Air Standard Cycles: Otto, Diesel, and Rankine

Introduction to Air Standard Cycles

Air standard cycles are theoretical models used to analyze the performance of internal combustion engines and steam power plants. They simplify the complex processes occurring in these machines by making several assumptions. These assumptions allow us to focus on the fundamental thermodynamics and predict the ideal efficiency.

The primary assumptions for air standard cycles are:

• The working fluid is air, which behaves as an ideal gas.
• All processes are reversible (i.e., no friction or dissipative effects).
• Heat is added or rejected instantaneously at specific points in the cycle.
• The specific heats of air are constant, although in reality, they vary with temperature.
• There are no chemical reactions, and no exhaust gases are considered.

By analyzing these idealized cycles, we can understand the factors affecting thermal efficiency and how to improve engine performance. The most common air standard cycles are the Otto cycle, the Diesel cycle, and the Rankine cycle.

The Otto Cycle

Description and Processes

The Otto cycle is the ideal cycle for spark-ignition (SI) internal combustion engines, commonly found in gasoline cars. It consists of four reversible processes:

1. Isentropic Compression (1-2): The air-fuel mixture is compressed adiabatically. Both pressure and temperature increase.
2. Constant Volume Heat Addition (2-3): Heat is added to the system at constant volume, simulating the combustion process. Both pressure and temperature increase.
3. Isentropic Expansion (3-4): The hot gases expand adiabatically, doing work on the piston. Both pressure and temperature decrease. This is the power stroke.
4. Constant Volume Heat Rejection (4-1): Heat is rejected from the system at constant volume, simulating the exhaust process. Both pressure and temperature decrease, returning the system to its initial state.

Analysis of the Otto Cycle

The thermal efficiency of the Otto cycle is primarily dependent on the compression ratio. The compression ratio ($r$) is defined as the ratio of the volume at the beginning of compression to the volume at the end of compression.

Let $V_1$ be the volume at the start of compression and $V_2$ be the volume at the end of compression. Then, $r = V_1 / V_2$.

For an ideal gas undergoing an isentropic process, we have $T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$, where $\gamma$ is the ratio of specific heats ($c_p / c_v$).

From the compression process (1-2), we get $T_2 / T_1 = (V_1 / V_2)^{\gamma-1} = r^{\gamma-1}$.

The heat added ($Q_{in}$) occurs at constant volume (process 2-3): $Q_{in} = m c_v (T_3 - T_2)$.

The heat rejected ($Q_{out}$) occurs at constant volume (process 4-1): $Q_{out} = m c_v (T_4 - T_1)$.

The net work done ($W_{net}$) is $Q_{in} - Q_{out}$.

The thermal efficiency ($\eta_{Otto}$) is defined as the ratio of net work done to the heat supplied:

$\eta_{Otto} = \frac{W_{net}}{Q_{in}} = \frac{Q_{in} - Q_{out}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}$

Substituting the expressions for heat addition and rejection:

$\eta_{Otto} = 1 - \frac{m c_v (T_4 - T_1)}{m c_v (T_3 - T_2)} = 1 - \frac{T_4 - T_1}{T_3 - T_2}$

Using the isentropic relations, we can relate temperatures. For the expansion process (3-4), $T_3 / T_4 = (V_3 / V_4)^{\gamma-1}$. Since $V_3 = V_2$ and $V_4 = V_1$, we have $T_3 / T_4 = (V_2 / V_1)^{\gamma-1} = (1/r)^{\gamma-1}$. So, $T_3 = T_4 r^{\gamma-1}$.

Similarly, for compression (1-2), $T_2 / T_1 = r^{\gamma-1}$, so $T_2 = T_1 r^{\gamma-1}$.

Substituting these into the efficiency equation:

$\eta_{Otto} = 1 - \frac{T_1 (T_4/T_1 - 1)}{T_2 (T_3/T_2 - 1)}$

It can be shown that $\frac{T_4 - T_1}{T_3 - T_2} = \frac{T_1}{T_2}$. Therefore,

$\eta_{Otto} = 1 - \frac{T_1}{T_2} = 1 - \frac{1}{r^{\gamma-1}}$

The Diesel Cycle

Description and Processes

The Diesel cycle is the ideal cycle for compression-ignition (CI) internal combustion engines, commonly found in diesel trucks and buses. It differs from the Otto cycle in how heat is added.

1. Isentropic Compression (1-2): Air is compressed adiabatically. Unlike the Otto cycle, only air is compressed, and fuel is injected later. The compression ratio is higher than in SI engines.
2. Constant Pressure Heat Addition (2-3): Heat is added to the system at constant pressure, simulating the combustion of injected fuel. This process occurs as the piston moves outwards.
3. Isentropic Expansion (3-4): The hot gases expand adiabatically, doing work. This is the power stroke.
4. Constant Volume Heat Rejection (4-1): Heat is rejected from the system at constant volume, returning the system to its initial state.

Analysis of the Diesel Cycle

The Diesel cycle involves two volume ratios: the compression ratio ($r = V_1 / V_2$) and the cutoff ratio ($r_c = V_3 / V_2$). The cutoff ratio represents the fraction of the stroke during which heat is added at constant pressure.

For the isentropic compression (1-2): $T_2 = T_1 r^{\gamma-1}$.

For the constant pressure heat addition (2-3): Since pressure is constant, $P_2 = P_3$. For an ideal gas, $PV = mRT$. So, $m R T_2 / V_2 = m R T_3 / V_3$, which means $T_3 / T_2 = V_3 / V_2 = r_c$. Therefore, $T_3 = T_2 r_c$.

For the isentropic expansion (3-4): $T_3 / T_4 = (V_3 / V_4)^{\gamma-1}$. Note that $V_4 / V_1 = V_3 / V_2 = r_c$, so $V_4 = V_1 / r_c$. And $V_3 / V_4 = V_3 / (V_1 / r_c) = (V_3/V_1) r_c$. This approach is getting complicated. A simpler way is to relate $T_4$ to $T_3$ using the volume ratio.

A more direct approach for efficiency is using heat added and rejected.

Heat added ($Q_{in}$) at constant pressure (2-3): $Q_{in} = m c_p (T_3 - T_2)$.

Heat rejected ($Q_{out}$) at constant volume (4-1): $Q_{out} = m c_v (T_4 - T_1)$.

Thermal efficiency ($\eta_{Diesel}$) = $1 - \frac{Q_{out}}{Q_{in}} = 1 - \frac{m c_v (T_4 - T_1)}{m c_p (T_3 - T_2)} = 1 - \frac{1}{\gamma} \frac{T_4 - T_1}{T_3 - T_2}$.

To simplify this, we use the isentropic relations:

From (1-2): $T_2 = T_1 r^{\gamma-1}$.

From (2-3) at constant pressure: $T_3 = T_2 r_c = T_1 r^{\gamma-1} r_c$.

From (3-4) isentropic: $T_4 = T_3 (V_3 / V_4)^{\gamma-1} = T_3 (V_3 / V_1)^{\gamma-1}$. Note that $V_3 / V_1 = (V_3/V_2) \times (V_2/V_1) = r_c \times (1/r) = r_c/r$. So, $T_4 = T_3 (r_c/r)^{\gamma-1}$.

Substitute $T_3$: $T_4 = (T_1 r^{\gamma-1} r_c) (r_c/r)^{\gamma-1} = T_1 r^{\gamma-1} r_c \frac{r_c^{\gamma-1}}{r^{\gamma-1}} = T_1 r_c^{\gamma}$.

Now substitute these temperatures into the efficiency formula:

$\eta_{Diesel} = 1 - \frac{1}{\gamma} \frac{T_1 r_c^{\gamma} - T_1}{T_1 r^{\gamma-1} r_c - T_1 r^{\gamma-1}} = 1 - \frac{1}{\gamma} \frac{T_1 (r_c^{\gamma} - 1)}{T_1 r^{\gamma-1} (r_c - 1)}$

$\eta_{Diesel} = 1 - \frac{1}{\gamma} \frac{r_c^{\gamma} - 1}{r^{\gamma-1} (r_c - 1)}$

Comparison of Otto and Diesel Cycles

Both cycles represent idealizations of internal combustion engines. The key difference lies in the heat addition process: constant volume for Otto and constant pressure for Diesel.

Compression Ratio: Diesel engines typically have higher compression ratios ($r$) than Otto engines. This is because Diesel engines compress only air, and fuel is injected at the end of compression, avoiding the knocking (pre-ignition) that limits compression ratios in gasoline engines.

Efficiency: For the same compression ratio, the Otto cycle is more efficient. However, since Diesel engines can operate at higher compression ratios, they often achieve higher actual efficiencies.

Fuel Injection: Otto cycles assume uniform mixture before combustion, while Diesel cycles assume fuel is injected and burns over a period at constant pressure.

Specific Heat Ratio ($\gamma$): The value of $\gamma$ for air is approximately 1.4.

Cutoff Ratio ($r_c$): In the Diesel cycle, $r_c$ is the ratio of volumes at the end and beginning of the constant pressure heat addition. $r_c = V_3/V_2$. For an ideal Diesel cycle, $r_c$ is typically greater than 1.

If we consider the limiting case of the Diesel cycle where heat addition occurs instantaneously ($r_c \to 1$), the Diesel cycle efficiency approaches the Otto cycle efficiency.

The Rankine Cycle

Description and Processes

The Rankine cycle is the ideal cycle for steam power plants, such as those used in thermal power stations. It involves a phase change of the working fluid (water/steam).

1. Isentropic Pumping (1-2): Liquid water is pumped from a low pressure to a high pressure. This process requires work input.
2. Constant Pressure Heat Addition (2-3): Heat is added to the water in a boiler at constant pressure, converting it into high-pressure, high-temperature steam.
3. Isentropic Expansion (3-4): The high-pressure steam expands through a turbine, producing work. The steam becomes a low-pressure, wet mixture.
4. Constant Pressure Heat Rejection (4-1): Heat is rejected from the steam in a condenser at constant pressure, converting it back into liquid water.

Analysis of the Rankine Cycle

The analysis of the Rankine cycle typically involves enthalpy changes, as the working fluid undergoes phase changes.

Work Input by Pump ($W_p$): $W_p = v_f (P_2 - P_1)$, where $v_f$ is the specific volume of the liquid.

Heat Supplied ($Q_{in}$): $Q_{in} = h_3 - h_2$, where $h_3$ is the enthalpy of the steam entering the turbine and $h_2$ is the enthalpy of the water entering the boiler.

Work Output by Turbine ($W_t$): $W_t = h_3 - h_4$, where $h_4$ is the enthalpy of the steam leaving the turbine.

Heat Rejected ($Q_{out}$): $Q_{out} = h_4 - h_1$, where $h_1$ is the enthalpy of the condensate leaving the condenser. Note that $h_1$ is essentially equal to $h_f$ at the condenser pressure.

Net Work Output ($W_{net}$): $W_{net} = W_t - W_p = (h_3 - h_4) - v_f (P_2 - P_1)$.

Thermal Efficiency ($\eta_{Rankine}$):

$\eta_{Rankine} = \frac{W_{net}}{Q_{in}} = \frac{W_t - W_p}{Q_{in}} = \frac{(h_3 - h_4) - v_f (P_2 - P_1)}{h_3 - h_2}$

In many practical analyses, the pump work ($W_p$) is small compared to the turbine work ($W_t$) and heat supplied ($Q_{in}$), so it is often neglected for a simplified calculation:

$\eta_{Rankine} \approx \frac{h_3 - h_4}{h_3 - h_2}$

Improving Rankine Cycle Efficiency

Several modifications can be made to the basic Rankine cycle to improve its thermal efficiency:

• Reheating: Steam is expanded partially in a high-pressure turbine, then reheated in the boiler, and finally expanded in a low-pressure turbine. This increases the average temperature at which heat is added and reduces the moisture content in the exhaust.
• Regenerative Feedwater Heating: Some steam is extracted from the turbine at various stages and used to preheat the feedwater entering the boiler. This reduces the amount of heat that needs to be supplied in the boiler, thereby increasing efficiency.
• Superheating: Heating the steam above its saturation temperature at the boiler pressure before it enters the turbine increases the average temperature of heat addition and reduces moisture content in the turbine.
• Improving Condenser Vacuum: Lowering the condenser pressure (increasing the vacuum) increases the temperature difference across the turbine and reduces the heat rejected, thus improving efficiency.

Internal Combustion Engine Performance Parameters

Key Performance Indicators

The performance of an internal combustion (IC) engine is evaluated using several parameters that indicate its efficiency, power output, and fuel consumption.

1. Indicated Power (IP)

Indicated Power is the theoretical power developed in the cylinder as a result of the combustion of fuel. It is calculated from the indicator diagram (pressure-volume diagram).

For a single-cylinder, four-stroke engine:

$IP = \frac{P_m \times L \times A \times N}{60}$ (in Watts, if $P_m$ is in Pascals, L in meters, A in $m^2$, N in RPM)

Where:

• $P_m$ = Mean effective pressure (average pressure during the cycle)
• $L$ = Stroke length
• $A$ = Piston area
• $N$ = Engine speed in revolutions per minute (RPM)

For a multi-cylinder engine, $IP_{total} = n \times IP_{single}$, where $n$ is the number of cylinders.

For a two-stroke engine, the denominator is 120 instead of 60, as power is developed in every revolution.

2. Brake Power (BP)

Brake Power is the actual power delivered at the engine's crankshaft. It is the power measured by a dynamometer. BP is always less than IP due to frictional losses.

$BP = \frac{2 \pi N T}{60}$ (in Watts, if N is in RPM and T is in Newton-meters)

Where:

• $N$ = Engine speed in RPM
• $T$ = Brake torque measured by the dynamometer

3. Frictional Power (FP)

Frictional Power is the power lost due to friction between moving parts (piston rings, bearings, etc.) and pumping losses (in and out of cylinders).

$FP = IP - BP$

Efficiency Parameters

1. Indicated Thermal Efficiency ($\eta_{it}$ or $\eta_{th,i}$):

This is the ratio of indicated work output to the energy supplied by the fuel.

$\eta_{it} = \frac{Indicated Work Output}{Heat Supplied} = \frac{IP \times 3600}{m_f \times CV}$ (if IP is in kW, $m_f$ in kg/hr, CV in kJ/kg)

Where:

• $m_f$ = Fuel consumption rate
• $CV$ = Calorific value of the fuel

2. Brake Thermal Efficiency ($\eta_{bt}$ or $\eta_{th,b}$):

This is the ratio of brake power output to the energy supplied by the fuel. It represents the overall efficiency of the engine.

$\eta_{bt} = \frac{Brake Power Output}{Heat Supplied} = \frac{BP \times 3600}{m_f \times CV}$ (if BP is in kW, $m_f$ in kg/hr, CV in kJ/kg)

3. Mechanical Efficiency ($\eta_m$):

This is the ratio of brake power to indicated power. It indicates the efficiency of the engine in converting indicated work to brake work.

$\eta_m = \frac{BP}{IP}$

Note: $\eta_{bt} = \eta_{it} \times \eta_m$.

4. Relative Efficiency ($\eta_{rel}$):

This is the ratio of the brake thermal efficiency of the actual engine to the thermal efficiency of the ideal cycle (Otto or Diesel) operating under similar conditions.

$\eta_{rel} = \frac{\eta_{bt}}{\eta_{ideal\_cycle}}$

5. Volumetric Efficiency ($\eta_{v}$):

This applies to naturally aspirated engines and is the ratio of the actual volume of air drawn into the cylinder during the suction stroke to the swept volume of the cylinder.

$\eta_{v} = \frac{\text{Actual volume of air inducted}}{\text{Swept volume}}$

Volumetric efficiency is usually less than 100% due to flow restrictions and incomplete filling of the cylinder. Turbocharged and supercharged engines can achieve volumetric efficiencies greater than 100%.

Combustion in IC Engines

Combustion is the rapid chemical reaction between a fuel and an oxidant (usually air) that produces heat and light. In IC engines, controlled combustion is essential for efficient power generation.

Phases of Combustion in SI Engines (Otto Cycle)

Combustion in SI engines is typically divided into three main phases after ignition:

1. Ignition Delay Period: The time interval between the spark plug firing and the initial flame propagation. During this period, fuel-air mixture undergoes chemical changes, forming unstable intermediate compounds.
2. Flame Propagation Period: Once ignition occurs, a flame front starts to move across the combustion chamber, consuming the mixture. This phase is relatively fast.
3. Afterburning: This is a slower burning process that continues after the main flame front has passed, ensuring complete combustion of the remaining fuel.

Knocking in SI Engines

Knocking (or pinging) is an abnormal combustion phenomenon in SI engines. It occurs when pockets of the fuel-air mixture auto-ignite before the main flame front reaches them, causing rapid pressure rises and shock waves.

Factors promoting knock:

• High compression ratios
• High engine speeds and loads
• Lean fuel-air mixtures (though rich mixtures can also knock under certain conditions)
• High intake air temperatures
• Advanced ignition timing
• Engine design (combustion chamber shape, spark plug location)

Octane Number is a measure of a fuel's resistance to knocking. Higher octane fuels are more resistant.

Combustion in CI Engines (Diesel Cycle)

Diesel combustion is different due to the compression ignition process.

1. Ignition Delay Period: Similar to SI engines, there's a delay between fuel injection and the start of combustion. This delay is crucial as it determines the amount of fuel that accumulates before ignition.
2. Premixed Charge Combustion: During the initial part of the delay, some fuel mixes with air and burns rapidly, leading to a sharp pressure rise.
3. Diffusion Combustion: As combustion progresses, the rate is controlled by the rate at which fuel vapor diffuses into the oxygen-rich zones and mixes. This is a slower, more controlled burning process.
4. Afterburning: Complete combustion of any remaining fuel.

Cetane Number

Cetane Number is the measure of the ignition quality of diesel fuels. It indicates how readily the fuel ignites under compression. Higher cetane numbers mean shorter ignition delays.

Cooling Systems in IC Engines

Engine cooling systems are vital to prevent overheating, which can cause component damage, reduced efficiency, and engine failure. They remove excess heat generated during combustion.

Types of Cooling Systems

1. Air Cooling:
   • Description: Relies on airflow over the engine exterior, often enhanced by fins cast onto the cylinder block and cylinder head to increase surface area.
   • Advantages: Simpler, lighter, no coolant leaks, less maintenance.
   • Disadvantages: Less effective, especially at low speeds or high loads; can be noisy; difficult to maintain uniform temperature.
   • Applications: Small engines (motorcycles, lawnmowers), older car engines.
2. Liquid Cooling:
   • Description: Uses a liquid coolant (typically a mixture of water and antifreeze) circulated through passages (water jackets) in the engine block and head. The coolant then passes through a radiator, where it is cooled by airflow before being recirculated.
   • Components: Water pump, radiator, thermostat, fan, hoses, expansion tank, water jackets.
   • Advantages: More effective and consistent cooling; quieter operation; allows for tighter engine tolerances.
   • Disadvantages: More complex, heavier, potential for leaks, requires more maintenance (coolant checks, flushing).
   • Applications: Most modern automotive engines.

Key Components of Liquid Cooling Systems

• Water Jackets: Passages surrounding the cylinders and combustion chambers.
• Water Pump: Circulates the coolant.
• Radiator: Heat exchanger where coolant releases heat to the atmosphere.
• Thermostat: Controls coolant flow to the radiator, regulating engine temperature. It stays closed when the engine is cold to allow faster warm-up and opens as the engine reaches operating temperature.
• Cooling Fan: Draws air through the radiator, especially at low vehicle speeds.
• Hoses: Connect engine components to the radiator.
• Antifreeze: Prevents freezing in cold weather and raises the boiling point of the coolant.

Lubrication Systems in IC Engines

Lubrication is crucial for reducing friction and wear between moving parts, dissipating heat, cleaning the engine, and preventing corrosion.

Functions of Lubrication

• Reduces Friction and Wear: Forms a thin film between surfaces, preventing direct metal-to-metal contact.
• Cools Engine Parts: Carries away heat generated by friction and combustion.
• Cleans Engine: Suspends and carries away carbon deposits and metallic particles.
• Seals Gaps: Fills the small gaps between piston rings and cylinder walls, improving compression and preventing blow-by.
• Protects Against Corrosion: Prevents rust and corrosion on metal surfaces.

Types of Lubrication Systems

1. Splash Lubrication:
   • Description: Oil is contained in a sump. A dipper (often attached to the connecting rod) splashes oil onto moving parts. It's suitable for simpler engines with lower operating speeds.
   • Limitations: Inconsistent lubrication, especially at high speeds or angles.
2. Pressure Lubrication (Forced Lubrication):
   • Description: An oil pump draws oil from the sump and forces it through oil galleries to lubricate critical components like crankshaft bearings, camshaft bearings, and piston pins.
   • Components: Oil sump, oil pump, oil filter, oil galleries, pressure relief valve.
   • Advantages: Provides a constant supply of oil under pressure to all vital parts, ensuring reliable lubrication.
   • Applications: Most modern automotive and industrial engines.
3. Wet Sump Lubrication:
   • Description: The oil reservoir (sump) is an integral part of the engine crankcase. The oil pump picks oil directly from this sump.
   • Advantages: Simple, compact.
   • Disadvantages: Oil supply can be interrupted during extreme acceleration/deceleration or steep inclines; oil can become contaminated with combustion byproducts.
4. Dry Sump Lubrication:
   • Description: The oil is stored in a separate tank. Multiple oil pumps scavenge oil from the crankcase and deliver it to the tank, and another pump delivers oil from the tank to the engine components.
   • Advantages: Ensures consistent oil supply regardless of engine orientation or acceleration; allows the engine to be mounted lower, reducing the center of gravity; provides better cooling of the oil.
   • Applications: High-performance engines (sports cars, racing engines), aircraft engines, large industrial engines.

Lubricating Oil Properties

• Viscosity: Resistance to flow. It's crucial for maintaining an oil film. Multi-grade oils (e.g., 10W-40) are designed to have different viscosities at different temperatures. The 'W' stands for 'Winter', indicating viscosity at low temperatures.
• Viscosity Index (VI): A measure of how much the viscosity of oil changes with temperature. A higher VI means less change in viscosity with temperature.
• Flash Point: The lowest temperature at which oil vapor will ignite momentarily when exposed to an open flame. A higher flash point is desirable.
• Pour Point: The lowest temperature at which oil will still flow.
• Oxidation Stability: Resistance to breaking down due to heat and oxygen.
• Detergency: Ability to keep engine parts clean by preventing deposit formation.
• Dispersancy: Ability to keep contaminants (like soot) suspended in the oil, preventing them from forming sludge.