OPTIMIZATION OF CARNOT CYCLE ENGINE ! WITH IRREVERSIBILITIES BY USING THERMODYNAMICS WITH FINITE SPEED S. Petrescu', M. Feidt2, M. Costea' 'Polytechnic University of Bucharest, Department of Applied Thermodynamics Splaiul Iudependentei 313, Bucharest, Romania 'L.E.M.T.A., U.M.R. 7563, Universith "Henri Poincar6" Nancy 1 2, avenue de la Fori3 de Haye, 54516 Vandoeurn, France Abstract. The Carnot cycle engine with internal and external irreversibilities is studied by using Thermodynamics with Finite Speed (TFS) that is based on a new expression of the First Law for Processes with Finite Speed. Several causes of the internal irreversibility are taken into account in a direct manner in this expression, namely (1) the finite speed of the piston, (2) mechanical friction between the machine parts. The Direct Method is used and the First Law for processes with fmite speed is integrated for all processes of the Carnot cycle. The First Law Efficiency and Power output of the Camot cycle with Finite Speed are computed and illustrated versus the piston speed. Maximum values for each of them and correspondingly optimum values of the piston speed are indicated. Consequently, there are two optimum piston speed, one for the maximum efficiency (when the speed tends to zero) and the other, for the Maximum Power output of the engine (when the speed has a finite value depending on many parameters of the cycle). The Efficiency corresponding to the Maximum Power is smaller than the Efficiency computed with the "nice radical" in CurUm-Ahlborn optimization. Optimum Temperatures of the working fluid at the source and sink are also determined and compared to the Chambadal temperature (geometrical average of the heat thermal reservoirs temperatures). Both the temperature of the working fluid at the hot-end and the gas temperature at the coldend of the engine evolutions versus the piston speed determine the difference between the optimum values of the piston speed cOrreSpOnding to the Maximum Power output or Maximum Efficiency of the engine. The Maximum Efficiency comsponds to the Minimum Entropy Generation in the cycle, that occurs when the piston speed is tending to zero, while a certain Entropy Generation is needed for the Maximum Power regime. 1. Introduction A study of a Carnot Cycle engine with finite speed of the piston is presented. By considering internal and external irreversibilities, a technique for calculating the efficiency and the power output of the engine operating upon the Carnot cycle is developed. It is based on the First Law of Thermodynamics for processes with finite speed [l-81 and the Direct Method [9-141. To apply the Direct Method in the irreversible Carnot cycle analysis, direct integration of the equations based on the First Law for processes with finite speed is performed. Hence one directly expresses the efficiency and the power output of the engine as functions of the piston speed. We call this technique the Direct Method. The results of this analysis are compared to those of Chambadal approach [151 on useful energy associated to the thermal resistance of heat exchangers and Carnot cycle, and Cunon-Ahlborn approach on the Carnot cycle engine with finite time [161. Our analysis shows that there are two optimum values of the piston speed, one corresponding the maximum power output and the other, to the maximum efficiency -L’ -, DISCLAIMER .. Portions of this document may -be illegible in electronic image products. Images are produced from the best available original document. of the engine. Only one maximum appears in the other approaches and it corresponds to the maximum power of the Carnot cycle. A difference between the Chambadal optimized temperature of the working fluid and our optimized temperatures that are function of the piston speed is shown. It is due to the heat transfer between the heat reservoirs and the working fluid, that in our analysis is modeled closer to the real operation of the engine. 2. First Law of Thermodynamics for processes with finite speed applied to the Carnot cycle engine by using the Direct Method The computation of the efficiency and power output of the irreversible Carnot cycle engine is performed using the First Law of Thermodynamics for processes with Finite Speed [l-81. The First Law for closed systems operating with Finite Speed is where U is the internal energy (J), Q the heat exchanged with the surroundings, Pa, i the instantaneous average pressure in the system (Pa), w the piston speed (ds), c the average molecular speed (ds), a = & the coefficient depending of the nature of the gas (y the ratio of specific heats), f the fraction of heat generated by friction between the machine parts that remains in the system (0 c f C l), APf the pressure drop due to the friction between machine parts (Pa), V volume of the system (m3). Then the irreversible work for processes with finite speed in closed systems is expressed by when applied to processes with finite speed for the Carnot cycle engine as described in the PV diagram, Figure 1 (sign + corresponds to the compression and sign - corresponds to the expansion). Previous paper [12] have studied the effect of internal irreversibilities due to the adiabatic processes with finite speed on the performance of the Carnot cycle engine in comparison with the Cwon-Ahlborn analysis [la]. We neglect here the internal irreversibility of the adiabatic processes with tinite speed, 2-3' and 4'-1 in order to focus on the irreversible isothermal processes with finite speed heat transfer, 1-2 and 3-4. Hence, the ratio of the volumes will be kept constant like in the classical Thermodynamics approach of the Carnot cycle (working with perfect gas) that will simplify the mathematical treatment of the problem. where E is the volumetric compression ratio. By integrating eqs. (1) and (2) for the isothermal processes in the Carnot cycle with finite speed (see Fig. 1) the following expressions yield I with with zL= 1+- ' [ aw fMfl with with zL= I+- [ &+GA!f 1 where TH is the source temperature (K), TL the sink temperature (K), R the gas constant (Jkg K), W,, the work done when temperature difference at the thermal reservoirs and finite piston speed of the piston are considered. I !f >: Figure 1. Carnot cycle engine withfinite speed of the piston By introducing eqs. (4)-(7b) in the expressions of the Efficiency and Power output of the Carnot engine one gets tlcc,w = -Wview = mR In E (ZHTH- ZLTL)-[- ---.-"fi '! ") (8a) QH,~ iH*mRTHhE ZH ZH TH TL tlcc,w < tlcc =I-- (8b) TH where Ap is the piston area (m'). To determine the temperatures of the gas, TH and TL,the condition of equality of the heat computed in Thermodynamics with Finite Speed, eq. (4a)-(5b) and the heat determined by heat transfer considerations must be fulfilled: at the hot - end of the engine QH,~,.,.~ = UHAHATH- ZH = z;i * mRT' h E (10) 4 at the cold - end of the engine IQL,~-~=ULALATL-rL = i;,ernRTLIn E (1 1) where U is the overall heat transfer coefficient (W/m2K), A the heat transfer area (m'), t the contact time between the gas and the reservoir (s), H and L denote the source and sink respectively. From eq. (10) and (1 1) the temperature differences at the source and sink are expressed as: Furthermore, these temperature differences are used to express the gas temperatures ;;I- mRT' h E zi -mRTL In E TH = TH,S - 9 TL=TL,s+ A (1 3) UHAH* 7~ L L' L By- considering a constant speed equal to the average speed of the cycle (machine), w , the products of the contact areas between the gas and heat reservoirs and the contact time duration will be given by AL-~L=-[l.dTnDL2 D 1-118 4 W where D is the cylinder diameter (m), L maximum cylinder length (m). Obviously both products in eqs. (14) and (1 5) are dependent of dimension and operation data. The last one also contains the temperature ratio, TL/ TH that depends of heat transfer conditions through the heat transfer coefficients, UH(W~and U~(wg), so it cannot be directly computed. Equations (13) yield the expressions of the gas temperature at the hot-end and cold-end, respectively where mR = &V1, f q,, and TI, = TL,~, VI, = VI (17) Due to the above mentioned dependence, the computation of the Efficiency and Power output of the engine is done iteratively, by combining eqs. (14H16)with the following two equations: where The computation of the heat transfer coefficients have used the following correlations [171, where the average bulk temperature of the fluid is considered NUD = 0.023 Re;’ Pr” (21) for ReD > 3000 and n = 0.4 for heating; n = 0.3 for cooling; and for Reg c2100. The dynamic viscosity and thermal conductivity of the air are computed as follows [17]: p *lo7= 4.12235 + 0.7121 11 - T - 4.28173 .lo4 .T2+ 5.86408 -lo-*- T3 + +1.25726-10-10*T4-6.68760*10-14*T5 +1.05134-10-’7 .T6 (23) b103 = 0.81363 + 0.08083.T + 6.13727.10” .T2- 2.10686.10-’ .T3+ +2.06091.10-10.T4 -8.42775.10-’4.T5 +1.27292.10-” .T6 (24) 3. Discussions The results of the calculations are presented in Figs. 2 and 3. They are relative to a Carnot cycle with perfect gas (air) and takes into account the Finite Speed of the piston, heat transfer correlation at the source and sink, variable gas viscosity and variable conduction coefficient of the working fluid with the temperature, variable heat transfer areas and time of contact with the two heat reservoirs during the isothermal processes (because of the piston movement).