Thermodynamics, Power Cycles and IC Engines — ESE Mechanical

Weightage: Thermodynamics is the foundation of the Mechanical papers, and power cycles and engines carry a large share of the numerical questions in Prelims Paper II and the Mains. The laws themselves are short, so marks are won by picking the right cycle formula and keeping track of states.

1. The first law

For a closed system the first law is $Q-W=\Delta U$. For an open (steady-flow) system it takes the form:

$Q-W_s=\Delta h+\frac{\Delta V^2}{2}+g\Delta z$

where $h$ is the specific enthalpy. Standard applications: a nozzle (no work, no heat) converts enthalpy into kinetic energy; a turbine produces work and has negligible heat loss; a throttle is isenthalpic.

For an ideal gas, $pv=RT$, $h$ and $u$ depend on temperature only, and $c_p-c_v=R$, with $\gamma =c_p/c_v$.

2. The second law and entropy

The Kelvin-Planck statement denies a cyclic engine that converts heat entirely to work, and the Clausius statement denies a cyclic device that moves heat from cold to hot with no work input. They are equivalent.

The Carnot efficiency is the highest possible between two reservoirs:

${\eta }_{Carnot}=1-\frac{T_L}{T_H}$

Entropy satisfies the Clausius inequality, $\oint \delta Q/T\le 0$, and for an isolated system entropy never decreases. For an ideal gas:

$\Delta s=c_p\ln \frac{T_2}{T_1}-R\ln \frac{p_2}{p_1}$

An availability (exergy) analysis gives the maximum useful work from a state relative to the surroundings: $\psi =(h-h_0)-T_0(s-s_0)$ plus kinetic and potential terms. Irreversibility is $I=T_0{S}_{gen}$.

Worked example. An engine operates between 600 K and 300 K. The Carnot efficiency is $1-300/600=50\%$. A claim of 60 percent violates the second law.

3. Air-standard cycles

Otto cycle (spark-ignition): two isentropes and two constant-volume processes.

${\eta }_{Otto}=1-\frac{1}{r^{\gamma -1}}$

with compression ratio $r$. For $r=8$ and $\gamma =1.4$, $\eta =1-8^{-0.4}=56.5\%$.

Diesel cycle (compression-ignition) adds heat at constant pressure. With cut-off ratio $\rho$:

${\eta }_{Diesel}=1-\frac{1}{r^{\gamma -1}}\cdot \frac{{\rho }^{\gamma }-1}{\gamma (\rho -1)}$

For the same compression ratio, Otto is more efficient. For the same maximum pressure and temperature, Diesel is more efficient, which is why diesel engines can run at higher compression.

Brayton cycle (gas turbine):

${\eta }_{Brayton}=1-\frac{1}{r_p^{(\gamma -1)/\gamma }}$

with pressure ratio $r_p$. For $r_p=10$ it is $1-10^{-0.2857}=48.2\%$. Intercooling, reheating and regeneration raise net work or efficiency. The back-work ratio of a gas turbine is high (around 40 to 60 percent), unlike a steam plant.

4. Steam power: the Rankine cycle

The ideal Rankine cycle has four steps: isentropic pump, constant-pressure boiler, isentropic turbine and constant-pressure condenser. The efficiency is $\eta =(w_T-w_P)/q_{in}$, and the pump work is tiny, $v\Delta p$.

Ways to raise efficiency:

• Raise boiler pressure and temperature. Higher pressure increases the mean heat-addition temperature but raises moisture at the turbine exit.
• Lower the condenser pressure.
• Reheat to limit exit moisture, with a small gain in efficiency.
• Regeneration (feed-water heating) extracts steam to heat feed water, raising the mean heat-addition temperature. An open heater mixes streams, while a closed heater keeps them separate.

The steam rate is $3600/w_{net}$ in kg per kWh, and the heat rate is the heat input per kWh. Remember that dryness fraction at the turbine exit must stay above roughly 0.88 to limit blade erosion.

5. Refrigeration cycle principle

The vapour compression cycle reverses a power cycle. The coefficient of performance is $COP=q_L/w$. For a reversible refrigerator, $COP=T_L/(T_H-T_L)$, and for a heat pump $COP=T_H/(T_H-T_L)$, always one more than that of a refrigerator between the same limits. Details of the cycle are in the heat-transfer and refrigeration chapter.

6. Internal combustion engines

For a reciprocating engine with $N$ rpm and $n_c$ cylinders:

• Indicated power $IP=p_{mi}LA{n}_c{N}_p$, with $N_p=N/2$ for a four-stroke engine and $N$ for a two-stroke.
• Brake power $BP=2\pi NT/60$.
• Friction power $FP=IP-BP$.
• Mechanical efficiency ${\eta }_m=BP/IP$.
• Brake specific fuel consumption $={\dot{m}}_f/BP$.
• Brake thermal efficiency ${\eta }_{bth}=BP/({\dot{m}}_f\times CV)$.

Knocking in an SI engine is auto-ignition of the end gas. It is reduced by a higher octane number, a lower compression ratio and better cooling. In a CI engine, knock is a long ignition delay, reduced by a higher cetane number.

Worked example. A four-stroke, single-cylinder engine has an indicated mean effective pressure of 7 bar, bore 100 mm, stroke 120 mm and speed 1500 rpm. Then $IP=7\times 10^5\times 0.12\times (\pi /4)(0.1{)}^2\times (750/60)=8.25\text{kW}$, since a four-stroke engine fires 750 times a minute.

Emissions to know are CO, unburnt hydrocarbons, NOx and particulates, and the controls are catalytic converters, EGR and filters. Indian vehicles follow the Bharat Stage standards, so check which stage is current.

Common traps

• Using Celsius in Carnot or entropy formulas. Use kelvin.
• Mixing which cycle is more efficient at equal compression ratio and at equal peak conditions.
• Forgetting that a throttle is isenthalpic, not isentropic.
• Using $N$ instead of $N/2$ for four-stroke power strokes.
• Counting pump work as negligible for a non-ideal or high-pressure cycle without checking.

Memory aids

• "1 minus T low over T high": Carnot.
• "Otto wins at equal r, Diesel at equal peak": cycle comparison.
• "Regenerate for the mean temperature": why feed-water heating helps.

Summary

The first law covers closed and open systems. The second law sets the Carnot limit and defines entropy and availability. Otto, Diesel and Brayton air-standard cycles each have a closed efficiency formula, and the Rankine cycle is improved by higher pressure, lower condenser pressure, reheat and regeneration.

Engine questions come down to indicated and brake power, efficiencies and the knock rules for SI and CI engines.

Exam protocol

• Convert to kelvin and absolute pressure first.
• Sketch the $T$-$s$ or $p$-$v$ diagram and label the states.
• Pick the cycle formula that matches the stated constraint.
• Check that no result violates the Carnot limit.