Heat can be used far more usefully than by simple transfer from one temperature level to a lower one. Indeed, useful work is produced by countless engines that employ the flow of heat as their energy source. The most common examples are the internal-combustion engine and the steam power plant. Collectively, these are heat engines. They all rely on a high-temperature source of heat, and all discard heat to the environment.
The second law imposes restrictions on how much of their heat intake can be converted into work, and our object now is to establish quantitatively this relationship. We imagine that the engine receives heat from a higher-temperature heat reservoir at TH and discards heat to a lower-temperature reservoir TC. The engine is taken as the system and the two heat reservoirs comprise the surroundings. The work and heat quantities in relation to both the engine and the heat reservoirs are shown in Fig. 5.1(a).
With respect to the engine, the first law as given by Eq. (2.3) becomes:
ΔU = Q + W = Q H + Q C + W
Because the engine inevitably operates in cycles, its properties over a cycle do not change. Therefore ΔU = 0, and W = −QH− QC.
The entropy change of the surroundings equals the sum of the entropy changes of the reservoirs. Because the entropy change of the engine over a cycle is zero, the total entropy change is that of the heat reservoirs. Therefore
Δ S total = − ___ Q H
T H − ___ Q C T C
Note that QC with respect to the engine is a negative number, whereas QH is positive. Combining this equation with the equation for W to eliminate QH yields:
W = T H Δ S total + Q C ( T ______H − T C T C )
This result gives the work output of a heat engine within two limits. If the engine is totally ineffective, W = 0; the equation then reduces to the result obtained for simple heat transfer between the two heat reservoirs, i.e.:
Δ S total = − Q C ( T ______H − T C T H T C )
The difference in sign here simply reflects the fact that QC is with respect to the engine, whereas previously it was with respect to the lower-temperature reservoir.
If the process is reversible in all respects, then ΔStotal = 0, and the equation reduces to:
W = Q C ( ___ T H
T C − 1) (5.3)
A heat engine operating as described in a completely reversible manner is very special and is called a Carnot engine. The characteristics of such an ideal engine were first described by N. L. S. Carnot3 in 1824.
Note again that QC is a negative number, as it represents heat transferred from the engine.
This makes W negative, in accord with the fact that work is not added to, but is produced by, the engine. Clearly, for any finite value of W, QC is also finite. This means that a portion of the heat transferred from the higher-temperature reservoir must inevitably be exhausted to the lower-temperature reservoir. This observation can be given formal statement:
It is impossible to construct an engine that, operating in a cycle, produces no effect (in system and surroundings) other than the extraction of heat from a reservoir and the performance of an equivalent amount of work.
Figure 5.1: Schematic diagrams: (a) Carnot engine; (b) Carnot heat pump or refrigerator.
(a) (b)
Hot Reservoir Hot Reservoir
Cold Reservoir Cold Reservoir
QH
W W
QC QC
QH
3Nicolas Leonard Sadi Carnot (1796−1832), French engineer. See http://en.wikipedia.org/wiki/Nicolas_Leonard_
5.1. Axiomatic Statements of the Second Law 177 The second law does not prohibit the continuous production of work from heat, but it does place a limit on how much of the heat taken into a cyclic process can be converted into work.
Combining this equation with W = −QH− QC to eliminate first W and then QC leads to Carnot’s equations:
____− Q C
T C = Q ___H
T H (5.4)
___W
Q H = ___ T C
T H − 1 (5.5)
Note that in application to a Carnot engine QH, representing heat transferred to the engine, is a positive number, making the work produced (W) negative. In Eq. (5.4) the small-est possible value of QC is zero; the corresponding value of TC is absolute zero on the Kelvin scale, which corresponds to −273.15°C.
The thermal efficiency of a heat engine is defined as the ratio of the work produced to the heat supplied to the engine. With respect to the engine, the work W is negative. Thus,
η ≡ ___− W
Q H (5.6)
In view of Eq. (5.5) the thermal efficiency of a Carnot engine is:
η Carnot = 1 − ___ T C
T H (5.7)
Although a Carnot engine operates reversibly in all respects, and cannot be improved, its efficiency approaches unity only when TH approaches infinity or TC approaches zero.
Neither condition exists on earth; all terrestrial heat engines therefore operate with thermal efficiencies less than unity. The cold reservoirs available on earth are the atmosphere, lakes, rivers, and oceans, for which TC ≃ 300 K. Hot reservoirs are objects such as furnaces where the temperature is maintained by combustion of fossil fuels or by fission of radioactive elements, and for which TH ≃ 600 K. With these values, η = 1 − 300/600 = 0.5, an approximate realistic limit for the thermal efficiency of a Carnot engine. Actual heat engines are irreversible, and η rarely exceeds 0.35.
Example 5.1
A central power plant, rated at 800,000 kW, generates steam at 585 K and discards heat to a river at 295 K. If the thermal efficiency of the plant is 70% of the maximum possible value, how much heat is discarded to the river at rated power?
Solution 5.1
The maximum possible thermal efficiency is given by Eq. (5.7). With TH as the steam-generation temperature and TC as the river temperature:
η Carnot = 1 − 295____
585 = 0.4957 and η = (0.7)(0.4957) = 0.3470
where η is the actual thermal efficiency. Combining Eq. (5.6) with the first law, written W = −QH− QC, to eliminate QH, yields:
Q C = ( 1 − η___
η ) W = ( _______1 − 0.347
0.347 ) (− 800,000) = −1,505,475 kW This rate of heat transfer to a modest river would cause a temperature rise of several °C.
5.2 HEAT ENGINES AND HEAT PUMPS
The following steps make up the cycle of any Carnot engine:
∙ Step 1: A system at an initial temperature of a cold reservoir at TC undergoes a reversible adiabatic process that causes its temperature to rise to that of a hot reservoir at TH. ∙ Step 2: The system maintains contact with the hot reservoir at TH and undergoes a
reversible isothermal process during which heat QH is absorbed from the hot reservoir.
∙ Step 3: The system undergoes a reversible adiabatic process in the opposite direction of Step 1 that brings its temperature back to that of the cold reservoir at TC.
∙ Step 4: The system maintains contact with the reservoir at TC, and undergoes a reversible isothermal process in the opposite direction of Step 2 that returns it to its initial state with rejection of heat QC to the cold reservoir.
This set of processes can in principle be performed on any kind of system, but only a few, yet to be described, are of practical interest.
A Carnot engine operates between two heat reservoirs in such a way that all heat absorbed is transferred at the constant temperature of the hot reservoir and all heat rejected is transferred 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 reversi-ble. Two important conclusions inevitably follow from the nature of the Carnot engine:
∙ Its efficiency depends only on the temperature levels and not upon the working substance of the engine.
∙ For two given heat reservoirs no engine can have a thermal efficiency higher than that of a Carnot engine.
We provide further treatment of practical heat engines in Chapter 8.
Because a Carnot engine is reversible, it may be operated in reverse; the Carnot cycle is then traversed in the opposite direction, and it becomes a reversible heat pump operating between the same temperature levels and with the same quantities QH, QC, and W as for the engine but reversed in direction, as illustrated in Fig. 5.1(b). Here, work is required, and it is used to “pump” heat from the lower-temperature heat reservoir to the higher-temperature heat reservoir. Refrigerators are heat pumps with the “cold box” as the lower-temperature reservoir and some portion of the environment as the higher-temperature reservoir. The usual measure
5.3. Carnot Engine with Ideal-Gas-State Working Fluid 179 of quality of a heat pump is the coefficient of performance, defined as the heat extracted at the lower temperature divided by the work required, both of which are positive quantities with respect to the heat pump:
ω ≡ ___ Q C
W (5.8)
For a Carnot heat pump this coefficient can be obtained by combining Eq. (5.4) and Eq. (5.5) to eliminate QH:
ω Carnot ≡ ______ T C
T H − T C (5.9) For a refrigerator at 4°C and heat transfer to an environment at 24°C, Eq. (5.9) yields:
ω Carnot = 4 + 273.15________
24 − 4 = 13.86
Any actual refrigerator would operate irreversibly with a lower value of ω . The practical aspects of refrigeration are treated in Chapter 9.
5.3 CARNOT ENGINE WITH IDEAL-GAS-STATE WORKING FLUID
The cycle traversed by a working fluid in its ideal-gas state in a Carnot engine is shown on a PV diagram in Fig. 5.2. It consists of four reversible processes corresponding to Steps 1 through 4 of the general Carnot cycle described in the preceding section:
∙ a → b Adiabatic compression with temperature rising from TC to TH. ∙ b → c Isothermal expansion to arbitrary point c with absorption of heat QH. ∙ c → d Adiabatic expansion with temperature decreasing to TC.
∙ d → a Isothermal compression to the initial state with rejection of heat QC.
In this analysis the ideal-gas-state working fluid is regarded as the system. For the iso-thermal steps b → c and d → a, Eq. (3.20) yields:
Q H = RT H ln ___ V cig
V big and Q C = RT C ln ___ V aig V dig Dividing the first equation by the second gives:
Q ___H
Q C =
T H ln ( V cig / V big )
____________
T C ln ( V aig / V dig ) For an adiabatic process Eq. (3.16) with dQ = 0 becomes,
− ____ C Vig
R ___dT T =
d V ig
____ V ig
For steps a → b and c → d, integration gives:
∫ T C T H C ___Vig R dT___
T = ln ___ V aig
V big and ∫ T C T H ___ C Vig R ___dT
T = ln ___ V dig V cig
Because the left sides of these two equations are the same, the adiabatic steps are related by:
ln ___ V dig
V cig = ln V ___aig
V big or ln ___ V cig
V big = − ln ___ V aig V dig
Combining the second expression with the equation relating the two isothermal steps gives:
___ Q H
Q C = − ___ T H
T C or ____− Q C T C = ___ Q H
T H This last equation is identical with Eq. (5.4).
5.4 ENTROPY
Points A and B on the PVt diagram of Fig. 5.3 represent two equilibrium states of a particular fluid, and paths ACB and ADB represent two arbitrary reversible processes connecting these points. Integration of Eq. (5.1) for each path gives:
Δ S t = ∫ d_____ Q rev
T and Δ S t = ∫ d_____ Q rev T
Figure 5.2: PV diagram showing a Carnot cycle for a working fluid in the ideal-gas state.
P
a b
c
d QH
QC TH
TC
TC TH
V
5.4. Entropy 181
Because ΔSt is a property change, it is independent of path and is given by S Bt − S At . If the fluid is changed from state A to state B by an irreversible process, the entropy change is again Δ S t = S Bt − S At , but experiment shows that this result is not given by ∫dQ / T evaluated for the irreversible process itself, because the calculation of entropy changes by this integral must, in general, be along reversible paths.
The entropy change of a heat reservoir, however, is always given by Q/T, where Q is the quantity of heat transferred to or from the reservoir at temperature T, whether the transfer is reversible or irreversible. As noted earlier, the effect of heat transfer on a heat reservoir is the same regardless of the temperature of the source or sink of the heat.
If a process is reversible and adiabatic, dQrev = 0; then by Eq. (5.1), dSt= 0. Thus the entropy of a system is constant during a reversible adiabatic process, and the process is said to be isentropic.
The characteristics of entropy may be summarized as follows:
∙ Entropy relates to the second law in much the same way that internal energy relates to the first law. Equation (5.1) is the ultimate source of all equations that connect entropy to measurable quantities. It does not represent a definition of entropy; there is none in the context of classical thermodynamics. What it provides is the means for calculating changes in this property.
∙ The change in entropy of any system undergoing a finite reversible process is given by the integral form of Eq. (5.1). When a system undergoes an irreversible process between two equilibrium states, the entropy change of the system Δ S t is evaluated by application of Eq. (5.1) to an arbitrarily chosen reversible process that accomplishes the same change of state as the actual process. Integration is not carried out for the irreversible path.
Because entropy is a state function, the entropy changes of the irreversible and reversible processes are identical.
∙ In the special case of a mechanically reversible process (Sec. 2.8), the entropy change of the system is correctly evaluated from ∫dQ / T applied to the actual process, even though the heat transfer between system and surroundings represents an external irreversibility. The reason is that it is immaterial, as far as the system is concerned,
Figure 5.3: Two reversible paths joining equilibrium states A and B.
A
D
C
B
Vt P