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# Theromodynamics

## An Investigation of the Thermodynamic Properties of Engine Cycles

##### Authors: Michael Kalan, Luke Pecha, Thaddeus Howard

September 30, 2014

I. Abstract
The objective of this project was to create a model for three different types of engine cycles: Carnot Cycle, Otto Cycle, and Brayton Cycle. These models will simulate the different steps of each process and calculate the energy and work done. The models were all initially computed using hydrogen as a fuel source. Real data was then obtained that allowed the models to be adjusted to match more realistic situations.

II. Introduction
Since the beginning of the 19th century, man has been working to harness the potential power that can be produced from the heat engine. In 1823, Nicolas Leonard Sadi Carnot first proposed the theoretical thermodynamic cycle, which produces heat that can be harnessed to do work. The general idea is that a system is heated expanding and doing work and then cooled and compressed to its original state. Hence, it is a cyclic process. This cycle, which became known as the Carnot cycle, is strictly theoretical and has never been utilized in an actual engine. However, it is useful to study because it is the foundation for all other cycles. Shortly after Nicolas Carnot proposed the Carnot Cycle, two Italians Eugenio Barsanti and Felice Matteucci worked on an engine that would utilize four strokes to produce power. The patent was obtained in 1861 by Alphonse Beau de Rochas. It was not until 1864 that a German engineer Nicolaus Otto built a working four-stroke engine. This engine became known as the Otto cycle engine. Over the years, numerous different engine cycles have been created. They go through different steps to do work, but they all utilize the same basic concept as described in Figure 1. In this project we have chosen to look at three different cycles: the Carnot cycle, Otto cycle, and Brayton cycle.

III. Theoretical Background
The first law of thermodynamics governs the processes that can be used to obtain work from an engine. The change in energy is equal to the heat minus the work done by the system (see equation 1).

(1) There are four basic processes that occur to get work from an engine: isothermal, adiabatic, isochoric (commonly called isovolumetric), and isobaric. An isothermal process is one that occurs under constant temperature. Thus if the change in temperature is zero, the change in E is also zero so Q=W. An isobaric process occurs under constant pressure. This leads to a situation in which heat is equal to the change in internal energy plus the work (equation 2).

(2) An isochoric process occurs under constant volume. When the change in volume is zero, the area under the curve is zero; thus, the work done is zero. The change in energy is all heat. Finally, an adiabatic process is characterized by having zero loss in heat (Q=O). That change in energy is all converted to work. The following section describes each of the three specific engine cycles we modeled in detail and how they utilize the above four basic processes.

Carnot Cycle Theory
The Carnot cycle has four steps: 1) isothermal expansion, 2) adiabatic expansion, 3) isothermal expansion, and 4) adiabatic compression as shown in Figure 2. In the first isothermal expansion, a compressed gas draws heat from the heat reservoir to remain at a constant temperature. This is followed by an adiabatic expansion in which the gas no longer draws heat from the hot reservoir; this is the power stroke. The gas is then compressed isothermally in the exhaust stroke, and draws heat away from the system into the cold reservoir. Otto Cycle Theory
The Otto cycle is the cycle used in gasoline engines. It incorporates four strokes. The first stroke is the power stroke in which an air-fuel mixture is ignited under maximum compression. The hot, pressurized gas drives the piston down, causing an adiabatic expansion as the gas expands. The second stroke starts after the piston reaches the end of the expansion stroke. The piston exhaust valve opens, causing an isochoric decrease in pressure. The piston then begins moving up, pushing out the burnt air-fuel mixture in an isobaric process. The third stroke begins when the piston reaches the top of its stroke. At this time, the exhaust valve closes, the intake valve opens, and the piston moves back down. This draws in new air-fuel mixture in another isobaric process. When the piston reaches the bottom, the intake valve closes, the piston moves back up, and begins adiabatically compressing the air-fuel mixture until it reaches the top of its stroke, ready for the spark plug to ignite the air-fuel mixture and restart the process. These processes are represented in Figure 3. Brayton Cycle Theory
The Brayton cycle is used in gas turbines, such as jet engines. It consists of four steps. In the first step, air is drawn in and compressed adiabatically. The second step ignites the air-fuel mixture in the combustion chamber, where it expands isobarically. Work is harnessed by the engine in the third step, when gas enters the turbine and nozzle. The turbine blades direct the flow of gas through the nozzle. In the nozzle, the pressure decreases, the volume increases, and there is no change in heat or entropy simply a change in work. The fourth step is isobaric, at a lower pressure. The gas is subsequently ejected.

In practice, the Brayton cycle is applied in two different ways: as an open cycle, or a closed cycle. The open cycle, in which fuel is injected at the beginning and later ejected, is used in jet engines. The closed cycle, in which the fuel is recirculated through a closed loop, is commonly used in power generators. (See Figures 4 and 5) IV. Implementation
The following section describes in detail what each of the three programs do and how they calculate their corresponding PV and TS diagrams. The Carnot cycle operates on monoatomic hydrogen as its fuel. Ideal Otto and Brayton cycles were also calculated using hydrogen. Subsequently, the Otto cycle and Brayton cycle programs were adjusted to simulate running on the appropriate fuels, i.e. gasoline and jet fuel. For each engine, two types of data are used. First, the programs are fed idealized conditions of a monatomic hydrogen gas with initial volumes and pressures that allow for clear visualization that can match theoretical PV and TS diagrams. Next, actual data is fed into the program to determine some numerical values for entropy, energy exchange, heat, work, temperatures, etc. This data, especially in the case of the Brayton cycle, helps to illuminate how far the models are stretched in reality, and gives approximations of work and energy output.

Carnot Cycle Calculations
The Carnot Cycle program generates a model of a cycle given an initial temperature and pressure. User-defined parameters for To and Po are used to calculate isothermal and adiabatic processes. The simulation was run at a compression ratio of 4:1, using one mole of monatomic hydrogen gas. The program implements the Van Der Waals equation of state 3 (equation 3) and the standard formula for an adiabatic process 3 (equation 4) to plot a pressure volume diagram for the engine. The volume is incremented and the corresponding pressures are plotted. The temperature-entropy diagram is plotted using the data calculated for the changes in entropy and temperature. The initial volume is used to calculate the remaining volume values to achieve the 4:1 compression ratio. Changing the initial temperature and volume changes the cycle's ability to achieve work.

Otto Cycle Calculations
The Otto Cycle program is initialized with a user-specified pressure and compression ratio. In this situation the pressure was set at 1 atmosphere (to simulate an engine that would run at sea level) and a standard compression ratio of 8:1 was used. Once these parameters are defined the program starts at the intake stroke. This is an isobaric process. The change in volume is incremented and plotted against the pressure. The next step is to calculate the compression stroke, which is adiabatic. The general equation for calculating an adiabatic process is shown in equation 4 above. Next comes another isochoric stroke reducing the system to the initial pressure. Finally comes an isobaric stroke which returns the cycle to its starting position. For each step the change in temperature, energy, heat, work, and entropy were calculated from the 1st Law. The appropriate equations were found in Stowe(263)3.

Brayton Cycle Calculations
The Brayton model steps through different volumes in order to produce a working model. As the cycle has two different types of changes adiabatic and isobaric there are two essential functions necessary to properly run the code3. For the adiabatic portion, equation 4 is used to step through a changing volume and obtain the appropriate pressure. For the isobaric portion, a final volume is calculated using the input values, such that the cycle will be complete. Finally, for each segment, the change in work, efficiency, energy, entropy, enthalpy, and temperature is calculated and tabulated.

Next, a few adjustments are made to the program in order to make it more realistic. First, the degrees of freedom value is changed to 134 in order to account for an engine using a long hydrocarbon chain (kerosene), instead of hydrogen. Second, the temperatures of the different parts of a jet engine are input. The initial adiabatic inlet and compressor portion decreases the temperature from an atmospheric 332K to 518K1. The isobaric combustor increases the temperature from 518K to 2553K1. The adiabatic turbine and nozzle section decreases the temperature from 2553K to 1133K1. The isobaric rejection of the fuel into the atmosphere decreases the temperature to 3321. Using this data, one can use the ideal gas equation to solve for the pressure and volume values to initiate the process, and run the cycle.

V. Results and Analysis

Carnot Cycle Results
Sample graphs for the output of the Carnot Cycle program are shown below. The graphs for dependence on initial volume and temperature are also shown. Work done has exponential decreasing dependence on volume and a positive linear dependence on temperature (see figure 7). Raw data is shown in Tables 1 and 2. Efficiency remains constant for any V0, but shows a slight degradation as T0 approaches large values. PV diagram and dependence graphs are shown below. TS diagram is not shown, since it is simply a rectangle for the Carnot Cycle.   Otto Cycle Results
The output from the Otto Cycle program is shown in figure 8 and table 3.
Figure 1 shows a PV diagram for an Otto cycle at an 8:1 compression and Figure 2 shows the corresponding PV diagram. The results in the table 3 show that, as would be expected, delta E is zero, in agreement with the 1st law. It may be seen from the work data for the 8:1 compression vs. the 9:1 compression that the higher compression ration results in higher net work output. However, there is a maximum compression ratio reached before the net work begins to drop off. From the graph, the maximum achievable work is at a compression ratio of 15:1.    Brayton Results
The ideal Brayton cycle using one mole of monatomic gas outputted a net zero change in energy, with work done by the system of 3 Joules. When real data is input, the total energy exchange is zero. The total work done by one cycle is 1.92 kJ.
Figure 10 shows the Pressure-Volume diagram for an idealized situation of one mole of monatomic hydrogen gas. The adiabatic portions are in blue, and the isobaric are in green. Many of the numbers for this cycle were tailored in order to provide a clear picture of the Brayton cycle. Figure 10 also shows the Temperature-Entropy diagram for the same idealized cycle. In the same way as the first, the adiabatic portions are in blue, and the isobaric portions are in green.  Figure 11 shows the Pressure-Volume diagram for the realistic Brayton cycle. The shape of the cycle is much more elongated than the idealized version, as the temperature differences are much more drastic. Again, the adiabatic portions are in blue, and the isobaric are in green.  VI. Conclusion and or Future Work
The purpose of this project was to model the various engine cycles and their thermodynamic properties. The results from the programs showed that real world efficiency cannot approach that of the idealized models. Work output is shown to be dependent on the initial values and operating temperatures. When implementing the actual data, we obtained reasonable approximations of 1930.4 Joules and 67.4 Joules, respectively, for the work output of the Brayton and Otto cycles. Overall the models created seemed to accurately model each of the three engines. Future work would include modeling different engine cycles, and combining portions of these three cycles to find a method for further increasing net work output and efficiency.

Works Cited

Spakovszky, Z. S., Prof. "3.7 Brayton Cycle." 3.7 Brayton Cycle. Massachusetts Institute of Technology, 03 Sept. 2007. Web. 11 Dec. 2013. <http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node28.html>.

engel, Yunus A., and Michael A. Boles. "9-8." Thermodynamics: An Engineering Approach. 7th ed. New York: McGraw-Hill, 2011. 508-09. Print.

Stowe, Keith, Prof. "Engines." An Introduction to Thermodynamics and Statistical Mechanics. New York: Cambridge UP, 2007. 252-86. Print.

William B.R. (1976). Scientific Engineering and Problem Solving with Computers. Englewood, NJ: Prentice Hall. Print.

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