Thermal barrier coatings (TBCs) applied to in-cylinder surfaces of a low temperature combustion (LTC) engine provide an opportunity for enhanced efficiency via two mechanisms: (i) positive impact on thermodynamic cycle efficiency due to combustion/expansion heat loss reduction, and (ii) enhanced combustion efficiency. Heat released during combustion increases the temperature gradient within the TBC layer, elevating surface temperature over combustion-relevant crank angles. Thorough characterization of this dynamic temperature “swing” at the TBC–gas interface is required to ensure accurate determination of heat transfer and the associated impact(s) on engine performance, emissions, and efficiencies. This paper employs an inverse heat conduction solver based on the sequential function specification method (SFSM) to estimate TBC surface temperature and heat flux profiles using sub-TBC temperature measurements. The authors first assess the robustness of the solution methodology ex situ, utilizing an inert, quiescent environment and a known heat flux boundary condition. The inverse solver is extended in situ to evaluate surface thermal phenomena within a TBC-treated single-cylinder, gasoline-fueled, homogeneous charge compression ignition (HCCI) engine. The resultant analysis provides crank angle resolved TBC surface temperature and heat flux profiles over a host of operational conditions. Insight derived from this work may be correlated with TBC thermophysical properties to determine the impact(s) of material selection on engine performance, emissions, heat transfer, and efficiencies. These efforts will guide next-generation TBC design.

## Introduction

Prior research has demonstrated a fundamental link between in-cylinder surface temperature and the thermal composition of the trapped mass [1,2]. The turbulent entrainment of “cool” eddies derived from the near-wall layer at the gas–wall interface have a significant impact on the thermal stratification of the interior charge. The chemical kinetics governing HCCI autoignition are largely temperature-driven, and as such remain sensitive to the localized charge temperature distribution. As a result, HCCI combustion is heavily influenced by the temperature profile (and by extension, heat transfer) at the gas–wall boundary [1–3].

From a thermodynamic perspective, thin (∼150 *μ*m or less) TBCs have been shown by the authors to reduce heat transfer within an LTC engine over combustion-relevant crank angles, increasing thermal efficiency [4]. Decreased closed-cycle heat transfer also facilitates more complete combustion, reducing unburned hydrocarbons (UBHC) and carbon monoxide (CO) while increasing combustion efficiency.

However, increasing the mean in-cylinder surface temperature across the entire engine cycle has been shown to heat incoming charge, degrading volumetric efficiency and power density [5,6]. Thus, from a conceptual standpoint, an idealized TBC should provide a “dynamic” surface temperature profile, which avoids charge heating during intake while minimizing heat transfer during late compression, combustion, and expansion. This desired temperature swing is qualitatively characterized relative to metal and “traditional” TBC surface temperature profiles in Fig. 1. A more complete evaluation of TBCs for LTC may be found in Ref. [4], while related TBC analysis for diesel applications may be found in Refs. [7–9].

Given the impact of in-cylinder surface temperature on autoignition, engine performance, emission formation, and cycle efficiencies, a thorough experimental evaluation must include measurements of temperature and heat flux phenomena at the gas–wall interface. As mentioned, a variety of pre-established measurement techniques exist for a non-TBC-treated engine [10–12]. Furthermore, as the authors' current experimental setup does not include optical access, image-based techniques such as those utilized in Refs. [8] and [13] do not apply.

In this work, the application of TBCs to in-cylinder surfaces relegates the placement of temperature sensors to subsurface locations (i.e., below the coating). As such, measurements recorded at these locations no longer represent surface phenomena, providing instead subcoating temperature information. Estimation of the unknown temperature and heat flux profiles at the gas–wall boundary (via subsurface measurements) constitutes an inverse heat conduction problem (IHCP) [14–16], whereby interior measurements are used to determine a domain's unknown boundary condition. Solving this IHCP is critical for understanding the underlying mechanism(s) and evaluating their impact(s) on LTC engine performance and efficiencies.

This paper is organized as follows: (1) A concise overview of inverse solution methodology, particularly the SFSM, provides requisite background for the included heat transfer analysis. (2) The experimental apparatus is overviewed, as is the yttria-stabilized zirconia (YSZ) TBC. The YSZ-treated piston/probe is evaluated using both ex situ and in situ experiments—ultimately correlating TBC surface phenomena with engine combustion and efficiency metrics. (3) Finally, in-cylinder measurements are used to evaluate instantaneous heat transfer rates and the associated cumulative heat loss over combustion-relevant crank angles.

## Background

A brief contextual summary will be provided for each of the main topics discussed in this paper. An overview of inverse solution methodology, air plasma sprayed YSZ, and the ex situ/in situ experimental devices are included.

### Inverse Analysis.

As discussed, the application of TBCs to instrumented surfaces within a non-optically accessible combustion chamber necessitates the implementation of inverse solution methodology—to link subsurface temperature measurements with surface phenomena [14–17]. This paper utilizes a modified form of the SFSM originally developed by Beck to conduct the inverse analysis [18]. This solution strategy estimates surface heat flux by minimizing the error between experimentally measured temperature profiles (at subsurface sensor locations) and a corresponding finite difference-derived temperature profile. In essence, the solver manipulates the magnitude of the unknown surface heat flux at each time-step such that the modeled temperature (at the senor location) converges upon the corresponding measured temperature. In this way, the functional form of the unknown heat flux boundary condition is “sequentially” estimated across the sampled data array. In the current work, modeled temperature profiles are derived from a discretized of the governing heat diffusion equation. The thermal system itself is assumed 1D as shown in Fig. 2.

### In Situ Temperature Measurements.

A variety of well-established measurement techniques exist for acquiring in situ temperature profiles [20]. This study utilizes fast-response, triaxial thermocouples [21] to measure sub–crank angle degree (CAD) temperature transients over a fired cycle. Within the test engine, probes are mounted through the cylinder head, flush with the combustion chamber surface (as shown in Fig. 4). Complete details of the test engine and associated test fuel are overviewed in Tables 1 and 2 in Appendix A. An ex situ radiation chamber utilizes the same sensors as the in situ apparatus. In both cases, non-TBC-treated probes (i.e., “metal” probes) provide direct measurements of surface temperature phenomena. Conversely, TBC-treated probes simultaneously record temperature profiles below the YSZ coating.

Surface and subsurface temperature signals are preprocessed to enhance signal fidelity prior to use within the heat flux postprocessing routine. Inverse methods are particularly sensitive to contaminants such as signal noise and measurement error. As such, a concerted effort is made to condition subsurface temperature traces to better enhance the quality of the inverse solver's surface temperature and heat flux estimates. These efforts include the use of phase-averaged data, targeted digital filtering, and mild smoothing of the input temperature profiles [22]. A complete discussion of measurement statistics, confidence intervals, and uncertainty of SFSM-derived heat flux estimates is provided in Appendix B.

### Thermal Barrier Coating.

The current study utilizes yttria-stabilized zirconia as the primary TBC layer (∼100μm). An intermediate nickel-chrome “stress relief” coat (∼50μm) ensures adhesion of the YSZ layer to the underlying substrate. Coatings were produced by a standard air plasma spray (APS) coating process in which the spray gun motion is done using a 6 axis robot in a standard spray booth enclosure [23,24]. The coating manufacturer reports 1.3–1.7 W/m K and 0.50–0.64 mm^{2}/s as the expected ranges for APS–YSZ thermal conductivity and diffusivity, respectively. In general, the reduced thermal conductivity and heat capacity of the YSZ coating enhance thermal insulation while reducing heat storage relative to metal and “traditional” TBC surfaces. As will be shown, these properties have a direct impact on LTC combustion, engine performance, exhaust composition, and thermal efficiency.

## Experimental Setup

A concise overview of both ex situ (radiation chamber) and in situ (test engine) experimental apparatus is provided. Modified operational procedures are also discussed.

### Radiation Chamber.

A custom-fabricated radiation chamber capable of generating HCCI-like heat flux (0.25–1.0 MW/m^{2}) is used for the nondestructive evaluation of coating performance. Unlike the reacting turbulent in-cylinder environment, the radiation chamber provides a spatially uniform heat flux of known magnitude. This ensures a shared boundary condition at each probe location, enabling direct comparison between surface-based (i.e., “direct”) and subsurface-based (i.e., “inverse”) heat transfer calculations. Detailed discussion of the apparatus and its capabilities is available from Refs. [25,26]. A simplified operational schematic is provided in Fig. 3.

It should be noted that emissivity and (partial) transparency vary between evaluated materials. “Clean” metal probes are likely to exhibit different radiative characteristics than their TBC-treated counterparts. Thus, a thin aerosol-based graphite layer is applied to both metal and TBC probes to ensure uniform absorption and emissivity characteristics.

### Test Engine.

In situ data are derived from a single cylinder, gasoline-fueled, low temperature combustion test engine—see Fig. 4. The engine is operated in homogeneous charge compression ignition mode and is fully instrumented to monitor engine performance and emissions while ensuring constant operational conditions. Fuel is delivered via direct injection at 333 CAD before top dead center firing (bTDC_{Firing}), ensuring homogeneity of the air/fuel mixture. A rebreathe lobe on the exhaust camshaft combined with controlled backpressure maintains ∼45% hot residual gas in-cylinder, facilitating autoignition. Additional residual is available through an external exhaust gas recirculation (EGR) circuit. Emission components and EGR percentage are sampled/analyzed using a Horiba “MEXA-7100D EGR” system.

### “Fuel Match, Phase Match” Engine Operational Procedure.

Given fixed operational set points—(including fuel mass, air fuel ratio (AFR), *T*_{Intake}, etc.)—TBCs have been shown to advance combustion phasing relative to their metal engine counterpart [4]. Variability in combustion phasing should be minimized in order to isolate the strictly “thermal” impact(s) of TBCs on engine performance and heat transfer. The authors have developed a “fuel match, phase match” (FMPM) operational procedure with this goal in mind. A brief overview follows.

Beginning with the metal engine configuration, fueling is adjusted to enable “robust” HCCI operation. Cycle-to-cycle variation, as measured by COV_{IMEP}, remains below 3%. Ringing intensity (RI) is likewise limited to 5 MW/m^{2}. Fueling rates satisfying these conditions are established at each engine speed, and become the “set points” for subsequent evaluation.

With fueling established, the temperature of the air mass entering the metal engine is adjusted until combustion phasing is positioned at 7 CAD aTDC (i.e., CA50 = 7 deg aTDC). Five hundred consecutive cycles are recorded under the aforementioned operational conditions, generating a “baseline” metal engine dataset for each engine speed of interest.

With metal engine data acquisition complete, the engine is “rebuilt” to include a TBC-treated piston and probe. To counteract the resultant phase-advance of the TBC configured engine, the external EGR circuit is activated to effectively replace some of the hot residual (re-inducted during rebreathing) with cooled residual. This substitution re-positions CA50 at 7 deg aTDC. It should be noted that all other operational set points remain fixed—ensuring similar operational parameters relative to the baseline metal engine case. The only difference is the fact that the metal engine does not utilize external EGR, while the TBC treated engine inducts air + EGR_{External}.

Once complete, the TBC engine results are evaluated against the metal engine data. An example of this process, as seen in phase-averaged cylinder pressure data, is included in Fig. 5.

A final note regarding FMPM operation: While the abovementioned procedure is designed to elucidate the thermal influence of TBCs upon relevant combustion/heat transfer metrics, the authors recognize that secondary impacts likely exist. For example, the use of external EGR to retard combustion phasing within the TBC-treated engine may impact the overall composition of the in-cylinder charge. If compositional variations between metal and TBC engine configurations exist, they may influence the chemical kinetics underlying HCCI autoignition. Even if total EGR percentage remains constant during FMPM operation, the external EGR utilized by the TBC-treated engine enters the combustion chamber at a much lower temperature than the internal EGR associated with the rebreathe event. This would likely affect the in-cylinder temperature distribution during closed-cycle CAD, impacting heat transfer and by extension, autoignition. Given the potential for such “secondary effects” of the current fuel-match/phase-match operating procedure, the authors continue to evaluate “best practices” relative to the goals stated at the onset of this section.

## Results

Radiation chamber and test engine results are examined herein. Comparison of the SFSM-based inverse solver against established solution methodology is discussed first. Efforts to more accurately define the key thermophysical parameters within the finite difference model are also considered. Finally, the inverse solver is extended to evaluate engine-derived temperature measurements, where surface temperature and heat flux profiles at the gas–wall interface are compared (metal versus TBC) and correlated with engine performance and emission metrics.

### Ex Situ Temperature and Heat Flux Measurements.

As mentioned, the radiation chamber's spatially invariant heat flux enables comparison between direct and inverse solution methodology. In this study, temperature measurements from the uncoated (i.e., metal) probe utilize a direct Fourier method [27,28] to calculate the incident heat flux. In contrast, sub-TBC temperature measurements employ the SFSM-based analysis to estimate the functional form of the surface heat flux [19].

The thermophysical properties and spatial dimensions of the composite TBC layer are adjusted within the tolerances provided by the coating supplier to further optimize agreement between solution methodologies. These efforts ensure that the discretized mathematical representation of the governing equation is consistent with empirical observations.

Note: This process will eventually be formalized, using the statistical techniques employed by parameter estimation [17]. However, comprehensive analysis of thermophysical parameters exceeds the targeted scope of the current paper.

Simply put, when a fixed heat flux *q*″ is applied to both metal and TBC probe surface boundaries, the magnitude of the resultant spatial temperature gradient at the surface is determined by the material's thermal conductivity, *k*. Thus, the reduced conductivity of the YSZ layer amplifies the magnitude of its surface temperature profile when compared against the metal probe.

*α*” of the governing heat diffusion equation

The inverse solver utilizes a discretized form of the partial differential equation described in Eq. (2), capturing the diffusive behavior specific to the TBC's thermophysical properties.

### In-Cylinder Temperature and Heat Flux Measurements.

With the validity of the inverse solution methodology established, the SFSM-based solver is extended in situ to determine TBC surface temperature and heat flux profiles under engine-firing conditions.

In this section, the FMPM procedure is utilized to ensure comparable operating conditions and combustion phasing between the metal and TBC engine configurations. Analog signals from the head-side probe locations are sampled at 0.5 CAD intervals, providing high-resolution measurements of fired-cycle temperature transients. These profiles enable calculation of heat flux using Fourier (metal probe) or SFSM (TBC probe) methods.

Figure 7 summarizes the temperature and heat flux profiles during FMPM operation over a sweep of engine speeds. When plotted against metal-engine data taken under identical FMPM operational parameters, the SFSM surface heat flux estimates show clear evidence of reduced heat loss, particularly during late compression, combustion, and early expansion. It should be noted that surface heat flux magnitude within the engine environment is greatly impacted by fueling. Thus, FMPM set points associated with higher overall fueling rates typically experience larger surface heat flux. Such is the case for 1200 rpm, where fuel mass injected is greatest relative to other operational speeds (see Fig. 7 for specific FMPM operational parameters). A more quantitative comparison of metal versus TBC heat transfer is provided in the Heat Transfer Analysis section.

As was shown with radiation chamber data, the inverse solver also provides estimates of TBC surface temperature. These results enable evaluation of YSZ coating performance in-cylinder, under firing conditions. The TBC surface temperature profiles plotted in Fig. 7 indicate that the YSZ-based coating closely mimics the desired surface temperature swing behavior. It is also observed that TBC surface temperature estimates over gas exchange (i.e., exhaust/intake) are of similar magnitude relative to their metal counterparts.

Further restricting analysis to intake-stroke CADs, the YSZ surface temperature exhibits similar magnitude relative to the metal surface profiles. As discussed, this is an important detail, as elevated TBC surfaces temperatures during intake have been shown to heat the incoming charge, reducing density, volumetric efficiency, and by extension, engine power [5,6].

Examination of the closed portion of the engine cycle reveals a substantial increase in TBC surface temperature during compression and expansion strokes relative to metal profiles. Again, this characteristic of the YSZ layer closely mimics the idealized coating behavior conceptualized at the onset. From a combustion standpoint, elevated closed-cycle wall temperature has a fundamental impact on heat transfer, effectively reducing heat loss, and thus retaining more thermal energy in-cylinder during expansion. This reduced heat loss during combustion and expansion increases thermal efficiency over the fired cycle. In addition, HCCI engine experiments with the YSZ layer have exhibited more complete fuel oxidation, increasing combustion efficiency, while reducing CO and UBHC in the exhaust stream. Evidence of these trends, as measured experimentally under FMPM conditions at 1600 rpm, is summarized in Fig. 8.

In a physical sense, it is helpful to remember that seconds rather than crank angle degrees are the ‘natural’ temporal metric for heat transfer. Thus, as engine speed increases, the absolute cycle duration, as measured in seconds, decreases, providing less time for the TBC layer to “shed” stored heat. As a consequence, the “bulk” wall temperature trends for both metal and YSZ coated engine configurations exhibit an upward shift. Closer inspection reveals that the absolute shift of mean temperature appears to be of similar magnitude between metal and TBC engines. This further validates, via in situ analysis, the YSZ-based TBC as an attractive coating material.

### Heat Transfer Analysis Using In Situ Heat Flux Profiles.

The crank angle resolved surface heat flux profiles may be utilized to provide a gross estimate of total heat loss through the combustion chamber surface during a fired cycle.

A few critical assumptions are necessary to transform the heat flux profile derived at a single probe location into a representative “global” parameter. Given the sparse in situ temperature sensor configuration, such assumptions are required. As a first-order approximation, spatial invariance is assumed. While this is not true in a rigorous sense, the early work by Change et al. [29] demonstrated a relatively high degree of uniformity between heat flux measured at nine different locations in the HCCI engine, thus lending merit to the aforementioned assumption.

For this analysis, the heat flux measured at the pulley-side head probe sensor location (in both metal and TBC engines) is assumed to be the boundary condition for all exposed combustion chamber surfaces. Furthermore, in the case of the YSZ coated probe, this assumption still holds since the entire piston surface is also coated with the YSZ TBC, accounting for a substantial portion of the chamber surface area during combustion.

The conversion of heat flux (W/m^{2}) to a heat transfer rate (Joules/CAD) requires calculation of crank angle resolved combustion chamber surface area. With the piston and head surface areas fixed, cyclic variation in exposed cylinder wall surface area relative to crank position can be calculated from the known engine geometry [11]. The sum of these individual components at each CAD represents the total instantaneous surface area. This crank angle resolved surface area is then multiplied by the aforementioned global heat flux to calculate the heat transfer rate at each crank angle in Joules per second. Finally, this global heat transfer rate is multiplied by the RPM-specific period (i.e., second/CAD)—expressing the instantaneous heat loss rate in Joules/CAD across the fired cycle.

To reiterate, the current analysis targets the combustion-relevant portion of the engine cycle. Variations in combustion phasing have been eliminated to ensure equitable comparison of the heat transfer characteristics between the engine configurations. As such, the following analysis utilizes FMPM results, and is limited to the crank angles spanning the combustion and expansion processes.

Figure 9 shows both instantaneous and cumulative heat transfer results across the crank angle range of interest (CADs spanning ∼360–390 deg). Inspection reveals consistently lower heat loss for the TBC treated engine. Combustion is entirely contained within this crank angle range (recall that CA50=7 deg aTDC for the FMPM data), and the temperature swing on the TBC surface dramatically reduces the peak heat flux during that time. The heat transfer traces at different speeds display similar features, but the phasing of the peak heat transfer rate for the TBC-treated engine exhibits a significant delay as engine speed increases. This is consistent with the peak surface temperature phasing estimates shown in Fig. 7, and the concept of “heat storage and release” due to the TBC thermal capacitance. As the surface temperature swing retards away from the combustion event due to increasing engine speed (heat transfer into the coating is a real-time event), YSZ's impact on the peak rate of heat loss decreases and shifts away from TDC. Recall, the gas–wall temperature differential is minimized, and associated reduction of heat loss is maximized, when peak TBC surface temperature tracks the peak bulk gas temperature closely.

The instantaneous surface heat flux traces for the TBC-coated engine eventually cross the corresponding profiles obtained in the metal engine. This happens during late expansion, and it is interesting to assess the overall effect of such behavior on the cumulative heat transfer results (see Fig. 9(b)). Clearly, the cumulative heat loss is significantly reduced during combustion and early expansion, but the two traces, one obtained for the metal engine and the other for the YSZ-coated piston, eventually converge. This validates the hypothesis that it is the dynamic effect of the surface temperature swing that produces a positive impact on thermal efficiency, rather than brute-force insulation and reduction of cumulative heat loss in an absolute sense.

For reference, Fig. 10 provides closed-cycle cylinder pressure and bulk gas temperatures profiles for the 2400RPM FMPM operating point. This regime is scrutinized since the metal engine and the YSZ peak heat flux phrasings exhibit their largest separation at this engine speed. The bulk gas temperature lines are virtually on top of each other; hence, surface phenomena is solely responsible for the observed heat transfer trends. The time scales associated with the cyclic events obviously influence the phasing and magnitude of peak heat loss rates and lead to trends shown in Fig. 9.

The authors wish to add a final comment regarding inverse solvers and signal containments follows—with particular respect to the preceding heat loss analysis. As discussed, inverse solvers are inherently vulnerable to measurement error and signal noise. Much effort has been invested in the development of effective regularization of the IHCP solution routine [15,16]. In general, these techniques seek to stabilize the solver in the face of noise prone data while maintaining sufficient fidelity of the estimated surface phenomena. Beck's SFSM solution methodology incorporates “future time-steps,” essential holding heat flux constant over a finite, forward-looking, time window to aid solver stability. Considerable subjectivity must be employed when defining an appropriate number of future time-steps. Too few future steps and the coherency of the solution rapidly degrades. In contrast, the inclusion of too many future steps excessively “smooths” solution estimates, masking higher frequency components of transient surface activity. The results presented in this paper represent the authors' best efforts at optimizing this, and other solver parameters. Nevertheless, signal noise and solver artifacts, although greatly attenuated, remain embedded within inverse estimates. Full automation and optimization of this regularization procedure is ongoing and remains a topic of continued research.

As the optimization of the inverse solver continues, a more complete characterization of the solution methodology will be included in future efforts. Accordingly, the authors intend to identify any “nonphysical” (i.e., “solver-induced”) artifacts as they relate to the estimated temperature and heat flux profiles. It should be noted that the current analysis assumes negligible solver bias within the magnitude and phasing of the temperature and heat flux results.

## Conclusions

Thermal conditions at the gas-wall boundary have a profound influence on the chemical kinetics underlying the HCCI combustion process. The application of thermal barrier coatings to combustion chamber surfaces elevates surface temperatures at the gas wall boundary, particularly during combustion and the subsequent expansion processes. These elevated surface temperature profiles reduce heat transfer when it matters the most, i.e., close to TDC, thus increasing the thermal energy during expansion and elevating thermal efficiency across the engine cycle. In addition, higher surface temperatures affect the near-wall zones, and lead to more complete oxidation of the air-fuel mixture, with lower concentrations of unburned hydrocarbon and CO in the exhaust.

Determining boundary conditions within the TBC treated engine sheds light on the underlying physical processes. An inverse technique which employs a modified form of the SFSM has been developed and employed to estimate TBC surface temperature and heat flux profiles from sub-TBC temperature measurements. These estimates, combined with direct measurements obtained in the metal engine, enable deeper insights into the mechanisms responsible for observed effects of the TBC on combustion, emission, and cycle efficiency. Ultimately, the surface temperature estimates for a given TBC allows correlating the coating thermal property with impacts on HCCI engine combustion and performance metrics.

Specifically, the application of ∼100 mm of yttria stabilized zirconia (in conjunction with a ∼50 mm bondcoat) has been shown to dramatically increase the surface temperature swing relative to non-TBC treated (i.e., metal) surfaces. This temperature swing reduces surface heat flux during early expansion, increasing both combustion and thermal efficiencies in the TBC-treated engine. Results obtained through the SFSM analysis exhibit large temperature swings on the YSZ coating's surface. The amplitudes are an order of magnitude higher than the swings measured on the surface of the metal piston. The profiles shown as function of the crank angle are somewhat “stretched” as the engine speed increases, due to the effect of time scales.

Close examination of the heat flux profiles measured on the metal surface and those estimated at the surface of the YSZ coating reveals significant reduction of the peak heat flux at the ceramic layer. The phasing of the peak heat flux, expressed in crank angle degrees, is retarded when the piston is covered with a YSZ coating. This phase delay grows with increasing engine speed. Examination of the instantaneous heat loss rate, and analysis of its integral, offer additional insight. Cumulative heat flux is reduced during combustion and early expansion when TBC is applied. However, metal and TBC traces converge towards the end of expansion. This validates the hypothesis that it is the dynamic effect of the surface temperature swing that produces a positive impact on thermal efficiency, rather than brute-force insulation and reduction of the cumulative heat loss over the whole cycle.

The insight from these efforts can be used to guide systematic thermal barrier coating development and optimization with a goal of maximizing the HCCI engine efficiency improvements and expansion of the low-load operating limit.

## Acknowledgment

The authors would like to thank General Motors (GM) Research and Development for support with engine hardware. Funding for this work was provided by the National Science Foundation (NSF)/U.S. Department of Energy (DOE) Partnership on Advance Combustion Engines: “Thermal Barrier Coatings for the LTC Engine—Heat Loss, Combustion, Thermal versus Catalytic Effect, Emissions and Exhaust Heat” —Award No. 1258714.

## Nomenclature

- AFR =
air fuel ratio

- bTDC =
before top dead center

- CAD =
crank angle degree

- CO =
carbon monoxide

- CA
_{50}=crank angle degree corresponding to 50% MFB location

- COV
_{IMEP}=coefficient of variation of indicated mean effective pressure

- EGR =
exhaust gas recirculation

- FMPM =
fuel match phase match

- HCCI =
homogeneous charge compression ignition

- IHCP =
inverse heat conduction problem

*k*=thermal conductivity (W/m K)

- LTC =
low temperature combustion

*q″*=heat flux (W/m

^{2})*q*_{Surf}=surface heat flux (W/m

^{2})- RI =
ringing intensity

- SFSM =
sequential function specification method

*t*=time

*T*=temperature

*T*_{Intake}=intake temperature (°C)

*T*_{Surf}=surface temperature (°C)

- TBC =
thermal barrier coating

- UBHC =
unburned hydrocarbons

*x*=spatial unit

- YSZ =
yttria stabilized zirconia

*α*=thermal diffusivity (m

^{2}/s)

### Appendix A: Specifications of the Single Cylinder Test Engine and Research Fuel

### Appendix B: Measurement Statistics and Uncertainty Temperature Measurements

The statistics of both in situ and ex situ temperature measurements are presented below. Potential error/uncertainty associated with the measurement device is overviewed, followed by a statistical description of measurement errors. Finally, a brief discussion of error/uncertainty propagation relative to both inverse and direct heat flux calculations is included.

The sub-TBC radiation chamber temperature measurements used during heat transfer analysis are summarized in Fig. 11. These data are collected over 400 consecutive cycles. These measurements are collected using the triaxial J-type thermocouple described in more detail in the manufacturer's documentation [21]. The probe's time constant is on the order of ∼10^{−6} s.

In general, the wide temperature range of the J-type calibration (0–750 °C) can result in a ‘bulk’ measurement error on the order of ±2 °C. However, in an effort to minimize measurement error, the authors calibrate probe-derived temperatures with independent measurements prior to each acquisition period.

The “raw” data shown in Fig. 11 are sampled at 0.1 CAD intervals, and are phase averaged over 400 consecutive cycles. Additional preprocessing includes digital filtering and mild smoothing. The resultant “preprocessed/filtered” trace is plotted against the 95% confidence interval in Fig. 12. Error is assumed to be additive, with zero mean, and is unbiased. With these assumptions, variations in temperature measurements are modeled using a Gaussian (i.e., “normal”) distribution. The standard deviations for the radiation chamber temperature measurements range between ±0.13 °C at 120.6 CAD and ±0.69 °C at 157.9 CAD with a cycle-wide σ_{mean} = ±0.17 °C.

A similar summary of raw in situ measurements collected over 500 consecutive fired engine cycles is provided in Fig. 13. Engine data are sampled at 0.5 CAD intervals. These traces are phase averaged and preprocessed using the same digital filtering/smoothing techniques discussed above. The preprocessed/filtered trace is again plotted against the 95% confidence interval—see Fig. 14. The standard deviation of engine temperature measurements ranges from 0.24 °C at 227.5 CAD to 0.49 °C at 462 CAD with a cycle wide σ_{mean} = ±0.33 °C. It should be noted that measurement statistics for the remaining engine operational speeds exhibit similar measurement distributions and uncertainties.

This section provides an overview of error and/or uncertainty propagation within the heat flux calculations. Both direct and inverse solvers utilize the phase-averaged temperature traces discussed above within their calculation routines. The direct (i.e., “Fourier”) method decomposes the preprocessed/filtered trace into steady and transient temperature components, and the resultant solution is exact insofar as it relies upon a truncated Fourier series. The inverse (i.e., SFSM) method uses the preprocess/filtered temperature traces within an error minimization routine—which results in the modeled temperature “exactly matching” the aforementioned measured temperature (see Ref. [15] for more complete details). In this way, the error/uncertainty associated with raw temperature measurements propagates throughout the respective heat flux calculation routines.

#### Cylinder Pressure Measurements

Potential error/uncertainty associated with the pressure transducer is overviewed, followed by a statistical description of the measurement variability. Finally, a brief discussion of the error/uncertainty propagation relative to pressure-derived metrics (including cycle work and thermal efficiency) is included.

In-cylinder pressure measurements recorded under the fuel match/phase match operating conditions (at 1200 rpm) are summarized in Fig. 15. These data are collected over 500 consecutive cycles and are sampled at 0.5 CAD increments. Measurements are derived from a Kistler 6125a piezoelectric pressure transducer [29]. The manufacturer reports measurement error due to thermal shock for pressures ranging from 9 to 300 bar to be on the order of 1%.

These raw traces are phase averaged and preprocessed using a zero-phase, low-pass digital filter. The resultant preprocessed/filtered trace is likewise plotted against the 95% confidence interval in Fig. 16. High pressure samples are magnified to highlight detail. The standard deviation of cylinder pressure measurements across the 500 raw cycles ranges from 0.006 bar at 147.5 CAD to 2.696 bar at 367.0 CAD with a cycle wide σ_{mean} = ±0.052 bar. As was found with temperature measurements, the cylinder pressure statistics for the remaining engine operational speeds exhibit similar measurement distributions and uncertainties.

The preprocessed/filtered pressure trace is integrated over the displaced volume to calculate engine-cycle work. This result is used in conjunction with metered fuel quantity to determine the thermal efficiency of the cycle—an example of which is reported in Fig. 8 (fuel flow is measured using a Max Machinery Type 213 flow meter, with manufacturer reported accuracy of ±0.2% of the measured value).

#### Uncertainty Analysis of SFSM-Derived Heat Flux Estimates

Evaluation of the uncertainty associated with the heat flux estimates derived from the SFSM utilizes the “filter form” of the IHCP problem. Expanded details of this approach can be found in Refs. [15] and [30–32]. In general, this approach to the IHCP utilizes the superposition principle, and, as such, is restricted to linear (in temperature) heat diffusion systems. Using superposition, the solution of the governing equation is decomposed into an arbitrary number of temperature components—each with an associated heat flux. The total surface heat flux at time *t*_{m} is then represented by the sum of individual heat flux components from *t*_{o} → *t*_{m}, plus those associated with future time-steps. As was required in the preceding analysis, a few statistical assumptions are necessary. These include: (i) random/additive temperature measurements and (ii) zero mean measurement errors which (iii) remain uncorrelated.

The superposition described above enables derivation of a compact mathematical expression which can be used to describe the change in heat flux per unit temperature error at each time-step in the measured data. This metric is commonly referred to as the IHCP filter coefficient and is unique to the thermophysical parameters of the heat transfer system. (For a more complete/detailed discussion of filter coefficients, including a thorough mathematical derivation, the interested reader is again referred to Refs. [15] and [30–32].)

Filter coefficients may be used in conjunction with the statistics of the raw temperature measurement to evaluate uncertainty of the SFSM-derived surface heat flux estimates. The results of these efforts for both ex situ radiation chamber and in situ engine analysis are summarized in Figs. 17 and 18.

A few qualitative observations pertaining to in-cylinder SFSM estimates follow. Temperature measurements obtained in situ generally contain more noise than their ex situ counterparts. This is particularly true for measurements spanning the gas exchange process, as the low magnitude incident heat flux signal must “compete” with the mechanical/electrical noise associated with engine operation. However, as the incident heat flux rises during late compression/combustion/early expansion, the signal to noise ratio also increases. This effectively “tightens” the statistical spread of closed-cycle temperature measurements—decreasing the uncertainty of the associated SFSM analysis. This trend is quite advantageous, as much of the analysis of TBC surface heat flux and temperature trends are isolated to closed-cycle crank angle degrees.

For brevity, SFSM uncertainty analysis is reported at the 1200 rpm FMPM data point. However, closed-cycle “tightening” of the estimated TBC surface heat flux uncertainty is observed across engine speeds. Additionally, the peak TBC surface heat flux remains statistically less than the associated metal surface heat flux—further supporting the insulative nature of the TBC-treated piston/probe.