Steam Jet Refrigeration Cycle

Chemical Engineering and Processing 41 (2002) 551– 561 www. elsevier. com/locate/cep Evaluation of steam jet ejectors Hisham El-Dessouky *, Hisham Ettouney, Imad Alatiqi, Ghada Al-Nuwaibit Department of Chemical Engineering, College of Engineering and Petroleum, Kuwait Uni6ersity, P. O. Box 5969, Safat 13060, Kuwait Received 4 April 2001; received in revised form 26 September 2001; accepted 27 September 2001 Abstract Steam jet ejectors are an essential part in refrigeration and air conditioning, desalination, petroleum re? ning, petrochemical and chemical industries.

The ejectors form an integral part of distillation columns, condensers and other heat exchange processes. In this study, semi-empirical models are developed for design and rating of steam jet ejectors. The model gives the entrainment ratio as a function of the expansion ratio and the pressures of the entrained vapor, motive steam and compressed vapor. Also, correlations are developed for the motive steam pressure at the nozzle exit as a function of the evaporator and condenser pressures and the area ratios as a function of the entrainment ratio and the stream pressures. This allows for full design of the ejector, where de? ing the ejector load and the pressures of the motive steam, evaporator and condenser gives the entrainment ratio, the motive steam pressure at the nozzle outlet and the cross section areas of the diffuser and the nozzle. The developed correlations are based on large database that includes manufacturer design data and experimental data. The model includes correlations for the choked ? ow with compression ratios above 1. 8. In addition, a correlation is provided for the non-choked ? ow with compression ratios below 1. 8. The values of the coef? cient of determination (R 2) are 0. 85 and 0. 78 for the choked and non-choked ? w correlations, respectively. As for the correlations for the motive steam pressure at the nozzle outlet and the area ratios, all have R 2 values above 0. 99. © 2002 Elsevier Science B. V. All rights reserved. Keywords: Steam jet ejectors; Choked ? ow; Heat pumps; Thermal vapor compression 1. Introduction Currently, most of the conventional cooling and refrigeration systems are based on mechanical vapor compression (MVC). These cycles are powered by a high quality form of energy, electrical energy. The inef? cient use of the energy required to operate such a process can be generated by the combustion of fossil uels and thus contributes to an increase in greenhouse gases and the generation of air pollutants, such as NOx, SOx, particulates and ozone. These pollutants have adverse effects on human health and the environment. In addition, MVC refrigeration and cooling cycles use unfriendly chloro-? oro-carbon compounds (CFCs), which, upon release, contributes to the destruction of the protective ozone layer in the upper atmosphere. * Corresponding author. Tel. : + 965-4811188x5613; fax: + 9654839498. E -mail address: eldessouky@kuc01. kuniv. edu. kw (H. El-Dessouky). Environmental considerations and the need for ef? cient se of available energy call for the development of processes based on the use of low grade heat. These processes adopt entrainment and compression of low pressure vapor to higher pressures suitable for different systems. The compression process takes place in absorption, adsorption, chemical or jet ejector vapor compression cycles. Jet ejectors have the simplest con? guration among various vapor compression cycles. In contrast to other processes, ejectors are formed of a single unit connected to tubing of motive, entrained and mixture streams. Also, ejectors do not include valves, rotors or other moving parts and are available ommercially in various sizes and for different applications. Jet ejectors have lower capital and maintenance cost than the other con? gurations. On the other hand, the main drawbacks of jet ejectors include the following: ? Ejectors are designed to operate at a single optimum point. Deviation from this optimum results in dramatic deterioration of the ejector performance. 0255-2701/02/$ - see front 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 Processing 41 (2002) 551 – 561 Ejectors have very low thermal ef? iency. Applications of jet ejectors include refrigeration, air conditioning, removal of non-condensable gases, transport of solids and gas recovery. The function of the jet ejector differs considerably in these processes. For example, in refrigeration and air conditioning cycles, the ejector compresses the entrained vapor to higher pressure, which allows for condensation at a higher temperature. Also, the ejector entrainment process sustains the low pressure on the evaporator side, which allows evaporation at low temperature. As a result, the cold evaporator ? uid can be used 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 increases the condensation temperature because of their low thermal conductivity. Also, the presence of these gases enhances corrosion reactions. However, the ejector cycle for cooling and refrigeration has lower ef? ciency than the MVC units, but their merits are manifested upon the use of low grade energy that has limited effect on the environment and lower ooling and heating unit cost. Although the construction and operation principles of jet ejectors are well known, the following sections provide a brief summary of the major 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 nozzle; (2) the suction chamber; and (3) the diffuser (Fig. 1). The nozzle and the diffuser have the geometry of converging/diverging venturi. The diameters and lengths of various parts forming the nozzle, the diffuser and the suction chamber, together with the stream ? ow rate and properties, de? e the ejector capacity and performance. The ejector capacity is de? ned in terms of the ? ow rates of the motive steam and the entrained vapor. The sum of the motive and entrained vapor mass ? ow rates gives the mass ? ow rate of the compressed vapor. As for the ejector performance, it is de? ned in terms of entrainment, expansion and compression ratios. The entrainment ratio (w ) is the ? ow rate of the entrained vapor Fig. 1. Variation in stream pressure and velocity as a function of location along the ejector. H. El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 – 561 divided by the flow rate of the motive steam.

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As for the expansion ratio (Er), it is de? ned as the ratio of the motive steam pressure to the entrained vapor pressure. The compression ratio (Cr) gives the pressure ratio of the compressed vapor to the entrained vapor. Variations in the stream velocity and pressure as a function of location inside the ejector, which are shown in Fig. 1, are explained below: ? The motive steam enters the ejector at point (p ) with a subsonic velocity. ? As the stream ? ows in the converging part of the ejector, its pressure is reduced and its velocity increases. The stream reaches sonic velocity at the nozzle throat, where its Mach number is equal to one. The increase in the cross section area in the diverging part of the nozzle results in a decrease of the shock wave pressure and an increase in its velocity to supersonic conditions. ? At the nozzle outlet plane, point (2), the motive steam pressure becomes lower than the entrained vapor pressure and its velocity ranges between 900 and 1200 m/s. ? The entrained vapor at point (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 and the converging section of the diffuser or it may ? ow as two separate treams as it enters the constant cross section area of the diffuser, where mixing occurs. ? In either case, the mixture goes through a shock inside the constant cross section area of the diffuser. The shock is associated with an increase in the mixture pressure and reduction of the mixture velocity to subsonic conditions, point (4). The shock occurs because of the back pressure resistance of the condenser. ? As the subsonic mixture emerges from the constant 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 converted into pressure.

The pressure of the emerging ? uid is slightly higher than the condenser pressure, point (c ). Summary for a number of literature studies on ejector design and performance evaluation is shown in Table 1. The following outlines the main ? ndings of these studies: ? Optimum ejector operation occurs at the critical condition. The condenser pressure controls the location of the shock wave, where an increase in the condenser pressure above the critical 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 critical points has negligible effect on the ejector entrainment ratio. 553 ? At the critical condition, the ejector entrainment ratio increases at lower pressure for the boiler and condenser. Also, higher temperature for the evaporator increases the entrainment ratio. ? Use of a variable position nozzle can maintain the optimum conditions for ejector operation. As a result, the ejector can be maintained at critical conditions even if the operating conditions are varied. ? Multi-ejector system increases the operating range and improves the overall system ef? ciency. Ejector modeling is essential for better understanding of the compression process, system design and performance evaluation. Models include empirical correlations, such as those by Ludwig [1], Power [2] and El-Dessouky and Ettouney [3]. Such models 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 models give more ? exibility in ejector design and performance evaluation [4,5]. Other ejector models are based on fundamental balance equations [6]. This study is motivated by the need for a simple mpirical model that can be used 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 literature studies. As will be discussed later, the model is simple to use and it eliminates the need for iterative procedures. 2. Mathematical model The review by Sun and Eames [7] outlined the developments in mathematical modeling and design of jet ejectors. The review shows that there are two basic approaches for ejector analysis. These include mixing of the motive steam and entrained vapor, either at constant ressure or at constant area. Design models of stream mixing at constant pressure are more common in literature because the performance of the ejectors designed by this method is more superior to the constant area method and it compares favorably against experimental data. The basis for modeling the constant pressure design procedure was initially developed by Keenan [6]. Subsequently, several investigators have used the model for design and performance evaluation of various types of jet ejectors. This involved a number of modi? cations in the model, especially losses within the ejector and mixing of the primary 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 Processing 41 (2002) 551 – 561 554 Table 1 Summary of literature studies on ejector design and performance Reference Fluid Boiler, evaporator and condenser temperature (°C) Conclusion [19] R-113 60–100; 5–18; 40–50 Basis for refrigerant selection for solar system, system performance increased with increasing boiler and evaporator temperatures and decreasing condenser temperature. 20] R-113; R-114; R-142b; R-718 80–95; 5–13; 25–45 Comparison of ejector and refrigerant performance. Dry, wet and isentropic ?uids. Wet ? uid damage ejectors due phase change during isentropic expansion. R-113 (dry) has the best performance and R142b (wet) has the poorest performance. [21,22] R-114 86; ? 8; 30 Increase in ejector performance using mechanical compression booster. [8] Water 120–140; 5–10; 30–65 Choking of the entrained ? uid in the mixing chamber affects system performance. Maximum COP is obtained at the critical ? ow condition. [13] Water 120–140; 5–10; 30–60

Effect of varying the nozzle position to meet operating condition. Increase in COP and cooling capacity by 100%. [23] R-113 70–100; 6–25; 42–50 Entrainment ratio is highly affected by the condenser temperature especially at low evaporator temperature. [24] R-11 82. 2–182. 2; 10; 43. 3 Entrainment ratio is proportional 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 cold 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 expansion valve using high-pressure motive liquid. [28] Water 100–165; 10; 30–45 Combined solar collector, refrigeration and seawater desalination system. Performance depends 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 primary ? 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 formation losses. [24] R11; R113; R114 93. 3; 10; 43. 3 Measure and calculate 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 120–140; 65–80 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 theory in which the entrained ? uid is choked, the plant scale results agree with this theory. Steam jet refrigeration should be designed for the most often prevailing conditions rather than the most severe to achieve greater overall ef? ciency. [5] Mathematical model use empirical parameters that depend solely on geometry. The parameters are obtained experimentally 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 expansion 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. Friction 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*2

Te/Tp p e ' M 2(k + 1) M 2(k ? 1) + 2 (8) Eq. (8) is used to calculate M*2, M*2, M4 e p Mach number of the mixed ? 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 lift 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 enthalpy 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 primary ? uid in the nozzle is expressed 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 given by this equation M* = Compression ratio Cr = Pc/Pe ? ? ' The mixing process is modeled 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 ?

Two solution procedures for the above model are shown in Fig. 2. Either procedure requires iterative calculations. The ? rst procedure is used for system design, where the system pressures and the entrainment ratio is de? ned. Iterations are made to determine the pressure of the motive steam at the nozzle outlet (P2) that gives the same back pressure (Pc). The iteration 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 saturation temperatures for the compressed vapor, entrained vapor and motive steam, which include Tc, Tp, Te, using the saturation temperature correlation given in the appendix. As for the universal gas constant and the speci? c heat ratio for steam, their values are taken as 0. 462 and 1. 3. The ? ow rates of the entrained vapor (me) and motive steam (mp) are calculated from Eqs. (1) and (2). A value for the pressure at point 2 (P2) is estimated and Eqs. (5) – (11) are solved sequentially to obtain the ressure of the compressed vapor (Pc). The calculated pressure of the compressed vapor is compared to the design value. A new value for P2 is estimated and the previous step is repeated until the desired value for the pressure of the compressed vapor is reached. Fig. 2. Solution algorithms of the mathematical model. (a) Design procedure to calculate 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 solution 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 iteration 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 in the appendix. ? 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 Development of the semi-empirical model is thought to provide a simple method for designing or rating of steam jet ejectors. As shown above, solution 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 widely 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 major ejector manufacturers, which includes Croll Reynolds, Graham 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. 7–1 0. 8–1 0. 85–0. 98 0. 85 0. 75 0. 75 0. 8 0. 85 0. 7–1 0. 8–1 0. 65–0. 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 ratio 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 reported 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 ratio 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. ? Sun [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 measure- 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 satisfactory for entrainment ratios between 0. 2 and 1.

This is because the major part of the data is found between entrainment ratios clustered 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 correlated 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 Entrainment 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 and

The R values for the above three correlations are above 0. 99. The semi-empirical ejector design procedure involves sequential solution of Eqs. (1) – (14) together 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 3

Range of design and experimental data used in model development Source Er Cr Pe (kPa) Pc (kPa) Pp (kPa) w Experimental Schutte–Koerting Croll–Rynolds Graham Power 1. 4–6. 19 1. 008–3. 73 1. 25–4. 24 1. 174–4. 04 1. 047–5. 018 1. 6–526. 1 1. 36–32. 45 4. 3–429. 4 4. 644–53. 7 2–1000 0. 872–121. 3 66. 85–2100. 8 3. 447–124. 1 27. 58–170. 27 2. 76–172. 37 2. 3–224. 1 790. 8–2859. 22 446. 06–1480. 27 790. 8–1480. 27 3. 72–510. 2 38. 6–1720 84. 09–2132. 27 6. 2–248. 2 34. 47–301. 27 344. 74–2757. 9 0. 11–1. 132 0. 1–4 0. 1818–2. 5 0. 18–3. 23 0. 2–4 H. El -Dessouky et al. / Chemical Engineering and Processing 41 (2002) 551 – 561 559 Table 4

Summary 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 acknowledge funding support of the Kuwait University Research Administration, Project No. EC084 entitled ‘Multiple Effect Evaporation and Absorption/Adsorption Heat Pumps’. Appendix A. Nomenclature A COP Cr Er m M M* Fig. 4. Fitting of the entrainment 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 1–7 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) here 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 percentage 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

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