Chapter 8, Problem 8.12P

(a)

Interpretation Introduction

Interpretation:

Show that the thermal efficiency of the air standard Diesel cycle can be expressed as:

  $\eta =1-{\left(\frac{1}{r}\right)}^{\gamma -1}\frac{r_c{}^{\gamma }-1}{\gamma \left(r_c-1\right)}$

Where $r_c$ is the cutoff ratio and is defined as $r_c=\frac{V_A}{V_D}$, $r$ is the compression ratio.

Concept Introduction:

The efficiency of the air standard Diesel cycle can be calculated by following formula,

  $\eta =1-\frac{1}{\gamma }\left[\frac{{\left(1/r_e\right)}^{\gamma }-{\left(1/r\right)}^{\gamma }}{1/r_e-1/r}\right]$  .......(1)

Where $r$ is the compression ratio and $r_e$ is the expansion ratio.

(b)

Interpretation Introduction

Interpretation:

Show that for the same compression ratio the thermal efficiency of the air standard Otto engine is greater than the thermal efficiency of the air-standard Diesel cycle.

Concept Introduction:

The efficiency of air standard Diesel cycle can be less only if the term $\frac{r_c{}^{\gamma }-1}{\gamma \left(r_c-1\right)}>1$

(c)

Interpretation Introduction

Interpretation:

How does the thermal efficiency of an air-standard Otto cycle with a compression ratio of 8 compare with the thermal efficiency of an air-standard Diesel cycle with the same compression ratio and a cut off ratio of 2? How is the comparison changed if the cutoff ratio is 3.

Concept Introduction:

The efficiency of an air-standard Otto cycle is

  ${\eta }_{Otto}=1-{\left(\frac{1}{r}\right)}^{\gamma -1}$  .......(a)