Question

A well-insulated rigid tank contains \(6 \mathrm{lbm}\) of saturated liquid- vapor mixture of water at 35 psia. Initially, three-quarters of the mass is in the liquid phase. An electric resistance heater placed in the tank is turned on and kept on until all the liquid in the tank is vaporized. Assuming the surroundings to be at \(75^{\circ} \mathrm{F}\) and 14.7 psia, determine \((a)\) the exergy destruction and ( \(b\) ) the second-law efficiency for this process.

Short answer

Based on the given step-by-step solution, the problem involves finding the exergy destruction and the second-law efficiency for a rigid, well-insulated tank containing a saturated liquid-vapor mixture of water at 35 psia as it is heated until all the liquid is vaporized. The procedure includes finding the initial and final states of the system, calculating the initial and final exergies, determining the exergy change, calculating the exergy destruction, and finally finding the second-law efficiency.

Step-by-step solution

Find the initial and final states of the system

To analyze the initial state, we know the system has a saturated liquid-vapor mixture at 35 psia. We can find the specific volumes, specific enthalpies, and specific internal energies of both the liquid and vapor phases at this pressure from the steam tables. Since three-quarters of the mass is in the liquid phase, the initial state can be determined using mass fractions. In the final state, the system has only vapor because all liquid has been vaporized. We can find the final state properties by knowing that it's a saturated vapor at 35 psia, and then using steam tables again.

Calculate the initial and final exergies of the system

Using the initial and final state properties, we can calculate the exergy for both states. We need to use the environment temperature \(T_{0} = 75^{\circ} F\) (537.6 R) and pressure \(P_{0} = 14.7\:psia\) for this. The exergy for each state can be calculated using the equation: \(e = u - u_{0} + P_{0}(v - v_{0}) - T_{0}(s - s_{0})\) Where \(u\), \(v\), and \(s\) are the specific internal energy, specific volume, and specific entropy from the initial and final states, and \(u_{0}\), \(v_{0}\), and \(s_{0}\) are the specific internal energy, specific volume, and specific entropy at the environment reference state.

Determine the exergy change of the system

The exergy change of the system is the difference between the final and initial exergies: \(\Delta e = e_{f} - e_{i}\).

Calculate the exergy destruction

The exergy balance equation for the process is given by: \(\Delta e = W_{elec} - X_{destr}\), where \(W_{elec}\) is the electrical work input and \(X_{destr}\) is the exergy destruction. The electrical work can be calculated from the energy balance equation: \(W_{elec} = m (u_{f} - u_{i})\), where \(m\) is the mass of the system. Exergy destruction can now be calculated as: \(X_{destr} = W_{elec} - \Delta e\)

Calculate the second-law efficiency

The second-law efficiency is the ratio of minimum useful work to the actual work done. In this case, the minimum useful work is the reversible work (negative exergy destruction), and the actual work done is the electrical work input. Second-law efficiency: \(\eta_{II} = \frac{W_{elec} - X_{destr}}{W_{elec}}\)

Key concepts

Saturated Liquid-Vapor Mixture

When actually dealing with a substance like water, there's a specific state that occurs during the phase change between liquid and vapor called a 'saturated liquid-vapor mixture'. In this state, both phases coexist and are in equilibrium with each other. This particular phase equilibrium is crucial for various engineering applications, including refrigeration and power generation. For example, in our problem, the tank initially contains a mixture with three-quarters of the mass as liquid and the remainder as vapor. Understanding the characteristics of this mixture helps us in evaluating system properties such as pressure, temperature, volume, enthalpy, and entropy, which are essential for the subsequent thermodynamic analysis.

Second-Law Efficiency

The 'second-law efficiency' is an evaluating criterion often used in thermodynamics to assess the effectiveness of thermodynamic processes. It is defined as the ratio of the actual useful work output to the maximum possible useful work output that could be achieved if the process were reversible. This concept is grounded in the second law of thermodynamics, which indicates that not all input energy can be converted into useful work due to irreversibilities like friction or heat loss. Regarding our exercise, the second-law efficiency provides insight into how well the electric heater performed in vaporizing the water relative to an ideal, reversible process.

Exergy Destruction

Exergy destruction refers to the irretrievable loss of work potential as energy is transformed from one form to another, often manifesting as an increase in entropy. It is a measure of the inefficiencies or irreversibilities within a thermodynamic system. According to the second law of thermodynamics, exergy destruction is always non-negative and occurs in real processes, contrasting with ideal reversible processes where exergy is conserved. In our problem, the exergy destruction quantifies the amount of energy that could not be used for vaporization due to these irreversibilities.

Electric Resistance Heater

An 'electric resistance heater' converts electrical energy into heat through the resistance encountered by electric current flowing through the heater's material. This type of heater is widely used for its simplicity and high efficiency at the point of use, and in our problem, it serves to vaporize the liquid in the tank. The heat generated by the resistance heater increases the energy of the water molecules, transitioning the liquid into vapor, all while considering the conservation of energy within the system.

Specific Enthalpy and Entropy

Two fundamental thermodynamic properties important in exergy analysis are 'specific enthalpy' and 'specific entropy'. Specific enthalpy is a measure of the total heat content of a system per unit mass, comprising both the internal energy and the work done by the system's pressure. On the other hand, specific entropy quantifies the randomness or disorder of a system. Changes in enthalpy and entropy are critical in determining the work required or produced during a process. For our system of the water mixture, these properties allow us to analyze the energy and entropy changes as the electric heater vaporizes the liquid.

Steam Tables

The 'steam tables' are an essential reference for engineers dealing with steam and water calculations. These tables contain detailed property information for water in all possible phases: solid, liquid, gas (steam), and mixtures like our saturated liquid-vapor scenario. The tables list values for temperature, pressure, specific volume, enthalpy, entropy, and internal energy — allowing for a comprehensive analysis of water-related thermodynamic systems. To solve our exercise, you'll reference these tables to find the specific properties of the saturated liquid-vapor mixture before and after the heating process.