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Chapter 4: Problem 64

A well-insulated rigid tank of volume \(10 \mathrm{~m}^{3}\) is connected to a large steam line through which steam flows at 15 bar and \(280^{\circ} \mathrm{C}\). The tank is initially evacuated. Steam is allowed to flow into the tank until the pressure inside is \(p\). (a) Determine the amount of mass in the tank, in \(\mathrm{kg}\), and the temperature in the tank, in \({ }^{\circ} \mathrm{C}\), when \(p=15\) bar. (b) Plot the quantities of part (a) versus \(p\) ranging from \(0.1\) to 15 bar.

Short Answer

Expert verified
The mass in the tank is roughly 48.54 kg and the temperature is 280掳C when the pressure is 15 bar.

Step by step solution

01

Understand the Problem

Given a well-insulated rigid tank of volume 10 m^3 initially evacuated, and connected to a large steam line where steam flows at 15 bar and 280掳C, we need to determine the mass and temperature inside the tank when the pressure reaches 15 bar.
02

Assume Final Conditions

When the pressure inside the tank reaches 15 bar, thermodynamic equilibrium is reached. Since the tank is well-insulated, there's no heat transfer, and since it's rigid, volume does not change.
03

Determine Steam Properties at 15 bar and 280掳C

Using steam tables, find the specific volume ( v) and other properties of steam at 15 bar and 280掳C. From the steam tables, we see that at 15 bar and 280掳C, the specific volume of steam is approximately 0.206 m^3/kg.
04

Calculate the Mass

Using the formula: m = V / specific volume, where V = 10 m^3 and specific volume is 0.206 m^3/kg. The mass m inside the tank can be calculated as m = 10 m^3 / 0.206 m^3/kg 鈮� 48.54 kg.
05

Temperature in the Tank

For the given conditions (15 bar and 280掳C), the temperature inside the tank remains at 280掳C because it matches the steam properties from the steam line.
06

Plot the Quantities Versus Pressure

Vary the pressure from 0.1 to 15 bar, and for each pressure, use the steam tables to determine the corresponding specific volume and temperature. Calculate the mass for each pressure and plot the mass and temperature versus pressure. For the sake of this exercise, detailed steps for each value are omitted, only the workflow is explained.

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Key Concepts

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

Thermodynamic Equilibrium
In thermodynamics, equilibrium refers to a state where there are no net changes in the macroscopic properties of a system. For the given problem, when the steam flows into the tank and the pressure inside reaches 15 bar, the system reaches thermodynamic equilibrium. This means that both pressure and temperature inside the tank stabilize as the system achieves a balance between the incoming steam and the existing conditions inside the tank. Since the tank is well-insulated and rigid, other forms of energy transfer such as heat cannot influence the system, and the volume remains constant. Thus, the steam entering the tank eventually balances with the steam already in the system, reaching an equilibrium state.
Specific Volume
Specific volume is defined as the volume occupied by a unit mass of a substance. Mathematically, it is represented as the reciprocal of density, and it helps in understanding how much space a particular mass of the substance will occupy. In the problem, when we find the specific volume of steam at 15 bar and 280掳C, we use the steam tables to determine it. For the given conditions, the specific volume of steam is approximately 0.206 m鲁/kg. This specific volume is crucial in calculating the mass of the steam inside the tank. Since we know the volume of the tank and the specific volume, we can easily find the mass using the formula: \( m = \frac{V}{\text{specific volume}} \).
Steam Tables
Steam tables are essential tools in thermodynamics, used to determine the properties of water and steam at various temperatures and pressures. They list properties like specific volume, enthalpy, and entropy for different states of water and steam. In solving our tank problem, we rely on steam tables to find the specific volume of steam at specific conditions (15 bar and 280掳C). By referring to the steam tables, we get precise values for these properties, which we then use to perform calculations. For instance, knowing the specific volume helps us determine the amount of mass inside the tank.
Mass Calculation
To find the mass of steam inside the tank, which is initially empty, we use the relationship between volume, specific volume, and mass. Given that the tank's volume is 10 m鲁 and the specific volume of steam at 15 bar and 280掳C is 0.206 m鲁/kg, we apply the formula: \( m = \frac{V}{\text{specific volume}} \). Inserting the known values, we get: \( m = \frac{10\, \text{m}鲁}{0.206\, \text{m}鲁/\text{kg}} \approx 48.54\, \text{kg} \). This formula simplifies the process, allowing us to determine that approximately 48.54 kg of steam is present in the tank at the specified conditions.

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