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Chapter 3: Problem 46

Water, initially saturated vapor at 4 bar, fills a closed, rigid container. The water is heated until its temperature is \(400^{\circ} \mathrm{C}\). For the water, determine the heat transfer, in kJ per kg of water. Kinetic and potential energy effects can be ignored.

Short Answer

Expert verified
The heat transfer required is 466.3 kJ/kg.

Step by step solution

01

Identify initial state properties

The water is initially a saturated vapor at 4 bar. Using steam tables, determine the specific volume ( nu _vapor) at this state. For water at 4 bar, the saturation temperature is approximately 143.6掳C, and the specific volume of saturated vapor, nu _vapor, is 0.4624 m^3/kg.
02

Determine the final state properties

The water is heated until its temperature reaches 400掳C. Since the volume is rigid, the specific volume remains constant. Using steam tables for 400掳C, find the specific enthalpy ( h ) for the state matching the specific volume 0.4624 m^3/kg. We use superheated steam tables for 400掳C and a specific volume of 0.4624 m^3/kg to find h_f = 3214.3 kJ/kg.
03

Calculate the initial specific enthalpy

For the initial state, the specific enthalpy of saturated vapor at 4 bar (denoted as h _g) can be found in the steam tables. At 4 bar, h _g is approximately 2748.0 kJ/kg.
04

Determine the heat transfer

The heat transfer (q) required to change the state from the initial to the final can be determined by the change in specific enthalpy. Thus, q = h _final - h _initial. Substituting the values: q = 3214.3 - 2748.0 = 466.3 kJ/kg.

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Key Concepts

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

specific volume
Specific volume is one of the fundamental properties in thermodynamics. It is defined as the volume occupied by a unit mass of a substance. For example, for 1 kg of water vapor at a certain state, the specific volume tells us how much space that vapor takes up.

In our problem, the specific volume (u) is crucial because the container is rigid, meaning its volume cannot change. Given that the water initially starts as a saturated vapor at 4 bar, we use the steam tables to find the initial specific volume. In this case, u is 0.4624 m鲁/kg. This value remains consistent through the heating process, as the specific volume is an intrinsic property when the volume of the container does not change.
saturated vapor
A saturated vapor is a state where the steam exists at the temperature and pressure right at the boiling point. For instance, it is on the edge of condensing without actually turning into liquid.

In this exercise, the water starts as saturated vapor at 4 bar of pressure. At this pressure, the corresponding saturation temperature is approximately 143.6掳C. This means that before heating, the water is about to condense but still entirely in vapor form. Identifying this state enables us to use the steam tables to look up other thermodynamic properties like specific volume and specific enthalpy.
specific enthalpy
Specific enthalpy (h) represents the total energy content per unit mass. It includes internal energy and the product of pressure and volume.

To find the heat transfer required to heat the water, we need to determine the specific enthalpy at both the initial and final states. Initially, the specific enthalpy of the saturated vapor at 4 bar (h_g) is 2748.0 kJ/kg, obtained from the steam tables. After heating to 400掳C, the specific enthalpy (using superheated steam tables at the same specific volume of 0.4624 m鲁/kg) is found to be 3214.3 kJ/kg.

The change in specific enthalpy (initial to final state) helps calculate the heat transfer, with the formula:
q = h_final - h_initial.
heat transfer
Heat transfer is the process of adding or removing thermal energy from a system. In this context, it's the energy required to heat the water from its initial state to the final state.

To find the heat transfer (q) in this problem, we make use of the change in specific enthalpy. Initial specific enthalpy is 2748.0 kJ/kg, and the final specific enthalpy is 3214.3 kJ/kg. The difference gives the heat transfer per unit mass of water:
\( q = 3214.3 - 2748.0 = 466.3 \text{kJ/kg} \)

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Most popular questions from this chapter

A piston-cylinder assembly contains 2 lb of water, initially at \(100 \mathrm{lbf} / \mathrm{in} .^{2}\) and \(400^{\circ} \mathrm{F}\). The water undergoes two processes in series: a constant-pressure process followed by a constantvolume process. At the end of the constant-volume process, the temperature is \(300^{\circ} \mathrm{F}\) and the water is a two-phase liquid- vapor mixture with a quality of \(60 \%\). Neglect kinetic and potential energy effects. (a) Sketch \(T-v\) and \(p-v\) diagrams showing the key states and the processes. (b) Determine the work and heat transfer for each process, all in Btu.

A closed, rigid tank is filled with water. Initially, the tank holds \(9.9 \mathrm{ft}^{3}\) saturated vapor and \(0.1 \mathrm{ft}^{3}\) saturated liquid, each at \(212^{\circ} \mathrm{F}\). The water is heated until the tank contains only saturated vapor. For the water, determine (a) the quality at the initial state, (b) the temperature at the final state, in \({ }^{\circ} \mathrm{F}\), and (c) the heat transfer, in Btu. Kinetic and potential energy effects can be ignored.

Five \(\mathrm{kg}\) of butane \(\left(\mathrm{C}_{4} \mathrm{H}_{10}\right)\) in a piston-cylinder assembly undergoes a process from \(p_{1}=5 \mathrm{MPa}, T_{1}=500 \mathrm{~K}\) to \(p_{2}=3 \mathrm{MPa}\), during which the relationship between pressure and specific volume is \(p v=\) constant. Determine the work, in kJ.

From an initial state where the pressure is \(p_{1}\), the temperature is \(T_{1}\), and the volume is \(V_{1}\), water vapor contained in a piston-cylinder assembly undergoes each of the following processes: Process 1-2: Constant-temperature to \(p_{2}=2 p_{1}\). Process 1-3: Constant volume to \(p_{3}=2 p_{1}\). Process 1-4: Constant pressure to \(V_{4}=2 V_{1}\). Process 1-5: Constant temperature to \(V_{5}=2 V_{1}\). On a \(p-V\) diagram, sketch each process, identify the work by an area on the diagram, and indicate whether the work is done by, or on, the water vapor.

Two kg of oxygen fills the cylinder of a piston-cylinder assembly. The initial volume and pressure are \(2 \mathrm{~m}^{3}\) and 1 bar, respectively. Heat transfer to the oxygen occurs at constant pressure until the volume is doubled. Determine the heat transfer for the process, in kJ, assuming the specific heat ratio is constant, \(k=1.35\). Kinetic and potential energy effects can be ignored.
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