Uncertainty Analysis of an Engine Test Cell

Home , Linear approximation

Thesis

Presented in Partial Fulfillment of the Requirements for the Degree Master in Science in

the Graduate School of The Ohio State University

Kepin Kavathia

Graduate Program in Mechanical Engineering

The Ohio State University

2018

Thesis Committee

Dr. Shawn Midlam-Mohler, Advisor

Dr. Marcello Canova

Dr. Punit Tulpule

Dr. Raju Dantuluri

Copyrighted by

Kepin Kavathia

2018

Abstract

Durability test has become increasingly important due to demand for the extended life of the vehicles and hence vehicle’s powertrain. When these long, time-consuming durability tests are running, there is a need for automated checks to assess if they are running as per the requirement. This requires us to know acceptable error range for each of the measured variables and performance parameters. Hence, the main objective is to find the exhaustive error band of the system’s important performance parameters through measured variables.

A detailed uncertainty analysis, consisting of systematic and random uncertainty is calculated for performance parameters like Brake Mean Effective Pressure (BMEP), Brake

Specific Fuel Consumption (BSFC), Air-Fuel Ratio (AFR), Indicative Mean Effective Pressure

(IMEP) and air flow rate into the engine. The uncertainty in the performance parameters is calculated from the uncertainty of the measured variables. The effect of the environmental conditions like air temperature, humidity, and variation of part-to-part are also taken into consideration for calculating the uncertainty.

The systematic uncertainty is calculated by the understanding of the system, i.e. the sensors and the actuators of the engine cell, while the random uncertainty is calculated using the data from GT-Power model simulations. Student’s t-distribution is used to calculate the uncertainty as for the true value of the mean and standard deviation of the data received is not known. ii

Error bounds of the performance parameters are found using the GT-Power simulations. A list of performance parameters with the major contributing factors to their uncertainties is listed. Root cause analysis of the performance parameter is shown at the end of the literature.

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Dedication

Dedicated to my mom and dad for their continuous support and standing up by my side

throughout my education.

सरस्वति नमस्िुभ्यं वरदे काम셂तिति ।

तवद्यारम्भं कररष्यातम तसतिभभविु मे सदा ॥

ज्ञानं िरम ं बलम ्

Acknowledgments

Thanks and acknowledgments are owed to several people and institutions, without whom this research would not have been possible. I would especially like to thank my advisor,

Dr. Shawn Midlam-Mohler. He provided me with the opportunity and support to conduct my graduate research at the Simulation, Innovation and Modeling Center (SIMCenter).

I would also like to thank my other committee members, Dr. Marcello Canova, Dr. Punit

Tulpule and Dr. Raju Dantuluri for their assistance throughout the project. Dr. Tulpule for help in the uncertainty analysis, Dr. Dantuluri for help with GT-Power models and analysis, and all other affiliates and individuals at the SIMCenter who made this project possible.

Finally, I would like to thank my father and mother, who supported me entirely throughout my study at The Ohio State University. It would not have been possible without their constant help.

Vita

October 1992 Born – Modasa, Gujarat, India

May 2014 B.E. (Hons.) Mechanical Engineering, BITS Pilani

July 2014 – June 2016 Senior Executive, Fiat India Automotives Ltd.

Jan 2017 - Present Graduate Research Associate, The Ohio State University SIMCenter

Publications

Kavathia, Kepin, Manoj Settipalli, and Samikkannu Raja. "Active-Passive Damping

Characteristics of Extension and Shear Mode Piezoelectric Fiber Nano-Reinforced

Composite (PFNRC)." ASME 2017 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2017.

Fields of Study

Major Field: Mechanical Engineering

Table of Contents

Abstract ...... ii Dedication ...... iv Acknowledgments...... v Vita ...... vi List of Tables ...... x List of Figures ...... xi Chapter 1. Introduction ...... 1 1.1 Motivation ...... 1 1.2 Current Practice ...... 2 1.3 Research Objective ...... 2 1.4 Approach ...... 3 1.5 Thesis Outline ...... 4 Chapter 2. Literature Review ...... 6 2.1 Introduction ...... 6 2.2 Theory of Uncertainty Analysis ...... 6 2.3 Air-Fuel Ratio ...... 7 2.4 Determination of TDC ...... 8 2.5 Pegging Method for Cylinder Pressure ...... 11 Chapter 3. Tools, Method and Experimental Setup ...... 16 3.1 Student’s t-distribution...... 16 3.2 Design-Stage Uncertainty Analysis ...... 19 3.3 Advanced-Stage Uncertainty Analysis ...... 21 3.3.1 Zero-Order Uncertainty ...... 21 3.3.2 First-Order Uncertainty ...... 22 3.3.3 Mth-Order Uncertainty ...... 23 3.4 Multiple-Measurement Uncertainty Analysis ...... 23

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3.5 Uncertainty Analysis: Error Propagation of the Performance Parameters ...... 25 3.5.1 Propagation of Error ...... 26 3.5.2 Sequential Perturbation ...... 27 3.6 Correction for Correlated Errors ...... 29 3.7 Test Cell of an Engine ...... 30 3.8 GT-Power Model and Simulations ...... 31 Chapter 4. Uncertainty in Air-Fuel Ratio ...... 34 4.1 Holl’s Algorithm ...... 34 4.2. Heywood’s Algorithm ...... 36 4.3 Brettschneider-Spindt Algorithm ...... 40 4.4 Results ...... 43 Chapter 5. Uncertainty in Air Flow Rate Estimation ...... 49 5.1 Uncertainty in Environmental Conditions...... 49 5.2 Uncertainty in Density of Air ...... 55 5.3 Calculation of Air Flow Rate ...... 57 5.4 Results ...... 58 Chapter 6. Uncertainty in IMEP ...... 64 6.1 Uncertainty due to an In-cylinder Pressure Measuring System ...... 66 6.2 Uncertainty in TDC Determination ...... 69 6.3 Uncertainty due to Misalignment of the Angle Encoder ...... 71 6.4 Uncertainty in Compression Ratio ...... 72 6.5 Results ...... 74 Chapter 7. Uncertainty in Other Performance Parameters ...... 77 7.1 Uncertainty in Engine ...... 77 7.2 Uncertainty in Torque ...... 80 7.3 Uncertainty in BMEP ...... 82 7.4 Uncertainty in BSFC ...... 85 7.5 Uncertainty in Power ...... 89 7.6 Uncertainty in SAE Corrected Factor ...... 93 7.6.1 Uncertainty in SAE Correction Factor ...... 93 7.6.2 Uncertainty in SAE Corrected Torque ...... 94 7.6.3 Uncertainty in SAE Corrected Power ...... 98

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Chapter 8. Conclusion ...... 102 8.1 Conclusion ...... 102 8.2 Future Work ...... 104 Bibliography ...... 105 Appendix A. Landscaped Page ...... 107

List of Tables

Table 1. Uncertainty in IMEP due to pressure measuring system ...... 69

Table 2. Uncertainty in IMEP due to an error in TDC determination ...... 70

Table 3. Uncertainty in IMEP due to misalignment of the angle encoder ...... 71

Table 4. Uncertainty in IMEP due to uncertainty in compression ratio ...... 73

List of Figures

Figure 1. Sketch of the arbitrary points for calculating the pressure difference [7] ...... 9

Figure 2. The pressure difference map from the simulated pressure diagrams of GT-

Power model [7] ...... 10

Figure 3. Effect of ±0.1 bar absolute pressure errors [9] ...... 12

Figure 4. Estimated polytropic coefficient [10] ...... 14

Figure 5. Sensor offset calculation error [10] ...... 14

Figure 6. Student's t-distribution coefficient against the degree of freedom and confidence interval ...... 18

Figure 7. Confidence interval against a standard normal distribution ...... 19

Figure 8. Flow diagram for DOE simulations ...... 31

Figure 9. DOE setup for simulating the results of the test cell ...... 33

Figure 10. Equivalence ratio predicted by different algorithms ...... 44

Figure 11. Uncertainty in AFR due to different algorithms...... 45

Figure 12. Random uncertainty in AFR ...... 46

Figure 13. Total uncertainty in AFR ...... 47

Figure 14. Uncertainty due to different factors of AFR...... 48

Figure 15. Uncertainty in atmospheric pressure ...... 50

Figure 16. Uncertainty in dry temperature ...... 51

Figure 17. Uncertainty in relative humidity ...... 52

Figure 18. Uncertainty in wet temperature ...... 54

Figure 19. Uncertainty in the density of air ...... 56

Figure 20. Random uncertainty in fuel flow rate ...... 59

Figure 21. Total uncertainty in fuel flow rate ...... 60

Figure 22. Systematic uncertainty in air flow rate ...... 61

Figure 23. Random uncertainty in air flow rate ...... 62

Figure 24. Total uncertainty in air flow rate ...... 63

Figure 25. Uncertainty calculation for IMEP ...... 65

Figure 26.P-V diagram showing noise at different frequencies ...... 66

Figure 27. Flow diagram for calculating uncertainty at a particular frequency ...... 67

Figure 28. Flow Diagram for calculating uncertainty at a particular engine speed ...... 68

Figure 29. Uncertainty in IMEP...... 74

Figure 30. Random uncertainty in IMEP ...... 75

Figure 31. Systematic uncertainty in engine speed ...... 78

Figure 32. Random uncertainty in engine speed ...... 79

Figure 33. Total uncertainty in engine speed ...... 80

Figure 34. Random uncertainty in engine torque...... 81

Figure 35. Total uncertainty in engine torque ...... 82

Figure 36. Random uncertainty in BMEP...... 84

Figure 37. Total uncertainty in BMEP ...... 85

Figure 38. Systematic uncertainty in BSFC ...... 87

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Figure 39. Random uncertainty in BSFC ...... 88

Figure 40. Total uncertainty in BSFC ...... 89

Figure 41. Systematic uncertainty in power generated by the engine ...... 90

Figure 42. Random uncertainty in power generated by the engine ...... 91

Figure 43. Total uncertainty in power generated by the engine ...... 92

Figure 44. Uncertainty in the correction factor ...... 94

Figure 45. Systematic uncertainty in SAE corrected torque ...... 95

Figure 46. Random uncertainty in corrected torque ...... 96

Figure 47. Total uncertainty in corrected torque ...... 97

Figure 48. Systematic uncertainty in corrected power...... 98

Figure 49. Random uncertainty in corrected power ...... 99

Figure 50. Total uncertainty in corrected power ...... 100

Figure 51. Root cause analysis of air flow rate ...... 103

Figure 52. Flow diagram of the engine test cell ...... 108

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Chapter 1. Introduction

1.1 Motivation

An automotive vehicle needs to withstand years of the duty cycle, making the auto researchers study and examine ways of extending the life of a vehicle. Durability test is therefore important as it can enable researchers to understand the behavior of the vehicle in a long-term point of view. When we talk about the durability test of an engine, automakers generally run tests for months, unattended by any person. Hence, the system needs to have a self-check to assess if the tests are running as per the requirement. This requires error bound of the system to be known, which can be calculated from the uncertainties inherent to the system.

There are always uncertainties in the data collected from the test cell of an engine. This uncertainty can be attributed to a number of different factors like environmental factors, sensor measurement noise, human error, approximations, etc. Unaccounted uncertainties can cause unexplained variation in performance parameters when multiple tests are run.

The accurate knowledge of the uncertainty of different performance parameters is important to understand the variation between different test runs. This analysis also provides a means to understand the root cause of an anomaly in the data.

1.2 Current Practice

The durability tests generally run for months without any human intervention. Once tests are completed, an engineer analyzes the tests and brings in the results. At times, it can happen that the tests did not run according to the test guides for the reasons unknown.

Since there are no individual checks for the performance metrics going out of the bound, the system does not give any alert. The main reason behind no such checks being in place is the unavailability of the error bounds for different measured variables and performance parameters.

1.3 Research Objective

The high-level objective of this study is to improve understanding of how different anomalies from nominal (or expected) operating conditions affect the functioning of an engine test cell during durability tests. In this research, we compute the exhaustive error bounds of the system’s performance parameters originating from various sources of errors.

To understand the error band, the sensitivity of the performance parameters for the engine in the test cell is identified. It is important to understand the uncertainty bounds specified in specification sheets of different engine and test cell components. Specification sheets for sensors provide the values of systematic uncertainty in the measured variable.

This research also focuses on analyzing the data collected from the test cell to calculate the random uncertainty in the measured variables. The last part of the research will deal with determining the impact of uncertainties in each measured variable on the performance parameters of the engine.

1.4 Approach

Initially, the important measurements and sources of uncertainties that affect each measurement are identified. This includes the uncertainty from the engine, test cell, ECU, and facility control. The uncertainty in measurement variables like speed, torque, exhaust gases, fuel flow rate, in-cylinder pressure, manifold pressure, crank angle, ambient temperature and pressure and the dry temperature is calculated through specification sheets of the sensors used in the test cell. The uncertainty was defined as twice of the standard deviation of the sample taken.

A detailed uncertainty analysis is done for performance parameters like Brake Mean

Effective Pressure (BMEP), Brake Specific Fuel Consumption (BSFC), Air-Fuel Ratio

(AFR), Indicative Mean Effective Pressure (IMEP) and air flow rate into the engine. The symbolic computation is used to calculate the uncertainty in the performance parameters using the uncertainty calculated for the measured variables.

There are certain factors like air temperature and humidity, etc., whose impact on the performance parameters are difficult to calculate through symbolic analysis since analytically modeling the relationship is challenging. As a direct relation between them is not available numerical analysis approach is used to understand the role of these variables on the performance parameters of the engine. A calibrated GT-Power model of the engine is used to generate the data required to compute uncertainties numerically. This part of the calculation gives the systematic uncertainty of the parameters.

The random uncertainty of the variables can be calculated using the data from the test cell.

Student’s t-distribution is used to calculate the random uncertainty in the variables. Adding up the random and systematic uncertainty gives us the overall uncertainty of a variable due to all the factors in the test cell. The uncertainty analysis is further used to develop root cause analysis of the errors in the performance parameters.

1.5 Thesis Outline

The subsequent portion of the thesis is organized as follows:

 Chapter 2 introduces a broad literature review on the relation of uncertainty in some

measured variables due to the performance parameter of the engine. It is used to

establish a relationship between the some of the measured variables and

performance parameters.

 Chapter 3 describes the propagation of error and sequential perturbation theory used

to calculate the uncertainty in performance parameters. It also describes the

student’s t-distribution to calculate the random uncertainty of a variable, strategy

for finding the error bounds and the test cell used for obtaining the experimental

data.

 Chapter 4 describes uncertainty in the air-fuel ratio by using three different

algorithms for calculating the air-fuel ratio.

 Chapter 5 describes uncertainty in the measurement of air flow rate calculated using

the air-fuel ratio, fuel flow rate, and air density.

 Chapter 6 describes uncertainty in the indicative system i.e. IMEP.

 Chapter 7 describes uncertainty in parameters like BMEP, BSFC, power, SAE

corrected power and SAE corrected torque.

 Chapter 8 is the concluding portion, highlighting the accomplishments and the

challenges faced during the project.

Chapter 2. Literature Review

2.1 Introduction

The literature review covers the topic in error bound and uncertainty analysis of a gasoline engine and models used to predict such factors. The focus of the literature review is in to identify the methods used to calibrate various factors for the accurate running of the engine like the determination of TDC, calculate AFR and pegging of the in-cylinder pressure transducer. This section is divided into four sections covering the theory of uncertainty analysis, determination of air-fuel ratio, determination of AFR and pegging method for cylinder pressure. The goal of the literature review is to find measured variables that affect the performance parameters and to identify any relationship between them.

2.2 Theory of Uncertainty Analysis

Uncertainty analysis is an important aspect as whenever a test is conducted, we need to know details about the quality of the results. Uncertainty analysis allows us to estimate the error bar around the value obtained during the tests. The theory of uncertainty analysis is explained in the previous literature [1]. It first explains the measurement errors, mainly systematic and random error.

The author also explains different levels of uncertainties, starting with design-state uncertainty, then to first-order uncertainty and lastly Mth order uncertainty analysis. There

6 are 2 ways of finding the error – numerically and symbolically. The book explains both the methods and describes the propagation of elemental errors to the results.

2.3 Air-Fuel Ratio

AFR is an important measurement which could be translated into engine fuel efficiency and emissions. Too lean or too rich mixture may result in an exhaust outside the permitted limits by the government agency EPA. One way of calculating AFR is by measuring the air flow rate through MAF (Manifold Air Flow) sensor and fuel flow rate. Air flow rate sensor isn’t sensitive to the direction of the flow of air. When a reversion pulse comes back out through the MAF sensor, this air movement is still measured. This causes an error in

AFR as well as running the engine at the desired conditions.

Another way of determining AFR is through direct measurement of important exhaust gas species. The disadvantage of using this method is that past literature does not agree on a single relation between the exhaust gases and AFR when the mole fraction of each species of the exhaust gas is known.

There have been various attempts at calculating AFR from the measured exhaust gas species. One such attempt was using CO2, CO and unburnt hydrocarbons [2]. This analysis assumes that there is no presence of oxygen in the fuel. The fuel only consists of hydrogen and carbon atoms. It also assumes that the leftover hydrocarbons has the same H/C ratio as that of the fuel. It also assumes the water-gas equilibrium coefficient to be constant. As this method did not use oxygen balance for the calculation, it is suspected that AFR estimation may be off by some margin.

Another attempt to calculate AFR was with 5 exhaust gases - CO2, CO, hydrocarbons, O2 and NOx. The literature attempted to use carbon, oxygen as well as hydrogen balance to calculate AFR [3]. The author still took the assumptions of leftover hydrocarbons to have the same H/C ratio as that of the fuel and water-gas equilibrium coefficient to be constant.

The author noted that varying the water-gas equilibrium coefficient from 1.5 to 5.5 had only 2 percent effect on the result. It was noted that the value of the coefficient generally lies between 3.2 and 3.8 [4]. Similar attempts were made in other literature [5][6]. They also used oxygen, carbon, and hydrogen balance to calculate the AFR from exhaust gases.

They took the similar assumption of water-gas shift and constant H/C ratio.

2.4 Determination of TDC

The determination of the top dead center (TDC) is particularly important while calculating the IMEP and other parameters that depend on the measured in-cylinder pressure diagram.

Literature indicative that 1°CA error in TDC position can cause 5-8% error in the indicated pressure and 10% deviation in the accumulative heat release rate [7]. The author al devised a simulation technique for determination of TDC by finding the angle between TDC and maximum pressure in the cylinder called loss angle. Because of the heat release losses and gas leakage from the cylinder, the maximum pressure point occurs before the TDC, instead of at the TDC.

A mathematical model of pressure change of compression and expansion process during the engine’s motoring was devised. The paper calculates the equivalence of the heat transfer and gas leakage’s effects the maximal pressure point position relative to TDC. It

8 proves that the effect of gas leakage is the same as the heat transfer from the engine cylinder. The unsymmetrical property of the pressure diagram on each side of the maximal pressure point is applied to determine the TDC [7]. Two points are arbitrarily selected on the pressure curve such that they are symmetrical to the maximum pressure point on the curve.

Figure 1. Sketch of the arbitrary points for calculating the pressure difference [7]

The pressure difference between the points a1 and a2 is directly linked to amount of heat transfer and gas leakage from the engine. A GT-Power model is used to obtain different

9 pressure vs. crank angle graphs under different heat transfer rate. So, at different heat transfer rate, the difference between maximal pressure point and TDC can be established.

Figure 2. The pressure difference map from the simulated pressure diagrams of GT- Power model [7]

The simulated pressure difference is compared with the experimental pressure difference.

The simulated curve that comes closest to the experimental pressure difference determines the combined effect of heat release and gas leakage. Vipre model is also used to check the certainty of the result. There is an error of 0.05°CA between the two models.

Another approach for determination of TDC is using a thermodynamic method, whose input is in-cylinder pressure sampled during the compression and expansion strokes [8].

The base theory for the method is finding the loss angle, the angle difference between the location of peak pressure (LPP) and TDC. The loss angle is derived through loss function.

2 휇 1 훿퐹 휗 = [ ] (1) 푙표푠푠 휌 − 1 휇 + 1 푐 훿휗 푝 퐿푃푃

This function is used to calculate the loss angle for a given engine. The author also puts the method into testing by thermodynamic simulations. The simulations show that the method proposed is intrinsically robust towards the entity of both heat transfer and mass leakage because it weighs up the effect of both losses in two particular crank positions and then estimates the entity of the two losses at the peak pressure position, which in turn allows evaluating the loss angle [8].

2.5 Pegging Method for Cylinder Pressure

Pegging method for the in-cylinder pressure sensor is important because the in-cylinder pressure transducer gives the relative pressure to the manifold pressure. To find the absolute pressure of the cylinder, the pressure needs a reference pressure, hence needs pegging. There are various ways of pegging the sensor but due to wave dynamics in the intake manifold, often there can be errors. Figure 3 shows error in the estimation of the polytropic coefficient of the process happening after the valves of the cylinders is closed.

Figure 3. Effect of ±0.1 bar absolute pressure errors [9]

The pegging can be done using the inlet pressure referencing, outlet pressure referencing, two-points referencing, three-points referencing, least squares method and the modified least square method [10] [11]. Inlet pressure referencing sets the reference pressure at

IBDC (Intake Bottom Dead Center) while outlet pressure referencing set it as exhaust back pressure during the exhaust stroke. However, these two reference techniques are limited to low engine speed and are prone to error at high engine speed [10] [11]. Reference pressure at IBDC works best the engine runs at low engine speeds while exhaust back pressure works best for part-load conditions [12].

Two-point referencing method assumes compression of the charge (air + fuel) is a polytropic process with a fixed polytropic coefficient. The sensor output at two points 12 during the compression is taken and the sensor offset is calculated. In the three-point referencing, the fixed polytropic coefficient is replaced with a variable one. This method is analyzed because two-point referencing suffers accuracy due to the assumption of the fixed polytropic coefficient.

Least-Square Method (LSM) uses several measurements and regression calculations to calculate the sensor offset. Again, an assumption of the fixed polytropic coefficient is taken while modified LSM does not make this assumption. Figure 4 shows the simulated and estimated polytropic coefficient of the system. Modified LSM shows much better estimation when compared to the 2-point method. Figure 5 shows the error in sensor offset for the two-point method, the three-point method, LSM and modified LSM. While the two- point method and LSM have a sinusoidal behavior over the cycles in error in estimation of sensor offset, modified LSM has the least error in the estimation.

Figure 4. Estimated polytropic coefficient [10]

Figure 5. Sensor offset calculation error [10]

Two other methods of pegging the pressure transducer are inlet manifold pressure referencing (IMPR) and polytropic index pressure referencing (PIPR) [9]. IMPR relies on getting accurate manifold pressure during BDC of the intake stroke and assuming the cylinder pressure is equal to the inlet manifold pressure. The data show that changing the position for referencing produces a sizeable change in the absolute pressure. The main concern with IMPR is that the referencing is carried out at the lowest pressure point in the cycle and linearity errors and signal noise are very large mainly at part load conditions.

Increase in the speed can again increase the shock waves in the manifold, causing inaccurate readings.

PIPR is basically a two-point referencing method. It uses a constant polytropic index to define the relationship 푃푉훾 when the valves of the cylinder are closed, where 푃 is the pressure inside the cylinder and 푉 is the instantaneous volume of the cylinder. This method is found to be affected by signal noise and generally produces higher cyclic variability.

Chapter 3. Tools, Method and Experimental Setup

The statistical tools and procedures used throughout the literature are presented in this chapter. This chapter also gives a short introduction to the experimental setup for collecting the data. A brief overview of Student’s t-distribution is supplied with the details about the uncertainty theory used in the literature [1]. The test cell setup for the experiments and the model developed of the test cell using the commercial software GT-Power by Gamma

Technologies is described in brief. Lastly, the methodology used to compute the systematic and random uncertainty is explained in brief.

3.1 Student’s t-distribution

Student’s t-distribution is a continuous probability distribution that is used when the sample size of the experiment is small, and the population standard deviation is unknown.

Student’s t-distribution is used in the analysis here because the data set that can be obtained from the experiment is limited i.e., the sample size is small. The standard deviation of the population is not known. This makes Student’s t-distribution the best choice to estimate the mean and sample standard deviation of the system.

Let 푥1, 푥2, … . , 푥푛 be a sample of size 푛 from a normally distributed population. Then

2 sample mean value (푥̅), the sample variance (푠푥 ) and the sample standard deviation (푠푥) is defined by

푁 1 푥̅ = ∑ 푥 (2) 푁 푖 푖=1

푁 1 푠2 = ∑(푥 − 푥̅)2 (3) 푥 푁 − 1 푖 푖=1

푁 1/2 1 푠 = ( ∑(푥 − 푥̅)2) (4) 푥 푁 − 1 푖 푖=1

The sample mean provides the most probable estimate of the true mean value of a measured variable and sample variance provides a probable estimate of the variation that could potentially be found in the dataset of the experiment. The degree of freedom, 휈 in a statistical value, equal to the number of data points minus the number of previously determined statistical parameters used in estimating that value [1]. Hence, for calculating the sample variance, the degree of freedom in 푁 − 1.

If we consider that the population is normally distributed, we can assume that a measured variable is normally distributed about the sample mean and we can statistically state that

푥푖 = 푥̅ ± 푡휈,푃푠푥 (5) where the variable 푡휈,푃 is Student’s t-distribution coefficient which depends on the degree of freedom of the sample variance and the confidence interval that we are targeting.

Basically, the interval ±푡휈,푃푠푥 represents a precision interval at the confidence of P%.

Figure 6. Student's t-distribution coefficient against the degree of freedom and confidence interval

Figure 6 shows the coefficient of the t-distribution against the degree of freedom and the confidence interval selected. The black dotted line represents the value of the degree of freedom after which the value of the coefficient becomes more or less constant. Carrying out experiments to increase the degree of freedom and decrease the value of the coefficient may not be practical. Figure 7 shows the range for 50, 90, 95 and 99% confidence interval.

Figure 7. Confidence interval against a standard normal distribution

3.2 Design-Stage Uncertainty Analysis

Design-stage uncertainty analysis presents only a minimum estimate of the uncertainty of the system, based on the method of measurement and instruments chosen. This type of analysis is used when the project is in the design phase when system data is not available yet. It is useful for selecting instruments and selecting measurement techniques. In the design stage, differentiating between systematic and random errors is too difficult. A systematic uncertainty is one of the primary uncertainty which remains constant when multiple measurements are taken at a given operating point. A systematic uncertainty is 19 generally an offset from the true value due to the instruments used in the experiment’s measurements. A random uncertainty causes the measured data to be scattered when repeated measurements are made at a given operating point. Random uncertainty can be caused due to lack of repeatability, resolution of the measuring instruments or change in the environmental conditions.

Principally, u0 is a rudimentary estimate of the random uncertainty due to instrument resolution.

1 푢 = ± 푟푒푠표푙푢푡𝑖표푛 (95%) (6) 0 2 This value represents approximately 95% confidence interval of the uncertainty. The other piece of information that is initially available during the design phase is the manufacturer’s specification sheet concerning instrument error. This is considered as the systematic uncertainty, 푢푐 of that particular measured variable. This is the uncertainty caused by the instruments used.

Measurement uncertainty could be attributed to multiple sources of measurement errors.

Each individual measurement error interacts with other errors to affect the uncertainty of a measurement. This is called uncertainty propagation. Each individual error is called an elemental error [1].

퐾 2 (7) 푢푐 = √∑ 푢푘 푘=1

The root sum squared (RSS) method of combining uncertainties assumes that the square of an uncertainty is a measure of the variance (i.e., s2) assigned to an error, and the propagation of these variances yields a probable estimate of the total uncertainty [1]. The design stage uncertainty, 푢푑 is defined as,

2 2 푢푑 = √푢0 + 푢푐 (8)

This provides minimum value for design stage uncertainty. 푢0 is the zero-order uncertainty.

3.3 Advanced-Stage Uncertainty Analysis

Design-stage uncertainty accounts for resolution of the measurement system and instrument errors but cannot account for many other uncertainties. Design-state uncertainty, for example, cannot give an estimate of the error that can be caused due to change in the environmental conditions. Additional information about the system can give us a better estimate of the uncertainty of the variables. This additional come from the data collected from the system. This requires a thorough understanding of the system, mainly the correlation between different measured variables and calculated parameters.

Advanced-stage uncertainty analysis allows to take procedural and test control errors that can potentially affect the measurement of the desired variables. The goal is to find total uncertainty using the measurements available from the system.

3.3.1 Zero-Order Uncertainty

At zero-order uncertainty, every variable and parameter that can potentially affect the measurement is considered to be fixed, including time. Any chance of data measured is

21 considered to be the result of instrument resolution used to measure a particular variable.

Equation (6) gives the estimate of the extent that variable is expected to vary.

3.3.2 First-Order Uncertainty

First-order uncertainty is generally the effect of time on the measurement of that particular variable. It’s basically setting up the test equipment, then letting the test set up a run without changing any factor or variable of the test. A variation could probably be observed, which is due to external factors affecting process control or it may be an inherent behavior of the system that is being analyzed.

In simple terms, the uncertainty at first level would be estimated for a particular variable by running the test cell at a single operating condition. Typically the operating point would be within the range of operating conditions where the test cell is actually run. The first- order uncertainty can be expressed as –

√ 2 2 푢푑 = 푢1 + 푢푐 (9) where,

푢 = 푡 푠 1 휈,푃 푥̅ (10)

푠푥 푠푥̅ = √푁 (11)

The uncertainty calculated by 푢1 includes the uncertainty due to the resolution of the instrument that is calculated by 푢0. The first-order uncertainty is generally inadequate for reporting the complete uncertainty of test results.

3.3.3 Mth-Order Uncertainty

With increasing in the order of the uncertainty calculation, we can increase the number of factors analyzed. A higher number of factors considered can give a higher and a more realistic estimate of the uncertainty of the system. Mth-order uncertainty depends on the feasibility of the test operating conditions and the variability of more measured variables

[1]. The drawback of this calculation is that it requires increasingly more tests to be conducted and data to be collected to get a higher order of uncertainty.

th The M -order uncertainty 푢푀 is given by –

1 푁−1 2 2 2 푢푀 = [푢푐 + (∑ 푢푖 )] (12) 1

3.4 Multiple-Measurement Uncertainty Analysis

Often in the case of higher-order uncertainty, there will be multiple sets of measurement that will be taken to evaluate the uncertainty in the measured variable. The procedure to calculate the uncertainty when there are such multiple measurements is discussed in the section.

The procedure to identify the uncertainty starts with identifying the elemental errors in the described measurement. Next, we need to estimate the magnitude of systematic and random error uncertainty for each of these elemental errors.

The total uncertainty in a measurement due to K random errors (elemental errors) if the measured variable is given by the measurement of random standard uncertainty, 푠푥̅, as estimated by

1 2 2 2 푠푥̅ = [(푠푥̅)1 + (푠푥̅)2 + ⋯ + (푠푥̅)퐾]2 (13) where,

푠푥푘 (푠푥̅)푘 = √푁푘

This random uncertainty represents a measurement uncertainty due to the elemental errors affecting the variability of the measured variable at a confidence interval of one standard deviation. The degrees of freedom, n, in the standard random uncertainty, is estimated using the Welch–Satterthwaite formula [1]

퐾 2 2 (∑푘=1(푠푥̅ )푘) 휈 = 4 퐾 (푠푥̅ )푘 ∑푘=1 ( ) 휈푘 where,

휈푘 = 푁푘 − 1

The uncertainty due to K systematic errors from the elemental errors of the measured variable is given by the measurement of systematic standard uncertainty, 푏푥̅, as estimated by

1 2 2 2 푏푥̅ = [(푏푥̅)1 + (푏푥̅)2 + ⋯ + (푏푥̅)퐾]2 (14)

The confidence interval for the calculated random and systematic uncertainty is one standard deviation. To calculate the total uncertainty at another confidence interval, we can use an appropriate t value which depends on the confidence interval needed and the degrees of freedom for the system. This general form of the equation is –

푢 = 푡 (푏2 + 푠2)1/2 푥 휈,푃 푥̅ 푥̅ (15) where,

퐾 2 2 2 (∑푘=1(푠푥̅ )푘 + (푏푥̅ )푘) 휈 = 4 4 퐾 (푠푥̅ )푘 퐾 (푏푥̅ )푘 ∑푘=1 ( ) + ∑푘=1 ( ) 휈푘 휈푘

3.5 Uncertainty Analysis: Error Propagation of the Performance Parameters

This section develops a method of estimating the uncertainty in a performance parameter based on a set of measured variables obtained under fixed operating condition. It will help us calculate the uncertainty of performance parameters that are dependent on measured variables. There are mainly two methods of calculating the error propagation to the performance parameters. One of the methods is a symbolic analysis, namely propagation

25 of error, used when the measured variables and performance parameter have a definite relationship that defines the impact on each other. The second is a numerical analysis, named sequential perturbation, used when the exact relationship between the measured variables and the performance parameter is not known.

3.5.1 Propagation of Error

This method is used when there is a definite relationship between the performance parameter y and a measured variable x. First of all, we measure the measured variable x for a certain number of times at a particular operating condition. This helps in calculating its sample mean and the uncertainty due to random error in the mean value, 푡휈,푃푠푥̅, also written as 푡푠푥̅ . Hence, for the measured value of x, y can be defined as –

푦̅ ± 훿푦 = 푓(푥̅ ± 푡 푠 ) 휈,푃 푥̅ (16)

By Taylor series,

2 푑푓(푥) 1 푑 푓(푥̅) 2 푦̅ ± 훿푦 = 푓(푥̅) ± [( ) 푡 푠 + ( ) (푡 푠 ) + ⋯ ] 푑푥 휈,푃 푥̅ 2 푑푥2 휈,푃 푥̅ (17) 푥=푥̅ 푥=푥̅

A linear approximation can be made as 푡휈,푃푠푥̅ is small

푑푓(푥) 훿푦 ≈ ( ) 푡휈,푃푠푥̅ (18) 푑푥 푥=푥̅ 푑푓(푥) ∴ 푢푦 = ( ) 푢푥 (19) 푑푥 푥=푥̅ This concept can be used for multivariable relationships –

푅 = 푓(푥1, 푥2, … , 푥퐿) (20) where L is the number of independent variables that R depends on. Hence, the best estimate of the true value of 푅, 푅’ is –

′ 푅 = 푅̅ + 푢푅 (21)

퐿 1/2 2 푢 = [∑(휃 푢 ) ] 푅 푖 푥̅푖 (22) 푖=1 where,

휕푅 휃푖 = 𝑖 = 1,2, … , 퐿 휕푥푖

This sensitivity can also be estimated numerically using finite differencing methods. The index is evaluated using either the mean values or the expected nominal values of the variables.

3.5.2 Sequential Perturbation

Sequential perturbation is a numerical approach used to estimate the uncertainty through a result that avoids the direct requirement of analytical relations. The approach is used when the data is already available in a discrete form.

The method uses a finite difference method to approximate the derivatives:

1. Based on a fixed operating condition and fixed measured independent variables,

calculate 푅0 = 푓(푥1, 푥2, 푥3, … , 푥퐿) 27

2. Increase the independently measured variables by their respective uncertainties

calculated through the specification sheets and recalculate the result based on each of

these new values.

+ 푅1 = 푓(푥1 + 푢푥1, 푥2, 푥3, … , 푥퐿)

+ 푅2 = 푓(푥1, 푥2 + 푢푥2, 푥3, … , 푥퐿)

+ (23) 푅퐿 = 푓(푥1, 푥2, 푥3, … , 푥퐿 + 푢푥퐿)

3. Decrease the independently measured variables by their respective uncertainties

calculated through the specification sheets and recalculate the result based on each of

these new values.

4. Calculate the difference.

+ + 휕푅푖 = 푅푖 − 푅0

− − (24) 휕푅푖 = 푅푖 − 푅0

5. Take an average of the two values calculated in the above steps.

휕푅+ − 휕푅− 휕푅 = 푖 푖 ≈ 휃 푢 푖 2 푖 푖 (25)

퐿 1/2 2 푢푅 = [∑ 휕푅푖 ] (26) 푖=1

3.6 Correction for Correlated Errors

In the previous section, we assumed that all the measured variables that a performance parameter depends on are independent of each other and their elemental errors are therefore not dependent on each other. This may not be true in all of the cases of calculating the uncertainty of the performance parameters. If any of the two errors are dependent on each other, they are said to be correlated with each other.

One example of the correlation is when two different measured variables are measured by the same instrument. In this case, the systematic uncertainty of those variables is correlated with each other. The total systematic uncertainty for the performance parameter is given by –

1/2 퐿 퐿−1 퐿 2 푏 = (∑(휃 푏 ) + 2 ∑ ∑ 휃 휃 푏 ) 푅 푖 푥̅푖 푖 푗 푥̅푖푥̅푗 (27) 푖=1 푖=1 푗=푖+1 where,

퐻

푏푥̅ 푥̅ = ∑(푏푥̅ ) (푏푥̅ ) 푖 푗 푖 ℎ 푗 ℎ ℎ=1 where 퐻 is the total number of elemental errors that are correlated between the measured variables 푥푖 and 푥푗.

3.7 Test Cell of an Engine

The analysis done in this literature is that of a 4-cylinder SI engine with turbocharger. One of the components that test cell of the engine has is an air handling unit. The air handling provides the manifold of the engine with the air at a controlled temperature. Before the manifold, the test cell has an air filter, air mass flow sensor, air pressure sensor and sensor to measure relative humidity and dry temperature. The mass flow rate sensor also measures the temperature of the air.

A throttle valve controls the amount of air into the system. Subsequently, the air enters the manifold of the engine. The manifold of the engine consists of a manifold air pressure sensor (MAP). The compressor of the turbocharger compresses the air to increase the volumetric efficiency and the intercooler cools the air after being compressed by the compressor. The fuel is mixed with fuel through gasoline direct injection (GDI).

The dynamometer controls the speed that the engine runs at. It also uses a load cell sensor to measure the torque produced by the engine. The engine has variable cam timing, which acts as an internal exhaust gas recirculation (EGR). The test cell has a coolant conditioning system to maintain the temperature of the water that is used as a coolant for the engine.

Test cell also has oil cooler to maintain the temperature of oil in the oil sump used for lubrication.

An underground tank lets the fuel through a fuel inlet module, into the mass flow meter sensor of fuel. The fuel then passes through fuel temperature control to maintain the

30 temperature of the fuel. The fuel then passes through fuel temperature and pressure sensor into the fuel rail of the engine.

After combustion, a small part is sent to the soot filter, through to the water trap and chiller, into the exhaust gas analyzer. The rest of the exhaust gas is sent to the turbine of the turbocharger, then to the catalytic converter. A lambda sensor is installed before the turbocharger to measure how rich or lean is the mixture. While air flow rate sensor works as feedforward for the fuel to be injected, lambda sensor works as the feedback sensor for the amount of the fuel to be injected.

3.8 GT-Power Model and Simulations

A GT-Power model is developed, which is calibrated against the actual 4-cylinder gasoline engine. This model is used to collect the data for finding the random uncertainty in the measured variables and performance parameters. The simulations are run for varying the dry temperature of the air, engine speed and torque produced by the engine.

Figure 8. Flow diagram for DOE simulations

Figure 8. Flow diagram for DOE simulations shows the summary of the calculation of the simulations. Design of Experiment (DOE) is set up against the varying environmental conditions and different gravity points. DOE is used to evaluate the random uncertainty in the variables and parameters. The simulations are run at 5 different dry air temperatures.

To simulate the real environmental conditions, random Gaussian numbers are added to the dry temperature. 10 such runs are done for each of the five values of dry temperature. These simulations are done for 13 unique values of engine speed and torque. This document refers to results against the operating conditions 1 to 13. Point to note is that engine speed increases from 1st to 13th while the torque demanded decreases.

These simulations give random uncertainty. Adding them with systematic uncertainty gives the final uncertainty in the parameter. The analysis is a 1st order uncertainty analysis with operating condition consisting of air temperature, engine speed and torque produced by the engine.

Figure 9. DOE setup for simulating the results of the test cell

Figure 9 explains the 1st order uncertainty analysis used to calculate the random uncertainty in measured variables, while specification sheets are used to calculate the systematic uncertainty. The uncertainty in performance parameters is calculated through the propagation of error.

Chapter 4. Uncertainty in Air-Fuel Ratio

Uncertainty in AFR is difficult to find as the exact relationship between the exhaust gas species namely, CO2, CO, total hydrocarbons (THC), O2 and NOx, and AFR is not known.

There have been several attempts at establishing a relationship between these exhaust gases and AFR. Three such attempts will be discussed in this literature.

Uncertainty in the AFR can be due to changes in the exhaust gases and the composition of the fuel. The exhaust gases out of the engine can change every cycle even with the same operating conditions as the pressure and combustion inside the cylinder changes every cycle. So, the change in the exhaust gases is one of the two factors that cause the uncertainty in the AFR. The other factor being the uncertainty in the ratio of hydrogen to carbon and oxygen to carbon in the fuel. The reason behind inaccurate H/C and O/C ratio could be the uncertainty in the actual calculation of these ratios as well as human error in entering the correct ratios into the test cell system when the tests are being carried out.

4.1 Holl’s Algorithm

First of the three methods of calculating the AFR uses only the reading of CO2, CO and

THC of all the exhaust gases readings [2]. This method is useful when the NOx and O2 readings are not available. The formula for calculating the AFR with the discussed method is –

28.97 퐴퐹푅 = 퐻 푂 12.01 + 1.008 ( ) + 16 ( ) 퐶 푟푎푡푖표 퐶 푟푎푡푖표 퐻 [ ] 0.5[퐶푂] (1 − 0.5 (퐶) ) 푇퐻퐶 (1 − − 푟푎푡푖표 ) 100 106

∗ [퐶푂] [퐶푂 ] [푇퐻퐶] + 2 + 100 100 106

[ 퐻 ( ) (28) 퐶 푟푎푡푖표 3.8[퐶푂2] 2[퐶푂] 4 ( 100 − 100 ) + 3.8[퐶푂2] [퐶푂] 100 + 100 퐻 (퐶) 3.8[퐶푂 ] 3[푇퐻퐶] ( 푟푎푡푖표 ∗ 2 ∗ ) 4 100 106 푂 (퐶) − − 푟푎푡푖표 3.8[퐶푂 ] [퐶푂] [퐶푂] [퐶푂 ] [푇퐻퐶] 2 ( 2 + ) ( + 2 + ) 100 100 100 100 106

] where,

[퐶푂2] is the concentration of CO2 in the exhaust in percentage

[퐶푂] is the concentration of CO in the exhaust in percentage

[푇퐻퐶] is the concentration of THC in the exhaust in ppm

퐻 ( ) is the ratio of hydrogen to carbon in the fuel used for combustion 퐶 푟푎푡푖표

푂 ( ) is the ratio of oxygen to carbon in the fuel used for combustion 퐶 푟푎푡푖표

The overall uncertainty due to elemental error uncertainty is again calculated via systematic and random uncertainty. The systematic uncertainty is calculated through the specification

35 sheet of the exhaust bench used to calculate the exhaust gases out of the engine. Random uncertainty needs to be calculated in the same way as the random uncertainty of the variables in the air flow rate is calculated. Using the residual sum of squares (RSS), the

systematic and random uncertainty gives us the value of 푢퐶푂, 푢퐶푂2 and 푢푇퐻퐶. Uncertainty in AFR is –

휕퐴퐹푅 2 휕퐴퐹푅 2 휕퐴퐹푅 2 푢퐴퐹푅 = (( 푢퐶푂) + ( 푢퐶푂2) + ( 푢푇퐻퐶) 휕[퐶푂] 휕[퐶푂2] 휕[푇퐻퐶] (29) 1/2 휕퐴퐹푅 2 휕퐴퐹푅 2 + ( 푢 ) + ( 푢 ) ) 휕[퐻/퐶] 퐻/퐶 휕[푂/퐶] 푂/퐶

4.2. Heywood’s Algorithm

The second method for calculating AFR uses all the five readings of the exhaust gases from the emission bench. This estimate is more accurate in the literature as it takes care of the oxygen in the fuel as with O2 and NOx readings, oxygen balance calculation is possible [3].

There are two basic assumptions of this method –

1. It uses water-gas equilibrium to estimate the amount of water in the exhaust gas. The

value of the constant, K is assumed to be a constant at 3.5.

2. It assumes that the ratio of carbon to hydrogen and carbon to oxygen in the exhaust gas

is the same as the fuel.

The formulation for estimating the AFR is described here. The equation (30) shows the estimated reaction between the fuel and air inside the engine cylinder.

푛푂 퐶 퐻 푂 + 2 (푂 + 3.773푁 ) 푛 푚 푟 휙 2 2 (30) = 푛푝(푥̃푇퐻퐶퐶푛퐻푚 + 푥̃퐶푂퐶푂 + 푥̃퐶푂2퐶푂2 + 푥̃푂2푂2 + 푥̃푁2푁2

+ 푥̃푁푂푁푂 + 푥̃퐻2푂퐻2푂 + 푥̃퐻2퐻2) where,

푥̃ is the mole fraction of that particular compound

The number of moles of oxygen required is –

푦 푧 푛 = 푥 + − (31) 푂2 4 2

Carbon balance –

푛 = 푛푝(푛푥̃푇퐻퐶 + 푥̃퐶푂 + 푥̃퐶푂2) (32)

Hydrogen balance –

푚 = 푛푝(푚푥̃푇퐻퐶 + 2푥̃퐻2푂 + 2푥̃퐻2) (33)

Oxygen balance –

2푛푂 푟 + 2 = 푛 (푥̃ + 2푥̃ + 푥̃ + 2푥̃ + 푥̃ ) (34) 휙 푝 퐶푂 퐶푂2 푁푂 푂2 퐻2푂

Nitrogen balance –

7.546푛푂 2 = 푛 (2푥̃ + 푥̃ ) (35) 휙 푝 푁2 푁푂

Mole fraction adds up to 1:

푥̃푇퐻퐶 + 푥̃퐶푂 + 푥̃퐶푂2 + 푥̃푂2 + 푥̃푁2 + 푥̃푁푂 + 푥̃퐻2푂 + 푥̃퐻2 = 1 (36)

Assuming a constant water-gas constant –

푥̃퐶푂푥̃퐻 푂 2 = 퐾 (37) 푥̃퐶푂2푥̃퐻2

The equivalence ratio of the fuel is calculated as –

2푛푂 휙 = 2 ∗ ∗ ∗ ∗ ∗ (38) 푛푝푥̃퐻2푂 + 푛푝(1 − 푥̃퐻2푂)(푥̃ 퐶푂 + 2푥̃ 퐶푂2 + 2푥̃ 푂2 + 푥̃ 푁푂 + 2푥̃ 푁푂2) − 푟 where,

푛 푛푝 = ∗ ∗ 푥̃푇퐻퐶 + (1 − 푥̃퐻2푂)(푥̃ 퐶푂 + 푥̃ 퐶푂2 )

Please note that the 푥̃∗ denotes the mole fraction of that compound in dry conditions. It can be

found by dividing the mole fraction by (1 − 푥̃퐻2푂).

The stoichiometric air-fuel ratio needs to be calculated in order to get the uncertainty in the

AFR. The H/C and O/C ratio will affect the stoichiometric AFR of the fuel.

Let’s assume that there is a complete combustion of the fuel via the reaction –

푥 푦 푥 퐶퐻 푂 + ( − + 1) 푂 → 퐶푂 + 퐻 푂 (39) 푥 푦 4 2 2 2 2 2

1 mole of fuel weighs –

12.01 + 1.008푥 + 16푦

while 1 mole of O2 weighs 32.

Hence, the oxygen-fuel mass ratio is –

푥 푦 32 ( − + 1) ∗ 4 2 12.01 + 1.008푥 + 16푦

As 100 kg of air will have 23.16 kg of oxygen,

푥 푦 32 100 퐴퐹푅 = ( − + 1) ∗ ∗ 푠푡표푖 4 2 12.01 + 1.008푥 + 16푦 23.16

퐴퐹푅 퐴퐹푅 = 푠푡표푖 (40) 휙

The overall uncertainty due to elemental error uncertainty is calculated via systematic and random uncertainty. The systematic uncertainty is calculated through the specification sheet of the exhaust bench used to calculate the exhaust gases out of the engine. Random uncertainty needs to be calculated in the same way as the random uncertainty of the variables in the air flow rate is calculated. Using the RSS, the systematic and random

uncertainty gives us the value of 푢퐶푂, 푢퐶푂2, 푢푇퐻퐶, 푢푂2, 푢푁푂, 푢퐻/퐶 and 푢푂/퐶. Uncertainty in AFR is –

휕퐴퐹푅 2 휕퐴퐹푅 2 휕퐴퐹푅 2 푢퐴퐹푅 = (( 푢퐶푂) + ( 푢퐶푂2) + ( 푢푇퐻퐶) 휕[퐶푂] 휕[퐶푂2] 휕[푇퐻퐶]

휕퐴퐹푅 2 휕퐴퐹푅 2 휕퐴퐹푅 2 + ( 푢푂2) + ( 푢푁푂) + ( 푢퐻/퐶) (41) 휕[푂2] 휕[푁푂] 휕[퐻/퐶]

1 휕퐴퐹푅 2 2 + ( 푢 ) ) 휕[푂/퐶] 푂/퐶

4.3 Brettschneider-Spindt Algorithm

The Brettschneider-Spindt algorithm is an iterative method to calculate the AFR using the readings of five exhaust gases [6] [5].

The basic assumption of this method is –

3. It uses water-gas equilibrium to estimate the amount of water in the exhaust gas. The

value of the constant, K is assumed to be a constant at 3.5.

4. It assumes that the ratio of carbon to hydrogen and carbon to oxygen in the exhaust gas

is the same as the fuel.

5. We here assume the water cooler temperature to be 5°C.

The equation (42) shows the estimated reaction between the fuel and air inside the engine cylinder.

퐶푥퐻푦푂푧 + 푛(푂2 + 퐴 푁2 + 퐵 퐶푂2 + 퐶퐻푎푏푠 퐻2푂)

→ 푎 퐶푂2 + 푏 퐶푂 + 푐 퐻2 + 푑 퐻2푂 + 푒 푂2 + 푓 푁2 (42)

+ 푔 푁푂푥 + ℎ 퐶푥푝퐻푦푝푂푧푝

1 퐶표표푙푒푟 퐻 푂 = (((((0.000000004182푇 + 0.000000033206)푇 2 101.3 푐 푐

+ 0.0000352248)푇푐 + 0.001244) 푇푐 + 0.0459058) 푇푐 (43)

+ 0.6048346)

where,

푇푐 is the analyzer cooler temperature in deg. C

푁2 퐴퐴 = 푎푚푏 푂2푎푚푏

퐶푂2 퐵퐵 = 푎푚푏 푂 2푎푚푏 (44) 0.00160757 ∗ 퐻 ∗ 100 퐶퐶 = 푎푏푠 푂2푎푚푏

푥 + 퐵퐵 ∗ 푛 푛푡표푡푎푙 = (퐶푂2 + 퐶푂 + 푇퐻퐶)(1 − 퐶표표푙푒푟 퐻2푂)

The concentration of the emissions is in mole fractions

퐶푂2 푎 = 푛푡표푡푎푙 1 − 퐶표표푙푒푟 퐻2푂

퐶푂 (45) 푏 = 푛푡표푡푎푙 1 − 퐶표표푙푒푟 퐻2푂

푇퐻퐶 ℎ = 푛푡표푡푎푙 1 − 퐶표표푙푒푟 퐻2푂

푁푂푥 푔 = 푛푡표푡푎푙 1 − 퐶표표푙푒푟 퐻2푂

푦 − 푦푝 ∗ ℎ 푐 = + 푛 ∗ 퐶퐶 − 푑 2

푂2 푒 = 푛푡표푡푎푙 1 − 퐶표표푙푒푟 퐻2푂

푦 − 푦푝 ∗ ℎ + 2 ∗ 푛 ∗ 퐶퐶 푑 = 푏 2 (푎퐾 + 1)

푓 = 푛 ∗ 퐴퐴 − 푔/2 where,

퐾 is the water gas equilibrium constant

The total number of moles is given by –

2푎 + 푏 + 푑 + 2푒 + 푔 + 푧푝 ∗ ℎ − 푧 푛 = (46) 2 + 퐶퐶 + 2 ∗ 퐵퐵

The calculated n is compared with the n calculated from the previous iteration. If the difference in the value is more than 10-6, a new iteration loop would run with the new value of n calculated by the above equation.

By the mass balance of the moles,

푛 푦 푧 (푥 + − ) 100 푛푂 4 2 28.97 퐴퐹푅 = 2 ∗ (47) 푂2푎푚푏 12.011푥 + 1.008푦 + 15.9994푧

This method uses the similar approve as that of Heywood’s. The value of AFR calculated through this method matches with the value of AFR calculated from Heywood’s method

42 to the second decimal. Hence, we don’t calculate the uncertainty again. The uncertainty in the AFR is the same as the one calculated by Heywood’s method.

4.4 Results

First, we need to check the how effectively does each algorithm predict the AFR of the combustion in the engine cylinder. The data for the exhaust gases were obtained from the test cell of the engine. Only one reading for each operating point is available. It is known that GT-Power is poor with the estimation of the exhaust gases. As a good estimate of exhaust gases are not available for different environmental conditions, the uncertainty due to the environmental conditions is not considered.

퐴퐹푅 퐸푞푢𝑖푣푎푙푒푛푐푒 푟푎푡𝑖표, 휙 = 푠푡표푖 (48) 퐴퐹푅

Figure 10. Equivalence ratio predicted by different algorithms

It is clear from Figure 10 that Heywood’s and Brettschneider-Spindt’s algorithm agrees with each other, with very slight variation in the 휙, while Holl’s algorithm has a significant deviation from the other two algorithms. The reason behind the deviation is because Holl’s algorithm ignores the readings from O2 and NOx.

Figure 11 shows the systematic uncertainty due to exhaust gases, H/C, and O/C ratio. The uncertainty in Brettschneider-Spindt’s algorithm is assumed to be same as that of

Heywood’s as the value and the logic used behind finding the AFR is the same. It is also assumed that the systematic uncertainty calculated from the test cell of the engine will be

44 the same if systematic uncertainty was calculated from the GT-Power data if there was a way to calculate the exhaust gases efficiently.

Figure 11. Uncertainty in AFR due to different algorithms

Though the uncertainty in less in case of Holl’s algorithm than Heywood’s, Holl’s algorithm suffers from the wrong prediction of the AFR. For further calculations of the performance parameters, only the AFR and uncertainty in AFR from Heywood’s algorithm is considered. The uncertainty in AFR increases with engine speed. The main reason behind

45 increasing uncertainty is the rich mixture of the charge inside the cylinder. We can see from the Figure 10 that with increasing speed, the engine is running at a richer mixture. Though the value of uncertainty almost remains the same, due to a decrease in the value of AFR, the uncertainty in terms of percentage goes up.

Figure 12. Random uncertainty in AFR

The random uncertainty in calculated through the data obtained from the GT-Power model.

We see that the uncertainty decreases with the increase in speed and increases with the

46 increase in the air temperature. The increase in uncertainty with temperature can be attributed to the increase in random uncertainty in fuel flow rate and the air inside the cylinder. We will see in the next chapter that the uncertainty of the density of air increases with an increase in air temperature. As the uncertainty of the density of air increases, the uncertainty in air and fuel inside the cylinder increases. This causes an increase in uncertainty in AFR.

Figure 13. Total uncertainty in AFR

Figure 13 shows the total uncertainty in the AFR estimation. We see a maximum of 2.78% error and a minimum of 2.68% error in the estimation of the AFR.

Figure 14. Uncertainty due to different factors of AFR

Figure 14 shows the individual contributions from different factors of AFR. It is clear that the graph that H/C and O/C ratios contribute more than 80% of the uncertainty in AFR.

Chapter 5. Uncertainty in Air Flow Rate Estimation

This chapter focuses on understanding the uncertainty in air flow rate in the engine’s cylinders. Air flow rate is an important parameter as it determines the efficiency of the engine. IC engine runs on the combustion of the mixture of fuel and air in the cylinder chamber. Having the right proportion of the air into the fuel is very important for efficient burning and ensuring exhaust gases are within the norms as regulated by the government agency.

5.1 Uncertainty in Environmental Conditions

This section discusses the uncertainty in dry temperature, wet temperature, atmospheric pressure and relative humidity of the air. These are the factors that have an influence on factors like the density of air, which in turn decides the air flow rate in the manifold of the engine. To simulate the random uncertainty, random numbers with normal distribution are added to the dry air temperature, atmospheric pressure and relative humidity of the air. The systematic uncertainty of these factors was calculated through the specification sheet of the sensors used to calculate these factor.

The wet temperature is calculated from dry temperature, atmospheric pressure, and relative humidity. The algorithm and the uncertainty calculation is discussed in this section.

Figure 15. Uncertainty in atmospheric pressure

Figure 15 shows the uncertainty in atmospheric pressure. As the atmospheric pressure does not depend on the operating condition of the engine, the uncertainty in atmospheric pressure is plotted against the normalized dry temperature of the air. Three curves show the systematic, random and total uncertainty in the atmospheric pressure estimation. The systematic uncertainty is a constant value of 0.4% while random uncertainty is due to the random number added to the pressure.

Figure 16. Uncertainty in dry temperature

The systematic uncertainty in air temperature is a constant value. Hence, with an increase in temperature, the percent uncertainty decreases as seen in Figure 16. Similarly, random numbers added to the pressure have a smaller effect as the atmospheric pressure increases.

This effect can be seen in the random uncertainty except for the first point.

Figure 17. Uncertainty in relative humidity

The uncertainty in relative humidity completely based on the random number generated through normal distribution. As the humidity is not varied with temperature and randomness is added to simulate the actual test cell conditions. Hence, we see the randomness in the uncertainty of the relative humidity.

The wet temperature of the air is calculated using the dry temperature, atmospheric pressure, and relative humidity of the air.

푇푑푟푦 17.27 퐸푠 = 6.108 ∗ 푒 237.3+푇푑푟푦 (49) 푅퐻 퐸 = 퐸푠 ∗ 100 where,

푅퐻 is the relative humidity of the air

Initially, we randomly estimate the wet temperature of the air. Let’s call it 푥1. We apply the Newton-Raphson method to estimate a new value of the wet temperature of the air, let’s call it 푥2. If the difference between 푥1 and 푥2 is more than 0.01, we reassign the value of 푥1 as current 푥2 and recalculate the value of 푥2. We continue the iteration till the difference goes below 0.01.

17.27푥1 237.3+푥 푓1 = 6.108푒 1 − 0.00066푃푎푡푚 ∗ 7.500616 ∗ 1.33324

∗ ((1 + 0.00115푥1)(푇퐷푟푦 − 푥1)) − 퐸

17.27푥 17.27(237.3 + 푥1) 1 237.3+푥1 푓2 = 2 6.108푒 − 0.00066푃푎푡푚 ∗ 7.500616 (50) (237.3 + 푥1)

∗ 1.33324 ∗ 0.00115푥1(푇퐷푟푦 − 푥1) − (1 + 0.00115푥1)

푓1 푥 = 푥 − 2 1 푓2

푓2 is differentiation of the function, 푓1. The overall uncertainty in wet temperature is –

2 2 2 휕(푓1) 휕(푓1) 휕(푓1) 푢푇푊푒푡 = (( 푢푇퐷푟푦) + ( 푢푃푎푡푚 ) + ( 푢푅퐻) 휕푇푑푟푦 휕푃푎푡푚 휕푅퐻 (51) 1/2 휕(푓1) 휕(푓1) + 2 푢푇퐷푟푦 푢푅퐻) 휕푇푑푟푦 휕푅퐻

Figure 18. Uncertainty in wet temperature

The uncertainty in the wet temperature of the air depends on the uncertainties in dry temperature, relative humidity, and atmospheric pressure. The uncertainty in wet

54 temperature varies from 0.29% to 0.25% as the dry temperature of the air increases. As the uncertainty in dry temperature and atmospheric pressure decreases with an increase in dry temperature, the uncertainty in wet temperature also decreases with an increase in air temperature.

5.2 Uncertainty in Density of Air

The density of the air is calculated through the dry, wet temperature and the atmospheric pressure of the air.

1.188푃푎푡푚 293 푃푤 휌퐴푖푟 = 1.1014 (1 − ) (52) 760 273 + 푇푑푟푦 푃푎푡푚 where,

푃푎푡푚 is the atmospheric pressure (mm Hg)

푇푑푟푦 is the dry temperature of the air (°C)

푃푤 is water pressure of the air (mm Hg)

2252000 푃 푃 = − 0.5(푇 − 푇 ) 푎푡푚 푤 (푇 −210)2 374−푇 퐷푟푦 푊푒푡 7.21523+ 푤푒푡 ∗ 푊푒푡 755 13.595푒 83324+53.5426(273.15+푇푊푒푡) 273.15+푇푊푒푡 where,

푇푊푒푡 is the wet temperature of the air (°C)

The uncertainty in the density of the air can be calculated in the same way as the overall uncertainty in the air flow rate is calculated.

2 휕휌 2 휕휌 휕휌 2 (53) √ 퐴푖푟 퐴푖푟 퐴푖푟 푢휌퐴푖푟 = ( 푢푃푎푡푚 ) + ( 푢푇퐷푟푦) + ( 푢푇푊푒푡 ) 휕푃푎푡푚 휕푇퐷푟푦 휕푇푊푒푡

Figure 19. Uncertainty in the density of air

Figure 19 shows uncertainty in the density of the air. We see that the dry and wet temperature terms are in the denominator in the formula of the density of air. Hence, we

56 see the opposite trend to that of dry temperature. As the temperature increases, the uncertainty in the density of the air also increases.

5.3 Calculation of Air Flow Rate

Air flow rate is one of the most important performance parameters when it comes to testing an engine in the test cell. The air flow rate sensor in the intake manifold of the engine is not accurate, as discussed earlier. Due to this inaccuracy in measurement through the manifold air flow rate sensor, the air flow rate estimation is done differently. The air flow rate is estimated using the calculated AFR and the measured fuel flow rate. Fuel flow rate sensor is used as the accuracy of the sensor is high with uncertainty in the measurement to be generally less than even 0.5%. Air flow rate in l/s is given by –

퐴퐹푅 ∗ 푚̇ 푓푢푒푙 푄퐴푖푟 = (54) 휌퐴푖푟 where,

푚̇ 푓푢푒푙 is the fuel flow rate (g/s)

휌퐴푖푟 is the density of air (g/l)

The uncertainty in the air flow rate dictated by the uncertainty in the 3 variables, air-fuel ratio (AFR), fuel mass flow rate and density of air.

푚̇ 2 퐴퐹푅 2 퐴퐹푅 ∗ 푚̇ 2 √ 푓푢푒푙 푓푢푒푙 (55) 푢푄퐴푖푟 = ( 푢퐴퐹푅) + ( 푢푚̇ 푓푢푒푙) + ( 2 푢휌퐴푖푟) 휌퐴푖푟 휌퐴푖푟 휌퐴푖푟

57 where,

푢퐴퐹푅 is the overall uncertainty in the calculation of AFR calculation

푢푚̇ 푓푢푒푙 is the overall uncertainty in the fuel flow rate

푢휌퐴푖푟 is the overall uncertainty in the calculation of density of the air

5.4 Results

The uncertainty in AFR and density of air was calculated in the previous sections. Fuel flow rate is a measured variable whose elemental error uncertainty will be divided into systematic and random error. The systematic error of the fuel is calculated using the specification sheet of the sensor used in the test cell to calculate the fuel flow rate. The systematic uncertainty is 0.1% for all varying conditions. The random uncertainty is calculated for first-level uncertainty.

Figure 20. Random uncertainty in fuel flow rate

Uncertainty in the fuel flow rate is plotted against the operating conditions and varying air temperature. The random uncertainty increases with the increase in the temperature. The uncertainty in the density of the air increases with the increase in air temperature. As the air temperature changes, it changes the density of the air, hence, amount of the air in the cylinder. The oxygen sensor of the engine then adjusts the fuel flow into the engine to maintain the engine speed and the torque produced, causing the random uncertainty in the fuel flow rate.

Figure 21. Total uncertainty in fuel flow rate

The total uncertainty is caused due to instruments used to measure the fuel flow rate

(systematic uncertainty) and change in the environmental condition (random uncertainty).

The uncertainty varies from 0.1% at low temperature to 0.22% at high temperature.

Figure 22. Systematic uncertainty in air flow rate

The systematic uncertainty in air flow rate increase with an increase in engine speed and increase in air temperature. The uncertainty increases with the increase in engine speed as the uncertainty in AFR increases with an increase in engine speed. The uncertainty increases with increasing air temperature as uncertainty in density in air and fuel flow rate increases. As AFR is the dominant factor, the systematic uncertainty follows the trend of the AFR.

Figure 23. Random uncertainty in air flow rate

The random uncertainty in air flow rate increases with the decrease in engine speed and increase air temperature. The uncertainty decreases with the increase in engine speed as the uncertainty in AFR decreases with increase in engine speed. The uncertainty increases with increasing air temperature as uncertainty in density in air and fuel flow rate increases.

Figure 24. Total uncertainty in air flow rate

The uncertainty in air flow rate into the engine depends on the uncertainty in AFR, fuel flow rate, atmospheric pressure, dry and wet temperature. Uncertainty in air flow rate varies from 2.81% to 2.97% as the engine speed and air temperature increases.

Chapter 6. Uncertainty in IMEP

Indicative Mean Effective Pressure (IMEP) is an important parameter as it indicates the effective pressure generated in the cylinder of an engine. The effective pressure basically indicates the energy generated inside the cylinder. IMEP is one of the parameters used by the automakers to evaluate the engine’s performance and it is also used to calculate the efficiency of the engine. As it is an important performance parameter, calculating the uncertainty in IMEP is important.

IMEP is calculated by equation (56) in kPa

360 ∫ 푃푑푉 퐼푀퐸푃 = −360 (56) 푉푑 where,

푃 is the pressure reading from the pressure transducer (kPa)

3 푉푑 is the swept volume of the cylinder (m )

Uncertainty in IMEP can be caused due to multiple reasons listed below –

1. Uncertainty in the indicative system (in-cylinder pressure measuring system)

2. Uncertainty in the Top Dead Center (TDC) determination

3. Uncertainty in angle encoder due to eccentric mounting, crankshaft deflection and

vibration (Misalignment of the angle encoder)

4. Uncertainty in compression ratio

Figure 25. Uncertainty calculation for IMEP

The figure shows the approach to calculating the uncertainty in the IMEP of the engine.

This approach adopted because the calculation of IMEP consists of an integral, hence calculation of uncertainty is not as straightforward as for the other performance parameter.

The grey part of the diagram shows the systematic part of the uncertainty while red shows the random part of the uncertainty.

6.1 Uncertainty due to an In-cylinder Pressure Measuring System

IMEP involves an integral over the cycle, which means that the uncertainty depends on the frequency and phase of the noise in the pressure signal. The figure illustrates how the noise frequency affects uncertainty in IMEP. At low frequencies, the area of the P-V curve varies depending on the phase. At very high frequencies, the averaging process should remove the effect of noise.

Figure 26.P-V diagram showing noise at different frequencies

In order to understand the effect of noise frequency in the pressure signal on uncertainty in

IMEP, the uncertainty is computed at a range of frequencies. This will give the uncertainty due to the sensor noise as against standard deviation in calculated IMEP over many cycles, which gives us the combined uncertainty due to sensor noise and running engine.

Figure 27. Flow diagram for calculating uncertainty at a particular frequency

The magnitude of sensor noise was found from the FFT (Fast Fourier Transform) analysis of the pressure readings from the non-firing non-running cylinder. The uncertainty propagation from pressure to IMEP is frequency dependent (as seen in the previous graphs), and the amplitude of sensor noise is also frequency dependent, but in order to obtain a single number representing uncertainty, a weighted average of the uncertainties over the frequency range is taken.

Since the unit of the magnitude of the non-firing non-running cylinder data is not exactly known, the magnitude of the data was used as weights to compute the weighted average to calculate the uncertainty at the particular engine speed. Figure 28 explains the algorithm developed.

Figure 28. Flow Diagram for calculating uncertainty at a particular engine speed

Table 1 shows the value of error in IMEP, gross IMEP and pumping IMEP of the engine.

Gross IMEP Pumping IMEP error Speed (RPM) IMEP error (%) error (%) (%) 0.15 1.42 0.98 75.75 0.22 1.70 1.14 76.60 0.29 1.55 0.94 62.01 0.37 1.59 1.05 52.28 0.44 1.58 1.25 48.38 0.51 1.66 1.09 43.95 0.59 1.83 1.07 31.43 0.66 1.87 1.10 24.82 0.68 1.91 1.16 23.33 0.69 1.95 1.14 22.25 0.71 1.90 1.07 20.75 0.72 1.86 1.06 20.04 0.74 1.92 1.06 22.32 0.74 1.79 1.00 19.96 0.81 1.73 1.00 15.58 0.81 1.74 1.02 15.01 0.88 1.80 1.07 12.67 0.90 1.77 1.06 11.93 0.91 1.78 1.12 11.80 0.93 1.85 1.09 12.03 0.94 1.85 1.11 12.01 0.96 1.93 1.18 12.80 1.00 1.97 1.18 13.70

Table 1. Uncertainty in IMEP due to pressure measuring system

6.2 Uncertainty in TDC Determination

It is known to the automakers that the exact determination when the piston is at TDC is of crucial importance for IMEP analysis. According to the existing literature [8], the uncertainty in the loss angle method used to determine the TDC is ±0.1 CA. If there is an error in TDC alignment, the whole of the integration of the pressure will be shifted by a

69 value of the angle. Hence, by shifting the integral by 0.1 CA, we can find the error in the

IMEP calculation due to uncertainty in TDC determination. Table 2 shows the percentage error in IMEP calculation when the TDC determination is off by ±0.1°CA.

+0.1 CA error -0.1 CA error IMEP Gross Pumping IMEP Gross Pumping Normalized error IMEP IMEP error IMEP IMEP Speed (%) error (%) error (%) (%) error (%) error (%) 0.15 0.48 0.48 0.32 -0.21 -0.21 -0.18 0.22 0.48 0.48 0.16 -0.23 -0.23 -0.09 0.29 0.51 0.50 0.12 -0.27 -0.26 -0.06 0.37 0.52 0.51 0.07 -0.29 -0.28 -0.04 0.44 0.52 0.51 0.06 -0.29 -0.28 0.00 0.51 0.53 0.51 0.08 -0.30 -0.29 0.02 0.59 0.52 0.50 0.08 -0.29 -0.28 0.03 0.66 0.54 0.51 0.06 -0.31 -0.29 0.02 0.68 0.52 0.49 0.05 -0.29 -0.27 0.02 0.69 0.55 0.52 0.04 -0.32 -0.30 0.02 0.71 0.55 0.52 0.02 -0.32 -0.30 0.02 0.72 0.55 0.52 0.01 -0.32 -0.30 0.02 0.74 0.55 0.52 0.01 -0.32 -0.30 0.01 0.74 0.55 0.52 -0.01 -0.32 -0.30 0.02 0.81 0.58 0.54 0.00 -0.34 -0.32 0.00 0.81 0.53 0.49 -0.01 -0.29 -0.27 0.00 0.88 0.55 0.50 -0.07 -0.30 -0.27 0.02 0.90 0.56 0.50 -0.08 -0.31 -0.27 0.02 0.91 0.57 0.51 -0.09 -0.31 -0.28 0.03 0.93 0.58 0.51 -0.10 -0.32 -0.29 0.03 0.94 0.60 0.53 -0.12 -0.34 -0.31 0.03 0.96 0.58 0.51 -0.13 -0.33 -0.29 0.03 1.00 0.57 0.50 -0.19 -0.31 -0.28 0.04

Table 2. Uncertainty in IMEP due to an error in TDC determination

6.3 Uncertainty due to Misalignment of the Angle Encoder

Uncertainty in IMEP by angle encoder is due to eccentricity in the mounting of the device.

+0.03 CA error -0.03 CA error Gross Pumping Gross Pumping IMEP IMEP IMEP IMEP IMEP IMEP Speed (RPM) error error error error error error (%) (%) (%) (%) (%) (%) 0.15 0.2389 0.2379 0.1436 0.0313 0.0309 -0.0071 0.22 0.2346 0.2323 0.0680 0.0204 0.0200 -0.0070 0.29 0.2380 0.2350 0.0552 0.0048 0.0047 0.0006 0.37 0.2397 0.2355 0.0345 -0.0036 -0.0034 0.0024 0.44 0.2371 0.2323 0.0415 -0.0050 -0.0044 0.0214 0.51 0.2386 0.2332 0.0563 -0.0094 -0.0080 0.0377 0.59 0.2363 0.2296 0.0602 -0.0074 -0.0053 0.0457 0.66 0.2409 0.2312 0.0445 -0.0114 -0.0091 0.0346 0.68 0.2373 0.2270 0.0380 -0.0044 -0.0027 0.0297 0.69 0.2444 0.2330 0.0290 -0.0137 -0.0118 0.0232 0.71 0.2451 0.2323 0.0184 -0.0135 -0.0117 0.0183 0.72 0.2462 0.2325 0.0111 -0.0131 -0.0114 0.0149 0.74 0.2458 0.2329 0.0098 -0.0145 -0.0133 0.0078 0.74 0.2476 0.2330 0.0020 -0.0136 -0.0122 0.0095 0.81 0.2575 0.2387 -0.0019 -0.0203 -0.0188 -0.0006 0.81 0.2419 0.2237 -0.0046 -0.0020 -0.0021 -0.0034 0.88 0.2545 0.2277 -0.0370 -0.0030 -0.0038 -0.0118 0.90 0.2569 0.2283 -0.0428 -0.0026 -0.0035 -0.0120 0.91 0.2592 0.2299 -0.0475 -0.0054 -0.0061 -0.0129 0.93 0.2629 0.2317 -0.0558 -0.0074 -0.0083 -0.0164 0.94 0.2680 0.2357 -0.0643 -0.0138 -0.0144 -0.0197 0.96 0.2650 0.2322 -0.0751 -0.0088 -0.0103 -0.0248 1.00 0.2634 0.2275 -0.1053 -0.0029 -0.0062 -0.0371

Table 3. Uncertainty in IMEP due to misalignment of the angle encoder

Eccentric mounting of the shaft was calculated through the runout test, running the shaft at low angular speed. The specification sheet of the angle encoder was used to find the error in crank angle linked to the account of eccentricity of the shaft. Table 3 shows the uncertainty in IMEP due to eccentricity in the shaft of the mounting of the angle encoder.

6.4 Uncertainty in Compression Ratio

IMEP inversely depends on the swept volume. The volume of a cylinder can be divided into two parts – swept volume and clearance volume. As the compression ratio decreases, the clearance volume increases. The uncertainty in compression ratio changes the volume of the cylinder at an instant as well as changes the pressure inside the cylinder. We can assume that the heat released by the fuel at a given instant in the same. Heat release at a given instant is given by –

푑푄 1 푑푃 훾 푑푉 (57) = ( ) 푉 + ( ) 푃 푑휃 훾 − 1 푑휃 훾 − 1 푑휃

훾 is the heat capacity ratio (= 1.3)

The profile of heat release is obtained from the nominal compression ratio and the pressure readings from the engine test cell. Then we assume the compression ratio to be different.

The new volume is calculated by calculating the new clearance volume due to new compression ratio. The original equation of heat release rate is rearranged –

푑푃 훾 − 1 푑푄 훾 푑푉 = − 푃 (58) 푑휃 푉푛푒푤 푑휃 푉푛푒푤 푑휃

This equation is a differential equation of 1st order in in-cylinder pressure against the crank angle. Solving equation (58) gives new pressure values.

+delta in -delta in CR CR Speed IMEP IMEP (RPM) error (%) error (%)

1000 3.7741 0.2183 1500 3.8857 0.7570 2000 3.8244 0.3137 2500 3.8965 0.2486 3000 3.8036 0.4544 3500 3.9107 0.1645 4000 3.9552 0.6806 4500 3.5210 0.4076 4600 3.9557 0.6285 4700 4.1534 0.6433 4800 4.1959 0.3769 4900 4.1228 0.5597 5000 4.2660 0.3357 5000 4.2083 0.2712 5500 3.9341 0.7428 5500 4.0255 0.5645 6000 4.3306 0.8843 6100 4.6275 0.6161 6200 4.4802 0.0488 6300 4.6108 0.8905 6400 4.7721 0.9111 6500 4.4909 0.9063 6800 4.7255 0.9309

Table 4. Uncertainty in IMEP due to uncertainty in compression ratio

6.5 Results

The previous four sub-sections calculate the systematic uncertainty of IMEP due to different factors affecting the calculation. The calculated systematic uncertainty needs to be added to random uncertainty from the GT-Power data.

Figure 29. Uncertainty in IMEP

Figure 29 shows the comparison of uncertainties due to different factors. It is clear that an increase in compression ratio of the cylinder contributes to the maximum uncertainty, followed by the indicating system.

Figure 30. Random uncertainty in IMEP

Figure 30. Random uncertainty in IMEP shows the random uncertainty in IMEP that are obtained from the GT-Power simulations. The uncertainty increases with temperature as the uncertainty in torque produced increases with the temperature. Uncertainty in torque

75 shows more uncertain combustion in the engine at higher air temperature, which also affects the IMEP of the engine. Increase in uncertainty with increasing speed can again be attributed to the effect of the torque.

Chapter 7. Uncertainty in Other Performance Parameters

We have discussed two of the important parameters in the previous chapters, namely, air flow rate and IMEP of the engine. The other performance parameters like torque, Brake

Mean Effective Pressure (BMEP), Brake Specific Fuel Consumption (BSFC), Power, SAE corrected torque and SAE corrected power, are discussed in this section.

7.1 Uncertainty in Engine

The GT-Power model was run with the constraint of running at the desired engine speed mentioned by the user. A random number that is normally distributed is added to actual engine speed to simulate the random uncertainty of the actual test cell engine.

Figure 31. Systematic uncertainty in engine speed

Systematic uncertainty in speed is a constant value based on the specification sheet of the dynamometer. As it is a constant value, with an increase in the engine speed, the systematic uncertainty of the engine speed decreases. It is not affected by air temperature by any noticeable margin.

Figure 32. Random uncertainty in engine speed

The random uncertainty in the engine speed is because of the random number generated by a normal distribution, added to the actual engine speed. As the magnitude have the lesser impact of the percent of uncertainty in engine speed, random uncertainty in engine speed decreases with an increase in engine speed. Effect of air temperature on the uncertainty of air is completely due to random number generation.

Figure 33. Total uncertainty in engine speed

The uncertainty in the engine speed exponentially decreases as the engine speed increases.

The uncertainty decreases from 0.072% to 0.015% as speed increases.

7.2 Uncertainty in Torque

Torque is one of the parameters that an engine is marketed with, other being the power of the engine. Hence, it is important to calculate the uncertainty in torque. The uncertainty in torque is caused by systematic and random uncertainty. The systematic uncertainty is calculated from the specification while the random uncertainty is calculated from the GT-

Power simulations.

The systematic uncertainty in engine torque comes out to be a constant value across different operating conditions and temperature range the engine test cell is run at. The systematic uncertainty in torque is 0.15% for the test cell temperature of 298K.

Figure 34. Random uncertainty in engine torque

The random uncertainty increases with the increase in engine speed and increase in air temperature. As the random uncertainty in the air and fuel in the cylinder increase, the uncertainty in torque produced by the engine also increases.

Figure 35. Total uncertainty in engine torque

The total uncertainty is lowest at 0.16% for low speed and low temperature and highest at

0.5% for high speed and high temperature.

7.3 Uncertainty in BMEP

BMEP is important because it is needed in the calculation of the efficiency of the engine.

The ratio of BMEP over IMEP gives the efficiency of a given engine. BMEP of an engine depends on the brake torque produced.

4휋휏 퐵푀퐸푃 = (59) 푉푑 where,

휏 is the brake torque produced by the engine

Hence, the uncertainty in BMEP is given by –

1 2 휏 2 √ (60) 푢퐵푀퐸푃 = 4휋 ( 푢휏) + ( 2 푢푉푑) 푉푑 푉푑 where,

푢휏 is uncertainty in torque (%)

푢푉푑 is uncertainty in swept volume (%)

The swept volume of the cylinder generally does not change. Only the clearance volume has uncertainty. The systematic uncertainty in engine torque comes out to be a constant value across different operating conditions and temperature range the engine test cell is run at. The systematic uncertainty in torque is 0.15% for the test cell temperature of 298K.

Figure 36. Random uncertainty in BMEP

The random uncertainty in BMEP increases with an increase in engine speed and an increase in air temperature. As the random uncertainty in the torque produced by the engine increases, the uncertainty in BMEP of the engine also increases.

Figure 37. Total uncertainty in BMEP

The only contribution to the uncertainty of the BMEP is uncertainty in torque. Therefore, the uncertainty in BMEP follows the same trend as that of uncertainty in torque. The uncertainty is 0.16% at the lowest speed and air temperature and goes up to 0.5% at the highest speed and air temperature.

7.4 Uncertainty in BSFC

BSFC indicates the amount of the fuel used against the brake power generated in the combustion. Hence, it is a measure of fuel efficiency of the engine. The advantage of using

BSFC over actual efficiency is that the energy density of the fuel burned is not required.

BSFC is defined as –

푚̇ 푓푢푒푙 퐵푆퐹퐶 = (61) 휏휔 where,

휔 is the engine speed (rad/s)

Since we know the exact relation of BMEP with other variables, it is recommended to use the Propagation of Error method for calculation of uncertainty in BMEP.

2 2 2 1 푚̇ 푓푢푒푙 푚̇ 푓푢푒푙 푢 = √( 푢 ) + ( 푢 ) + ( 푢 ) (62) 퐵푆퐹퐶 휏휔 푚̇ 푓푢푒푙 휏2휔 휏 휏휔2 휔 where,

푢휔 is the uncertainty in engine speed

Figure 38. Systematic uncertainty in BSFC

Uncertainty in BSFC depends on the uncertainty in engine speed, the torque produced and fuel flow rate. As the systematic uncertainty in torque and fuel flow rate are a constant percent, systematic uncertainty in BSFC follows the trend of systematic uncertainty in engine speed. Hence, systematic uncertainty in BSFC decreases with increase in engine speed.

Figure 39. Random uncertainty in BSFC

The random uncertainty in engine speed is not affected by engine speed or air temperature while the random uncertainty in fuel flow rate is not affected by the engine speed. Hence, random uncertainty in BSFC mainly follows the trend of the random uncertainty in torque.

It increases with increase in engine speed and air temperature.

Figure 40. Total uncertainty in BSFC

Out of the three factors affecting the uncertainty in BSFC, uncertainty in torque is the most influential one due to its larger value in comparison to uncertainty in engine speed and fuel flow rate. Hence, uncertainty in BSFC behaves similarly to the uncertainty in torque. It goes from a maximum value of 1.35% to a minimum value of 0.3%.

7.5 Uncertainty in Power

Power is one of the two parameters used by the automakers to evaluate the engine. So, it becomes important to calculate the uncertainty in power. Power of an engine is defined as

–

푃 = 휏휔 (63)

The uncertainty in power is –

2 2 푢푃 = √(휔푢휏) + (휏푢휔) (64)

Figure 41. Systematic uncertainty in power generated by the engine

Uncertainty in power depends on the uncertainty in engine speed and the torque produced.

As the systematic uncertainty in torque is a constant percent, systematic uncertainty in

90 power follows the trend of systematic uncertainty in engine speed. Hence, systematic uncertainty in power decreases with increase in engine speed.

Figure 42. Random uncertainty in power generated by the engine

The random uncertainty in engine speed is not affected by engine speed or air temperature.

Hence, random uncertainty in power follows the trend of the random uncertainty in torque.

It increases with increase in engine speed and air temperature.

Figure 43. Total uncertainty in power generated by the engine

Again, out of the two factors affecting the uncertainty in power produced by the engine, uncertainty in torque is the most influential one due to its larger value in comparison to uncertainty in engine speed. Hence, uncertainty in power behaves similarly to the uncertainty in torque. It goes from 0.16% to 0.5% as the engine speed and air temperature increases.

7.6 Uncertainty in SAE Corrected Factor

7.6.1 Uncertainty in SAE Correction Factor

The performance of the engine is affected by the density of the air in the inlet of a cylinder as well as by the characteristics of the fuel. The performance is also affected by the atmospheric conditions of the test cell. The standard operating condition for the test cell in

100 kPa pressure for the inlet air, 99 kPa pressure for the dry air and inlet air temperature to be at 25 °C.

Since these conditions are not always possible, a correction factor is used to provide for a common basis of the comparison. This factor is generally applied to the torque and power for comparison. The correction factor is defined by (65).

2 99 (273.15 + 푇퐷푟푦) 퐶퐴 = ( ) (65) (푃푎푡푚 − 푃푤) 298

Section 5.2 shows us that water vapor pressure, 푃푤 is a function of atmospheric pressure, dry and wet temperature Wet temperature in turn depends on air temperature, atmospheric pressure and relative humidity. Since, the uncertainty in correction factor, CA is –

휕퐶퐴 2 휕퐶퐴 2 휕퐶퐴 2 푢퐶퐴 = (( 푢푃푎푡푚) + ( 푢푇퐷푟푦) + ( 푢푅퐻) 휕푃푎푡푚 휕푇퐷푟푦 휕푅퐻 (66) 1/2 휕퐶퐴 휕퐶퐴 + 2 푢푇퐷푟푦 푢푅퐻) 휕푇푑푟푦 휕푅퐻

Figure 44. Uncertainty in the correction factor

Uncertainty in SAE correction factor depends on the uncertainty in atmospheric pressure, dry and wet temperature. As the pressure in water vapor, which depends on wet temperature, is in the denominator, the trend is opposite to that of dry and wet temperature.

Hence, the uncertainty in SAE correction factor increase with an increase in dry air temperature.

7.6.2 Uncertainty in SAE Corrected Torque

The uncertainty in SAE corrected torque is defined as –

휏퐶퐴 = (1.176퐶퐴 − 0.176)휏 (67)

The uncertainty in torque is –

2 √ 2 (68) 푢휏퐶퐴 = ((1.176퐶퐴 − 0.176)푢휏) + (1.176휏 ∗ 푢퐶퐴)

Figure 45. Systematic uncertainty in SAE corrected torque

Systematic uncertainty in SAE corrected torque depends on systematic uncertainty in torque and correction factor. As systematic uncertainty in torque is a constant, the trend in the systematic uncertainty of corrected torque follows that of the correction factor.

Figure 46. Random uncertainty in corrected torque

Random uncertainty in corrected torque increases with the increase in air temperature while it remains almost constant with a change in engine speed. This is because random

96 uncertainty in correction factor and torque increases with an increase in air temperature while random uncertainty in torque changes in small value with changing engine speed.

Figure 47. Total uncertainty in corrected torque

As the uncertainty in correction factor remains constant with the change in engine speed, the uncertainty in SAE corrected torque follows a similar trend to that of torque while it follows the trend of correction factor with increasing air temperature. The uncertainty increases from 0.19% to 2.65% as the engine speed increases.

7.6.3 Uncertainty in SAE Corrected Power

SAE corrected power is defined as –

푃퐶퐴 = (1.176퐶퐴 − 0.176)푃 (69)

The uncertainty in power is –

2 √ 2 (70) 푢푃퐶퐴 = ((1.176퐶퐴 − 0.176)푢푃) + (1.176푃 ∗ 푢퐶퐴)

Figure 48. Systematic uncertainty in corrected power

Systematic uncertainty in SAE corrected power depends on systematic uncertainty in power and correction factor. As systematic uncertainty in power follows the trend of systematic uncertainty in speed but the value is small when compared to systematic uncertainty due to correction factor. Since the trend in the systematic uncertainty of corrected power follows that of the correction factor.

Figure 49. Random uncertainty in corrected power

Random uncertainty in corrected power increases with the increase in air temperature while it increases slightly with the change in engine speed. This is because random uncertainty in correction factor and power increases with an increase in air temperature while random uncertainty in power causes a small change with changing engine speed.

Figure 50. Total uncertainty in corrected power

The uncertainty in SAE corrected power depends on uncertainty in correction factor, engine speed and torque produced by the engine. Uncertainty in torque is the most

100 dominant factor with the increase in engine speed while SAE correction factor is the dominant factor when the uncertainty is compared against the air temperature. The uncertainty increases from 0.2% to 2.65% as the engine speed increases.

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Chapter 8. Conclusion

8.1 Conclusion

An uncertainty analysis for a complete engine’s test cell is presented for the understanding of the factors that affect performance parameters of an engine. The research initially used the relationship between the measured variables and performance parameters of an engine.

An extensive list of performance parameters is considered to calculate the error bound and equations were explicitly written down to relate uncertainty in a variable due to a parameter. In the case of AFR, where the exact relation between the exhaust gases and

AFR is not known, the available estimation algorithms in the existing literature are used to establish the relationship.

The systematic uncertainty of the variables was found from the specification sheets and manuals of the sensors used in the engine test cell. In order to compute random uncertainty, a full factorial DOE is used for variations in uncertain parameters. The DOE was executed using the GT-Power model. Propagation of error is used to calculate the uncertainty in each of the performance parameters with respect to the measured variables. It gives us good estimations of the variation that can be expected from the engine test cell, with the available sensors, actuators, and control strategy.

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Figure 51. Root cause analysis of air flow rate

The analysis can also be used for root cause analysis. Figure 51 shows the factors that the air flow rate of the engine depends on. From the analysis, we know that AFR is a major factor contributing to the uncertainty in air flow rate. The analysis of the AFR gives the maximum uncertainty due to H/C and O/C ratios. Hence, the major contributing factors to the uncertainty in air flow rate is H/C and O/C ratios.

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In the case of IMEP, the major factor is an increase in compression ratio, followed by an indicating system of the test cell, while torque is the biggest factor for BMEP, BSFC, and power, which in turn is caused due to varying environmental conditions.

As the error bound of the parameters are known, this also gives the idea about the possible variation that the test cell can show. This can be used to evaluate the design changes of the test cell as well as for engine. If any design change in the test cell or engine shows an improvement in a performance parameter less than the error bound i.e. 2 times the standard deviation, the improvement shown may not be an actual improvement. It might just be a random chance of getting the improved value of the performance parameter.

8.2 Future Work

The uncertainty analysis currently carried out is with respect to the data available with the

GT-Power model of a 4-cylinder engine. A detailed uncertainty analysis on the engine test bench data from yearlong tests can be performed. With the availability of the data from the test cell, a comparative study can be done between the simulations and test data to assess the magnitude of the performance of the GT-Power model.

This uncertainty analysis can further be used to determine the testing equipment or engine degradation over time. The data can also be used to carry out diagnostics of the test cell and coming up with replacement criteria for the sensors used in the test cell.

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Appendix A. Landscaped Page

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Figure 52. Flow diagram of the engine test cell 108