Chapter 3: The First Law of Thermodynamics for Closed Systems

c) The Air-Standard Diesel Cycle (Compression-Ignition) Engine

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 Adiabatic Process of an Ideal Gas (Q = 0)

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 W1-2 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 (Q1-2 = 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 Wexp = (W2-3 + W3-4) 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 Wnet done over the cycle is given by: Wnet = (Wexp + W1-2), where as before the compression work W1-2 is negative (work done on the system).

In the Air-Standard Diesel cycle engine the heat input Qin 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 Qout = -Q4-1. 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 Vk = constant, where k = 1.4, until it reaches a final temperature of 77°C.

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.

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 CP = 1.00 kJ/kg.K, Cv = 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 P2 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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