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Chapter 11: Problem 72

A Brayton cycle produces \(14 \mathrm{MW}\) with an inlet state of \(17^{\circ} \mathrm{C}, 100 \mathrm{kPa}\), and a compression ratio of \(16: 1 .\) The heat added in the combustion is \(960 \mathrm{kJ} / \mathrm{kg} .\) What is the highest temperature and the mass flow rate of air, assuming cold air properties?

Short Answer

Expert verified
The highest temperature in the Brayton cycle can be calculated using the provided compression ratio and heat added during combustion, and the mass flow rate can be calculated using the power output and the heat added.

Step by step solution

01

Calculate the temperature after the isentropic compression

Since we know the compression ratio \(r_c\), we can calculate the temperature \(T_2\) at the end of the isentropic compression process using the relation:\[T_2=T_1 \cdot (r_c)^{(\gamma -1)/\gamma}\]Here, \(T_2\) is the temperature at the end of the compression state in Kelvin, \(T_1\) is the initial temperature (17掳C) which should be converted into Kelvin by adding 273.15, \(r_c\) is the compression ratio which is 16, and \(纬\) (gamma) is the specific heat ratio, which is 1.4 for air.
02

Calculate the highest temperature

The heat input during the isobaric combustion process can be related to the temperature at the end of this process (which is also the maximum temperature in the cycle). The heat added \(q_{in}\) can be expressed as:\[q_{in}=c_p(T_3-T_2)\]where \(c_p\) (1.005 kJ/kgK for air) is the specific heat of the substance at constant pressure, \(T_3\) is the temperature at the end of heat addition or the highest temperature, and \(T_2\) is the temperature at the end of compression. Solving this for \(T_3\),\[T_3=q_{in}/c_p + T_2\]
03

Calculate the mass flow rate

The power produced \(W_{net}\) is related to the mass flow rate \(\.\dot{m}\) and the heat properties of the air as follows:\[W_{net} = \.\dot{m} \cdot (q_{in}-q_{out})\]Here \(q_{out}\) is the heat rejected, but since \(q_{out}\) is not provided we can simplify this to:\[W_{net} = \.\dot{m} \cdot q_{in}\]Solving for mass flow rate,\[\.\dot{m} = W_{net} / q_{in}\]

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Key Concepts

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

Isentropic Compression
Understanding the principle of isentropic compression is essential to grasp the workings of the Brayton cycle. Isentropic compression refers to a process where a gas, such as air, is compressed without any heat transfer occurring to or from the environment; in other words, it's an adiabatic and reversible process. This stage is crucial in the Brayton cycle as it increases the pressure and temperature of the air before it enters the combustion chamber.

In the context of the exercise, the temperature after isentropic compression is calculated using the initial temperature and the compression ratio, alongside the specific heat ratio for air. It's important because it sets the stage for the energy to be added during the combustion process. The formula \(T_2=T_1 \cdot (r_c)^{(\gamma -1)/\gamma}\) is derived from the ideal gas law and principles of thermodynamics, reflecting how the air temperature changes as a result of the compression.
Isobaric Combustion
Isobaric combustion occurs when fuel is burned at a constant pressure to add energy to the system. In this phase of the Brayton cycle, the temperature of the air rises significantly without any change in pressure, hence the term 'isobaric' which means 'constant pressure'.

The formula \(q_{in}=c_p(T_3-T_2)\) represents the relationship between the heat added to the system and the temperature change of the air. The maximum temperature achieved at the end of this process not only determines the efficiency of the cycle but also the thermal stresses that the engine components will endure. For students, decoding this step involves understanding how heat transfer at constant pressure can specifically affect the temperature of a gas and the overall energy flow within the Brayton cycle.
Mass Flow Rate
The mass flow rate is a measure of the amount of mass moving through a particular point in the cycle per unit of time. It is critical for quantifying the overall energy conversion and performance of the Brayton cycle. In terms of the exercise, the mass flow rate connects the power output of the cycle with the energy added during combustion.

Using the formula \(\dot{m} = W_{net} / q_{in}\), with \(W_{net}\) representing the net power output and \(q_{in}\) being the heat added per unit mass, students can calculate the flow rate of air required to produce a specified amount of power. Understanding this concept is important not only for solving mathematical problems but also for practical applications, such as designing engines and predicting their performance under various operating conditions.

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Most popular questions from this chapter

A gas-turbine cycle has two stages of compression, with an intercooler between the stages. Air enters the first stage at \(100 \mathrm{kPa}, 300 \mathrm{K}\). The pressure ratio across each compressor stage is 5 to \(1,\) and each stage has an isentropic efficiency of \(82 \%\). Air exits the intercooler at \(330 \mathrm{K}\). The maximum cycle temperature is \(1500 \mathrm{K},\) and the cycle has a single-turbine stage with an isentropic efficiency of \(86 \%\). The cycle also includes a regenerator with an efficiency of \(80 \%\). Calculate the temperature at the exit of each compressor stage, the second-law efficiency of the turbine, and the cycle thermal efficiency.

A gasoline engine has a volumetric compression ratio of 10 and before compression has air at \(290 \mathrm{K}, 85 \mathrm{kPa},\) in the cylinder. The combustion peak pressure is 6000 kPa. Assume cold air properties. What is the highest temperature in the cycle? Find the temperature at the beginning of the exhaust (heat rejection) and the overall cycle efficiency.

A power plant with one closed feedwater heater has a condenser temperature of \(45^{\circ} \mathrm{C}\), a maximum pressure of \(5 \mathrm{MPa}\), and boiler exit temperature of \(900^{\circ} \mathrm{C} .\) Extraction steam at \(1 \mathrm{MPa}\) to the feedwater heater condenses and is pumped up to the \(5 \mathrm{MPa}\) feedwater line where all the water goes to the boiler at \(200^{\circ} \mathrm{C}\). Find the fraction of extraction steam flow and the two specific pump work inputs.

A Rankine cycle uses ammonia as the working substance and powered by solar energy. It heats the ammonia to \(140^{\circ} \mathrm{C}\) at \(5000 \mathrm{kPa}\) in the boiler/superheater. The condenser is water cooled, and the exit is kept at \(25^{\circ} \mathrm{C}\). Find \((T, P\) and \(x\) if applicable for all four states in the cycle.

A four-stroke gasoline engine has a compression ratio of 10: 1 with 4 cylinders of total displacement 2.3 L. The inlet state is \(280 \mathrm{K}, 70 \mathrm{kPa}\), and the engine is running at 2100 RPM with the fuel adding \(1800 \mathrm{kJ} / \mathrm{kg}\) in the combustion process. What is the net work in the cycle, and how much power is produced?
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