Consider a Diesel cycle that starts (at point a in Fig. 20.7) with air at temperature Ta. The air may be treated as an ideal gas. (a) If the temperature at point c is Tc, derive an expression for the efficiency of the cycle in terms of the compression ratio r. (b) What is the efficiency if Ta = 300 K, Tc = 950 K, ? = 1.40, and r = 21.0?

Solution 64CP Step 1: A)The analysis results in the following three general forms representing an adiabatic Process PV diagram T vk1= constant T p(1k)/kconstant P vk = constant where k is the ratio of heat capacities and has a nominal value of 1.4 at 300K for air. Step 2: The adiabatic compression process. The temperature of the air increases during the compression process and with a large compression ratio and reaches the ignition temperature of the injected fuel. P2 V2 k V2 (P ) = ( V ) = r [r = V compression ratio] 1 1 1 T V k1 (T2) = ( V2) = rk1 1 1 To compress the gas P-V cure, and is evaluated as follows. 2 2 w = Pdv = cost V dv = cost( v1) = P.V [ K v1]2 12 1k 1k 1 1 P V P V T T w = ( P.V ) = ( 2 2 1 1) = m.R[ 2 1] 12 1k 1k 1k Since for ideal gas P.V = m.R T Step 3 : The fuel is injected and combusted, this is represented by a constant pressure expansion process. At state 3 ("fuel cutoff") the expansion process continues adiabatically with the temperature decreasing until the expansion is Complete. The adiabatic expansion process.The total expansion work is w exp = w 23 + w 34W exp and is shown as the area under the P-V diagram and is analysed Step 4: 3 w = Pdv = P .(V V ) 23 2 3 2 2 Q 34 w 34 = m.u = m.C .Tv w 34 = m.C (v 3 ) 4 w exp= w + w 23 34