Welcome to Thermodynamics and Engines!
In this chapter, we are going to explore how heat energy can be turned into useful work. This is the science behind everything from the engine in a car to the refrigerator in your kitchen. Thermodynamics might sound intimidating, but it is really just a way of keeping track of energy. Think of it like a bank account for heat! By the end of these notes, you will understand how engines squeeze, burn, and expand gases to keep the world moving.
1. The First Law of Thermodynamics
The First Law of Thermodynamics is essentially the Law of Conservation of Energy, but specifically for thermal systems. It tells us that energy cannot be created or destroyed, only moved around.
The core equation is:
\(Q = \Delta U + W\)
Let's break that down:
- \(Q\) is the energy transferred to the system by heating.
- \(\Delta U\) is the change (increase) in internal energy (the kinetic and potential energy of the gas molecules).
- \(W\) is the work done by the system (e.g., the gas pushing a piston up).
An Everyday Analogy:
Imagine you receive £100 (Heat \(Q\)). You put £40 into your savings account (Internal Energy \(\Delta U\)) and spend £60 on a new pair of shoes (Work \(W\)). The total amount you received equals what you saved plus what you spent. Simple, right?
Quick Review Box:
- If \(Q\) is positive, heat is entering the gas.
- If \(\Delta U\) is positive, the temperature of the gas is rising.
- If \(W\) is positive, the gas is expanding and doing work on its surroundings.
Key Takeaway: The energy you put in as heat must either make the gas hotter or make the gas do work.
2. Non-Flow Processes
A "non-flow" process just means we are looking at a fixed mass of gas (like gas trapped in a cylinder) that isn't flowing in or out. There are four main types you need to know:
Isothermal Change:
The temperature stays constant (\(\Delta U = 0\)).
Equation: \(pV = \text{constant}\)
Since the temperature doesn't change, all the heat you put in must be converted directly into work (\(Q = W\)).
Adiabatic Change:
No heat enters or leaves the system (\(Q = 0\)).
Equation: \(pV^{\gamma} = \text{constant}\) (where \(\gamma\) is the adiabatic index).
In this process, if the gas does work, it has to "pay" for it using its own internal energy, so it cools down (\(\Delta U = -W\)). This happens during very fast expansions or compressions.
Constant Pressure (Isobaric):
The pressure stays the same.
Work done: \(W = p\Delta V\).
If you heat a gas at constant pressure, it will expand and get hotter at the same time.
Constant Volume (Isovolumetric):
The volume doesn't change, so the gas can't push anything.
Work done: \(W = 0\).
All the heat you add goes straight into increasing the internal energy (\(Q = \Delta U\)), meaning the temperature and pressure rise quickly.
Don't worry if the adiabatic equation looks scary! Just remember it describes a process where the gas is "isolated" from heat exchange.
Key Takeaway: Different conditions (constant T, P, V, or no heat) change how the First Law is applied.
3. The p–V Diagram
A p–V diagram is a graph of Pressure (\(p\)) against Volume (\(V\)). It is the most important tool for an engineer because it visually shows the work being done.
Calculating Work:
- The area under a single line on a p–V diagram represents the work done during that process.
- For a cyclic process (where the gas returns to its starting state), the area of the loop represents the net work done per cycle.
Step-by-Step: Understanding the Loop
1. The top part of the loop usually shows expansion (work being done by the gas).
2. The bottom part usually shows compression (work being done on the gas).
3. The difference between these two areas—the bit inside the loop—is the "profit" or net work you get out of the engine.
Common Mistake to Avoid:
Check the units on the axes! Pressure is often in \(kPa\) (\(10^{3} Pa\)) and volume in \(cm^{3}\) or \(litres\). Always convert to \(Pa\) and \(m^{3}\) before calculating work to get your answer in Joules.
Key Takeaway: Area = Work. A bigger loop means the engine is doing more work per cycle.
4. Engine Cycles: Petrol and Diesel
Engineers use indicator diagrams to show what is actually happening inside a real engine. You need to compare these to the theoretical "ideal" cycles.
The Four-Stroke Cycle:
A simple way to remember the stages is: Induction, Compression, Expansion (Power), and Exhaust.
Memory Aid: Suck, Squeeze, Bang, Blow!
Petrol vs. Diesel:
- Petrol engines use a spark plug to ignite the fuel-air mixture. In a theoretical cycle, we assume this happens at constant volume.
- Diesel engines compress air until it is hot enough to ignite the fuel automatically. In a theoretical cycle, the combustion happens at constant pressure.
Power Calculations:
1. Input Power: The energy stored in the fuel.
\(P_{\text{input}} = \text{calorific value} \times \text{fuel flow rate}\)
2. Indicated Power: The power generated inside the cylinder (calculated from the p–V loop).
\(P_{\text{ind}} = (\text{area of loop}) \times (\text{cycles per second}) \times (\text{number of cylinders})\)
3. Brake Power: The actual useful power delivered to the crankshaft (output).
\(P_{\text{brake}} = T\omega\) (where \(T\) is torque and \(\omega\) is angular velocity).
4. Friction Power: The power lost to moving parts.
\(P_{\text{friction}} = P_{\text{ind}} - P_{\text{brake}}\)
Key Takeaway: No engine is perfect; we lose power to heat and friction. We track this using Indicated, Brake, and Friction power.
5. The Second Law and Efficiency
The Second Law of Thermodynamics tells us that you cannot simply turn all heat into work. You must have a cold place (a sink) to dump some waste heat.
Efficiency Formulas:
The general efficiency of any engine is:
\(\eta = \frac{W}{Q_{H}} = \frac{Q_{H} - Q_{C}}{Q_{H}}\)
Where \(Q_{H}\) is heat from the hot source and \(Q_{C}\) is heat rejected to the cold sink.
Maximum Theoretical Efficiency:
Even a perfect, frictionless engine has a limit. This is called the Carnot Efficiency:
\(\eta_{\text{max}} = \frac{T_{H} - T_{C}}{T_{H}}\)
Important: Temperatures MUST be in Kelvin! (\(K = ^{\circ}C + 273\))
Did you know?
To make an engine more efficient, you either need to make the source much hotter or the sink much colder. This is why car engines get so hot—they are trying to reach higher efficiencies!
Quick Review: The Three Efficiencies
- Overall Efficiency: Brake Power / Input Power.
- Thermal Efficiency: Indicated Power / Input Power.
- Mechanical Efficiency: Brake Power / Indicated Power.
Key Takeaway: You can't win! The Second Law says you will always lose some energy to the surroundings.
6. Reversed Heat Engines
What if we run the cycle backward? Instead of using heat to make work, we use work to move heat from a cold place to a hot place. This is how refrigerators and heat pumps work.
Instead of "efficiency," we use the Coefficient of Performance (COP). Because these devices move more heat energy than the work put in, COPs are usually greater than 1.
Refrigerator COP:
We care about the heat removed from the cold space (\(Q_{C}\)).
\(COP_{\text{ref}} = \frac{Q_{C}}{W} = \frac{T_{C}}{T_{H} - T_{C}}\)
Heat Pump COP:
We care about the heat delivered to the hot house (\(Q_{H}\)).
\(COP_{\text{hp}} = \frac{Q_{H}}{W} = \frac{T_{H}}{T_{H} - T_{C}}\)
Analogy:
A refrigerator is like a boat with a leak. The work you do bailing out water (heat) keeps the boat dry (cold), but you have to throw the water over the side into the ocean (hot surroundings).
Key Takeaway: Reversed engines don't create "cold"; they just pump heat from one place to another.