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Chapter 2: Problem 74

The following table gives data, in \(\mathrm{kJ}\), for a system undergoing a power cycle consisting of four processes in series. Determine, the (a) missing table entries, each in kJ, and (b) the thermal efficiency. $$ \begin{array}{crrr} \text { Process } & \Delta E & Q & W \\ \hline 1-2 & -1200 & 0 & \\ 2-3 & & 800 & \\ 3-4 & & -200 & -200 \\ 4-1 & 400 & & 600 \end{array} $$

Short Answer

Expert verified
All missing entries in kJ: \( W_{1-2} = 1200, \Delta E_{2-3} = 800, W_{2-3} = 0, Q_{4-1} = 1000 \). Thermal efficiency is 1 (or 100%).

Step by step solution

01

- Apply Energy Balance Equation

For each process in the cycle, use the energy balance equation: \[ \ Delta E = Q - W \ \]
02

- Determine Missing Work for Process 1-2

Using \( \Delta E = Q - W \) for process 1-2: \[ -1200 = 0 - W \] \( W_{1-2} = 1200 \mathrm{kJ} \)
03

- Determine Missing Energy Change for Process 2-3

Using \( \Delta E = Q - W \) for process 2-3: Rearrange to find \( \Delta E \). Since \( Q = 800 \mathrm{kJ} \) and \( W \) is missing, we need more information to solve \( \Delta E \). This will be filled in later based on the total cycle balance.
04

- Determine Work for Process 3-4

Using \( \Delta E = Q - W \) for process 3-4: \[ \Delta E_{3-4} = -200 - (-200) = 0 \]
05

- Determine Heat for Process 4-1

Using \( \Delta E = Q - W \) for process 4-1: \[ 400 = Q - 600 \] \( Q_{4-1} = 1000 \mathrm{kJ} \)
06

- Check Energy Balance for Entire Cycle

Sum of \( \Delta E \) for entire cycle should equal zero as it returns to initial state: \[ -1200 + \Delta E_{2-3} + 0 + 400 = 0 \] Solve for \( \Delta E_{2-3} \): \( \Delta E_{2-3} = 800 \mathrm{kJ} \).\[ W_{2-3} = Q_{2-3} - \Delta E_{2-3} = 800 \mathrm{kJ} - 800 \mathrm{kJ} = 0 \]
07

- Determine Thermal Efficiency

Use the formula for thermal efficiency: \[ \eta = \frac{W_{cycle}}{Q_{in}} \] where total work, \( W_{cycle} = 600 + 1200 \), and total input heat \( Q_{in} \) includes positive heat transfers, which are 800 + 1000. Thus, \[ \eta = \frac{1800 \mathrm{kJ}}{1800 \mathrm{kJ}} = 1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Balance Equation
The energy balance equation is fundamental in thermodynamics. It helps us analyze different processes within a cycle. For any thermodynamic process, the energy change can be expressed using the equation:
\( \Delta E = Q - W \)

This means the change in internal energy (\

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