The **Air
Standard Diesel cycle** is the ideal
cycle for **Compression-Ignition** (CI) reciprocating engines, first proposed by Rudolph
Diesel over 100 years ago. The following link by the **Kruse
Technology Partnership** describes the **four-stroke
diesel cycle** operation including a short history of Rudolf
Diesel. The four-stroke diesel engine is usually used in motor
vehicle systems, whereas larger marine systems usually use the
**two-stroke
diesel cycle**. Once again we have an excellent animation
produced by **Matt
Keveney** presenting the operation of the **four-stroke
diesel cycle**.

The actual CI cycle is extremely complex, thus in initial analysis we use an ideal "air-standard" assumption, in which the working fluid is a fixed mass of air undergoing the complete cycle which is treated throughout as an ideal gas. All processes are ideal, combustion is replaced by heat addition to the air, and exhaust is replaced by a heat rejection process which restores the air to the initial state.

The ideal air-standard diesel engine undergoes
4 distinct processes, each one of which can be separately analysed,
as shown in the *P-V* diagrams below. Two of the four processes
of the cycle are **adiabatic** processes (adiabatic = no transfer of heat), thus
before we can continue we need to develop equations for an ideal
gas adiabatic process as follows:

The analysis results in the following three general forms representing an adiabatic process:

where k is the ratio of heat capacities and has a nominal value of 1.4 at 300K for air.

Process 1-2 is the adiabatic compression process. Thus the temperature of the air increases during the compression process, and with a large compression ratio (usually > 16:1) it will reach the ignition temperature of the injected fuel. Thus given the conditions at state 1 and the compression ratio of the engine, in order to determine the pressure and temperature at state 2 (at the end of the adiabatic compression process) we have:

Work W_{1-2} required to compress the
gas is shown as the area under the *P-V* curve, and is evaluated
as follows.

An alternative approach using the energy equation
takes advantage of the adiabatic process (Q_{1-2} = 0) results in a much simpler process:

(thanks to student Nichole Blackmore for making me aware of this alternative approach)

During process 2-3 the fuel is injected and combusted and this is represented by a constant pressure expansion process. At state 3 ("fuel cutoff") the expansion process continues adiabatically with the temperature decreasing until the expansion is complete.

Process 3-4 is thus the adiabatic expansion
process. The total expansion work is W_{exp}
= (W_{2-3}
+ W_{3-4}) and
is shown as the area under the *P-V* diagram and is analysed
as follows:

Finally, process 4-1 represents the constant volume heat rejection process. In an actual Diesel engine the gas is simply exhausted from the cylinder and a fresh charge of air is introduced.

The net work W_{net} done over the
cycle is given by: W_{net} = (W_{exp} + W_{1-2}),
where as before the compression work W_{1-2} is negative
(work done *on* the system).

In the Air-Standard Diesel cycle engine the
heat input Q_{in} occurs by combusting the fuel which
is injected in a controlled manner, ideally resulting in a constant
pressure expansion process 2-3 as shown below. At maximum volume
(bottom dead center) the burnt gasses are simply exhausted and
replaced by a fresh charge of air. This is represented by the
equivalent constant volume heat rejection process Q_{out}
= -Q_{4-1}.
Both processes are analyzed as follows:

At this stage we can conveniently determine the engine efficiency in terms of the heat flow as follows:

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The following problems summarize this section:

**Problem 3.4 - **A frictionless piston-cylinder device contains 0.2
kg of air at 100 kPa and 27°C. The air is now compressed slowly
according to the relation *P V*^{}k = constant, where
k = 1.4, until it reaches a final temperature of 77°C.

- a) Sketch the
*P-V*diagram of the process with respect to the relevant constant temperature lines, and indicate the work done on this diagram. - b) Using the basic definition of boundary
work done determine the boundary
**work done**during the process [-7.18 kJ]. - c) Using the energy equation determine the
**heat transferred**during the process [0 kJ], and verify that the process is in fact adiabatic.

*Derive* all equations used starting with the basic energy
equation for a non-flow system, the equation for internal energy
change for an ideal gas (Δu), the basic equation for boundary
work done, and the ideal gas equation of state [*P.V = m.R.T*].
Use values of specific heat capacity defined at 300K for the entire
process.

**Problem 3.5 - **Consider the expansion stroke only of a typical Air
Standard Diesel cycle engine which has a compression ratio of
20 and a cutoff ratio of 2. At the beginning of the process (fuel
injection) the initial temperature is 627°C, and the air expands
at a constant pressure of 6.2 MPa until cutoff (volume ratio 2:1).
Subsequently the air expands adiabatically (no heat transfer)
until it reaches the maximum volume.

- a) Sketch this process on a
*P-v*diagram showing clearly all three states. Indicate on the diagram the total work done during the entire expansion process. - b) Determine the temperatures reached at
the end of the constant pressure (fuel injection) process [1800K], as
well as at the end of the expansion process [830K], and draw the three
relevant constant temperature lines on the
*P-v*diagram. - c) Determine the total work done during the expansion stroke [1087 kJ/kg].
- d) Determine the total heat supplied to the air during the expansion stroke [1028 kJ/kg].

*Derive* all equations
used starting from the ideal gas equation of state and adiabatic
process relations, the basic energy equation for a closed system,
the internal energy and enthalpy change relations for an ideal
gas, and the basic definition of boundary work done by a system
(if required). Use the specific heat values defined at 1000K for
the entire expansion process, obtained from the table of **Specific Heat
Capacities of Air**.

**Solved Problem 3.6
- **An ideal air-standard Diesel cycle
engine has a compression ratio of 18 and a cutoff ratio of 2.
At the beginning of the compression process, the working fluid
is at 100 kPa, 27°C (300 K). Determine the temperature and
pressure of the air at the end of each process, the net work output
per cycle [kJ/kg], and the thermal efficiency.

Note that the nominal specific heat capacity
values used for air at 300K are C_{P} = 1.00 kJ/kg.K,
C_{v} = 0.717 kJ/kg.K,, and k = 1.4. However they are
all functions of temperature, and with the extremely high temperature
range experienced in Diesel engines one can obtain significant
errors. One approach (that we will adopt in this example) is to
use a typical average temperature throughout the cycle.

**Solution Approach:**

The first step is to draw a diagram representing
the problem, including all the relevant information. We notice
that neither volume nor mass is given, hence the diagram and solution
will be in terms of specific quantities. The most useful diagram
for a heat engine is the *P-v* diagram of the complete cycle:

The next step is to define the working fluid
and decide on the basic equations or tables to use. In this case
the working fluid is air, and we have decided to use an average
temperature of 900K throughout the cycle to define the specific
heat capacity values as presented in the table of **Specific
Heat Capacities of Air**.

We now go through all four processes in order to determine the temperature and pressure at the end of each process.

Note that an alternative method of evaluating
pressure P_{2} is to simply use the ideal gas equation
of state, as follows:

Either approach is satisfactory - choose whichever you are more comfortable with. We now continue with the fuel injection constant pressure process:

Notice that even though the problem requests "net work output per cycle" we have only calculated the heat in and heat out. In the case of a Diesel engine it is much simpler to evaluate the heat values, and we can easily obtain the net work from the energy balance over a complete cycle, as follows:

You may wonder at the unrealistically high
thermal efficiency obtained. In this idealized analysis we have
ignored many loss effects that exist in practical heat engines.
We will begin to understand some of these loss mechanisms when
we study the Second Law in **Chapter
5**.

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**On to Part d) of
The First Law - Otto Cycle Engines**

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Engineering Thermodynamics by Israel Urieli is licensed under a
Creative Commons Attribution-Noncommercial-Share
Alike 3.0 United States License