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1、化工热力学英文版化工热力学英文版chemical and engineering thermodynamicsChapter 5 the second law of thermodynamicsContents5.1 statements of the second law5.2 heat engines5.3 thermodynamic temperature scales5.4 entropy5.5 entropy changes of an ideal gas5.6 mathematical statement of the second law5.7 entropy balance f
2、or open systems5.8 calculation of ideal work5.9 lost work5.10 the third law of thermodynamics5.11 entropy from the microscopic viewpointEmphasis5.2 heat engines5.3 thermodynamic temperature scales5.4 entropy5.5 entropy changes of an ideal gas5.6 mathematical statement of the second lawDifficultiesEn
3、tropy, mathematical statement of the second lawThermodynamics is concerned with transformations of energy, and the laws of thermodynamics describe the bounds within which these transformations are observed to occur. The first law reflects the observation that energy is conserved, but it imposes no r
4、estriction on the process direction. Yet, all experience indicates the existence of such a restriction, the concise statement of whichconstitutes the second law.The difference between the two forms of energy, heat and work, provide some insight into the second law. In an energy balance, both work an
5、d heat are included as additive terms, implying that one unit of heat, a joule, is equivalent to the same unit of work. Although this is true with respect to an energy balance, experience teaches that there is a difference of kind between heat and work. This experience is summarized by the following
6、 facts.Work is readily transformed into other forms of energy: for example, into potential energy by elevation of a weight, into kinetic energy by acceleration of a mass, into electrical energy by operation of a generator. These processes can be made to approach a conversion efficiency of 100% by el
7、imination of friction, a dissipative process that transforms work into heat. Indeed, work is readily transformed completely into heat, as demonstrated by joules experiments.On the other hand, all efforts to devise a process for the continuous conversion of heat completely into work or into mechanica
8、l or electrical energy have failed. Regardless of improvements to the devices employed, conversion efficiencies do not exceed 40%. Evidently, heat is a form of energy intrinsically less useful and hence less valuable than an equal quantity of work or mechanical or electrical energy.Drawing further o
9、n our experience ,we know that the flow of heat between two bodies always takes place from the hotter to the cooler body, and never in the reverse direction. This fact is of such significance that its restatement serves as an acceptable expression of the second law.5.1statements of the second lawThe
10、 observations just described suggest a general restriction on processes beyond that imposed by the first law. The second law is equally well expressed in two statements that described this restriction:Statement 1: no apparatus can operate in such a way that its only effect (in system and surrounding
11、s) is to convert heat absorbed by a system completely Into work done by the system.Statement 2: no process is possible which consists solely in the transfer of heat from one temperature level to a higher one.Statement 1 does not say that heat cannot be converted into work; only that the process cann
12、ot leave both the system and its surroundings unchanged. Consider a system consisting of an ideal gas in a piston/cylinder assembly expanding reversibly at constant temperature. According to Eq(2.3), Ut=Q+W. for an ideal gas, Ut=0, and therefore, Q=-W. the heat absorbed by the gas from the surroundi
13、ngs is equal to the work transferred to the surroundings by the reversible expansion of the gas. At first this might seem a contradiction of statement 1, since in the surroundings the result is the complete conversion of heat into work. However, this statement requires in addition that no change occ
14、ur in the system (expanding), a requirement that is notmet. This process is limited in another way, because the pressure of the gas soon reaches that of the surroundings, and expansion ceases. Therefore, the continuous production of work from heat by this method is impossible. If the original state
15、of the system is restored in order to comply with the requirements of statement 1, energy from the surroundings in the form of heat is transferred to the system to maintain constant temperature. This reverse process requires at least the amount of work gained from the expansion; hence no network is
16、produced. Evidently, statement 1 may be expressed in an alternative way:Statement 1a: it is impossible by a cyclic process to convert the heat absorbed by a system completely into work done by the system.The word cyclic requires that the system be restored periodically to its original state. In the
17、case of a gas in a piston/cylinder assembly, its initial expansion and recompression to the original state constitute a complete cycle. If the process is repeated, it becomes a cyclic process. The restriction to a cyclic process in statement 1a amounts to the same limitation as that introduced by th
18、e words only effect in statement 1.The second law does not prohibit the production of work from heat, but it does place a limit on how much of the heat directed into a cyclic process can be converted into work done by the process. With the exception of water and wind power, the partial conversion of
19、 heat into work is the basis for nearly all commercial production of power. The development of a quantitative expression for the efficiencyof this conversion is the next step in the treatment of the second law.5.2 heat enginesThe classical approach to the second law is based on a macroscopic viewpoi
20、nt of properties, independent of any knowledge of the structure or behavior of molecules. It arose from the study of heat engines, devices or machines that produce work from heat in a cyclic process. An example is a steam power plant in which the working fluid( steam) periodically returns to its ori
21、ginal state. In such a power plant the cycle (in its simplest form) consists of the following steps:Liquid water at ambient temperature is pumped into a boiler at high pressure.Heat from fuel (heat of combustion of a fossil fuel or heat from a nuclear reaction) is transferred in the boiler to the wa
22、ter, converting it to high-temperature steam at the boiler pressure.Energy is transferred as shaft work from the steam to the surroundings by a device such as a turbine, in which the steam expands to reduced pressure andtemperature.Exhaust steam from the turbine is condensed by transfer of heat to t
23、he surroundings, producing liquid water for return to the boiler, thus completing by cycle.Essential to all heat-engine cycles are absorption of heat into the system at a high temperature, rejection of heat to the surroundings at a lower temperature, and production of work. In the theoretical treatm
24、ent of heat engines, the two temperature level which characterize their operation are maintained by heat reservoirs, bodies imagined capable of absorbing or rejecting an infinite quantity of heat without temperature change. In operation, the working fluidWith eq(5.1) this becomes:Or on the degree of
25、 reversibility of its operation. Indeed, a heat engine operating in a completely reversible manner is very special, and is called a carnot engine. The characteristic of such an ideal engine were first described by N.L.S. Carnot in 1824. the four steps that make up a carnot cycle are performed in the
26、 following order:Step1: a system at the temperature of a cold reservoir TC undergoes a reversible adiabatic process that caused its temperature to rise to that of a hot reservoir at THStep2: the system maintains contact with the hot reservoir at TH, and undergoes a reversible isothermal process duri
27、ng which heat |QH | is absorbed from the hot reservoir.Step3: the system undergoes a reversible adiabatic process in the opposite direction of step1 that brings its temperature back to that of the cold reservoir at TCStep4: the system maintains contact with the reservoir at TC, and undergoes a rever
28、sible isothermal process in the opposite direction of step2 that returns it to its initial state with rejection of heat |QC | to the cold reservoir.A carnot engine operates between two heat reservoirs in such a way that all heat absorbed is absorbed at the constant temperature of the hot reservoir a
29、nd all heat rejected is rejected at the constant temperature of the cold reservoir. Any reversible engine operating between two heat reservoirs is a carnot engine; an engine operating on a different cycle must necessarily transfer heat across finite temperature differences and therefore cannot be re
30、versible.Carnots theoremStatement 2 of the second law is the basis for carnots theorem:For two given heat reservoirs no engine can have a thermal efficiency higher than that of a carnot engine.To prove carnots theorem assume the existence of an engine E with a thermal efficiency greater than that of
31、 a carnot engine which absorbs heat |QH | from the hot reservoir, produces work |W | , and discards heat |QH | - |W | to the cold reservoir. Engine E absorbs heat |QH | from the same hot reservoir, produces the same work |W | , and discards heat |QH | - |W | to the same cold reservoir. If engine E h
32、as the greater efficiency,Since a carnot engine is reversible, it may be operated in reverse; the carnot cycle is then traversed in the opposite direction, andit becomes a reversible refrigeration cycle for which the quantities |QH |, |QC | and |W | are the same as for the engine cycle but are rever
33、sed in direction. Let engine E drive the carnot engine backward as a carnot refrigerator, as shown schematically in fig5.1. for the engine/refrigerator combination, the net heat extracted from the cold reservoir is:The net heat delivered to the hot reservoir is also |QH | - |QH | . Thus, the sole re
34、sult of the engine/refrigerator combination is the transfer of heat from temperature TC to the higher temperature TH. Since this is in violation of statement 2 of the second law, the original premise that engine E has a greater thermal efficiency than the carnot engine is false, and carnots theorem
35、is proved. In similar fashion, one can prove that all carnot engines operating between heat reservoirs at the same two temperatures have the same thermal efficiency. Thus a corollary to carnots theorem states:The thermal efficiency of a carnot engine depends only on the temperature levels and not up
36、on the working substances of the engine.5.3 ideal-gas temperature scales; carnots equationsThe cycle traversed by an ideal gas serving as the working fluid in a carnot engine is shown by a PV diagram in fig5.3. it consists of four reversible steps:ab adiabatic compression until the temperature rises
37、 from TC to THbc isothermal expansion to arbitrary point c with absorption of heat |QH |.cd adiabatic expansion until the temperature decreases to TCda isothermal compression to the initial state with rejection of heat |QC |.ThereforeSince the left side of these two equations are the same,Equations(
38、5.7) and(5.8) are known as carnots equations. In eq(5.7) the smallest possible value of |QC | is zero; the corresponding value of TC is the absolute zero of temperature on the kelvin scale. As mentioned in Sec1.5, this occurs at -273.15.equation(5.8) shows that the thermal efficiency of a carnot eng
39、ine can approach unity only when TH approaches infinity or TC approaches zero. Neither of these conditions is attainable; all heat engines therefore operate with thermal efficiency less than unity. The cold reservoirs naturally available on earth are the atmosphere, lakes and rivers, and the oceans,
40、 for which TC300K. Hot reservoirs are objects such as furnaces where the temperature is maintained by combustion of fossil fuel and nuclear reactors where the temperature is maintained by fission of radioactive elements. For these practical heat sources, TH 600K. With these values,This is a rough pr
41、actical limit for the thermal efficiency of a carnot engine; actual heat engines are irreversible, and their thermal efficiencies rarely exceed 0.35 5.4 ENTROPYEquation(5.7)for a carnot engine may be written:If the heat quantities refer to the engine( rather than to the heat reservoirs), the numeric
42、al value of QH is positive and that QC is negative. The equivalent equation written without absolute-value signs is thereforeorThus for a complete cycle of a carnot engine, the two quantities Q/T associated with the absorption and rejection of heat by the working fluid of the engine sum to zero. The
43、 working fluid of cyclic engine periodically returns to its initial state, and its properties, e.g., temperature, pressure, and internal energy, return to their initial values. Indeed, a primary characteristic of a property is that the sum of its changes is zero for any complete cycle. Thus for a ca
44、rnot cycle, eq(5.9)suggests the existence of a property whose changes are given by the quantities Q/T.Our purpose now is to show that eq(5.9), applicable to the reversible carnot cycle, also applies to other reversible cycles. The closed curve on the PV diagram of fig5.4 represents an arbitrary reve
45、rsible cycle traversed by an arbitrary fluid. Divide the enclosed area by a series of reversible adiabatic curves. Since such curves cannot intersect, they may be drawn arbitrarily close to one another. Several such curves are shown on the figures as long dashed lines. Connect adjacent adiabatic cur
46、ves by two short reversible isotherms which approximate the curve ofthe arbitrary cycle as closely as possible. The approximation clearly improves as the adiabatic curves are more closely spaced. When the separation becomes arbitrarily small, the original cycle is faithfully represented. Each pair o
47、f adjacent adiabatic curves and their isothermal connecting curves represent a carnot cycle for which eq(5.9) applies.Each carnot cycle has its own pair of isotherms TH and TC and associated heat quantities QH and QC. These are indicated on fig5.4 for a representative cycle. When the adiabatic curve
48、s are so closely spaced that the isothermal steps are infinitesimal, the heat quantities become dQH and dQC ,and eq(5.9)for each carnot cycle is written:In this equation TH and TC, absolute temperatures of the working fluid of the carnot engines, are also the temperatures traversed by the working fl
49、uid of the arbitrary cycle. Summation of all quantities dQ/T for the carnot engines leads to integral:Where the circle in the integral sign signifies integration over the arbitrary cycle, and the subscript “rev” indicates that the cycle is reversible.Thus the quantities dQrev/T sum to zero for the a
50、rbitrary cycle, exhibiting the characteristic of a property. We therefore infer the existence of a property whose differential changes for the arbitrary cycle are given by these quantities. The property is called entropy, and its differential changes are:Points A and B on the PV diagram of fig5.5 re
51、present two equilibrium states of a particular fluid, and paths ACB and ADB show two arbitrary reversible processes connecting these points. Integration of eq(5.11)for each path gives:Where in view of eq(5.10) the two integrals must be equal. We therefore conclude that St is independent of path and
52、is a property change given by SBt- SAt .For a irreversible process, the entropy change is not given by the integration of dQ/T, it should be integrated by a reversible process.But for the entropy change of a heat reservoir, is always given by Q/T, where Q is the quantity of heat transferred to or fr
53、om the reservoir at temperature T, whether the transfer is reversible.If a process is reversible and adiabatic, dQrev=0; then by eq(5.11), dSt=0. thus the entropy of a system is constant during a reversible and adiabatic process, and the process is said to be isentropic.This discussion of entropy ca
54、n be summarized as follows:Entropy owes its existence to the second law, from which it arises in much the same way as internal energy does from the first law. Eq(5.11) is the ultimate source of all equations that relate the entropy to measurable quantities. It does not represent a definition of entr
55、opy; there is none in the context of classical thermodynamics. What it provides is the means for the calculating changes in this property. Its essential nature is summarized by the following axiom:There exists a property called entropy S, which is an intrinsic property of a system, functionally rela
56、ted to the measurable coordinates which characterize the system. For a reversible process, changes in this property are given by Eq(5.11)the change in entropy of any system undergoing a finite reversible process is:When a system undergoes an irreversible process between two equilibrium states, the e
57、ntropy change of the system St is evaluated by application of eq(5.13) to an arbitrary chosen reversible process that accomplishes the same change of state as the actual process. Integration is not carried out for the irreversible path. In the special case of a mechanically reversible process(sec2.8
58、) , the entropy change of the system is correctly evaluated from the integration of dQ/Tapplied to actual process, even though the heat transfer is irreversible. The reason is that it is immaterial, as far as the system is concerned, whether the temperature difference causing the heat transfer is di
59、fferential or finite. The entropy change of a system caused by the transfer of heat can always be calculated by ,whether the heat transfer is accomplished reversibly or irreversibly. However, when a process is irreversible on account of finite differences in other driving forces, such as pressure ,
60、the entropy change is not caused by the heat transfer, and for its calculation one must devise a reversible means of accomplishing the same change of state.5.5 entropy changes of an ideal gasFor one mole or a unit mass of fluid undergoing a mechanically reversible process in a closed system, the fir
61、st law, eq(2.8), becomes:differentiation of the defining equation for enthalpy, H=U+PV, yields: Eliminating dU gives:Or As a result of eq(5.11), this becomes:Where S is the molar entropy of an ideal gas. Integration from an initial state at conditions T0 and P0 to a final state at conditions T and P
62、 gives:Although derived for a mechanically reversible process, this equation relates properties only, and is independent of the process causing the change of state. It is a general equation for the calculation of entropy changes of an ideal gas.5.6 mathematical statement of the second lawConsider tw
63、o heat reservoirs, one at temperature TH and a second at the lower TC. Let a quantity of heat |Q| be transferred from the hotter to the cooler reservoir. The entropy changes of the reservoirs at TH and TC are:These two entropy changes are added to giveSince THTC, the total entropy change as a result
64、 of this irreversible process is positive. Also, Stotal becomes smaller as the difference TH-TC gets smaller. When TH is only infinitesimally higher than TC, the heat transfer is reversible, and Stotal approaches zero. Thus for the process of irreversible heat transfer, Stotal is always positive, ap
65、proaching zero as the process becomes reversible.Consider now an irreversible process in a closed system wherein no heat transfer occurs. Such a process is represented on the PV diagram of fig5.6, which shows an irreversible, adiabatic expansion of 1 mole of fluid from an initial equilibrium state a
66、t point A to a final state at point B. Now suppose the fluid is restored to its initial state by a reversible process consisting of two steps: first, the reversible , adiabatic( constant-entropy) compression of the fluid to the initial pressure, and second, a reversible, constant-pressure step that
67、restores the initial volume. If the initial process results in an entropy change of the fluid, then there must be heat transfer during the reversible, constant-pressure second step such that:The original irreversible process and the reversible restoration process constitute a cycle for which U=0 and
68、 for which the work is therefore:However, according to statement 1a of the second law, Qrev cannot be directed into the system, for the cycle would then be a process for the complete conversion into work of the heat absorbed. Thus, is negative, and it follows that SAt-SBt is also negative; whence SA
69、tSBt. Since the original irreversible process is adiabatic( Ssurr=0) ,the total entropy change of the system and surroundings as a result of this process is Stotal= SBt-SAt0. In arriving at this result, our presumption is that the original irreversible process results in an entropy change of the flu
70、id. If the original process is in fact isentropic, then the system can be restored to its initial state by a simple reversible adiabatic process. This cycle is accomplished with no heat transfer and therefore with no net work. Thus the system is restored without leaving any change elsewhere ,and thi
71、s implies that the original process is reversible rather than irreversible. Thus the same result is found for adiabatic process as for direct heat transfer: Stotal is always positive, approaching zero as a limit when the process becomes reversible. This same conclusion can be demonstrated for any pr
72、ocess whatever, leading to the general equation: Stotal 0 5.19This mathematical statement of the second law affirms that every process proceeds in such a direction that the total entropy change associated with it is positive, the limiting value of zero being attained onlyby a reversible process. No
73、process is possible for which the total entropy decreases. For a cyclic heat engine, assuming that takes in heat |QH| from a heat reservoir at TH, and discards heat |QC| to another heat reservoir at TC. Since the engine operates in cycles, it undergoes no net changes in its properties. The total ent
74、ropy change of the process is therefore the sum of the entropy changes of the heat reservoirs:The work produced by the engine is:Elimination of |QC| between these two equations and solution for |W| gives:5.7entropy balance for open systemsJust as an energy can be written for processes in which fluid
75、 enters, exists, or flows through a control volume(sec2.12), so too can an entropy balance be written. The different thing is that entropy is not conserved. The second law states that the total entropy change associated with any process must be positive, with a limiting value of zero for a reversibl
76、e process. This requirement is taken into account by writing the entropy balance for both the system and the surroundings, considered together, and by including an entropy-generation term to account for the irreversibilities of the process. This term is the sum of three others: one for entropy chang
77、es in the streams flowing in and out of the control volume, one for entropy changes within the control volume, and one for entropy changes in the surroundings. If the process is reversible, these three terms sum to zero so that Stotal =0 . If the process is irreversible , they sum to a positive quan
78、tity, the entropy-generation term.The statement of balance, expressed as rates, is therefore:The equivalent equation of entropy balance is:Where is the rate of entropy generation. This equation is the general rate form of the entropy balance, applicable at any instant. Each term can vary with time.
79、The first term is simply the net rate of gain in entropy of the flowing streams, i.e., the difference between the total entropy transported out by exit steams and the total entropy transported in by entrance streams. The second term is the time rate of change of the total entropy of the fluid contai
80、ned within the control volume. The third term accounts for entropy changes in the surroundings, the result of heat transfer between system and surroundings.Let rate of heat transfer with respect to a particular part of the control surface be associated with where subscript denotes a temperature in t
81、he surroundings. The rate of entropy change in the surroundings as a result of this transfer is then the minus sign converts defined with respect to the system, to a heat rate with respect to the surroundings. The third term in eq(5.20) is therefore the sum of all such quantities:Equation(5.20) is n
82、ow written:The final term, representing the rate of entropy generation reflects the second law requirement that it be positive for irreversible processes. There are two sources of irreversibility: (a) those within the control volume, i.e., internal irreversibilities, and (b) those resulting from hea
83、t transfer across finite temperature differences between system and surroundings, i.e.,external irreversibilities. In the limiting case where =0, the process must be completely reversible, implying that both internal and external are in reversible change. for a steady-flow process the mass and entro
84、py of the fluid in the control volume are constant, and is zero, eq(5.21) then becomes:If in addition there is but one entrance and one exit, with the same for both streams, dividing through by yields:5.8 calculation of ideal workIn any steady-state flow process requiring work, there is an absolute
85、minimum amount which must be expended to accomplish the desired change of state of the fluid flowing through the control volume. In a process producing work, there is an absolute maximum amount which may be accomplished as the result of a given change of the fluid flowing through the control volume.
86、 In either case, the limiting value obtains when the change of state associated with the process is accomplished completely reversibly. For Such a process, the entropy generation is zero, and eq(5.22), written for the uniform surroundings temperature T,becomes:orSubstitution this expression in energ
87、y balance eq(2.30):A completely reversible process is hypothetical, devised solely for determination of the ideal work associated with a given change of state. The objective is to compare the actual work of a process with the work of the hypothetical reversible process. No description is ever requir
88、ed of hypothetical processes devised for the calculation of ideal work. One need only realize that such processes may always be imagined. Eq(5.24) through eq(5.27) give the work of a completely reversible process associated with given property changes in the flowing streams. When the same change occ
89、urs for the actual process. Fixed relation can be found between actual work and ideal work.For a producing work system:A thermodynamic efficiency is defined as the ration of the actual work to the ideal work.For a receiving work system:A thermodynamic efficiency is defined as the ration of the ideal
90、 work to the actual work.5.9 lost workWork that is wasted as the result of irreversibilities in a process is called lost work, Wlost, and is defined as the difference between the actual work of a process and the ideal work for the process. Thus by definition,The engineering significance of this resu
91、lt is clear: the greater the irreversibility of a process, the greater the rate of the entropy production and the greater the amount of energy that becomes unavailable for work. Thus every irreversibility carries with it price.5.10 The third law of thermodynamicsMeasurements of heat capacities at ve
92、ry low temperatures provide data for the calculation from eq(5.13) of entropy changes from to 0K. When these calculations are made for different crystalline forms of the same chemical species ,the entropy at 0K appears to be the same for all forms. When the form is non-crystalline, e.g., amorphous o
93、r glassy, calculations show that the entropy of the more random form is greater than that of the crystalline form. Such calculations, which are summarized elsewhere, lead to the postulate that the absolute entropy is zero for all perfect crystalline substances at absolute zero temperature. If the entropy is zero at 0K, then eq(5.13) lends itself to the calculation of absolute entropies. With T=0 as the lower limit of integration, the absolute entropy of a gas at temperature T based on calorimetric data is:
