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- 1. i_HtrFr' =f- I f.' Gontents Preface to first edition Preface to second edition Preface to third edition Preface to fourth edition Acknowledgements 1 General introduction 2 Systems *]."The laws of thermodynamics 4 Steam and two-phase systems 5 Gases and single-phase systems 6 Thermodynamic reversibility 7 Entropy 8-Steam plant 9 The steam engine 10 Nozzles 11 Steam turbines 12 Air and gas compressors 13 Ideal gas power cycles . 14 Intemal combustion engines $15 Engine trials te-Combustion 17 Refrigeration 18 Heat transfer Appendix I Summary of the International System of Units (SI) Appendix 2 Thermometry Appendix 3 Pressure and its measurement Appendix 4 Two-phase property tables Appendix 5 Note on gravitational potential energy and also kinetic energy Index iu Preface to first edition This book has been written to cover the work necessary for the subject of Heat Engines set in the Ordinary National Certificate and Ordinary National Diploma examinations and also examinations of similar ,, standard. The book has been divided into separate chapters covering various branches of the study. So far as possible, it has been arranged so that it can be worked through progressively from the beginning to the end although certain self-contained chapters may be read separately. Ideas are developed assuming a minimum knowledge, and every attempt has been made to present the work so that it does not appear simply as a collection of facts. It is hoped that in this way it has been made more interesting and readable. There are numerous worked the text. Where possible the examples are taken from past examination papers of various examin- ing institutions. These past examination questions have been used in order that the student will appreciate the standard required at the earliest opportunity. The author has made a step-by-step solution of the worked examples, thus piloting the student through the work without making undue assumptions. It must be remembered that whereas many steps may appear obvious to some, and could therefore be left out of the text, these same steps are not so clear to others. It is hoped that the book will cater for all. There are questions, with answers, set for private working at the end of each chapter. These are similarly, in the main, taken from past examination papers. Examples are marked as to their origin, but where unmarked they are original. The book has been illustrated throughout and every effort has been ln- ist, lled or in ni- v. vii, viii xi xii I s7 1t- 82 139 t99 209. 2!3 303 338 359 385 420 515 582 624 703 723 748 754 770 785 and line has to first her first 788 791
- 2. 1, Aoknowledgements The author wishes to express thanks to the following: M anufacturing organisations for their illustrations and information Babcock International PLC Figs. 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 8.10, 8.11, 8.13, 8.14; Ford Motor Company Figs. 14.12, 14.L3,14.14,14.15, 14.16,14.17,14.18; E. Readerand Sons, Limited Fig.9.1,9.10; Rolls Royce Limited Figs. 14.23, 14.24; Ruston and Hornsby Limited Fig. 14.30; Information on super-critical boilers from Simon-Carves Limited. Table in Appendix 4 This table is taken from a papercalled'Properties of water of importance in Thermometry and Calorimetry', by Professor E. J. Lefevre of London University, given at the National Conference on the Change to the International System (SI) Units for Energy held at Church House, Westminster, London in 1968. The Conference was sponsored by the Ministry of Technology, now the Department of Trade and Industry. General introduction 1.1 The word thermodynamlcs is made up from two words: Thermo - from a Greek word meaning hot, or heat, and, Dynamics - the study of matter in motion - again from a Greek word meaning power, or Powerful. Thus, the wo rdthermodynamicsmeansthe study of heat related to matter in motion. Much of the study of engineering (or applied) thermodynamics is concerned with work producing or utilising machines such as engines, turbines and compressors together with the working substances used in such machines. There are certain topics which are common and fundamental to many sections of the study of engineering thermodynamics' It is the purpose of this general introduction to investigate these common fundamentals. In the chapters which follow, it will be seen how they apply and are related to other sections of the study. 1.2 Working substance All thermodynamic systems require some working substance in order that the various operations required of each system can be carried out. The working subitances are, in general, fluids which are capable of deformationln that they can readily be expanded and compressed. The working substance also takes part in energy transfer. For example it can receive -or reject heat energy or it can be the means by which work is done. Common examples of working substances used in thermodynamic systems zre air and steam' I ., - xu
- 3. SPECIFIC QUANTITIES If a value can be assigned to a property then it function because its value can be plotted on a graph- Properties which are independent of mass, such pressure, are said lo be intensiue properties. to be a point Properties which are dependent upon mass, such as energy in its various forms, are called extensiue properties' volume and If a property can be varied at will, quite independently of other properties, then the property is said to be an independent property. The temperature and pressure of a gas, for example, can be varied quite independently of each other and thus, in this case, temperature and pressure are independent properties. It will be found, however, when discussing the formation of a vapour, that the temperature at which a liquid boils depends upon the pressure at which the formation of the vapour is occurring. Here, if the pressure is fixed then the temperature becomes dependent upon the pressure. Hence the pressure is an independent property but the temperature is a dependent property. A knowledge of the various thermostatic properties of a substance delrnes the state ofthe substance. Ifa property, or properties, are changed, then the state is changed. Properties are independent of any process which any particular substance may have passed through from one state to another, being dependent only upon the end states. In fact, a property can be identified if it is observed to be a function of state only. Since, at a particular state, a substance will have certain properties which are functions of that state, then there will be certain relationships which exist between these properties. These relationships will be investigated in the text. A property which includes a function of time, used to deline a rate at which some interaction can occur, such as the transfer of mass, momentum or energy, as examples, is referred to as a transport property. Examples of such properties are thermal conductiuity and uiscosity. 1.6 Specific quantity In the discussion of properties it was suggested that those properties which were associated with the mass of a substance are called extensive properties. For convenience, at times, it is useful to discuss the properties of unit mass of a substance. To indicate that this is the case, the word speci/ic is used to prefix the property. as temperature and li,'lJl: ls that by also re, to be and are the the of lnd
- 4. a cHNFrAl, tN'rlot)r,('1'loN . Thur, lhc rpocific volume of a substlnce at some particular state, is the volume whlch unit mass of the-substance wil occupy at that particu- lrr ltrto, Othor rpocific quanrities will be discussed in the text. 1,7 Temperature To undcrrtand the meaning of temperature it is necessary, first, to rcfor to the human sense of feeling. It is common experience to talk aboul iome things feeling hot and other rhings feeling cold. No special training in ncccssary to make the decision whether one thing is hotter sr coldor than another. The decision is made as a result of the natural rogctlon of the sense of feeling. . ln mmc cnses, this_ sense of feeling is a satisfactory method for the tlotormlnrtion of whether something ii hot or cold enorigh. Roi ixampte, hforo gotting into a bath, it is quite good enough to iect the water to tlocidc whcther it is too hot or too cold. At each subsequent bath it woulel bc possible to make the water about as hot or cold as before. Tho watcr would probably not be exactly as hot or cold as previously urcd, howcver, since the sense of feeling is not sufliciently accurate to admit of exact repetition. A further important point about this sense of fccling is that it differs slightly from person to person. what is 'too hot' for one pcrson may be Just right' for another and'too cold'for somebody clsc, Fccling, therefore, is not particularly satisfactory where a high degree of accuracy is required. Accuracy is, of course, not the sort of thing re- quirod when taking a bath. There are, however, many things which requiro a high dcgree of accuracy when determining their degree of hotn"r, or coldncss. Thc sense of feeling is not good enough in thise cases. It is thoroforc nocoriary to.dlevise some method otherlhan personal feeling for tho dotorminntiorfof hotness or coldness. A grort mrny dcviccs, both ancient and modern, have been designed and mrdo for thir purpose. Each device uses the ellect that hotness or coldnorr hm on romc particular substance as the means by which the dclrcc of hotnur or coldness is measured. For example, all substances, to r $mtff or lomor oxtcnt, expand when heated and contract when oooltd, Tbr frtlnt lo which this expansion or contraction occurs can bc urd $ I nuilure sf tho change in hotness or coldness. A scale of hotnru ot ooldnil murt tro dcvised, however, such that each device will rooord thr nnU dojroc of hotncss or coldness when used in the same condltionr, Hrvlnl llrod r rsrlo, it is useful to have a single word to denote roforoncc to thlr mlc, Tho word ls temperature and the scale is called the temperatun tca/l, Thr rubJoot of temperature investigation TEMPERATURE 5 is called thermomety. The methods used in thermometry will be dis- cussed in Appendix 2 under that name. It is necessary here, however, to investigate the temperature scales used. Many attempts have been made in the past to lay down a scale of temperature. The work has culminated in the generally accepted use of two temperature scales. These are the Fahrenheit and the celsius scales. The Fahrenheit scale is named after Daniel Gabriel Fahrenheit (1696- 1736), born in Danzig, who devised this scale of temperature. The Celsius scale (often referred to as the centigrade scale) is named after Anders Celsius (1701-44), born at Uppsala, who is described as the inventor of the scale. The Celsius scale is the temperature scale which is most bommonly used world wide. The Fahrenheit scale is generally becoming progressively phased out. The customary temperature scale adopted for use with the SI system of units is the Celsius scale. For customary use, the lower fixed point is the temperature of the melting of pure ice, commonly referred to as the freezing point. This point is designated 0'C. The upper fixed point is the temperature at which pure water boils and this is designated 100'C. In the past, this customary temperature scale has been referred to as the Centigrade scale. The use of the word Centigrade is now discouraged, the accepted reference now being that of the Celsius scale. - of interest, the freezing and boiling points of pure water are designated 32'F and 212'F, respectively, on the Fahrenheit scale. It will be shown later, when dealing with the properties of solids, liquids and vapours, that the temperature at which a liquid freezes or boils depends upon the pressure exerted at the surface of the liquid. This temperature increases as the pressure increases in the case of boiling and slightly decreases with increase of pressure in the case of freezing. To standardise the freezing and boiling temperature on a thermometric scale, therefore, requires that the pressure at which the freezing or boiling occurs be also standardised. This pressure is taken as 760 mm of mercury which is calledthe stanilard atmospheric pressure or the standard atmosphere, being a mean representative pressure of the atmosphere. Figure 1.1 shows the way the customary Celsius scale is divided up. The lower fixed point being 0'C and the upper hxed point being 100.C, then there are 100 Celsius degrees between these two hxed points. This 100 Celsius degrees is called the fundamental interual. In the above discussion it will be noted that the choicb of the fixed
- 5. Jff (IBNNNAL I NTRODTICTION Fl3. l.l points was of an arbitrary nature. The freezing and boiling points of water were chosen for convenience. Other points on the International Temperature Scale are then chosen and referred to the originally con- ceived scale. Since the original choice of fixed points is of an arbitrary nature, the Celsius scale is sometimes referred to as the normal or empirical or customary or practical temperature scale. Since the Celsius scale is only a part of the more extensive thermo- dynamic, or absolute temperature scale, it is sometimes called a truncated t hermod.y na,mic scale. Subsequent work will show that there is the possibility of an absolute zero of temperature which will then suggest an absolute tempetatute scale. An absolute zero of temperature would be the lowest temperature possible and this would, therefore, be a more reasonable temperature to adopt as the zero for a temperature scale. The absolute thermodynamic temperature scale is called the Keloin scale. This scale was devised by Lord Kelvin, a British scientist, in about 1851. The Kelvin unit of temperature is called the keluin and is given the symbol K. A temperature, f, on the Kelvin scale is written TK, not ?oK. The kclvin degree has the same magnitude as the Celsius degree for 'bll practical purposcs. Thc abeoluta zero of temperature appears impossible to reach in practicc. Howcvcr, itr idontity is dofinod by giving to the triple point of water a value of 273,16 kohin (273.16 K), The triplo point ir dofincd in Chapter 4, Steam and Two-Phase Systems. With thc abroluto zoro ro dcfrncd, the zero of the Celsius thermodynamic scale is dcftnod ar 0'C - 273.15 K. Thus, t : T-273'15 PRESSURE 7 where, t : temperature on the Celsius thermodynamic scale : f"C T: temperature on the Kelvin thermodynamic scale : TK From equation (1), T: t*273'15 (can use 273 for most calculations) (2) By choosing the zero of the Celsius thermodynamic scale as OoC : 273'15 K, this approximates very closely to the customary celsius scale and thus 0"c on the customary celsius scale is very nearly equal to 0.c on the celsius thermodynamic scale. Also, 100"c on the customary celsius scale is very nearly equal to 100"c on the celsius thermodynamic scale. Practical consideration of the measurement of temperature is given in Appendix 2, Thermometry. 1.8 Pressure Pressure is defined as force per unit area. Thus, if a force F is applied to an area A, and if this force is uniformly distributed over the area, then the pressure P exerted is given by the equation P :: (r) A If, F : force in newtons (N) A = area in metres2 (m2) then the unit of pressure becomes the newton/metre2 (N/m2), which is the basic unit of pressure in the SI system of units. This unit of pressure is sometimes called the pas cal (Pa). . Common multiples of this basic unit of pressure will be the kilonewton/ metre2 (1kf/m,: t03N/m2) and the meganewtonfmetre2 (1MN/mi : 106 N/m2). The bar inay also be commonly used (l bu. : ior Nimry as may also the hectobar (l hbar : 102 bar : 107 N/m2) but it should be noted that these are not preferred units in that they are not multiples of the power of 3. (See Appendix 1.) If a force is applied to a solid then it will be transmitted through the solid in the direction of application of the force. In Fig. 1.2 (a) is shown a solid being.pressed against a fixed wall by rReans of a force F. If the contact area is A then the pressure set up at the contact surface : FIA and it is normal to the contact surface. , on the other hand, Fig. 1.2 (b) shows a piston of area.4 enclosing a fluid in a cylinder. If a force F is now applied to the piston then a prcssure P : FIA will be set up in the fluid. Unlike the solid, howevei, where (1)
- 6. OENERAL INTRODUCTION Fig. 1.2 this pressure would be transmitted in the line of action of the applied force, in the fluid, this pressure P : FIA is set up in all directions in the cylinder. Any vessel then, in which there is a fluid under pressure, must be capable of withstanding the pressure in all directions. The fact that the pressure distribution in a fluid does occur in all directions is easily demonstrated by blowing up a balloon which swells up in all directions. Practical consideration of the measurement of pressure is given in Appendix 3. 1.9 Volume Volume is a property, being that property which is associated with cubic measure. The unit of volume is the cubic metre (m3) together with its multiples and submultiples. Sometimes the litre (l) may be used. 1 litre : I cubic decimetre (1dm3: [10-trn]t).Although in the SI system of units the litre is a non-preferred unit, it is already in such common usage that its use will, no doubt, continue. If the volume of a substance increases then the substance is said to have been expanded. If the volume of a substance decreases then the substance is said to have been compressed. Specific volume is given the symbol u. The volume of any mass, other than unity, is given the symbol I/. f.10 Phase When a substance is of the same nature throughout its mass it is said to be in a phase. Matter can exist in three phases, solid, liquid and vapour or gas. If the matter exists in only one of these forms then it is in a single phase. lf lwo phases exist together then the substance is in the form of a two-phase mixture. Examnles of this are when a solid is being melted into a liquid or when a liquid is being transformed into a vapour. In a THE CONSTANT TEMPERATURE PROCESS., 9 single phase the substance is said to be homogenous. If it is two-phase it is said to be heterogenous. A heterogenous mixture of three phases can exist. This is discussed under the heading of the Triple Point during the discussion on steam. 1.11 Two-property rule Now that the concept of properties, state and phase have been indicated, it is now possible to write down the two-property rule. Iftwo independent properties ofa pure substance are defined, then all other properties, or the state ofthe substance, are also defined. Ifthe state of the substance is known then the phase or mixture of phases of the substance are also known. The idea of the two-property rule was suggested in section 1.3. l.l2 Process When the state of a substance is changed by means of an operation or operations having been carried out on the substance, then the substance is said to have undergone a process. Typical processes are the expansion and compression of a gas or the conversion of water into steam. A process can be analysed by an investigation into the changes which occur in the properties of a substance, and the energy transfers which may have taken place. 1.13 Cycle If processes are carried out on a substance such that, at the end, the substance is returned to its original state, then the substance is said to have been taken through a cycle. This is commonly required in many. engines. A sequence of events takes place which must be repeated and repeated. In this way the engine continues to operate. Each repeated sequence ofevents is'called a cycle. l.l4 The constant temperature process This is a process carried out such that the temperature remains constant throughout the process. It is often referred to as an isothermal process. Particular cases of the constant temperature process will be dealt with in the text. PISTON AREA A -
- 7. l0 (IIINERALTNTRODUCTrON l.lt The constant pressure process Thin is a process carried out such that the pressure remains constant lhroughout the process. It is often referred to as an isobaric or isopiestic proccss. Particular cases of the constant pressure process will be dealt with in the text. l.16 The constant volume process This is a process carried out such that the volume remains constant throughout the process. It is often referred to as an isometric or isochoric process. Particular cases of the constant volume process will be dealt with in the text. l.l7 EnerBJ Energy is defined as that capacity a body or substance possesses which can result in the performance of work. Here, work is defined, as in mecha- nics, as the result of moving a force through a distance. The presence of energy can only be observed by its effects and these can appear in many'diflerent forms. An example where some of the forms in which energy can appear is in the motor car. The petrol put into the petrol tank must contain a potential chemical form ofenergy because by burning it in the engine, the motor car, through various mechanisms, is propelled along the road. Thus, work, by dehnition, is being done because a force is being moved through a distance. As a result of burning the petrol in the engine, the general tempera- tures of the working substances in the engine, and the engine, will be increased and this increase in temperature must initially have been responsible for propelling the motor car. Due to ths increase in temperature of the working substances then, since the motor car is moved and work is done, the working substance at the increased temperature must have contained a form of energy resultant frorn this increased temperature. This energy content resultant from the consideration of the temperature of a substance is called internal energy (see section 1.23). Some of this internal energy in the working substances of the engine will transfer to the cooling system of the engine because the cooling water becomes hot. A transfer of energy in this way, because of temperature dilferences, is called heat-transter (scc scction 1.24). WORK AND THE PRESSURE_VOLUME DIAGRAM II The motor car engine will probably have an electric generator, or alternator, which is rotated by the engine and is used to charye the battery. The battery, by its construction and chemical nature, stores energy which can appear at the battery terminals as electricity. The electricity from the battery can be used to rotate the engine starter which, in turn, r-otates and starts the engine. By rotating the engine to start it, the electric motor must be doing work and thus, electricity must have the capacity for doing work, and hence is a form of energy. To stop the motor car the brakes are applied. After the motor car has stopped the brake drums are hot and thus, as discussed above, the internal energy of the brake drum materials must have been increased. This internal energy increase resulted from the stopping of the motor car and hence there must have been a type of energy which the motor car possessed while it was in motion. This energy of motion is called kinetic energy (see Appendix 5). From this discussion it will be seen that energy can appear in many forms and further, it appears that energy, through the action of various devices, can be converted from one form into another. All the possible forms of energy have not been discussed here. More will be said about energy, and its various forms, later in the text. l.l8 Work If a system exists in which a force at the boundary of the system is moved through a distance, then work is done by or on the system (see chapter 3, Systems). Note that as soon as the force eeases to be moved then any work which was being done also ceases. work is therefore a transient quantity being descriptive of that process by which a force is moved thiough a distance. work, being a transient quantity, is therefore not a property. work is given the symbol w.lf it is required to indicate a rate'ai which work is being done then a dot is placed over the symbol IZ. Thus W : work done/unit time 1.19 Work and the pressure-volume diagram consider Fig. 1.3. In the lower half of the diagram is shown a cylinder in which a fluid at pressure P is trapped by means of a piston of area A. The fluid here is the system. From this, Force on piston : pressure xarea: pA (l) Let this force PA be just sufficient to overcome some external load. -
- 8. t2 CENERAL INTRODUCTION Fig. r.3 Now let the piston move back a distance L along the cylinder while at the same time the pressure of the fluid remains constant. The force on the piston will have remained constant. Work done : Force x distance : PAx L (2) This equation could be rearranged to read Workdone:PxAL ,,{L : volume swept out by the piston, called the'swept'or'stroke' volume : (v2_v) .'. Work done : P(V2-VL) (3) Above the diagram of the piston and cylinder is shown a graph of the operation plotted with the axes of Pressure and Volume. Such a graph is called a P*V diagram or graph, sometimes said to be illustrated on a P-V plane. The graph appears as horizontal straight line ab whose height is at pressure P and whose length is from original volume Z, to frnal volume 7r. Now consider the area abcd under this graph. Area : P(V2-V) But this is the same as the worl6 done equation (3). WORK AND THE PRESSURE_VOLUME DIAGRAM 13 Hence it follows that the area under a P-V diagram gives the work done. This can be shown to be true by an analysis of the units. If the pressure is in newtons/metre2 (N/m2) and the volume is in cubic metres (m3) then, by equation (3), the work done is given by the product of pressure and change in volume. ... units of work done : #r -, : Nm which is the unit of work. Now the graph shown in Fig. 1.3 illustrates the particular case of constant pressure expansion. Fig. 1.4 Consider Fig. 1.4. Here is shown a P-V diagram of the type usually obtained when an expansion takes place in a thermal engine. It will be noted that it is now a curve with original pressure and volume P, and I/r, respectively. The final pressure and volume are Prand Vr, respectively. It will be noted that both the pressure and the volume have changed in this case. What of the area under this graph and will this area still give the work done? Figure 1.5 shows the same graph but this time it has been divided up into small rectangles. The area of each small rectangle represents work done as has been shown. The sum of all the areas of these small rectangles But AREA I {vr- v, ) PRES5URE A - (4) Flg. r.5
- 9. 14 oENERAL tNTRoDUcrroN would, therefore, approximate very closely to the area under the graph and hcnce the work done. The greater the number of rectangles then the more nearly does the sum equal the actual area and hence the actual work done. lf the number of rectangles was made infinitely great, therefore, the sum would, in fact, equal the actual area which would then give the actual work done. Now this is exactly what happens when the area is solved by the use ofthe integral calculus. Consider Fig. 1.6. Here is illustrated the same P-V diagram. Consider sorne point in the expansion, X say, where the pressure is P and the volunre is V.Let there be an elemental expansion dlz from this volume Z. Then, Work done during the elemental expansion : P 6V (4) The total work done will be obtained by summing all the elemental strips of width 5V from volume V, to volume Vr. ... Total work done :'--f'r 6, (5) y=yt Now, if an infinite number of strips are taken, in which case 6V becomes inlinitely small, or as it is written, as 6V --- 0, then THE pol-yrRoplc pRocEss pt-:c, A CONSTANT 15 compression. The compression curve plotted on a p-v diagram has the same general shape as that of an expansion except that it is in a reverse direction. This means that the volume decreaseJ while the pressure in- creases. Thearea under thecurve,given byequation(6), gives the work done. It should be noted that when the area for an expaision is determined it is positive, indicating that'work is obtained froman expa.nsion. when the area for a compression is determined it is negative, indicating that work must be done on the working fluid in order to compress it. EXAMPLE 1 A fluid in a cylinder is at a pressure of 700 kN/m2. It is expanded at constant pressure from a volume of 0.28 m3 to a volume of l.6g m3. Determine the work done. Work done : W: P(Vz-V) : 700 x 103(1.69 -0.2s) :7x105x1.4 : (9.8 x 105) Nm : (9.8 x 105)J : 980 kJ : 0'98 MJ Work done : 1", o, J,, (6) Fig. 1.6 Nothing more will be said about this equation at present. It will be met later when it will be solved for particular cases. Its limitations will be discussed in Chaptcr 6 on Thermodynamic Reversibility. As an important point it will be noted that the discussion has con- centrated on expansion. In engines, however, many cases of compression are encountered. The comprcssion is really the reverse of an cxpansion. What has been said about expansion, therefore, applies equilly well to 1.20 The polytropic process PVo : C, a constant changes of state of working substances in thermodynamic systems are often brought about by the expansion or compression of the working substance. Suppose, then, that an experiment is conducted on a mass of working substance such that an expansion takes place changing the state lrom state 1 to state 2.Letthe pressure change from p, to p, and the volume froy't v, to trzr. Assume that arrangements are made to re-cord the pressure and volume as the experiment proceeds. If, then, from the results o'btained, values of pressure and volume are plotted on a p-v graph then a smooth curve results as showir in Fig. 1.7. simply by inspection of the curve, it is not directry possibre to teil whether there is a law connecting pressure and volum. ioi tt. expansion carried out. However, suppose now that log p is plotted againsi log lz. The graph obtained is as shown in Fig. t.g. rhis is much b"etter for the graph appears as a straight line and is of th-eJorm logP: -nlogV+logC I - (1)
- 10. t6 OENERAL INTRODUCTION Fig. 1.7 Fig. l.E where, - n : slope of the line, log C - intercept on the log P axis. Now equation (1) can be rewritten log P+nlogV:logC or taking antilogs, PV' : C, a constant (2) Further experiments on dilTerent substances taking different quan- tities of substance and also including the case of compression as well as expansion will yield a similar result. Equation (2) may therefore be considered as the law for the general case of expansion or compression of a substance. This geneqal case of expansion or compression of a substance according to the law PY" : C is called a polytropic expansion or compression ot a polytrope. It should be noted that the value of the constant C will change with each change of condition, so also will the value of n which is called the index of the expansion or compression or the polytropic exponent. Since all conditions of state during the expansion or compression lie : o.tt2' 'J# :0.fi2' 'J; --.-Fi,t;ryF: EXAMPLES 11 on the curve PV": C, then it follows that PrVl' : PzVi : P3V! : P4Vt :..., etc. where 1, 2,3,4, etc., represent different conditions of state taken during the expansion or compression. To give an idea of the value of the index n, it will generally lie within the range I to 1.7. For most cases, however, it will probably lie more closely within the range 1.2 to 1.5. Further, note, that if n : 0 then the equation becomes, PVo: C,aconstant and since Vo : I then equation (4) becomes, P:C,aconstant which indicates a constant pressure process. Also, the equation PVo : C can be rearranged to read, Prl'V: C,aconstant (6) This is obtained by taking the nth root of both sides. Now if n: @, then Plln : ptta : Po : l, in which base, equation (6) becomes, V:C,aconstant which indicates a constant volume process. EXAMPLE 2 0'll2m3 of gas has a pressure of 138kN/m2. It is compressed to 690 kN/m2 according to the law PVr'4 - C. Determine the new volumc of the gas. Since the gas is compressed according to the law PVr'a : C, then, (3) (4) (5) (7',) PrVrt'n: PzVz 'i: (2)" o,2: from which. (t)"" vz: ,,(3)"' : vr 'l'i
- 11. 18 GENERAL INTRODUCTION 0.r12 0.112 :- '{s 3.157 Vz: O'035 5m3 EXAMpLES 19 l.2l Work and the polytropic process. If a substance is to be compressed from a lower pressure to a higher pressure then work will be required in order to carry out the compression. when the substance is at the new high pressure, it now has the potential to expand and, in expanding, do some work. It is important to determine the magnitude of this quantity of work. It has already been shown that ' work done is given by the area under a p-v diagram of"n ""punrion or compression and.hence the problem now is to determine the area under a curve of the form PV" : C. consider Fig. 1.9. Here is shown a p-v graphof an expansion according to the law PV" : C from state pr, I/, to new state pr, Vr. Now PZ' : C, or P : CV-" Substituting equation (2) in equation (l), work done : cf" v-'dv lntcgrating J'' work done : -9. I v-,.r1" -n+ I L J", :firrrn+t _v;n+7f C _ n + I LVr- nVr.- Vr- "Vrf _ P"V._ P.V, L-n+,1- from equation (2), nnd multiplying top and bottom Uy - l: work done : P*tv'- PLv' n-t g) Equation (4) will apply equally well to an expansion or a compression. By reading Pr, V, as the original conditions and pr, Z, as the final conditions it will be found that for an expansion, the work done is positive, meaning that work is done by the substance. For a compression, however, again readinE pr Vr as the original conditions and P r, V, as the final conditions it will be found that the work done is negative, meaning that the work must be done on the substance. EXAMPLE 3 0'014 m3 gas at a pressure of 2070 kN/m2 expands to a pressure of 207 kN/m2 according to the law PVl'35 : C. Determine the work done by the gas during the expansion. The work done during a polytropic expansion is given by the expression: Work done _ Prvt- P2V2 n-l In this problem Z2 is, as yet, unknown and must therefore be calculated. (2) (3) Fig. 1.9 consider a point on the curve at which the pressure is p and the volume is z Let the gas expand from this point by very small volume dlzaccording to the law PV': c- The work done during this very small expansion is very nearly equal to p6v rn the lirnit, as ,6v--* e the area, "ni h"no the work done : P dV. For the whole expansion from I and 2. Work done : I:,' PdV (t)
- 12. r 20 GENERAL INTRODUCTION Now P,Vi = pzV!, .'. Vz: rrG)''" or vz : 0'014" /2ozo trt'rs - ) : 0'014" t'rs"/10: o'014x 5'505 Vr:0'077 ^t ... Work done : {(2070 x 103 x 0'014) - (207 x 103 x 0.077)} 1.35 - 1 103 _ - 103 : 0.35 Q9 - ts'95) : fij x 13.05 : (37.3 x 103) Nm : (37.3 x 103) J (l Nm : lJ) :37'3kJ 1.22 Work and the hyperbolic process Tl_"_ hyp:.-bolic process is a particular case of the polytropic process, PV" : C, being the case when n : l. Thus, the law for the hyperbolic process is, This law, if plotted on a p-V diagram will appear as a rectangular hyper- bola and hence its name. For a hyperbolic change from state 1 to state 2, from eouation (l) PV:C P1V1 : PrY, (1) (2) (3) substituting An expression for the work done during a polytropic process has already been determined. This has been shownio be, Work done _ PrVL-P2V2 n-l In the case of an hyperbolic process p1V1 : prV, and this in equation (3), together with the fact that n : l, then, Work done - Prvr- P2V2 -0n-l 0 This is indeterminate- EXAMPLES 2I Now if the law PV : C is plotted on a P-V graph there is a definite area beneath the curve. This being the case, then, it appears that a start from first principles is necessary to determine this area. Consider Fig. 1.10. By similar analysis to that given for the polytropic expansion, workdone : [n,'ra, (4) Fig. 1.10 In this case, however, PV: C, hence P : CIV and substituting this in equation (4), then, work done : ,1.,, ff : ,l^ ,frr,, : C[ln Vr-,| Vrf : C ln ft: ,rnft since PIz = C. Here ln represents log.. Now VrlV, is the expansion ratio which'is often letter r. .'. Work done : PV ln r EXAMPLE 4 A gas is compressed hyperbolically from a pressure and volume of 100 kN/m2 and 0.056 m3, respectively, to a volume of 0.007 m3. Determine the hnal pressure and the work done on fhe gas. (5) designated by the (6)
- 13. 22 cENERAL rNTRoDUcrroN Since the gas is compressed hyperbolically, then, PrVr: PrVror Pr: Pr|: 100 x ffi : tOO -5.6x103In8 - 5'6 x I03 x2{79 -(Ll.64x 103)Nm : -(tt.64x 103)J - 1r.64kJ i.e. Work done on the gas : 11.64 kJ 1.23 Internal energy If a hot body is placed in contact with a cold body then the temperature of the hot body begins to fall while at the same time the temperature of the cold body .begins to rise. To account for this it is said that the hot body gives up heat and hence its temperature falls, while the cold body receives this heat and its temperature rises. observing this fact some early investigators, around the eighteenth century, considered that heat must have properties similar to that of a fluid. One such fluid was called'caloric'. Thus, if a body was heated, then caloric was said to have passed from the source of heat supply into the body and hence it became hot. Conversely, if a body cooled, it lost some of its caloiic. Since the weight of the body was unafiected by either being heated or cooled it was considered that caloric was a weightless fluid and was said to hll the minute spaces or pores of the body. Another such fluid was called 'frigoric' which wap used to explain the phenomena of cold. It was said to be composed of minute darts of . INTERNAL ENERGY 23 frost, If one's hand is placed on a piece of ice, for example, not only does it fccl cold but it can feel somewhat painful. It was suggested that the rcsson for this was that the dart-like particles of frigoric were being lrunsferred from the ice into the hand and thus the hand became cold nccompanied by a sense of pain. Yet another fluid which accounted for the heating effect produced by n fire was called 'phlogiston'. To produce its heating effect the hre was raid to have given up phlogiston to the bodies being heated. As a matter of interest, when it was found that a gas, which is now known as oxygen, wus directly associated with the phenomena of combustion, it was called 'dephlogisticated air'. The reason for this was that this gas, being the runly one associated with the production of fire, was said therefore to be the only gas without phlogiston and hence, as it received phlogiston, fire rcsulted and heat was produced. Nearly all the known phenomena of heating and cooling could be oxplained by the introduction of such fluids as caloric, frigoric and phlogiston. These theories were eventually found to be false. Notable among ihose who showed them to be wrong was Count Rumford of Munich. (lount Rumford was really an American citizen, born near Boston in 1753. His real name was Benjamin Thompson and he had to leave America for his part in the rebellion of the British Colonies. He settled in Bavaria where he became the superintendent of an arsenal in Munich. He was rewarded with the title'Count Rumford' for his services in this respest. During his work in the arsenal, he noticed that when boring a cannon, the material of the cannon became extremely hot and this did not seem to tie up very well with the caloric theory which was widely accepted at this time. He accordingly conducted an experiment in an attempt to settle this rnatter. Instead of using a sharp boring tool he used a blunt one which was rubbed against a mass of about 51 kg of gunmetal. After some 960 revolutions the temperature of the gunmetal had risen by itbout 39'C. A minute quantity of gunmetal had been rubbed off during the experiment. Htre then was an experiment in which there had been no hot source to supply heat by yielding up some of its caloric in heating the gunmetal, and yet the gunmetal had, in fact, become extremely hot. It might have been suggested that the caloric which was originally in the metallic dust which had been rubbed off had been left behind in the main bulk of gunmetal and hence had produced the temperature rise. The amount of metallic dust was so small and the quantity of heat developed so large that Count Rumford concluded that this was impossible. It alio appeared that so long as the rubbing, or friction, of the boring tool continued, then heat would continire to be produced.and hence the supply of heat gener- x8 : 800 kN/m2 Work done : PVlnr: pVhYi vl 0.007 : 100 x 103 x 0.056In 0.056 : - 100 x 103 x O.OSO tn 0'056 0.007
- 14. 24 GENERAL INTRODUCTION atcd in this way was inexhaustible. He therefore suggested that since the hcat was generated as a result of motion, then heat was in some way the result of the motion of the particles which make up a body. There was nothing really new about this idea. It had been the view taken by some philosophers from very early times. After its short accepted life, in and about the eighteenth century, the supposed flriid nature of heat was dropped. It then became the accepted theory that heat was a manifestation of the degree of agitation of the very minute particles (atoms and molecules, to be discussed later) which make up a body. Part of the old philosophy remains, however, for it is common practice to refer to such things as 'the flow of heat' and .quantity of heat' which still suggest a fluid nature of heat. It is important, however, to think a little more about the theory that heat is a result of the degree of agitation of the particles which make up a body. If a particle is in motion, then it will possess kinetic energy, which is a function of the velocity at which the particle is moving (see Appendix 5). It appears that the greater the kinetic energy which can be imparted to the particles which make up a body then, in general, the higher the temperature of the body will become. It has now become clear that the store of energy which results from the random motion of the atoms and molecules of a body would be far better referred to as internal energy, leaving the term heat to be used to describe that energy transfer process which results from a temperature difference. At any one particular state, the atoms and molecules will have a particular overall degree of random motion and, in a pure substance, this degree of random motion will be the same each time the substance returns to that state. The degree of random motion must therefore be a property. Since internal energy is a function of the degree of random motion, then internal energy must be a property. Count Rumford's experiment showed that, in his particular case, an increase in internal energy content resulted in an increase in temperature. This is always the case in a single-phase system. count Rumford's single- phase system was the gunmetal cannon, and note here that the internal energy increase was the result ofa blunt boring tool being rubbed against the cannon. The energy transfer in this case was really a work transfer, work having been done against friction. It must be noted, however, that an internal energy increase does not always result in an increase in temperature. It will be shown during the discussion on two-phase systems that when the phase is being changed from one to another, such as Water into steam, the temperature will remain constant. Here, the internal energy increases at constant tem- perature, the increase in internal energy being necessary to carry out the HEAT 25 degree of separation of the molecules to change the water into steam. The same general situation arises during the change of a solid into a liquid. Another point arises out of the discussion of internal energy. It has been stated that the internal energy of a substance results from the motion of its atoms or molecules. In a fluid, the atoms and molecules have rather greater motions than solids, and do, in fact* move freely about (rather more freely in the case of gases). This means that the atoms and molecules will be constantly impinging upon the walls of any con- taining vessel. Now the impact of a particle on a wall means that a force will be imparted to the wall. The constant bombardment of the walls of the containing vessel results in a total average force on each wall. When this average force is reduced to that which occurs on unit area of the wall it is called the pressure on the wall. Again, in the above discussion it has been noted that the internal cnergy content is the result of the motion of the atoms and molecules which make up a body. Further, it was noted that an increase in internal cnergy content in general results in an increase in temperature. As the temperature of a body falls, therefore, the motion of the atoms and molecules reduces, and the internal energy content also reduces. From this, then, it is reasonable to assume that a condition of state exists in which the atoms and molecules of a body are completely at rest, in which case the internal energy content would then be zero and the temperature would have reached its absolute zero. The idea of an absolute zero of tcmperature has already been mentioned in section 1.17. In the case of internal energy, specific internal energy is designated u, the internal energy of any mass, other than unity, is designated U. 1.24 Heat l)uring the discussion on internal energy it was suggested that at one time it was considered that a body could contain heat. This is now not con- sidered as being the case, the internal store of energy being now referred lo as internal energy, which is a property. However, it was further suggested that, during an energy transfer process which results from the temperature difference between one body and another, the energy so transferred is called heat. The heat, having been transferred, will then disperse into other forms of energy such as internal energy or work, the dispersal being a function of the $ystem employed. Note that heat is a transient quantity, it being simply descriptive of the energy transfer process through a system boundary resulting from
- 15. 26 GENERAL rNTRoDUcrIoN temperature difference. If there is no temperature difference, then there is no heat transfer. Note, also, that since the term heat is used to describe a transfer process, then heat energy ceases to exist when the process ceases. Thus heat is not a property. Heat energy is given the symbol Q. To indicate a rate of heat transfer, a dot is placed over the symbol Q. Thus ' A: heat transfer/unit time 1.25 Specific heat capacity For unit mass of a particular substance at a temperature t, let there be a change of temperature dr brought about by a transfer of heat 6e. The specific heat capacity, c, of the substance at temperature r is defined by the ratio 6Ql6t. Thus, In the limit, as df --+Q, 1fie1, ,:dQdt Specific heat capacity is generally found to vary with temperature. For example, the specific heat capacity of water falls slightly from a temperature of 0'C to a minimum at about 35"C and then begins to rise again (see Appendix 4). Specific heat capacity can also vary with pressure and volume. This is particularly true of compressible fluids such as gases (see Chapter 5). It is common practice to use an average value of specific heat capacity within a given temperature range. This average value is then used as being constant within the temperature range. This being the case, equation (2) can be rewritten where, Q : heat transfer/unit mass, /kg, At: change in temperature, K. Note that from equation (3) the basic unit for specific heat capacity is jouleslkilogram keluin or fkg K. Such a multiple as kilojouleslkilogramkeluin (kJlkgK) may also be used. EXAMPLES 27 A particular case in the use ofspecihc heat capacity arises in the use of water as a measuring device in calorirnetry. Such measurements of tcmp€rature as are made during a calorimetric experiment are made while the pressure of the water remains constant. A process in which the pressure remains constant is said to be isobaric. The specific heat capacity in this case is therefore said to be the isobaric rycciJic heat capacity and is written cr. Tables of values of c, for water from 0"C to 100'C are given in Appendix 4. Table I gives a few examples of average speciltc heat capacities of some solids and liquids. The specihc heat capacity of gases is dealt with sepa- rately in Chapter 5. TAsr-n l. rnsrn oF AvERAGE SPECIFIC mAT cAPACITIES Solids Liquids ^ _6Q "- at (1) (2) Specific heat Substance capaclty J/ke K Specific heat Substance capaclty JlkeK Specific heat Substance capaclty J/ke K Aluminium Brass Cast iron Copper Crown glass Lead Nickel Steel Tin Zinc 915 375 500 390 670 130 460 450 230 390 Benzene 1700 Ether 2300 Ethyl alcohol 2 500 Paraffin 2130 Mercury 140 (3)._a-- Lt EXAMPLE 5 5kg of steel, specific heat capacity 450JSK, is heated from 15"C to 100'C. How much heat is required? From equation (1): Heat require ^ =T;'iiiL'* - "' , : lgi2skl
- 16. 28 cENERAL rNrRoDUcrroN EXAMPLE 6 A copper vessel of mass 2 kg contains 6 kg of water. If the initial temperature of the vessel plus water is 20'C and the final temperature is 90'C, how much heat is transferred to accomplish this change, assuming that there is no heat loss? From Table 1: Specilic heat capacity of copper : 390 Jr&g K Heat required by copper vessel : 2x390 x (90-20) :2x3X)x70 :54600J From Appendix 4: Specific heat capacity of water at 20'C: 4 181.6 J/kg K Specific heat capacity of water at 90"C :4204.8JlkgK Average specific heat capacity of water in the temperature range 20.C to 90"c 4 181.6 + 4204.8 8 386.4 : 2 : 2 :4t93.2JAeK .'. heat transfer required by water : 6x 4193.2x (90 -20) :6x4193.2x70 :1761144J (see section 1.26, Caloimetry, (4), for an alternative, more method) .'. heat transferred to vessel+water : 54600* L76ll44 :1815744J : 1815'744kJ : 1.815 744MJ equation accurate EXAMPLE 7 An iron casting of mass 10 kg has an original temperature of 200"C. If the casting loses heat to the value 715.5kJ, what is the final temperature? From Table 1: Specific heat capacity ofcast iron:500J/kgK Heat transferred from casting : mc(tz-t) (l) CALORIMETRY 29 Heat transfetred : -715'5 kJ : -715 s00J Notc the negative sign, indicating a heat loss. lirom equation (1), therefore, -715500: 10x 500x (tr-200) 7 15 500 tz : 200- r r ra, : 2AO - l43'l tz: 56'9'C l:XAMPLE 8 A liquid of mass 4kg has its temperature increased from 15"C to 100"C. Heat transfer into the liquid to the value 7l4kJ is required to accomplish the increase in temperature. Determine the specific heat capacity of the liquid. Heat transfer required : Q: mc(t.t-tr) .o.. c------:- and Q:714kJ:714000J m(tt-tt) - 714000 714000 :4(1oo-lt: 4x 85 714000 :- 340 : 2 100 J/kg K :2.tkJkeK 1,26 Calorimetry 'l'hc subject of calorimetry is concerned with the determination of standard thcrmal quantities, such as specific heat capacity and the calorific value of I'ucls (see section 1.30 and Chapter 16). The principle can be explained in the.simple type of experiment used for the determination of the specific heat of a solid. Figure 1.11 shows a sketch ofan apparatus used. It consists ofan outer can A inside which is contained a smaller can B which is supported on $upports G. This small can B is called a calorimeter and is very often made of copper. The calorimeter is first weighed and is then partly ltlled
- 17. 'l ii I 30 GENERAL INTRODUCTION Fig. 1.ll with water. It is then reweighed. From the difference between the two weights the mass of water in the calorimeter is determined. The calorimeter with its water content is then placed on its supports in the outer can and a cover plate F is placed over the top. A thermometer C and a stirrer D are placed through this cover plate such that they have their ends immersed in the water. This assembly is then left to attain a steady temperature, which can be observed on the thermometer. In the meantime a piece of the solid, E, to be tested, is weighed and then heated to some known temperature. For example, it could be placed in the steam over boiling water. It could then be assumed that after a reasonable period of time the piece of solid would have reached the temp€rature corresponding to boiling point. When this is so, and the temperature of the apparatus is steady, the cover plate is quickly lifted and the solid very quickly transferred into the water. of the calorimeter, making sure that surplus water is not carried over with it from the heating vessel. The cover plate is quickly replaced and stirring immediately commenced in order to ensure as even a distribution of temperature as possible. A careful check is now kept of the thermometer which will record an increase in temperature. This temperature increase will continue until a maximum has been reached. This maximum tem- perature is recorded. Now consider what has happened inside the calorimeter after the hot solid has been introduced. Due to its higher temperature it will, immediately, upon introduction, transfer some heat to its surroundings. Hence its temperature will begin to fall. On the other hand, the surround- ings, receiving this heat, will have their temperature elevated. This transfer of heat will continue, and can only continue until a common temperature has been reached by the whole apparatus. This linal common temperature will lie somewhere between the original solid and apparatus temperatures. Since at this final common temperature all heat exchange oeases, then at N this condition it can b€ stated that, Heat transfer loss by solid : CALORIMETRY Heat transfer gain bY water 3l (l) mass of solid x specific heat capacity of solid x temperature drop of solid : mass of water x specific heat capacity of water x temperature rise of water flw : trl&SS of water zs : lrl&SS of solid c : specific heat capacity ofsolid cp : sPeciltc heat capacitY of water f r : original temperature of water fz : original temperature of solid t : final common temperature then, from equation (2), ms x c x (t z- t) : fr. Y c o x (t - t t) from which, m*c,(t - t r) c:- (3) m,(t t- t) Now it has already been stated that the specihc heat capacity ofwater vtries with temperature and fuence equation (3) assumes that the value of the specilic heat capacity of water, co, used is the average within the lcmperature range tt to t. A more accurate way to determine the heat transfer into or out of water is as follows. It will be shown in subsequent work (see Chapter 4, Steam and Two- l)hase Systems, for example), that, if a process is carried out at constant prossure, then the heat transfer appears as a change of enthalpy in the substance at constant pressure. An initial definition of enthalpy is given in scction 1.29. Specific enthalpy is given the symbol /r. Now the water in the experiment already described is at constant prcssure and hence, in this case, heat transfer will appear in the water as a change of enthalpy. If, flw : rrlsSS of water h : specific enthalpy of water at temperature / hr : sPecific enthalpy of water at temperature tr (2)
- 18. 1i 32 cENERAL rNTRoDUcrroN Change of enthalpy : m*(h-h) Equation (3), now becomes, m"(h-hr) L-- m"(t r- t) Appendix 4 gives values of specific enthalpy for water in joules/kilogram (J/ke). Note that in example 6, the heat transfer required by the water could have been more accurately determined by the use of equation (4). Using the table given in Appendix 4, Specific enthalpy of water at 20"C : 83949 Jkg Specific enthalpy of water at 90'C : 376934JkC .'. using equation (4), Change of enthalpy : heat transfer required by water : 6(376934-83949) :6x292985 : 1757 9l0J This more accurate value is slightly less than that obtained in example 6, in which the average value of specific heat capacity of water within the given temperature range was used. A further point which needs correction is that, in equations (3) and (5), no account has been taken ofthe fact that, as well as the water being heated by the hot solid, so also has the apparatus such as the calorimeter, the stirrer and the thermometer. They have also been heated through exactly the same temperature range and have thus received some of the heat transferred from the solid. A correction for this may be made as follows: Let, m1 : mass of calorimeter m2 : mass of thermometer immersed n3 : mass of stirrer immersed cl : specilic heat capacity of calorimeter cz : specific heat capacity of thermometer cg : specific heat capacity of stirrer then since, Heat transfer gained : Heat transfer lost ryc o(t - t ) + m, c r(t - t r) + m rc r(t - t r) + m rc r(t - t r) : m,c (t, - t) I 'i l iI E CALORIMETRY (m*c o * mrc, + m 2c 2 + mtc t)(t - t ) : m"c(t z - t) Now before continuing, consider the heating of the calorimeter alone. The amount of heat transferred into the calorimeter is givenby mrcr(t - tr). Let the mass of water which would be raised through the same tcmperature change (t-t) by this same amount of heat transfer be m,r. Then, since both heat quantities are equal it follows that: m"tc r(t - t r) : mtc t(t - t t) 33 (6) (s) lfl"lco: lll1c1 Note that from equation (7) m"t:Y!2ep The mass, r?i"1, iS called the water equioalent of the calorimeter. Note that co in this case is the average value within the range f, F rom equation (7), in a similar way, m62cp: m2C2 mscp: m3Ca nnd substituting back into equation (6) (m*c o* m"rc o * m"2c ol mgc r)(t - t ) : m"c(tz - t) or (m* * m", * m"2 * m"r)c e(t - t ) : m"c(t z - t) which can be writtqr (m* + m")c oQ- /r) : m"c(t, - t r) (7) (8) to t. ffie: ffietIm"2*tn"3 (e) (10) (1 l) (12) rn" : the total water equivalent of the apparatus. This can be estimated hy substituting equation (8) in equation (12) thus, mtcrlm2c2lm3ca cp frr: (13)
- 19. { li ili ll I rl iili I r; ll lrl ill il{ ;i 34 GENERAL INTRoDUCTIoN Also, by the use of the enthalpy table for water, equation (11) becomes, (m, + m.)(h - h r) : m"c(t, - t) (14) Instead of finding the water equivalent of the separate parts of the apparatus, the water equivalent of the whole calorimeter system can be directly determined by conducting an experiment using a substance of known specific heat capacity. Thus, in equation (14), the only unknown would be the water equivalent of the calorimeter system, m", which may therefore be determined. The value m" then becomes a standard for the apparatus as used and can then be used in the determination of unknown specific heats using the same apparatus. A second point which needs correction is that of heat transfer loss. As soon as the temperature of the apparatus begins to rise above that of the surroundings then heat transfer will commence to be lost to the surroundings. As a result of this heat transfer loss, the maximum temperature, recorded as t above, will be slightly low. This will evidently result in an error in the determination of specific heat capacity. Many methods have been devised as means to correct for this heat transfer loss. A satisfactory cooling correction method is obtained by using Newton's Law of cooling. This law states that the 'rate of cooling of a body is proportional to the excess temperature of the body above its surroundings'. The law is reasonably accurate so long as the excess temperature above the surroundings is not too high. Large temperature changes are not usually obtained during experiments in calorimetry and hence a cooling correction using Newton's Law of Cooling can be obtained. According to this law, then, if the rate of cooling, xc say, is known at some temperature, te say, the temperature of the surroundings being /o,.then if this cooling rate is plotted on a graph, a straight line joining this point with ro on the temperature axis will give the graph of the rate of cooling at any temperature. Note that at to the rate of cbofing is zero since the apparatus is then at the same temperature as that of the surroundings. The graph is shown in Fig. 1.12. ' The problem now is to determine the actual loss of temperature which has occurred from a knowledge of the rate of cooling. Consider Fig. 1.13. This shows a graph of the rate of cooling plotted against time. Consider a point on the graph where the rate of cooling : x and the time = 0. From this point let the time increase by elemental time period 60, Then the loss of temperature during this small period of time : x60. From this, then, 02 Total loss of temperature from time 0, and 02:lxi? This is the area under the graph. . " CALORIMETRY 35 l'lg. l.r2 l'ltr. l.l3 Lct the graph between time 0, and time 92 be linear, then, if Rate of cooling at time 0r: xr tfld Rate of cooling at time 0z : xz 'l'he area under the graph X'*X".^ :-;(0r_0) : loss of temperature due to cooling during this period It (02- 0 ) : l, i.e. unit time, then Loss of temperature due to cooling : (x1*x)12 :X^ 'l'lris is equal to the average rate of cooling during unit time. (16) ) (ls) (t7)
- 20. { I x GENERAL INTRODUCTION TRUE MAXIMUM TEMPERATURE Fig. 1.14 Now if, during the progress of a calorimetry experiment, the temperature is recorded against time, then from the results a temperature-time graph can be plotted as shown in Fig. 1.14. If unit time is small enough, then during each separate unit interval of time, a rate of cooling-time graph may be considered as being sensibly straight. Ifthis is the case then, the loss of temperature due to cooling during each unit interval of time will be given by (xr+xr)12, from equation (17). Now the rate of cooling will not have been plotted on a time base but on a temperature base as shown in Fig. 1.12. However, when the rate of cooling is xr, at the beginning of unit time interval, the temperature is rr. At the end of unit time interval, the rate of cooling is x, and the temperature is rr. Since the rate of cooling-temperature graph is linear then the temperature loss during unit time interval, which numerically equals the mean rate of change of temperature (xr*xr)12: x. will be given at the mean temperature tn between f, and /r. From this the temperature-time graph can be corrected for temperature loss. A calorimetry experiment in which cooling correction is included is conducted as follows. The calorimeter assembly, before the introduction of the hot solid is left to attain steady atmospheric temperature /o. The hot solid is then introduced and the temperature is recorded at subsequent intervals of one minute. Maximum temperature will sodn be reached and from then CALORIMETRY 31 on the temperature will begin to fall. Temperature readings are con- tinued for a period while the temperature is falling A temperature-time graph is then plotted from the results. The general shape of the graph will be as shown in Fig. 1.14. OA will be the period while the temperature is rising and AB will be the recorded period during cooling. From this graph the rate of cooling-temperature graph is obtained as follows. Select a short section of the graph during the cooling period AB where the graph is straight or approximately straight. This is shown as XY on Fig. l.l4.Let the temperature and time at X be f" and 0" respectively. Let the temperature and time at Y be ty and 0, respectively. The fall of tcmperature will be given by (t,-tr) and this will have occurred in a time (0 r- 0,). Thus the rate of fall of temperature per unit time during this period u IG U E, f F G, uo- =u F u )&. F U G l F c U c E U F t,-tn or-o, "e (18) 'l'his rate of fall of temperature can then be assumed to equal the rate of fall of temperature actually occurring at the mean temperature during the selected period. - (r"+ tr)12: t" (19) 'T'he point xe,te can then be plotted as indicated in Fig. 1.12, and thus the nrte of cooling*temperature graph is obtained. From this, the temperature-time graph is corrected as follows. The mean temperature during the first minute is determined and the rute of cooling at this mean temperature is determined from the rate of cooling-temperature graph. Then, if the actual temperature at the end of the hrst minute : /r ?rd the mean rate of cooling during this period * .xr, the temperature which would have been attained at the end of the ' lirst minute, had there been no cooling : 1tt + x1). The mean temperature is now determined during the second minute. 'l'hc rate of cooling at this temperature is determined from the rate of cooling-temperature graph : x2. If the actual temperature at the end of lhc second minute: fz, then the corrected temperature : t2*(x1*x). Note that the correction in this case: (xr*x2), since this is the total loss which has occurred in 2 minutes. Similarly, at the end of 3 minutes the correct temperature will be 1.1 *(xt *x2*xr), and so on. A new temperature-time graph can now bc plotted which will flatten out at the top. This is as it should be because ln a case where there is no cooling taking place, when all heat exchange has occurred and the maximum temperature has been reached then the lcmperature will remain constant. /
- 21. 38 GENERAL INTRoDUCTIoN The true temperature rise can now be extracted from the new graph which is illustrated in Fig. 1.14. EXAMPLE 9 A calorimeter of mass 0.llkg contains 0.34kg of water. The initial temperature of the water and calorimeter is 16'C. A piece of solid of mass 0'41kg is heated to a temperature of 100"C and at this temperature it is immersed in the water of the calorimeter. The temperature of the calorimeter system rises to 32'C. Neglecting heat loss and the water- equivalent of the caloriineter, estimate the specific heat capacity of the solid. From equation (5): , _m*(h-ht) m"(tr- t) 0.34(t34 t0t-67 2t8) [Enthalpy values from Appendix 4] 0.41(100 - 32) EXAMPLE 11 During the experiment conducted in the above examples, temperature was recorded against time. The following results were obtained. EXAMPLES 39 I From equation (14), t- (m** m")(h -hr) m"(tr- t (0'34 +0.010 4x 134 101 - 67 2 18) 0.41(100 - 32) 0'3504x66883 l.ii lll iii 1l ill 041 x 68 : 840.6 Jr&g K i I 0'34 x 66 883 : 815.65 Jr&g K EXAMPLE 10 If the specific heat capacity of the material of the calorimeter in the above example is 394JlkgK, estimate the specific heat capacity of the solid including the water equivalent of the calorimeter. c, for water at 16'C : 4184'6 J/kg K co for water at 32"C : 4178.0 Vkg K fValues from Appendix 4] Average c, for water in the temperature range l6"C to 32"C 4184.6+4178.0 8362.6 : 2 : 2 :4181'3J/kgK From equation (8), 394 m": o'l1 " aIft : o'o1o4 kg Time 0 I 2 3 4 5 6 7 8 9 t0 ll 12 13 t4 (min) Temp. 'c 19.4 22.9 26.3 29.4 3t.s 32.r 32.1 31.8 31.4 31.0 30.5 30.0 29.6 29.0 Using these results, determine the specific heat capacity of the solid, including the water equivalent of the calorimeter and the cooling correc- tion. The results of this experiment and the determined cooling corrections are best set out in tabular form. From the results, the temperature*time graph is plotted as shown in Fig. I 15. A straight portion of the cooling section of this graph is now selected in order to determine the rate of cooling at the mean temperature within the selected section. The graph is relatively straight within the section 9 to l3 minutes. At 9 minutes the temperature : 3l.4oC, from the graph. At 13 minutes the temperature : 29.5.C, from the graph. Fall of temperature during this interval : 31.4-29.5: 1.9.C This fall of temperature has taken place over a period of four minutes. .'. rate of fall of temperature during this period : 1.914: 0.475'C/min: water equivalent of calorimeter.'
- 22. GENERAL INTRODUCTION o r E F E U 4 U F 345 678 TIME MINUTES ffil ii il$ tll tl ilt t{ {'l ,i ilil ':i'l 1il - Fie. 1.15 N EXAMPLES 4I Mean temperature during this interval 31.4+29.5 60.9 22 : 30'45'C This point is plotted on the rate of cooling-temperature graph, Fig. 1.16, and it is joined to 16'l'c on the temperature axis. 16.1"c is the temperature at commencement when the rate of cooling is zero. The correction of the temperature-time graph can now proceed. The mean temperature during each one-minute interval is deter- mined and column 3 of the table completed. At each mean temperature, Time (min) Temp. ('c) Mean Temp. temp. correction ('c) ("c) Total temp. Corrected correction temp. ('c) ("c) 0 I 2 3 4 5 6 7 8 9 r0 il t2 r3 t4 16.1 t9.4 22.9 26.3 29.4 31.5 32.1 32.1 31.8 3t.4 31.0 30.5 30.0 29.s 29.0 17.75 21.t5 24.6 27.85 30.45 31.8 32.1 31.95 31.6 31.2 30.75 30.25 29'8 29.3 0.055 0.165 0.280 0.382 0.475 0.520 0.580 0.575 0.s70 0.500 0.485 0.470 0.45s 0.435 *t 0.220 0.500 0.882 1.357 r'877 2.457 3.032 3.602 4.102 4.s87 5.057 5.512 5.947 16.1 t9'455 23.120 26.8 30.282 32.857 33.977 34.557 34.832 35.002 35.102 35.087 35.057 35.012 34'947 thc rate of cooling in oc/min is determined from the rate of cooring- lomperature graph, shown in Fig. 1.16. Column 4 is completed in this way. 'l'he total temperature correction is now completed from column 4 by rtlcling the individual temperature corrections and thus column 5 is 0rltnpleted. ('olumn 6 is now completed, which gives the actual temperatures, by etlding the temperature correction to the original temperature. These c!'lual temperatures are now plotted on the temperature-time graph as thown in Fig. 1.15. From this corrected graph, the true maximum tem- pcnrlure, which would have occurred had there been no cooling, can now be tlctermined.
- 23. o GENERAL INTRODUCTION RATE OF FALL OF TEMPERATURE 'CIMIN o o(' o 9do(' ol ! @ (o N o N oo 9s5{oo oo 5 z z+ N) t) N o) -{MN <51t m trp >(Jl {c I r*, o N { N o t) (,o (, o (j) ot N) o, Ct) (t 5 (^, ol ii iri Irili iilil ril il,i rt rl 11 {il lilr ill ilil1 { Fig. 1.16 EXAMPLES From the graph, the corrected maximum temperature :35.12"C tJsing values obtained from Appendix 4, At 35.12'C c p : 4 177.9 +0.12(4 178.0- 4 177.9) J lkgK Note that, 4 178.0 /kgK : cp at 36"C 4177.9 JlkgK : c, at 35oC co at 3112"C : 4177'9 -t0'12(0.1) : 4177.9 +0.012 : 4177-912 JkeK At 16.1'C c p : 4 184'6 +0' 1(4 183'7 - 4 184.6) : 4 184.6+0.1(-0.9) : 4 184.6 -0.09 : 4 184.51 J/ke K average co in the range 16.1"C to 35'12"C 4l84.sl +4177.912 8362.422 . :----^ :4181'2IlJ7'kgK. At 35.12'C h : 146635 +0'12(150813 - 146635) : 146635 + 0'12(4178) : 146635+ 501.36 : r47 t36.36Jke Ar 16.1"C h : 67 218+0'1(71 402 - 67 218) : 67 218 + 0'1(4 184) : 67 218 + 418.4 : 67 636.4 Jke 394 n,:O'lI" +tgl2fi:0.0104kg, from equation (8) (m* * m") (h - U,from equation (14) ': ^r1r-, _ (0.34 + 0.010 4) (147 136'36 - 67 636'4) 0.41(100- 35.12) 0'3504 x79499'96 0'41 x 64.88 : 1048 Vkg K
- 24. 4 GENERAL INTRODUCTION EXAMPLE T2 An experiment was carried out on a calorimeter system to determine its water equivalent by using a piece of copper of known specific heat : 390 J/kg K. The calorimeter contained 0.25 kg of water, initially at a temperature of 13'C. The mass of the piece of copper was 0.545 kg and its temperature was raised to 95'C. At this temperature it was immersed in the water in the calorimeter. After cooling correction had been carried out, the final temperature of the calorimeter system was found to be 24'C. Determine the water equivalent of the calorimeter system. From equation (14) (m* + m")(h - h,,) : m,c(t, - t) "' (ry*m") -m"9Gz-t)h-hr From which m"c(tr- t) lll": --;---;--lll* h-hr Using the table in Appendix 4, At 24"C, h: 100672lks At 13'C h : 54659 Jke 0'545x 39Ox(95-24) - 0.25ffir: 100672-s4659 0'545x 39Ox7l -0.25460t3 : 0'328 -0'25 : 0'078 kg 1.27 The adiabatic process Ifa process is carried out in a system such that there is no heat transferred into or out of the system (i.e. Q :0) then the process is said to be adiabatic. Such a process is not really possible in practice although it can be closely approached. RELATIONSHIP BETWEEN HEAT AND WORK 45 lf a system is sufliciently thermally insulated then heat transfer can be considered as negligible and the process or processes within the system can be considered as being adiabatic. Alternatively, if a process is carried out with sufficient rapidity, there will bc little time for heat, transfer. Thus if a process is rapid enough, it can be considered as being effectively adiabatic. The implications of any particular process being considered as adiatic will be dealt with in the text. 1.28 Relationship between heat and work Figure 1.17 shows two containers each containing a mass of water: m and each having a thermometer inserted such that temperature measure- ment can be made. In each case, the mass of water is the system. It should be noted that any other fluid of mass : rz could also be considered as the system in this discussion. THERMOMETER SYSTEM BOUNDARY SYSTEM BOUNDARY Fig.l.17 HEATER RNAL (e, (b) can transfer heat energy immersed in the water done when the wheel is At (a) it is arranged that an external heater Q through the system boundary into the water. At (b) it is arranged that a paddle wheel is such that external paddle or stirring work lTis rotated. In each case it is assumed that there is no energy loss from the system. Consider the arrangement at (a). It is common experience to heat water in some containing vessel by means of some external heating device. Let the initial temperature as recorded on the thermometer be fr, and after heating in which heat energy : Q is transferred into the water, let the final temperature be t2. Consider, now, the arrangement at (b). The container once again con- tains a mass of water : ru but in this case a paddle wheel is introduced WATER ! MASS:m i
- 25. I tri1' 6 GENERAL INTRODUCTIoN into the water. It is common experience that friction makes things warm. The simple experience of rubbing one's hands together in a brisk manner will show this. In the case under consideration it is possible to rotate the paddle wheel against the frictional resistance of the water. Assume that the initial temperature of the water is f , and, after doing an amount of work : W on the paddle wheel, the final temperature is f2. Now a similar effect has been produced in both cases (a) and (b) in that a mass of water : m starting at a temperature : f, has experienced a rise in temperature : (tr-tt) On the one hand, however, it was heat which was transferred to produce the effect but on the other, it was a work transfer which produced an exactly similar e{Tect. The conclusion from this rnust therefore be that there is a relationship between heat and work. If the unit of energy for both work and heat is the same, then, since the same effect was produced in each case, the relationship is of the form W:Q The unit of energy in the SI system of units is the joule (J) (see Appendix l). 1J : 1Nm which has the units kg$ t r" - kgg From equation (1), since 17: Q, then the unit of energy for both work and heat is the joule, named after James Prescott Joule (1819-89), an English physicist. In older systems of units, heat energy was defined using water as a reference substance. This is now abandoned and in its place is the energy unit, the joule. In some calorimetric devices, however, water is used as a means of measurement. This is discussed in section 1.26, Caloimetry. 1.29 Enthalpy It has been shown that internal energy, pressure and volume are properties. During subsequent discussion a particular combination of these properties will often appear. The combination is in the form ulPu and because this combination has a particular significance in some processes, it is given a name. The name is enthalpy and is given the symbol h. Thus, h: u*Pa. Note that, since pressure, volume and temperature arelroperties, then the combina- (1) THE PRINcIPLE oF THE THERMoDYNAMIC ENGINS 47 tion of these properties in the form of enthalpy makes enthalpy a property ulso. Specific enthalpy is designated h. The enthalpy of any mass, other than unity is designated fI. 1.30 The principle of the thermodynamic engine The thermodynamic engine is a device in which energy is supplied in the form of heat and some of this energy is transformed into work. It would be ideal, of course, if all the energy supplied was transformed into work. Unfortunately no such complete transformation process lrlg. l.l8 cxists or, as will be shown in Chapter 6 on Reversibility, can possibly exist. The usual process in the engine can be followed by reference to Fig. 1.18. With all engines there must be a source of supply of heat and, with uny quantity of heat Q supplied from the source to the engine, an amount W will successfully be converted into work. This will leave a quantity of hcat (p - W) to be rejected by the engine into the sink. The ratio W Work done Heat received lr called the thermal fficiency and determines what fraction of the heat input has actually been successfully converted into work output. It will bc evident that the object in all engines is, or should be, to make the thcrmal efliciency as near unity as is possible. Now all engines use a working substance as the means of carrying out the conversion ofheat energy into work. The heat energy is usually obtained hy burning a'fuel or 6y ttt".rnonuclear reaction. This heat energy is ruitably transferred into the working substance and, as a consequence of this transfer, the pressure and temperature of the substance are usually (1)
- 26. 48 cENERAL rNTRoDUcrroN raised above that of the surroundings. In this condition the substance is capable of doing work. For example, it could be enclosed by means of a piston in a cylinder and if the piston were free to move, it would be pushed down the cylinder and work would be done as the substance expands. The substance would lose some of its energy in doing this work. When the substance has performed as much work as is practically possible, it could be removed from the cylinder and rejected to the sink. By returning the piston to its original position and then introducing some more high energy-containing substance, the process could be repeated. This is basically what happens in any piston-type engine. It will be noted that the intake and rejection processes of the working substance are intermittent in this case. In the majority of turbine-type engines, however, the working substance passes through in a continuous-flow process. There are two possibilities with regard to the introduction of the energy into the working substance which, in most cases, is either a vapour or a gas. The first possibility is that of transferring heat into the substance outside the engine and then passing the high energy containing substance over into the engine. This is the usual process carried out when using steam as the working substance which is formed outside the engine in a boiler and is then passed to the engine. This is a case of a vapour being used as the working substance. The second possibility is that of introducing the energy directly into the working substance in the engine. This is the usual process carried out in petrol, oil and gas engines in which the fuel is introduced directly into, and burnt in, the engine cylinders. When this is the case, the engines are referred lo as internal combustion engines, abbreviated 'I.C engines'. Each method naturally has its own complexity. More will be said about it in the chapters devoted to these types ofengines. Now a further note about thermal efliciency. It has already been stated that the process in the engine is that of receiving heat, converting some of it into work and then rejecting the remainder. From this then it app€ars that, neglecting losses, the difference between the heat received and the heat rejected is equal to the work done or, Heat received-Heat rejected : Work done (2) Now, Thermal ry: . Work done fsee equation (1)] Heat received (ry, Greek letter eta, is the symbol usually ilsed for efliciency.) (3) THE PRINCIPLE OF THE THERMODYNAMIC ENGINE .'. using equation (2) in (3), Thermal4: Heat received - Heat rejected Heat received :1- Heat rejected Heat received From equation (3), Work done : Heat received x Thermal r 49 (4) (5) (6) (7) Work done Heat received : -'fhermal 4 Now from equation (4), Thermal 4 : Heat received -Heat rejected Heat received .'. Heat received x Thermal 4 : Heat received-Heat rejected from which, Heat rejected : Heat received - (Heat received x Thermal 4) : (l-Thermal 4) Heat received (8) Strictly, equations (1) to (8) will only apply to a system in which heat and work only transfer across the system boundary, a heat engine, in fact (see section 1.31). For other systems, such as an I.C. engine, for example, the thermal efficiency may be defined as, Work done Thermal 4 : Energy received It is useful to note here that the amount of energy liberated by a fuel when burnt is defined by its calorific ualue (see Chapter 16). The calorific value of a fuel is defined as the amount of energy liberated by burning unit mass or volume of the fuel. Thus, if by burning 1kg of petrol, 43MJ/kC are liberated then the calorific value of the petrol is a3N{Jlkg. Special calorimeters have been developed for the determination of the calorific value of fuels. They are described in Chapter 16, Combustion. (e)
- 27. 50 GENERAL rNTRoDUcrIoN EXAMPLE 13 A petrol engine uses 20.4 kg of petrol/h of calorihc value 43 MJ/kg. The thermal efficiency of the engine is 20/,. Determine the power output of the engine and the energy rejected/min. 2}'4kspetrol/h : # o*U Energy liberated petrol : (m x +: x ro6 )r/s Of this, only 20 is successfully transformed into power output. Power outn.t : (Zt ' 3600"43x 106x0'2)Jls lzo.q x 43 xo.2 : - 'ixlor/w _ (20.4 x 43 x 0.2 . -_.:(. * /u* : 48'7 kW Energy rejected : (r _ thermat q) x Energy *Tr,J"":o""tion (8)l : (1-0.2)(#'a3 x ro6)r/min : (o r "#"+: x ro6)r/min : (or "t#"or): 11'7 MJ/min It might be useful to note here that it may be thought that a thermal efficiency of 20/" is extremely low. It is unfortunately the case, however, that engine thermal efliciencies are very low. More will be said on this subject later when dealing with engines and plant. EXAMPLE 14 A steam plant uses 3'045 tonne of coal/h. The steam is fed to a turbine the output of which is 4'1MW. The calorific value of the coal is 28 MJ/kg. Determine the thermal efliciency of the plant. MECHANICAL POWER 3'045 tonne/h : 3'045 x 106 megagramlh : 3045 I 304s <Elh:3600:0.846kg/s Energy liberated by coal : (0.846 x 28 x 106) J/s Power output from turbine: (4.1x 106)W Thermal 4 : : (4.1 x 106) J/s Power output Energy liberated by coal 4'1x 106 - 0'846 x28 t 10u : O'174 :0'174x 100 :17.4% l.3l The heat engine It is convenient here to add a note on the term the heat engine. Since heat is defined as that transfer of energy which results from a difference in temperature, then a heat engine must be an engine in which tu transfer of heat occurs. If heat is introduced into a system and as the rcsult of a cyclic process some work appears from that system, togethor with some heat rejection from the system, then this is a heat engine. 'l'his is illustrated in Fig. 1.18. In practice, such an engine, or plant really, ic thc closed-circuit steam turbine plant of a power station. On the other hand, the open-circuit, internal combustion engine, ruch as a petrol engine is strictly not a heat engine, for fuel and air arc udmitted, which niust cross the system boundary, combustion is internal, us the name implies, and combustion products and heat are rejectod, with some work crossing the system boundary. However, thermodynamic engines are mostly all colloquially referrcd lo as heat engines. 1.32 ilIechanical power Power is defined as the rate of doing work, or, 3l - Work done Power - Time taken (l)
- 28. 52 GENERAL INTRODUCTION If the unit of work is the joule (J) and thq time taken is in seconds (s), then the unit of power, from equation (1) is, I Power : I or joule/second. s The rate of doing work of l joule/second is called the watt (W) Thus, lW : 1J/s (see also Appendix 1) The name watt is after James Watt (1736-1819) of steam engine fame. In the next section (1.33) it will be shown that the electrical unit of power is also the watt, thus giving a very convenient comparison between mechanical and electrical power. EXAMPLE 15 At a speed of 50 km/h the resistance to motion of a car is 900 N. Neglecting losses, what is the power output of the engine of the car at this speed? . Speed : 5o km/h : !H# -n Power : Work done/s : Resistance to motion (N) x speed (m/s) /gmx 5o x 103 _ l_ lNm,is 3600 / : (12'5 x 103) Nm/s : (12.5 x 103) fs : (12.5 x 103) W : 12'5 kW : Power output of engine. 1.33 Electrical power The use of electricity is now so widespread that it is essential to have a knowledge of electrical power. The fact that electrical energy can be converted into mechanical energy can be readily observed in the electric motor. Again, electrical energy can be converted into thermal energy by means of the commonly used electric heater. Since electrical energy can readily be converted into work, electrical energy input to an electrical circuit is sometimes referred to as electric work transfer. The effort which drives electricity through an electric circuit is called the potential dffirence, symbol I/. This effort is usually supplied by means (2) ELECTRICAL POWER 53 of a generator or a battery. The unit of potential difference is called the uolt, abbreviation V. An instrument called the voltmeter is made to measure potential difference. To measure the potential dilTerence of a gcnerator or battery the voltmeter is connected across the terminals of thc generator or battery. The quantity of electricity being driven round a circuit.is called the current. The unit of current is the ampere, abbreviated to amp or A, rymbol I. An instrument called the ammeter is made to measure electric current. In order to measure the current, the ammeter is connected in the circuit such that the current must flow through it. Figure 1.19 illustrates the connections of the voltmeter and ammeter into tn electric circuit. Note that the voltmeter is connected across the circuit, lhus measuring the potential difference. If any electrical device is con- nccted across a circuit in this manner it is said to be connected in parallel. Fh.l.le The ammeter, on the other hand, is connected actually in the circuit luch that the current must flow through it and hence the ammeter will mcasure the current. If any device is connected actually in an electric circuit, such as in the case of the ammeter, it is said to be connected in the circuit in series. ln some generators,,and in all batteries, the current delivered to any circuit is always in the same direction. The connections to either the Icnerator or battery are made by means of terminals. Current flowing ln one direction is said to be ilirect current, abbreviated d.c. One of the lcrminals of either the generator or battery is said to be positive, marked '*, and the other is said to be negative, marked -. Direct current is always considered as flowing from the positive to the negative terminal. The generator ofdirect current electricity is referred to as a d.c. generator. There are other generators, however, which generate electricity in which the current'is continuously changing its direction. Such current lr referred to as alternating current, abbreviated a.c. In this case, neither tcrminal can be designated as either being positive or negative since they AMilT TER
- 29. 54 GENERAL INTRoDUCTIoN are both continuously changing in polarity, as it is called. The type of meters to measure potential difference and current are different in design in this case but they measure potential difference in volts and current in amps, as before. The generator to develop alternating current electricity is referred to as an alternator. Most electric power developed in power- stations is a.c., and in Gt. Britain the standard number of current direction changes is 50 times/s. Each change from positive to negative and back again is called a cycle. Thus, in the above case, the current is said to have 50 cycles/s, which is called the current frequency. Now a frequency of I cycle/s is.called t hertz (Hz). Hence a frequency of 50 cycles/s : 50 c/s : 50 Hz. The unit of power in an electric circuit is called the watt, and this is the rate of working in an electrical circuit whose potential difference is 1 volt with a current flow of I amp. Thus for any circuit, vr:w (1) where I/ : volts, I : &rnpS, 17 - watts. The unit of electric power, the watt, has the same magnitude as that of the unit of mechanical power (1 W : 1 fs). This is arranged by the choice of the units of potential difference and current. EXAMPLE 16 An engine drives an electric generator and 8/, of its power is lost in the transmission to the generator. The generator has an efficiency of 95/, and its electrical output is at 230 volts. It delivers a current of 60 amps. Determine the power output of the engine. Power delivered by generator : I/f watts : (230 x 60) : 13 800W _ 13.8 kW But the generator is only 95/"effrcient. .'. Power input from engine : # : 14.53 kW - u.95 Also, 8 of the engine power is lost in transmission. Hence, 14'55 kW represents 92/. of the available engine output. EXAMPLES !5 .'. Power output from engine : # : 1s.79 kw EXAMPLE 17 A 4-kilowatt heater operates at 230 volts. Determine the current taken in amps. VI : W where 17 is in watts . t _W _4 x 1000 11.^ ^__^.'. l:V: 2n :l/.4&[lpS EXAMPLE 18 A power station output is 500 megawatts. During a test it is found that this rcpresents 28/, of the eneiffp*ilTilt'dTfiE planq by means of burning coal in the boilers. The coal used liberat*es 29'5 MVkg. Determine the mass of coal burnt by the power station in I hour. 500 Mw : (500 x 106) w This.represents 28/. of the energy available from the coal. .'. Energy from coal : %#9 * (500 x 106) -,:1;* '7t (soo!_oj x:0cP)nn The coal liberates 29'5MJlke: (29.5 x 106) rykg .'. mass of coal u 5oo x 106 x 36oo sed/h : 0.28t 2t.s']tr- 5xl02x3'6x103 0'28x29'5 : (2.18 x 10s) kg :218000k9 : 218 t (tonne) l megagram : 103 kg]lNote: 1 tonne :
- 30. 56 GENERAL INTRODUCTION QUESTTONS 1 t. the temperature of 4'5 kg of water is raised from 15'C to 100'C at constant atmospheric pressure. Determine the heat transfer required. [1602'00e kJ] 2. A piece of steel of mass 0.35 kg is at a temperature of 98"C. It is immersed in 0'2 kg of water which is contained in a calorimeter. The initial temperature of the calorimeter and water is 18'C. The final temperature of the calorimetet system after the immersion of the steel, and after a cooling correction has been taken into account, is 26'C. If the specific heat capacity of the steel is 450 J/kg K, determine the water equivalent of the calorimeter system. [0'13e kg] 3. A car has a mass of 1600 kg. It has an engine which develops 35 kW when travelling at a speed of 70 km/h. Neglecting losses, determine the resistance to motion in N/kg. U'I25N,4gl 4. A power station has an output of 800 MW and the thermal e{ficiency is 28/,. Determine the coal consumed in tonne/h if the calorific value of the coal is 31MJ/kg. [331'8 tonne/h] 5. A diesel engine uses 54'5 kg of fuel oil/h of calorific value 45 MJ,/kg. The thermal efficiency of the engine 1s25%. Determine the power output of the engine in kilowatts. [170.3 kw] 6. An engine rejects 1,26DlNdllh when running at a thermal efficiency of 22/,.The calorific value of the fuel used is42MJlkg. Determine the power output of the engine in kilowatts and the mass of fuel uped/h. [98'7 kw; 38'a6kglh] 7. l4'slitres of gas at a pressure of 1720 kN/m2 is contained in a cylinder. It is expanded at constant pressure until its volume becomes 130'5litres. Determine the work done by the gas. uee's kJl 8. A quantity of steam the original pressure and volume being 140 kN/m2 and 150 litres, respectively, is compressed to a volume of 30 litres, the law of compression being PVr'z : C. Determine the final pressure and the work done. [966kN/m2; -39'9kJ] 9. A quantity of gas has an initial pressure of 2"12 MN/m2 and a volume of 5'6dm3(l). It is expanded according to the law PVr'15: C down to a pressure of 340 kN/m2. Determine the final volume and the work done. [26'15 dm3; 18'2 kJ] Systems 6 u(t {} 2,1 Generalintroduction All physical things in nature have some form of boundary whose shape ln general identihes it as the object that it is. Inside its boundary there erc certain things with particular functions to carry out. This inside srrangement is called a system. Outside the boundary of the object are lhc surroundings and the reaction between the system and surroundings ln general controls the behaviour pattern of the object. A human being end a tiee are systems. Heat engines and allied arrangements, which are lhc concern here, are other systems. It is not necessary that at any one time a complete object need be under investigation. Only part may be rrnder study and this part may then be considered as the system. In other words, a system can be defined as a particular region which is under study. It is identifibd by its boundary around which are the surroundings. The boundary need not be fixed. For example, a mass of gas (the rystem) may expand and hence the boundary in this case will modify nnd interactions will occur with the surroundings at the boundary. If lhc mass of a system remains constant then the system is said to be a tkised svstem. If, on the other hand, the mass ofa system changes, or is continuously changing, then the system is said to be an open system. for example, an air compressor is an open system since air is continuously rlrcaming into and out of the machinq, in other words, air mass is crossing ils boundary. This is called a two-flow open system. Another example is air leaving a compressed air tank. This would bc a one-flow apen system since air is only leaving the tank and none is ln any system, energies such as work and heat could be arranged to oross the boundary. Closed and open systems are illustrated in Fig. 2.1.
- 31. BOUNDARY 58 SYSTEMS FIXED MASS CLOSED SYSTEM. ENERGY LEAVING BOTJNDARY CAN CHANGE SHAPE SURROUNDINGS MASS LEAVING COULD ALSO ENTER} Fig.2.1 EN€RGY ENTERING ONE-FLOW OPEN SYSTEM. 2.2 Control Yolume If the volume of a system under study remains constant then this volume is called the control volume. The control volume is bounded by the control surface. A control volume and its surface are illustrated in Fig. 2.2.It is shown as a fixed volume enclosing a steam turbine and condenser. Va.rious masses and energies can be investigated as they cross the control surface into, or out of, the control volume. The control volume is similar in concept to the open system. In the case of the control volume, however, both the volume and position are fixed whereas with an open system the volume could change both in size and position. The air compressor given as an example of an open system in section 2.1 could also be considered as a control volume. HEAT LOST TO SURROUNDINGS r--*----'ENERGY I -------- INPUT FROM! STEAM I WORK OUTPUT COOLING WATER OUT HEAT LOST TO COOLING WATER COOTING WATER IN ENERGY LOST CONDENSA1E OUT TWO-FLOW OPEN SYSTEM. TO CONDENSATE THE oBSERVATIoN oF ENERGY 59 2,3 The conservation of energy ln section 1.17 the concept of energy was discussed. From this discussion it appears that, by designing suitable devices, then one form of energy can be transformed into another. In a power- station, the potential chemical energy in the fuel produces a high-tem- perature furnace. From the furnace, heat energy is transferred into the nteam being formed, which is passed into a turbine where some of it is converted into work. The work is put into an alternator where some is converted into electrical energy. The electricity generated is then passed out of the station to the public, who use it in various devices to produce heat, light'and power. Actually, not all the energy which is put into the furnaces of the power-station ultimately appears as electrical energy. There are many losses through the plant, as indeed there are in any power plant. However, it is found that, in any energy transformation system, if all the energy forms are added up, and including any energy losses which may have occurred, then the sum is always equal to the energy input. Written as an equation, this becomes, Initial energy of the system*Energy entering the system : Final energy of the system*Energy leaving the system Naturally, all the various energies must appear with the same units in order to do this. The fact that the total energy in any one energy system remains constant in called the principle of the Conseruation of Energy. This states that cncrgy can neither be created nor destroyed but can only be changed in form. As a further point, from work carried out in the field of nuclear physics, it appears that there is some relationship between energy and matter. 'l'his has been made manifest by the fact that during a nuclear reaction, rome of the energy released can only be accounted for by reference to thc loss of nuclear matter which has occurred during the reaction. Thus it appears that matter and energy are, in some way, related. From this, lhe Conservation of Energy theory should strictly be modified to the Conservation of Energy and Matter theory. Any matter-energy trans- formation which occurs in any reaction outside the nuclear field, if it occurs at all, is extremely small, however. Thus in the absence of nuclear rcaction, all energy transformation is discussed using the principle of the ('onservation of Energy.
- 32. 60 sYsrEMS 2.4 Energy forms in thermodynamic systems Various energy forms can exist in thermodynamic systems. In some systems they may all be present. In other systems only some may be present. The various forms of energy appearing in thermodynamic systems are listed below. The basic unit of energy, in all forms, is the joule (J). Multiples such as the kilojoule (kJ) or the megajoule (MJ) are often used (see Appendix 1). (i) Grauitational potential energy If the fluid is at some height Z above a given datum level, then, as a result'of its mass it possesses gravitational potential energy with respect to that datum. Thus, for unit mass of fluid, in the close vicinity of the earth, Potential energy : gZ x 9.812 (See Appendix 5) (ii) Kinetic energy If the fluid is in motion then it possesses kinetic energy. If the fluid flowing with.velocity C then, for unit mass of fluid, Kinetic energy - (See Appendix 5) (iii) lnternal energy All fluids store energy. The store of energy within any fluid can be increased or decreased as the result of various prooesses carried out on or by the fluid. The energy stored within a fluid which results from the internal motion of its atoms and molecules is called its internal energy and was discussed in section l.23.lt is usually designated by the letter U. If the internal energy of unit mass of fluid is being discussed then this is called the specific internal energy and is designated by z. (iu) Flow or displacement energy Any volume of fluid entering or leaving a system must displace an equal volume ahead of itself in order to enter or leave the system, as the case may be. The displacing mass must do work on the mass being displaced, since the movement of any mass can only be achieved at the expense of work. ls C2 T ENERGY FORMS IN THERMODYNAMIC SYSTEMS Flow entering# +Flowleaving r- ---- flg.2.3 Specitic volume=v=Al Figure 2.3 illustrates a part of a system which can be considered as being lhc entry or exit of the system. Consider unit mass of fluid entering or leaving the system. Let the fluid bo at uniform pressure P and enter or leave the system a distance I over a uniform area A. Let the dpecific volume of the fluid : u : volume of unit mass. Work done : force x distance lnd force : pressure x area :PxA :PA .'. Work done: PAxl :PxAl : Pu since Al: u 'l'his is called flow or displacement work. At entry it is energy received by the system. At exit it is energy lost by the system. lpl Heat receiued or rejected ln uny system a fluid can have a direct reception or rejection ofheat energy lrunsferred through the system boundary. It is designatedby Q. This must bc taken in its algebraic sense. Thus if, Heat is received then Q is a 6l Pressure P jL

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