IDEAL OTTO CYCLE - P-V DIAGRAM VISUALIZER

AIM:

• To visualise a P-V Diagram for an Ideal Otto Cycle.
• To compute engine's thermal efficiency.

OTTO CYCLE

Otto Cycle is one of most common thermodynamic cycles that can be found in automobile engines. Otto cycle describes the working of a four stroke spark ignition engine (Petrol Engines).

An ideal Otto cycle consists of a series of four reversible processes:

• two isentropic (reversible adiabatic) processes and two isochoric (constant volume) processes.

P  - Pressure             V - Volume                  T - Temperature           S - Entropy

Qin   - Heat Added    Qout - Heat rejected     Win   - Work input        Wout - Work output

PROCESS FLOW:

 Process 1-2  -  ISENTROPIC COMPRESSION

• As the piston moves from Bottom Dead Centre (BDC) to Top Dead Centre (TDC), the Air-fuel mixture inside the cylinder is compressed adiabatically from state 1 to state 2 (P1 to P2)
• Entropy of the system during the comprssion remoians constant ( S1 = S2)
• A change in volume of the Air-fuel mixture inside the cylinder occurs from v1 to v2, which is known as the compression ratio.

 Process 2-3 - CONSTANT VOLUME HEAT ADDITION

• When the piston reaches at Top Dead Centre (TDC), a spark is provided to ignite the air-fuel mixture compressed  to the clearance volume of the cylinder. There will be an initial rise in temperature during compression in addition to the heat transfer obtained through the induction of spark.
• This causes a rapid combustion of the air-fuel mixture leading to a rise in pressure P2 to P3

Process 3-4  -  ISENTROPIC EXPANSION

• The pressure rise caused due to the rapid combustion pushes the piston from Top Dead Centre (TDC) to Bottom Dead Centre (BDC) thereby undergoing expansion of the gas inside the cylinder thereby producing some form of useful work in pushing the piston from TDC to BDC. 
• Since some work output is produced during this process, it is called as Power Stroke.

Process 4-1 - CONSTANT VOLUME HEAT REJECTION

• The combustion products inside the cylinder is exhausted to the atmosphere in this phase of the cycle when the piston moves from Bottom Dead Centre (BDC) to Top Dead Centre (TDC)
• in this process the pressure decreases from P4 to P1, Temperature reduces to T1 while volume remains constant V4 = V1.

 Efficiency of Ideal OTTO CYCLE

      η = WORK / Heat Input.

         = Heat Output + Heat Input / Heat Input 

         = 1 +(Heat Output/Heat Input)

Heat Input is the heat absorbed by the system during Process 2 - 3

Heat Output is the heat rejected by the system during Process 4 - 1

Since Heat input is the heat flowing to the system, it is positive whereas Heat Output is flowing out of the system , it should be negative.

      Therefore, η  = 1 - (Heat Output/Heat input)

Heat Output  = CvΔT 

                      = Cv(T4 -T1)

Heat Input    = CvΔT 

                      = Cv(T3 -T2)

    Therefore, η  = 1 - [Cv(T4 -T1)/Cv(T3-T2)]

for isentropic process 1-2 and 3-4

the ratio V₁/V₂ is known as the compression ratio, r   

Thermal Efficiency       

CODE

To Visualize P-V Diagram and Calculate Efficiency of the engine:

Part 1 

 Defining the function to trace the volumetric change when the piston moves inside the cylinder

function [V] = OTTO_Fn(D,L_s,L_r,cr,crank_a,crank_b)

x = L_s / 2;
R = L_r/x;
v_s = (pi/4)*D^2*L_s;
v_c = v_s/(cr-1);

Deg = linspace(crank_a,crank_b,100);

x1 = (0.5*(cr-1));
x2 = (R+1-cosd(Deg));
x3 = ((R^2-sind(Deg).^2).^0.5);

V = (1+(x1*(x2-x3)))*v_c;

end

Part 2

 Plotting the P-V diagram and Calculating the Thermal Efficiency.

% PLOT - OTTO CYCLE
%CALCULATE - THERMAL EFFICIENCY

clear all
close all
clc

%----------------------------------------------------

%INPUTS
%1 atm = 101325 Pa
%UNITS 
%Pressure - Pa
%Volume - m^3
%Temperature - K
%Bore, stroke, Connecting rod length - m

p1 = 101325;
t1 = 700;
gamma = 1.4;
t3 = 2000;
m=1;

%----------------------------------------------------

%Fixing Engine Parameters to obtain Actual Volume
D = 0.1;
L_s = 0.2;
L_r = 0.25;
cr = 4;

%----------------------------------------------------

%swept volume = volume from top dead centre to bottom dead centre
v_s = (pi/4)*D^2*L_s;
%clearance volume = volume of gap between Top dead centre and Cylinder head
v_c = v_s/(cr-1);

%----------------------------------------------------
%Executng the variable at different states
%----------------------------------------------------

% STATE 1
v1 = v_s+v_c;
%volume left before compression
%p1 and t1 is defined already.
%now we have p1 t1 v1.

%----------------------------------------------------

%STATE 2
v2 = v_c;
% volume left after compression
%p1v1^gamma = p2v2^gamma;
%p2 = p1v1^gamma/v2^gamma
p2 = p1*(cr^gamma);
%p1v1/t1 = p2v2/t2
t2 = (p2*v2*t1)/(p1*v1);
%now we have p2 v2 t2.
%calling function to obtain curve state 1 to state2
%Since pv^gamma = Constant
Const_c = p1*(v1^gamma);
CMP_vol = OTTO_Fn(D,L_s,L_r,cr,180,0);
CMP_pres = Const_c./(CMP_vol.^gamma);

%----------------------------------------------------

%STATE 3
v3 = v2;
%p3v3/t3 = p2v2/t2
%p3 = p2v2t3/v3t2
p3 = (p2*t3)/t2;
%now we have p3 v3 t3.

%----------------------------------------------------

%STATE 4
v4 = v1;
%p3v3^gamma = p4v4^gamma
%p4 = p3v3^gamma/v4^gamma
p4 = p3*((v3/v4)^gamma);
%p3v3/t3 = p4v4/t4
t4 = (p4*v4*t3)/(p3*v3);
%now we have p4 v4 t4.

%----------------------------------------------------

%Calling function to obtain curve state 3 to state 4
%Since pv^gamma = Constant
Const_c = p3*(v3^gamma);
EXP_vol = OTTO_Fn(D,L_s,L_r,cr,0,180);
EXP_pres = Const_c./(EXP_vol.^gamma);

%----------------------------------------------------

%Thermal Efficiency
 T_EF = 1-(1/(cr^gamma-1));
 Thermal_efficiency = T_EF
 Thermal_efficiency_percentage = T_EF*100
 

%----------------------------------------------------
 
%Plotting the OTTO Cycle Curve
figure(m)
hold on
grid on
plot(v1,p1,'*','color','r')
plot(v2,p2,'*','color','r')
plot(CMP_vol,CMP_pres, 'color','r','linewidth',2)
plot([v2 v3],[p2 p3],'color','g','linewidth',2)
plot(v3,p3,'*','color','r')
plot(v4,p4,'*','color','r')
plot(EXP_vol,EXP_pres,'color','b','linewidth',2)
plot([v4 v1],[p4 p1],'color','g','linewidth',2)
xlabel('VOLUME (m^3)    >>>>','color','b')
ylabel('PRESSURE (Pa)    >>>>','color','b')
%----------------------------------------------------

RESULT:

PLOT :

Efficiency:

Thermal Efficiency = (Work Done/Heat Added)

                            = (Heat Added - Heat Rejected)/Heat Added

                            = 1 - (heat rejected/heat added)

                            = 1 - [(t4-t1)/(t3-t2)]

 for isentropic process 1to2 and 3to4

                            t2 = t1*(cr^(gamma-1))

similarly,

                            t3 = t4*(cr^(gamma-1))

Thermal_efficiency = 1 - (1/cr^(gamma-1))

                            =83.2339%

Reference:

CONCLUSION:

Otto Cycle is the sequence of processes that take place in Petrol Engines.

The compression ratio is a factor that influences the performance characteristics of internal combustion engines.  The total Work Ouput from the system is the area inside the P-V plot.

Since Thermal Efficiency = 1 -  (1 / (comp. ratio^(gamma -1)))

This implies that the Efficiency is proportional to the compression ratio of the engine,ie, as compression ratio increases , the thermal efficincy of the engine also increases.