Introduction

Parametric tests are designed for analysing data from a known distribution, and most of these tests assume the distribution is normal. Although parametric tests are quite robust to departures from normality and a transformation can often be used to normalise data, there are some cases where the population is so grossly non-normal that parametric testing is unwise. In these cases, a powerful analysis can often still be done by using a non-parametric test.

Non-parametric tests are not just alternatives to the parametric procedures for analysing ratio, interval and ordinal data described in Chapters 9 to 18. Often life scientists measure data on a nominal scale. For example, Table 3.4 gave the numbers of basal cell carcinomas detected and removed from different areas of the human body. This is a sample containing frequencies in several discrete and mutually exclusive categories and there are non-parametric tests for analysing this type of data (Chapter 20).

Introduction

So far, this book has only covered tests for one and two samples. But often you are likely to have univariate data for three or more samples and need to test whether there are differences among them.

For example, you might have data for the concentration of cholesterol in the blood of seven adult humans in each of five different dietary treatments and need to test whether there are differences among these treatments. The null hypothesis is that these five treatment samples have come from the same population. You could test it by doing a lot of independent samples t tests (Chapter 9) to compare all of the possible pairs of means (e.g. mean 1 compared to mean 2, mean 1 compared to mean 3, mean 2 compared to mean 3 etc.), but this causes a problem. Every time you do a two-sample test and the null hypothesis applies (so the samples are from the same population), you run a 5% risk of a Type 1 error. As you do more and more tests on two samples from the same population, the risk of a Type 1 error rises rapidly.

Introduction

When you use a single-factor ANOVA to compare the means in an experiment with three or more treatments, a significant result only indicates that one or more appear to be from different populations. It does not identify which particular treatments appear to be the same or different.

For example, a significant difference among the means of three treatments, A, B and C can occur in several ways. Mean A may be greater (or less) than B and C, mean B may be greater (or less) than A and C, mean C may be greater (or less) than A and B and finally means A, B and C may all be different to each other.

Introduction

Often life scientists obtain data for two or more variables measured on the same set of subjects or experimental units because they are interested in whether these variables are related and, if so, the type of functional relationship between them.

If two variables are related, they vary together – as the value of one variable increases or decreases, the other also changes in a consistent way.

Introduction

Statisticians and life scientists who teach statistics are often visited in their offices by a researcher or student they may never have met before, who is clutching a thick pile of paper and perhaps a couple of flash drives or CDs with labels such as ‘Experiment 1’ or ‘Trial 2’. The visitor drops everything heavily on the desk and says ‘Here are my results. What stats do I need?’

This is not a good thing to do. First, the person whose advice you are seeking may not have the time to work out exactly what you have done, so they may give you bad advice. Second, the answer can be a very nasty surprise like ‘There are problems with your experimental design.’

Overview

Thermodynamics is unquestionably the most powerful and most elegant of the engineering sciences. Its power arises from the fact that it can be applied to any discipline, technology, application, or process. The origins of thermodynamics can be traced to the development of the steam engine in the 1700's, and thermodynamic principles do govern the performance of these types of machines. However, the power of thermodynamics lies in its generality. Thermodynamics is used to understand the energy exchanges accompanying a wide range of mechanical, chemical, and biological processes that bear little resemblance to the engines that gave birth to the discipline. Thermodynamics has even been used to study the energy exchanges that are involved in nuclear phenomena and it has been helpful in identifying sub-atomic particles. The elegance of thermodynamics is the simplicity of its basic postulates. There are two primary ‘laws’ of thermodynamics, the First Law and the Second Law, and they always apply with no exceptions. No other engineering science achieves such a broad range of applicability based on such a simple set of postulates.

So, what is thermodynamics? We can begin to answer this question by dissecting the word into its roots: ‘thermo’ and ‘dynamics’. The term ‘thermo’ originates from a Greek word meaning warm or hot, which is related to temperature. This suggests a concept that is related to temperature and referred to as heat. The concept of heat will receive much attention in this text. ‘Dynamics’ suggests motion or movement. Thus the term ‘thermodynamics’ may be loosely interpreted as ‘heat motion’. This interpretation of the word reflects the origins of the science. Thermodynamics was developed in order to explain how heat, usually generated from combusting a fuel, can be provided to a machine in order to generate mechanical power or ‘motion’. However, as noted above, thermodynamics has since matured into a more general science that can be applied to a wide range of situations, including those for which heat is not involved at all. The term thermodynamics is sometimes criticized because the science of thermodynamics is ordinarily limited to systems that are in equilibrium. Systems in equilibrium are not ‘dynamic’. This fact has prompted some to suggest that the science would be better named ‘thermostatics’ (Tribus, 1961).

Combustion is the reaction of a fuel with oxygen. It is a subset of the more general subject of chemical equilibrium, which is considered in Chapter 14. Combustion is treated as a separate topic because combustion reactions tend to progress until the fuel is completely consumed. Combustion of various hydrocarbon fuels is the major source of useful energy (i.e., exergy) for transportation, electrical generation, space conditioning, water heating, and industrial processes.

Introduction to Combustion

Table 13-1 summarizes the sources of the energy that are consumed in the United States. The nation consumes about 102 quads (1.02 × 1017 Btu or 1.08 × 1011 GJ) of useful energy each year, which represents about 25% of the world energy consumption. The majority of this energy is provided by combustion. There are two major concerns with this current situation. First, nearly all combustible fuels contain carbon, which results in the generation of carbon dioxide during combustion, as shown in this chapter. Combustion in the U.S. alone resulted in the production of 5,990 million metric tons (5.9 × 1012 kg) of carbon dioxide in 2007 according to the U.S. Department of Energy (2009). Carbon dioxide in the atmosphere absorbs a portion of the thermal energy that is re-radiated from the earth and thereby contributes to global warming. Second, reserves of petroleum and natural gas, which account for over 60% of our useful energy supply, are finite. Although the extent of these reserves is a subject of debate, it is likely that they will become depleted in less than 50 years at the present rate of use. There are hundreds of years of coal reserves, but coal generates more carbon dioxide per unit energy than natural gas and coal also produces other contaminants when combusted. The sustainability of our energy supply and the associated problem of global warming are perhaps the most serious problems that the human race has ever faced.

The specific internal energy, enthalpy and entropy at 1 atm pressure for common combustion gases are provided as a function of temperature in the following tables.

1. Table F-1: Ideal gas properties of CO2

2. Table F-2: Ideal gas properties of CO

3. Table F-3: Ideal gas properties of O2

4. Table F-4: Ideal gas properties of N2

5. Table F-5: Ideal gas properties of H2O

The data in these tables were obtained from EES. The reference state for specific enthalpy is based on the enthalpy of formation relative to the elements at 25°C. The reference state for specific entropy is based on the Third Law of Thermodynamics. The reference values are from:

1. Bonnie J. McBride, Michael J. Zehe, and Sanford Gordon

2. “NASA Glenn Coefficients for CalculatingThermodynamic Properties of Individual Species”

3. NASA/TP-2002-211556, Sept. 2002

4. http://www.lerc.nasa.gov/WWW/CEAWeb/

Note that these tables can be printed from the website associated with this text, www.cambridge.org/kleinandnellis, for use during closed book examinations.

As noted in Chapter 1, there are two very different methods to describe a thermodynamic system corresponding to the microscopic approach and the macroscopic approach. So far, only the macroscopic (or classical) approach has been used in this book. The macroscopic approach is based on empirical laws and it describes the equilibrium state of a system in terms of a relatively small number of properties such as temperature, pressure, internal energy, and entropy. The macroscopic approach can be applied to any system and it is mathematically much simpler than the microscopic approach. At no point in the development of the First or Second Laws is it necessary to know that matter consists of individual particles (molecules).

The disadvantage of the macroscopic approach to thermodynamics is that it does not provide any physical insight into the First and Second Laws upon which it is based. More importantly, classical thermodynamics does not provide any means to directly calculate the thermodynamic properties that are needed to apply the First and Second Laws. For example, we know that the specific heat capacities, cP and cv , are needed to evaluate specific enthalpy and specific internal energy. Macroscopic thermodynamics shows us that cP and cv are related (e.g., cP – cv = R for an ideal gas) and it can also explain how these specific heat capacities vary with pressure at a given temperature, as discussed in Chapter 10. However, macroscopic thermodynamics offers no information on how cP and cv vary with temperature at constant pressure. Indeed, it is even difficult to understand what the property temperature refers to using only concepts from macroscopic thermodynamics. Statistical thermodynamics takes a microscopic approach in an attempt to advance our knowledge in these areas.