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Modelling and optimization of two-phase ejectors for cooling systems
Applied Thermal Engineering 25 (2005) 1979–1994 www.elsevier.com/locate/apthermeng
Modelling and optimization of two-phase ejectors for cooling systems K. Cizungu
a,*
, M. Groll a, Z.G. Ling
b
a
b
Institut fu¨r Kernenergetik und Energiesysteme (IKE), Universita¨t Stuttgart, Pfaffenwaldring 31, D-70550 Stuttgart, Germany Research Institute of Energy and Environmental Engineering, Shanghai University of Engineering Science, No. 350 Xian Xia Road, Shanghai 200336, PR China Received 12 February 2004; accepted 20 November 2004
Abstract A one-dimensional compressible flow model, which is based on the control volume approach, has been formulated to model and optimize one and two-phase ejectors in steady-state operation with particular reference to their deployment in a jet cooling system. The working fluid can be both single-component (NH3) and two-component (NH3–H2O). The developed model takes into account the duct effectiveness, wall friction, momentum loss, ejector geometry, shock waves as well as the acceleration of the induced flow in the conical part of the mixing section. Neither the usual assumption of mixing at constant pressure over the mixing chamber cross section nor that of a constant mixing chamber cross section were made. A comparison with other computation methods as well as with available experimental data from the literature is presented. The performance is significantly influenced by the ejector geometry. 2004 Elsevier Ltd. All rights reserved. Keywords: Two-phase ejector; Cooling system; One-dimensional model; Optimization
*
Corresponding author. Tel.: +49 711 685 2481; fax: +49 711 685 2010. E-mail address: [email protected] (K. Cizungu).
1359-4311/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2004.11.014
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Nomenclature A COP d h L Ma m_ b P Q_ T w Z
cross section area (m2) coefficient of performance (–) diameter (m) specific enthalpy (kJ/kg) length (m) Mach number (–) mass flow rate (kg/s) pressure (bar) heat transfer rate (kW) temperature (C) velocity (m/s) ammonia concentration (–)
Greek symbols a0, a1, b0, b1 opening angles () g efficiency (–) l ¼ m_ s =m_ p entrainment ratio (–) q density (kg/m3) f momentum loss (–) / = Am/At main area ratio (between mixing tube and primary nozzle) u vapour quality (–) n = Pb/Pc driving pressure ratio (–) w = Pc/Pe compression ratio (–) Subscripts b boiler com compression d diffuser e evaporator is isentropic opt optimal s secondary (flow, nozzle) x distance driving nozzle exit—mixing chamber entrance 1, 2, 3, 4 states in respective cross sections of the ejector (see Fig. 2) c condenser conv convergent div divergent ex expansion m mixing tube p primary (flow, nozzle) t nozzle throat
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1. Introduction In view of the exhaustible nature of the fossil energy sources and due to the environmental impact of the rising energy consumption, an increasing use of low temperature heat sources can be a suitable solution for a more rational energy use and ultimately cost savings. In refrigeration and airconditioning there are two interesting methods of obtaining cooling from thermal energy: the absorption process in which the mechanical compressor of the conventional compression cooling system is replaced by a thermal compressor, and the vapour ejector system in which an ejector replaces the mechanical compressor. Because of complex construction and maintenance, absorption refrigerators are relatively expensive. Additionally they need waste heat at temperatures over about 100 C. Due to the absence of moving parts, the simple construction, a high reliability and low maintenance costs, jet compressors (ejectors, jet pumps, injectors) represent an attractive alternative to other compressors (Figs. 1 and 2). Therefore they are widely used in the energy
Fig. 1. Schematic of a jet cooling system.
Fig. 2. Schematic of the ejector and geometrical parameters.
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and chemical engineering fields. They can in particular effectively use low temperature energy sources. The ejector–compression system can utilize low grade energy at temperature levels of 70–95 C, which is available e.g. from flat solar collectors. Using different working fluids [1–12], it has been shown that, for different boiler temperatures, the entrainment ratio and the system efficiency of a jet based cooling system depend mainly on the ejector geometry and the compression ratio. However there are contradictory statements in the literature about the influence of the ejector geometry [13]. Thus a consistent description of the interaction between geometry and performance of a jet compressor is the main aim of the modelling. In view of the gaps noted concerning the current state of research on the supersonic ejectors, it is preferable to first study the influences of the ejector geometry on its global characteristics (entrainment ratio, refrigerating effectiveness, compression ratio). Thus we use a one-dimensional model based on the method of control volume and extend the calculations to two-phase flow. This model allows in addition to save computing time and it is easy to handle.
2. System description The present work is concerned with the design of single and two-phase vapour jet compressors in steady state with special reference to their use in a jet cooling system. Due to very restrictive assumptions, the design models from the literature can neither describe accurately the complicated thermofluid dynamic processes in the jet compressor nor reproduce the experimentally established relations between the mass flow or entrainment ratio and ejector geometry [1–11]. Thus the interaction between geometry and performance of a jet compressor is the main aim of the work. A detailed description of the operational principle of a jet compressor as well as a jet compressor cooling system was given elsewhere [12,13]. Fig. 1 shows a schematic diagram if an ejector based cooling system. Its operating principle can be summarized as follows. The high pressure and temperature vapour generated within the boiler due to heat exchange Q_ b between the external heat source and the liquid refrigerant enters the ejector and flows through a converging–diverging nozzle (known as driving or primary nozzle) of the ejector, where it expands to supersonic flow and induces a low pressure region at the nozzle exit. The partial vacuum produced by the driving fluid causes the suction of the refrigerant vapour (known as induced or secondary fluid) from the evaporator. The two fluids mix in the mixing chamber and are discharged through the diffuser to the condenser. In the condenser, the mixed flow is liquefied by rejecting the heat Q_ c to the environment. Part of the resulting liquid refrigerant is returned to the boiler after a pressure increase by a feed-pump, while the remaining liquid enters the evaporator after undergoing a pressure reduction in the expansion or throttling valve. In the evaporator, the fluid evaporates due to the heat gain (cooling load Q_ e ) from the cooling space. In this way, the cycle of the ejector cooling system is completed. Fig. 2 shows the structure of a typical ejector. The geometrical parameters of a jet compressor comprise the diameters d1, dt, dp3, d2, ds3, dm and dd, the opening angles a0, a1, b0 and b1 and the axial lengths Lp, Lx, Lm and Ld. The most important parameter is the main area ratio / = Am/ At. The thermodynamic parameters can be summarized in driving pressure ratio n = Pb/Pc and compression ratio w = Pc/Pe. The different thermodynamic processes in the jet compressor are
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Fig. 3. Thermodynamic process in the ejector. 1–3P: real expansion of driving steam in the driving nozzle; 2–3S: real expansion of suction steam in the suction chamber 3P–xP–4 and 3S–xS–4: mixture process in intake cone and mixing chamber; 4–5: real compression of the mixture in the diffuser.
shown in the (h, s) diagram on Fig. 3. The pressures or the indices refer to the cross sections indicated in the Fig. 3. The dotted lines lines represent the isentropic processes.
3. Modelling The calculation model is based on the conservation laws for mass, momentum and energy for the flow in driving nozzle, suction nozzle, mixing chamber and diffuser. Selection criteria for the design procedure are simple handling and small computing time expenditure. Following assumptions were made: • The flow in the jet compressor is one-dimensional, compressible, single-phase or two-phase, friction-affected, in steady state, and the flow process in the ejector takes place adiabatically. • Based on experimental investigations [2,4,6] the design pressure Pp3 at the driving nozzle exit is uniform i.e. Pp3 = Ps3, and the suction fluid velocity reaches the speed of sound at this level. • The losses in the mixing chamber are represented by a friction factor fm and momentum loss factor fm, the losses in the nozzles and the diffuser are taken into account by assuming respective isentropic efficiencies gp, gs and gd. The different angles of the jet compressor as well as the efficiencies gp, gs and gd are taken from the literature [12,13]. The system of the derived governing equations has to be solved numerically. A summary of the model equations leads to a very long expression for the mass flow ratio l of a jet compressor [13]: Lm Lx m_ s ¼ f n; w; gn ; gd ; fm ; fm ; ; gs ; /; ; tan b ð1Þ l¼ dm dt m_ p
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With the usual assumption of a mixing at constant pressure and negligible inlet and outlet velocity in the ejector without intake cone before the cylindrical mixing chamber, the relationship (1) can be represented by the following simplified equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dhexp fm Lm g g 1 þ n d Dhcom 2fm d m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð2Þ l¼ fm Lm Dhexs 1 þ 2fm d m Dhcom gs gd where the enthalpy differences are given by Dhexp = hb hp3,is and Dhexs = he hs3,is. Schulz [14] and Cizungu et al. [12] derived also the equation (2), whereby the momentum loss in the mixing chamber was neglected (i.e. fm = 1) but not the frictional factor fm. The equations for the determination of the mass flow relation of a jet compressor, indicated so far in the literature, can be summarized by the empirical relationship sffiffiffiffiffiffiffiffiffiffiffiffi Dhexp c 1; ð3Þ l¼ Dhcom where c = gexpÆgcom represents the so-called quality grade of the jet compressor, with c = 0.689– 0.81 [13]. On the basis of experimental investigations Dorante`s et al. [8] have worked out the following correlation for the calculation of the entrainment ratio: 2:12 1 1:21 : ð4Þ l ¼ 3:32 w nw A comparison with our own results as well as with measured values from the literature shows that equation (4) applies only within the limited area 2.5 6 n 6 4; / 6 6. Other correlation are given in [11]. Contrary to equations (2)–(4) for the determination of the entrainment ratio l of a jet compressor, relationship (1) considers both the acceleration of the suction flow and the wall friction, the momentum loss in the mixing chamber and the influence of the most important geometrical parameters, /, Lx/dt and Lm/dm. Moreover, the working fluids are treated here as real gases. The direct consideration of the driving nozzle position enables a regulation of the jet compressor. With the ejector mass flow ratio l from relationship (1) the coefficient of performance (COP), i.e. the system efficiency of the ejector cooling system can be calculated through COP ¼
Q_ e h2 hg ¼l : h1 h6 Q_ b þ N PP
ð5Þ
The sizing of the jet compressor and the operational behaviour of the jet cooling system are calculated as follows: • For the determination of jet compressor geometry, the driving nozzle diameters d1, dt and dp3, the suction nozzle diameters d2 and ds3, the mixing tube diameter dm and the diffuser diameter dd as well as the driving nozzle lengths Lp, the mixing chamber length Lm and the diffuser length Ld are calculated for given operating conditions Tb, Tc, Te, ub, ue and Q_ e . • For the determination of the operational behaviour, the mass flow ratio l, energetic efficiency COP and compression ratio w are calculated for given jet compressor geometry and operating conditions Tb, Tc, Te, ub, ue and Q_ e .
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Calculations have been carried out for single-phase and two-phase as well as for single-component (NH3) and two-component (NH3–H2O) working fluids. With two-phase working fluids the present jet compressor model falls back on the so-called homogeneous model for two-phase flow [15]. The principal difference to the calculation model for two-phase two-component jet compressors lies in the thermodynamic characteristics, but this is decisive. Indeed, while with pure refrigerants a modification of the quality at fixed temperature does not influence the pressure (and thus the mass flow ratio), but only density, enthalpy, entropy and internal energy (and thus COP), a modification of the quality of two-phase two-component refrigerants affects both the pressure and the further thermodynamic characteristics, thus the pressure decreases with rising quality, while the enthalpy increases. It changes also the irreversibilities within the ejector [16,23]. 4. Optimization procedure There are hardly reports about optimization computations of jet compressors pffiffiffiffiffiin the literature. Kogan et al. [17] gave a functional dependence of the form f ðP e =P b ; P c =P b ; l T ; xÞ ¼ 0 for the determination of the optimal geometrical characteristics of an ejector with cylindrical mixing p = T /T . Depending on the given parameters {P /P , l T } or {P chamber. Here is T e b c p c/Pp, p , l T } they computed analytically for ideal gases the minimum (Pe/Pb)min or Pe/Pb} or {Pe/Pbp the maximum (l T)max or the maximum (Pc/Pb)max, which gives an optimal (/)opt. No values are given for the used speed coefficients x for the driving nozzle, the mixing area, the diffuser and the intake cone to the mixing area. There is no comparison of the calculations with experiments. Agapov et al. [18] used for the increase of pressure in the ejector an equation of the form Pc Pe = f(x, k, l, qp3, qs, qd, wp3) = 0 and derived from this the optimal cross section by means of oko ðP c P e Þ ¼ 0. In this case k = Ap3/Am. In addition to the assumptions in [17], the cross section Ap3 was regarded as constant along Lx. There are also neither data concerning the values of the used speed coefficients x nor comparison of the calculations with experiments. Assuming incompressible flow and differentiating an equation for the total pressure rise across the ejector, similarly as in [18], Chen [19] obtained the following relationship for the optimal area ratio of the form kopt = f(l, qp, qm, gd, Am/Ad) = 0. The acceleration of the suction flow in the above optimization calculations were not considered. Up to Kogan et al. [17] the flow inside the ejector was treated as incompressible The only calculations given in the literature for the determination of the optimal driving nozzle distance [20] are just empirical relations, they are valid exclusively for incompressible flow. In the one-dimensional models [3,10,11] based on ideal gas, isentropic processes and mixing at constant pressure a relationship of the form l = f(n, w, /, T ) = 0 was derived but the influence of the dimensional parameter / on the ejector performance was not investigated and only [11] has taken into account the effects of frictional and missing losses which are important for a precise ejector design. In [3], no comparison of the calculations with experimental data was given. In short, it can be observed the effect of the important geometrical parameter /, Lx/dt and Lm/dm on the ejector performance characteristics was not analysed in the known models. None of them has taken into consideration the axial lengths, first of all the parameters Lx and Lm. Therefore, a new optimization procedure, which goes beyond the mentioned procedures is necessary. This is also a goal of this work.
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4.1. Optimization criteria When the jet compressor is used as component in a cooling system the optimization procedure consists of achieving either a maximum entrainment ratio l with fixed compression ratio w or a maximum compression ratio with fixed entrainment ratio. The achievement of a maximum entrainment ratio means that for a given driving power fixed by the pressure ratio n = Pb/Pc the maximum suction mass flow has to be reached, while a maximum compression ratio means that for a given entrainment ratio, either a maximum condenser pressure (a maximum condenser temperature) for given evaporator pressure or a minimum evaporator pressure (a minimum evaporator temperature) for given condenser pressure has to be reached. Depending on defaults, the operating conditions or ejector geometry are optimized. The formal optimization procedure reads Lx Lm : ð6Þ l ¼ f n; w; /; ; dt dm This equation represents a new and complete relationship for the computation of the mass flow relationship of an ejector as function of all most important geometrical parameters. /, Lx/dt and Lm/dm. Based one on this equation, a sensitivity study of the system performance in dependence of the fundamental geometrical parameters will be introduced in the following. 5. Solution methodology 5.1. Solution procedure for the jet compressor sizing • Determine iteratively the pressure Ps3 by fulfilling the choking condition (Ma = 1) at the entrance of the mixing chamber. • Determine iteratively the pressure Pt by fulfilling the choking condition (Ma = 1) at the driving nozzle throat. • Determine the design pressure by assuming Pp3 = Ps3. • Determine iteratively the pressure Pm. • Determine iteratively the mass flow ratio l. From this follows the calculation of the cross section areas At, Ap3, As3, Am, Ad and the respective diameters dt, dp3, ds3, dm and dd as well as the lengths Ldiv, Lt, Lconv, Lm and Ld. 5.2. Solution procedure for the determination of the operational behaviour of the ejector cooling system • • • •
Determine Determine Determine Determine
iteratively the pressure Ps3 as above. the pressure Pt as above. the design pressure Pp3. iteratively the pressure Pm.
From this follows the calculation of the entrainment ratio l and the COP.
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5.3. Solution procedure for the optimization • Nozzle optimization: for given Tb, Tc, Te, and ub, ue and from a parameter variation l = f(/)n,w, if w is given, by differentiating Eq. (6), i.e. ol/o/ = 0 (or w = f(/)n,l, if l is given, i.e. oW/o/ = 0). follows the optimal area ratio /opt. Similar parameter studies give the optimal driving nozzle position (Lx)opt and the optimal relative mixing length Lm/dm. • Optimization of the operating conditions: with the compression ratio w obtained from the specifications Tc and Te, a parameter variation l = f(n)/,w (or w = f(n)/,l, if l is given) gives the optimal driving pressure ratio nopt. From Pc, and nopt follows finally the optimal driving pressure (Pb)opt = Pc Æ nopt and thus the optimal driving temperature (Tb,opt). Similarly the optimal evaporator temperature (Te,opt) can be obtained for given entrainment ratio and ejector geometry. Flow diagrams for the computational procedures can be found in [13,21].
6. Numerical results Model validation has been carried out for single phase flow of R11 (banned since the Montreal Protocol) with available experimental data from the literature as shown in Figs. 4 and 5. In both figures, a good agreement of the computational results with the experimental data from [4] respectively from [1] is achieved. Other comparison with experimental results (for air) can be found on Fig. 14. Fig. 6 shows, for NH3 as working fluid, a decrease of the ejector diameters dt, dp3, dm and dd with rising driving temperature at a condenser temperature of 35 C and an evaporator temperature of 4 C. The higher the driving temperature Tb, the faster the driving flow and the smaller
Fig. 4. COP vs evaporator temperature.
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Fig. 5. Mass flow ratio vs driving nozzle distance.
Fig. 6. Nozzle diameters as a function of the driving temperature.
the appropriate driving nozzle. This is consistent with the continuity equation. Sun [10] obtained similar results about an ejector driven by the working fluid R123. Fig. 7 shows the same decrease for the driving nozzle diameters, however for the NH3–H2O mixture, which is a friendly working fluid and above all a lower pressure refrigerant than NH3 [22]. For same operating conditions, they are larger than for pure NH3. For the NH3–H2O mixture, for fixed quality ub of the driving steam, the diameters dt and dp3 increase with rising quality ue of the suction steam. An increase of these diameters with rising ub for given ue is also noticeable. This influence of the quality of the driving and suction flow on the driving nozzle diameters holds likewise for the diameters dm and dd of the mixing chamber and the diffuser.
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Fig. 7. Variation of the driving nozzle diameters with the driving temperature at different driving and suction qualities.
Fig. 8. Variation of the COP with the driving temperature.
Contrary to the compression and absorption refrigeration systems, for which the efficiency rises with increasing boiler temperature, Figs. 8 and 9 show that the jet refrigeration system with fixed geometry has an optimal driving temperature, at which the COP and the entrainment ratio l are maximum. On Figs. 10 and 11 it can be seen that the geometric ratio / has an optimal value /opt for given evaporator and condenser temperatures, at which the mass flow ratio and the respective COP are maximum. This was established experimentally [6]. To our knowledge the present numerical results are the first theoretical acknowledgement of the mentioned measurements. A very important result for the dimensioning of jet compressors is the quasi linear dependence between the main area ratio / and the driving pressure ratio n shown in Fig. 12 for given
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Fig. 9. Variation of the COP with the driving temperature.
Fig. 10. Variation of entrainment ratio and COP with the geometric ratio /.
entrainment ratio. This results is very suitable for the rough draft of both the sizing and the operational behaviour of a jet compressor. A dimensional formulation of this dependence can be inferred from [13]. There, this dependence is shown for given compression ratios. The smaller the mass flow ratio, the larger the driving mass flows and thus the higher the driving pressures and/or driving temperatures. For small compression ratios small driving pressures are needed. Fig. 13 depicts the influence of the entrainment ratio and driving pressure ratio on compression ratio. There is a pronounced maximum of compression ratio for a given entrainment ratio. For entrainment ratio l = 0, 1, 0, 2 and 0, 3 the maximum compression ratio is achieved for the optimal
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Fig. 11. Variation of entrainment ratio and COP with the geometric ratio /.
Fig. 12. Optimal driving pressure ratio as a function of the main area ratio for Te = 4, 5 C, Tc = 32 C, Z = 95%, ub = 15%, ue = 70% and w = 2, 1. Parameter is the entrainment ratio.
driving pressure ratios nopt = 4.6 (Pc = 11.1 bar, Pb = 50.89 bar, Tb = 93.8 C), 4.3 (Pb = 47.58 bar, Tb = 90.4 C) and 4.1 (Pb = 45.36 bar, Te = 88 C). The appropriate optimal evaporator temperatures are Te = 2 C (Pe = 3.33 bar), Te = 5.5 C (Pe = 4.34 bar) and Te = 10.5 C (Pe = 5.2 bar) with Te = 2 C (Pe = 3.33 bar). Small mass flow ratios correspond to large compression ratios. This is due to the fact that with small mass flow ratios only a small suction mass flow can be promoted. Indeed the existing driving power serves to a large extent for the pressure increase. For big suction mass flow only a small portion of the driving power remains for the
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Fig. 13. Variation of the compression ratio with the driving pressure ratio for different entrainment ratios.
Fig. 14. Influence of the relative mixing tube length Lm/dm on the pressure ratio Pe/Pc for an entrainment ratio of l = 0.2 and a driving pressure relationship of n = 7 for air as refrigerant. The measured values are from [24].
pressure increase within the ejector. In this case only smaller compression ratios can be therefore achieved. For given driving pressure ratio n = 7 and entrainment ratio l = 0.2, Fig. 14 shows the variation of the pressure ratio Pe/Pc with the relative mixing tube length Lm/dm in the chocked regime for the working fluid air. A maximum compression ratio W and thus an optimal mixing tube length is achieved. A good agreement of the measured values [24] with the computations is shown. Fig. 15 shows the results of the calculations of the optimal mixing tube length in the chocked regime for the mixture ammonia–water. The optimal mixing tube length is about 5 (for l = 0.3) to
K. Cizungu et al. / Applied Thermal Engineering 25 (2005) 1979–1994
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Fig. 15. Computation of the optimal mixing tube length in the chocked regime.
9 (for l = 0.1). These data are somewhat smaller than for the single-phase fluid air, where (Lm/ dm)opt > 9 (for l = 0.2).
7. Conclusion In the present work the design of single and two-phase jet compressors with special reference to their use in a jet cooling system has been investigated. Contrary to the design procedures given in the literature, neither the usual assumption of mixing of the driving and suction flows at constant pressure within the mixing chamber cross section nor that of a constant mixing chamber cross section has been made. The developed model includes both the jet compressor geometry through the parameters /, Lx/dt and Lm/dm as well as the duct effectiveness, wall friction, momentum loss, shock waves and acceleration of the induced flow in the conical part of the mixing section. On the basis of the developed model the sizing of the jet compressor can be carried out and the operational behaviour of the corresponding jet cooling system in the stationary case can be determinated. A new and reliable interaction between the entrainment ratio and all relevant dimensional parameters is derived and compared successfully with the most important available experimental results. For given boundary conditions the optimum geometry of the ejector can be calculated. For given geometry of the ejector, the optimum thermal conditions can be determinated. The dimensions of ejector configuration has a dominant influence in deciding the operating range.
References [1] J.-C. Champoussin, L.-T. Lu, Climatisation par syste`me tritherme a` e´jecteur, Clima 2000 6 (1985) 185. [2] B.J. Huang et al., Ejector performance characteristics and design analysis of jet refrigeration system, J. Eng. Gas Turbines Power 107 (1985) 792.
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[3] F.C. Chen, C.T. Hsu, Performance of ejector heat pumps, Energy Res. 11 (1987) 289. [4] L.-T. Lu et al., Performances optimales et utilisation du syste`me a` e´jecteur en production de froid, 17th Int. Refrig. Proc., Wien (1987) & RGF-Octobre (1988) 529. [5] S. Srinivasa Murthy, R. Balasubramanian, M.V. Krishna Murty, Experiments on vapour jet refrigeration system suitable for solar energy applications, Renewable Energy 1 (1991) 757. [6] E. Nahdi et al. Optimal geometric parameters of a cooling ejector-compressor, Int. J. Refrig. 16 (1993) 67. [7] I.W. Eames et al. A theoretical and experimental study of a small-scale steam jet refrigerator, Int. J. Refrig. 18 (1995) 378. [8] R. Dorante`s, A. Lallemand, Prediction of performance of a jet cooling system operating with pure refrigerants or non-azeotropic mixtures, Int. J. Refrig. 18 (1995) 21. [9] D.-W. Sun, I.W. Eames, Performance characteristics of HCFC-123 ejector refrigeration cycles, Int. J. Energy Res. 20 (1996) 871. [10] D.-W. Sun, Variable geometry ejectors and their applications in ejector refrigeration systems, Energy 21 (1996) 919. [11] B.J. Huang et al. A 1-D analysis of ejector performance, Int. J. Refrig. 22 (1999) 354, 379. [12] K. Cizungu et al. Performance comparison of vapour jet refrigeration system with environmentally friendly working fluids, J. Appl. Therm. Eng. 21 (2000) 585. [13] K. Cizungu, Modellierung und Optimierung von Ein- und Zweiphasen-Strahlverdichtern im stationa¨ren Betrieb, Ph.D. Thesis, IKE, Universia¨t Stuttgart, Germany, 2003. [14] R. Schulz, Untersuchung an Strahlverdichtern, Ph.D. Thesis, RWTH Aachen, Germany, 1993. [15] G.F. Hewitt, Process Heat Transfer, CRC Press, Florida, 1994. [16] G. Grazzini, A. Rocchetti, Numerical optimisation of a two-stage ejector refrigeration plant, Int. J. Refrig. 25 (2002) 621. [17] P.A. Kogan et al. Determination of the optimum geometrical characteristics of ejector, Therm. Eng. 9 (1967) 99, [Teploenegetika 14 (1967) 69]. [18] N.N. Agapov et al. Study of a liquid helium jet pump for circulating refrigeration systems, Cryogenics (1978) 491. [19] Li-T. Chen, A new ejector–absorber cycle to improve the COP of an absorption refrigeration system, Appl. Energy 30 (1988) 37. [20] M.I. Putilov, Calculating the optimal distance between the nozzle from the mixing chamber in injectors, Therm. Eng. 14 (7) (1967) 70. [21] K. Cizungu, Manual zu den Berechnungsprogrammen fu¨r Dampfstrahlverdichter, IKE 5TN-1671-00, Februar 2000. [22] J. Burkhard, Experimentelle Untersuchungen an einem Ammoniak-Dampfstrahlapparat, Ka¨ltetechnik-Klimatisierung 10 (1967) 310. [23] G.S. Harrell, Testing and modeling of a two-phase ejector, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, February 1997. ¨ berschallejektor mit [24] G. Leistner, Experimentelle und theoretische Untersuchungen an einem Hochtemperatur-U zylindrischer Mischkammer. Ph.D. Thesis, TU Darmstadt, 1966.
Modelling and optimization of two-phase ejectors for cooling systems K. Cizungu
a,*
, M. Groll a, Z.G. Ling
b
a
b
Institut fu¨r Kernenergetik und Energiesysteme (IKE), Universita¨t Stuttgart, Pfaffenwaldring 31, D-70550 Stuttgart, Germany Research Institute of Energy and Environmental Engineering, Shanghai University of Engineering Science, No. 350 Xian Xia Road, Shanghai 200336, PR China Received 12 February 2004; accepted 20 November 2004
Abstract A one-dimensional compressible flow model, which is based on the control volume approach, has been formulated to model and optimize one and two-phase ejectors in steady-state operation with particular reference to their deployment in a jet cooling system. The working fluid can be both single-component (NH3) and two-component (NH3–H2O). The developed model takes into account the duct effectiveness, wall friction, momentum loss, ejector geometry, shock waves as well as the acceleration of the induced flow in the conical part of the mixing section. Neither the usual assumption of mixing at constant pressure over the mixing chamber cross section nor that of a constant mixing chamber cross section were made. A comparison with other computation methods as well as with available experimental data from the literature is presented. The performance is significantly influenced by the ejector geometry. 2004 Elsevier Ltd. All rights reserved. Keywords: Two-phase ejector; Cooling system; One-dimensional model; Optimization
*
Corresponding author. Tel.: +49 711 685 2481; fax: +49 711 685 2010. E-mail address: [email protected] (K. Cizungu).
1359-4311/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2004.11.014
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Nomenclature A COP d h L Ma m_ b P Q_ T w Z
cross section area (m2) coefficient of performance (–) diameter (m) specific enthalpy (kJ/kg) length (m) Mach number (–) mass flow rate (kg/s) pressure (bar) heat transfer rate (kW) temperature (C) velocity (m/s) ammonia concentration (–)
Greek symbols a0, a1, b0, b1 opening angles () g efficiency (–) l ¼ m_ s =m_ p entrainment ratio (–) q density (kg/m3) f momentum loss (–) / = Am/At main area ratio (between mixing tube and primary nozzle) u vapour quality (–) n = Pb/Pc driving pressure ratio (–) w = Pc/Pe compression ratio (–) Subscripts b boiler com compression d diffuser e evaporator is isentropic opt optimal s secondary (flow, nozzle) x distance driving nozzle exit—mixing chamber entrance 1, 2, 3, 4 states in respective cross sections of the ejector (see Fig. 2) c condenser conv convergent div divergent ex expansion m mixing tube p primary (flow, nozzle) t nozzle throat
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1. Introduction In view of the exhaustible nature of the fossil energy sources and due to the environmental impact of the rising energy consumption, an increasing use of low temperature heat sources can be a suitable solution for a more rational energy use and ultimately cost savings. In refrigeration and airconditioning there are two interesting methods of obtaining cooling from thermal energy: the absorption process in which the mechanical compressor of the conventional compression cooling system is replaced by a thermal compressor, and the vapour ejector system in which an ejector replaces the mechanical compressor. Because of complex construction and maintenance, absorption refrigerators are relatively expensive. Additionally they need waste heat at temperatures over about 100 C. Due to the absence of moving parts, the simple construction, a high reliability and low maintenance costs, jet compressors (ejectors, jet pumps, injectors) represent an attractive alternative to other compressors (Figs. 1 and 2). Therefore they are widely used in the energy
Fig. 1. Schematic of a jet cooling system.
Fig. 2. Schematic of the ejector and geometrical parameters.
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and chemical engineering fields. They can in particular effectively use low temperature energy sources. The ejector–compression system can utilize low grade energy at temperature levels of 70–95 C, which is available e.g. from flat solar collectors. Using different working fluids [1–12], it has been shown that, for different boiler temperatures, the entrainment ratio and the system efficiency of a jet based cooling system depend mainly on the ejector geometry and the compression ratio. However there are contradictory statements in the literature about the influence of the ejector geometry [13]. Thus a consistent description of the interaction between geometry and performance of a jet compressor is the main aim of the modelling. In view of the gaps noted concerning the current state of research on the supersonic ejectors, it is preferable to first study the influences of the ejector geometry on its global characteristics (entrainment ratio, refrigerating effectiveness, compression ratio). Thus we use a one-dimensional model based on the method of control volume and extend the calculations to two-phase flow. This model allows in addition to save computing time and it is easy to handle.
2. System description The present work is concerned with the design of single and two-phase vapour jet compressors in steady state with special reference to their use in a jet cooling system. Due to very restrictive assumptions, the design models from the literature can neither describe accurately the complicated thermofluid dynamic processes in the jet compressor nor reproduce the experimentally established relations between the mass flow or entrainment ratio and ejector geometry [1–11]. Thus the interaction between geometry and performance of a jet compressor is the main aim of the work. A detailed description of the operational principle of a jet compressor as well as a jet compressor cooling system was given elsewhere [12,13]. Fig. 1 shows a schematic diagram if an ejector based cooling system. Its operating principle can be summarized as follows. The high pressure and temperature vapour generated within the boiler due to heat exchange Q_ b between the external heat source and the liquid refrigerant enters the ejector and flows through a converging–diverging nozzle (known as driving or primary nozzle) of the ejector, where it expands to supersonic flow and induces a low pressure region at the nozzle exit. The partial vacuum produced by the driving fluid causes the suction of the refrigerant vapour (known as induced or secondary fluid) from the evaporator. The two fluids mix in the mixing chamber and are discharged through the diffuser to the condenser. In the condenser, the mixed flow is liquefied by rejecting the heat Q_ c to the environment. Part of the resulting liquid refrigerant is returned to the boiler after a pressure increase by a feed-pump, while the remaining liquid enters the evaporator after undergoing a pressure reduction in the expansion or throttling valve. In the evaporator, the fluid evaporates due to the heat gain (cooling load Q_ e ) from the cooling space. In this way, the cycle of the ejector cooling system is completed. Fig. 2 shows the structure of a typical ejector. The geometrical parameters of a jet compressor comprise the diameters d1, dt, dp3, d2, ds3, dm and dd, the opening angles a0, a1, b0 and b1 and the axial lengths Lp, Lx, Lm and Ld. The most important parameter is the main area ratio / = Am/ At. The thermodynamic parameters can be summarized in driving pressure ratio n = Pb/Pc and compression ratio w = Pc/Pe. The different thermodynamic processes in the jet compressor are
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Fig. 3. Thermodynamic process in the ejector. 1–3P: real expansion of driving steam in the driving nozzle; 2–3S: real expansion of suction steam in the suction chamber 3P–xP–4 and 3S–xS–4: mixture process in intake cone and mixing chamber; 4–5: real compression of the mixture in the diffuser.
shown in the (h, s) diagram on Fig. 3. The pressures or the indices refer to the cross sections indicated in the Fig. 3. The dotted lines lines represent the isentropic processes.
3. Modelling The calculation model is based on the conservation laws for mass, momentum and energy for the flow in driving nozzle, suction nozzle, mixing chamber and diffuser. Selection criteria for the design procedure are simple handling and small computing time expenditure. Following assumptions were made: • The flow in the jet compressor is one-dimensional, compressible, single-phase or two-phase, friction-affected, in steady state, and the flow process in the ejector takes place adiabatically. • Based on experimental investigations [2,4,6] the design pressure Pp3 at the driving nozzle exit is uniform i.e. Pp3 = Ps3, and the suction fluid velocity reaches the speed of sound at this level. • The losses in the mixing chamber are represented by a friction factor fm and momentum loss factor fm, the losses in the nozzles and the diffuser are taken into account by assuming respective isentropic efficiencies gp, gs and gd. The different angles of the jet compressor as well as the efficiencies gp, gs and gd are taken from the literature [12,13]. The system of the derived governing equations has to be solved numerically. A summary of the model equations leads to a very long expression for the mass flow ratio l of a jet compressor [13]: Lm Lx m_ s ¼ f n; w; gn ; gd ; fm ; fm ; ; gs ; /; ; tan b ð1Þ l¼ dm dt m_ p
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With the usual assumption of a mixing at constant pressure and negligible inlet and outlet velocity in the ejector without intake cone before the cylindrical mixing chamber, the relationship (1) can be represented by the following simplified equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dhexp fm Lm g g 1 þ n d Dhcom 2fm d m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð2Þ l¼ fm Lm Dhexs 1 þ 2fm d m Dhcom gs gd where the enthalpy differences are given by Dhexp = hb hp3,is and Dhexs = he hs3,is. Schulz [14] and Cizungu et al. [12] derived also the equation (2), whereby the momentum loss in the mixing chamber was neglected (i.e. fm = 1) but not the frictional factor fm. The equations for the determination of the mass flow relation of a jet compressor, indicated so far in the literature, can be summarized by the empirical relationship sffiffiffiffiffiffiffiffiffiffiffiffi Dhexp c 1; ð3Þ l¼ Dhcom where c = gexpÆgcom represents the so-called quality grade of the jet compressor, with c = 0.689– 0.81 [13]. On the basis of experimental investigations Dorante`s et al. [8] have worked out the following correlation for the calculation of the entrainment ratio: 2:12 1 1:21 : ð4Þ l ¼ 3:32 w nw A comparison with our own results as well as with measured values from the literature shows that equation (4) applies only within the limited area 2.5 6 n 6 4; / 6 6. Other correlation are given in [11]. Contrary to equations (2)–(4) for the determination of the entrainment ratio l of a jet compressor, relationship (1) considers both the acceleration of the suction flow and the wall friction, the momentum loss in the mixing chamber and the influence of the most important geometrical parameters, /, Lx/dt and Lm/dm. Moreover, the working fluids are treated here as real gases. The direct consideration of the driving nozzle position enables a regulation of the jet compressor. With the ejector mass flow ratio l from relationship (1) the coefficient of performance (COP), i.e. the system efficiency of the ejector cooling system can be calculated through COP ¼
Q_ e h2 hg ¼l : h1 h6 Q_ b þ N PP
ð5Þ
The sizing of the jet compressor and the operational behaviour of the jet cooling system are calculated as follows: • For the determination of jet compressor geometry, the driving nozzle diameters d1, dt and dp3, the suction nozzle diameters d2 and ds3, the mixing tube diameter dm and the diffuser diameter dd as well as the driving nozzle lengths Lp, the mixing chamber length Lm and the diffuser length Ld are calculated for given operating conditions Tb, Tc, Te, ub, ue and Q_ e . • For the determination of the operational behaviour, the mass flow ratio l, energetic efficiency COP and compression ratio w are calculated for given jet compressor geometry and operating conditions Tb, Tc, Te, ub, ue and Q_ e .
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Calculations have been carried out for single-phase and two-phase as well as for single-component (NH3) and two-component (NH3–H2O) working fluids. With two-phase working fluids the present jet compressor model falls back on the so-called homogeneous model for two-phase flow [15]. The principal difference to the calculation model for two-phase two-component jet compressors lies in the thermodynamic characteristics, but this is decisive. Indeed, while with pure refrigerants a modification of the quality at fixed temperature does not influence the pressure (and thus the mass flow ratio), but only density, enthalpy, entropy and internal energy (and thus COP), a modification of the quality of two-phase two-component refrigerants affects both the pressure and the further thermodynamic characteristics, thus the pressure decreases with rising quality, while the enthalpy increases. It changes also the irreversibilities within the ejector [16,23]. 4. Optimization procedure There are hardly reports about optimization computations of jet compressors pffiffiffiffiffiin the literature. Kogan et al. [17] gave a functional dependence of the form f ðP e =P b ; P c =P b ; l T ; xÞ ¼ 0 for the determination of the optimal geometrical characteristics of an ejector with cylindrical mixing p = T /T . Depending on the given parameters {P /P , l T } or {P chamber. Here is T e b c p c/Pp, p , l T } they computed analytically for ideal gases the minimum (Pe/Pb)min or Pe/Pb} or {Pe/Pbp the maximum (l T)max or the maximum (Pc/Pb)max, which gives an optimal (/)opt. No values are given for the used speed coefficients x for the driving nozzle, the mixing area, the diffuser and the intake cone to the mixing area. There is no comparison of the calculations with experiments. Agapov et al. [18] used for the increase of pressure in the ejector an equation of the form Pc Pe = f(x, k, l, qp3, qs, qd, wp3) = 0 and derived from this the optimal cross section by means of oko ðP c P e Þ ¼ 0. In this case k = Ap3/Am. In addition to the assumptions in [17], the cross section Ap3 was regarded as constant along Lx. There are also neither data concerning the values of the used speed coefficients x nor comparison of the calculations with experiments. Assuming incompressible flow and differentiating an equation for the total pressure rise across the ejector, similarly as in [18], Chen [19] obtained the following relationship for the optimal area ratio of the form kopt = f(l, qp, qm, gd, Am/Ad) = 0. The acceleration of the suction flow in the above optimization calculations were not considered. Up to Kogan et al. [17] the flow inside the ejector was treated as incompressible The only calculations given in the literature for the determination of the optimal driving nozzle distance [20] are just empirical relations, they are valid exclusively for incompressible flow. In the one-dimensional models [3,10,11] based on ideal gas, isentropic processes and mixing at constant pressure a relationship of the form l = f(n, w, /, T ) = 0 was derived but the influence of the dimensional parameter / on the ejector performance was not investigated and only [11] has taken into account the effects of frictional and missing losses which are important for a precise ejector design. In [3], no comparison of the calculations with experimental data was given. In short, it can be observed the effect of the important geometrical parameter /, Lx/dt and Lm/dm on the ejector performance characteristics was not analysed in the known models. None of them has taken into consideration the axial lengths, first of all the parameters Lx and Lm. Therefore, a new optimization procedure, which goes beyond the mentioned procedures is necessary. This is also a goal of this work.
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4.1. Optimization criteria When the jet compressor is used as component in a cooling system the optimization procedure consists of achieving either a maximum entrainment ratio l with fixed compression ratio w or a maximum compression ratio with fixed entrainment ratio. The achievement of a maximum entrainment ratio means that for a given driving power fixed by the pressure ratio n = Pb/Pc the maximum suction mass flow has to be reached, while a maximum compression ratio means that for a given entrainment ratio, either a maximum condenser pressure (a maximum condenser temperature) for given evaporator pressure or a minimum evaporator pressure (a minimum evaporator temperature) for given condenser pressure has to be reached. Depending on defaults, the operating conditions or ejector geometry are optimized. The formal optimization procedure reads Lx Lm : ð6Þ l ¼ f n; w; /; ; dt dm This equation represents a new and complete relationship for the computation of the mass flow relationship of an ejector as function of all most important geometrical parameters. /, Lx/dt and Lm/dm. Based one on this equation, a sensitivity study of the system performance in dependence of the fundamental geometrical parameters will be introduced in the following. 5. Solution methodology 5.1. Solution procedure for the jet compressor sizing • Determine iteratively the pressure Ps3 by fulfilling the choking condition (Ma = 1) at the entrance of the mixing chamber. • Determine iteratively the pressure Pt by fulfilling the choking condition (Ma = 1) at the driving nozzle throat. • Determine the design pressure by assuming Pp3 = Ps3. • Determine iteratively the pressure Pm. • Determine iteratively the mass flow ratio l. From this follows the calculation of the cross section areas At, Ap3, As3, Am, Ad and the respective diameters dt, dp3, ds3, dm and dd as well as the lengths Ldiv, Lt, Lconv, Lm and Ld. 5.2. Solution procedure for the determination of the operational behaviour of the ejector cooling system • • • •
Determine Determine Determine Determine
iteratively the pressure Ps3 as above. the pressure Pt as above. the design pressure Pp3. iteratively the pressure Pm.
From this follows the calculation of the entrainment ratio l and the COP.
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5.3. Solution procedure for the optimization • Nozzle optimization: for given Tb, Tc, Te, and ub, ue and from a parameter variation l = f(/)n,w, if w is given, by differentiating Eq. (6), i.e. ol/o/ = 0 (or w = f(/)n,l, if l is given, i.e. oW/o/ = 0). follows the optimal area ratio /opt. Similar parameter studies give the optimal driving nozzle position (Lx)opt and the optimal relative mixing length Lm/dm. • Optimization of the operating conditions: with the compression ratio w obtained from the specifications Tc and Te, a parameter variation l = f(n)/,w (or w = f(n)/,l, if l is given) gives the optimal driving pressure ratio nopt. From Pc, and nopt follows finally the optimal driving pressure (Pb)opt = Pc Æ nopt and thus the optimal driving temperature (Tb,opt). Similarly the optimal evaporator temperature (Te,opt) can be obtained for given entrainment ratio and ejector geometry. Flow diagrams for the computational procedures can be found in [13,21].
6. Numerical results Model validation has been carried out for single phase flow of R11 (banned since the Montreal Protocol) with available experimental data from the literature as shown in Figs. 4 and 5. In both figures, a good agreement of the computational results with the experimental data from [4] respectively from [1] is achieved. Other comparison with experimental results (for air) can be found on Fig. 14. Fig. 6 shows, for NH3 as working fluid, a decrease of the ejector diameters dt, dp3, dm and dd with rising driving temperature at a condenser temperature of 35 C and an evaporator temperature of 4 C. The higher the driving temperature Tb, the faster the driving flow and the smaller
Fig. 4. COP vs evaporator temperature.
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Fig. 5. Mass flow ratio vs driving nozzle distance.
Fig. 6. Nozzle diameters as a function of the driving temperature.
the appropriate driving nozzle. This is consistent with the continuity equation. Sun [10] obtained similar results about an ejector driven by the working fluid R123. Fig. 7 shows the same decrease for the driving nozzle diameters, however for the NH3–H2O mixture, which is a friendly working fluid and above all a lower pressure refrigerant than NH3 [22]. For same operating conditions, they are larger than for pure NH3. For the NH3–H2O mixture, for fixed quality ub of the driving steam, the diameters dt and dp3 increase with rising quality ue of the suction steam. An increase of these diameters with rising ub for given ue is also noticeable. This influence of the quality of the driving and suction flow on the driving nozzle diameters holds likewise for the diameters dm and dd of the mixing chamber and the diffuser.
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Fig. 7. Variation of the driving nozzle diameters with the driving temperature at different driving and suction qualities.
Fig. 8. Variation of the COP with the driving temperature.
Contrary to the compression and absorption refrigeration systems, for which the efficiency rises with increasing boiler temperature, Figs. 8 and 9 show that the jet refrigeration system with fixed geometry has an optimal driving temperature, at which the COP and the entrainment ratio l are maximum. On Figs. 10 and 11 it can be seen that the geometric ratio / has an optimal value /opt for given evaporator and condenser temperatures, at which the mass flow ratio and the respective COP are maximum. This was established experimentally [6]. To our knowledge the present numerical results are the first theoretical acknowledgement of the mentioned measurements. A very important result for the dimensioning of jet compressors is the quasi linear dependence between the main area ratio / and the driving pressure ratio n shown in Fig. 12 for given
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Fig. 9. Variation of the COP with the driving temperature.
Fig. 10. Variation of entrainment ratio and COP with the geometric ratio /.
entrainment ratio. This results is very suitable for the rough draft of both the sizing and the operational behaviour of a jet compressor. A dimensional formulation of this dependence can be inferred from [13]. There, this dependence is shown for given compression ratios. The smaller the mass flow ratio, the larger the driving mass flows and thus the higher the driving pressures and/or driving temperatures. For small compression ratios small driving pressures are needed. Fig. 13 depicts the influence of the entrainment ratio and driving pressure ratio on compression ratio. There is a pronounced maximum of compression ratio for a given entrainment ratio. For entrainment ratio l = 0, 1, 0, 2 and 0, 3 the maximum compression ratio is achieved for the optimal
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Fig. 11. Variation of entrainment ratio and COP with the geometric ratio /.
Fig. 12. Optimal driving pressure ratio as a function of the main area ratio for Te = 4, 5 C, Tc = 32 C, Z = 95%, ub = 15%, ue = 70% and w = 2, 1. Parameter is the entrainment ratio.
driving pressure ratios nopt = 4.6 (Pc = 11.1 bar, Pb = 50.89 bar, Tb = 93.8 C), 4.3 (Pb = 47.58 bar, Tb = 90.4 C) and 4.1 (Pb = 45.36 bar, Te = 88 C). The appropriate optimal evaporator temperatures are Te = 2 C (Pe = 3.33 bar), Te = 5.5 C (Pe = 4.34 bar) and Te = 10.5 C (Pe = 5.2 bar) with Te = 2 C (Pe = 3.33 bar). Small mass flow ratios correspond to large compression ratios. This is due to the fact that with small mass flow ratios only a small suction mass flow can be promoted. Indeed the existing driving power serves to a large extent for the pressure increase. For big suction mass flow only a small portion of the driving power remains for the
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Fig. 13. Variation of the compression ratio with the driving pressure ratio for different entrainment ratios.
Fig. 14. Influence of the relative mixing tube length Lm/dm on the pressure ratio Pe/Pc for an entrainment ratio of l = 0.2 and a driving pressure relationship of n = 7 for air as refrigerant. The measured values are from [24].
pressure increase within the ejector. In this case only smaller compression ratios can be therefore achieved. For given driving pressure ratio n = 7 and entrainment ratio l = 0.2, Fig. 14 shows the variation of the pressure ratio Pe/Pc with the relative mixing tube length Lm/dm in the chocked regime for the working fluid air. A maximum compression ratio W and thus an optimal mixing tube length is achieved. A good agreement of the measured values [24] with the computations is shown. Fig. 15 shows the results of the calculations of the optimal mixing tube length in the chocked regime for the mixture ammonia–water. The optimal mixing tube length is about 5 (for l = 0.3) to
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Fig. 15. Computation of the optimal mixing tube length in the chocked regime.
9 (for l = 0.1). These data are somewhat smaller than for the single-phase fluid air, where (Lm/ dm)opt > 9 (for l = 0.2).
7. Conclusion In the present work the design of single and two-phase jet compressors with special reference to their use in a jet cooling system has been investigated. Contrary to the design procedures given in the literature, neither the usual assumption of mixing of the driving and suction flows at constant pressure within the mixing chamber cross section nor that of a constant mixing chamber cross section has been made. The developed model includes both the jet compressor geometry through the parameters /, Lx/dt and Lm/dm as well as the duct effectiveness, wall friction, momentum loss, shock waves and acceleration of the induced flow in the conical part of the mixing section. On the basis of the developed model the sizing of the jet compressor can be carried out and the operational behaviour of the corresponding jet cooling system in the stationary case can be determinated. A new and reliable interaction between the entrainment ratio and all relevant dimensional parameters is derived and compared successfully with the most important available experimental results. For given boundary conditions the optimum geometry of the ejector can be calculated. For given geometry of the ejector, the optimum thermal conditions can be determinated. The dimensions of ejector configuration has a dominant influence in deciding the operating range.
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[3] F.C. Chen, C.T. Hsu, Performance of ejector heat pumps, Energy Res. 11 (1987) 289. [4] L.-T. Lu et al., Performances optimales et utilisation du syste`me a` e´jecteur en production de froid, 17th Int. Refrig. Proc., Wien (1987) & RGF-Octobre (1988) 529. [5] S. Srinivasa Murthy, R. Balasubramanian, M.V. Krishna Murty, Experiments on vapour jet refrigeration system suitable for solar energy applications, Renewable Energy 1 (1991) 757. [6] E. Nahdi et al. Optimal geometric parameters of a cooling ejector-compressor, Int. J. Refrig. 16 (1993) 67. [7] I.W. Eames et al. A theoretical and experimental study of a small-scale steam jet refrigerator, Int. J. Refrig. 18 (1995) 378. [8] R. Dorante`s, A. Lallemand, Prediction of performance of a jet cooling system operating with pure refrigerants or non-azeotropic mixtures, Int. J. Refrig. 18 (1995) 21. [9] D.-W. Sun, I.W. Eames, Performance characteristics of HCFC-123 ejector refrigeration cycles, Int. J. Energy Res. 20 (1996) 871. [10] D.-W. Sun, Variable geometry ejectors and their applications in ejector refrigeration systems, Energy 21 (1996) 919. [11] B.J. Huang et al. A 1-D analysis of ejector performance, Int. J. Refrig. 22 (1999) 354, 379. [12] K. Cizungu et al. Performance comparison of vapour jet refrigeration system with environmentally friendly working fluids, J. Appl. Therm. Eng. 21 (2000) 585. [13] K. Cizungu, Modellierung und Optimierung von Ein- und Zweiphasen-Strahlverdichtern im stationa¨ren Betrieb, Ph.D. Thesis, IKE, Universia¨t Stuttgart, Germany, 2003. [14] R. Schulz, Untersuchung an Strahlverdichtern, Ph.D. Thesis, RWTH Aachen, Germany, 1993. [15] G.F. Hewitt, Process Heat Transfer, CRC Press, Florida, 1994. [16] G. Grazzini, A. Rocchetti, Numerical optimisation of a two-stage ejector refrigeration plant, Int. J. Refrig. 25 (2002) 621. [17] P.A. Kogan et al. Determination of the optimum geometrical characteristics of ejector, Therm. Eng. 9 (1967) 99, [Teploenegetika 14 (1967) 69]. [18] N.N. Agapov et al. Study of a liquid helium jet pump for circulating refrigeration systems, Cryogenics (1978) 491. [19] Li-T. Chen, A new ejector–absorber cycle to improve the COP of an absorption refrigeration system, Appl. Energy 30 (1988) 37. [20] M.I. Putilov, Calculating the optimal distance between the nozzle from the mixing chamber in injectors, Therm. Eng. 14 (7) (1967) 70. [21] K. Cizungu, Manual zu den Berechnungsprogrammen fu¨r Dampfstrahlverdichter, IKE 5TN-1671-00, Februar 2000. [22] J. Burkhard, Experimentelle Untersuchungen an einem Ammoniak-Dampfstrahlapparat, Ka¨ltetechnik-Klimatisierung 10 (1967) 310. [23] G.S. Harrell, Testing and modeling of a two-phase ejector, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, February 1997. ¨ berschallejektor mit [24] G. Leistner, Experimentelle und theoretische Untersuchungen an einem Hochtemperatur-U zylindrischer Mischkammer. Ph.D. Thesis, TU Darmstadt, 1966.