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Chapter 7: Problem 32

One-half pound of air is contained in a closed, rigid, insulated tank. Initially the temperature is \(520^{\circ} \mathrm{R}\) and the pressure is \(14.7\) psia. The air is stirred by a paddle wheel until its temperature is \(600^{\circ} \mathrm{R}\). Using the ideal gas model, determine for the air the change in exergy, the transfer of exergy accompanying work, and the exergy destruction, all in Btu. Ignore the effects of motion and gravity and let \(T_{0}=537^{\circ} \mathrm{R}, p_{0}=14.7\) psia.

Short Answer

Expert verified
Exergy change, transfer and destruction can be determined by applying ideal gas laws, work and exergy relations. Calculations yield respective values in Btu.

Step by step solution

01

- Define the system and state properties

Identify that the air is in a closed, rigid, and insulated tank. The mass of air is 0.5 lb. The initial temperature is 520掳R and the pressure is 14.7 psia.
02

- Apply the Ideal Gas Law

Assume the ideal gas model, and use the relation for specific volume: \[ v = \frac{RT}{P} \] Calculate the specific volumes at initial and final states using this relation.
03

- Calculate Initial and Final States

Since the tank is rigid, specific volume remains constant. Calculate specific volumes as: \[ v_1 = \frac{RT_1}{P_1}, \ v_2 = v_1 \] where \[ T_1 = 520R, \ P_1 = 14.7 psia \]
04

- Determine Initial and Final Exergies

Use the exergy formula for an ideal gas: \[ \text{Exergy} = h - h_0 - T_0(s - s_0) \] where \[ h = c_pT \] and \[ s = c_p \text{ln} \frac{T}{T_0} - R \text{ln} \frac{P}{P_0} \] Calculate initial and final exergies.
05

- Calculate Exergy Change

Calculate the change in exergy: \[ \text{Change in Exergy} = \text{Exergy}_{\text{final}} - \text{Exergy}_{\text{initial}} \]
06

- Determine Work Transfer Exergy

For a paddle wheel, the work done is mechanical stirring. Use: \[ \text{Exergy Transfer} = W \] where work can be related to change in internal energy in this isolated system.
07

- Calculate Exergy Destruction

Use the exergy balance equation for a closed system: \[ \text{Change in Exergy} = \text{Exergy Transfer} + \text{Exergy Destruction} \] Rearrange to solve for exergy destruction.
08

- Plug In Constants and Values

Use given constants like T0 = 537 R, P0 = 14.7 psia, and previously calculated intermediate variables. Calculate the numerical values.

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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 principle in thermodynamics that relates the pressure, volume, and temperature of an ideal gas. It is expressed as:
\[ PV = nRT \]
where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature.
For air contained in a rigid, insulated tank, like in the given problem, the volume remains constant. This means that the specific volume (v) is unchanged during the process. We can use a simplified expression for specific volume:
\[ v = \frac{RT}{P} \]
This relationship aids in determining the state properties of the air both before and after the process driven by the paddle wheel.
Specific Volume
Specific volume is defined as the volume occupied by a unit mass of a substance. It is the reciprocal of density and can be crucial in thermodynamic calculations, especially in ideal gas scenarios.
Using the Ideal Gas Law, specific volume for a gas is given by:
\[ v = \frac{RT}{P} \]
In this exercise, since the tank is rigid, the specific volume remains constant throughout the process. Initially, the specific volume is calculated at 520掳R and 14.7 psia. Since there鈥檚 no change in volume, the same specific volume applies at 600掳R:
Exergy
Exergy is a measure of the maximum useful work possible during a thermodynamic process. It considers the system's state relative to a reference environment.
In the context of an ideal gas, exergy can be calculated using the formula:
\[ \text{Exergy} = h - h_0 - T_0(s - s_0) \]
where h is the specific enthalpy, s is the specific entropy, and the subscript 0 denotes properties at the reference environment. For an ideal gas:
\[ h = c_pT \]
and:
\[ s = c_p \text{ln} \frac{T}{T_0} - R \text{ln} \frac{P}{P_0} \]
By applying these relations, you can calculate the initial and final exergy states of the air in the tank.
Paddle Wheel Work
In thermodynamics, a paddle wheel is a common device used to stir the gas in a closed system, doing work on the gas. This work is a form of mechanical energy input.
For this specific problem, paddle wheel work results in an increase in internal energy, which manifests as an increase in temperature of the gas.
This work can be associated with the change in exergy and is accounted for in the exergy transfer term:
\[ \text{Exergy Transfer} = W \]
Here, the energy input by the paddle wheel is directly related to the increase in internal energy of the air.
Exergy Destruction
Exergy destruction represents the loss of useful work potential due to irreversibilities in a process, often converted to heat. It鈥檚 a key concept in understanding the efficiency of thermodynamic processes.
The exergy balance equation for this closed system is:
\[ \text{Change in Exergy} = \text{Exergy Transfer} + \text{Exergy Destruction} \]
Rearranging the formula to solve for exergy destruction, we get:
\[ \text{Exergy Destruction} = \text{Change in Exergy} - \text{Exergy Transfer} \]
By calculating the initial and final exergy, and knowing the exergy transferred by the paddle wheel, we can determine the amount of exergy destroyed due to irreversibilities.

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Most popular questions from this chapter

Air enters a counterflow heat exchanger operating at steady state at \(27^{\circ} \mathrm{C}, 0.3 \mathrm{MPa}\) and exits at \(12^{\circ} \mathrm{C}\). Refrigerant 134 a enters at \(0.4 \mathrm{MPa}\), a quality of \(0.3\), and a mass flow rate of \(35 \mathrm{~kg} / \mathrm{h}\). Refrigerant exits at \(10^{\circ} \mathrm{C}\). Stray heat transfer is negligible and there is no significant change in pressure for either stream. (a) For the Refrigerant 134a stream, determine the rate of heat transfer, in \(\mathrm{kJ} / \mathrm{h}\). (b) For each of the streams, evaluate the change in flow exergy rate, in \(\mathrm{kJ} / \mathrm{h}\), and interpret its value and sign. Let \(T_{0}=22^{\circ} \mathrm{C}, p_{0}=0.1 \mathrm{MPa}\), and ignore the effects of motion and gravity.

Steam at \(450 \mathrm{lbf} / \mathrm{in}^{2}, 700^{\circ} \mathrm{F}\) enters a well-insulated turbine operating at steady state and exits as saturated vapor at a pressure \(p\). (a) For \(p=50 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}\), determine the exergy destruction rate, in Btu per lb of steam expanding through the turbine, and the turbine exergetic and isentropic efficiencies. (b) Plot the exergy destruction rate, in Btu per lb of steam flowing, and the exergetic efficiency and isentropic efficiency, each versus pressure \(p\) ranging from 1 to \(50 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}\) Ignore the effects of motion and gravity and let \(T_{0}=70^{\circ} \mathrm{F}\), \(p_{0}=1 \mathrm{~atm} .\)

One lbmol of carbon monoxide gas is contained in a \(90-\mathrm{ft}^{3}\) rigid, insulated vessel initially at 5 atm. An electric resistor of negligible mass transfers energy to the gas at a constant rate of \(10 \mathrm{Btu} / \mathrm{s}\) for \(2 \mathrm{~min}\). Employing the ideal gas model and ignoring the effects of motion and gravity, determine (a) the change in exergy of the gas, (b) the electrical work, and (c) the exergy destruction, each in Btu. Let \(T_{0}=70^{\circ} \mathrm{F}\), \(p_{0}=1 \mathrm{~atm}\).

Air initially at 1 atm and \(500^{\circ} \mathrm{R}\) with a mass of \(2.5 \mathrm{lb}\) is contained within a closed, rigid tank. The air is slowly warmed, receiving 100 Btu by heat transfer through a wall separating the gas from a thermal reservoir at \(800^{\circ} \mathrm{R}\). This is the only energy transfer. Assuming the air undergoes an internally reversible process and using the ideal gas model, (a) determine the change in exergy and the exergy transfer accompanying heat, each in Btu, for the air as the system. (b) determine the exergy transfer accompanying heat and the exergy destruction, each in Btu, for an enlarged system that includes the air and the wall, assuming that the state of the wall remains unchanged. Compare with part (a) and comment. Let \(T_{0}=90^{\circ} \mathrm{F}, p_{0}=1 \mathrm{~atm}\).

Refrigerant \(134 a\) as saturated vapor at \(-10^{\circ} \mathrm{C}\) enters a compressor operating at steady state with a mass flow rate of \(0.3 \mathrm{~kg} / \mathrm{s}\). At the compressor exit the pressure of the refrigerant is 5 bar. Stray heat transfer and the effects of motion and gravity can be ignored. If the rate of exergy destruction within the compressor must be kept less than \(2.4 \mathrm{~kW}\), determine the allowed ranges for (a) the power required by the compressor, in \(k W\), and (b) the exergetic compressor efficiency. Let \(T_{0}=298 \mathrm{~K}, p_{0}=1\) bar.
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