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Chapter 6: Problem 98

Methane \(\left(\mathrm{CH}_{4}\right)\) undergoes an isentropic expansion from an initial state where the temperature is \(1000 \mathrm{~K}\) and the pressure is 5 bar to a final state where the temperature is \(T\) and the pressure is \(p\). Using the ideal gas model together with \(\bar{c}_{\mathrm{p}}(T)\) from Table A-21, determine (a) \(p\) when \(T=500 \mathrm{~K}\) (b) \(T\) when \(p=1\) bar. (c) Check the results of parts (a) and (b) using \(I T\).

Short Answer

Expert verified
(a) p = 4.365 bar, (b) T = 45 K

Step by step solution

01

Identify Initial and Final States

The initial state of the methane is given as: \(T_1 = 1000 \mathrm{~K}\) and \(p_1 = 5\) bar. The final state variables are \(T\) and \(p\). Two scenarios are provided: one where \(T_2 = 500 \mathrm{~K}\) and another where \(p_2 = 1\) bar.
02

Understand Isentropic Relation

For an isentropic process of an ideal gas, we can use the relation: \[\left(\frac{T_2}{T_1}\right)^{\frac{\bar{R}}{\bar{c}_p(T)}} = \frac{p_2}{p_1}\] where \(\bar{R}=8.314 \frac{J}{mol \cdot K}\) for methane and \(\bar{c}_p(T)\) is the specific heat capacity at constant pressure, which is a function of temperature.
03

Calculate \(p_2\) for \(T_2 = 500 \mathrm{~K}\)

Using the specific heat capacity from Table A-21, we find \(\bar{c}_p(1000 \mathrm{~K}) = 35.1 \frac{J}{mol \cdot K}\) and \(\bar{c}_p(500 \mathrm{~K}) = 25.1 \frac{J}{mol \cdot K}\). For an approximation, using average \(\bar{c}_p\), we can take \(\bar{c}_p = 30.1 \frac{J}{mol \cdot K}\). Substitute into the isentropic relation: \[ \left( \frac{500}{1000} \right)^{\frac{8.314}{30.1}} = \frac{p_2}{5}\]. Therefore, solving for \(p_2\): \[ p_2 = 5 \times \left( \frac{1}{2} \right)^{\frac{8.314}{30.1}} = 5 \times 0.873 = 4.365 \text{ bar} \]
04

Calculate \(T_2\) for \(p_2 = 1 \text{ bar}\)

Using the same isentropic relation: \[\left( \frac{T_2}{1000} \right)^{\frac{8.314}{30.1}} = \frac{1}{5}\]. Solving for \(T_2\): \[ \left( \frac{T_2}{1000} \right)^{0.276} = 0.2 \]. Thus: \[\frac{T_2}{1000} = 0.2^{3.62}\]. So, \[ T_2 = 1000 \times 0.0448 = 45 \text{ K} \].
05

Validate Results using Information Technology (I T)

Use software or online calculators for thermodynamic properties of methane and the isentropic process to validate intermediate steps. Compare the calculated pressure and temperature with the result from the steps.

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Key Concepts

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

ideal gas model
To understand the isentropic expansion in this exercise, we first need to grasp the *Ideal Gas Model*. This model assumes that gases consist of a vast number of very small particles moving in random directions and that these particles do not interact except for perfectly elastic collisions.
In the context of the Ideal Gas Law, the equation is: \[ PV = nRT \] where
  • \(P\) is the pressure,
  • \(V\) is the volume,
  • \(n\) is the amount of substance,
  • \(R\) is the ideal gas constant,
  • \(T\) is the absolute temperature.
Here, methane can be treated as an ideal gas, making use of this law crucial for solving related problems.
specific heat capacity
*Specific Heat Capacity* is another essential concept for this exercise. It quantifies the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. For gases, we often talk about the specific heat capacity at constant pressure, denoted as \( \bar{c}_p(T) \).
For methane, \( \bar{c}_p(T) \) can vary with temperature. For example, in the given solution:
  • \( \bar{c}_p(1000 \text{ K}) = 35.1 \frac{J}{mol \cdot K} \)
  • \( \bar{c}_p(500 \text{ K}) = 25.1 \frac{J}{mol \cdot K} \)
By averaging these values, we use \( \bar{c}_p = 30.1 \frac{J}{mol \cdot K} \). This approximation simplifies calculations for problems involving temperature changes and isentropic relations.
thermodynamic properties
*Thermodynamic Properties* describe the fundamental aspects of substances that dictate the behavior in thermodynamic systems. These include temperature, pressure, volume, internal energy, and entropy.
For this problem, we focus on the properties of methane during the isentropic expansion process. Key properties are:
  • Initial and final temperatures \( T_1 \) and \( T_2 \)
  • Initial and final pressures \( p_1 \) and \( p_2 \)
Thermodynamic relationships help us connect these properties, as seen in the isentropic process equations.
isentropic process
Finally, understanding the *Isentropic Process* is vital. This is a thermodynamic process where entropy remains constant. For an ideal gas undergoing an isentropic process, relations between temperature and pressure are described by the equation: \[ \frac{T_2}{T_1} = \frac{p_2}{p_1}^{\frac{\bar{R}}{\bar{c}_p(T)}} \textrm{with } \bar{R} = 8.314 \frac{J}{mol \bullet K} \] In this exercise involving methane, this equation allowed us to solve for unknown pressures and temperatures in different cases:
  • When \( T_2 = 500 \text{ K} \)
  • When \( p_2 = 1 \text{ bar} \)
Such calculations underline the importance of understanding isentropic processes in thermodynamics.

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Most popular questions from this chapter

Air enters a compressor operating at steady state at 1 bar, \(22^{\circ} \mathrm{C}\) with a volumetric flow rate of \(1 \mathrm{~m}^{3} / \mathrm{min}\) and is compressed to 4 bar, \(177^{\circ} \mathrm{C}\). The power input is \(3.5 \mathrm{~kW}\). Employing the ideal gas model and ignoring kinetic and potential energy effects, obtain the following results: (a) For a control volume enclosing the compressor only, determine the heat transfer rate, in \(\mathrm{kW}\), and the change in specific entropy from inlet to exit, in \(\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\). What additional information would be required to evaluate the rate of entropy production? (b) Calculate the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\), for an enlarged control volume enclosing the compressor and a portion of its immediate surroundings so that heat transfer occurs at the ambient temperature, \(22^{\circ} \mathrm{C}\).

Air is compressed in an axial-flow compressor operating at steady state from \(27^{\circ} \mathrm{C}, 1\) bar to a pressure of \(2.1\) bar. The work input required is \(94.6 \mathrm{~kJ}\) per \(\mathrm{kg}\) of air flowing through the compressor. Heat transfer from the compressor occurs at the rate of \(14 \mathrm{~kJ}\) per \(\mathrm{kg}\) at a location on the compressor's surface where the temperature is \(40^{\circ} \mathrm{C}\). Kinetic and potential energy changes can be ignored. Determine (a) the temperature of the air at the exit, in \({ }^{\circ} \mathrm{C}\). (b) the rate at which entropy is produced within the compressor, in \(\mathrm{kJ} / \mathrm{K}\) per \(\mathrm{kg}\) of air flowing.

An insulated cylinder is initially divided into halves by a frictionless, thermally conducting piston. On one side of the piston is \(1 \mathrm{~m}^{3}\) of a gas at \(300 \mathrm{~K}, 2\) bar. On the other side is \(1 \mathrm{~m}^{3}\) of the same gas at \(300 \mathrm{~K}, 1\) bar. The piston is released and equilibrium is attained, with the piston experiencing no change of state. Employing the ideal gas model for the gas, determine (a) the final temperature, in \(\mathrm{K}\). (b) the final pressure, in bar. (c) the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{kg}\).

Steam enters a horizontal \(15-\mathrm{cm}\)-diameter pipe as a saturated vapor at 5 bar with a velocity of \(10 \mathrm{~m} / \mathrm{s}\) and exits at \(4.5\) bar with a quality of \(95 \%\). Heat transfer from the pipe to the surroundings at \(300 \mathrm{~K}\) takes place at an average outer surface temperature of \(400 \mathrm{~K}\). For operation at steady state, determine (a) the velocity at the exit, in \(\mathrm{m} / \mathrm{s}\). (b) the rate of heat transfer from the pipe, in \(\mathrm{kW}\). (c) the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\), for a control volume comprising only the pipe and its contents. (d) the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\), for an enlarged control volume that includes the pipe and enough of its immediate surroundings so that heat transfer from the control volume occurs at \(300 \mathrm{~K}\). Why do the answers of parts (c) and (d) differ?

Water vapor enters an insulated nozzle operating at steady state at \(0.7 \mathrm{MPa}, 320^{\circ} \mathrm{C}, 35 \mathrm{~m} / \mathrm{s}\) and expands to \(0.15 \mathrm{MPa}\). If the isentropic nozzle efficiency is \(94 \%\), determine the velocity at the exit, in \(\mathrm{m} / \mathrm{s}\).
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