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Chapter 3: Problem 69

Water contained in a piston-cylinder assembly, initially at \(300^{\circ} \mathrm{F}\), a quality of \(90 \%\), and a volume of \(6 \mathrm{ft}^{3}\), is heated at constant temperature to saturated vapor. If the rate of heat transfer is \(0.3 \mathrm{Btu} / \mathrm{s}\), determine the time, in min, for this process of the water to occur. Kinetic and potential energy effects are negligible.

Short Answer

Expert verified
Solve for mass using specific volumes and then use heat transfer rate and specific enthalpy change to determine the time.

Step by step solution

01

Understand Initial Conditions

The water is initially at \(300^{\text{F}}\) with a quality \( x_1 = 0.90 \). The initial volume \( V_1 \) is 6 ft\(^3\).
02

Find Properties at Initial State

From steam tables, at \( 300^{\text{F}} \), find the saturated liquid and vapor specific volumes, \( v_f \) and \( v_g \). Calculate the initial specific volume \(v_1\) using the formula: \[ v_1 = x_1 v_g + (1 - x_1) v_f \]
03

Calculate Initial Mass

Use the relationship between volume, specific volume, and mass: \[ V_1 = m v_1 \] Solve for mass \( m \): \[ m = \frac{V_1}{v_1} \]
04

Determine Final State

At the final state, the water is a saturated vapor, so quality \( x_2 = 1 \) and specific volume \( v_2 = v_g = 1.0362 \frac{ft^3}{lbm} \).
05

Calculate Heat Required

The heat required to change from initial to final state can be calculated using the formula: \[ Q = m (h_2 - h_1) \] where \( h_2 \) and \( h_1 \) are the specific enthalpies of the final and initial states respectively.
06

Calculate Time Required for Heating

Given the rate of heat transfer \( \.Q = 0.3 \ \text{Btu/s} \), calculate the time required for heating using: \[ time = \frac{Q}{\.Q} \] Convert the time from seconds to minutes

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Key Concepts

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

Piston-Cylinder Assembly
A piston-cylinder assembly is a classic setup in thermodynamics used to understand various processes involving gases and vapors. It consists of a cylinder with a movable piston. The piston can move up or down within the cylinder, allowing the volume inside the cylinder to change.
This assembly is often used to study processes like heating, cooling, compression, and expansion of gases and vapors.
In the given problem, water is contained in a piston-cylinder assembly. The volume changes during the heating process while maintaining constant temperature, demonstrating a typical thermodynamic scenario.
Quality of Steam
The quality of steam, often represented by the symbol **x**, is a measure of the proportion of the mass of steam that is in the vapor phase.
In technical terms, quality is the ratio of the mass of vapor to the total mass of the mixture of liquid and vapor. It ranges from 0 to 1 (or 0% to 100%).
For instance, in the problem, the initial quality is 90%, meaning the water is 90% vapor and 10% liquid at the start. By the end of the process, the quality becomes 100%, indicating that the water is completely in the vapor phase (saturated vapor).
Specific Volume
Specific volume (v) is an important property in thermodynamics, defined as the volume occupied by a unit mass of a substance.
It is usually expressed in units like ft鲁/lbm or m鲁/kg. You can think of it as the 'volume per mass'.
In this problem, the specific volume helps to determine the initial and final volumes of the water mixture. We calculated the initial specific volume using steam tables and the given quality of the initial state.
Specific volume can vary with temperature and pressure, and it is crucial in analyzing and solving thermodynamic problems.
Heat Transfer Rate
The heat transfer rate is the amount of heat energy transferred per unit time.
In the given problem, the rate of heat transfer is provided as 0.3 Btu/s. This tells us how quickly heat is being added to the water in the piston-cylinder assembly.
The heat transfer rate is crucial for determining how long a process will take. In this case, we used it to calculate the time required to transform the water from its initial state to saturated vapor at constant temperature.
The formula used was: \[ time = \frac{Q}{\text{Heat Transfer Rate}} \]
Saturated Vapor
Saturated vapor is a term used to describe a state where a substance is at its boiling point and the liquid and vapor phases coexist in equilibrium.
At this point, any additional heat will cause the liquid to convert into vapor without raising the temperature.
In the problem, the final state of the water is a saturated vapor, meaning it has all converted to the vapor phase at the saturation temperature.
Understanding the concept of saturated vapor is crucial in thermodynamic calculations, especially when dealing with phase changes.

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Most popular questions from this chapter

Three lb of water is contained in a piston-cylinder assembly, initially occupying a volume \(V_{1}=30 \mathrm{ft}^{3}\) at \(T_{1}=\) \(300^{\circ} \mathrm{F}\). The water undergoes two processes in series: Process 1-2: Constant-temperature compression to \(V_{2}=11.19 \mathrm{ft}^{3}\), during which there is an energy transfer by heat from the water of \(1275 \mathrm{Btu}\). Process 2-3: Constant-volume heating to \(p_{3}=120 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}\) Sketch the two processes in series on a \(T-v\) diagram. Neglecting kinetic and potential energy effects, determine the work in Process 1-2 and the heat transfer in Process 2-3, each in Btu.

A rigid, well-insulated container with a volume of \(2 \mathrm{ft}^{3}\) holds \(0.12 \mathrm{lb}\) of ammonia initially at a pressure of \(20 \mathrm{lbf} / \mathrm{in}^{2}\) The ammonia is stirred by a paddle wheel, resulting in an energy transfer to the ammonia with a magnitude of 1 Btu. For the ammonia, determine the initial and final temperatures, each in \({ }^{\circ} \mathrm{R}\), and the final pressure, in lbf/in. \({ }^{2}\) Neglect kinetic and potential energy effects.

Air undergoes a polytropic process in a piston-cylinder assembly from \(p_{1}=1\) bar, \(T_{1}=295 \mathrm{~K}\) to \(p_{2}=7\) bar. The air is modeled as an ideal gas and kinetic and potential energy effects are negligible. For a polytropic exponent of \(1.6\), determine the work and heat transfer, each in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of air, (a) assuming constant \(c_{v}\) evaluated at \(300 \mathrm{~K}\). (b) assuming variable specific heats. Using \(I T\), plot the work and heat transfer per unit mass of air for polytropic exponents ranging from \(1.0\) to \(1.6 .\) Investigate the error in the heat transfer introduced by assuming constant \(c_{v}\).

A tank contains \(2 \mathrm{~m}^{3}\) of air at \(-93^{\circ} \mathrm{C}\) and a gage pressure of \(1.4 \mathrm{MPa}\). Determine the mass of air, in \(\mathrm{kg}\). The local atmospheric pressure is \(1 \mathrm{~atm}\).

Two-tenths kmol of nitrogen \(\left(\mathrm{N}_{2}\right)\) in a piston-cylinder assembly undergoes two processes in series as follows: Process 1-2: Constant pressure at 5 bar from \(V_{1}=1.33 \mathrm{~m}^{3}\) to \(V_{2}=1 \mathrm{~m}^{3}\). Process 2-3: Constant volume to \(p_{3}=4\) bar. Assuming ideal gas behavior and neglecting kinetic and potential energy effects, determine the work and heat transfer for each process, in kJ.
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