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

The aim of the present paper is the determination of engine temperatures in dependence of the operating conditions, in particular of the engine application data version. That means different engine mappings like ignition time or air–fuel ratio variations are investigated with regard to solid temperatures. One distinguishes, therefore, between 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:
$$\begin{aligned} \underline{M}(t)= \left( n_{\text {engine}}(t),m_{\text {air}}(t),t_{\text {int}}(t),T_{\text {i}}(t),m_{\text {fuel}}(t) \right) . \end{aligned}$$
(1)
In this paper, however, we consider the special case of stationary engine states. The entries are engine speed, air mass flow rate, inlet air temperature, induced torque, and fuel mass flow rate. Obviously, these outer boundary conditions have to be seen as 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.
Fig. 1

Method overview for generating boundary conditions for a detailed three-dimensional CFD–CHT simulation with the help of pressure indication measurement \(p(\alpha _{cr})\) at a given engine speed\(n_{\text {engine}}\)

An overview of the method is shown in Fig.1. An own implemented code in MATLAB© 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 \(\alpha\):
$$\begin{aligned} \varvec{ q}=-\,\alpha \left( T_{\mathrm{{ref}}} - T_s \right) \varvec{ n}. \end{aligned}$$
(2)
\(\varvec{ n}\) is the surface normal vector and \(T_s\) is the solid temperature. Using the Buckingham Pi-Theorem, the instantaneous heat transfer coefficient \(\alpha\) and the corresponding reference temperature \(T_{\text {ref}}\) are determined. The idea is to interpret these quantities as random variables. In this case, however, one has to use conditional probability density functions. In the case of the heat transfer coefficient, this is \(p_{\alpha |\underline{M}}\) for a given engine state \(\underline{M}\).
Physically speaking, the heat transfer coefficient \(\alpha\) strongly depends on the engine state \(\underline{M}\). The description as a random variable is quite useful because one can take into account the strong scattering nature of the cylinder pressure. Especially, combustion with high \(\lambda _{\text {cmb}}\) values tends to scatter. It describes the ratio between actual air mass and stoichiometric air mass. For calculating stationary temperature fields, one has to integrate the approximated density functions to get expectation values of heat transfer coefficients and the corresponding reference temperatures. Strictly spoken, an engine does not have a stationary state as a result of the oscillating four-stroke process. Nevertheless, one can consider quasi-stationary conditions if one imagines a time-scale separation realized by a statistical description. This is a great advantage because of the non-linearity in Eq. (2). Neither for quasi-stationary conditions nor for transient simulations, one can use the Robin boundary condition in the following way:
$$\begin{aligned} \langle \varvec{ q} \rangle \ne - \langle \alpha \rangle \left( \langle T_{\text {ref}} \rangle - \langle T_s \rangle \right) \varvec{ n}. \end{aligned}$$
(3)
Note that \(\langle \cdot \rangle\) is a mathematical operator for the expectation value. \(\langle T_s \rangle\) is the searched solid temperature which can be a function of time in transient simulations. Instead of Eq. (3), one has to use
$$\begin{aligned} \langle \varvec{ q} \rangle= & {} - \langle \alpha \left( T_{\text {ref}} - T_s \right) \rangle \varvec{ n}\nonumber \\= & {} -\left( \langle \alpha T_{\text {ref}} \rangle - \langle \alpha T_s \rangle \right) \varvec{ n}\nonumber \\= & {} - \langle \alpha \rangle \left( \frac{\langle \alpha T_{\text {ref}} \rangle }{\langle \alpha \rangle } - \langle T_s \rangle \right) \varvec{ n}\nonumber \\= & {} - \langle \alpha \rangle \left( T_{\text {mod}} - \langle T_s \rangle \right) \varvec{ n}. \end{aligned}$$
(4)
The term \({T_{\text {mod}}=\langle \alpha T_{\text {ref}} \rangle / \langle \alpha \rangle }\) will be called the modified reference temperature. Whenever it will be spoken of a reference temperature, this modified version is meant. Notice that Eq. (4) can be rewritten as
$$\begin{aligned} \langle \varvec{ q} \rangle= & {} - \langle \alpha \left( T_{\text {ref}} - T_s \right) \rangle \varvec{ n}\nonumber \\= & {} -\left( \langle \alpha T_{\text {ref}} \rangle - \langle \alpha T_s \rangle \right) \varvec{ n}\nonumber \\= & {} -\left( \langle \alpha \rangle \langle T_{\text {ref}} \rangle + \langle \alpha ^{\prime } T_{\text {ref}}^{\prime } \rangle - \langle \alpha T_s \rangle \right) \varvec{ n}. \end{aligned}$$
(5)
Therefore, the modified gas temperature \(T_{\text {mod}}\) implies the effect due to the correlation \({\langle \alpha ^{\prime } T_{\text {ref}}^{\prime } \rangle }\). The superscript \((\cdot )^{\prime }\) describes fluctuations. In Eq. (4), it was assumed that the correlation of the heat transfer coefficient \(\alpha\) and the solid temperature \(T_s\) is negligibly small:
$$\begin{aligned} {\langle \alpha ^{\prime }T_s^{\prime } \rangle \ll \langle \alpha \rangle \langle T_s \rangle .} \end{aligned}$$
(6)
Because of large heat capacities of solids, their temperatures behave much slower than variations of heat transfer coefficients. Inequality (6) can be checked with the following conservative assumptions: temperature fluctuations \(T_s^{\prime }\) and HTC fluctuations \(\alpha ^{\prime }\) are inphase with a maximum temperature amplitude of 30 K during an engine cycle [35]. For typical engine operating conditions with solid temperatures in the range of 400–500 K, the left-hand side of Eq. (6) is two orders of magnitude smaller than the right-hand side. In the subsequent FVM simulation, \({\langle \alpha \rangle }\) and \(T_{\text {mod}}\) are input parameters describing Robin boundary conditions. Remember that one can estimate the correlation between two random variables with the derivation of Eq. (4). For the correlation of the HTC and ACT in the combustion chamber, it follows:
$$\begin{aligned} {\langle \alpha ^{\prime } T_{\text {ref}}^{\prime } \rangle =\langle \alpha T_{\text {ref}} \rangle -\langle \alpha \rangle \langle T_{\text {ref}} \rangle .} \end{aligned}$$
(7)
Of course, this correlation is not zero and, hence, cannot be neglected in the case of the combustion chamber. Pressure and temperature peaks occur simultaneously. For this reason, the HTC and the ACT have a strong correlation. Summarizing the above-mentioned statistical description with the example of the HTC in the combustion chamber, one gets for any arbitrary random variable \(\tilde{f}(\alpha )\) the following expression for its expectation value:
$$\begin{aligned} \langle \tilde{f}(\alpha ) \rangle = \int \limits _{\mathbb {R}_{\ge 0}} \tilde{f}(A) p_{\alpha |\underline{M}}(A) \,dA. \end{aligned}$$
(8)
In this case, the variable 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

The correlation according to [39]
$$\begin{aligned} N\!u_v = 1,84Re_v^{0.58} (D_v/l_v)^{0.2} \end{aligned}$$
(9)
is used for the outlet valve stem. \(Re_v\) and \(N\!u_v\) are the Reynolds number and Nusselt number based on the valve lift \(l_v\) and the exhaust jet gas velocity. \(D_v\) is the diameter of the valve. For the inlet valve stem, a similar expression can be assumed [40]. Therefore, the coefficient in Eq. (9) has to be reduced by 40%. For the intake port, a simple form of
$$\begin{aligned} N\!u = c Re^m, \end{aligned}$$
(10)
with a coefficient 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]:
$$\begin{aligned} N\!u=\sqrt{ \left( 8Re_{\text {j}}Pr/\pi \right) }. \end{aligned}$$
(11)
\(N\!u\) and \(Re_{\text {j}}\) are based on the duct diameter and the exhaust jet gas velocity. \(Pr=\nu /a\) is the Prandtl number and describes the ratio between the kinematic viscosity \(\nu\) and the temperature conductivity a.

2.2 Heat transfer modeling in the combustion chamber

Regarding the sensitivity to different engine mappings, three different models are investigated. Radiation is generally neglected. Depending on the channel geometry swirl, flow structures can occur in engine cylinders. It is important to note that the investigated engine does not have any significant swirl motion. For the reference temperature in Eq. (2), the cylinder-average gas temperature
$$\begin{aligned} \overline{T}_g=\frac{pV}{NR} \end{aligned}$$
(12)
is used. 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.
Table 1

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

Mesh studies have been performed to ensure mesh independence regarding several thermal quantities. First, local mesh size in the cylinder head was varied in the region of the high-speed channel flow around the valve seats and cylinder liner cooling areas.
Fig. 2

Overview about local mesh size. Left: qualitative fluid velocity field in the water channel. Right: qualitative temperature field for fluid and solid regions

Second, local mesh size of the water region was varied at the same domain. The boundary layer, however, was kept constant with three cells. The first cell lied either in the viscous sublayer (dimensionless wall distance \(y^{+} < 5\)) or in the logarithmic layer (dimensionless wall distance \(y^{+} <35\)). The results are shown in Figs. 3 and 4. An overview about the mesh can be seen in Fig. 2.
Fig. 3

Mesh study for the cylinder head, solid region. Local mesh was varied in the region of the high-speed channel flow around the valve seats and cylinder liner cooling areas. a Surface average solid temperature at the combustion chamber wall. b Heat flux of the wetted surfaces

Fig. 4

Mesh study for the cylinder head, water region. Local mesh was varied in the region of the high-speed channel flow around the valve seats and cylinder liner cooling areas. Heat flux of the wetted surfaces

3.2 Spatial distribution of heat transfer coefficients and turbulence modeling

To take account of the spatial distribution of the HTC within the combustion chamber, detailed 3D-CFD simulation results from [13] are used. The relative distribution between different components is listed in Table 2. In addition, the spatial dependence of the HTC on the cylinder head and piston surface is assumed as a linear function with respect to the radial position. Regarding the liner, a linear distribution with respect to axial position is modeled. Therefore, only the part is considered which is available for the propagating flame. The radial and axial distributions in the combustion chamber are implemented in such a way that the spatial course is continuous. It is important to note that the global HTC, according to Eq. (8), serves as a preexponential factor for each component.
Table 2

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/(m2K) 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/(m2K) 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

The spark plug is modeled as a perfectly isolated steel layer on the one side which has a perfect heat conduction to the cylinder head on the other side. At the bottom surface which projects into the combustion chamber, the heat transfer is also given by a heat transfer coefficient and the average gas temperature. However, due to the fact that the spark plug projects deeper into the combustion chamber, one has to modify the heat transfer coefficient resulting from the Pi-Theorem. A heat increasing factor \(\alpha _{\text {increase}}\) was, therefore, introduced resulting in a modified heat transfer coefficient for the spark plug \(\alpha _{\text {spark}}\):
$$\begin{aligned} \alpha _{\text {spark}} = \alpha _{\text {increase}} \alpha _{\pi }. \end{aligned}$$
(13)
\(\alpha _{\pi }\) is the heat transfer coefficient resulting from the Pi-Theorem. The factor \(\alpha _{\text {increase}}\) was calibrated once.

4 Results and discussion

A 0.5 L turbocharged single cylinder SI engine was investigated under varying boundary conditions. Figure 5 gives an overview about the parameter set and the corresponding in-cylinder pressure curves.
Fig. 5

Averaged in-cylinder pressure curves for three different thermodynamic variations. Within one variation, the pressure curves are normed to the maximum value. All three variations represent full load conditions. a Ignition time variation with constant fuel mass flow and air–fuel ratio: \(n_{\text {engine}}=6000 ~\text {rpm}\). b Boost pressure variation with constant air–fuel ratio. Ignition timing was set for an optimum thermal efficiency: \(n_{\text {engine}}=8000 ~\text {rpm}\). c Lambda variation with constant fuel mass flow. Ignition timing was set for an optimum thermal efficiency: \(n_{\text {engine}}=8000 ~\text {rpm}\)

Type K Thermocouples with an uncertainty of 1 K were used for the measurement of solid temperatures. Heat-conducting paste with a thermal conductivity of 2.5 W/(mK) was put between the solid surface and the thermocouples. Fluid temperatures were measured with Pt100 sensors which have an uncertainty of 0.053 K. Measuring points were installed inside of the spark plug coat, at the combustion chamber wall of the cylinder head and the exhaust valve seat (seat ring). A thermocouple was, therefore, welded into the spark plug which ended at the last thread. For the combustion chamber wall, a hole until 1 mm above the surface was drilled. The position was located at the outer diameter of the combustion chamber. For each measuring point, 60 combustion cycles were recorded with the help of a piezo-quartz pressure transducer mounted on the cylinder head. A sampling rate of 0.1 °CA was, therefore, used. The crankshaft position was measured by a crankshaft angle sensor to determine the in-cylinder gas pressure as a function of crank angle. An overview about the method validation is given in Fig. 6. The simulated solid temperature difference between different engine settings, which is based on the thermal boundary conditions of the suggested calculation technique and various heat transfer models, is compared with the measured temperature difference.
Fig. 6

Overview about the method validation

4.1 Statistics of maximum in-cylinder pressure

A brief summary of pressure statistics is given in Fig. 7. A Gaussian distribution was, therefore, assumed for the maximum pressure \(p_{\text {max}}\):
$$\begin{aligned} p_{p_{\text {max}}}=\frac{1}{\sqrt{2\pi \sigma ^2}} \exp \left( -\frac{\left( p_{\text {max}}-\xi \right) ^2}{2\sigma ^2}\right) . \end{aligned}$$
(14)
According to Table 1 and a model constant of \(m=0.8\), the maximum HTC is approximately linear in the maximum pressure. Remember that the average gas temperature \(\overline{T}_g\) is nearly proportional to the pressure for a constant amount of molecules which is the reason why the maximum heat flux is approximately in quadratic proportion to the maximum pressure (Eq. (2)). Regarding mechanical loads in the high cycle fatigue regime, the maximum pressure is a representative quantity. Note that ignition time and boost pressure variations have a strong effect on expectation values \(\xi\) and standard deviations \(\sigma\), respectively. For different loads, by adjusting boost pressure, a quite linear behaviour can be seen.
Fig. 7

Statistics of maximum in-cylinder pressure. Gaussian distribution on the left side. Pressure values are normed to the maximum average value. Expectation value \(\xi\) and standard deviation \(\sigma\) on the right side. Within one variation \(\xi\) and \(\sigma\) are normed to the maximum value. a, b Ignition time variation. c, d Boost pressure variation. e, f Lambda variation

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

There are lots of other reasons to change the ignition angle during a race: in response to different torque requirements because of traction control or due to preceded knocking events, for example. With more spark advance, the thermal efficiency of the engine increases, because more fuel is chemically converted at the top dead center. This results, however, in a higher heat transfer rate to the combustion chamber wall (Fig. 5a). In Table 3, the most important engine parameters are summarized. \(t_{\text {amb}}\) and \(\alpha _{\text {ign}}\) are the ambient air temperature and ignition angle, respectively.
Table 3

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

For this experiment, two different positions are investigated: the spark plug and the combustion chamber wall. In Fig. 8, the variations of the heat transfer coefficients and the modified average gas temperatures are shown.
Fig. 8

Progressions of the heat transfer coefficients and the modified average gas temperatures: a HTC increase in the combustion chamber due to more spark advance for three different models. b Corresponding ACT rise with increasing ignition angle.

The dashed lines are linear regression lines for a better comparison. Remember that due to the non-linearity of Eq. (4), one has to use modified average gas temperatures: \({\langle \alpha T_{ref} \rangle / \langle \alpha \rangle }\). These values are not independent on the models due to the influence of the heat transfer coefficient \(\alpha\). The most significant differences can be seen in Fig. 8a): whereas the model according to Hohenberg does not change significantly the heat transfer coefficients, there is a huge increase predicted by the Woschni model. According to Fig. 8b), the change in the modified average gas temperature is quite similar for all three models. There is a slightly higher increase for the Woschni model. In Fig. 9, the simulation and experimental results are shown for the solid temperatures.
Fig. 9

Comparison between simulation and experimental results for ignition time variation: a Temperature increase at the combustion chamber wall due to more spark advance. b Corresponding temperature rise at the spark plug with increasing ignition angle.

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/(m2K). The reported peak value is about 75 W/(m2K). 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/(m2K).

4.3 Boost pressure variation with constant lambda

In this experiment, the fuel mass flow and the air mass flow were varied keeping the air–fuel ratio constant. The ignition time was adjusted manually to the individual knocking limit. This experiment can be seen as a load variation at a constant engine speed. The boost pressure was varied between 0.8 and 1.4 bar resulting in a throttled and turbocharged engine state. The engine parameters are summarized in Table 4.
Table 4

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

The measuring points were the same as for the ignition time experiment. In Fig. 10, the variations of the heat transfer coefficients and the modified average gas temperatures are shown. The dashed lines are cubic and linear regression lines. For comparative purposes, the average gas temperature (not modified) is plotted. It is just the resulting temperature after the ideal gas law. Of course, it is the same for every model. The difference results from the non-linearity of Eq. (4) and reflects the great importance of correlations like Eq. (7).
Fig. 10

Progressions of the heat transfer coefficients and the modified average gas temperatures: a HTC increase in the combustion chamber due to more boost pressure for three different models. b Corresponding ACT development with increasing boost pressure. For comparative purposes, the average temperature (not modified) is plotted

Fig. 11

Comparison between simulation and experimental results for boost pressure variation: a Temperature increase at the combustion chamber wall due to higher loads. b Corresponding temperature rise at the spark plug with increasing boost pressure

In Fig. 11, the simulation and experimental results are shown for solid temperatures. Again, dashed lines are added for a better evaluation of sensitivities. The results are comparable with [37]: increasing load also resulted in an approximately linear temperature increase in the same order of magnitude. Given the small disparities between simulation and experiment in Fig. 11a, it is a remarkable result that every model can predict sensitivities to boost pressure variations within a small range. Eichelberg’s model is not as effective as the two other. The models according to Woschni and Hohenberg are within the measurement uncertainty of the thermocouples. Beyond the throttled condition, the temperature difference at the combustion chamber wall is in good agreement with the experimental results. Besides a variation of the induced torque, boost pressure variations with a constant air–fuel ratio result in a continuous change in the amount of gas within the combustion chamber. That is the reason for this kind of experiment to be more challenging than pure ignition time variations. The progress of the modified average gas temperature is much more complicated, see Fig. 10b. The striking similarity between the models according to Woschni and Hohenberg, in this case, can be seen in Figs. 10a and 11a. The differences between the average heat transfer coefficients and the resulting temperatures at the combustion chamber wall are quite small. For load variations at constant engine speed, this behaviour can also be found in [26]: plotting the isolines for the time averaged heat transfer ratio between the models according to Woschni and Hohenberg within a load–speed diagram both models behaves quite similar with increasing engine load. This means that the isolines are almost perpendicular to the engine speed axis. Other investigated models like Bargende [15] or Kleinschmidt [47] show larger modifications. The model according to Bargende includes the turbulent kinetic energy and describes the combustion process with the help of a two-zone model. Kleinschmidt uses partial differential equations for the description of physical processes within a combustion chamber. The lower gradient of the heat transfer coefficient according to Eichelberg’s model is balanced with the higher sensitivity to the modified average gas temperature resulting in similar solid temperature predictions. As can be seen in Fig. 11b, each model overpredicts significantly the spark plug temperatures with increasing boost pressure. At the highest boost pressure of 1400 mbar, the temperature difference between experiment and simulation is equivalent to the measured temperature increase. It seems that the simplified spark plug modeling approach according to Eq. (13) overestimates the higher heat transfer to spark plugs. A more reasonable approach should not only account for the additional heat resulting from another position relative to the combustion chamber but also from the ignition process itself. This would decrease the sensitivity to changes within the combustion process. According to Eq. (2), a new simplified approach could be
$$\begin{aligned} \varvec{ q}= -\,\alpha _{\text {increase}} \alpha _{\pi } \left( T_\mathrm{{ref}}-T_{\text {spark}} \right) \varvec{ n}- \varvec{ q}_{\text {Ign}}. \end{aligned}$$
(15)
The nomenclature is the same as in Eq. (13). \(\varvec{ q}_{\text {Ign}}\) is the additional heat flux vector due to the ignition process. Its offset function reduces the sensitivity to changes within the combustion process. However, this vector is probably a quite complicated function of the local thermodynamical state around the spark plug and the required ignition voltage. Since modeling of such a formation of free radicals is not that easy and the spark plug is not in the main focus of this paper, Eq. (15) will not be further investigated.

4.4 Lambda variation by adjusting boost pressure

While keeping the engine speed and the fuel mass flow constant, the boost pressure was varied resulting in different values for the air–fuel ratio. The ignition time was adjusted manually to a specific thermal efficiency and knocking level. Besides the variation of the heat transfer coefficients in the combustion chamber and the modified average gas temperature, the differences in the boost pressure at a constant fuel mass flow rate have another effect which results in a variation of the heat transfer coefficients in the outlet channel and the outlet valve stem. In addition, the exhaust temperature varies because of different thermal efficiencies. Moreover, the heat capacity of the exhaust gas varies with different air–fuel ratios. All in all, an experiment with a variation of the mixture is a problem which contains lots of different aspects: changes in the thermochemical status in the combustion chamber, in the absolute number of molecules and in the pressure curves during the working cycle. The differences in the exhaust heat transfer coefficients and temperatures come along. In Table 5, the engine parameters are summarized.
Table 5

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

In Fig. 12, the resulting differences in the heat transfer coefficients and modified average gas temperatures for each model are presented. The highest value for \(\lambda _{\text {cmb}}\) serves as reference point.
Fig. 12

Progressions of the heat transfer coefficients and the modified average gas temperatures as a function of air–fuel ratio. The dashed lines are fitted polynomials of the fourth order using the method of least squares: a HTC increase in the combustion chamber due to higher air–fuel ratios for three different models. b Corresponding ACT development with increasing air–fuel ratios. For comparative purposes, the average temperature (not modified) is plotted.

Again, one can see large differences between the Eichelberg model and the two other models. The tendency is the same as in Fig. 10. Remember that now the highest value for the air–fuel ratio was set as a reference point. According to Fig. 12a, the Eichelberg model predicts little changes in the modified average gas temperature during \(\lambda _{\text {cmb}}\)—variations. In addition, this model shows the highest increase in the modified average gas temperature with increasing air–fuel ratio. Both result in a complete overestimation of solid temperature differences, as can be seen in Fig. 13a. The two other models behave quite similar concerning the development of the heat transfer coefficient in the combustion chamber. Hohenberg’s prediction for the average gas temperature is, however, too weak, resulting in an underestimation of solid temperatures, as can be seen in Fig. 13a. The prediction according to Woschni is the best compared to the other two models which is the result of interaction between the change in the heat transfer coefficient and the average gas temperature. While the first falls continuously with increasing air–fuel ratio, the second has a maximum in rich combustion conditions. This example of different air–fuel ratios shows clear differences in the models under investigation. In Fig. 12, the average gas temperature (not modified) is plotted. Again, one can see the large importance of the correlation term (7). The modified gas temperature is clearly larger than the average gas temperature resulting in a higher heat flux to the wall. This means that the correlation must be positive: \({\langle \alpha ^{\prime } T_\mathrm{{ref}}^{\prime } \rangle \ge 0 }\). This is physically meaningful because, during the combustion stroke, the average gas temperature and the heat transfer coefficient have its maximum shortly after top dead center. In addition, both maximum values show a particularly strong amplitude with an enormous contribution to the correlation term. In Fig. 12b, one can also see that the physical average gas temperature, e.g., not modified, has its maximum at \(\lambda _{\text {cmb}} \approx 1\). Actually, it is a bit lower than one. This is also physically meaningful, because it is well known that the laminar flame speed and the adiabatic flame temperature have their maximum values when \(\lambda _{\text {cmb}}\) is a little bit lower than one [5]. This is also the reason why the maximum value of the exhaust ring temperature occurs at the stoichiometric region (see Fig. 13a).
Fig. 13

Progression of the exhaust valve seat temperature as a function of air–fuel ratio. The highest value for \(\lambda _{\text {cmb}}\) was set as a reference point. The dashed lines are fitted polynomials of the fourth order using the method of least squares: a Simulated and measured temperatures for three different models. b Simulated and measured temperatures for the Woschni model with three different exhaust pipe heat transfer coefficients

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

a

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

m2/s

B

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

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

N

Amount of substance

mol

\(N\!u\)

Nusslet number

Dimensionless

p

Static pressure

Pa

\(p_{\text {m}}\)

Motored cylinder pressure

Pa

\(p_{\text {int}}\)

Boost pressure

mbar

\(p_n\)

Probability density function on n

Dimensionless

\(p_{\text {max}}\)

Maximum pressure within one engine cycle

Pa

Pr

Prandtl number \(\nu /a\)

Dimensionless

\(\varvec{ q}\)

Heat flux vector

W/m2

R

Universal gas constant

J/(molK)

Re

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

t

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

v

Characteristic velocity

m/s

\(v_{\text {j}}\)

Exhaust jet velocity through valve opening

m/s

\(v_{\text {piston}}\)

Mean piston speed

m/s

V

Volume

m3

\(V_{\text {d}}\)

Displaced volume

m3

\(y^{+}\)

Dimensionless wall distance

Dimensionless

Greek symbols

Symbol

Description

Unit

\(\alpha\)

Heat transfer coefficient

W/(m2K)

\(\alpha _{\text {cr}}\)

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 \(\mu /\rho\)

m2/s

\(\rho\)

Mass density

kg/m3

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

PDF

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Porsche AGPorsche MotorsportWeissachGermany
  2. 2.Chair for Continuum MechanicsInstitute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT)KarlsruheGermany

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