# Efficieny of a diesel engine

1. Jan 25, 2010

1. The problem statement, all variables and given/known data
The diagram below shows the thermodynamic cycle of a diesel engine. The compression ratio is the ratio of maximum to minimum volume; $r=\frac{V_{1}}{V_{2}}$. In addition, the so-called cutoff ratio is defined by $r_{c} = \frac{V_{3}}{V_{2}}$. Find an expression for the engine's efficiency, in terms of $r$, $r_{c}$ and the specific heat ratio $\gamma$.
2. Relevant equations
$$PV=nRT$$
PV^gamma = constant
$$W_{adiabatic} = \frac{P_{1}V_{1} - P_{2}V_{2}}{\gamma-1}$$
$$\epsilon = \frac{W_{net}}{Q_h}$$
$$P_{2} = P_{3}$$
3. The attempt at a solution
$$W_{net} = W_{12} + W_{23} + W_{34} + W_{41}$$
And I think I must only use negative values for Q_h because the efficiency is the ratio of work done to heat absorbed. That is a little bit against what I would think because it would make more sense to me if Q_h was just the sum of Q_12, Q_23, Q_34, Q_41.
First of all
$$W_{net} = \frac{P_{1}V_{1}-P_{1}r^\gamma V_{2}}{\gamma-1}+P_{1}r^\gamma(V_{3}-V_{2}) + \frac{P_{1}r^\gamma V_{3} - P_{1}r_{c}^\gamma V_{1}}{\gamma-1}$$
The only negative Q_h I think is Q_41 which is
$$Q_{h} = Q_{41} = n C_{v} (T_{1}-T_{4}) = \frac{P_{1}V_{1}C_{v}}{R}(1-r_{c}^\gamma)$$
And when I calculate $\epsilon$, low and behold out pops the wrong answer (and a huge mess of V1's V2's and so on) which I cant seem to get rid of. So it seems I have plucked out the wrong values to calculate Q_h or W_net but I'm not sure which combination to pull out.

The book manages to squeeze out
$$\epsilon = 1-r^{1-\gamma}(\frac{r_{c}^\gamma-1}{\gamma(r_{c}-1)})$$

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