Question

Determine the exergy destruction associated with the heat addition process and the expansion process in Prob. \(10-37 .\) Assume a source temperature of \(1600 \mathrm{K}\) and a sink temperature of 285 K. Also, determine the exergy of the steam at the boiler exit. Take \(P_{0}=100 \mathrm{kPa} .\)

Short answer

- The heat transfer during the heat addition process is \(60.98 kJ/kg\). 2) What is the exergy destruction associated with the heat addition process? - The exergy destruction associated with the heat addition process is \(42.75 kJ/kg\). 3) What is the work output during the expansion process? - The work output during the expansion process is \(7.98 kJ/kg\). 4) What is the exergy destruction associated with the expansion process? - The exergy destruction associated with the expansion process is \(-53 kJ/kg\). 5) What is the exergy of the steam at the boiler exit? - The exergy of the steam at the boiler exit is \(957.17 kJ/kg\).

Step-by-step solution

Heat Transfer During Heat Addition Process

First, we have to find the heat transfer during the heat addition process. It is given as (from Problem 10-37): $$Q_{in}=60.98 kJ/kg$$

Exergy Destruction during Heat Addition Process

Now, we can find the exergy destruction during the heat addition process using the following formula: $$Ed_{in} = Q_{in} \left(1-\frac{T_0}{T_{source}}\right)$$ where \(Ed_{in}\) is the exergy destruction during heat addition process, \(T_0\) is the ambient temperature, and \(T_{source}\) is the source temperature. Substitute the given values: $$Ed_{in} = 60.98 \times (1- \frac{285}{1600})$$ $$Ed_{in} \approx 42.75 kJ/kg$$

Work during Expansion Process

From Problem 10-37, it is given that: $$W_{out} = 7.98 kJ/kg$$

Exergy Destruction during Expansion Process

Now, we can find the exergy destruction during the expansion process using the following formula: $$Ed_{out} = W_{out} - W_{ideal} = W_{out} - Q_{in}$$ where \(Ed_{out}\) is the exergy destruction during expansion process and \(W_{ideal}\) is the ideal work output. Substitute the given values: $$Ed_{out} = 7.98 - 60.98$$ $$Ed_{out} = -53 kJ/kg$$

Exergy at the Boiler Exit

Now, we will find the exergy at the boiler exit, which is given by the following formula: $$E_{exit} = u - u_0 - T_0(s - s_0) + P_0(v - v_0)$$ Here, \(E_{exit}\) is the exergy at the boiler exit, \(u, u_0, s, s_0, v, \text{ and } v_0\) are values from table A-2 (superheated steam): \(u = 2776.1 kJ/kg, u_0 = 429.94 kJ/kg, s = 6.5408 kJ/(kg\cdot K), s_0 = 1.6315 kJ/(kg\cdot K), v = 0.5166 m^3/kg, v_0 = 0.001043 m^3/kg\). Now substitute the given values: $$E_{exit} = 2776.1 - 429.94 - (285)(6.5408 - 1.6315) + (100)(0.5166 - 0.001043)$$ $$E_{exit} \approx 957.17 kJ/kg$$ The exergy destruction associated with the heat addition process is \(42.75 kJ/kg\), with the expansion process is \(-53 kJ/kg\), and the exergy of the steam at the boiler exit is \(957.17 kJ/kg\).

Key concepts

Heat Transfer in Thermodynamics

Understanding heat transfer is fundamental to grasping the concepts of energy efficiency and conservation in thermodynamics. Heat transfer is the movement of thermal energy from one object or substance to another as a result of a temperature difference. This transfer occurs in three main ways: conduction, convection, and radiation.

In thermodynamic systems, such as a steam boiler, heat transfer plays a crucial role during the heat addition process. For instance, in the problem given, heat is added to the system at a rate of 60.98 kJ/kg, increasing the energy within the steam. However, not all of this energy can be converted into useful work. The efficiency of this conversion is affected by the temperature differential between the heat source and the environment, which in the example provided, has a dramatic effect on the exergy destruction during the process.

Understanding how heat transfer affects energy and exergy can help students identify opportunities to improve the efficiency of energy conversion systems. This could involve anything from optimizing temperatures within the system to using insulation to prevent heat loss. The key takeaway is that minimizing exergy destruction is essential for maximizing the efficiency of thermodynamic processes.

Expansion Process in Engines and Turbines

The expansion process is a fundamental concept in thermodynamics, commonly observed in engines and turbines where a working fluid, often steam or gas, expands and does work on the surrounding system. Essentially, as the steam expands, it pushes against the turbine blades or the pistons, converting thermal energy from the heat into mechanical work.

In the given problem, the steam undergoes an expansion process where its volume increases, and it does a work output of 7.98 kJ/kg. An ideal expansion process, often depicted by an isentropic process in idealized thermodynamic cycles, would convert the entire energy available from the heat added into work. However, real-world processes are not perfectly efficient; they are subject to irreversibilities that lead to exergy destruction, which is a measure of lost opportunity to do work.

By analyzing the expansion process in detail, students can gain insights into the real-world efficiency of thermodynamic systems and the importance of reducing irreversibilities to get closer to the ideal behavior. This understanding is crucial when it comes to the design and optimization of engines and turbines for maximum performance and energy conservation.

The Fundamentals of Thermodynamics

Thermodynamics is the science of energy, particularly heat and work, and how they transform within a system, governed by four fundamental laws. Energy can neither be created nor destroyed according to the First Law, also known as the law of energy conservation. The Second Law introduces the concept of entropy, a measure of disorder, and explains that natural processes tend to increase the overall entropy of the universe.

In the scope of our problem, these laws help determine the exergy or 'available energy' of the steam at different points in the process. Exergy is the maximum useful work possible during a process that brings a system into equilibrium with a heat reservoir, and the destruction of exergy is a consequence of inefficiencies and irreversibilities within a system.

The calculations provided in the problem, using thermodynamic properties from standard tables, show how engineers use these principles to predict the performance of thermal systems. For students, grasping the concepts of thermodynamics implies understanding how energy is transferred and transformed, and how irreversibilities due to friction, unrestrained expansions, and non-ideal heat transfer lead to the inevitable destruction of exergy.