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Chapter 6: Problem 102

A rigid tank is filled initially with \(5.0 \mathrm{~kg}\) of air at a pressure of \(0.5 \mathrm{MPa}\) and a temperature of \(500 \mathrm{~K}\). The air is allowed to discharge through a turbine into the atmosphere, developing work until the pressure in the tank has fallen to the atmospheric level of \(0.1 \mathrm{MPa}\). Employing the ideal gas model for the air, determine the maximum theoretical amount of work that could be developed, in \(\mathrm{kJ}\). Ignore heat transfer with the atmosphere and changes in kinetic and potential energy.

Short Answer

Expert verified
The maximum theoretical amount of work developed is \( \frac{mR (T_f - T_i)}{M} \).

Step by step solution

01

Define initial state conditions

The initial mass of the air is given as \(m_i = 5.0 \mathrm{kg}\). The initial pressure \(P_i\) and temperature \(T_i\) of the air are \(0.5 \mathrm{MPa}\) and \(500 \mathrm{K}\), respectively.
02

Define final state conditions

The final pressure \(P_f\) in the tank is given as \(0.1 \mathrm{MPa}\). We aim to find the work done as the air discharges until this condition.
03

Use the ideal gas law

Using the ideal gas law for state 1 (initial condition): \ PV = nRT \[ P_iV = n_iR T_i \ n_i = \frac{m}{M} \] where \(M\) is the molar mass of air ( \(M = 28.97 \mathrm{g/mol}\)).
04

Calculate initial volume

Rearrange the ideal gas law for volume: \[ V = \frac{n_iRT_i}{P_i} = \frac{m_i \times R \times T_i}{M \times P_i} \] where \(R\) is the universal gas constant ( \(R = 8.314 \frac{J}{mol\cdot K}\)). Plug in the values to find the initial volume.
05

Apply the energy conservation principle

For a closed system, use the first law of thermodynamics ignoring heat transfer and changes in kinetic or potential energy. Assuming a quasistatic process: \[ W = - \frac{mR (T_f - T_i)}{M} \]
06

Use given conditions and equation to solve for work

Since pressure drops to atmospheric level, rearrange: \[ T_i = T_f \] \[ Simplifying this equation for maximum theoretical work we will get the final answer considering the values. \]

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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 equation in thermodynamics that describes the relationship between pressure, volume, and temperature of an ideal gas. It is expressed as \[ PV = nRT \] where
  • P is the pressure of the gas,
  • V is the volume,
  • n is the number of moles,
  • R is the universal gas constant (8.314 \frac{J}{mol\bullet K}
  • T is the temperature in Kelvin.
In the given exercise, you use the ideal gas law to determine the volume of air in the tank initially by rearranging the formula to solve for volume (V = \frac{nRT}{P}). This relationship helps us understand how the pressure, volume, and temperature are interrelated when air (assumed to be an ideal gas) is discharged through the turbine.
First Law of Thermodynamics
The first law of thermodynamics is also known as the principle of energy conservation. It states that energy cannot be created or destroyed, only transformed from one form to another. Mathematically, it is expressed as \[ \text{ΔU} = Q - W \] where:
  • ΔU is the change in internal energy,
  • Q is the heat added to the system, and
  • W is the work done by the system.
In this exercise, we assume no heat transfer with the surroundings and ignore changes in kinetic and potential energy. Therefore, the work done by the system as air discharges through the turbine is derived purely from internal energy changes, leading us to use
  • W = -\frac{mR (T_f - T_i)}{M}
This equation simplifies the process of finding the maximum theoretical work output.
Quasistatic Process
A quasistatic process is an idealized or hypothetical process that happens infinitely slowly, so the system remains in thermodynamic equilibrium at all times during the process. Such a process requires small, incremental changes that ensure the state variables are always well-defined and reversible.

In the given exercise, assuming the discharge process through the turbine to be quasistatic means that the air has time to adjust its pressure and volume incrementally, ensuring equilibrium is maintained. This assumption simplifies the analysis, allowing us to apply the ideal gas law and derive the expression for work done in the theoretical maximum case.
Energy Conservation
Energy conservation is a key principle in thermodynamics and forms the foundation for understanding how energy changes and transfers in a closed or isolated system. It asserts that the total energy of an isolated system remains constant, implying energy cannot be created or destroyed but can only change forms.

In the context of our problem, energy conservation is applied through the first law of thermodynamics to calculate the amount of work done by the system. By knowing that no heat is transferred with the surroundings and ignoring kinetic and potential energy changes, the focus is purely on converting internal energy into work. This approach uses the relations derived from the ideal gas law and ensures the calculations are consistent with the overall principle of energy conservation.

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Most popular questions from this chapter

A system consists of \(2 \mathrm{~m}^{3}\) of hydrogen gas \(\left(\mathrm{H}_{2}\right)\), initially at \(35^{\circ} \mathrm{C}, 215 \mathrm{kPa}\), contained in a closed rigid tank. Energy is transferred to the system from a reservoir at \(300^{\circ} \mathrm{C}\) until the temperature of the hydrogen is \(160^{\circ} \mathrm{C}\). The temperature at the system boundary where heat transfer occurs is \(300^{\circ} \mathrm{C}\). Modeling the hydrogen as an ideal gas, determine the heat transfer, in \(\mathrm{kJ}\), the change in entropy, in \(\mathrm{kJ} / \mathrm{K}\), and the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{K}\). For the reservoir, determine the change in entropy, in \(\mathrm{kJ} / \mathrm{K}\). Why do these two entropy changes differ?

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.

Air enters a turbine operating at steady state at 6 bar and \(1100 \mathrm{~K}\) and expands isentropically to a state where the temperature is \(700 \mathrm{~K}\). Employing the ideal gas model and ignoring kinetic and potential energy changes, determine the pressure at the exit, in bar, and the work, in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of air flowing, using (a) data from Table A-22. (b) \(I T\). (c) a constant specific heat ratio from Table A-20 at the mean temperature, \(900 \mathrm{~K}\). (d) a constant specific heat ratio from Table A-20 at \(300 \mathrm{~K}\).

A patent application describes a device for chilling water. At steady state, the device receives energy by heat transfer at a location on its surface where the temperature is \(540^{\circ} \mathrm{F}\) and discharges energy by heat transfer to the surroundings at another location on its surface where the temperature is \(100^{\circ} \mathrm{F}\). A warm liquid water stream enters at \(100^{\circ} \mathrm{F}, 1 \mathrm{~atm}\) and a cool stream exits at temperature \(T\) and \(1 \mathrm{~atm}\). The device requires no power input to operate, there are no significant effects of kinetic and potential energy, and the water can be modeled as incompressible. Plot the minimum theoretical heat addition required, in Btu per \(\mathrm{lb}\) of cool water exiting the device, versus \(T\) ranging from 60 to \(100^{\circ} \mathrm{F}\).

A well-insulated rigid tank of volume \(10 \mathrm{~m}^{3}\) is connected by a valve to a large-diameter supply line carrying air at \(227^{\circ} \mathrm{C}\) and 10 bar. The tank is initially evacuated. Air is allowed to flow into the tank until the tank pressure is \(p\). Using the ideal gas model with constant specific heat ratio \(k\), plot tank temperature, in \(\mathrm{K}\), the mass of air in the tank, in \(\mathrm{kg}\), and the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{K}\), versus \(p\) in bar.
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