Modeling of the PV Diagram - Term Paper Example

PV DIAGRAMS By Student’s name Course code and name Professor’s name University name City, State Date of submission 1.0 Introduction A P-V diagram is simply a graph of gas pressure versus volume plotted to show the state of gas and the amount of work done during a process of heat transfer. This is emphasized by the ideal gas laws which are used to indicate the amount of work done by the kinetic and potential changes that take place during heat transfer within a combustion chamber. The microscopic energy that is involved during the heat transfer eventually transforms into a large amount of macroscopic mechanical energy which is the main force of interest to a mechanical engineer. Considering that a system is closed, the ideal gas is exposed to basic concepts of thermodynamics. During the process of heating, the gas is made to expand slowly to reach an equilibrium pressure and temperature in a process that is quasi-static. The gas pressure is usually equal to the amount of pressure exerted by external forces only that this is given as a negative energy. In an isobaric process for example, gas expands at a constant temperature and many other processes that cannot be mentioned herein due to the limited scope of this assignment. It is however important to note that pressure of a given piston is given as a ratio between the exerted force divided by cross sectional area (Fullerton, 2011). An example of a P-V diagram is shown in figure 1 below with the area under the graph being the work done by the gas. This is however an example of a process that is quasi-static in which the curve area has a direct approximation formula. In other types of P-V diagrams, the work done is calculated using the trapezoidal process which is highlighted herein. This report shall therefore be looking at the approach used in formulating P-V diagrams and also the way of calculating the work done by use of simple Microsoft Excel programming. This study emerges successful in finding out the missing formulas in this assignment and eventual modelling of the P-V diagram to suit the maximum energy requirements of both diesel and petrol engines. Examples are illustrated prior to the real assignment to create an understanding of the required concepts for the success of this study. Figure 1: Example of a P-V diagram (isobaric process). 2.0 Estimating Work Done In order to estimate the work done in an expansion system, it is important to note that it will be equal to the area covered by the curve. This example shows how the calculation is carried out for a circle using Microsoft Excel program in the instruction file. It is assumed that the points around the circle are known and that the rest can be calculated through simple mathematics. The figure below shows the example of the circle for which the area was calculated in the video example. Figure 2: How to calculate the area of an enclosed curve. Letting 2 be the starting point for this approximation, the points are spread within angles of 45º and their coordinate values calculated accordingly. In order to calculate the x-coordinate fields for the unknown points, the formula 2-cos (angle in radians) while the y-coordinate is calculated using the formula 2+sin (angle in radians). These values were then copied from the Excel sheet and accordingly presented in table 1 shown below. Degrees Radians x y 0 0 1.00 2.00 45 0.785398 1.29 2.71 90 1.570796 2.00 3.00 135 2.356194 2.71 2.71 180 3.141593 3.00 2.00 225 3.926991 2.71 1.29 270 4.712389 2.00 1.00 315 5.497787 1.29 1.29 360 6.283185 1.00 2.00 Table 1: Exact coordinate values of the points identified for calculation of area. These points were then projected to the x-axis in order for area calculation to be carried out using the trapezoidal rule. These points were effectively subtracted to find out the length of height of the trapezoids and eventually summing the negative and positive areas up in order to come up with the total area as shown in table 2 below. Degrees Radians x y dA 0 0 1.00 2.00 0 45 0.785398 1.29 2.71 0.68934 90 1.570796 2.00 3.00 2.017767 135 2.356194 2.71 2.71 2.017767 180 3.141593 3.00 2.00 0.68934 225 3.926991 2.71 1.29 -0.48223 270 4.712389 2.00 1.00 -0.81066 315 5.497787 1.29 1.29 -0.81066 360 6.283185 1.00 2.00 -0.48223 Total Area 2.828427 Table 2: A summation of the areas (dA) gives the total area of the circle. 3.0 Plotting a P-V Diagram In order to understand the concept of plotting P-V diagrams, an example was provided within the instruction as way of learning. For this purpose, the following P-V diagram was obtained while gauging various levels of injection in way of making the combustion process more efficient for a petrol engine. Figure 3: P-V diagram obtained during acquainting exercise. In order to plot a P-V diagram, some of the essential points that form the crank circle are important to be determined. The equations that are responsible for the angular position are calculated using the same approach as the one used by the example shown above though this is a little bit advanced. The position of the crank angle is therefore calculated using the equations derived below. The layout shown in figure 3 below is important in determining these equations for the sake of finding these positions. Figure 4: A layout showing various principle variables of crank geometry. Using the triangle relation, it is possible to come up with a formula to determine the position of the piston round the crank. This equation is however rearranged resulting to equation (1) which is straight forward in getting the piston position. (1) Where - Crank angle in radians - Connecting rod length - Crank radius (crank throw) - Top piston position For the purpose of this assignment, the variables used are shown in figure (5) below ranging from the piston diameter, to power of the given combustion system. Figure 5: P-V diagram in the instruction file. In order to calculate the swept volume, the equation (2) below shall be applied. (Connecting rod length) = 0.15m (Compression ratio) = 11:1 (Crank throw) = 0.05m (Diameter of piston) = 0.08m (Length of stroke) = 0.1m (Piston Position) = variable. The original fuel injection profile maintains a power limit of 1.748 units which is the indicated value on the example. The graphs below show the initial condition P-V and an evident for the values obtained during this exercise. Figure 6: Fuel injection profile prior to tuning for respective engines. Figure 7: A P-V diagram showing a combustion system whose injection profile is yet to be tuned. The exercise required tuning of injection profile as a way of improving the efficiency of the engine in terms of power output. This was first carried out for the petrol engine whose compression ratio was 8:1 as a comparison for the diesel engine whose compression is 11:1. The graphs shown below were obtained when the fuel profile was tuned and the combustion pressure maintained within a practical limit of 20 bars. The maximum attainable power by a petrol engine of compression ratio 8:1 becomes 1.692. This proves that the lower the engine compression ratio, the lower the energy that is likely to be produced. Figure 8: Fuel injection profile of a petrol engine. Figure 9: P-V diagram for a fine tuned petrol engine. This process was also carried out for the diesel engine of compression ratio 11:1 in order to establish the maximum attainable energy. Beyond this study there are many identifiable factors that attribute towards the high efficiency of diesel engines and vice versa for petrol engines. While study dwelled on the compression ratios the following graphs were obtained at a maximum attainable power of 1.848 units. Figure 10: Fuel injection profile of a diesel engine. Figure 11: P-V diagram for a fine tuned diesel engine. 4.0 Conclusion This study successfully depicts the process of coming up with P-V diagrams for the sake of combustion engines analysis and optimization. It is established that; maintaining the pressure in the engine at a given critical point results to higher amount of power in diesel engines than in petrol engines. This is also attributed to the level of compression which automatically varies between the two types of engines. 5.0 Reference List Fullerton, D. (2011) Honors Physics Essentials: An Aplusphysics Guide, 1st edition, New York: Silly Beagle Productions. Read More