Applicability of high dimensional model representation correlations for ignition delay times of n-heptane/air mixtures

Abstract

It is difficult to predict the ignition delay times for fuels with the two-stage ignition tendency because of the existence of the nonlinear negative temperature coefficient (NTC) phenomenon at low temperature regimes. In this paper, the random sampling-high dimensional model representation (RS-HDMR) methods were employed to predict the ignition delay times of n-heptane/air mixtures, which exhibits the NTC phenomenon, over a range of initial conditions. A detailed n-heptane chemical mechanism was used to calculate the fuel ignition delay times in the adiabatic constant-pressure system, and two HDMR correlations, the global correlation and the stepwise correlations, were then constructed. Besides, the ignition delay times predicted by both types of correlations were validated against those calculated using the detailed chemical mechanism. The results showed that both correlations had a satisfactory prediction accuracy in general for the ignition delay times of the n-heptane/air mixtures and the stepwise correlations exhibited a better performance than the global correlation in each subdomain. Therefore, it is concluded that HDMR correlations are capable of predicting the ignition delay times for fuels with two-stage ignition behaviors at low-to-intermediate temperature conditions.

Keywords

ignition delay / random sampling / high dimensional model representation / n-heptane / fuel kinetics

Introduction

The limited fossil fuel resources behoove us to develop highly-efficient combustion strategies for internal combustion (IC) engines. Advanced engine combustion strategies, e.g., homogeneous charge compression ignition (HCCI) [¹], low temperature combustion [²], partially premixed compression ignition (PPCI) [³,⁴], and reactivity-controlled compression ignition (RCCI) [⁵,⁶], have always been proposed for engine efficiency improvement. However, the combustion processes in these advanced combustion engines were highly dependent on the fuel chemical reactivity, which poses challenges to engine combustion phase control [⁷,⁸]. Therefore, the reliable prediction of fuel auto-ignition tendency at engine-like conditions of high pressures and intermediate temperatures was of increasing importance in the development and optimization of advanced engine combustion strategies [⁹,¹⁰]. Ignition delay (ID) time [¹¹–¹⁴] is one of the most important parameters to evaluate the fuel auto-ignition tendency in the oxidation and combustion processes. In the past decades, ID times of different fuels have been widely measured using perfectly stirred reactors [¹⁵], jet-stirred reactor [¹⁶], shock tubes [¹⁷], and rapid compression machines [¹⁸]. However, experimental measurements of fuel ID times over a wide range are unrealistic and inefficient, and as a result, many parameters such as equivalence ratio, charge composition, temperature, and pressure vary in practical conditions. Although numerical simulations based on detailed fuel chemical mechanisms might be an alternative for fuel ID times determination, the detailed mechanisms of practical fuels generally contain hundreds of species and thousands of reactions, thus leading to a considerable calculation time. From an engineering viewpoint, it is necessary to propose a time-efficient method for prediction of fuel ID times across a wide range of operation conditions based on limited data points.

Studies have been conducted to develop simplified methods for fuel ignition tendency prediction. As early as 1950s, Livengood and Wu proposed a predictive integral method to correlate the auto-ignition instants in IC engines with those in rapid compression machines [¹⁹], and this integral methodology was recently extended to the prediction of the two-stage fuel ignition [²⁰]. In addition, Donato and Petersen [²¹] established a simplified correlation for syngas chemical reaction times based on the nonlinear regression in the context of changed equivalence ratio, temperature, and pressure. Zhou et al. also developed a correlation to determine the chemical timescales of various surrogate fuels in the kinetically controlled combustion processes [²²]. These simplified correlations well reproduced the fuel ignition tendency within a given range of conditions, but with an insignificant computational cost.

The high dimensional model representation (HDMR) method [²³], a tool to capture the relationships between the input and output parameters of high dimensional models, was also used to construct correlations for the prediction of fundamental engine combustion behaviors [²⁴,²⁵]. Zhao et al. [²⁴] constructed two types of HDMR correlations, the global and stepwise correlations, for the ID times prediction of hydrogen/air mixtures and found that the stepwise correlations had a higher prediction accuracy at the transition temperature and pressure conditions. However, it could be seen that up to the present the HDMR method has only been used in the ID time prediction for small-molecule fuels as hydrogen, but has not yet been validated in the ignition prediction of large-size fuels that may exhibit nonmonotonic behaviors in auto-ignition, i.e., the negative temperature coefficient (NTC) phenomenon [²⁶–²⁸].

In this paper, the HDMR method was employed to predict the ID times of n-heptane, which exhibits an NTC phenomenon under low-to-intermediate temperature conditions. Also, n-heptane has a similar cetane number with the conventional diesel fuel and is commonly used to represent diesel fuel under certain conditions. The ID times of n-heptane/air mixtures under adiabatic constant-pressure conditions were calculated using a detailed n-heptane chemical mechanism. The calculation conditions cover a wide range of temperatures from 650 K to 1250 K, pressures from 5 atm to 50 atm and equivalence ratios from 0.3 to 1, respectively. Both global and stepwise HDMR correlations were constructed and their prediction results for the n-heptane ID times over the domain of temperature, pressure, and equivalence ratio were compared and analyzed. Finally, an HDMR correlation was constructed based on the experimentally measured ID times from literature and its prediction performance was compared with the one constructed using the simulated ID times.

Methodology

RS-HDMR methods

The HDMR series methods, developed by Rabitz et al. [²⁹,³⁰], have been widely used to deal with the high-dimensional situations [³¹,³²]. These analytical tools could establish straight forward and simple correlations between the input and output information based on a limited number of sample data. In the HDMR methods, the output variable f(x) is expressed as a finite hierarchical correlation function expansion with input variables x = (x₁, x₂,…, x_n) in the following forms [²⁸,²⁹]:

(1)

where the zeroth-order (i = 0) component function f₀ reflects the average response of all the input variables to f(x), the first-order component function f_i(x_i) denotes the individual effect of the ith input variable to the output f(x), and the second-order component function f_ij(x_i, x_j) indicates the combined effects of the input variables x_i and x_j on the output f(x) and so on. Finally, the nth-order component function f_{1, 2,…, n} (x_1, x₂,…, x_n) represents the correlated effect contributed by all the input variables to the output f(x).

Practical approaches and optimization procedures for the HDMR component functions construction were also of importance and interests [³³,³⁴]. Li et al. [³⁵–³⁷] combined random sampling with the HDMR method to deal with the high dimensional problems more efficiently. Due to its advantages in establishing the input-output correlations from the complex situations, the RS-HDMR method was used in the present paper here to construct correlations for the ID times of n-heptane/air mixtures. To reduce sample numbers, the RS-HDMR component functions were expanded by optimal weighted orthonormal polynomials {j} [³⁶]

(2)f_i(x_i)\approx {\displaystyle {\sum }_{r=\mathrm{1}}^k{\alpha }_r^{(\mathrm{0})i}}{\phi }_r^i(x_i),

(3)f_{ij}(x_i,x_j)\approx {\displaystyle {\sum }_{r=\mathrm{1}}^k[{\alpha }_r^{(ij)i}{\phi }_r^i(x_i)+{\alpha }_r^{(ij)j}{\phi }_r^j(x_j)]+}{\displaystyle {\sum }_{p=\mathrm{1}}^l}{\displaystyle {\sum }_{q=\mathrm{1}}^{l^{\prime }}}{\beta }_{pq}^{(\mathrm{0})ij}{\phi }_p^i(x_i){\phi }_q^j(x_j),

(4)\begin{array}{l}{f}_{ijk}(x_i,x_j,x_k)\approx {\displaystyle {\sum }_{r=\mathrm{1}}^k}[{\alpha }_r^{(ijk)i}{\phi }_r^i(x_i)+{\alpha }_r^{(ijk)j}{\phi }_r^j(x_j)+{\alpha }_r^{(ijk)k}{\phi }_r^k(x_k)]+\\ {\displaystyle {\sum }_{p=\mathrm{1}}^l}{\displaystyle {\sum }_{q=\mathrm{1}}^{l^{\prime }}}[{\beta }_{pq}^{(ijk)ij}{\phi }_p^i(x_i){\phi }_q^j(x_j)+{\beta }_{pq}^{(ijk)ik}{\phi }_p^i(x_i){\phi }_q^k(x_k)+\\ {\beta }_{pq}^{(ijk)jk}{\phi }_p^j(x_j){\phi }_q^k(x_k)]+{\displaystyle {\sum }_{p=\mathrm{1}}^m}{\displaystyle {\sum }_{q=\mathrm{1}}^{m^{\prime }}}{\displaystyle {\sum }_{r=\mathrm{1}}^{m^{″}}}{\gamma }_{pqr}^{(\mathrm{0})ijk}{\phi }_p^i(x_i){\phi }_q^j(x_j){\phi }_r^k(x_k),\end{array}

where k, l, l’, m, m’, and m’’ are integers, g, {\beta }_{pq}, and {\gamma }_{pqr}are constant coefficients to be determined by

(5){\alpha }_r^i\approx {\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle {\sum }_{s=\mathrm{1}}^N}f(x^{(s)}){\phi }_r^i(x_i^{(s)}),

(6){\beta }_{pq}^{ij}\approx {\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle {\sum }_{s=\mathrm{1}}^N}f(x^{(s)}){\phi }_p^i(x_i^{(s)}){\phi }_q^j(x_j^{(s)}),

(7){\gamma }_{pqr}^{ijk}\approx {\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle {\sum }_{s=\mathrm{1}}^N}f(x^{(s)}){\phi }_p^i(x_i^{(s)}){\phi }_q^j(x_j^{(s)}){\phi }_r^k(x_k^{(s)}),

and {j} are defined as

(8){\phi }_{\mathrm{1}}^i(x_i)=a_{\mathrm{1}}{x}_i+a_{\mathrm{0}},

(9){\phi }_{\mathrm{2}}^i(x_i)=b_{\mathrm{2}}{x}_i^{\mathrm{2}}+b_{\mathrm{1}}{x}_i+b_{\mathrm{0}},

(10){\phi }_{\mathrm{3}}^i(x_i)=c_{\mathrm{3}}{x}_i^{\mathrm{3}}+c_{\mathrm{2}}{x}_i^{\mathrm{2}}{c}_{\mathrm{1}}{x}_i+c_{\mathrm{0}}.

The constant coefficients a₀, a₁, b₀, b₁,…and c₃ were decided to generate orthonormal polynomials {j} using a given series of random sample points [³⁶].

Due to the nonlinear features of the ID times of n-heptane, the component functions of RS-HDMR may need higher-order polynomial terms to obtain accurate results. However, higher polynomial orders may increase errors of RS-HDMR. It is, therefore, necessary to determine the optimal polynomial order of each component function and a set of optimization procedures developed by Ziehn and Tomlin [³²] were used here. In addition, the control variate method developed by Li et al. [³⁰] was used here to decrease the requirement of sample size without significant sacrifice of accuracy, and the truncated RS-HDMR expansion h(x), as shown in Eq. (11), was used instead of Eq. (1).

(11)

where the coefficients {\overline{{\alpha }_r^i}, \overline{{\beta }_{pq}^{ij}}, \overline{{\gamma }_{pqr}^{ijk}}} are determined by direct Monte Carlo integration given in Eqs. (5)–(7).

Ignition delay times calculation

ID times (t) of the homogeneous n-heptane/air mixtures over a broad range of initial conditions (initial temperatures T₀, initial pressure P₀ and equivalence ratio φ were calculated to generate the ID times database for the HDMR correlations construction. The calculation ranges of the initial temperature, initial pressure and equivalence ratio were from 650 K to 1250 K, 5 atm to 50 atm, and 0.3 to 1, respectively. 2000 computer-runs were performed to generate the ID times with the initial temperature, pressure, and equivalence ratio uniformly distributed in the test domain. The CHEMKIN software [³⁸] and a detailed n-heptane mechanism (550 species and 2450 reactions) [³⁹] widely used and verified [³⁹–⁴¹] were used to simulate n-heptane homogenous auto-ignition processes. The auto-ignition instant is defined as the time when the OH concentration reaches its peak value [⁴²], and the interval between the start of calculation and the ignition timing was set as the ID time.

As the ID time could be correlated with the temperature, pressure, and equivalence ratio in the Arrhenius form [⁴³], the HDMR correlations are as such constructed based on

(12){\log }_{1\mathrm{0}}\tau =f(\mathrm{1}/T_{\mathrm{0}},{\log }_{1\mathrm{0}}P,\phi ).

The inputs 1/T₀, log₁₀P and φ were then randomly sampled in a range of 1/1250<1/T₀<1/650, 0.7<log₁₀P<1.7 and 0.3<φ <1 for HDMR correlations construction. For the global correlation, different sample sizes out of 2000 computer generated points were used to construct correlations and the rest of the data points were then used to validate their prediction performances. For the stepwise correlations, 300–600 data points, depending on the complexity of the curve, were sampled in each subdomain and the remaining points in the same subdomain were used for the prediction performance validation.

Results and discussion

ID times of n-heptane under various initial conditions

Figure 1 shows the ID times of the n-heptane/air mixtures at different initial temperature, pressure, and equivalence ratios. It is observed that the ID time is not monotonically decreased with the increased temperature, but slightly increases with increased temperature within the temperature range of around 750–1000 K. This so-called NTC behavior is caused by the competition between the chain propagation and the chain branching reactions of the peroxy radicals. Given the initial temperature and pressure, the ID times are extended with a reduced equivalence ratio. As the equivalence ratio is reduced to 0.3, a significant ID time extension is observed, especially at low temperature and NTC regions, while the ID times at equivalence ratios of 0.6 and 1.0 are close to each other at low and high temperature regions. It could be also seen that the NTC behavior is more pronounced as the equivalence ratio increases, while no clear NTC region but only a turning point is observed in the ID time curve at the equivalence ratio of 0.3. Increased pressure shortens the ID time at given temperatures and equivalence ratios, and shifts the NTC region to a region of higher temperatures. These nonlinear behaviors of the ID times versus temperature, pressure, and equivalence ratio increase the difficulty for accurate ID times prediction.

Figure 2 compares ID times versus initial pressures at different initial temperatures and equivalence ratios. Generally, the ID time versus initial pressure is close to a linear relation, while the slope of decreased ID times versus increased pressure are different at changed temperature and equivalence ratio conditions. At a equivalence ratio of 0.3, the slopes of ID time versus initial pressure are close for changed initial temperatures, but as the equivalence ratio increases, the magnitudes of slopes at initial temperatures from 800 to 1100 K are higher than those at the initial temperature of 700 K. At high pressure conditions, the ID times with initial temperatures within 800–1000 K are quite close. Decreasing or increasing the initial temperature from this range will obviously extend or shorten the ID times.

Prediction performance using a global correlation

The HDMR methods based on the global correlations were first utilized in the ID times prediction. The coefficient of determination, R², of the ID times predicted by using the global correlation ({\tau }_{\mathrm{HDMR}}) and those calculated by using detailed n-heptane/air chemistry ({\tau }_{\mathrm{CHEM}}), is demonstrated in Fig. 3 to present the accuracy of the prediction. The prediction performance is examined over the entire range of initial conditions, with the temperature, pressure, and equivalence ratio ranging between 650 and 1250 K, 5–50 atm and 0.3–1, respectively. Specifically, Fig. 3(a) depicts the prediction performance of the constructed global correlation using 450 sample points. It is noticed that the R² is only 0.6552 which is unsatisfactory for the prediction. Therefore, a larger sample size of 1000 points was used for the global correlation construction and R² increases to 0.9667, as displayed in Fig. 3(b). This implies that an increase in sample size is beneficial for the prediction performance of the global correlation.

However, as exhibited in Fig. 4, the improved global correlation is still not sufficiently accurate to capture the trends of ID times calculated by using the detailed chemical mechanism, especially near the turning points at a low initial pressure. The possible reason for this is that the global correlation covers such a wide sample range and can only predict the overall trend of the ID time but fails to capture the local details. As mentioned above, the correlation depends on the optimal weighted orthonormal polynomials {j} and constant coefficients g, β_pq, and γ_pqr and these terms are obtained by the iteration of outputs and inputs of sample points. Once the sample size is sufficiently large, the iterative errors tend to be reduced accordingly. As a result, the global correlation performs more accurately with a larger sample size. Nevertheless, it should be pointed out that although the global correlation fails to satisfy the accuracy requirement at some specific conditions, its advantages are still obvious as only one function is needed to cover the entire initial condition range, especially when the prediction accuracy is not highly concerned.

Prediction performance using stepwise correlations

To establish a more accurate correlation, the domain of the initial temperatures and pressures were divided into subdomains while specific stepwise correlations were constructed based on different subdomains. The division of subdomains is first based on the fuel ignition characteristics with changed temperature in Fig. 1. In this study, due to the existence of the NTC region, the ID time curve is more complicated and exhibits the characteristics of high-order function, which increases the difficulties of constructing the correlation by using the above-mentioned HDMR method [²⁸–³²]. To reduce the difficulty of constructing the correlation, we here divide the whole temperature domain into two subdomains and the ID time curve becomes two simpler ones. 900 K is selected as the boundary point here. In addition, it is found that the NTC region could shift to the higher temperatures with increased pressure. To reduce the influences of the NTC shift, the pressure range is also divided and finally four subdomains were generated as listed in Table 1.

Figure 5 compares the ID times calculated by using CHEMKIN and the prediction by the stepwise correlations for initial conditions in Subdomain 2 of low temperature and high pressure and Subdomain 3 of high temperature and low pressure. The coefficients of determination, R², of both stepwise correlations are close to unity. To further compare the prediction performances of the global and the stepwise correlations, Table 2 tabulates the coefficients of the determination generated by using the global correlation and stepwise correlations at the four subdomains. The R²s of the stepwise correlations are much closer to unity than those of the global correlation. Hence, the stepwise correlations are able to improve the prediction accuracy over the entire domain of the initial conditions. This is mainly due to the changed ranges of initial conditions for the construction of the global correlation and the stepwise correlations. As the coefficients in the component functions of the correlations are determined by the iteration of the sampled inputs and outputs, smaller ranges of sample data could produce higher prediction performances. In Fig. 4, the results predicted by using the stepwise correlations are also plotted as the circle points, and it is observed that the prediction by using the stepwise correlations agree better with the calculation results by using the detailed mechanism.

Figure 6 presents the traces of the ID times versus pressure at different temperatures. In Fig. 6(a), the global correlation fails to capture the nonlinear trend and shows large deviations at the intermediate pressure zone, and the stepwise correlations agree well with the calculation results obtained by using CHEMKIN. The reason for this is that the coefficients of the component functions in the global correlation are constructed by the iteration of the large number of samples. As shown in Fig. 2, most trends of the ID times versus pressure exhibit high linearity extent, and as such the global correlation constructed by these samples can only capture the overall linear trend but fail to predict the special cases. Similarly, as shown in Fig. 6(b), the global correlation has a good accuracy at the low-pressure zone, while there are large deviations at the high-pressure zone. In contrast, the stepwise correlations show a higher accuracy over the entire pressure domain.

Prediction performance of the global correlation constructed by using experimental data

Further, an attempt was made in this paper to establish an HDMR correlation based on the experimental data and the correlation was applied for the prediction of ID times in practical conditions. Here 300 measured ID times from Refs. [⁴⁴–⁵⁰] were used to construct the global correlation (2) over a range of the initial temperature from 650 K to 1250 K, the initial pressure from 12 atm to 50 atm and the equivalence ratio from 0.3 to 1. It should be mentioned that the sample data are not randomly distributed within the range of initial conditions. The prediction performances of global correlation (1) constructed in Subsection 3.1 and global correlation (2) are compared at a range of initial temperatures, as shown in Table 3. It is seen that the ID times calculated by using global correlation (1) generally agree well with the simulation results obtained by using CHEMKIN, with relative errors below 5%. However, the relative errors of the ID times prediction using global correlation (2) are higher because the limited sample data of ID times could be obtained from literature for correlation construction. The consistent with the results in Subsection 3.1, the global correlation constructed using 450 sample points also have a low coefficient of determination. Additionally, the experimental data from the literature are not randomly distributed within the pressure and equivalence ratio range, also leading to increased prediction errors of the constructed correlation. In spite of the higher relative errors of global correlation (2), it is believed that its prediction performance could be improved as long as more sample data are provided for correlation construction.

Conclusions

In this study, correlations for ID times of n-heptane/air mixtures were constructed by using the RS-HDMR methods across a wide range of initial temperatures, pressures, and equivalence ratios. Two kinds of HDMR correlations, the global and stepwise correlations, were constructed and their prediction performances for the ID times were compared with the results obtained by using the detailed chemical mechanism. The global correlation could provide a generally satisfactory prediction performance while failed to capture the local features of ID times at the NTC region due to its wider sample range. The stepwise correlations constructed for each subdomain showed more accurate prediction results, especially at the transition locations of the NTC region. Finally, a global correlation based on the experimentally measured ID times was constructed and its prediction performance was examined. Therefore, HDMR correlations could be used to efficiently and accurately predict the ID times for fuels of two-stage auto-ignition behaviors with limited samples in certain conditions.

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