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Chapter 6: Problem 51

One \(\mathrm{kg}\) of propane initially at 8 bar and \(50^{\circ} \mathrm{C}\) undergoes a process to 3 bar, \(20^{\circ} \mathrm{C}\) while being rapidly expanded in a piston-cylinder assembly. Heat transfer between the propane and its surroundings occurs at an average temperature of \(35^{\circ} \mathrm{C}\). The work done by the propane is measured as \(42.4 \mathrm{~kJ}\). Kinetic and potential energy effects can be ignored. Determine whether it is possible for the work measurement to be correct.

Short Answer

Expert verified
Use the first law: \( Q = W + \triangle U \). Verify if calculated \(Q\) aligns with process conditions to check validity.

Step by step solution

01

Understand the Initial and Final States

The propane undergoes a process from an initial state of 8 bar and 50掳C to a final state of 3 bar and 20掳C. These states need to be analyzed to find properties such as internal energy.
02

Identify Process Type and Given Data

The propane expands rapidly in a piston-cylinder assembly, suggesting an adiabatic process (no heat transfer) or an isothermal process (constant temperature). The given data includes heat transfer at an average temperature of 35掳C and work done as 42.4 kJ.
03

Apply the First Law of Thermodynamics

The first law of thermodynamics for a closed system (piston-cylinder) is given by: \[ Q - W = \triangle U \]. Where \(Q\) is the heat transfer, \(W\) is the work done by the system (42.4 kJ), and \(\triangle U\) is the change in internal energy. Rearrange this formula to solve for \(Q\): \[ Q = W + \triangle U \]
04

Determine Change in Internal Energy

To find \(\triangle U\), use thermodynamic tables or software for propane to find internal energy at initial and final states. Denote initial internal energy as \(U_1\) and final internal energy as \(U_2\), and compute \(\triangle U\) using the relation: \[ \triangle U = U_2 - U_1 \]
05

Evaluate Heat Transfer

Using the average temperature for heat transfer (35掳C) and substituting \(Q\) into the first law equation: \[ Q = W + (U_2 - U_1) \]. Assess if the value of \(Q\) is reasonable given the process conditions.
06

Validate Work Measurement

Compare the calculated heat transfer \(Q\) with physical expectations from the process. If the calculated \(Q\) aligns closely with the physical scenario and constraints, the measured work of 42.4 kJ can be considered valid; else, re-evaluate the assumptions.

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Key Concepts

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

First Law of Thermodynamics
The First Law of Thermodynamics is a fundamental concept in physics and thermodynamics. It states that energy cannot be created or destroyed, only transferred or converted from one form to another. This principle can be expressed by the equation: \[ Q - W = \triangle U \] Where:
  • Q: Heat added to the system
  • W: Work done by the system
  • \triangle U: Change in internal energy of the system
In this problem, the propane in the piston-cylinder assembly undergoes a change. By applying the First Law, we can understand how much energy is transferred as heat, and how much work is done by the system. It helps us check if the work measurement is reasonable given the process conditions.
Piston-cylinder assembly
A piston-cylinder assembly is a common setup in thermodynamics. It consists of a piston that moves within a cylinder, allowing the gas inside to expand or compress. This setup can be used to study different thermodynamic processes, such as isothermal (constant temperature), adiabatic (no heat transfer), and isobaric (constant pressure) processes.In the given problem, propane expands rapidly in a piston-cylinder assembly, which means the piston moves to allow the gas to expand. The key characteristics of this setup are:
  • The volume change is controlled by the movement of the piston.
  • It can be either insulated (adiabatic) or allow for heat exchange (non-adiabatic).
  • The pressure and temperature can vary, impacting the internal energy and work done.
Understanding these properties helps us analyze the behavior of the gas and calculate the related work and heat.
Internal Energy
Internal energy refers to the total energy contained within a system due to molecular motion and interactions. For a gas, it depends on the temperature and pressure. In this exercise, the internal energy of propane changes as it transitions from one state to another (from 8 bar and 50掳C to 3 bar and 20掳C).Calculating the change in internal energy (\triangle U) involves the following steps:
  • Identify the initial and final states of the system.
  • Use thermodynamic tables or software to find the internal energy values (U_1 for initial state and U_2 for final state).
  • Calculate the difference: \[ \triangle U = U_2 - U_1 \]
In the exercise, this change helps in applying the First Law of Thermodynamics to ascertain if the provided work done by the propane (42.4 kJ) is possible. By evaluating \[ Q = W + \triangle U \], we can check the consistency of the solution with physical laws.

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Most popular questions from this chapter

Steam enters a turbine operating at steady state at \(6 \mathrm{MPa}, 600^{\circ} \mathrm{C}\) with a mass flow rate of \(125 \mathrm{~kg} / \mathrm{min}\) and exits as saturated vapor at \(20 \mathrm{kPa}\), producing power at a rate of 2 MW. Kinetic and potential energy effects can be ignored. Determine (a) the rate of heat transfer, in \(\mathrm{kW}\), for a control volume including the turbine and its contents, and (b) the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\), for an enlarged control volume that includes the turbine and enough of its surroundings that heat transfer occurs at the ambient temperature, \(27^{\circ} \mathrm{C}\).

One kg of air contained in a piston-cylinder assembly undergoes a process from an initial state where \(T_{1}=300 \mathrm{~K}\), \(v_{1}=0.8 \mathrm{~m}^{3} / \mathrm{kg}\) to a final state where \(T_{2}=420 \mathrm{~K}, v_{2}=\) \(0.2 \mathrm{~m}^{3} / \mathrm{kg}\). Can this process occur adiabatically? If yes, determine the work, in \(\mathrm{kJ}\), for an adiabatic process between these states. If no, determine the direction of the heat transfer. Assume the ideal gas model for air.

Refrigerant 22 enters the heat exchanger of an airconditioning system at \(80 \mathrm{lbf} / \mathrm{in}^{2}\) with a quality of \(0.2\). The refrigerant stream exits at \(80 \mathrm{lbf} / \mathrm{in}^{2}, 60^{\circ} \mathrm{F}\). Air flows in counterflow through the heat exchanger, entering at \(14.9 \mathrm{lbf}\) in. \(^{2}, 80^{\circ} \mathrm{F}\), with a volumetric flow rate of \(100,000 \mathrm{ft}^{3} / \mathrm{min}\) and exiting at \(14.5 \mathrm{lbf} / \mathrm{in}^{2}, 65^{\circ} \mathrm{F}\). Operation is at steady state, stray heat transfer from the outside of the heat exchanger to the surroundings can be neglected, and kinetic and potential energy effects are negligible. Assuming ideal gas behavior for the air, determine the rate of entropy production in the heat exchanger, in Btu/min \({ }^{\circ}{ }^{\circ} \mathrm{R}\).

One-tenth kmol of carbon monoxide \((\mathrm{CO})\) in a pistoncylinder assembly undergoes a process from \(p_{1}=150 \mathrm{kPa}\), \(T_{1}=300 \mathrm{~K}\) to \(p_{2}=500 \mathrm{kPa}, T_{2}=370 \mathrm{~K}\). For the process, \(\mathrm{W}=-300 \mathrm{~kJ}\). Employing the ideal gas model, determine (a) the heat transfer, in kJ. (b) the change in entropy, in \(\mathrm{kJ} / \mathrm{K}\). Show the process on a sketch of the \(T-s\) diagram.

One lb of water contained in a piston-cylinder assembly, initially saturated vapor at \(1 \mathrm{~atm}\), is condensed at constant pressure to saturated liquid. Evaluate the heat transfer, in Btu, and the entropy production, in Btu/ \({ }^{\circ} \mathrm{R}\), for (a) the water as the system. (b) an enlarged system consisting of the water and enough of the nearby surroundings that heat transfer occurs only at the ambient temperature, \(80^{\circ} \mathrm{F}\). Assume the state of the nearby surroundings does not change during the process of the water, and ignore kinetic and potential energy.
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