Derive Otto cycle efficiency and Mean Effective Pressure relation.
To derive the efficiency of the Otto cycle and the relation between the efficiency and mean effective pressure (MEP), we need to understand the concept and equations involved.
The Otto cycle is a thermodynamic cycle that represents the idealized behavior of gasoline engines. It consists of four processes: intake, compression, combustion/expansion, and exhaust. The cycle assumes the following idealizations:
1. Adiabatic processes: Compression and expansion processes are assumed to be adiabatic, meaning no heat enters or leaves the system.
2. Constant volume combustion: The combustion process occurs at constant volume, which is an idealization of the spark ignition process in an engine.
The efficiency of the Otto cycle (η) is defined as the ratio of the net work done by the system (W_net) to the heat input (Q_in):
η = W_net / Q_in
The net work done can be calculated by subtracting the work done during the compression process (W_comp) from the work done during the expansion process (W_exp):
W_net = W_exp - W_comp
To derive the efficiency, we need to calculate both W_exp and W_comp.
1. Work done during the expansion process (W_exp):
During the expansion process, the working fluid (air-fuel mixture) expands adiabatically and does work on the piston. The work done during this process is given by the equation:
W_exp = h3 - h4
where h3 is the enthalpy at the end of the combustion process (state 3) and h4 is the enthalpy at the end of the expansion process (state 4). These values can be obtained from the specific heat ratio (gamma, γ) and temperatures at states 3 and 4.
2. Work done during the compression process (W_comp):
During the compression process, the working fluid is compressed adiabatically, and work is done on the fluid. The work done during this process is given by the equation:
W_comp = h2 - h1
where h2 is the enthalpy at the end of the compression process (state 2) and h1 is the enthalpy at the beginning of the compression process (state 1).
The heat input (Q_in) can be calculated by subtracting the heat rejected during the exhaust process from the heat added during the combustion process:
Q_in = Q_comb - Q_rej
where Q_comb is the heat added during combustion, and Q_rej is the heat rejected during the exhaust process.
Once we have obtained the values for W_net and Q_in, we can substitute them into the efficiency equation to calculate the efficiency.
Now, let's discuss the relation between the efficiency and mean effective pressure (MEP). The MEP is defined as the average pressure acting on the piston during the power stroke. It is related to the net work done during the cycle.
The net work done (W_net) is directly proportional to the MEP. The relation between the two can be expressed as:
W_net = MEP * V_displacement
where V_displacement is the displacement volume of the engine.
Substituting this relationship into the efficiency equation, we get:
η = MEP * V_displacement / Q_in
Therefore, the efficiency of the Otto cycle is directly proportional to the MEP. An increase in MEP leads to an increase in efficiency, provided the heat input remains constant.
To summarize, to derive the efficiency of the Otto cycle, we need to calculate the work done during the expansion and compression processes and the heat input. Once we have these values, we can substitute them into the efficiency equation to obtain the efficiency. The efficiency is directly proportional to the mean effective pressure, indicating that an increase in MEP leads to an increase in efficiency, given a constant heat input.