Exercises: Heat engines

Name the four strokes of a petrol engine and describe what happens in each one: position of the piston, state of the valves, and what happens to the fuel-air mixture.

The figure schematically shows the four strokes of an internal combustion engine, but one is covered. State which stroke is missing, what occurs during it, and how you deduce it from the position of the piston and the valves of the adjacent strokes.

Explain the function of these internal combustion engine components: spark plug, injector or carburettor, crankshaft, connecting rod, and valves. Which of them distinguishes a petrol engine from a Diesel engine?

Define the cylinder bore and the piston stroke, and write the formula for unit displacement in terms of both. Why can two engines with the same displacement behave very differently if one is 'long-stroke' (stroke greater than bore) and the other is 'oversquare'?

The figure shows a cylinder with its bore and stroke dimensioned. Calculate the unit displacement $V_u = \pi \cdot \frac{D^2}{4} \cdot c$ using the data from the figure, express it in $\text{cm}^3$, and calculate the total displacement if the engine has three cylinders.

Explain the basic differences between a two-stroke engine and a four-stroke engine: how many revolutions of the crankshaft each cycle requires, how gases enter and leave, and why two-stroke engines are usually lighter but more polluting.

Define top dead centre (TDC) and bottom dead centre (BDC) of the piston and explain what volume remains above the piston at each position. At which of the two does ignition of the mixture occur?

Explain what a turbocharger does: what waste energy it uses (that of the exhaust gases), what it compresses, and why a small turbocharged engine can perform like a larger naturally aspirated engine. What is 'turbo lag'?

An engine has four cylinders of unit displacement $500\ \text{cm}^3$. a) Calculate the total engine displacement. b) If the volume of the combustion chamber of each cylinder (with the piston at top dead centre) is $50\ \text{cm}^3$, calculate the compression ratio $r = \frac{V_u + V_c}{V_c}$. c) Explain what improvement and what risk a high compression ratio involves.

Compare the Otto cycle (petrol) with the Diesel cycle: how fuel is introduced, how ignition occurs, and what compression ratio each uses. Reason why Diesel engines are usually more efficient at partial load.

Of the chemical $100\ \text{kJ}$ in petrol, a spark-ignition engine delivers about mechanical $30\ \text{kJ}$ to the wheels. Explain where the remaining $70\ \text{kJ}$ are lost (exhaust heat, cooling, friction) and cite two technological improvements that reduce these losses.

An engine delivers a torque of $180\ \text{N}\cdot\text{m}$ at $3\,000\ \text{rpm}$. a) Express the angular velocity in $\text{rad/s}$ ($\omega = \frac{2 \pi n}{60}$). b) Calculate the power output $P = M \cdot \omega$ and express it in $\text{kW}$. c) Calculate the work produced by each cylinder explosion if the engine is a four-cylinder, four-stroke engine.

The specific fuel consumption of an engine is $240\ \text{g/kWh}$. a) Calculate how much petrol it consumes in one hour operating at $50\ \text{kW}$. b) Using the calorific value of petrol ($\approx 12\ \text{kWh/kg}$), calculate the thermal efficiency of the engine. c) Comment on the result in light of the typical loss balance.

Explain why a Diesel engine does not need spark plugs: what happens to the air temperature when it is highly compressed, when the Diesel fuel is injected, and how auto-ignition occurs. Relate this to its higher compression ratio.

The valve timing mechanism controls the opening of the valves. Explain what the camshaft is, what it means for an engine to be 'DOHC' (double overhead camshaft), and why modern engines feature variable valve timing (varying the opening timing according to engine speed).

List the main pollutants from a combustion engine (CO, NOx, particulates, unburnt hydrocarbons) and explain which exhaust aftertreatment system reduces each one (three-way catalytic converter, particulate filter, EGR gas recirculation). Why can CO₂ not be 'filtered' and can only be reduced by consuming less fuel?

On the test bench, the following data are obtained for a four-cylinder, four-stroke engine ($1\,600\ \text{cm}^3$ total displacement): at $2\,500\ \text{rpm}$ it delivers a torque of $145\ \text{N}\cdot\text{m}$ while consuming $6,8\ \text{kg/h}$ of gasoline ($P_c = 44\ \text{MJ/kg}$); at $6\,000\ \text{rpm}$ its torque drops to $110\ \text{N}\cdot\text{m}$ while consuming $19,2\ \text{kg/h}$. a) Determine the effective power at both engine speeds and calculate the engine's brake thermal efficiency at each of them. b) A mechanic states: "At $6\,000\ \text{rpm}$, the engine draws in more air per second, so cylinder filling and combustion are better, and engine torque should be at its maximum." Refute this statement technically and thermodynamically, justifying the torque drop at high engine speeds based on volumetric efficiency (throttling at the valves) and mechanical friction losses.

The energy behavior of a $1\,400\ \text{kg}$ passenger car is analyzed in an urban cycle with heavy traffic in two versions: conventional internal combustion and hybrid (with regenerative braking with an overall efficiency of $65\,\%$ to the battery). a) During braking from $50\ \text{km/h}$ to $0\ \text{km/h}$, calculate how much kinetic energy is converted into waste heat in the brakes of the internal combustion vehicle and how much useful electrical energy the hybrid manages to store in its battery. b) In the subsequent start-up to $50\ \text{km/h}$, the conventional internal combustion engine operates at low load with an efficiency of barely $18\,\%$. Justify how the hybrid manages this phase (torque split between the internal combustion engine and the electric motor) to prevent the combustion engine from working in that low-efficiency zone. c) On a highway trip at a steady $120\ \text{km/h}$ on flat ground, reason why the hybrid's fuel economy advantage practically disappears and may even be penalized compared to an equivalent internal combustion vehicle.

You have a four-stroke single-cylinder engine in the laboratory coupled to an eddy current dynamometer brake whose lever arm measures $r = 0,35\ \text{m}$ and rests on a load cell. a) In a steady-state test at $3\,600\ \text{rpm}$, the cell registers a force of $52\ \text{N}$ and the engine consumes a graduated cylinder with $50\ \text{cm}^3$ of gasoline (density $\rho = 0,74\ \text{g/cm}^3$) in exactly $42\ \text{s}$. Calculate the effective power of the engine in $\text{kW}$ and its specific fuel consumption ($g_e$) expressed in $\text{g}/(\text{kW}\cdot\text{h})$. b) Design the complete experimental protocol to obtain the specific fuel consumption curve versus engine speed at full load between $1\,500$ and $5\,000\ \text{rpm}$. Specify which input parameter you would control on the brake, which magnitudes you would measure directly and indirectly, and what thermal and mechanical safety precautions you should take during the test.

A manufacturer replaces a naturally aspirated $2,0\ \text{L}$ 4-cylinder engine with a turbocharged 3-cylinder $1,0\ \text{L}$ engine ("downsizing") with identical maximum power ($110\ \text{kW}$). a) In urban driving at low load, to deliver $15\ \text{kW}$, the naturally aspirated engine operates with the intake throttle almost closed, whereas the turbo engine operates at higher relative pressures. Explain thermodynamically how this affects the pumping loop of the $p\text{-}V$ diagram and the volumetric efficiency. b) On a prolonged uphill highway drive at full load, the fuel consumption of the turbocharged engine shoots up and even exceeds that of the $2,0\ \text{L}$ engine. Justify this phenomenon by analyzing the risk of self-detonation (engine knock), the effective compression ratio, and the need to enrich the fuel mixture ($\lambda < 1$) for internal cooling. c) Assess why downsizing showed major advantages in the old NEDC homologation cycle and lost a large part of its relative advantage with the WLTP and RDE tests.

The overall Well-to-Wheel $\text{CO}_2$ emissions of two equivalent passenger cars are evaluated for a $100\ \text{km}$ journey: a gasoline car that consumes $6,2\ \text{L}/100\ \text{km}$ (direct emission: $2,35\ \text{kg CO}_2/\text{L}$; extraction and refining costs: $+20\,\%$ additional) and an electric car that consumes $16\ \text{kWh}/100\ \text{km}$ (with grid and charging losses of $15\,\%$). a) Calculate the total mass of $\text{CO}_2$ emitted by the gasoline car over the $100\ \text{km}$. b) Determine the threshold emission intensity value of the electricity mix (in $\text{g CO}_2/\text{kWh}$ generated) above which the electric car ceases to be a cleaner option than the gasoline car. c) If the electric car is recharged in country A (coal-based mix: $750\ \text{g CO}_2/\text{kWh}$) and in country B (renewable/nuclear mix: $80\ \text{g CO}_2/\text{kWh}$), calculate the overall emissions in each case and technically distinguish with rigor between the impact on urban air quality (local emissions) and the impact on global climate change.

Many internal combustion engines in hybrid vehicles use the Atkinson/Miller cycle instead of the conventional Otto cycle. a) Sketch and explain on a schematic $p\text{-}V$ diagram how the expansion stroke is made larger than the effective compression stroke by delaying the closure of the intake valves (variable valve timing VVT). What impact does this phase shift have on the utilization of gas energy and on the exhaust temperature? b) If the Atkinson cycle offers superior thermal efficiency, justify why it is not widely used in conventional passenger cars without hybridization or supercharging, paying attention to specific power and cylinder filling at low engine speeds. c) Justify how the hybrid system synergistically solves and compensates for this intrinsic limitation of the Atkinson engine.

An internal combustion engine develops an effective power of $80\ \text{kW}$ with an overall effective efficiency of $32\,\%$. The liquid cooling system must dissipate $30\,\%$ of the heat released during combustion using a coolant with an average specific heat $c_e = 3,6\ \text{kJ}/(\text{kg}\cdot\text{K})$. a) Calculate the mass flow rate of coolant ($\text{kg/s}$) required so that the temperature difference between the inlet and outlet of the radiator does not exceed $8\ \text{K}$. b) In a winter test with an outside temperature of $2\ ^\circ\text{C}$, the engine temperature gauge does not exceed $60\ ^\circ\text{C}$ (when its optimal value is $90\ ^\circ\text{C}$), increasing fuel consumption and emissions. Reasonably diagnose which element of the circuit has failed, what physical state it is in, and what thermodynamic consequences it causes on the engine. c) Justify why the cooling circuit operates pressurized to approximately $1,4\ \text{bar}$ and technically argue what would happen if the coolant were replaced with tap water at atmospheric pressure.

A vehicle tested on a chassis dynamometer under the WLTP cycle homologates a fuel consumption of $5,0\ \text{L}/100\ \text{km}$, but in an RDE test (real driving emissions with a PEMS system) its consumption rises to $6,6\ \text{L}/100\ \text{km}$. a) Analyze three independent physical causes why laboratory conditions underestimate the driving resistance of the real vehicle (include aerodynamic, topographical, and auxiliary consumer effects). b) During the mountain section of the RDE test, the $1\,300\ \text{kg}$ vehicle climbs a net elevation gain of $550\ \text{m}$ over $10\ \text{km}$. Considering $g = 9,8\ \text{m/s}^2$, calculate the increase in potential energy. If during that climb the average engine efficiency is $25\,\%$, determine how many extra liters of gasoline ($P_c = 44\ \text{MJ/kg}$, $\rho = 0,75\ \text{kg/L}$) have been consumed solely to overcome the elevation gain. c) Design the technical requirements and representativeness limits that a valid RDE test must meet (speed profile, altitude ranges, and temperature ranges) to prevent a manufacturer from manipulating test conditions.

Unit challenge. Is it energetically feasible to replace a fossil-fuel boiler with an aerothermal heat pump in your local environment? Act as a domestic energy consultant to find out.

1. Data collection: Look up the average winter temperature for your municipality ($T_F$) on AEMET and set the indoor comfort temperature to $T_C = 21\text{ }^\circ\text{C}$ (express both in Kelvin). Note the data for a standard domestic boiler (thermal efficiency $\eta = 85\,\%$).

2. Modeling: Create a sheet in LibreOffice Calc to model the reverse cycle. Calculate the theoretical Carnot $\text{COP}_{\text{max}}$ using $\text{COP}_{\text{Carnot}} = \frac{T_C}{T_C - T_F}$ and estimate the actual $\text{COP}$ assuming a relative efficiency of $50\,\%$ of the ideal.

3. Balance and comparison: For a typical heat demand of $Q_C = 20\text{ kWh/day}$, apply the first and second laws of thermodynamics ($Q_C = W + Q_F$) to determine the electrical energy ($W$) required by the refrigerating machine versus the heat generated by combustion.

4. Reasoned proposal: Write a brief report (one page) with a comparative chart justifying the most efficient solution based on the limitations of the Carnot cycle in your climate.

Evaluation criteria:

• Rigor in thermodynamic calculations (absolute temperature scales, $\text{COP}$, $\eta$, and energy balances).

• Correct integration of real climatic data in LibreOffice Calc.

• Technical justification grounded in the laws of thermodynamics.