# CFD–CHT calculation method using Buckingham Pi-Theorem for complex fluid–solid heat transfer problems with scattering boundary conditions

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## Abstract

A three-dimensional CFD–CHT simulation method is presented and validated with a turbocharged single cylinder SI engine. Various ignition time and lambda strategies as well as variations of boost pressure are investigated with regard to cycle averaged component temperatures. This complements existing published works which experimentally studied crank angle resolved heat fluxes or temperature swings rather than averaged temperatures. Cyclical fluctuations in the pressure curves were measured and processed statistically using probability density functions for the heat transfer coefficient and the cylinder gas temperature. The corresponding joint probability density function considers their strong correlation. The interpretation as random variables enables a time-scale separation with a low-pass filter function. The thermomechanical problem of heat transfer is addressed with simplified models according to Woschni, Eichelberg, and Hohenberg. Previous investigations primarily focused on their predictive quality of instantaneous in-cylinder heat fluxes. In this paper, their effect on cycle averaged component temperatures is investigated and the corresponding different sensitivities to specific engine settings are presented and compared with measurements. It is shown that, by choosing the right model, the suggested simulation approach is an alternative to prevailing experimental methods in temperature analysis: all thermodynamic variations examined are in good agreement with theoretical predictions.

## Keywords

Similarity mechanics Buckingham Pi-Theorem Conjugate heat transfer Engine heat transfer Scattering boundary conditions## 1 Introduction

### 1.1 State of the art

One of the aims of the present engine development is increasing the thermodynamic efficiency with optimized combustion processes. For this purpose, lots of engine parameters are available resulting in a complex interaction and, therefore, in a conflict of objectives. As an example, super-lean combustion is a strategy to decrease heat loss by reducing cylinder gas temperatures: lambda values near two are reported in [1]. However, the exhaust gas treatment has to be modified regarding the NOx reduction process: a detailed investigation of lean NOx trap (LNT) strategies can be found in [2]. This possibility of variation requires a better understanding how the engine settings influence component temperatures. Considering, for example, aluminium alloys, the resistance against fatigue is very sensitive to temperature [3].

One big challenge is the scattering nature of combustion processes. According to [4], some reasons for the typical scattering nature of SI turbo engines are fluctuations in turbulent flow pattern, local differences in gas composition, or decreased local laminar flame speed because of rich or lean combustion. Local flame quenching within turbulent flow can also be one reason [5]. Very late ignition times due to knocking control or complex interactions between dynamic boost pressure and exhaust back pressure, resulting in different residual gas, could be other reasons for the scattering behaviour [4].

In the area of engine heat transfer, early works addressed the problem with dimensional analysis and pronounced experimental studies: [6, 7, 8]. Based on these more phenomenological results, improved physical models are suggested with simple global turbulence modeling: in [9], a characteristic gas velocity is contained which includes the turbulent kinetic energy. Similarly, [10] developed a complete heat transfer model with a Reynolds–Colburn analogy. For the determination of heat transfer coefficients, a global *k*-\(\varepsilon\) model was used. For example, details to global turbulence modeling within engine combustion chambers can be found in [11]. In contrast, there also exist many works which uses detailed CFD in-cylinder flow simulations, including heat transfer processes: [12, 13], and [14]. In [15], a method is proposed which couples detailed CFD techniques with a simplified engine working process analysis to ensure the overall heat transfer rate: much better results can be obtained in comparison with state-of-the-art wall function heat transfer models. In addition, specific models were developed for different kinds of engines or flow structures: relating to heat transfer, HCCI (Homogeneous Charge Compression Ignition) engines are investigated in [16, 17], and [18]. Similarly, hydrogen engines are studied in [19, 20], or [21].

Regarding heat transfer, the dependence on different engine settings, like ignition time, air–fuel ratio or boost pressure, is of great interest. An excellent review about such sensitivities, in motored and fired engine operation, can be found in [22]. In this context, a design of experiments method is applied in [23]: various engine settings like, e.g., ignition timing, air–fuel ratio, fuel, or compression ratio, are investigated. In a more fundamental manner, the Polhausen equation \(N\!u \propto Re^m Pr^n\) in seven different operating regimes is verified. Therefore, via different exponents *m* and *n*, the Nusselt number \(N\!u\) is correlated with the Reynolds number *Re* and the Prandtl number *Pr*. Many works address the dependence on engine settings, including experimental measurements of heat fluxes and the prediction accuracy of different models: [24, 25, 26, 27, 28, 29]. As an example, [30] investigated heat fluxes as a function of ignition timing, air–fuel ratio, and mixture preparation.

Nevertheless, the questions that arise in this context relate to resulting component temperatures. Most of the previous works investigated only crank angle resolved temperature curves: [31, 32, 33, 34], or [35]. The influence of different engine settings on the cycle averaged component temperatures is not studied there. It is rare to find publications like [36] or [37]. The first one investigated experimentally averaged cylinder liner temperatures of a diesel engine as a function of ignition time and engine speed, whereas the second one measured cylinder head and piston mean temperatures of a SI engine in dependence of engine speed and load. In addition, in [38], the cylinder head mean temperature is experimentally investigated with regard to three different air–fuel ratios. In all aforementioned references, it can be observed that the cycle averaged mean temperatures reacted more sensitively to engine settings, in contrast to the corresponding temperature swings. However, a general calculation method is missing. In particular, according to inverse problems, the identification of suitable thermal boundary conditions which can account for different engine settings is challenging and it is not clear how expedient already existing heat transfer models are. That is exactly the focus of the present paper.

### 1.2 Outline of the paper

In this paper, a simplified calculation method is proposed to calculate cycle averaged temperatures as a function of different engine settings. The research question can be formulated as follows: with regard to averaged solid temperatures, knowing full well that instantaneous in-cylinder heat transfer is a highly complex problem, is it possible to use simplified, statistical calculation methods without simulating crank angle resolved heat fluxes? In particular, with respect to these integral heat fluxes, how purposeful are fast and simplified heat transfer models regarding to different engine settings? Previous investigations of the aforementioned works showed that the instantaneous heat flux of these models can strongly differ from each other and from experimental measurements.

In relation with the statistical description and calculation method, cyclical fluctuations in the pressure curves were measured and described statistically using probability density functions: gas pressure, corresponding heat transfer coefficients, cylinder gas temperature, and their product, in particular, are described as random variables. This interpretation enables a time-scale separation with a low-pass filter function.

Different operating conditions like varying air–fuel ratios, adjustments of ignition times, or variations in boost pressure are, therefore, experimentally investigated with a 0.5 L turbocharged single cylinder SI engine and compared to simulation results. In this paper, the pressure signal from the combustion chamber is used to approximate the heat transfer to different components like cylinder head, piston, valves, or the cylinder liner. With the help of older methods using the Buckingham Pi-Theorem, like correlations after Woschni [7], Hohenberg [8], or Eichelberg [6], the averaged heat transfer coefficient can then be approximated. The spatial distribution can be obtained approximately from detailed CFD simulations performed for one operating condition.

### 1.3 Method used in this work

*inner*and

*outer*boundary conditions for an engine. The outer boundary conditions can be defined as a in general time-dependent five-dimensional engine state matrix:

*concentrated parameters*that means they are no functions of place. Assuming that this state matrix is known, one has to determine the inner boundary conditions. In the context of conjugate heat transfer simulations, this includes the determination of thermal boundary conditions: temperature (Dirichlet BC), heat flux (Neumann BC), or heat transfer coefficients (Robin BC). In the following, a detailed three-dimensional CFD–CHT simulation is presented. A method is suggested how one can translate

*outer, concentrated*boundary conditions into

*inner, distributed*boundary conditions of an engine.

*Distributed*means that these BCs have to be functions of place. To control the outer BCs to a certain point, lots of variables are available: injection and ignition time or boost pressure, only to name a few. The inner BCs, for example, are results of the resulting turbulent flow of the water cooling system or the turbulent flow with chemical reactions in the cylinder. Especially, this internal flow is subjected to scattering heat releases and pressure developments.

^{©}is used to determine boundary conditions for the subsequent detailed 3D-FVM simulation. One needs, therefore, a high-pressure indication measurement \(p(\alpha _{\text {cr}})\) of the engine for the used application data version. \(\alpha _{\text {cr}}\) is the crank angle of the crankshaft. It is important to notice that lots of information about heat transfer is hidden in the pressure signal which was resolved with 0.1 \(^{\circ }\)CA. Most of the following heat transfer boundary conditions are Robin BC. Assuming that all internal flows can be described by forced convection, one can assume a heat flux of the Newton form with temperature independent heat transfer coefficient \(\alpha\):

*A*is the realization of the random variables \(\alpha\) describing the HTC for a given engine state matrix \(\underline{M}\). \(p_{\alpha |\underline{M}}(A)\) is the conditional PDF. Of course, the kind of integration in Eq. (8) can be used to calculate any type of statistical moment. The probability density functions are implemented as normed histograms. For the numerical integration, trapezoidal rule was chosen.

## 2 Determination of boundary condition

In the following, a brief summary of modeling thermal boundary conditions is given. In this paper, a combination of a detailed CFD RANS simulation for the water channel and a simplified approach using the Buckingham Pi-Theorem is used. The last one offers boundary conditions for the surfaces which are wetted with gas. According to a CHT method, the solid and fluid regions within the FVM simulation are solved simultaneously.

### 2.1 Heat transfer modeling in the inlet and outlet systems

*c*and exponent

*m*is used. According to a fully developed flow, in this paper, a value of one was used for the exponent

*m*. The coefficient

*c*is used as a calibration parameter. For the exhaust port, the following model is proposed [39]:

*a*.

### 2.2 Heat transfer modeling in the combustion chamber

*N*is the amount of substance,

*R*is the universal gas constant, and

*V*is the total volume. In Table 1, a summary of the investigated models is given.

Heat transfer coefficients in the combustion chamber

Model | HTC | Characteristic velocity | References |
---|---|---|---|

Eichelberg | \({\alpha =C_{\text {Eichelberg}} v^{1/3} \left( p \overline{T}_g\right) ^{1/2}}\) | \({ v=v_{\text {piston}}}\) | [6] |

Woschni | \({ \alpha =C_{\text {Woschni}} B^{m-1} p^m v^m}\) | \({ v=c_1v_{\text {piston}}}{ +c_2V_{\text {d}}\left( \overline{T}_g/\left( pV\right) \right) _{\text {IVC}}\left( p-p_m \right) }\) | [7] |

\({ ~~~~~~\overline{T}_g^{0.75-1.62m} }\) | |||

Hohenberg | \({ \alpha =C_{\text {Hohenberg}} d_{\text {s}}^{m-1} p^m v^m}\) | \({ v=\left( {\overline{T}_g /~ \text {K}}\right) ^{-0.1625}}\) | [8] |

\({ ~~~~~~\overline{T}_g^{0.75-1.62m} }\) | \({ ~~~~~\left( v_{\text {piston}}+1.4~ \text {m/s} \right) }\) |

A value of \(m=0.8\) is typical. The model parameters \(C_{\text {Eichelberg}}\), \(C_{\text {Woschni}}\), and \(C_{\text {Hohenberg}}\) are calibration values. *B* and \(d_{\text {s}}\) describe the engine bore and the diameter of a sphere with the same volume as the instantaneous cylinder volume. \(V_{\text {d}}\) is the displaced volume and \(p_{m}\) is the motored pressure. \(v_{\text {piston}}\) is the mean piston speed. For the Woschni model, the constant \(c_2\) is zero except in the combustion and expansion period. In this paper, a value of \(c_2=3.24\text {e}^{-3} ~ \text {m/(sK)}\) is used. For the constant \(c_1\), a value of 6.18 in the gas exchange period and 2.28 else is used. The unit of the static pressure *p* is usually *bar*.

## 3 Finite volume model

The commercial software StarCCM+ by Siemens (release v11.06) was used for calculating solid temperatures according to a CHT simulation. The model contains diverse engine components, e.g., the crankcase, cylinder head, cylinder liner, piston, cam cover, valves as well as the valve rings.

### 3.1 Mesh study

### 3.2 Spatial distribution of heat transfer coefficients and turbulence modeling

Relative distribution of the HTC between different components in the combustion chamber [13]

Component | Weighting factor |
---|---|

Cylinder liner | 1.0 |

Cylinder head | 3.25 |

Piston | 2.5 |

Intake and exhaust valves | 3.6 |

Special attention is required for the contact pair valve ring (inlet and exhaust side). For the thermal resistance, respectively, heat transfer coefficient, a value of 35000 W/(m^{2}K) was used [41]. An analogous adaption of the thermal resistance is necessary for the contact pair valve–valve guide. The empirical value of 170 W/(m^{2}K) was, therefore, chosen [42].

Increased heat transfer due to turbulence was calculated with the help of the SST *k*-\(\omega\) turbulence model by Menter [43]. The increased heat transfer was modeled according to \({\text {div}}\,({\langle { \varvec{ u}^{\prime } T^{\prime } }\rangle }) = {\text {div}}\,({ \rho c_p a_t \mathrm{grad}\left( \langle { T }\rangle \right) })\). In this case, \(\varvec{ u}\) and *T* are the fluid velocity vector and its temperature. \(\rho\) describes the density and \(c_p\) the heat capacity at constant pressure. For the turbulent Prandtl number \(Pr_t=\nu _t/a_t\), the value was set to 0.9 [44]. \(\nu _t\) and \(a_t\) are the turbulent viscosity and temperature conductivity, respectively.

### 3.3 Spark plug modeling

## 4 Results and discussion

### 4.1 Statistics of maximum in-cylinder pressure

In contrast to that, lambda variations have only a meaningful effect on \(\sigma\). The average maximum pressure only reduces about 4% with increasing lambda values. Because of current racing regulations with a limited maximum fuel flow, it is important to increase the thermal efficiency for optimizing the overall performance. Only conditions with excess air are, therefore, investigated. Details of engine conditions are summarized in the following chapters.

### 4.2 Ignition time variation

Engine parameters during variation of ignition angle

Engine parameter | Value | Engine parameter | Value |
---|---|---|---|

\(n_{\text {engine}}\) | 6000 rpm | \(t_{\text {oilin}}\) | 388.15 K |

\(t_{\text {int}}\) | 312.15 K | \(t_{\text {waterin}}\) | 387.5 K |

\(t_{\text {amb}}\) | 294.15 K | \(m_{\text {fuel}}\) | 115.5 mg/stroke |

\(\lambda _{\text {cmb}}\) | 1.2 | \(\alpha _{\text {ign}}\) | 20.5–25.25 \(^{\circ}\)CA |

The lowest ignition angle point was used for calibration. Compared with variations in the air–fuel ratio or the boost pressure, the temperature differences are small. This is in accordance with [20]: varying ignition timing caused a time-delayed heat flux, whereas the temperature differences were rather secondary. In addition, a linear regression line is added for each model and the experimental result to evaluate the sensitivities to ignition time variations. Crank angle-based measurements show that the model according to Woschni tends to overpredict the heat transfer at the top dead center region, but underestimates the heat flux during compression and exhaust phases [28]. According to [45], Hohenberg’s model reveals some improvements in this aspect. Analogous comparisons done by [46] show that the Eichelberg model predicts lower values than the Woschni model. According to measurements done by [26], one can summarize that the Woschni model predicts higher time averaged heat fluxes and total heat releases for most engine states in comparison with the Hohenberg model. This ratio is a function of engine load and speed. Regarding crank angle-based heat flux measurements, one can guess that the Woschni model overpredicts the time averaged heat flux. Hohenberg’s model seems to be better. This tendency, however, does not provide information about heat flux expectation values when using calibrated model equations. The results in Fig. 9 show that the Woschni model overpredicts the sensitivity to ignition angle variations for both, combustion chamber wall and spark plug. The other two models give too low gradients for the combustion chamber wall temperatures in comparison with the experimental values. True values probably lie somewhere in between. The temperatures for the spark plug are, however, in good agreement with the measurements. It must be pointed out that the spark plug modeling according to Eq. (13) is a strong simplification. It is also possible that this model overpredicts the heat increase to the spark plug in general: a slightly underestimation of the HTC would then result in a good agreement for the spark plug heat transfer. One interesting observation is the noticeable temporary decrease of both temperatures at an ignition angle of 22.75\(^{\circ }\)CA. It seems that this phenomenon is not a measuring error because every model predicts this temperature drop. It is satisfying that the models can predict such an unpredictable event using the information hidden in the pressure signal. Remarkably, the power of the engine does not decrease at this measuring point: the positive gradient of thermal efficiency with increasing ignition angle does not show any abnormality. In comparison with [28], the predicted differences in the *averaged* HTCs are correctly lower than the corresponding peak values during an engine cycle. For example, consider the prediction by Woschni which gives a difference of the averaged value in the order of 45 W/(m^{2}K). The reported peak value is about 75 W/(m^{2}K). Both results are based on an ignition time interval of 5 \(^{\circ }\)CA. Compared with the Hohenberg model, the resulting difference is only about 6 W/(m^{2}K).

### 4.3 Boost pressure variation with constant lambda

Engine parameters during variation of boost pressure at a constant air–fuel ratio

Engine parameter | Value | Engine parameter | Value |
---|---|---|---|

\(n_{\text {engine}}\) | 8000 rpm | \(t_{\text {oilin}}\) | 387.15 K |

\(t_{\text {int}}\) | 301.15 K | \(t_{\text {waterin}}\) | 386.5 K |

\(t_{\text {amb}}\) | 294.65 K | \(m_{\text {fuel}}\) | 27–52 mg/stroke |

\(\lambda _{\text {cmb}}\) | 1.21 | \(\alpha _{\text {ign}}\) | 37.5–44.0 °CA |

### 4.4 Lambda variation by adjusting boost pressure

Engine parameters during variation of air–fuel ratio at a constant mass flow rate

Engine parameter | Value | Engine parameter | Value |
---|---|---|---|

\(n_{\text {engine}}\) | 8000 rpm | \(t_{\text {oilin}}\) | 364.15 K |

\(t_{\text {int}}\) | 306.15 K | \(t_{\text {waterin}}\) | 356.5 K |

\(t_{\text {amb}}\) | 300.15 K | \(m_{\text {fuel}}\) | 91.25 mg/stroke |

\(\lambda _{\text {cmb}}\) | 0.9–1.4 | \(\alpha _{\text {ign}}\) | 26.0–42 \(^{\circ }\)CA |

Another interesting aspect can be seen in Fig. 13b. For comparison reasons, simulations have been performed with different heat transfer coefficients in the exhaust pipe. Therefore, a slightly larger and lower HTC value was investigated. Of course, the heat transfer coefficient in the outlet pipe has an influence on solids which are located next to the outlet. Lower values result in a less sensitivity to air–fuel ratios. This can be explained with the influence of the exhaust gas temperature which is a complicated function of thermal efficiency and effective heat capacity of the exhaust gas mixture. The progression of the exhaust gas temperature is qualitatively similar to the temperature curve in Fig.13b.

## 5 Conclusions

With regard to component temperatures and their dependence on different engine settings, a simplified, statistical CFD–CHT calculation method is presented. Concerning the initial research question, if such a method without simulating crank angle resolved heat fluxes is purposeful for determining cycle averaged component temperatures, the result is as follows.

In spite of the difference concerning the instantaneous heat flux, discussed in numerous publications, the proposed simulation technique, which uses integrated boundary conditions, together with the Buckingham Pi-Theorem, more precisely, with simplified engine specific correlations, is in general able to predict relative differences in cycle averaged component temperatures as a function of the engine setting. If the calculation time is an important factor, it is appropriate to use these simplified models.

One has, however, to pay attention which model in detail is used. The Woschni model slightly overpredicts the increased heat transfer with increasing ignition time, whereas the other two models underpredict the effect. The differences to measurements were in the range of 2 K. This is small compared to a measurement inaccuracy of 1 K. For boost pressure variations, all models describe a similar sensitivity to load changes even if individual temperature differences up to 4 K can be identified. In contrast, the correlation according to Woschni is clearly the best choice for Lambda variations: the other two models incorrectly calculated the maximum temperature by the amount of 8 K, whereas the model according to Woschni was within the range of the measurement error.

Another main conclusion is the importance of significant non-linearity effects in heat transfer problems when engine cycles are not simulated in detail. Concerning gas temperatures, the difference between the pure expectation value and the statistically modified average value, which takes the correlation between the heat transfer coefficient and the gas temperature into account, can be up to 300 K, depending on the model.

## 6 Nomenclature

Symbol | Description | Unit |
---|---|---|

| Temperature conductivity \(a=\lambda / \left( \rho c_p\right)\) | m |

| Engine bore | m |

\(c_{\text {p}}\) | Specific heat at constant pressure | J/(kgK) |

\(d_{\text {s}}\) | Diameter of a sphere with the same volume as the instantaneous cylinder volume | m |

| Re exponent | Dimensionless |

\(m_{\text {air}}\) | Mass flow of air | mg/stroke |

\(m_{\text {fuel}}\) | Mass flow of fuel | mg/stroke |

\(\underline{M}\) | Five-dimensional engine state matrix of outer boundary conditions | Various |

\(n_{\text {engine}}\) | Engine speed | Rounds per minute [rpm] |

\(\varvec{ n}\) | Boundary normal vector | Dimensionless |

| Amount of substance | mol |

\(N\!u\) | Nusslet number | Dimensionless |

| Static pressure | Pa |

\(p_{\text {m}}\) | Motored cylinder pressure | Pa |

\(p_{\text {int}}\) | Boost pressure | mbar |

\(p_n\) | Probability density function on | Dimensionless |

\(p_{\text {max}}\) | Maximum pressure within one engine cycle | Pa |

| Prandtl number \(\nu /a\) | Dimensionless |

\(\varvec{ q}\) | Heat flux vector | W/m |

| Universal gas constant | J/(molK) |

| Reynolds number \(lv/\nu\) | Dimensionless |

\(Re_{\text {j}}\) | Exhaust jet Reynolds number \(l v_{\text {j}}/\nu\) | Dimensionless |

\(t_{\text {int}}\) | Inlet temperature of air | K |

\(t_{\text {amb}}\) | Ambient temperature of air | K |

\(t_{\text {waterin}}\) | Inlet water temperature | K |

\(t_{\text {waterout}}\) | Outlet water temperature | K |

\(t_{\text {oilin}}\) | Inlet oil temperature | K |

\(t_{\text {oilout}}\) | Outlet oil temperature | K |

| Physical time | s |

\(T_{\text {ref}}\) | Reference temperature | K |

\(\overline{T}_g\) | Cylinder-average gas temperature | K |

\(T_s\) | Solid temperature | K |

\(T_{\text {mod}}\) | Statistically modified temperature | K |

\(T_{\text {i}}\) | Indicated torque by combustion | Nm |

\(\varvec{ u}\) | Velocity vector | m/s |

| Characteristic velocity | m/s |

\(v_{\text {j}}\) | Exhaust jet velocity through valve opening | m/s |

\(v_{\text {piston}}\) | Mean piston speed | m/s |

| Volume | m |

\(V_{\text {d}}\) | Displaced volume | m |

\(y^{+}\) | Dimensionless wall distance | Dimensionless |

**Greek symbols**

Symbol | Description | Unit |
---|---|---|

\(\alpha\) | Heat transfer coefficient | W/(m |

\(\alpha _{\text {cr}}\) | Crank angle | \(^{\circ}\)CA |

\(\alpha _{\text {ign}}\) | Ignition crank angle | \(^{\circ}\)CA |

\(\lambda\) | Thermal conductivity | W/(mK) |

\(\lambda _{\text {cmb}}\) | Ratio between actual air mass and stoichiometric air mass | dimensionless |

\(\mu\) | Dynamic viscosity | kg/(ms) |

\(\nu\) | Kinematic viscosity \(\mu /\rho\) | m |

\(\rho\) | Mass density | kg/m |

**Mathematical notation**

Symbol | Description |
---|---|

\(\langle {\cdot }\rangle\) | Expectation value regarding time |

\(\left( \cdot \right) ^{\prime }\) | Fluctuation value regarding time |

**Abbreviations**

Abbreviation | Description |
---|---|

CFD | Computational fluid dynamics |

CHT | Conjugate heat transfer |

RANS | Raynolds average navier stokes |

BC | Boundary condition |

FVM | Finite volume method |

HTC | Heat transfer coeffcient |

ACT | Average cylinder temperature |

IVC | Inlet valve closing |

| Probability density function |

## Notes

### Acknowledgements

P. Hölz would like to thank Tobias Möllenhof and Christian Eifrig, both Porsche Motorsport, for their experimental support.

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