Plotting of ideal otto cycle p-v diagram four stroke engine

Aim : Plotting of ideal otto cycle p-v diagram four stroke engine.

Objective : Got to know about four stroke otto cycle basics, and algorithm to plot p-v diagram using matlab.

Theory of Ideal otto cycle :

                     otto cycle is an air standard cycle used to analysis for petrol engine.               

                     At first processes occuring in otto cycle is important to analyse the engine.

                    An air standard cycle , because in ideal engines air is assumed as a charge.

Assumptions for ideal cycles compared to real cycels :

1. Closed cycle.
2. Instantaneous heat addition.
3. No heat transfer.
4. No pumping work.
5. No blowdown losses.
6. No blowby losses.
7. No friction losses.
8. Complete combustion.                

Reversible Processes occuring in pertol engine (otto cycle) -

1. Adiabatic compression.
2. Constant volume heat addition.
3. Adiabatic expansion.
4. Constant volume heat rejection.

Engine Nomenclture : 

1. Bore - Internal diameter of the cylinder.
2. Stroke - distance travelled by piston during one stroke.
3. Dead centers - Extreme positions of the piston.
4. swept volume - Volume travlled by the piston.
5. clearance volume - volume between piston at TDC and cylinder.
6. Total volume = clearance volume + swept volume.
7. compression ratio = ratio of volume between before combustion and after combustion.

Representation of strokes in four stroke petrol engine:

      From the above figure , observed that 4 - different strokes 

1.  Suction stroke.
2.  compression stroke.
3.  power stroke.
4.  exhaust stroke.

  In suction stroke air occupy the volume of the cylinder volume in ideal engine , in real engine air + fuel mixture with proper stichometric ratio through intake valve.

  In compression stroke the air get compressed by the piston , pressure increases in combustion chamber(closed place where the combustion takes place).

  After compression , piston at TDC combustion takes place due to spark produces by the spark plug.

  In power stroke , due to combustion of charge the high pressure forces developed on piston crown . due to this we get the power and so called it as power stroke.

  In exhaust stroke , the combusted gases expelled through the exhaust valve.

An important concept in engine - Valve timing diagram

 Above two diagrams describe valve timings of engine , first one is theoritical practically not possible.

 Valve timing diagram depends on the type of engine , and that controlled by PCM or ECM electronic components.

 Explanation of theoritical valve timing diagram :

   1. Intake valve opens when the piston is at TDC and closes when piston is at BDC, these valves are opened at suction stroke.

   2. Compression stroke begines when piston at BDC and completes when it reaches at TDC.

   3. Instantaneous Ignition takes places that can be conclude by the valve timing diagram.

   4. Then power stroke starts Piston starts at TDC and ends when piston at BDC.

   5. Exhaust stroke - expelling all combusted gases from combustion chamber , when piston moving from BDC to TDC.

 Basic description About practical valve timing diagram:

     Inlet and exhaust valves does not open immediately when piston reaches TDC and BDC , valves takes some time to open and close , so that can observe positions 

     by the help of diagram , various angles mentioned represents the crank angle.

     And can observe that ignition process is not instantaneous.

    Valve timing diagram varies for every engine , that depends on the clearances , load conditoins .... etc.., (many parameters).

P-V diagram of ideal otto cycle :

At first let's describe some differences between ideal and real cycle :  

Engine performance parameters :

1. Indicated mean effective pressure.
2. Brake mean effective pressure.
3. Mean piston speed.
4. Stichometeric ratio. (air : fuel)
5. calorific value of fuel.
6. Specific fuel consumption.
7. Friction mean effective pressure.
8. Indicated thermal effeciencies.
9. Brake thermal effeciencies.
10. Mechanical efficiency.
11. Relative efficiency.
12. Volumetric efficiency.

Relations used in otto cycle :

1.     p-v relation for adiabatic process 

        `pv^gamma` = constant`

          p - pressure of the system at a point.

          v - volume of the system at a point.

   2.     Ideal gas eqaution. - used because ideal gas considered as a charge for ideal engines.

           `pv = constant`

            p - pressure of the system at a point.

            v - volume of the system at a point.

   3.     Swept volume (`V_s`) :

                  `V_s = pi/4 * B^2 * S`

               B - internal diameter of the cylinder.

               S  - stoke of the cylinder.

   4. Variation of cylinder volume with respect to crank angle : (v')

             `(v')/v_c = 1 + (1/2)*(r_c -1)*(n+1-cos(theta)-sqrt (n^2 - sin^2(theta)))`.

            `v_c` - clearance volume.

             `r_c` - compression ratio.

              n    -  obliquity ratio. -ratio of connecting rod length and crank radius.

              theta in degress considered.

  5. Thermal efficiency `eta_(thermal)`:

           `eta_(thermal) = 1/(r_c)^(gamma-1)`

Code snippet for plotting p-v diagram : 

% p-v diagram of ideal otto cycle
close all
clear,clc 

%Inputs
gamma = 1.4;
% t3 state variable property - temparature  at end of combustion(constant volume heat addition).
t3 = 3500 ;   % in kelvin.
compression_ratio = 8 ; % cr ranges 6:1 to 10:1 for petrol engine.  = v1/v2 = v4/v2

% engine geometric parameters in S.I units.
connecting_rod_length = 0.6;   
crank_length =  0.4;
bore_radius = 0.4;

stroke_length = crank_length/2;
% calculation of swept volume 
v_swept = (pi/4)*bore_radius^2*stroke_length;

%claculation of clearance volume
%compression ratio - ratio of volume before combustion and after combustion.

v_clearance = v_swept/(compression_ratio - 1);

% defining state variables of state-1 at the start of combustion.
p1 = 155000;      % pascals
t1 = 900;   % in kelvin.
% v1 volume at BDC sum of _clearance and v_swept.
v1 = v_clearance + v_swept;


%state varaibles at state point 2

% v2 = v_clearance
v2 = v_clearance;

% as compression undergoes adiabatic compression p-v relation p*v^gamma
%p1*v1^gamma = p2*v2^gamma;
%v1/v2 = compression ratio
p2 = p1*compression_ratio^gamma;

% using ideal gas equation as we are using ideal gas in ideal otto cycle
% claculating v2(volume at start of combustion)
%p1*v1/t1 = p2*v2/t2

t2 = (p2*v2*t1)/(p1*v1);

% state variables at state point 3
v3 = v2;
p3 = (p2*t3)/t2;
% t3 defined as a input

% state variables at state point 4
v4 = v1;
p4 = p3/(compression_ratio)^gamma;
t4 = (p4*t1)/p1;

thermal_efficiency = 1/(compression_ratio)^(gamma-1);

%function volume_variable = volume_cylinder(cr,stroke,bore,con_rod,start_crank,end_crank)

v_compression = volume_cylinder(compression_ratio,stroke_length,bore_radius,connecting_rod_length,180,360);
constant1 = p1*v1^gamma;
p_compression = constant1./(v_compression.^gamma);

v_expansion = volume_cylinder(compression_ratio,stroke_length,bore_radius,connecting_rod_length,0,180);
constant2 = p3*v3^gamma;
p_expansion = constant2./(v_expansion.^gamma);



plot(v1,p1,'marker','*','color','r');
hold on
plot(v_compression,p_compression,'color','c');
plot([v2,v3],[p2,p3],'color','r');
plot(v2,p2,'marker','*','color','y');
plot(v3,p3,'marker','*','color','m');
plot(v_expansion,p_expansion,'color','g');
plot([v1,v4],[p1,p4],'color','r');
plot(v4,p4,'marker','*','color','b');

Explanation of code step-wise : 

  1. At first using the comman comands   

  close all - Clearing all existing plots and figures.

  clear  - Clearing all data in the work piece.

  clc    - clearing the command window.

   2. At first initialised the required inputs ,can onderved in code. 

     like t3,crank radius,connecting rod length, gamma,p1,t1...

 3. Then used relations to calculate the swept volume and clearance volume.

     relation defined at the code itself as comments.

4. v1 - volume at state 1 is equal to sum of clearance volume and swept volume.

5. state 2 ,state 3 and state 4 variables are calculated using p-v relations of respective process.

6. Process and some required data mentioned in the code as comments.

function volume_variable = volume_cylinder(cr,stroke,bore,con_rod,start_crank,end_crank)

a = stroke/2;
n = con_rod/a;

v_s = (pi/4)*bore^2*stroke;
v_c = v_s/(cr-1);

theta = linspace(start_crank,end_crank,200);

t1 = 0.5*(cr-1);
t2 = n + 1 - cosd(theta);
t3 = (n^2 - sind(theta).^2).^0.5;

volume_variable =   (1 + t1*(t2 - t3)).*v_c;
end

7. This function snippet used for calculation of variable volume during compression and expansion processes.

8. using the relation from engine kinematics , mentined in the code as array variable volume_variable.

8. stores volumes data of exapansion and compression process stored in repective array object .

9.using ' plot ' command plotted the figure.

Outputs :

Conclusion :

1. plotted the p-v diagram of ideal otto cycle and studied and analysed on the ideal otto cycle

2. Made an required calculations.