How Is the Efficiency of the Diesel Cycle Derived Using Compression Ratios?

[xspook]

Homework Statement

Derive a formula for the efficiency of the Diesel cycle, in terms of the compression ratio V1/V2

Homework Equations

e=\frac{W}{Q_{h}}
w= ∫pdV

The Attempt at a Solution

Now I know I should have used e=1-\frac{Q_{c}}{Q_{h}} to get it but he said it is possible using e=\frac{W}{Q_{h}}

I know that W_{4-1} is equal to zero and W_{2-3} is equal to P_{2}(V_{3}-V_{2})

what I don't know is how he got

W_{1-2} = \frac{e}{1-δ}*cv^{-δ+1} = \frac{1}{1-δ}*(P_{2}V_{2}-P_{1}V_{1})

and

W_{3-1} = \frac{e}{1-δ}*cv^{-δ+1} = \frac{1}{1-δ}*(P_{4}V_{4}-P_{3}V_{3})

 

[DrClaude]

What are points 1,2,3, and 4? What is $\delta$?

 

[xspook]

1,2,3 and 4 are the strokes of the cycle. Well the different points on the picture. and δ is supposed to be gamma, which is the adiabatic exponent.

I thought I could use

W=∫pdV = C_{v}(T_{1}-T_{2}
for the compression stroke...but I guess that is incorrect

 

[DrClaude]

1,2,3 and 4 are the strokes of the cycle. Well the different points on the picture.

You have to be more specific. The numbering of points on a cycle is arbitrary. For each part of the cycle, state something like: 1→2 isothermal expansion at $T_2$.

 

[xspook]

From 1→2 is compression
From 2→3 is fuel injection/combustion
From 3→4 is the power stroke
From 4→1 is exhaust

 

[Andrew Mason]

1,2,3 and 4 are the strokes of the cycle. Well the different points on the picture. and δ is supposed to be gamma, which is the adiabatic exponent.

I thought I could use

W=∫pdV = C_{v}(T_{1}-T_{2}
for the compression stroke...but I guess that is incorrect

The ideal diesel cycle consists of a constant pressure expansion (2-3) followed by an adiabatic expansion (3-4) followed by constant volume cooling (4-1) followed by adiabatic compression (1-2).

So heat goes in only from 2-3 and heat goes out only from 4-1. Since W = Qh-Qc and η = W/Qh = 1-Qc/Qh you just have to deal with the two parts in which heat flows (ie. 4-1 and 2-3).

Can you work out Qh and Qc? (hint: it involves temperature change and heat capacity).

AM