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Chapter 17: Problem 55

Find the work, in kJ/kg, needed to compress air isentropically from \(20^{\circ} \mathrm{C}\) and \(100 \mathrm{kPa}\) to \(6 \mathrm{MPa}\). a) \(-523\) c) \(-423\) b) \(-466\) d) \(-392\)

Short Answer

Expert verified
Answer: The work needed to compress the air isentropically is approximately -466 kJ/kg.

Step by step solution

01

Determine the initial and final states

Before we start calculating the work done, we must know the initial and final states of the air. The initial state is with the temperature at 20°C (293.15K) and pressure at 100 kPa. The final state has a pressure of 6 MPa.
02

Use the relation for isentropic process in terms of pressure and temperature

To determine the relationship between temperature and pressure in an isentropic process, use the expression: \((\frac{P_2}{P_1})^{(\gamma-1)/\gamma} = (\frac{T_2}{T_1})\), where \(P_1\) and \(T_1\) are initial pressure and temperature, \(P_2\) and \(T_2\) are final pressure and temperature, and \(\gamma\) is the heat capacity ratio.
03

Calculate the final temperature (\(T_2\))

Using the above relation, we can find the final temperature \(T_2\). For air, the heat capacity ratio \(\gamma = 1.4\). $$ \left( \frac{6000}{100} \right) ^{\frac{1.4-1}{1.4}} = \left( \frac{T_2}{293.15} \right). $$ Solving for \(T_2\), we find the final temperature to be approximately 665.97 K.
04

Find the work done using the heat capacity ratio

We can calculate the work done using the specific heat capacity at constant volume (\(c_v\)), initial and final temperatures, and heat capacity ratio \(\gamma\). The work done (in kJ/kg) during the isentropic compression is given by: $$ W = -cv(T_2 - T_1)/(\gamma - 1). $$ The specific heat capacity at constant volume (\(c_v\)) for air is approximately 0.718 kJ/(kg·K).
05

Calculate the work done using given values

Now, we can plug the values in to find the work done: $$ W = -0.718(665.97 - 293.15)/(1.4 - 1). $$ Solving this expression, we find the work done to be approximately -466 kJ/kg. The correct answer is b) \(-466\).

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Most popular questions from this chapter

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