In a motorcycle engine, a piston is forced down toward the crankshaft when the f

Question

In a motorcycle engine, a piston is forced down toward the crankshaft when the fuel in the top of the piston’s cylinder undergoes combustion. The mixture of gaseous combustion products then expands adiabatically as the piston descends. Find the average power in (a) watts and (b) horsepower that is involved in this expansion when the engine is running at 4300 rpm, assuming that the gauge pressure immediately after combustion is 16 atm, the initial volume is 47 cm3, and the volume of the mixture at the bottom of the stroke is 250 cm3. Assume that the gases are diatomic and that the time involved in the expansion is one-half that of the total cycle.

Explanation / Answer

pf = piVi/Vf = pi [Vi/Vf]

The work done W for adiabatic process is defined as, W = [pfVf - piVi]/(-1)

Here, initial pressure is pi, initial volume is Vi and final volume is Vf and adiabatic constant is .

Solution:-

First we have to find out final pressure pf. To obtain pf, substitute 16.0 atm for pi, 47.0 cm3 for Vi, 250 cm3 for Vf and 1.40 for in the equation pf = pi [Vi/Vf],

pf = pi [Vi/Vf]

    =(16.0 atm) [47/250]1.40 = 1.54 atm

To obtain the work done W by the gas during the expansion, substitute 16.0 atm for pi, 47.0 cm3 for Vi,1.54 atm forpf, 250 cm3 for Vf and 1.40 for in the equation W = [pfVf - piVi]/(-1), we get,

W = [pfVf - piVi]/(-1) = [(16.0 atm) (47.0 cm3) – (1.54 atm) (250 cm3)]/(1.40-1)

   = [(16.0 atm×(1.01×105 Pa/1 atm)) (47.0 cm3×(10-6 m3/1 cm3))] –[ (1.54 atm×(1.01×105 Pa/1 atm)) (250 cm3×(10-6m3/1 cm3))]/(1.40-1)

=92.67 Pa.m3

= (92.67 Pa.m3) (1 N/m2/ 1 Pa)

= (92.67 N.m) (1 J/ 1 N.m)

= 92.67 J

The above process happens 4300 times per minute, but the actual time to complete the process is half of the cycle, or 1/8600 of a minute.

Thus the average power P involved in this expansion when the engine is running at 4300 rpm will be,

P = (92.67 J) (8600)/(60 s)

   = 13.28 ×103 J/s

   = (13.28×103 J/s) (1 W/1 J/s)

= 13.28 ×103 W

From the above observation we conclude that, the average power P involved in this expansion when the engine is running at 4300 rpm would be 13.28 kW = 17.8 hp

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