There are four main types of bonds, split into two groups. The strong primary bonds (Ionic, macromolecular covalent & metallic), and the weaker secondary bonds (molecular covalent and van der Waals): Ionic This is the electrostatic force of attraction between oppositely charged ions It is formed by electron transfer: metal lose electrons while non-metals gain them It is non-directional High melting/boiling points Easily soluble Poor conductivity when solid, but when molten, charged ions are free to move around Generally crystalline solids at room temperature & pressure Covalent This is formed from a shared pair of electrons Typically occurs if an atom’s outer shell is about half empty (gaining/losing ~four electrons requires too much energy) It is directional: the formation and orientation affects the overall molecular shape There are two types: Molecular Bonds (simple covalent): These have a low melting and boiling point, due to the weak intermolecular forces They have poor solubility in water Conductivity is also poor, as there are no ions, and all electrons are fixed Generally gaseous or liquidous at room temperature and pressure Macromolecular Bonds (giant covalent): These have very high melting and boiling points, as the bonds themselves are very strong and there are very many of them, so a vast amount of energy is required to break these Insoluble in water Mostly do not conduct, but graphite does Generally solid at room temperature and pressure Metallic This is the electrostatic force between positive metal ions (cations) and a sea of delocalised electrons (which are negatively charged) The cations are in regular rows, with the electrons free to move (hence conduct) around them Melting/boiling points are high, as there are strong electrostatic forces between cations and electrons Insoluble in water Very good conduction Generally shiny, malleable solid at room temperature and pressure Van der Waals Dipoles form when an atom has a net charge, due to an imbalance in protons and electrons Dipoles can be temporary or permanent Once a dipole forms, it encourages the neighbouring atom to become an oppositely charged dipole This +/- force of attraction is the van der Waals bond A hydrogen bond is a particular type of van der Waals bond. Since the hydrogen atom is the smallest (with just 1 electron in one shell), when it combines with much larger atoms such as nitrogen, oxygen and fluorine, there is a major electromagnetic imbalance. This causes permanent dipoles to form. These give rise to far stronger bonds than temporary dipoles. A common example is water. It is the hydrogen bonds that cause a solvent to be polar. As you can see from the tetrahedron above, most materials have predominantly one type of bond throughout their structure, but some (like ceramics and polymers) have a mixture. Polymers typically consist of long chains of carbon atoms that are covalently bonded with other atoms (hydrogen, boron etc). These bonds are extremely strong and rigid, with deformation only occurring in extreme conditions. It is the weaker van der Waals bonds that also exist between molecules in polymers that make them flexible and easy to break. Interatomic Forces & Bond Energies When two oppositely charged atoms (or ions) are far apart, their attraction is negligible. As they get closer, however, the attractive force between them increases. Yet at a certain point, the atoms get so close that the electron clouds start to overlap. At this point, the negatively charged electrons repel each other, causing a repulsive force, between the oppositely charged atoms. The repulsive force only acts at a very short range The net force between the atoms, F, will therefore be given by: The potential energy in the bond is calculated as the integral of the resultant force with respect to separation: There is a certain separation, r₀, where the resultant force is equal to zero and the bond energy is at a minimum Potential energy of a bond is often also defined using power laws. A common example is: The first term is from the attractive force, the second from the repulsive force A, B, m & n are system constants Typical values for m and n are: m = 1 for ionic bonds m = 2, n = 9 for covalent bonds 1 ≤ m ≤ 4 for metallic bonds Differentiating this gives an expression for the force: This equation can be used to determine the equilibrium spacing, by setting F to equal zero. Potential Energy – Interatomic Separation Graphs At the equilibrium: the atoms are a distance of r₀ apart the force is zero the energy is at a minimum: E₀ As r < r₀, there is an immense amount of energy in the strongly repulsive bond. When r > r₀, there is never a positive energy (it tends to zero) Interatomic Force – Separation Graphs These are the exact opposite of the energy graph. Finding Metallic Young’s Modulus We can approximate the Young’s Modulus for metals very accurately from the force in the metallic bonds: Young’s Modulus, E, is directly proportional to the gradient of the straight-line segment immediately after r₀: This does not apply to non-metals, as their structure is fundamentally different. This must not be used to estimate UTS, as it will give a far higher value: Predictions of UTS using this method will be in the range of E/10 or E/15 The actual UTS of metals is more like E/100 or E/1000 Thermal Expansion We can use the energy-separation graph to explain thermal expansion: r₀ is given as the midpoint on the horizontal line connecting the two points on the curve that have the same energy As the temperature increases, so does the energy in the system Therefore, the horizontal line connecting the two points gets longer Since the gradient on the right of the minimum point is shallower, the horizontal line grows more on the right than it does on the left Therefore, the midpoint, r₀, moves to the right as the energy increases As the temperature increases, the equilibrium spacing between atoms increases. This is thermal expansion.

In this notes sheet... Internal Flow Fully Developed Laminar Flow Turbulent Flow in Pipes Pump Head An internal flow is any flow that is fully enclosed by fixed surfaces, and is predominantly determined by the viscous forces at the surfaces. The most common example is pipe flow. As flow enters a pipe, you can see the effect of the viscous force coming into play: Initially, before the fluid enters the pipe, the flow is inviscid: the velocity field is uniform. Throughout the entrance length, the velocity decreases closer to the surfaces. The flow is fully developed in a constant velocity field, with zero relative velocity at the boundaries (no-slip condition). The entrance length is defined as the length of pipe for the flow to fully develop. Fully Developed Laminar Flow The most general form of this is between two flat, parallel, infinite plates: In laminar flow, all velocities are parallel to the net direction of flow. Taking the infinitely small fluid particle above, we can apply conservation of momentum to the horizontal forces acting on in: The left and right forces in grey are pressure forces The top and bottom forces in red are viscous forces We do not know the directions of these, only that they point in opposite directions The Following Steps show how to derive the velocity profile for fully developed laminar flow between two infinite plates, but the same steps should be followed to derive any similar velocity profile. Pressure Forces Pressure force on the left: Difference in pressure over distance δx: Pressure force on the right: Viscous Forces We know that the viscous force is given as: See notes sheet on forces in fluids Therefore, bottom viscous force is given as: The velocity on the top surface is given as: So, the top viscous force is: We know that the top viscous force must be positive, and te bottom force negative (see notes sheet on force in fluids). Newton’s First Law & Conservation of Momentum Since the velocity is independent of x-position in laminar flow, there is no change in velocity in this direction. This means there is no net horizontal force acting on the particle, so all the forces must cancel out: Both parts of the equation must be equal to the same constant then: Integrating the first of these over a pipe length L: Plugging this into the second equation: Substituting in the boundary conditions from the no-slip condition: This is the velocity profile for laminar, fully developed flow between to infinite, parallel plates. It is known as Poiseuille Flow. The maximum velocity is the bit outside the brackets: Circular Pipes The profile for a circular pipe is similar, known as the Hagen-Poiseuille Law: Turbulent Flow in Pipes Laminar flow is disappointingly rare in real life. Instead, the vast majority of flow is turbulent. Turbulent flow is unsteady and chaotic. We can define flow, and the extent of its turbulence, using the Reynold’s Number, Re: ρ is the density of the fluid u is the average velocity of the fluid particles d is the pipe diameter μ Is the viscosity ν is the kinematic viscosity Reynold’s Number is one of three dimensionless quantities. The greater the Reynold’s Number, the easier it is for the flow to become turbulent. Generally, laminar flow occurs at Re < 2000, turbulent flow at Re > 3000. The intermediate stage is the 'transition region', where flow is hard to define. Though the instantaneous velocities of fluid particles are not steady in turbulent flow, we can model the mean velocity as statistically steady. The mean velocity is given as the average cross-sectional velocity: u is the average cross-sectional velocity Q is the volumetric flow rate A is the cross-sectional area This means we can use the pipe flow energy equation: A slight correction factor is required for kinetic energy when making this approximation, but it is close to 1 that we can ignore it for turbulent flow (though you cannot ignore it for laminar flow). Lost Energy We need to know the lost energy term (lost height) in the pipe flow energy equation. This is made up of two factors: major losses and minor losses: Major Losses Major Losses are caused by friction with the pipe wall in long, straight sections of pipe. They are given as a proportion of the overall kinetic energy loss, and are directly proportional to the length of the pipe, and inversely proportional to the diameter: Where f is another dimensionless quantity: friction factor (sometimes called the Darcy friction factor). The third and final dimensionless quantity is relative roughness, r. This is given as the ratio of the average height of wall roughness to diameter: All three dimensionless quantities are related via the Moody Chart: Minor Losses Minor losses are much harder to calculate and need to be found empirically. They are caused by pipe bends, inlets/outlets, changes in cross-sectional area etc. The minor losses are often greater than the major losses. We can approximate the minor losses in terms of a component coefficient, k: The constant k must be determined for the component in question. For example, a particular valve may have a coefficient of 0.001 or 7. Pump Head The pump head term in the pipe flow energy equation can be found in terms of the work output of the pump being used. This can be expressed in terms of the pump work rate (or power): Which is given as the pressure rise caused by the pump multiplied by the volumetric flow rate:

In this notes sheet: The Hard Sphere Model Packing in Two Dimensions Packing in Three Dimensions Simple Cubic Body-Centred Cubic (BCC) Face-Centred Cubic (FCC) Hexagonal Close-Packed (HCP) Atomic Density Understanding the molecular structure of a material is crucial in understanding how that material behaves. Many of its physical properties are directly caused and explained by its crystalline structure (known as a lattice). We define this as: A crystalline lattice is a regular repeating pattern of atoms or molecules. An amorphous material is one that does not display a crystalline structure. Instead, the structure is random – examples include some polymers and ceramics. Some other polymers are known as semi-crystalline The Hard Sphere Model The Hard Sphere Model allows us to model atoms or molecules as, you guessed it, hard spheres. There are a number of conditions for this to apply: The outer electron shell must be full The remaining electrons must be distributed rotationally symmetrically Electron clouds of neighbouring atoms do not overlap much or at all For the outer shell of electrons to be full, metals must lose electrons, while non-metals must gain electrons. This means they are actually ions. Packing in Two Dimensions It is impossible to arrange circles in such a way that there are no unfilled spaces between them: Squares, rectangles, triangles, and hexagons, however, can be arranged into a complete lattice: The primitive unit cell is the smallest cell that can be regularly repeated throughout the structure – the hexagon can be split into six equilateral triangles. The unit cell is the easiest cell to repeat: the hexagonal is more practical, as it does not require rotating, whereas the triangle does. The two arrangements of circles above can, however, be modelled as square and rectangular patterns respectively, where the centre of an atom is at each vertex: Shown in red on the diagrams are the close-packed directions of each structure. These are the lines along which ions continuously touch (the line never passes through empty space). As you can see: The square lattice has two close-packed dimensions The hexagonal lattice has three close-packed directions Since the hexagonal lattice is more densely packed, it is known as the close-packed arrangement. Packing in Three Dimensions If we look at the same arrangements in 3D, is clear that, as well as close-packed directions, there are close-packed planes. These are planes in which all the ions are arranged in a hexagonal lattice. There are four typical unit cells in three dimensions: Simple Cubic This is simply a 2D square lattice in 3D: 1 atom per unit cell (⅛ on each vertex) 6 nearest neighbours (each ion touches six others - also known as the coordination number) Lattice parameter: a = 2r No close-packed planes No slip systems Atomic packing factor: 52.4% Body-Centred Cubic (BCC) In the simple cubic cell above, there is a gap in the centre. When the outer atom eights are moved apart enough to fit another atom in the middle, you have the BCC arrangement: 2 atoms per unit cell (⅛ at each vertex, one in the centre) 8 nearest neighbours Lattice parameter a = ⁴/₃ √3 r No close-packed planes 2 close-packed directions No slip systems Atomic packing factor: 68.2% Face-Centred Cubic (FCC) This is the most densely packed cubic arrangement. In this instance, there are three close-packed planes stacked on top of one another, in an ABC arrangement: Again, the unit cell is a cube, but the atoms on the vertices are further apart. This allows 4 half atoms to fit in the centre: 4 atoms per unit cell (⅛ on each vertex, ½ on each side) 12 nearest neighbours Lattice parameter a = √2 r 4 close-packed planes 3 close-packed directions 12 slip systems Atomic packing factor: 74% Hexagonal Close-Packed (HCP) This is where there are two close-packed planes stacked one on top of the other, this time in an ABAB arrangement: This leads to a hexagonal unit cell: 6 atoms per unit cell (⅙ on each vertex, ½ on top and bottom, 3 in centre) 12 nearest neighbours Lattice parameter: a = 2r 1 non-parallel close-packed plane 3 close-packed directions 3 slip systems Atomic packing factor: 74% The height of the hexagonal close-packed unit cell is given as: Note that HCP and FCC both have the same atomic packing factor, 74% Atomic Density If we know the atomic radius, mass and the type of unit cell, we can estimate the density of a material: N is the number of atoms per unit cell M is the atomic mass V is the volume of the unit cell N(A) is Avogadro’s Number, 9.02x10²³

In this notes sheet: Slip & Slip Systems Calculating Deformations Defects in Crystal Structures Strengthening There are two types of deformation of crystalline materials: elastic and plastic. The former is fully reversible, whereas in the latter, atomic planes slip over one another to form a new, different formation. This kind of deformation is irreversible. Note that the elastic deformation graph could be non-linear (e.g. rubber) Slip & Slip Systems The slipping of atomic planes (slip planes) happens in whatever direction requires the least energy. It is brought about by the application of an external force, resulting in shear stresses within the crystal structure. In a perfect sample, the slip would occur over the whole plane at once, however imperfections in the crystal structure prevent this: slip is a gradual process. Close-packed planes are the most susceptible to slip. This is because more energy is required for a non-close-packed plane to slip, as there is a greater distance for each displaced atom to move. The number of slip systems a crystal lattice has reflects how likely slip in the structure is. The more slip systems present, the more susceptible to slip. This is determined as the product of close-packed directions and non-parallel close-packed planes: Lattices with few slip systems are said to show brittle deformation, while those that have many undergo ductile deformation. HCP Slip Systems There are three close-packed directions per close-packed plane, however all close-packed planes are parallel. Therefore, the number of slip systems is 3, so the deformation is brittle: FCC Slip Systems The close-packed planes in an FCC lattice are along the diagonal-corner, and there are four: Each plane has three close-packed directions, so the number of slip systems is 12, so the deformation is ductile. BCC Slip Systems There are no close-packed planes, but there are 6 nearly-close-packed planes, each with two close-packed directions: Therefore, the number of slip systems is 12, so the deformation is ductile. Calculating Deformation The applied stress must be resolved in terms of the slip plane, φ, and direction, λ: σ is the normal stress, given by F/A Under the application of a shear stress, slip will occur in the most favourably orientated slip system. This is the system with the largest value of resolved shear stress: The value of shear stress required for this slip to occur in the most favourable direction is the critical resolved shear stress: For single crystals, slip occurs when φ = λ = 45°. Defects in Crystal Structures Crystals are never perfect – they are littered with defects that allow slip to occur at significantly lower values of applied stress than you would calculate from the equations above. There are three types of defects: Point Defects – defects with atomic dimensions in all three directions, e.g. a missing atom (a vacancy) Line Defects – defects in two atomic dimensions but are normal in the third direction. Area Defects – defects in only one dimension, e.g. a grain boundary All defects distort the lattice, increasing the energy stored inside it. Line Defects Line defects, also known as dislocations, is when a row of atoms is added or removed. These allow lattices to slip in steps, instead of all at once. Two types of dislocations are edge and screw dislocations: An edge dislocation is when a row of atoms is removed: A screw dislocation is when a set of rows slips a certain amount to one side: In both cases, the Burgers vector, b, measures the slip. As you can see from this schematic, edge dislocations happen in steps, until the edge of the crystal has been reached and no more slip can occur. This is because far less energy is required to move only a small amount every time. Area Defects & Grain Boundaries Grain boundaries occur between uniform regions of crystal structures. These could be crystals of the same atom, or different. Either way, they provide a blockage to slip, as slip cannot occur across such a boundary. A material with multiple crystal grains is known as a polycrystal. The smaller the grains, the higher the stress required to cause slip. This is expressed by the Hall-Petch equation: σ(y) is the yield strength σ(i) is the intrinsic lattice strength k is a material constant d is the average diameter of grains Strengthening It may seem intuitive that more dislocations = less strong material, but this is not the case. Yes, a perfect material with no defects at all would be very strong, but this is impossible to achieve. A small number of dislocations reduces the strength, but as you increase the number of dislocations, the strength actually increases. This is because the defects then interact and obstruct each other, reducing the likeliness of slip. Therefore, the four main ways of strengthening a material are: Reduce the grain size Generate more dislocations Adding smaller atoms into the lattice Adding larger atoms into the lattice The easiest methods of generating more dislocations to a structure are work hardening or melting and solidifying. It is important to note, however, that you cannot add infinite dislocations to a structure. This is because two opposite dislocations will meet and annihilate: This leads to a resolved shear stress-strain curve like this:

In this notes sheet... Definitions Types of Alloying Complete Solid Solutions Partial Solid Solutions Solid Solution Hardening Precipitation Hardening Alloys are metals mixed with other materials.These could be metal or non-metal. They are commonly used instead of pure metals, as pure metals generally have low strength and high ductility – generally in engineering, we want the opposite. There are a number of key definitions: A component is one of the elements mixed in the alloy A phase is a region within a material with uniform chemical and physical properties The solvent is the main component of the allow: the element/compound in highest concentration The solutes are the elements/compounds that are mixed in with the solvent: they are in lower concentration A solid solution is a homogeneous single-phase mixture of solute in solvent The composition of a component is the percentage mass of that component in an alloy or phase The constitution is the overall sum of phases Types of Alloying There are a number of different types of alloys, with different methods used to make them. Substitution This is when solute atoms replace some solvent atoms. For this to work, the Hume-Rothery rules must apply. According to these, the solute and solvent atoms should: be similar in size (diameters within ~15%) have the same general crystal structure have the same number of valence electrons have similar electronegativity If these rules do not apply, there will be too much lattice distortion and an alternative crystal structure will form. Interstitial Addition This is when smaller solute atoms fit between larger solvent atoms. In order for this to occur, the diameter of the solute atoms must be smaller than about 0.6 the diameter of the solvent atoms. Generally, the elements that are small enough to fit in the interstitial sites of solvent structures are: Boron, B Carbon, C Chlorine, Cl Hydrogen, H Nitrogen, N Oxygen, O Phosphorus, P Sulphur, S When all the interstitial spaces are filled, the alloy is said to have reached its limit of solubility. Solid Solutions This is when solvent and solute atoms mix to form a single, homogenous structure. Complete solid solubility occurs when two metals with similar atomic diameter are melted together. As they cool slowly, the solute atoms diffuse throughout the solvent creating a uniform single phase. Some compounds mixtures exhibit partial (limited) solid solubility. This means that they can only form a single phase up to a certain composition. Beyond this, no solute can dissolve in the solvent, and so settles in a separate phase. Sugar in water is a common example: initially, all the sugar dissolves. Keep adding more, however, and it will settle on the bottom – the limit of solubility has been reached. Complete Solid Solutions Equilibrium phase diagrams represent the different phases present at different compositions of alloys. They are different for complete solid solutions and ferrous and non-ferrous partial solid solutions. The cooling process that forms solid solutions is, obviously, a function of temperature. We can plot this with respect to time for different compositions of the alloy on cooling curves: For pure substances, there is no change in temperature as the change in state occurs. In the solidification process (freezing), this is due to the latent heat of fusion: during phase-change, there is no change in kinetic energy in the particles, only the potential energy changes. However, for a solid solution the solidification process occurs over a range of temperatures. Between these two temperatures, there is a two-phase region where some of the compound with the higher melting point starts to precipitate in the still-liquid compound with the lower melting point. The extent of this two-phase region depends on the composition of the alloy, and is shown on an equilibrium phase diagram: Linear Interpolation (Lever Rule) Linear interpolation can be used to find the ratio of liquid to solid in the two-phase region, at a given composition and temperature: When in the two-phase region, the total mass of the substance is given as the sum of the masses of the liquid and solid phases: The mass of liquid phase is given as: The mass of solid phase is given as: Sometimes, this is called the ‘Lever Rule’ Partial Solid Solutions Pure substance are not mixed, and as such there is no difference in complete and partial solubility cooling diagrams when there is no mixture: However, when two components are mix as a partial solid solution, the change of state occurs at a lower temperature range: This leads to a drastically different equilibrium phase diagram (this example is tin, Sn in lead, Pb): There are three single-phase regions, shaded red: α phase: solid solution of Sn in Pb β phase: solid solution of Pb in Sn L phase: liquid solution of Pb & Sn There are also three two-phase regions, shaded grey: L+α Phase: liquid solution of Pb & Sn with some solid Pb in it L+β Phase: liquid solution of Pb & Sn with some solid Sn in it α+β Phase: mixture of non-uniform Pb & Sn The eutectic point is the composition and temperature at which the liquid solution is in equilibrium with multiple solid phases. Eutectic Microstructures As you can see, if you cool a non-ferrous partial solid solution alloy at the eutectic composition, you pass through two distinct microstructures. The eutectic structure is a lamellar form of uniform layers of α and β. The isotherm (horizontal constant temperature line) on which this lies is the eutectic line: At the eutectic point, the structure is 100% eutectic As you go away from the eutectic point, the percentage of the structure that is eutectic decreases At either end of the eutectic line, the structure is 0% eutectic The same linear interpolation as seen above for complete solid solutions can be used to find the amount of eutectic structure present at a given composition. Low-Solute Composition Microstructures When slow cooling at low solute compositions, four microstructural stages are passed through. Again, at each stage, the proportional microstructures can be calculated using interpolation. Solid Solution Hardening In a completely solid solution, there is only one phase, consisting of solvent and solute atoms. Generally, the solute atoms are slightly larger or smaller than the solvent atoms: When the solute atoms are smaller than the solvent atoms, a tensile strain field is induced in the crystal lattice. The solute atoms will migrate to a compressive strain field in the lattice, reducing the overall strain. When the solute atoms are larger than the solvent atoms, a compressive strain field is induced in the crystal lattice. The solute atoms will move to a tensile strain field in the lattice, reducing the overall strain. When the solute atoms settle in the dislocation sites, they impede dislocation movement. This means more energy is required to deform the material: it has been strengthened. Precipitation Hardening Unlike solid solution hardening, precipitation hardening introduces a second phase to the lattice. This second phase impedes dislocation movement, and the finer and more evenly distributed the second phase precipitates are, the better the hardening effect. As the slow cooling equilibrium phase diagram above (Figure 5.13) shows, the β precipitates that form in stage 4 are coarse and concentrated along the α grain boundaries. This makes the alloy quite soft. Precipitation hardening therefore consists of rapid cooling: it is a non-equilibrium process and leads to far finer β precipitates spread evenly throughout the α grains. There are four steps involved: Solution treatment – the alloy is heated to form a single-phase solid solution Quenching – the alloy is cooled so quickly that no β precipitates can form: the solution is supersaturated Heat treatment – the solid solution is heated to a specific temperature for a specific amount of time, allowing gradual diffusion to occur: fine β precipitates form throughout the α grains Cooling – the two-phase alloy is cooled to room temperature without altering the microstructure Precipitation hardening only works if the second phase gets more soluble as temperature increases. Nucleation & Diffusion Nucleation is the process by which the β precipitates form. Diffusion is the process by which the newly formed β precipitates disperse throughout the α grains. To achieve the evenly distributed, fine β grains throughout the structure, a balance between nucleation and diffusion is required: The optimal temperature for the heat treatment is at peak rate of formation. The rate of formation of the second phase, D, is modelled by an Arrhenius rate law: D is the rate of formation Q is the activation energy required R is the universal gas constant, 8.31 T is the temperature in Kelvin This can also be represented on a time-temperature-transformation (TTT) curve:

In this notes sheet... Introduction to Ferrous Alloys Pure Iron Fe-C Alloys (Steels) Equilibrium (Slow) Cooling Cast Irons Heat Treatments Ferrous alloys are alloys whose primary component is iron: they are generally mixed with carbon, and split into two groups: Steels have < 1.7% Carbon Cast Irons have 1.7 - 4% Carbon Each group is further split into subgroups: Steels Low Alloys ≤ 0.3% C These are low carbon, high strength steels used predominantly in construction 0.3 – 0.7% C These are medium carbon, heat treatable and so can be made very hard – used for railway tracks 0.7 – 1.7% C These are high-carbon, high-hardness steels, used for cutting tools High Alloys These are steels mixed with more atoms to form complex alloys Stainless steel is an example: 18% chromium and 8% nickel Cast Irons 1.7 -4% Greater than 4% C, cast iron becomes far too brittle for practical use There are many versions of cast irons: white iron only consists of iron and carbon, grey iron may include silicon, magnesium, cerium etc. Pure Iron To understand the microstructures and mechanical properties of ferrous alloys, we need to understand pure iron (iron with less than 0.03% C by weight). There are three forms of pure iron: Ferrite, α-Fe: This occurs at low temperatures, below 910°C, has a BCC structure and is magnetic Austenite, γ-Fe: This occurs at medium temperatures, between 910 & 1391°C, has an FCC structure, and is not magnetic δ-Fe: This occurs at high temperatures, above 1391°C. It has a BCC structure Fe-C Alloys These three different forms of pure iron give rise to different alloyed forms depending on temperature. For this reason, the equilibrium phase diagram for Fe-C alloys is slightly different to those of non-ferrous alloys (there is an additional layer). The interstitial solid solutions of C in Fe on the equilibrium phase diagram are also called α, γ, δ because the crystal structures are retained. These are not pure iron, however, but are Fe-C alloys. δ -Fe does is not generally of interest. Cementite, Fe₃C Cementite is a hard, brittle compound that forms in addition to the ferrite and austenite phases in Fe-C alloys. It has a fixed carbon content of 6.7% C. Equilibrium (Slow) Cooling Single phase regions are labelled in red, two-phase regions in grey Because of the second layer in the equilibrium phase diagram, there is a eutectoid point as well as a eutectic point. At the eutectoid point, three solid phases are in equilibrium (α, γ & Fe₃C) At the eutectic point, a liquid is in equilibrium with two solid phases (γ & Fe₃C) The eutectoid point lies at 0.76% C at a temperature of 723°C The eutectic point lies at 4.3% C at a temperature of 1130°C Eutectoid Structure When austenite is slow-cooled at the eutectoid composition (0.76% C), the single solid phase turns into a two-phase lamellar ferrite-cementite structure at the eutectoid point, 723°C. When it is further cooled, the eutectoid structure remains unchanged. The eutectoid structure consists of laminated sheets of cementite in ferrite. At 723°C, the structure is zero % eutectoid at 0.22% C and 6.7%C, but 100% eutectoid at 0.76% C. Interpolation (the lever rule) can be used to find the ratio of eutectoid to non-eutectoid microstructure along this isotherm, or any isotherm below 723°C. Hypo-eutectoid Structure Hypo-eutectoid refers to compositions below the eutectoid composition (0.76% C). When austenite is slow cooled at below 0.76% C, the α+γ phase is passed through, and the end result is not purely eutectoid. The end result, point 4, is a two-phase equilibrium structure of pro-eutectoid ferrite (α) along the former austenite (γ) grain boundaries and pearlite in between – a lamellar structure of eutectoid ferrite and cementite. Pearlite is what makes the alloy hard: cementite is an extremely hard (but brittle) compound, and the fine lamellas obstruct dislocation movement, giving the material an increased resistance to wear and indentation (hardness). Hyper-eutectoid Structure As you might expect, hyper-eutectoid refers to compositions above the eutectoid point. When austenite is slow cooled, it passes through the γ +Fe₃C region before passing the eutectoid temperature (723°C). The end result is a two-phase equilibrium structure of cementite (Fe₃C) along the former austenite (γ) grain boundaries and pearlite in between – a lamellar structure of ferrite (α) and cementite. Cast Irons Cast irons all have between 1.7 and 4%C, giving them a hyper-eutectoid structure. White cast iron has only carbon as a solute, and this is found throughout the structure as cementite and in pearlite. Adding 2% silicon, magnesium, and cerium as well as carbon as solutes creates different microstructures: Ferritic grey cast iron: graphite flakes suspended within a ferrite matrix lead to a high number of stress concentrations, making the material very brittle. Spheroidal cast iron: graphite spheres suspended within a ferrite matrix reduce stress concentrations, making the material stronger and less susceptible to fracture. Heat Treatment Heat treatments are used to make carbon steels with superior mechanical properties to simple slow-cooled steels or cast iron. Normalising The pearlite that forms in equilibrium cooling (as described above) is very coarse, because of the very slow cooling process (annealing). Using faster air-cooling leads to much finer pearlite, with a far denser lamellar structure. This is because the atoms do not have time to diffuse over a long distance and is known as normalising. Fine pearlite is much stronger, because there are far more, tightly packed grain boundaries. These are very good at impeding dislocations, however this increase in strength comes at the expense of ductility. Moderate Quenching Quenching is the process of rapid cooling. If the transformation of austenite (γ) occurs fairly quickly, bainite forms instead of pearlite. The two are very similar, but bainite forms as very small needles of cementite in a ferrite matrix, rather than regular laminated layers of each: Bainite forms at lower temperatures than pearlite – around 250-500°C, as opposed to ~700°C Rapid Quenching If the austenite (γ) is cooled very rapidly (faster than 100°C/s), there is no time for any diffusion of atoms to occur, and so the austenite does not transform into ferrite. Below ~550°C, austenite is so unstable that it deforms via displacement (not diffusion) into a body-centred tetragonal arrangement (BCT). This is a single-phase structure known as martensite – a supersaturated solution of C in Fe. It is very strong and hard, but incredibly brittle. Martensite only forms below 0.7% C This is because the temperature at which it forms decreases as carbon content increases, and at greater than 0.7% C, the temperature is below room temperature. Generally, not all the austenite will transform into martensite. This leads to a two-phase structure of thin martensite needles surrounded by lighter γ: Martensite left as it is far too brittle for most engineering uses. Therefore, it needs to be tempered to restore some ductility. This is done by heating it to 250-300°C or 500-650°C and holding it there for a certain amount of time, allowing controlled diffusion to occur. Tempered martensite consists of small, smooth cementite phases within a ferrite matrix. The size and smoothness of these particles depends on the temperature and duration of tempering: If the particles are too small, too large, or too smooth, they are ineffective at impeding dislocation movement If the particles are not smooth enough, they cause stress concentrations that can lead to fracture TTT Diagram of Heat Treatments This diagram is for steel at the eutectoid composition. Note that the martensite is at around zero degrees at this composition, which is not practical – this graph is for demonstration only, and you would not find martensite at this composition in the real world.

In this notes sheet... Basics Properties Manufacturing Ceramics are inorganic, non-metal compounds with very lough toughness but incredibly high hardness and low temperature and corrosion resistance. There are basic ceramics that have been used for thousands of years (for bricks, vases etc.), and today there are complex high-performance ceramics with very specific applications. Crystal Structure Ceramics exhibit both ionic and covalent bonding between ions, and the extent of this is described by the degree of ionic character: MgO is 73% ionic, whereas SiC is only 12% ionic. The crystal structure of a ceramic is typically far more complex than that of metals, as there are multiple cations (positive ions) and anions (negative ions). Most ceramics have a highly organised structure, but glass is an exception: it is not crystalline: Glass is amorphous. Properties of Ceramics The stress-strain properties of ceramics are not investigated using a typical tensile test, as they are hard to grip: Grip too hard, and they fracture Don’t grip hard enough, and they move out of alignment Instead, a bending test is performed: Fracture occurs at the lower side, as this is in tension. Ceramics are better in compression than in tension. Steels, on the other hand, are better in tension. This is why reinforced concrete is such an excellent construction material. Young’s Modulus Very high, as the ionic bonds are stiffer than metallic bonds. Strength Very high strength, as the ionic bonds require significantly more energy to be broken than metallic bonds. Ceramics very rarely break due to tensile yielding, however, as they are likely to fracture before reaching their UTS Ductility Not ductile at all: slip is very difficult in ionic structures, as like-charged ions would need to pass each other. Overcoming these repulsive forces would require an extremely high shear stress. Toughness Generally pretty low toughness, due to the many internal defects that form in the manufacturing process. These stress concentrations lead to brittle fracture. Hardness Very high hardness, so ceramics are often used as abrasives or cutting surfaces. Creep Resistance Good – ceramics only creep at temperatures significantly higher than metals. Fatigue Resistance Generally alright – fatigue requires a crack to initiate and propagate, but the reduced slipping capacity from the ionic bonding makes propagation difficult. Generally, a crack will either break the ceramic right away or not at all. Density Lighter than most metals – around 2.5 to 6 Mg/m³ Conductivity Excellent thermal and electrical insulators – ionic and covalent bonding has no free electrons. Corrosion Resistance Excellent, as ceramics have already undergone corrosion. Melting Point Very high, due to high bond energies: between 1,900 and 2,800°C. Manufacturing Ceramics The melting point of ceramics is far too high to melt them and pour them into a mould: what do you even make the mould out of!? Pressing Instead, ceramic particles are mixed with a bonding agent (typically water) and pressed into shape: Uniaxial powder pressing applies a force in one direction only and is the most common form. It leaves many irregularities in microstructure and density. Isotactic powder pressing applies a uniform force from all directions, creating a far more uniform structure and density. However, it is more complex and expensive. Sintering After pressing, ceramics are sintered (heated to allow them to set/cure). This causes the grain boundaries between neighbouring particles to coalesce: This leaves spherical voids/pores in the structure – these are what cause the low toughness in ceramics.

In this notes sheet... Polymeric Structure Thermoplastics Thermosets Elastomers Tensile Properties Temperature-Dependent Properties Creep in Polymers Failure in Polymers Broadly, polymers (also known as plastics) are split into three types: Thermoplastics Thermosets Elastomers Each type has a specific set of properties that are dictated by its structure and the bonding within the polymer. Polymeric Structure All polymers consist of long macromolecules which contain chains of covalently bonded atoms. The process of turning single monomer molecules into long chains of polymers is known as polymerisation: Polymerisation is a chemical reaction that generally happens at high temperatures with the help of a catalyst (this is to break the double bond between the carbon atoms). The schematic above shows the polymerisation of ethylene, C₂H₄ into polyethylene (PE). Chains are typically between 10³ and 10⁵ monomers long. The molecular weight of the polymer is the product of the chain length and monomer weight. Chains vary hugely in weight, and so commercial polymers are classified in terms of mean molecular weight: The chains can be linear, branched, or cross-linked networks: This is what dictates their properties. Bonding Monomers are bonded as chains with covalent bonds Chains are connected at branches and cross-links with Van der Waals bonds Often, hydrogen bonds form here too Thermoplastics As seen above, these can be regular semi-crystalline structures or amorphous (no ordered structure). The more branches there are to a chain, the less regular the structure. We say semi-crystalline, as no polymers are truly ordered. All thermoplastics contain crystalline and amorphous regions. The Van der Waals bonds between chains break when heated at lower temperatures than the covalent bonds holding the monomers as chains. This means they easily melt and can be formed into different shapes over and over again. Semi-Crystalline Thermoplastics Some of the most common examples include: Polythene (PE) Very cheap Easy to mould Tough Generally used in bottles, packaging, pipes Polypropylene (PP) Same properties as PE, but stiffer Good UV resistance Fatigue resistance Used as fibres, outdoor furniture, rope Polytetrafluoroethylene (PTFE) Excellent temperature and chemical resistance Good non-stick properties Used in lubricants, chemical containers & bottles, non-stick surfaces Amorphous Thermoplastics Common examples are: Polystyrene (PS) Transparent Very cheap Very mouldable Brittle Can be expanded into foam Used for stationary, packaging, food containers, electrical insulation Polyvinylchloride (PVC) Cheap Stiff but brittle Can be expanded into foam Used in window frames, sheeting, artificial leather, fibres Polymethylmethacrylate (PMMA) Transparent Water resistant Used for windows, laminates, surgical instruments Thermosets Thermosets are highly crosslinked and amorphous. They are generally formed by mixing two compounds which undergo a chemical reaction. This reaction is irreversible, forming strong crosslinks between chains. Therefore, they do not soften or melt when heated, and cannot be reformed. Uses for thermosets are specialist: Epoxies are used as a matrix for fibres in composites and adhesives Phenolics are used in motor housing, telephones, and electrical fixtures Thermosetting polyesters are used for composites, helmets, and automotive bodies Elastomers Many double carbon bonds remain intact in elastomers, so there are few crosslinks between chains. This makes them extremely elastic. The more cross links there are, the more brittle and less elastic the elastomer becomes. Elastomers do not soften or melt. They burn. Examples of Elastomers Natural Rubber (polyisoprene) Harvested from the sap of the Hevea tree Used in erasers and for latex Synthetic Rubber (polybutadiene) Synthesised via polymerisation Used in car tyres Neoprene (polychloroprene) Used in oil resistant seals Tensile Properties of Polymers When a load is applied to a polymer: Bonds may rotate Chains may slip & disentangle Chains may stretch The slipping and disentanglement of chains is very easy in elastomers and thermoplastics, giving both very low stiffness and fairly low strength. The crosslinks in thermosets, however, provide more stiffness, but at the cost of reduced toughness. Tangent & Secant Modulus Polymers rarely have linear sections on a stress-strain curve. As a result, the modulus cannot easily be measured as the gradient. Instead, there are two methods used: Polymers & Temperature The tensile properties of polymers are often hugely dependent on temperature. Take this graph of PMMA (amorphous thermoplastic), for example: The change in properties occurs because as temperature increases, there is a change in specific volume: At the glass transition temperature, secondary bonds between chains break apart allowing molecules to rotate (movement of side groups) In an amorphous solid, there is free space in which the molecules can rotate, leading to a change in physical properties. Throughout the transition region, it is soft and rubbery. Crystalline structures are incredibly dense, with little free space. The molecules have nowhere to rotate, so properties do not change in the transition region. Semi-crystalline solids show a slight change. Amorphous solids show a change in properties at glass transition, crystalline solids do not. Temperature-Dependent Properties of Amorphous Plastics Modulus of Elasticity in Thermoplastics Modulus of Elasticity in Thermosets & Elastomers Both thermosets and elastomers are amorphous They show a glass transition region, but do not melt The more cross-links there are, the higher and more temperature-stable the elastic modulus The process of adding crosslinks is called vulcanisation – this is done by adding sulphur Creep in Polymers Polymers are viscoelastic solids. This means their response to instantaneous stress is two-fold: there is an immediate strain followed by a time-dependent strain. When an instantaneous load like this is applied to both an elastic and a viscoelastic solid, the strain responses vary: Creep in Polymers Polymers creep at significantly lower temperatures than metals. In fact, polymeric creep is a serious issue even at room temperature, as the polymer chains uncoil so easily. Creep is dependent on both temperature and applied stress: The higher the temperature or stress, the faster the creep. Unlike metals, there is no secondary steady-state creep region: This strain response is defined by the polymeric creep law: A and n are material constants Creep Modulus Since creep is dependent on stress, we can define a creep modulus as the constant applied stress divided by the strain with respect to time: The lower the creep modulus, the easier the polymer creeps. Failure in Polymers Tensile Failure The tensile failure mechanism in a polymer depends on its temperature – specifically, its temperature in relation to the glass transition temperature, Tg: Below 0.75 Tg (in Kelvin), polymers are in the brittle, glassy region so fail by brittle fracture This is likely to initiate at a surface defect, like a scratch of machining line Above (Tg - 50 K), thermoplastics become plastic: Initially, the thermoplastic deforms linearly elastically Next, the thermoplastic yields The thermoplastic is drawn out until all the chains are uncoiled This massively increases the fracture strength, as at high stresses the chains are so spread out that they act as reinforcement fibres. Plastics that do not draw at room temperature (e.g., polystyrene) undergo crazing: A microvoid in the polymer is held together with ligaments across the surface Beyond the linear elastic region, these ligaments are drawn out Eventually, the ligaments snap, and the craze becomes a crack Crazing is what makes plastics appear white when bent: this is known as stress whitening. Impact Failure The energy required to break a polymer varies hugely with temperature, and is investigated using a Charpy impact test: Note that modifying brittle polymers like polystyrene (PS) massively increases the energy required to break at standard temperatures: the modified plastic is called high-impact polystyrene (HIPS). Fatigue Failure Different polymers have vastly different fatigue behaviours. Some have fatigue limits, others do not:

In this notes sheet... Composite Structure Composite Types Critical Fibre Length Fibre Volume Fraction Loading Fibre Composites Pure materials and alloys are good for standard applications, but often a very specific set of properties is required from a material. In this instance, a composite may be used or even designed in order to maximise the best properties of a number of materials combined. A composite is a material that employs multiple different phases to attain better specific properties than either phase alone. There has to be a noticeable boundary between the two phases, and one needs to be introduced to the other rather than the two phases forming simultaneously – a two-phase slow cooled metal alloy is not a composite, then. Composites occur naturally as well as artificially. Good examples are bones and carbon fibre respectively. Composite Structure There are two main phases in a composite: the continuous matrix phase is the weaker phase that transfers applied loads to the reinforcement. It also acts as protection for the reinforcement. the dispersed reinforcement phase adds the desired property (strength, stiffness, hardness etc.) to the material by delaying crack dispersion. The matrix could be a metal, ceramic or a polymer. The reinforcement is generally a ceramic (carbon is especially common), but could take a number of forms: Particulate reinforced composites use small particles, like spheres or flakes. Fibre reinforced composites contain fibres. These could be very short, fairly long, or even continuous throughout the structure. Structural Composites may use particles, fibres, or even both for reinforcement, arranged in specific forms to maximise strength: honeycombs, layers and sheets are common. Types of Composites Metal Matrix Composites (MMCs) A cermet is a ceramic-metal composite. Typically used for cutting tools and dies. Advantages include high elastic modulus, toughness, and ductility. Disadvantages include the high density and expense. Standard matrix materials include: Aluminium & aluminium-lithium alloys Magnesium Copper Titanium Super-alloys. Standard reinforcement materials include: Graphite Alumina (aluminium oxide) Silicon carbide Boron Tungsten carbide Ceramic Matrix Composites (CMCs) Generally used for temperature and corrosion sensitive applications, like engine components and deep-sea mining, or extremely hard cutting tools. Advantages include the excellent corrosion and temperature resistance. Disadvantages are the extreme brittleness and expense. Standard matrix materials include: Silicon carbide Silicon nitride Aluminium oxide Standard reinforcement materials include: Carbon Aluminium oxide Polymer Matrix Composites (PMCs) Generally used for lightweight structures, like aircraft, vehicles, sporting, and marine equipment. Advantages include low density, easy processing, and specific properties Disadvantages include poor temperature and chemical resistance. Standard matrix materials include: Nylon Polypropylene (PP) Epoxy Phenolic Polyester Standard reinforcement materials include: Glass Carbon Boron Aramid (Kevlar) Critical Fibre Length The length of fibres makes a huge difference to the composite’s performance. Too short, and the adhesive bond between the fibre and the matrix will break before the fibre itself breaks, so the fibre pulls out of the structure Too long, and the adhesive bond between matrix and fibre transmits too much load, breaking the fibre. As with most things, a fine balance is required between the two. This is known as the critical fibre length. For a uniform circular cross-section fibre of half-length x, the shear force at the fibre-matrix boundary is given by: The force required to break the fibre is given by: The critical half-length is when these two forces equal one another: The critical length is therefore: Standard fibre lengths are: 0.2mm for a carbon fibre in an epoxy matrix 0.5mm for a glass fibre in a polyester matrix 1.8mm for a glass fibre in a polypropylene matrix Fibre Volume Fraction Fibre composites are generally described in terms of fibre volume fraction: this is (you guessed it) the proportion of the materials volume that is taken up by fibres not matrix. For fully aligned fibres, the volume fraction could be up to around 65%. Generally, it is lower. For uniform cross-sectional and continuous fibres, the area fraction is the same as the volume fraction. Loading Fibre Composites Isostrain When the fibres are parallel to the applied load, and the fibre-matrix adhesive bond does not break, the matrix and fibres experience the same strains: The total force is therefore the sum of the forces in the fibres and matrix: When both the fibres and the matrix are in the linear elastic region, Hooke’s law applies to both: Therefore: Isostress When the fibres are perpendicular to the applied stress, the stress in the fibres and matrix are the same: Therefore: Comparing Isostrain and Isostress Stress-Strain Behaviour It is harder to predict the strength of a composite without empirical investigation, however we can predict the behaviour by assuming it will initially perform similar to the fibre, but deform uniformly once the matrix begins to undergo plastic deformation.

This is absolutely not a cheat sheet... We do not endorse cheating at all! It's just a better name than "List of vital but forgettable equations for A Level Maths" The Formula Book for Edexcel A-Level Maths and Further Maths can be seen here and downloaded here: Trigonometry Cosine Rule a² = b² + c² - 2bc cosA Area of a Triangle Area = ½ ab sinC Radians 1° = π/180 1 rad = 180/π Small Angle Approximations sinθ ≈ θ tanθ ≈ θ cosθ ≈ 1 - θ²/2 Trig Identities sin² θ + cos² θ ≡ 1 tan θ ≡ sin θ / cos θ 1 + tan² x ≡ sec² x 1 + cot² x ≡ cosec² x Angle Addition Formulae sin(A ± B) ≡ sinA cosB ± cosA sinB cos(A ± B) ≡ cosA cos B ∓ sinA sinB tan(A ± B) ≡ (tanA ± tanB) / (1 ∓ tanA tanB) Double Angle Formulae sin2A ≡ 2 sinA cosA cos2A ≡ cos²A - sin²A ≡ 2cos²A - 1 ≡ 1 - 2sin²A tan2A ≡ (2 tanA) / (1 - tan²A) R-Addition Formula a sinx ± b sinx can be expressed as R sin(x ± α) a cosx ± b sinx can be expressed as R cos(x ∓ α) Where: R cos α = a R sin α = b R = √(a² + b²) Differentiation & Integration From First Principles: Product Rule: Quotient Rule: Integration by Parts Trigonometric Integration ∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + c ∫ e ˣ dx = e ˣ + c ∫ 1/x dx = ln|x| + c ∫ cos(x) dx = sin(x) + c ∫ sin(x) dx = -cos(x) + c ∫ tan(x) dx = ln|sec(x)| + c ∫ sec²(x) dx = tan(x) + c ∫ cosec(x) cot(x) dx = -cosec(x) + c ∫ cosec²(x) dx = -cot(x) + c ∫ sec(x) tan(x) dx = sec(x) + c ∫ f'(ax+b) dx = f(ax+b) / a + c Series Expansions The Binomial Expansion: The Taylor Series: The Maclaurin Expansion: Common Expansions:

In this notes sheet... Reversible & Irreversible Processes The Second Law Heat Engines & Efficiency Clausius' Statement Actual vs Ideal Efficiencies Improving Efficiency The Carnot Cycle The second law is used for a number of important calculations: finding temperature differences, reversibility, and efficiency of reversible vs irreversible heat engines. Reversible & Irreversible Processes Let us take the classic example of a quasi-equilibrium expansion: If the mass on the piston is made up of a vast number of minuscule weights, like sand, removing a single grain pushes the piston up ever so slightly. The change is so small, that the return to equilibrium is almost instant. For each grain of sand removed, we have an additional point on the P-V diagram – as pressure decreases, volume increases. This process is reversible It is reversible, because if we return the grains of sand one at a time, the piston will move down by the same amount. We neglect friction or leakage. If, however, the weight consist of one large mass, and this is removed, the process is not in quasi-equilibrium. The process is irreversible In this example, the expansion itself is quasi-equilibrium, as we have returned to the grains of sand. However, when the piston hits the top or bottom stoppers, it makes a sound, releasing energy. This energy cannot be gotten back, so The process is irreversible What makes a process irreversible? Friction Unresisted expansion Heat transfer to surroundings Combustion Mixing of different fluids Clearly, then, the majority of real-life processes are irreversible. The Second Law Take the example of a spindle rotating in a container: Spindle rotates Gas in container heats up Heat from gas is conducted through container and heats up the surroundings If we reverse this: Surroundings are heated Gas inside container heats up Spindle does not start rotating We know from experience that this is the case, but why? This is where the second law comes in: Heat cannot be transferred from a cold body to a hotter body without a work input. This is the all-important idea of direction: some things are allowed backwards as well as forwards, but many things are only allowed forwards. Heat Engines & Efficiency A heat engine is a device that connects a hot and cold reservoir with a work input or output. From the first law: The second law is often also used to describe efficiency, η: Using the equation from the first law above, the efficiency can also be written as: Reversed Engine For a reversed engine, like a refrigerator, work is the input not the output. The efficiency it known as the ‘Coefficient of Performance’ (COP). In a fridge, the output is the cold reservoir, so COP is given as: Clausius’ Statement The Clausius statement of the second law tells us that it is only possible to reverse a heat engine if there is a net work input from its surroundings: A reversible engine is always more efficient than an irreversible engine. We can prove this by contradiction: if we take two engines, one reversible and one irreversible, and assume that the former has an efficiency of 10% and the latter 20%: If we now reverse the reversible engine: Modelling both engines as a single control volume: According to the Clausius statement, this is impossible. Similarly, we can use Clausius’ statement to show the 100% efficient engines are impossible: You cannot have an E100 engine Taking a 100% efficient engine and connecting its output to a reversed engine looks like this: Modelling these two engines as one control volume once again violates the Clausius statement: This is summarised by the Kelvin-Planck statement: No heat engine can deliver a work output equivalent to the heat input from a single reservoir. Actual vs Ideal Efficiencies Since no heat engine can be 100% efficient, it is often unhelpful to talk about an engine’s actual efficiency. Instead, when talking about its performance, we compare its efficiency to that of its ideal (reversible) counterpart: The efficiency of a reversible engine is a function of the temperatures between which the engine operates. This leads to the equation: This is due to the absolute temperature scale, as defined by Kelvin: the scale that sets the triple point of water (0°C) as 273.16 K. This scale is identical to the ideal gas temperature scale, from which the relationship Pv=RT is derived. This can be reworked to show that for a reversible engine, heat is a function of temperature. Therefore: Since the increment of the scale is 1 K, this simply becomes: Remember this is only the case for ideal, reversible engines. There is a huge difference in the actual efficiency of a heat engine and its efficiency compared to the maximum achievable efficiency from its reversible counterpart: The actual efficiency appears to be only 10%, but really it is 50%: Reversed Engine Back to the fridge example, the ideal Coefficient of Performance (COP) for a reversible reversed engine is given as a function of the temperatures, not heats: Improving Efficiency As you can probably tell from the numbers in the example above, efficiencies of heat engines can often be quite low. Therefore, there is a constant push to increase this to boost the performance of a system, to make it more economical, or to make it more environmentally friendly. There are a few key ways of boosting a heat engine’s efficiency: Find a lower temperature cold reservoir This is difficult, however. Generally, rivers, lakes or the ocean are used for cold reservoirs, but these are rarely lower than around 10 °C Increase the temperature of the hot reservoir Increasing the temperature difference between the two reservoirs increases the maximum possible efficiency Minimise losses Insulation, less friction etc. brings the actual engine closer to its reversible counterpart Find a use for the losses Can you use the output heat for a practical purpose, such as heating? Generally, this is difficult because the cold reservoir is around 20 °C, which is not helpful for anyone. Carnot Cycle The Carnot cycle is an unobtainable, ideal cycle of compression and expansion. An ideal petrol piston engine can be modelled as a Carnot engine. It consists of four stages: 1) Isothermal Compression There is a heat output Since the process is isothermal, there is no change in internal energy. Therefore, according to the first law, ΔQ = ΔW The piston moves down The temperature does not change, so T₁ = T₂, which equals the cold reservoir temperature 2) Adiabatic Compression There is a work input Adiabatic, so first law becomes -ΔW = ΔU The piston still moves down T₃ = the hot reservoir temperature 3) Isothermal Expansion There is a heat input Isothermal, so ΔQ = ΔW The piston moves up T₃ = T₄ = hot reservoir temperature 4) Adiabatic Expansion There is a work output Adiabatic, so first law becomes -ΔW = ΔU The piston continues to move up T₁ = T₂ = the cold reservoir temperature All four processes are fully reversible

In this notes sheet... The Clausius Inequality Entropy T-S Diagrams Combining 1st & 2nd Laws Perfect Gas Processes Isentropic/Adiabatic Efficiency Entropy & Steam As seen in the notes sheet on the second law, we can compare a heat engine’s actual efficiency with that of its reversible counterpart. If we want to compare these in greater detail, we need to use entropy. The Clausius Inequality In the notes sheet on the second law, we looked at heat engine connected to two thermal reservoirs, modelling two heat transfers only: In reality, however, this is not the case. Instead, each of these heat transfers occurs in infinitely many infinitesimally small steps, dQ: We can integrate to sum the total positive and negative heat transfers in one complete cycle: However, this on its own is not particularly helpful. Instead, we want to be able to find the heat transfer at a specific temperature. Therefore, the Clausius inequality looks at Q/T instead: Reversible Cycles The Carnot cycle is the ideal reversible cycle: Applying the Clausius inequality: Since the process is reversible, the two ratios must equal each other. Therefore, for a reversible process: Irreversible Cycles We know that the efficiency of the reversible engine is greater than that of the irreversible one, so the heat output of the irreversible engine is greater than that of the reversible: This means that But since they are both taking heat from the same hot reservoir: Therefore, plugging these into the equation derived above: We find that for an irreversible cycle: Entropy Applying Clausius’ inequality to the two reversible cycles shown above shows that the integral of dQ/T equals zero in both. This means that the change in Q/T is path independent: it is a property. This property is called entropy, S, and the change in it is given as: The units of entropy are J/K – Joules per Kelvin If we take a cycle with one reversible and one irreversible process, we know the total change in entropy must be negative (see Clausius’ inequality above): The right-hand side is the definition of change in enthalpy, so: Whenever an irreversible process occurs, the enthalpy increases. The entropy in the system itself may decrease, but then that of the surroundings will increase significantly more. Temperature-Entropy Diagrams The area under a reversible temperature-entropy curve is the heat transfer that takes place in that process: Adiabatic Process The reversible process is isentropic, as its enthalpy change equals its heat transfer, zero The irreversible process experiences an increase in entropy. Heat Addition The Process is not adiabatic, so there is a heat transfer to the system. Both the reversible and irreversible processes increase in entropy. The irreversible process increases more in entropy. Heat Rejection There is a negative heat transfer in heat rejection This means the change in entropy for a reversible process will also be negative The change in entropy for the irreversible process could be either positive or negative, depending on how irreversible it is Carnot Cycle The Carnot cycle is a rectangle on a T-S diagram: Isothermal & reversible heat rejection, so reduction in entropy. Adiabatic & reversible, so isentropic. Temperature increases. Isothermal & reversible heat addition, so increase in entropy. Adiabatic & reversible, so isentropic. Temperature decreases. Combining First & Second Laws According to the first law, for a smaller part of a reversible process: Taking the heat transfer as the area under the T-S graph and work as the area under the P-V diagram: This equation applies to both reversible and irreversible processes. Alternatively, we can rewrite the first law in terms of enthalpy: This equation also applies to both reversible and irreversible processes. Isothermal, Reversible Processes In an isothermal process, the first law becomes: Applying this to the equation for the change in entropy in a reversible process: Substituting in the ideal gas law (Pv = RT): This only applies to reversible, isothermal processes Perfect Gas Processes For an ideal gas: For a perfect gas, Cv is constant. Therefore, the integrated entropy equation becomes: This can alternatively be expressed in terms of temperature and pressure: Or in terms of pressure and volume: Isentropic Processes For an isentropic (reversible, adiabatic) process, setting the left-hand side to zero shows us that: All of these equations only apply to perfect gasses. For steam, see below. Isentropic (Adiabatic) Efficiency In an adiabatic process, such as a turbine, compressor or pump, the ideal reversible efficiency is that of the isentropic process. Turbine Efficiency For a turbine, the ideal process is a vertical line downwards on an enthalpy-entropy diagram, whereas the actual process is a diagonal: Since the end points of both processes (reversible and irreversible) are on the same isobar, there is a difference in enthalpy. This gives the isentropic efficiency: Compressor/Pump Efficiency For a compressor or pump, the directions are reversed: Therefore the isentropic efficiency is given as: Entropy & Steam T-s Diagram for Steam Similar to the vapour dome, the T-s diagram for steam is bell-shaped: The isobars inside the dome are the same as the isotherms Pressure increases diagonally from bottom right to top left To find the limits of the dome at each temperature/pressure, steam tables need to be used Interpolation is used to find the quality of the wet steam, just like dryness fraction. h-s Diagram for Steam This is an odd one. There are some similarities to the T-s diagram above, but not many: The critical point is not on the top of the curve, it is far on the left (not even shown on this diagram). This means the sub-cooled liquid region is rarely seen Isobars and isotherms are not horizontal in the wet vapour region, but are diagonal straight lines Isobars in the superheated vapour region are curved As s increases far beyond the wet vapour region, isotherms become near horizontal (the steam comes close to being an ideal gas at high entropies)