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Chapter 7: Problem 57

Carbon monoxide at \(250 \mathrm{lbf}^{2} / \mathrm{in}^{2}, 850^{\circ} \mathrm{R}\), and a volumetric flow rate of \(75 \mathrm{ft}^{3} / \mathrm{min}\) enters a valve operating at steady state and undergoes a throttling process. Assuming ideal gas behavior (a) determine the rate of exergy destruction, in Btu/min, for an exit pressure of \(30 \mathrm{lbf} / \mathrm{in}^{2}\) (b) plot the exergy destruction rate, in Btu/min, versus exit pressure ranging from 30 to \(250 \mathrm{lbf} / \mathrm{in}^{2}\) Let \(T_{0}=530^{\circ} \mathrm{R}, p_{0}=15 \mathrm{lbf} / \mathrm{in}^{2}\)

Short Answer

Expert verified
Use the specific exergy formula to find \( \psi_1 \) and \( \psi_2 \), calculate \( \dot{m} \), and find \( \dot{E}_d \) as the product of mass flow rate and exergy change. Plot \( \dot{E}_d \) against exit pressure.

Step by step solution

01

- Understanding the Throttling Process

In a throttling process, the enthalpy remains constant: \[ h_{in} = h_{out} \]. Given the ideal gas assumption, specific enthalpy of an ideal gas depends only on temperature. Therefore, the temperature at the exit is the same as at the entrance, \(T = 850^\circ R\).
02

- Define the Inlet and Exit Conditions

At the inlet: \[p_1 = 250 \text{ lbf/in}^2, T_1 = 850^\circ R\]. At the exit: \[p_2 = 30 \text{ lbf/in}^2, T_2 = 850^\circ R\]. The surrounding conditions are \(T_0 = 530^\circ R\) and \(p_0 = 15 \text{ lbf/in}^2\).
03

- Calculate Specific Exergy at Inlet and Exit

The specific exergy of an ideal gas is given by: \[ \psi = c_p(T - T_0) - R T_0 \ln\left(\frac{p}{p_0}\right) \]. For carbon monoxide, assume specific heat at constant pressure \(c_p = 0.249 \text{ Btu/lb}\cdot\text{掳R}\) and gas constant \(R = 0.068 \text{ Btu/lb}\cdot\text{掳R}\).
04

- Calculate Specific Exergy at the Inlet

Using the formula: \[ \psi_1 = c_p(T_1 - T_0) - R T_0 \ln\left(\frac{p_1}{p_0}\right) \], Substitute \(T_1 = 850^{\circ}R\), \(T_0 = 530^{\circ}R\), \(p_1 = 250 \text{ lbf/in}^2\), and \(p_0 = 15 \text{ lbf/in}^2\): \[ \psi_1 = 0.249(850 - 530) - 0.068 \cdot 530 \cdot \ln\left(\frac{250}{15}\right) \].
05

- Calculate Specific Exergy at the Exit

Using the same formula: \[ \psi_2 = c_p(T_2 - T_0) - R T_0 \ln\left(\frac{p_2}{p_0}\right) \], Substitute \(T_2 = 850^{\circ}R\), \(T_0 = 530^{\circ}R\), \(p_2 = 30 \text{ lbf/in}^2\), and \(p_0 = 15 \text{ lbf/in}^2\): \[ \psi_2 = 0.249(850 - 530) - 0.068 \cdot 530 \cdot \ln\left(\frac{30}{15}\right) \].
06

- Calculate the Mass Flow Rate

Using the ideal gas equation for the volume flow rate: \[ \dot{m} = \frac{\dot{V} p}{R T} \], substitute \(\dot{V} = 75 \text{ ft}^3/\text{min}\), \(p = 250 \text{ lbf/in}^2\), and \(T = 850^{\circ}R\): \[ \dot{m} = \frac{75 \cdot 250}{0.068 \cdot 850} \text{ lb/min} \].
07

- Calculate the Rate of Exergy Destruction

The rate of exergy destruction is given by: \[ \dot{E}_{d} = \dot{m}(\psi_1 - \psi_2) \]. Substitute the values of \(\dot{m}\), \(\psi_1\), and \(\psi_2\) to find \( \dot{E}_{d} \).
08

- Plot Exergy Destruction Rate vs. Exit Pressure

Vary the exit pressure from 30 to 250 lb/in^2 and calculate \( \psi_2 \) for each value. Plot \( \dot{E}_{d} \) against the range of exit pressures.

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Key Concepts

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

Throttling Process
In thermodynamics, a throttling process is one where a fluid's enthalpy remains constant as it passes through a restriction, such as a valve. This process is considered adiabatic, meaning no heat is exchanged with the surroundings. In simpler terms, when a gas undergoes throttling, the product remains at the same temperature but experiences a drop in pressure. This is crucial in understanding the rest of the concepts because temperature consistency simplifies our calculations.
Ideal Gas Behavior
Ideal gas behavior refers to the assumption that gases follow the ideal gas law perfectly: \(PV = nRT\). This assumption simplifies calculations because we don't have to account for complexities like intermolecular forces. For carbon monoxide in this exercise, treating it as an ideal gas makes it easier to use the specific enthalpy and pressure relations directly.
Specific Exergy
Specific exergy is a measure of the useful work potential of a substance, relative to its environment. For an ideal gas, it can be described by the formula: \(蠄 = c_p(T - T_0) - R T_0 \text{ln}( \frac{p}{p_0} )\). Here, - \(c_p\) is the specific heat at constant pressure- \(T\) is the initial temperature - \(T_0\) is the surrounding temperature or reference state- \(p\) is the initial pressure - \(p_0\) is the surrounding pressure or reference state. This formula helps in determining the energy that can be extracted from the gas, considering its deviation from the surrounding conditions.
Exergy Destruction Rate
Exergy destruction represents the loss of potential work during a process, primarily due to irreversibilities like friction or mixing. The rate of exergy destruction in a throttling process can be found using the mass flow rate and the difference in specific exergy between the inlet and outlet states: \( \text{\textdot{E}}_d = \text{\textdot{m}}(蠄_1 - 蠄_2) \). Here, - \( \text{\textdot{m}} \) is the mass flow rate- \( 蠄_1 \) and \( 蠄_2 \) are the specific exergy values at the inlet and exit, respectively. By calculating these values, we can quantify how much energy potential is lost as the gas passes through the valve.
Steady State Thermodynamics
Steady state thermodynamics refers to systems where properties do not change over time. In this exercise, the temperature, pressure, and mass flow rate of carbon monoxide remain consistent throughout the throttling process. This makes the analysis simpler because the energy and mass balances can be straightforwardly applied without accounting for time-dependent changes. This fundamentally helps in modeling real-world engineering systems where stability is crucial.

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Most popular questions from this chapter

A high-pressure (HP) boiler and a low-pressure (LP) boiler will be added to a plant's steam-generating system. Both boilers use the same fuel and at steady state have approximately the same rate of energy loss by heat transfer. The average temperature of the combustion gases is less in the LP boiler than in the HP boiler. In comparison to the LP boiler, might you spend more, the same, or less to insulate the HP boiler? Explain.

A domestic water heater holds \(189 \mathrm{~L}\) of water at \(60^{\circ} \mathrm{C}\), \(1 \mathrm{~atm}\). Determine the exergy of the hot water, in \(\mathrm{kJ}\). To what elevation, in \(m\), would a \(1000-\mathrm{kg}\) mass have to be raised from zero elevation relative to the reference environment for its exergy to equal that of the hot water? Let \(T_{0}=298 \mathrm{~K}\), \(p_{0}=1 \mathrm{~atm}, \mathrm{~g}=9.81 \mathrm{~m} / \mathrm{s}^{2}\).

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 \(T_{0}=70^{\circ} \mathrm{F}\), \(p_{0}=1 \mathrm{~atm}\).

Two kilograms of carbon monoxide in a piston-cylinder assembly, initially at 1 bar and \(27^{\circ} \mathrm{C}\), is heated at constant pressure with no internal irreversibilities to a final temperature of \(227^{\circ} \mathrm{C}\). Employing the ideal gas model, determine the work, the heat transfer, and the amounts of exergy transfer accompanying work and heat transfer, each in kJ. Let \(T_{0}=300 \mathrm{~K}, p_{0}=1\) bar and ignore the effects of motion and gravity.

A vessel contains \(1 \mathrm{lb}\) of air at pressure \(p\) and \(200^{\circ} \mathrm{F}\). Using the ideal gas model, plot the specific exergy of the air, in Btu/lb, for \(p\) ranging from \(0.5\) to 2 atm. The air is at rest and negligible elevation relative to an exergy reference environment for which \(T_{0}=60^{\circ} \mathrm{F}, p_{0}=1 \mathrm{~atm}\).
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