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A numerical procedure to model heat transfer in radial turbines for automotive engines
Accepted Manuscript A Numerical Procedure to Model Heat Transfer in Radial Turbines for Automotive Engines Xunan Gao, Bojan Savic, Roland Baar PII: DOI: Reference:
S1359-4311(18)34701-X https://doi.org/10.1016/j.applthermaleng.2019.03.014 ATE 13443
To appear in:
Applied Thermal Engineering
Received Date: Accepted Date:
31 July 2018 4 March 2019
Please cite this article as: X. Gao, B. Savic, R. Baar, A Numerical Procedure to Model Heat Transfer in Radial Turbines for Automotive Engines, Applied Thermal Engineering (2019), doi: https://doi.org/10.1016/ j.applthermaleng.2019.03.014
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A Numerical Procedure to Model Heat Transfer in Radial Turbines for Automotive Engines Xunan Gao1, 2 , Bojan Savic1 , and Roland Baar1 Abstract—The thermal condition of a turbocharger considerably differs between different applications, which makes it difficult to model the heat transfer. In this study, a general procedure to characterize the heat transfer model for a radial turbine was established using 3D Conjugate Heat Transfer (CHT) simulation. The external convective and radiation heat losses were modelled by an equivalent convection heat transfer coefficient and an equivalent metal emissivity, following an analysis of the heat transfer mechanisms. The established model was validated by measurements under adiabatic and diabatic conditions. The characterization procedure was further simplified to extend its applications. The number of required operating points for simulation and measurements were significantly reduced, which enabled it to be used under different operating conditions, e.g., engine test and in-vehicle operations, where a constant turbo speed line could hardly be regulated. Then a comparison was conducted with a reference model based on the geometric simplification. The proposed model was observed to give a slightly better result regarding turbine housing temperatures under low and high turbine inlet temperatures. In the end, an analysis of turbine heat flow was conducted, which indicated that on average 80.4% of the turbine internal heat transfer occurred at the volute and 17.3% at the diffuser. Besides, the external heat losses can be up to 2.511 times of the turbine mechanical power. Keywords—Turbocharger, turbine, heat transfer, convection, radiation.
F
N OMENCLATURE
1
Cp P T η κ h q 0 1 12s 2 3 34s 4 act BH C conv ef f is mea mean oil rad s st T t TH
Increasing the fuel economy and decreasing the emissions are always among the most popular topics of the automotive industry. Turbocharging, together with other relevant technologies, offers a promising possibility to achieve these goals by improving the power density as well as the combustion processes. To take full advantage of these technologies, the matching of the turbocharging system to the engine becomes a critical step in engine development processes. For this task, the 1D simulation tools are usually employed because of its relatively low computational cost. Here the modelling of turbochargers relies on the turbine and compressor maps provided by the manufactures. These maps are measured on the hot gas test benches under characterization conditions described in the standards, such as SAE J922 [1] and SAE J1826 [2]. The turbine and compressor efficiencies are derived from the measured pressures and gas temperatures at the outlet and the inlet of the turbine and the compressor (Equation 1 and 2).
Specific heat capacity at constant pressure Power Temperature Efficiency Isentropic exponent of gas Enthalpy Heat flow per unit time Environment Compressor inlet Isentropic compression process of compressor Compressor outlet Turbine inlet Isentropic expansion process of turbine Turbine outlet Actual value Bearing housing Compressor Convective Effective Isentropic Measured value Mean value Lubrication oil Radiation Solid (temperature) Static value (temperature, pressure) Turbine Total value (temperature, pressure) Turbine housing
1 Chair of Powertrain Technologies, Technische Universit¨at Berlin (TU Berlin), Carnotstr. 1a, 10587 Berlin, Germany. 2 Corresponding author. E-mail:[email protected]
I NTRODUCTION
κ−1
ηC,is
( P2t ) κ − 1 ∆h12s T2st − T1t = = = P1tT2t ∆h12 T2t − T1t T1t − 1
ηT,is =
1 − TT4t ∆h34 T4t − T3t 3t = = κ−1 ∆h34s T4s − T3t 1 − ( PP4s ) κ 3t
(1)
(2)
During the characterization process, a certain amount of heat transfer occurs inside the turbocharger as well as between the turbocharger and the surroundings (Figure 1). Baar et al. [3] indicated that even with insulation, heat transfer in a hot gas test could not be neglected. It superimposes
2
would arise. The heat transfer parameters obtained at the turbocharger test bench could hardly represent the thermal conditions of the engine experiments or the in-vehicle operations. Being exposed to hot gases, the turbine is usually the main subject of heat transfer in a turbocharger. The goal of this study is to establish a general easy-to-use characterization procedure for the turbine, with which the heat transfer parameters could be determined, regardless of the thermal conditions of the turbocharger. In this paper, a detailed characterization procedure for the model of turbine heat transfer is proposed after analyzing the heat transfer mechanisms of turbochargers. Then the model is validated by measurement under adiabatic and diabatic conditions. To extend the scope of its application, the characterization procedure is further simplified, which enables to be applied for the turbocharger in engine tests and in-vehicle operations, where a constant turbo speed line is difficult to be regulated. After that, a comparison is conducted with a reference model based on the geometric simplification. In the end, an analysis of the turbine heat flow is carried out.
Fig. 1. Turbocharger heat flows
(a) Compressor
(b) Turbine
Fig. 2. Evolution processes of turbocharger
the measured turbine and compressor outlet temperatures (T2 and T4 ) from the actual values after the work transfer processes as shown in Figure 2. Since T10 , T20 , T30 and T40 can hardly be spatially identified or correctly recorded, the work transfer and heat transfer cannot be separated by the standard characterization process. This eventually leads to an underestimate of the compressor efficiency and an overestimate of the turbine efficiency in the turbine and compressor maps. The effects of heat transfer become especially significant under low-speed operating conditions with high gas temperatures. Under these conditions, the measured isentropic turbine efficiency can be even higher than one. Cormerais et al. [4] reported an underestimation of the isentropic compressor efficiency of up to 15% under a turbine inlet ¨ temperature of 500°C even with insulation. Ozdemir [5] conducted the CHT simulation for a turbocharger turbine and indicated that the total turbine heat loss could account for up to 82% of the total turbine enthalpy drop under the standard characterization condition (T3 =600°C). Therefore, heat transfer plays a significant role in the turbocharger characterization process. These maps contain a non-negligible fraction of heat transfer effects, and could not reflect the actual mechanical transfer performance of turbochargers. Using the traditional maps directly in a 1D simulation will deteriorate the matching of turbocharged engines. One should bear in mind that the degree of heat transfer is highly dependent on the surrounding thermal conditions. Serrano et al. [6] indicated that if the turbocharger operated under similar thermal conditions, with which the maps were measured, a reliable prediction of turbine outlet temperatures and turbo speeds could be obtained. However, once the thermal conditions deviate, an erroneous prediction
2
B IBLIOGRAPHY
In recent years, many studies were done to obtain a deep insight into the heat transfer phenomenon of the turbocharger. Bohn et al. [7] established a complex 3D CHT simulation model for the complete turbocharger, which contained the turbine, the compressor, and the oil-cooled bearing housing. The external heat transfer was modelled based on the temperature distribution of the casing surfaces derived from the measurement by a thermography camera. As a result, the temperature distribution of the casing and detailed quantification of the heat flux of the compressor were obtained. Roclawski et al. [8] conducted a CHT simulation for a centrifugal compressor. The simulated metal temperatures were observed to be in good agreement with experimental data. Serrano et al. [9] developed a special test bench in which the incompressible thermal oil flowed through the turbocharger. It enabled the determination of metal thermal properties since the fluid temperature change was only due to the heat transfer. Payri et al. [10] employed an analytical method to model the radiation heat transfer of turbocharger. The turbocharger was geometrically simplified as a dumbbell. The view factors between surfaces were calculated by analytical equations. The proposed model was then validated by engine tests with the maximum turbine inlet temperature of around 900°C. Results showed that, on the turbine side, most of the heat losses occurred through radiation. Besides, the most considerable contribution of the radiation heat loss came from the one from the turbine to the ambience, while the radiations from the turbine to other components were negligible in comparison. On the compressor side, though the radiation from the turbine to the compressor accounts for the most critical part of the total radiation heat flows, external convective heat transfer could not be neglected. Serrano et al. [11][12] then used this model to establish a 1D predictable model for turbochargers. A reliable prediction of the turbo speeds and the outlet gas temperatures of the
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turbine and the compressor were obtained. Nevertheless, this analytical method could only be applied with certain conditions, i.e., turbocharger test bench, where the surrounding hot surfaces could be clearly identified. Another attempt of the studies in this area is to embed the heat transfer model into engine simulations. One of the most widely used approaches is to build a thermal network model for 1D engine models. The 1D engine simulations are typically carried out based on a common assumption that the compression process of the compressor and the expansion process of the turbine are regarded as adiabatic processes. Some studies locate the heat transfer before and after the expansion and the compression processes, respectively. Additionally, the area ratio of convective surfaces before and after these work transfer processes should be specified. Burke et al. [13] assumed that 85% of the total turbine wetted area located before the expansion process and 15% after it. A more aggressive assumption was made by Romagnoli and Martinez-Botas [14] and Olmeda et al. [15] that all the heat transfer took place before the expansion process of the turbine and after the compression process of the compressor. Aghaali [16] included the turbocharger heat transfer in engine simulation by adding a heat tank before the turbine and a heat source after the compressor. The simulation results showed a reliable prediction of the turbine and compressor outlet temperatures without the implementation of the mass flow coefficient and the efficiency coefficient.
3 3.1
E XPERIMENT SETUP Test bench
All the measurements were conducted on the turbocharger hot gas test bench at the Chair of Powertrain Technologies at the Technical University of Berlin. Figure 3 shows the layout of the turbocharger test bench. Three electronic compressors are used to supply the mass flow for the turbine. The gas is heated by either an electronic heater or a combustion chamber. The exhaust and turbocharger metal temperatures are recorded by thermocouples. PT100 sensors are used to measure the temperatures of air, lubrication oil and cooling water. To take into account the non-uniform flow field, at each of the measuring positions before and after the turbomachinery, temperature and pressure values are measured by three thermocouples and four pressure pipes which are azimuthally installed. This test bench enables to measure over a wide range of operating conditions regarding turbine inlet temperatures (ambient to 1050°C), mass flow rates (up to 1400kg/h) and pressures (maximal electronic compressor pressure up to 4 bar). The wastegate valve is closed during the measurements in this study. 3.2
Adiabatic measurement
Under standard characterization conditions, the turbine is exposed to hot gas heated by the combustion chamber. The measured enthalpy changes of the turbine and the compressor are the combinations of work transfer and heat transfer. Thus the obtained efficiency values do not reflect the actual aerodynamic performance of turbocharger. To
eliminate the effects of heat transfer, experiments were first performed under the adiabatic condition employed by Baar et al. [3] as shown in Equation 3.
T3 = Toil,mean and T4 = T0
(3)
where, T3 , T4 and T0 represent the gas temperatures of the turbine inlet, the turbine outlet and the ambience. Toil,mean is the mean temperature of lubrication oil. The intention behind is to minimize the heat transfer from the turbine to the bearing housing as well as to the ambience. Furthermore, all of the turbocharger surfaces and measurement pipes are fully insulated. Figure 4 shows the turbocharger setup at the hot gas test bench. Thus the question is how to quantify the heat flows that are still left. To do this, a power-based criterion was applied, which was firstly proposed by Baar et al. [17]. Zimmermann et al. [18] tested it on a couple of different turbochargers with different sizes and configurations to verify its reliability. Based on the previous studies, Savic et al. [19] extended it, which enabled a direct determination of the isentropic turbine efficiencies and isentropic compressor efficiencies from hot gas test data. Savic et al. [20] modified the approach and showed a comparison to a standard CHT-simulation regarding isentropic turbine efficiencies. The modification was carried out because other heat flow influences were observed to have non-negligible influences. According to this criterion, the measurement data is plotted in Figure 5. The x-axis is the isentropic compressor power (Pc,is ) derived from Equation 4 and the y-axis is the effective turbine power (Pt,ef f ) determined through the temperature difference of the turbine inlet and outlet flow (Equation 5). k−1 Pc,is = m ˙ c · Cp,c · T1 · πc k − 1 (4)
Pt,ef f = m ˙ t · Cp,t · (T3 − T4 )
(5)
Pc,is is chosen as x-axis because it is the most reliable value in the measurement. It can be seen from Equation 4 that, at a certain compressor operating point with a defined mass flow rate (m ˙ c ) and pressure ratio (πc ), Pc,is is the function of compressor inlet temperature (T1 ). T1 is barely affected by the thermal condition of turbine side. This is justified by the work done by Bohn et al. [7], where a CHT simulation was conducted for the complete turbocharger. The heat flux through compressor internal surfaces was invested on a longitudinal section under different compressor and turbine operating conditions, which indicated that a negligible amount of heat flux occurred before the compressor impeller. This also helps explain the assumption employed by Olmeda et al. [15] and Baines et al. [21] that all of the compressor heat transfer was located after compressor impeller when modeling turbocharger. On the compressor side, Shaaban [22] indicated that at a given operating point the heat transfer to compressor led to an increase in compressor power demand (i.e. Pc,ef f ). The increase of Pc,ef f depends on the heat flow to compressor as well as the fraction of heat transfer before impeller. Since the amount of heat transfer before compressor impeller is negligible, the actual Pc,ef f can be assumed to be unchanged
4
Fig. 3. Layout of the turbocharger test bench
under different thermal conditions. Therefore, the deviation of the measured isentropic compressor efficiency from the actual value results only from the effects of heat transfer. On the turbine side, the measured turbine power can be divided into two parts. One part is transferred to mechanical power to drive the shaft. The other part is associated with the heat flows within the turbocharger as well as from the turbine to the environment. If the turbocharger operates under ideal adiabatic conditions, the turbine enthalpy drop should be entirely converted to shaft power. In Figure 5 a straight line is added by fitting the operating points with maximum Pc,is of each speed line. This fitting line is then extended to cut the axis of ordinates. Since the ordinate intercept also means a speed of zero, there is no friction power or turbine power at this point. The ordinate intercept thereby results from heat losses of the turbine side. The ordinate intercept of the adiabatic measurement in this study is 0.2561kW, which indicates a good quality of the adiabatic measurement. Here, the points with maximum Pc,is are chosen to generate the fitting line because they are at the middle region of a turbine map, which would not be restricted by the limitations of the test bench. It is worth mentioning that, we have currently found that the maximum Pc,is are not perfectly linear. Nevertheless, adopting a straight line here is reasonable considering that it is the very first step of this approach and the goal is to develop an easy-to-use model. 3.3
Fig. 4. Turbocharger setup at test bench
Diabatic measurement
The second phase of the measurements was conducted under the standard conditions according to SAE J922 [1] and
Fig. 5. Turbine power-based criterion
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TABLE 1 Mesh statistics
Fig. 6. thermocouple setup on the turbine housing
Fluid domain Component Inlet extend Volute Impeller (Fluid) Diffuser Mixing device Outlet extend
Mesh type Hex. Tet.+ Prism Hex. Tet.+ Prism Tet.+ Prism Hex.
Number of elements 0.454 mil. 1.622 mil. 1.901 mil. 1.687 mil. 0.919 mil. 0.215 mil.
Solid domain Component Impeller (Solid) Turbine housing
Mesh type Tet. Tet.
Number of elements 0.576 mil. 1.732 mil.
Total
9.106 mil.
Fig. 7. CHT simulation model
J1826 [2], but with different turbine inlet gas temperatures (T3 =400°C, 600°C, 800°C and 950°C). The mean temperature of cooling water and lubrication oil was regulated to 90°C for all the operating points. In order to record the solid temperatures, 5 thermocouples (Ts,1 , Ts,2 , Ts,3 , Ts,4 and Ts,5 ) were installed on the turbine housing. 3 additional thermocouples (TF lange,aT H , TF lange,bT H and TF lange,T HLH ) were fixed at the flanges which connected the turbine housing with the inlet pipe, the bearing housing and the outlet pipe respectively to serve as boundary conditions for the CHT simulation. Figure 6 depicts the installation of thermocouples on the turbine housing.
4
N UMERICAL MODELLING
The commercial code ANSYS CFX 17.2 was employed as the numerical solver, which used the element-based finite volume method. The CHT model consists of fluid and solid domains as shown in Figure 7. 4.1
Grid generation
In the fluid domain, the rotational region was discretized by a structural hexahedral grid. Figure 8 shows the mesh of a single passage. For the volute and diffuser regions, an unstructured tetrahedral grid was employed along with 10 prism layers in the boundary layers. To obtain a high resolution of boundary layer flow and heat transfer phenomena, the dimensionless wall distance (y + ) was regulated to be less than 10 for fluid domains. The turbine housing was meshed by a tetrahedral grid. A grid independence study was carried out to avoid grid-induced errors. To eliminate the effect of the computational boundaries on the simulation results, the computational domains of the turbine inlet and outlet were extended.
Fig. 8. Grid of the fluid domain of impeller
4.2
General settings
In the fluid domains, the steady-state RANS (Reynolds Averaged Navier-stokes) equations were solved. The closure of the equations was brought by the SST (Menter‘s Shear Stress Transport) turbulence model, which is a combination of the k –ω model and the k –ε model. Specifically, in the near-wall region the k –ω model is used to ensure a reliable prediction of the onset and the amount of flow separation under adverse pressure gradients. In the free-shear layer flows, the k –ε model is employed since it is the most widely used turbulence model for flows containing relatively small pressure gradients for industrial applications. In the solid domains, the energy equation was simultaneously solved. The Conservative Interface Flux algorithm was set for heat transfer at the interface between the solid and fluid domains. With this method, the temperatures and the heat flows of interfaces are the direct results of the simulation. The specification of the convective heat transfer coefficient is superfluous. For the turbine inlet, the total pressure and total temperature were set as boundary conditions. For the turbine outlet, the static pressure was specified in accordance with experimental data. Three additional thermocouples fixed at the flanges of turbine housing serve as boundary conditions for the CHT simulation. Table 2 shows an overview of the boundary conditions and their ranges in this study.
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TABLE 2 Boundary conditions of the CHT model
Location Inlet Outlet Impeller Flange Ambience
4.3
Boundary condition Total temperature (T3t ) Total Pressure (P3t ) Static pressure (P4 ) Speed (nT C ) Temperature (TF lange ) Temperature (Tamb )
°C kPa kPa 103 /min °C °C
44.9 - 79.5 162 - 330 101 - 102 123 - 186 20.4 - 73.0 22.4
Modelling of turbine heat transfer
Heat transfer occurs in the turbocharger by conduction, convection and radiation. When establishing the heat transfer model for the turbine, heat transfer boundary conditions should be carefully considered. As for the conduction heat transfer between the turbine housing and the bearing housing as well as the connected pipes, a constant temperature boundary condition was applied. The temperatures were provided by three additional thermocouples installed at the flanges of the turbine housing. As for the convective heat transfer, the internal convective heat transfer, which took place between the turbine housing and the working gas, was the direct results of simulation since the fluid and solid domains were simultaneously solved and a conservative heat flux boundary condition was applied to the interface. The external convective heat transfer was modelled by a userspecified heat transfer coefficient based on the Newton‘s Law of Cooling (Equation 6). In this study, we assume that the surrounding flow is dominated by forced convection. Therefore, the area-averaged equivalent convective heat transfer coefficient he is constant under a certain application condition. A is the area of the object. Ts is the solid surface temperature. Tf is the ambient temperature. Obviously, here he is the control parameter.
qconv = he · A · (Ts − Tf )
(6)
The radiation heat loss of turbine is modelled by the Stefan-Boltzmann Law. Accordingly, the radiation energy per unit time of a grey body can be expressed by:
qrad = ε · σ · A · T 4
(7)
where ε is the metal emissivity, σ is the Stefan-Boltzmann constant, A is the area of the body, T is the absolute temperature of the grey body in Kelvin. The net radiation heat loss rate of a grey body with the cold surroundings can be expressed by:
qrad = ε · σ · A · T 4 − Tc 4
Range Diabatic test
Adiabatic test
(8)
where Tc is the absolute temperature of the cold surroundings in Kelvin. The radiation heat transfer rate between two grey-body surfaces is given by: σ · T1 4 − T2 4 qrad,1→2 = 1−ε1 (9) 1−ε2 1 A1 ·ε1 + A1 ·F1→2 + A2 ·ε2
400 142 - 335 100 - 101 121 - 217 116.2 - 331.5 18.4
600 122 - 360 100 - 103 92 - 234 171.1 - 484.3 18.9
800 123 - 332 101 - 103 92 - 234 219.5 - 617.6 21.2
950 121 - 313 100 - 101 92 - 234 254.1 - 710.6 22.9
where ε1 and ε2 are the metal emissivity of two surfaces. F1→2 is the view factor from surface 1 to surface 2, which is determined by the spatial configuration of the two surfaces. Therefore, the emissivity ε and the view factor F1→2 are the control parameters here. As mentioned before, the heat transfer conditions of the turbine differ among various applications, e.g., turbocharger test bench, engine test bench, in-vehicle operations, which results in different heat transfer parameters. The net radiation heat loss of the turbine housing, for example, is strongly dependent on the hot surfaces around as well as the ambient temperature. Their influences are difficult to be quantified. In this study, we intend to develop a general characterization procedure of the heat transfer parameters for a radial turbine that could be used in a wide range of applications. To do this, we modified Equation 8 to be as Equation 10, where εe is the equivalent emissivity, which combines the effects of metal emissivity and view factor. To keep it as simple as possible, the ambient temperature Tc was removed from the equation as well. Here, the equivalent emissivity εe becomes the control parameter for the radiation heat transfer.
qrad = εe · σ · A · T 4
(10)
Therefore, establishing the turbine heat transfer model can be represented by determining the characteristic parameters (i.e., he and εe ) of the convective and radiation heat transfer. A complete characterization procedure of these parameters is carried out in the next section.
5 5.1
C HARACTERIZATION PROCEDURE Determining convective and radiation parameters
Based on the analysis in previous sections, the independent heat transfer parameters of this study are the equivalent convective heat transfer coefficient he and the equivalent metal emissivity εe . It is obvious from Equations 6 and 10 that their influences on the calculated heat transfer differ among various thermal conditions. To get a comprehensive understanding, we first conducted the CHT simulations with different he and εe . Under each turbine inlet gas temperature T3 , one representative operating point is selected from the mediumspeed line as shown in Table 3. The selected operating points have the maximum isentropic compressor power (Pc,is ) at their speed lines. This is to ensure that these points are at
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TABLE 3 Representative operating points
No. 1 2 3 4
Speed (1/min) 184515 184408 186373 186441
T3 (°C) 400 600 800 950
Pressure ratio (-) 2.463 2.313 2.207 2.145
the middle region of each speed line, which will not be influenced by the test bench limitations. Figure 9 depicts the mean deviation of the simulated turbine housing temperatures from the measurement data against turbine inlet gas temperatures under different he . ¯ s is defined by EquaThe mean temperature deviation ∆T tion 11. Ts,i,sim and Ts,i,mea represent the turbine housing temperatures from simulation and measurement data at five different measuring positions as shown in Figure 6.
(a) he = 30 W/(m2 · K)
5
X ¯s=1· ∆T (Ts,i,sim − Ts,i,mea ) 5 i=1
(11)
The line of εe = 0 represents the simulation without taking into account the radiation heat transfer. The mean deviation significantly increases with the increase of turbine inlet temperature. It also shows the necessity to take into consideration the radiation heat transfer by CHT simulation. The line of εe = 1 represents the simulation considering the turbine housing as a black-body and ignoring the effects of surrounding surfaces. Also, a considerable deviation occurs. It is clear that he is the crucial factor in determining the solid temperature deviation under low turbine inlet temperature. While under high turbine inlet temperatures the equivalent emissivity εe plays a significant role. It is also expected that an optimal combination of he and εe exists, which results in the minimum mean deviation of turbine housing temperatures under all turbine inlet temperatures (T3 ). To deduce this optimal combination, the simulation results were reprocessed as Figure 10, which shows the ¯ s against the equivalent mean temperature deviation ∆T emissivity εe . For clarity, only the simulation results under T3 = 400°C and 950°C are plotted. A linear and a secondorder curve are created by fitting to the simulation results under T3 = 400°C and T3 = 950°C, respectively. The fitting lines of T3 = 400°C and T3 = 950°C with the same he intercept at a certain εe . At this interception point, the mean temperature deviations under both T3 are equal. Then, a second-order curve (red, dash) is generated by fitting to all the interception points of different h3 . Along this curve, the mean temperature deviations under different T3 remain ¯ s is equal. Thus, at its intersection with the x-axis the ∆T expected to be zero for all the T3 . In this study, the line cuts the x-axis with an intersection (Equivalent emissivity, εe ) of 0.122. To get the corresponding he of the intersection point, all of the intersection points of different he in Figure 10 are plotted in Figure 11 , which shows the mean temperature deviation against he . It is clear that a straight fitting line can be generated, which cuts the
(b) he = 35 W/(m2 · K)
(c) he = 40 W/(m2 · K)
(d) he = 70 W/(m2 · K) Fig. 9. Mean solid temperature deviation of turbine housing
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outlet gas temperature and isentropic turbine efficiency between simulation and measurement, respectively. The simulation results correspond well with the experimental data, which indicates the good reliability of the simulation model in aerodynamics.
Fig. 10. Mean temperature deviation vs. equivalent emissivity
(a) M ass f low parameter
Fig. 11. Mean temperature deviation vs. equivalent heat transfer coefficient
x-axis with an intersection (he ) of 48.974 W/(m2 · K). Here we obtain a combination of εe and he directly by plotting, which is expected to result in the minimum mean temperature deviations under all T3 . A preliminary evaluation was carried out by running the simulation with the resultant combination of εe and he under different T3 . Figure 12 depicts the result of the preliminary evaluation, which is in good agreement with the expectation. In the next section, a detailed validation will be conducted.
(b) T urbine outlet temperature
(c) Isentropic turbine ef f iciency Fig. 13. Comparison of simulation results and measurement data under adiabatic condition Fig. 12. Preliminary evaluation of the resultant parameters
5.2
Validation
5.2.1 Aerodynamic validation Simulations under adiabatic conditions were carried out, with which the turbine external surfaces were set as adiabatic. Figure 13 shows the comparison of mass flow, turbine
5.2.2 Thermodynamic validation The optimal combination of εe and he previously obtained was applied to the CHT simulation under diabatic conditions. Simulations were performed at all of the operating points under different turbine inlet temperatures. An evaluation was made regarding heat flow rate. According to the previously mentioned power-based criterion, the measurement data under different turbine inlet
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temperatures are depicted in Figure 14. Additionally, the adiabatic measurement data is added as a reference. Here the power-based criterion offers the unique possibility to quantify the total turbine heat losses. As marked in Figure 14, the vertical difference of diabatic and adiabatic measurement data is the total heat loss of the turbine. Figure 15 shows the comparison of the simulated turbine heat losses with the measurement data, which are derived from the power-based criterion. For clarity, only the operating points with the largest isentropic compressor power of each speed lines are plotted. It is shown that the simulation results correspond well with the measurement data. Nevertheless, one should bear in mind that the fitting curves employed by the power-based criterion in this study are straight lines. However, we currently find that the points with the maximum Pc,is are not perfectly linear. This may have a slight influence on the derived turbine heat losses. It is especially obvious under high T3 (e.g., 950°C) that the points with maximum Pc,is at both low and high speeds are slightly below the fitting line. A higher-degree polynomial is expected to deliver a better approximation. We are conducting intensive investigations on this topic. More details may be found in our future publications. 5.3
(a) T3 = 400°C
Simplification of the proposed procedure
In the previous sections, a characterization procedure of heat transfer parameters for the turbine was established and validated. It enables reliable modelling of the turbine heat transfer. Table 4 shows the number of operating points for the simulations and measurements that were used in the characterization procedure. It is clear that a large effort has to be devoted to both the simulation and the measurements. In this section, the proposed procedure is further simplified to extend applications to various thermal conditions, i.e., turbocharger hot gas test bench, engine test bench, invehicle operations, where the requirements of operating points may hardly be fulfilled.
(b) T3 = 600°C
TABLE 4 Operating points of the characterization procedure
ITEM
T3
LEVELS
4
ITEM
T3
LEVELS
4
Measurement Speed lines Operating points (OPs) (per T3 ) (per Speed line) 8 8 OPs (per T3 ) 1
Simulation he (per OP) 4
εe (per he ) 8
Total OPs
(c) T3 = 800°C
256
Total OPs 128
To simplify the characterization procedure, we first reconsider the simulation campaigns. In the previous original procedure, the combination of heat transfer parameters (εe and he ) were derived from Figure 10 and Figure 11, in which only the simulation results under the highest (950°C) and the lowest T3 (400°C) were used. Figure 12 indicates that the simulations under these two T3 are sufficient to deliver a reliable result for different T3 . As for each T3 , a second-order curve (red, dash) was generated in Figure 10 by fitting to the intercept points of each he . Here, the levels
(d) T3 = 950°C Fig. 14. Measurement data according to the power-based criterion
10
TABLE 6 Validation of the simplified characterization procedure
Procedure Original Simplified
6
Fig. 15. Comparison of the simulated turbine heat losses with measurement data
of he can be reduced from 4 to 3, comparing to the original characterization procedure. For each he , in Figure 10 a straight line (400°C) and a second-order curve (950°C) are generated by fitting to the points of different εe . Therefore, 3 levels of equivalent emissivity εe (i.e., 0, 0.5 and 1) are sufficient to produce the fitting curves. As a conclusion, with the simplified procedure, the simulation should be carried out at 18 operating points. Accordingly, measurement is required to be carried out at only two operating points with different T3 (low and high temperatures). Table 5 shows the operating points for simulations and measurements of the simplified procedure. It is clear that the simplified procedure significantly reduces the number of operating points of the simulation and measurement. Furthermore, since the measurement data is only required at two operating points, it enables the proposed characterization procedure to be used in a wide range of applications, i.e., engine test or in-vehicle operations, where a constant turbo speed line is difficult to be regulated. The evaluation of its applications for other thermal conditions will be discussed in our future studies. TABLE 5 Operating points of the simplified characterization procedure
ITEM
T3
LEVELS
2
ITEM
T3
LEVELS
2
Measurement Speed lines Operating points (OPs) (per T3 ) (per Speed line) 1 1 OPs (per T3 ) 1
Simulation he (per OP) 3
εe (per he ) 3
Total OPs 2
Total OPs 18
To validate its reliability, the simplified procedure was then employed for the characterization of heat transfer parameters. Table 6 shows the heat transfer parameters (εe and he ) as well as the mean temperature deviations derived from the original and the simplified procedures. The simplified procedure delivers reliable results with a temperature deviation which is comparable to that of the original characterization procedure.
Parameters he εe 48.974 0.122 49.028 0.125
Mean solid temperature deviation (°C) T3 =400°C 600°C 800°C 950°C -0.387 -1.662 -1.897 -1.444 -0.458 -1.807 -2.154 -1.842
C OMPARISON WITH THE REFERENCE MODEL
In this section, to further evaluate the reliability of the established characterization procedure, a reference model is established based on the CHT model in this study. The internal heat transfer, between the turbine housing and the working gas, is resolved in the same way as was mentioned in previous sections. With regards to the external heat transfer, the parameters for the radiation and convection heat transfer are determined by a method proposed by Payri et al. [10]. The modelling of the external heat transfer is explained in section 6.1. And a comparison of the simulated turbine housing temperatures is presented in section 6.2. 6.1 Modelling of the external heat transfer (reference model) The external heat transfer to be modelled here includes radiation and convection. Both of them rely on a geometric simplification of the turbocharger. The radiation is modelled by determining the view factor from the turbine housing to the ambience using analytic equations. The convection is modelled by determining its non-dimensional parameters. 6.1.1 External radiation According to the Stefan-Boltzmann Law, the heat flow rate between two grey-body surfaces is given by Equation 9. In the reference model, the emissivity ε is specified consistently with the material property. The key factor of modelling radiation heat transfer is the determination of the the view factors between different surfaces. As is explained in section 2 and section 4.3, here only the radiation between the turbine housing and the ambience is modelled. Firstly, the turbocharger is geometrically simplified as three cylinders (Figure 16), similarly as done by Payri et al. [10]. The turbine surfaces include one interior surface and two exterior surfaces. The view factor from turbine exterior surfaces and to the ambience is equal to unity. As for the turbine interior surface, possible radiation connections are: • • •
Between interior surfaces of turbine and compressor (Ft→c ). Turbine interior surface with bearing housing surface (Ft→b ). Turbine interior surface with ambience (Ft→amb ).
These view factors can be calculated using the analytic expressions in Table 7 as described by Payri et al. [10]: 1)
Ft→c : Being calculated by the equation for view factor between two concentric discs separated by a concentric cylinder (case 1).
11
turbine cylinder. g = 9.8m/s2 is the acceleration due to Earth’s gravity, β = 2/(Tw + Tamb ) is the coefficient of thermal expansion, u is the air velocity in test bench, ν is the kinematic viscosity. Tw and Tamb are temperatures of the turbine housing and the ambience, respectively. The range of Ri indicates that in this case neither natural nor forced convection is predominant. A combined convection heat transfer should be considered. The heat transfer coefficient of the combined convection can be calculated by Equation 17. Fig. 16. Geometric simplification of turbocharger
2)
Ft→b : Firstly, the view factor from the bearing housing surface to the turbine interior surface (Fb→t ) can be calculated by the equation for view factor between a ring and a lateral surface (case 2). Then, Ft→b can be obtained by applying the reciprocity property (Equation 12) : Ft→b = Fb→t ·
3)
3
Ab 4 · Lb · Db = Fb→t · (12) At,inner Dt 2 − Db 2
where, Ab is the area of the bearing housing surface. At,inner is the area of the turbine housing interior surface. Lb is the length of the bearing housing. Dt and Db are the diameters of the turbine housing and the bearing housing, respectively. Ft→amb : Being calculated by the summation property given by Equation 13:
Ft→amb = 1 − Ft→c − Ft→b
(13)
Therefore, the radiation heat flux from turbine interior surface to the ambience can be computed by Equation 14. The total radiation heat flux from the turbine to the ambience is calculated by Equation 15.
q˙rad,t,inner =
q˙rad,t =
ε · Ft→amb ε · (1 − Ft→amb ) + Ft→amb · At,inner · σ · (Tw 4 − Tamb 4 )
(14)
1 · [qrad,t,inner · At,inner + (At − At,inner ) At (15) ·σ · (Tw 4 − Tamb 4 )
6.1.2 External convection The geometric simplification used in this section is the same as for the modelling of external radiation. Firstly, a preliminary estimate of Richardson Number Ri is performed as Equation 16:
0.6214 ≤ Ri = GrD /ReD 2 ≤ 1.0441
3
N uD = N uD,N C + N uD,F C
(16)
where Grashof number GrD = g · β · (Tw − Tamb ) · D3 /ν 2 and Reynolds number ReD = u · D/ν are non-dimensional parameters based on the diameter D of the simplified
3
(17)
where N uD,N C and N uD,F C are the area-averaged Nusselt numbers of natural and forced convection, respectively. N uD,N C is determined by Equation 18, which is correspond to the natural convection of a horizontal cylinder.
( N uD,N C =
0.387 · RaD 1/6 0.60 + [1 + (0.559/P r)9/26 ]8/27
)2 (18)
where RaD = GrD · P r is the Rayleigh number, P r = ν/α is the Prandtl number. α is the thermal diffusivity. N uD,F C can be calculated by the Churchill–Bernstein correlation (Equation 19), which is used to estimate the areaaveraged Nusselt number of a cylinder in cross flow.
0.62 · ReD 1/2 · P r1/3 ReD 5/8 4/5 ·[1+( ) ] 2/3 1/4 282000 [1 + (0.4/P r) ] (19) The obtained parameters of radiation and convection are set as boundary conditions for the CHT simulation. A comparison between the proposed and the reference models is carried out in the next section. N uD,F C = 0.3+
6.2 Comparison of simulated turbine housing temperatures Figure 17–20 show the comparison of measured and simulated turbine housing temperatures under T3 =400°C, 600°C, 800°C and 950°C, respectively. The positions of the thermocouples are defined by Figure 6. Two series of simulation results presented here are derived from the proposed and the reference model, respectively. Both of the simulation models are observed to give a comparable result, which is in good agreement with the measurement data. Under T3 = 400°C and T3 = 950°C, a slightly better result can be obtained by applying the proposed model. Another factor should be considered is the reliability of the simulation model when being used under different application conditions. The characteristic parameters for the proposed model are determined by using CHT simulation and measurement data regardless of the thermal conditions, under which the turbocharger operates. For this reason, the proposed model is estimated to capture the heat transfer behaviour accurately. Its reliability under different application conditions will be evaluated in our following publications.
12
TABLE 7 Analytic expressions of the view factors
Case
Layout
View factor
Parameters r1 h R2 = rh2 Rc = rhc A = R1 2
R1 = h
A 2
cos−1
Rc R2
− (1 +
C 2 )(1
+
1 π·A
1
Rc B cos−1 R + 2Rc (tan−1 Y − tan−1 2 1 h i 1/2 (1+C 2 )(Y 2 −D 2 ) 2 −1 D ) · tan (1+D 2 )(C 2 −Y 2 )
+
A1/2 − tan−1 B 1/2 )
1/2 [1+(R1 +Rc )2 ](R1 −Rc ) 2 [1+(R1 −Rc ) ](R1 +Rc ) 1/2 # 1/2 [1+(R2 +Rc )2 ](R2 −Rc ) 2 2 −1 + 1 + (R2 + Rc ) 1 + (R2 − Rc ) · tan [1+(R2 −Rc )2 ](R2 +Rc )
+
1 + (R1 + Rc )2
1 + (R1 − Rc )2
1/2
· tan−1
− Rc 2
2
B = R2 − Rc 2 C = R2 + R1 D = R2 − R1 Y = A1/2 + B 1/2 r1 r2 H = rh 2 A = H2
R=
2
B 8RH
+
1 2π
cos−1
A B
−
1 2H
h
(A+2)2 R2
i1/2 −4 cos−1 AR − B
A 2RH
sin−1 R
+ R2 − 1
B = H 2 − R2 + 1
7
S IMULATION RESULTS
In this section, two phases of analyses are carried out regarding the turbine heat flow derived from the proposed model. The first one focuses on the influence of turbine inlet temperatures on the distribution of heat losses of the turbine housing internal surfaces. The second one shows the amount of turbine external heat losses under different T3 as well as its compositions. 7.1
Distribution of the turbine internal heat flow
Figure 21 shows the temperature distribution of turbine housing at one operating point under T3 = 600°C. The heat transfer between working gas and turbine housing occurs at the turbine internal surfaces before (Volute to TH), during (Rotor to TH) and after (Diffuser to TH) the expansion process. Figure 22 shows the proportion of turbine internal heat flow (Q/Qinternal ) against the turbine inlet temperature (T3 ). Qinternal is the total heat flow at turbine internal surfaces. Most of the turbine internal heat transfer takes place at the volute, accounting for an average of 80.4% of total internal heat flow under different T3 and up to 97.9% at high-speed conditions under T3 = 400°C. The proportion of heat transfer at the rotor is negligible (below 7%) under all T3 . 7.2
Turbine external heat losses
Figure 23 shows the ratio of the external heat losses (Q) to the turbine mechanical power (Wt ) against T3 . The total external heat losses increase significantly with the increase of T3 , rising from an average of 0.180 to up to 2.511 times of the turbine mechanical power. The maximum value corresponds to the operating condition with the highest pressure ratio of the lowest speed line under T3 = 950°C. Besides, the range of Q/Wt has a strong relationship with the turbine inlet temperature, varying from 0.442 under T3 = 400°C to 2.332 under T3 = 950°C. This also indicates the importance of taking into consideration the effects of heat transfer when modelling the turbocharger.
8
C ONCLUSIONS
A general characterization procedure to model the turbine heat transfer was established using 3D CHT simulation. The conduction heat loss of turbine housing was modelled by specifying the flange temperatures according to the measurements. After analyzing the heat transfer mechanism, the modelling of external convective and radiation heat transfer was represented by determining the equivalent convective heat transfer coefficient (he ) and the equivalent metal emissivity (εe ). The proposed model was then validated by measurements conducted at the turbocharger hot gas test bench under both adiabatic and diabatic conditions. A previously introduced power-based criterion was employed to verify the quality of the adiabatic measurement as well as to quantify the turbine heat loss of diabatic measurements. The proposed characterization procedure was further simplified. The number of required operating points was significantly reduced (from 128 to 18 for the simulation and from 256 to 2 for the measurement). It made the proposed procedure to be a general procedure which is suitable for a wide range of applications, e.g., engine test and in-vehicle operations, where a constant turbo speed line is difficult to be adjusted. A reference model based on geometric simplification was introduced, with which the external radiation was modelled using analytic equations. The external convection was modelled by determining the non-dimensional parameters. Both models gave comparable results regarding turbine housing temperatures. A slightly better result under low and high turbine inlet temperatures was observed with the proposed model. The analysis of turbine heat flow indicated that the turbine internal heat transfer mainly occurred at the volute (80.4% on average), comparing to the diffuser with an average of 17.3%. The external heat flow, which increases significantly with the increase of turbine inlet temperature, can be up to 2.511 times of the turbine mechanical power.
13
(a) Ts,1
(a) Ts,1
(b) Ts,2
(b) Ts,2
(c) Ts,3
(c) Ts,3
(d) Ts,4
(d) Ts,4
(e) Ts,5
(e) Ts,5
Fig. 17. Comparison of measured and simulated turbine housing temperatures (T3 =400°C)
Fig. 18. Comparison of measured and simulated turbine housing temperatures (T3 =600°C)
14
(a) Ts,1
(a) Ts,1
(b) Ts,2
(b) Ts,2
(c) Ts,3
(c) Ts,3
(d) Ts,4
(d) Ts,4
(e) Ts,5
(e) Ts,5
Fig. 19. Comparison of measured and simulated turbine housing temperatures (T3 =800°C)
Fig. 20. Comparison of measured and simulated turbine housing temperatures (T3 =950°C)
15
Fig. 21. Temperature distribution of turbine housing Fig. 23. Turbine external heat losses to the ambience
[8]
[9]
[10] Fig. 22. Distribution of turbine internal heat flow
[11]
ACKNOWLEDGMENTS The authors would like to thank the support from the ANSYS Germany GmbH. Xunan Gao is supported by a scholarship from the China Scholarship Council (CSC).
R EFERENCES [1] [2] [3]
[4]
[5]
[6]
[7]
SAE, “J922–turbocharger nomenclature and terminology,” SAE standard, 2011. ——, “J1826–turbocharger gas stand test code,” SAE standard, 1995. R. Baar, C. Biet, V. Boxberger, and R. Zimmermann, “New evaluation of turbocharger components based on turbine outlet temperature measurements in adiabatic conditions,” IQPC-Tagung DOWNSIZING & TURBOCHARGING. Dusseldorf, Germany, 2014. M. Cormerais, J.-F. Hetet, P. Chesse, and A. Maiboom, “Heat transfer analysis in a turbocharger compressor: modeling and experiments,” SAE Technical Paper, Tech. Rep., 2006. ¨ S. Ozdemir, “W¨armestromanalyse der Radialturbinenstufe eines PKW-Abgasturboladers mittels numerischer Simulation,” Ph.D. dissertation, Technische ¨ Universit¨ at Berlin, 2017. J. R. Serrano, P. Olmeda, F. J. Arnau, A. Dombrovsky, and L. Smith, “Analysis and Methodology to Characterize Heat Transfer Phenomena in Automotive Turbochargers,” Journal of Engineering for Gas Turbines and Power, 2014. D. Bohn, T. Heuer, and K. Kusterer, “Conjugate Flow and Heat Transfer Investigation of a Turbo Charger: Part I Numerical Results,” in Volume 3: Turbo Expo 2003. ASME, jan 2003, pp. 715–722.
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
¨ H. Roclawski, C. Oberste-Brandenburg, and M. Bohle, “Conjugate Heat Transfer Analysis of a Centrifugal Compressor for Turbocharger Applications,” 2016. J. Serrano, P. Olmeda, A. Paez, and F. Vidal, “An experimental procedure to determine heat transfer properties of turbochargers,” Measurement Science and Technology, vol. 21, no. 3, p. 035109, 2010. F. Payri, P. Olmeda, F. J. Arnau, A. Dombrovsky, and L. Smith, “External heat losses in small turbochargers: Model and experiments,” Energy, 2014. ´ ReyesJ. R. Serrano, F. J. Arnau, R. Novella, and M. A. Belmonte, “A procedure to achieve 1D predictive modeling of turbochargers under hot and pulsating flow conditions at the turbine inlet,” Tech. Rep., 2014. J. Serrano, P. Olmeda, F. Arnau, and A. Dombrovsky, “General procedure for the determination of heat transfer properties in small automotive turbochargers,” SAE International Journal of Engines, vol. 8, no. 1, pp. 30–41, 2015. R. Burke, C. Vagg, D. Chalet, and P. Chesse, “Heat transfer in turbocharger turbines under steady, pulsating and transient conditions,” International Journal of Heat and Fluid Flow, vol. 52, pp. 185–197, apr 2015. A. Romagnoli and R. Martinez-Botas, “Heat transfer analysis in a turbocharger turbine: An experimental and computational evaluation,” Applied Thermal Engineering, vol. 38, pp. 58–77, may 2012. P. Olmeda, V. Dolz, F. J. Arnau, and M. A. ReyesBelmonte, “Determination of heat flows inside turbochargers by means of a one dimensional lumped model,” Mathematical and computer modelling, vol. 57, no. 7-8, pp. 1847–1852, 2013. H. Aghaali, “On-engine turbocharger performance considering heat transfer,” Ph.D. dissertation, KTH Royal Institute of Technology, 2012. R. Baar, C. Biet, and R. Zimmermann, “Experimental modelling of adiabatic turbocharger conditions to investigate the isentropic turbine efficiency,” in 2nd Conference on Engine Processes, Berlin, 2015. R. Zimmermann, R. Baar, and C. Biet, “Determination of the isentropic turbine efficiency due to adiabatic measurements and the validation of the conditions via a new criterion,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, p. 095440621667068, sep 2016. B. Savic, R. Zimmermann, B. Jander, and R. Baar, “New
16
Phenomenological and Power-Based Approach for Determining the Heat Flows of a Turbocharger directly from hot gas test data,” in 12th European Conference on turbomachinery, ETC, Stockholm, Sweden, 2017. [20] B. Savic, X. Gao, and R. Baar, “Turbocharger Heat Transfer Determination with a Power Based Phenomenological Approach and a CHT Validation,” in ISROMAC 17 International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Maui, Hawaii, 2017. [21] N. Baines, K. D. Wygant, and A. Dris, “The analysis of heat transfer in automotive turbochargers,” in ASME Turbo Expo 2009: Power for Land, Sea, and Air. American Society of Mechanical Engineers, 2009, pp. 115–126. [22] S. Shaaban, “Experimental investigation and extended simulation of turbocharger non-adiabatic performance,” Ph.D. dissertation, Fachbereich Maschinenbau, Universitt Hannover, Germany., 2004.
Highlights
• • • • •
A general numerical procedure for the determination of heat transfer parameters of radial turbines is proposed. Turbocharger test is carried out under different thermal conditions. A power-based method is employed to quantify the turbine total heat loss. The simulated turbine housing metal temperatures are in good agreement with experiments. The simplified procedure allows a reduction of computational and experimental efforts while maintaining good accuracy.
S1359-4311(18)34701-X https://doi.org/10.1016/j.applthermaleng.2019.03.014 ATE 13443
To appear in:
Applied Thermal Engineering
Received Date: Accepted Date:
31 July 2018 4 March 2019
Please cite this article as: X. Gao, B. Savic, R. Baar, A Numerical Procedure to Model Heat Transfer in Radial Turbines for Automotive Engines, Applied Thermal Engineering (2019), doi: https://doi.org/10.1016/ j.applthermaleng.2019.03.014
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
A Numerical Procedure to Model Heat Transfer in Radial Turbines for Automotive Engines Xunan Gao1, 2 , Bojan Savic1 , and Roland Baar1 Abstract—The thermal condition of a turbocharger considerably differs between different applications, which makes it difficult to model the heat transfer. In this study, a general procedure to characterize the heat transfer model for a radial turbine was established using 3D Conjugate Heat Transfer (CHT) simulation. The external convective and radiation heat losses were modelled by an equivalent convection heat transfer coefficient and an equivalent metal emissivity, following an analysis of the heat transfer mechanisms. The established model was validated by measurements under adiabatic and diabatic conditions. The characterization procedure was further simplified to extend its applications. The number of required operating points for simulation and measurements were significantly reduced, which enabled it to be used under different operating conditions, e.g., engine test and in-vehicle operations, where a constant turbo speed line could hardly be regulated. Then a comparison was conducted with a reference model based on the geometric simplification. The proposed model was observed to give a slightly better result regarding turbine housing temperatures under low and high turbine inlet temperatures. In the end, an analysis of turbine heat flow was conducted, which indicated that on average 80.4% of the turbine internal heat transfer occurred at the volute and 17.3% at the diffuser. Besides, the external heat losses can be up to 2.511 times of the turbine mechanical power. Keywords—Turbocharger, turbine, heat transfer, convection, radiation.
F
N OMENCLATURE
1
Cp P T η κ h q 0 1 12s 2 3 34s 4 act BH C conv ef f is mea mean oil rad s st T t TH
Increasing the fuel economy and decreasing the emissions are always among the most popular topics of the automotive industry. Turbocharging, together with other relevant technologies, offers a promising possibility to achieve these goals by improving the power density as well as the combustion processes. To take full advantage of these technologies, the matching of the turbocharging system to the engine becomes a critical step in engine development processes. For this task, the 1D simulation tools are usually employed because of its relatively low computational cost. Here the modelling of turbochargers relies on the turbine and compressor maps provided by the manufactures. These maps are measured on the hot gas test benches under characterization conditions described in the standards, such as SAE J922 [1] and SAE J1826 [2]. The turbine and compressor efficiencies are derived from the measured pressures and gas temperatures at the outlet and the inlet of the turbine and the compressor (Equation 1 and 2).
Specific heat capacity at constant pressure Power Temperature Efficiency Isentropic exponent of gas Enthalpy Heat flow per unit time Environment Compressor inlet Isentropic compression process of compressor Compressor outlet Turbine inlet Isentropic expansion process of turbine Turbine outlet Actual value Bearing housing Compressor Convective Effective Isentropic Measured value Mean value Lubrication oil Radiation Solid (temperature) Static value (temperature, pressure) Turbine Total value (temperature, pressure) Turbine housing
1 Chair of Powertrain Technologies, Technische Universit¨at Berlin (TU Berlin), Carnotstr. 1a, 10587 Berlin, Germany. 2 Corresponding author. E-mail:[email protected]
I NTRODUCTION
κ−1
ηC,is
( P2t ) κ − 1 ∆h12s T2st − T1t = = = P1tT2t ∆h12 T2t − T1t T1t − 1
ηT,is =
1 − TT4t ∆h34 T4t − T3t 3t = = κ−1 ∆h34s T4s − T3t 1 − ( PP4s ) κ 3t
(1)
(2)
During the characterization process, a certain amount of heat transfer occurs inside the turbocharger as well as between the turbocharger and the surroundings (Figure 1). Baar et al. [3] indicated that even with insulation, heat transfer in a hot gas test could not be neglected. It superimposes
2
would arise. The heat transfer parameters obtained at the turbocharger test bench could hardly represent the thermal conditions of the engine experiments or the in-vehicle operations. Being exposed to hot gases, the turbine is usually the main subject of heat transfer in a turbocharger. The goal of this study is to establish a general easy-to-use characterization procedure for the turbine, with which the heat transfer parameters could be determined, regardless of the thermal conditions of the turbocharger. In this paper, a detailed characterization procedure for the model of turbine heat transfer is proposed after analyzing the heat transfer mechanisms of turbochargers. Then the model is validated by measurement under adiabatic and diabatic conditions. To extend the scope of its application, the characterization procedure is further simplified, which enables to be applied for the turbocharger in engine tests and in-vehicle operations, where a constant turbo speed line is difficult to be regulated. After that, a comparison is conducted with a reference model based on the geometric simplification. In the end, an analysis of the turbine heat flow is carried out.
Fig. 1. Turbocharger heat flows
(a) Compressor
(b) Turbine
Fig. 2. Evolution processes of turbocharger
the measured turbine and compressor outlet temperatures (T2 and T4 ) from the actual values after the work transfer processes as shown in Figure 2. Since T10 , T20 , T30 and T40 can hardly be spatially identified or correctly recorded, the work transfer and heat transfer cannot be separated by the standard characterization process. This eventually leads to an underestimate of the compressor efficiency and an overestimate of the turbine efficiency in the turbine and compressor maps. The effects of heat transfer become especially significant under low-speed operating conditions with high gas temperatures. Under these conditions, the measured isentropic turbine efficiency can be even higher than one. Cormerais et al. [4] reported an underestimation of the isentropic compressor efficiency of up to 15% under a turbine inlet ¨ temperature of 500°C even with insulation. Ozdemir [5] conducted the CHT simulation for a turbocharger turbine and indicated that the total turbine heat loss could account for up to 82% of the total turbine enthalpy drop under the standard characterization condition (T3 =600°C). Therefore, heat transfer plays a significant role in the turbocharger characterization process. These maps contain a non-negligible fraction of heat transfer effects, and could not reflect the actual mechanical transfer performance of turbochargers. Using the traditional maps directly in a 1D simulation will deteriorate the matching of turbocharged engines. One should bear in mind that the degree of heat transfer is highly dependent on the surrounding thermal conditions. Serrano et al. [6] indicated that if the turbocharger operated under similar thermal conditions, with which the maps were measured, a reliable prediction of turbine outlet temperatures and turbo speeds could be obtained. However, once the thermal conditions deviate, an erroneous prediction
2
B IBLIOGRAPHY
In recent years, many studies were done to obtain a deep insight into the heat transfer phenomenon of the turbocharger. Bohn et al. [7] established a complex 3D CHT simulation model for the complete turbocharger, which contained the turbine, the compressor, and the oil-cooled bearing housing. The external heat transfer was modelled based on the temperature distribution of the casing surfaces derived from the measurement by a thermography camera. As a result, the temperature distribution of the casing and detailed quantification of the heat flux of the compressor were obtained. Roclawski et al. [8] conducted a CHT simulation for a centrifugal compressor. The simulated metal temperatures were observed to be in good agreement with experimental data. Serrano et al. [9] developed a special test bench in which the incompressible thermal oil flowed through the turbocharger. It enabled the determination of metal thermal properties since the fluid temperature change was only due to the heat transfer. Payri et al. [10] employed an analytical method to model the radiation heat transfer of turbocharger. The turbocharger was geometrically simplified as a dumbbell. The view factors between surfaces were calculated by analytical equations. The proposed model was then validated by engine tests with the maximum turbine inlet temperature of around 900°C. Results showed that, on the turbine side, most of the heat losses occurred through radiation. Besides, the most considerable contribution of the radiation heat loss came from the one from the turbine to the ambience, while the radiations from the turbine to other components were negligible in comparison. On the compressor side, though the radiation from the turbine to the compressor accounts for the most critical part of the total radiation heat flows, external convective heat transfer could not be neglected. Serrano et al. [11][12] then used this model to establish a 1D predictable model for turbochargers. A reliable prediction of the turbo speeds and the outlet gas temperatures of the
3
turbine and the compressor were obtained. Nevertheless, this analytical method could only be applied with certain conditions, i.e., turbocharger test bench, where the surrounding hot surfaces could be clearly identified. Another attempt of the studies in this area is to embed the heat transfer model into engine simulations. One of the most widely used approaches is to build a thermal network model for 1D engine models. The 1D engine simulations are typically carried out based on a common assumption that the compression process of the compressor and the expansion process of the turbine are regarded as adiabatic processes. Some studies locate the heat transfer before and after the expansion and the compression processes, respectively. Additionally, the area ratio of convective surfaces before and after these work transfer processes should be specified. Burke et al. [13] assumed that 85% of the total turbine wetted area located before the expansion process and 15% after it. A more aggressive assumption was made by Romagnoli and Martinez-Botas [14] and Olmeda et al. [15] that all the heat transfer took place before the expansion process of the turbine and after the compression process of the compressor. Aghaali [16] included the turbocharger heat transfer in engine simulation by adding a heat tank before the turbine and a heat source after the compressor. The simulation results showed a reliable prediction of the turbine and compressor outlet temperatures without the implementation of the mass flow coefficient and the efficiency coefficient.
3 3.1
E XPERIMENT SETUP Test bench
All the measurements were conducted on the turbocharger hot gas test bench at the Chair of Powertrain Technologies at the Technical University of Berlin. Figure 3 shows the layout of the turbocharger test bench. Three electronic compressors are used to supply the mass flow for the turbine. The gas is heated by either an electronic heater or a combustion chamber. The exhaust and turbocharger metal temperatures are recorded by thermocouples. PT100 sensors are used to measure the temperatures of air, lubrication oil and cooling water. To take into account the non-uniform flow field, at each of the measuring positions before and after the turbomachinery, temperature and pressure values are measured by three thermocouples and four pressure pipes which are azimuthally installed. This test bench enables to measure over a wide range of operating conditions regarding turbine inlet temperatures (ambient to 1050°C), mass flow rates (up to 1400kg/h) and pressures (maximal electronic compressor pressure up to 4 bar). The wastegate valve is closed during the measurements in this study. 3.2
Adiabatic measurement
Under standard characterization conditions, the turbine is exposed to hot gas heated by the combustion chamber. The measured enthalpy changes of the turbine and the compressor are the combinations of work transfer and heat transfer. Thus the obtained efficiency values do not reflect the actual aerodynamic performance of turbocharger. To
eliminate the effects of heat transfer, experiments were first performed under the adiabatic condition employed by Baar et al. [3] as shown in Equation 3.
T3 = Toil,mean and T4 = T0
(3)
where, T3 , T4 and T0 represent the gas temperatures of the turbine inlet, the turbine outlet and the ambience. Toil,mean is the mean temperature of lubrication oil. The intention behind is to minimize the heat transfer from the turbine to the bearing housing as well as to the ambience. Furthermore, all of the turbocharger surfaces and measurement pipes are fully insulated. Figure 4 shows the turbocharger setup at the hot gas test bench. Thus the question is how to quantify the heat flows that are still left. To do this, a power-based criterion was applied, which was firstly proposed by Baar et al. [17]. Zimmermann et al. [18] tested it on a couple of different turbochargers with different sizes and configurations to verify its reliability. Based on the previous studies, Savic et al. [19] extended it, which enabled a direct determination of the isentropic turbine efficiencies and isentropic compressor efficiencies from hot gas test data. Savic et al. [20] modified the approach and showed a comparison to a standard CHT-simulation regarding isentropic turbine efficiencies. The modification was carried out because other heat flow influences were observed to have non-negligible influences. According to this criterion, the measurement data is plotted in Figure 5. The x-axis is the isentropic compressor power (Pc,is ) derived from Equation 4 and the y-axis is the effective turbine power (Pt,ef f ) determined through the temperature difference of the turbine inlet and outlet flow (Equation 5). k−1 Pc,is = m ˙ c · Cp,c · T1 · πc k − 1 (4)
Pt,ef f = m ˙ t · Cp,t · (T3 − T4 )
(5)
Pc,is is chosen as x-axis because it is the most reliable value in the measurement. It can be seen from Equation 4 that, at a certain compressor operating point with a defined mass flow rate (m ˙ c ) and pressure ratio (πc ), Pc,is is the function of compressor inlet temperature (T1 ). T1 is barely affected by the thermal condition of turbine side. This is justified by the work done by Bohn et al. [7], where a CHT simulation was conducted for the complete turbocharger. The heat flux through compressor internal surfaces was invested on a longitudinal section under different compressor and turbine operating conditions, which indicated that a negligible amount of heat flux occurred before the compressor impeller. This also helps explain the assumption employed by Olmeda et al. [15] and Baines et al. [21] that all of the compressor heat transfer was located after compressor impeller when modeling turbocharger. On the compressor side, Shaaban [22] indicated that at a given operating point the heat transfer to compressor led to an increase in compressor power demand (i.e. Pc,ef f ). The increase of Pc,ef f depends on the heat flow to compressor as well as the fraction of heat transfer before impeller. Since the amount of heat transfer before compressor impeller is negligible, the actual Pc,ef f can be assumed to be unchanged
4
Fig. 3. Layout of the turbocharger test bench
under different thermal conditions. Therefore, the deviation of the measured isentropic compressor efficiency from the actual value results only from the effects of heat transfer. On the turbine side, the measured turbine power can be divided into two parts. One part is transferred to mechanical power to drive the shaft. The other part is associated with the heat flows within the turbocharger as well as from the turbine to the environment. If the turbocharger operates under ideal adiabatic conditions, the turbine enthalpy drop should be entirely converted to shaft power. In Figure 5 a straight line is added by fitting the operating points with maximum Pc,is of each speed line. This fitting line is then extended to cut the axis of ordinates. Since the ordinate intercept also means a speed of zero, there is no friction power or turbine power at this point. The ordinate intercept thereby results from heat losses of the turbine side. The ordinate intercept of the adiabatic measurement in this study is 0.2561kW, which indicates a good quality of the adiabatic measurement. Here, the points with maximum Pc,is are chosen to generate the fitting line because they are at the middle region of a turbine map, which would not be restricted by the limitations of the test bench. It is worth mentioning that, we have currently found that the maximum Pc,is are not perfectly linear. Nevertheless, adopting a straight line here is reasonable considering that it is the very first step of this approach and the goal is to develop an easy-to-use model. 3.3
Fig. 4. Turbocharger setup at test bench
Diabatic measurement
The second phase of the measurements was conducted under the standard conditions according to SAE J922 [1] and
Fig. 5. Turbine power-based criterion
5
TABLE 1 Mesh statistics
Fig. 6. thermocouple setup on the turbine housing
Fluid domain Component Inlet extend Volute Impeller (Fluid) Diffuser Mixing device Outlet extend
Mesh type Hex. Tet.+ Prism Hex. Tet.+ Prism Tet.+ Prism Hex.
Number of elements 0.454 mil. 1.622 mil. 1.901 mil. 1.687 mil. 0.919 mil. 0.215 mil.
Solid domain Component Impeller (Solid) Turbine housing
Mesh type Tet. Tet.
Number of elements 0.576 mil. 1.732 mil.
Total
9.106 mil.
Fig. 7. CHT simulation model
J1826 [2], but with different turbine inlet gas temperatures (T3 =400°C, 600°C, 800°C and 950°C). The mean temperature of cooling water and lubrication oil was regulated to 90°C for all the operating points. In order to record the solid temperatures, 5 thermocouples (Ts,1 , Ts,2 , Ts,3 , Ts,4 and Ts,5 ) were installed on the turbine housing. 3 additional thermocouples (TF lange,aT H , TF lange,bT H and TF lange,T HLH ) were fixed at the flanges which connected the turbine housing with the inlet pipe, the bearing housing and the outlet pipe respectively to serve as boundary conditions for the CHT simulation. Figure 6 depicts the installation of thermocouples on the turbine housing.
4
N UMERICAL MODELLING
The commercial code ANSYS CFX 17.2 was employed as the numerical solver, which used the element-based finite volume method. The CHT model consists of fluid and solid domains as shown in Figure 7. 4.1
Grid generation
In the fluid domain, the rotational region was discretized by a structural hexahedral grid. Figure 8 shows the mesh of a single passage. For the volute and diffuser regions, an unstructured tetrahedral grid was employed along with 10 prism layers in the boundary layers. To obtain a high resolution of boundary layer flow and heat transfer phenomena, the dimensionless wall distance (y + ) was regulated to be less than 10 for fluid domains. The turbine housing was meshed by a tetrahedral grid. A grid independence study was carried out to avoid grid-induced errors. To eliminate the effect of the computational boundaries on the simulation results, the computational domains of the turbine inlet and outlet were extended.
Fig. 8. Grid of the fluid domain of impeller
4.2
General settings
In the fluid domains, the steady-state RANS (Reynolds Averaged Navier-stokes) equations were solved. The closure of the equations was brought by the SST (Menter‘s Shear Stress Transport) turbulence model, which is a combination of the k –ω model and the k –ε model. Specifically, in the near-wall region the k –ω model is used to ensure a reliable prediction of the onset and the amount of flow separation under adverse pressure gradients. In the free-shear layer flows, the k –ε model is employed since it is the most widely used turbulence model for flows containing relatively small pressure gradients for industrial applications. In the solid domains, the energy equation was simultaneously solved. The Conservative Interface Flux algorithm was set for heat transfer at the interface between the solid and fluid domains. With this method, the temperatures and the heat flows of interfaces are the direct results of the simulation. The specification of the convective heat transfer coefficient is superfluous. For the turbine inlet, the total pressure and total temperature were set as boundary conditions. For the turbine outlet, the static pressure was specified in accordance with experimental data. Three additional thermocouples fixed at the flanges of turbine housing serve as boundary conditions for the CHT simulation. Table 2 shows an overview of the boundary conditions and their ranges in this study.
6
TABLE 2 Boundary conditions of the CHT model
Location Inlet Outlet Impeller Flange Ambience
4.3
Boundary condition Total temperature (T3t ) Total Pressure (P3t ) Static pressure (P4 ) Speed (nT C ) Temperature (TF lange ) Temperature (Tamb )
°C kPa kPa 103 /min °C °C
44.9 - 79.5 162 - 330 101 - 102 123 - 186 20.4 - 73.0 22.4
Modelling of turbine heat transfer
Heat transfer occurs in the turbocharger by conduction, convection and radiation. When establishing the heat transfer model for the turbine, heat transfer boundary conditions should be carefully considered. As for the conduction heat transfer between the turbine housing and the bearing housing as well as the connected pipes, a constant temperature boundary condition was applied. The temperatures were provided by three additional thermocouples installed at the flanges of the turbine housing. As for the convective heat transfer, the internal convective heat transfer, which took place between the turbine housing and the working gas, was the direct results of simulation since the fluid and solid domains were simultaneously solved and a conservative heat flux boundary condition was applied to the interface. The external convective heat transfer was modelled by a userspecified heat transfer coefficient based on the Newton‘s Law of Cooling (Equation 6). In this study, we assume that the surrounding flow is dominated by forced convection. Therefore, the area-averaged equivalent convective heat transfer coefficient he is constant under a certain application condition. A is the area of the object. Ts is the solid surface temperature. Tf is the ambient temperature. Obviously, here he is the control parameter.
qconv = he · A · (Ts − Tf )
(6)
The radiation heat loss of turbine is modelled by the Stefan-Boltzmann Law. Accordingly, the radiation energy per unit time of a grey body can be expressed by:
qrad = ε · σ · A · T 4
(7)
where ε is the metal emissivity, σ is the Stefan-Boltzmann constant, A is the area of the body, T is the absolute temperature of the grey body in Kelvin. The net radiation heat loss rate of a grey body with the cold surroundings can be expressed by:
qrad = ε · σ · A · T 4 − Tc 4
Range Diabatic test
Adiabatic test
(8)
where Tc is the absolute temperature of the cold surroundings in Kelvin. The radiation heat transfer rate between two grey-body surfaces is given by: σ · T1 4 − T2 4 qrad,1→2 = 1−ε1 (9) 1−ε2 1 A1 ·ε1 + A1 ·F1→2 + A2 ·ε2
400 142 - 335 100 - 101 121 - 217 116.2 - 331.5 18.4
600 122 - 360 100 - 103 92 - 234 171.1 - 484.3 18.9
800 123 - 332 101 - 103 92 - 234 219.5 - 617.6 21.2
950 121 - 313 100 - 101 92 - 234 254.1 - 710.6 22.9
where ε1 and ε2 are the metal emissivity of two surfaces. F1→2 is the view factor from surface 1 to surface 2, which is determined by the spatial configuration of the two surfaces. Therefore, the emissivity ε and the view factor F1→2 are the control parameters here. As mentioned before, the heat transfer conditions of the turbine differ among various applications, e.g., turbocharger test bench, engine test bench, in-vehicle operations, which results in different heat transfer parameters. The net radiation heat loss of the turbine housing, for example, is strongly dependent on the hot surfaces around as well as the ambient temperature. Their influences are difficult to be quantified. In this study, we intend to develop a general characterization procedure of the heat transfer parameters for a radial turbine that could be used in a wide range of applications. To do this, we modified Equation 8 to be as Equation 10, where εe is the equivalent emissivity, which combines the effects of metal emissivity and view factor. To keep it as simple as possible, the ambient temperature Tc was removed from the equation as well. Here, the equivalent emissivity εe becomes the control parameter for the radiation heat transfer.
qrad = εe · σ · A · T 4
(10)
Therefore, establishing the turbine heat transfer model can be represented by determining the characteristic parameters (i.e., he and εe ) of the convective and radiation heat transfer. A complete characterization procedure of these parameters is carried out in the next section.
5 5.1
C HARACTERIZATION PROCEDURE Determining convective and radiation parameters
Based on the analysis in previous sections, the independent heat transfer parameters of this study are the equivalent convective heat transfer coefficient he and the equivalent metal emissivity εe . It is obvious from Equations 6 and 10 that their influences on the calculated heat transfer differ among various thermal conditions. To get a comprehensive understanding, we first conducted the CHT simulations with different he and εe . Under each turbine inlet gas temperature T3 , one representative operating point is selected from the mediumspeed line as shown in Table 3. The selected operating points have the maximum isentropic compressor power (Pc,is ) at their speed lines. This is to ensure that these points are at
7
TABLE 3 Representative operating points
No. 1 2 3 4
Speed (1/min) 184515 184408 186373 186441
T3 (°C) 400 600 800 950
Pressure ratio (-) 2.463 2.313 2.207 2.145
the middle region of each speed line, which will not be influenced by the test bench limitations. Figure 9 depicts the mean deviation of the simulated turbine housing temperatures from the measurement data against turbine inlet gas temperatures under different he . ¯ s is defined by EquaThe mean temperature deviation ∆T tion 11. Ts,i,sim and Ts,i,mea represent the turbine housing temperatures from simulation and measurement data at five different measuring positions as shown in Figure 6.
(a) he = 30 W/(m2 · K)
5
X ¯s=1· ∆T (Ts,i,sim − Ts,i,mea ) 5 i=1
(11)
The line of εe = 0 represents the simulation without taking into account the radiation heat transfer. The mean deviation significantly increases with the increase of turbine inlet temperature. It also shows the necessity to take into consideration the radiation heat transfer by CHT simulation. The line of εe = 1 represents the simulation considering the turbine housing as a black-body and ignoring the effects of surrounding surfaces. Also, a considerable deviation occurs. It is clear that he is the crucial factor in determining the solid temperature deviation under low turbine inlet temperature. While under high turbine inlet temperatures the equivalent emissivity εe plays a significant role. It is also expected that an optimal combination of he and εe exists, which results in the minimum mean deviation of turbine housing temperatures under all turbine inlet temperatures (T3 ). To deduce this optimal combination, the simulation results were reprocessed as Figure 10, which shows the ¯ s against the equivalent mean temperature deviation ∆T emissivity εe . For clarity, only the simulation results under T3 = 400°C and 950°C are plotted. A linear and a secondorder curve are created by fitting to the simulation results under T3 = 400°C and T3 = 950°C, respectively. The fitting lines of T3 = 400°C and T3 = 950°C with the same he intercept at a certain εe . At this interception point, the mean temperature deviations under both T3 are equal. Then, a second-order curve (red, dash) is generated by fitting to all the interception points of different h3 . Along this curve, the mean temperature deviations under different T3 remain ¯ s is equal. Thus, at its intersection with the x-axis the ∆T expected to be zero for all the T3 . In this study, the line cuts the x-axis with an intersection (Equivalent emissivity, εe ) of 0.122. To get the corresponding he of the intersection point, all of the intersection points of different he in Figure 10 are plotted in Figure 11 , which shows the mean temperature deviation against he . It is clear that a straight fitting line can be generated, which cuts the
(b) he = 35 W/(m2 · K)
(c) he = 40 W/(m2 · K)
(d) he = 70 W/(m2 · K) Fig. 9. Mean solid temperature deviation of turbine housing
8
outlet gas temperature and isentropic turbine efficiency between simulation and measurement, respectively. The simulation results correspond well with the experimental data, which indicates the good reliability of the simulation model in aerodynamics.
Fig. 10. Mean temperature deviation vs. equivalent emissivity
(a) M ass f low parameter
Fig. 11. Mean temperature deviation vs. equivalent heat transfer coefficient
x-axis with an intersection (he ) of 48.974 W/(m2 · K). Here we obtain a combination of εe and he directly by plotting, which is expected to result in the minimum mean temperature deviations under all T3 . A preliminary evaluation was carried out by running the simulation with the resultant combination of εe and he under different T3 . Figure 12 depicts the result of the preliminary evaluation, which is in good agreement with the expectation. In the next section, a detailed validation will be conducted.
(b) T urbine outlet temperature
(c) Isentropic turbine ef f iciency Fig. 13. Comparison of simulation results and measurement data under adiabatic condition Fig. 12. Preliminary evaluation of the resultant parameters
5.2
Validation
5.2.1 Aerodynamic validation Simulations under adiabatic conditions were carried out, with which the turbine external surfaces were set as adiabatic. Figure 13 shows the comparison of mass flow, turbine
5.2.2 Thermodynamic validation The optimal combination of εe and he previously obtained was applied to the CHT simulation under diabatic conditions. Simulations were performed at all of the operating points under different turbine inlet temperatures. An evaluation was made regarding heat flow rate. According to the previously mentioned power-based criterion, the measurement data under different turbine inlet
9
temperatures are depicted in Figure 14. Additionally, the adiabatic measurement data is added as a reference. Here the power-based criterion offers the unique possibility to quantify the total turbine heat losses. As marked in Figure 14, the vertical difference of diabatic and adiabatic measurement data is the total heat loss of the turbine. Figure 15 shows the comparison of the simulated turbine heat losses with the measurement data, which are derived from the power-based criterion. For clarity, only the operating points with the largest isentropic compressor power of each speed lines are plotted. It is shown that the simulation results correspond well with the measurement data. Nevertheless, one should bear in mind that the fitting curves employed by the power-based criterion in this study are straight lines. However, we currently find that the points with the maximum Pc,is are not perfectly linear. This may have a slight influence on the derived turbine heat losses. It is especially obvious under high T3 (e.g., 950°C) that the points with maximum Pc,is at both low and high speeds are slightly below the fitting line. A higher-degree polynomial is expected to deliver a better approximation. We are conducting intensive investigations on this topic. More details may be found in our future publications. 5.3
(a) T3 = 400°C
Simplification of the proposed procedure
In the previous sections, a characterization procedure of heat transfer parameters for the turbine was established and validated. It enables reliable modelling of the turbine heat transfer. Table 4 shows the number of operating points for the simulations and measurements that were used in the characterization procedure. It is clear that a large effort has to be devoted to both the simulation and the measurements. In this section, the proposed procedure is further simplified to extend applications to various thermal conditions, i.e., turbocharger hot gas test bench, engine test bench, invehicle operations, where the requirements of operating points may hardly be fulfilled.
(b) T3 = 600°C
TABLE 4 Operating points of the characterization procedure
ITEM
T3
LEVELS
4
ITEM
T3
LEVELS
4
Measurement Speed lines Operating points (OPs) (per T3 ) (per Speed line) 8 8 OPs (per T3 ) 1
Simulation he (per OP) 4
εe (per he ) 8
Total OPs
(c) T3 = 800°C
256
Total OPs 128
To simplify the characterization procedure, we first reconsider the simulation campaigns. In the previous original procedure, the combination of heat transfer parameters (εe and he ) were derived from Figure 10 and Figure 11, in which only the simulation results under the highest (950°C) and the lowest T3 (400°C) were used. Figure 12 indicates that the simulations under these two T3 are sufficient to deliver a reliable result for different T3 . As for each T3 , a second-order curve (red, dash) was generated in Figure 10 by fitting to the intercept points of each he . Here, the levels
(d) T3 = 950°C Fig. 14. Measurement data according to the power-based criterion
10
TABLE 6 Validation of the simplified characterization procedure
Procedure Original Simplified
6
Fig. 15. Comparison of the simulated turbine heat losses with measurement data
of he can be reduced from 4 to 3, comparing to the original characterization procedure. For each he , in Figure 10 a straight line (400°C) and a second-order curve (950°C) are generated by fitting to the points of different εe . Therefore, 3 levels of equivalent emissivity εe (i.e., 0, 0.5 and 1) are sufficient to produce the fitting curves. As a conclusion, with the simplified procedure, the simulation should be carried out at 18 operating points. Accordingly, measurement is required to be carried out at only two operating points with different T3 (low and high temperatures). Table 5 shows the operating points for simulations and measurements of the simplified procedure. It is clear that the simplified procedure significantly reduces the number of operating points of the simulation and measurement. Furthermore, since the measurement data is only required at two operating points, it enables the proposed characterization procedure to be used in a wide range of applications, i.e., engine test or in-vehicle operations, where a constant turbo speed line is difficult to be regulated. The evaluation of its applications for other thermal conditions will be discussed in our future studies. TABLE 5 Operating points of the simplified characterization procedure
ITEM
T3
LEVELS
2
ITEM
T3
LEVELS
2
Measurement Speed lines Operating points (OPs) (per T3 ) (per Speed line) 1 1 OPs (per T3 ) 1
Simulation he (per OP) 3
εe (per he ) 3
Total OPs 2
Total OPs 18
To validate its reliability, the simplified procedure was then employed for the characterization of heat transfer parameters. Table 6 shows the heat transfer parameters (εe and he ) as well as the mean temperature deviations derived from the original and the simplified procedures. The simplified procedure delivers reliable results with a temperature deviation which is comparable to that of the original characterization procedure.
Parameters he εe 48.974 0.122 49.028 0.125
Mean solid temperature deviation (°C) T3 =400°C 600°C 800°C 950°C -0.387 -1.662 -1.897 -1.444 -0.458 -1.807 -2.154 -1.842
C OMPARISON WITH THE REFERENCE MODEL
In this section, to further evaluate the reliability of the established characterization procedure, a reference model is established based on the CHT model in this study. The internal heat transfer, between the turbine housing and the working gas, is resolved in the same way as was mentioned in previous sections. With regards to the external heat transfer, the parameters for the radiation and convection heat transfer are determined by a method proposed by Payri et al. [10]. The modelling of the external heat transfer is explained in section 6.1. And a comparison of the simulated turbine housing temperatures is presented in section 6.2. 6.1 Modelling of the external heat transfer (reference model) The external heat transfer to be modelled here includes radiation and convection. Both of them rely on a geometric simplification of the turbocharger. The radiation is modelled by determining the view factor from the turbine housing to the ambience using analytic equations. The convection is modelled by determining its non-dimensional parameters. 6.1.1 External radiation According to the Stefan-Boltzmann Law, the heat flow rate between two grey-body surfaces is given by Equation 9. In the reference model, the emissivity ε is specified consistently with the material property. The key factor of modelling radiation heat transfer is the determination of the the view factors between different surfaces. As is explained in section 2 and section 4.3, here only the radiation between the turbine housing and the ambience is modelled. Firstly, the turbocharger is geometrically simplified as three cylinders (Figure 16), similarly as done by Payri et al. [10]. The turbine surfaces include one interior surface and two exterior surfaces. The view factor from turbine exterior surfaces and to the ambience is equal to unity. As for the turbine interior surface, possible radiation connections are: • • •
Between interior surfaces of turbine and compressor (Ft→c ). Turbine interior surface with bearing housing surface (Ft→b ). Turbine interior surface with ambience (Ft→amb ).
These view factors can be calculated using the analytic expressions in Table 7 as described by Payri et al. [10]: 1)
Ft→c : Being calculated by the equation for view factor between two concentric discs separated by a concentric cylinder (case 1).
11
turbine cylinder. g = 9.8m/s2 is the acceleration due to Earth’s gravity, β = 2/(Tw + Tamb ) is the coefficient of thermal expansion, u is the air velocity in test bench, ν is the kinematic viscosity. Tw and Tamb are temperatures of the turbine housing and the ambience, respectively. The range of Ri indicates that in this case neither natural nor forced convection is predominant. A combined convection heat transfer should be considered. The heat transfer coefficient of the combined convection can be calculated by Equation 17. Fig. 16. Geometric simplification of turbocharger
2)
Ft→b : Firstly, the view factor from the bearing housing surface to the turbine interior surface (Fb→t ) can be calculated by the equation for view factor between a ring and a lateral surface (case 2). Then, Ft→b can be obtained by applying the reciprocity property (Equation 12) : Ft→b = Fb→t ·
3)
3
Ab 4 · Lb · Db = Fb→t · (12) At,inner Dt 2 − Db 2
where, Ab is the area of the bearing housing surface. At,inner is the area of the turbine housing interior surface. Lb is the length of the bearing housing. Dt and Db are the diameters of the turbine housing and the bearing housing, respectively. Ft→amb : Being calculated by the summation property given by Equation 13:
Ft→amb = 1 − Ft→c − Ft→b
(13)
Therefore, the radiation heat flux from turbine interior surface to the ambience can be computed by Equation 14. The total radiation heat flux from the turbine to the ambience is calculated by Equation 15.
q˙rad,t,inner =
q˙rad,t =
ε · Ft→amb ε · (1 − Ft→amb ) + Ft→amb · At,inner · σ · (Tw 4 − Tamb 4 )
(14)
1 · [qrad,t,inner · At,inner + (At − At,inner ) At (15) ·σ · (Tw 4 − Tamb 4 )
6.1.2 External convection The geometric simplification used in this section is the same as for the modelling of external radiation. Firstly, a preliminary estimate of Richardson Number Ri is performed as Equation 16:
0.6214 ≤ Ri = GrD /ReD 2 ≤ 1.0441
3
N uD = N uD,N C + N uD,F C
(16)
where Grashof number GrD = g · β · (Tw − Tamb ) · D3 /ν 2 and Reynolds number ReD = u · D/ν are non-dimensional parameters based on the diameter D of the simplified
3
(17)
where N uD,N C and N uD,F C are the area-averaged Nusselt numbers of natural and forced convection, respectively. N uD,N C is determined by Equation 18, which is correspond to the natural convection of a horizontal cylinder.
( N uD,N C =
0.387 · RaD 1/6 0.60 + [1 + (0.559/P r)9/26 ]8/27
)2 (18)
where RaD = GrD · P r is the Rayleigh number, P r = ν/α is the Prandtl number. α is the thermal diffusivity. N uD,F C can be calculated by the Churchill–Bernstein correlation (Equation 19), which is used to estimate the areaaveraged Nusselt number of a cylinder in cross flow.
0.62 · ReD 1/2 · P r1/3 ReD 5/8 4/5 ·[1+( ) ] 2/3 1/4 282000 [1 + (0.4/P r) ] (19) The obtained parameters of radiation and convection are set as boundary conditions for the CHT simulation. A comparison between the proposed and the reference models is carried out in the next section. N uD,F C = 0.3+
6.2 Comparison of simulated turbine housing temperatures Figure 17–20 show the comparison of measured and simulated turbine housing temperatures under T3 =400°C, 600°C, 800°C and 950°C, respectively. The positions of the thermocouples are defined by Figure 6. Two series of simulation results presented here are derived from the proposed and the reference model, respectively. Both of the simulation models are observed to give a comparable result, which is in good agreement with the measurement data. Under T3 = 400°C and T3 = 950°C, a slightly better result can be obtained by applying the proposed model. Another factor should be considered is the reliability of the simulation model when being used under different application conditions. The characteristic parameters for the proposed model are determined by using CHT simulation and measurement data regardless of the thermal conditions, under which the turbocharger operates. For this reason, the proposed model is estimated to capture the heat transfer behaviour accurately. Its reliability under different application conditions will be evaluated in our following publications.
12
TABLE 7 Analytic expressions of the view factors
Case
Layout
View factor
Parameters r1 h R2 = rh2 Rc = rhc A = R1 2
R1 = h
A 2
cos−1
Rc R2
− (1 +
C 2 )(1
+
1 π·A
1
Rc B cos−1 R + 2Rc (tan−1 Y − tan−1 2 1 h i 1/2 (1+C 2 )(Y 2 −D 2 ) 2 −1 D ) · tan (1+D 2 )(C 2 −Y 2 )
+
A1/2 − tan−1 B 1/2 )
1/2 [1+(R1 +Rc )2 ](R1 −Rc ) 2 [1+(R1 −Rc ) ](R1 +Rc ) 1/2 # 1/2 [1+(R2 +Rc )2 ](R2 −Rc ) 2 2 −1 + 1 + (R2 + Rc ) 1 + (R2 − Rc ) · tan [1+(R2 −Rc )2 ](R2 +Rc )
+
1 + (R1 + Rc )2
1 + (R1 − Rc )2
1/2
· tan−1
− Rc 2
2
B = R2 − Rc 2 C = R2 + R1 D = R2 − R1 Y = A1/2 + B 1/2 r1 r2 H = rh 2 A = H2
R=
2
B 8RH
+
1 2π
cos−1
A B
−
1 2H
h
(A+2)2 R2
i1/2 −4 cos−1 AR − B
A 2RH
sin−1 R
+ R2 − 1
B = H 2 − R2 + 1
7
S IMULATION RESULTS
In this section, two phases of analyses are carried out regarding the turbine heat flow derived from the proposed model. The first one focuses on the influence of turbine inlet temperatures on the distribution of heat losses of the turbine housing internal surfaces. The second one shows the amount of turbine external heat losses under different T3 as well as its compositions. 7.1
Distribution of the turbine internal heat flow
Figure 21 shows the temperature distribution of turbine housing at one operating point under T3 = 600°C. The heat transfer between working gas and turbine housing occurs at the turbine internal surfaces before (Volute to TH), during (Rotor to TH) and after (Diffuser to TH) the expansion process. Figure 22 shows the proportion of turbine internal heat flow (Q/Qinternal ) against the turbine inlet temperature (T3 ). Qinternal is the total heat flow at turbine internal surfaces. Most of the turbine internal heat transfer takes place at the volute, accounting for an average of 80.4% of total internal heat flow under different T3 and up to 97.9% at high-speed conditions under T3 = 400°C. The proportion of heat transfer at the rotor is negligible (below 7%) under all T3 . 7.2
Turbine external heat losses
Figure 23 shows the ratio of the external heat losses (Q) to the turbine mechanical power (Wt ) against T3 . The total external heat losses increase significantly with the increase of T3 , rising from an average of 0.180 to up to 2.511 times of the turbine mechanical power. The maximum value corresponds to the operating condition with the highest pressure ratio of the lowest speed line under T3 = 950°C. Besides, the range of Q/Wt has a strong relationship with the turbine inlet temperature, varying from 0.442 under T3 = 400°C to 2.332 under T3 = 950°C. This also indicates the importance of taking into consideration the effects of heat transfer when modelling the turbocharger.
8
C ONCLUSIONS
A general characterization procedure to model the turbine heat transfer was established using 3D CHT simulation. The conduction heat loss of turbine housing was modelled by specifying the flange temperatures according to the measurements. After analyzing the heat transfer mechanism, the modelling of external convective and radiation heat transfer was represented by determining the equivalent convective heat transfer coefficient (he ) and the equivalent metal emissivity (εe ). The proposed model was then validated by measurements conducted at the turbocharger hot gas test bench under both adiabatic and diabatic conditions. A previously introduced power-based criterion was employed to verify the quality of the adiabatic measurement as well as to quantify the turbine heat loss of diabatic measurements. The proposed characterization procedure was further simplified. The number of required operating points was significantly reduced (from 128 to 18 for the simulation and from 256 to 2 for the measurement). It made the proposed procedure to be a general procedure which is suitable for a wide range of applications, e.g., engine test and in-vehicle operations, where a constant turbo speed line is difficult to be adjusted. A reference model based on geometric simplification was introduced, with which the external radiation was modelled using analytic equations. The external convection was modelled by determining the non-dimensional parameters. Both models gave comparable results regarding turbine housing temperatures. A slightly better result under low and high turbine inlet temperatures was observed with the proposed model. The analysis of turbine heat flow indicated that the turbine internal heat transfer mainly occurred at the volute (80.4% on average), comparing to the diffuser with an average of 17.3%. The external heat flow, which increases significantly with the increase of turbine inlet temperature, can be up to 2.511 times of the turbine mechanical power.
13
(a) Ts,1
(a) Ts,1
(b) Ts,2
(b) Ts,2
(c) Ts,3
(c) Ts,3
(d) Ts,4
(d) Ts,4
(e) Ts,5
(e) Ts,5
Fig. 17. Comparison of measured and simulated turbine housing temperatures (T3 =400°C)
Fig. 18. Comparison of measured and simulated turbine housing temperatures (T3 =600°C)
14
(a) Ts,1
(a) Ts,1
(b) Ts,2
(b) Ts,2
(c) Ts,3
(c) Ts,3
(d) Ts,4
(d) Ts,4
(e) Ts,5
(e) Ts,5
Fig. 19. Comparison of measured and simulated turbine housing temperatures (T3 =800°C)
Fig. 20. Comparison of measured and simulated turbine housing temperatures (T3 =950°C)
15
Fig. 21. Temperature distribution of turbine housing Fig. 23. Turbine external heat losses to the ambience
[8]
[9]
[10] Fig. 22. Distribution of turbine internal heat flow
[11]
ACKNOWLEDGMENTS The authors would like to thank the support from the ANSYS Germany GmbH. Xunan Gao is supported by a scholarship from the China Scholarship Council (CSC).
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Highlights
• • • • •
A general numerical procedure for the determination of heat transfer parameters of radial turbines is proposed. Turbocharger test is carried out under different thermal conditions. A power-based method is employed to quantify the turbine total heat loss. The simulated turbine housing metal temperatures are in good agreement with experiments. The simplified procedure allows a reduction of computational and experimental efforts while maintaining good accuracy.