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Chapter 3: Problem 25

In each of the following cases, ammonia contained in a closed, rigid tank is heated from an initial saturated vapor state at temperature \(T_{1}\) to the final temperature, \(T_{2}\) : (a) \(T_{1}=20^{\circ} \mathrm{C}, T_{2}=40^{\circ} \mathrm{C}\). Using \(I T\), determine the final pressure, in bar. (b) \(T_{1}=70^{\circ} \mathrm{F}, T_{2}=120^{\circ} \mathrm{F}\). Using \(I T\), determine the final pressure, in lbf/in. \({ }^{2}\) Compare the pressure values determined using \(I T\) with those obtained using the appropriate Appendix tables for ammonia.

Short Answer

Expert verified
Calculate initial saturation pressures. Apply the ideal gas law to find \( P_2 \). Compare with table values.

Step by step solution

01

Title - Identify initial conditions

For both parts (a) and (b), start by identifying the initial conditions of the ammonia. In part (a), the initial temperature is given as \( T_1 = 20^\text{掳C} \). In part (b), the initial temperature is given as \( T_1 = 70^\text{掳F} \). Convert to the same unit system if necessary.
02

Title - Determine initial pressure

Look up the saturation pressure corresponding to the initial temperature \( T_1 \). For \( T_1 = 20^\text{掳C} \), find the saturation pressure \( P_1 \) from the ammonia tables (or use a steam table if appropriate). Similarly, find the saturation pressure for \( T_1 = 70^\text{掳F} \).
03

Title - Apply the Ideal Gas Law (if applicable)

For a closed, rigid tank, volume remains constant. Assuming the gas behaves ideally at higher temperatures, use the relation: \[ \frac{P_2}{P_1} = \frac{T_2}{T_1} \] to find the new pressure at the final temperature.
04

Title - Determine the final temperatures

Convert the final temperatures into the same units as the initial temperatures. For part (a), the final temperature is already in Celsius, \( T_2 = 40^\text{掳C} \). For part (b), convert the final temperature from Fahrenheit to Rankine: \[ T_2 = 120^\text{掳F} + 459.67 \] using \( T_\text{R} = T_\text{F} + 459.67 \).
05

Title - Calculate the final pressures

Using the ideal gas law relation from Step 3, calculate the final pressures: \[ P_2 = P_1 \times \frac{T_2}{T_1} \] for both parts (a) and (b) using respective units.
06

Title - Compare with Appendix tables

To ensure accuracy, compare the calculated final pressures with values from appropriate ammonia tables. For the given final temperatures, look up the saturation pressures and compare them to the calculated values to check for consistency.

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Key Concepts

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

saturated vapor state
A saturated vapor state occurs when a substance exists entirely in its vapor form at a temperature and pressure that corresponds to its boiling point. In this state, the vapor is at equilibrium with its liquid phase. This means any addition of heat or reduction in pressure will cause the vapor to condense into a liquid. For ammonia, at the given initial conditions in the exercise, the substance is in a saturated vapor state. This is important because it determines the initial pressure and temperature conditions that we need to solve the exercise.
saturation pressure
The saturation pressure is the pressure at which a substance will transition between its liquid and vapor states at a given temperature. For any substance, this is a unique value and can be found on saturation tables specific to the substance, such as ammonia in this case. When ammonia is in a saturated vapor state at 20掳C, it has a specific saturation pressure that we must use to find the final pressure when the temperature changes. Saturation pressure is critical in thermodynamics because it helps us understand phase transitions and the conditions required for these changes.
ideal gas law
The ideal gas law is a fundamental equation in thermodynamics, given by

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Most popular questions from this chapter

Ten kg of hydrogen \(\left(\mathrm{H}_{2}\right)\), initially at \(20^{\circ} \mathrm{C}\), fills a closed, rigid tank. Heat transfer to the hydrogen occurs at the rate \(400 \mathrm{~W}\) for one hour. Assuming the ideal gas model with \(k=1.405\) for the hydrogen, determine its final temperature, in \({ }^{\circ} \mathrm{C}\).

A piston-cylinder assembly fitted with a slowly rotating paddle wheel contains \(0.13 \mathrm{~kg}\) of air, initially at \(300 \mathrm{~K}\). The air undergoes a constant-pressure process to a final temperature of \(400 \mathrm{~K}\). During the process, energy is gradually transferred to the air by heat transfer in the amount \(12 \mathrm{~kJ}\). Assuming the ideal gas model with \(k=1.4\) and negligible changes in kinetic and potential energy for the air, determine the work done (a) by the paddle wheel on the air and (b) by the air to displace the piston, each in kJ.

The following table lists temperatures and specific volumes of ammonia vapor at two pressures: $$ \begin{array}{lccc} {}{}{p=50 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}} & {}{c}{p=60 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}} \\ { 4 } T\left({ }^{\circ} \mathrm{F}\right) & v\left(\mathrm{ft}{ }^{3} / \mathrm{lb}\right) & T\left({ }^{\circ} \mathrm{F}\right) & v\left(\mathrm{ft}^{3} / \mathrm{lb}\right) \\ \hline 100 & 6.836 & 100 & 5.659 \\ 120 & 7.110 & 120 & 5.891 \\ 140 & 7.380 & 140 & 6.120 \end{array} $$ Data encountered in solving problems often do not fall exactly on the grid of values provided by property tables, and linear interpolation between adjacent table entries becomes necessary. Using the data provided here, estimate (a) the specific volume at \(T=120^{\circ} \mathrm{F}, p=54 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}\), in \(\mathrm{ft}^{3} / \mathrm{lb}\). (b) the temperature at \(p=60 \mathrm{lbf} / \mathrm{in}^{2}, v=5.982 \mathrm{ft}^{3} / \mathrm{lb}\), in \({ }^{\circ} \mathrm{F}\). (c) the specific volume at \(T=110^{\circ} \mathrm{F}, p=58 \mathrm{lbf} / \mathrm{in}^{2}\), in \(\mathrm{ft}^{3} / \mathrm{lb}\).

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.

A gallon of milk at \(68^{\circ} \mathrm{F}\) is placed in a refrigerator. If energy is removed from the milk by heat transfer at a constant rate of \(0.08 \mathrm{Btu} / \mathrm{s}\), how long would it take, in minutes, for the milk to cool to \(40^{\circ} \mathrm{F}\) ? The specific heat and density of the milk are \(0.94 \mathrm{Btu} / \mathrm{lb} \cdot{ }^{\circ} \mathrm{R}\) and \(64 \mathrm{lb} / \mathrm{ft}^{3}\), respectively.
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