A Pressure Verses Volume Diagrams - Case Study Example

PV DIAGRAMS Name Institution Instructor Date Contents Introduction 3 Aim of the experiment 4 Experimental procedure 4 Theory 5 7 The diagram above describes the geometry of the crankshaft where there is an illustration of the piston pin layout of the crank pins as well as crank centre (Srinivasan, 2004). 7 Where; 7 l = length of the rod 7 r = radius of the crank at half of the stroke 7 A = angle of the crank from the top dead centre 7 X = piston position in relation to the crank centre 7 P = pin of the piston 7 N = pin of the crank 7 O = centre of the crank 8 V= velocity of the piston in the upward direction along the centreline of the bore cylinder from the crank centre 8 a = acceleration of the piston pin in the upward direction from along the cinders bore of the centreline 8 ω= the crank angular velocity in radian s per second 8 Angular velocity 8 The angular velocity of the crankshaft has a relationship with the revolutions per minute of the engine and this is demonstrated through the following equation 8 Triangle equation 8 As indicated in the above diagram, the centre, of the crank, the pin of the crank as well as the piston pin results in the formation of triangle (PON). Introducing the cosine rule results in the equation 8 8 Considering the equations in relation to the angular position , the equations are derived as a representation of the and description of the piston’s reciprocating motion in relation to the angle of the crank (Demirel, 2012). 9 Rearranging the formula relating the crank angle with respect to its position we have: 9 Velocity 9 The derivation of the velocity equation with respect to the angle of the crank employs the use of the chain rule in which case the derivative I first considered as follows 9 Acceleration 10 The derivation of the acceleration formula in relation to the angle of the crank employs the use of the quotient rule and the chain rule to obtain the second derivative as indicated 10 10 In obtaining equations in relation to the time domain, the derivative of the angular velocity is obtained as shown: 10 For a constant angular velocity, the area A is obtained as 10 10 Results 11 Discussion 15 Conclusion 16 References 17 Introduction A pressure verses volume (PV) diagram represents a plot of pressure change denoted by P in relation volume demonstrated by V for a given process or processes. Theses set of process leads to the formation of typical thermodynamic cycles. When the cycle has operated to completion, the state of the system remains the same. This means that the device in the system goes back to the initial volume and pressure. The pressure volume diagrams were initially referred to as the indicator diagram. It was developed for enhancing the operating efficiencies of engines. The pressure volume diagrams are used in wide applications in the estimation of the net work done in a thermodynamic cycle (Demirel, 2012). The net work done represents the area under the enclosure of the pressure volume curve in the diagram. The usage of this finds derivation through developing indicator diagrams, which are useful in the estimation of the operation of a steam engine. The diagram is used to carry out specific recording of the volume of the stream verses the pressure of the stream for a cylinder throughout the piston’s motion cycle in an engine stream. The diagram is useful in the computation of the work done and therefore offers a measure of power released in the engine (Rathakrishnan, 2005). Aim of the experiment The aim of the experiment was to practically evaluate the operational features of four-stroke diesel cycle through the characteristics of the pressure volume diagram and compare with that of the four-stroke petrol cycle. The experiment also aimed at the establishment and identification of the relationship newton the two as well as the parameters that are significant in the functionalities. Experimental procedure The turning of the engine was done in accordance with the following procedural steps: The first step involved the reproduction of the spread sheet, which indicated a single cycle for a diesel engine with four strokes. The products spread sheet was the then subjected to some enhancements with regard to its labelling and layout in order to improve its clarity. This step also involved the derivation of the derivation of the formulas that were later used in obtaining power output. In the second step of the experimental procedure, the tuning of the diesel engine was done through the alteration of the profile of the heat released. The power output was then increased as much as possible using an amount of fuel that measured 0.9 units. The profile of the inaction was indicated on the upper graph. There was also an increase in pressure to a value that made sure that the maximum combustion chamber pressure did not go beyond 20 bar. This was in order to prevent the blowing up of the engine. In the third step of the experimental procedure, modification of the engine was done through conversion from a diesel engine to a petrol engine. This entailed the reduction of the ration of compression while maintaining the maximum limit of pressure at 20 bar. The amount of fuel used here was the same as the amount of fuel used on the diesel engine. Tuning of the petrol engine was done through the alteration of the profile for heat release. This resulted in the changing of the V diagram, which stimulated a spark explosion. The amount of power produced was then investigated. Step four of the experimental procedure entailed an explanation of all the results obtained in this particular experiment and drawing of conclusions as well as recommendations. Theory PV diagram The pressure volume diagram below indicate the characteristics of a commonly used pressure volume diagram. Several states that are numbered from 1 to 4 are indicated. The path existing between each two states comprises of several processes labelled A through to D. These processes brings about the alteration of the of the volume or pressure in the system (Demirel, 2012). Typical PV diagram A significant feature of the diagram is that the extent of energy subjected to expansion or gained in the system as work done can be found or estimated through the calculation of the area that is covered by the graph in the chart. In the case of cyclic graphs, the net work done in the system is obtained by computing the area covered by the curve. In the figure shown above, the processes represented by 1-2-3 are responsible for the production of the work output. However, the processes represented by 3-4-1 need a small amount of energy input energy input to go back to the initial state or position. Therefore, the work done can be computed from the difference between the two processes (Rathakrishnan, 2005). Derivation of equation The diagram above describes the geometry of the crankshaft where there is an illustration of the piston pin layout of the crank pins as well as crank centre (Srinivasan, 2004). Where; l = length of the rod r = radius of the crank at half of the stroke A = angle of the crank from the top dead centre X = piston position in relation to the crank centre P = pin of the piston N = pin of the crank O = centre of the crank V= velocity of the piston in the upward direction along the centreline of the bore cylinder from the crank centre a = acceleration of the piston pin in the upward direction from along the cinders bore of the centreline ω= the crank angular velocity in radian s per second Angular velocity The angular velocity of the crankshaft has a relationship with the revolutions per minute of the engine and this is demonstrated through the following equation Triangle equation As indicated in the above diagram, the centre, of the crank, the pin of the crank as well as the piston pin results in the formation of triangle (PON). Introducing the cosine rule results in the equation Considering the equations in relation to the angular position , the equations are derived as a representation of the and description of the piston’s reciprocating motion in relation to the angle of the crank (Demirel, 2012). Rearranging the formula relating the crank angle with respect to its position we have: Velocity The derivation of the velocity equation with respect to the angle of the crank employs the use of the chain rule in which case the derivative I first considered as follows Acceleration The derivation of the acceleration formula in relation to the angle of the crank employs the use of the quotient rule and the chain rule to obtain the second derivative as indicated In obtaining equations in relation to the time domain, the derivative of the angular velocity is obtained as shown: For a constant angular velocity, the area A is obtained as Then the relationship applies as follow Results The following values were used in determining the behaviour of the curve for the diesel engine cycle with the compression ration maintained at 16.5 to 1 Table indicating the results obtained from experimental performance for the crank angle and the heat released in the case of a diesel engine cycle. Angle Heat Release 355 0.01 360 0.1 365 0.12 370 0.14 375 0.05 380 0.01 385 0.03 390 0.3 395 0.001 400 0.861 Total 0.542 The compression ratio was set as at a value that did not exceed 0.9 and the area of the pv diagram was maximized as much as possible. Graph of heat release against the crank angle for the diesel engine cycle The nature and feature of the pressure volume diagram for diesel engine cycle The following values were used in determining the behaviour of the curve for the petrol engine cycle obtain following the modification of the diesel engine cycle with the compression ration maintained at 20 to 1 Table indicating the results obtained from experimental performance for the crank angle and the heat released in the case of a petrol engine cycle. Angle Heat Release 355 0.008 360 0.009 365 0.1 370 0.12 375 0.14 380 0.003 385 0.01 390 0.03 395 0.0012 400 0.45 Total 0.871 Graph of heat release against the crank angle for the petrol engine cycle The nature and feature of the pressure volume diagram for petrol engine cycle Discussion The typical diesel and petrol cycles are represented in the curves obtained in the results section above and they comprise of four stages, which combine to form a closed cycle. Two isochoric and two isothermal processes.  The processes are indicated on both a pressure-volume (P-V) curve and heat release verses crank angle diagram. The area that is covered by the path of the process path in the pressure volume diagram for diagram for the two different cycles is a representation of the area and the work done. The work and heat are not gained or released by the system. Work is done is released through the cycle only when the isothermal process are carried out. It is important to incorporate a flywheel into the system design in order ensure facilitation of the work release from the system (Rathakrishnan, 2005). In this experimental case, the thermal efficiency for the systems used could be regarded as output ratio to that of the energy input into the system in terms of supply of fuel. This efficiency usually finds basis on the indicated of the brake output. It is an exact indication of the performance efficiency through which the energy from the fuel that is subjected to conversion into mechanical work. Thermal efficiency represents the efficiency of combustion. This is due to the reason that the entire chemical energy that is obtained from the fuel is subjected to conversion into heat energy during the process off combustion. The thermal efficiency of the brake is obtained through the consideration of parameters such as the calorific value of the fuel, the mass of fuel as well as the brake power. The energy input that is directed into the engine assumes a variety of forms upon entry into the engine system. A portion of it becomes converted into brake output, which is released through the exhaust. The other content is consumed through utilisation into cooling of the lubrication oil and water. The total energy break up into various different portions is a very important process sin heat balance (Srinivasan, 2004). Conclusion In conclusion, the performance of the experiment and the evaluation of the results played an important role in understanding the features and operation conditions of both diesel and petrol engines. This was facilitated through an understanding of the nature of the pressure volume diagrams displayed by the operation of the two cycles. The experiment played an important role in enlightening the participants with regard to different formulas that are applicable for analysing the performances and features of the cycles involved. References Demirel, Y. (2012). Energy production, conversion, storage, conservation, and coupling. London, Springer. Rathakrishnan, E. (2005). Fundamentals of engineering thermodynamics. New Delhi, Prentice Hall. Srinivasan, S. (2004). Automotive engines. New Delhi, Tata McGraw-Hill Pub. Read More