Assessment of a One Dimensional Loss Model For The Compressor of a Turbocharger

(1)

INOM

EXAMENSARBETE

TEKNIK,

GRUNDNIVÅ, 15 HP

,

STOCKHOLM SVERIGE 2020

Assessment of a One Dimensional

Loss Model For The Compressor

of a Turbocharger

HUGO ESSINGER

LASZLO VIMLATI

KTH

(2)

Assessment of a One Dimensional Loss

Model For The Compressor of a Turbocharger

Hugo Essinger

1

and L ´aszl ´

o Viml ´ati

2

1_{hugoes@kth.se} 2_{vimlati@kth.se}

ABSTRACT

This report sets out to implement and asses a one-dimensional loss model for centrifugal compressors. The loss model which has been evaluated is the model by Oh et al. (1997) [1]. The implementation was completed using iterative methods implemented in Matlab. It was shown that the performance prediction using the Oh et al method produced usable estimates in the higher rotational speed region used in this work, at design conditions. The pressure ratio estimates had a difference compared to the measured reference data of 5-10% while the isentropic efficiency had slightly higher differences. Furthermore, it was shown that the best estimates of the pressure ratio came from the lowest rotational speeds, and successively flattened out from the expected curvature as the speed increased. The isentropic efficiency did not have the same property, giving the best consistent estimate at a higher rotational speed of 143 kRPM. The conclusion which was drawn from this work was that the model by Oh et al. is a useful model for prediction of performance at design points in the lower RPM region, while requiring complementary calculations at off-design conditions at high RPM.

It was also pointed out that there are several areas that require further work within performance prediction to make it more implementation-friendly.

Keywords: Turbocharger, Compressor, Single-zone, Modelling, Loss Models, Pressure Ratio, Isentropic Efficiency, Performance Prediction

SAMMANFATTNING

(3)

1 NOMENCLATURE

1.1 Notation

A - Area

b - Hub-to-shroud passage width

b∗ - Ratio of vaneless diffuser inlet width to impeller exit C - Absolute velocity

cf - Skin friction coefficient

cp - Specific heat at constant pressure

Df - Diffusion factor

D - Diameter

Dhyd - Impeller average hydraulic diameter

fd f - Disc friction coefficient

finc - Incidence coefficient

h - Enthalpy L - Length

Lb - Impeller flow length

Lθ - Impeller meridional length

m - Meridional coordinate M - Mach number ˙ m - Mass flow N - Rotational speed P - Pressure PR - Pressure ratio r - Radius Re - Reynolds number T - Temperature U - Tangential velocity W - Relative velocity Z - Number of blades

α - Angle between absolute flow and tangential direction β - Angle between relative flow and tangential direction γ - Heat capacity ratio

δ - Relative difference ε - Clearance

εwake - Wake fraction of blade-to-blade space

µ - Dynamic viscosity ν - Kinematic viscosity ηs - Isentropic efficiency ρ - Fluid density ω - Angular velocity 1.2 Subscripts 0 - Total/Stagnation condition 1 - Impeller inlet

1t - Impeller inlet tip

2 - Impeller outlet / diffuser inlet 3 - Diffuser outlet

b - Blade property m - Meridional component

m1m - Meridional direction at impeller inlet rm - Root mean squared position

(5)

2 INTRODUCTION

Internal combustion engines have since their dawn become increasingly refined with revolutionising inventions quickly making their way in to the market, such as fuel injection (instead of using a carburettor) and the turbocharger. The turbocharger, by pressurising the inlet air, allows for drastically higher performance for an engine with given displacement. If greater performance is not of interest, the turbocharger can be used to enable engine downsizing. By downsizing the engine losses from friction etc can be reduced, which is of great interest in the market of today. The compressor side of a turbo consists of the following parts:

Figure 1. Turbo description (Adapted from [2]).

It turns out that the turbocharger is from a fluid mechanics perspective a very complex system, often utilizing quite complex geometries and sporting high rotational speeds (sometimes upwards of and above 100,000 RPM). Thus, in order to construct well performing turbochargers some rather complex and advanced techniques can be (and are being) utilized, such as Computational Fluid Dynamics (CFD). CFD software calculates the flow properties throughout the complex 2D/3D geometry by numerically solving variations of the Navier-Stokes equations, which require a dense mesh to give reliable results. Although CFD can give good results, it is both computationally expensive and requires vast knowledge of the subject. Because of these problems more simplistic methods are often of interest. The initial 3-dimensional problem can be solved approximately in one dimension in a much more simple and cost effective way, by formulating performance based on rotational speed, mass flow rate and compressor geometry. The formulation of the 1D problem is a reduction of the actual flow and originates mainly from empirical correlations from experiments. Owing to this, many actual phenomena are lost in the models, which narrow down the usage of the models to fewer operating conditions. Possible effects of this, as well as further details about the restrictions and when other modeling options may be considered are discussed further in 5 Discussion. The result of such a solution may not be perfect but can be sufficient for many applications, as well as narrowing down the possibilities for deeper analysis of the design

(6)

using CFD e.g later on in order to refine the parameters withheld using the 1-dimensional model. These 1D models have been developed and refined by researchers of the subject for quite some time. In this project a 1D model for the compressor-side of a turbocharger, by Oh et al [1], will be assessed for a given set of design and operation parameters. One way to categorize the 1D models is by number of flow zones that form their basis. The number of flow zones varies between zero and two. Zero-zone models do not require a breakdown of the turbocharger into its different parts (impeller, diffuser, volute), but treats the whole structure based on flow- and work coefficients. It is assumed that well designed compressors have similar designs and therefore the performance of a compressor can be extrapolated from the already known design conditions.

Single-zone models, which is the main focus of this work, treat the flow as a mean flowline by assuming uniform flow through all parts of the compressor of the turbocharger. Conservation of energy, mass flow and rothalpy are utilized to calculate the losses in the different parts. Losses originate from experimental correlation and are calculated throughout the turbo in order to acquire the real flow conditions. Besides these losses, the flow is assumed to be isentropic. These factors increase the geometric requirements and model complexity compared to the zero-zone models, with a significant increase in accuracy and freedom of analysis in separate parts/losses.

Two-zone models consist of a jet-wake flow approach, as the flow is divided into two parts. It consists of an isentropic jet flow region with high velocities and a low-momentum wake flow zone on the suction side where all the losses occur. The two different zones are treated separately and combined through jet-wake mixing. Two-zone modelling, related to this work, is touched upon by Oh et al [1], which uses an empirical correlation between the wake and jet mass flow fractions according to [3].

In order to properly analyze the turbo with one- and two-zone approaches, the different geometries are required. Flow velocities are dependent on the geometry of the different parts, which are represented with velocity triangles:

U1t

C1

W1t

β1 α1

(a) Velocity triangle at inlet tip.

Cu2 Cm2 C2 U2 W2 α2b β2b

(b) Velocity triangle at outlet. Figure 2. Velocity triangles for the impeller.

These velocity components and angles play a key role in the models used throughout this work. For this to be valid, some simplifications are required. It is here assumed that the tangential component of the absolute velocity is zero at the impeller inlet tip, meaning that it is purely meridional. It is also assumed that the absolute velocity is the same in the impeller inlet tip as in the inlet of the compressor.

The volute of the compressor is not included in this work for simplification purposes.

3 METHODS AND MATERIALS

3.1 General method

The modeling was conducted numerically using Matlab for a given set of design parameters. The implementation of the model by Oh et al [1] was completed using the following iterative method:

1. Calculate thermodynamic conditions and velocity triangles in the inlet of the part. The inlet of each part is the outlet of the previous section of the turbo, meaning that the diffuser inlet is the same as the impeller outlet. 2. Outlet velocity is guessed, whereafter the losses are calculated in order to obtain the outlet conditions.

3. Compare the estimated mass flow rate, computed through isentropic relations based on the outlet conditions, to the actual mass flow rate to check convergence.

4. If the mass flow rates differ, increment the guessed velocity and start over from step 2.

(7)

Velocities in different parts of the compressor were all formulated by using the relations presented in Figure 2. A graphical illustration of the solution procedure is given below:

Start, inlet conditions

Guess impeller outlet meridional velocity

Calculate losses and resulting PR over impeller Determine resulting mass flow rate, using

isentropic relations

Increment velocity

Is ˙mcalc= ˙m?

Guess diffuser outlet velocity, use impeller outlet as diffuser inlet

Calculate losses and resulting total PR Yes

Determine resulting mass flow rate, using

isentropic relations Increment velocity Is ˙mcalc= ˙m? Done, calculate ηs no no yes

Figure 3. Iterative solution procedure for Oh et al [1]. 3.2 Oh et al.’s set of loss models

The loss models according to [1] are defined as enthalpy differences, which are later combined to calculate the resulting pressure loss and isentropic efficiency. The losses consist of internal- and parasitic losses, where the internal losses cause a pressure loss, while the parasitic losses only contribute to a loss in isentropic efficiency, as pointed out by Oh [1]. The internal losses are combined to obtain the pressure ratio using the following equation:

PR= " ∆hEuler− ∑ ∆hint cpT01 ! + 1 #γ /(γ −1) , (1)

where ∆hEuleris the ideal enthalpy change. In order to calculate the tangential component of the absolute velocity at

the impeller outlet, which differs from the ideal case due to slip. The slip factor formulation of Wiesner [4] was used without modification. This is also the reason why the flow angles α2and β2differ from the blade angles α2band β2b

that give the ideal velocity components, see Figure 2b.

The initial compressor inlet velocity was obtained from the mass flow rate: C1=

˙ m ρ01A1

, (2)

while the tangential velocities were obtained from the rotational speed:

U= ωr, (3)

which makes the mass flow rate, the rotational speed and the geometry the governing variables for the pressure ratio and the isentropic efficiency.

All the enthalpy differences are described in the following sections, for each part of the compressor.

3.2.1 Impeller internal loss models

Incidence loss:

Incidence loss occurs due to the direction of the flow diffusing from the impeller blade angle, which will make the flow

(8)

direction differ from the fixed blade angle. This can be modelled in several ways, but Oh [1] suggested the use of the following formula, found in [5]:

∆hinc= finc

W_u12

2 , (4)

where fincis an incidence coefficient between 0.5-0.7 according to the authors’ work.

Blade loading loss:

The boundary layer in the impeller grows highly dependently on the diffusion of the fluid in the impeller. A diffusion factor can be used to calculate the subsequent losses which can be calculated as follows, according to Coppage [6]:

∆hbl= 0.05D2fU22, (5)

where the diffusion factor Df is expressed as:

Df = 1 − W2 W1t + 0.75∆hEuler/U 2 2 (W1t/W2) h (Z/π)(1 − D1t/D2) + 2D1t/D2 i . (6)

The enthalpy rise throughout the impeller can be derived from Euler’s equation of turbomachinery combined with constant Rothalpy, which gives:

∆hEuler= 1 2(U 2 2−U1t2) + 1 2(W 2 1t−W22). (7)

Skin friction loss:

Friction that is acting between the wall and the airflow causes a loss in pressure, which is modelled according to the work of Jansen [7]: ∆hs f = 2cf Lb Dhyd ¯ W2, (8)

where ¯W is defined as: ¯

W=C1t+C2+W1t+ 2W1h+ 3W2

8 (9)

and the skin friction coefficient is determined in this work by: cf=

0.455

log10(Re1)2.58(1 + 0.144M1)

, (10)

to include compressibility effects and where Re1is the Reynolds number in the impeller inlet.

Clearance loss:

Clearance loss is a loss caused by the fluid flowing through the clearance between the blades and the shroud in the impeller. This is modelled using Jansen’s [7] formula:

∆hcl= 0.6 ε b2 Cu2 ( 4π b2Z " r2_1t− r2 1h (r2− r1t)(1 + ρ2/ρ1) # Cu2Cmlm )1₂ . (11) Mixing loss:

The mixing of the wake and jet result in further losses, as proposed by Johnston and Dean [8]: ∆hmix= 1 1 + tan2_α 2m 1 − εwake− b∗ 1 − εwake !2 C2₂ 2 . (12)

Here εwakeis calculated using the iterative methodology described by Yoon [3], where the diffusion ratio is set to a

(9)

3.2.2 Diffuser internal loss model

Vaneless diffuser loss:

Losses occurring in the diffuser are modelled according to the formulation of Stanitz [10], slightly modified from [1]: ∆hvd= cpT02 " P3 P02 !(γ−1)/γ) − P3 P03 !(γ−1)/γ)# (13) When including the contribution of the vaneless diffuser loss to the pressure ratio using (1), the ideal enthalpy change ∆hEulerhas to contain the contribution from the isentropic diffusion in the diffuser as:

∆hEuler= ∆hEuler,impeller+ 1 2(C 2 2m−C3m2 ) = 1 2(U 2 2−U1t2) + 1 2(W 2 1t−W22) + 1 2(C 2 2m−C3m2 ). (14) 3.2.3 Impeller parasitic loss models

Disc friction loss:

The friction of the fluid that flows in the clearance between the blades and the shroud of the impeller contribute to a work input to the fluid, which doesn’t lead to a decrease in pressure. It is modelled in accordance to the formulation of Daily and Nece [11]:

∆hd f= fd f ¯ ρ r₂2U₂3 4 ˙m , (15) where: ¯ ρ = ρ1+ ρ2 2 , fd f =    2.67 Re0.5_{d f}, if Red f< 3 × 10 5 0.0622 Re0.2_{d f} , if Red f≥ 3 × 10 5_, Red f = U2r2 ν2 =ρ2U2r2 µ2 . Recirculation loss:

Recirculation loss occurs at low mass flows, when some of the air flows back onto the impeller tip, which will in turn increase the work input to the fluid, leading to reduction in efficiency. This is modelled using the formulation of Oh [1]: ∆hrc= 8 × 10−5sinh(3.5α2m3 )D2fU22, (16)

where α2mis in radians.

Leakage loss:

The flow which leaks between the blade and shroud gap loses energy which later is re-energized again, leading to a loss in efficiency. This loss is modelled with Aungier’s [12] formulation:

(10)

˙

mcl= ρ2ZεLθUcl.

These parasitic losses only contribute to the isentropic efficiency, which is computed using the following equation: ηs= ∆hEuler− ∑ ∆hint ∆hact = ∆hEuler− ∑ ∆hint ∆hEuler+ ∑ ∆hint+ ∑ ∆hpst , (18)

where the term ∆hact is the actual input enthalpy change including the contribution from the parasitic losses.

4 RESULTS

The result from the implementation of the model by Oh et al [1] is given in the comparative performance maps below, where the estimated pressure ratio and efficiency is plotted as a function of the corrected mass flow rate with the measured data as reference. The pressure ratios are compared in the following figure:

Figure 4. Performance map comparison between Oh et al model and measured data, with regards to pressure ratio.

(11)

Figure 5. Performance map comparison between Oh et al model and measured data, with regards to isentropic

efficiency.

To quantify the prediction match to the measurements, a comparative analysis was conducted for Oh et al, through calculating differences of the predicted points compared to the reference data, using the following formula:

δ = |estimate − measured value|

measured value , (19)

with the following result:

(12)

Figure 6. Comparative analysis of Oh et al compared to reference data with regards to pressure ratio, including 5%

and 10% reference lines.

Figure 7. Comparative analysis of Oh et al compared to reference data with regards to isentropic efficiency, including

5% and 10% reference lines.

5 DISCUSSION

(13)

From Figure 4 it is also visible that the model by Oh et al produces an underestimate of the pressure ratio, with the model difference dominated by the 5-10% range as visualized by Figure 6. It can also be observed that the estimated values follow the same curvature/trend in the lower speed regions, as it gradually shifts to predict a straight line in the higher speed regions, as seen in the deviation development for higher rotational speeds in Figure 6. This leads to the conclusion that the model by Oh et al is to be used foremost in the low speed region, as done in the work of Oh et al [1], while still producing useful estimates in the higher RPM region.

It can also be seen that the model produced a representative estimate of the isentropic efficiency with slightly greater deviation, see Figures 5 and 7. It does not show the same trend as was the case with the pressure ratio, as the best estimate is produced in the mid range of the rotational speed, around 140-161 kRPM, as supported by Figure 5. Though, the same flattening of the efficiency can be observed, as was observed in the pressure ratio estimate at the highest rotational speed. The isentropic efficiency is generally underestimated at the lowest and the highest mass flow rate for each rotational speed until the transition to a flat estimate is reached. This is similar to the behaviour of the pressure ratio estimate, as well as the representative prediction of the curvature of each speed line.

The origin of this decrease in accuracy at rising rotational speed might be the simplifying assumptions made during the implementation, which have been mentioned previously. Due to the experimental empirical formulation of the loss terms, a wide range of actual losses and phenomena are not handled by the model. One such possible source of deviation might be the effect of the volute on the pressure ratio and isentropic efficiency, as that isn’t included in this model. The model by Oh et al doesn’t handle supersonic flow properties such as shocks, contrary to the similar model by Aungier [12], which might further decrease the model precision at higher velocities and rotational speeds, as used in this work. It is also seen that as the map moves away from the design points, the prediction drifts further away from the measured data, indicating off-design problems with the model. This drift of agreement is due to the relation between the loss model and the design point, where the model is fitted to agree with the design point. The pressure ratio of the worst prediction case is in the region of the ceiling where the loss terms have been verified, as stated in [1], which might contribute to the increase in difference to the reference data. Furthermore, the mass flow in the work by Oh et al [1] is significantly higher than in the present work, which might be another source for the differences compared to the measured reference data. These factors contribute to certain restrictions in the usability of this model. The difference between measured performance and the performance predicted by the model increases with operating conditions further from the design conditions. As such, performance predictions from the model will not be useful for many real world applications where off-design performance is of importance. In other words, other performance analysis methods (such as CFD) are to be recommended for off-design conditions. As a conclusion, the model by Oh et al can be considered a good model for performance prediction for compressors, even in the high RPM range compared to the values which it was developed for in [1], while CFD or other alternative methods are recommended for off-design analysis.

The implementation of the model by Oh et al presented a big flaw within the compressor performance prediction field, being the sparse information regarding many of the key concepts of the modelling. One of the most prominent examples is the Diffusion rate, which is a key parameter in determining the mixing loss (see section 3.2.1), where no information is given in the literature regarding the calculation process. The vaneless diffuser loss in the model was also given in the wrong form, causing the iterative processes to diverge. Equation (13) had to be reversed from the original loss term provided in [1], as the pressure in the diffuser outlet is greater than the pressure its inlet, consequently producing negative enthalpy changes and thereby increased pressure ratio in the iteration. Once the loss term was reversed, by swapping the two terms in the bracket, convergence to a feasible pressure ratio value was achieved and the loss term became positive, as it should be to provide a lower pressure ratio than the ideal. Most of the loss terms depend on the outlet conditions, as they are derived from experimental data, which makes the problem iterative. In many cases it was unclear which variables require iterative methods to obtain and which can be obtained analytically, without prior knowledge of performance modelling. Furthermore, there are numerous articles within this topic that use CFD as a base for obtaining analytically unknown parameters, which deviates from the purpose of the 1D models of avoiding the use of numerically expensive CFD in early design stages. These deficiencies lead to a less implementation friendly basis within this field of performance prediction, which can be improved by including further explanations of key factors and concepts. There are also a number of important references that are unavailable, as [5], due to them being presentations held in the past. Their content is only briefly touched upon by the more recent works, leaving out some essential information. This leads to increased hardship for the implementer, as some information is unobtainable or require combination of multiple articles to access, as was the case in this work with the parameter fincin Equation (4).

(14)

The models are described with different types of equations separately and not in the order in which they are used in the implementation, which made the implementation more difficult than it had to be. A lot of time has been spent laying a puzzle of equations as well as searching for the actual pieces of the puzzle.

ACKNOWLEDGMENTS

We would like to use this section to thank our mentor Mihai Mihaescu and assistant mentor Emelie Trigell for their continuous support and well meaning throughout the project. Also, the work of Sergio Sanz Solaesa [13] has been of great help, describing this complicated and somewhat mystified subject in a clearer, more direct perspective. We he have been fortunate to get the opportunity to work with and be mentored by bright and friendly minds as well as being introduced to the work being done at the Department of Mechanics at KTH Royal Institute of Technology.

REFERENCES

[1] _{H.W. Oh, E.S. Yoon, and M.K. Chung. An optimum set of loss models for performance prediction of centrifugal}

compressors. Proceedings of the Institution of Mechanical Engineers Part A Journal of Power and Energy, (211):331–338, 1997.

[2] _{NASA. Turbocharger. URL:https://commons.wikimedia.org/wiki/File:Turbocharger.jpg (online: accessed may 4,}

2020). 2003.

[3] _{H.W. Oh, E.S. Yoon, and M.K. Chung. Systematic two-zone modelling for performance prediction of centrifugal}

compressors. Proceedings of the Institution of Mechanical Engineers Part A Journal of Power and Energy, (216):75–87, 2002.

[4] _{F.J. Wiesner. A review of slip factors for centrifugal impellers. ASME J. Engng For Power, (89):558–572, 1967.} [5] _{Conrad O., Raif K., and Wessels M. The calculation of performance maps for centrifugal compressors with}

vane-island diffusers. In ASME Twenty-fifth Annual International Gas Turbine Conference and Twenty-second Annual Fluids Engineering Conference on Performance Prediction of Centrifugal Pumps and Compressors, New Orleans, Louisiana, pp. 227-234. New Orleans, Louisiana, March 1980, pp. 227-234, March 1980.

[6] _{J.E Coppage, F. Dallenbach, H. Eichenberger, G.E. Hlavaka, E.M Knoernschild, and N. Van Lee. Study of}

supersonic radial compressors for refrigeration and pressurization systems, WADC report 55-257. WADC report 55-257, 1956.

[7] _{W.; Jansen. A method for calculating the flow in a cenrifugal impeller when entropy gradients are present. Royal}

Society Conference on Internal Aerodynamics (Turbomachinery), 1967 (IME).

[8] _{Johnson J.P and R.C. Dean Jr. Losses in vaneless diffusers of centrifugal compressors and pumps: Analysis,}

experiment, and design. ASME J. Engng for Power, (88(1)):49–62, 1966.

[9] _{R.A van den Braembussche. Design and Analysis of Centrifugal compressors Chapter 3 Impeller Flow Calculation,}

von Karman Institute of Fluid Dynamics, Belgium, 2019, PP. 76. von Karman Institute of Fluid Dynamics, 2019.

[10] _{J.D. Stanitz. One-dimensional compressible flow in vaneless diffusers of radial- and mixed-flow centrifugal}

compressors, including effects of friction , heat transfer and area change. NACA-TN-2563. NACA-TN-2563, 1952.

[11] _{Daily J.W and Nece R.E. Chamber dimension effects on induced flow and frictional resistance of enclosed rotating}

disks. ASME J. Basic Engng, (82):217–232, 1960.

[12] _{Ronald H Aungier. Centrifugal Compressors - A Strategy for Aerodymanic Design and Analysis. The American}

Society of Mechanical Engineers, 2000.

[13] _{S. Sanz Solaesa. Analytical prediction of turbocharger compressor performance: A comparison of loss models}

(15)