# Off-Design Considerations through the Properties of Some Pressure-Ratio Line of Radial Inflow Turbines

Authors: N. Binder; X. Carbonneau; P. Chassaing
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

Radial turbines are commonly used in applications involving operation through severe off-design conditions. The emergence of variable-geometry systems leads to the distinction between two off-design concepts: operational and geometric off-designs. Both of these operating constraints should be integrated in the design procedure. Recent developments in prediction and optimization methods allowed such an integration, but involving complex algorithms is coupled with semiempiric loss models. This paper provides a basis to obtain simple information from an existing or predesigned machine, for both operational and geometric off-design conditions. An alternative turbine map is defined using loading and flow coefficients. A one-dimensional analysis shows that the constant pressure-ratio lines are straight lines whose slope is remarkably correlated with the pressure-ratio value and geometrical characteristics. This theoretical approach is validated against the experimentation of two machines, the linearity is observed in both cases. The direct influence of the stator configuration on the pressure-ratio lines confirms the applicability of this work to variable-geometry stages. A dimensionless cross-section of the stator is thus defined. However, the unexpected displacement of the intercept of the pressure-ratio lines limits the application field of this method. Nevertheless, a simple performance prediction analysis is proposed for blocked mass flow operation.


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## Body

## 1. Introduction

Nowadays radial-inflow turbines are preferred to axial
machines in an important number of applications.
Their low-cost manufacturing and compactness are appreciated in fields such as turbocharging, air-conditioning systems, or cryogenics. In addition to those
industrial qualities, radial-turbine stages have interesting operating abilities.
Compared to axial geometries, the peak efficiency is generally lower, and the
stage integration does not ease multistage functioning. But the radius change
across the rotor gives a positive contribution in the Euler equation, resulting
in higher specific power. This balance is thoroughly discussed in [1]
where charts are presented, based on dimensionless parameters of the design
point (specific diameter and speed). Regions of best efficiency are defined for
the different architectures, and first-design rules are proposed. However, this
does not warrant an adequate off-design performance. Actually the compactness
of the impeller of a radial turbine allows very high rotational speed, and the
stage can sustain high pressure ratios. This flexibility is appreciated in case
of off-design functioning necessity. Even for such cases, the design of the
machines has long been focused on a single specification, the performance range
being evaluated subsequently. Numbers of efficient procedures derived from one-dimension analysis have been proposed with that purpose, for example, by
Rohlik [2], Rogers [3], or Whitfield and Baines [4]. Those
procedures improved in time with accurate loss, deviation, and blockage models
based on physical considerations, and supported by data over a large number of experiments.
See, for instance, the reviews by Moustapha et al. [5], Aungier [6], or Ebaid et al. [7] where a unified
approach is proposed. These procedures now involve elaborated codes allowing
off-design predictions, made so far with the use of specific routines such as
Wasserbauer and Glassmann [8]. The improvement of CFD has given the
opportunity to analyze a stage at minimal cost at the end of the design
procedure. The use of optimization algorithms also becomes accessible. For
example, Harinck et al. [9] optimized
a radial turbine by means of genetic algorithm, and Demeulenaere and Hirsh [10]
proposed a multipoint optimization procedure to insure best off-design
performances. All these methods are efficient and give accurate results, but require
a specific involvement of the users for their implementation.This assessment gets more complex when considering
variable-geometry stages. The operating range of such stages is widened, by
adapting the flow capacity to the inlet conditions. Thus in addition to
“operating off-design” the geometry variations shift the stator from
its design configuration, creating a “geometric off-design” (Figure1). Its effect on classical design-parameters has
been presented in [11]. Apart from specific losses due to technological effects (mechanical
clearance, partial admission at rotor inlet, spacers), the nozzle configuration
is likely to take extreme positions, which may prove critical when using
classical models.Figure 1
Off-design configurations.The present paper provides a basis to obtain simple off-design
information from existing or predesigned machines on either fixed- or variable-geometry
stages. It is based on an alternative presentation of the turbine map which gives
a new insight into the global functioning of the machine and unifies the
influences of operational and geometric off-designs. This observation has led to introduce
a one-dimension analysis whose hypotheses are not based on design-point
simplification. The alternative diagram is detailed in the first part of this
paper, through the properties of a pressure-ratio line. The results of the
analysis will be compared with experimental results for validation in the
second part. In the last part, first applications for design and prediction are proposed.


## 2. The Pressure-Ratio Line

The choice of aero-thermodynamic parameters used to describe
turbomachines has been long discussed in the literature (see, e.g., [12, 13]).
Similarity analysis shows that this description can be summarized by the
evolution of four nondimensional parameters if the Reynolds number and gas
effects are neglected. The classic representation of the operating field of
compressor or turbine stages is given in a map of reduced mass flow against
pressure ratio, for reduced-speed lines and efficiency regions. An alternative
is found using other nondimensional quantities such as the flow and loading
coefficients. Those quantities are currently used during the first steps of
design, to obtain the main characteristics of the stage through correlations
(see [14]), but can easily be expressed as combination of
the classic parameters. Consequently a ψ−ϕ map should express more than correlations, and
actually presents interesting characteristics. More specifically, a
one-dimension approach gives a simple equation for the trajectory of a constant
pressure-ratio line in this map. This analysis is presented as follows.To ease the integration of the Euler equation at the inlet and outlet of
the rotor, two hypotheses on the flow are generally considered as follows:(i)
the homogeneity of
the flow along the span of the blade at the inlet;(ii)
the
conservation of the angular moment at the outlet.
Thus, considering the first law of thermodynamics, the
Euler equation gives for total enthalpy variation:(1)ΔH=U4Vθ4−U5(U5−Wθ5). The second hypothesis insures the possibility to write
this equation at any rotor outlet radius. No clear convention is adopted in the
literature even if the RMS radius is often chosen. The outlet tip radius will here
be considered in order to preserve the expression of the global trim of the
rotor in the equations. The above expression can then be written as follows:(2)ΔHU42=rTi1U4(Vθ4rTi1+D5D4Wθ5rTi1)−(D5D4)2. Considering isentropic process in the stator, conservation
of mass at rotor inlet gives (3)Vθ4rTi1=m˙rTi1Pi1tan(α4)S4(Pi1P4)1/γ. Considering total to static efficiency definition,
conservation of mass at the rotor outlet gives (4)Wθ5rTi1=m˙rTi1Pi1tan(β5)S5πts×{1−ηts[1−(1πts)(γ−1)/γ]}(Pi5P5)(1−γ)/γ. Equations (2), (3), and (4) give (5)ΔHU42=rTi1m˙U4Pi1D42{[Pi1P4]1/γD42tan(α4)S4⁢+[Pi5P5](1−γ)/γD5D4D42tan(β5)S5πts⁢×[1−ηts[1−(1πts)(γ−1)/γ]]}−(D5D4)2. Loading and flow coefficients are defined as follows:(6)ψ=ΔHU42,ϕ=rTi1m˙U4Pi1(π/4)D42.  For simplicity's sake, we will also define an
intermediate pressure ratio and two geometric factors: (7)ξ=P5P4,S4*=S4D42tan(α4),S5*=S5D42tan(β5).   The use of the sine rule to estimate α3 and the assumption
of small changes of this angle through the free space allows one to express S4* after the stator
characteristics at the throat: (8)S4*=S3*=S3D42(r3/r4)sin(α3).S3* and S5*, with these
definitions, are nondimensional parameters characterizing both the stator and
rotor geometries and can be considered as dimensionless cross-sections. In that
condition (5) becomes as follows: (9)ψ=ϕ⁢⋅π4{[ξ⁢⋅πts]1/γ(S3*)−1+[Pi5P5](1−γ)/γD5D4(S5*)−1×πts[1−ηts[1−(1πts)(γ−1)/γ]]}−(D5D4)2. At constant pressure ratio, small variations of ξ    are expected. The influences of ηts and Pi5/P5 can be shown to
be negligible compared with those
of πts, S3* and S5*. Therefore, the trajectory
of a pressure-ratio line in a ψ−ϕ map, for a given geometry, is expected to be linear.
The intercept is fixed and depends on the trim of the rotor. The slope
increases with the pressure ratio and decreases with S3* and S5* as illustrated in Figure 2(a).ψ−ϕ map.
(a)(b)In this diagram, the effects
of operating and geometric off-designs are unified. Both are reflected by the
displacement of the operating point along the pressure-ratio line, or the modification
of its slope. Other properties of theψ−ϕ diagram can be further detailed in order to
locate classic features as speed lines and best-efficiency regions. The flow
coefficient is then written using reduced rotational speed and mass flow, the loading
coefficient using efficiency and reduced tip speed: (10)ϕ=240π2QrtNrt,(11)ψ=12ηts(CsU4)2. Consequently the shape of the speed lines can easily
be deduced. For a given value of Nrt, ϕ has the same variation as Qrt and will increase with pressure ratio. This is
observed at each intersection with a pressure-ratio line, until the choking
value is reached. As Nrt increases, the
choking value of ϕ decreases, hence defining the
stack of speed lines quoted on Figure 2(b). Besides, for fixed inlet temperature, pressure-ratio lines are also adiabatic-speed lines by definition. An
intersection between a pressure-ratio line and a speed line defines the value
of the reduced tip speed (U4/Cs). Assuming small
variations of efficiency when U4/Cs is fixed, relation
(11) insures the conservation of this value along a ψ  line. Considering that best efficiency is
reached near U4/Cs=0.7, the best-efficiency region
can be expected at ψ≈ηmax as illustrated in
Figure 2(b).The potential of theψ−ϕ diagram has therefore been demonstrated. This
alternative turbine map represents off-design behavior in a quite simple
transcription. Moreover, it is theoretically possible to model the whole
performance of a stage with straight lines, from its predesign characteristics,
including influence of nozzle modifications for variable-geometry stages. These
properties are obtained subject to restrictive hypothesis of a one-dimension
approach and have thus to be confronted to real operation. Experimental results
of two turbine stages will now be analyzed in that context, focusing on the influence
of the pressure ratio for the first stage and stator geometry for the second.


## 3. Experimental Verification

The above analysis is compared to the results of experiments performed
with two different turbine stages, representing either operational or geometric
off-designs. The properties of the pressure-ratio lines in aψ−ϕ diagram have been presented previously.
According to relation (9), the slopes of the lines are strongly linked to the
pressure-ratio value and both the dimensionless sections of the stator and the
rotor. As the rotor geometry is obviously fixed when functioning, the influences
of the first two parameters only will be checked. The first turbine stage (TS1) is part of an aircraft
air-conditioning pack. It has been tested for a wide range of pressure-ratio
values. The second one (TS2) is a
variable-geometry turbocharger turbine in which the modification of blade's angle
of attack allows variations of the dimensionless stator section.The characteristics of the stages are presented
in Table1, which together with (9) gives the
expected pressure-ratio lines for the two stages. The values of the efficiency
and intermediate pressure-ratio (ηts and ξ) are kept constant for each stage and set according to the design
specification. Four values of πt (1.8, 2.5, 3, and 3.5) are considered for TS1, three opening positions at πt=1.6 for TS2 (Figure 3).Table 1TS1TS2D4 (mm)14147N  (rpm at design)45 000150 000(D5/D4)20.320.67S3*0.0380.074 to 0.52S5*0.570.91Pressure-ratio
line prediction for (a)TS1 and (b) TS2.
(a)(b)As relation (11) imposes the order of magnitude of ψ for an efficient operation, a tenfold
difference is observed between the values of ϕ for the two stages. The
initial stage specification and the choice of the rotor diameter set this
proportion. Consequently, the slopes of the pressure-ratio lines are quite
different from one stage to another (≈40 forTS1; ≈5 forTS2). As a result, the dimensionless
stator section (S3*)
also presents large differences (Table 1). Relation (9) actually determines the
correlation between the slope of pressure-ratio lines and the dimensionless
stator section, thus expressing the link between operating behavior and
geometry configuration of the stage.The experimental results of these two stages are now
considered. The four dimensionless parameters (Qrt, Nrt, πt, and η) are obtained through mass flow,
pressure, and temperature-drop measurements. Rotational speed is given by count
on the compressor stage; the values of ψ and ϕ follow. To avoid perturbation of temperature-drop
measurement by conduction, tests were conducted under ambient inlet temperature
conditions. For detailed information about the experimental facilities see
[15] for TS1, and [11]
for TS2. The results are plotted in the ψ−ϕ map (see Figure 4).Experimental pressure-ratio line for (a)TS1 and (b) TS2.
(a)(b)The linearity of the evolution is remarkable. It
supports the initial assumption according to which the influence ofηts and ξ variations along
the line is small. The expected influences of pressure ratio and S3* on the slope
of the line are confirmed.The magnitude of the gradient is consistent with what
was expected in Figure3. On the contrary, the displacement of the intercept along the axis is
unexpected. According to (9), it was supposed to be fixed by the rotor
geometry, with no possible evolution. The origin of this mismatch should be
found in the second hypothesis considered for the integration of Euler equation
presented in the first part of the paper. Actually, the angular momentum is not
conserved at the outlet of a turbine stage, particularly in off-design conditions.
Five-hole-probes measurements are available for the stage TS1. An analysis of these results should give indications of the
correction to bring, and help to choose the correct outlet radius considered
for the integration.The location of other characteristic lines
completes this verification. Speed, reduced tip speed, and efficiency lines of
the stageTS1 are interpoled on the
diagram (Figures 5(a), 5(b), and 5(c)). The shape of the speed lines
is confirmed with respect to Figure
2(b). The reduced tip-speed lines do not strictly
coincide with ψ lines due to effective efficiency variations.
Despite these variations, the best-efficiency region is found as expected at ψ≈ηmax.(a) Interpolation of speed, (b) reduced speed,
and (c) efficiency lines.
(a)(b)(c)This experimental verification leads to a partial
validation of the one-dimension analysis. The pressure-ratio lines are actually
straight lines in aψ−ϕ map. The unification of geometric and
operational off-design transcriptions has been observed. The best-efficiency regions
are found where expected. However, the displacement of the intercept was unexpected. It actually needs more
attention. At this state this phenomenon limits the predictive potential
of the diagram. Still, applications of this work can now be considered, in
particular a first approach to design considerations and to performance
predictions.


## 4. Application for Design and Prediction

### 4.1. Design

The first application of this work concerns the design
of vaned nozzles. Equation (9) shows the importance of the stator
dimensionless section S3*.
In a one-dimension approach, the conservation of its value when changing the
stator configuration should not influence the stage behavior (either at design
point or at off-design conditions). The designer is free to define the right
set of radii, outlet angle, and geometric section for best efficiency, using
classical criteria (such as found, e.g., in [16]).
Blockage, deviation, and losses will certainly affect this assessment, but most
likely in a lesser measure. This point will be discussed in a forthcoming
paper.If conservation ofS3* preserves the stage behavior, its variation logically modifies the operating
characteristics of the stage. Through this modification, adaptation of the
stage to inlet changing conditions is possible, thus defining the “variable-geometry”
concept. S3* is
therefore identified as the determining parameter when variable-geometry
effects are considered.Finally, for fixed-geometry stages, an early
evaluation ofS3* should prove useful to estimate off-design requirements feasibility. In design
process, the value of loading coefficient is set in order to maximize
efficiency, according to relation (11). The specification and the choice of
inlet diameter of the rotor will determine the flow-coefficient value at the design
point. This choice can be conducted for performance (Chen and Baines [14] recommend 0.2–0.4 for better peak efficiency)
or be constrained by mechanical or economic limitations (some designers may
want to use existing basis for the new specification). The order of magnitude
for the slope of pressure-ratio line of the stage is then frozen and  (9) can give a first evaluation of the
stator configuration through S3*.
This value allows a first estimation of the stage's ability to reach specific
operating points.


### 4.2. Prediction

A second application lies in prediction of
performances. As straight lines, pressure-ratio lines are easily modelled in
theψ−ϕ map. But the displacement of the intercept
presented in the second part of this paper coupled with influence of deviation,
blockage, and losses forbids
the full prediction of the ψ−ϕ diagram. However, the interpolation of the
complete map should be possible with two points of each pressure-ratio line. At
this point the difficulty lies in the back deduction of the four classical
dimensionless parameters (Qrt, Nrt, πt, and η). The following two cases are
considered.(i)
Choking is not reached:
four parameters have to be determined, out of three equations based on the
knowledge ofψ or ϕ, πt and the pressure-ratio
line equation. The problem is open and cannot be solved simply.(ii)
Choking is reached or small variations of mass
flow are expected:Qrt is frozen, removing
the need of an extra equation. Two operating points of a pressure-ratio line
allow the prediction of the whole performance line.
In the first case, a separation procedure is being
developed and needs more attention. For the second case, the evolution of
efficiency against rotational speed can be deduced. Rewriting (11) with Nrt, πt, and η, and using the definition (10)
along a pressure-ratio line trajectory, the following equation can be
established: (12)ηNrt2=A⁢⋅QrtNrt+B, where A and B are determined after the
pressure-ratio line constants and the value of πt. Thereby the
evolution of efficiency against rotational speed at constant pressure ratio and
constant mass flow is quadratic. An example is proposed in Figure 6. Three pressure-ratio lines (one for TS1, and two for TS2 in closed-nozzle configuration) are presented; small mass flow
variations are observed (Figure 6(a)). Two points of each line are selected, giving the
coefficients of the ψ−ϕ line (Figure 6(b)). Equation (12) expresses the expected evolution of
efficiency against rotational speed, the prediction and the results are plotted
on Figure
6(c).Interpolation process verification.
(a)(b)(c)A very good agreement is achieved between the
predicted interpolation and the actual measurements. The evolutions of
rotational speed are reproduced for both stages, and the influence of pressure ratio
forTS2 is consistent. The
conservation of the mass flow along the pressure-ratio line has direct
influence on the accuracy. Thereby, the prediction is much better for stage TS1 at blockage than for TS2 where blockage is not reached. But,
in both cases, it has been possible to obtain the whole pressure-ratio line
from only two operating points. This process is likely to reduce the number of
points necessary to obtain the full map of a stage.


## 4.1. Design

The first application of this work concerns the design
of vaned nozzles. Equation (9) shows the importance of the stator
dimensionless section S3*.
In a one-dimension approach, the conservation of its value when changing the
stator configuration should not influence the stage behavior (either at design
point or at off-design conditions). The designer is free to define the right
set of radii, outlet angle, and geometric section for best efficiency, using
classical criteria (such as found, e.g., in [16]).
Blockage, deviation, and losses will certainly affect this assessment, but most
likely in a lesser measure. This point will be discussed in a forthcoming
paper.If conservation ofS3* preserves the stage behavior, its variation logically modifies the operating
characteristics of the stage. Through this modification, adaptation of the
stage to inlet changing conditions is possible, thus defining the “variable-geometry”
concept. S3* is
therefore identified as the determining parameter when variable-geometry
effects are considered.Finally, for fixed-geometry stages, an early
evaluation ofS3* should prove useful to estimate off-design requirements feasibility. In design
process, the value of loading coefficient is set in order to maximize
efficiency, according to relation (11). The specification and the choice of
inlet diameter of the rotor will determine the flow-coefficient value at the design
point. This choice can be conducted for performance (Chen and Baines [14] recommend 0.2–0.4 for better peak efficiency)
or be constrained by mechanical or economic limitations (some designers may
want to use existing basis for the new specification). The order of magnitude
for the slope of pressure-ratio line of the stage is then frozen and  (9) can give a first evaluation of the
stator configuration through S3*.
This value allows a first estimation of the stage's ability to reach specific
operating points.


## 4.2. Prediction

A second application lies in prediction of
performances. As straight lines, pressure-ratio lines are easily modelled in
theψ−ϕ map. But the displacement of the intercept
presented in the second part of this paper coupled with influence of deviation,
blockage, and losses forbids
the full prediction of the ψ−ϕ diagram. However, the interpolation of the
complete map should be possible with two points of each pressure-ratio line. At
this point the difficulty lies in the back deduction of the four classical
dimensionless parameters (Qrt, Nrt, πt, and η). The following two cases are
considered.(i)
Choking is not reached:
four parameters have to be determined, out of three equations based on the
knowledge ofψ or ϕ, πt and the pressure-ratio
line equation. The problem is open and cannot be solved simply.(ii)
Choking is reached or small variations of mass
flow are expected:Qrt is frozen, removing
the need of an extra equation. Two operating points of a pressure-ratio line
allow the prediction of the whole performance line.
In the first case, a separation procedure is being
developed and needs more attention. For the second case, the evolution of
efficiency against rotational speed can be deduced. Rewriting (11) with Nrt, πt, and η, and using the definition (10)
along a pressure-ratio line trajectory, the following equation can be
established: (12)ηNrt2=A⁢⋅QrtNrt+B, where A and B are determined after the
pressure-ratio line constants and the value of πt. Thereby the
evolution of efficiency against rotational speed at constant pressure ratio and
constant mass flow is quadratic. An example is proposed in Figure 6. Three pressure-ratio lines (one for TS1, and two for TS2 in closed-nozzle configuration) are presented; small mass flow
variations are observed (Figure 6(a)). Two points of each line are selected, giving the
coefficients of the ψ−ϕ line (Figure 6(b)). Equation (12) expresses the expected evolution of
efficiency against rotational speed, the prediction and the results are plotted
on Figure
6(c).Interpolation process verification.
(a)(b)(c)A very good agreement is achieved between the
predicted interpolation and the actual measurements. The evolutions of
rotational speed are reproduced for both stages, and the influence of pressure ratio
forTS2 is consistent. The
conservation of the mass flow along the pressure-ratio line has direct
influence on the accuracy. Thereby, the prediction is much better for stage TS1 at blockage than for TS2 where blockage is not reached. But,
in both cases, it has been possible to obtain the whole pressure-ratio line
from only two operating points. This process is likely to reduce the number of
points necessary to obtain the full map of a stage.


## 5. Conclusion

A new definition of a turbine map has been presented.
Based on the properties of the pressure-ratio lines, this alternative map is
supported by a one-dimension analysis and experimental verifications. This
diagram not only allows simple representation of the stage behavior through
straight lines, but also unifies geometric and operational off-design effects. The
slope of the pressure-ratio lines is significantly influenced by the pressure-ratio
itself and by a dimensionless parameter for characterizing the stator.
Experimental results for two stages showed similar influences in good agreement
with the analysis. However, the intercept of the line is not fixed as was expected.
More work is currently conducted to better understand this effect and take it into
account. The importance of the stator dimensionless cross-section has been shown.
Its fundamental role in the definition of stage behavior has been assessed in
the case of variable-geometry devices, for which it proved particularly well
adapted. Finally, a first application to the prediction of performances has
been performed for choked configurations. In this case, it has been shown that
a full pressure-ratio line can accurately be obtained from two of its points. Forthcoming
work will focus on three axes: (i) the influence of dimensionless stator section
and its applications in a variable-geometry context, (ii) the influence of
rotor outlet flow on the displacement of the intercept of the pressure-ratio
line, (iii) the setup of a nonrestrictive interpolation process for nonchocked
stage functioning.



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*Source: 273296-2009-02-05.xml*