Empirical correlation for ejector design

(1)

Empirical correlation for ejector design

B.J. Huang*, J.M. Chang

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan Received 4 May 1998; received in revised form 24 November 1998; accepted 23 December 1998

Abstract

In the present study, two empirical correlations from the test results of 15 ejectors are derived for the performance prediction of ejectors using R141b as the working fluid. The ratio of the hypothetical throat area of the entrained flow to the nozzle throat area Ae/At, the geometric design parameter of the ejector A3/At, and the pressure ratios Pg/Peand Pc*/Peare used to correlate the performance of the ejector. The prediction of the entrainment ratiov using the correlations is within ^ 10% error. A method of calculation for the ejector design using the correlations is also developed. R141b is shown in the present study to be a good working fluid for an ejector. The measuredv for the ejectors used in the present study can reach as high as 0.54 at Pg 0.465 MPa (848C), Pc* 0.087 MPa (288C) and Pe 0.040 MPa (88C). For Pg 0.538 MPa (908C), Pc* 0.101 MPa (328C) and Pe 0.040 MPa (88C),v reaches 0.45. q 1999 Elsevier Science Ltd and IIR. All rights reserved.

Keywords: Refrigerating system; Ejector system; Refrigerant; R141b; Ejector

Corre´lation empirique pour la conception des e´jecteurs

Re´sume´

Dans cette e´tude, on a e´tabli deux corre´lations empiriques a` partir des re´sultats expe´rimentaux obtenus utilisant 15 e´jecteurs; ces corre´lations ont e´te´ utilise´es ensuite pour pre´dire la performance d’e´jecteurs utilisant le R141b comme fluide frigorige`ne. Les rapports Ae/At(section de passage du fluide entraıˆne´ rapporte´ a` la section the´orique du col de l’e´jecteur),

et A3/At(section de sortie de l’e´jecteur rapporte´ a` la section the´orique du col de l’e´jecteur) et les relations entre pressions Pg/Pe

et Pc*/Pesont utilise´s pour trouver la corre´lation de la performance de l’e´jecteur. La pre´vision du taux d’entraıˆnement a` partir

des corre´lations est pre´cise a` la hauteur de ^ 10%. Les auteurs ont e´galement de´veloppe´ une me´thode de calcul permettant de concevoir des e´jecteurs a` partir des corre´lations. On a montre´ dans cette e´tude que le R141b s’ave`re eˆtre un fluide actif efficace pour cette utilisation. Lev mesure´ des e´jecteurs utilise´s dans cette e´tude peuvent atteindre 0.54 a` Pg 0.465 MPa (848C),

Pc* 0.087 MPa (288C) et Pe 0.040 MPa (88C). Pour Pg 0.538 MPa (908C), Pc* 0.101 MPa (328C) et Pe 0.040 MPa

Nomenclature

A area, m2

d diameter, m

G mass flow rate per unit area, kg s21m22

h enthalpy, kJ kg21

_m mass flow rate, kg s21

P pressure, MPa

T temperature,8C

PII: S 0 1 4 0 - 7 0 0 7 ( 9 9 ) 0 0 0 0 2 - X

* Corresponding author. Tel.:1 2-2363-4790; fax: 1 886-2-2363-0549.

(2)

3 exit of the constant-area mixing chamber

1. Introduction

Since its invention in the early twentieth century, the gas-to-gas, or vapor-to-vapor ejector, has found wide applica-tion in industries for the processes of evacuaapplica-tion, refrigera-tion, and solid powder transportation etc., or in modern jet planes for thrust augmentation. Air and steam are the common working fluids of an ejector. The study of refrig-erant (CFCs, HCFCs and HFCs) ejectors for air-condition-ing or refrigeration applications started in the mid-1950s for utilizing low-grade energy such as solar or waste heat energy as the heat source.

The operation of a gas-to-gas or vapor-to-vapor ejector results mainly from the gas-dynamic effect and the momen-tum exchange of two gaseous streams (primary and

second-2. Experimental setup

2.1. Test facility and operation

Fig. 1 shows the schematic diagram of the test facility. R141b is selected as the working fluid in the present study since R141b has a positive-slope saturated-vapor line in the thermodynamic T–s diagram. This will not produce conden-sation of the vapor during an isentropic expansion in the ejector, and thus reduces losses.

The working fluid is heated and evaporated in the genera-tor by the circulated glycol which is heated by a 5-kW electrical heater installed in the heating tank. The glycol temperature is controlled to within ^ 0.5 K accuracy by a PID controller so that the primary flow of the ejector from the generator is maintained at fixed temperature. The condenser is cooled by a direct expansion of liquid R22

Fig. 1. Schematic diagram of ejector test facility. Fig. 1. Sche´ma du dispositif utilise´ pour tester des e´jecteurs.

(3)

refrigerant generated from a heat pump using a vapor compression cycle. The condenser temperature, as well as the back pressure of the ejector can thus be adjusted between 108C and 308C and the experiment can be performed in any season. Cold water circulates through the evaporator to remove the evaporation heat.

The mass flow rate of the primary and the entrained flows is measured using two rotameters which are calibrated to within ^ 5% uncertainty. The primary flow is determined from the mass balance of the ejector. The mass flow rate through the nozzle of the ejector can also be calculated from the choked flow relation in gas-dynamics using the measured inlet temperature and pressure, and the properties of R141b. It was found that the calculated value coincides with the measured one to within ^ 5% uncertainty.

Three strain-gage type pressure transducers are used to measure the pressures of the primary and entrained flows, and the back pressure. The uncertainty is within ^ 5%. Vapor temperatures at the nozzle inlet, the suction port and the exit port of the ejector are measured by T-type thermocouples with an uncertainty of ^ 0.7 K.

The temperature and pressure signals are recorded by a YOKOGAWA hybrid recorder HR1300. All the measured data are transferred to a PC586 through a IEEE488 interface for data processing.

To obtain a steady-state operation during the test, the flow rates of the R22 coolant flow for the condenser, the cold water for the evaporator and the glycol solution are kept steady. To adjust the flow rate to the generator, we installed a bypass line at the exit of the circulation pump. It takes about 1 h to warm up the test facility and about 30 min for each steady-state run.

2.2. Ejector specifications

Fig. 2 is the schematic diagram of an ejector. To derive an empirical correlation for ejector performance, we have to test as many different ejectors as possible. The ejector is thus designed in three major parts: nozzle, suction chamber body, and constant-area mixing chamber (with diffuser). The connection between the different parts is to standard specifications so that the parts are interchangeable. Three nozzles are designed and fabricated in the experiment. The specifications of nozzles are listed in Table 1. We designed five different sizes of the constant-area mixing chambers (including diffuser) as listed in Table 2. The ejector area Fig. 2. Schematic diagram of ejector design.

Fig. 2. Sche´ma de la conception de l’e´jecteur.

Table 1

Dimensions of nozzles Tableau 1

Dimensions des tuye`res

Nozzle Throat diameter, dt.(mm) Exit diameter, dp1(mm)

A 2.64 4.50

B 2.75 4.66

D 2.93 4.46

Table 2

Dimensions of constant-area mixing chambers Tableau 2

Dimensions des chambres des me´lange a` superficie constante Mixing

chamber

d3(mm) Inlet converging

angle (8)

Ejector area ratio (with Nozzle A), A3/At A 6.70 68 6.441 B 6.98 60 6.990 G 7.34 60 7.730 C 7.60 67 8.287 D 8.10 68 9.414

(4)

ratios provided for the tests thus range from 6.4 to 9.4 if combined with Nozzle A.

3. Experimental analysis of R141b ejector

3.1. Performance modes of an ejector

In addition to the choking in the nozzle, the ejector has another choking phenomenon. As pointed out by Huang et

al. [1], the second choking of an ejector results from the acceleration of the entrained flow from a stagnant state at the suction port to a supersonic flow in the mixing chamber. Fig. 3 shows the variation of entrainment ratio v with discharge or back pressure Pcat fixed suction pressure Pe and fixed primary flow pressure Pg.

The ejector performance can then be divided into three operational modes, according to the back pressure Pc (refer-ring to Fig. 3): (1) double-choking or critical mode at Pc# P*c, while the primary and the entrained flows are both Fig. 3. Operational modes of an ejector.

Fig. 3. Modes de fonctionnement d’un e´jecteur.

Fig. 4. Entrainment ratio measurement for Ejector A-D. Fig. 4. Mesures de la relation d’entraıˆnement pour l’e´jecteur A-D.

(5)

choking and the entrainment ratio is constant, i.e. v constant; (2) single-choking or subcritical mode at P*c, Pc , Pc, while only the primary flow is choking and v changes with the back pressure Pc; and (3) back-flow or malfunction mode at Pc# Pco, while both the primary and the secondary flows are not choking and the entrained flow is reversed (malfunction), i.e.v # 0.

The performance mode of an ejector depends on the ejec-tor design and the operating conditions. The operational mode essentially results from the shock movement inside the ejector [1]. For better performance of an ejector cooling system, the ejector should be designed and operated in the double-choking or critical mode whose performance char-acteristics are our main interest in the present study. The critical mode is determined by the critical back pressure P*c, which is related to the pressures of the primary and the entrained flows and the design of the ejector. Hence, P*c is an important variable in an ejector’s performance.

The aforementioned ejector performance can be illu-strated by the entrainment ratio measured for the ejector assembly of Nozzle A and Mixing Chamber D (Ejector A-D), as shown in Fig. 4. For easy understanding in an air-conditioning application, we also present the saturated vapor temperature in the figures throughout this article.

Fig. 5 shows that the operational modes of Ejector A-D can be characterized using the pressure ratios, Pg/Pe, Pc/Peand Pco/Pe. For the same typical ejector, the corre-lation between Pg/Pe and P*c/Pe or Pco/Pe exists, irre-spective of the operation conditions, as shown in Fig. 5. However, the functional relationship changes for differ-ent ejectors. That is, a universal correlation cannot be obtained simply from the functional relation of Pg/Peand Pc/

Peor Pco/Pe. In the present study, we use the hypothetical throat area of the entrained flow in the critical mode as a key variable for deriving a universal correlation for ejector performance.

3.2. Hypothetical throat area of entrained flow at critical mode operation

As choking of the entrained flow results from the accel-eration of the entrained flow from a stagnant state at the suction port to a supersonic flow in the mixing chamber. A hypothetical throat area, “effective area Ae” [1,5], can be defined for the entrained flow at critical operation mode. The hypothetical throat area of an ejector Aecan be determined from the measured mass flow rate of the entrained flow _ms.

Fig. 2 shows the schematic of the mixing process of the two streams in the ejector. It is postulated [1,3] that after exhausting from the nozzle, the primary flow fans out without mixing with the entrained flow and induces a converging duct for the entrained flow. This duct acts as a converging nozzle such that the entrained flow is accelerated to a sonic velocity at some cross-section yy (hypothetical throat). After that, mixing of the two streams starts. The position of the hypothetical throat (cross-section yy) can be either in the suction chamber or in the constant-area mixing chamber depending on the nozzle position, i.e. the distance x from the constant-area mixing chamber. The present study found experimentally that the ejector with a nozzle located inside the suction chamber has a better performance.

Fig. 5. Measured operation mode for Ejector A-D. Fig. 5. Mode de fonctionnement (mesures) de l’e´jecteur A-D.

(6)

3.3. Calculation of hypothetical throat area Ae

The computation of Aeis carried out using an iterative scheme and a thermodynamic table to take into account the real gas properties of R141b. For given inlet conditions (Pe, Te) at the suction port, the computational procedures are as follows:

1. Given a pressure at the hypothetical throat Py, find the enthalpy at the hypothetical throat for an isentropic process from the thermodynamic table hys  (Se, Py), where Se f(Pe, Te).

2. Compute the gas velocity at the hypothetical throat from the energy balance relation

Vy

2hshe2 hys

q

1

wherehs (he2 hy)/(he2 hys) is the isentropic effi-ciency of the entrained flow. Here, we takehs 0.85. 3. Compute the enthalpy at the hypothetical throat for an

adiabatic process using the energy balance relation

hy he2 V 2

y=2: 2

4. Find the specific volume of gas at the hypothetical throat from the thermodynamic table

yyyPy; hy: 3

5. Compute the mass flow rate per unit area at the hypothe-tical throatGy Gy _ms Ae Vy yy: 4

6. Return to Step (1) and change the pressure at the hypothetical throat Py: Repeat the above computation until Gyreaches a maximum value (a choking condition).

7. Compute the hypothetical throat area Ae using the

measured entrained flow rate _msfrom the relation

Ae _ms=Gy: 5

3.4. Performance function of R141b ejector

The hypothetical throat area of an ejector Aecan be treated

as a characteristic parameter of the ejector. The entrained flow rate of an ejector can be easily determined if Aeis known for a

given suction condition. Further, we can use the area ratio

Ae=Atto represent the entrainment ratio of an ejectorv. Combining the correlation shown in Fig. 5, we found that the entrainment ratiov of an ejector in the critical mode operation is a function of ejector design (represented by the area ratio A3=At), and the pressure ratios (Pg=Peand P*c=Pe). That is, the following correlation exists for the ejector performance Ae At  f A3 At ; Pg Pe ; P*c Pe : 6

Fig. 6. Variation of hypothetical throat area ratio with Pg=Pefor various ejectors at Pe 0:040 MPa. Fig. 6. Variation du rapport A3=Aten fonction de Pg=Pepour divers e´jecteurs a` Pe 0:040 MPa.

(7)

3.5. Empirical correlations of R141b ejector

In the present study, we tested five different ejectors (Ejector A-A, A-B, A-C, A-D, A-G) at different

Pgcorresponding to the saturated vapor temperature from

788C to 958C which is in the application range. It is very interesting to note from Figs. 6 and 7 that, the hypothetical area ratio Ae=Atis approximately independent of Pg=Pe: The value of Ae=Atdepends on the ejector design (A3=At). Plot-ting Ae=Atversus A3=At, we found that Ae=Atis approximately Fig. 7. Variation of hypothetical throat area ratio with Pg=Pefor various ejectors at Pe 0:0473 MPa.

Fig. 7. Variation du rapport A3=Aten fonction de Pg=Pepour divers e´jecteurs a` Pe 0:0473 MPa.

Fig. 8. Variation of hypothetical throat area ratio with ejector area ratio A3=At. Fig. 8. Variation du rapport A3=At.

(8)

Fig. 9. Variation of ejector design area ratio A3=Atwith pressure ratios Pg=Peand P*c=Pe. Fig. 9. Variations des relations des superficies d’e´jecteurs A3=Aten fonction des relations Pg=Peet P*c=Pe.

Fig. 10. Comparison of measured and calculatedv. Fig. 10. Comparaison desvmesure´s et calcule´s.

(9)

a function of A3=Atonly as shown in Fig. 8. An empirical correlation can then be obtained:

Ae At 20:0517 A3 At 2 11:4362 A3 At 2 4:1734: 7

From the above relationship, the performance function of an ejector, Eq. (6), can be simplified to

A3 At  f Pg Pe ;P*c Pe : 8

Eq. (8) indicates that, for a fixed ejector (A3=Atfixed), the ejector performance follows the relationship:

f Pg Pe ; P*c Pe constant 9

which was shown to exist in practice, as shown in Fig. 5. The test results for five different ejectors are plotted in Fig. 9 from which another empirical correlation can be determined as A3 At  bo1 b1rc1 b2r 2 c1 b3rg1 b4rcrg1 b5r 2 crg1 b6r 2 g 1 b7rcr 2 g1 b8r 2 cr 2 g 10

where rc P*c=Pe; rg Pg=Pe; bo 5:4497;

b1 26:7759; b2 1:4952; b3 2:3116; b4 20:590;

b5 0:018105; b6 20:03786; b7 0:012983; b8

20:000812145:

Eqs. (7) and (10) are the two empirical correlations for an R141b ejector’s performance. A comparison between the measured and the calculation using the correlation for the five tested ejectors is presented in Fig. 10. The errors are all within ^ 10%.

The above empirical correlations can also be extrapolated to predict the performance of ejector with different nozzles. We tested another 10 different ejectors, which were assembled from two different nozzles (Nozzle D and Nozzle E) and five constant-area mixing chambers (A, B, C, D, G). The tests results are compared with the calculations using the empirical correlations. Fig. 11 shows that the errors are all within ^ 10%. This further verifies the empirical corre-lations for the performance prediction of ejectors using R141b as the working fluid.

4. Application of empirical correlations in ejector design The two empirical correlations for R141b ejector perfor-mance, Eqs. (7) and (10), are very useful in the design application. The procedure of ejector design analysis is as follows:

1. Given the operating conditions: Pg; Tg; Pe; Te; P*c; T*c: 2. Determine the ejector area ratio A3=Atfrom Eq. (10). 3. Determine the hypothetical area ratio Ae=Atfrom Eq. (7). 4. Compute the mass flow rate per unit area Gy _ms=Ae through the hypothetical throat of the entrained flow Fig. 11. Comparison of measured and calculatedvfrom extrapolation of the empirical correlations.

(10)

correlations can result in a prediction error , ^ 10% However, it should be noted that the performance of an ejector is also affected by the quality of hardware machin-ing, such as finishmachin-ing, interior surface roughness, centerline alignment, and the material used. The empirical correlations were derived from the test results of 15 ejectors with good quality in machining.

The ejector area ratio depends on the ratio of back pres-sure to entrained prespres-sure P*c=Pe (compression ratio) and also the refrigerant used. The area ratios of the ejectors used in the present experiments range from 6.4 to 9.4 which is adequate for 141b only. As can be seen from Fig. 9, the compression ratio decreases with increasing ejector area ratio. For the area ratio ranging from 6.4 to 9.4, the compres-sion ratio is in the range of 1.5–3.5, which is the operating range of air-conditioning systems.

Superheating of the primary flow may improve the ejector performance, as was noted by several researchers. However, we did not superheat the primary flow and still obtained a high entrainment ratio. This is due to the fact that R141b has a positive-slope saturated-vapor line in the thermodynamic

T–s diagram and the vapor will not condense during an

isentropic expansion in the ejector.

For example, for Ejector A-D, the measured entrain-ment ratio v < 0:54 at Pg 0:465 MPa 848C, P*c

0:087 MPa 288C and Pe 0:040 MPa 88C as shown

in Fig. 10. For Pg 0:538 MPa 908C, P*c

0:101 MPa 328C and Pe 0:040 MPa 88C, the

measured entrainment ratiov < 0:45:

This also indicates that R141b is a very good working fluid for an ejector as was found by Dorantes and Lallemand [6]. Although R123 has a similar boiling point to R141b, the performance of an R123 ejector is worse according to theo-retical calculation [7].

The application of a refrigerant ejector in refrigeration and air-conditioning is not as successful as a steam or air ejector because of its poor performance compared with that of a conventional device. However, we have shown that the ejector performance can be further improved through care-ful design and good manufacturing technique. Further

^ 10% error. A method of calculation for the ejector design

using the present correlations is also presented. In addition, R141b is shown to be a good working fluid for an ejector. The measuredv for the ejectors used in the present study can reach as high as 0.54 at Pg 0:465 MPa 848C, P*c

0:087 MPa 288C and Pe 0:040 MPa 88C: For

Pg 0:538 MPa 908C, P*c 0:101 MPa 328C and

Pe 0:040 MPa 88C;v reaches 0.45.

Acknowledgements

The present study was supported by the National Science Council, ROC, Taiwan, through Grant No. NSC86-2811-E-002-004R.

References

[1] Huang BJ, Jiang CB, Fu FL. Ejector performance characteris-tics and design analysis of jet refrigeration system. ASME J. Engng Gas Turbines and Power 1985;107:792–802. [2] Sun D-W. Recent developments in the design theories and

applications of ejectors—a review. J. Inst. Energy 1995;68:65–79.

[3] Sun D-W. Variable geometry ejectors and their applications in ejector refrigeration systems. Energy 1996;21:919–929. [4] Sun D-W. Experimental investigation of the performance

char-acteristics of a steam jet refrigeration system. Energy Sources 1997;19:349–367.

[5] Munday JT, Bagster DF. A new ejector theory applied to steam jet refrigeration. Ind. Engng Chem. Process Des. Dev. 1977;16(4):442–449.

[6] Dorantes R, Lallemand A. Prediction of performance of a jet cooling system operating with pure refrigerants or non-azeo-tropic mixtures. Int. J. Refrig. 1995;18(1):21–30.

[7] Sun D-W, Emes IW. Performance characteristics of HCFC-123 ejector refrigeration cycles. Int. J. Energy Res. 1996;20:871– 885.

[8] Huang BJ, Petrenko VA, Chang JM, Zhuk KB. A high-perfor-mance solar ejector cooling system. Proceedings ISES 1997 Solar World Congress. Tajeon, Korea, 24–30 August 1997, Vol. 4, pp. 274–277.