Question

(a) How much net work does this engine do per cycle? 

(b) Assuming that the efficiency of the engine is 0.444, what is the heat input into the gas per cycle? 

(c) How much heat is exhausted per cycle? 

(d) It takes 3.0 s for the engine to go through each cycle. The engine is used to drive a turbine that spins at 3000 rev/min. What is the average torque exerted on the turbine?

Answer 1

(a)

Calculate the net work done by calculating the area under the PV diagram.

Wnet = ar(ΔABC) + ar(rect.ACDE)

          = (1/2 × AB × AC) + (AC × AD)

         = (1/2 × 4 atm. × 1.5 m3) + (1.5 m3 × 1 atm.)

         = (3 × 101325)N∙m + (1.5 × 101325)N∙m

Wnet = 455.962 kN∙m

         = 455.962 kJ

 

Hence, the net work done by the engine per cycle is 455.962 kJ.

 

(b)

Calculate the heat input into the gas per cycle.

Qi = Wnet/η

 

Here, Qi is the heat input, and η is the efficiency of engine.

Substitute 455.962 kJ for Wnet, and 0.444 for η.

Qi = 455.962 kJ/0.444

     = 1026.94 kJ

 

Hence, the heat input into the gas per cycle is 1026.94 kJ.

 

(c)

Calculate the heat rejected by the gas per cycle.

Qr = Qi - Wnet

 

Here, Qr is the heat rejected by the gas.

Substitute 455.962 kJ for Wnet, and 1026.94 kJ for Qi.

Qr = (1026.94 – 455.962) kJ

     = 570.98 kJ

 

Hence, the heat rejected by the gas per cycle is 470.98 kJ.

 

(d)

Calculate the power of engine.

P = Wnet/Δt

 

Here, Δt is the time elapsed.

 

Substitute 3 s for Δt, and 455.962 kJ for Wnet.

P = 455.962 kJ/3 s

   = 151.987 kW

 

Calculate the torque exerted on the turbine.

T = P × 60/2πN

 

Here, T is the torque exerted, P is the power of engine, N is the rpm of turbine.

 

Substitute 151.987 kW for P, and 3000 rpm for N.

T = (151.987 kW × 60 s)/(2 × 3.142 × 3000)

   = 0.4837 kN∙m

   = 483.7 N∙m

Hence, the torque exerted is 483.7 N∙m.

(a) Hence, the net work done by the engine per cycle is 455.962 kJ.

(b) Hence, the heat input into the gas per cycle is 1026.94 kJ.

(c) Hence, the heat rejected by the gas per cycle is 470.98 kJ.

(d) Hence, the torque exerted is 483.7 N∙m.