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A one-dimensional performance model for turbocharger turbine under pulsating inlet condition
A one-dimensional performance model for turbocharger turbine under pulsating inlet condition H Chen1, D E Winterbone2 1 National Laboratory of Engine Turbocharging Technologies, China 2 Formerly UMIST, UK
ABSTRACT This paper describes a new 1-D model of turbocharger turbine working under pulsating inlet condition. The flow inside the turbine volute is treated as onedimensional and unsteady, but with mass removal or addition to simulate the flow into and from the rotor. The flow in each rotor passage is treated as onedimensional and unsteady, and is coupled to the volute flow through a sliding rotorstator interface. This model is solved by the method of characteristics. The results show that the model captures important unsteady features such as hysteresis of turbine mass flow and efficiency. Possible improvements and extensions to the model are also discussed. NOMENCLATURE A a b Cf CFD Cis Cp f h L m Mf
m N n1, n2 nb P Q R Re s Str t T U
Flow passage area (m2) Speed of sound (m/s) Flow passage width (m) Coefficient in disk friction torque expression (22) Computational Fluid Dynamics Isentropic expansion (spout) speed (m/s) Specific heat at constant pressure (J/kg) Pulse frequency (Hz) Specific enthalpy (J/kg) Wet periphery of flow passage (m) Index in angular momentum equation (14) Disk friction torque (N-m) Mass flow rate (kg/s) Turbine rotating speed (rev/s) Indices in incidence loss equation (17) Number of rotor blades Pressure (N/m2) Heat transfer through the walls of flow passage (J/kg-s) Radius (m) Reynolds number Clearance between rotor backdisc and heat shield (m) Strouhal number Time (s) Temperature (oK) Rotor peripheral (tip) speed (m/s)
_______________________________________ © The author(s) and/or their employer(s), 2014
113
V W x
Absolute velocity (m/s) Relative velocity (m/s) Coordinate along flow direction in the volute (m)
Greek Symbols Angle between absolute and peripheral velocities Angle between relative velocity and radial direction Right-hand term of compatibility conditions along characteristics Ratio of specific heats Right-running, left-running Math-lines and path-line (characteristics) Cycle mean, total-to-static efficiency Kinematic viscosity (m2/s) Density (kg/m3) Time scale, wall shear stress (N/m2) Azimuth angle Mass withdraw from/addition to the volute per unit length per unit time (kg/s-m) Rotor angular speed (rad/s) Subscripts 0 Stagnation or total state, turbine inlet 1 Centroid of turbine housing 2 Volute exit 3 Rotor inlet (after incidence) a Ambient f Fluid m Mean p Pulse r Rotor s Steady flow u Unsteady flow w Wall 1. INTRODUCTION Turbocharging turbines for internal combustion engines often work under pulsating flow conditions. It has long been recognised that the performance of these turbines can be quite different from that under steady flow conditions. Yet there is still a lack of understanding of the phenomena, and there is no simple method that can be used with confidence to predict the performance of the turbine under such conditions. The relative importance of unsteady effects produced by pulsating flow might be estimated by a Strouhal number which is the ratio of two timescales (1): Strf-p = f/ p
(1)
where f is the time for fluid particles to be transported through the turbine components, and p is the time scale of the unsteadiness of pulsating flow. The following rough guides hold, if Strf-p << 1, quasi-steady effects dominate; Strf-p >>1, unsteady effects dominate; Strf-p ~ 1, both quasi-steady and unsteady effects are important.
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Take for example, a four-stroke, four cylinder engine rotating at 2000rpm and a turbine housing with averaged flow passage length of 150mm and an averaged flow velocity of 100m/s; these would give f, p and Strf-p values of 0.0015s, 0.015s, and 0.1 respectively. In this case both unsteady and quasi-steady effects exist in the housing. Averaged flow passage length of turbine rotor is about a quarter of that of the housing, so quasi-steady effects should dominate the rotor flow. Another important time scale for turbines is the time scale for the rotor to rotate one revolution. This time scale r is largely independent of f, and there is evidence (2-3) that with the reduction of r or the increase of turbine speed, turbine behaviour becomes increasingly steady. So another Strouhal number may be needed: Strr-p= r/ p
(2)
and when Strr-p is small, steady effects may dominate. Existing low dimensional methods for predicting turbine performance under unsteady flow fall into three categories: steady, quasi-steady and unsteady flow models. The steady flow method uses cycle mean inlet conditions and measured steady flow performance maps; while the quasi-steady flow method uses the same performance maps in a quasi-steady manner to predict instantaneous performance from the instantaneous inlet conditions. The quasi-steady flow method may produce poorer cycle-mean performance results than the simpler steady flow method. The first unsteady flow model was perhaps presented by Wallace and Adgey (4) in 1967. In this model, the nozzled turbine housing is simulated by a convergent nozzle and the one-dimensional unsteady flow equations can then be applied to the nozzle. The rotor is treated as a single, two-dimensional duct. Results obtained in this work (5) showed that it is possible to trace pressure waves through the turbine albeit in a relatively crude manner. The Wallace model was modified by Mizumachi and his co-workers (6), who reformed the equations, and simplified the rotor flow passage to a one-dimensional duct. The modified model was used for full admission, and showed fairly good agreement in mass flow between predictions and experiment. They also developed a new model for partial admission of sector divided, nozzled, twin-entry housing. In the new model, the stator and the rotor were each divided peripherally into six sections, a typical streamline in one sector of the scroll was branched twice and the branching points were treated as 'constant pressure' junctions. Rotation of the rotor was simulated by transferring the connection of the rotor passages with the nozzles one by one at a time interval of one-sixth of a rotor revolution. The mass flow rate and the pressure waves at the turbine entries were well predicted, but the efficiency was overestimated. Chen and Winterbone (7) proposed a housing model that replaces the volute by an equivalent length of nozzle, and applied one-dimensional unsteady flow equations to this nozzle. The flow in the rotor is treated as quasi-steady, and simulated by a meanline model. The model was later (2) applied to a mixed flow turbine. The results showed it captured unsteady effects better than a quasi-steady model, but it did not show any improvement on predicted cycle-mean performance versus a steady flow method. Baines et al. (8) described a turbine model in which the volute is represented as a volume between the turbine inlet pipe and the turbine rotor entry. They showed that this approach, which is effectively a zero-dimensional model, can predict some of the measured features of unsteady flow.
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The difficulty in modelling the unsteady flow in the volute housing is that the volute is not a simple pipe and one-dimensional gas flow equations are difficult to apply. Treating it as a volume ignores the gas momentum equation and will inevitably lead to error. In the remainder of this paper, a new one-dimensional unsteady flow model of the turbine volute housing is proposed to overcome this difficulty. The model was originally devised between 1987-88 but has not been formally published. 2. PHYSICAL MODEL OF TURBINE 2.1 Curved slot model of volute with mass removal/addition Under steady flow conditions, flow continuously leaves the volute along the circumferential direction of turbine volute and enters the rotor. The same happens under unsteady flow condition, the only difference being that the flow is a function of time and may reverse, entering the volute from the rotor. This flow leaving/entering the volute may be considered as mass removal/addition to the flow inside the volute, so the volute flow can still be treated as one-dimensional, but with additional terms to take into account this mass removal/addition, see Figure 1. The unsteady, one-dimensional gas flow equations (9-10), as applied to the volute model in Figure 1, are: Continuity
1 1V1 1V1 dA1 t x1 A1 A1 dx1
(3)
where is the mass withdrawn from or added to the volute per unit length per unit time, and is expressed as:
2V2 sin 2 b2 R2 /( R1 0.5
dR1 ) d
(4)
Momentum
1V1 1V1 p V dA1 L1 V2 cos( 2 1 ) 1 1 sign (V1 ) w t x1 A1 A1 dx1 A1 2
2
(5)
where w is the wall shear stress, and is expressed as:
w 0.0025 1V1 2
(6)
Energy
1h01 p1 1V1h01 V h dA h02 1 1 01 1 1Q t x1 A1 A1 dx1
(7)
where Q is the heat transfer through the wall, and is expressed as:
Q 0.0025
116
L1 2 V1C p (Tw T1 0.424V1 / C p ) A1
(8)
P1 1 V1 A1 R1
dx1 y
1
P1+dP1 1+d1 V1+dV1 A1+dA1 R1+dR1 +d
x x
y
P2, 2, V2 2, A2, R2
1 R1
R2
Fig. 1 Curved slot model of turbine volute 2.2 Method of characteristics and boundary conditions The method of characteristics (11) is used to give necessary equations for various boundaries:
dp1 a1 1 dV1 a1 1 a1 2 3 dt 2
(9)
along Mach lines and ,
dx1 V1 a1 dt
(10)
and
dp1 a1 d1 3 dt
(11)
2
along path line
dx1 V1 dt
(12)
where
1 11 12 ,
11
A1
, 12
2 11 V2 cos( 2 1 ) V1
1V1 dA1 A1 dx1
L1 sign (V1 ) w , A1
3 ( 1)11 (h02 h01 ) V1 2 Q 1 .
,
(13a)
(13b)
(13c)
For the entry region of the turbine housing from the turbine inlet flange to the tongue ( = 0o), equations (3) to (13) are used with = 0. The instantaneous static pressure and time mean stagnation temperature at the inlet flange are assumed to be known from experimental measurements, instantaneous static temperature is then calculated using the method of (12) for inflow to the inlet. The velocity at the inlet is calculated through the characteristic, and for outflow from the inlet, the density at the inlet is calculated through the path line characteristic. The volute end ( = 360o) is simplified to a closed end with V1 = V2 = 0. Two Mach line characteristics are used to obtain the pressure and density at this point.
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The flow between the volute exit or station 2 and the volute central line or station 1 is assumed quasi-steady, for outflow from the volute to the rotor (W3 > 0), angular momentum equation
V2 cos 2 V1 cos 1 R1 / R2
m
(14)
is applied, where m is an index and may be estimated from experimental data (m = 1 implies angular momentum conservation), and following assumptions are used: P2 = P1
(15)
(16)
At rotor inlet, a modification of NASA's incidence loss model (13-14) is used for inflow:
W3 W2 cos n1 2 , for 2 0 ; 2
2
W3 W2 1 sin n2 2 , for 2 0 , 2
2
(17a) (17b)
where W2 and W3 are the relative velocities at rotor entry before and after incidence, as shown in Figure 2. The indices n1 and n2 were selected from test data. Energy and mass conservations and the compatibility condition along the characteristic running from inner rotor toward the rotor entry are also used: 2
W3 P3 W2 P2 2 1 3 2 1 2
(18)
3W3 A3 2W2 sin 2 A2
(19)
2
dp adW 2 W dA dR L L a sign(W ) w 2 R ( 1) Q W sign(W ) w dt a A dx A R A (20) where Q and w are the heat transfer term and wall shear stress of the rotor passages respectively. For reversed flow from the rotor to the volute ( W3
0 ), the same equations as (14)
to (20) are used except that equation (17) is replaced by the compatibility condition along the path line:
L dp a 2 d ( 1) Q W sign (W ) w dt A
V2
W2 2<0
V3 W3
2>0 U2
U3
Fig. 2 Velocity triangles before and after incidence at rotor inlet
118
(21)
2.3 Rotor model and solution method Equations similar to those in (6) are used for the rotor passages, with wall friction and heat transfer terms included. The disc friction is estimated by the following formula (15) 5 M f 0.5C f a 2 R3 ,
(22)
where Mf is the friction torque, Cf is given by
C f 0.080Re
1/ 4
/( S / R3 ) 1 / 6 , and Re R3 / 2
(23)
The Lax-Friedrichs algorithm (16) has been used to solve governing equations for internal points, and the Courant-Isaacson-Rees algorithm (17) has been applied to solve the characteristic equations for boundary points. The volute between = 0o to 360o was divided into nb sections with each section extending over the same angle = 2/nb, where nb is the number of rotor passages. Connection between these sections and the rotor passages is carried out in every time step. 3. RESULTS 3.1 Steady flow results Steady flow calculations were first carried out to test the model. Turbine data used was supplied by Holset and are shown in Table 1. The values of indices m, n1 and n2 were chosen at different rotor speeds to give agreement between the predicted and measured performances. These steady state results are shown in Figures 3-4. The maximum error in the predicted mass flow is about 5%. Given the fact that only a very simple friction loss model was used for the rotor passages, and some important losses such as clearance loss and bearing loss were not included, the over-predicted efficiency is encouraging. Table 1. Main input data Housing inlet area
4.71 x10-3m2
Housing end ( =0) area
0.163 x10-3m2
Width of volute exit (station 2)
0.0171m
Number of rotor blades
12
Rotor inlet radius
0.0485m
Rotor inlet area
4.125 x10-3m2
Rotor exit radius
0.0295m
Blade angle (trailing edge shroud)
63.5o
Rotor exit area
1.73 x10-3m2
Gas constant
287.1 J/kg-oK
Gas specific heat ratio
1.4
Gas viscosity
1.5 x10-5m2/s
Turbine inlet total temperature
400 oK
Wall temperature
320 oK
Ambient pressure Pa
1.0133 x105N/m
Ambient density a
1.225 kg/m3
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8 5
6
7
P00 / Pa 2
2
3
4
P00 / Pa 1.2
1
N / T00 0
10
20
30
40
50
Fig. 3 Steady mass flow prediction
0
10
20
30
40
50
Fig. 4 Steady efficiency prediction
3.2 Unsteady flow results Computations were carried out to investigate the influence of pulsating flow on turbine performance. The results are shown in Figures 5-10. Figures 5-6 show the mass flow and efficiency characteristics under a pulsating inlet condition of P00/Pa = 1.4-0.2sin(270t) and
N / T00,m 30 . The turbine operating loci show hysteresis
between expansion pressure ratio and non-dimensional mass flow rate, and hysteresis between efficiency and blade speed ratio as obtained by Dale and Watson (20). Figure 7 shows the variation of flow angle at volute exit with time at two different circumferential locations. Strong pulsating of the angle at the tongue region, Figure 7a, is damped at the middle of the volute as shown in Figure 7b. Waves with different shapes and periods were used to investigate the influence of such parameters as frequency, amplitude and shape on the turbine performance. The results are given in Figures 8-10, where unsteady flow results are plotted against steady flow results. Figure 8 shows a slight increase of mass flow rate and a slight decrease in efficiency with increasing pulse frequency. These results agree with those obtained from experiment in (12). Figure 8b also suggests that the steady flow method might overestimate as well as underestimate both the efficiency and power depending on non-dimensional turbine speed N / T0 . The figure further suggests that the steady flow method might give a better prediction of turbine efficiency at higher turbine speeds. This is consistent with the influence of the Strouhal number Strr-p as discussed earlier: for the highest pulse frequency of 100Hz modelled, the values of Strr-p as expressed by equation (2), are 0.5, 0. 167 and 0.106 respectively for the three turbine speeds in the figures. Increasing the pulse amplitude reduces the mass flow rate as shown in Figure 9a. The relationship between turbine efficiency and the pulse amplitude, indicated in Figures 9b, shows a strong dependence on turbine speed. The influence of pulse shape is shown in Figure 10. The stronger the pulsating is, the lower the turbine efficiency tends to be. The figure also suggests that a steep wave-front might result in a lower turbine efficiency.
120
1 .0 1 .1 1 .2
1.9 1 1.1
1.2
1.3
1.4
0 .3 0 .4 0 .5 0 .6 0 .7 0 .8
1.5
1.6
0 .9
1.7 1.8
P00 ( t ) / Pa
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
U / C is , m
0.2
6.5
0.6
0.8
1.0
1.2
Fig. 6 Unsteady efficiency locus
58
28
Fig. 5 Unsteady mass flow locus
0.4
2, 180 (t )
54 56
27 26
N / T00,m 30
23
40 42 44
24
46 48
25
50
52
22
N / T00,m 30 0
2
4
6
8
10
12
14
0
2
(a) =330deg
4
6
8
10
12
14
(b) =180deg
1
Fig. 7 Flow angle variation with time and location
P00 / Pa 1 . 6 0 . 6 sin( 2 ft )
t s ,u / t s , s
0
f (Hz)
20
40
60
80
(a) m u / m s
100
0.75 0.8
0.9
0.92
0.94
0.85 0.9 0.95
0.96
1
0.98
1.05
mu/ ms
0
20
40
(b)
60 f (Hz)
80
100
t s ,u /t s , s
Fig. 8 Effect of pulse frequency on mass flow and efficiency
121
1.1
s
0.7
0.82 0.86
0.8
0.90
0.9
0.94
m u/ m
t s,u /t s,s
1.0
0.98 1
0
0.2
0.4
0.6
X
0.6
X 0.8
0
1.0
0.2
0.4
(b)
(a) m u / m s
0.6
0.8
1.0
t s , u / t s , s
Fig. 9 Effect of pulse amplitude X on mass flow and efficiency
0.8
t s ,u
0.5
3.0 3.2 3.4
0.6
0.7
3.6 3.8 4.0
m T00,m / P00,m *105
Steady flow
N / 0
5
10
(a)
15
20
25
T 00 , m 30
m T00, m / P00, m *105
35
0.1 0.2
2.6 2.8
0.3
0.4
P00 / Pa 1.2 0.2 sin( 2 50t )
N / 0
5
10
15
(b)
T 00 , m 20
25
30
35
t s
Fig. 10 Effect of pulse shape on mass flow and efficiency 4. DISCUSSION The model presented here treats turbine housing more rationally than in other previous models, and takes into consideration more geometrical parameters of the housing. However, since the model was first conceived over two decades ago, improvements are now appropriate and possible. Firstly, equations (15) and (16) for volute exit should be replaced, perhaps by adiabatic flow assumption and mass conservation between stations 1 and 2; secondly, the closed end treatment of the volute end may be replaced by a junction model allowing nonzero velocity at the end; thirdly, better rotor loss models, perhaps adopting first the steady flow loss modelling technique, could be used to improve the accuracy of the performance prediction. Application of the current vaneless housing model to vaned housing is straight forward. Station 2 is now the inlet to the nozzles which can be treated similarly to the rotor inlet, but in the stationary frame. Extending the model to twin-entry turbine housing, whether meridionally divided or circumferentially divided, is
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possible. For the circumferentially divided housing, the extension is simpler and more straightforward; for the meridionally divided housing, additional modelling is required to allow for the interaction between the two branches of the volute, and between the volute and the rotor. A three-way junction may be suitable for this task. 5. REFERENCES 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Greitzer E. M., An introduction to unsteady flows. Vol. 2 of Thermodynamics and fluid mechanics of turbomachinery, edited by Ucer A. S., Stow P. and Hirsch C., NATO ASI Series, Martinus Nijhoff Publishers, 1985. Chen H., Hakeem I. and Martines-Botas R. F., Modelling of a turbocharger turbine under pulsating inlet conditions. Proc. IMechE, Pt. A, Vol. 210, pp. 397-408, 1996. Private communication with Honda simulation engineers in Japan, 2011. Wallace F. J. and Adgey J. F., Theoretical assessment of the non-steady flow performance of inward radial flow turbines. Proc. IMechE, Vol. 182, Pt. 3H, pp. 22-36, 1967-8. Wallace F. J., Adgey J. F. and Blair G. P., Performance of inward radial flow turbines under non-steady flow conditions. Proc. IMechE, Vol. 184, Pt. 1, No. 10, pp. 183-195, 1969-70. Mizumachi N., Yoshiki H. and Endoh T., A study on performance of radial turbine under unsteady flow conditions. Report of Inst. Indus. Sci., Uni. Tokyo, Vol. 28, No. 1, Dec. 1979. Chen H. and Winterbone D. E., A method to predict performance of vaneless radial turbines under steady and unsteady flow conditions. IMechE 4th Int. conf. Turbocharging and Turbochargers, Paper C405/008, Proc. IMechE, pp. 13-22, 1990. Baines N. C., Hajilouy-Benisi A. and Yeo J. H., The pulse flow performance and modelling of radial inflow turbines. IMechE 5th Int. conf. Turbocharging and Turbochargers, Paper C484/006, Proc. IMechE, pp. 209-18, 1994. Zucrow M. J. and Hoffman J. D., Gas dynamics. Vol. 2, John Wiley & Sons, Inc. 1977. Woods W. A. and Allison A., Unsteady compressible flow with gradual mass addition and area change. IMechE Sixth Thermodynamics and Fluid Mechanics Convention, IMechE conf. Pub. 1976-6, pp 179-186. Benson R. S., The thermodynamics and gas dynamics of internal combustion engines. Vol. I, edited by Horlock J. H. and Winterbone D. E., Oxford Uni. Press, 1982. Benson R. S., Non-steady flow in a turbocharger nozzless radial gas turbine. SAE Meeting Milwaukee, SAE Paper 740739, Sept. 1974. Wasserbauer C. A. and Glassman A. J., Fortran program for predicting offdesign performance of radial-inflow turbines. NASA Technical Note D-8063, 1975. Meitner P. L. and Glassman A. J., Off-design loss model for radial turbine with pivoting, variable-area stators. NASA Technical Report 80-C-13, 1980. Glassman A. J. et al, Turbine design and application. NASA SP-290, 1972-75. Lax P. D., Weak solution of non-linear hyperbolic equations and their numerical computation. Commun. Pure and Appl. Math., Vol. 7, pp. 159-193, 1954. Courant R., Isaacson E. and Rees M., On the solution of non-linear hyperbolic differential equations by finite differences. Commun. Pure and Appl. Math., Vol. 5, pp. 243-355, 1952. Dale A. and Watson N., Vaneless diffuser turbocharger performance. IMechE 3rd Int. conf. Turbocharging and Turbochargers, Paper C110/86, Proc. IMechE, pp. 65-76, 1986-4.
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ABSTRACT This paper describes a new 1-D model of turbocharger turbine working under pulsating inlet condition. The flow inside the turbine volute is treated as onedimensional and unsteady, but with mass removal or addition to simulate the flow into and from the rotor. The flow in each rotor passage is treated as onedimensional and unsteady, and is coupled to the volute flow through a sliding rotorstator interface. This model is solved by the method of characteristics. The results show that the model captures important unsteady features such as hysteresis of turbine mass flow and efficiency. Possible improvements and extensions to the model are also discussed. NOMENCLATURE A a b Cf CFD Cis Cp f h L m Mf
m N n1, n2 nb P Q R Re s Str t T U
Flow passage area (m2) Speed of sound (m/s) Flow passage width (m) Coefficient in disk friction torque expression (22) Computational Fluid Dynamics Isentropic expansion (spout) speed (m/s) Specific heat at constant pressure (J/kg) Pulse frequency (Hz) Specific enthalpy (J/kg) Wet periphery of flow passage (m) Index in angular momentum equation (14) Disk friction torque (N-m) Mass flow rate (kg/s) Turbine rotating speed (rev/s) Indices in incidence loss equation (17) Number of rotor blades Pressure (N/m2) Heat transfer through the walls of flow passage (J/kg-s) Radius (m) Reynolds number Clearance between rotor backdisc and heat shield (m) Strouhal number Time (s) Temperature (oK) Rotor peripheral (tip) speed (m/s)
_______________________________________ © The author(s) and/or their employer(s), 2014
113
V W x
Absolute velocity (m/s) Relative velocity (m/s) Coordinate along flow direction in the volute (m)
Greek Symbols Angle between absolute and peripheral velocities Angle between relative velocity and radial direction Right-hand term of compatibility conditions along characteristics Ratio of specific heats Right-running, left-running Math-lines and path-line (characteristics) Cycle mean, total-to-static efficiency Kinematic viscosity (m2/s) Density (kg/m3) Time scale, wall shear stress (N/m2) Azimuth angle Mass withdraw from/addition to the volute per unit length per unit time (kg/s-m) Rotor angular speed (rad/s) Subscripts 0 Stagnation or total state, turbine inlet 1 Centroid of turbine housing 2 Volute exit 3 Rotor inlet (after incidence) a Ambient f Fluid m Mean p Pulse r Rotor s Steady flow u Unsteady flow w Wall 1. INTRODUCTION Turbocharging turbines for internal combustion engines often work under pulsating flow conditions. It has long been recognised that the performance of these turbines can be quite different from that under steady flow conditions. Yet there is still a lack of understanding of the phenomena, and there is no simple method that can be used with confidence to predict the performance of the turbine under such conditions. The relative importance of unsteady effects produced by pulsating flow might be estimated by a Strouhal number which is the ratio of two timescales (1): Strf-p = f/ p
(1)
where f is the time for fluid particles to be transported through the turbine components, and p is the time scale of the unsteadiness of pulsating flow. The following rough guides hold, if Strf-p << 1, quasi-steady effects dominate; Strf-p >>1, unsteady effects dominate; Strf-p ~ 1, both quasi-steady and unsteady effects are important.
114
Take for example, a four-stroke, four cylinder engine rotating at 2000rpm and a turbine housing with averaged flow passage length of 150mm and an averaged flow velocity of 100m/s; these would give f, p and Strf-p values of 0.0015s, 0.015s, and 0.1 respectively. In this case both unsteady and quasi-steady effects exist in the housing. Averaged flow passage length of turbine rotor is about a quarter of that of the housing, so quasi-steady effects should dominate the rotor flow. Another important time scale for turbines is the time scale for the rotor to rotate one revolution. This time scale r is largely independent of f, and there is evidence (2-3) that with the reduction of r or the increase of turbine speed, turbine behaviour becomes increasingly steady. So another Strouhal number may be needed: Strr-p= r/ p
(2)
and when Strr-p is small, steady effects may dominate. Existing low dimensional methods for predicting turbine performance under unsteady flow fall into three categories: steady, quasi-steady and unsteady flow models. The steady flow method uses cycle mean inlet conditions and measured steady flow performance maps; while the quasi-steady flow method uses the same performance maps in a quasi-steady manner to predict instantaneous performance from the instantaneous inlet conditions. The quasi-steady flow method may produce poorer cycle-mean performance results than the simpler steady flow method. The first unsteady flow model was perhaps presented by Wallace and Adgey (4) in 1967. In this model, the nozzled turbine housing is simulated by a convergent nozzle and the one-dimensional unsteady flow equations can then be applied to the nozzle. The rotor is treated as a single, two-dimensional duct. Results obtained in this work (5) showed that it is possible to trace pressure waves through the turbine albeit in a relatively crude manner. The Wallace model was modified by Mizumachi and his co-workers (6), who reformed the equations, and simplified the rotor flow passage to a one-dimensional duct. The modified model was used for full admission, and showed fairly good agreement in mass flow between predictions and experiment. They also developed a new model for partial admission of sector divided, nozzled, twin-entry housing. In the new model, the stator and the rotor were each divided peripherally into six sections, a typical streamline in one sector of the scroll was branched twice and the branching points were treated as 'constant pressure' junctions. Rotation of the rotor was simulated by transferring the connection of the rotor passages with the nozzles one by one at a time interval of one-sixth of a rotor revolution. The mass flow rate and the pressure waves at the turbine entries were well predicted, but the efficiency was overestimated. Chen and Winterbone (7) proposed a housing model that replaces the volute by an equivalent length of nozzle, and applied one-dimensional unsteady flow equations to this nozzle. The flow in the rotor is treated as quasi-steady, and simulated by a meanline model. The model was later (2) applied to a mixed flow turbine. The results showed it captured unsteady effects better than a quasi-steady model, but it did not show any improvement on predicted cycle-mean performance versus a steady flow method. Baines et al. (8) described a turbine model in which the volute is represented as a volume between the turbine inlet pipe and the turbine rotor entry. They showed that this approach, which is effectively a zero-dimensional model, can predict some of the measured features of unsteady flow.
115
The difficulty in modelling the unsteady flow in the volute housing is that the volute is not a simple pipe and one-dimensional gas flow equations are difficult to apply. Treating it as a volume ignores the gas momentum equation and will inevitably lead to error. In the remainder of this paper, a new one-dimensional unsteady flow model of the turbine volute housing is proposed to overcome this difficulty. The model was originally devised between 1987-88 but has not been formally published. 2. PHYSICAL MODEL OF TURBINE 2.1 Curved slot model of volute with mass removal/addition Under steady flow conditions, flow continuously leaves the volute along the circumferential direction of turbine volute and enters the rotor. The same happens under unsteady flow condition, the only difference being that the flow is a function of time and may reverse, entering the volute from the rotor. This flow leaving/entering the volute may be considered as mass removal/addition to the flow inside the volute, so the volute flow can still be treated as one-dimensional, but with additional terms to take into account this mass removal/addition, see Figure 1. The unsteady, one-dimensional gas flow equations (9-10), as applied to the volute model in Figure 1, are: Continuity
1 1V1 1V1 dA1 t x1 A1 A1 dx1
(3)
where is the mass withdrawn from or added to the volute per unit length per unit time, and is expressed as:
2V2 sin 2 b2 R2 /( R1 0.5
dR1 ) d
(4)
Momentum
1V1 1V1 p V dA1 L1 V2 cos( 2 1 ) 1 1 sign (V1 ) w t x1 A1 A1 dx1 A1 2
2
(5)
where w is the wall shear stress, and is expressed as:
w 0.0025 1V1 2
(6)
Energy
1h01 p1 1V1h01 V h dA h02 1 1 01 1 1Q t x1 A1 A1 dx1
(7)
where Q is the heat transfer through the wall, and is expressed as:
Q 0.0025
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L1 2 V1C p (Tw T1 0.424V1 / C p ) A1
(8)
P1 1 V1 A1 R1
dx1 y
1
P1+dP1 1+d1 V1+dV1 A1+dA1 R1+dR1 +d
x x
y
P2, 2, V2 2, A2, R2
1 R1
R2
Fig. 1 Curved slot model of turbine volute 2.2 Method of characteristics and boundary conditions The method of characteristics (11) is used to give necessary equations for various boundaries:
dp1 a1 1 dV1 a1 1 a1 2 3 dt 2
(9)
along Mach lines and ,
dx1 V1 a1 dt
(10)
and
dp1 a1 d1 3 dt
(11)
2
along path line
dx1 V1 dt
(12)
where
1 11 12 ,
11
A1
, 12
2 11 V2 cos( 2 1 ) V1
1V1 dA1 A1 dx1
L1 sign (V1 ) w , A1
3 ( 1)11 (h02 h01 ) V1 2 Q 1 .
,
(13a)
(13b)
(13c)
For the entry region of the turbine housing from the turbine inlet flange to the tongue ( = 0o), equations (3) to (13) are used with = 0. The instantaneous static pressure and time mean stagnation temperature at the inlet flange are assumed to be known from experimental measurements, instantaneous static temperature is then calculated using the method of (12) for inflow to the inlet. The velocity at the inlet is calculated through the characteristic, and for outflow from the inlet, the density at the inlet is calculated through the path line characteristic. The volute end ( = 360o) is simplified to a closed end with V1 = V2 = 0. Two Mach line characteristics are used to obtain the pressure and density at this point.
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The flow between the volute exit or station 2 and the volute central line or station 1 is assumed quasi-steady, for outflow from the volute to the rotor (W3 > 0), angular momentum equation
V2 cos 2 V1 cos 1 R1 / R2
m
(14)
is applied, where m is an index and may be estimated from experimental data (m = 1 implies angular momentum conservation), and following assumptions are used: P2 = P1
(15)
(16)
At rotor inlet, a modification of NASA's incidence loss model (13-14) is used for inflow:
W3 W2 cos n1 2 , for 2 0 ; 2
2
W3 W2 1 sin n2 2 , for 2 0 , 2
2
(17a) (17b)
where W2 and W3 are the relative velocities at rotor entry before and after incidence, as shown in Figure 2. The indices n1 and n2 were selected from test data. Energy and mass conservations and the compatibility condition along the characteristic running from inner rotor toward the rotor entry are also used: 2
W3 P3 W2 P2 2 1 3 2 1 2
(18)
3W3 A3 2W2 sin 2 A2
(19)
2
dp adW 2 W dA dR L L a sign(W ) w 2 R ( 1) Q W sign(W ) w dt a A dx A R A (20) where Q and w are the heat transfer term and wall shear stress of the rotor passages respectively. For reversed flow from the rotor to the volute ( W3
0 ), the same equations as (14)
to (20) are used except that equation (17) is replaced by the compatibility condition along the path line:
L dp a 2 d ( 1) Q W sign (W ) w dt A
V2
W2 2<0
V3 W3
2>0 U2
U3
Fig. 2 Velocity triangles before and after incidence at rotor inlet
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(21)
2.3 Rotor model and solution method Equations similar to those in (6) are used for the rotor passages, with wall friction and heat transfer terms included. The disc friction is estimated by the following formula (15) 5 M f 0.5C f a 2 R3 ,
(22)
where Mf is the friction torque, Cf is given by
C f 0.080Re
1/ 4
/( S / R3 ) 1 / 6 , and Re R3 / 2
(23)
The Lax-Friedrichs algorithm (16) has been used to solve governing equations for internal points, and the Courant-Isaacson-Rees algorithm (17) has been applied to solve the characteristic equations for boundary points. The volute between = 0o to 360o was divided into nb sections with each section extending over the same angle = 2/nb, where nb is the number of rotor passages. Connection between these sections and the rotor passages is carried out in every time step. 3. RESULTS 3.1 Steady flow results Steady flow calculations were first carried out to test the model. Turbine data used was supplied by Holset and are shown in Table 1. The values of indices m, n1 and n2 were chosen at different rotor speeds to give agreement between the predicted and measured performances. These steady state results are shown in Figures 3-4. The maximum error in the predicted mass flow is about 5%. Given the fact that only a very simple friction loss model was used for the rotor passages, and some important losses such as clearance loss and bearing loss were not included, the over-predicted efficiency is encouraging. Table 1. Main input data Housing inlet area
4.71 x10-3m2
Housing end ( =0) area
0.163 x10-3m2
Width of volute exit (station 2)
0.0171m
Number of rotor blades
12
Rotor inlet radius
0.0485m
Rotor inlet area
4.125 x10-3m2
Rotor exit radius
0.0295m
Blade angle (trailing edge shroud)
63.5o
Rotor exit area
1.73 x10-3m2
Gas constant
287.1 J/kg-oK
Gas specific heat ratio
1.4
Gas viscosity
1.5 x10-5m2/s
Turbine inlet total temperature
400 oK
Wall temperature
320 oK
Ambient pressure Pa
1.0133 x105N/m
Ambient density a
1.225 kg/m3
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8 5
6
7
P00 / Pa 2
2
3
4
P00 / Pa 1.2
1
N / T00 0
10
20
30
40
50
Fig. 3 Steady mass flow prediction
0
10
20
30
40
50
Fig. 4 Steady efficiency prediction
3.2 Unsteady flow results Computations were carried out to investigate the influence of pulsating flow on turbine performance. The results are shown in Figures 5-10. Figures 5-6 show the mass flow and efficiency characteristics under a pulsating inlet condition of P00/Pa = 1.4-0.2sin(270t) and
N / T00,m 30 . The turbine operating loci show hysteresis
between expansion pressure ratio and non-dimensional mass flow rate, and hysteresis between efficiency and blade speed ratio as obtained by Dale and Watson (20). Figure 7 shows the variation of flow angle at volute exit with time at two different circumferential locations. Strong pulsating of the angle at the tongue region, Figure 7a, is damped at the middle of the volute as shown in Figure 7b. Waves with different shapes and periods were used to investigate the influence of such parameters as frequency, amplitude and shape on the turbine performance. The results are given in Figures 8-10, where unsteady flow results are plotted against steady flow results. Figure 8 shows a slight increase of mass flow rate and a slight decrease in efficiency with increasing pulse frequency. These results agree with those obtained from experiment in (12). Figure 8b also suggests that the steady flow method might overestimate as well as underestimate both the efficiency and power depending on non-dimensional turbine speed N / T0 . The figure further suggests that the steady flow method might give a better prediction of turbine efficiency at higher turbine speeds. This is consistent with the influence of the Strouhal number Strr-p as discussed earlier: for the highest pulse frequency of 100Hz modelled, the values of Strr-p as expressed by equation (2), are 0.5, 0. 167 and 0.106 respectively for the three turbine speeds in the figures. Increasing the pulse amplitude reduces the mass flow rate as shown in Figure 9a. The relationship between turbine efficiency and the pulse amplitude, indicated in Figures 9b, shows a strong dependence on turbine speed. The influence of pulse shape is shown in Figure 10. The stronger the pulsating is, the lower the turbine efficiency tends to be. The figure also suggests that a steep wave-front might result in a lower turbine efficiency.
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1 .0 1 .1 1 .2
1.9 1 1.1
1.2
1.3
1.4
0 .3 0 .4 0 .5 0 .6 0 .7 0 .8
1.5
1.6
0 .9
1.7 1.8
P00 ( t ) / Pa
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
U / C is , m
0.2
6.5
0.6
0.8
1.0
1.2
Fig. 6 Unsteady efficiency locus
58
28
Fig. 5 Unsteady mass flow locus
0.4
2, 180 (t )
54 56
27 26
N / T00,m 30
23
40 42 44
24
46 48
25
50
52
22
N / T00,m 30 0
2
4
6
8
10
12
14
0
2
(a) =330deg
4
6
8
10
12
14
(b) =180deg
1
Fig. 7 Flow angle variation with time and location
P00 / Pa 1 . 6 0 . 6 sin( 2 ft )
t s ,u / t s , s
0
f (Hz)
20
40
60
80
(a) m u / m s
100
0.75 0.8
0.9
0.92
0.94
0.85 0.9 0.95
0.96
1
0.98
1.05
mu/ ms
0
20
40
(b)
60 f (Hz)
80
100
t s ,u /t s , s
Fig. 8 Effect of pulse frequency on mass flow and efficiency
121
1.1
s
0.7
0.82 0.86
0.8
0.90
0.9
0.94
m u/ m
t s,u /t s,s
1.0
0.98 1
0
0.2
0.4
0.6
X
0.6
X 0.8
0
1.0
0.2
0.4
(b)
(a) m u / m s
0.6
0.8
1.0
t s , u / t s , s
Fig. 9 Effect of pulse amplitude X on mass flow and efficiency
0.8
t s ,u
0.5
3.0 3.2 3.4
0.6
0.7
3.6 3.8 4.0
m T00,m / P00,m *105
Steady flow
N / 0
5
10
(a)
15
20
25
T 00 , m 30
m T00, m / P00, m *105
35
0.1 0.2
2.6 2.8
0.3
0.4
P00 / Pa 1.2 0.2 sin( 2 50t )
N / 0
5
10
15
(b)
T 00 , m 20
25
30
35
t s
Fig. 10 Effect of pulse shape on mass flow and efficiency 4. DISCUSSION The model presented here treats turbine housing more rationally than in other previous models, and takes into consideration more geometrical parameters of the housing. However, since the model was first conceived over two decades ago, improvements are now appropriate and possible. Firstly, equations (15) and (16) for volute exit should be replaced, perhaps by adiabatic flow assumption and mass conservation between stations 1 and 2; secondly, the closed end treatment of the volute end may be replaced by a junction model allowing nonzero velocity at the end; thirdly, better rotor loss models, perhaps adopting first the steady flow loss modelling technique, could be used to improve the accuracy of the performance prediction. Application of the current vaneless housing model to vaned housing is straight forward. Station 2 is now the inlet to the nozzles which can be treated similarly to the rotor inlet, but in the stationary frame. Extending the model to twin-entry turbine housing, whether meridionally divided or circumferentially divided, is
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possible. For the circumferentially divided housing, the extension is simpler and more straightforward; for the meridionally divided housing, additional modelling is required to allow for the interaction between the two branches of the volute, and between the volute and the rotor. A three-way junction may be suitable for this task. 5. REFERENCES 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Greitzer E. M., An introduction to unsteady flows. Vol. 2 of Thermodynamics and fluid mechanics of turbomachinery, edited by Ucer A. S., Stow P. and Hirsch C., NATO ASI Series, Martinus Nijhoff Publishers, 1985. Chen H., Hakeem I. and Martines-Botas R. F., Modelling of a turbocharger turbine under pulsating inlet conditions. Proc. IMechE, Pt. A, Vol. 210, pp. 397-408, 1996. Private communication with Honda simulation engineers in Japan, 2011. Wallace F. J. and Adgey J. F., Theoretical assessment of the non-steady flow performance of inward radial flow turbines. Proc. IMechE, Vol. 182, Pt. 3H, pp. 22-36, 1967-8. Wallace F. J., Adgey J. F. and Blair G. P., Performance of inward radial flow turbines under non-steady flow conditions. Proc. IMechE, Vol. 184, Pt. 1, No. 10, pp. 183-195, 1969-70. Mizumachi N., Yoshiki H. and Endoh T., A study on performance of radial turbine under unsteady flow conditions. Report of Inst. Indus. Sci., Uni. Tokyo, Vol. 28, No. 1, Dec. 1979. Chen H. and Winterbone D. E., A method to predict performance of vaneless radial turbines under steady and unsteady flow conditions. IMechE 4th Int. conf. Turbocharging and Turbochargers, Paper C405/008, Proc. IMechE, pp. 13-22, 1990. Baines N. C., Hajilouy-Benisi A. and Yeo J. H., The pulse flow performance and modelling of radial inflow turbines. IMechE 5th Int. conf. Turbocharging and Turbochargers, Paper C484/006, Proc. IMechE, pp. 209-18, 1994. Zucrow M. J. and Hoffman J. D., Gas dynamics. Vol. 2, John Wiley & Sons, Inc. 1977. Woods W. A. and Allison A., Unsteady compressible flow with gradual mass addition and area change. IMechE Sixth Thermodynamics and Fluid Mechanics Convention, IMechE conf. Pub. 1976-6, pp 179-186. Benson R. S., The thermodynamics and gas dynamics of internal combustion engines. Vol. I, edited by Horlock J. H. and Winterbone D. E., Oxford Uni. Press, 1982. Benson R. S., Non-steady flow in a turbocharger nozzless radial gas turbine. SAE Meeting Milwaukee, SAE Paper 740739, Sept. 1974. Wasserbauer C. A. and Glassman A. J., Fortran program for predicting offdesign performance of radial-inflow turbines. NASA Technical Note D-8063, 1975. Meitner P. L. and Glassman A. J., Off-design loss model for radial turbine with pivoting, variable-area stators. NASA Technical Report 80-C-13, 1980. Glassman A. J. et al, Turbine design and application. NASA SP-290, 1972-75. Lax P. D., Weak solution of non-linear hyperbolic equations and their numerical computation. Commun. Pure and Appl. Math., Vol. 7, pp. 159-193, 1954. Courant R., Isaacson E. and Rees M., On the solution of non-linear hyperbolic differential equations by finite differences. Commun. Pure and Appl. Math., Vol. 5, pp. 243-355, 1952. Dale A. and Watson N., Vaneless diffuser turbocharger performance. IMechE 3rd Int. conf. Turbocharging and Turbochargers, Paper C110/86, Proc. IMechE, pp. 65-76, 1986-4.
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