STABILTY AND PERFORMANCE OF A LOW SPEED COMPRESSOR WITH MODIFIED CASING

Fayez M. Wassef, Ahmed S. Hassan, Hany A. Mohamed, and Mohamed A. ZakiMechanical Engineering Department, Assiut, University, Assiut, Egypt.

Abstract:- The essential of the present wok is to increase the limit of stability and improve the performance of the low speed compressor utilizing modified casing. Two schemes were used for the modification of the casing in the vaneless region at the shroud side. The first, circumferential groove with different depths was made and the second, ring with different thickness was fitted. The experiments were carried out on a low speed compressor with the modified cases at different operating conditions. The experimental data were processed using the Fast Fourier Transformation analysis (FFT) to estimate the Power Spectrum Density (PSD) for detecting the initiation of rotating stall and surge. The pressure distribution at inlet and exit of a diffuser passage were obtained. The speed and direction of the stall cell rotation relative to the impeller speed beside the number of the stall cells were investigated. The present measurements show that the inception of unsteady flow appears at vaneless zone between the impeller exit and diffuser vane leading edge. It is concluded that modified casing by one way of the two presented schemes can be used to increase the limit of stability for low speed compressor at different operating conditions.

Keywords: Stall and surge detection, Pressure transducers, Time variation of static pressure, Power Spectrum Density.

1 IntroductionCompression systems are widely used in engineering applications, such as turbojet engines, air conditioning systems; industrial compressors. The function of compressors in these systems is to increase the pressure of the fluid and they are designed to operate in steady axisymmetric flow. As one decreases the mass flow rate through the turbomachine, the pressure rise increases and this improves the performance. However, below a critical value of mass flow rate, the steady design flow is no longer stable and oscillations occur, whose main types can be classified as (deep) surge, rotating stall and classic surge. Roughly speaking, surge is a large-amplitude, low frequency (compared to rotating stall) oscillation of the total, annulus-averaged mass flow rate and may be encountered in compressor systems. The vibration can be so severe that the flow reverses for a part of the period, which is often called deep surge. Rotating stall is an abrupt fall of pressure rise below a critical mass flow rate associated with the rotation of one or several stalled blade passages around the annulus. In this case, one finds that the annulus-averaged mass flow rate remains constant in time. Classic surge is a combination of surge and rotating stall, when during a surge cycle, the compressor passes in and out of rotating stall. All of these instabilities limit the operation severely and may lead to large penalties in performance. Hence, there has been an extensive search for a low-order model capable of describing the essential of the dynamics in order to gain a parametric understanding of how to avoid these instabilities [1-11]. Research into the prediction and control of rotating stall and its detrimental effects has been pursued for many years. Among these, casing treatment and active control techniques are noteworthy. Casing treatments have been popular since long time, as reported by [12-19]. The idea of extending the compressor operating range through the use of the casing treatment techniques have been experiments studied by [20- 23]. Most of them require complicated mechanisms and utilize additional machinery that eventually decreases the overall efficiency and reliability. Recently, shallow grooves mounted on a casing wall or diffuser wall parallel to the pressure gradient called J-grooves treatment are proposed by [24-25].The above-mentioned works show that the casing treatment has limitations such as an increase in stall or stability margin up to only a few percent and insufficient to affect the entire flow range.

Therefore, more casing treatment techniques in the centrifugal compressors are still needed. In the present work, circumferential groove or protrude or combined of the both through the vaneless region are proposed as new casing treatments for the centrifugal compressor. The stall and surge margins, limit of stability based on both the stall and surge triggering and pressure coefficient with the flow coefficient are experimentally studied. The flow angles, that are representing the stall initiation for the original compressor, are studied theoretically and experimentally. Both the theoretical and the experimental results are compared with those experimentally obtained by another author. 2. Experimental Setup and Test Facility Figure 1 shows the schematic drawing and the photographic picture of the open loop centrifugal compressor test facility used in the experimental work. The compressor is released from an actual aircraft turbocharger (Allis Chalmers type, AN D 132, licensed under general electric company of serial No. 57153 CH, A.F. model 7S-B22-A6, made in USA). A 3.7 kW variable speeds AC motor with speed up to 6000 rpm is coupled with the compressor instead of turbocharger turbine. The motor speed is measured with accuracy of ±10 rpm by using available tachometer. This compressor is constructed from radial blade rotor, parabolic vanes diffuser and volute casing. The compressor draws air at atmospheric conditions and discharges into a large tank followed by an orifice flow meter and control valve for measuring and controlling the amount of the mass flow rate.

Fig.1 Compressor test facility

Conventional pressure and temperature tapes are located through the compressor flow system for securing the overall compressor performance characteristics. Other wall pressure taps are jointed into the front casing of the compressor as shown in Fig.2. The taps points are located as follow:points 1 – 9: along the impeller passage, points 10 – 17: in the vaneless region at R = 1.1,points 18 – 26: along the diffuser passages at

different three radial sections, andpoints 27 – 39: into the diffuser exit.

Fig.2 Locations of pressure taps on the compressor casing

Three high sensitivity semiconductor-type pressure transducers of omega type, PX-236-100GV silicon diaphragm with full bridge are jointed to the compressor casing. Two transducers are jointed into the vaneless region with 90o shifted through the circumference at R = 1.1 and the third is jointed into the diffuser exit at R = 1.6. A DC amplifier (SENSOTEC'S SA-BII) receives the output signals from the pressure transducers and provides a 16-bit A/D converter board (multi-function acquisition card) supported into PC-SCOPE software for simultaneously pressure for one second at a rate of 1 kHz. The board is supported by PC-SCOPE software, which turns the computer to oscilloscope and stores the pressure waveforms in ASCII file. Subsequently the data in the file were processed using the Fast Fourier Transformation Analysis (FFT) to estimate the Power Spectrum Density (PSD) by Welch’s averaged, modified periodogram method for discrete-time signal vector. The

arrangements of the pressure signals from the pressure transducers and the recording system, which were used for studying the steady and the unsteady flow phenomena are shown in Fig. 3.

Fig.3 Pressure recording system

2.1 Rotating Stall and Surge DeterminationThe surge in centrifugal compressor commonly determine from the experimental as known from the pervious works [3-5 and 11] at amplitude and frequency of pressure coefficient wave higher than ±30% from the mean value and less than 10 Hz at maximum power spectra density (PSD) respectively. Also, the stall commonly determine from the experimental as known from the pervious works [3-6 and 11] at amplitude and frequency of pressure coefficient wave approach from ±30% relative to the mean value and ranged from 10 to 50 Hz at maximum PSD respectively. Two pressure waves should be recorded simultaneously at same radius with different circumferential positions in between to determine number of the stall cells. However, at surge phenomena, the pressure fluctuations recorded at the two positions have not any phase shift meaning one-dimensional fluctuation in flow rate as concluded by [3-5].The number of stall cells, , and the propagation speed of the stall cells, s, are evaluated using the following equations: = , and

where is the period of rotating stall cells, is the time difference between signals, is the angular gap between sensors, and is angular velocity of impeller.

One experiment test is chosen to show the surge phenomena at = 0.068 and = 1.343 as in Fig. 4. It is clearly shown that the amplitude of the pressure fluctuation exceeded ±30% of the pressure coefficient and low frequency of 5 Hz at maximum PSD. It is also shown that the pressure oscillations occurred simultaneously in the circumferential direction as recorded from two couple of sensors mounted at the exit of impeller 90-degree apart peripherally from each other. Thus, the compressor is run in surge at this flow condition of = 0.068 with one-dimensional fluctuation in flow rate.

Fig. 4 Time variation of static pressure and PSD at

flow coefficient =0.068 One experiment test is also chosen to show the rotating stall phenomena at = 0.072 and = 1.325 as in Fig. 5. It is clearly shown that the

amplitude of the pressure fluctuations exceeded ±30% of the pressure coefficient with a phase shift as recorded from two couple of sensors mounted at the exit of impeller 90-degree apart peripherally from each other. At maximum values of PSD, it is found from the analysis of the signals achieved from the two sensors that frequencies of about 4, 17, 32 and 37 Hz from the first sensor and 17 Hz from the second through one-second interval are obtained. These results meaning a presence of rotating stall and surge in this operating condition. To explain this complicated flow phenomena, a reverse flow would takes place through some positions in vaneless region causing the surge and stall cells triggering at certain positions in vaneless region. According to the above-mentioned equations, the number of stall cells is three and the propagation speed of the stall cells is 26%.

Fig.5 Fluctuation of pressure coefficient and Power spectrum at two measured points of impeller exit at

=0.072 with presence of stall and surge

2.2 Experimental Schemes:The experiments works were carried out using the original compressor without any modification in all the compressor parts (Fig. 6-a) and then the compressor casing was treated for next tests as follow:

i. A circumferential groove was machined in the compressor casing through the vaneless region as shown in Fig. 6-b.

ii. The experiments were done with circumferential groove of three different depths, tg, relative to the impeller exit width, b2, of Tg = tg/b2 = 0.05, 0.12, 0.2 at groove radial height, hg, relative to the impeller exit radius, r2, of Hg = hg/r2 = 0.2. The groove radial height is measured from the impeller exit edge through the vaneless region.

iii. The experiments were done with circumferential groove of three different groove radial height relative to the impeller exit radius of Hg = 0.065, 0.13, 0.2 at groove depth relative to the impeller exit width of Tg = 0.2.

iv. A circumferential protrude with a ring form was fixed on the compressor casing through the vaneless region as shown in Fig. 6-c.

v. The experiments were done with circumferential protrudes of a three different depths, tp, relative to the impeller exit width, b2 of Tp = tp/b2 = 0.04, 0.08, 0.14 at protrude radial height, hp, relative to the impeller exit radius, r2, of Hp = hp/r2 = 0.2. The radial height of protrude is measured from the impeller exit edge through the vaneless region.

vi. The experiments were done with circumferential protrude of radial height relative to the impeller exit radius of Hp = 0.13 and 0.2 at protrude depth relative to the impeller exit width of Tp = 0.14.

vii. A combined of circumferential groove and protrude was made on the compressor casing through the vaneless region. The circumferential groove was machined at the impeller exit edge then the circumferential protrude with a ring form was fixed radial upper it as shown in Fig. 6-d. The experiments were done with this combination at Hg = 0.13, 0.06, and Hp = 0.13, 0.06 at Tg = 0.2, Tp= 0.14.

Fig.6 Original and different modifications in thetested compressor casing

2.3 Uncertainty AnalysisMeasurements of rotational speed were made with a tachometer with maximum uncertainty of ±10 rpm that covers full scale of 3500 to 4000 rpm. Individual temperature measurements were made with mercury thermometers with uncertainty of ± 0.5 C. Measurement of the air flow rate in kg/s, , was made using a calibrated orifice meter of correlated equation

where pup and pdown are the pressures at the orifice meter upstream and downstream in Pa with maximum uncertainty of ±10 Pa and T1 is the air temperature at orifice meter upstream. The pressure transducers combined with the amplifier and PC used for measuring the rest static pressures were calibrated using a dead weight gage with maximum uncertainty of ±15 Pa.

Following method presented by Moffat [17], then the maximum uncertainty of the flow coefficient , the pressure coefficient and the flow angle through this study were calculated as ± 0.0015, ± 0.0094 and ± 0.85o respectively, which in

percentage values will be ± 1.6 %, ±1.25 % and 5 % respectively.

3 Theoretical work

The tangential velocity component at impeller exit is usually decreased with the decrease in flow coefficient. The rotating stall in the vaneless region generally occurs in the low flow rate where is much smaller than (referring to Fig. 7)

meaning small flow angle r .

In the vaneless region of a vaned diffuser, the radially outward positive pressure gradient exists due to the centrifugal force. The main flow processes a strong swirl component and, therefore, large angular momentum when the rotating stall occurs. Assuming << in the vaneless region at constant discharge i.e.

. Thus, the equation of angular momentum balance at can be written as:

(1)

Where, bar and double bar denoted the area average (over z = 0 to z = b) and the mass average value, respectively. The tangential component of the shearing stress u in Eq. (1) can be approximately express, as in [25] as follows by assuming the flow distortion parameter .

(2)

where Re is the Reynolds number, , and is the kinematics viscosity of the of air. According to this approximately expression of the tangential component of the shearing stress u , Eq. (1) becomes

(2)

Then

Since the flow angle, r through the vaneless region can be obtained in a simple form in terms of the flow angle at impeller exit 2 as:

(3)

Fig. 7 Flow through the vaneless region

2. Theoretical and Experimental ValidationsThe validations in this section make use the results obtained for the tested compressor without any casing modification at R = r/r2 = 1.1. Both the experimental and theoretical results for flow angle r through the vaneless region at different mean flow ratio compared with the experimental results obtained by [25] are present in Fig. 8. It is clearly shown a good agreement between the present experimental results with those obtained by [25]. In addition, the trend of the theoretical results agrees with the excremental values. Although the theoretical values less than the corresponding experimental results with maximum deviation of about 15%, it can be accepted to predict the flow angle through the vaneless region at different mean flow ratio. This gives a confidence in both the present experimental and theoretical results. A linear decrease in the flow angle through the vaneless region with decrease the mean flow rate can be observed from Fig. 8. The surge at R = 1.1 is occurred at zero flow angle although it may be occurred at small mean flow ratio in practical.

The fluctuations of pressure coefficient () at R = 1.1 obtained from the

present work at flow coefficient of = 0.068 and from [25] at = 0.05 are presented in Figs 9(a) and 9(b) respectively. It is shown that the waveform of the oscillation recorded from both the present and the previous [25] works have approximately similar triangular shape. It can be observed that the average value of the fluctuation for both works equals 0.6 although they have different amplitudes. The difference in the amplitude values of the pressure coefficient would be attributed to the differences on the geometric and performance characteristics of the two compressors. The difference in the number of cycles per second recorded from the two works as shown from Figs.

9(a) and 9(b) is induced from difference impeller tested speeds. It can be observe that the above comparisons give good validation for the present experiments work.

Fig.8. Comparisons between the experimental and

theoretical results for the original tested compressor with the experimental results obtained

by [25].

(a) (b)Fig. 9 Fluctuations of pressure coefficient at low

flow rates of the present test compressor and previous work of [25].

5 Experimental Results and Discussions5.1 Limit of stable operationFigures 10-a to 10-d show the fluctuations of pressure coefficient and power spectrum density (PSD) for the original compressor at different flow conditions; = 0.305, 0.128, 0.121 and 0.108. These flow conditions were selected (from 25 flow conditions) to describe the flow during its progressing from maximum flow or fully open throttle ( = 0.305) to the end of flow stability or initiation of stall and triggering of surge ( = 0.108). At flow rate coefficient, = 0.305, Fig.10-a,

where the compressor runs at its maximum flow rate or fully open throttle, the amplitude of the pressure fluctuations coefficient is very small at the impeller exit. The frequency at this operating point is very high and the PSD of the pressure coefficient is very small leading to steady state compressor operation.

At the operating points between the maximum flow rate, = 0.305 to just before = 0.128, the compressor shows relatively simple fluctuations. The amplitude of fluctuations of pressure coefficient increases with decrease the flow rate.

At flow coefficient = 0.128, as shown in Fig.10-b, the impeller exit station shows unstable

operation, the fluctuation amplitude of pressure at the diffuser inlet increases with relatively low frequency, 30 Hz, due to beginning of the rotating stall appearance at the vaneless region.

At flow coefficient of = 0.121 as shown in Fig. 10-c, the instability at the diffuser inlet occurred with two main predominate frequencies of 15 Hz and 32 Hz and full span stall may be occurred. Since, the amplitude of pressure fluctuation reached about 20% of compressor maximum pressure coefficient while the frequencies, 15 Hz and 32 Hz, lies in the range of stall frequency. At this operating point also, there are low frequencies of 4 to 10 Hz, which lay in the range of surge frequencies, at low amplitudes of PSD. Therefore, the present flow situation can be classified as a rotating stall and triggering of surge.

At flow condition of = 0.108, high amplitude of pressure fluctuation at the diffuser inlet, exceeded 40% of compressor pressure coefficient, as shown in Fig.10-d. The frequency of these pressure fluctuations correspond to different predominant of 4 Hz and higher. Thus, the compressor operation at this condition can be categorized as deep surge, as obtained by [5].

( a )

( b )

( c )

( d )

Fig.10 Time variation of pressure coefficient for the compressor without casing treatments for

different flow conditionsThe remarkable effect of casing treatment is shown in Fig. 11. This figure shows time variation of pressure coefficient at different flow rates of the modified compressor (circumferential groove of g = 0.12 and Hg = 0.2), as shown in the right hand side, compared with original compressor, as shown in the left hand side. The left hand side of the figure shows large amplitude of pressure fluctuations in ranges of stall and surge at the different flow conditions. Small amplitudes of pressure fluctuations on the right hand side of the figure were observed at the different operating points. An increase in the stable flow limit of about 10% of maximum flow rate could be obtained using this modification. This indicate that using circumferential groove of g = 0.12 and Hg = 0.2 as a casing treatment can be suppressed stall and surge and increases the compressor stable flow operation.

Fig.11 suppression of stall and surge

5.2Compressor performance characteristics with different casing treatments The characteristics curves of the tested compressor with and without casing treatments are presented in Figs. 12 to 14. The ordinates are the pressure coefficient, and the abscissa is the flow coefficient, . According to the experimental results as explained in the above section (5.1), the stall margin at which rotating stall starts in appearance, are indicated in the figures.

Figure 12 shows the effect of different circumferential groove and protrude depths at constant height ratio of Hg = Hp = 0.2 on the compressor performance characteristics. The performance characteristics for the original compressor without any casing treatment are also presented in Fig. 12. It is shown that at groove of Tg = 0.2 the compressor gives noticeable increasing in the stall margin and decrease in compressor pressure coefficient at low flow rate. While increasing the protrude thickness increases the compressor pressure coefficient and protrude of Tp

= 0.14 gives the highest-pressure coefficient while it gives low-pressure coefficient at maximum flow rates. It is clear that the stall margin increases by increasing the groove or protrude depth. The treatment causes this enhancement in the stall margin may be due to retard the flow separations, which would be induced through the vaneless region at stall initiation. It is clearly shown that using protrude of Hp = 0.2 and Tp = 0.04 gives the best suitable performance compared to the original performance of the compressor where the large stall margin, highest-pressure coefficient at low flow rates and slightly low-pressure coefficient at maximum flow.

0.6

0.8

1

1.2

1.4

1.6

0 0.08 0.16 0.24 0.32Flow coefficient,

Pres

sure

coe

ffici

ent,

OriginalTp = 0.04Tp = 0.08Tp = 0.14Initiation of stall

Hp = 0.2

0.6

0.8

1

1.2

1.4

0 0.08 0.16 0.24 0.32Flow coefficient,

Pres

sure

coe

ffici

ent,

OriginalTg = 0.05Tg = 0.12Tg = 0.2Initiation of stall

Hg = 0.2

Fig.12 Performance of the original and modified compressors at Hg =Hp = 0.2.

Figure 13 shows effects of the radial heights of the circumferential groove and protrude on the compressor characteristics at constant depth. It is clearly shown that increasing the groove height at constant depth achieves a decrease in the pressure coefficient at low flow rate and increases it at higher flow rate. It is clearly shown that the stall margin increases by increasing the groove or protrude depth. This enhancement may be due to retard the flow separations, which would be induced through the vaneless region at stall initiation. It can be observed that increase the depth and height ratios of protrude improve the stall margin leading to useful recommendation should be considered when a low flow rate is required.

Also, effects of combination of the circumferential grooves and protrudes together on the compressor characteristics at Tg = 0.2 and Tp = 0.14 are shown in Fig. 14. At Hg = 0.13 and Hp=

0.06, it is clearly shown that there are a decrease in the pressure coefficient at low flow rate and no improving in the stall margin. On the other hand, modifying the casing with heights of groove and protrude Hg = 0.06 and Hp= 0.13 increases the stall margin while slightly increase the pressure coefficient at low flow rate and decrease it at higher flow rate. The enhancement obtained in this case may be due to restrict the flow separations, which would be induced through the vaneless region at stall initiation.

0.6

0.9

1.2

1.5

0 0.08 0.16 0.24 0.32Flow coefficient,

Pres

sure

coe

ffici

ent,

OriginalHg = 0.065Hg = 0.13Hg = 0.2Initiation of stall

0.6

0.8

1

1.2

1.4

1.6

0 0.08 0.16 0.24 0.32Flow coefficient,

Pres

sure

coe

ffici

ent,

OriginalHp =0.13Hp =0.2Initiation of stall

Tg = 0.2 Tp = 0.14

Fig.13 Performance of the original and modified

compressors at Tg= 0.2 and Tp= 0.14.

0.6

0.9

1.2

1.5

0 0.08 0.16 0.24 0.32Flow coefficient,

Pres

sure

coe

ffici

ent,

OriginalHg=0.13, Hp=0.06Hg=0.06, Hp=0.13Initiation of stall

Tg = 0.2, Tp =0.14

Fig.14 Effect of combined groove and protrude

height on compressor performanceThe improvements of stall margin, surge margin, maximum pressure coefficients and range of stable operation based on stall and surge appearance due to the casing modification for the results set in Figs. 12 to 14 are presented in Figs. 15 to 17.Figures 15 shows that the improvements in stall and surge margins are increased by increasing the groove or protrude depth ratios. Figure 16 shows that the improvements in stall are increased by increasing the groove or protrude depth ratio. While the improvements in surge margins are increased by increasing the groove depth ratio and its have a peak value by increasing the protrude height ratio. These improvements contracted with decrease the pressure coefficient in groove treatment. The present work at protrude modification achieves of about 12% improvement in stall margin at Hp = 0.13 and Tp = 0.14. On the other hand, increasing the groove depth ratios lead to highest improvements in stable operation based on stall and surge appearance as indicated by stable limit in Fig. 15. While optimum depth and height ratios of protrude is required for the best

improvement in case the protrude modification. Improvement of about 55% and 39% in the stability limits based on stall and surge occurrences are achieved using groove modification at Hp = 0.2 and Tp = 0.2. Nevertheless, according to the above figures (12 and 13), this modification gives low-pressure coefficient at low flow rates. Thus, modification utilizing protrude is recommended for using the centrifugal compressor at low flow rate and high-pressure ratio. However, modification-utilizing groove is recommended for using the centrifugal compressor when low-pressure ratio and high flow rate are required.

Fig.15 Improvements on compressor performance

-20

0

20

40

60

0 0.07 0.14 0.21Groove height ratio, Hg

Impr

ovem

ent,

%

Stall marginSurge marginPressure coeff.Stab. lim it, stall Stab. lim it, surge

Tg = 0.2

0

10

20

30

0 0.07 0.14 0.21Protrude height ratio, Hp

Impr

ovem

ent,

%

Stall marginSurge marginPressure coeff.Stab. limit, stall Stab. limit, surge

Tp = 0.14

Fig.16 Improvements on compressor performance

Modification utilizing combination of groove and protrude at Tg = 0.2, Tp = 0.14, Hg = 0.06 and Hp= 0.13, achieves improvements in stall margin, surge margin, maximum pressure coefficients and range of stable operation based on stall and surge occurrences of about 28%, 22%, 4%, 39% and 22% respectively as shown in Fig. 17. The enhancement obtained in this case may be due to restrict the flow separations, which would be induced through the vaneless region at stall initiation. According to Fig. 17, noticeable improvements would be obtained by the combination treatment when Hp >> Hg for equal groove and protrude depths.

Fig.17 Improvements on compressor performance

6 ConclusionsIn a low speed centrifugal compressor, variation of static pressure were measured using three pressure sensors with high frequency response (mounted on the compressor casing: two at the mid space between the rotor exit (R=1.1), 90 degree apart in circumferential direction and the third at the diffuser exit) at the steady and unsteady conditions. The measured pressures are discussed in the relation to stall and /or surge using power spectrum analysis. Two types of typical vaneless region rotating stall were detected, one with three cells at relatively higher mass flow rate, they rotated at 26% relative to rotor speed and the other with one stall cell at lower flow rate near surge point with speed of 78% relative to rotor speed. The rotating stall occurs at relatively high flow rate and is very complicated phenomenon in three dimensional flow fluctuations, while the compressor surge occurs at low flow rate with one-dimensional fluctuation. The measured is quite evident and agrees that the underlying cause of surge is aerodynamic stall.In order to control and suppress rotating stall and surge in the present compressor, passive method of utilized groove and protrudes were manufactured with the compressor front casing and their effects on compressor performance are experimentally studied. The results show radial groove of 0.2 relative the diffuser width (31 mm) can suppress rotting stall and surge and increases the limits of stability. The effect of circumferential groove on suppressing rotating stall is theoretically investigated. The mechanism of suppressing rotating stall by the circumferential groove is analyzed theoretically. Making protrude at the vaneless area with specified dimensions can suppress surge for entire compressor flow range. In addition, the remarkable effect of the circumferential groove is to decrease the tangential velocity of the main flow. The present theory is seen to give good prediction and is more accurate

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