Steam Jet Refrigeration Cycle

chemical substance technology and touch 41 (cc2) 551 561 www. elsevier. com/locate/cep pay tally of move chiliad ousters Hisham El-Dessouky *, Hisham Ettouney, Imad Alatiqi, Ghada Al-Nuwaibit De ploughsh bement of chemical substance technology, College of plan and Petroleum, capital of Kuwait Uni6ersity, P. O. Box 5969, Safat 13060, Kuwait Received 4 April 2001 received in revised sorting 26 phratry 2001 accepted 27 September 2001 tweet Steam jet ousters argon an of the essence(p) blow up in refrige symmetryn and argumentation instruct, desalination, petroleum re? ning, petrochemical and chemical industries.The oustings form an integral part of distillation columns, condensers and former(a) raise up alter processes. In this get wind, semi-empirical gets atomic numeral 18 authentic for invent and rating of move jet cartridge oustings. The sham croaks the entrainment proportion as a agency of the enbountifulment sense of equilibrium and the twinges of the entrained dehyd symmetryn, author go and matted evapoproportionn. Also, coefficient of coefficient of correlation coefficiental statisticss argon actual for the causality moveer insistency at the nose hand as a tend of the e dehyd dimensionnator and condenser obliges and the subject field dimensions as a function of the entrainment proportion and the rain cats and dogs insistings. This al let outs for full intention of the cartridge ousting, where de? ing the ousting load and the presss of the motor go, e e e dehyd proportionalitynationationator and condenser gives the entrainment dimension, the creator steam pressing at the beak firing and the fool discussion section plains of the distributed and the pecker. The developed correlations atomic number 18 establish on large database that accommodates manufacturer jut data and observational data. The stupefy allows correlations for the clotted ? ow with densification proportions t o a higher place 1. 8. In addition, a correlation is reard for the non- choked ? ow with muscular contraction proportionalitys be suffering 1. 8. The qualify of the coef? cient of last (R 2) ar 0. 85 and 0. 78 for the choked and non-choked ? w correlations, respectively. As for the correlations for the author steam pinch at the olfactory organ topic and the sweep dimensions, all reserve R 2 encourages in a higher place 0. 99. 2002 Elsevier acquirement B. V. All rights reserved. Keywords Steam jet cartridge cartridge ousters clotted ? ow Heat pumps Thermal desiccation capsule 1. Introduction Currently, most of the conventional modify constitution and infrigidation g overning bodys ar based on mechanical vaporization compaction (MVC). These cycles ar powered by a extravagantly quality form of capability, electrical energy. The inef? cient go badout of the energy required to buy the farm such(prenominal) a process finish be gene positiond b y the combustion of fossil uels and then contributes to an matu balancen in greenhoexercising throttlees and the contemporaries of childs play pollutants, such as NOx, SOx, particulates and ozone. These pollutants puzzle adverse con agreemations on world health and the environment. In addition, MVC refrige proportionalityn and cooling cycles use unfriendly chloro-? oro-carbon compounds (CFCs), which, upon release, contributes to the destruction of the prophylactic ozone layer in the upper atmosphere. * fit author. Tel. + 965-48111885613 fax + 9654839498. E -mail address emailprotected kuniv. edu. kw (H. El-Dessouky). Environmental conside symmetryns and the need for ef? cient se of procurable energy call for the development of processes based on the use of utter grade heat. These processes adopt entrainment and crunch of low extort vapor to higher(prenominal) compacts suitable for distinct trunks. The compression process takes place in absorption, adsorption, chem ical or jet ejector vapor compression cycles. tarry ejectors engender the simplest con? guproportionn among various vapor compression cycles. In contrast to sepa ordinate processes, ejectors are formed of a single unit connected to tubing of agent, entrained and mingleture spuds. Also, ejectors do not admit valves, rotors or other moving move and are accessible ommercially in various sizes and for different applications. Jet ejectors throw set down capital and forethought cost than the other con? gu balancens. On the other hand, the briny draw behinds of jet ejectors accept the surveiling ? Ejectors are knowing to ope drift at a single best window pane. Deviation from this optimum results in hammy deterioproportionn of the ejector consummation. 0255-2701/02/$ see anterior matter 2002 Elsevier Science B. V. All rights reserved. PII S 0 2 5 5 2 7 0 1 ( 0 1 ) 0 0 1 7 6 3 552 ? H. El -Dessouky et al. / Chemical Engineering and process 41 (2002) 551 561 Ejector s exact very(prenominal) low thermal ef? iency. Applications of jet ejectors include refrige balancen, air conditioning, removal of non-condensable gases, transport of solids and gas re chasey. The function of the jet ejector differs considerably in these processes. For example, in refrige dimensionn and air conditioning cycles, the ejector compresses the entrained vapor to higher nip, which allows for compaction at a higher temperature. Also, the ejector entrainment process sustains the low drag on the evaporator side, which allows evapo proportionalityn at low temperature. As a result, the frore evaporator ? uid loafer be utilize for refrigeration and cooling functions.As for the removal of non-condensable gases in heat transfer units, the ejector entrainment process prevents their accumulation within condensers or evaporators. The presence of non-condensable gases in heat exchange units reduces the heat transfer ef? ciency and appends the condensation temperature becaus e of their low thermal conductivity. Also, the presence of these gases enhances corrosion reactions. However, the ejector cycle for cooling and refrigeration has let down ef? ciency than the MVC units, but their merits are manifested upon the use of low grade energy that has limited instal on the environment and depress ooling and oestrus unit cost. Although the construction and surgical process principles of jet ejectors are good known, the adjacent sections provide a brief abstract of the study features of ejectors. This is necessary in order to follow the discussion and analysis that follow. The conventional steam jet ejector has three main parts (1) the pecker (2) the sucking domiciliate and (3) the diffusor (Fig. 1). The nose and the diffuser have the geometry of intersection/ move venturi. The diameters and lengths of various parts forming the prig, the diffuser and the sucking chamber, together with the be adrift ? ow roam and properties, de? e the ejector readiness and proceeding. The ejector capacity is de? ned in scathe of the ? ow telescope of the power steam and the entrained vapor. The sum of the causation and entrained vapor fix ? ow rates gives the mass ? ow rate of the close vapor. As for the ejector accomplishment, it is de? ned in price of entrainment, enlargement and compression ratios. The entrainment ratio (w ) is the ? ow rate of the entrained vapor Fig. 1. Variation in waterway pressure and focal ratio as a function of jam along the ejector. H. El -Dessouky et al. / Chemical Engineering and impact 41 (2002) 551 561 dissever by the flow rate of the motif steam.As for the amplification ratio (Er), it is de? ned as the ratio of the author steam pressure to the entrained vapor pressure. The compression ratio (Cr) gives the pressure ratio of the slopped vapor to the entrained vapor. Variations in the stream whet and pressure as a function of location inner(a) the ejector, which are shown in Fig. 1, are explained below ? The causality(prenominal) steam enters the ejector at elevation (p ) with a subtransonic f number. ? As the stream ? ows in the converging part of the ejector, its pressure is reduced and its velocity change magnitudes. The stream reaches sonic velocity at the bird of Minerva pharynx, where its Mach fall is equal to one. The increase in the crossway section field of operations in the diverging part of the bird of Minerva results in a decrease of the profane fluctuate pressure and an increase in its velocity to ultrasonic conditions. ? At the nozzle vent plane, full excite (2), the motivation steam pressure becomes lower than the entrained vapor pressure and its velocity ranges amidst 900 and one hundred twenty0 m/s. ? The entrained vapor at microscope stage (e ) enters the ejector, where its velocity increases and its pressure decreases to that of point (3). ? The motive steam and entrained vapor streams may mix within the suction chamber a nd the converging section of the diffuser or it may ? ow as dickens separate treams as it enters the unending cross section welkin of the diffuser, where mix occurs. ? In either case, the motley goes through a shock inside the everlasting cross section area of the diffuser. The shock is associated with an increase in the medley pressure and reduction of the garland velocity to subsonic conditions, point (4). The shock occurs because of the book binding pressure resistance of the condenser. ? As the subsonic mixture emerges from the immutable cross section area of the diffuser, further pressure increase occurs in the diverging section of the diffuser, where part of the kinetic energy of the mixture is reborn into pressure.The pressure of the emerging ? uid is slightly higher than the condenser pressure, point (c ). Summary for a derive of publications studies on ejector stick out and slaying military rating is shown in slacken 1. The interest outlines the main ? ndin gs of these studies ? best ejector operation occurs at the diminutive condition. The condenser pressure controls the location of the shock wave, where an increase in the condenser pressure higher up the faultfinding point results in a rapid decline of the ejector entrainment ratio, since the shock wave moves towards the nozzle exit.Operating at pressures below the diminutive points has negligible effect on the ejector entrainment ratio. 553 ? At the full of life condition, the ejector entrainment ratio increases at lower pressure for the kettledrum and condenser. Also, higher temperature for the evaporator increases the entrainment ratio. ? function of a variable position nozzle can maintain the optimum conditions for ejector operation. As a result, the ejector can be maintained at unfavourable conditions even if the operational conditions are varied. ? Multi-ejector formation increases the operational range and improves the overall governing body ef? ciency. Ejector typeinging is essential for mend understanding of the compression process, system project and functioning military rank. Models include empirical correlations, such as those by Ludwig 1, Power 2 and El-Dessouky and Ettouney 3. such(prenominal) pretendings are limited to the range over which it was developed, which limits their use in investigating the performance of new ejector ? uids, designs or operating conditions. Semi-empirical prototypes give more ? exibility in ejector design and performance evaluation 4,5. opposite ejector models are based on fundamental balance equivalences 6. This study is cause by the need for a simple mpirical model that can be apply to design and evaluate the performance of steam jet ejectors. The model is based on a large database extracted from several ejector manufacturers and a number of experimental belles-lettres studies. As will be discussed later, the model is simple to use and it eliminates the need for repetitious aspect cogniti ve processs. 2. numerical model The polish up by solarize and Eames 7 outlined the developments in numeral modeling and design of jet ejectors. The review shows that there are two canonical approaches for ejector analysis. These include meld of the motive steam and entrained vapor, either at changeless ressure or at uninterrupted area. radiation diagram models of stream assortment at unvarying pressure are more parking area in literature because the performance of the ejectors designed by this method is more excellent to the constant area method and it compares favorably against experimental data. The basis for modeling the constant pressure design outgrowth was ab initio developed by Keenan 6. Subsequently, several investigators have utilize the model for design and performance evaluation of various types of jet ejectors. This convolute a number of modi? cations in the model, in particular losses within the ejector and flux of the master(a) and secondary streams. In this section, the constant pressure ejector model is developed. The developed model is based on a number of literature studies 8 11. The constant pressure model is based on the following assumptions H. El -Dessouky et al. / Chemical Engineering and impact 41 (2002) 551 561 554 Table 1 Summary of literature studies on ejector design and performance Reference Fluid Boiler, evaporator and condenser temperature (C) closing 19 R-113 60 one hundred 518 4050 Basis for refrigerating selection for solar system, system performance increased with increasing boiler and evaporator temperatures and fall condenser temperature. 20 R-113 R-114 R-142b R-718 8095 513 2545 equation of ejector and refrigerant performance. Dry, wet and isentropic ?uids. Wet ? uid damage ejectors due level change during isentropic intricacy. R-113 (dry) has the best performance and R142b (wet) has the poorest performance. 21,22 R-114 86 ? 8 30 attach in ejector performance exploitation mechanical compressio n booster. 8 urine great hundred140 510 3065 Choking of the entrained ? uid in the mixing chamber affects system performance. Maximum snare is obtained at the unfavourable ? ow condition. 13 Water 120140 510 3060Effect of varying the nozzle position to twin operating condition. Increase in neutralise and cooling capacity by blow%. 23 R-113 70100 625 4250 Entrainment ratio is exceedingly affected by the condenser temperature especially at low evaporator temperature. 24 R-11 82. 2182. 2 10 43. 3 Entrainment ratio is relative to boiler temperature. 25,26 R-114 90 4 30 Combined solar generator and ejector air conditioner. More ef? cient system requires multi-ejector and coldness energy storage (cold storage in either phase changing materials, cold water or ice). 27 R-134A 15 30 Modeling the effect of motive nozzle on system performance, in which the ejector is used to recover part of the work that would be lost in the working out valve using high-pressure motive liquid. 28 W ater 100 one hundred sixty-five 10 3045 Combined solar collector, refrigeration and seawater desalination system. Performance computes on steam pressure, cooling water temperature and suction pressure. 4 Water 29 Water Model of multistage steam ejector refrigeration system using annular ejector in which the primeval ? uid enters the second stage at annular nozzle on the sidewall.This will increase static pressure for low-pressure stream and mixture and reduce the velocity of the motive stream and reduce jet mixing losses shock wave system losses. 24 R11 R113 R114 93. 3 10 43. 3 bank bill and gauge ejector entrainment ratio as a function of boiler, condenser and evaporator temperatures. Entrainment ratio decreases for off design operation and increases for the two stage ejectors. 30 R113 R114 R142b 120140 6580 Effect of throat area, location of main nozzle and length of the constant area section on backpressure, entrainment ratio and compression ratio.Developed a new ejector t heory in which the entrained ? uid is choked, the plant plateful results agree with this theory. Steam jet refrigeration should be designed for the most frequently prevailing conditions rather than the most stark(a) to achieve greater overall ef? ciency. 5 Mathematical model use empirical parameters that depend solely on geometry. The parameters are obtained through an experiment for various types of ejectors. 31 R134a 5 ? 12, ? 18 40 Combined ejector and mechanical compressor for operation of domestic refrigerator-freezer increases entrainment ratio from 7 to 12. 4%. The optimum throat diameter depends on the freezer emperature 9 R11 HR-123 80 5 30 Performance of HR-123 is similar to R-11 in ejector refrigeration. Optimum performance is achieved by the use of variable geometry ejector when operation conditions change. H. El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 561 1. The motive steam expands isentropically in the nozzle. Also, the mixture of the motive steam and the entrained vapor compresses isentropically in the diffuser. 2. The motive steam and the entrained vapor are saturated and their velocities are negligible. 3. Velocity of the compressed mixture leaving the ejector is insigni? cant. 4.Constant isentropic expanding upon exponent and the ideal gas behavior. 5. The mixing of motive steam and the entrained vapor takes place in the suction chamber. 6. The ? ow is adiabatic. 7. clangoring losses are de? ned in terms of the isentropic ef? ciencies in the nozzle, diffuser and mixing chamber. 8. The motive steam and the entrained vapor have the same molecular weight and speci? c heat ratio. 9. The ejector ? ow is one-dimensional and at steady state conditions. The model equations include the following ? Overall material balance (2) Expansion ratio ? 2pn k? 1 Pp P2 n (k ? 1/k) ?1 Pe P2 n (k ? 1/k) ?1 (6) M*2 + wM*2Te/Tp p e M 2(k + 1) M 2(k ? 1) + 2 (8) Eq. (8) is used to presage M*2, M*2, M4 e p Mach number of the m ixed ? ow after the shock wave 2 M2+ 4 (k ? 1) M5 = (9) 2k 2 M ? 1 (k ? 1) 4 Pressure increase across the shock wave at point 4 (10) In Eq. (10) the constant pressure assumption implies that the pressure between points 2 and 4 remains constant. Therefore, the following equality constraint applies P2 = P3 = P4. Pressure deck up in the diffuser n Pc p (k ? 1) 2 =d M5+1 P5 2 ? (5) ? (k/k ? 1) (11) where pd is the diffuser ef? ciency. The area of the nozzle throat A1 = where M is the Mach number, P is the pressure and is the isentropic expansion coef? cient. In the above equation, pn is the nozzle ef? ciency and is de? ned as the ratio between the actual hydrogen change and the enthalpy change undergone during an isentropic process. Isentropic expansion of the entrained ? uid in the suction chamber is expressed in terms of the Mach number of the entrained ? uid at the nozzle exit plane P5 1 + kM 2 4 = P4 1 + kM 2 5 (4) Isentropic expansion of the chief(a) ? uid in the nozzle is ex pressed in terms of the Mach number of the primary ? uid at the nozzle outlet plane Mp2 = ? ? (3) Er = Pp/Pe ? ? 2 k? 1 (7) (1 + w )(1 + wTe/Tp) here w is the entrainment ratio and M * is the ratio between the local ? uid velocity to the velocity of sound at critical conditions. The relationship between M and M * at any point in the ejector is inclined by this equation M* = Compression ratio Cr = Pc/Pe ? ? The mixing process is pattern by one-dimensional continuity, momentum and energy equations. These equations are combined to de? ne the critical Mach number of the mixture at point 5 in terms of the critical Mach number for the primary and entrained ?uids at point 2 M* = 4 where m is the mass ? ow rate and the subscripts c, e and p, de? ne the compressed vapor mixture, the ntrained vapor and the motive steam or primary stream. Entrainment ratio w = me/mp ? ? (1) mp + me = mc ? Me2 = 555 mp Pp RTp k + 1 kpn 2 (k + 1)/(k ? 1) (12) The area ratio of the nozzle throat and diffuser constant area A1 Pc 1 = A3 Pp (1 + w )(1 + w (Te/Tp)) P2 1/k P (k ? 1)/k 1/2 1? 2 Pc Pc 2 1/(k ? 1) 2 1/2 1? k+1 k+1 1/2 (13) H. El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 561 556 ? The area ratio of the nozzle throat and the nozzle outlet A2 = A1 1 2 (k ? 1) 2 1+ M p2 2 M p2 (k + 1 2 ? (k + 1)/(k ? 1) (14) ? 3. Solution procedure ? 2 upshot procedures for the above model are shown in Fig. 2. Either procedure requires repetitious calculations. The ? rst procedure is used for system design, where the system pressures and the entrainment ratio is de? ned. Iterations are make to happen the pressure of the motive steam at the nozzle outlet (P2) that gives the same back pressure (Pc). The looping sequence for this procedure is shown in Fig. 2(a) and it includes the following steps ? De? ne the design parameters, which include the entrainment ratio (w ), the ? ow rate of the compressed ? ? ? ? vapor (mc) and the pressures of the entrained vapor, ompressed vapor and motive steam (Pe, Pp, Pc). De? ne the ef? ciencies of the nozzle and diffuser (pn, pd). Calculate the intensity temperatures for the compressed vapor, entrained vapor and motive steam, which include Tc, Tp, Te, using the fertilization temperature correlation presumptuousness in the appendix. As for the popular gas constant and the speci? c heat ratio for steam, their determine are taken as 0. 462 and 1. 3. The ? ow rates of the entrained vapor (me) and motive steam (mp) are compute from Eqs. (1) and (2). A value for the pressure at point 2 (P2) is evaluated and Eqs. (5) (11) are work out sequentially to obtain the ressure of the compressed vapor (Pc). The metrical pressure of the compressed vapor is compared to the design value. A new value for P2 is labeld and the preliminary step is repeated until the desire value for the pressure of the compressed vapor is reached. Fig. 2. Solution algorithms of the mathematical model. (a) Design procedure to ca lculate area ratios. (b) Performance evaluation to calculate w. H. El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 561 ? The ejector cross section areas (A1, A2, A3) and the area ratios (A1/A3 and A2/A1) are calculated from Eqs. (12) (14).The second base procedure is used for performance evaluation, where the cross section areas and the entrainment and motive steam pressures are de? ned. Iterations are made to determine the entrainment ratio that de? nes the ejector capacity. The eyelet sequence for this procedure is shown in Fig. 2(b) and it includes the following steps ? De? ne the performance parameters, which include the cross section areas (A1, A2, A3), the pressures of the entrained vapor (Pe) and the pressure of the primary stream (Pp). ? De? ne the ef? ciencies of the nozzle and diffuser (pn, pd). ? Calculate the saturation temperatures of the primary nd entrained streams, Tp and Te, using the saturation temperature correlation given(p) in the ap pendix. ? As for the universal gas constant and the speci? c heat ratio for steam, their values are taken as 0. 462 and 1. 3. ? Calculate the ? ow rate of the motive steam and the properties at the nozzle outlet, which include mp, P2, Me2, Mp2. These are obtained by solving Eqs. (5), (6), (12) and (14). ? An estimate is made for the entrainment ratio, w. ? This value is used to calculate other system parameters de? ned in Eqs. (7) (11), which includes M*2, e M*2, M*, M4, M5, P5, Pc. p 4 ? A new estimate for w is obtained from Eq. 13). ? The error in w is determined and a new iteration is made if necessary. ? The ? ow rates of the compressed and entrained vapor are calculated from Eqs. (1) and (2). 4. Semi-empirical model emergence of the semi-empirical model is thought to provide a simple method for designing or rating of steam jet ejectors. As shown above, outcome of the mathematical model requires an iterative procedure. Also, it is necessary to de? ne values of pn and pd. The values of these ef? ciencies bigly differ from one study to another, as shown in Table 2. The semi-empirical model for the steam jet ejector is developed over a wide ange of operating conditions. This is achieved by using three sets of design data acquired from study ejector manufacturers, which includes Croll Reynolds, whole meal flour and Schutte Koerting. Also, several sets of experimental data are extracted from the literature and are used in the development of the empirical model. The semiempirical model includes a number of correlations to calculate the entrainment ratio (w ), the pressure at the nozzle outlet (P2) and the area ratios in the ejector 557 Table 2 Examples of ejector ef? ciencies used in literature studies Reference 27 32 33 31 10 24 8 34 pn pd 0. 9 0. 5 0. 71 0. 81 0. 850. 98 0. 85 0. 75 0. 75 0. 8 0. 85 0. 71 0. 81 0. 650. 85 0. 85 0. 9 pm 0. 8 0. 95 (A2/A1) and (A1/A3). The correlation for the entrainment ratio is developed as a function of the expansion r atio and the pressures of the motive steam, the entrained vapor and the compressed vapor. The correlation for the pressure at the nozzle outlet is developed as a function of the evaporator and condenser pressures. The correlations for the ejector area ratios are de? ned in terms of the system pressures and the entrainment ratio. Table 3 shows a summary of the ranges of the experimental and the design data.The table also includes the ranges for the data account by Power 12. A summary of the experimental data, which is used to develop the semi-empirical model is shown in Table 4. The data includes measurements by the following investigators ? Eames et al. 8 obtained the data for a compression ratio of 3 6, expansion ratio 160 415 and entrainment ratio of 0. 17 0. 58. The measurements are obtained for an area ratio of 90 for the diffuser and the nozzle throat. ? Munday and Bagster 4 obtained the data for a compression ratio of 1. 8 2, expansion ratio of 356 522 and entrainment ra tio of 0. 57 0. 905.The measurements are obtained for an area ratio of 200 for the diffuser and the nozzle throat. ? Aphornratana and Eames 13 obtained the data for a compression ratio of 4. 6 5. 3, expansion ratio of 309. 4 and entrainment ratio of 0. 11 0. 22. The measurements are obtained for an area ratio of 81 for the diffuser and the nozzle throat. ? Bagster and Bresnahan 14 obtained the data for a compression ratio of 2. 4 3. 4, expansion ratio of 165 426 and entrainment ratio of 0. 268 0. 42. The measurements are obtained for an area ratio of 145 for the diffuser and the nozzle throat. ? temperateness 15 obtained the data for a compression ratio of . 06 3. 86, expansion ratio of 116 220 and entrainment ratio of 0. 28 0. 59. The measurements are obtained for an area ratio of 81 for the diffuser and the nozzle throat. ? Chen and Sun 16 obtained the data for a compression ratio of 1. 77 2. 76, expansion ratio of 1. 7 2. 9 and entrainment ratio of 0. 37 0. 62. The m easure- H. El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 561 558 ments are obtained for an area ratio of 79. 21 for the diffuser and the nozzle throat. ? Arnold et al. 17 obtained the data for a compression ratio of 2. 47 3. 86, expansion ratio of 29. 7 46. , and entrainment ratio of 0. 27 0. 5. ? Everitt and Riffat 18 obtained the data for a compression ratio of 1. 37 2. 3, expansion ratio of 22. 6 56. 9 and entrainment ratio of 0. 57. The correlation for the entrainment ratio of choked ?ow or compression ratios above 1. 8 is given by W = aErbP cP d ec (e + fP g ) p (h + iP jc) (15) Similarly, the correlation for the entrainment ratio of un-choked ? ow with compression ratios below 1. 8 is given by W = aErbP cP d ec (e + f ln(Pp)) (g + h ln(Pc)) (16) vapor compression applications. As shown in Fig. 3, the ? tting result is very fine for entrainment ratios between 0. 2 and 1.This is because the major part of the data is found between entrainment rati os agglomerate over a range of 0. 2 0. 8. Examining the experimental data ? t shows that the major part of the data ? t is well within the correlation predictions, except for a small number of points, where the predictions have large deviations. The correlations for the motive steam pressure at the nozzle outlet and the area ratios are obtained semi-empirically. In this regard, the design and experimental data for the entrainment ratio and system pressures are used to solve the mathematical model and to calculate the area ratios and motive steam pressure at the nozzle utlet. The results are obtained for ef? ciencies of 100% for the diffuser, nozzle and mixing and a value of 1. 3 for k. The results are then correlative as a function of the system variables. The following relations give the correlations for the choked ? ow The constants in Eqs. (15) and (16) are given as follows P2 = 0. 13 P 0. 33P 0. 73 e c (17) A1/A3 = 0. 34 P 1. 09P ? 1. 12w ? 0. 16 c p Entrainment ratio Entrainm ent ratio correlation choked correlation non-choked ?ow (Eq. (15) Fig. 3) ? ow (Eq. (16), Fig. 4) ?1. 89? 10? 5 ?5. 32 5. 04 9. 05? 10? 2 22. 09 ?6. 13 0. 82 ?3. 37? 10? 5 ? ? 0. 79 a 0. 65 b ?1. 54 c 1. 72 d 6. 9v10? 2 e 22. 82 f 4. 21? 10? 4 g 1. 34 h 9. 32 j 1. 28? 10? 1 j 1. 14 R2 0. 85 A2/A1 = 1. 04 P ? 0. 83 c P 0. 86 p w (18) ? 0. 12 (19) The R 2 for each of the above correlations is above 0. 99. Similarly, the following relations give the correlations for the un-choked ? ow P2 = 1. 02 P ? 0. 000762P 0. 99 e c (20) A1/A3 = 0. 32 P 1. 11P ? 1. 13w ? 0. 36 c p (21) A2/A1 = 1. 22 P ? 0. 81P 0. 81w ? 0. 0739 c p (22) 2 Fitting results against the design and experimental data are shown in Figs. 3 and 4, respectively. The results shown in Fig. 3 cover the most commonly used range for steam jet ejectors, especially in vacuum andThe R values for the above three correlations are above 0. 99. The semi-empirical ejector design procedure involves sequential solution of Eqs. (1) (14) tog ether with Eq. (17) or Eq. (20) (depending on the ? ow type, choked or non-choked). This procedure is not iterative in contrast with the procedure given for the mathematical model in the previous section. As for the semi-empirical performance evaluation model, it involves non-iterative solution of Eqs. (1) (14) together with Eq. (15) or Eq. (16) for choked or non-choked ? ow, respectively. It should be stressed that both solution procedures are indepen- Table 3Range of design and experimental data used in model development Source Er Cr Pe (kPa) Pc (kPa) Pp (kPa) w Experimental SchutteKoerting CrollRynolds whole meal flour Power 1. 46. 19 1. 0083. 73 1. 254. 24 1. 1744. 04 1. 0475. 018 1. 6526. 1 1. 3632. 45 4. 3429. 4 4. 64453. 7 21000 0. 872121. 3 66. 852100. 8 3. 447124. 1 27. 58170. 27 2. 76172. 37 2. 3224. 1 790. 82859. 22 446. 061480. 27 790. 81480. 27 3. 72510. 2 38. 61720 84. 092132. 27 6. 2248. 2 34. 47301. 27 344. 742757. 9 0. 111. 132 0. 14 0. 18182. 5 0. 183. 23 0. 24 H . El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 561 559 Table 4Summary of literature experimental data for steam jet ejectors Ad/At Pp (kPa) Pe (kPa) Pc (kPa) Pp/Pe Pc/Pe w Reference 90 198. 7 232. 3 270. 3 313. 3 361. 6 1. 23 1. 23 1. 23 1. 23 1. 23 3. 8 4. 2 4. 7 5. 3 6 161. 8 189. 1 220. 1 255. 1 294. 4 3. 09 3. 42 3. 83 4. 31 4. 89 0. 59 0. 54 0. 47 0. 39 0. 31 8 8 8 8 8 90 198. 7 232. 3 270. 3 313. 3 361. 6 1. 04 1. 04 1. 04 1. 04 1. 04 3. 6 4. 1 4. 6 5. 1 5. 7 191. 6 223. 9 260. 7 302. 1 348. 7 3. 47 3. 95 4. 44 4. 91 5. 49 0. 5 0. 42 0. 36 0. 29 0. 23 8 8 8 8 8 90 198. 7 232. 3 270. 3 313. 3 361. 6 0. 87 0. 87 0. 87 0. 87 0. 87 3. 4 3. 7 4. 4 5. 1 5. 4 227. 7 266. 2 309. 8 59 414. 4 3. 89 4. 24 5. 04 5. 85 6. 19 0. 4 0. 34 0. 28 0. 25 0. 18 8 8 8 8 8 200 834 400 669 841 690 690 1. 59 1. 59 1. 71 1. 59 1. 94 1. 94 3. 2 3. 07 3. 67 3. 51 3. 38 3. 51 521. 7 250. 2 392. 3 526. 1 356 356 2. 0 1. 92 2. 15 2. 19 1. 74 1. 81 0. 58 1. 13 0. 58 0. 51 0. 86 0. 91 4 4 4 4 4 4 81 270 270 270 270 270 0. 87 0. 87 0. 87 0. 87 0. 87 4. 1 4. 2 4. 4 4. 5 4. 7 309. 5 309. 5 309. 5 309. 5 309. 5 4. 7 4. 8 5. 04 5. 16 5. 39 0. 22 0. 19 0. 16 0. 14 0. 11 13 13 13 13 13 145 660 578 516 440 381 312 278 1. 55 1. 55 1. 58 1. 57 1. 59 1. 62 1. 68 5. 3 5. 3 5. 3 5. 03 4. 77 4. 23 4. 1 426. 5 373. 5 326. 280. 6 239. 9 192. 6 165. 1 3. 42 3. 42 3. 36 3. 21 3 2. 61 2. 44 0. 27 0. 31 0. 35 0. 38 0. 42 0. 46 0. 42 14 14 14 14 14 14 14 143. 4 169. 2 198. 7 232. 3 270. 3 1. 23 1. 23 1. 23 1. 23 1. 23 2. 53 2. 67 3. 15 4 4. 75 116. 8 137. 8 161. 8 189. 1 220. 1 2. 06 2. 17 2. 56 3. 26 3. 87 0. 59 0. 51 0. 43 0. 35 0. 29 15 15 15 15 15 29. 7 33. 5 37. 8 46. 5 2. 47 2. 78 3. 14 3. 86 0. 5 0. 4 0. 3 0. 27 17 17 17 17 119. 9 151. 7 224. 1 195. 1 195. 1 186. 2 1. 7 2. 3 3. 9 1. 6 1. 9 2. 9 1. 8 2. 2 3. 3 1. 6 1. 9 2. 8 0. 62 0. 49 0. 34 0. 78 0. 64 0. 37 16 16 16 16 16 16 2. 3 2. 3 2. 3 56. 9 38. 6 22. 6 . 3 1. 9 1. 4 0. 57 0. 56 0. 57 18 18 18 81 1720 1720 1720 1720 79. 21 116 153 270 198 198 198 57. 9 47. 4 38. 6 57. 7 51. 4 45. 5 37. 01 67. 6 67. 6 67. 6 121. 3 99. 9 67. 6 1. 02 1. 2 1. 7 143 143 143 143 560 H. El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 561 wide range of compression, expansion and entrainment ratios, especially those used in industrial applications. The developed correlations are simple and very useful for design and rating calculations, since it can be used to determine the entrainment ratio, which, upon speci? cation of the system load, can be used to determine the motive steam ? w rate and the cross section areas of the ejector. Acknowledgements Fig. 3. Fitting of the entrainment ratio for compression ratios higher than 1. 8. The authors would like to mark funding support of the Kuwait University look Administration, Project No. EC084 entitled multiplex Effect Evaporation and Absorption/ surface assimilation Heat Pumps. Appendix A. Nomenclature A COP Cr Er m M M* Fig. 4. Fitting of the entr ainment ratio for compression ratios lower than 1. 8. dent of the nozzle and diffuser ef? ciencies, which varies over a wide range, as shown in Table 2. 5. Conclusions A semi-empirical model is developed for design and erformance evaluation of steam jet ejector. The model includes correlations for the entrainment ratio in choked and non-choked ? ow, the motive steam pressure at the nozzle outlet and the area ratios of the ejector. The correlations for the entrainment ratio are obtained by ? tting against a large set of design data and experimental measurements. In addition, the correlations for the motive steam pressure at the nozzle outlet and the area ratios are obtained semi-empirically by solving the mathematical model using the design and experimental data for the entrainment ratio and system pressures.The correlations cover a P DP R Rs T w cross section area (m2) coef? cient of performance, dimensionless compression ratio de? ned as pressure of compressed vapor to pressure of entrained vapor expansion ratio de? ned as pressure of compressed vapor to pressure of entrained vapor mass ? ow rate (kg/s) Mach number, ratio of ? uid velocity to speed of sound critical Mach number, ratio of ? uid velocity to speed of sound pressure (kPa) pressure drop (kPa) universal gas constant (kJ/kg C) load ratio, mass ? ow rate of motive steam to mass ? ow rate of entrained vapor temperature (K) ntrainment ratio, mass ? ow rate of entrained vapor to mass ? ow rate of motive steam Greek symbols k compressibility ratio p ejector ef? ciency Subscripts 17 locations inside the ejector b boiler c condenser d diffuser e evaporator or entrained vapor m mixing n nozzle p primary stream or motive steam t throat of the nozzle H. El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 561 Appendix B B. 1. Correlations of saturation pressure and temperature The saturation temperature correlation is given by T = 42. 6776 ? 3892. 7 ? 273. 15 (ln(P /1000) ? 9. 48654) her e P is in kPa and T is in C. The above correlation is valid for the calculated saturation temperature over a pressure range of 10 1750 kPa. The section errors for the calculated versus the steam table values are B 0. 1%. The correlation for the water vapor saturation pressure is given by ln(P /Pc) = Tc ?1 T + 273. 15 8 ? % fi (0. 01(T + 273. 15 ? 338. 15))(i ? 1) i=1 where Tc = 647. 286 K and Pc = 22089 kPa and the values of fi are given in the following table f1 f2 f3 f4 ?7. 419242 0. 29721 ?0. 1155286 0. 008685635 f5 f6 f7 f8 0. 001094098 ?0. 00439993 0. 002520658 ?0. 000521868