Chapter 4: Q. 4.18 (page 133)

Derive equation $4.10$ for the efficiency of the Otto cycle.

Short Answer

Expert verified

The efficiency of the Otto cycle is $η = 1 - {(\frac{V_{2}}{V_{1}})}^{γ - 1}$

Step by step solution

Introduction

Efficiency $(η) = \frac{Net work done (W_{net})}{Heat supplied (Q_{1})}$

Heat supplied:

$Q_{1} = m C_{v} (T_{3} - T_{2}) - - - (1)$

Hear rejected:

$\begin{aligned} Q_{2} = m C_{v} (T_{4} - T_{1}) - - - (2) \\ η = \frac{Q_{1} - Q_{2}}{Q_{1}} = 1 - \frac{Q_{2}}{Q_{1}} - - (3) \\ From (1) and (2) \\ η = 1 - \frac{m C_{v} (T_{4} - T_{1})}{m C_{4} (T_{3} - T_{2})} \\ η = 1 - \frac{(T_{4} - T_{1})}{(T_{3} - T_{2})} - - (4) \end{aligned}$

Calculation

As it is an adiabatic process:

$\begin{aligned} T_{2} V_{2}^{γ - 1} = T_{1} V_{1}^{γ - 1} \\ \frac{T_{2}}{T_{1}} = {(\frac{V_{1}}{V_{2}})}^{γ - 1} \end{aligned}$

Similarly,

$\begin{aligned} T_{2} V_{2}^{γ - 1} = T_{1} V_{1}^{γ - 1} \\ \frac{T_{3}}{T_{4}} = {(\frac{V_{4}}{V_{3}})}^{γ - 1} = {(\frac{V_{1}}{V_{2}})}^{γ - 1} \\ ∴ \frac{T_{2}}{T_{1}} = \frac{T_{3}}{T_{4}} \end{aligned}$

$\frac{T_{3}}{T_{2}} = \frac{T_{4}}{T_{1}} - - - - (5)$

Subtract the above equation by 1 both sides:

$\begin{aligned} \frac{T_{3}}{T_{2}} - 1 = \frac{T_{4}}{T_{1}} - 1 \\ \frac{T_{3} - T_{2}}{T_{2}} = \frac{T_{4} - T_{1}}{T_{1}} \\ \frac{T_{1}}{T_{2}} = \frac{T_{4} - T_{1}}{T_{3} - T_{2}} = {(\frac{V_{2}}{V_{1}})}^{γ - 1} \end{aligned}$

Substitute the above equation in (4):

$η = 1 - {(\frac{V_{2}}{V_{1}})}^{γ - 1}$

Conclusion

Efficiency of the Otto cycle is $η = 1 - {(\frac{V_{2}}{V_{1}})}^{γ - 1}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Derive equation $4.10$ for the efficiency of the Otto cycle.

Short Answer

Step by step solution

Introduction

Calculation

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Astrophysics

Linear Momentum

Atoms and Radioactivity

Conservation of Energy and Momentum

Nuclear Physics

Study anywhere. Anytime. Across all devices.

Derive equation 4.10for the efficiency of the Otto cycle.

Short Answer

Step by step solution

Introduction

Calculation

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Astrophysics

Linear Momentum

Atoms and Radioactivity

Conservation of Energy and Momentum

Nuclear Physics

Study anywhere. Anytime. Across all devices.

Derive equation $4.10$ for the efficiency of the Otto cycle.