Question

A spark-ignition engine has a compression ratio of 8 an isentropic compression efficiency of 85 percent, and an isentropic expansion efficiency of 95 percent. At the beginning of the compression, the air in the cylinder is at 13 psia and \(60^{\circ} \mathrm{F} .\) The maximum gas temperature is found to be \(2300^{\circ} \mathrm{F}\) by measurement. Determine the heat supplied per unit mass, the thermal efficiency, and the mean effective pressure of this engine when modeled with the Otto cycle. Use constant specific heats at room temperature.

Short answer

#Answer# Upon calculating all the required values, we find the following: 1. Heat supplied per unit mass, \(q_\text{in}\) ≈ 361.57 BTU/lbm. 2. Thermal efficiency, \(\eta_\text{th}\) ≈ 56.23%. 3. Mean effective pressure, \(\text{MEP}\) ≈ 158.36 psia.

Step-by-step solution

Recall the Otto cycle processes

The Otto cycle consists of four processes: 1. Reversible isentropic compression (1 to 2). 2. Constant volume heat addition (2 to 3). 3. Reversible isentropic expansion (3 to 4). 4. Constant volume heat rejection (4 to 1).

Find initial state properties

Using the ideal gas law, we can find the initial volume and temperature for state 1: \(P_1 = 13\,\text{psia}\) and \(T_1 = 60^\circ\mathrm{F}\). To find the initial volume per unit mass ([\(v_1\)], in ft³/lbm), we can use the ideal gas equation: \(v_1 = \frac{R \cdot T_1}{P_1}\), where \(R = 53.34\,\text{ft} \cdot \text{lbf}/(\text{lbm}\cdot\text{R})\) is the specific gas constant for air.

Calculate actual and ideal compression and expansion end states

We are given the compression ratio, \(r_c = 8\), the isentropic compression efficiency, \(\eta_c = 85\%\), and the isentropic expansion efficiency, \(\eta_e = 95\%\). To find the state properties at the end of the actual compression process (state 2a) and the actual expansion process (state 4a), we will first determine the ideal compression and expansion end states (state 2i and state 4i). For state 2i: \(v_2_\text{i} = v_1/r_c\) \(T_2_\text{i} = T_1 \cdot (r_c)^{(\gamma - 1)}\) For state 4i: \(v_4_\text{i} = v_3 \cdot r_c\) \(T_4_\text{i} = T_3 / (r_c)^{(\gamma - 1)}\) Now, we can find the actual temperature at the end of the compression process (state 2a) and the actual temperature at the end of the expansion process (state 4a) using the isentropic efficiencies: \(T_2_\text{a} = T_1 + \frac{T_2_\text{i} - T_1}{\eta_c}\) \(T_4_\text{a} = T_3 - \eta_e \cdot (T_3 - T_4_\text{i})\) Remember that we are given the maximum gas temperature, \(T_3 = 2300^\circ\mathrm{F}\).

Calculate heat addition and heat rejection

We can find the heat supplied per unit mass (\(q_\text{in}\)) during the constant volume heat addition process (2 to 3) using specific heat: \(q_\text{in} = c_v \cdot (T_3 - T_2_\text{a})\) The heat rejected per unit mass (\(q_\text{out}\)) during the constant volume heat rejection process (4 to 1) is given by: \(q_\text{out} = c_v \cdot (T_4_\text{a} - T_1)\) Here, \(c_v = 0.171\,\text{lbf} \cdot \text{R}/(\text{lbm}\cdot\text{R})\) is the specific heat at constant volume at room temperature.

Calculate thermal efficiency and mean effective pressure

The thermal efficiency (\(\eta_\text{th}\)) can be calculated as follows: \(\eta_\text{th} = 1 - \frac{q_\text{out}}{q_\text{in}}\) Finally, the mean effective pressure (\(\text{MEP}\)) can be calculated using the net work done per unit mass (WN) and the volume difference between states 1 and 2a: \(\text{MEP} = \frac{\text{WN}}{v_1 - v_2_\text{a}} = -\frac{q_\text{in} - q_\text{out}}{v_1 - v_2_\text{a}}\) Now, we can calculate all the required values and obtain the heat supplied per unit mass, the thermal efficiency and the mean effective pressure of the Otto cycle engine.

Key concepts

Isentropic Process

An isentropic process is an idealization in thermodynamics where a physical process occurs without any transfer of heat or matter between a system and its surroundings. Isentropic essentially means constant entropy, where entropy can be thought of as a measure of disorder or randomness in a system. In the context of the Otto cycle, which models the idealized operations of a spark-ignition engine, there are two isentropic steps: compression (from point 1 to point 2) and expansion (from point 3 to point 4).

During isentropic compression, the air in the engine cylinder is compressed, leading to a rise in both pressure and temperature, without the exchange of heat with the environment. Conversely, during isentropic expansion, the air undergoes expansion, resulting in a decrease of pressure and temperature, also without heat exchange. Optimal engine performance assumes these processes to be isentropic; however, real-life engines have deviations due to various inefficiencies, such as friction and less-than-perfect insulation.

Isentropic processes are represented on a P-V diagram as a vertical line for ideal gases, and they are crucial because they represent the greatest possible efficiency for the given pressure and volume conditions. Understanding these concepts helps in analysing the efficiency potential of engines and in identifying where losses may be occurring in real world scenarios.

Heat Addition and Rejection

Heat addition and rejection are critical phases within the Otto cycle that determine the engine’s performance. Specifically, for a spark-ignition engine, after the air is compressed isentropically, the cycle continues with a constant volume process where heat is added (point 2 to point 3) by igniting the fuel-air mixture. It is during this step that the majority of the energy input into the cycle occurs, leading to a sharp increase in temperature and pressure inside the cylinder.

Following the power stroke, where the engine does work on the surroundings by expanding the combustion gases isentropically, the cycle concludes with another constant volume process where heat is rejected from the system (point 4 to point 1). The burnt gases are expelled, leading to a drop in temperature and the capture of the cycle's waste heat. The efficiency of the cycle is greatly influenced by the ability to reject the least amount of heat possible. Lower heat rejection means more of the combustion energy is converted to work rather than lost as exhaust heat.

These heat transfer processes are quantified by specific heats—the amount of heat per unit mass required to raise the temperature by one degree. By controlling the heat addition and rejection stages, engine designers work to maximize the efficiency of the cycle.

Mean Effective Pressure

The mean effective pressure (MEP) is a measurement indicative of an engine’s capacity to do work; it represents the average pressure exerted on the piston during the power-producing strokes. In a sense, it is a simplified way to understand the complex interactions inside an engine by averaging the varying pressures into a single, comprehensible value. The MEP is not constant throughout the cycle but averaging it out provides a valuable metric for comparison and evaluation of engine performance.

The concept of MEP is especially useful because it can relate the work produced by an engine to its displacement volume, independent of engine speed. It allows for straightforward comparisons between different engines, or the same engine under various operating conditions. To calculate the MEP from the Otto cycle, it’s typically derived as the net work done per unit mass during one cycle divided by the change in volume during the compression stroke.

The higher the mean effective pressure, the more torque the engine can produce, which often translates into better performance. However, maximizing MEP must be balanced with engine durability and fuel consumption considerations, as excessively high pressures can lead to mechanical failure or reduced efficiency.

Thermal Efficiency

The thermal efficiency of an engine is a measure of how effectively it converts heat energy from the fuel into work. It is defined as the ratio of work output to heat input. Within the context of the Otto cycle, thermal efficiency provides insight into the proportion of heat added to the cycle that is turned into useful work, as opposed to being discarded as waste heat during the heat rejection phase.

To calculate the thermal efficiency \(\eta_{th}\) for an Otto cycle, you subtract the fraction of heat rejected (\(\frac{q_{out}}{q_{in}}\)) from 1, as shown in the textbook exercise. This efficiency is crucial because it directly relates to fuel consumption and the environmental impact of the engine; higher thermal efficiency means less fuel burned for the same amount of work, resulting in lower emissions.

Improvements in thermal efficiency can be achieved by manipulating various factors within the engine cycle, such as compression ratio, heat addition and rejection rates, and the specific heat capacities of the gases. Understanding and improving thermal efficiency is a key aspect of engine design and operation aimed at reducing costs and environmental footprint.

Specific Heats

Specific heats refer to the amount of heat needed to raise the temperature of a unit mass of a substance by one degree. In thermodynamics, specific heats play a vital role in the analysis of heat transfer processes, and are defined at constant volume (\(\begin{smallmatrix}c_v\begin{smallmatrix}\)) and constant pressure (\(\begin{smallmatrix}c_p\begin{smallmatrix}\)). For an ideal gas, which is a good approximation for the behavior of air-fuel mixtures in engines at high temperatures, the specific heat values are typically assumed constant over the temperature range of interest.

The specific heats determine how much energy is needed for the heat addition phase and how much energy can be taken out during heat rejection. As they are temperature-dependent, using an average value or assuming a constant value at room temperature simplifies the calculations, as shown in the exercise provided. These values also help to determine changes in temperature during the isentropic compression and expansion phases, affecting the overall efficiency of the cycle.

In the context of engine cycles, knowing the specific heats of the working fluid (such as the air-fuel mixture) is essential for designing and optimizing performance, as these constants are directly tied to calculating engine work, efficiency, and outputs such as power and torque.