Analyzing Engine Exhaust Gas Temperature Pulsations and Gas-Dynamics Using Thin-Wire Thermocouples

Exhaust gas temperature is a vital parameter in the design, development, and monitoring of ICEs, enabling high-efficiency operation of turbocharging and exhaust aftertreatment systems. The mean EGT is typically measured using robust sheathed invasive thermal sensors with limited bandwidth, such as thermocouples and resistance thermometers. The sensors' thermal inertia and heat losses (conduction/radiation) cause errors in the measured EGT requiring correction by sensor design and/or additional measurements. Furthermore, the unsteadiness of ICE exhaust gas flows shifts the mean measured EGT by invasive thermal sensors [1]. Increasing the bandwidth of EGT measurements addresses these errors [1,2]. Papaioannou et al. [3] indicated a reduction in the mean measured EGT when reducing the wire diameter of exposed junction thermocouples from 254 μm to 50.8 μm. The mean measured EGT reduction was especially significant (40-80 K) between the exposed junction and (3 mm) sheathed thermocouple constructions, similar to observations by Bajwa et al. [4] (50 K reduction). Bajwa et al. also modeled the engine conditions and showed a further reduction of 57 K in the mean EGT between the 50 μm thermocouple and the gas. Venkataraman et al. [5] showed that reducing the thermocouple wire diameter (76 μm–51 μm) at the exhaust port significantly improved the response with a 92-K peak–peak increase, a 26-K mean EGT drop, and a twofold increase (to ±12 K) in the sensitivity to mean EGT cycle-to-cycle variations. Therefore, improving the accuracy of both instantaneous and mean EGTs in ICEs necessitates increasing the bandwidth of EGT measurements.

The EGT pulse is conventionally estimated using one-dimensional gas-dynamic system simulations. Model calibration and validation utilizes high-bandwidth intake, in-cylinder, and exhaust pressure measurements for three pressure analysis. However, reference EGT measurements are limited to mean values using sheathed thermal sensors [6] as the conflicting requirements of sensor robustness, commercial availability, and cost [7] constrain high-bandwidth EGT measurements. Optical temperature measurement techniques based on laser diagnostics and radiation thermometry [8] are confined to special test facilities involving significant setup modification and complexity. While promising, exposed thin-wire thermal sensors with reduced wire diameters and an increased bandwidth [9] are impeded by a robustness-to-response tradeoff. Engine experiments with tungsten resistance wire thermometers (RWTs) down to 6.9 μm [10] indicated several limitations. These include a significant static calibration drift (50% [11]), requirements for dynamic error compensation through dynamic calibration [11] or using multiwire signal reconstruction [10], and a short test duration due to thin-wire fragility [12]. To overcome these limitations, Mollenhauer [13], Kar et al. [14], and Kee et al. [15] used thicker multiwire thermocouples to reconstruct the temperature signal by estimating the instantaneous thermocouple time constants. While the efficacy of temperature signal reconstruction techniques was demonstrated under controlled conditions [14,15], they remain to be established in engine exhaust conditions and lack a reference EGT measurement.

While the literature highlights the measurement challenge and limitations of high-bandwidth EGT in ICEs, drawbacks in the numerical estimation of the EGT pulse need further investigation. Gas-dynamic studies focused on model development [16] and ICE exhaust tuning for performance [17] based on finite wave theory predominantly rely on high-bandwidth pressure measurements alone for validation. Benson and Galloway [18] adequately validated the modeled exhaust pressure pulses in a two-stroke diesel engine. Nevertheless, deviations in the modeled EGT pulse amplitude from the tungsten RWT measurements were attributed to heat transfer with the colder residual gases and pipe walls [18]. Caton and Heywood [11] improved the accuracy of the modeled EGT pulse by validating exhaust port heat transfer models with tungsten RWTs. Although, the coefficients and exponents of the correlations vary widely in the literature and are not universally applicable [19]. When considering thermal sensor modeling, Franzke et al. [20] improved the accuracy of the mean modeled EGT of a sheathed thermocouple to within 10 K of measurements through a one-dimensional model representation over 0D. However, sensor model validation was applied only to the mean EGT. Bajwa et al. [4] calibrated a 50-μm thermocouple model to the measured cycle averaged EGT pulse through multipliers to the thermocouple's convective heat transfer coefficient and emissivity. This approach lumps model calibration on the thermal sensor while the accuracy of the modeled EGT pulse remains unknown.

Notwithstanding the benefits of high-bandwidth EGT measurements for experimental and numerical analysis in ICEs, measurement technique limitations present a significant barrier. Combining an underutilized attribute of thin-wire thermal sensors (the signal's time derivative) with the drift insensitivity and robust sensor designs of Type-K thin-wire thermocouples reported by Venkataraman et al. [5] shows promise in furthering the adoption of increased bandwidth EGT measurements. Gardiner et al. [21,22] applied the thermocouple signal derivative without response compensation to monitor the combustion process for cycle-to-cycle variations and misfires using stepped-sheathed thermocouples (similar to [23]). Mavropoulos and Hountalas [24] qualitatively depicted the blowdown and scavenge phases of the exhaust discharge process using the derivative of exhaust manifold surface thermocouples. As EGT signal reconstruction utilizes the thermocouple signal derivative to account for the dynamic error, Kar et al. [14] highlighted rapid cooling before the EGT pulse and attributed it to mixing and valve choked flow (early blowdown) induced heat losses. While Papaioannou et al. [3] independently reaffirmed these observations, further insight into the underlying physics that determines the phase and amplitude of the cooling needs examining.

This study investigates the need for increased bandwidth EGT measurements alongside pressure measurements to improve the accuracy of gas exchange assessments in ICEs. While the literature typically compares simulated pressure and EGT pulses to measurements, this work contrasts the measured EGT and exhaust pressure pulses to elucidate differences and identify unique features in the EGT pulse.

This section begins by describing the adopted experimental setup and test methodology. Motivated by the literature in the Introduction (Sec. 1), EGT pulse measurements utilized a bead welded 50.8-μm Type-K thin-wire thermocouple and its signal derivative. The theoretical background concerning the thermal sensors' heat balance in Sec. 2.2 highlights the potential and limitations of assessing the EGT pulse through the thermocouple signal derivative. Subsequently, finite wave theory is discussed in Sec. 2.3 to qualitatively isolate trends in the thermocouple signal derivative to the EGT using the measured static pressure pulses.

Experiments were performed on an inline-6 HDD engine with the exhaust manifold modified to accommodate a single-pipe with a measurement section (Fig. 1) that bypassed the turbocharger turbine and the EAT unit (refer to [5] for a more detailed description of the experimental setup). Table 1 presents the base engine specifications. The split exhaust enabled cylinder-6 with the single-pipe to have no backpressure, while the remaining five cylinders provided the compressor work (boost pressure) for all the cylinders. The single-pipe exhaust setup isolated the exhaust pulse from interactions and permitted assessments with engine speed and load sweeps that varied the characteristics of exhaust valve discharge and gas dynamics.

The measurement plane of the static pressure and thin-wire thermocouple in the measurement section was located at 11D (11× pipe diameter, 440 mm) downstream port exit. The development length allowed for minimizing gradients in the exhaust flow velocity and temperature profiles. Although limited by practical constraints, the test setup was within guidelines for complete hydrodynamic development of steady turbulent flows in circular pipes with an L/D of 10–60 [25,26]. The single-pipe and the measurement section were not thermally insulated to maintain accessibility.

An AVL Dynodur 500 dynamometer controlled the engine speed and load. A partially open electronic control unit provided control of the timing and quantity of injected fuel, the fuel rail pressure, and the variable geometry turbine (VGT) rack position. Mean pressure and temperature levels for monitoring the engine boundary conditions at various locations used GEMS 3500 strain gauge pressure transducers and 3-mm Type-K Inconel sheathed thermocouples. The outer wall temperature measurement of the measurement section used bolt-on washer Type-K thermocouples on either side. The exhaust gas mass flow in the single-pipe was indirectly estimated using the fuel flow measured using an AVL 733 s fuel scale and lambda estimates from a Horiba MEXA-7100DEGR using the no O₂ method. The total fuel flow was scaled to one cylinder, assuming equal fuel distribution across cylinders.

Crank angle resolved pressure measurements at cylinder-6 with the single-pipe exhaust included the intake (Keller M5HB), in-cylinder (Kistler 7061B with a Kistler type 5011 charge amplifier), and the static pressure at the exhaust port and measurement section (Keller M8coolHB). The Keller pressure sensors had a bandwidth of 50 kHz. The 50.8 μm bead-welded Type-K thermocouple was procured from Omega Engineering Inc. and fabricated in-house with inputs from the literature [3,14,15] (refer to sensor “Slack-3” in Ref. [5] for more details). The exposed wire length-to-diameter ratio $(l/d)$ was estimated to be 115. The raw thermocouple output was conditioned by an AD597 (Type-K) amplifier module with cold junction compensation and a scaling factor of 10 mV/$°$C. A passive lowpass filter with a cutoff frequency of 10 kHz conditioned the amplifier output. The thin-wire thermocouple was inserted 15 mm into the pipe (5 mm above the pipe centerline) and was aligned parallel to the single-pipe axis.

All the pressure and temperature sensors, excluding the Keller pressure transducers calibrated by the manufacturer, were calibrated in-house. The 50.8- μm thermocouple showed good linearity $(R2>0.99)$ between 303 and 623 K when calibrated against an Isotech Quick-Cal calibrator oven with a gain increase of $0.77%$ (from $10 mV/°$C) and an offset of $−0.89 °$C. Crank angle-resolved pressure and thin-wire thermocouple measurements were sampled at 0.1 crank angle degree (CAD) for 300 cycles using an 8-channel 12-bit PowerDAQ analog to digital converter card. This corresponded to a sampling frequency between 42 kHz at 700 rpm and 114 kHz at 1900 rpm. Simultaneous engine boundary condition measurements were sampled at 1 Hz over 1 min using an in-house-developed code. Data were logged on a warm engine after the measurement section walls attained the thermal equilibrium criterion (discussed subsequently). The cycle-resolved parameters in this study are represented by the cycle average (ensemble average) of 300 engine cycles after conditioning with a polynomial smoothing filter (second order Savitzky–Golay filter [27]) using 101 samples (10 CAD). This filter is commonly adopted in temperature signal reconstruction studies (e.g., Ref. [14]). The thermocouple signal derivative was computed using a central difference scheme over a length of 20 + 1 samples.

Building on idealized dynamic valve discharge experiments in a rig [28], the experimental design focused on the implications of the valve discharge process on the exhaust pressure and temperature pulse waveforms subject to the valve opening (engine) speed and the pressure ratio between the cylinder and exhaust port exit $(PREV)$ at exhaust valve open (EVO) $(PREVO)$⁠. Winroth [28] showed that the instantaneous mass flowrate from the pressurized cylinder to the exhaust pipe increased with the $PREVO$ (fixed valve opening speed) and with the valve opening speed (fixed $PREVO$⁠). The experimental design was realized through a load sweep (labeled LS1-4) at a constant engine speed (1500 rpm) and a speed sweep (SS) (labeled SS1-4). For comparable test conditions across operating points, the single-pipe gas temperature (3-mm sheathed thermocouple) was maintained around 640±10 K. Additionally, the estimated trapped mass in cylinder-6 was regulated across the speed sweep points for a $PREVO≈4$ to isolate the valve opening speed. Table 2 summarizes the engine operating conditions with the mean and two-standard deviation spread where applicable. The LS2 and SS3 operating points ensured repeatability checks within the test. The thin-wire thermocouple was mounted on a warm engine to avoid undesired exposure to the warm-up phase. The measurement section wall's thermal equilibrium criterion was set to a temperature change within 1 K/minute, ensuring at least 15 min of steady operation at each test point.

The formulation in Eq. (3) represents the thermocouple's dynamic error $(τdTjdt)$ caused by the thermal inertia $(mCp)$ of the junction. The measured signal derivative and the junction's time constant $(τ)$ can compensate for the dynamic error. However, τ in the engine exhaust context is complex given the time-varying nature of the exhaust flow (affecting h), the EGT $(Tg)$⁠, and the thermophysical properties of the wire $(Cp)$ over an engine cycle. Equation (3) shows that the thermocouple signal derivative is dependent on both the instantaneous temperature difference between the gas and the junction $(Tg−Tj)$ along with the heat transfer coefficient (h). $(Tg−Tj)$ also determines the direction of heat flow. Associating changes in the thermocouple signal derivative exclusively to the EGT (T_g in Eq. (3)) is limited by the flow dependency through h. Additionally, the relative significance of the dynamic error $(τdTjdt)$ to changes in the junction temperature (T_j) also determines the causal factor affecting the EGT (T_g) when assessing it through the thermocouple derivative. The thermocouple signal derivative is referenced in this study as the indicative EGT pulse waveform that represents the unscaled dynamic error.

As indicated in the literature [16,17], analyzing and optimizing ICE exhaust gas dynamics primarily rely on resolving the exhaust pressure pulses. Theoretical formulations of the exhaust gas dynamics provide insight into the pressure wave-induced gas molecular velocity. The assumptions used in such formulations limit the accuracy of the absolute flow velocity levels, which lack spatial resolution. However, an indication of the flow velocity based on the measured pressure pulses aids in isolating trends in the thermocouple signal derivative to the convective heat transfer coefficient (h) and hence the EGT.

Annand and Roe [16] state that in contrast to weak sound waves, engine exhaust pressure pulses cause a significant displacement in the gas molecules, inducing their flow with the passing of the (finite) pressure waves. When considering an undisturbed gas at a pressure p_o and a temperature T_o, the ratio of the instantaneous wave-to-undisturbed gas pressure (⁠$PRo=p/po$⁠) indicates the nature of the wave as a compression or expansion wave. A compression wave increases the pressure with a $PRo>1$ upon passage through an undisturbed gas, while an expansion wave decreases the pressure with a $PRo<1$⁠. The induced gas molecular velocity is in the same direction as the propagating compression waves, while the opposite occurs for expansion waves. The pressure differential between gas molecules on the wave and that of the undisturbed gas causes and determines the direction of the induced flow of gas molecules.

Using equations Eqs. (4) and (5), the implications of $PRo$ on the wave and gas molecular velocities C and u can be assessed. Table 3 denotes the computed wave and gas molecular velocities considering a $γ=1.33$ for the exhaust gas [16] while varying the $PRo$ between 0.6 and 1.4 to represent points on an expansion and compression wave, respectively. Pearson et al. [17] highlighted the nonlinearity (directional dependence) in wave propagation velocities of compression and expansion waves (also seen in Table 3) and emphasized avoiding the erroneous assumption of a constant wave propagation velocity $(C=co)$ independent of the direction of wave propagation. The nonlinearity refers to expansion waves propagating disproportionately slower than compression waves for a given change of $PRo$ from unity. It is also apparent from Table 3 that the nonlinearity applies to the magnitude of the induced gas molecular velocity (u) with expansion waves inducing greater gas molecular velocities in the opposite direction (negative sign) of wave propagation.

This section analyzes the exhaust pulse characteristics over the engine load and speed sweeps. The measured static pressure and $PREV$ evolution from EVO provide the conventional basis for investigating exhaust pulsations to identify the blowdown and scavenge phases of the exhaust discharge process and gas-dynamic induced compression and expansion waves. The thin-wire thermocouple signal and its time derivative extend the assessment to the measured and indicative EGT pulses, including the prepulse fluctuation before the conventional blowdown phase of the exhaust pulse at the measurement section. Trends of the pressure and indicative EGT pulse waveforms are first delineated and subsequently examined across the operating points.

Figure 2 shows the exhaust pulse characteristics over the LS points. The exhaust pulses are characterized by the cycle averaged static pressure (Fig. 2(a)), the measured thin-wire thermocouple signal (Fig. 2(b)), and its time derivative (Fig. 2(c)) along with the $PREV$ evolution from EVO (Fig. 2(d)).

From the cycle-averaged static pressure in Fig. 2(a), it is evident that the pressure rise rate and peak pressure of the exhaust pulse increase from LS1-4 (around 170–175 CAD), illustrating the blowdown phase with pressure-driven exhaust flow. With an increasing $PREVO$ over LS1-4, the $PREV$ evolution from EVO to $PREV=1$ in Fig. 2(d) provides insight into the amplitude and duration of the blowdown phase across the LS points observed in Fig. 2(a). Following the first peak associated with the blowdown phase, the static pressure drops until it rises around 270 CAD to form a second peak representing the scavenge phase (displacement phase) with piston-driven exhaust flow. Subsequently, a significant pressure drop between 300 and 480 CAD indicates an expansion wave followed by a compression wave that peaks at around 640 CAD. The expansion and compression waves represent the interacting forward and reverse (reflected) traveling waves in the single-pipe exhaust. With a consistent fundamental pulse frequency of 12.5 Hz (1500 rpm) over the LS points, the traveling waves are comparable in phase and amplitude, resulting in similar static pressure levels at the measurement section at EVO.

In Fig. 2(b), the cycle-averaged thermocouple signals are compared over the LS test points. Unlike the pressure pulses presented in Fig. 2(a), the distinct phases of the exhaust pulse are not discernible in the measured EGT pulse due to the limited bandwidth of the 50.8 μm thermocouple. However, it is observable in Fig. 2(b) that the ascent of the measured EGT pulse advances in phase and becomes steeper from around 180 CAD for LS1-2 to 170 CAD for LS3-4. The measured EGT peaks around 360 CAD for LS1-2 and marginally advances for LS3-4. The peak-to-peak of the measured EGT pulse increases from LS1-4, typically leading to greater peak values except for LS2 due to its higher mean level (refer to Table 2). Similarly, except for LS2, the measured EGT around EVO decreases with an increase in load. Prepulse EGT fluctuations are visible in the cycle-resolved EGT in Fig. 2(b) and the inset between 160 and 180 CAD across the LS points and appear insignificant with a peak-to-peak of 1 K.

Unlike the measured EGT signals in Fig. 2(b), the thermocouple signal time derivatives over the LS points in Fig. 2(c) showcase two peaks of the indicative EGT waveform between EVO and 360 CAD. The peaks represent the blowdown and scavenge phases of the EGT pulse resulting from the exhaust discharge process as observed in the pressure pulse waveforms in Fig. 2(a). Additionally, Fig. 2(c) shows that the thermocouple signal derivative is sensitive to the expansion and compression wave-induced EGT changes beyond 300 CAD. The thermocouple signal derivative shows clear trends over LS1-4 with an increasing indicative EGT peak over the blowdown phase, while the scavenge phase shows the contrary. These trends differ from the pressure pulses in Fig. 2(a), wherein the blowdown and scavenge pressure levels increased with the load, although the pressure levels converged toward the scavenge phase. It is also evident from the thermocouple derivative signal at LS1 that the indicative scavenge EGT peak is at a higher level than the blowdown in contrast to other LS points. The prepulse EGT fluctuation in Fig. 2(c) is amplified in the thermocouple signal derivative despite its low amplitude in the measured signal, indicating its high-frequency content. From LS1-4, a marginal phase advance and a decreasing peak-to-peak in the prepulse fluctuation is observable. Notably, the prepulse fluctuation is only present in the EGT pulse, while absent in the pressure pulses in Fig. 2(a). The observed trends in the indicative EGT pulse waveforms are analyzed in Sec. 3.3.

Figure 3 illustrates the exhaust pulse characteristics over the SS points similar to the depiction over the LS points. This includes the cycle averaged static pressure (Fig. 3(a)), the measured EGT (Fig. 3(b)), and its time derivative (Fig. 3(c)) along with the $PREV$ evolution from EVO (Fig. 3(d)).

The cycle-averaged static pressure over the SS points in Fig. 3(a) shows that a combination of differences in exhaust gas dynamics and valve discharge increase the complexity of characterizing the pulse compared to the LS cases. The blowdown phase in Fig. 3(a) shows an increasing pressure rise rate and peak pressure of the exhaust pulse from SS1-3, while these attributes are comparable for SS3-4. A delay of around 10 CAD in the arrival of the blowdown pressure pulse at the measurement section is also evident between SS1 and SS4. The $PREV$ evolution from a comparable level at EVO in Fig. 3(d) shows that the blowdown duration increases from SS1-4 as the $PREV$ reached unity around 170 CAD for SS1 and 200 CAD for SS4. It is also evident in Fig. 3(a) that the scavenge phase peak for SS3-4 at around 300–330 CAD is consistent with observations over the LS points. However, at SS1 and SS2, the pressure pulse drops to an expansion wave around 200 CAD and beyond 270 CAD. The advance in the expansion wave from SS4-1 into the exhaust valve open period in Fig. 3(a) can be attributed to the decreasing engine speed and corresponding valve and piston velocities that delay cylinder scavenging relative to the wave propagation velocity in the single pipe. Figure 3(e) illustrates the differences in the wave reflections between successive pulses by depicting two engine cycles at SS1, which provides the lengthiest timescale at the lowest tested engine speed of 700 rpm. Figure 3(e) shows that from SS1-4, the number of exhaust pulses in the same time scale (343 ms) increases with an advance in EVO and a decrease in time for the traveling waves to attenuate toward the equilibrium pressure level in the pipe. This leads to the EVO of the subsequent cycle occurring earlier in the reflection pulse train from SS1-4, which can benefit from the appropriate tuning of the exhaust pipe geometry. The nature of the exhaust pulse determines that of the reflected waves. The pulse reflections at SS1-2 with predominant blowdown pulses appear sinusoidal, while the pulses at SS3-4 with a blowdown and scavenge phase appear trapezoidal. These factors reduced the static pressure around EVO at the measurement section in the single pipe from SS1-4.

Similar to observations over the LS points, the cycle-averaged thermocouple signals over the SS test points in Fig. 3(b) do not capture the distinct phases of the exhaust process that are observable in the pressure signals in Fig. 3(a). When increasing the engine speed from SS1-4, the ascent in the measured EGT pulse shows a phase delay in the signal similar to the pressure pulses in Fig. 3(a) with a weakening gradient and peak-to-peak. While SS1 indicates two peaks in the measured EGT, SS3-4 only indicates one delayed peak that matches with the second peak of SS1 around 360 CAD. SS2 shows an intermediate-measured EGT pulse with a slower ascent than SS1 but faster than SS3-4 and peaks at around 300 CAD. The measured EGT around EVO increases from SS1-4, with the prepulse EGT fluctuations in Fig. 3(b) and the inset between 140 and 180 CAD weakening from a peak-to-peak of 3 K at SS1 to 1 K at SS4, remaining insignificant relative to the measured EGT pulses.

As observed over the LS points, the thermocouple signal time derivatives over the SS points in Fig. 3(c) reveal waveform features of the EGT pulse such as the blowdown and scavenge peaks that are not discernible in the measured EGT pulses in Fig. 3(b). However, unlike the LS test points, the indicative EGT pulses show significant differences in their waveforms compared to the corresponding pressure waveforms in Fig. 3(a). From SS1-4, the indicative EGT waveform shows a weakening blowdown phase and a strengthening scavenge phase, although SS2-3 appear qualitatively comparable to their corresponding pressure pulse waveforms in Fig. 3(a). The trends in the blowdown phase amplitude are opposing for the thermocouple signal derivative and the static pressure signals. Expansion and compression waves are evident in the thermocouple signal derivatives of SS3-4 beyond 300 CAD, while wave effects in SS1-2 are less sensitive and occur earlier in the cycle as shown by the pressure signals in Fig. 3(a) between 200 and 270 CAD. The prepulse EGT fluctuation is amplified in the thermocouple signal derivative with retention of the signals' phase delay and peak-to-peak reduction over SS1-4 test points from the measured EGT pulses in Fig. 3(b). The prepulse fluctuation is again limited to the measured EGT pulses and is not visible in the pressure pulses over the SS points. Section 3.3 examines the observed trends in the indicative EGT pulse waveforms.

When comparing the measured static pressure pulses and the indicative EGT waveforms over the LS and SS test points in Secs. 3.1 and 3.2, the disparities observed included the presence of a prepulse fluctuation exclusively in the EGT waveform and opposing trends in the scavenge phase of the LS points and blowdown phase of the SS points. As discussed in Sec. 2.2, the thermocouple signal derivative cannot be isolated to the EGT due to the combined dependence on the EGT and the flow velocity (impacting h). When normalizing the measured pressure signals over the LS and SS test points with the backpressure in Table 2 as the undisturbed single-pipe pressure, the pressure ratio $PRo$ helps estimate the wave (C) and induced gas molecular velocity (u) over the engine cycle as discussed in Sec. 2.3. The trends of C and u were consistent with the static pressure trends over the test points even when computing γ for the exhaust gas composition and the measured EGT. As variations in $Po$ are within 2% across the operating points (refer to Table 2), the trends in the static pressure and $PRo$ are used interchangeably to qualitatively represent the C and u waveforms enabling isolation of trends in the thermocouple signal derivative to the EGT.

To analyze the prepulse fluctuation in the indicative EGT illustrated in Figs. 2(c) and 3(c), Fig. 4 focuses on the thermocouple derivative within ±4 K/ms in Figs. 4(a) and 4(b) between 125 CAD (EVO) and 200 CAD over the LS and SS test points. The $PRo$ within the same CAD interval are showcased in Figs. 4(c) and 4(d).

From Figs. 4(c) and 4(d), it is evident that $PRo$ increases over the LS and SS test points beyond EVO with a delay due to the downstream location of the measurement plane from the exhaust port. While $PRo$ is comparable at EVO for the LS test points, it decreases from around 1 to 0.9 from SS1-4. The disparity in $PRo$ between the LS and SS test points was attributed to differences in the reflected waves with a varying engine speed (refer to Sec. 3.2.1). From Table 3, a higher $PRo$ corresponds to a higher wave propagation velocity, which implies that the phase of the prepulse fluctuation in the indicative EGT pulse advances over LS1-4 where $PRo$ diverges and over SS1-4 from EVO as seen in Figs. 4(a) and 4(b). The greater significance of the phase delay over SS1-4 relates to the lower wave propagation velocity as the $PRo$ decreases below unity from EVO.

A heating stage (⁠$dTj↑$⁠) in the prepulse EGT fluctuation is observable across all operating points in this study (Figs. 4(a) and 4(b)) with only a cooling stage reported in the literature [3,14]. The negative dT_j at EVO across all operating points indicates that the thermocouple junction temperature $(Tj)$ is decreasing ahead of the subsequent cycle's exhaust pulse due to conductive and radiative heat loss. Upon EVO, the passing compression wave increases the local gas pressure and hence temperature by the equation of state for an ideal gas. With the measurement plane located 440 mm downstream of the exhaust port exit, the significantly higher wave propagation velocity over the gas molecular velocity for a given $PRo>1$ (refer to Table 3) leads to a heating stage dominated by the compression wave's pressure rise rate, which determines the EGT rise rate. This explanation supports the observations over LS1-4 and SS1-3, while the deviation in SS4 is purported to a greater significance of wave-induced disturbances (nonstationary gas) at this operating condition. Subsequent to the heating stage, an EGT drop (cooling stage with $dTj↓$⁠) is evident across all operating points (Figs. 4(a) and 4(b)) despite the continuing increase in $PRo$ (Figs. 4(c) and 4(d)). The EGT drop denotes the arrival of the upstream exhaust gas molecules that loose heat due to mixing and heat convection with the walls. However, the EGT drop decreases in peak–peak from LS1-4 and SS1-3 (SS3-4 are comparable) contrary to the wave-induced gas molecular velocity trends (⁠$PRo$ trends), which are increasing. The cooling stage discrepancy is supported by trends in the measurement section wall temperature that increase from LS1-4 and SS1-4 (refer to Table 2). The increasing wall temperature trend indicates that the upstream gas molecules over the cooling stage have a relatively higher EGT over LS1-4 and SS1-4 due to conditioning by the wall. In Fig. 4(a), LS3–4 show a reduced heating rather than a real cooling stage as dT_j remains positive despite the dip.

Following the rapid prepulse fluctuation, the blowdown and scavenge phases in the indicative EGT waveform represent the passage of the high-temperature bulk gas molecules discharged from the cylinder to the exhaust pipe. Over the LS points (Figs. 2(a) and 2(c)), the trends in the pressure and indicative EGT pulse waveforms aligned over the blowdown phase while they were opposing through the scavenge phase. Contrary to the LS points, the trends reversed over the SS points (Figs. 3(a) and 3(c)) with an opposing blowdown phase and comparable scavenge phase trend. Table 4 provides a qualitative assessment of the significance of the dynamic error $(τdTjdt)$ to ascertain the tendency of the indicative EGT waveform to represent the gas temperature over the blowdown and scavenge phases of the LS and SS points.

In Table 4, the trend in the indicative EGT waveform is represented by (⁠$⇑$/$⇓$⁠), while trends in the pressure (p), convective heat transfer coefficient (h), and junction time constant (τ) are denoted by (↑/↓). Table 4 shows that over all test conditions, the trends in p increased, leading to greater wave-induced velocities (u) that caused the h to increase (refer to Table 3 and Eq. (2)). From Eq. (3), $τ∼1h$ leads to a τ reduction over LS1-4 and SS1-4 blowdown and scavenge phases. Combining the τ and $dTjdt$ trends provides the trend of the dynamic error in Table 4, which has a greater significance over the LS1-4 scavenge phase and the SS1-4 blowdown phase as the trends align. Notably, these cases revealed a discrepancy between the pressure and indicative EGT waveforms implying that the observed deviations will amplify in the true EGT pulse when correcting for the dynamic error. The translation of these trends to the true EGT pulse depends on the relative significance of the variation of the thermocouple's junction temperature across the operating points when correcting for the dynamic error (refer to Eq. (3)).

The results of this study elucidate disparities between the pressure and indicative EGT waveforms through the thermocouple signal derivative. Existing assessments in the literature typically contrast the measured pressure [16,17] and EGT pulses [11,18] to their respective model estimates but overlook comparing the measurements. The results indicate that pressure pulse measurement-based EGT estimation with sheathed thermal sensors considering an isentropic state change [13] can be erroneous as the estimated EGT waveform retains the pressure waveform. As heat transfer is a crucial parameter that distinguishes the EGT pulse from the pressure pulse, the results reiterate the importance of EGT pulse measurements for model calibration and validation. High-bandwidth EGT measurements enable accuracy improvements of the modeled EGT pulse [11,19] across engine types, measurement locations, and operating conditions.

While the 50.8-μm thin-wire thermocouple used in this study was bandwidth limited to sufficiently resolve the EGT pulse, the signal derivative revealed attenuated features in the measured EGT pulse. The thermocouple signal derivative was adequate to qualitatively establish fundamental differences between the pressure and EGT pulse waveforms. Nevertheless, bandwidth limitations constrain quantitative EGT pulse assessments. For quantitative assessments, the straightforward approach would be to increase the bandwidth of the thin-wire thermocouple by reducing its diameter to the lowest reported value of 12.5 μm [15]. The reduced wire diameter combined with a taut wire design and coated weld faces improves its robustness-to-response tradeoff [5]. Fabrication limitations dictate that wire diameters below 12.7 μm require tungsten RWTs [9]. However, drift and dynamic calibration remain crucial [11] for RWTs. This highlights the need for further efforts to minimize the propensity to drift (wire material) and the dynamic error (sensor design).

The alternative approach for quantitative EGT pulse assessments is multiwire-based EGT signal reconstruction. Herein, the results of this study indicate the significance and implication of the prepulse EGT fluctuation and the gas-dynamic-induced temperature changes as perturbing factors for existing on-engine signal reconstruction techniques. When considering the highly unsteady exhaust of single-cylinder engines, the flow velocity can cross no flow zones (zero velocity) with occurrences of flow reversal. These result from the valve discharge process and exhaust gas dynamics and are sensitive to the engine operating condition and backpressure. The sensors' heat balance changes significantly from the simplified first-order system notation in Eq. (3) in these low-flow regions. The greater significance of the conduction and radiation heat transfer modes at low flow velocities implies that they cannot be neglected. Such exhaust conditions have been the conventional test environment for on-engine EGT signal reconstruction, which fundamentally deviates from the controlled environment test conditions. The flow rigs used to establish the reconstruction techniques by Kar et al. [14] and Kee et al. [15] ensured a near-constant flow velocity (21 m/s [14]) while emulating EGT pulsations. However, engine exhaust pulsations have varying flow velocities in tandem with the EGT. Nevertheless, existing reconstruction techniques remain promising if appropriately designed sensors are tested in engine exhaust conditions with adequately high mean flow velocities and reduced unsteadiness that ideally avoids the low flow velocity region below 20 m/s.

Further investigation is required to employ the benefits of high-bandwidth EGT measurements and on-engine EGT reconstruction. Contrarily, thin-wire thermocouples facilitate accuracy improvements of the measured and modeled EGT. Bajwa et al. [4] presented the benefits of validating the cycle-resolved EGT with thin-wire thermocouples over conventional mean EGT validation from sheathed thermal sensors. However, independently validated thermal sensor models are required to avoid lumping model uncertainties while assuming the modeled EGT pulse to be accurate. Furthermore, the thermocouple signal derivative supports validating high-frequency components of the EGT pulse. Nevertheless, the significance of noise amplification when computing the signal derivative remains a constraining factor for this approach.

This study investigated the need for increased bandwidth exhaust gas temperature (EGT) measurements in ICEs by contrasting the measured EGT with high-bandwidth exhaust pressure measurements. Theoretical considerations, including the thermocouple's heat balance and finite wave theory, assessed unique and contrary features between the EGT and pressure waveforms. A 50.8-μm Type-K exposed bead thermocouple provided EGT pulse measurements and indicated the EGT waveform through the signal derivative. The signal derivative represents the unscaled dynamic error of the thermocouple. Experiments on an idealized single-cylinder exhaust of a heavy-duty diesel engine varied the characteristics of exhaust valve discharge and gas dynamics. The results present significant differences between the indicative EGT waveform and the exhaust pressure waveforms that necessitate high-bandwidth EGT measurements to improve the accuracy of experimental analysis and model validation. The results show that:

• Unlike the measured thermocouple signal, the signal derivative reveals the blowdown and scavenge phases of the EGT pulse. The signal derivative is also sensitive to gas-dynamic induced temperature changes from EVC to EVO that are not discernible in the measured signal.

• While the pressure waveforms depend on the exhaust valve discharge process and gas dynamics, deviations in the indicative EGT waveform were caused by heat transfer from gases escaping the cylinder and residual gases in the exhaust pipe conditioned by the walls along the exhaust flow path.

• Contrary trends over the blowdown (speed sweep points) and scavenge phases (load sweep points) between the pressure pulse and the indicative EGT waveform were qualitatively supported by trends in the thermocouple's dynamic error.

• The phase and amplitude of the rapid pre-pulse EGT fluctuation in the thermocouple signal derivative were explained through a heating and cooling stage. While pre-pulse heating was attributed to the passage of compression waves, pre-pulse cooling was additionally subject to heat transfer of the wave-induced flow of residual gas molecules conditioned by the pipe walls.

The authors acknowledge Professor Ulf Olofsson (KTH) for his support and feedback through this work. This study was financed by the Competence Center for Gas Exchange (CCGEx) at KTH and the Swedish Energy Agency.

• Energimyndigheten (Award No. P33834-3; Funder ID: 10.13039/501100004527).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

A =

area, $m2$

C =

wave propagation velocity, $m s−1$

C_p =

specific heat capacity, $J kg−1 K−1$

d =

thin-wire diameter, $m$

D =

pipe diameter, $m$

h =

convective heat transfer coefficient, $W m−2 K−1$

k =

thermal conductivity, $W m−1 K−1$

l/d =

thin-wire length-to-diameter ratio

L/D =

pipe length-to-diameter ratio

m =

thermocouple junction mass, $kg$

p =

absolute pressure, $kPa$

R =

specific gas constant, $J kg−1 K−1$

R² =

coefficient of determination

t =

time, $s$

T =

temperature, $K$

u =

wave induced gas molecular/flow velocity, $m s−1$

V =

thermocouple voltage, $V$

x =

length scale, $m$

Abbreviations

ADC =

analog to digital converter

ATDCf =

after firing top dead center

B0 =

0% biodiesel/neat diesel

CAD =

crank angle degree

CoV =

coefficient of variation

EAT =

exhaust aftertreatment

ECU =

electronic control unit

EGT =

exhaust gas temperature

EVC =

exhaust valve close

EVO =

exhaust valve open

HDD =

heavy duty diesel

ICE =

internal combustion engine

IMEP =

indicated mean effective pressure

$LS1−4$ =

load sweep test points 1–4

MIMS =

mineral insulated metal sheathed

PR =

pressure ratio

RWT =

resistance wire thermometers

$SS1−4$ =

speed sweep test points 1–4

TPA =

three pressure analysis

VGT =

variable geometry turbine

$0/1D$ =

0/1-dimensional

Greek Symbols

γ =

ratio of specific heat capacity

ε =

emissivity

ν =

kinematic viscosity, $m2s−1$

σ =

Stefan–Boltzmann constant, $Wm−2K−1$

τ =

thermocouple time constant, $s$

Subscripts and Superscripts

c =

cross-section

EV =

exhaust valve

g =

gas

j =

junction

o =

undisturbed condition

s =

surface

w =

wall