Irreversible Carnot cycles

Consequently for the output work A A of an irreversible Carnot cycle we have... 

For an irreversible Carnot cycle with the working temperatures T y and Tq, where T(v>To > 0, it follows from the definition of the (transformation) efficiency, denoted as 7, that... 

In the case of an irreversible Carnot cycle O, Clausius integral, the algebraic sum of all the by-temperature-reduced heats (34) [both delivered (directly) to the cycle and drained off (directly) from the cycle], is less than 0. This is caused by the heat AQox being drained off from the medium C (into B) when its temperature is Tq. This is a consequence of the requirement that the thermodynamic path O in C must be cyclical, and, thus, that the whole process be repeatable. 

The exergy equation (2.26) enables useful information on the irreversibilities and lost work to be obtained, in comparison with a Carnot cycle operating within the same temperature limits (T ,ax = Ey and T in = To). Note first that if the heat supplied is the same to each of the two cycles (Carnot and IJB), then the work output from the Carnot engine (Wcar) is greater than that of the IJB cycle (Wijg), and the heat rejected from the former is less than that rejected by the latter. 

For an irreversible Carnot type cycle (ICAR) with all heat supplied at the top temperature and all heat rejected at the lowest temperature (Tmax = rmi, = To, / UT = 0, icAR=l). but with irreversible compression and expansion (rxicAR = < 1). Eqs. (2.33) and (1.17) yield... 

Show that the efficiency of a Carnot cycle in which any step is carried out irreversibly cannot be greater than that of a reversible Camot cycle. 

In the development of the second law and the definition of the entropy function, we use the phenomenological approach as we did for the first law. First, the concept of reversible and irreversible processes is developed. The Carnot cycle is used as an example of a reversible heat engine, and the results obtained from the study of the Carnot cycle are generalized and shown to be the same for all reversible heat engines. The relations obtained permit the definition of a thermodynamic temperature scale. Finally, the entropy function is defined and its properties are discussed. 

This chapter establishes a direct relation between lost work and the fluxes and driving forces of a process. The Carnot cycle is revisited to investigate how the Carnot efficiency is affected by the irreversibilities in the process. We show to what extent the constraints of finite size and finite time reduce the efficiency of the process, but we also show that these constraints still allow a most favorable operation mode, the thermodynamic optimum, where the entropy generation and thus the lost work are at a minimum. Attention is given to the equipartitioning principle, which seems to be a universal characteristic of optimal operation in both animate and inanimate dynamic systems. 

The above relate to Figure A.2, which shows an enhanced version of Figure A.l, designed to allow operation of the cell at any selected high temperature and pressure. Isentropic circulators are incorporated to generate the increased conditions. The cell generates heat which is passed without temperature difference to a Carnot cycle to generate power, a reversible process free from the irreversibility of combustion. 

A prime lesson in irreversible process theory is based on Figure 3.1, illustrating Joule s experiment. In that experiment, shaft power was dissipated irreversibly by a rotating paddle, to become energy in a tank of near ambient temperature water. The chaotically interactive translation, vibration and rotation of fluid molecules is energy.Energy is accessible to generate power, only by cyclic processes (heat cycles) as defined by Carnot. (Carnot cycle theory is outlined in Chapter 1.)... 

Much more would have to be done in the laboratory to investigate the possibility of a practical Faradaic reformer choice of electrode and electrolyte the possibility of irreversible electrode reactions the need for an electrocatalyst. It can be concluded safely that a basis for fuel chemical exergy efficiency calculations exists, namely the Faradaic reformer, fuel cell combination at standard conditions. The reduced performance of the reformer fuel cell combination, at temperature and pressure, can be left as a major exercise for the reader by adding isentropic circulators and a Carnot cycle to Figure A.2. 

By raising the operating pressure and temperature, the output of equilibrium fuel cells is shown to be reduced. The output includes electrical power from the fuel cell, power from the isen-tropic and isothermal circulators, and power from the Carnot cycle. No artificial credit is given for producing combined heat and power by the popular but erroneous addition of power and heat, which have different units, to produce mythical high efficiencies. The addition of similar units (e.g. power plus power, or heat plus heat) is the valid procedure, in line with Joule s irreversible experiment, from which 1W s 1J. The 3> sign can never mean =. In particular the stirred liquid of the experiment cannot be unstirred. 

For given values of Tc and Tu, the highest possible value of co is attained for Carnot-cycle refrigeration. The lower values for the vapor-eompression eycle result from irreversible expansion in a tlrrottle valve and irreversible compression. The following example provides an indication of typical values for eoefficients of perfomianee. 

If the reversed Carnot cycle were coupled to the irreversible machine and driven by it, there would be a net amount of heat <2i Qi > 0 flowing into the reservoir from this coupled machine. There being no net work production, this amount of heat is obtained from the low-temperature reservoir at T2. The coupled machine would be pumping heat from a lower temperature to a higher temperature while producing no other changes in the surroundings. But such a result is impossible by the second law. 

The result that a spontaneous (and therefore irreversible) process occurs with an overall increase of entropy in the system and its surroundings is universally true. The proof of this is based on the fact that the efficiency of a Carnot cycle in which some of the steps are irreversible must be less than that of a purely reversible cycle, since the maximum work is performed by systems which are undergoing reversible processes. Thus in place of equation (5.23) we have, for an irreversible cycle,... 

The above value is not bad. In reality much more exergy is destroyed because of irreversibilities, particularly due to the cooling. For example, if we take into account only the hot utility for separation (feed preheating and reboiler) the total heat is g, = 138 + 1147 = 1285 kW. This could be extracted virtually from surroundings and converted by means of a Carnot cycle working between 298 and 406.15 K, for which the following work can been obtained ... 

But, generally, such a cycle with adiabatic and isothermal irreversible processes may be realized with real gas (or even liquid). Those with real gas approximate the reversible Carnot cycle with ideal gas by a double limiting process as follows (i.e., we form the ideal cyclic process from set A (and also B and C), see motivation of postulate U2 in Sect. 1.2) running this cycle slower and slower... 

The efficiency of an irreversible cycle operating between temperatures Th and Tl, is less than that of the Carnot cycle between the same two temperatures. This condition is equivalent to the inequality... 

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