Coursework Sample: Thermodynamics Problem Portfolio: Heat Engines and Cycles

Model Answer

Introduction

This portfolio applies the first and second laws of thermodynamics to three idealised heat-engine cycles that underpin the majority of practical power generation and propulsion systems: the Carnot cycle, which sets the theoretical efficiency ceiling for any engine operating between two fixed temperatures; the air-standard Otto cycle, which approximates the behaviour of a spark-ignition petrol engine; and the Rankine cycle, which describes a conventional steam power plant. For each problem the relevant governing equations are stated, full numerical working is shown, and the assumptions underlying the idealisation are made explicit. A comparison table and a pressure–volume diagram are provided to support the analysis, and the portfolio concludes by discussing why real engines always fall short of the efficiencies calculated here.

Problem 1: Carnot Engine Efficiency

Given: A Carnot engine operates between a hot reservoir at T_H = 800 K and a cold reservoir at T_C = 300 K, receiving Q_H = 500 kJ of heat per cycle from the hot reservoir.

Assumptions: Both heat transfer processes are reversible and isothermal, both work processes are reversible and adiabatic, and there are no irreversibilities such as friction or heat loss to the surroundings (Cengel and Boles, 2019).

The Carnot efficiency depends only on the two reservoir temperatures:

η_Carnot = 1 − T_C/T_H = 1 − 300/800 = 1 − 0.375 = 0.625, or 62.5%.

The work output per cycle follows directly:

W = η × Q_H = 0.625 × 500 kJ = 312.5 kJ.

By the first law, the heat rejected to the cold reservoir is the heat supplied less the work extracted:

Q_C = Q_H − W = 500 − 312.5 = 187.5 kJ.

As a check, the ratio Q_C/Q_H should equal T_C/T_H for a reversible Carnot cycle: 187.5/500 = 0.375, which matches 300/800 = 0.375 exactly, confirming the working is internally consistent (Moran et al., 2018). No real engine operating between these two temperatures could exceed 62.5% efficiency; this figure represents an unattainable upper bound that is useful as a benchmark against which real designs are judged, rather than a design target in its own right.

Problem 2: Otto Cycle Analysis

Given: An air-standard Otto cycle, representing a four-stroke petrol engine, has a compression ratio r = 9 and receives q_in = 1,800 kJ/kg of heat during the constant-volume combustion process, as shown in Figure 1.

Figure 1: Pressure–volume (P–V) diagram of the air-standard Otto cycle analysed in Problem 2 (compression ratio r = 9), showing the two isentropic processes (1→2 compression, 3→4 expansion) and the two constant-volume heat-transfer processes (2→3 addition, 4→1 rejection). The shaded area represents the net specific work output calculated below.

Assumptions: The working fluid is modelled as air behaving as an ideal gas with a constant ratio of specific heats γ = 1.4 (the “cold-air standard” assumption); compression (1→2) and expansion (3→4) are isentropic; heat addition (2→3) and rejection (4→1) occur instantaneously at constant volume; and there are no losses due to friction, blow-by past the piston rings, or incomplete combustion (Rajput, 2010).

The air-standard Otto cycle efficiency depends only on the compression ratio:

η_Otto = 1 − r^1−γ = 1 − 9^−0.4.

Evaluating 9^0.4: ln 9 = 2.1972, so 0.4 × ln 9 = 0.8789, and e^0.8789 = 2.408. Therefore:

η_Otto = 1 − 1/2.408 = 1 − 0.4153 = 0.5847, or 58.5%.

The net specific work output is:

w_net = η_Otto × q_in = 0.5847 × 1,800 kJ/kg = 1,052.5 kJ/kg.

The heat rejected during the exhaust stroke follows from the first law:

q_out = q_in − w_net = 1,800 − 1,052.5 = 747.5 kJ/kg.

As a check, q_out/q_in = 747.5/1,800 = 0.4153, which equals 1 − η_Otto as required. This confirms that raising the compression ratio further would raise theoretical efficiency, which is precisely why modern petrol engines have progressively increased compression ratios subject to the practical limit imposed by engine knock, the uncontrolled auto-ignition of the fuel–air mixture ahead of the flame front (Heywood, 2018).

Problem 3: Rankine Cycle for a Steam Power Plant

Given: A simple Rankine cycle operates with the boiler producing superheated steam at 8 MPa and 500°C, and the condenser operating at 10 kPa.

Assumptions: The pump and turbine are treated as isentropic; pressure drops in the boiler, condenser and connecting pipework are neglected; the condenser produces saturated liquid at its exit; and property values are taken from standard steam tables (Rogers and Mayhew, 1994).

State 1 (condenser exit, saturated liquid at 10 kPa): h₁ = 191.8 kJ/kg, specific volume v₁ = 0.001010 m³/kg.

State 2 (pump exit, 8 MPa): because the feedwater is essentially incompressible, the pump work can be approximated as w_p = v₁(P₂ − P₁) = 0.001010 × (8,000 − 10) kPa = 8.1 kJ/kg. Therefore h₂ = h₁ + w_p = 191.8 + 8.1 = 199.9 kJ/kg.

State 3 (boiler exit, 8 MPa, 500°C): from the superheated steam tables, h₃ = 3,399.5 kJ/kg and s₃ = 6.7266 kJ/kg·K.

State 4 (turbine exit, 10 kPa): the expansion is isentropic, so s₄ = s₃ = 6.7266 kJ/kg·K. At 10 kPa the saturation properties are s_f = 0.6493 kJ/kg·K, s_fg = 7.5009 kJ/kg·K, h_f = 191.8 kJ/kg and h_fg = 2,392.8 kJ/kg. The dryness fraction (quality) of the exhaust steam is:

x₄ = (s₃ − s_f) ÷ s_fg = (6.7266 − 0.6493) ÷ 7.5009 = 6.0773 ÷ 7.5009 = 0.810, i.e. 81.0% dry.

h₄ = h_f + x₄ × h_fg = 191.8 + (0.810 × 2,392.8) = 191.8 + 1,938.6 = 2,130.4 kJ/kg.

Cycle performance: turbine work w_t = h₃ − h₄ = 3,399.5 − 2,130.4 = 1,269.1 kJ/kg. Heat added in the boiler q_in = h₃ − h₂ = 3,399.5 − 199.9 = 3,199.6 kJ/kg. Net work w_net = w_t − w_p = 1,269.1 − 8.1 = 1,261.0 kJ/kg. Thermal efficiency:

η_Rankine = w_net ÷ q_in = 1,261.0 ÷ 3,199.6 = 0.394, or 39.4%.

As a consistency check, the heat rejected in the condenser can be found two ways. Directly, q_out = h₄ − h₁ = 2,130.4 − 191.8 = 1,938.6 kJ/kg. Independently, applying the first law to the whole cycle gives q_out = q_in − w_net = 3,199.6 − 1,261.0 = 1,938.6 kJ/kg. The two routes agree exactly, confirming that the four state points and the resulting efficiency figure are internally consistent (Cengel and Boles, 2019).

This figure is realistic for a simple, non-reheated Rankine plant and illustrates why real power stations add reheat and regenerative feedwater heating stages to raise cycle efficiency closer to 45–48% (Dixon and Hall, 2014). The 81.0% exit quality calculated for state 4 is also of direct engineering relevance: steam below roughly 88–90% dryness carries a significant proportion of liquid droplets, which erode the last-stage turbine blades over time, so a plant designer would normally add a reheat stage specifically to keep the exit quality above this threshold rather than accepting the simple-cycle value calculated here (Eastop and McConkey, 1993).

Comparing the Three Cycles

Table 1 summarises the operating conditions and results calculated above. The Carnot result represents an unattainable theoretical ceiling; the Otto and Rankine results represent achievable idealisations of real engines, each still well above what an actual machine can deliver once irreversibilities are included.

Table 1: Comparison of operating parameters and calculated thermal efficiency for the three cycles.

The comparison shows a clear trend: efficiency falls as the working fluid and cycle become more representative of a practical machine. The Carnot cycle, having no internal irreversibility and operating between only two temperatures, gives the highest figure. The Otto cycle, though idealised, at least reflects a realistic compression ratio and constant-volume combustion process. The Rankine cycle produces the lowest efficiency of the three because a large amount of heat must be rejected as latent heat in the condenser, which cannot be avoided within a simple vapour cycle regardless of how well individual components are designed (Moran et al., 2018).

Discussion: Second Law Limits and Real Engine Losses

The second law of thermodynamics guarantees that no heat engine can convert all the heat it receives into work; some heat must always be rejected to a lower-temperature sink, which is why none of the efficiencies calculated above can ever reach unity, even under ideal, frictionless assumptions (Cengel and Boles, 2019). Real engines fall further short of these idealised figures for several additional reasons. Friction between moving parts, such as the piston and cylinder wall in a petrol engine or the bearings in a steam turbine, converts useful work into heat and dissipates it irreversibly, generating entropy and destroying the potential to do useful work in a way the idealised cycle does not capture (Eastop and McConkey, 1993). Combustion in a real Otto-cycle engine is neither instantaneous nor complete, so heat is released over a finite crank angle rather than at a single point of constant volume, and some fuel energy is lost as unburned hydrocarbons in the exhaust, unlike the instantaneous constant-volume heat addition assumed in Problem 2 (Heywood, 2018). Heat is also lost directly to the cylinder walls, exhaust and surroundings rather than being fully confined to the working fluid, and mechanical accessories such as oil pumps, water pumps and cooling fans consume a share of the gross work produced before it ever reaches the output shaft.

For the Rankine plant, additional losses arise from pressure drops in long runs of pipework, incomplete expansion in the turbine due to blade friction and moisture formation in the low-pressure stages, and heat loss from the boiler and condenser to the environment (Rogers and Mayhew, 1994). As a result, a typical road petrol engine achieves a real-world thermal efficiency closer to 25–30%, well below the 58.5% calculated for the idealised air-standard Otto cycle, while a modern coal or gas-fired steam plant with reheat and regeneration typically achieves 40–45%, still short of the theoretical Carnot ceiling for the temperatures involved (Institution of Mechanical Engineers, 2020). While the calculations in this portfolio are essential for establishing the theoretical performance limits of each cycle, real engineering designs are always evaluated against a lower, achievable target efficiency, and the gap between the idealised and actual figure is itself a useful diagnostic of where design improvements, such as reheat, regeneration or reduced friction losses, would have the greatest practical impact.

A related way of framing these losses is through exergy, or the maximum useful work theoretically extractable from a given heat source relative to the surrounding environment. Every irreversibility described above, whether friction, uncontrolled heat transfer across a finite temperature difference, or throttling, destroys exergy rather than simply “losing” energy, since the first law alone cannot explain why a given quantity of heat becomes progressively less useful for producing work (Moran et al., 2018). Exergy analysis is therefore a more diagnostic tool than a simple energy balance for locating where, within a real plant, redesign would recover the most lost work potential; a poorly insulated boiler and an inefficient turbine stage may both “lose” similar amounts of energy, yet the turbine stage typically destroys far more exergy and represents the better target for investment.

Conclusion

This portfolio has calculated and compared the performance of three idealised heat-engine cycles. The Carnot engine achieves 62.5% efficiency, setting an unattainable benchmark for any device operating between 800 K and 300 K. The air-standard Otto cycle, more representative of a petrol engine with a compression ratio of 9, achieves 58.5% efficiency under cold-air assumptions. The Rankine cycle, representative of a simple steam power plant operating at 8 MPa and 500°C against a 10 kPa condenser, achieves 39.4% efficiency, with an exhaust steam quality of 81.0% that has direct implications for turbine blade design. In every case, the second law of thermodynamics places an unavoidable ceiling on performance, and real machines fall further below the idealised figures once friction, incomplete combustion, heat loss and mechanical inefficiencies are taken into account. Understanding both the idealised cycle and its real-world losses is essential for identifying where engineering improvements can meaningfully raise efficiency.

References

Cengel, Y.A. and Boles, M.A. (2019) Thermodynamics: An Engineering Approach. 9th edn. New York: McGraw-Hill Education.

Dixon, S.L. and Hall, C.A. (2014) Fluid Mechanics and Thermodynamics of Turbomachinery. 7th edn. Oxford: Butterworth-Heinemann.

Eastop, T.D. and McConkey, A. (1993) Applied Thermodynamics for Engineering Technologists. 5th edn. Harlow: Longman.

Heywood, J.B. (2018) Internal Combustion Engine Fundamentals. 2nd edn. New York: McGraw-Hill.

Institution of Mechanical Engineers (2020) Automotive Engine Efficiency: Technical Briefing. London: IMechE.

Moran, M.J., Shapiro, H.N., Boettner, D.D. and Bailey, M.B. (2018) Fundamentals of Engineering Thermodynamics. 9th edn. Hoboken: Wiley.

Rajput, R.K. (2010) Engineering Thermodynamics. 3rd edn. Sudbury: Jones & Bartlett.

Rogers, G.F.C. and Mayhew, Y.R. (1994) Thermodynamic and Transport Properties of Fluids: SI Units. 5th edn. Oxford: Blackwell.