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Superheated steam at 40 bar absolute and \(500^{\circ} \mathrm{C}\) flows at a rate of \(250 \mathrm{kg} / \mathrm{min}\) to an adiabatic turbine, where it expands to 5 bar. The turbine develops \(1500 \mathrm{kW}\). From the turbine the steam flows to a heater, where it is reheated isobarically to its initial temperature. Neglect kinetic energy changes. (a) Write an energy balance on the turbine and use it to determine the outlet stream temperature. (b) Write an energy balance on the heater and use it to determine the required input (kW) to the steam. (c) Verify that an overall energy balance on the two-unit process is satisfied. (d) Suppose the turbine inlet and outlet pipes both have diameters of 0.5 meter. Show that it is reasonable to neglect the change in kinetic energy for this unit.

Short Answer

Expert verified
From these steps, we can find the outlet steam temperature of the turbine, the required input to the steam on the heater, and also verify the overall energy balance on the system. Besides, it is justifiable to neglect the kinetic energy changes based on relative quantities.

Step by step solution

01

Energy Balance on the Turbine

The adiabatic turbine can be described with the steady state energy balance, as such: \(h_{in}-h_{out}=W\). First, look up the steam enthalpy \(h_{in}\) at \(40 \,bar, 500^{\circ}C\) in steam tables. Now, we have to determine \(h_{out}\). Use the fact that the work done by the turbine \(W\) is given as \(1500\, kW\) and the flow rate is \(250 \, kg/min = 4.167 \, kg/s\), we then solve the energy balance for \(h_{out}\), resulting lower enthalpy. Look up this value in the steam tables to find the corresponding temperature.
02

Energy Balance on the Heater

Once the outlet stream temperature is known from the turbine, an energy balance can be performed on the heater. We have \(h_{in}-h_{out}= -Q_{in}\), where \(Q_{in}\) is the energy input we need to find. From previous step, we find \(h_{in}\), use the initial enthalpy as (\(h_{out})\ will be the same. By rearranging and putting the identified values, get \(Q_{in}\).
03

Overall Energy Balance of the System

Verify the overall energy balance on this two-unit system by checking work done by the turbine equals heat added in the heater.
04

Kinetic Energy Change

To verify whether it is reasonable to neglect kinetic energy changes, calculate the velocity of the stream entering and leaving the turbine using the volumetric flow rate equation \(Q=vA\). Here \(A\) stands for the cross-sectional area of the pipe \(A= \pi (d/2)^2\), and substituted with given diameter and mass flow rate. Then calculate the kinetic energy \(KE=1/2 mv^2\). If the kinetic energy is small as compared to the overall energy changes in the system, it can be neglected.

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Key Concepts

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

Superheated Steam Thermodynamics
Superheated steam plays a crucial role in numerous industrial processes due to its high thermal energy content. It's characterized by having a temperature higher than its boiling point at a given pressure, which means it doesn't contain moisture. Understanding its properties requires delving into steam thermodynamics. It's worth noting that this form of steam is essential for minimizing energy loss and improving efficiency.
To analyze such systems, we often refer to steam tables, which provide property data of water and steam, including enthalpy, temperature, and pressure. These tables are invaluable in determining steam conditions and guiding calculations in thermodynamic cycles, such as those involving turbines or heaters. The exercise problem unfolds within these parameters, utilizing steam tables to find the initial enthalpy of the steam entering a turbine and subsequently the exit conditions after expansion.
Recognizing temperature and pressure conditions in superheated steam allows for precise control in engineering applications. This leads to efficient energy use, reducing waste and improving the overall efficiency of thermal processes.
Adiabatic Turbine Calculations
An adiabatic turbine functions by expanding steam without any heat exchange with its surroundings, thereby maximizing the conversion of steam's internal energy to work. In the context of the exercise, we focus on applying an energy balance around the turbine. The key idea here is that any decrease in steam's enthalpy directly corresponds to the mechanical work output. The power output of the turbine is given as 1500 kW, and combined with the mass flow of steam ( 4.167 kg/s), it allows us to compute changes in enthalpy across the turbine. The enthalpy at the start and end helps pinpoint the steam's exit conditions via steam tables. This method, while relatively straightforward, highlights the importance of understanding energy transformations in thermodynamic systems. It illustrates how efficiently energy initially stored in high-pressure, high-temperature steam can perform mechanical work, a fundamental aspect in power generation.
Enthalpy and Steam Tables
To solve energy balance problems like the one in the given exercise, enthalpy calculations are paramount. Enthalpy, a measure of energy in a thermodynamic system, combines internal energy with pressure and volume impacts. In steam processes, it helps quantify energy changes as steam undergoes various processes.
Steam tables act as the go-to resource for finding the enthalpy of steam at specific pressure and temperature points. When steam enters the turbine at a specified pressure and temperature, these tables provide the enthalpy value needed for further calculations.
In this exercise, steam tables are used twice: first to find the enthalpy of superheated steam entering the turbine, and second to determine downstream conditions after expansion. By checking values in steam tables at each stage, we ensure accurate results, establishing a fundamental understanding of thermodynamics essential for designing thermal systems.
Isobaric Heating Processes
Isobaric processes, where pressure stays constant while other conditions change, are an important concept in thermodynamics. In the exercise, steam after expansion is reheated back to its original temperature under a constant pressure of 5 bar. This reheating requires input energy, illustrated through an energy balance calculation
Reheating under constant pressure enables precise reconditioning of steam properties, ensuring readiness for use in subsequent processes. The amount of heat added is determined by calculating the change in enthalpy during heating.
This part of the exercise underscores the significance of controlled heating processes in maintaining system efficiency and stability. By understanding isobaric heating, we can optimize energy use, enhance process sustainability, and improve the performance of systems utilizing steam in various applications.

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Most popular questions from this chapter

Steam produced in a boiler is frequently "wet"-that is, it is a mist composed of saturated water vapor and entrained liquid droplets. The quality of a wet steam is defined as the fraction of the mixture by mass that is vapor. A wet steam at a pressure of 5.0 bar with a quality of 0.85 is isothermally "dried" by evaporating the entrained liquid. The flow rate of the dried steam is \(52.5 \mathrm{m}^{3} / \mathrm{h}\). (a) Use the steam tables to determine the temperature at which this operation occurs, the specific enthalpies of the wet and dry steams, and the total mass flow rate of the process stream. (b) Calculate the heat input (kW) required for the evaporation process. (c) Suppose leaks developed in the feed pipe to the dryer and in the dryer exit pipe. Speculate on what you would see at each location.

The specific enthalpy of liquid \(n\) -hexane at 1 atm varies linearly with temperature and equals \(25.8 \mathrm{kJ} / \mathrm{kg}\) at \(30^{\circ} \mathrm{C}\) and \(129.8 \mathrm{kJ} / \mathrm{kg}\) at \(50^{\circ} \mathrm{C}\) (a) Determine the equation that relates \(\hat{H}(\mathrm{kJ} / \mathrm{kg})\) to \(T\left(^{\circ} \mathrm{C}\right)\) and calculate the reference temperature on which the given enthalpies are based. Then derive an equation for \(\hat{U}(T)(\mathrm{kJ} / \mathrm{kg})\) at 1 atm. (b) Calculate the heat transfer rate required to cool liquid \(n\) -hexane flowing at a rate of \(20 \mathrm{kg} / \mathrm{min}\) from \(60^{\circ} \mathrm{C}\) to \(25^{\circ} \mathrm{C}\) at a constant pressure of 1 atm. Estimate the change in specific internal energy \((\mathrm{kJ} / \mathrm{kg})\) as the n-hexane is cooled at the given conditions.

Arsenic contamination of aquifers is a major health problem in much of the world and is particularly severe in Bangladesh. One method of removing the arsenic is to pump water from an aquifer to the surface and through a bed packed with granular material containing iron oxide, which binds the arsenic. The purified water is then either used or allowed to seep back through the ground into the aquifer. In an installation of the type just described, a pump draws 69.1 gallons per minute of contaminated water from an aquifer through a 3 -inch ID pipe and then discharges the water through a 2-inch ID pipe to an open overhead tank filled with granular material. The water leaves the end of the discharge line 80 feet above the water in the aquifer. The friction losses in the piping system are \(10 \mathrm{ft} \cdot \mathrm{lb}_{\mathrm{f}} / \mathrm{lb}_{\mathrm{m}}\) (a) If the pump is \(70 \%\) efficient (i.e., \(30 \%\) of the electrical energy delivered to the pump is not used in pumping the water), what is the required pump horsepower? (b) Even if we assume that the iron oxide binds \(100 \%\) of the arsenic, what other factors limit the effectiveness of this operation?

Liquid water at 60 bar and \(250^{\circ} \mathrm{C}\) passes through an adiabatic expansion valve, emerging at a pressure \(P_{\mathrm{f}}\) and temperature \(T_{\mathrm{f}} .\) If \(P_{\mathrm{f}}\) is low enough, some of the liquid evaporates. (a) If \(P_{\mathrm{f}}=1.0\) bar, determine the temperature of the final mixture \(\left(T_{\mathrm{f}}\right)\) and the fraction of the liquid feed that evaporates \(\left(y_{\mathrm{v}}\right)\) by writing an energy balance about the valve and neglecting \(\Delta \dot{E}_{\mathrm{k}}\) (b) If you took \(\Delta \dot{E}_{\mathrm{k}}\) into account in Part (a), how would the calculated outlet temperature compare with the value you determined? What about the calculated value of \(y_{\mathrm{v}} ?\) Explain. (c) What is the value of \(P_{\mathrm{f}}\) above which no evaporation would occur? (d) Sketch the shapes of plots of \(T_{\mathrm{f}}\) versus \(P_{\mathrm{f}}\) and \(y_{\mathrm{v}}\) versus \(P_{\mathrm{f}}\) for 1 bar \(\leq P_{\mathrm{f}} \leq 60\) bar. Briefly explain your reasoning.

Energy may be produced from solid waste in two ways: (1) generate methane from anaerobic decomposition of the waste and burn it (landfill-gas-to-energy, or LFGTE) or(2) burn the waste directly (waste-to-energy, or WTE). The heat generated by either method can be used to produce steam, which impinges on a turbine rotor connected to a generator to produce electricity. LFGTE produces about 215 k Wh electricity/ton of waste, and WTE produces roughly 600 kWh/ton of waste. The average output of a large power plant is 1 GW, which is enough to supply the annual residential energy consumption of a city of roughly 800,000 people. (a) The current rate of municipal solid-waste generation in the United States is approximately 413 million tons per year. If all of it were used for energy recovery, how many \(1 \mathrm{GW}\) power plants could LFGTE supply? How many if WTE is used? A useful source of information regarding LFGTE is the U.S. EPA Landfill Methane Outreach Program, http://www.epa.gov//mop/; the Waste-to-Energy Research and Technology Council at Columbia University provides useful information on WTE, http://www.seas.columbia.edu/earth/wtert/; and information on natural gas can be obtained from the U.S. Energy Information Administration, http:// www.eia.doe.gov/oil_gas/natural_gas/info_glance/natural_gas.html.

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