the vapor content or quality, extensive state variables.

7.6.1 Quantitative T-s-diagram and log(p)-1/T-diagram

Here we wish to show two additional diagrams, a quantitative T-s-diagram and a log(p)-h-diagram, seefigures 7.15 and 7.16.

7.7 Properties of state of real fluids

Now we know already the property behavior of real fluids, that is real liquids and real gases, we know theevaporation and condensing processes, and the two phase region. On the other hand we are familiar withthe ideal gas equation, pV = mRT, that is the thermal equation of state for ideal gases. We might wonder,if it could be possible, to derive a thermal equation of state (thermal EOS) which explains ideal gases and

real gases in the liquid-vapor region simultaneously.1

Johannes Diderik van der Waals, 1837-1923, see figure 7.17 found such an equation, and it is simple and itis based on mending the funamental assumptions we made for an ideal gas. The so-called van the Waalsequation is simple and physically justified. The idea is as follows: There are two basic assumptions, whichled to the ideal gas equation:

1. The molecules have no interaction force between them, if they are distant. This corresponds to thefreedom of attractive or repulsive forces between the molecules. If we would consider interaction forcesthis would lead to an additional internal pressure.

11) (source: wikimedia commons, the copy-right of this image has expired because it was published more than 70 years agowithout a public claim of authorship (anonymous or pseudonymous), and no subsequent claim of authorship was made in the70 years following its first publication, so the photo is public domain)


Properties of state of real fluids13


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1000kJ/kg 2000kJ/kg

h=3500kJ/kg

3000kJ/kg

100bar

500bar1000 bar

10bar

1,0 bar

50bar

0,1bar

0,01m3/kg

10m3/kg

0,1m3/kg

T-s-diagram water, quantitative, with isobaric, isochoric, isothermic lines

free according to H.D. Baehr, Thermodynamik, Springer Verlag)

Figure 7.15: Quantitative T s-diagram for water with isobaric lines, isochoric lines, and lines for constantenthalpy.

log(p)-h-diagram for pure substance

liquid

2phase

gas

Figure 7.16: Qualitative ln(p)-h-diagram for a pure substance, which is used in cooling technology.




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Properties of state of real fluids 15

Figure 7.17: Johannes Diderik van der Waals, 1837-1923, Quelle: wikimedia commons, copyright: seefootnote 1

2. The sum of the volumes of the individual molecules is small in relation to the total volume of thecontainer:

i

Vi V

. It might be good idea to include the molecules volume into the thermal EOS.

Consequentely the van der Waals EOS is:

p +

a

v2

(v b) = RT, (7.8)

The term a/v2 accounts for (weak) attractive or repulsive forces between the molecules, b accounts for the(small) volume of the molecules. The rsulting curves are shown in figure 7.18 and 7.19 as black isobaric lines,which continue into the two phase region as broken green curves. Values for a and b are tabulated for manysubstances and there are approximations for their determination from the critical values: (pcr, Tcr, Vcr).

The parts a-b and d-e in the two phase region can be observed, the parts b-d is thermodynamically notfeasible. We want to discuss the resulting curves:

1. Curve part a -> b, best see T-v-dia: imagine a liquid which is heated over a flame, say in a test tubein a students chemical lab. The liquid gets hotter and hotter untilit should start to evaporate, whichist refuses to do. Say the liquid is water at 1 bar, it gets e.g. to t = 105 Centigrade, and suddenly itseparates in a boiling liquid, a saturated liquid, and vapor bubbles, which rapidly accelerate the liquidfrom the test tube, like in an explosion. Therefore it is a must to wear always safety gogles whenworking in chemistry lab. The liquid at 1 bar and 105 C ist called superheated liquid (from a-b isliquid) and can exit only under very undisturbed circumstances.

2. Curve part b -> d: best see p-v-dia: if by a small disturbance the volume increases by dv >0 thepressure would get larger by dp > 0, thus the expansion would increase which is a positive feed backloop and makes this region instable, does not exist.


Properties of state of real fluids15


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van der Waals p-v-diagram

T2

T1< T

2

a

b

c e

d

van der Waals equation of state for real gasess

Figure 7.18: Van der Waals equation in p-v-diagram, green broken lines are van der Waals, straight blackline is thermodynamic equilibrium.

van der Waals T-v-diagram

p1 < p2

p2

a

b

ce

d

van der Waals equation of state for real gasess

Figure 7.19: Van der Waals equation in t-v-diagram, green broken lines are van der Waals, straight blackline is thermodynamic equilibrium.




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Equations of state (EOS) with virial coefficients 17

3. Curve part d -> e, this branch is called under cooled vapor and is made use of e.g. in the Wilsoncloud chamber and plays a very important role in meteorology and in the atmosphere. If we have airin between e d, it should have build rain droplets, but does not so, so it is still undercooled vapor.Until a disturbance happens and it separates in rain droplets and saturated vapor. This effect can showdifferences of several degree on Centrigrade from under cooled point to saturated point.

The values for a and b within the van der Waals equation are tabulated for different substances and one canestimate these also from the critical parameters (pkr, Tkr, Vkr). The van that Waals equation of state hasgreat theoretical importance particularly in statistic thermodynamics. The knowledge of the thermodynamicproperties of materials is fundamental for the solution of engineering tasks, especially in the chemical industryand process engineering. For practical use mostly other EOS (not the vdW EOS) are applied.

7.8 Equations of state (EOS) with virial coefficients

For practical use the van der Waals equation is not so often employed but one uses equations of state (EOS)

with so called virial coefficients. These are the result of more or less physically justified series expansions ofp depending on powers of 1/v = which are fitted to experimental results. For pure materials one makes aseries expansion, e.g. the Kamerlingh Onnes equation of state:

pv = A +B

v+

C

v2+

D

v4+

E

v6+

F

v8(7.9)

with the so-called virial coefficients A = RT, B = b1T + b2 + b3/T + b4/T2 + ., C =, D =, E =, F =.

Values for the coefficients are tabulated. There are many further equations of state in the literature, a goodintroduction is the work of Wagner and Span, Bochum, Germany, and of Kretzschmar, Zittau, Germany. For

water EOS is published within the VDI Heat Atlas. Some more important EOS are:

Beattie Bridgeman equation

Benedict Webb ruby equation

fair Kwong equation

fair Kwong Soave equation

Peng Robinson equation

7.9 Realgasfaktor

The deviation of the behavior of a real gas from that of an ideal gas can easily be estimeted with the helpof the so called real gas factor:

Z =pv

RT(7.10)

For ideal gases Z = 1.An example of Z as a function of T and p for water is shown in figure 7.20. As result from the picture one cansee that with low pressure or at high temperature Z 1, i.e. then the gas behaves with good approximationlike an ideal gas. Low pressure and high temperature means in comparison to the critical values pkr, Tkr.


Equations of state (EOS) with virial coefficients17


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Z =pv

RTfor water

real gas factor

ref: Perry)

H. Perry, Chemical

Engineers

Handbook, 5, 1973,according to

K.F. Knoche,

TechnischeThermodynamik,

Vieweg Verlag

Figure 7.20: Real gas factor for water in dependence on temperature and pressure, T and p are given as ratioto the critical values.

7.10 Liquid vapor region

In the liquid and gaseous state in thermodynamic equilibrium a material is homogeneous, each state is thencalled a phase.In the liquid vapor region the material heterogeneous, it consists of two separated phases.Here we want to study only those conditions in which the two phases are in thermodynamic equilibrium, thatis they have the same pressure and the same temperature.The specific variables of state, e.g. u,h,v,s, are naturally different for the two phases.In this section we limit our study to the liquid vapor region and to pure water.We learned that for a pure substance in thermodynamic equilibrium we need exactly to independent statevariables to describe the state completely. But because in the liquid vapor region T and p are coupled, weneed another state variable, that is the quality or vapor content from above, x.Lets first reconsider our definition from above:See figure 7.21, definition:

1. () stands for boiling liquid, thats saturated liquid, left boundary curve from the left hand side up tothe critical point, v, h, s, m.

2. () stands for saturated vapor, right boundary curve from the critical point to the right, v, h, s, m,etc.

The definition of vapor content or quality is:

x = mvml + mv

x =m

m + m.




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Liquid vapor region 19

liquid-vapor area

` ``

3

7

Figure 7.21: Liquid vapor region for water in p-v-diagram. See definitions of saturated liquid as () and assaturated vapor as ().

Wet vapor is a mixture from saturated liquid and saturated vapor, which are in thermodynamic equilibrium.

x =m

m + m=

m

mwith m = m + m (7.11)

And for example the volume as extensive variable of state is the sum of the partial volumes of the liquid andvapor phase:

V = V + V = mv + mv (7.12)

or for an arbitrarily selected state (3):

V3 = V

3+ V

3= m

3v3

+ m3

v3

(7.13)

And also

v =V

m=

m

m + mv +

m

m + mv (7.14)

v = Vm

= (1 x)v + xv = v + x(v v) (7.15)

with


Liquid vapor region19


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liquid-vapor area: Law of opposite lever armsGeometrical interpretation in p - v - diagram:

Law of the opposite lever arms

Figure 7.22: Rule of the opposite lever arms, a/b = m/m.

x = m

m + m = v

v

v v = h

h

h h = s

s

s s (7.16)

From this we can see, the rule of the opposite lever arms:

v v

v v=

x

1 x=

m

m=

A

b(7.17)

See figure 7.22.Because in the h s, T s, p v-diagrams the mixture state is on the straight line between the and

points these formulas are identical also for s and h, e.g. see figure 7.23.

Thus:

v v

v v=

x

1 x=

m

m=

A

b=

h h

h h=

s s

s s(7.18)

Thus from the diagrams one can directly read distances with the help of a ruler and from that get the ratioof the liquid to the vapor mass and the enthalpy and entropy quantitatively. The values on the boader lines() and () are given also in tables for the liquid vapor region.

7.11 Sketch of the most important h-s-diagram

It is often required that one should draw a process quickly and qualitatively into a h-s-diagram. For this oneshould know how the isothermal and isobaric lines run in the diagram, see figure 7.24. Note especially theisothermal line behavior above the critical point and in the liquid region.




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Sketch of the most important h-s-diagram 21

A

b

p = const

liquid-vapor area: Law of opposite lever arms

Figure 7.23: Rule of the opposite lever arms in h-s-diagram.

T

T

T

p

p

p

x

h-s-diagram of water with p and t lines

Notice especially the

course of the isothermal

lines.

Figure 7.24: Sketch of h-s-diagram with isobaric and isthermal lines for water.


Sketch of the most important h-s-diagram21


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7.12 T-s-diagram with evaporation heat

Definition of the heat of vaporization (evaporation enthalpy):

r(T) = h(T) h(T) (7.19)

r(p) = h(p) h(p) (7.20)

or

r = h h = (u u) +p(v v) (7.21)

with u u - internal heat of vaporization, and p(v v) - external heat of vaporization.

A nice interpretation of this is to be seen in the T s-diagram. Generally according to the Gibbs relation itis:

T ds = dh vdp (7.22)

If is p=const is valid:

T ds = dh vdp = dh (7.23)

if T = is const is valid:

T ds = dh vdp = dh = T(s s) = h h = r (7.24)

The evaporation enthalpy in the T s-diagram is the rectangle surface under the isotherms T, see figure7.25.

7.13 Mathematics: linear interpolation

Often we have to interpolate values from tables, that means intermediate values have to be found betweenthe fixed table values. Here is a simple example, see 7.26, 7.27 and 7.28, for more look at mathematics textbooks.

7.14 Clausius Rankine process, classical power plant stationprocess

Modern life depends on electricity, think of the skript I am just writing, or all the pumps which drain thewaste water continuously. Electricity in large quantities, say Giga Watts come from power plant stations,

fueled with hard coal, brown coal, oil, natural gas or nuclear fuel.Most of these power plant stations, gas fueled are one exemption, heat and evaporate water in a cycle whichthen drives a steam turbine which then turn an electric generator which delivers electricity in a range of GW.A typical modern hard coal fired power plant station delievers about 05 GW electricity.




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Clausius Rankine process, classical power plant station process 23

liquid-vapor area: Heat of vaporization

p

Surface = r = h ``- h `

Figure 7.25: T-s-diagram with heat of evaporation area.

linear interpolation

Figure 7.26: Mathematics, linear interpolation, I


Clausius Rankine process, classical power plant station process23


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Figure 7.27: Mathematics, linear interpolation, II


Figure 7.28: Mathematics, linear interpolation, III




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Elements of a simple vapor power plant

boiler

Condenser

feed waterpump

Clausius Rankine process

air +

fuel

exhaust gas

ash

Figure 7.29: Clausius Rankine process, configuration diagram.

The main parts of a modern power plant station are the boiler, the water cycle, the exhaust gas aftertreatment and the electrical turbine.

Here we will condsider the main water cycle, the so called primary water cycle. A fundamental and basicwater cycle, in which the important features can be studied, is the so called Clausius Rankine Cycle (CRC).It consists of just four parts and several assumptions are made for the principle cycle. In a real power plantstation we have much more apparatus, but with our simple approach here, we can later on extend our analysisto the more complex systems without much trouble.

We call a Clausius Rankine Process (CRP) the clock wise cyclic process which can be seen in figure 7.29.The corresponding h-s-diagram can be seen in the next figure 7.30. And an example of some real moderndata, is given in figure 7.31. Rostock power plant station is currently the most modern hard coal station inGermany.

First see figure 7.29.

The principle clock wise cycle consists of four main aggregates working with pure water. All data are justexample data from the Rostock power plant. Smaller and bigger plants exist.

Starting from state point (1), which is undercooled liquid water at a pressure p1 < 1bar, the feed pumptakes in the undercooled liquid water and pumps it to a pressure of e.g. p2 = 250bar, state point (2).

In the vapor generator, or boiler, the liquid water first is heated until it is saturated water (), is thenevaporated until it is saturated vapor () and then it is over heated until state point (3).

The heat inflow e.g. is Q23 +1.5GW.

A typical state (3) can be: p = 260bar, t = 545C, which is overheated water vapor.

The hot vapor (3) flows into the turbine which creates mechanical power by rotation of the axle and thenby use of a generator electrical power, w

t34 0.5GW < 0.

The water flow with low pressure and low temperature, state (4) then enters the condenser, where it is cooleddown to the state (1) again. State (4) typically is in the liquid vapor region with a high vapor quality x.

The heat flow Q41 < 0 goes as waste heat flow to the environment, e.g. to a river or in a cooling tower to




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h-s-diagram for Clausius Rankine process

Figure 7.30: Clausius Rankine process, h-s-diagram.

Some technical data to hard coal power plant in Rostock

Electrical output power (gross/net): 553/509 MWNet efficiency: 43,2%boiler firing thermal input: 1370 MWvapor flow rate: 1650 t/hLive vapor pressure (3): 262 barvapor temperature (3): 545 Cadditionally long-distance heat supply: max 300 MWEfficiency with max. long-distance heating: 62%

Source: www. kraftwerk-rostock. de

On this site there are also beautiful functional diagrams

Figure 7.31: Clausius Rankine process, some data of an example of the hard coal fired power plant stationin Rostock, Germany.




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the atmosphere.Thus the process is closed.We want to draw a h s-diagram of the cyclic process now, let us see figure 7.30.First we draw an empty h-s-diagram of water with the boudary line of the saturated liquid, the critical pointand the boundary line of the saturated vapor, and then isobaric lines, one for the high pressure, p2 and onefor the low pressure, p4.

How many isobaric lines do we need?That depends on what assumptions we make for the pressure drop by dissipation in the heat exchangers from(2->3) and (4->1). If we consider these as frictionless, then we have no pressure drop there, and then weonly need two isobaric lines, that is what we will do here.In principle we have the following informations about the cycle:

1st law for stationary flow processes (SFP): dh = q+ wt.

2st law for stationary flow processes (SFP): ds = (q)/T + (wr/T = (q)/T + sirr.

Table values and diagram values for the material variables like p, T, h, s, v, h, s, v, h, s and v.

Thus, lets follow the process:

process 1->2, feed pump:If the pump works adiabatically and frictionless, then ds = 0. If there is friction, then point (2) lieson the right hand side of point (1) and because of the 1st law for SFP, state (2) is over state (1).T ds = q+ wdiss, dh = q+ wt.

Process 2->3, boiler or vapor generator:q > 0, wt = 0 => dh > 0.If no friction arises, the pressure remains constant, if friction arises, the pressure drops slightly. With aClausius Rankine process mostly one regards the boiler as frictionless. If the water vapor is overheated,then point (3) lies on the right hand side and above point (2) on the isobaric line of p2 = p3.

process 3->4: turbine:See also chapter 4 in which we explained such an expansion in a turbine process in very detail. If theturbine is adiabatic and frictionless, then ds34 = 0. If some friction arises, then point (4) is on theright hand side of point (3), and because of the 1st law for SFP (4) is below point (3) dh < 0.

process 4->1: Condenser:q41 < 0, wt41 = 0, dh < 0. If no friction arises, the pressure remains constant, if some friction arises,the pressure drops slightly. With the classical Clausius Rankine process one regards the condenser asfrictionless. The vapor (4) becomes full condensed and somewhat undercooled (1).

That meany the typical assumptions for the Clausius Rankine process are: the pressure drop in the vaporgenerator and condenser is neglected, the irreversibilities in the pump and in the turbine are considered.In the next figure ?? some typical data of the hard coal power plant station in Rostock, Germany, are shown.This is currently the most modern hard coal station in Germany.Now we can do some calculations with the help of the 1st and 2nd law.The total cyclic Clausius Rankine processIf we regard the entire process as closed system, it follows:

wt12 + wt34 = q23 + q41 (7.25)

The delivered work is the turbine work (wt34 < 0) plus the pump work (wt23 > 0) and is altogether negative:




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wt,ges =W

m= wt12 + wt34 < 0 (7.26)

For all individual aggregates the 1st law for SFP is valid:

0 = h1 h2 + wt12 + q12 (7.27)

And the 2nd law for stationary flow processes reads:

0 = s1 s2 +q12T

+ serr (7.28)

With the frictionless evaporator and frictionless condenser T as thermodynamic mean temperature is:

s3s2

T ds = Tm23(s3 s2) (7.29)

The feed pump (1->2):The pump work is:

wt12 = h2 h1 > 0 (7.30)

In oder to get q quality measure for the pump, we define the isentropic pump efficiency as ratio of the pumps

real work to the minimum work which would be required in a reversible process (ds = 0, see also figure 7.30:

p12 =minimumwork

realwork=

h2 h1h2 h1

1 (7.31)

The pump work is small in the comparison with the turbine work:

|wt12| < |wt34| (7.32)

The vapor generator:The vapor generator is assumed to be frictionless and thus an isobaric process. In reality the pressure dropcan be several % of the input pressure.

q23 = h3 h2 > 0 (7.33)

The adiabatic turbineThe adiabatic turbine work is:

wt34 = h4 h3 < 0 (7.34)

We define here also an isentropic turbine efficiency, in order to measure the quality of the turbine, see againfigure 7.30:




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T34 =realworkdone

maximumworkpossible=

h4 h3h4 h3

1 (7.35)

The condenser:In the condenser the remainder of the enthalpy of the flow is delivered as waste heat to the environment in

oder to close the cyclic process. The condenser is assumed as frictionlessly and thus as isobaric:

q41 = h4 h1 < 0 (7.36)

Clausius Rankine process altogether:The duty for the Clausius Rankine process altogether:

wt,ges = wt34 + wt12 = (|wt34| |wt12|) < 0 (7.37)

wt,ges = wt34 + wt12 = h4 h3 + h2 h1 (7.38)

The thermal efficiency of the entire cyclic process is:

th =|wt,ges|

|q23|=

|h4 h3| |h2 h1|

|h3 h2|< 1 (7.39)

7.14.1 Exergy efficiency

As we have seen, the exergy is the availability of the energy, that is the part of an energy that is freelyconvertible into any other form of energy. Thus an exergy efficiency is the ratio of the exergy coming out ofa process or a machine in comparison to the exergy which is put in.Thus we define the exergy efficiency for the CRP as technical output work flow Wt in relation to the exergyflow which the boiler takes in, that is the exergy flow of stream flow (2) minus exergy flow of stream flow(1): m(e2 e1):

p = Wt

m (e2 e1)=

wte2 e1

(7.40)

e2 is the specific exergie of the water flow at (2), e1 is the specific exergie of the water flow at (1),

e1 = h1 hu Tu (s1 su) (7.41)

e2 = h2 hu Tu (s2 su) (7.42)

e2 e1 = h2 h1 Tu (s2 s1) (7.43)

wt,ges = wt34 + wt12 = h4 h3 + h2 h1 (7.44)

p = h4 h3 + h2 h1

h2 h1 Tu (s2 s1)(7.45)

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