Efficiency of a Diesel Engine

akan

1. The problem statement, all variables and given/known data

The figure shows the thermodynamic cycle of a diesel engine. This cycle differs from that of a gasoline engine in that combustion takes place isobarically. The compression ratio r is the ratio of maximum to minimum volume: r = V_1 /V_2. In addition, the so-called cutoff ratio is defined by r_c = V_3 /V_2.

Find an expression for the engine's efficiency, in terms of the ratios r and r_c and the specific-heat ratio y. Although your expression suggests that the diesel engine might be less efficient than the gasoline engine , the diesel's higher compression ratio more than compensates, giving it a higher efficiency (the ratio of the work delivered to the heat extracted during the combustion phase).



2. Relevant equations
W = (p1v1 - p2v2) / (y-1)
e = W / Q_h

3. The attempt at a solution
Q_h = p(V_3 - V_2)
W_12 = (p_1 V_1 - p_2 V_2) / (y - 1)
W_23 = p(V_3 - V_2) (though I think this one is not work delivered)
W_34 = (p_3 V_3 - p_4 V_4) / (y - 1)
W_41 = 0

W = (p_1 V_1 - p_2 V_2) / (y - 1) + (p_3 V_3 - p_4 V_4) / (y - 1)
= (p_1 V_1 - p_2 V_2 + p_3 V_3 - p_4 V_4) / (y - 1)

e = (p_1 V_1 - p_2 V_2 + p_3 V_3 - p_4 V_4) / [(y - 1) (p (V_3 - V_2))]

Since p_2 and p_3 are the same, this simplifies to...

e = (p_1 V_1 + p (V_3 - V_2) - p_4 V_4) / [(y - 1) (p (V_3 - V_2))]

Now, p_4 is not given and I cannot eliminate it so I don't know how to proceed from here. Please help.