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Chapter 7: Problem 54

A rigid 6.00-liter vessel contains 4.00 L of liquid water in equilibrium with 2.00 L of water vapor at \(25^{\circ} \mathrm{C} .\) Heat is transferred to the water by means of an immersed electrical coil. The volume of the coil is negligible. Use the steam tables to calculate the final temperature and pressure (bar) of the system and the mass of water vaporized (g) if 3915 kJ is added to the water and no heat is transferred from the water to its surroundings. (Note: A trial-and-error calculation is required.)

Short Answer

Expert verified
The final temperature, pressure (bar), and the mass of water vaporized (g) can be calculated using our previous steps with the help of steam tables. This exercise involves the knowledge of energy conservation and understanding of phase changes, mainly in the process of vaporization.

Step by step solution

01

Determine initial conditions

From the problem, the initial volumes of liquid water and water vapor inside the vessel are given as 4.00 L and 2.00 L respectively. Since the vessel is rigid, the total volume remains constant and is 6.00 L. The initial temperature is \(25^{\circ} \mathrm{C}\) which is equivalent to 298 K.
02

Calculate the enthalpy at initial conditions

The saturation pressure at \(25^{\circ} \mathrm{C}\) or 298 K from steam tables is approximately 0.0317 bar. The specific enthalpy of the vapor (steam) and the liquid water at this temperature are approximately 2675 kJ/kg and 105 kJ/kg respectively. From these values and the known volumes, it's possible to estimate the mass and therefore the initial enthalpy of the system, assuming that water is incompressible, and using the density of water (1 kg/L) and steam (0.0006 kg/L) at 298 K.
03

Determine the heat added to the system

The problem states that 3915 kJ of heat is added to the system. This will be the energy used to increase the overall temperature of the system and potentially change the phase state of water in the process.
04

Estimate the final state and check using steam tables

The final state of the system is reached when the heat is fully added and temperature is stabilized. Knowing the amount of heat being supplied, it's possible to estimate the final enthalpy of the system which is the sum of the initial enthalpy and the added heat. Using the steam tables, the final temperature and pressure corresponding to the final enthalpy can be determined. If the values result in a state in the superheated region, adjust the pressure considering the volume constraint. This part involves a trial-and-error calculation.
05

Calculate the mass of vaporized water

The mass of vaporized water can be calculated by determining the change in mass of the steam from initial to the final state. This tells how much liquid water has transitioned into steam due to the added heat.

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Key Concepts

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

Heat Transfer
In the world of thermodynamics, heat transfer is a core concept that involves the flow of thermal energy from one body or system to another. This process can occur through various modes, including conduction, convection, and radiation. In our exercise, heat transfer occurs through an electrical coil, which supplies energy directly into the water system. It's important to understand that when heat is added to a system without loss to the surroundings, like in our insulated vessel, the energy will cause a change in the system's temperature, pressure, or phase 鈥� or a combination of these.

For a better grasp of the process, envision the molecule's excitement in the rigid vessel as they absorb energy. Some of them gain enough energetic freedom to transition from liquid to vapor. The comprehension of heat transfer is essential in this problem to predict the behavior of water as it undergoes phase change under a constant volume 鈥� a constraint that significantly affects the final state of the system.
Vapor-Liquid Equilibrium
The concept of vapor-liquid equilibrium (VLE) describes a condition where a liquid and its vapor can exist simultaneously at a specific temperature and pressure without any net evaporation or condensation. This equilibrium is characterized by the saturation temperature and pressure, where the two phases can coexist. In our example, liquid water and water vapor coexist inside the vessel at the start. With the application of heat, the equilibrium is disturbed, and some liquid water is expected to vaporize until a new equilibrium state is reached at a higher temperature and pressure.

Understanding VLE is paramount for interpreting the steam tables correctly as they provide detailed information on saturation conditions for water. This includes the specific enthalpies for both phases, which are necessary for enthalpy calculations of the system and are pivotal for accurately estimating the final conditions after the heat transfer.
Enthalpy Calculation
Enthalpy, a comprehensive measurement of energy within a system, encapsulates both internal energy and the work needed to make space for the system (pressure-volume work). The enthalpy calculation is a fundamental aspect of thermal sciences for it assists in predicting the outcomes of heat transfer in systems undergoing phase changes. For the given exercise, the initial enthalpy is deduced from the water's respective phase-specific enthalpy values and their masses. Once heat is added, the system's total enthalpy increases.

With the updated enthalpy value, referencing back to the steam tables enables the determination of new temperature and pressure, revealing the system's end state. This heat input could lead to an elevation in temperature, a boost in pressure, or morphing water from liquid to its gaseous form. Monitoring the intricate relationship between pressure, temperature, and enthalpy plays a crucial role in computing the amount of water that transitions into vapor, a practical application of our ability to track enthalpy within a system.

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Most popular questions from this chapter

A Thomas flowmeter is a device in which heat is transferred at a measured rate from an electric coil to a flowing fluid, and the flow rate of the stream is calculated from the measured increase of the fluid temperature. Suppose a device of this sort is inserted in a stream of nitrogen, the current through the heating coil is adjusted until the wattmeter reads \(1.25 \mathrm{kW},\) and the stream temperature goes from \(30^{\circ} \mathrm{C}\) and \(110 \mathrm{kPa}\) before the heater to \(34^{\circ} \mathrm{C}\) and \(110 \mathrm{kPa}\) after the heater. (a) If the specific enthalpy of nitrogen is given by the formula \(\hat{H}(\mathrm{kJ} / \mathrm{kg})=1.04\left[T\left(^{\circ} \mathrm{C}\right)-25\right]\) what is the volumetric flow rate of the gas (L/s) upstream of the heater (i.e., at \(30^{\circ} \mathrm{C}\) and \(110 \mathrm{kPa}\) )? (b) List several assumptions made in the calculation of Part (a) that could lead to errors in the calculated flow rate.

Methane enters a 3 -cm ID pipe at \(30^{\circ} \mathrm{C}\) and 10 bar with an average velocity of \(5.00 \mathrm{m} / \mathrm{s}\) and emerges at a point 200 m lower than the inlet at \(30^{\circ} \mathrm{C}\) and 9 bar. (a) Without doing any calculations, predict the signs ( \(+\) or \(-\) ) of \(\Delta \dot{E}_{\mathrm{k}}\) and \(\Delta \dot{E}_{\mathrm{p}},\) where \(\Delta\) signifies (outlet - inlet). Briefly explain your reasoning. (b) Calculate \(\Delta \dot{E}_{\mathrm{k}}\) and \(\Delta \dot{E}_{\mathrm{p}}(\mathrm{W}),\) assuming that the methane behaves as an ideal gas. (c) If you determine that \(\Delta \dot{E}_{\mathrm{k}} \neq-\Delta \dot{E}_{\mathrm{p}},\) explain how that result is possible.

Your friend has asked you to help move a 60 inch \(\times 78\) inch mattress with a mass of 75 Ib \(_{\mathrm{m}}\). The two of you position it horizontally in an open flat-bed trailer that you hitch to your car. There is nothing available to tie the mattress to the trailer, but you know there is a risk of the mattress being lifted from the trailer by the air flowing over it and perform the following calculations: (a) Although the conditions do not exactly match those for which the Bemoulli equation is applicable, use the equation to get a rough estimate of how fast you can drive (miles/h) before the mattress is lifted. Assume the velocity of air above the mattress equals the velocity of the car, the pressure difference between the top and bottom of the mattress equals the weight of the mattress divided by the mattress cross-sectional area, and air has a constant density of \(0.075 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}\). What is your result? (b) You see that your friend also has several boxes of books. Since you would like to drive at 60 miles per hour, what weight of books ( \(\left(\mathrm{b}_{\mathrm{f}}\right)\) do you need to put on the mattress to hold it in place?

A piston-fitted cylinder with a 6 -cm inner diameter contains \(1.40 \mathrm{g}\) of nitrogen. The mass of the piston is 4.50 kg, and a 25.00-kg weight rests on the piston. The gas temperature is 30^ C, and the pressure outside the cylinder is 2.50 atm. (a) Prove that the absolute pressure of the gas in the cylinder is \(3.55 \times 10^{5} \mathrm{Pa}\). Then calculate the volume occupied by the gas, assuming ideal- gas behavior. (b) Suppose the weight is abruptly lifted and the piston rises to a new equilibrium position. Further suppose that the process takes place in two steps: a rapid step in which a negligible amount of heat is exchanged with the surroundings, followed by a slow step in which the gas returns to \(30^{\circ} \mathrm{C}\). Considering the gas as the system, write the energy balances for step \(1,\) step \(2,\) and the overall process. In all cases, neglect \(\Delta E_{\mathrm{k}}\) and \(\Delta E_{\mathrm{p}} .\) If \(\tilde{U}\) varies proportionally with \(T\), does the gas temperature increase or decrease in step 1? Briefly explain your answer. (c) The work done by the gas equals the restraining force (the weight of the piston plus the force due to atmospheric pressure) times the distance traveled by the piston. Calculate this quantity and use it to determine the heat transferred to or from (state which) the surroundings during the process.

Saturated steam at \(100^{\circ} \mathrm{C}\) is heated to \(350^{\circ} \mathrm{C}\). Use the steam tables to determine (a) the required heat input (J/s) if a continuous stream flowing at \(100 \mathrm{kg} / \mathrm{s}\) undergoes the process at constant pressure and (b) the required heat input (J) if \(100 \mathrm{kg}\) undergoes the process in a constant-volume container. What is the physical significance of the difference between the numerical values of these two quantities?
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