Diesel cycle
The ideal cycle of the diesel engine (as opposed to the fast cycle, which is closer to reality) of four strokes is an idealization of the gauge diagram of a diesel engine, omitting the charge renewal phases, and it is assumed that the thermodynamic fluid that evolves is a perfect gas, generally air. Furthermore, it is accepted that all processes are ideal and reversible, and that they are carried out on the same fluid. Although all this leads to a very approximate model of the actual behavior of the engine, it allows at least a series of qualitative conclusions to be drawn with respect to this type of engine. It should not be forgotten that the large marine and railway traction engines are from the 2-stroke diesel cycle.
Phases
1. Compression, process 1-2: is a process adiabatic compression reversible (isentropic), that is without heat exchange with the outside and with a work done to the system to compress it. The piston, being at the dead point, starts its promotional career, compressing the air contained in the cylinder. This elevates the thermodynamic state of the fluid, increasing its pressure, temperature and decreasing its specific volume. In idealization, the process is governed by the equation of the adiabatic process P⋅ ⋅ vk=cte{displaystyle Pcdot v^{k}=cte}. The pressure at point 2 will be worth:
- P1⋅ ⋅ V1k=P2⋅ ⋅ V2k{displaystyle P_{1}cdot V_{1}^{k}=P_{2}cdot V_{2}^{k}}{k}}
- P2=P1⋅ ⋅ V1kV2k{displaystyle P_{2}={frac {P_{1}cdot V_{1⁄4}{k}}{V_{2}{2}{k}}}}}}}}
as
- V1kV2k=roga roga k{displaystyle {frac {V_{1}{k}}{V_{2}{k}}}}}}=xi ^{k}}{k}}
- P2=P1⋅ ⋅ roga roga k{displaystyle P_{2}=P_{1}cdot xi ^{k}
and the temperature:
- T1⋅ ⋅ V1k− − 1=T2⋅ ⋅ V2k− − 1{displaystyle T_{1}cdot V_{1}^{k-1}=T_{2}cdot V_{2}^{k-1}}}
- T2=T1⋅ ⋅ V1k− − 1V2k− − 1{displaystyle T_{2}={frac {T_{1}cdot V_{1}{k}}{V_{2}{2}{k-1}}}}}}
- T2=T1⋅ ⋅ roga roga k− − 1{displaystyle T_{2}=T_{1}cdot xi ^{k-1}
roga roga {displaystyle xi } = compression ratio: is the relationship between the final and initial volumes.
k{displaystyle k} = adiabatic exponent: is the reason between constant pressure heat capacity (Cp{displaystyle C_{p}}) and constant volume heat capacity (Cv{displaystyle C_{v}}).
2. Combustion, process 2-3: In this idealization, the heat input Qp is simplified by an isobaric process (at constant pressure). However, Diesel combustion is much more complex: in the area of top dead center (TDC) (generally a little before reaching it due to problems related to the thermal inertia of the fluids, that is, the delay between the injection and spontaneous ignition), fuel injection begins (in car engines, diesel, although it is enough that the fuel is sufficiently self-igniting and not very volatile). The injector sprays and beads "atomizes" the fuel, which, in contact with the atmosphere inside the cylinder, begins to evaporate. Since the fuel of a diesel engine has to be very self-igniting (high knocking power, high Cetane number), it happens that, long before the injection of all the fuel has finished, the first drops of injected fuel self-ignite and they start a first combustion characterized by being very turbulent and imperfect, as the mixture of air and fuel has not had enough time to become homogeneous. This stage is very fast, and in the present cycle it is ignored, but not so in the so-called fast Diesel cycle, in which it is symbolized as an isochoric compression at the end of the compression. Subsequently, on the fresh mass that has not been burned, a second combustion is given, called combustion by diffusion, much slower and more perfect, which is what is simplified here by an isobaric process. In this combustion by diffusion, around 80% of the fresh mass is usually burned, hence the previous stage is usually ignored. However, it is also true that the vast majority of the pressure work and the losses and irreversibilities of the cycle occur in the initial combustion, so simply omitting it will only lead to an imperfect model of the Diesel cycle. The consequence of combustion is the sudden rise in the thermodynamic state of the fluid, actually due to the chemical energy released in the combustion, and which in this model must be interpreted as heat that the thermodynamic fluid receives, and as a result of which it expands in a reversible isobaric process.
- Pb=Pc{displaystyle Pb=Pc}
- VbTb=VcTc{displaystyle {frac {Vb}{Tb}}}={frac {Vc}{Tc}}}}}}{
- rc=VcVb{displaystyle rc={frac {Vc}{Vb}}}}}
- Qa=mCp(Tc− − Tb){displaystyle Qa=mCp(Tc-Tb)}
- W=Qa− − Qr{displaystyle W=Qa-Qr}
- 3. Explosion/Expansion, process 3-4: simplified by a istropic expansion (adiabatic) of the thermodynamic fluid, to the specific volume that was at the beginning of the compression. In reality, the expansion occurs as a result of the high thermodynamic state of gases after combustion, which push the piston from the PMS to the PMI, producing a job. Note how, as in every engine cycle of four times or two times, only in this race, in expansion, is produces a job.
Pd⋅ ⋅ Vdk=Pc⋅ ⋅ Vck{displaystyle Pdcdot Vd^{k}=Pccdot Vc^{k}}}
Expansion ratio:
- re=VaVc{displaystyle re={frac {Va}{Vc}}}
Compression ratio:
- rx=(Rc)(Re){displaystyle rx=(Rc)}
4. Last stage, process 4-1: this stage is an isochoric process (exhaust) that is to say at constant volume. From the final expansion pressure to the initial compression pressure. Strictly speaking, it lacks any physical meaning, and is simply used ad hoc, in order to close the ideal cycle. However, there are authors who are not satisfied with all the idealizations made, insist on giving a physical meaning to this stage, and associate it with the renewal of the load, because, they reason, this is what occurs in the two preceding races. compression and follow expansion: burnt-mass exhaust and fresh-mass intake. However, the escape is a process that requires much more work than that involved in this process (none), and also neither of the two processes occurs, even remotely, at a constant specific volume.
- Vd=Va{displaystyle Vd=Va}
- PdTd=PaTa{displaystyle {frac {Pd}{Td}}}{frac {Pa}{Ta}}}}}
- Qr=mCr(Td− − Ta){displaystyle Qr=mCr(Td-Ta)}
Ways to calculate Fraction of Stroke at Closing:
- Fc=PaPme{displaystyle Fc={frac {Pa}{Pme}}}
- Fc=(rc)(Vb)− − Vb(rk)(Vb)− − Vb{displaystyle Fc={frac {(rc)(Vb)-Vb}{(rk)(Vb)-Vb}}}}}
Where:
Pa: Initial pressure
Pme: mean effective pressure
rc: Intake Close Ratio
rk: Compression ratio
Vb: Volume b
It is important to note how, in the diesel cycle, the four times of the engine should never be confused with the thermodynamic cycle that idealizes it, which only refers to two of the times: the compression stroke and the expansion stroke. The charge renewal process falls outside of the diesel cycle processes, and is not even a thermodynamic process in the strict sense.
Maximum thermal efficiency
The maximum thermal efficiency in a diesel cycle depends on the compression ratio and the cut-off ratio. It is governed by the following formula under standard cold air analysis:
MIL MIL th=1− − 1rγ γ − − 1(α α γ γ − − 1γ γ (α α − − 1)){displaystyle eta _{th}=1-{frac {1}{r^{gamma -1}}}}}{left({alphac {alpha ^{gamma }-1}{gamma (alpha-1)}}{right)}
Where;
- MIL MIL th{displaystyle eta _{th}} is thermal efficiency
- α α {displaystyle alpha } is the cut relationship V3V2{displaystyle {frac {V_{3}}{V_{2}}}}}} (relation between the final and initial volume in the combustion phase)
- r is the compression ratio V1V2{displaystyle {frac {V_{1}}{V_{2}}}}}}
- γ γ {displaystyle gamma } is the specific heat ratio (Cp/Cv)
The shear ratio can be expressed in terms of temperature as shown below:
- T2T1=(V1V2)γ γ − − 1=rγ γ − − 1{displaystyle {frac {T_{2}}{T_{1}}}}}={left({frac {V_{1}}{V_{2}}}}}{right)^{gamma -1}=r^{gamma -1}}}
- T2=T1rγ γ − − 1{displaystyle displaystyle {T_{2}}}={T_{1}}r^{gamma -1}}
- V3V2=T3T2{displaystyle {frac {V_{3}}{V_{2}}}}}}{{frac {T_{3}}{T_{2}}}}}}}}}}
- α α =(T3T1)(1rγ γ − − 1){displaystyle alpha =left({frac {T_{3}}}{T_{1}}}}}}{right)left({frac {1}{r^{gamma -1}}}}}}{right)}
T3{displaystyle T_{3}} can be approximated to the temperature of the flame of the fuel used. The temperature of the flame can be approximated to the diabatic temperature of fuel flame with the corresponding air-fuel ratio and compression pressure, p3{displaystyle p_{3}}. T1{displaystyle T_{1}} can approach the temperature of the inlet air.
This formula only calculates the ideal thermal efficiency. Actual thermal efficiency will be much lower due to heat and friction losses. The formula is more complex than that of the Otto cycle (gasoline engines) which has the following formula:
MIL MIL orttor,th=1− − 1rγ γ − − 1{displaystyle eta _{otto,th}=1-{frac {1}{r^{gamma -1}}}}}
