Notes for introduction to thermodynamics

Thermodynamics is defined as the "science of the relationship between heat and mechanical work" (Pocket Oxford Dictionary).

Thermodynamics concerns the conversion of heat into and from other forms of energy - most notably (for Engineers) mechanical work. In these notes we shall investigate the laws of Thermodynamics - principles that constrain the performance of machines. We shall touch upon the efficiency of machines and engines - for example, for a given input of electricity to an air conditioner, what is the cooling effect? The topic finds itself in the undergraduate syllabus of not only engineers, but chemists, physicists, material scientists, physicians and mathematicians.

There are some useful historical markers. We note the work of Boyle and Charles on the pressure and temperature of gases. Toy heat engines, known as early as the second century, preceded the practical machines of Newcomen and Watt built in the 18th century. All these machines were constructed without an adequate theoretical knowledge of the topic and in consequence their very low mechanical efficiency - maybe only 1 % or 2% - was not fully appreciated at the time. Indeed heat and work were considered to be different entities; the legacy is that heat is still measured in both Calories and Joules. Through the conversion of “paddle wheel work" into a temperature rise, Joule (1843) demonstrated that both were forms of energy.

Steam and air engines were further developed in the nineteenth century, culminating in the steam turbine. The theoretical strides were made in terms of understanding efficiency - the ratio of work output to heat input. Sadi Carnot's theoretical cycle (1824) has never been realised but in representing the maximum achievable efficiency it sets the benchmark for all practical engines. Clausius’s ideas of classical thermodynamics included the First Law, a restatement of the principle of energy conservation. Lord Kelvin’s statement of the Second Law states that not only must an engine accept heat, it must also reject some of this heat, the minimum possible waste heat being rejected by Carnot's cycle. Another insight from Carnot, that hotter sources of heat promote more efficient engines, has motivated materials scientists to study and develop temperature resistance, and experts in heat transfer to develop cooling circuits for machine internals. Modern turbines, for example, extract energy from gases hotter than 1200°C by means of steam- or air- cooled turbine blades, possibly ceramic.

Further topics include refrigeration and air conditioning, heat transfer, combustion, gas angular momentum in turbines and compressors, compressible flows (in jets), fuel cells, and hydgrogen technology.

2.) Solving Problems

Our knowledge of science is imprecise, and the design engineer must justify his/ her mental image of the problem, assumptions, and choice of physical laws to many stakeholders. A model answer should include the following.

3.) Dimensions and Units

Treat units of measure carefully. In the past, incorrect handling of units has resulted in inoperative and dangerous equipment.

Definition: Dimensional homogenuity is a necessary condition that holds true when units on the LHS and RHS of an equation are identical. For example, consider the equation for kinetic energy $$ E_k = \frac{m v^2}{2} $$ Writing the SI units for each side we get $$ [J] = kg \, m^2 \, s $$

(Some would use symbols MLT). Recalling the definition of the joule, we observe that the equation is dimensionally homogenous. Note that if the factor had been other than 0.5, the equation would still have been dimensionally homogenous but nonetheless wrong.

Data are not always provided in SI units - in these notes we shall often use kilojoules for energy and bar for pressure, in conformity with frequent industrial practice and many textbooks. This is resolved through unit conversion factors.

Some engineers may feel more comfortable changing the units of individual quantities. This is acceptable, although more time consuming. The important thing is that units are used consistently.

$$ m = 2 \, lb \times [\frac{1kg}{2.205lb} ]=0.907 kg $$ $$ v = 4\frac{ft}{s} \times [\frac{1m}{3.2808ft}]=1.219 m/s $$ $$ E_k = \frac{1}{2} m\,v^2 = \frac{1}{2} \times 0.907 \times 1.219^2 = 0.0674 kg\,m^2\,s^{-2} = 0.674 J $$

4) A System and its Properties

For the time being, a closed system is a space containing a fluid with a boundary between said fluid and its surroundings.

To determine the state of a system we need to know properties such as temperature and pressure. The value of a property must be independent of the path through which a fluid has passed to reach its state. The property belongs to and describes its system. Let us consider a mechanical property, altitude. Consider a cyclist travelling from a village centre, up a hill to a picnic area. This route might be directly up a straight main road, or via a more circuitous path. The change in altitude is always the same - it is a property - although the work done depends on the path chosen. The properties of a thermodynamic system determine its energy content.

There are six properties of particular interest. Temperature, pressure and density are well known. We shall introduce internal energy, enthalpy and entropy later. From the state postulate , in a simple system two of these independent properties determine the state of a simple system (thereby fixing the other four).

Figure 1 Abstract view of a system, its boundary, and its surroundings. The system properties include pressure (p), specific volume(v), temperature (T), specific internal energy (u), specific enthalpy (h), and specific entropy (s). (a) As drawn, work transferred from surroundings to system has positive sign (b) as drawn, work transferred from system to surroundings has positive sign.

A simple system is pure - there is only one chemical component. We shall treat air as pure, because there is no significant chemical interaction between its constituents. A simple system comprises only one phase (solid, liquid or gas). Let us now attend to the well-known properties, temperature and pressure.

A gas or liquid applies a force per unit area - a pressure - to any bounding surface (real or imaginary). Pressure is isotropic: at any point in the fluid system it applies equally in all directions. Note two concepts, gauge pressure is defined with reference to atmospheric pressure; hence a gauge pressure of 0.1 bar means "0.1 bar more than the standard atmospheric pressure of 1.013 bar" and corresponds to an absolute pressure of 1.103 bar. Absolute pressure uses vaccuum as its datum. Generally, pressures written in bar should be taken as absolute pressures.

For temperature , Celcius formulated a material, two-point scale: the melting temperature of ice and the boiling point of water at an international standard atmosphere. Kelvin’s thermodynamic scale has no dependence on the behaviour of a particular substance. It has an absolute zero at T = 0 K. (We shall revisit this concept in more detail.)

Consider systems A, B and C, where A is a special system called a "thermometer". The Zeroth Law of Thermodynamics states that if A is in equilibrium with both B and C, it follows that B and C are in equilibrium. One concludes that if the state of A indicates the (equal) temperatures of B and C, two objects at the same temperature are at thermal equilibrium with each other.

A process transforms a system from one equilibrium state to another - e.g. from T1, p1 to T2, p2. The transition between start and end states follows a path ; a quasi-equilibrium process is such that the system is always very close to equilibrium as it moves between start and end points. This is achieved by the path covering an infinite number of equilibrium states, with infinitessimally small net driving forces driving the system from state to state. At equilibrium all driving forces acting on an object sum to zero. (We shall, later on, replace the term “quasi-equilibrium” with “reversible”, which is defined from a different, more profound viewpoint but has the same practical implications).

5) Work and Heat

Work and heat are not properties of a system. They are energy in transit from one place to another.

Work and heat are transferred from the surroundings to the system, thereby crossing the system boundary and changing the total energy of the system. The transfer of work and heat takes part in a process.

Work is defined as the product of a force (F) and the distance over which the force is applied (s). Depending on sign convention, the work definition is $$W = \pm Fs $$

With regard to sign convention work transfer from its surroundings to a system is positive. (See Figure 1 and Section 2.11 of the IUPAC Green book, linked here .) The same sign convention applies to heat. Heat is transferred by a combination of three mechanisms: conduction, convection and radiation. Burning a fuel releases heat according to the fuels calorific value (in kJ/kg) and some or all of that heat can be transferred to an engine.

In thermodynamic processes, there several variables (properties) describe states 1 and 2. It is conventional to use just the state numbers 1 and 2 to identify limits. $$W = \pm \int_{s1}^{s2} F(s) ds $$ becomes \begin{equation} W = \pm \int_{1}^{2} F(s) ds \end{equation}

There are some variations on the work definition. In angular motion work is the product of applied torque (M) and the angle of rotation, thus $$ W = \pm M \theta $$

If an electric charge $ Q_{elec} $ is moved so that it experiences a difference in electrical potential V $$ W = \pm V Q_{elec} $$

Importantly, if a gas expands at constant pressure in a piston-cylinder, so that the volume changes by an amount $ \Delta v $, the reversible work is, $$ W = \pm p \Delta V $$

The dependence of force on displacement means that the function F(s) must be integrated. We look for the area underneath a line.

Problem Statement: Work required to extend a spring.
Schematics:

Figure 1.010 Spring loading (a) change from zero loading to initial loading to additional loading (b) plot of force versus displacement. The work is the area of the trapezium bounded by displacements s1 and s2
Assumptions: Linear spring characteristics - allows Hooke ’ s Law.
Physical Laws: Hooke ’ s Law relates force to displacement. The work definition gives work - the area under the line in the above figure.
Calculation: If the force were constant, one would use the work defininition in the form $W=F(s_2-s_1)$. However, force varies linearly with displacement, $F=ks$ , between locations 1 and 2. Integration is required. $$ W = \int^2_1 F(s) ds = \int^2_1 k s ds = k \frac{s_2^2-s_1^2}{2} $$ where $k= 200 kN/m $. If we put in numbers, $$ W = 200 kN/m \times \frac{0.07^2-0.02^2}{2}m^2=0.450 kJ $$
Discussion: A simpler approach is to envisage the area under the line as a trapezium. Then the work is $$ W = \frac{(F_2 + F_1)\times(s_2-s_1)}{2} = k \frac{s_2+s_1}{2} \times(s_2-s_1) = 200kN/m \frac{0.09m}{2} \times 0.05m =0.450 kJ$$ The identical answer is unsurprising because $ (s_2+s_1)(s_2-s_1) = s_2^2-s_1^2 $. It is also worth commenting on sign convention . In these notes I use the IUPAC convention that positive work is applied to the system; in the above a positive number makes the spring act as the system. Were the system were an actuator, acting on the spring and transferring work, the sign would be negative.

One of the simplest concepts of heat transfer is Newton's Law of Cooling . A simple Newton Heat Transfer from surroundings to system is, $$ Q = K_{heat} (T_{surrounding}-T_{system}) t $$ where $ K_{heat} $ is a constant of proportionality and t is time. The temperature difference is often termed the "temperature driving force". This is instructive; a system and surroundings at the same temperature are in thermal equilibrium.

6) Energy Conservation, the non-Flow Energy Equation

The well known Non-Flow Energy Equation (NFEE) equates the change of microscopic energy in a closed system to the addition of energy work and heat.

A stationary system is one where any change in kinetic or potential energy is small.

Internal energy U is the combination of all forms of microscopic energy in a fluid. This includes nuclear energy, chemical energy, and energy of phase change. With regard to the kinetic theory of gases, it includes sensible energy, in the form of the translation, rotation and vibration of molecules.

A closed system is one where no material passes through the system boundary.

Now consider a closed stationary system. The only energy of interest is internal energy, and any process that adds heat and work to the system increases its internal energy. From the principle of energy conservation it follows that, $$ U_2 −U_1 =Q+W $$

Note that here I use Q and W to sum all forms of heat and work. Subscript 1 refers to the start condition of the system and subscript 2 to its end condition. It will be shown in the next section that the NFEE follows from the First Law of Thermodynamics.

Problem Statement: Internal energy change in a process.
Schematics:

Figure 1.020 Showing mixer with N=5, rotations, a torque of M = 4 kN m and a heat loss of 50 kJ
Assumptions: Stationary fluid - ignore changes in kinetic energy and potential energy. NFEE can be used.
Physical Laws: Work definition, gives work. First Law - enables use of NFEE
Calculation: The work is the product of torque and angular displacement. Noting $2\pi$ radians per revolution, the angular dispacement is, $$ \theta = 2 \pi N = 2 \pi \times 5 = 31.4 \; rad $$ The work is transferred from the surroundings to the system (the liquid) and is hence positive. $$ W = M \theta = 4 kN m \times 31.4 \; rad= +125.6 kJ $$ The Non-Flow Energy Equation yields, $$ U_2-U_1 = Q + W = 125.6 + (-50) = 75.6 kJ $$
Discussion: If one uses the older (pre-IUPAC 1992) sign convention, $$ U_2 - U_1 = Q - W_{older} $$ This time the work transfer counts as negative - so $$ U_2 - U_1 = -50 - (-125.6) = 75.6 kJ $$ The calculated change in identical energy is identical.

7) The First Law of Thermodynamics and the Cyclic Process

In a cyclic process (for example an engine cycle), the net heat input equals the net work output.

A closed system undergoes a cycle when it passes through a series of events which leave its final state equal to all aspects of its initial state. A cycle can involve the transfer of both work and heat across the system boundary. Consider the simple case of a mass being raised by a string and pulley and then being allowed to fall to its original position under the restraint of friction from a brake shoe.

If the brake shoe is lifted and a force, F, applied to raise the weight from its original position to a new position, then work is done upon the system in order to raise the weight, that is a positive quantity of work is transferred across the boundary.

If the original force, F, is then removed and the brake applied carefully, allowing the weight to fall slowly to its new position, no work crosses the boundary. However, the brake shoe will get hot and the system can only be returned to its original condition if heat can be transferred across the boundary, until the temperature shoe returns to its original value. When this has happened the system will have completed a cycle and both work and heat will have been transferred across the boundary; in this particular example the IUPAC convention sets heat transfer as negative and work transfer as positive.

Experiments were carried out by Joule, 1843, and from his work and the results of other workers the First Law has been evolved, namely:- “when any closed system is taken through a cycle the net work done by the systems upon the surroundings is equal to the net heat supplied to the system from the surroundings.” Expressed in mathematical symbols it states \begin{equation} \sum_{cycle} \delta Q + \sum_{cycle} \delta W = 0 \end{equation}

The use of small increments ( $ \delta Q, \delta W $ ) indicates that distribution of work throughout the cycle may well appear at numerous outlets, the net heat and work being the sum of a series of said increments. The subscript "cycle" on the summation signs signifies that the result is true only for cyclic processes.

The First Law is an axiom - it cannot be proved, but clearly no cases exist which disprove it. We shall establish three important corollaries.

If a cycle is incomplete it is clear that $$ \sum \delta Q + \sum \delta W \neq 0 $$ The left hand side of the above inequality forms a finite quantity. This leads to the first corollary, which expresses a useful form of the First Law.

8) Internal Energy and the NFEE

The NFEE is shown here to be the first of three corollaries of the First Law. The following discussion appears in numerous text books.

During a cyclic process the state of the system, that is to say the value of the properties associated with it, is continually changing, only reverting to its initial value at the completion of the cycle. We could represent a cycle on a graph for which the two axes are independent properties (for instance, gas pressure and gas volume). Suppose the cycle is represented by the paths A and C, point 2 being some intermediate state and point 1 the initial (and final) state. The First Law equation (Equation 3) is repeated below, $$ \sum_{cycle} \delta Q + \sum_{cycle} \delta W = 0 $$

Consider cycles AC and BC. To satisfy the above law for cycles, for cycle AC path A represents the "forward path" from 1-to-2 and path C represents the "reverse path" from 2-to-1. Then to satisfy the First Law (as described above).

$$ \sum_{A} \delta Q + \sum_{A} \delta W + \sum_{C} \delta Q + \sum_{C} \delta W = 0 $$

$$ \sum_{B} \delta Q + \sum_{B} \delta W + \sum_{C} \delta Q + \sum_{C} \delta W = 0 $$

Importantly, from state 1 to state 2 the group, $$ \sum \delta Q + \sum \delta W $$

is independent of the path taken - paths A and B produce the same outcoume. The group is therefore a property. We shall call it $ \Delta U $, or the change in internal energy. (It becomes internal energy when state 1 occurs at a datum value and is renamed (typically) as state 0).

and one often seen the following version, written in more general terms $$ \Delta U = Q + W $$

The heat supplied to a closed system plus the work done on the system equals the increase in internal energy. This particular energy balance excludes mechanical forms of energy which might be superimposed upon the system, such as overall kinetic energy and changes in potential energy.

The above equation is known as the Non-Flow Energy Equation. The units of internal energy are the same as those of heat and work, usually expressed in kJ, but frequently specific internal energy is used,u kJ/ kg, that is internal energy per unit mass. Note that the NFEE relates to changes in internal energy. For practical purposes we can assume that at some arbitrary state the internal energy is zero, so that it possesses a finite numerical value at any other state - and being a property its value will depend only upon that state and be independent of work and heat transfer processes that led up to that state (save that the value of (Q + W) will be the same whatever the paths involved). This means that we shall be able to calculate U in terms of other properties and tabulate them, subject to an arbitrary datum value.

For the time being, we shall not consider systems in which nuclear or chemical reactions are taking place, so that hidden forms of energy due to nuclear and chemical bonds remains constant and so does not enter into our calculations. Changes of phase, such as from water to steam, will be dealt with in Topic 2.

Two further corollaries are of interest, but they are not as central to our calculations as the first.

Corollary 2: The internal energy of a closed system remains unchanged if the system is isolated from its surroundings.

This follows the principle of the conservation of energy. Clearly, if a system is isolated from its surroundings no work or heat can cross the boundary, that is Q = W =0. Hence $ \Delta U = 0 $ from the None Flow Energy Equation.

Introduction to Thermodynamics

Topic 1. Introduction

1.) Preamble