Engine In-cylinder model - Willans Line Method

The map based modelling of the engine is possible when the experimental data are available. In alternative the map of the specific fuel consumption can be generated starting form data of a known engine with Willans line approach. This approach is powerful for the analysis of new technologies, not present in available engine. With the Willans line method [156, 297], the in-cylinder processes and friction losses of the ICE are modelled by a parametric approach. Willans line approach consists of an affine representation relating the available energy (i.e. the energy theoretically available for conversion) to the useful energy that is actually present at the output of the energy converter. There are many advantages of adopting such a formulation to describe both conventional and hybrid vehicles. Among others the most important are: the ability to make the model independent of the power rating or displacement of the device, thus permitting easy scaling; the compactness of the model, requiring only two parameters; its straightforward definition

191 and implementation; and the ability to easily compose models, thanks to the unified method of representation.

Nevertheless, some drawbacks should be considered when applying such an affine relationship to modern powertrain configurations, such as those featuring turbocharging, downsizing or both advanced design solutions.

The Willans line modeling technique was initially applied to describe the relationship, approximately linear, linking mean effective pressure to brake and engine fuel consumption. Rizzoni et al. in [157]

extended this model to describe the energy converters, such as engines and electric machines.

In fact, in stationary conditions, it is possible to estimate the efficiency of a new machine belonging to the same category, using a scaling approach [158]. Furthermore, this approach is well-suited for immediate and approximate assessment of new design configurations and new engineering solutions for those classes of prototype machines or, in general, for those machines for which there are not enough data on efficiencies.

The key factor of this unique representation is that for a fixed speed, there is a linear relationship between output and input variables. That method suggests a close correlation between the total available energy (the fuel chemical energy) and the real used energy (the energy available at the crankshaft). The resulting representation is described by two factors: the slope, or intrinsic energy conversion efficiency e, and the vertical axis intercept Ploss, which represents the losses due to mechanical friction, alternator, auxiliaries etc., as shown in Figure 10-9.

Figure 10-9 Linear relation between P_out and P_in at constant speed, adapted from [157]

The energy conversion efficiency and the relationship between input and output powers are thus expressed by the following equation:

𝑃_𝑜𝑢𝑡 = 𝑒 ∙ 𝑃_𝑖𝑛− 𝑃_{𝑙𝑜𝑠𝑠}

(10.27) where e has been considered obtained as the product of thermodynamic ηt and combustion ηb efficiencies.

This means that e already accounts for losses due to pumping work. Equation (10.27) can also be expressed in terms of torque:

𝑇_𝑒= 𝑒 ∙ 𝐻_𝐿𝐻𝑉∙𝑚̇_𝑓

𝜔 − 𝑇_{𝑙𝑜𝑠𝑠}= 𝑒 ∙ 𝑇_𝑎 − 𝑇_{𝑙𝑜𝑠𝑠}

(10.28)

192 with Ta the available torque, HLHV as the fuel lower heating value, 𝑚̇_𝑓 the fuel flow rate and Tloss loss torque due to frictions.

To achieve scalable relations, it is necessary to introduce the concept of mean effective pressure BMEP (which can also be seen as the engine's ability to provide mechanical work), the available mean effective pressure AMEP (that represents the available chemical energy by fuel supply) and the average piston speed vpist, corresponding respectively to the output torque, the available torque and engine speed. For a four stroke engine, they can be obtained: Where 𝑉_𝑑 is the engine displacement an S is the piston stroke. The engine efficiency can then be defined as follows:

𝜂_𝐼𝐶𝐸=𝑃_𝑜𝑢𝑡

𝑃_𝑖𝑛 = 𝑇_𝑒∙ 𝜔 𝑚̇_𝑓∙ 𝐻_𝐿𝐻𝑉

(10.30) By means of the relationships given by equations (10.29) and substituting they into equations (10.28) and (10.30), the definition of engine dimensionless efficiency is

𝐵𝑀𝐸𝑃 = 𝑒 ∙ 𝐴𝑀𝐸𝑃 − 𝐹𝑀𝐸𝑃 ⟹ 𝜂_𝐼𝐶𝐸 =𝐵𝑀𝐸𝑃 𝐴𝑀𝐸𝑃

(10.31) where FMEP is the mean effective pressure lost and is defined in a similar way to BMEP and AMEP.

Parameters e and FMEP are functions of load and piston average speed. The following parameters, selected by referring to Willans line parameter correlations proposed in [157], were experimentally validated on different engines. It is worth mentioning here that the efficiency-related parameters listed in equations (10.32) and (10.33) were curve-fitted as a multi-linear regression model, here obtained by substituting such equations in equation (10.31). As aforementioned, the pumping work produces losses that are considered into the

193 efficiency term e. For several applications, it needs to decouple the pumping losses from the other members of the equation (10.31), and the result is expressed by the following equation:

𝐵𝑀𝐸𝑃 = 𝑒 ∙ 𝐴𝑀𝐸𝑃 − 𝐹𝑀𝐸𝑃 − 𝑃𝑀𝐸𝑃 turbocharging or of other devices and technologies that could affect the backpressure and the boosting of the engine. The efficiency related parameters should be adapted in such a way as to decouple purely thermodynamic from turbocharging effects.

Starting from the equation (10.34) the torque delivered can be re-calculated with following equation, by splitting the terms related to both mechanical and pumping losses from the available torque generated by the fuel: where the coefficients 𝑘_𝑒, 𝑘_𝑓 and 𝑘_𝑝 have been introduced to consider the effects linked to the different engine technologies presented in the previous Chapter # 6.

In particular ketakes into account the correction on the thermodynamic efficiency (e.g. for lower/higher heat losses, MFB50 position, knock phenomena), kf is the correction for the friction losses (e.g. for heavier/lighter crank-train, bigger/smaller bearings, etc..), kp is the correction for the pumping losses (e.g.

for part load operations throttle adoption or not, turbocharger improvements, VGT or WG adoption, etc..).

Moreover, the coefficient e of equation (10.34) is re-formulated highlighting the ideal thermodynamic efficiency 𝜂_𝑖𝑑𝑐, dependent on compression ratio. In this work, a further modification to theoretical efficiency has been introduced to take into account the Miller cycle, as proposed in [88]:

𝜂_𝑖𝑑𝑐 = 1 − 1

𝑟_𝑔^𝛾−1∙ 𝑓(𝜎)

(10.37) where  = rg/rtr is the expansion – compression ratio, with 𝑟_𝑔 geometrical and 𝑟_𝑡𝑟 trapped compression ratio, respectively. Moreover, f(σ) takes into the account the effect of the Miller cycle on the theoretical efficiency and in [88] is evaluated as:

where AFR is the air to fuel ratio and Tair the temperature of intake air. Finally the equation (10.36) can be re-written introducing the parameter , the volumetric efficiency ηv and the air density  as:

194 Summarizing, Willans line main concepts and applications are:

• to scale engine characteristics from a reference one, with the constraints that they should belong to the same engine class, such as SI engines or CI engines, either aspirated or turbocharged;

• since they aim at generating scale coefficients, the curves should adapt to a wide range of vpist values, to avoid extrapolation when Willans-based engine model is scaled up/down or used for other engine configurations;

• to evaluate the impact of additional technologies on the engine efficiency.

Analysis of Technologies to improve a Turbo GDI Engine

The equation (10.39) has been used to evaluate the achievable improvement on BSFC of 1.4L 4 cylinder, Turbo GDI engine, with a VVT and Compression Ratio (CR) equal to 10, chosen as baseline engine.

The corrective parameters kf , ke and kp have been evaluated starting from literature analysis and by means of a 1-D engine model, calibrated and validated against experimental data, as described in a published work dedicated to activity research on WI system [217]. Specifically, the 1-D model contains, in addition to the air path with turbocharger and supercharger, a predictive combustion modelling, which can also predict MAPO statistical distribution. In case of water injection, as an example, the corrective parameters can be found by firstly establishing a trade-off between MAPO percentile and indicated efficiency for knock limited engine operating points [76]. Subsequently, water injection is simulated and the new trade-off can be defined; ke is therefore calculated and extrapolated for the nearest points, obtaining the static map to be used in the 0-D model. In Table 10-2, the effects of the ICE investigated technologies on the parameters are illustrated:

External EGR k_e for the lower heat losses and knock mitigation, k_p,

Lean Combustion  , k_e, k_p especially at part low operations and knock resistance

Water Injection k_e, and knock resistance higher r_g allowed  higher η_idc

Variable Compression R. ηidc, k_f

2stage-Air Charging Extreme downsizing  V_d and pme_req

195 In Figure 10-10 and Figure 10-11, the BSFC map and the correlations between measured and estimated values of BMEP and BSFC for the baseline engine are shown, demonstrating the feasibility of the approach.

Figure 10-10 BSFC evaluation for 1.4L Turbo GDI engine (baseline)

Figure 10-11 Correlation of BMEP and BSFC estimated vs. measured

The advantage of the approach is its modularity. Indeed, from the baseline case, the BSFC curve derived for any combination of technologies can be estimated. As an example, Figure 10-12 shows the BSFC map for the baseline case, adding the Miller cycle with an increased CR (12.5), cylinder displacement 1.5L and friction reduction of 20%. The results are quite similar than those shown in Figure 10-13, describing the measured BSFC map of the baseline engine improved with the previous technologies, as investigated in [159].

196 Figure 10-12 Simulated BSFC of 1.5L Turbo GDI with Miller cycle

Figure 10-13 Measured BSFC map from [159] of the engine used for demonstrating the approach feasibility; the results are similar to the ones shown in Figure 10-12 obtained from the model