1. Develop an expression for cutoff ratio r_c which expresses it in terms of q_in/(c_pT₁r^k-1 ) for an air-standard Diesel cycle.

2. An air-standard cycle, called the dual cycle, with constant specific heats is executed in a closed piston-cylinder system and is composed of the following five processes:

1-2 Isentropic compression with a compression ratio, r = V₁/V₂

2-3 Constant volume heat addition with a pressure ratio, r_p = P₃/P₂

3-4 Constant pressure heat addition with a volume ratio, r_c = V₄/V₃

4-5 Isentropic expansion while work is done until V₅ = V₁

5-1 Constant volume heat rejection to the initial state

(a) Sketch the P-v and T-s diagrams for this cycle.

(b) Obtain an expression for the cycle thermal efficiency as a function of k, r, r_c, and r_p.

(c) Evaluate the limit of the efficiency as r_p approaches unity and compare your answer with the expression for the Diesel cycle efficiency.

(d) Evaluate the limit of the efficiency as r_c approaches unity and compare your answer with the expression for the Otto cycle efficiency.