Reversible cycles

The second law of thermodynamics may be used to show that a cyclic heat power plant (or cyclic heat engine) achieves maximum efficiency by operating on a reversible cycle called the Carnot cycle for a given (maximum) temperature of supply (T ax) and given (minimum) temperature of heat rejection (T jn). Such a Carnot power plant receives all its heat (Qq) at the maximum temperature (i.e. Tq = and rejects all its heat (Q ) at the minimum temperature (i.e. 7 = 7, in) the other processes are reversible and adiabatic and therefore isentropic (see the temperature-entropy diagram of Fig. 1.8). Its thermal efficiency is... 

In the ultimate version of the reheated and intercooled reversible cycle [CICICIC- HTHTHT- XJr, both the compression and expansion are divided into a large number of small processes, and a heat exchanger is also used (Fig. 3.6). Then the efficiency approaches that of a Carnot cycle since all the heat is supplied at the maximum temperature Tr = T ax and all the heat is rejected at the minimum temperature = r,nin. 

The important point here is that the efficiency is a function of the temperature ratio 6 as well as the pressure ratio r (and d, whereas it is a function of pressure ratio only for the reversible cycle, [CHT]r. 

The arguments of this section are developed sequentially, starting with internally reversible cycles and then considering irreversibilities. Here we concentrate on the gas turbine with simple closed or open cycle (CHT, CBT). 

Thus the cooled reversible cycle [CHT]rci with a first rotor inlet temperature, Tj, will have an internal thermal efficiency exactly the same as that of the uncooled cycle [CHTJru with a higher turbine entry temperature 3 = Tr, and the same pressure ratio. There is no penalty on efficiency in cooling the turbine gases at entry but note that the specific work output, w = (wj — wc)/CpT = [(0 /x) — 11(j — 1), is reduced, since 0 < 0. 

A reversible cycle with turbine expansion split into two steps (high pressure, HP, and low pressure, LP) is illustrated in the T, s diagram of Fig. 4.3. The mass flow through the heater is still unity and the temperature rises from T2 to Tt, = Tq hence the heat supplied (3b is unchanged, as is the overall isentropic temperature ratio (x). But cooling air of mass flow i//H is used at entry to the first HP turbine (of isentropic temperature ratio. xh) and additional cooling of mass flow is introduced subsequently into the LP turbine (of isentropic temperature ratio Xl)- The total cooling flow is then i/( = i/ h + >h.-... 

Fig. 4.4. Temperature-entropy diagram for multi-step cooling—reversible cycle 1CHT]r< m (after Ref. [5 ).

The heat supplied is = 6p[7 3 — Tj], and for each of these reversible cycles the heat rejected will be + [Pg.54]

For the various reversible cycles described in Section 4.2.1, the thermal efficiency was the same, independent of the number of cooling. steps. This is not the case for the irreversible cycles described in this section. Both the thermal efficiency and the turbine exit temperature depend on the number and nature of cooling steps (whether the cooling air is throttled or not). 

One of these alternate models, postulated by Gunter Wachtershanser, involves an archaic version of the TCA cycle running in the reverse (reductive) direction. Reversal of the TCA cycle results in assimilation of CO9 and fixation of carbon as shown. For each turn of the reversed cycle, two carbons are fixed in the formation of isocitrate and two more are fixed in the reductive transformation of acetyl-CoA to oxaloacetate. Thus, for every succinate that enters the reversed cycle, two succinates are returned, making the cycle highly antocatalytic. Because TCA cycle intermediates are involved in many biosynthetic pathways (see Section 20.13), a reversed TCA cycle would be a bountiful and broad source of metabolic substrates. 

No thermodynamic cycle can be more efficient than a reversible cycle operating between the same temperature limits. 

The efficiency of all reversible cycles absorbing heat from a single-constant higher temperature and rejecting heat at a single-constant lower temperature must be the same. 

The maximum theoretical work Wn, obtainable from a system was derived by Carnot who considered the transformation of heat energy into work when a perfect gas in a cylinder with a piston was taken through a reversible cycle (the Carnot cycle), in which the system was almost at equilibrium during each step of the cycle. It was shown that... 

Reverse cycle. The direction of flow of the refrigerant is reversed to make the evaporator act as a condenser. Heat storage or another evaporator are needed as a heat source. 

Winter heating items fitted within room air-conditioners may be electric resistance elements, hot water or steam coils, or reverse cycle (heat pump). One model of water-cooled unit operates with a condenser water temperature high enough to be used also in the heating coil. 

The heat reclaim packaged unit system comprises water-cooled room units with reverse cycle valves in the refrigeration circuits. The water circuit is maintained at 21-26°C, and may be used as a heat source or sink, depending on whether the individual unit is heating or cooling. (See Figure 28.11.)... 

Provide mid-season heating from condenser heat or heat pump (reverse-cycle) operation... 

With this aim, Carnot proceeded to introduce the novel conception of a reversible cycle of operations, and arrived at the exceedingly important result that the motive power capacity for doing wrork] of heat is independent of the agents employed to develop it its quantity is determined solely by the temperatures of the bodies between which, in the final result [i.e., after the completion of the cycle], the transfer of heat occurs. Let there be given a source, and a refrigerator, at temperatures Ti, T2 respectively, where Tx > T2. In order that finite quantities of heat may be added to or taken from these without change of their temperatures, we may suppose them to consist of... 

There is now performed a reversible cycle called Carnot s cycle, and consisting of four operations ... 

We will now consider the changes produced in the direct and reversed cycles. 

From this we deduce the following important corollary If a quantity of heat Q is absorbed in a reversible cycle, with given temperatures of source and refrigerator, the quantity of work A obtained from it is independent of the arrangement used in performing the cycle. 

Corollary 1.—The area enclosed by the circuit representing an isothermal reversible cycle on the indicator diagram is zero if, therefore, the curve is not a segment of a line transversed from A to B and then from B to A (Fig. 4), it must form two loops of equal areas but traversed in opposite senses, or else such a system of positive and negative loops that the total area is zero. 

Corollary 2.—The algebraic sum of the quantities of heat withdrawn from or given to the constant temperature reservoir in an isothermal reversible cycle is zero. 

Equation (6) was obtained in a much less direct manner by Clausius in December, 1854, and is Usually known as the Equality of Clausius. It applies only to reversible cycles. 

For if AMB, ANB are any two such reversible paths (Fig. 12), these taken together constitute a reversible cycle AMBN, for which... 

The area is positive if traced out clockwise. Since the heat absorbed in the cycle is equal to the work done, the areas of the Carnot s cycle on the (p, v) and (S, T) diagrams are equal. This may be generalised to apply to any reversible cycle where the only external work is done by expansion. 

In a Carnot s cycle, the entropy Qi/Ti is taken from the hot reservoir, and the entropy Q2/T2 is given up to the cold reservoir, and no other entropy change occurs anywhere else. Since these two quantities of entropy are equal and opposite, the entropy. change in the hot reservoir is exactly balanced, or, to use an expression of Clausius, is compensated by an equivalent change in the cold reservoir. Again, in any reversible cycle there is on the whole no production of entropy so that all the changes are compensated. 

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