Chapter 6: Problem 3
How might you explain the entropy production concept in terms a child would understand?
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chapter 6: Problem 3
How might you explain the entropy production concept in terms a child would understand?
These are the key concepts you need to understand to accurately answer the question.
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Get started for freeAn isolated system of total mass \(m\) is formed by mixing two equal masses of the same liquid initially at the temperatures \(T_{1}\) and \(T_{2}\). Eventually, the system attains an equilibrium state. Each mass is incompressible with constant specific heat \(c\). (a) Show that the amount of entropy produced is $$ \sigma=m c \ln \left[\frac{T_{1}+T_{2}}{2\left(T_{1} T_{2}\right)^{1 / 2}}\right] $$ (b) Demonstrate that \(\sigma\) must be positive.
The theoretical steam rate is the quantity of steam required to produce a unit amount of work in an ideal turbine. The Theoretical Steam Rate Tables published by The American Society of Mechanical Engineers give the theoretical steam rate in lb per \(\mathrm{kW} \cdot \mathrm{h}\). To determine the actual steam rate, the theoretical steam rate is divided by the isentropic turbine efficiency. Why is the steam rate a significant quantity? Discuss how the steam rate is used in practice.
A cylindrical rod of length \(L\) insulated on its lateral surface is initially in contact at one end with a wall at temperature \(T_{\mathrm{H}}\) and at the other end with a wall at a lower temperature \(T_{\mathrm{C}}\). The temperature within the rod initially varies linearly with position \(z\) according to $$ T(z)=T_{\mathrm{H}}-\left(\frac{T_{\mathrm{H}}-T_{\mathrm{C}}}{L}\right) z $$ The rod is then insulated on its ends and eventually comes to a final equilibrium state where the temperature is \(T_{\mathrm{f}}\). Evaluate \(T_{\mathrm{f}}\) in terms of \(T_{\mathrm{H}}\) and \(T_{\mathrm{C}}\) and show that the amount of entropy produced is $$ \sigma=m c\left(1+\ln T_{\mathrm{f}}+\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}-T_{\mathrm{C}}} \ln T_{\mathrm{C}}-\frac{T_{\mathrm{H}}}{T_{\mathrm{H}}-T_{\mathrm{C}}} \ln T_{\mathrm{H}}\right) $$ where \(c\) is the specific heat of the rod.
Answer the following true or false. If false, explain why. (a) The change of entropy of a closed system is the same for every process between two specified states. (b) The entropy of a fixed amount of an ideal gas increases in every isothermal compression. (c) The specific internal energy and enthalpy of an ideal gas are each functions of temperature alone but its specific entropy depends on two independent intensive properties. (d) One of the \(T d s\) equations has the form \(T d s=d u-p d v\). (e) The entropy of a fixed amount of an incompressible substance increases in every process in which temperature decreases.
Nitrogen \(\left(\mathrm{N}_{2}\right)\) at \(3.8 \mathrm{~atm}\) and \(170^{\circ} \mathrm{C}\) enters an insulated turbine operating at steady state and expands to \(1 \mathrm{~atm}\). If the isentropic turbine efficiency is \(83.2 \%\), determine the temperature at the turbine exit, in \({ }^{\circ} \mathrm{C}\), using the ideal gas model for the nitrogen and ignoring kinetic and potential energy changes.
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