Maximum entropy generation rate in a heat exchanger at constant inlet parameters

Transcription

1 Maximum entroy generation rate in a heat exhanger at onstant inlet arameters Rafał LASKOWSKI, Maiej JAWORSKI Online: htt:// df Cite this artile as: Laskowski R., Jaworski M. Maximum entroy generation rate in a heat exhanger at onstant inlet arameters. Journal of Mehanial and Energy Engineering, Vol. (4), No., 7, Journal of Mehanial and Energy Engineering ISSN (Print): ISSN (Online): Volume: (4) Number: Year: 7 Pages: Artile Info: Reeived 5 Aril 7 Aeted 7 May 7 Oen Aess This artile is distributed under the terms of the Creative Commons Attribution 4. (CC BY 4.) International Liense (htt://reativeommons.org/lienses/by/4./), whih ermits unrestrited use, distribution, and rerodution in any medium, rovided you give aroriate redit to the original author(s) and the soure, rovide a link to the Creative Commons liense, and indiate if hanges were made.

2 ISSN: Vol. No. (4) June MAXIMUM ENTROPY GENERATION RATE IN A HEAT EXCHANGER AT CONSTANT INLET PARAMETERS Rafał LASKOWSKI *, Maiej JAWORSKI * Institute of Heat Engineering, /5 Nowowiejska Str., -665, Warsaw, Poland, rlask@it.w.edu.l Institute of Heat Engineering, /5 Nowowiejska Str., -665, Warsaw, Poland, maiej.jaworski@it.w.edu.l (Reeived 5 Aril 7, Aeted 7 May 7) Abstrat: The main goal of the aer is to rovide a ondition for whih a imum entroy generation ours in a heat exhanger at onstant inlet arameters (temeratures and mass flow rates). Knowing this ondition is essential during the design of the heat exhanger as it allows designers to avoid one of its most unfavourable oerating onditions in terms of thermodynamis. Entroy generation resulting from the resistane of heat-transferring fluids to flow was not taken into aount. Entroy generation was analysed as a funtion of a heat flow rate at onstant arameters at the inlet of a ondenser and a ounter-flow double-ie heat exhanger. The analysis showed that for the ondenser the entroy generation rate inreases with the inrease in the heat flow rate. The imum entroy generation rate ours for the imum flow rate of the heat that an be transferred aording to the definition of heat transfer effetiveness. For the ounterflow heat exhanger, the entroy generation as a funtion of the heat flow rate reahes imum at onstant inlet arameters (temeratures and mass flow rates). It aeared that the eak entroy generation, or the largest exergy loss, ours when the outlet temeratures of the fluids are equal. This assertion was verified against data obtained from a simulator of the ounter-flow heat exhanger for two different relations between heat aaity rates. Keywords: imum entroy generation rate, heat exhanger. INTRODUCTION Heat exhangers are widely used in industrial aliations to transfer heat from a higher to a lower temerature fluid. Heat transfer effetiveness is ommonly used to omare heat exhangers and assess their erformane, it is defined as the ratio of the atual to the imum rate of heat flow whih an be transferred in the heat exhanger [ 4]: ε =. () & Q The heat transfer effetiveness, whih is used as an indiator in the omarison of heat exhangers, is a funtion of two arameters: NTU (number of heat transfer units) whih is equal to a rodut of an overall heat transfer oeffiient (U) and a heat transfer surfae area (A), divided by the smaller fluid heat aaity rate UA NTU =, () C ( C ) min,c and the ratio of heat aaity rates: C C ( C, C ) ( C, C ) min C r =, (3) where the heat aaity rate is a rodut of the fluid mass flow rate and its seifi heat at onstant ressure: C = m &. (4) Sine irreversible roesses ourring during the heat transfer are not diretly taken into aount in the definition of the heat transfer effetiveness, the seond

3 8 Laskowski R., Jaworski M. Journal of Mehanial and Energy Engineering, Vol. (4), No., 7, law of thermodynamis was alied to enable a more omrehensive assessment of the heat exhanger erformane. Aording to the seond law of thermodynamis, irreversible roesses take lae in the heat exhanger during the heat flow, their measure being the rate of entroy generation. The entroy generation rate in the heat exhanger results from the heat flow and the resistane of heat-transferring fluids to flow (ressure losses). The heat exhangers should be designed so that the losses due to irreversible roesses aomanying the heat flow are ket as small as ossible. MaClintok [5] and Prigogine [6] were the first to assess heat exhanger erformane by introduing the minimization of the entroy generation. Bejan [7, 8] develoed the aroah of the entroy generation minimization (EGM) and roosed an entroy generation number N S defined as the entroy generation rate (Ṡ) divided by the lower heat aaity rate: N s S& =, (5) C min where the entroy generation rate for the heat exhanger in whih the heat is transferred between two fluids is equal to: S & =. (6) In his aroah, Bejan onsidered two tyes of irreversibility: one resulting from the heat transfer, and one assoiated with the resistane of heat-transferring fluids to flow. For variables (temeratures and mass flow rates) at the heat exhanger inlet and for known heat exhanger geometry, Bejan introdued a arameter N S as a funtion of heat transfer effetiveness ε, and observed that the heat transfer effetiveness does not always inrease with a derease in entroy and does not always reah imum. Bejan named suh a behaviour of N S with reset to the heat transfer effetiveness a aradox. This aradox was exlained in [9, ]. To exlain it, some researhers introdued a revised arameter N S. For examle, the arameter roosed by Bejan (N S ) was develoed in [-] wherein another definition was given under the name of an entroy generation index, equal to the sum of entroy generation rates (Ṡ) divided by a rodut of an overall heat transfer oeffiient (U) and the heat transfer surfae area (A): S& Γ =. (7) UA A arameter roosed by Hesselgreaves [3] is equal to the sum of entroy generation rates for the heat exhanger divided by the heat flow rate onsidering the inlet temerature of the older fluid so that the roosed arameter is dimensionless (the revised entroy generation number [4]): T S N = & i. (8) s Shah and Skieko [5] resented relations between the heat transfer effetiveness and entroy generation for various tyes of heat exhangers. They demonstrated that for a minimum entroy generation the effetiveness an have an intermediate imum or minimum value. In the literature a number of aers [6-3] an be found wherein the entroy generation minimization is analysed regarding various tyes of heat exhangers. The EGM method is mainly emloyed in seleting otimum geometri arameters of the heat exhanger, suh as the tube inner diameter [4-8] Mohammed [9] introdued the arameter N S as a funtion of the heat transfer effetiveness, the ratio of heat aaity rates and temerature ratio at the heat exhanger inlet. He demonstrated that the heat transfer effetiveness of a heat exhanger should aroah one, as in suh a ase irreversible roesses of a smaller extent (lower entroy generation) are exeted. Mohammed [9] roved that the entroy generation assoiated with the resistane of flow is muh lower than that resulting from the heat transfer in the heat exhanger. Ahmad Fakheri [3] indiated that the entroy minimization should not be an objetive funtion for the heat exhanger, and roosed an entroy flux whih is defined by Eq. (6). In most of the aers ited, the entroy generation minimization was analysed for variable arameters (temeratures and mass flow rates) at the heat exhanger inlet and for known heat exhanger geometry, or for onstant inlet and outlet arameters (temeratures and mass flow rates) so that an otimal geometry, e.g. the tube diameter or the ies ith an be hosen. In the author s view, it is imortant to determine not only the otimal arameters but also those oerating onditions that are the most unfavourable in terms of thermodynamis so that they an be avoided during the design. For this urose a heat flow rate was sought for whih the largest entroy generation rate (the largest exergy loss) ours at known (onstant) inlet arameters. The largest exergy loss ours for the largest entroy generation rate. Aording to the Guoy-Stodola theorem, the exergy loss equals the rodut of the entroy generation rate and the ambient temerature [3, 3]: δ B = T S. (9) The entroy generation was analysed as a funtion of the heat flow rate for known arameters (temera- o &

4 Laskowski R., Jaworski M. Journal of Mehanial and Energy Engineering, Vol. (4), No., 7, tures, mass flow rates) at the inlet in the ase of a heat exhanger with hase hange (a ondenser) and without it (a ounter-flow double-tube heat exhanger).. ANALYSIS OF ENTROPY GENERA- TION: A PHASE-CHANGE HEAT EX- CHANGER (A CONDENSER) In a ondenser, the older fluid reeives heat from the ondensing fluid whih, deending on the state of steam, gives u some or all heat of evaoration. In the ondenser under onsideration, water flows through tubes and reeives the heat from the ondensing steam flowing by the outer surfae of the tubes. Pressure losses were not taken into aount on the water and steam sides. The temerature and mass flow rate of ooling water and the saturation temerature of steam were assumed as the known arameters. The entroy hange due to the hase transition of saturated steam is [-3, 3]: r =. () The entroy generation rate for water equals [-3, 3]: T s o = w ln. () Ti The sum of entroy generation rates for both the fluids an be exressed by Eq. (6). By inserting Eqs. () and () into Eq. (6), it an be obtained: mr o S & & = ln w. () T On onsidering the equations for the heat flow rate: Eq. () takes the form: & = r, (3) Q ( T T ) &, (4) Q = w o i S& = ln + w. (5) wt Eq. (5) shows a relation between the entroy generation rate in the ondenser and the heat flow rate. In order to determine the imum value of the entroy generation rate for the ondenser at onstant temerature and mass flow rate of ooling water and saturation temerature of steam, one has to alulate a derivative: ds& w = + d T T s = w i. (6) When the ondition (6) is taken into aount, the value of the heat flow rate for the imum entroy generation rate in the ondenser is obtained as equal to the imum heat flow rate aording to the definition of the heat transfer effetiveness (): = =. (7) S w( Ti ) By inserting (7) into (5), it an be written: m ( ) w T i S & & = ln w. (8) T i The first term in Eq. (8) is the imum entroy generation rate on the steam side: i s = w = T. (9) s The seond term in Eq. (8) is the imum entroy generation rate on the water side: s = w ln. () Ti On onsidering Eqs. (9) and (), Eq. (8) takes the form: S & = () In the ase of a hase-hange heat exhanger (ondenser) at onstant temerature and mass flow rate of ooling water and steam saturation temerature, with the inrease in the heat flow rate, the entroy generation rate inreases and reahes the imum value for the heat flow rate equal to Q aording to Eq. (7), being the imum flow rate of the heat that an be transferred in the heat exhanger. Then, aording to Eq. (), the heat transfer effetiveness equals one (ε = ), while the outlet temerature of ooling water equals the saturation temerature of steam (T o =T s ). 3. ANALYSIS OF ENTROPY GENERA- TION: A HEAT EXCHANGER WITH- OUT PHASE CHANGE The entroy generation in the ase of a heat exhanger without hase hange was analysed for a ounter-flow double-tube heat exhanger with water as heat transferring fluids. To simlify the analysis, no ressure losses for both fluids were taken into aount. The temeratures at the heat exhanger inlet and mass flow rates of both the fluids were assumed as the known arameters. The entroy generation rate in the ase of the ounter-flow heat exhanger in question has the form of Eq. (6).

5 8 Laskowski R., Jaworski M. Journal of Mehanial and Energy Engineering, Vol. (4), No., 7, The entroy generation rate for the hotter fluid has the form: o = ln. () T The entroy generation rate for the older fluid has the form: o = ln. (3) Ti Combining Eqs. () and (3) the total entroy generation rate in heat exhanger an be written as: or: i ln ln + T o S& =, (4) T o T o ln ln + T o S& =. (5) T T On onsidering the equations for the heat flow rate: Eq. (5) takes the form: ( T T ) &, (6) Q = o i ( T T ) &, (7) Q = o i At onstant inlet temeratures and mass flow rates of both fluids, in order to evaluate the imum value of the entroy generation rate for the heat flow rate, one has to alulate the derivative of Eq. (8). Taking into aount the ondition for the extremum of entroy generation rate ( d S& / d = ), the value of the heat flow rate for the imum entroy generation rate is obtained in the form: S& = S ln + + T i ln + T i. (8) ( T T ) i i =. (9) Eq. (9) an be exressed as a funtion of the imum heat flow rate aording to Eq. () and the ratio of heat aaity rates: r = S + C. (3) The heat transfer effetiveness for the imum entroy generation rate equals: ε =. (3) + C r By inserting (9) into (8), we obtain the imum entroy generation rate: S& =. (3) By omaring Eqs. (5) and (3), we obtain two onditions: T i T i T o =, (33) ( ) T o T i T i ln ( ) T i + T i T i ln ( ) T i T i T i =. (34) ( ) The omarison between the onditions (33) and (34) indiates that the imum entroy generation rate in the heat exhanger (without onsidering ressure losses), at onstant temeratures at the inlet and mass flow rates of both the fluids, ours when their outlet temeratures are equal: T o = T o. (35) 4. A DESCRIPTION OF A COUNTER- FLOW DOUBLE-PIPE HEAT- EXCHANGER SIMULATOR The above derived relations were verified against data obtained from a simulator of the ounter-flow heat exhanger. A diagram of the ounter-flow double-ie heat exhanger with ommonly used symbols is shown in Fig.. It was assumed that the hotter fluid flows in a tube of smaller diameter d =.3 m. The older fluid flows between two tubes. The diameter of the outer tube is D =.43 m. The heat transferring fluids were assumed to be water. The temerature of the water giving u heat at the heat exhanger inlet is 5 C. The temerature of the heated water at the heat exhanger inlet is 7 C. T o T, i m & T o Fig.. A diagram of the ounter-flow double-ie heat exhanger with ommonly used symbols T, m & i

6 Laskowski R., Jaworski M. Journal of Mehanial and Energy Engineering, Vol. (4), No., 7, The simulator inut data were temeratures at the inlet and mass flow rates of both fluids, and geometrial data (inner and outer diameters, length of the heat exhanger). The outut (alulated) data were temeratures of both fluids at the heat exhanger outlet, overall heat transmission oeffiient, heat flow rate and the heat transfer effetiveness. Calulations were erformed for two relations between the heat aaity rates: C >C and C >C. For the first one, the mass flow rates were ṁ =. kg/s for the older fluid, and ṁ =.3 kg/s for the hotter fluid. For the seond one, ṁ =. kg/s and ṁ =.3 kg/s were assumed. The alulations were erformed for onstant inlet temeratures and mass flow rates of both fluids for a heat exhanger length from to m. 5. RESULTS The outut data, i.e. heat flow rate, and heat exhanger effetiveness, obtained from the ounter-flow heat-exhanger simulation, are shown in the following figures for the two relations between heat aaity rates: C >C and C >C. The heat flow rate as a funtion of the heat exhanger length is dislayed in Fig.. the heat flow rate kw Fig.. The heat flow rate as a funtion of the heat exhanger length The heat transfer effetiveness of the heat exhanger as a funtion of the heat exhanger length is shown in Fig. 3. the heat transfer effetiveness the heat exhanger length m C>C C>C the heat exhanger length m C>C C>C Fig. 3. The heat transfer effetiveness of the heat exhanger as a funtion of the heat exhanger length At onstant arameters (temeratures and mass flow rates) at the heat exhanger inlet in Figs. and 3, a regularity of hanges in the heat flow rate and heat transfer effetiveness an be observed. With inreasing heat transfer surfae area the heat flow rate and the heat transfer effetiveness of the heat exhanger inreases. The overall heat transfer oeffiient for the relation C >C is greater than C >C, this is the reason why the erformane (heat flow rate, effetiveness) of the heat exhanger for the relation C >C is greater. The entroy generation rate as a funtion of the heat exhanger length is dislayed in Fig. 4 with lear imum values. the entroy generation rate W/K the heat exhanger length m C>C C>C Fig. 4. Entroy generation rate as a funtion of the heat exhanger length Based on data shown in Fig. 4, the imum entroy generation rate in the heat exhanger an be found. For short heat exhanger ies a substantial rise of entroy generation rate is observed with inreasing heat transfer surfae area until the imum value is reahed. Following the imum value of the entroy generation rate, further inrease in the heat transfer surfae area results in a slow derease in the entroy generation rate in the heat exhanger. Due to the higher the overall heat transfer oeffiient for the relation C >C the shorter length of the heat exhanger in order to ahieve equal exit temerature is needed. Therefore, the imum entroy generation rate for relation C >C ours for a smaller length of the heat exhanger. For the relation between the heat aaity rates C >C the imum entroy generation rate was Ṡ =.64 W/K for the heat exhanger length of L = m. For this length of the heat exhanger, the heat flow rate equals Q & = kw, the heat transfer effetiveness is ε =.769, while the temeratures at the heat exhanger outlet are equal: t o = t o = 5.68 C. For the relation between the heat aaity rates C >C the imum entroy generation rate was Ṡ =.6 W/K for the heat exhanger length of L = m. For this length of the heat exhanger, the temeratures at its outlet are equal: t o = t o = 43.9 C; the values of heat transfer rate and effetiveness are

7 84 Laskowski R., Jaworski M. Journal of Mehanial and Energy Engineering, Vol. (4), No., 7, the same as in the ase of the relation between the heat aaity rates C >C, whih follows from the assumed values of mass flow rates and Eqs. (3) and (3). The hange in arameter Γ as defined by Eq. (7) with inreasing heat exhanger length is shown in Fig. 5. S/UA Fig. 5. The hange in arameter Γ as defined by Eq. (7) as a funtion of the heat exhanger length The hange in arameter Ns as defined by Eq. (8) with inreasing heat exhanger length is shown in Fig. 6. Ns the heat exhanger length m C>C C>C the heat exhanger length m C>C C>C Fig. 6. The hange in arameter Ns as defined by Eq. (8) as a funtion of the heat exhanger length For arameters defined by Eqs. (7) and (8), no extremum exists with inreasing heat transfer surfae area for onstant arameters (temeratures and mass flow rates) of both the fluids at the heat exhanger inlet for the two relations between the heat aaity rates (C >C and C >C ). Exergy loss (Eq. (9)) with lear imum values is shown in Fig. 7. Referene temerature (T o ) was assumed to be equal to the temerature of the older fluid (water) (T i ) at the heat exhanger inlet. As for entroy generation rate (Fig. 4) heat transfer surfae area at whih exergy losses reah imum deends on the ratio of C and C. This extremum an be easily determined from the harts resented or, more reisely, with the use of heat exhanger omuta-tional simulator. exergy loss W Fig. 7. Exergy loss (9) as a funtion of the heat exhanger length 6. CONCLUSIONS the heat exhanger length m C>C C>C The main goal of the aer is to rovide a ondition for whih a imum entroy generation ours in a heat exhanger at onstant inlet arameters (temeratures and mass flow rates). If suh a working oint is known, one of the most unfavourable oerating onditions an be avoided during the design of the heat exhanger. The aer resents an analysis of the entroy generation for a ondenser and a ounter-flow double-ie heat exhanger. In the aer, the entroy generation resulting from the heat flow was onsidered, whereas the entroy generation due to the resistane of heattransferring fluids to flow was not taken into aount. The entroy generation rates were resented as funtions of the heat flow rate. For onstant arameters (temeratures and mass flow rates) at the heat exhanger inlet, a heat flow rate for whih the largest entroy generation rate ours and the onditions for suh a rate were sought. For the ondenser, the imum entroy generation rate ours for the imum flow rate of the heat that an be transferred aording to the definition of heat transfer effetiveness (), whih orresonds to the equality of the outlet temeratures of the oolant and ondensing fluid. For the ounter-flow heat exhanger, the entroy generation reahes imum as a funtion of the heat flow rate. It aeared that at onstant arameters (temeratures and mass flow rates) at the heat exhanger inlet the imum entroy generation, or the largest exergy loss, ours when the outlet temeratures of the fluids are equal. This feature was verified against data obtained from a simulator of the ounterflow heat exhanger. At onstant inlet arameters (temeratures and mass flow rates), heat transfer surfae areas were inreased, and alulations were erformed. The data from the ounter-flow heatexhanger simulator onfirmed that the imum entroy generation rate (the largest exergy loss) ours under the ondition that the outlet temeratures of the fluids are equal. The entroy generation for the ounter-flow heat exhanger was analysed for two relations

8 Laskowski R., Jaworski M. Journal of Mehanial and Energy Engineering, Vol. (4), No., 7, between the heat aaity rates of fluids: C >C and C >C. Nomenlature Symbols A heat transfer area, m C heat aaity rate, W/K C r ratio of heat aaity rates C r = C min/c, seifi heat at onstant ressure, J/(kg K) w seifi heat of water, J/(kg K) d diameter of the smaller tube, m D diameter of the larger tube, m L length of the heat exhanger, m mass flow rate, kg/s NTU the number of heat transfer units NTU = UA/C min, N s the revised entroy generation number, r hase transition heat, J/kg s seifi entroy, J/kg/K Ṡ rate of entroy generation, W/K T temerature, K t temerature, C U overall heat transfer oeffiient, W/(m K) Q & heat flow rate, W δ B exergy loss, W ε heat transfer effetiveness, - Γ the entroy generation index, entroy flux, Indies hot fluid old fluid i inlet o outlet, referene onditions s saturated onditions imum value min minimum value Referenes. Cengel Y. A. (998). Heat transfer. MGraw-Hill, New York.. Cengel Y. A. (). Heat and mass transfer. MGraw- Hill, New York. 3. Holman J. P. (). Heat transfer. MGraw-Hill, New York.. 4. Kostowski E. (). Wymiana ieła. Wydawnitwo Politehniki Śląskiej, Gliwie. (in Polish) 5. MClintok F. A. (95). The design of heat exhangers for minimum irreversibility. Annual Meeting of the ASME, New York, USA, aer No. 5 l-a Prigogine I. (967). Introdution to Thermodynamis of Irreversible Proesses (3rd Edition). Wiley, New York, Bejan A. (977). The onet of irreversibility in heat exhanger design: ounterflow heat exhangers for gasto-gas aliations. Journal of Heat Transfer, Vol. 99, No. 3, Bejan A. (98). Seond-law analysis in heat transfer and thermal design, Advanes in Heat Transfer, Vol. 5, Sekuli D. P. (986). Entroy generation in a heat exhanger. Heat Transfer Engineering, Vol. 7, Ogiso K. (3). Duality of heat exhanger erformane in balaned ounter-flow systems, Journal of Heat Transfer, Vol. 5, No. 3, Xu Z., Yang S., Chen Z. (996). A modified entroy generation number for heat exhangers. Journal of Thermal Siene, Vol. 5, No. 4, Daxi X., Zhixin L., Zengyuan G. (996). On effetiveness and entroy generation in heat exhange. Journal of Thermal Siene, Vol. 5, No.4, Hesselgreaves J. E. (). Rationalization of seond law analysis of heat exhangers. International Journal of Heat and Mass Transfer, Vol. 43, No., Guo J., Xu M., Cheng L. (9). The aliation of field synergy number in shell-and-tube heat exhanger otimization design. Alied Energy, Vol. 83, No., Shah R. K., Skieko T. (4). Entroy generation extrema and their relationshi with heat exhanger effetiveness number of transfer unit behavior for omlex flow arrangements. Jounal of Heat Transfer, Vol. 6, No. 6, Sahiti N., Krasniqi F., Fejzullahu Xh., Bunjaku J., Muriqi A. (8). Entroy generation minimization of a double-ie in fin heat exhange. Alied Thermal Engineering, Vol. 8, No. 7-8, Ordonez J., Bejan A. (). Entroy generation minimization in arallel-lates ounterfow heat exhangers. International Journal of Energy Researh, Vol. 4, No., Mishra M., Das P. K., Sarangi S. (9). Seond law based otimisation of rossflow late-fin heat exhanger design using geneti algorithm. Alied Thermal Engineering, Vol. 9, No. 4-5, Ogulata R. T., Doba F., Yilmaz T. (). Irreversibility analysis of ross flow heat exhangers. Energy Conversion and Management, Vol. 4, No. 5, Guo, J., Cheng, L., Xu, M. (). Multi-objetive otimization of heat exhanger design by entroy generation minimization. Journal of Heat Transfer, Vol. 3, No. 8, Kolenda Z., Donizak J., Hubert J. (4). On the minimum entroy rodution in steady state heat ondution roesses. Energy, Vol. 9, No. -5, Kolenda Z. (6). Analysis of the ossibility to redue the imerfetions of the thermodynami roesses of the suly of eletriity, heat and ooling in the ontext of sustainable develoment of the ountry. In: Exergy analysis and entroy generation minimization method (Ziębik A., Szargut J., Stanek W., Eds.). Publiation of Polish Aademy of Sienes. 3. Szargut J. (998). Problems of thermodynamis otimization. Arhives of Thermodynamis, Vol. 9, No.3-4, Laskowski R. Smyk A., Rusowiz A., Grzebiele A. (6). Determining the otimum inner Diameter of Condenser Tubes Based on Thermodynami Objetive Funtions and an Eonomi Analysis. Entroy, Vol. 8, No., ( ages) 5. Laskowski R., Rusowiz A., Grzebiele A. (5). Estimation of a tube diameter in a hurh window ondenser based on entroy generation minimization. Arhives of Thermodynamis, Vol. 36, No. 3, Laskowski R., Rusowiz A., Grzebiele A. (5). Minimizing the entroy inrease as a tool for otimization of the inner diameter of the ondenser tube. Przemysł Chemizny, Vol. 94, No., Laskowski R., Smyk A., Rusowiz A. (6). Dobór odowiedniej średniy rurek skralaza na odstawie minimum liniowego ooru ielnego i liniowego strumienia generaji entroii. Chłodnitwo, Vol., No. 9,. 8-. (in Polish)

9 86 Laskowski R., Jaworski M. Journal of Mehanial and Energy Engineering, Vol. (4), No., 7, Laskowski R., Rusowiz A., Grzebiele A., Jaworski M. (6). Oena konstrukji skralaza na odstawie minimum strumienia generaji entroii. Aaratura Badawza i Dydaktyzna, No., (in Polish) 9. Mohamed H. A. (6). Entroy generation in ounter flow heat exhangers. Journal of Heat Transfer, Vol. 8, No., Fakheri A. (). Seond law analysis of heat exhangers. Journal of Heat Transfer, Vol. 3, No., Szargut J., Petela R. (965). Egzergia. Wydawnitwa Naukowo-Tehnizne, Warszawa. (in Polish) 3. Szargut J. (998). Termodynamika (Wyd. VI). Państwowe Wydawnitwo Naukowe, Warszawa. (in Polish) Biograhial notes Maiej Jaworski reeived his M.S. and Ph.D. degrees in Mehanis and D.S. in Power Engineering from Warsaw University of Tehnology (WUT) in 983, 998 and 5 resetively. Currently he works as Assoiate Professor in the Institute of Heat Engineering, at the Faulty of Power and Aeronautial Engineering, WUT. His sientifi interests fous on thermodynamis (e.g. analysis of thermal devies on the base of the seond law), heat transfer (e.g. interation of laser radiation energy of extremely high intensity with matter) and thermal energy storage roblems (eseially the aliations of hase hange materials in different engineering areas). He has artiiated in many international and national researh rojets. Currently he is involved in H rojet aiming at the develoment of PhD rogram on Thermal Energy Storage. He has ublished more than sientifi aers in various international and national journals and onferene roeedings, he is o-author of several books on thermodynamis and heat transfer. Rafał Laskowski reeived his M.S. degree in Mehanial Engineering and next Ph.D degree at the Warsaw University of Tehnology, in and 9, resetively. He is a researher at Faulty of Power and Aeronautial Engineering at the Warsaw University of Tehnology, where urrently he works as an assistant rofessor. His sientifi interests fous on roblems onerning mathematial modeling of ower lant and their omonents like heat exhangers, gas and steam turbines. He has ublished more than 6 sientifi aers in various international and national journals, book haters, and onferene roeedings.