Dynamic Otto Cycle Analysis

Engineering students encounter the Otto cycle in their first course in thermodynamics (usually during the sophomore year). This cycle is the theoretical basis for the spark ignition (SI) internal combustion engine (ICE). The traditional analysis (the air-standard analysis) of the Otto cycle is a static thermodynamic analysis that cannot be used to predict the dynamic performance of a SI ICE. Given sufficient information, the work per cycle for a particular engine can be computed. However, by making three simple modifications, the air-standard analysis can be extended to include a computation of the dynamic performance of a SI ICE. The first of these modifications is the selection of representative values of specific heats and specific heat ratios for the working fluid during each process. This improves the accuracy of the analysis. The second is an equation relating the heat release during combustion to pertinent engine parameters (the fuel-air ratio and the compression ratio). The third is the inclusion of an equation for the volumetric efficiency of the engine as a function of engine speed. This incorporates into the analysis the single most significant loss and results in performance that is dependent on engine speed. The resulting analysis predicts the dynamic performance (power and torque as a function of engine speed) of contemporary SI ICE engines with reasonable accuracy. Most importantly, this analysis can be easily understood and conducted by engineering students in their first thermodynamics course. Students have used this analysis, with excellent results, to analyze typical engines for a variety of applications (various types of passenger cars, pick-up trucks, SUV’s, Formula 1 vehicles and, even, “monster” trucks). Background The engine used for most contemporary motor vehicles is the four-stroke spark-ignition (SI) internal combustion engine (ICE). The engine typically has 4, 6 or 8 cylinders. The SI ICE combines non-flow and semiflow thermodynamic processes. The four strokes, which occur for each cylinder over two revolutions of the engine’s crankshaft, are the intake stroke, the compression stroke, the power (expansion) stroke and the exhaust stroke. Combustion of fuel and air occurs as the compression stroke ends and the power stroke begins. These processes and their thermodynamic modeling are discussed in detail in books on thermodynamics and internal combustion engines. The theoretical thermodynamic model for the SI ICE is the Otto cycle. The Otto cycle is shown on pressure-volume coordinates in Figure 1. It is a stationary, closed thermodynamic cycle consisting of the following four internally reversible processes: isentropic compression (1-2), constant volume heat addition (2-3), isentropic expansion (3-4) and constant volume heat rejection (4-1). The idealized Otto cycle includes the following five assumptions (referred to P ge 540.1 as the “cold-air-standard” assumptions) that are made to simplify the analysis: 1. the working fluid is air (at ambient temperature and pressure at state 1), 2. air behaves as an ideal gas, 3. air has constant specific heats, determined at 25C, 4. the combustion process is replaced by external heating and 5. the exhaust/intake processes are replaced by external cooling. The net work produced by the idealized Otto cycle can be computed through a First Law of Thermodynamics analysis. The mass of air contained in a cylinder of an engine modeled by the idealized Otto cycle is a fixed value (independent of engine speed), dependent only on engine geometry and the ambient temperature and pressure. As a consequence, the work done per cycle is a constant and the power output of such an engine varies linearly with the engine speed (i.e., the number of cycles per second). The Dynamic Otto Cycle Analysis The goal of this study was to develop a simple First Law of Thermodynamics analysis that would predict, with reasonable accuracy, performance curves (power vs. engine speed and torque vs. engine speed) for contemporary automotive engines. The dynamic Otto cycle analysis developed in this study uses the assumptions of the idealized Otto cycle described above, with just two exceptions. First, and most importantly, the mass of air in the cylinder is dependent on the engine speed as well as engine geometry and ambient temperature and pressure. Second, the specific heats used in the analysis are assumed constant for each process but their numerical values are determined at the approximate mean temperatures for each process. In addition, the quantity of heat transferred during the heating process is related to engine parameters. The power output of a four-stroke SI ICE is ( ) c net c W N N W , 2 . = (1) where . W = the power output of engine, kW, c N = the number of cylinders in the engine, N = the engine speed (crankshaft rotations per second), Hz, and c net W , = the net work produced by one cylinder during two revolutions of the crankshaft (i.e., for one power stroke), kJ . Figure 1. The Otto Cycle P-V Diagram Volume P re ss ur e