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Dynamic analysis of the combustion cycle.

During the combustion process of I.C. engine the fuel heat energy is converted to mechanical work. The working cycle of the I.C.engine is evaluated in terms of indicated power .  Empirical values of gas pressure force versus the crank angle are tabulated as shown below  to calculate the torque generated during the working cycle in conventional engine and oscillator engine.

Crank angle gas pressure Crank angle gas pressure force MPa. force 0 0.018 370 5.402 30 -0.015 390 3.420 60 -0.015 420 1.350 90 -0.015 450 0.720 120 -0.015 480 0.450 150 -0.015 510 0.280 180 -0.015 540 0.150 210 -0.015 570 0.025 240 -0.015 600 0.018 270 0.020 630 0.018 300 0.150 660 0.018 330 0.720 690 0.018 360 1.923 720 0.018

Conventional piston/cylinder arrangement

In Cylinder/piston with crank gear arrangement the gas pressure force on piston is converted into rotary motion and torque of crank shaft. The actual driving force is the tangential component of the force acting on crank pin, which is equivalent to  T = P. sin(A+B)/ cosB . wherein the gas pressure forces exerted on the piston are brought on to crank pin are replaced with radial and tangential components of force. ( B is the crank angle counted off from the piston axis and A is the angle through which the connecting rod diverges from cylinder axis.)

Acting upon the crank pin, force directed from crank rod produces two components of forces.  Let the force directed along the crank radius be K.

       Then K=P cos{A+B}/ cosB  ( This massive component of force is spent as heat losses and frictional losses)

and let the force tangent to the crank radius circumference be T. Then    T = P sin(A+B)/ cosB ( This is the only component of force available to drive the crank gear)

The numerical values of the trigonometrically functions included for lambda = 0.27, if connecting rod length/ crank radius= lambda = 0.27

In this analysis we are deriving the torque generated to drive out put power and total value of work done per cycle event. Torsional moment(torque) of one cylinder (MN m) is determined by the value of  M= TxR ( Where R is the crank radius)

So, at any instance, Work done will be equal to:

                          torque x circumferential distance traveled by the crank gear

To compare the total productive power imparted on drive shaft, we correlate  conventional engine and oscillator engine having similar swept volume of 1791822.37 cubic mm . ( This value is taken in to account as the equivalent volume of the selected oscillator engine)

For piston/ cylinder engine assume the Bore = Stroke.   Then the piston stroke (2R)

where Volume : 1791822.37 =R^3 x(22/7) x 2   so that R=65.814mm 

Now if we analyze the characteristics of conventional engine having crank radius of 65.81mm , we be able to compare with the characteristics of  oscillator engine.

The curve of Torque verses  the crank angle is plotted to compare the Total tortional moment generated during a combustion cycle.

Combustion chart for conventional engine

Crank angle	Gas pressure force Mpa.	sin(A+B) cos B	Piston area,mm2	Total force Mpa..mm2	Distance traveled by crank pin-mm	Work done Mpa. mm3
0	0.018	0.000	13,613.23102	0.00000	0	0.00000
30	-0.015	0.625	13,613.23102	-127.62404	34.474	-2,199.85559
60	-0.015	0.993	13,613.23102	-202.76908	34.474	-5,694.98615
90	-0.015	1.000	13,613.23102	-204.19847	34.474	-7,014.89951
120	-0.015	0.740	13,613.23102	-151.10686	34.474	-6,124.39796
150	-0.015	0.376	13,613.23102	-76.77862	34.474	-3,928.06214
180	-0.015	0.000	13,613.23102	0.00000	34.474	-1,323.43312
210	-0.015	0.376	13,613.23102	-76.77862	34.474	-1,323.43312
240	-0.015	0.740	13,613.23102	-151.10686	34.474	-3,928.06214
270	0.020	-1.000	13,613.23102	-272.26462	34.474	-7,297.65428
300	0.150	-0.993	13,613.23102	-2,027.69076	34.474	-39,644.33089
330	0.720	0.625	13,613.23102	6,125.95396	34.474	70,641.76273
360	1.923	0.000	13,613.23102	0.00000	34.474	105,593.06836
370	5.402	0.222	13,613.23102	16,325.58562	34.474	281,404.11927
390	3.420	0.625	13,613.23102	29,098.28130	34.474	782,971.19398
420	1.350	0.993	13,613.23102	18,249.21684	34.474	816,128.82536
450	0.720	1.000	13,613.23102	9,801.52633	34.474	483,510.66002
480	0.450	0.740	13,613.23102	4,533.20593	34.474	247,087.77996
510	0.280	0.376	13,613.23102	1,433.20096	34.474	102,842.95556
540	0.150	0.000	13,613.23102	0.00000	34.474	24,704.08497
570	0.025	-0.376	13,613.23102	-127.96437	34.474	-2,205.72187
600	0.018	-0.740	13,613.23102	-181.32824	34.474	-5,331.27670
630	0.018	-1.000	13,613.23102	-245.03816	34.474	-7,349.27756
660	0.018	-0.993	13,613.23102	-243.32289	34.474	-8,417.87941
690	0.018	-0.625	13,613.23102	-153.14885	34.474	-6,833.98338
720	0.018	0.000	13,613.23102	0.00000	34.474	-2,639.82671

Cumulative work done by piston in a working cycle( two revolutions). =2803627.37

This graph shows the variation of value sin(A+B)/cosB with respect to gas pressure Vs crank angle. It clearly shows that involving this factor can decrease the efficiency by great amount.

An example showing how the work done is calculated in the above chart

Crank angle	Gas pressure	sin(A+B)/cosB	Piston surface area	F=PxA	distance traveled by crank pin	Average work done in 30degree
370	5.402	0.222	13,613.23102	16,325.58562	34.474	281,404.11927
390	3.420	0.625	13,613.23102	29,098.28130	34.474	782,971.19398

Take the example of about data: during the crank angle turning from 370 degrees to 390 degree gas pressure is assumed as 5.4002 MPa; and the Total force acted upon the connecting rod small end = Gas pressure x piston area

There fore the tangential component of this force acting on crank pin to drive the shaft is  5.402 x.222 x  sin(A+B)/cosB

= 5.402 x 13613.23 x 0.222 = 16325.58 MPa. mm2

Within a 30 degrees angle Crank pin will travel a distance = 34.4743

Work done F s = 16325.58 x 34.474  within the specified turn of 30 degrees.

Oscillator engine arrangement

Equivalent oscillator engine having mobile flap surface area = 8450 mm2; distance from center to the point of action of resultant force = 67.4723;   distance traveled by this point within 30 degrees = 35.337mm. Tabulated data for the work done during a complete working cycle, due to gas pressure exerted on the flap surface is given below:

Combustion chart for oscillator engine
Flap   angle	gas pressure force	Flap surface area	Force on flap	lever arm	Distance traveled	Work done Mpa.MM3
0	0.018	8,450.	152.100	67.4623	35.337	0.00000
30	-0.015	8,4500	-126.750	67.4623	35.337	447.89648
60	-0.015	8,4500	-126.750	67.4623	35.337	-4,478.96475
90	-0.015	8,450	-126.750	67.4623	35.337	-4,478.96475
120	-0.015	8,450	-126.750	67.4623	35.337	-4,478.96475
150	-0.015	8,450	-126.750	67.4623	35.337	-4,478.96475
180	-0.015	8,450	-126.750	67.4623	35.337	-4,478.96475
210	-0.015	8,450	-126.750	67.4623	35.337	-4,478.96475
240	-0.015	8,450	-126.750	67.4623	35.337	-4,478.96475
270	0.020	8,450	169.000	67.4623	35.337	746.49413
300	0.150	8,450	1,267.500	67.4623	35.337	25,380.80025
330	0.720	8,450	6,084.000	67.4623	35.337	129,889.97775
360	1.923	8,450	16,249.350	67.4623	35.337	394,596.79448
370	5.402	8,450	45,646.900	67.4623	35.337	1,093,613.89313
390	3.420	8,450	28,899.000	67.4623	35.337	1,317,114.23415
420	1.350	8,450	11,407.500	67.4623	35.337	712,155.39525
450	0.720	8,450	6,084.000	67.4623	35.337	309,048.56775
480	0.450	8,450	3,802.500	67.4623	35.337	174,679.62525
510	0.280	8,450	2,366.000	67.4623	35.337	108,988.14225
540	0.150	8,450	1,267.500	67.4623	35.337	64,198.49475
570	0.025	8,450	211.250	67.4623	35.337	26,127.29438
600	0.018	8,450	152.100	67.4623	35.337	6,419.84948
630	0.018	8,450	152.100	67.4623	35.337	5,374.75770
660	0.018	8,450	152.100	67.4623	35.337	5,374.75770
690	0.018	8,450	152.100	67.4623	35.337	5,374.75770
720	0.018	8,450	152.100	67.4623	35.337	5,374.75770
4353553.737

Graphical representation for Work done during a working cycle for both engine arrangements are plotted as shown below:

4353

Torque curves

The graph shows the torque generated during a working cycle in conventional engine and oscillator engine. Accordingly 1.55 times higher torque is generated in oscillator engine , for equal quantities of fuel burnt in both models.

                 Accordingly Work done in Oscillator engine during a working cycle   = 4353553.737

                 Work done in Conventional engine during a working cycle                  =    2803627.37

So that, 1.55 times higher torque will be generated in Oscillator engine arrangement, for same fuel consumption.

We are pleased to receive your comments.

Last Updated on 10/11/2007
By .LJayasuriya