Peter Newton Department of Mechanical Engineering, Imperial College, London SW7 2BX, UK e-mail:
[email protected]
Alessandro Romagnoli Department of Mechanical Engineering, Imperial College, London SW7 2BX, UK e-mail:
[email protected]
Ricardo Martinez-Botas Department of Mechanical Engineering, Imperial College, London SW7 2BX, UK e-mail:
[email protected]
Colin Copeland Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK e-mail:
[email protected]
Martin Seiler ABB Turbo Systems, Baden 5401, Switzerland e-mail:
[email protected]
1
A Method of Map Extrapolation for Unequal and Partial Admission in a Double Entry Turbine This paper presents a method for prediction of the unequal admission performance of a double entry turbine based on the full admission turbine maps and a minimal number of unequal admission points. The double entry turbine has two separate inlet ports which feed a single turbine wheel: this arrangement can be beneficial in a turbocharger application; however the additional entry does add complexity in producing a complete turbine map which includes unequal admission behavior. When a double entry turbine is operated under full admission conditions, with both entries feeding the turbine equally, this will act effectively as a single entry device and the turbine performance can be represented by a standard turbine map. In reality a multiple entry turbine will spend the majority of time operating under varying degrees of unequal admission, with each entry feeding the turbine different amounts; the extent of this inequality can have a considerable impact on turbine performance. In order to produce a full map which extends from full admission through to the partial admission case (where one inlet has no flow) a large number of unequal admission data points are required. The paper starts by discussing previous attempts to describe the partial and unequal admission performance of a double entry turbine. The full unequal admission performance is then presented for a nozzled, double entry turbine. The impact of unequal admission on turbine performance is demonstrated. Under some conditions of operation, the turbine efficiency may be less than half that of the equivalent full admission case based on the average turbine velocity ratio. A method of using the steady, equal admission maps, with a limited number of unequal admission data points, to predict the full unequal admission behavior is presented. A good agreement is found when the map extension method is validated against the full unequal admission turbine performance measured on a test stand. In the prediction of efficiency a mean error of approximately 0.39% is found between the test stand data and the proposed extrapolation method, with a standard deviation of 2.79%. A better agreement is generally found at conditions of higher power. [DOI: 10.1115/1.4025763]
Introduction
The past five decades have seen a marked increase in the use of turbocharging technology applied to the internal combustion engine. This is not surprising considering the relatively low cost of these devices and the substantial benefits they can offer in terms of engine performance and efficiency. One of the main problems with this technology is in correctly matching the turbocharger to the internal combustion engine which can have a considerable impact on the benefits gained. This problem is inherent when trying to connect a rotor dynamic device, which is designed to accept a steady flow of working fluid, to a positive displacement internal combustion engine which produces a pulsating exhaust flow used to drive the turbine. To this end, it is not just the performance of the turbocharger that is of concern but also of its impact on the performance of the combustion engine. The application of a turbine in the exhaust flow of a reciprocating engine will have an inevitable influence on the gas wave dynamics in the exhaust manifold. In a conventional single entry device all of the exhaust pipes converge to a single turbine entry allowing mixing between the different exhaust flows, which, depending upon the valve timing, can have a negative impact on the scavenging and blow-down characteristics of the combustion engine [1]. In order to lessen this effect the double entry turbine has two inlets which remain separated, each feeding 180 deg of Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received August 28, 2013; final manuscript received September 2, 2013; published online November 28, 2013. Editor: Ronald Bunker.
Journal of Turbomachinery
the rotor wheel, Fig. 1 shows the double entry volute used in this study, the two inlets have been designated as inner and outer entries. This volute arrangement offers a twofold benefit in terms of turbocharger performance: firstly, with the engine cylinders split into two banks, each feeding a separate volute entry, it is easier to avoid exhaust pulse overlap, which could otherwise have a detrimental effect on the engine performance. Secondly, with less chance for mixing, more of the pulsating exhaust flow energy will be preserved so that this is available at the turbine wheel [1]. Clearly, the double entry device displays tangible benefits over a conventional single entry turbine in the turbocharging application: however, the addition of a second inlet to the volute also brings extra complexity in determining the steady state turbine performance. When both volute entries are feeding the turbine equally (the full admission condition) the turbine will act in the same manner as a single entry device. In this case determining the turbine performance from a conventional turbine map is straightforward. In reality the exhaust pulses feeding each limb of the turbine will be timed so that they are out of phase with each other; as a result the turbocharger turbine will rarely act in full admission and will spend the majority of time with unequal flows driving each 180 deg sector of the turbine wheel, in the extreme one limb may be flowing while the other has no flow, a condition known as partial admission. Copeland et al. [2–4] and Copeland [5] conducted the most complete experimental study of double entry performance, characterizing the whole steady state operating envelope of the turbine from full to partial admission. When the operating point of the turbine was fixed in terms of average velocity ratio they found a
C 2014 by ASME Copyright V
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Fig. 1
The double entry turbine
significant efficiency penalty as the flow inequality was increased across the two entries, this effect was more pronounced towards the lower power end of turbine operation. They also discussed the effect of flow inequality on the mass flow characteristics of the turbine. If the two scroll entries were kept completely separate then you may expect that the mass flow characteristics of one limb would be independent of the other: Copeland [5] found a definite interaction between the two. If the pressure ratio across one of the limbs was fixed while the pressure ratio across the adjacent limb was decreased then the mass flow though the fixed pressure limb was found to increase, conversely, increasing the pressure ratio in the adjacent limb was found to restrict the mass flow through the constant pressure side. These findings, in terms of both mass flow and efficiency, echoed those of previous researchers [6–8]. More recently, Copeland et al. [4] used a three-dimensional computational model to investigate the flow structure within an unequally admitted double entry turbine. This model captured the significant interaction between the two limbs which was previously established experimentally, this interaction was found to originate in the interspace region between the nozzle row exit and the rotor wheel. The model showed how the gas from the high pressure entry will over expand into this area, effectively throttling the flow from the low pressure limb. This region was also subject to significant mixing where the higher velocity flow from one limb met the lower velocity flow from the adjacent limb leading to considerable dissipation. Newton et al. [9] subsequently carried out a further, more thorough computational study of the losses in a partially admitted double entry turbine. Here they saw that the loss generation in the rotor wheel through the nonflowing section of the partially admitted turbine was of the same magnitude as that in the flowing section of the rotor. On top of this they found that the generation of vortices in the rotor passage as it traversed the nonflowing region compromised the development of the flow when the passage entered the flowing section of the volute leading to further losses. Because of these considerable effects, knowledge of the unequal admission performance of the turbine is useful in correctly matching an engine to a turbocharger, a process which relies on the steady state turbine maps. Unfortunately the complete unequal admission maps are not always as readily available as the standard (full admission) turbine maps due to the fact that
many gas test stands are not capable of operating in unequal admission. Even when they are, populating a full, unequal admission map requires a large number of data points. Romagnoli et al. [10] started to address this issue, proposing a method of predicting the unequal admission mass flow for both a double and a twin entry volute from the full admission maps with only a few extra data points taken in unequal admission. In this study they found that the twin entry turbine characteristics agreed reasonably well with their approach however, the correlation with the performance of the double entry turbine was less satisfactory. This provides the rationale for the current study which proposes a method of extrapolating the full admission map for a given double entry turbine to obtain the unequal admission performance, of both mass flow and efficiency, given a minimal number of additional data points taken under unequal admission. The study is based entirely upon experimental data taken from the turbocharger test facility at Imperial College London and relates to the performance of a nozzled, double entry, mixed flow turbine. The rotor wheel and volute geometry are the same as those used by Copeland et al. [2–4] and Copeland [5] however, the nozzle throat area is 12.5% larger. The measured turbine performance will be presented, initially to show the full admission performance characteristics and then the effect of unequal admission will be discussed. Following this a method will be presented to allow extrapolation from the full admission turbine maps to the unequal admission performance using three different constants which are obtained through the processing of a minimal number of unequal admission data points.
2
Experimental Setup
The turbocharger testing facility has been described in detail by several researchers [5,11,12], therefore, only a brief overview is given here. The test facility layout is shown in Fig. 2. Air is supplied to the facility by a centrally housed compressor system; this has the capability to deliver up to 1 kg/s of air at 4 barA. The air enters through a single inlet pipe where the flow can be regulated by the master control valve. From here the air goes through a heater stack, which can supply up to 72 kW of heat to the flow. Although the facility is designed, through similitude, to be a cold
Fig. 2 Test facility layout
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flow facility a heater section is needed to allow some control over the gas temperature and also to avoid condensation after the flow has expanded through the turbine. Following the heater section the flow is split into two separate pipes which can be controlled separately to allow unequal flows through each. V-cone type flowmeters are used to measure the mass flow rate through each limb. These are differential pressure type flow-meters and have been shown to allow more accurate measurement of mass flow over a wider operating range than the equivalent orifice plate system which was previously used by Copeland [5]. Downstream of the flow-meters the air traverses through the pulse generator. This is a rotating chopper plate type pulse generator and facilitates unsteady testing, however, since this work was entirely based upon steady operating data the chopper plate could be left in the fully open position. From here there is a guillotine safety valve: this is a spring loaded cut-off valve which is activated if any of the operating parameters go outside of a predefined safety limit. Finally, before the air enters the turbine scroll it passes through a measurement plane where temperature and pressure are measured so that it is possible to calculate the isentropic energy available to the turbine. The turbine is directly connected to a high-speed eddy current dynamometer. This is capable of speeds up to 60 krpm and can dissipate more than 60 kW of shaft power. The dynamometer is gimballed such that the entire turbine loading is reacted against a load cell. This loading includes the losses associated with the turbine bearings and so gives a very true value of the aerodynamic work delivered by the turbine. The shaft speed of the turbine is also measured within the dynamometer; this is done by a digital counter system which measures the time between the pulses of an optical sensor which is interfered by a 10-toothed wheel attached to the rotor shaft. With these two measurements the actual power delivered by the turbine can be calculated and the full turbine performance evaluated. Upon processing the experimental data it was discovered that one of the impulse lines leading to the differential pressure transducer on one of the v-cone flow meters had suffered a small leakage. A correction was applied to amend this fault however the uncertainty in the data is increased accordingly. The 95% confidence interval for uncertainty in mass flow rate ranged from around 61% at the very high mass flow rate conditions to around 610% for some of the very low mass flow rates measured during unequal admission testing. The uncertainty in the measured power produced by the turbine spanned a similar range to the uncertainty in mass flow rate. A more detailed discussion of the experimental uncertainties associated with this facility can be found in Copeland [5].
3
Performance Parameters
Presentation of the performance of a double entry turbine is more complex than for a single entry device. When the double entry turbine is operated in full admission, its performance can be represented by the same means as a single entry device, where each operating variable can be characterized by a single value: this is not the case when the turbine is operated in unequal admission. In this case we must define the operating condition of each volute inlet separately. This issue was discussed in greater depth by Copeland [5], from where we use the same performance parameters to define unequal admission operation; they are repeated here for completeness: PRinner ¼
ðP01 Þouter P4 pffiffiffiffiffiffiffi m_ T01 ¼ P01 inner
PRouter ¼ MPinner
ðP01 Þinner P4
Journal of Turbomachinery
(1) (2) (3)
MPouter ¼
pffiffiffiffiffiffiffi m_ T01 P01 outer
w_ actual w_ is inner þ w_ is outer U2 pd2 N ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cis _ wis inner þ w_ is outer 2 m_ total gts ¼
(4) (5) (6)
Two further parameters, useful for the following analysis, are the mass flow ratio and the effective area. The mass flow ratio defines the degree of unequal admission from 1 in full admission to 0 in partial admission MFR ¼
m_ LP m_ HP
(7)
where m_ LP represents the mass flow rate through the low pressure volute entry and m_ HP that of the mass flow through the high pressure entry, whether that is the inner or outer entry. Also used in the following analysis is the parameter ð1 MFRÞ This is, in one sense, a more intuitive number, it is bounded from 0 in full admission, where there is no effect of flow inequality, to 1 in partial admission where the unequal admission effect is at a maximum. Finally, the effective area is defined as the effective throat area of a fixed, isentropic nozzle such that, for a given expansion ratio, its mass flow will match that of the turbine [1]. The effective area expresses the swallowing capacity of a turbine; its use in turbochargers is particularly relevant as for many simple 1D engine modeling codes it is often easiest to treat the turbine as a simple nozzle [1] Aeff inner ¼
m_ inner qis inner cis inner
(8)
Aeff outer ¼
m_ outer qis outer cis outer
(9)
Although, the effective area can often be treated as a constant it does show some dependency on rotational speed and the expansion ratio.
4
Turbine Performance
4.1 Full Admission Results. Since it is not the principal intent of the paper, a long discussion of the measured turbine performance will not be given. The aim of this section is primarily in demonstrating the effect that unequal admission can have on the turbine performance. Firstly the equal admission performance of the turbine will be presented. Five different speed lines were tested at pseudo nondimensional pffiffiffiffi speeds corresponding to 26.9, 32.3, 37.8, 43.0, and 48.3 rps= K. The efficiency and mass flow parameter values have been normalized by those of the peak effipffiffiffiffi ciency point on the 43:0 rps= K speed line which occurred at an isentropic velocity ratio of 0.648. Figures 3 and 4 show the efficiency and mass flow characteristics, respectively, for the turbine. As discussed in the introduction, the current study was carried out on a mixed flow turbine and the measured trends are typical of such a device. When the efficiency is plotted against the isentropic velocity ratio, the curves for each different speed line almost collapse to a single characteristic. The peak efficiency point for each different speed line is within 5% of the peak recorded efficiency. The peak efficiency velocity ratio varies from 0.58–0.65 for the different speed lines with a mean of 0.621. Determination of the true peak efficiency velocity ratio is not necessarily straightforward due to the flatness of the efficiency curves in this region and also because of the discrete nature of the measured data. Nevertheless, the peak efficiency velocity ratio JUNE 2014, Vol. 136 / 061019-3
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seems to lie below 0.7, this is the value which would be expected for a purely radial inflow turbine. Perhaps the most notable feature of this characteristic is the width of the available data, for the lowest speed line this spans from a velocity ratio of 1.069 to 0.388. This demonstrates the capability of the eddy current dynamometer in coping with a very broad operation of the turbine. The mass flow characteristic shows the typical dependence on speed for a turbine of this type. The mass flow parameter curve shifts to the right as the turbine speed increases due to the increasing centrifugal pressure field opposing the flow that is driving the turbine. The width of the map again reflects the capability of the dynamometer; the data ranges from a total to static pressure ratio of 1.103 to 2.454 which is approaching the choking condition for this turbine geometry.
Fig. 3 Full admission turbine efficiency p characteristic for five ffiffiffiffi different speed lines from 26.9 – 48.3 rps= K
Fig. 4 Full admission mass flow characteristic for the double entry turbine
4.2 Unequal Admission Results. Testing in unequal admission is more complex than that of equal admission since controlling the ratio of mass through each limb adds an extra degree of freedom. Here, the work of Copeland [5] and Copeland et al. [2–4] was useful; they found that the effect of flow inequality was symmetric about the inner and outer scroll entries. This reduced the required data set by a half so that the flow control valve could be kept fully open for one limb and the flow varied on the other limb to achieve different levels of inequality. The results could be mirrored to obtain the unequal admission performance if the flow inequality were reversed. In this series of testing the outer limb control valve was kept fully open while the inner limb control valve was set to different levels of opening. Over 50 unequal admission results were taken at three pffiffiffiffi different speeds, corresponding to 26.9, 37.8 and 43:0 rps= K, this was equivalent to over 100 points when the data were mirrored. Although it is possible to show the full operating characteristics of the turbine under full and unequal admission on a series of 2D charts, this requires many different plots and large amount of paper space. The reason for this is that, in order to define the operating point of a multiple entry turbine, at least three separate parameters are required. Copeland et al. [2] discussed this issue and went on to show a three-dimensional color coded plot in order to present the performance of the turbine, over the full range of equal and unequal operation, on a single plot. On this plot, the expansion ratio across each limb was plotted on two axes while the velocity ratio was on a third. An equivalent plot is shown here in Fig. 5 for the current turbine nozzle configuration. The contours in this plot show the turbine efficiency normalized by the peak efpffiffiffi ffi ficiency for the 43:0 rps= K speed line (at a velocity ratio of 0.648).
Fig. 5 A three-dimensional color coded plot showing the efficiency of the turbine over the full range of equal and unequal operation. One plane is shown at U/CIS – 0.65, corresponding the peak efficiency velocity ratio in equal admission, the other plane dissects the plot through PR inner 5 PR outer.
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Figure 5 shows several aspects of double entry turbine performance both in terms of turbine behavior and also of the difficulty in characterizing it. As expected the region of highest efficiency corresponds to an area along the equal admission axis (where PRinner ¼ PRouter ) at a velocity ratio of around 0.65. As you move from this region the efficiency is observed to decline in all three directions showing its dependence on the three parameters, PRinner , PRouter and U=Cis . The discrete blue markers on the chart show the experimental data points (including those that have been mirrored in unequal admission), this gives an idea of the density of data required to build up the full operating map for the turbine. A linear triangle based method was used to interpolate the turbine performance between these discrete data points. Although the current data set allows a good representation of the turbine performance in unequal admission it is also apparent that the unequal admission data points are sparser than those in equal admission. This demonstrates effectively the challenge of producing a full unequal admission map for a multiple entry turbine based on experimental data; to populate the unequal admission section of the map as densely as the equal admission data points would require a very large experimental effort. A more extensive collection of data would not have been possible within the testing schedule on the current turbine. While the three-dimensional approach of Fig. 5 permits certain qualitative trends to be observed and, perhaps as importantly, offers an impression of the three-dimensional operating envelope covered by a multiple entry turbine, a quantitative analysis is difficult. To address this, the analysis must revert to two-dimensional plots. Because the overall effect of unequal admission was qualitatively similar for each different speed, in p the ffiffiffiffi interest of brevity only a single speed condition of 43:0 rps= K will be discussed here. Figure 6 shows the observed efficiency trends for the turbine in both full and partial admission, again the efficiency data are normalized by the peak efficiency case for full admission. The partial admission data shows a significant efficiency penalty compared to the full admission data. It also shows a movement of the characteristic to the left; the curve has still not reached the peak efficiency point at a velocity ratio of around 0.55 compared to a peak at 0.648 in the full admission case. It is worthwhile to note that the velocity ratio referred to here is the average velocity ratio as calculated by Eq. (6). The full and partial admission conditions, shown in Fig. 6, represent two extremes with varying conditions of unequal admission lying between the two. On an engine the turbocharger may act in partial admission momentarily as the flow in one sector drops to zero. As the flow in this sector rises again during an exhaust pulse, the flow in the adjoining sector may then drop to zero, reversing the partial admission and so it is not uncommon that a double
Fig. 6 Comparison of the full admission efficiency characteristic to that of partial admission, a considerable impact of partial admission operation is observed
Journal of Turbomachinery
Fig. 7 A plot illustrating the effect of unequal admission on the turbine efficiency for several different loading conditions of the turbine corresponding to average velocity ratios of0.55, 065 pffiffiffiffi and 0.8 at a constant speed of 43:0 rps= K
entry turbocharger on a real engine would be subject to the full range of these conditions. Figure 7 shows how the turbine efficiency is affected by increasing levels of unequal admission towards being partially admitted. The abscissa shows (1-MFR) so that flow inequality is increasing from left to right. The ordinate shows the overall isentropic efficiency of the turbine, again normalized by the peak efficiency case in full admission. Three data series are plotted, each one corresponding to a different value of velocity ratio, as calculated by Eq. (6). A downward trend in efficiency is clearly evident with increasing inequality between the two scroll entries. It is also apparent that the effect of flow inequality becomes increasingly severe as the velocity ratio is increased; the partial admission efficiency for a velocity ratio of 0.8 is almost 50% lower than the full admission case, at a velocity ratio of 0.55 the drop in efficiency due to partial admission is less than 25%. In order to demonstrate the effect of unequal admission on massflow, previous researchers have used the idea of an effective area [2–6] as defined in Eqs. (8) and (9). Using this concept Fig. 8 illustrates the extent to which the swallowing capacity of each entry can be influenced by the flow in the other. In this figure the pressure ratio on the abscissa corresponds to that across each individual volute entry and the green circles show the swallowing characteristic of one limb during equal admission; this exhibits a clear relationship with pressure ratio, for the given speed of
Fig. 8 Mass flow characteristics of the turbine under inequal admission, the full admission swallowing capacity of one turbine limb is compared to the swallowing capacity of each limb under inequal admission
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pffiffiffiffi 43:0 rps= K. The other data series (blue diamonds and red squares) show the effective area calculated for each limb under various degrees of unequal admission. In unequal admission the red squares represent the swallowing capacity of the entry which has the higher pressure ratio while the blue diamonds show the effective area calculated for the entry with a lower pressure ratio. It is clear that the effect of unequal admission is to increase the swallowing capacity of the high pressure limb and reduce the swallowing capacity of the low pressure limb. It is also evident that the swallowing characteristics of the high pressure limb remain closer to the characteristics for one entry in equal admission than those for the low pressure limb, which are quite adversely affected during unequal admission. It was found that as the level of unequal admission was increased the mass flow characteristics moved further from the full admission characteristic, this led to a significant dispersion of the data which is evident in this figure. Several of the points for the low pressure limb are seen to sit on the abscissa of Fig. 8, these show the partial admission points where the effective area for the low pressure limb is 0. A positive pressure ratio in this case is still seen due to the centrifugal head of the rotating turbine wheel and due to leakage from the higher pressure limb.
5
Fig. 9 Plot showing a comparison of the extrapolation function for the full admission efficiency characteristic compared to the measured full admission performance
Extrapolation Model
5.1 Full Admission Extrapolation. In order to define a method of predicting the unequal admission performance of a double entry turbine, it is first necessary to define a means of extrapolation and interpolation for the full admission turbine performance. In this work, the method for map extrapolation was based on that of Salim et al. [13]. The method is based on the normalization of all data to the peak efficiency point for each speed line. A separate extrapolation was applied for each different speed line however pffiffiffiffi this section will concentrate only on that for the 43:0 rps= K speed line for brevity since the procedure is the same in each case. The normalized efficiency characteristic will peak at a value of unity on both the efficiency and velocity ratio axes. The following function can then be fitted to the data: U Cisnorm U Cisnorm
A U 1 : gtsnorm ¼ 1 1 Cis norm 2 U > 1 : gtsnorm ¼ 1 B 1 Cis norm
(10)
Fig. 10 Plot showing a comparison of the extrapolation function for the turbine mass flow parameter compared to the measured full admission performance against velocity ratio
(11)
Using this method the mean error in efficiency between the measured experimental data and the extrapolation function was 1.13% with a standard deviation of 2.75%. Figure 9 shows the extrapolation function plotted against the measured experimental data pffiffiffiffi points for the 43:0 rps= K speed line, on this plot the abscissa shows the actual velocity ratio, whereas the ordinate shows the normalized efficiency. Contrary to intuition, which would attempt to describe the mass flow parameter as a function of the expansion ratio, the normalized mass flow parameter was correlated against the velocity ratio. A simple quadratic polynomial function was fitted to this data of the form 2 2 U2 U2 þb þc MPnorm ¼ a Cis norm Cis norm
(12)
The fit to experimental data in this case was better than for efficiency, the mean error was 0.002% with a standard deviation of 0.55%. Figure 10 shows a comparison of the extrapolation function to the measured data for the normalized mass flow parameter plotted against the velocity ratio, Fig. 11 shows the same data for
Fig. 11 Plot showing a comparison of the extrapolation function for the turbine mass flow parameter compared to the measured full admission performance against pressure ratio
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normalized mass flow parameter plotted in the more usual manner, against pressure ratio. 5.2 Unequal Admission. 5.2.1 Mass Flow Characteristic. In order to estimate the mass flow through each entry of the turbine under a condition of unequal admission it is useful to start from a prediction of the mass flow based on the full admission data. It is possible to treat each entry to the turbine as a completely separate machine, neglecting any influence between the two. Using this assumption, for a given turbine speed and (different) expansion ratio across each volute entry, the mass flow through each limb can be calculated as half that of the turbine when operating in full admission over the same expansion ratio and turbine speed. From Sec. 4.2, it was clear that the flow through one limb of the double entry turbine will affect the flow in the other; specifically it was shown that the higher pressure entry will tend to over expand, leading to an increase in mass flow compared to the full admission characteristic while the lower pressure entry will suffer a decrease in swallowing capacity. Considering the opposing effect on the swallowing capacity through each limb it is not surprising that, for a given unequal admission operating condition, when the predicted and measured values of total mass flow rate through both limbs are compared a reasonable agreement is found. Figure 12 shows the agreement between the predicted and measured total mass flow rates through the turbine. This plot shows all of the unequal admission data points taken for all different speed lines however all the values in this figure have again been normalized by those pffiffiffiof ffi the full admission peak efficiency point on the 43:0 rps= K speed line. An overall RMS error was found to be 6.5% however, this figure is somewhat misleading since the errors at lower values of total mass flow, which are of less significance, were found to be much larger than the errors obtained at greater mass flows. Although it is possible to obtain a satisfactory prediction of the total turbine mass flow using only the turbine full admission characteristics, it is clear that using this analysis the calculated mass flow ratio between the two limbs will not be predicted accurately. A reliable estimate of the mass flow ratio is important as this can have a significant bearing on the turbine performance, as shown in Sec. 4.2. Plotting the measured mass flow ratio between the two limbs against that calculated from the full admission data as described above, a relationship is found between the two. Figure 13 shows the mass flow ratio calculated from the full admission turbine map pffiffiffiffiagainst the measured mass flow ratio for a speed of 43:0 rps= K, the data for the other speed lines followed a qualitatively similar trend but are not plotted here.
Fig. 13 A comparison of the mass flow ratio (Eq. (7)) calculated using the full admission turbine characteristics for each limb to that measured on the test facility
If the double entry turbine were to behave like two separate turbines, each with half the full admission mass flow, as described above, then the predicted and measured mass flow ratios would be equal and all the data points would align with the red line, y ¼ x. In reality, the data points in Fig. 13 lie beneath this line showing that the measured mass flow ratio is lower than the calculated mass flow ratio. Several points are seen to lie on the abscissa of this figure; these are partial admission points where the measured mass flow ratio was 0. It is also seen that the calculated mass flow ratio in some cases is negative, suggesting that the measured static pressure in the low pressure limb is below the centrifugal head for 0 mass flow predicted by the extrapolation of the full admission data. In reality, the prediction of mass flow in this region is highly dependent on the pressure ratio (see Fig. 11) and so a small change in pressure ratio can have a large impact on the predicted mass flow leading to these unrealistic results. This, of course, also relies upon the extrapolation of the full admission mass flow characteristic. In Fig. 13, the values of mass flow ratio calculated from the full admission characteristics are seen to follow a predominantly linear relationship with the measured values of mass flow ratio however, an offset is present at both ends of this line. The line intercepts 0 on the measured mass flow ratio (corresponding to partial admission) at a calculated mass flow ratio of around 0.2, while the linear trend crosses a calculated mass flow ratio of unity at a measured mass flow ratio of around 0.96. Considering this, the relation between the calculated and measured values of mass flow ratio could be described by a function MFRcalc
