The piston-cylinder device shown on the left is a cylinder in a spark ignition e

Question

The piston-cylinder device shown on the left is a cylinder in a spark ignition engine. Below are engine specs for a BMW M5 Sedan with a Twin Turbo V-8: cylinder stroke 8.9 cm cylinder bore 8.9 cm compression ratio 10 number of cylinders 8 The engine is designed to operate with surroundings at STP conditions and a cycle reaching a maximum pressure of 90 atm in the cylinder. Model the cycle as an ideal Otto cycle with the cold air standard and create a table listing the pressure, temperature and volume (extensive) for the working fluid at each stage (ignore exhaust & intake strokes).

Explanation / Answer

For ideal Otto cycle, these are the different states.

State 1-2: Isentropic Compression

State 2-3: Constant volume Combustion

State 3-4: Isentropic expansion

State 4-1: Constant volume blowdown

For STP conditions, P1 = 100 kPa, T1 = 25 deg C = 298 K

For each cylinder, displaced volume Vdisp = pi/4 * Bore2 * Stroke

= 3.14/4 * 8.92 * 8.9

Vdisp = 553.4 cm3

Compression ratio r = (Vdisp + Vc) / Vc........where Vc = Clearance volume

10 = (553.4 + Vc) / Vc

Vc = 61.49 cm3

Volume at state 1, V1 = Vdisp + Vc = 553.4 + 61.49 = 614.89 cm3

Volume at state 2, V2 = Vc = 61.49 cm3

Since process 2-3 is constant volume, V3 = V2 = 61.49 cm3

Since process 4-1 is constant volume, V4 = V1 = 614.89 cm3

Now for isentropic compression, P2 / P1 = (V1 / V2)n............where n = 1.4 (Specific heat ratio for ideal air)

P2 / 100 = (614.89 / 61.49)1.4

P2 = 2511.88 kPa

By ideal gas equation, P1*V1 / T1 = P2*V2 / T2

T2 = (P2 / P1)*(V2 / V1)*T1

T2 = (2511.88 / 100)*(61.49 / 614.89)*298

T2 = 748.5 K = 475.4 deg C

Given, P3 = 90 atm = 9119.25 kPa

For constant volume combustion process, P3/T3 = P2 / T2

T3 = (P3 / P2)*T2

T3 = (9119.25 / 2511.88) * 748.5

T3 = 2717.4 K = 2444.2 deg C

For isentropic expansion, P4 / P3 = (V3 / V4)n

P4 = (61.49 / 614.89)1.4 * 9119.25

P4 = 363 kPa

Using ideal gas relationship, P3*V3 / T3 = P4*V4 / T4

T4 = (P4 / P3)*(V4 / V3)*T3

T4 = (363 / 9119.25)*(614.89 / 61.49)*2717.4

T4 = 1081.8 K = 808.7 deg C

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