Answer to Question #207223 in Molecular Physics | Thermodynamics for shiny

Question

An ideal diesel engine has a compression ratio of 20 and uses air as the working fluid. The state of air at

the beginning of the compression process is 95kPa and 20

0C. If the maximum temperature in the cycle is

not exceeding 2200K, determine (a) the thermal efficiency and (b) the mean effective pressure.

Answer 1

(a) First the cutoff ratio is determined by manipulating the ideal gas relations at states 2 and 3 where the temperature at state 2 is expressed through the temperature at state 1 and the compression ratio:

$r_c = \frac{α_3}{α_2} \\ = \frac{T_3}{T_2} \\ = \frac{T_3}{T_1r^{k-1}} \\ = \frac{2200}{293 \times 10^{(1.4-1)}}\\ = 2.27$

The efficiency then is:

$η= 1 - \frac{1}{r^{k-1}} \times \frac{r_c^k-1}{k(r_c-1)} \\ = 1 - \frac{1}{20^{(1.4-1)}} \times \frac{2.27^{1.4}-1}{1.4(2.27-1)} \\ = 0.635$

(b) The mean effective pressure is determined from the standard relation where the work is expressed through the efficiency and the heat input is 2-3:

$MEP = \frac{w}{α_1-α_2} \\ = \frac{ηq_{in}}{α_1(1- \frac{1}{r})} \\ = \frac{rP_1ηc_p(T_3-T_1r^{k-1})}{RT_1(r-1)} \\ = \frac{20 \times 95 \times 0.635 \times 1.005(2200-93 \times 20^{1.4-1})}{0.287 \times 293(20-1)} \\ = 933 \;kPa$