## Atkinson Cycle – Processes

In an **Atkinson cycle** (modified Otto cycle), the system executing the cycle undergoes a series of four processes: two isentropic (reversible adiabatic) processes alternated with one isochoric process and one isobaric process:

**Isentropic compression**(compression stroke) – The gas (fuel-air mixture) is compressed adiabatically from state 1 to state 2, as the piston moves from intake valve closing point (1) to top dead center. The surroundings do work on the gas, increasing its internal energy (temperature) and compressing it. On the other hand the entropy remains unchanged. The changes in volumes and its ratio (*V*_{1}*/ V*) is known as the compression ratio. The compression ratio is smaller than the expansion ratio._{2}**Isochoric compression**(ignition phase) – In this phase (between state 2 and state 3) there is a constant-volume (the piston is at rest ) heat transfer to the air from an external source while the piston is at rest at top dead center. This process is similar to the isochoric process in the Otto cycle. It is intended to represent the ignition of the fuel–air mixture injected into the chamber and the subsequent rapid burning. The pressure rises and the ratio (*P*_{3}*/ P*) is known as the “explosion ratio”._{2}**Isentropic expansion**(power stroke) – The gas expands adiabatically from state 3 to state 4, as the piston moves from top dead center to bottom dead center. The gas does work on the surroundings (piston) and loses an amount of internal energy equal to the work that leaves the system. Again the entropy remains unchanged. The volume ratio (*V*_{4}*/ V*) is known as the isentropic expansion ratio._{3}**Isobaric exhaust (exhaust stroke)**– The main goal of the modern Atkinson cycle is to allow the pressure in the combustion chamber at the end of the power stroke to be equal to atmospheric pressure. Since there can be atmospheric pressure in the chamber, then there is no decompression as in an Otto cycle. The piston moves from bottom dead center (BDC) to top dead center (TDC) and the cycle passes points**4 → 1 → 0.**In this stroke the exhaust valve is open while the piston pulls an exhaust gases out of the chamber.

During the **Atkinson cycle**, work is done on the gas by the piston between states 1 and 2 (**isentropic compression**). Work is done by the gas on the piston between stages 3 and 4 (**isentropic expansion**). The difference between the work done by the gas and the work done on the gas is the net work produced by the cycle and it corresponds to the area enclosed by the cycle curve. The work produced by the cycle times the rate of the cycle (cycles per second) is equal to the power produced by the Atkinson engine.

## Isentropic Process

An **isentropic process** is a** thermodynamic process**, in which the **entropy** of the fluid or gas remains constant. It means the **isentropic process** is a special case of an **adiabatic process** in which there is no transfer of heat or matter. It is a **reversible adiabatic process**. The assumption of no heat transfer is very important, since we can use the adiabatic approximation only in **very rapid processes**.

**Isentropic Process and the First Law**

For a closed system, we can write the **first law of thermodynamics in terms of enthalpy**:

**dH = dQ + Vdp**

**or**

**dH = TdS + Vdp**

**Isentropic process (dQ = 0):**

**dH = Vdp → W = H**_{2}** – H**_{1}** → H**_{2}** – H**_{1}** = C**

_{p}

*(T*

_{2}

*– T*

_{1}

*)**(for ideal gas)*

**Isentropic Process of the Ideal Gas**

The **isentropic process** (a special case of adiabatic process) can be expressed with the **ideal gas law** as:

*pV ^{κ} = constant*

or

**p _{1}V_{1}^{κ} = p_{2}V_{2}^{κ}**

in which **κ = c _{p}/c_{v}** is the ratio of the

**specific heats**(or

**heat capacities**) for the gas. One for

**constant pressure (c**

_{p}**)**and one for

**constant volume (c**

_{v}**)**. Note that, this ratio

**κ**

**= c**is a factor in determining the speed of sound in a gas and other adiabatic processes.

_{p}/c_{v}## Isochoric Process

An** isochoric process** is a thermodynamic process, in which the **volume** of the closed system **remains constant** (V = const). It describes the behavior of gas inside the container, that cannot be deformed. Since the volume remains constant, the heat transfer into or out of the system does not the p∆V work, but only changes the** internal energy** (the temperature) of the system.

**Isochoric Process and the First Law**

The classical form of the first law of thermodynamics is the following equation:

**dU = dQ – dW**

In this equation dW is equal to **dW = pdV** and is known as the boundary work. Then:

**dU = dQ – pdV**

In **isochoric process** and the **ideal gas**, all of heat added to the system will be used to increase the internal energy.

**Isochoric process (pdV = 0):**

**dU = dQ ***(for ideal gas)*

**dU = 0 = Q – W → W = Q **

*(for ideal gas)*

**Isochoric Process of the Ideal Gas**

The **isochoric process** can be expressed with the **ideal gas law** as:

or

On a **p-V diagram**, the process occurs along a horizontal line that has the equation V = constant.

See also: Guy-Lussac’s Law

## Isobaric Process

An** isobaric process** is a thermodynamic process, in which the **pressure** of the system **remains constant** (p = const). The heat transfer into or out of the system does work, but also changes the internal energy of the system.

Since there are changes in internal energy (dU) and changes in system volume (∆V), engineers often use the** enthalpy** of the system, which is defined as:

**H = U + pV**

**Isobaric Process and the First Law**

The classical form of the first law of thermodynamics is the following equation:

**dU = dQ – dW**

In this equation dW is equal to **dW = pdV** and is known as the boundary work. In an isobaric process and the ideal gas, **part of heat added** to the system will be used to **do work** and **part of heat** added will increase the **internal energy** (increase the temperature). Therefore it is convenient to use the **enthalpy** instead of the internal energy.

**Isobaric process (Vdp = 0):**

**dH = dQ → Q = H**_{2}** – H**_{1}

**At constant entropy**, i.e. in isentropic process, the **enthalpy change** equals the **flow process work** done on or by the system.

**Isobaric Process of the Ideal Gas**

The **isobaric process** can be expressed with the **ideal gas law** as:

or

On a **p-V diagram**, the process occurs along a horizontal line (called an isobar) that has the equation p = constant.

See also: Charles’s Law

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