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Chapter 6: Problem 43

Air as an ideal gas is compressed from a state where the pressure is \(0.1 \mathrm{MPa}\) and the temperature is \(27^{\circ} \mathrm{C}\) to a state where the pressure is \(0.5 \mathrm{MPa}\) and the temperature is \(207^{\circ} \mathrm{C}\). Can this process occur adiabatically? If yes, determine the work per unit mass of air, in \(\mathrm{kJ} / \mathrm{kg}\), for an adiabatic process between these states. If no, determine the direction of the heat transfer.

Short Answer

Expert verified
The process cannot be adiabatic; heat must be added to the system.

Step by step solution

01

- Determine initial and final temperatures in Kelvin

Convert the given temperatures from degrees Celsius to Kelvin using the formula \( T(K) = T(掳C) + 273.15 \). Initial temperature: \( T_1 = 27 + 273.15 = 300.15 \mathrm{K} \). Final temperature: \( T_2 = 207 + 273.15 = 480.15 \mathrm{K} \).
02

- Calculate specific heat ratio (纬) for air

For air, the specific heat ratio (纬) is typically given as \( \frac{C_p}{C_v} = 1.4 \).
03

- Check if the process is adiabatic

For an adiabatic process, the relation \( \frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma - 1}{\gamma}} \) holds true. substituting the given values: \( \frac{480.15}{300.15} = \left( \frac{0.5}{0.1} \right)^{\frac{1.4-1}{1.4}} \).Simplifying this, \( 1.6 = 5^{0.2857} \), which results in \( 1.6 \approx 1.8 \). Since the values do not match exactly, the process cannot be adiabatic.
04

- Determining the direction of heat transfer

Since the process cannot be adiabatic, heat is transferred. The temperature of the air increases meaning heat is added to the system.

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Key Concepts

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

ideal gas law
The Ideal Gas Law is a fundamental equation in thermodynamics that describes the behavior of an ideal gas. It is expressed as:
\[ PV = nRT \]
Where:
- \( P \) is the pressure of the gas
- \( V \) is the volume of the gas
- \( n \) is the number of moles of gas
- \( R \) is the universal gas constant
- \( T \) is the temperature of the gas in Kelvin
It helps us understand how changes in temperature, pressure, and volume affect a gas. In the problem, we use the ideal gas law to track changes in states of air, assuming air behaves like an ideal gas. This equation is crucial when we assess how gases transition from one state to another under different conditions. Because the process involves changes in temperature and pressure, we utilize the ideal gas law to understand these interactions deeply.
specific heat ratio
The specific heat ratio, often denoted by \( \gamma \) (gamma), is the ratio of the specific heat capacity at constant pressure \( C_p \) to the specific heat capacity at constant volume \( C_v \). Mathematically, it is represented as:
\[ \gamma = \frac{C_p}{C_v} \]
For air and many other diatomic gases, the value of \( \gamma \) is typically around 1.4. This ratio is significant in determining how a gas behaves during different thermodynamic processes, especially adiabatic processes.
In our exercise, we employ this ratio to check if given pressures and temperatures of air illustrate an adiabatic process:
\[ \frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^\frac{\gamma - 1}{\gamma} \]
By substituting our values, we find that the expected ratio does not align perfectly. This discrepancy suggests the process is not entirely adiabatic, indicating heat transfer is involved.
heat transfer
Heat transfer is the process by which thermal energy flows from one body or substance to another due to temperature differences. In our exercise, we analyze if heat transfer occurs by contrasting the ideal ratios for an adiabatic process.
Since the provided conditions do not match the requirements for an adiabatic process:
\[ \frac{480.15}{300.15} eq \left(\frac{0.5}{0.1}\right)^{0.2857} \approx 1.8 \]
It's clear that the process is not adiabatic. The mismatch reveals that the system must have some heat transfer. Since the air's temperature increases, we can deduce that heat is added to the system. Tracking heat flow helps us understand energy conversion and the efficiency of processes in thermodynamics.

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Most popular questions from this chapter

An electric motor operating at steady state draws a current of 10 amp with a voltage of \(220 \mathrm{~V}\). The output shaft rotates at 1000 RPM with a torque of \(16 \mathrm{~N} \cdot \mathrm{m}\) applied to an external load. The rate of heat transfer from the motor to its surroundings is related to the surface temperature \(T_{\mathrm{b}}\) and the ambient temperature \(T_{0}\) by \(\mathrm{hA}\left(T_{\mathrm{b}}-T_{0}\right)\), where \(\mathrm{h}=100 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}, \mathrm{A}=0.195 \mathrm{~m}^{2}\), and \(T_{0}=293 \mathrm{~K}\). Energy transfers are considered positive in the directions indicated by the arrows on Fig. P6.51. (a) Determine the temperature \(T_{\mathrm{b}}\), in \(\mathrm{K}\). (b) For the motor as the system, determine the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\). (c) If the system boundary is located to take in enough of the nearby surroundings for heat transfer to take place at temperature \(T_{0}\), determine the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\), for the enlarged system.

One-half kilogram of propane initially at 4 bar, \(30^{\circ} \mathrm{C}\) undergoes a process to 14 bar, \(100^{\circ} \mathrm{C}\) while being rapidly compressed in a piston-cylinder assembly. Heat transfer with the surroundings at \(20^{\circ} \mathrm{C}\) occurs through a thin wall. The net work is measured as \(-72.5 \mathrm{~kJ}\). Kinetic and potential energy effects can be ignored. Determine whether it is possible for the work measurement to be correct.

Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) enters a nozzle operating at steady state at 28 bar, \(267^{\circ} \mathrm{C}\), and \(50 \mathrm{~m} / \mathrm{s}\). At the nozzle exit, the conditions are \(1.2\) bar, \(67^{\circ} \mathrm{C}, 580 \mathrm{~m} / \mathrm{s}\), respectively. (a) For a control volume enclosing the nozzle only, determine the heat transfer, in \(\mathrm{kJ}\), and the change in specific entropy, in \(\mathrm{kJ} / \mathrm{K}\), each per \(\mathrm{kg}\) of carbon dioxide flowing through the nozzle. What additional information would be required to evaluate the rate of entropy production? (b) Evaluate the rate of entropy production, in \(\mathrm{kJ} / \mathrm{K}\) per \(\mathrm{kg}\) of carbon dioxide flowing, for an enlarged control volume enclosing the nozzle and a portion of its immediate surroundings so that the heat transfer occurs at the ambient temperature, \(25^{\circ} \mathrm{C}\).

Air enters a compressor operating at steady state at \(17^{\circ} \mathrm{C}\), 1 bar and exits at a pressure of 5 bar. Kinetic and potential energy changes can be ignored. If there are no internal irreversibilities, evaluate the work and heat transfer, each in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of air flowing, for the following cases: (a) isothermal compression. (b) polytropic compression with \(n=1.3\). (c) adiabatic compression. Sketch the processes on \(p-v\) and \(T-s\) coordinates and associate areas on the diagrams with the work and heat transfer in each case. Referring to your sketches, compare for these cases the magnitudes of the work, heat transfer, and final temperatures, respectively.

A system undergoes a thermodynamic power cycle while receiving energy by heat transfer from an incompressible body of mass \(m\) and specific heat \(c\) initially at temperature \(T_{\mathrm{H}}\). The system undergoing the cycle discharges energy by heat transfer to another incompressible body of mass \(m\) and specific heat \(c\) initially at a lower temperature \(T_{\mathrm{C}}\). Work is developed by the cycle until the temperature of each of the two bodies is the same, \(T^{\prime}\). (a) Develop an expression for the minimum theoretical final temperature, \(T^{\prime}\), in terms of \(m, c, T_{\mathrm{H}}\), and \(T_{\mathrm{C}}\), as required. (b) Develop an expression for the maximum theoretical amount of work that can be developed, \(W_{\max }\), in terms of \(m, c, T_{\mathrm{H}}\), and \(T_{\mathrm{C}}\), as required. (c) What is the minimum theoretical work input that would be required by a refrigeration cycle to restore the two bodies from temperature \(T^{\prime}\) to their respective initial temperatures, \(T_{\mathrm{H}}\) and \(T_{\mathrm{C}} ?\)
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