Question

An ideal Diesel cycle has a maximum cycle temperature of 2000 deg C. The state of the air at the beginning of the compression is 95 kPa, and 15 deg C. This cycle is executed in a 4stroke, 8-cylinder engine with a bore of 10 cm and a piston stroke of 12 cm. The compression ratio is 20. Determine the cutoff ratio and the power produced by this engine when it is run at 2400 rpm. using variable specific heats

Answer 1

Given:

Ideal Diesel cycle, 4 stroke - 8 cylinder

Tmax = T₃ = 2000^oC = 2273 ^oK

Compression Ratio, r = 20

P₁ = 95 kPa,

T₁ = 15^oC = 288 ^oK

Bore = 10 cm,

Stroke = 12 cm

Rpm = 2400 = 40 rps

V₄ = V₁ = V_BDC = π/4 * d² * L = π/4(0.1)²(0.12) = 9.425 × 10^-4 m³

and,   

  

In 1-2; Ideal gas, isentropic process:

where, k = ratio of specific heat;

Assumption for ideal gas k = 1.4

Also, a Compression ratio is the ratio of V₁ to V₂

So, V₂ = V₁ / r = 4.7125 × 10^-5 m³

By, Putting r = 20; k = 1.4; and T₁ = 288 ^oK

T₂ = 954.562 ^oK

In 2-3; Ideal gas, Constant pressure Heat Addition:

Cutoff Ratio is ratio of Volume of V₃ to V₂.

So, putting the value of T₃ and T₂ ;

r_c = 2.381;

So, the cutoff ratio is r_c = 2.381.

V₃ = r_c × V₂ = 2.381 × 4.7125 × 10^-5 m³ = 1.122 × 10^-4 m³

Also if we use variable specific heat, then k value is unknown if it is not ideal

So,   

In 3-4; Ideal gas, isentropic process:

Also V₄ = V₁

So, T₄ = 2273 × 0.119^0.4 = 970.26 ^oK

Now, the work output per cycle per cylinder;

For ideal gas C_v  = 0.717 kJ/kgK ;  C_p =1.004 kJ/kgK

Constant Volume Heat transfer; q_4-1 = C_v(T₄ - T₁) = 0.717 (970.26 - 288) = 489.18 kJ/kg

Constant Pressure Heat transfer;q_2-3 = C_p(T₃ - T₂) = 1.004 (2273 - 954.562) = 1323.71 kJ/kg

By conservation of energy,

Work Output = q_2-3  - q_4-1

= 1323.71 - 489.18

= 834.53 kJ/kg

m' = m × N/2 = 1.08 × 10^-3 × 40/2 = 0.0216 kg/sec

Power Output = Work Output × m'

= 834.53 kJ/kg × 0.0216 kg/sec

= 18.025 kW

So, for 8 cylinder power output will be 8 times.

So, Power Output = 8 × 18.025 = 144.206 kW

Remarks - Just consider the process one by one and find the values of temperature and volume at different points.

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