Week 2 Air standard Cycle

Air-Standard Cycle – Otto Cycle

Aim:-

Development of code for air standard Otto cycle to extract the pressure vs volume diagram and to calculate the thermal efficiency of otto cycle.

Objective:-

• To study the working of otto cycle.
• To develop the algorithm for calculation of state variable in otto cycle.
• To plot the state variable in the graph called pv-diagram.
• To calculate the thermal efficiency of cycle.

PROCESS IN OTTO CYCLE:-

• Process 0–1 a mass of air is drawn into piston/cylinder arrangement at constant pressure.
• Process 1–2 is an adiabatic (isentropic) compression of the charge as the piston moves from the bottom dead center(BDC) to the top dead center (TDC).
• Process 2–3 is a constant-volume heat transfer to the working gas from an external source while the piston is at the top dead center. This process is intended to represent the ignition of the fuel-air mixture and the subsequent rapid burning.
• Process 3–4 is an adiabatic (isentropic) expansion (power stroke).
• Process 4–1 completes the cycle by a constant-volume process in which heat is rejected from the air while the piston is at the bottom dead center.
• Process 1–0 the mass of air is released to the atmosphere in a constant pressure process.

The Otto cycle consists of isentropic compression, heat addition at constant volume, isentropic expansion, and rejection of heat at constant volume. In the case of a four-stroke Otto cycle, technically there are two additional processes: one for the exhaust of waste heat and combustion products at constant pressure (isobaric), and one for the intake of cool oxygen-rich air also at constant pressure; however, these are often omitted in a simplified analysis. Even though those two processes are critical to the functioning of a real engine, wherein the details of heat transfer and combustion chemistry are relevant, for the simplified analysis of the thermodynamic cycle, it is more convenient to assume that all of the waste heat is removed during a single volume change.

Description & Governing equations:

An engine is a machine which generate power through the combustion air-fuel mixture by providing heat at utmost pressure and temperature. During this combustion process the piston which is main part of engine goes back and forth to repeat the cycle. This cycle of intake, compression, combustion and expansion are governed by the specific air standard cycles. This air standard cycle provides the information about the process how they are happing inside the cylinder. The type engine decides which kind of cycle it is running in the process. As if it is petrol engine then it performs Otto cycle and if it is diesel engine then it performs diesel engine. Both the process is different but involve under the same piston-cylinder of engine.

Otto cycle is called constant volume cycle as the heat addition process happens at constant volume and heat rejection also. The process involves isentropic compression and expansion of air inside the cylinder. The heat addition is done by the spark plug in case of petrol engine and in case of diesel engine it is done by high pressured fuel spray following by auto-ignition phenomenon. The PV diagram tells us the pressure and volume condition at which the heat added and rejected during the process. There is T-S diagram which is temperature and entropy diagram for same PV diagram which tell us about the temperature changes with respect to constant entropy process. The actual PV diagram is totally different than that of air-standard cycle because of neglection of friction, mass factors of components and inertia of air flow. The main most important aspect of air-standard cycle and actual cycle is that in air-standard the working medium is ideal gas.

From the above PV diagram, we can see that the process is in cyclic nature. The process always starts from the heat rejection which means from point 4-1 and during this process the new volume of medium is taken inside the cylinder in case of actual cycle, but in above case the process starts from the point 0 which is consider as intake point and at point 1 the piston is supposed to be at BDC (Bottom Dead Center). In process 1-2 the isentropic compression process is done where fluid is compressed whose pressure and temperature is get increased. After the peak pressure and temperature are reached during the process 2-3 is heat addition process is done which is given by:

Qs=Mcv(T3-T2)

After adding this heat at peak pressure condition the air get ignited and expand spontaneously which result in generation of high temperature and piston traveled from top dead center to bottom dead center. This process eventually generates the power which in turn gives the torque to crankshaft resulting in repeating of cycle. After the combustion isentropic expansion process is carried out from 3-4. The constant heat rejection process from 4-1 is done when exhaust valve is open which release the by-product of combustion which is given as:

Qr=Mcv(T4-T1)

From these two equations we can find the efficiency of otto cycle as:

During isentropic process the relation between temperature and volume can be written as:

Similarly, for process 3-4

To calculate each and every state point in PV diagram we need to know some basic terminology factor of engine such as bore, stroke, connecting rod length and compression ratio. Similarly, we need to calculate the clearance volume and swept volume which is given by:

And clearance volume is given as:

The relation from which pressure is calculate as:

PROCEDURE:-

• Since the otto cycle is an air standard cycle we will take the gamma value as 1.4.
• At the initial condition, we will consider the T1 and P1 values as 500K and 101.325KPa respectively.
• The geometrical parameters are stoke, diameter, connecting rod length, crank start and end angle are defined to the function in the program.
• From the above parameters, the v_c and v_s are calculated accordingly.
• The other states are calculated using adiabatic and ideal gas equations.
• Similarly, the thermal efficiency is calculated using the above formulas.
• For plotting p-v diagram plt.plot command is used from matplotlib.pyplot module.
• For the compression stage, the starting and ending crank angles are 180 and 0.
• For the expansion stage, the starting and ending crank angles are 0 and 180.
• Let us consider the value to be iterated is 50 accordingly v null set is created to store the value using append command.

Explanation of Code:-

• The code starts from the basic inputs for calculation of state points which are bore, stroke, connecting rod length and compression ratio. The code is divided in two different parts one in which we calculate the state point and in other we calculate the volume swept by the piston thought the whole cycle.
• We assume some atmospheric condition for pressure and temperature which are very much required for the calculation of other parameters. From these parameters we calculate the pressure at state point 2 and after that using ideal gas law equation, we calculate the temperature at point 2. As volume at point 1 is addition of swept volume and clearance volume which is further calculated from the bore, stroke and compression ratio. Similarly, volume at point 2 is nothing but clearance volume.
• After getting the pressure, temperature and volume at point 1 & 2 we calculate the further state point by assuming the temperature at point 3 which is around 2300 K. We also know that as process 2-3 is constant volume heat addition the volume at point 2 is equals to volume at point. So, again by using ideal gas law we can calculate the pressure at point 3 which should be higher of any other pressure point in the otto cycle.
• Considering another isentropic process to calculate the state point 4 where the volume at point 4 is equal to volume at point 1 due constant volume heat rejection process. The rest of parameter can be calculated by ideal gas law.
• After calculation all the point, the engine kinematic need to be taken into consideration to calculate the actual volume swept by the piston. This is important because, if we plot all the state point it will show a linear line connection between each of the pressure and volume data which is not suitable.
• To calculate the swept volume by the piston we are going to call the function called ‘pistion_kinematics’ which have one output variable and 6 input parameters which are defined earlier. The relation by which the swept volume is calculated is given as:
• In which R is the ratio of connecting rod length to the crack radius. The above equation will give the swept volume from piston but still we need to calculate the swept volume for compression stroke and expansion stroke so that we can plot in the PV diagram.
• For the compression stroke we can calculate the volume from the changing the crank angle from 180 to 0. Similarly, we know that PV^γ is constant from which we can calculate the pressure variation for compression stroke. Similar terminology can be used for expansion stroke in which crank angle changes from 0 to 180 degree.
• After calculation of all this equation we put the plot accordingly. The code for whole process is given as below.

Program:-

Output:-

Conclusion:-

In this program we have calculated the each and every state point in PV diagram also we have plotted the same. The maximum pressure in the engine is around 60.7 Mpa and thermal efficiency of engine is around 62.98%. The thermal efficiency of otto cycle is function of compression ratio and ratio of specific heat. As specific heat is assumed to be a constant for any working fluid, the efficiency is increased by increasing the compression ratio.