Air-Standard Analysis of Diesel Cycle: An Ideal Power Cycle for Diesel Engine

An ideal power cycle that studies the diesel engine under the cold air-standard assumptions (as articulated in general in our previous articles) is called the ideal diesel cycle. It is shown in the P-V and T-s diagrams below:

P V Diagram
T s diagram

An ideal diesel cycle is approximated to consist of four internally reversible processes, which are mentioned below:

  1. Process 1-2: Isentropic Compression of the fixed mass of air when the piston moves from the bottom dead center to the top dead center. This process is the same as in the Otto cycle.
  2. Process 2-3: Isobaric Heat Addition of the fixed mass of air from an external source when the piston is at the top dead center. It signifies the ignition of the hot air in the real diesel engine and the subsequent rapid burning.
  3. Process 3-4: Isentropic expansion during the power stroke. This process is the same as in the Otto cycle.
  4. Process 4-1: Isochoric Heat Rejection from the hot air when the piston is at the bottom dead center. It represents the end of the complete thermodynamic power cycle of an ideal diesel engine.

In the ideal diesel engine, heat is approximated to be added at constant pressure. Why? It is because the combustion process in the real diesel engine is not instantaneous as in the case of an ideal Otto engine, due to which there exists a finite combustion duration, which typically lasts for several degrees of crankshaft rotation, rendering its simulation possible at constant pressure.

Along the same lines, one can ask: why heat rejection at constant volume, as in the case of the Otto cycle? It can be responded by relating it to the existence of the real-time constant volume of the combustion chamber when the piston is at the top dead center when burned gases leave the opened exhaust valve/s during the exhaust process.

Likewise, isentropic compression and expansion of air in the ideal diesel cycle imply that both compression and expansion are adiabatic and fully reversible processes that represent the accurate thermodynamic behavior of the ideal gas.

Interpretation of the P-V and T-s Diagram of an Air-Standard Diesel Cycle

The area under the P-V and T-s diagram shows the work done and net heat supplied during the cycle, respectively.

PV-Diagram: As shown on the P-V plot, the area under curve 1-2-a-b-1 represents the work done per unit of mass during the compression process, whereas the area under curve 2-3-4-b-a-2 shows the work done per unit of mass during the expansion process.

The enclosed area of the P-V diagram shows the net work done per unit of mass.

T-s Diagram: As shown on the T-s plot, the area under the curve, 2-3-a-b-2, shows the heat supplied per unit of mass, whereas the area below the curve, 1-4-a-b-1, shows the net heat rejected per unit of mass.

The enclosed area of the T-s diagram shows the net heat supplied per unit of mass.

Because the heat addition process takes place at the constant pressure rather than at constant volume, as in the Otto cycle, the added heat performs two functions: it does some work on air by displacing it. It increases the internal energy of the molecules of the air.

q_{in}=w_b+\triangle u=P_2\triangle v+\triangle u=P_2\left(v_3-v_2\right)+\left(u_3-u_2\right)=\left(P_2v_3+u_3\right)-\left(P_2v_2+u_2\right)
q_{in}=h_3-h_2=c_p\left(T_3-T_2\right)

Where P2=P3 and h stands for enthalpy at the given state.

Likewise, the heat rejection process takes place at a constant volume akin to that of an ideal Otto cycle; the rejection of heat lowers the internal energy of the air. Thus;

q_{out}=u_4-u_1=c_v\left(T_4-T_1\right)

The following expression gives the thermal efficiency:

n_{otto\;cycle}=\frac{W_{net}}{q_{in}}=\frac{q_{in}-q_{out}}{q_{in}}=1-\frac{q_{out}}{q_{in}}=1-\frac{T_4-T_1}{k\left(T_3-T_2\right)}

(Eq. 1)

Let’s take the assumptions of the diesel cycle process-wise to simplify the above expression in order to find out the values of T1, T3, and T4 in terms of T2.

Process 1-2: Isentropic Compression

\frac{T_1}{T_2}=\left(\frac{V_2}{V_1}\right)^{k-1}

or

T_1=T_2\left(\frac1{r^{k-1}}\right)

(Eq. 1a)

where

r=\frac{v_1}{v_2}

Process 2-3: Isobaric Heat Addition

\frac{v_3}{T_3}=\frac{v_2}{T_2}

or

T_3=T_2\left(\frac{v_3}{v_2}\right)=T_2\;r_c

(Eq. 2a)

Where rc is called the cutoff ratio and is defined as the ratio of cylinder volume after and before the combustion.

Process 3-4: Isentropic Expansion

\frac{T_4}{T_3}=\left(\frac{v_3}{v_4}\right)^{k-1}=\left(\frac{v_3}{v_2}\times\frac{v_2}{v_4}\right)^{k-1}=\left(\frac{r_c}r\right)^{k-1}

or

T_4={T_3\;\left(\frac{r_c}r\right)^{k-1}}

(Eq. 3a)

Putting all the values of T1, T3, and T4 from equations 1a, 2a, and 3a in equation 1’s final expression and solving gives the following form of the thermal efficiency of the diesel engine.

\eta_{diesel}=1-\frac1{r^{k-1}}\left[\frac{r_c^r-1}{k\left(r_c-1\right)}\right]

Analysis

The final expression of the thermal efficiency of an ideal diesel engine, as formulated above, shows the thermal performance or efficiency of a real diesel engine as the function of the compression ratio and the cutoff ratio of the engine cylinder. The higher the values of r and RC, the higher the thermal efficiency of the diesel engine.

Upon close reading of the above expression, one confirms that the efficiency of the diesel engine differs from that of a petrol engine by the expression in the bracket, whose value is always greater than 1. To this, one can estimate that for a set of petrol engines and diesel engines of equal size (equal cylinder volume):

\eta_{otto}>\eta_{diesel}

For an exceptional yet limiting case, if rc=1, then the equation of the thermal efficiency of a diesel engine takes the following expression.

\eta_{diesel}=\eta_{otto}

Diesel engines usually operate at much higher compression ratios than their spark-ignited counterparts. Therefore, they prove to be more efficient if the comparison is made between the two, irrespective of the engine size.

In diesel engines, the fuel burns more completely and fully because combustion (and the overall entire cycle) takes place at relatively lower revolutions of crankshaft per minute.

The air-fuel ratio is higher in diesel engines than in petrol engines.

Due to the cheaper cost of diesel fuel and higher thermal efficiency, diesel engines are used where the demand for high power is an inevitable necessity, such as in the case of locomotives, electric generator engines used in emergencies, large ships, and so forth.